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LA-UR-2070
**Oscillations in the GSI electron capture experiment**
H. Burkhardt, J. Lowe and G.J. Stephenson Jr.
[*Physics and Astronomy Department, University of New Mexico, Albuquerque,\
NM 87131, USA*]{}
T. Goldman
[*Theoretical Division, Los Alamos National Laboratory, Los Alamos,\
NM 87545, USA*]{}
Bruce H.J. McKellar
[*School of Physics, University of Melbourne, Victoria 3010, Australia*]{}
31 March 2008
[**Abstract:**]{} In a recent paper, oscillations observed in the electron capture probability were attributed to the mixing of neutrino mass eigenstates. This paper is shown to be in error in two respects.
In a recent measurement[@ref:gsi] at GSI of the electron capture rate for hydrogen-like $^{140}Pr$ ions,
$$\begin{aligned}
^{140}Pr^{58+}\rightarrow ^{140}Ce^{58+}+\nu
\label{eq.1}\end{aligned}$$
the decaying ions were found to have a non-exponential time dependence. The approximately sinusoidal modulation superimposed on the exponential decay was tentatively interpreted as resulting from the oscillations of the recoil neutrino due to the presence of at least two mass eigenstates in the electron neutrino state.
Two papers[@ref:ivanov],[@ref:faber] have claimed to provide a theoretical explanation of this effect. We examine here the first of these and find that it has two errors which, when corrected, remove the predicted oscillations.
The two errors in the paper by Ivanov [*et al*]{}.[@ref:ivanov] are the following.
\(1) Ivanov [*et al*]{}.[@ref:ivanov] write the amplitude for the electron capture as
$$\begin{aligned}
A(^{140}Pr^{58+}\rightarrow ^{140}Ce^{58+}+\nu)(t)=
-i\Sigma_{j=1}^{3}\int_0^t\langle ^{140}Ce^{58+}\nu\mid H_W(\tau)\mid
^{140}Pr^{58+}\rangle d\tau
\label{eq.2}\end{aligned}$$
Thus the final neutrino state in the amplitude is a sum over neutrino mass eigenstates. However, when the neutrino is not observed, one should sum the probabilities, not the amplitudes, over a complete set of neutrino states. This can be done over mass eigenstates, flavour eigenstates, or even the states $|\nu_1 + \nu_2 + \nu_3 \rangle/\sqrt{3}$, $|\nu_1 - \nu_2 \rangle/\sqrt{2}$, and $|\nu_1 + \nu_2 -2 \nu_3 \rangle/\sqrt{6}$, the first of which is the correctly normalised version of the state used by Ivanov [*et al*]{}. The result is the same whichever complete set of states is used — there are no oscillations.
This is just another way of saying that the electron capture process always produces only the combination of mass eigenstates corresponding to an electron neutrino.
This point is essentially the same as that made by Giunti[@ref:giunti] in a recent comment, who shows that when the correct final state is used, there are no oscillations.
\(2) A further error in Ivanov [*et al*]{}.[@ref:ivanov] arises in the treatment of wave packeting. It is essential to include wave packet structure in the treatment, partly to provide a realistic description of the experimental set-up but also because oscillations cannot be observed without wave packeting. The latter point requires that the spacial width of a packet should be less than the wavelength of the oscillations.
Ivanov [*et al*]{}. therefore define the initial state as a wave packet, so that there is a spread of energies in the inital state. This approach appears to give difficulties in keeping track of 4-momentum conservation in the decay. Indeed, Ivanov [*et al*]{}. appear to conclude that 4-momentum is not conserved as a result of the spread required to produce a wave packet.
An alternative approach is to calculate for a plane-wave initial state, with a sharply defined 4-momentum. Then the wave-packeting is introduced by summing a set of such solutions to produce the required packet size. We used this procedure in a paper treating neutral kaon oscillations[@ref:usplb1] and also in two more general papers that additionally include neutrino oscillations from pion decay[@ref:usprd],[@ref:usplb2]. In this way, it is straightforward to include 4-momentum conservation exactly at all stages of the calculation, which is desirable because the decay interaction conserves 4-momentum. When this is done, the usual result follows simply, namely that the particle with mixed mass eigenstates shows oscillations when it is observed, but the particle recoiling against it does not.
It is important to treat the kinematics exactly, and not to ignore energy terms that are comparable with the mass difference between mass eigenstates or the width of the wave packet. In the course of our work, we showed that no approximation is necessary in the evaluation; in Ref.[@ref:usprd] we derived the oscillation expression exactly, to all orders and including the correct kinematics.
The situation proposed by Ivanov [*et al*]{}.[@ref:ivanov] is strikingly similar to that described in a series of papers by Widom and collaborators[@ref:widom] on the process $\pi {\rm}p\rightarrow \Lambda K^0$, in which it was predicted that oscillations should be observed in [*both*]{} the $K^0$ [*and*]{} the $\Lambda$ distributions (which is certainly not the case, experimentally) and that the usual expression relating the oscillation frequency to $m_L-m_S$ is in error. On examining the algebra of Ref. [@ref:widom], we found[@ref:usplb1],[@ref:usprd],[@ref:usplb2] a kinematic approximation in the evaluation which gave rise to these unexpected effects, which are quite inconsistent with experiment. With the exact treatment, these unexpected effects disappeared and the standard result was recovered. (For yet another exact momentum conservation treatment using plane waves, see Goldman[@ref:goldman].)
Finally, we recall that Aharonov and Moinester[@ref:Aharanov] examined this question long ago in connection with a pion decay experiment at the Los Alamos Meson Physics Facility in the early 1980’s. Initially, there appeared to be oscillations about exponential decay registered in the appearance of muons in the final state. This was eventually traced to an experimental problem involving cross-talk between timing cables[@ref:Moinester]. Aharonov and Moinester also concluded that these oscillations could not have had a basis in neutrino physics, due to the orthogonality of the neutrino mass eigenstates[@ref:Aharanov].
We conclude that the papers of Refs.[@ref:ivanov] and [@ref:faber] do not provide an explanation of the oscillations observed in the GSI experiment[@ref:gsi] in terms of neutrino mixing and that the experimental results are not a consequence of neutrino oscillations.
[99]{}
Yu.A. Litvinov [*et al*]{}., arxiv/0711.3709; [*Phys. Rev. Lett. *]{}[**99**]{}, 262501 (2007).
A.N. Ivanov [*et al*]{}., arxiv/0801.2121.
M. Faber, arxiv/0711.3262.
Carlo Giunti, arxiv/0801.4639. For a reply, see A.N. Ivanov [*et al*]{}., arxiv/0803.1289.
J. Lowe, B. Bassalleck, H. Burkhardt, A. Rusek, G.J. Stephenson Jr. and T. Goldman, hep-ph/9605234; [*Phys. Lett. *]{}[**B384**]{}, 288 (1996).
H. Burkhardt, T. Goldman, G.J. Stephenson Jr. and J. Lowe, [*Phys. Rev. D*]{}[**59**]{}, 054018 (1999).
H. Burkhardt, T. Goldman, G.J. Stephenson Jr. and J. Lowe, hep-ph/0302084; [*Phys. Lett. *]{}[**B566**]{}, 137 (2003).
Y.N. Srivastava, A. Widom and E. Sassaroli, [*Phys. Lett. *]{}[**B344**]{}, 436 (1995); Y.N. Srivastava, A. Widom and E. Sassaroli, [*Zeit. Phys. *]{}[**C66**]{}, 601 (1995).
T. Goldman, unpublished, hep-ph/9604357.
Y. Aharanov and M.A. Moinester, unpublished.
M.A. Moinester, private communication.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The generalized totally asymmetric exclusion process (TASEP) [\[]{}J. Stat. Mech. P05014 (2012)[\]]{} is an integrable generalization of the TASEP equipped with an interaction, which enhances the clustering of particles. The process interpolates between two extremal cases: the TASEP with parallel update and the process with all particles irreversibly merging into a single cluster moving as an isolated particle. We are interested in the large time behavior of this process on a ring in the whole range of the parameter $\lambda$ controlling the interaction. We study the stationary state correlations, the cluster size distribution and the large-time fluctuations of integrated particle current. When $\lambda$ is finite, we find the usual TASEP-like behavior: The correlation length is finite; there are only clusters of finite size in the stationary state and current fluctuations belong to the Kardar-Parisi-Zhang universality class. When $\lambda$ grows with the system size so does the correlation length. We find a nontrivial transition regime with clusters of all sizes on the lattice. We identify a crossover parameter and derive the large deviation function for particle current, which interpolates between the case considered by Derrida-Lebowitz and a single particle diffusion.'
author:
- 'A.E. Derbyshev'
- 'A.M.Povolotsky'
- 'V. B. Priezzhev'
title: Emergence of jams in the generalized totally asymmetric simple exclusion process
---
Introduction\[sec:Introduction\]
================================
The asymmetric simple exclusion process (ASEP) is one of the basic models of driven transport admitting an analytical treatment [@Spoh91; @Ligg99; @Gunter]. It is commonly accepted that different versions of ASEP provide an adequate description of statistical properties of one-dimensional diffusive and driven-diffusive systems. During the last decades the ASEP was a laboratory for obtaining the universal critical exponents and scaling functions of the Edward-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) universality classes [@EW; @KPZ]. The range of models that can be solved exactly is very limited, but the universality implies that results obtained from their solution apply to a wide range of stochastic systems, like interacting particle systems, growing interfaces, crystal facets, polymers in random media, etc. Among the results, which are believed to be universal, are the dynamical exponent of the KPZ class [@Gwa; @Spohn], the KPZ-EW crossover function for the relaxation time [@Doochul; @Kim], the large deviation function (LDF) for particle current in the systems with periodic [@Derrida; @Lebowitz] and open [@de; @Gier; @Essler; @GLMV] boundary conditions. More recently, consideration of these processes on the infinite lattice yielded plenty of results on the universal scaling functions for probability distributions and correlation functions characterizing the nonstationary time evolution.
The totally asymmetric exclusion process (TASEP) is the simplest version of the ASEP, possessing a special mathematical structure, which simplifies the solution significantly. Using this structure, Derrida and Lebowitz obtained the first exact expression for the LDF of particle current for an arbitrary lattice size, which yielded the universal scaling function in the scaling limit. Also, closed determinantal formulas for the Green’s functions were derived for the TASEP on both the infinite lattice [@Schutz] and the ring [@priezzhev]. Finally, all multipoint correlation functions for the process in the infinite system were constructed [@Sasamoto; @BFPS; @IS; @BFPS1; @BF; @BFS; @PovPriS; @PPP]. Remarkably, unlike the partially asymmetric case, the TASEP remains exactly solvable in a discrete time framework. The models with several different updates were solved: backward sequential [@BrPrS], parallel [@PP] and sublattice parallel [@PPS]. All these versions of the TASEP demonstrate the same universal KPZ behavior in the scaling limit. It is of interest, however, to examine possible mechanisms taking the system away from the KPZ class, to see how the KPZ universality breaks down.
To our knowledge, the generalized TASEP (gTASEP) studied here was first considered in [@Woelki], where without any reference to its integrable structure, it was used as an example of a traffic model with a stationary measure admitting factorized representation. It was later rediscovered in [@genTASEP], within a totally different context as an integrable generalization of the TASEP. Finally, it was shown to be a particular $q=0$ limiting case of the general three parametric Bethe ansatz-solvable stochastic chipping model [@chipping], also referred to as a q-Hahn or $(q,\mu\text{,\ensuremath{\nu}})-$boson process [@Corwin2; @BCPS]. In turn, it containes already known TASEPs with parallel and sequential update as particular cases. In the gTASEP an additional interaction between particles is introduced, which enhances the clustering of particles comparing to the usual TASEP. The dynamics of the model can be viewed as the TASEP-like process, where clusters of particles diffuse, breaking into parts and merging together. The relative frequency of these processes is controlled by an extra parameter $\lambda$. The bigger value of $\lambda$, the stronger is the effective attraction between particles and the larger is the size of clusters in typical particle configuration. A limiting case $\lambda\to\infty,$ which we refer to as the deterministic aggregation (DA) limit, produces the process, where particles stick together irreversibly, finally forming a single giant cluster, which moves as an ordinary random walk.
The main aim of the present paper is to study how the large-scale behavior of the steady state in gTASEP changes as the DA limit is approached. We concentrate on the stationary state correlations and fluctuations of particle current on the ring. For moderate interaction strength it is natural to expect that the scaling behavior of gTASEP will be similar to the usual TASEP, which belongs to the KPZ universality class. For the latter, it is well known that the stationary state is uncorrelated if looked at in the scale of the system size. Also the motion of particles in an infinite system is subdiffusive. Though it is still diffusive in a finite system, the diffusion coefficient decays as $\Delta\sim1/\sqrt{L}$ as the system size $L$ grows to infinity. Further details of the large time fluctuations of particle current can be extracted from the universal LDF obtained by Derrida and Lebowitz in [@Derrida; @Lebowitz] for the usual TASEP, and later proved to hold for several other systems [@lee; @kim; @PPH; @povolotsky; @Povolotsky; @Mendes]. On the other hand in the DA limit the particles form a single giant cluster, which moves as a single particle. This behavior obviously corresponds to correlation length unboundedly growing with the system size and to purely diffusive motion of each individual particle. As a result, there are many small particle clusters, finite range correlations, and KPZ-like fluctuations on the one end of the range of $\mbox{\ensuremath{\lambda}}$ and one macroscopic cluster with pure single-particle diffusion on the other. Then the natural question to ask is how many particle clusters can there be and how large can typical particle clusters be for the KPZ universality to survive and how the two regimes are connected to each other. Intuition says that at least at a finite density of finite clusters, which is maintained at finite values of $\lambda,$ we should be in the KPZ regime, as the finite clusters can be effectively treated as larger particles. The analysis below shows, however, that one can approach the DA limit much more closely keeping the universal KPZ form of the current LDF. We show that even when $\lambda$ and, hence, the typical size of clusters, grow with the system size, the LDF preserves its functional form, unless the order of $\lambda$ is as large as $L^{2}$. When $\lambda/L^{2}\to0$, the dependence on the value of $\lambda$ affects only the non-universal constants controlling the fluctuation scale but not the functional form of the distribution. At the scale $\lambda\sim L^{2}$ there is only a few (a finite number of) macroscopic clusters on the lattice and the correlation length is of order of system size $L$. At this scale, the transition from the KPZ to the DA limit takes place. We obtain the LDF that crosses over from the KPZ Derrida-Lebowitz form to pure Gaussian as $\lambda/L^{2}$ varies from zero to infinity.
To have a rough idea of where the scale $\lambda\sim L^{2}$ comes from, the following simple mean-field argument can be used, which should not be considered as a derivation, but can be viewed as a description of the scenario of the transtion regime. Let us think about the large number $M$ of interacting particles diffusing on the one-dimensional lattice with overall particle density being fixed, $M/L=c$, as about diffusing, aggregating and dissociating clusters (compact groups of particles). Two clusters merge when coming in contact, while any cluster can break down into two smaller clusters at any point with small rate $\alpha$. Then as $\alpha$ goes to zero, we expect to observe a transition from a finite density of finite clusters to a single cluster of size $M$. In the transition regime there is a finite number of clusters of any macroscopic size. Under this suggestion consider the conditions for the equilibrium to hold between merging and breaking up clusters at all scales. Let $P(n)$ be a global density (mean number per unit length) of $n$-particle clusters, which is supposed to be of the same order through the whole range of $n$ in the transition regime. The number of clusters of size $n$ in the system is equal to $LP(n)$, and the total number of clusters of any size in a typical configuration is given by the sum $L\sum_{1\leq n\leq M}P(n)$. For this number to be finite (of order of one), the value of $P(n)$ should be of order of $1/L^{2}$. As the cluster of $n$ particles can split into two smaller clusters at any of its points, the mean rate of decay of such clusters will be $n\alpha P(n)$. On the other hand, the number of clusters of size $n$ appearing per unit time is $\sum_{k}P(k)P(n-k)$, which is of order of $1/L^{3}.$ Equating these two expressions we find that $\alpha$ must be of order of $1/L^{2}.$ An analog of splitting rate $\alpha$ in our model is the inverse of the parameter $\lambda$. Our asymptotic analysis indeed shows that the scaling parameter controlling the transition occurring in the limit $\lambda\to\infty$ can be chosen proportional to $\lambda/L^{2}.$ Tuning this parameter one can obtain both the particle current LDFs for KPZ and DA regimes as limiting cases. We want to emphasize that the above description gives a qualitative picture, which can only illustrate the exact results obtained below.
It is worth mentioning other studies of models, where the particle clustering strength can be controlled. A version of the TASEP with next-nearest neighbor interaction was proposed in [@AntalSchutz]. The particle flow has the jamming tendency and for this reason the flow diagram is shifted in the region of large densities. The finding of a “fourth phase” in the mean-field approximation (approved by Monte Carlo simulations) demonstrates an unusual and nontrivial character of particle flow when it enters the jam regime.
The model which allows for the diffusion of clusters, aggregation on contacts between them and single-particle dissociation has been considered in [@Majumdar]. A mean field analysis of the model showed that the system undergoes the dynamical phase transition: The steady state mass distribution in one phase decays exponentially for large masses. In another phase, the model predicts an infinite aggregate in addition to a power-law mass decay.
Note that the mentioned models do not belong to the class of integrable models. The models like these are generally studied in the mean field approximation, or at best allow the exact characterization of the stationary state distribution; see, e.g., [@AntalSchutz]. Such an analysis provides the thermodynamical description, like the density-current relation, which is not universal and to large extent depends on particular dynamical rules. In contrast, in our case the integrability allows the exact treatment of the full dynamical problem, which contains information about universal fluctuations in the scaling limit.
Our paper is organized as follows. In Sec. \[sec:Model-definition-and\] we formulate the model and explain the zero-ramge rocess (ZRP)-ASEP mapping, which allows us to establish a relation between gTASEP and another zero-range type model with an unbounded number of particles in a site. While many quantities characterizing the two models coincide, the advantage of models like the ZRP is a factorized form of steady state distribution, which can be analyzed with the canonical partition function formalism.
In Sec. \[sec:Stationary-state\] we study the stationary state of both gTASEP and the corresponding ZRP-like model. For the ZRP-like model we obtain the exact expressions for the partition function on an arbitrary finite lattice and use it to derive the occupation number distribution. The latter can be reinterpreted as the cluster size distribution in the gTASEP. We also derive the generating function of particle jumps and, in particular, obtain the exact formula for the mean number of particle jumps per unit time. The exact partition function and particle current are represented as contour integrals, which, then, are explicitly evaluated in terms of the Gauss $_{2}F_{1}$ and Appell $F_{1}$ terminating hypergeometric series, respectively. Then we perform an asymptotic analysis of the integrals obtained, first, in the saddle point approximation, which is applicable when $\lambda/L^{2}\to0,$ and, second, in the limit $\lambda/L^{2}\to const>0,$ when the saddle point approximation fails. In the first case, we obtain the geometric finite (or subextensive) cluster size distribution and the thermodynamic formula for particle current (flow diagram) depending on two parameters and particle density. In the second case we obtain the distribution of cluster sizes on the system size scale, expressed in terms of the modified Bessel functions. In the last section of Sec. \[sec:Stationary-state\] we analyze the stationary state of the gTASEP directly in the grand-canonical ensemble exploiting the fact that the stationary measure of the gTASEP is similar to the Gibbs measure of a one-dimensional Ising model. We evaluate the two-point correlation function and discuss its behavior in both limits.
Section \[sec:Statistics-of-particle\] is devoted to the analysis of particle current fluctuations. We first remind the reader of the Bethe ansatz solution of the models discussed and then obtain the largest eigenvalue of the Markov matrix deformed by including parameter $\gamma$, counting the particle jumps. The eigenvalue, obtained in the parametric form as two series with coefficients expressed via $_{2}F_{1}$ and $F_{1}$, has a meaning of the rescaled cumulant generating function of the total number of particle jumps or of the Legendre transform of the corresponding LDF. In particular, in addition to the exact particle current obtained in Sec. \[sec:Stationary-state\] we derive the exact expression for the diffusion coefficient of a particle in gTASEP. The asymptotic analysis again consists of two parts: the saddle point approximation for $\lambda/L^{2}\to0$, which reproduces the universal function by Derrida and Lebowitz through a range of scales of $\lambda$, and the asymptotic analysis on the scale $\lambda\sim L^{2}$, describing the KPZ-to-Gauss transition.
The last section, Sec. \[sec:Universality-and-relation\], is intended to bind together the variety of the results obtained for the KPZ regime. We remind to the reader of the scaling theory, developed in [@krug; @meakin; @halpin-healy], which claims that many non-universal quantities characterizing the systems belonging to the KPZ universality class can be expressed in terms of only two dimensional invariants, which, in particular, are related to the dimensional constants in the KPZ equation. We show that this hypothesis is confirmed by our results and conversely express the non-universal scaling constants in the LDF in terms of the KPZ dimensional invariants.
In Appendix \[sec:Explicit-expressions-of\] we give explicit formulas for some model-dependent constants appearing in the calculations and establish relations between them, which confirm KPZ universality. The formulas used to work with the special functions are listed in Appendices \[sec:Hypergeometric-functions\] and \[sec:Modified-Bessel-function\].
Model definition and ZRP-ASEP mapping\[sec:Model-definition-and\]
=================================================================
Consider $M$ particles on the one-dimensional lattice of $L$ sites with periodic boundary conditions. Each lattice site can be occupied by, at most, one particle. Particle configurations are recorded as $N$-tuples of particle coordinates $\mathbf{x}=(1\leq x_{1}<x_{2}<\dots<x_{M}\leq L)$. Particle configurations evolve in discrete time with *clusterwise* backward-sequential update rules. We refer to a compact string of particles like $(x_{i-k}=x-k,...,x_{i}=x)$ surrounded by two empty sites as a cluster. The update of particle configuration at a given time step starts from the rightmost particle of any cluster. For definiteness, one can choose the cluster with the maximal coordinate $x_{i}\leq L$ of the rightmost particle. The particle tries to jump one step to the right, i.e., clockwise, succeeding, $(x\to x+1\mod L$), with probability $p$ or failing, $(x\to x)$, with probability $1-p$. In the case of success and if the cluster consists of more than one particle, the second particle tries to follow the first one with probability $\mu$, which is, in general, different from $p$. So do the third, fourth, etc., particles until either some particle of the cluster has failed to jump or the cluster has ended. In other words, for $k-$particle cluster with $k>1$ the following outcomes are possible:
1. all $k$ particles stay with probability $(1-p)$;
2. $m<k$ particles make a step with probability $p\mu^{m-1}\left(1-\mu\right)$;
3. all $k$ particles jump with probability $p\mu^{k-1}$
Then we go to the next cluster in a counterclockwise direction and continue the update until all clusters on the lattice have been updated. Note that the result clearly does not depend on what cluster we choose to start. The clusterwise backward-sequential update, i.e. the condition of starting from the rightmost particle of a cluster excludes the situation when the tail of a cluster is updated before its head, which would occur with the conventional sitewise backward update, when the sites $1$ and $L$ are inside the same cluster. It is also easy to see that the exclusion rule is automatically satisfied.
The above formulation uses two control parameters $p$ and $\mu$ having a meaning of probabilities, hence varying in the range $0\leq p,\mu\leq1$. The particular cases $\mu=0$ and $\mu=p$ correspond to the TASEP with parallel and backward-sequential updates, respectively. In the case $\mu=1$ the probability for all particles of a cluster to follow the first particle is equal to one. Therefore, clusters can only merge and no dissociation occurs in this limit.
Using so-called ZRP-ASEP mapping, the gTASEP can be related to a model of the ZRP type with an unbounded number of particles in a site. To establish the correspondence, we replace a string of sites occupied by a cluster of $n$ particles together with one empty side ahead by a single site with $n$ particles as shown in Fig. \[ASEP-ZRP\].
![ZRP-ASEP mapping. []{data-label="ASEP-ZRP"}](asep1){width="1\columnwidth"}
Thus, $M$ particles are placed on the lattice $\mathcal{L}$ consisting of $N=L-M$ sites allowed to hold any number of particles unlike at most one particle per site in the ASEP. The one-step jump of $m$ particles from a splitting cluster will be replaced with a jump of the same number of particles from the corresponding site to the next site on the right. According to the above dynamical rules, the jump of $m$ particles from a site with $n$ particles depends on both $n$ and $m$, and has the form $$\varphi(m|n)=\left\{ \begin{array}{ll}
(1-p), & m=0;\\
p\mu^{m-1}\left(1-\mu\right), & 0<m<n;\\
p\mu^{n-1} & m=n,
\end{array}\right.\label{eq: phi(m|n)}$$ for $n>0$ and $\varphi(0|0)=1$ with all the sites being updated simultaneously at a given time step, as in the parallel update scheme. As noted in Sec. \[sec:Introduction\], the hopping probabilities of our model are a particular $q=0$ limit of the general three-parametric hopping probabilities of the model proposed in [@chipping], where they depended on three parameters $q,\mu$ and $\nu$. Here we use the notations of [@chipping], which are different from those used in the first paper on the gTASEP [@genTASEP]. Specifically, the two parameters $\mu$ and $\nu$ of the present article (as well as of [@chipping]) correspond to $p(1+\nu)$ and $\nu p/(1-p)$ of [@genTASEP] respectively. The parameter $\nu$ used here is related to the parameters $p$ and $\mu$ defined above by $$\nu=\frac{\mu-p}{1-p}.\label{eq:nu}$$ In the following, where it is more appropriate for brevity of notations, we also use another parameter $$\begin{aligned}
\lambda & = & \frac{1}{1-\nu}.\label{eq:lambda}\end{aligned}$$ In particular, it is convenient for studying the DA limit, which corresponds to $\lambda\to\infty$.
Note that for periodic lattices, the ZRP-ASEP mapping is not the one-to-one correspondence between particle configurations, though they can be made equivalent up to the lattice rotations. In particular, there are more particle configurations in ASEP-like systems that in ZRP-like systems. Given two initial particle configurations in the ZRP and the ASEP related to one another by the mapping, the further processes are in the one-to-one correspondence in terms of relative distances between particles, but not in terms of particle coordinates. The essence of the difference is different translational symmetries of the lattices with different numbers of sites: A particle configuration returns to itself after $L$ unit translations in the ASEP and after $M$ unit translations in the ZRP. As seen from the Bethe ansatz solution below, a little modification is necessary to transform formulas using the coordinate notations in one system to those in the other. However, this difference does not affect translation invariant quantities, e.g. stationary state observables such as particle density and current. Once calculated in one system such a quantity can be easily related to similar quantity in the other. The densities of particles defined in the TASEP-like and and ZRP-like systems as $$c=M/L\,\,\,\mbox{and}\,\,\,\rho=M/N,$$ respectively, are related by $$c=\frac{\rho}{1+\rho}.$$ The total numbers of jumps made by all particles are the same in both systems. In particular, this is the case for the stationary state average number of jumps per one step $J$. Then the mean velocities of particles $v=J/M$ are the same in both systems, while the stationary state currents, i.e., average numbers of particles leaving one site per time step, $j^{ASEP}=J/L$ and $J^{ZRP}=J/N$, satisfy the following relation: $$\frac{j^{ASEP}}{c}=\frac{j^{ZRP}}{\rho}=v.$$ Below, we use these relations to express stationary characteristics of the gTASEP on terms of quantities obtained for corresponding ZRP-like system.
Stationary state\[sec:Stationary-state\]
========================================
Partition function formalism
----------------------------
The advantage of the ZRP-like system is that its stationary measure has a particularly simple form. Specifically, consider the one-dimensional periodic lattice $\mathcal{L}=\mathbb{Z}/N\mathbb{Z}$ consisting of $N$ sites with $M$ particles on it. Every site can hold any number of particles. It is convenient to specify particle configurations in the ZRP-like systems by $N$ occupation numbers of all sites $$\mathbf{n}=\left\{ n_{1},\ldots,n_{N}\right\} .\label{C}$$ The system configuration is updated at every time step by bringing any number $m_{i}\leq n_{i}$ of particles from every site $i=1,\dots,N$ to the next site $i+1$ with probability $\varphi(m_{i}|n_{i})$ defined in (\[eq: phi(m|n)\]). Therefore, the probability $P_{t}\left(\mathbf{n}\right)$ for the system to be in a configuration $\mathbf{n}$ at time step $t$ obeys the Markov equation $$P_{t+1}(\mathbf{n})=\sum_{\mathbf{n'}}\mathbf{M_{\mathbf{n,}n'}}P_{t}(\mathbf{n'}),\label{eq:Markov eq.}$$ with transition matrix **M** defined by matrix elements $$\mathbf{M_{n,n'}}=\sum_{\{m_{k}\in\mathbb{Z}_{\geq0}\}_{k\in\mathcal{L}}}\prod_{i\in\mathcal{L}}T_{n_{i},n_{i}'}^{m_{i-1},m_{i}},\label{eq:Markov matrix}$$ where $T_{n_{i},n_{i}'}^{m_{i-1},m_{i}}=\delta_{(n_{i}-n_{i}'),(m_{_{i-1}}-m_{i})}\varphi(m_{i}|n_{i}')$. It was proved in [@Evans; @Zia; @Majumdar] that if and only if the hopping probabilities $\varphi(m|n)$ have the functional form $$\varphi(m|n)=\frac{v(m)w(n-m)}{\sum_{i=0}^{n}v(i)w(n-i)},\label{eq: phi(m|n) - general}$$ with two arbitrary positive functions $v(k)$ and $w(k)$, the Markov equation (\[eq:Markov eq.\]) has a unique stationary solution, which belongs to the class of the so-called product measures; i.e., the probability of a configuration is given by the product of one-site factors $$P_{st}\left(\mathbf{n}\right)=\frac{1}{Z\left(M,N\right)}\prod_{i=1}^{N}f\left(n_{i}\right),\label{P_st}$$ where the one-site factor is given by $$f\left(n\right)=\sum_{i=0}^{n}v(i)w(n-i),\label{eq: f(n)}$$ and $Z\left(M,N\right)$ is the normalization constant, referred to as the partition function in statistical physics. In our case, the functions $v(k)$ and $w(k)$ that define the hopping probabilities (\[eq: phi(m|n)\]) can be chosen as $$\begin{aligned}
v(k) & = & \mu^{k}(\delta_{k,0}+(1-\delta_{k,0})(1-\nu/\mu)),\label{eq:v(k)}\\
w(k) & = & (\delta_{k,0}+(1-\delta_{k,0})(1-\mu)),\label{eq:w(k)}\end{aligned}$$ which, according to (\[eq: f(n)\]), yield an expression for the single site weight, $$f(n)=(\delta_{n,0}+(1-\delta_{n,0})(1-\nu))\label{eq: f(n) form}$$
With the product measure (\[P\_st\]) in hands, we are in position to use the partition function formalism [@evans; @burda] to calculate the stationary state observables. The partition function is the normalization constant of the stationary distribution (\[P\_st\]), given by the sum of unnormalized weights over all particle configurations, $$Z\left(M,N\right)=\sum_{n_{1},\dots,n_{N}\geq0}\delta_{\left\Vert n\right\Vert ,M}\prod_{i=1}^{N}f\left(n_{i}\right),$$ Here $f(n)$ is the one-site weight defined in (\[eq: f(n) form\]), $\left\Vert n\right\Vert =n_{1}+\dots+n_{N}$, and the Kronecker $\delta$ symbol constrains the summation to particle configurations with the number of particles fixed to $M$. The sum is given by the contour integral $$Z\left(M,N\right)=\oint_{\Gamma_{0}}\frac{\left[F(z)\right]^{N}}{z^{M+1}}\frac{dz}{2\pi i},\label{Z(N,M)}$$ where $$F\left(z\right)=\sum_{n=0}^{\infty}z^{n}f\left(n\right)$$ is the generating function of the one-site weights, and the contour of integration is a small circle closed around the point $z=0$ leaving all other singularities of $F(z)$ outside. The partition function contains information about the stationary state of the model. In particular, the probability for a site to be occupied by $n$ particles is $$P(n)=f(n)\frac{Z(M-n,N-1)}{Z(M,N)}.\label{eq:P(n)}$$
Another correlation function is the probability $\mathcal{H}(k)$ for $k$ particles on the lattice to hop simultaneously, $$\mathcal{H}(k)=\sum_{\mathbf{n,m}\in\mathbb{Z}_{\geq0}^{N}}\delta_{\left\Vert \mathbf{m}\right\Vert ,k}\delta_{\mathbf{\left\Vert n\right\Vert },M}\varphi(\mathbf{m}|\mathbf{n})P_{st}(\mathbf{n}),$$ where $\varphi(\mathbf{m}|\mathbf{n})=\prod_{1\leq i\leq L}\varphi(m_{i}|n_{i})$ and we suggest that $\varphi(m|n)=0,$ when $m>n$. The corresponding generating function, $$\Psi\left(x\right)\equiv\sum_{n=0}^{M}x^{n}\mathcal{H}(n).\label{Psi(x)}$$ can be obtained in the form of contour integral $$\Psi(x)=\frac{1}{Z\left(M,N\right)}\oint_{\Gamma_{0}}\frac{\left[\Phi\left(x,z\right)\right]^{N}}{z^{M+1}}\frac{dz}{2\pi i}$$ with the two-variable generating function $$\begin{aligned}
\Phi(x,z) & = & \sum_{n=0}^{\infty}\sum_{m=0}^{n}\varphi(m|n)f(n)x^{m}z^{n}.\end{aligned}$$ The values of $\mathcal{H}(n)$ can be then represented via contour integrals with $\Psi(x)$ around $x=0$, while for the moments we need the derivatives at $x=1$. In particular, the average total number of particles jumping per unit time can be evaluated as $$J=\Psi'(1).$$
Using the representation (\[eq: phi(m|n) - general\]) of $\varphi(m|n)$ and interchanging the order of summations, we obtain $$\Phi(x,z)=V(xz)W(z),$$ where $V(t)$ and $W(t)$ are the generating functions of the above sequences $v(k)$ and $w(k)$: $$\begin{aligned}
V(t) & = & \sum_{k=0}^{\infty}v(k)t^{k},\,\,\, W(t)=\sum_{k=0}^{\infty}w(k)t^{k}.\end{aligned}$$ Noticing that $F(z)=V(z)W(z)$, we obtain $$J=\frac{N}{Z\left(M,N\right)}\oint_{\Gamma_{0}}\frac{\left[F\left(z\right)\right]^{N}}{z^{M}}\frac{V'(z)}{V(z)}\frac{dz}{2\pi i}.\label{eq:J}$$ The generation functions of the sequences $w(k)$, $v(k)$, and $f(k)$ from (\[eq:v(k)\])–(\[eq: f(n) form\]) are $$V(t)=\frac{1-\nu t}{1-\mu t},\,\, W(t)=\frac{1-\mu t}{1-t},\,\, F(t)=\frac{1-\nu t}{1-t}.$$ Then the above integrals can be evaluated in terms of hypergeometric functions. Specifically, the grand canonical partition function $\left[F(z)\right]^{N}$ and the function $\left[F(z)\right]^{N}V'(z)/V(z)$, are generating functions of the $_{2}F_{1}$ Gauss hypergeometric functions and $F_{1}$ Appell function respectively. The integrals extract from these series the coefficients of the terms of order $M$ and $(M-1)$, respectively (see Appendix \[sec:Hypergeometric-functions\]). Then, for the partition function we have $$Z(M,N)=\left(\begin{array}{c}
L-1\\
M
\end{array}\right)\left._{2}F_{1}\right.(-M,-N;1-L;\nu),$$ while the average number of particles jumping per unit time is $$J=\frac{(\mu-\nu)NM}{(L-1)}\frac{F_{1}(1-M;1-N,1;2-L;\nu,\mu)}{\left._{2}F_{1}\right.(-M,-N;1-L;\nu)}.\label{eq:J-1}$$ When one of the arguments is zero, the $F_{1}$ Appell function is reduced to the $_{2}F_{1}$ Gauss function (see appendix \[sec:Hypergeometric-functions\]). Therefore, in the limit $\mu=0,$ i.e. the parallel update (PU) case, we recover the result obtained in [@Povolotsky; @Mendes]: $$J_{PU}=\frac{p}{1-p}\frac{NM}{(L-1)}\frac{\left._{2}F_{1}\right.(1-M;1-N,2-L;-\frac{p}{1-p})}{\left._{2}F_{1}\right.(-M,-N;1-L;-\frac{p}{1-p})}.$$ The $\nu=0$ case corresponds to the backward-sequential update (BSU), for which we have a formula $$J_{BSU}=\frac{pNM}{(L-1)}\left._{2}F_{1}\right.(1-M;1,2-L;p),$$ obtained in [@Brankov; @Papoyan; @Poghosyan; @Priezzhev]. [\[]{}There is a minor mistype in [@Brankov; @Papoyan; @Poghosyan; @Priezzhev]: the factor $z$ corresponding to our $p$ is missing from the final expression, formula (16).[\]]{} Also the $p=1$ limit of (\[eq:J-1\]) was obtained in [@Woelki]. It should be noted also that the $F_{1}$ Appell function is a two-variable reduction of the Lauricella hypergeometric function $F_{D}$, which depends on an arbitrary number of variables. The particle current in a particular example of ZRP, where, at most, one particle may jump from sites with $r\leq K$ particles with arbitrary probabilities $0<u(r)<1$ and from sites with $r>K$ with probability $u(r)=1$ was obtained in [@Kanai] in terms of Lauricella hypergeometric function $F_{D}$ of $K$ variables. Presumably, our case and the one studied in [@Kanai] can be unified within a larger class of processes.
The results we have just obtained give the exact formulas for the partition functions, from which we can also obtain the occupation number distribution, and the mean particle current on an arbitrary finite lattice. However, of physical interest is the thermodynamic limit, $$N\to\infty,M\to\infty,M/N=\rho=const\label{eq:thermodynamic}$$ It turns out that depending on the scale of the parameter $\lambda,$ two different regimes naturally appear.
Asymptotic analysis \[sub:Asymptotic-analysis\]
-----------------------------------------------
### Saddle point method, $\lambda/N^{2}\to0$ {#saddle-point-method-lambdan2to0 .unnumbered}
In the limit (\[eq:thermodynamic\]), we can try to evaluate the integrals (\[Z(N,M)\]) and (\[eq:J\]) in the saddle point approximation. The integrals have the form $$\mathcal{I}_{N}\left(h(z),g(z)\right)=\oint_{\Gamma_{0}}e^{Nh(z)}g(z)\frac{dz}{2\pi iz},\label{eq:INT}$$ where $$h(z)=\ln(1-\nu z)-\ln(1-z)-\rho\ln z.\label{eq:h(z)}$$ The critical points of the function $h(z)$ are defined by equation $h'(z)=0,$ which has two solutions $$z_{\pm}=1+\frac{(1-\nu)}{2c\nu}\left(1\pm\sqrt{1+\frac{4(1-c)c\nu}{1-\nu}}\right),\label{eq:z_pm}$$ where $c=\rho/(1+\rho)$ is the concentration of particles in the ASEP-like system. The simple analysis shows that $0<z_{-}<1$ and $z_{+}>1$, $\Re h(z_{+})<0$ and $\Re h(z_{-})>0.$ Therefore $$z=z_{-}$$ is the point which gives the dominant contribution to the integrals. We now choose the steepest descent contour being a circle of radius $z_{-}$with the center at the origin. Then the integral (\[eq:INT\]) asymptotically is
$$\mathcal{I}_{N}\left(h(z),g(z)\right)=\frac{e^{Nh_{0}}}{\sqrt{2\pi N|h_{2}|}}\left[g_{0}+\frac{1}{2N}\left(\frac{g_{2}}{|h_{2}|}+\frac{g_{1}h_{3}}{h_{2}^{2}}+\frac{g_{0}}{4}\left(\frac{h_{4}}{h_{2}^{2}}+\frac{5h_{3}^{2}}{3|h_{2}|^{3}}\right)\right)+O\left(N^{-2}\right)\right],\label{eq:INT ASYMP}$$
where $g_{k}=\left(\mbox{i}z\partial_{z}\right)^{k}g(z)|_{z=z_{-}}$ and $h_{k}=\left(\mbox{i}z\partial_{z}\right)^{k}h(z)|_{z=z_{-}}.$ Choosing $g(z)=1$ we obtain the leading order of the partition function $$Z(M,N)=\mathcal{I}_{N}\left(h(z),1\right)\simeq\frac{\exp\left(Nh_{0}\right)}{\sqrt{2\pi N|h_{2}|}}.$$ To obtain the occupation number distribution $P(n)$ using (\[eq:P(n)\]) we need also the value of $Z(M-n,N-1)$, which can be evaluated choosing $g(z)=z^{n}\exp\left[-h(z)\right].$ As a result, in the limit (\[eq:thermodynamic\]) for $n$ not too large $P(n)\simeq f(n)g(z_{-})$, from where we have a usual Gibbs-like form $$\begin{aligned}
P(n) & = & \lambda^{-1}\exp\left(-n/n^{*}-h(z_{-})\right),\,\,\, n>0\\
P(0) & = & \exp\left(-h(z_{-})\right),\end{aligned}$$ with $n^{*}=-1/\ln z_{-}$, obtained before in [@Woelki]. This form of the distribution suggests that only sites with finite (mainly $n\lesssim n^{*}$) occupation numbers have a chance to appear in a typical configuration in the stationary state.
To interpret this results in terms of the gTASEP, we note that conditioned to occupied sites, $n>0,$ the distribution obtained gives the cluster size distribution. It is the geometric distribution $P(l_{\mbox{cl}}=n)=\left(1-z_{-}\right)z_{-}^{n}$ , with the mean cluster size equal to $\left\langle l_{\mbox{cl}}\right\rangle =z_{-}\left(1-z_{-}\right)^{-1}.$ As $\lambda\to\infty,$ the mean cluster size grows as $\left\langle l_{\mbox{cl}}\right\rangle \sim\sqrt{\lambda/\rho}$, and, correspondingly, the mean number of clusters at the lattice is $M\left\langle l_{\mbox{cl}}\right\rangle ^{-1}\simeq Lc^{3/2}/\sqrt{\lambda(1-c)}$. As discussed in the end of this section, the saddle point approximation is valid as far as $\lambda L^{-2}\to0$. Therefore, these results hold as far as the number of clusters grows with the system size and their size is subextensive, i.e., much less than $L$.
To obtain the particle current we must also evaluate the ratio $$J=N\frac{\mathcal{I}_{N}\left(h(z),g(z)\right)}{\mathcal{I}_{N}\left(h(z),1\right)},$$ with the function $g(z)=zV'(z)/V(z)=\left(1-\mu z\right)^{-1}-\left(1-\nu z\right)^{-1}$. The next to leading order finite size correction to the particle current has a universal meaning in context of KPZ theory, which will be discussed below. Therefore, we keep the terms of the asymptotic expansion up to the next to leading order, which yields $$\begin{aligned}
J & = & Ng_{0}+\frac{1}{2}\left(\frac{g_{1}h_{3}}{\left|h_{2}\right|^{2}}+\frac{g_{2}}{\left|h_{2}\right|}\right)+O(N^{-1}).\label{eq:J asymp}\end{aligned}$$ From the leading order we obtain the current-density relation (so-called flow diagram) for the gTASEP, which being expressed in terms of probabilities $p$ and $\mu,$ reads as $$\begin{aligned}
j^{ASEP} & = & \frac{cp(1+(1-2c)\mu)}{2\mu+2c(p(1-\mu)-\mu)}\label{eq:j^TASEP}\\
& - & \frac{cp\sqrt{(1-\mu)(1-4(1-c)c(p-\mu)-\mu)}}{2\mu+2c(p(1-\mu)-\mu)},\nonumber \end{aligned}$$ and reproduces the formula obtained in [@Woelki]. The particular cases of this expression are:
[$\mu=0$,]{}
: well known current-density relation for the TASEP with PU first obtained in studies of traffic models [@SSNI], $$j_{PU}=\frac{1}{2}\left(1-\sqrt{1-4pc(1-c)}\right);$$
[$\mu=p$,]{}
: BSU [@RSSS], $$j_{BSU}=\frac{(1-c)cp}{1-cp};$$
[$\mu=1$,]{}
: the limit of DA in which all particles finally stick together into a single giant cluster, which performs ordinary Bernoulli random walk, $$j_{DA}=cp.$$
In the case of PU, the current-density plot is symmetric due to particle-hole symmetry. The bigger the value of $\mu,$ the more right skewed is the plot. In the DA limit, it degenerates into the linear function describing a random walk of a single particle making steps of length $M$.
The explicit expression of the next to leading order finite-size correction to the current given in (\[eq: current correction (z\_)\]) of Appendix \[sec:Explicit-expressions-of\] is rather cumbersome. However, it is informative to look at it close to DA limit: $$\begin{aligned}
\mu & \to & 1,\nu\to1,p=\frac{\mu-\nu}{1-\nu}=const.\end{aligned}$$ It is convenient to describe this limit in terms of the parameter $\lambda$ defined in (\[eq:lambda\]), for which the limit corresponds to $\lambda\to\infty$. Then the leading asymptotics in $\lambda$ is $$\begin{aligned}
j^{ASEP}(L) & - & j^{ASEP}(\infty)\label{eq:current correctioon (asymp)}\\
& = & \frac{1}{L}\left[\frac{3cp(1-p)}{4\left(1-c\right)}+O\left(\lambda^{-1/2}\right)\right]+O\left(L^{-2}\right).\nonumber \end{aligned}$$ Surprisingly the $1/L$ correction saturates to the finite limit as $\lambda\to\infty$. However, the next orders’ corrections diverge in this limit, so that the effective expansion parameter is $\sqrt{\lambda}/L,$ from where we can estimate the range of validity of the expressions obtained. One can see that the correction becomes non-neglectable as soon as $\lambda$ is of the order of $L^{2}.$ In fact, the very applicability of the saddle point method is violated at this scale. The reason for that is merging of the saddle points $z_{-}$and $z_{+}$ with the pole of the function under the integral, which makes all the terms of the expansion of the function $h(z)$ effectively of the same order. Indeed, though we implicitly assumed that all the parameters of $h(z)$ are constants as $N$ grows, the saddle point method can still be applied with $N-$dependent parameters as far as the limit $$\lim_{N\to\infty}\left|\frac{N^{1-k/2}h_{k}}{h_{2}^{k/2}}\right|=0$$ holds for $k\geq3$, where $h_{k}$ is the $k$-th derivative of $h(z)$ evaluated at the saddle point $z_{-}$. In the limit $\lambda\to\infty$, the saddle points are as close to $z=1$ as $z_{\pm}=1\pm\sqrt{1/\rho\lambda}+O(1/\mbox{\ensuremath{\lambda}})$ and $h_{k}$ grows as $h_{k}\sim\lambda^{\frac{k-1}{2}}$. Therefore, the above limit holds as far as $\lambda N^{-2}\to0$. The situation when $\lambda N^{-2}\to const>0$, corresponding to the transition regime, requires a separate asymptotic analysis.
### Transition regime, $\lambda N^{-2}=const.$ {#transition-regime-lambda-n-2const. .unnumbered}
Let us consider the integral (\[Z(N,M)\]) representing the partition function $Z(M,N)$. To evaluate the integral asymptotically we note that the function under the integral has only two singularities in the complex plane $z=0$ and $z=1$, and, in particular, is analytic at infinity. Therefore we can deform the integration contour $\Gamma_{0}$ closed around the origin by a contour $\Gamma_{1}$ closed around $z=1$:
$$Z(M,N)=-\oint_{\Gamma_{1}}e^{Nh(z)}\frac{dz}{2\pi iz}.\label{intc}$$
Apart from specifying position of the contour with respect to singularities, we can choose it of any form. It is convenient to integrate over a small circle centered at $z=1$ going close to the saddle points. Then, instead of the the function $h(z)$, one can use its asymptotic expansion at this contour. Choosing $$z=1+\frac{e^{\mbox{i}\varphi}}{\sqrt{\rho\lambda}}$$ we get $$h\left(z\right)=-2\sqrt{\frac{\rho}{\lambda}}\cos\varphi+O(1/\lambda)$$ and for the integral we obtain $$\begin{aligned}
Z(M,N) & = & \frac{-1}{\sqrt{\rho\lambda}}\int_{0}^{2\pi}e^{-2N\sqrt{\rho/\lambda}\cos\varphi+\mbox{i}\varphi}\frac{d\varphi}{2\pi}+O(N^{-2})\\
& \simeq & \frac{\theta}{2M}I_{1}(\theta)\end{aligned}$$ where $I_{k}(y)$ is the modified Bessel function of the first kind (for the definition and properties see appendix \[sec:Modified-Bessel-function\]) and we introduced the scaling parameter $$\theta=2N\sqrt{\frac{\rho}{\lambda}},\label{eq:theta}$$ which is finite in the limit under consideration and is supposed to control the transition from the KPZ to the DA regime. For the occupation number probability distribution $P(n)$ we also need $Z(M-n,N-1)$, which in the leading order is obtained from the above equation by replacing: $M\to M-n,$ $\rho\to\rho-n/N$ and $\theta\to\theta\sqrt{1-n/M}$. Then for the occupation number probability distribution we have $$\begin{aligned}
P(0) & \simeq & 1-\frac{\theta}{2N}\frac{I_{0}(\theta)}{I_{1}(\theta)},\label{cluster1}\\
P\left(n\right) & \simeq & \frac{\theta^{2}}{4NM}\frac{I_{1}\left(\theta\sqrt{1-\frac{n}{M}}\right)}{I_{1}\left(\theta\right)\sqrt{1-\frac{n}{M}}},\,\,\,0<n<M,\label{eq: cluster2}\\
P(M) & \simeq & \frac{\theta}{2N}\frac{1}{I_{1}(\theta)}.\label{eq:cluster3}\end{aligned}$$ Here we kept only the leading order of the expansion of $P(n)$ for $n>0$ and two highest orders in $P(0)$ (the latter can actually be obtained from the former by normalization). It is clear from the first line that the typical configuration contains only a finite number of sites are occupied. Consider now the distribution of the random variable $\chi=n/M\in[0,1]$ conditioned at $n>0,$ i.e. only the occupied sites are counted. To evaluate the conditional probability, we divide the right-hand side of (\[eq: cluster2\]) and (\[eq:cluster3\]) by $\mathrm{Prob}(n>0)=(1-P(0))$. The distribution obtained has well defined limiting behavior as $N\to\infty$ being parametrized by single parameter $\theta:$ $$\begin{aligned}
\mbox{Prob}(\chi=1) & = & \frac{1}{I_{0}(\theta)},\label{eq:Prob(n/M=00003D1|n>0)}\\
\mbox{Prob}(\chi<x) & = & \frac{\theta}{2I_{0}\left(\theta\right)}\int_{0}^{x}\frac{I_{1}\left(\theta\sqrt{1-y}\right)}{\sqrt{1-y}}dy.\label{eq:Prob(n/M<x|n>0)}\end{aligned}$$ A finite fraction of the distribution is concentrated at single point $\chi=1$ and the rest is the continuous distribution on $[0,1)$. In terms of the gTASEP the probability (\[eq:Prob(n/M=00003D1|n>0)\]) is exactly the limiting fraction of time, which all particles spend in a single cluster. The rest of the time finitely many clusters of macroscopic size, $n\sim M$, exist on a lattice. For $\chi<1$ the distribution of the fraction of all particles contained in a given cluster converges to the continuous distribution (\[eq:Prob(n/M<x|n>0)\]). The mean size of a cluster is $\left\langle l_{\mbox{cl}}\right\rangle \simeq2MI_{1}(\theta)\left[\theta I_{0}(\theta)\right]^{-1}$ and, $M/\left\langle l_{\mbox{cl}}\right\rangle $ is the expected number of clusters, which starting from one at $\theta=0$ approaches a linear growth $M/\left\langle l_{\mbox{cl}}\right\rangle \simeq\theta/2$ as $\theta\to\infty.$
Transfer matrix approach and correlation length\[sub:Transfer-matrix-approach\]
-------------------------------------------------------------------------------
In addition to the above calculations with the ZRP-like system, which is suitable for characterizing the current and cluster size distribution, one can look at the system directly in the ASEP formulation, which is more appropriate for for study of the stationary correlation functions. The stationary measure of particle configurations in the TASEP-like system is similar to that of the ZRP-like system up to the symmetry with respect to lattice rotation. Specifically, we must replace every site occupied with $n$ particles with a cluster of $n$ particles plus one empty site. Looking at formulas for the stationary measure of a ZRP-like system (\[P\_st\]) and (\[eq: f(n) form\]) we assign the weight $(1-\nu)$ to each cluster and the weight $1$ to each empty site. It is convenient to study the system in the grand canonical ensemble, where in addition to the above cluster weights we attach the fugacity $z$ to each particle. Then, the stationary probability of a particle configuration $\bm{\tau}=(\tau_{1},\dots,\tau_{L}),$ with occupation numbers $\tau_{1},\dots,\tau_{L}=0,1$, will be $$P_{st}(\bm{\tau})=\frac{1}{\mathcal{Z}_{L}(z)}T_{\tau_{1},\tau_{2}}\dots T_{\tau_{L-1},\tau_{L}}T_{\tau_{L},\tau_{1}},$$ where $T_{0,0}=1$, $T_{0,1}=T_{1,0}=\sqrt{z(1-\nu)},$ and $T_{1,1}=z$. This measure is similar to the Gibbs measure of the 1D Ising model, as was first observed in [@SSNI] in the context of the TASEP with PU. Correspondingly, for periodic boundary conditions the partition function is given by the trace of $L$-th power of the transfer matrix $$\mathcal{Z}_{L}(z)=\mbox{Tr}T^{L}=\lambda_{1}^{L}+\lambda_{2}^{L},$$ where $$\begin{aligned}
\lambda_{1} & = & \frac{1}{2}\left(1+z+\sqrt{(z+1)^{2}-4\nu z}\right),\\
\lambda_{2} & = & \frac{1}{2}\left(1+z-\sqrt{(z+1)^{2}-4\nu z}\right),\end{aligned}$$ are eigenvalues of the matrix $T$, defined so that $\lambda_{1}>\lambda_{2}\geq0$. The largest eigenvalue $\lambda_{1}$ defines the specific free energy, $$f(z)=-\lim_{L\to\infty}\frac{\ln\mathcal{Z}_{L}(z)}{L}=-\ln\lambda_{1}.$$ The density of particles is fixed by the thermodynamic relation $$c=-z\partial_{z}f(z).\label{eq:density-fugacity}$$ This is the quadratic equation for $z$ with two roots, which interchange under replacement $c\longleftrightarrow1-c$. Which one to choose is to be decided from direct evaluation of the particle density, i.e. of one-point correlation function. In general to evaluate $s-$point correlation functions of the form $\left\langle \tau_{k_{1}}\dots\tau_{k_{s}}\right\rangle $, where $\left\langle a\right\rangle $ is the notation for expectation value of the random variable $a$, one has to insert the matrix $$\widehat{\tau}=\left(\begin{array}{cc}
0 & 0\\
0 & 1
\end{array}\right)$$ into the product of transfer matrices in the places corresponding to sites $k_{1},\dots,k_{s}$: $$\left\langle \tau_{k_{1}}\dots\tau_{k_{s}}\right\rangle =\frac{\mbox{Tr}\left[T^{k_{1}}\widehat{\tau}T^{k_{1}+k_{2}}\widehat{\tau}\dots\widehat{\tau}T^{L-(k_{1}+\dots+k_{s})}\right]}{\mathcal{Z}_{L}(z)}.$$ To evaluate expressions of this kind we also need the eigenvectors of $T$, $$\begin{aligned}
\mathbf{v}_{1} & = & \left[\begin{array}{c}
\left(\frac{\sqrt{(z+1)^{2}-4\nu z}-z+1}{2\sqrt{(z+1)^{2}-4\nu z}}\right)^{\frac{1}{2}}\\
\\
\left(\frac{\sqrt{(z+1)^{2}-4\nu z}+z-1}{2\sqrt{(z+1)^{2}-4\nu z}}\right)^{\frac{1}{2}}
\end{array}\right],\\\end{aligned}$$ and $$\begin{aligned}
\mathbf{v}_{2} & = & \left[\begin{array}{c}
-\left(\frac{\sqrt{(z+1)^{2}-4\nu z}+z-1}{2\sqrt{(z+1)^{2}-4\nu z}}\right)^{\frac{1}{2}}\\
\\
\left(\frac{\sqrt{(z+1)^{2}-4\nu z}-z+1}{2\sqrt{(z+1)^{2}-4\nu z}}\right)^{\frac{1}{2}}
\end{array}\right],\\\end{aligned}$$ corresponding to $\lambda_{1}$ and $\lambda_{2}$ respectively, which are normalized to $\left\Vert v_{1}\right\Vert =\left\Vert v_{2}\right\Vert =1.$ Verifying the identity $c=\left\langle \tau\right\rangle =\mbox{Tr}(\widehat{\tau}T^{L})\simeq(\mathbf{v}_{1},\widehat{\tau}\mathbf{v}_{1}),$ we see that the root of (\[eq:density-fugacity\]) to be chosen is $$\begin{aligned}
\!\!\!\!\!\! z^{*} & = & 1-2\left(1+\frac{\sqrt{(1-\nu)\left(1-\nu(1-2c)^{2}\right)}}{(1-2c)(1-\nu)}\right)^{\!\!-1}\!\!\!\!\!\!.\label{eq:z^*}\end{aligned}$$ With the use of relations $(\mathbf{v}_{1},\widehat{\tau}\mathbf{v}_{1})=c,(\mathbf{v}_{2},\widehat{\tau}\mathbf{v}_{2})=(1-c),$ and $(\mathbf{v}_{1},\widehat{\tau}\mathbf{v}_{2})=\sqrt{c(1-c)}$ the two point correlation function is given by
$$\begin{aligned}
\left\langle \tau_{1}\tau_{1+k}\right\rangle & = & \frac{c^{2}+(1-c)^{2}e^{-L/\xi}+c(1-c)(e^{-k/\xi}+e^{-(L-k)/\xi})}{1+e^{-L/\xi}}\end{aligned}$$
where $$\xi\equiv\frac{1}{\ln(\lambda_{1}/\lambda_{2})}=-\left[\ln\left(1-\frac{2}{1+\sqrt{\frac{1-(1-2c)^{2}\nu}{1-\nu}}}\right)\right]^{-1}\label{eq:corr length}$$ is the correlation length. When the correlation length is finite, i.e. for $\nu<1,$ the two-point correlator becomes a product of one point correlators $\left\langle \tau_{1}\tau_{1+k}\right\rangle \to c^{2}$ at large distances, $L\gg k\to\infty.$ As expected, the covariance decays exponentially in the range $\xi\ll k\ll L$, $$C(k)\equiv\left\langle \tau_{1}\tau_{1+k}\right\rangle -\left\langle \tau_{1}\right\rangle \left\langle \tau_{1+k}\right\rangle \simeq c(1-c)e^{-k/\xi},\label{eq:covariance}$$ which justifies $\xi$ being the correlation length. As $\nu\to1$, i.e., $\lambda\to\infty$, the correlation length diverges as $$\xi\simeq\sqrt{\lambda c(1-c)}.$$ In the transition regime, $\lambda\sim L^{2},$ the correlation range is of the order of the system size $L$. Expressed in terms of the distance $r$ measured in units of the system size the covariance has the form $$C(Lr)=\frac{(1-2c)e^{-1/\widetilde{\xi}}+c(1-c)(e^{-r/\widetilde{\xi}}+e^{-(1-r)/\widetilde{\xi}})}{1+e^{-1/\widetilde{\xi}}},$$ where $\widetilde{\xi}=2c(1-c)/\theta$ is the effective correlation length in the system size scale, which depends on the transition parameter $\theta$ defined in (\[eq:theta\]).
Statistics of particle current\[sec:Statistics-of-particle\]
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Bethe ansatz and method by Derrida-Lebowitz
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To characterize current fluctuations we introduce a deformed Markov matrix $\mathbf{M}^{\gamma}$, depending on an auxiliary parameter $\gamma$, where every particle step is supplied with an extra weight $\exp\gamma$. Then the matrix elements are defined as follows $$\mathbf{M}_{\mathbf{n,n'}}^{\gamma}=\mathbf{M}_{\mathbf{n,n'}}\exp\left(\gamma\mathcal{N}(\mathbf{n,n'})\right),$$ where $\mathbf{M}_{\mathbf{n,n'}}$ — are matrix elements of the original Markov matrix and $\mathcal{N}(\mathbf{n,n'})$ is the number of particle jumps in the one-step transition from $\mathbf{n}'$ to $\mathbf{n}$. This matrix governs the evolution of the generating function $G_{t}\left(\mathbf{n,\gamma}\right)=\sum_{Y=0}^{\infty}e^{\gamma Y}P_{t}\left(\mathbf{n},Y\right),$ of the joint probability $P_{t}\left(\mathbf{n},Y\right)$ for the system to be in configuration $\mathbf{n}$ at time $t$, while the total distance $Y_{t}$ traveled by all particles is equal to $Y$, $$\mathbf{G}_{t+1}=\mathbf{M^{\gamma}}\mathbf{G}_{t},$$ where $\mathbf{G}_{t}$ is the column vector with components $G_{t}\left(\mathbf{n,\gamma}\right)$. The generating function $\left\langle e^{\gamma Y_{t}}\right\rangle $ of moments of $Y_{t}$ is a sum of $\sum_{\mathbf{n}}G_{t}\left(\mathbf{n,\gamma}\right)$ over all configurations. Its large time behavior is dominated by the eigenvalue $\Lambda_{0}\left(\gamma\right)$ of the matrix $\mathbf{M^{\gamma}}$ with the largest real part, and, hence, the logarithm of $\Lambda_{0}\left(\gamma\right)$ is the scaled generating function of cumulants of $Y_{t}$, $$\ln\Lambda_{0}\left(\gamma\right)=\lim_{t\rightarrow\infty}\frac{\ln\left\langle e^{\gamma Y_{t}}\right\rangle }{t}.\label{comulant}$$
The diagonalization of matrix $\mathbf{M^{\gamma}}$ using the Bethe ansatz technique was described in details in [@genTASEP; @chipping] for both ASEP and ZRP-like versions of our system. Let us first briefly discuss the latter. Alternatively to the set of occupation numbers $\mathbf{n}$, it is convenient to describe particle configurations in terms of coordinates of particles on the lattice $\mathcal{L}$, $$\mathbf{y}=(1\leq y_{1}\leq\dots\leq y_{M}\leq N)$$ in the same way as we do for the ASEP, except that the coordinates are weakly ordered, because many particles in a site are allowed. Then, the components of eigenvectors $\mathbf{\Psi}$ of $\mathbf{M^{\gamma}}$ are looked for in the form $$\mathbf{\Psi_{n}}(\mathbf{z})=P_{st}\left(\mathbf{n}\right)\Psi^{0}(\mathbf{y}|\mathbf{z}),\label{eq: eigenvector}$$ where $P_{st}\left(\mathbf{n}\right)$ is the stationary state weight of particle configuration, and $$\Psi^{0}(\mathbf{y}|\mathbf{z})=\sum_{\sigma\in S_{M}}A_{\sigma}z_{\sigma_{1}}^{y_{1}}\dots z_{\sigma_{M}}^{y_{M}}\label{eq: Bethe ansatz}$$ is the Bethe function depending on $M$-tuple $\mathbf{z}=(z_{1},\dots,z_{M})$ of complex numbers to be defined later. Here, $\mathbf{y}$ are weakly increasing coordinates of particles corresponding to the occupation numbers $\mathbf{n}$, the summation is over the permutations $\sigma=(\sigma_{1},\dots,\sigma_{M})$ of the numbers $(1,\dots,M)$, and $A_{\sigma}$ are the permutation-dependent coefficients defined by relation $$\frac{A\dots ij\dots}{A\dots ji\dots}=-\frac{\Big(1-e^{-\gamma}z_{i}\Big)\Big(\nu-e^{-\gamma}z_{j}\Big)}{\Big(1-e^{-\gamma}z_{j}\Big)\Big(\nu-e^{-\gamma}z_{i}\Big)}.\label{eq:A_ij/A_ji}$$ One can show that action of the matrix $\mathbf{M^{\gamma}}$ on $\mathbf{\Psi}$ is reduced to multiplication by the eigenvalue $$\Lambda(\gamma)=\prod_{i=1}^{M}\Big(e^{\gamma}pz_{i}^{-1}+(1-p)\Big),\label{eq:eigenvalue}$$ provided that the Bethe function satisfies periodic boundary conditions $\Psi^{0}(y_{1},\dots,y_{M}|\mathbf{z})=\Psi^{0}(y_{2},\dots,y_{M},y_{1}+N|\mathbf{z})$, which are equivalent to the system of $M$ algebraic Bethe ansatz equations (BAEs), $$z_{i}^{N}=\left(-1\right)^{M-1}\prod_{j=1}^{M}\frac{\Big(1-e^{-\gamma}z_{i}\Big)\Big(\nu-e^{-\gamma}z_{j}\Big)}{\Big(1-e^{-\gamma}z_{j}\Big)\Big(\nu-e^{-\gamma}z_{i}\Big)}.\label{eq:BAE ZRP}$$ for the numbers $z_{1},\dots,z_{M}$.
Though the above analysis is applied to the ZRP-like system, the minor modification is required to the ASEP. The eigenvector of the corresponding Markov matrix is that in (\[eq: Bethe ansatz\]), except that the particle coordinates are read off in a different way. Specifically, the ASEP coordinates $\mathbf{x}$ are obtained from the ZRP coordinates by a shift $$(x_{1},x_{2},\dots,x_{M})=(y_{1},y_{2}+1\dots,y_{M}+M-1),$$ which ensures them to be strictly increasing as necessary. One can also look for the eigenvector right in the form (\[eq: Bethe ansatz\]) in terms of the ASEP coordinates $\mathbf{x}$. Then we will have to multiply the ratio of amplitudes (\[eq:A\_ij/A\_ji\]) by the factor $z_{i}/z_{j}$. The form (\[eq:eigenvalue\]) of the eigenvalues stays the same and the periodic boundary conditions $\Psi^{0}(x_{1},\dots,x_{N}|\mathbf{z})=\Psi^{0}(x_{2},\dots,x_{N},x_{1}+L|\mathbf{z})$ yield the BAE $$z_{i}^{L}=\left(-1\right)^{M-1}\prod_{j=1}^{M}\frac{z_{i}\Big(1-e^{-\gamma}z_{i}\Big)\Big(\nu-e^{-\gamma}z_{j}\Big)}{z_{j}\Big(1-e^{-\gamma}z_{j}\Big)\Big(\nu-e^{-\gamma}z_{i}\Big)},\label{eq:BAE TASEP}$$ which are different from (\[eq:BAE ZRP\]) in a single factor $\prod_{j=1}^{M}z_{j}$.
The problem of finding the largest eigenvalue for both models is reduced to identifying a particular solution of the BAE corresponding to the ground state. To this end, we note that in the limit $\gamma\to0$ the matrix $\mathbf{M}^{\gamma}$ turns to the transition probability matrix $\mathbf{M}$ having the largest eigenvalue equal to one, so that we expect $\Lambda_{0}(\gamma)\to1$ as $\gamma\to0$. In addition, the corresponding eigenvector becomes the stationary state in this limit, which can be obtained from (\[eq: eigenvector\]) and (\[eq: Bethe ansatz\]) by setting $z_{1}=\dots=z_{M}=1$. Taking the product of all $M$ equations in both systems, we see that all solutions satisfy the constraints $\left(\prod_{j=1}^{M}z_{j}\right)^{N}=1$ and $\left(\prod_{j=1}^{M}z_{j}\right)^{L}=1$ for (\[eq:BAE ZRP\]) and (\[eq:BAE TASEP\]), respectively. Therefore, the sets of solutions of BAE can be classified into sectors, where the product of Bethe roots equals different roots of unity independent of $\gamma$. In particular, continuing the $\gamma=0$ limit to arbitrary values of $\gamma,$ we see that in both systems the ground states belong to the sector
$$\prod_{j=1}^{M}z_{j}=1,\label{eq:translation invariance}$$
where the systems (\[eq:BAE ZRP\]) and (\[eq:BAE TASEP\]) are identical and we can use either of them to obtain the eigenvalue $\Lambda_{0}(\gamma)$. The product of the Bethe roots is the factor that the Bethe eigenfunction multiplies by under the unit translation, i.e. the eigenvalue of the translation operator commuting with the matrix $\mathbf{M}^{\gamma}$. Its value reflects the translation invariance of the ground state mentioned before.
To find the solution of (\[eq:BAE ZRP\]), we first make a change of variables, $$z_{i}=e^{\gamma}\frac{1-\nu u_{i}}{1-u_{i}}.$$ In the variables $u_{i}$, the BAE and the eigenvalue $\Lambda(\gamma)$ simplify to the following form $$\begin{aligned}
\left(\frac{1-\nu u_{i}}{1-u_{i}}\right)^{N}e^{N\gamma} & = & (-1)^{M-1}\prod_{j=1}^{M}\frac{u_{i}}{u_{j}},\label{BAEy}\\
\Lambda\left(\gamma\right) & = & \prod_{i=1}^{M}\left(\frac{1-\mu u_{i}}{1-\nu u_{i}}\right).\label{Lambda(gamma)y}\end{aligned}$$ The method used by Derrida and Lebowitz to find eigenvalues for the TASEP is based on the observation that the solutions of BAE can be found among the roots of a single polynomial, $$\mathcal{P}(u)=\left(1-\nu u\right)^{N}B-(1-u)^{N}u^{M},$$ where $B=\left(-1\right)^{M-1}e^{\gamma N}\prod_{j=1}^{M}u_{j}$ is the parameter that itself is a function of the solution. What we actually need is to evaluate the sums of values that particular functions take on the roots from the solution of interest. These sums can be evaluated using the Cauchy theorem, $$\sum_{i=1}^{M}f(u_{j})=\oint_{\Gamma_{0}}f(u)\frac{\mathcal{P}'(u)}{\mathcal{P}(u)}\frac{du}{2\pi\mbox{i}},\label{eq:sum over roots}$$ where the integration is over the contour enclosing all the necessary roots and the function $f(u)$ is analytic inside the contour. In our case the roots from the solution corresponding to the ground state are those $M$ roots $z_{1},\dots,z_{M}$, which approach one as $\gamma\to0$ or zero in terms of the variables $u_{1},\dots,u_{M}.$ Choosing the function $f(u)$ in the form $f(u)=\ln\left[(1-\mu u)/(1-\nu u)\right]$ we obtain after the integration by parts the logarithm of the largest eigenvalue as function of $B$: $$\begin{aligned}
\ln\Lambda_{0}(\gamma) & =\left(\mu-\nu\right) & \oint_{\Gamma_{0}}\frac{\ln\left[1-\frac{B(1-\nu u)^{N}}{(1-u)^{N}u^{M}}\right]}{\left(1-\mu u\right)(1-\nu u)}\frac{du}{2\pi i}.\label{Labda(B)}\end{aligned}$$ Here the integration is over the contour satisfying to the condition $|B(1-\nu u)^{N}/(1-u)^{N}u^{M}|<1$ and enclosing $M$ roots of $\mathcal{P}(u)$ located near the origin. Note that the contour does not cross any branch cuts of the logarithm, which can be chosen connecting $M$ roots inside the contour to the origin and the other $N$ roots outside the contour to $u=1.$ Such a contour exists if $|B|$ is small enough. The relation of $\gamma$ and $B$ can be recovered from the translation invariance condition (\[eq:translation invariance\]), which after taking a logarithm and going to the variables $u_{i}$ yields $$\gamma=\frac{1-\nu}{M}\oint_{\Gamma_{0}}\frac{\ln\Bigg(1-\frac{B(1-\nu u)^{N}}{(1-u)^{N}u^{M}}\Bigg)}{\left(1-u\right)\left(1-\nu u\right)}\frac{du}{2\pi i}.\label{gamma(B)}$$ Note also, that what we actually integrated by parts to arrive at formulas (\[Labda(B)\]) and (\[gamma(B)\]) were not exactly the original expressions given by (\[eq:sum over roots\]), but the ones obtained by addition of terms analytic inside the contour of integration, which, hence, are integrated out to zero.
To evaluate the integrals we use a series expansion of the logarithms in powers of $B$ and integrate the resulting series term by term. Integrations can explicitly be performed in terms of the Appell and Gauss hypergeometric functions:
$$\begin{aligned}
\ln\Lambda_{0}(\gamma) & = & -(\mu-\nu)\sum_{n=1}^{\infty}\frac{B^{n}}{n}\binom{Ln-2}{Mn-1}F_{1}\left(1-nM;1-nN,1;2-nL;\nu,\mu\right),\label{eq:Lambda-ser}\\
\gamma & = & -\frac{1-\nu}{M}\sum_{n=1}^{\infty}\frac{B^{n}}{n}\binom{Ln-1}{Mn-1}\left._{2}F_{1}\right.\left(\begin{array}{c}
1-Mn,1-Nn\end{array};1-nL;\nu\right).\label{eq:gamma-ser}\end{aligned}$$
The scaled cumulants $$c_{n}\equiv\lim_{t\to\infty}\left.\frac{\partial^{n}}{t\partial\gamma^{n}}\left\langle e^{\gamma Y_{t}}\right\rangle \right|_{\gamma=0}=\left.\frac{\partial^{n}}{\partial\gamma^{n}}\ln\Lambda_{0}(\gamma)\right|_{\gamma=0}.$$ of the particle current $Y_{t}$ can be obtained as the coefficients of the power expansion of $\ln\Lambda_{0}(\gamma)$ in $\gamma$, which can be constructed to any finite order by eliminating $B$ between the two series. In particular, using the Euler transformation for hypergeometric functions (\[eq:Euler\]) the first scaled cumulant $c_{1}=\lim_{t\to\infty}t^{-1}\left\langle Y_{t}\right\rangle $, the number of particle jumps per unit time, can be shown to coincide with $J$ from (\[eq:J-1\]) obtained by averaging over the stationary state. The second cumulant, the scaled variance of $Y_{t}$, is related to the diffusion coefficient for a particle $\Delta=M^{-2}c_{2}$. The exact value of the latter is
$$\begin{aligned}
\Delta & =\lambda p\frac{\binom{2L-2}{2M-1}}{\binom{L-1}{M-1}^{2}} & \left[\frac{(2L-1)}{2(L-1)}\frac{F_{1}(1-M;1-N,1;2-L;\nu,\mu)\left._{2}F_{1}\right.\left(1-2M,1-2N,1-2L;\nu\right)}{\left[\left._{2}F_{1}\right.\left(1-M,1-N,1-L;\nu\right)\right]^{3}}\right.\\
& & \left.\hspace{60mm}-\frac{F_{1}(1-2M;1-2N,1;2-2L;\nu,\mu)}{\left[\left._{2}F_{1}\right.\left(1-M,1-N,1-L;\nu\right)\right]^{2}}\right],\end{aligned}$$
from which, using the identities for Gauss and Appell functions we can recover the corresponding quantities for particular cases of PU, $\mu=0$, BSU, $\nu=0$ and the DA limit $\mu\to\nu=1,p=const$. The exact formula of cumulant $c_{n}$ is already rather cumbersome for $n=2$, and it becomes more and more complicated as $n$ grows. Of major interest is the scaling behavior of the cumulants and the whole function $\Lambda_{0}(\gamma),$ which is also related to the LDF of particle current.
Scaling limits \[sub:Scaling-limits\]
-------------------------------------
We would like to investigate the thermodynamic limit $$M,N\to\infty,M/N=\rho.$$ The structure of terms of the series obtained for $\Lambda_{0}(\gamma)$ and $\gamma$ is very similar to that of the integrals analyzed above for the partition function and average current. Thus, the same asymptotic analysis is applicable. Again, depending on the scale of $\lambda$ there are two different regimes: the first, where the integrals can be analyzed in the saddle point approximation and the second where the integrals can be evaluated in terms of modified Bessel functions. It is worth emphasizing again that the saddle point approximation is valid at any scale of $\lambda$ satisfying $\lambda N^{-2}\to0.$ Thus, the universal KPZ scaling function obtained in this approximation holds through a range of scales, with the scale entering only to the non-universal scaling constants. Then, the as the parameter $\lambda N^{-2}$ varies from zero to infinity, the KPZ-Gauss transition takes place.
### KPZ regime, $\lambda N^{-2}\to0,\gamma\to0,\gamma\lambda^{1/4}N^{3/2}=const,$
Up to the $1/n$ factor and the common factors before the integrals the terms of the order $n$ of series (\[eq:Lambda-ser\]) and (\[eq:gamma-ser\]) are given by the integrals $\mathcal{I}_{nN}(h(z),g(z))$ of the form (\[eq:INT\]) with the function $h(z)$ defined in (\[eq:h(z)\]) and instead of the function $g(z)$ we substitute $r(z)=z\left[\left(1-\mu z\right)\left(1-\nu z\right)\right]^{-1}$ for $\Lambda_{0}(\gamma)$ series and $s(z)=z\left[\left(1-z\right)\left(1-\nu z\right)\right]^{-1}$ for the terms of $\gamma$ series. To obtain the meaningful precision it is enough to keep only leading order terms in asymptotics of $\gamma$ and the next to the leading order terms for the eigenvalue. Evaluating $\mathcal{I}_{nN}(h(z),r(z))$ and $\mathcal{I}_{nN}(h(z),s(z))$ with the help of (\[eq:INT ASYMP\]) in two leading orders we obtain the universal scaling form obtained first in [@Derrida; @Lebowitz], $$\ln\Lambda(\gamma)=J_{\infty}\gamma+aN^{-3/2}G(bN^{3/2}\gamma),\label{eq:ldf-kpz}$$ where $G(z)$ has a parametric representation $$\begin{aligned}
G(z) & = & -\mbox{Li}_{5/2}(t),\,\,\, z=-\mbox{Li}_{3/2}(t),\label{eq:Der-Lib}\end{aligned}$$ via the polylogarithm function $\mbox{Li}_{s}(x)=\sum_{i>0}x^{i}/i^{s}$. The infinite volume current $$J_{\infty}=Mpr_{0}/s_{0}=Lj^{ASEP}$$ coincides with the particle current obtained from the averaging over stationary state, the coefficients $a$ and $b,$ $$a=\frac{\mu-\nu}{2\sqrt{2\pi|h_{2}|}}\left(\frac{r_{2}-s_{2}/s_{0}}{|h_{2}|}+\frac{(r_{1}-s_{1}/s_{0})h_{3}}{h_{2}^{2}}\right)$$ and $$b=\frac{\sqrt{2\pi|h_{2}|}}{\rho s_{0}(1-\nu)}$$ are the non-universal model-dependent constants expressed via the coefficients of expansion of the functions $h(z),r(z),s(z)$ in the dominant saddle point $z_{-}$: $r_{k}=\left(\mbox{i}z\partial_{z}\right)^{k}r(z)|_{z=z_{-}}$, $s_{k}=\left(\mbox{i}z\partial_{z}\right)^{k}s(z)|_{z=z_{-}}$ and $h_{k}=\left(\mbox{i}z\partial_{z}\right)^{k}h(z)|_{z=z_{-}}$ . The explicit expressions of these constants can be found in Appendix \[sec:Explicit-expressions-of\] .
It is clear from (\[eq:ldf-kpz\]) that nontrivial scaling occurs, when $\gamma$ is of order of $b^{-1}N^{-3/2},$ which is of order of $\lambda^{-1/4}N^{-3/2}$ as $\lambda\to\infty.$ The scaling form (\[eq:ldf-kpz\]) suggests that at large time deviations of the time-averaged current from its thermodynamic value are of the form $$\mbox{Prob}\left(Y_{t}/t<y\right)\sim\exp\left[-atN^{-3/2}\widehat{G}\left(\frac{y-J_{\infty}}{ab}\right)\right],\label{eq:ldf}$$ where the scaling function $\widehat{G}(x)=\sup_{t}\left(xt-G(t)\right)$ is a Legendre transform of function $G(t)$. Appearance of the factor $tN^{-3/2}$ is the universal KPZ-specific feature, which is akin to the fact that the dynamical exponent of the KPZ class is $z=3/2.$ Specifically, the term $atN^{-3/2}$ is the parameter, which is supposed to be large, for the large deviation approximation to be good. This is in correspondence with the results of [@Derrida; @Lebowitz], up the fact that factor $tN^{-3/2}$ is corrected by the prefactor $a,$ which decays as $a\sim\lambda^{-1/4}$ as $\lambda\to\infty.$ When $\lambda$ grows with $N$ as $\mbox{\ensuremath{\lambda\sim}}N^{\alpha}$, we expect that the dynamical exponent varies continuously as $z=3/2+\alpha/4$ from KPZ, $z=3/2$, to diffusive, $z=2,$ value as $\alpha$ varies from $\alpha=0$ to $\alpha=2$. As was discussed above, the method of asymptotical analysis used is valid also for $\lambda$ growing with $N$ slower than $N^{2}.$ Hence, the applicability of the Derrida-Lebowitz scaling form of the LDF extends to systems with larger than KPZ characteristic time scales, until the scaling becomes diffusive, $t\sim N^{2}$. In the DA limit all particles stick together into the cluster, which performs an ordinary random walk making $M$-step jumps at a time. The central part of its LDF is expected to be pure Gaussian $\mbox{Prob}\left(Y_{t}/t<y\right)\sim\exp\left(-tM^{-2}y^{2}/2\right).$ This indicates that the transition regime is expected on the scale $\lambda\sim N^{2}.$
The cumulants of $Y_{t}$ can be obtained by differentiating the function (\[eq:ldf-kpz\]). In particular, from the first derivative we obtain the mean number of particle jumps per unit time up to the first order finite size correction that was already obtained in (\[eq:current correctioon (asymp)\]). Note that it is the value of this correction, $$\left(j^{ASEP}(L)-j^{ASEP}((\infty)\right)L=ab,\label{eq:current correction -2}$$ which is the denominator of the argument of the LDF $\hat{G}$ in (\[eq:ldf\]). As noted in [@Derrida; @Lebowitz] the applicability of the scaling form of the LDF obtained is limited by the condition that the argument of the function $\widehat{G}(x)$ is of the order of one. Therefore, the denominator plays the role of the scale for the deviation of time-averaged number of jumps from its mean value, in which the scaling form (\[eq:ldf\]) is valid. Remarkably, its value stays finite when $\lambda$ grows to infinity. The diffusion coefficient for one particle, related to the second cumulant, is $$\Delta=\frac{\left(1-c\right)^{3/2}}{c^{2}}\frac{b^{2}a}{2\sqrt{2L}},\label{eq:Delta exact}$$ decaying as $L^{-1/2}$, which is specific for the KPZ class. In the limit $\lambda\to\infty,$ we have $$\Delta\simeq\frac{\lambda^{1/4}}{\sqrt{L}}\frac{3}{4}\sqrt{\frac{\pi}{2}}\frac{p(1-p)}{\left[c(1-c)\right]^{1/4}},\label{eq:Delta lambda->infty}$$ which again signals that when $\lambda\sim L^{2}$ the motion of particles changes from subdiffusive to diffusive. However, as it was discussed above, the saddle point method fails at this scale of $\lambda$ and we should again use different asymptotic analysis.
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### The transition regime: $\lambda/N^{2}=const$, $\gamma N^{2}=const$
When $\lambda\sim N^{2}$, the integrals $\mathcal{I}_{nN}(h(z),r(z))$ and $\mathcal{I}_{nN}(h(z),s(z))$ can be evaluated in terms of the modified Bessel functions of the first kind as in the second part of Sec. \[sub:Asymptotic-analysis\]. As a result we obtain the parametrically defined function $\ln\Lambda(\gamma)$
$$\begin{aligned}
\ln\Lambda(\gamma) & =\gamma pM+N^{-2} & p(1-p)\mathcal{G}_{\theta}(N^{2}\rho\gamma),\label{eq:Lambda - transition}\end{aligned}$$
where the function $\mathcal{G}_{\theta}\left(t\right)$ depending on the transition parameter $\theta$ defined in (\[eq:theta\]) has the following parametric form: $$\mathcal{G}_{\theta}\left(t\right)=\frac{\theta^{2}}{4}\sum_{k=1}^{\infty}I_{2}(k\theta)\frac{B^{k}}{k},\,\,\, t=-\frac{\theta}{2}\sum_{k=1}^{\infty}I_{1}(k\theta)\frac{B^{k}}{k}.$$ To observe the nontrivial scaling, $\gamma$ must scale with $N$ so that the limit $\gamma N^{2}=const$ holds.
Using an asymptotic form of the modified Bessel functions (\[eq:Bessel large x\]) for large $\theta$ and small $t$, such that $t^{2}\theta=const,$ we find $$\mathcal{G_{\theta}}(t)\simeq-\frac{\theta t}{2}+\frac{3}{8}\sqrt{\frac{\theta}{2\pi}}G\left(t\sqrt{\frac{8\pi}{\theta}}\right),\,\,\,\theta\to\infty$$ where $G(x)$ is the Derrida-Lebowitz scaling function (\[eq:Der-Lib\]).
In the opposite DA limit the particles in a finite system form a single cluster of $M$ particles, which move together performing the Bernoulli random walk. In this case the exact cumulant generating function is $$\begin{aligned}
\lim_{t\to\infty}t^{-1}\ln\left\langle e^{\gamma Y_{t}}\right\rangle & = & \ln\left(1-p+pe^{\gamma M}\right).\\
& \simeq & Mp\gamma+M^{2}p(1-p)\frac{\gamma^{2}}{2},\end{aligned}$$ where in the second line we show the two first terms of the small $\gamma$ expansion. These are the only terms responsible for the limit $\lim_{M\to\infty}M^{2}\left(\lim_{t\to\infty}t^{-1}\ln\left\langle e^{\gamma Y_{t}}\right\rangle -Mp\gamma\right)=M^{4}\gamma^{2}$ to exist under condition $\gamma M^{2}=const$. This agrees with the behavior of (\[eq:Lambda - transition\]) at small $\theta$, which follows from the limiting form of function $\mathcal{G}_{\theta}\left(t\right)$: $$\mathcal{G}_{\theta}\left(t\right)\simeq\frac{t^{2}}{2}-\frac{\theta^{2}t}{8},\,\,\,\theta\to0.$$ The distribution of the time-averaged number of particle jumps corresponding to (\[eq:Lambda - transition\]) has the form $$\mbox{Prob}\left(Y_{t}/t<y\right)\sim\exp\left[-\frac{p(1-p)t}{N^{2}}\mathcal{\widehat{G}}_{\theta}\left(\frac{y-Mp}{\rho p(1-p)}\right)\right],$$ where LDF $\mathcal{\widehat{G}}_{\theta}(x)$ is the Legendre transform of $\mathcal{G}_{\theta}\left(t\right).$ The presence of the factor $tN^{-2}$ is specific for the diffusive systems, though the LDF has a nontrivial form, unlike purely quadratic Gaussian single particle case. It follows from the above analysis of the limiting behavior of $\mathcal{G}_{\theta}\left(t\right)$ that the LDF $\mathcal{\widehat{G}}_{\theta}(x)$ continuously interpolates between the Gaussian quadratic and the KPZ scaling function as $\theta$ varies from zero to infinity.
Differentiating $\Lambda(\gamma)$ we obtain the cumulants of this distribution. The first one is $$J\simeq Mp-p(1-p)\rho\frac{\theta}{2}\frac{I_{2}(\theta)}{I_{1}(\theta)},$$ i.e. a finite number of macroscopic clusters present on the lattice for a finite fraction of time results in a finite correction to the total number of jumps, which exactly equals to $Mp$ in the DA limit. In the small-$\theta$ limit this fraction is approximately $\theta^{2}/2$. Making a small-$\theta$ expansion one can see that this contribution, $p(1-p)\rho\theta^{2}/8$, is indeed of the same order also depending on particle density.
The one-particle diffusion coefficient obtained from the second cumulant,
$$\Delta=p(1-p)\left[\frac{I_{1}(2\theta)}{I_{1}^{2}(\theta)}\left(\frac{I_{2}(2\theta)}{I_{1}(2\theta)}-\frac{I_{2}(\theta)}{I_{1}(\theta)}\right)\right],\label{1}$$
is finite in the thermodynamic limit, similarly to the one-particle random walk, when it exactly equals $p(1-p)$. In the KPZ-DA transition regime, this value is corrected by the factor in the square brackets. In the limit $\theta\to0$ this factor saturates to one, recovering the free-particle diffusion coefficient. As $\theta\to\infty$ the diffusion coefficient behaves as $\Delta\simeq(3p(1-p)/4)\sqrt{\pi/\theta}$ indicating the transition to KPZ behavior. What is different from the DA limit as well as from the KPZ regime is the behavior of the higher cumulants. As follows from the formula (\[eq:Lambda - transition\]) they have the scaling $c_{n}\sim N^{2(n-1)}$ unlike $c_{n}\sim N^{3/2(n-1)}$ in the KPZ regime and $c_{n}\sim N^{n}$ in the DA limit. It is remarkable that in the transition regime the order of growth of the cumulants with $N$ is higher than in both the KPZ and the DA limit for $n\geq3$. Then, it is natural to expect that the cumulants, rescaled to remove the dependence on the system size and all the other parameters except $\theta$, $$\tilde{c}_{n}=\lim_{N\to\infty}c_{n}\times\left[N^{2(n-1)}\rho^{n}p(1-p)\right]^{-1}=\mathcal{G}_{\theta}^{(n)}(0)$$ will vanish in both $\theta\to0$ and $\theta\to\infty$ limits having an extrema at some finite values of $\theta$. Indeed, as seen from Fig. \[fig:The-rescaled-cumulants\] the third and fourth rescaled cumulants show the non-monotonous behavior having minimum and maximum at some finite values of $\theta,$ respectively. The quantity, which can be used as a measure of proximity to the KPZ regime, is the universal cumulant ratio, $$R(\theta)=\frac{c_{3}^{2}}{c_{2}c_{4}}=\frac{\left(\mathcal{G}_{\theta}^{(3)}(0)\right)^{2}}{\mathcal{G}_{\theta}^{''}(0)\mathcal{G}_{\theta}^{(4)}(0)},$$
![Universal cumulant ratio $R(\theta).$ The dashed line shows the limiting KPZ value $R(\infty).$\[fig:Universal-cumulant-ratio\]](ucr){width="0.9\columnwidth"}
depending solely on the parameter $\theta$. As shown in Fig. \[fig:Universal-cumulant-ratio\], starting from zero at $\theta=0$ the ratio $R(\theta)$ monotonously approaches its limiting universal KPZ value $$\lim_{\theta\to\infty}R(\theta)=\frac{2\left(3/2-8/3^{3/2}\right)^{2}}{15/2-24/\sqrt{3}+9/\sqrt{2}}\simeq0.41517,$$ first obtained in [@Derrida; @Appert].
Universality and relation to KPZ equation \[sec:Universality-and-relation\]
===========================================================================
In context of stochastic models, the concept of universality suggests that in the scaling limit a large class of models is characterized by probability distributions having the same universal functional form. The notion of the scaling limit implies that the temporal and spacial coordinates as well as the random variables of interest are measured in scales related to each other via simple power laws. Their exponents, usually referred to as critical exponents, is a fixed set of numbers specifying given universality class. In this way the scales are defined up to non-universal constants, which depend on parameters of the specific model. Correspondingly, numerical quantities characterizing the random variables, e.g., cumulants or correlators, depend on these constants only. As applied to the problem of KPZ interface growth in one-dimensional system of size $L$, the distribution of the height $h(x,t)$ of growing interface being a random function of the spatial and temporal coordinates $x$ and $t$ is characterized by two sets of amplitudes [@krug; @meakin; @halpin-healy],
$$\begin{aligned}
a_{n} & = & \lim_{t\to\infty}\lim_{L\to\infty}t^{-n/3}\left\langle \left(h(x,t)-\overline{h}\right)^{n}\right\rangle _{c}\label{eq:a_n}\\
b_{n} & = & \lim_{L\to\infty}\lim_{t\to\infty}L^{-n/2}\left\langle \left(h(x,t)-\overline{h}\right)^{n}\right\rangle _{c}\label{eq:b_n}\end{aligned}$$
for transient, $t\ll L^{3/2}$, and stationary, $t\gg L^{3/2},$ parts of evolution respectively, where $n\in\mathbb{N}$, $\left\langle x^{n}\right\rangle _{c}$ is the notation for $n$-th cumulant of the random variable $x$, and $\bar{h}=L^{-1}\int_{0}^{L}h(x,t)dx$ is the mean interface height for a given process realization. Also one can define finite time (size) corrections to the average interface velocity as compared to the one calculated at infinite time (in infinite system), $$\begin{aligned}
a_{v} & = & \lim_{t\to\infty}\lim_{L\to\infty}t^{2/3}\left(\left\langle \partial h/\partial t\right\rangle -v_{\infty}\right)\label{eq:a_v}\\
b_{v} & = & \lim_{L\to\infty}\lim_{t\to\infty}L\left(\left\langle \partial h/\partial t\right\rangle -v_{\infty}\right),\label{eq:b_v}\end{aligned}$$ where $v_{\infty}=\lim_{t,L\to\infty}\left\langle \partial h/\partial t\right\rangle .$ It was conjectured in [@krug; @meakin; @halpin-healy] that all these quantities can be expressed in terms of only two dimensional invariants. The conjecture was first proposed based on analysis of the KPZ equation itself, $$\frac{\partial h}{\partial t}=\widetilde{\nu}\Delta h+\widetilde{\lambda}\left(\nabla h\right)^{2}+\eta,\label{eq:KPZ}$$ which was the first prototypical model catching the universal features of the KPZ class. Here the notations for parameters $\widetilde{\lambda}$ and $\widetilde{\nu}$ have a tilde to keep the notations traditional for KPZ equation and distinguish them from the $\nu$ and $\lambda$ of our model. The white noise $\eta$ is fully characterized by the covariance $$\left\langle \eta(x,t)\eta(x',t')\right\rangle =D\delta(x-x')\delta(t-t').$$ For an interface described by (\[eq:KPZ\]) the two mentioned dimensional invariants are $\widetilde{\lambda}$ and $A=D/2\widetilde{\nu}.$ In terms of these constants the transient amplitudes (\[eq:a\_n\]) and (\[eq:a\_v\]) are given by $$a_{n}=\left(\left|\widetilde{\lambda}\right|A^{2}\right)^{n/3}\widetilde{a}_{n}\,\,\,\mbox{and}\,\,\, a_{v}=(|\widetilde{\lambda}|A^{2})\widetilde{a}_{v},$$ where $\widetilde{a}_{n}$ and $\widetilde{a}_{v}$ are universal numbers. These numbers, as known from the later development of the field, must be related to cumulants of the universal distributions like the Tracy-Widom distributions, dependent on global form (large-scale) of initial conditions. For the stationary amplitudes (\[eq:b\_n\]) and (\[eq:b\_v\]) in the system with periodic boundary conditions we have $$b_{2}=\frac{A}{12},\,\,\, b_{n}=0,\,\,\, n>2\label{eq:b_2}$$ and $$b_{v}=-\frac{A\widetilde{\lambda}}{2}.\label{eq:b_v univ}$$ Vanishing of all amplitudes $b_{n}$ except the second one is due to the Gaussian stationary height distribution of the KPZ interface. The universality conjectured in [@krug; @meakin; @halpin-healy] suggests that for an interface belonging to the KPZ class, the amplitudes (\[eq:a\_n\])–(\[eq:b\_v\]) have the same dependence on $\widetilde{\lambda}$ and $A$, which can in general be defined without appealing to the KPZ equation and measured experimentally. Namely, the parameter $\widetilde{\lambda}$, related to the response of the interface velocity to introducing a small tilt $h(x,t)\to h(x,t)+\kappa x$, is defined as $$\widetilde{\mbox{\ensuremath{\lambda}}}=\frac{\partial^{2}v_{\infty}}{\partial\kappa^{2}},\label{eq:lambda-def}$$ and the parameter $A$ is the amplitude of spacial correlation function $$\lim_{t\to\infty}\left\langle \left(h(x,t)-h(y,t)\right)^{2}\right\rangle _{c}=A\left|x-y\right|.\label{eq:A-def}$$
The above studied ASEP-like system can be related to the KPZ interface on the lattice by $$h_{i+1}-h_{i}=1-2\tau_{i},$$ where $\tau_{i}=0,1$ is the occupation number of the $i$th site and $h_{i}$ is the interface height above the bond connecting sites $i-1$ and $i$ of the lattice, $i=1,\dots,L.$ For this mapping being consistent with the number of particles on the lattice the interface must satisfy helicoidal boundary conditions $$h_{i+L}=h_{i}-(L-2M),$$ which gives a tilt $\kappa=1-2c$ to the interface. Then the change of the interface height $\left(h_{i}(t)-h_{i}(0)\right)$ in time is nothing but twice the number of particles that have traversed the bond $(i-1,i)$ by the time $t$. Correspondingly the limiting interface speed is twice the particle current in the ASEP-like system, $v_{\infty}=2j^{ASEP}$, where $j^{ASEP}$ was obtained in (\[eq:j\^TASEP\]). Then we have $$\widetilde{\lambda}=\frac{1}{2}\frac{\partial^{2}j^{ASEP}}{\partial c^{2}}.$$ Using (\[eq:covariance\]) we obtain for $i\ll j$ $$\begin{aligned}
\left\langle \left(h_{i}-h_{j}\right)^{2}\right\rangle _{c} & = & 4\sum_{i\leq k,l\leq j+1}\left(\left\langle \tau_{k}\tau_{l}\right\rangle -c^{2}\right)\\
& \simeq & 4c(1-c)\coth\left(\frac{1}{2\xi}\right)\left|i-j\right|;\end{aligned}$$ i.e., $A=4c(1-c)\coth(1/(2\xi))$, where $\xi$ is the correlation length (\[eq:corr length\]). The explicit expressions of $\widetilde{\lambda}$ and $A$ can be found in Appendix \[sec:Explicit-expressions-of\]. Also the finite size correction $b_{v}$ to the limiting interface velocity is twice the correction to the particle current given in (\[eq:J asymp\]) and (\[eq:current correction -2\]). One confirmation of the universality is the observation that the relation (\[eq:b\_v\]) between $b_{v}$ and the parameters $\widetilde{\lambda}$ and $A$, defined by (\[eq:lambda-def\]) and (\[eq:A-def\]) respectively, holds exactly (see appendix \[sec:Explicit-expressions-of\].).
Another demonstration of universality can be obtained using the results of Sec. \[sec:Statistics-of-particle\]. Note that the amplitudes (\[eq:a\_n\])–(\[eq:b\_v\]) characterize the form of the interface relative to its average position $\bar{h}.$ At the same time the absolute value of the interface height is dominated by the position of its center of mass, which, up to the bounded initial value, is $\overline{h}\simeq2L^{-1}Y_{t}$. Therefore, the universal LDF obtained for $Y_{t}$ also characterizes the statistics of the motion of the center of mass of interface. On the other hand, its scaling properties are expected to be defined by the dimensionful invariants $\widetilde{\lambda}$ and $A$ solely. In particular, simple dimensional arguments together with the scaling ansatz show [@Krug; @review] that the variance of $\overline{h},$ related to the diffusion coefficient of a particle by $\left\langle \overline{h}^{2}\right\rangle _{c}=4c^{2}\Delta t,$ has the form $$\left\langle \overline{h}^{2}\right\rangle _{c}=s_{0}A^{3/2}|\widetilde{\lambda}|L^{-1/2}t,\label{eq:h bar universal}$$ where $s_{0}=\sqrt{\pi}/4$ is a universal number first obtained in [@DEM]. Comparing this formula with the expression (\[eq:Delta exact\]) obtained for $\Delta$, we obtain relation (\[eq:b(A,c)\]) between $b$, $A$, and the density $c$, which is indeed confirmed from explicit calculations in Appendix \[sec:Explicit-expressions-of\]. The same arguments can be applied to the cumulants of an arbitrary order. In general, all the model dependence of the scaled cumulant generating function (and hence of the LDF) obtained in Sec. \[sec:Statistics-of-particle\] is incorporated into two constants $a$ and $b$. It takes some algebra to show that these constants can be reexpressed in terms of the dimensional invariants $A$ and $\widetilde{\lambda}$ of this section: $$\begin{aligned}
a & = & \frac{\sqrt{2A}\left|\widetilde{\lambda}\right|(1-c)^{3/2}}{4\sqrt{\pi}},\label{eq:a(A,lambda)}\\
b & \mbox{=}- & \mbox{sgn}\,\widetilde{\lambda}\,\frac{\sqrt{\pi A/2}}{(1-c)^{3/2}}.\label{eq:b(A,lambda)}\end{aligned}$$ In the spirit of universality we conjecture this relation to be universal. Up to our knowledge, it did not yet explicitly appear in the literature.
Finally it is informative to see how the system approaches the DA limit. As $\lambda\to\infty$, we asymptotically have $$A\simeq8\sqrt{\lambda}\left[c(1-c)\right]^{3/2}\,\,\,\mbox{and\,\,\,}\widetilde{\lambda}\simeq-\frac{3(1-p)p}{8(1-c)^{5/2}c^{1/2}}\sqrt{\frac{1}{\lambda}}.$$ As we saw, the KPZ regime (in particular, the universal scaling form of the LDF of interface height) holds until the value of $\lambda$ becomes of the order of $\lambda\sim N^{2},$ i.e., $A$ and $\widetilde{\lambda}$ being of order $L$ and $1/L$, respectively. Remarkably the product $\widetilde{\lambda}A$ proportional to $b_{v}$, which is related to the typical fluctuation range, stays finite in the limit $\lambda\to\infty$. Therefore, first, up to the scale $\lambda\sim N^{2},$ the increase of $\lambda$ affects only non-universal constants preserving the universal functional form (\[eq:ldf-kpz\]) of the LDF. Then, in the scale $\lambda\sim N^{2}$ the functional form of LDF starts to gradually change until reaching the purely Gaussian form.
We acknowledge the financial support from the Government of the Russian Federation within the framework of the implementation of the 5-100 Programme Roadmap of the National Research University Higher School of Economics. The work was also supported by the RFBR grant under Project 14-01-00474-a and by Heisenberg-Landau program.
Explicit expressions of model-dependent constants and universal relations\[sec:Explicit-expressions-of\]
========================================================================================================
Though the scaling functions found from the asymptotic analysis are of a rather simple form, the model-dependent constants expressed as functions of particle density are very cumbersome. It is much more efficient to consider them as functions of associated fugacities. In particular, the fugacity $z_{-}$ appears in the analysis of the stationary measure of the ZRP-like system in Sec. \[sub:Asymptotic-analysis\] as the saddle point, where the main contribution to the integrals comes from. Its relation to the particle density in the TASEP-like system is given in (\[eq:z\_pm\]), or conversely $$c=\frac{(1-\nu)z_{-}}{1-\nu\left(2-z_{-}\right)z_{-}}.\label{eq:c(z_)}$$ Then we obtain the mean number of particle jumps per unit time,
$$\begin{aligned}
J & =N\frac{z_{-}(\mu-\nu)}{\left(1-\mu z_{-}\right)\left(1-\nu z_{-}\right)}\\
& +\frac{(1-\mu)(\mu-\nu)}{(1-\nu)}\frac{\left(1-z_{-}\right)z_{-}\left(1-\nu z_{-}\right)\left(1-\mu\nu z_{-}^{3}\right)}{\left(1-\mu z_{-}\right){}^{3}\left(1-\nu z_{-}^{2}\right){}^{2}},\nonumber \end{aligned}$$
which being divided by size of the system $L$, yields the thermodynamic value of the mean particle current, $$\begin{aligned}
j_{\infty} & =(1-c)\frac{z_{-}(\mu-\nu)}{\left(1-\mu z_{-}\right)\left(1-\nu z_{-}\right)}\\
& =\frac{(\mu-\nu)\left(1-z_{-}\right)z_{-}}{\left(1-\mu z_{-}\right)\left(1-\nu\left(2-z_{-}\right)z_{-}\right)},\nonumber \end{aligned}$$ plus the $1/L$ finite-size correction
$$\begin{aligned}
L(j_{L}-j_{\infty}) & =\frac{(1-\mu)(\mu-\nu)}{(1-\nu)}\label{eq: current correction (z_)}\\
\times & \frac{\left(1-z_{-}\right)z_{-}\left(1-\nu z_{-}\right)\left(1-\mu\nu z_{-}^{3}\right)}{\left(1-\mu z_{-}\right){}^{3}\left(1-\nu z_{-}^{2}\right){}^{2}}.\nonumber \end{aligned}$$
These expressions show up again in Sec. \[sub:Scaling-limits\], where the latter one appears to be a product of the two model-dependent constants,
$$\begin{aligned}
a & =\frac{(1-\mu)(\mu-\nu)}{\sqrt{2\pi}(1-\nu)^{3/2}}\label{eq:a(z_)}\\
& \times\frac{\sqrt{z_{-}}\left(1-z_{-}\right){}^{2}\left(1-\nu z_{-}\right){}^{2}\left(1-\mu\nu z_{-}^{3}\right)}{\left(1-\mu z_{-}\right){}^{3}\left(1-\nu z_{-}^{2}\right){}^{5/2}},\nonumber \\
\nonumber \\
b & =\frac{\sqrt{2\pi(1-\nu)z_{-}\left(1-\nu z_{-}^{2}\right)}}{\left(1-z_{-}\right)\left(1-\nu z_{-}\right)},\label{eq:b(z_)}\end{aligned}$$
which also determine the scaling behavior of all higher cumulants, such as diffusion coefficient, as well as the characteristic temporal and fluctuation scales. Indeed, multiplying (\[eq:a(z\_)\]) and (\[eq:b(z\_)\]) we obtain exactly (\[eq: current correction (z\_)\]) confirming the announced relation (\[eq:current correction -2\]).
Another fugacity $z^{*},$ given in (\[eq:z\^\*\]), appears in Sec. \[sub:Transfer-matrix-approach\] from transfer-matrix analysis of the TASEP-like system. The two seemingly unrelated sets of results obtained from ZRP-like and TASEP-like systems turn out to be linked, when one checks the universal relations between the two-dimensional invariants obtained in KPZ theory. One of the invariants, the amplitude of the correlation function (\[eq:A-def\]), is given in terms of $z^{*}$ by $$\begin{aligned}
A & =4c(1-c)\frac{\lambda_{1}+\lambda_{2}}{\lambda_{1}-\lambda_{2}}\label{eq:A}\\
& =\frac{4(1-\nu)z^{*}(z^{*}+1)}{\left((z^{*}+1)^{2}-4\nu z^{*}\right)^{3/2}},\nonumber \end{aligned}$$ while the other one, the non-linearity coefficient from KPZ equation, obtained from the second derivative of particle current depending on $z_{-}$:
$$\begin{aligned}
\widetilde{\lambda} & =\frac{1}{2}\frac{\partial^{2}j_{\infty}}{\partial c^{2}}=\frac{1}{2}\left(\frac{1}{\partial c/\partial z_{-}}\frac{\partial}{\partial z_{-}}\right)^{2}j_{\infty}\\
& =-\frac{(1-\mu)(\mu-\nu)}{(1-\nu)^{2}}\frac{\left(1-\nu\left(2-z_{-}\right)z_{-}\right){}^{3}\left(1-\mu\nu z_{-}^{3}\right)}{\left(1-\mu z_{-}\right){}^{3}\left(1-\nu z_{-}^{2}\right){}^{3}}.\nonumber \end{aligned}$$
Noting that the fugacities are related by $$z^{*}=\frac{z_{-}\left(\nu z_{-}-1\right)}{z_{-}-1}\label{eq:z^* vs z_}$$ we find that the product of the dimensional invariants is equal to $$\widetilde{\lambda}A=-2b_{v},$$ where $b_{v}=2ab=2L(j_{L}-j_{\infty}),$ which is nothing but the relation (\[eq:b\_v\]). Another relation,
$$b=\frac{\sqrt{\pi A}}{\sqrt{2}\left(1-c\right){}^{3/2}},\label{eq:b(A,c)}$$
can be verified by direct examining formulas (\[eq:c(z\_)\],\[eq:b(z\_)\],\[eq:A\],\[eq:z\^\* vs z\_\]). This complies with another prediction of KPZ theory (\[eq:h bar universal\]) in conjunction with the formula (\[eq:Delta exact\]) obtained for the diffusion coefficient. Then the connection (\[eq:a(A,lambda)\],\[eq:b(A,lambda)\]) between $a,b$ and $\widetilde{\lambda},A$ is straightforward.
Hypergeometric functions\[sec:Hypergeometric-functions\]
=========================================================
Gauss hypergeometric functions
------------------------------
Series representation: $$_{2}F_{1}(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}n!}x^{n}$$ Generating function for terminating series $_{2}F_{1}(a,b;c;x)$ with $a$ negative integer: $$\begin{aligned}
G(x,t) & \equiv & \frac{(1-xt)^{\alpha}}{(1-t)^{\beta}}\\
& = & \sum_{n=0}^{\infty}\frac{(\beta)_{n}}{n!}\left._{2}F_{1}\right.(-n,-\alpha,-\beta-n+1;x)t^{n}\end{aligned}$$ Euler transformation: $$F(a,b;c;z)=\left(1-z\right)^{c-a-b}F(c-a,c-b;c;z)\label{eq:Euler}$$ Chu-Vandermonde identity: $$\left._{2}F_{1}\right.(-n,-\alpha,-\beta-n+1;1)=\frac{(\beta-\alpha)_{n}}{(\beta)_{n}}\label{eq:Chu}$$
Appell hypergeometric function $F_{1}$
--------------------------------------
Series representations: $$\begin{aligned}
F_{1}(\alpha;\beta,\beta';\gamma;x,y) & = & \sum_{n,m=0}^{\infty}\frac{(\alpha)_{m+n}(\beta)_{m}(\beta')_{n}}{(\gamma)_{m+n}m!n!}x^{m}y^{n}\end{aligned}$$ Generating function for terminating series $F_{1}(\alpha;\beta,\beta';\gamma;x,y)$ with $\alpha$ negative integer : $$\begin{aligned}
G(x,y,z) & \equiv & \left(1-z\right)^{\alpha}(1-xz)^{-\beta}(1-yz)^{-\beta'}\\
& = & \sum_{n=0}^{\infty}\frac{(-\alpha)_{n}}{n!}F_{1}(-n;\beta,\beta';\alpha-n+1;x,y)z^{n}\end{aligned}$$
One-variable reduction:
$$F_{1}(\alpha;\beta,\beta';\gamma;x,0)=\left._{2}F_{1}\right.(\alpha,\beta;\gamma;x)\label{eq:F_1(0)}$$
Generalized Chu-Vandermonde identity $$F_{1}(-n;\beta,\beta';\alpha-n+1;x,1)=\frac{\left(\beta'-\alpha\right)_{n}}{(-\alpha)_{n}}\left._{2}F_{1}\right.(-n,\beta;\alpha-\beta'-n+1;x)\label{eq:Chu-1}$$
Modified Bessel function\[sec:Modified-Bessel-function\]
========================================================
Integral representation: $$I_{k}(y)=\int_{0}^{2\pi}\exp\left(y\cos\varphi+ki\varphi\right)\frac{d\varphi}{2\pi}$$ Asymptotic behavior: $$\begin{aligned}
I_{\alpha}(x) & = & \frac{e^{x}}{\sqrt{2\pi x}}\Big(1-\frac{4\alpha^{2}-1}{8x}\Big),\,\,\, x\to\infty\label{eq:Bessel large x}\\
I_{\alpha}(x) & = & \frac{1}{\alpha!}\Big(\frac{x}{2}\Big)^{\alpha},\,\,\, x\to0\label{eq:Bessel small x}\end{aligned}$$
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\[Bibliography\]
[ ${\mathbb{R}}$ & the real numbers\
${\mathbb{R}}_{+}$ & the non-negative real numbers\
${\mathbb{R}}^m$ & the $m$-dimensional Euclidean space\
${\mathbb{N}}$ & the natural numbers: $1, 2, 3, . . .$\
$ {\mathbb{B}}_{r} (\bar{x}) $ & closed ball of radius $ r > 0 $ centred at $\bar{x}$\
$ {\mathbb{B}}$ & closed unit ball, $ {\mathbb{B}}_{1} (0) $\
$ \mathrm{int\,} C $ & interior of the set $C$\
$ \mathrm{cl\,} C $ & closure of the set $C$\
$ x_k \longrightarrow x $ & the sequence $ (x_k) $ is convergent to $x$\
$ t_k \downarrow 0 $ & a sequence of positive numbers $ t_k $ tending to $ 0 $\
$ d(x, C) $ & distance from $x$ to the set $C$\
$e(C, D) $ & excess of the set $C$ beyond the set $D$\
$ | x | $ & absolute value of $ x \in {\mathbb{R}}$\
${\left \| x \right \|}$ & norm of $x$\
$ {\langle\,x \, , y \, \rangle} $ & canonical inner product, bilinear form\
$ | H |^{+} $ & outer norm\
$| H |^{-}$ & inner norm\
$ T(\bar{x};\,C) $ & Bouligand-Severi tangent cone (contingent cone) to the set $C$ at $\bar{x}$\
$\widetilde{T}(\bar{x};\, C)$ & Bouligand paratingent cone to the set $C$ at $\bar{x}$\
$\widehat{N}(\bar{x}; \,C) $ & Fréchet normal cone (regular normal cone) to the set $C$ at $\bar{x}$\
$N(\bar{x}; \,C)$ & Mordukhovich normal cone (limiting normal cone) to the set $C$ at $\bar{x}$\
$ F : X \rightrightarrows Y $ & set-valued mapping from $X$ into the subsets of $Y$\
$ f : X \to Y $ & function $f$ from $X$ into $Y$\
$ A^T $ & transpose of the matrix $A$\
$ \mathrm{ker\,} A$ & kernel of the the linear operator $A$\
$ \mathrm{det\, } A$ & determinant of the matrix $A$\
$ { \mathrm{gph} \, F } $ & graph of the mapping $F$\
$\mathrm{dom\,} F $ & domain of the mapping $F$\
$\mathrm{rge\,} F $ & range of the mapping $F$\
$\partial_F f(\bar{x}) $ & Fréchet subdifferential (F-subdifferential) of the function $f$ at $\bar{x}$\
$ \partial f(\bar{x}) $ & Mordukhovich (limiting) subdifferential of the function $f$ at $\bar{x}$\
$ \partial_{>} f(\bar{x}) $ & outer subdifferential of the function $f$ at $ \bar{x} $\
$ \partial_B h (\bar{u}) $ & Bouligand’s limiting Jacobian of the function $ h $ at $\bar{u}$\
$ \partial h (\bar{u}) $ & Clarke’s generalized Jacobian of $h$ at $\bar{u}$\
$ \nabla f (\bar{x}) $ & Jacobian matrix of the function $f$ at $\bar{x}$\
$D f(\bar{x}) $ & derivative of the function $f$ at $\bar{x}$\
$ DF{ (\bar{x}\, | \, \bar{y}) } $ & graphical derivative of the mapping $F$ [at $ \bar{x} $ for $ \bar{y} $]{}\
$ D^* F{ (\bar{x}\, | \, \bar{y}) } $ & coderivative of the mapping $F$ [at $ \bar{x} $ for $ \bar{y} $]{}\
$ \widetilde{D} F { (\bar{x}\, | \, \bar{y}) } $ & strict graphical derivative of the mapping $F$ [at $ \bar{x} $ for $ \bar{y} $]{}\
$ \mathrm{clm\,}( f; \bar{x} ) $ & calmness modulus of the function $f$ at $\bar{x}$\
$ \mathrm{clm}_x ( f ; { (\bar{p}, \bar{x}) } ) $ & partial calmness modulus of $f$ with respect to $x$ at ${ (\bar{p}, \bar{x}) }$\
$ \widehat{\mathrm{clm}}_x \, (f; (\bar{p}, \bar{x})) $ & uniform partial calmness modulus of $f$ with respect to $x$\
$ \mathrm{clm\,}(S; \bar{y}|\bar{x})$ & calmness modulus of $S$ at $\bar{y}$ for $\bar{x}$\
$ \mathrm{lip\,}(f; \bar{x}) $ & Lipschitz modulus of the function $f$ at $\bar{x}$\
$ \widehat{\mathrm{lip}}_x \, (f; (\bar{p}, \bar{x})) $ & uniform partial Lipschitz modulus of $f$ with respect to $x$\
$ \mathrm{lip\,}(S; \bar{y}| \bar{x}) $ & Lipschitz modulus of $ S $ at $ \bar{y}$ for $\bar{x}$\
$ \mathrm{reg\,}(F; \bar{x}|\bar{y})$ & regularity modulus of $F$ at $\bar{x}$ for $\bar{y}$\
$ \mathrm{subreg\,} (F; \bar{x} | \bar{y}) $ & modulus of metric sub-regularity of $F$ at $\bar{x}$ for $\bar{y}$ ]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'New capacity upper bounds are presented for the discrete-time Poisson channel with no dark current and an average-power constraint. These bounds are a simple consequence of techniques developed for the seemingly unrelated problem of upper bounding the capacity of binary deletion and repetition channels. Previously, the best known capacity upper bound in the regime where the average-power constraint does not approach zero was due to Martinez (JOSA B, 2007), which is re-derived as a special case of the framework developed in this paper. Furthermore, this framework is carefully instantiated in order to obtain a closed-form bound that noticeably improves the result of Martinez everywhere. Finally, capacity-achieving distributions for the discrete-time Poisson channel are studied under an average-power constraint and/or a peak-power constraint and arbitrary dark current. In particular, it is shown that the support of the capacity-achieving distribution under an average-power constraint only must be countably infinite. This settles a conjecture of Shamai (IEE Proceedings I, 1990) in the affirmative. Previously, it was only known that the support must be unbounded.'
author:
- Mahdi Cheraghchi
- 'João Ribeiro[^1]'
bibliography:
- 'poisson.bib'
title: 'Structural Results and Improved Upper Bounds on the Capacity of the Discrete-Time Poisson Channel '
---
Introduction {#sec:intro}
============
We study the capacity of the classical discrete-time Poisson (DTP) channel, along with properties of its capacity-achieving distributions. Given an input $x \in {\mathbb{R}}^{\geq 0}$, the channel outputs a sample from Poisson distribution with mean $\lambda+x$, where $\lambda \geq 0$ is a channel parameter called the dark current. The DTP channel is motivated by applications in optical communication, involving a sender with a photon-emitting source and a receiver that observes the arrived photons (some of which may not have originated in the sender’s source, hence the dark current parameter) [@Sha90].
The capacity of the DTP channel is infinite if there are no constraints on the input distributions. For this reason, a power constraint should be imposed on the input distribution. The most typical choice, that we consider in this work, is an average-power constraint $\mu\in{\mathbb{R}}^{\geq 0}$, under which only input distributions $X$ satisfying ${\mathds{E}}[X]\leq \mu$ are allowed. Several works also consider the case where a peak-power constraint is imposed on $X$, i.e., $X\leq A$ for some fixed $A\in{\mathbb{R}}^{>0}$ with probability 1 (e.g., [@LM09; @LSVW11; @WW14; @SSEL15; @AAGNKM15]). Setting $A=\infty$ corresponds to the case where no peak-power constraint is present.
Currently, no expression for the capacity of the DTP channel under an average-power constraint is known. Consequently, there has been considerable interest in obtaining sharp bounds and in determining the asymptotic behavior of the DTP channel capacity in several settings, and in investigating properties of capacity-achieving distributions. We focus on upper bounds for the capacity of the DTP channel with $\lambda=0$ under an average-power constraint $\mu$. Note that any such upper bound is also a capacity upper bound for the DTP channel with $\lambda>0$, as such a channel can be simulated from the DTP channel with $\lambda=0$ by having the receiver add an independent Poisson random variable with parameter $\lambda$ to the output.
The problem of better understanding the properties of capacity-achieving distributions for a given channel has also received significant attention. Normally, one is interested in determining whether a capacity-achieving distribution has finite or discrete support. Besides the fact that studying properties of such distributions may provide more insight into the channel capacity, it is also of practical importance. In fact, showing that the optimal distribution can be finite or discrete reduces the complexity of the problem of finding or approximating such a distribution, and allows the application of a wider range of numerical methods. The finiteness and discreteness of capacity-achieving distributions is well-understood for very general classes of noise-additive channels. However, much less is known for non-additive channels, and in particular the DTP channel.
Previous work
-------------
The two main regimes for studying the asymptotic behavior of the DTP channel capacity are when $\mu\to0$ and $\mu\to\infty$. Brady and Verdú [@BV90] studied the asymptotic behavior of the capacity under an average-power constraint $\mu$ when $\mu\to\infty$ and $\mu/\lambda$ is kept fixed. Later, Lapidoth and Moser [@LM09] studied the same problem when $\lambda$ is constant, with and without an additional peak-power constraint. When $\mu\to 0$, Lapidoth et al. [@LSVW11] determined the first-order asymptotic behavior of the capacity when $\mu$ goes to zero, both when $\mu/\lambda$ is kept constant and when $\lambda$ is fixed, with and without a peak-power constraint. Later, Wang and Wornell [@WW14] improved their result when $\mu/\lambda$ is constant.
Obtaining capacity upper bounds for the DTP channel has been a major subject of interest. Explicit asymptotic capacity upper bounds for the DTP channel under an average-power constraint can be found in [@LM09; @LSVW11; @WW14; @AAGNKM15]. The current best non-asymptotic upper bound, which is in fact the best capacity upper bound outside the limiting case $\mu \to 0$, was derived by Martinez [@Mar07]. However, its proof contains a small gap, as mentioned in [@LM09], and is not considered completely rigorous. A more detailed discussion of these upper bounds and of the asymptotic behavior of the capacity can be found in Section \[sec:prevbounds\]. While we focus on capacity upper bounds, we mention that explicit (asymptotic and non-asymptotic) capacity lower bounds for several settings have been derived in [@Mar07; @LM09; @CHC10; @LSVW11; @WW14; @YZWD14].
There is a large amount of literature focusing on properties of capacity-achieving distributions for many classes of channels. As discussed before, one is mostly interested in determining whether such optimal distributions have finite or discrete support. The landscape of this problem is well-understood for quite general classes of noise-additive channels under several input constraints (see, e.g., the early works [@Smi71; @Sha90; @AFTS01] and the recent works [@ED18; @FAF18; @DGPS18])
The shape of capacity-achieving distributions for the DTP channel was first studied by Shamai [@Sha90], who showed that a capacity-achieving distribution for the DTP channel under a peak-power constraint must have finite support, and conjectured that the capacity-achieving distribution for the DTP channel under an average-power constraint only is discrete. He also gave conditions which ensure that distributions with two mass points are optimal. These results were extended by Cao, Hranilovic, and Chen [@CHC14a; @CHC14b]. In particular, they showed that a capacity-achieving distribution for the DTP channel under an average-power constraint only must have unbounded support. Moreover, they also proved that such a distribution must have some mass at $x=0$, and, if a peak-power constraint $A$ is present, some mass at $x=A$ as well. Unlike noise-additive channels, not much is known about the capacity-achieving distributions of the DTP channel when there is only an average-power constraint present.
Other aspects and settings of the DTP channel have also received attention recently. A generalization of the DTP channel was studied by Aminian et al. [@AAGNKM15], where simple and general capacity upper bounds in the presence of average- and peak-power constraints are also given for the classical DTP channel. Sutter et al. [@SSEL15] studied numerical algorithms for approximating the capacity of the DTP channel in the presence of both average- and peak-power constraints, and obtained sharp capacity bounds in this setting.
Our contributions and techniques
--------------------------------
In the first part of this work, we derive improved capacity upper bounds for the DTP channel with $\lambda=0$ under an average-power constraint. Our technique is based on a natural convex duality formulation developed by Cheraghchi [@Che17] for the seemingly unrelated problem of upper bounding the capacity of binary deletion and repetition channels. Furthermore, we prove new results on the shape of capacity-achieving distributions for the DTP channel.
We show that the result of Martinez [@Mar07] can be obtained as an immediate special (sub-optimal) case of our results, thus giving a simple and rigorous proof for this bound. Furthermore, we extract two improved bounds from our more general result (Theorem \[thm:mainbound\]); one involving the minimization of a smooth convex function over $(0,1)$, as well as a closed-form bound (Theorem \[thm:closedbound\]). Both of these bounds are strictly tighter than the bound by Martinez for all $\mu>0$. Thus, we obtain the current best capacity upper bounds for the DTP channel with $\lambda=0$ under an average-power constraint $\mu$ for all values of $\mu$ outside the limiting case $\mu \to 0$. An additional feature of our results is that they are simple to derive.
In the second part, we study properties of capacity-achieving distributions for the DTP channel. Notably, we show that a capacity-achieving distribution for the DTP channel under an average-power constraint must be discrete. This settles a conjecture of Shamai [@Sha90] in the affirmative. Previously, it was only known that the support was unbounded. In fact, we actually show the stronger result that the support must have finite intersection with all bounded intervals. This brings the state of knowledge on this topic for the DTP channel closer to that of noise-additive channels, which are much better understood. Our proof techniques are general and work under any dark current and any combination of average-power and peak-power constraints. In particular, we give an alternative proof that the capacity-achieving distribution under average- and peak-power constraints is finite, which was originally proved by Shamai [@Sha90].
The rest of the article is organized as follows: In Section \[sec:prem\] we introduce our notation. Further discussion of the best previously known bounds, along with the asymptotic behavior of the capacity when $\lambda=0$, appear in Section \[sec:prevbounds\]. The duality-based framework and the derivation of our upper bounds (including the bound by Martinez as a special case) are presented in Section \[sec:mainbound\]. Finally, we compare the bounds from Section \[sec:mainbound\] with those from Section \[sec:prevbounds\] in Section \[sec:comp\]. In Section \[sec:shape\], we present our results on the shape of capacity-achieving distributions for the DTP channel.
Notation {#sec:prem}
========
We denote the capacity of the DTP channel with average-power constraint $\mu$ and $\lambda=0$ by $C(\mu)$. We measure capacity in nats per channel use and denote the natural logarithm by $\log$. Random variables are usually denoted by uppercase letters such as $X$, $Y$, and $Z$. For a discrete random variable $X$, we denote by $X(x)$ the probability that $X$ takes on value $x$. The support of a random variable $X$ is denoted by $\mathsf{supp}(X)$, i.e., $\mathsf{supp}(X)$ is the smallest closed set $\mathcal{W}$ such that $\Pr[X\in \mathcal{W}]=1$. When the context is clear, we may at certain points confuse random variables and their associated cumulative distribution functions. The Kullback-Leibler divergence between $X$ and $Y$ is denoted by $D_{{\sf KL}}(X\|Y)$. In general, we use the convention that $0\log 0=0$.
Previously known bounds and asymptotic results {#sec:prevbounds}
==============================================
In this section, we survey the best previously known capacity upper bounds and the known results on the asymptotic behavior of $C(\mu)$. The asymptotic regimes considered in the literature are when $\mu\to 0$ and $\mu\to\infty$.
In the small $\mu$ regime, Lapidoth et al. [@LSVW11] showed that $$\lim_{\mu\to 0}\frac{C(\mu)}{\mu\log(1/\mu)}=1.$$ Moreover, they gave the following upper bound matching the asymptotic behavior [@LSVW11 expression (86)], $$\begin{aligned}
\label{bound:lapidoth}
C(\mu)&\leq -\mu\log p-\log(1-p)+\frac{\mu}{\beta}+\mu\cdot \max\bigg(0,\frac{1}{2}\log\beta+\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\bar{\Gamma}(1/2,1/\beta)}{\sqrt{\pi}}+\frac{1}{2\beta}{\aftergroup\egroup\originalright})\bigg),\end{aligned}$$ where $p\in(0,1)$ and $\beta>0$ are free constants, and $\bar{\Gamma}$ is the upper incomplete gamma function. It is easy to see that the optimal choice for $p$ is $p=\frac{\mu}{1+\mu}$.
Later, Wang and Wornell [@WW14] determined the higher-order asymptotic behavior of $C(\mu)$ in the small $\mu$ regime, where it was shown that $$C(\mu)= \mu\log(1/\mu)-\mu\log\log(1/\mu)+O(\mu)$$ when $\mu\to 0$. This was previously noted by Chung, Guha, and Zheng [@CGZ11], although they only proved the result for a more restricted set of input distributions (as mentioned in [@WW14]). Wang and Wornell [@WW14 expression (180)] gave an upper bound (valid for small enough $\mu$) matching this asymptotic behavior; namely, $$\begin{aligned}
\label{bound:ww}
C(\mu)&\leq \mu+\mu\log\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{\mu}{\aftergroup\egroup\originalright})+\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{1-\mu}{\aftergroup\egroup\originalright})+\mu\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{1-\frac{1}{\log(1/\mu)}}{\aftergroup\egroup\originalright})+\mu\cdot \sup_{x\geq 0}\phi_\mu(x),\end{aligned}$$ where $\phi_\mu(x):=\frac{1-e^{-x}}{x}\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{x}{\mu\log(1/\mu)}{\aftergroup\egroup\originalright})$.
In the large $\mu$ regime, Lapidoth and Moser [@LM09] showed that $$\lim_{\mu\to \infty}\frac{C(\mu)}{\log \mu}=\frac{1}{2}.$$ The best upper bound in this regime (and, in fact, anywhere outside the asymptotic limit $\mu\to 0$) was derived by Martinez [@Mar07 expression (10)] and is given by $$\begin{aligned}
\label{bound:martinez}
C(\mu)&\leq {\mathopen{}\mathclose\bgroup\originalleft}(\mu+\frac{1}{2}{\aftergroup\egroup\originalright})\log{\mathopen{}\mathclose\bgroup\originalleft}(\mu+\frac{1}{2}{\aftergroup\egroup\originalright})-\mu\log\mu-\frac{1}{2}+\log{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{\sqrt{2e}-1}{\sqrt{1+2\mu}}{\aftergroup\egroup\originalright}).\end{aligned}$$ It holds that attains the first-order asymptotic behavior of $C(\mu)$ both when $\mu\to 0$ and when $\mu\to\infty$, and is strictly better than for all $\mu>0$. However, as noted in [@LM09], the proof in [@Mar07] is not considered to be completely rigorous as it contains a gap (a certain equality is only shown numerically).
Aminian et al. [@AAGNKM15 Example 2] give the upper bound $$\sup_{X:\mathbb{E}[X]\leq \mu}\mathsf{Cov}(X+\lambda,\log(X+\lambda))$$ for the capacity of the DTP channel with an average-power constraint $\mu$ and dark current $\lambda$, where $\mathsf{Cov}(\cdot,\cdot)$ denotes the covariance. However, this bound is only useful when $\lambda$ is large.
Finally, we note that an analytical lower bound is also given in [@Mar07]. This lower bound is obtained by considering gamma distributions as the input to the DTP channel (and thus negative binomial distributions as the corresponding output). More precisely, we have $$\begin{aligned}
C(\mu)&\geq (\mu+\nu)\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\mu+\nu}{\nu}{\aftergroup\egroup\originalright})+\mu(\psi(v+1)-1)\nonumber\\&-\int_0^1 {\mathopen{}\mathclose\bgroup\originalleft}(1-{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\nu}{\nu+\mu(1-t)}{\aftergroup\egroup\originalright})^\nu{\aftergroup\egroup\originalright})\frac{t^{\nu-1}}{(1-t)\log t}-\frac{\mu}{\log t}dt\label{eq:martinezlb}\end{aligned}$$ for all $\nu>0$, where $\psi(y)=\frac{d}{d y}\log\Gamma(y)$ is the digamma function ($\Gamma$ denotes the gamma function). Martinez [@Mar07] also obtained the elementary lower bound $C(\mu)\geq \frac{1}{2}\log(1+\mu)$. These bounds behaves well when $\mu$ is large. In fact, the capacity is known to behave like $\frac{1}{2}\log\mu$ when $\mu\to \infty$.
The proposed upper bounds {#sec:mainbound}
=========================
In this section, we derive new upper bounds on $C(\mu)$. While previous upper bounds are mostly based on duality results from [@LM03], our derivation (although still duality based) follows from the application of a framework recently developed in [@Che17] in the context of binary deletion-type channels.
The convex duality formulation {#sec:highlevel}
------------------------------
In this section, we give a high-level overview of our approach towards obtaining improved capacity upper bounds.
We denote the DTP channel with dark current $\lambda$ under an *output* average-power constraint $\mu$ and an input peak-power constraint $A$ by ${{\sf DTP}}_{\lambda,A,\mu}$. In words, this channel accepts input distributions $X$ such that $\mathsf{supp}(X)\subseteq [0,A]$ and which have associated output distributions $Y$ satisfying $\mathds{E}[Y]\leq \mu$. We may set $A=\infty$, in which case there is no peak-power constraint. When $A=\infty$ and $\lambda=0$, we denote the corresponding channel by ${{\sf DTP}}_\mu$.
Note that imposing an average-power constraint $\mu$ on the output of the DTP channel is equivalent to imposing an average-power constraint $\mu-\lambda$ on its input. Because of this, one can easily move back and forth between input and output average-power constraints for the DTP channel. We may refer to “input average-power constraint" simply as “average-power constraint" throughout the paper.
A main component of our proofs is the following natural duality result for the DTP channel. This statement was originally proved for general channels with discrete input and output alphabets in [@Che17]. A proof of Theorem \[thm:dual\] for a general class of channels with continuous input under output average-power constraints and/or input peak-power constraints is presented in Appendix \[sec:proof1\].
\[thm:dual\] Suppose there exist a random variable $Y$, supported on $\mathbb{N}$, and parameters $\nu_0\in{\mathbb{R}}$ and $\nu_1\in\mathbb{R}^{\geq 0}$ such that $$\label{eq:KLreq}
D_{{\sf KL}}(Y_x\| Y)\leq \nu_0+\nu_1{\mathds{E}}[Y_x]$$ for every $x\in [0,A]$, where $Y_x$ denotes the output of the DTP channel with dark current $\lambda$ when $x$ is given as input, i.e., $Y_x$ follows a Poisson distribution with mean $\lambda+x$. Assume that $\mu>\lambda$ (otherwise the problem is trivial). Then, we have $$C({{\sf DTP}}_{\lambda,A,\mu})\leq\nu_0+ \nu_1\mu.$$ Moreover, an input distribution $X$ is capacity-achieving for ${{\sf DTP}}_{\lambda,A,\mu}$ and $$C({{\sf DTP}}_{\lambda,A,\mu})= \nu_0+\nu_1\mu$$ if and only if its corresponding output distribution $Y$ satisfies $\mathds{E}[Y]=\mu$ and $$D_{{\sf KL}}(Y_x\| Y)\leq \nu_0+\nu_1{\mathds{E}}[Y_x]$$ for every $x\in[0,A]$, with equality for all $x\in{\sf supp}(X)$.
Although there is a very simple correspondence between input and output average-power constraints for the DTP channel, this is not always the case. For general channels, considering the output mean as a parameter (as opposed to the input mean) leads to a more natural design of candidate distributions to be used in the analogue of Theorem \[thm:dual\].
We call distributions $Y$ satisfying in Theorem \[thm:dual\] for some parameters $\nu_0$ and $\nu_1$ *dual-feasible.* For the DTP channel with $\lambda=0$, we wish to find a dual-feasible distribution $Y$ and parameters $\nu_0, \nu_1 > 0$ such that $$D(Y_x||Y)\leq \nu_0+\nu_1 {\mathds{E}}[Y_x]=\nu_0+\nu_1x$$ for all $x\in\mathbb{R}^{\geq 0}$, and the inequality gap as small as possible. Using Theorem \[thm:dual\], we readily obtain an upper bound for $C({{\sf DTP}}_\mu)=C(\mu)$.
The digamma distribution {#sec:digamma}
------------------------
The result of Martinez [@Mar07] follows the common approach of a convex duality formulation that leads to capacity upper bounds given an appropriate distribution on the channel output alphabet. Indeed, this is also the approach that we take. The dual distribution chosen by [@Mar07] is a negative binomial distribution, which is a natural choice corresponding to a gamma distribution for the channel input. However, lengthy manipulations and certain adjustments are needed to obtain a closed-form capacity upper bound for this choice. We use a slightly different duality formulation, as discussed in Section \[sec:highlevel\]. Furthermore, for the dual output distribution, we use a distribution that we call the “digamma distribution” and is designed by Cheraghchi [@Che17] precisely for the purpose of use in the duality framework of [@Che17]. This distribution asymptotically behaves like the negative binomial distribution. However, it is constructed to automatically yield provable capacity upper bounds without need for any further manipulations or adjustments. This is the key to our refined bounds and dramatically simplified analysis[^2].
For a parameter $q\in(0,1)$, the digamma distribution $Y^{(q)}$ is defined over non-negative integers with probability mass function $$\label{eq:digamma}
Y^{(q)}(y):=y_0 \frac{\exp(y \psi(y))(q/e)^y}{y!},\quad y=0,1,\ldots,$$ where $y_0$ is a normalizing factor depending on $q$ (we omit this dependence in the notation for brevity), $\psi$ is the digamma function, and $y \psi(y)$ is understood to be zero for $y=0$. For positive integers $y$, we have $\psi(y)=-\gamma+\sum_{k=1}^{y-1}1/k$, where $\gamma\approx 0.5772$ is the Euler-Mascheroni constant.
We will need to control the normalizing factor $y_0$, which is accomplished by the following result.
\[lem:y0upper\] We have $$\begin{aligned}
\log{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{2}{e^{1+\gamma}}{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{\sqrt{1-q}}-1{\aftergroup\egroup\originalright}){\aftergroup\egroup\originalright})\leq-\log y_0\leq \log{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{1}{\sqrt{2e}}{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{\sqrt{1-q}}-1{\aftergroup\egroup\originalright}){\aftergroup\egroup\originalright})
\end{aligned}$$ for all $q\in(0,1)$.
Sharper bounds exist for $-\log y_0$ based on special functions (Lerch transcendent).
We will also be using the fact that the digamma distribution is closely related to the negative binomial distribution. We denote the negative binomial distribution with number of failures $r$ (note that $r$ is not necessarily an integer) and success probability $p$ by ${\sf NB}_{r,p}$. Its probability mass function is given by $$\mathsf{NB}_{r,p}(y)=\binom{y+x-1}{x}p^y(1-p)^r,\quad y=0,1,2,\dots.$$ We have the following result.
\[lem:negbin\] For all $y\geq 1$ and $q\in(0,1)$, $$\frac{2}{e^{1+\gamma}}{\sf NB}_{1/2,q}(y)\leq \frac{\sqrt{1-q}P_{Y^{(q)}}(y)}{y_0}\leq \frac{1}{\sqrt{2e}}{\sf NB}_{1/2,q}(y).$$
A first capacity upper bound {#sec:firstbound}
----------------------------
In this section, we use the digamma distribution and the approach outlined in Section \[sec:highlevel\] in order to derive an upper bound for $C(\mu)$.
The random variable $Y_x$ in this case satisfies $Y_x=\mathsf{Poi}(x)$, where $\mathsf{Poi}(\lambda)$ denotes a Poisson distribution with mean $\lambda$. Therefore, its probability mass function is given by $$Y_x(y)=e^{-x}\frac{x^y}{y!},\quad y=0,1,2,\dots.$$
We will now give a short proof that the digamma distribution given in is dual-feasible for the ${{\sf DTP}}_\mu$ channel by invoking well-known facts from the theory of special functions.
First, for $q\in(0,1)$ and some function $g$ satisfying $g(y)\leq y\log y+o(y)$, consider a general distribution $Y$ of the form $$Y(y)=y_0\frac{\exp(g(y))(q/e)^y}{y!},\quad y=0,1,2,\dots,$$ where $y_0$ is the normalizing factor. The upper bound on $g$ ensures that $Y$ is a valid probability distribution. In this case, the Kullback-Leibler divergence between $Y_x$ and $Y$ has a simple form for every $x$. We have $$\begin{aligned}
\label{eq:KL}
D_{\sf{KL}}{\mathopen{}\mathclose\bgroup\originalleft}(Y_x\big|\big|Y{\aftergroup\egroup\originalright})&=\sum_{y=0}^\infty Y_x(y)\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{Y_x(y)}{Y(y)}{\aftergroup\egroup\originalright})\nonumber\\
&= \sum_{y=0}^\infty Y_x(y)(-\log y_0+y(1-\log q )-g(y)-x+y\log x)\nonumber\\
&=-\log y_0-x\log q+x\log x-{\mathds{E}}[g(Y_x)].\end{aligned}$$
Via , it follows that $Y$ is dual feasible provided that we choose $g$ such that $$\label{eq:conddualfeasible}
{\mathds{E}}[g(Y_x)]=e^{-x}\sum_{y=0}^\infty \frac{g(y)}{y!}x^y\geq x\log x$$ for all $x\geq 0$.
From the theory of special functions (by instantiating the Tricomi confluent hypergeometric function $U(a,n+1,z)$ with approriate parameters: [@AS65 13.1.6, p. 505 with $a=n+1=1$] combined with [@AS65 13.6.12, p. 509] and [@AS65 13.6.30, p. 510]), we have the identity $$\label{eq:specfunc}
e^x E_1(x)= \sum_{y=0}^\infty \frac{\psi(1+y)}{y!}x^y-e^x\log x,$$ where $E_1(x)=\int_1^{\infty}e^{-xt} dt/t$ is the exponential integral function and $\psi$ is the digamma function. Multiplying both sides of by $xe^{-x}$ leads to $$e^{-x}\sum_{y=0}^\infty \frac{y\psi(y)}{y!}x^y = x\log x +xE_1(x)\geq x\log x.$$ Consequently, the choice $$\label{eq:choiceg}
g(y)=y\psi(y)$$ with the convention $g(0)=0$ satisfies and thus leads to a dual feasible distribution $Y$. Furthermore, $g(y)=y\log y+o(y)$, as desired.
We briefly give some intuition as to how shows naturally in [@Che17]. A possible approach towards tightly satisfying is to design $g^*$ such that $$\mathds{E}[g^*(Y_x)]=x\log x,\quad \forall x\geq 0.$$ It is possible to derive a formal solution $g^*$ to this functional equation of the form $g^*(y)=\int_0^\infty h(y,t)dt$ for some function $h(\cdot,\cdot)$. However, $g^*(y)$ is a divergent integral for all $y>0$. Therefore, $g^*$ does not exist. A possible solution to this problem is to truncate the integration bounds so that the integral converges. Using some identities from the theory of special functions, truncating the integration bounds of $g^*(y)$ appropriately leads to the choice .
Combining the choice of $g$ in with allows us to conclude that $$\label{ineq:klgap}
D_{\sf{KL}}{\mathopen{}\mathclose\bgroup\originalleft}(Y_x\big|\big|Y^{(q)}{\aftergroup\egroup\originalright})\leq -\log y_0-x\log q$$ for all $x\geq 0$. Applying Theorem \[thm:dual\], we conclude that $$\label{bound:mean-limited}
C(\mu)=C({{\sf DTP}}_\mu)\leq -\log y_0-\mu\log q,$$ which immediately leads to the following result.
\[thm:mainbound\] For all $\mu \geq 0$, we have $$\label{bound:our}
C(\mu)\leq \inf_{q\in(0,1)} (-\log y_0-\mu\log q).$$
Elementary bounds in a systematic way {#sec:qchoice}
-------------------------------------
While Theorem \[thm:mainbound\] gives an upper bound on $C(\mu)$, it involves minimizing a rather complicated function (for which we do not know an exact closed-form expression) over a bounded interval. Since it is of interest to have easy-to-compute but high quality upper bounds, we consider instantiating the parameter $q$ inside the infimum in with a simple function of $\mu$. In this section, we present a systematic way of deriving such a good choice $q(\mu)$. Finally, we upper bound $-\log(y_0)$ using Lemma \[lem:y0upper\], obtaining an improved closed-form bound for $C(\mu)$.
We determine a good choice $q(\mu)$ for the parameter $q$ in indirectly by instead choosing $q(\mu)$ so that the associated distribution $Y^{(q(\mu))}$ (given by ) has expected value close to $\mu$. The reasons for this are the following: First, a capacity-achieving distribution $X$ under an average-power constraint $\mu$ must satisfy ${\mathds{E}}[X]={\mathds{E}}[Y]=\mu$ (see Appendix \[sec:existoptimal\]). While a capacity-achieving $X$ does not necessarily induce a digamma distribution over the output, the digamma distribution seems to be close to optimal, since the gap between the two expressions in is $xE_1(x)$, which decays exponentially with $x$. Second, numerical computation suggests that the distribution $Y$ induced by the choice of $q$ that minimizes the bound from Theorem \[thm:mainbound\] has expected value very close (or equal) to $\mu$. While determining a choice $q(\mu)$ such that ${\mathds{E}}{\mathopen{}\mathclose\bgroup\originalleft}[Y^{(q(\mu))}{\aftergroup\egroup\originalright}]$ is very close to $\mu$ for all $\mu>0$ may be complicated, we settle for a choice $q(\mu)$ that behaves well when $\mu\to 0$ and $\mu\to\infty$.
We begin by studying how $q(\mu)$ should behave when $\mu\to \infty$. In this case, we should have $q(\mu)\to 1$. Lemma \[lem:y0upper\] implies that $$\frac{2}{e^{1+\gamma}}+{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{2}{e^{1+\gamma}}{\aftergroup\egroup\originalright})\sqrt{1-q}\leq \frac{\sqrt{1-q}}{y_0}\leq \frac{1}{\sqrt{2e}}+{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{1}{\sqrt{2e}}{\aftergroup\egroup\originalright})\sqrt{1-q},$$ from which we can conclude that $$\label{eq:asympqy0}
\frac{2}{e^{1+\gamma}}\leq \frac{\sqrt{1-q}}{y_0}\leq \frac{1}{\sqrt{2e}}+o(1)$$ when $q\to 1$. Combining with Lemma \[lem:negbin\], we obtain $$\frac{2\sqrt{2e}}{e^{1+\gamma}}-o(1)\leq \frac{{Y^{(q)}}(y)}{{\sf NB}_{1/2,q}(y)}\leq \frac{e^{1+\gamma}}{2\sqrt{2e}}\approx 1.038$$ for $y=0,1,\dots$, when $q\to 1$, and so we conclude that the digamma distribution is well-approximated by ${\sf NB}_{1/2,q}$ when $q$ is close to 1.
Recall that we want a choice of $q(\mu)$ such that $Y^{(q(\mu))}$ has expected value as close as possible to $\mu$ in the large $\mu$ regime. The choice of $q$ which ensures that ${\mathds{E}}{\mathopen{}\mathclose\bgroup\originalleft}[{\sf NB}_{1/2,q}{\aftergroup\egroup\originalright}]=\mu$ is $q=\frac{2\mu}{1+2\mu}$, and so we want $q(\mu)$ to satisfy $q(\mu)=\frac{2\mu}{1+2\mu}+o{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{\mu}{\aftergroup\egroup\originalright})$ when $\mu\to\infty$.
One could set $q(\mu)=\frac{2\mu}{1+2\mu}$ to obtain the desired behavior above, but we will show that we can correct this choice in order to achieve ${\mathds{E}}{\mathopen{}\mathclose\bgroup\originalleft}[Y^{(q(\mu))}{\aftergroup\egroup\originalright}]= \mu+o(\mu)$ when $\mu\to 0$. To make the derivation simpler, we will instead work with the quantity $\frac{1}{1-q(\mu)}$.
Consider a choice $q(\mu)$ satisfying $$\frac{1}{1-q(\mu)}=1+\alpha\mu+\frac{\beta\mu^2}{1+\mu}$$ for some constants $\alpha$ and $\beta$. It is easy to see that $\frac{1}{1-q(\mu)}$ behaves as $1+\alpha\mu+o(\mu)$ when $\mu\to 0$ and as $1+(\alpha+\beta)\mu+o(\mu)$ when $\mu\to\infty$, which means we can set its asymptotic behavior in both the small and large $\mu$ regimes independently of each other. Moreover, setting $\alpha+\beta=2$ leads to the desired behavior $q(\mu)=\frac{2\mu}{1+2\mu}+o{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{\mu}{\aftergroup\egroup\originalright})$ when $\mu\to\infty$.
We now proceed to choose $\alpha$. As mentioned before, we determine the choice of $\alpha$ which ensures that ${\mathds{E}}{\mathopen{}\mathclose\bgroup\originalleft}[Y^{(q(\mu))}{\aftergroup\egroup\originalright}]= \mu+o(\mu)$ when $\mu\to 0$. It is straightforward to see that, by construction, $q(\mu)=\alpha\mu+o(\mu)$ when $\mu\to 0$. We will need the following result.
\[lem:asympmean\] We have ${\mathds{E}}{\mathopen{}\mathclose\bgroup\originalleft}[Y^{(q)}{\aftergroup\egroup\originalright}]= e^{-(1+\gamma)}q+o(q)$ as $q\to 0$.
Recall that $g(y)=y\psi(y)$, and note that $$\label{eq:expval}
\frac{{\mathds{E}}{\mathopen{}\mathclose\bgroup\originalleft}[Y^{(q)}{\aftergroup\egroup\originalright}]}{q}=y_0e^{-(1+\gamma)}+y_0\sum_{y=2}^\infty y\cdot \frac{e^{g(y)-y}q^{y-1}}{y!}.$$ It is easy to see that $y_0$ approaches $1$ (using Lemma \[lem:y0upper\], for example) and the second term in the RHS of vanishes when $q\to 0$, and so the result follows.
The remarks above, combined with Lemma \[lem:asympmean\], imply that ${\mathds{E}}{\mathopen{}\mathclose\bgroup\originalleft}[Y^{(q(\mu))}{\aftergroup\egroup\originalright}]= e^{-(1+\gamma)}\alpha\mu+o(\mu)$ when $\mu\to 0$. Therefore, it suffices to set $\alpha=e^{1+\gamma}$ to have ${\mathds{E}}{\mathopen{}\mathclose\bgroup\originalleft}[Y^{(q(\mu))}{\aftergroup\egroup\originalright}]=\mu+o(\mu)$ when $\mu\to 0$. Based on this, we set $q(\mu)$ to be such that $$\label{eq:qchoice}
\frac{1}{1-q(\mu)}=1+e^{1+\gamma}\mu+\frac{(2-e^{1+\gamma})\mu^2}{1+\mu}.$$
Combining the previous discussion, Theorem \[thm:mainbound\], and Lemma \[lem:y0upper\], we obtain the following result.
\[thm:closedbound\] We have $$\label{bound:optquppery0}
C(\mu)\leq \inf_{q\in(0,1)} f(\mu,q),$$ where $ f(\mu,q):=-\mu\log q+\log{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{1}{\sqrt{2e}}{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{\sqrt{1-q}}-1{\aftergroup\egroup\originalright}){\aftergroup\egroup\originalright})$.
In particular, by instantiating $q$ with $q(\mu)$ defined in , $$\begin{aligned}
\label{bound:qchoice}
C(\mu)&\leq \mu \log {\mathopen{}\mathclose\bgroup\originalleft}(\frac{1+{\mathopen{}\mathclose\bgroup\originalleft}(1+e^{1+\gamma }{\aftergroup\egroup\originalright}) \mu+2\mu ^2}{e^{1+\gamma } \mu + 2 \mu ^2}{\aftergroup\egroup\originalright})+\log {\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{1}{\sqrt{2e}}{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{1+(1+e^{1+\gamma})\mu+2\mu^2}{1+\mu}}-1{\aftergroup\egroup\originalright}){\aftergroup\egroup\originalright}).\end{aligned}$$
Note that $f(\mu,\cdot)$ is an elementary, smooth, and convex function for every fixed $\mu\geq 0$. Therefore, can be easily approximated to any desired degree of accuracy.
The reasons why we base our choice of $q(\mu)$ on instead of are the following: First, $q(\mu)$ is still close to optimal when used in (see Figure \[fig:allbounds02\]). Second, the choice is independent of the upper bound on $-\log y_0$, and so can be reutilized if a better bound is used.
The result of Martinez as a special case {#sec:martinez}
----------------------------------------
In this section, we show that the bound by Martinez can be quite easily recovered through our techniques. More precisely, we show that this bound is a special case of with a sub-optimal choice of $q=2\mu/(1+2\mu)$. In particular, this implies that is strictly tighter than . In this section, we define $m(\mu)$ to be the right hand side of . Recall that $ f(\mu,q)=-\mu\log q+\log{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{1}{\sqrt{2e}}{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{\sqrt{1-q}}-1{\aftergroup\egroup\originalright}){\aftergroup\egroup\originalright})$.
\[thm:martinez\] We have $f{\mathopen{}\mathclose\bgroup\originalleft}(\mu,\frac{2\mu}{1+2\mu}{\aftergroup\egroup\originalright})=m(\mu)$ for all $\mu\geq 0$. Moreover, for every $\mu>0$ there is $q^*_\mu\in(0,1)$ such that $f(\mu,q^*_\mu)<m(\mu)$.
To prove the first statement of the theorem, we compute $$\begin{aligned}
&m(\mu)-f\!{\mathopen{}\mathclose\bgroup\originalleft}(\mu,\frac{2\mu}{1+2\mu}{\aftergroup\egroup\originalright})\\
&={\mathopen{}\mathclose\bgroup\originalleft}(\mu+\frac{1}{2}{\aftergroup\egroup\originalright})\log{\mathopen{}\mathclose\bgroup\originalleft}(\mu+\frac{1}{2}{\aftergroup\egroup\originalright})-\mu\log\mu-\frac{1}{2}+\log{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{\sqrt{2e}-1}{\sqrt{1+2\mu}}{\aftergroup\egroup\originalright}) \\&- \mu\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1+2\mu}{2\mu}{\aftergroup\egroup\originalright})-\log{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{1}{\sqrt{2e}}{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{1+2\mu}-1{\aftergroup\egroup\originalright}){\aftergroup\egroup\originalright})\\
&=\frac{1}{2}\log{\mathopen{}\mathclose\bgroup\originalleft}(\mu+\frac{1}{2}{\aftergroup\egroup\originalright})+\mu{\mathopen{}\mathclose\bgroup\originalleft}(\log{\mathopen{}\mathclose\bgroup\originalleft}(\mu+\frac{1}{2}{\aftergroup\egroup\originalright})-\log\mu{\aftergroup\egroup\originalright})-\frac{1}{2}+\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\sqrt{1+2\mu}+\sqrt{2e}-1}{\sqrt{1+2\mu}}{\aftergroup\egroup\originalright}) \\&- \mu\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1+2\mu}{2\mu}{\aftergroup\egroup\originalright})-\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\sqrt{1+2\mu}+\sqrt{2e}-1}{\sqrt{2e}}{\aftergroup\egroup\originalright})\\
&=\frac{1}{2}\log{\mathopen{}\mathclose\bgroup\originalleft}(\mu+\frac{1}{2}{\aftergroup\egroup\originalright})-\frac{1}{2}+\log{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{2e}{1+2\mu}}{\aftergroup\egroup\originalright})=0.
\end{aligned}$$
To see that the second statement holds, it suffices to show that $\frac{\partial f}{\partial q}{\mathopen{}\mathclose\bgroup\originalleft}(\mu,\frac{2\mu}{1+2\mu}{\aftergroup\egroup\originalright})\neq 0$ for all $\mu>0$. We have $$\label{eq:deriv}
\frac{\partial f}{\partial q}(\mu,q)=-\frac{\mu}{q}+\frac{1}{2{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{2e}+\frac{1}{\sqrt{1-q}}-1{\aftergroup\egroup\originalright})(1-q)^{3/2}}.$$ Instantiating with $q=\frac{2\mu}{1+2\mu}$ yields $$\frac{\partial f}{\partial q}{\mathopen{}\mathclose\bgroup\originalleft}(\mu,\frac{2\mu}{1+2\mu}{\aftergroup\egroup\originalright})=-\frac{1+2\mu}{2}+\frac{(1+2\mu)^{3/2}}{2{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{2e}+\sqrt{1+2\mu}-1{\aftergroup\egroup\originalright})},$$ and now it is enough to note that $$\begin{aligned}
& -(1+2\mu){\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{2e}+\sqrt{1+2\mu}-1{\aftergroup\egroup\originalright})+(1+2\mu)^{3/2}\\
& =(1+2\mu)(1-\sqrt{2e})< 0
\end{aligned}$$ for all $\mu\geq 0$.
Finally, we show that the explicit choice $q(\mu)$ from Section \[sec:qchoice\] yields a strictly better upper bound than the Martinez bound .
We have $f(\mu,q(\mu))<f{\mathopen{}\mathclose\bgroup\originalleft}(\mu,\frac{2\mu}{1+2\mu}{\aftergroup\egroup\originalright})=m(\mu)$ for all $\mu>0$.
We give short arguments that the statement holds whenever $\mu\geq 1.61$ and when $\mu$ is sufficiently small. The middle region can be verified numerically. Let $d(\mu):=m(\mu)-f(\mu,q(\mu))$.
We begin by noting that $q(\mu)>\frac{2\mu}{1+2\mu}$ for all $\mu>0$. This is because $$q(\mu)-\frac{2\mu}{1+2\mu}=\frac{(e^{1+\gamma}-2)\mu}{(1+2\mu)(1+\mu(1+e^{1+\gamma}+2\mu))}>0.$$ The inequality is true since the both the denominator and numerator are positive for all $\mu>0$ (observe that $e^{1+\gamma}>2$).
Due to the convexity of $f(\mu,\cdot)$, the desired statement holds for a given $\mu$ if $\frac{\partial f}{\partial q}(\mu,q(\mu))<0$. Recalling and , we have $$\label{eq:dfdq}
\frac{\partial f}{\partial q}(\mu,q(\mu))=-\mu-\frac{1+\mu}{e^{1+\gamma}+2\mu}+\frac{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\mu {\mathopen{}\mathclose\bgroup\originalleft}(2 \mu+e^{1+\gamma }+1{\aftergroup\egroup\originalright})+1}{\mu+1}{\aftergroup\egroup\originalright})^{3/2}}{\sqrt{\frac{\mu{\mathopen{}\mathclose\bgroup\originalleft}(2 \mu+e^{1+\gamma }+1{\aftergroup\egroup\originalright})+1}{\mu+1}}+\sqrt{2 e}-1}.$$ Let $T_1(\mu)=\mu+\frac{1+\mu}{e^{1+\gamma}+2\mu}$ and $T_2(\mu)=\frac{\mu {\mathopen{}\mathclose\bgroup\originalleft}(2 \mu+e^{1+\gamma }+1{\aftergroup\egroup\originalright})+1}{\mu+1}$. Then, taking into account , we have $\frac{\partial f}{\partial q}(\mu,q(\mu))<0$ if and only if $$\begin{aligned}
\frac{T_2(\mu)^{3/2}}{\sqrt{T_2(\mu)}+\sqrt{2e}-1}&<T_1(\mu),\end{aligned}$$ which is equivalent to $$\begin{aligned}
T_2(\mu)^{3/2}-T_1(\mu)\sqrt{T_2(\mu)}<T_1(\mu)(\sqrt{2e}-1.)\end{aligned}$$ Squaring both sides shows that $\frac{\partial f}{\partial q}(\mu,q(\mu))<0$ whenever $$\begin{aligned}
T_2^3(\mu)-2T_1(\mu)T_2^2(\mu)-T_1(\mu)^2T_2(\mu)^2<T_1(\mu)^2(\sqrt{2e}-1)^2.\end{aligned}$$ Observe that each term in the inequality is a rational function. As such, we can expand each term, and then compute and eliminate the common denominator to obtain an equivalent polynomial inequality, which turns out to be $$\begin{aligned}
e^{2+2 \gamma }-4 e^{1+\gamma }+8 \sqrt{2 e}-8 e+{\mathopen{}\mathclose\bgroup\originalleft}(24 \sqrt{2 e}-3 e^{2+2 \gamma }+e^{3+3 \gamma }-24 e-8{\aftergroup\egroup\originalright}) \mu\\+{\mathopen{}\mathclose\bgroup\originalleft}(24 \sqrt{2 e}-8 e^{1+\gamma }+2 e^{2+2 \gamma }-24 e-4{\aftergroup\egroup\originalright}) \mu^2+{\mathopen{}\mathclose\bgroup\originalleft}(8 \sqrt{2 e}-8 e-4{\aftergroup\egroup\originalright}) \mu^3<0.\end{aligned}$$ There are many known methods for determining the roots of degree-3 polynomials. We can use such a method to see that the largest root of the polynomial on the left-hand side is smaller than $1.61$, and so the inequality holds whenever $\mu\geq 1.61$.
To prove that $d(\mu)>0$ for $\mu$ small enough, we look at the limiting behavior of $d(\mu)$ when $\mu\to 0$. We have that $$\begin{aligned}
d(\mu)={\mathopen{}\mathclose\bgroup\originalleft}(1+\gamma+\frac{1}{\sqrt{2 e}}-\log 2 -\frac{e^{\frac{1}{2}+\gamma }}{2 \sqrt{2}}{\aftergroup\egroup\originalright})\mu+o(\mu)\approx 0.27\mu+o(\mu)\end{aligned}$$ when $\mu\to 0$, which implies that $d(\mu)>0$, and hence $m(\mu)>f(\mu,q(\mu))$, when $\mu$ is small enough.
When $\mu<1.61$ but it is not too small, one can show $d(\mu)>0$ by employing a computer algebra system. However, $d(\mu)$ is a complex expression, and so cannot be processed directly by such a system. We avoid this issue in the following way: For $\mu\in[0.3,1.61]$, we lower bound $d(\mu)$ by positive rational functions. This is done by replacing the logarithmic and square root terms of the expression by appropriate bounds (described below) which are themselves rational functions. Then, the question of whether $d(\mu)>0$ is reduced to showing that a certain polynomial is positive in the given interval, which can be formally checked by a computer algebra system with little effort. For $\mu<0.3$, our lower bounds for $d(\mu)$ are not good enough, and so we use the same reasoning to show that its second derivative $d''(\mu)$ is negative for $\mu<0.3$. This implies that $d(\mu)$ is concave in $[0,0.3]$, which, combined with the previous results, concludes the proof.
We do not explicitly write down the relevant lower bounds for $d(\mu)$ and upper bounds for the second derivative, as they feature high-degree polynomials. Instead, we describe the relevant bounds on the logarithmic and square root terms. Then, determining the corresponding rational function and formally checking whether it is positive in a given interval is a straightforward process.
The expression $d(\mu)$ features logarithmic terms, along with square root terms of the form $\sqrt{1+2\mu}$ and $\sqrt{(1 + (1 + e^{1+\gamma}) \mu + 2 \mu^2)/(1 + \mu)}$ (recall and ). For every $x\geq 1$, we have the bounds [@Top06] $$\frac{(x-1)(6+5(x-1))}{2(3+2(x-1))}\leq \log x\leq \frac{(x-1)(x+5)}{2x(2+x)}.$$ Furthermore, we can upper bound $\sqrt{1+2\mu}$ and $\sqrt{(1 + (1 + e^{1+\gamma}) \mu + 2 \mu^2)/(1 + \mu)}$ by their Taylor series of degree 5 and 3, respectively, around $\mu=1$. Replacing the relevant terms in $d(\mu)$ by their respective bounds described above yields a rational function lower bound which can be easily shown to be positive for $\mu\in[0.3,1.61]$ by a standard computer algebra system.
For $\mu<0.3$, the bounds above are not tight enough to show that $d(\mu)$ is positive, and so we focus on its second derivative $d''(\mu)$. However, $d''(\mu)$ cannot be processed directly by a computer algebra system either, and so we follow the same reasoning as before. The only terms of $d''(\mu)$ that need to be bounded are of the form $\sqrt{1+2\mu}$ and $\sqrt{(1 + (1 + e^{1+\gamma}) \mu + 2 \mu^2)/(1 + \mu)}$. It suffices to upper bound (resp., lower bound) $\sqrt{1+2\mu}$ by its Taylor series of degree 1 (resp., 2) around $\mu=0$. However, extra care is needed when dealing with $\sqrt{(1 + (1 + e^{1+\gamma}) \mu + 2 \mu^2)/(1 + \mu)}$. We split the interval $[0,0.3]$ into two intervals: First, in $(0,0.25]$ we lower bound the term by its Taylor series of degree 2 around $\mu=0$. Second, in $(0.25,0.3]$ we lower bound it by its Taylor series of degree 2 around $\mu=0.25$.
Replacing the relevant terms of $d''(\mu)$ by their respective bounds, we obtain a negative rational function upper bounding $d''(\mu)$ in each of $(0,0.25]$ and $(0.25,0.3]$, which can be formally checked to be negative with a computer algebra system. This implies that $d(\mu)$ is concave in $(0,0.3]$, and so, combined with the facts that $d(\mu)>0$ for $\mu$ small enough and $d(\mu)>0$ for $\mu\geq 0.3$, we conclude that $d(\mu)>0$ for all $\mu>0$.
Comparison with previously known upper bounds {#sec:comp}
=============================================
In this section, we compare the bounds from Theorem \[thm:closedbound\] with the previously known bounds described in Section \[sec:intro\]. Moreover, we investigate the loss incurred by using instead of .
Figure \[fig:allbounds02\] showcases a plot comparing the bounds from Theorem \[thm:closedbound\] to previously known bounds. The curve corresponding to the bound of Lapidoth et al. is actually the plot of $\mu\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1+\mu}{\mu}{\aftergroup\egroup\originalright})+\log(1+\mu)$, which lower bounds the RHS of . There is a noticeable improvement over the Martinez bound when $\mu$ is not very small, and one can see that is very close to and (with significant overlap), which confirms that the choice $q(\mu)$ from Section \[sec:qchoice\] is close to optimal. Table \[table:compbounds\] gives the numerical values attained by , , and for several values of $\mu$. Table \[table:compq\] compares the choice for $q(\mu)$ with the actual optimal value of $q$ for several values of $\mu$. As expected from the previous observations, the explicit choice is always quite close to the optimal value.
Due to the fact that our bounds are tighter than Martinez’s bound, both of them satisfy the first-order asymptotic behavior of $C(\mu)$ when $\mu\to 0$ and when $\mu\to\infty$. However, they do not exhibit the correct second order asymptotic term when $\mu\to 0$. In fact, the second-order asymptotic term of our bounds when $\mu\to 0$ is $-O(\mu)$, while the correct term is $-\mu\log\log(1/\mu)$. For this reason, our bounds do not improve on the Wang-Wornell bound when $\mu$ is sufficiently small (numerically, when $\mu<10^{-6}$), while they noticeably improve on every previous bound when $\mu$ is not too small.
![Comparison of upper bounds and the analytical lower bound with $\nu=0.05$ for $\mu\in[0,0.2]$.[]{data-label="fig:allbounds02"}](AllBounds02-withLB.pdf){width="60.00000%"}
$\mu$ Bound Bound Bound
------- -------- -------- --------
0.05 0.1280 0.1296 0.1406
0.1 0.1983 0.2010 0.2193
0.2 0.2951 0.2994 0.3262
0.5 0.4689 0.4753 0.5101
1 0.6367 0.6437 0.6785
5 1.1407 1.1492 1.1665
10 1.4005 1.4093 1.4187
20 1.6806 1.6886 1.6930
50 2.0756 2.0815 2.0829
: Comparison between the bound and the elementary bounds and in nats/channel use.[]{data-label="table:compbounds"}
$\mu$ Optimal $q$ $q(\mu)$ as in
------- ------------- ----------------
0.05 0.1851 0.1905
0.1 0.3025 0.3143
0.2 0.4482 0.4663
0.5 0.6447 0.6607
1 0.7676 0.7738
5 0.9309 0.9252
10 0.9617 0.9576
20 0.9794 0.9771
50 0.9912 0.9904
: Comparison between optimal $q$ in for each $\mu$ and the choice $q(\mu)$ as in .[]{data-label="table:compq"}
Figure \[fig:compmartinezdiff\] showcases the distance of Martinez’s bound to and . The plotted curves have similar shapes and are close to each other, which again shows that we do not lose much by replacing $-\log y_0$ by the upper bound of Lemma \[lem:y0upper\] and instantiating $q$ with the sub-optimal explicit choice $q(\mu)$ from Section \[sec:qchoice\].
![Comparison of difference between and , and between and for $\mu\in[0,10]$.[]{data-label="fig:compmartinezdiff"}](CompMartinezDiff-rev2.pdf){width="60.00000%"}
Figure \[fig:relimprov\] showcases the relative distance of the Martinez bound to and . In other words, if $m(\cdot)$ denotes the Martinez bound and $b(\cdot)$ is either the RHS of or of , then the plot shows the quantity $(m(\mu)-b(\mu))/m(\mu)$. Observe that, using , we obtain an improvement of up to $8.2\%$ over , while we can get improvements close to $9.5\%$ using . Note that the two curves are close to each other and similar shape, reinforcing the fact that the loss incurred by using instead of is small.
![Relative difference between and , and between and for $\mu\in[0,1]$.[]{data-label="fig:relimprov"}](relative-improv-rev2.pdf){width="60.00000%"}
The shape of capacity-achieving distributions {#sec:shape}
=============================================
Besides understanding the capacity of communications channels, there has also been a significant amount of work towards determining the properties of capacity-achieving distributions. In particular, one is normally interested in knowing whether a capacity-achieving distribution has finite or discrete support, even though the input alphabet may not be a discrete set.
The study of capacity-achieving distributions for the DTP channel was initiated by Shamai [@Sha90], who proved that capacity-achieving distributions for the DTP channel with both average- and peak-power constraints have finite support. More recently, Cao, Hranilovic, and Chen [@CHC14a; @CHC14b] derived more properties of such distributions. Notably, they show that a capacity-achieving distribution must be supported at $0$ and at $A$ if a peak-power constraint $X\leq A$ is present. Furthermore, they show that distributions with bounded support are not capacity-achieving for the DTP channel with only an average-power constraint. For completeness, we show that there exist capacity-achieving distributions for the DTP channel under an average-power constraint in Appendix \[sec:existoptimal\].
In this section, we show that a capacity-achieving distribution for the DTP channel with arbitrary dark current $\lambda\geq 0$ under an average-power constraint and/or a peak-power constraint must be discrete. As mentioned before, this settles a conjecture of Shamai [@Sha90]. In fact, we show the stronger result that the support of a capacity-achieving distribution $X$ for the DTP channel under an average-power constraint and/or a peak-power constraint must have finite intersection with every bounded interval. Our techniques are general, and we recover Shamai’s original result [@Sha90] for the DTP channel under a peak-power constraint ($A<\infty$) with an alternative proof.
Consider a discrete probability distribution $Y$ supported on the non-negative integers. For our results, it suffices to consider $Y$ with full support. This is because all optimal output distributions of the DTP channel have full support. In fact, the only input distribution which does not induce an output distribution with full support is the distribution which assigns probability 1 to $x=0$, which is clearly not optimal. The following result gives a characterization of optimal output distributions for the DTP channel (which we might also call *capacity-achieving* at times) that will be useful in later proofs.
\[lem:specform\] Consider a distribution $Y$ with full support over the non-negative integers. Furthermore, for a given function $g$ define its (real-valued) exponential generating function $G$ as $$\label{eq:expgen}
G(z)=\sum_{i=0}^\infty \frac{g(i)}{i!}z^i.$$ Let $Y_x=\mathsf{Poi}(\lambda+x)$. Then,
1. $Y$ can be written as $$\label{eq:specform}
Y(y)=y_0 \frac{\exp(g(y))(q/e)^y}{y!}, \quad y=0,1,2,...,$$ for any constants $y_0,q>0$ and some $g$ satisfying $g(y)\leq y\log y + O(y)$ when $y\to\infty$. Moreover, we can always choose $q\in(0,1]$ and $g(y)\leq y\log y +o(y)$ simultaneously.
2. If $Y$ satisfies for some $y_0$, $q$, and $g$, then $$\label{eq:specKL}
{D_\mathsf{KL}}(Y_x||Y)=-\log y_0 -\mathds{E}[Y_x]\log q + (\lambda+x)\log (\lambda+x)-e^{-(\lambda+x)}G(\lambda+x)$$ for all $x\geq 0$;
3. Suppose $X$ is capacity-achieving for the DTP channel with dark current $\lambda$ under an average-power constraint $\mu$ and peak-power constraint $A$ (we may have $A=\infty$). Furthermore, let $Y$ be the associated output distribution. Then, we can choose $y_0$, $q$, and $g$ in such that $$\label{eq:condG}
G(\lambda+x)\geq (\lambda+x)e^{\lambda+x}\log(\lambda+x),\quad \forall x\in [0,A]$$ with equality for all $x\in\mathsf{supp}(X)$.
We begin with the first point. Fix $y_0,q>0$, and consider $g$ defined as $$g(y)=\log y! + y - y\log q -\log y_0+\log Y(y).$$ It is clear that $$Y(y)=y_0 \frac{\exp(g(y))(q/e)^y}{y!},$$ for all $y\geq 0$. Moreover, we have $-\infty<\log Y(y)<0$. Thus, it follows that $g$ is defined and $$g(y)<\log y! + y - y\log q -\log y_0=y\log y + O(y),$$ as desired. It remains to see that we can actually have $q\in (0,1]$ and $g(y)\leq y\log y+o(y)$ at the same time. This follows immediately from the observation that, if $q=1$, then $$g(y)=\log y! + y -\log y_0+\log Y(y)\leq y\log y +\frac{1}{2}\log y + O(1)=y\log y+o(y).$$
For the second point, write $Y$ as in . Then, noting that $Y_x=\mathsf{Poi}(\lambda+x)$, $$\begin{aligned}
{D_\mathsf{KL}}(Y_x||Y)&=-H(Y_x)-\mathds{E}[\log Y(Y_x)]\\
&= (\lambda+x)(\log(\lambda+x)-1)-\mathds{E}[\log Y_x!]-\mathds{E}[\log y_0 + g(Y_x)+Y_x\log q -Y_x-\log Y_x!]\\
&= -\log y_0-\mathds{E}[Y_x]\log q +(\lambda+x)\log(\lambda+x)-\mathds{E}[g(Y_x)],
\end{aligned}$$ with the convention that $0\log 0=0$. The result follows by observing that $\mathds{E}[g(Y_x)]=e^{-(\lambda+x)}G(\lambda+x)$.
Regarding the third point, let $X$ be as in the theorem statement, and let $\mu_X=\mathds{E}[X]$. In particular, $X$ is capacity-achieving among all input distributions with support contained in $[0,A]$ and expected value at most $\mu_X$. Equivalently, $X$ is capacity-achieving among all input distributions with support contained in $[0,A]$ and output expected value at most $\mu_X+\lambda$. According to Theorem \[thm:dual\], we know there exist $a\in\mathbb{R}^{\geq 0}$ and $b\in\mathbb{R}$ such that $$\label{eq:KLineqopt}
{D_\mathsf{KL}}(Y_x||Y)\leq a\mathds{E}[Y_x]+b$$ for all $x\in [0,A]$, with equality if $x\in \mathsf{supp}(X)$.
Choose $y_0=e^{-b}$ and $q=e^{-a}$. Then, there is $g$ satisfying $g(y)\leq y\log y+O(y)$ and such that holds for $Y$ with these choices of $y_0$ and $q$. According to , we have $$\label{eq:KLopt}
{D_\mathsf{KL}}(Y_x||Y)=a\mathds{E}[Y_x]+b+(\lambda+x)\log (\lambda+x)-\mathds{E}[g(Y_x)].$$ Note that $\mathds{E}[g(Y_x)]=e^{-(\lambda+x)}G(\lambda+x)$. Then, from and it follows that $$\mathds{E}[g(Y_x)]-(\lambda+x)\log (\lambda+x)=e^{-(\lambda+x)}G(\lambda+x)-(\lambda+x)\log (\lambda+x)\geq 0$$ with equality for all $x\in \mathsf{supp}(X)$. This concludes the proof.
We will also need the following concentration bound for the Poisson distribution, which is a consequence of Bennett’s inequality (see [@Can17]).
\[lem:concpoisson\] For $0\leq\delta\leq 1$, we have $$\Pr[|\mathsf{Poi}(\lambda)-\lambda|\leq \delta\lambda]\geq 1-2\exp{\mathopen{}\mathclose\bgroup\originalleft}(-\frac{\delta^2\lambda}{4}{\aftergroup\egroup\originalright}).$$
For completeness, we now show that the support of a capacity-achieving input distribution for the DTP channel under an average-power constraint only must be unbounded. This result was originally proved in [@CHC14a]. Our proof follows a similar technique to the proof in [@Sha90] that the support of a capacity-achieving distribution for the DTP channel under a peak-power constraint $A<\infty$ is finite.
\[thm:unboundedsupp\] Suppose $X$ is a capacity-achieving distribution for the DTP channel with dark current $\lambda$ under an average-power constraint $\mu$ and no peak-power constraint. Then, $\mathsf{supp}(X)$ is unbounded.
Fix $X$ as in the theorem statement, and let $Y$ be the corresponding output distribution. Furthermore, let $\mu_X=\mathds{E}[X]$. In particular, $X$ is capacity-achieving among all input distributions with expected value at most $\mu_X$, and all such input distributions are exactly those whose corresponding output distributions have expected value at most $\mu_X+\lambda$. Then, by Theorem \[thm:dual\] we know there exist $a,b\in\mathbb{R}$ such that $$\label{eq:KLineqsupp}
{D_\mathsf{KL}}(Y_x||Y)\leq a\mathds{E}[Y_x]+b$$ for all $x\geq 0$.
Suppose that $\mathsf{supp}(X)\subseteq [0,x_0]$ for some $x_0$. Let $F$ be the cumulative distribution function of $X$. Then, we have $$\begin{aligned}
Y(y)&=\int_{0}^{x_0}e^{-(\lambda+x)}\frac{(\lambda+x)^y}{y!}dF(x)\\
&\leq \int_0^{x_0} e^{-\lambda}\frac{(\lambda+x_0)^y}{y!}dF(x)\\
&=e^{-\lambda}\frac{(\lambda+x_0)^y}{y!}.
\end{aligned}$$ It follows that $$-\log Y(y)\geq \log y! +\lambda-y\log(\lambda+x_0),$$ and so we have $$\label{eq:logy}
-\log Y(y)\geq (1-o(1))y\log y$$ when $y\to\infty$. As a consequence, $$\begin{aligned}
-\mathds{E}[\log Y(Y_x)]&=-\sum_{y=0}^\infty Y_x(y)\log Y(y)\nonumber\\
&\geq \Pr[Y_x\geq (1-(\lambda+x)^{-1/3})(\lambda+x)](1-o(1))(\lambda+x-(\lambda+x)^{2/3})\log(\lambda+x-(\lambda+x)^{2/3})\nonumber\\
&\geq (1-2\exp(-x^{1/3}/4))(1-o(1))(\lambda+x-(\lambda+x)^{2/3})\log(\lambda+x-(\lambda+x)^{2/3})\nonumber\\
&\geq (1-o(1))(\lambda+x)\log(\lambda+x)\label{eq:logyopt}
\end{aligned}$$ when $x\to \infty$. The first inequality holds when $x\to\infty$ due to . The second inequality follows from Lemma \[lem:concpoisson\] with $\delta=(\lambda+x)^{-1/3}$ .
On the other hand, $$\label{eq:entopt}
H(Y_x)=O(\log(\lambda +x))$$ when $x\to\infty$. This holds since $H(Y_x)$ is upper bounded by the entropy of a geometric distribution with expected value $\lambda+x$, as it maximizes the entropy over all distributions over the non-negative integers with fixed expected value. Therefore, if we let $h$ denote the binary entropy function, $$H(Y_x)\leq (\lambda+x)h{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{\lambda+x}{\aftergroup\egroup\originalright})=O(\log(\lambda+x))$$ when $x\to\infty$, as desired.
From and it follows that $${D_\mathsf{KL}}(Y_x||Y)=-H(Y_x)-\mathds{E}[\log Y(Y_x)]=\Omega((\lambda+x)\log(\lambda+x)).$$ However, if this holds there cannot be constants $a,b\in\mathbb{R}$ such that holds, since $\mathds{E}[Y_x]=\lambda+x$. This is a contradiction, as we assumed that $Y$ was dual feasible.
To conclude this section, we show that capacity-achieving input distributions for the DTP channel under an average-power constraint and/or a peak-power constraint must be discrete. We actually prove that the support of a capacity-achieving distribution $X$ under an average-power constraint and/or a peak-power constraint must have finite intersection with every bounded interval. In particular, our techniques also recover Shamai’s result for the DTP channel under a peak-power constraint [@Sha90] in an alternative way.
\[thm:discretesupp\] Suppose $X$ is a capacity-achieving distribution for the DTP channel with dark current $\lambda$ under an average-power constraint $\mu>0$ and/or a peak-power constraint $A$ (we may have $A=\infty$). Then, $\mathsf{supp}(X)\cap I$ is finite for every bounded interval $I$. In particular, $\mathsf{supp}(X)$ is countably infinite when $A=\infty$ and finite when $A<\infty$.
The statement for an average-power constraint $\mu=0$ is trivial, so we assume $\mu>0$. Fix $X$ as in the theorem statement, and let $Y$ be the corresponding output distribution. Define $\mu_X=\mathds{E}[X]$. Then, $X$ is optimal over all distributions with support in $[0,A]$ and mean at most $\mu_X$ (regardless of whether there is an average-power constraint in place or not). As a result, Lemma \[lem:specform\] guarantees the existence of a function $g$ such that its exponential generating function $G$ satisfies $$G(\lambda+x)\geq (\lambda+x)e^{\lambda+x}\log(\lambda+x),\quad \forall x\in [0,A]$$ with equality for $x\in\mathsf{supp}(X)$. Under a change of variables, this is equivalent to $$G(x)\geq xe^{x}\log x=:f(x),\quad \forall x\in [\lambda, A+\lambda],$$ with equality for $x\in S=\mathsf{supp}(X)+\lambda$.
Suppose there exists a bounded interval $I$ such that $\mathsf{supp}(X)\cap I$ is infinite. As a result, we have that $S'=S\cap (I+\lambda)$ is also infinite.
Since $Y$ is an output distribution of the DTP channel and $\mathds{E}[X]>0$ necessarily (otherwise $I(X;Y)=0$ and $X$ is not capacity-achieving), we have that $Y$ has full support. Combining this with the fact that $-\log Y(y)=O(y\log y)$ when $y\to\infty$ for any output distribution $Y$ of the DTP channel, it follows that ${D_\mathsf{KL}}(Y_x||Y)$ is finite for every $x\geq 0$. Recalling , we have that $G(z)$ is finite for every $z\geq \lambda$, and hence for every $z\in\mathbb{R}$. Therefore, since $G$ is a power series, it follows that $G$ is real analytic in $(-\infty,\infty)$. Moreover, we have that $f$ is real analytic in $(0,\infty)$.
Since $G$ and $f$ are both real analytic in $(0,\infty)$ and agree on an infinite set $S'$ in this interval, it follows that $G(x)=f(x)$ for all $x\in(0,\infty)$ provided that $S'$ has a limit point in $(0,\infty)$ (via the identity theorem for real analytic functions [@KP02 Corollary 1.2.6]).
Assume that indeed $S'$ has a limit point in $(0,\infty)$. Then, it follows that $G(x)=f(x)$ for all $x\in(0,\infty)$. We show that this leads to a contradiction. In fact, note that, according to , it follows that $G$ is real analytic with finite $i$-th derivative $g(i)$ at $x=0$. On the other hand, the first right-derivative of $f$ at $x=0$ is infinite. This means that we cannot have $G(x)=f(x)$ for $0<x< \infty$. As a result, we conclude that $\mathsf{supp}(X)\cap I$ must be finite, as desired.
We now prove that $S'$ must have a limit point in $(0,\infty)$. Suppose that $S'$ has no limit points in $(0,\infty)$. Then, since $S'$ is a bounded infinite set, it must be the case that $0$ is a limit point of $S'$ (bounded infinite sets have at least one limit point). We show that $0$ cannot be a limit point of $S'$. If $\lambda>0$ this is trivially true since $S'\subseteq I+\lambda$ and so its limit points are at least as large as $\lambda$. We therefore assume $\lambda=0$.
Suppose that $0$ is a limit point of $S'$. Then, there exists a sequence $(x_i)$ such that $x_i\in S'$ and $x_i\neq 0$ for all $i$, and $x_i\to 0$. In particular, we have $G(x_i)=f(x_i)$ for all $i$. We prove that this cannot hold. Observe that $$\lim_{i\to\infty} f(x_i)/x_i =\lim_{i\to\infty} e^{x_i}\log x_i=-\infty.$$ On the other hand, recalling , $$G(x_i)/x_i=g(0)/x_i + g(1)+o(1),$$ when $i\to\infty$ (and hence $x_i\to 0$). Recalling with $x=0$, we must have $G(0)=g(0)\geq 0$. As a result, $G(x_i)/x_i$ is bounded from below by a constant for $i$ large enough, and so it must be the case that $G(x_i)\neq f(x_i)$ for $i$ large enough. Therefore, $0$ cannot be a limit point of $S'$.
The proof concludes by noting that $$\mathsf{supp}(X)=\bigcup_{i=0}^{A-1}(\mathsf{supp}(X)\cap [i,i+1]).$$ If $A$ is finite, then so is $\mathsf{supp}(X)$. On the other hand, if $A=\infty$, then $\mathsf{supp}(X)$ is countable, and thus countably infinite by invoking Theorem \[thm:unboundedsupp\].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Shlomo Shamai for asking them whether the capacity-achieving input distribution for the DTP channel under an average-power constraint must be discrete. This led them to the result of Section \[sec:shape\] that answers the question in the affirmative.
Existence of capacity-achieving distributions for the DTP channel under an average-power constraint {#sec:existoptimal}
===================================================================================================
In this section, we argue that capacity-achieving distributions exist for the DTP channel under an average-power constraint. For simplicity, we will assume that $\lambda=0$. Every result holds for arbitrary $\lambda\geq 0$ and under an additional peak-power constraint as well. Our approach follows that of [@AFTS01 Appendix I] closely.
We need to deal with the weak-\* topology on the set $\mathcal{F}$ of probability distributions on $\mathbb{R}$. We do not define the associated concepts of weak-\* compactness and continuity, but instead refer the reader to [@Lue97 Section 5.10] and the introduction of [@AFTS01 Appendix I] for the relevant background. We focus only on the parts where our approach necessarily differs from that of [@AFTS01 Appendix I].
Let $$\Omega_\mu={\mathopen{}\mathclose\bgroup\originalleft}\{F\in \mathcal{F}: F(0^-)=0,\int_0^\infty x dF(x)\leq \mu{\aftergroup\egroup\originalright}\}.$$ In other words, $\Omega_\mu$ is the set of probability distributions with support in $\mathbb{R}^{\geq 0}$ and bounded expected value. We also define $$\Omega^=_\mu={\mathopen{}\mathclose\bgroup\originalleft}\{F\in \mathcal{F}: F(0^-)=0,\int_0^\infty x dF(x)= \mu{\aftergroup\egroup\originalright}\}.$$
Given some distribution $F\in\Omega_\mu$, we denote by $I(F)$ the functional which maps $F$ to the mutual information $I(X_F;Y_F)$, where $X_F$, distributed according to $F$, is the input to the DTP channel and $Y_F$ is the corresponding output distribution. Then, we can write $$\label{eq:capsup}
C(\mu)=\sup_{F\in\Omega_\mu}I(F).$$
We begin by showing that capacity-achieving distributions exist for every $\mu\geq 0$. In other words, the supremum in is actually a maximum. In order to see this, we employ the following general lemma.
\[lem:max\] If $J:\Omega\to\mathbb{R}$ is weak-\* continuous on a weak-\* compact set $\Omega\subseteq X$, where $X$ is a linear vector space, then $J$ achieves its maximum in $\Omega$.
In our case, $J$ is the mutual information $I(\cdot)$ and $\Omega=\Omega_\mu$. It is easy to see that $\Omega_\mu$ is convex. One can then follow the same reasoning as in [@AFTS01 Appendix I.A] to show that $\Omega_\mu$ is weak-\* compact. It remains to show that $I(\cdot)$ is weak-\* continuous.
$I(\cdot)$ is weak-\* continuous in $\Omega_\mu$.
For $F\in\Omega_\mu$, we have $$I(F)=H(Y_F)-\int_0^\infty H(Y_x)dF(x),$$ where $Y_F$ is the output distribution induced by $F$. We show that the two terms in the right hand side are weak-\* continuous.
The proof that $\int_0^\infty H(Y_x)dF(x)$ is a weak-\* continuous function of $F\in\Omega_\mu$ follows in the same way as the analogous result in [@AFTS01 Appendix I.B]. This is because, for the DTP channel, $H(Y_x)$ is continuous in $x$ for all $x\geq 0$, $H(Y_x)=O(\log(1+x))$, and we have the constraint $\int_0^\infty xdF(x)\leq \mu$ for all $F\in\Omega_\mu$.
It remains to show that $H(Y_F)=-\sum_{y=0}^\infty Y_F(y)\log Y_F(y)$ is a weak-\* continuous function of $F\in\Omega_\mu$. Fix a sequence of distributions $(F_n)$ that converges weakly to some $F$, denoted by $F_n\xrightarrow{w} F$. In order to show that a functional $f$ is weak-\* continuous, in this case it suffices to show that $f(F_n)\to f(F)$ in the usual Euclidean metric in $\mathbb{R}$ as $n\to\infty$.
We have $$\begin{aligned}
\lim_{n\to \infty} H(Y_{F_n})&=-\lim_{n\to \infty} \sum_{y=0}^\infty Y_{F_n}(y)\log Y_{F_n}(y)\label{eq:hy1}\\
&=-\sum_{y=0}^\infty \lim_{n\to \infty} Y_{F_n}(y)\log Y_{F_n}(y)\label{eq:hy2}\\
&=-\sum_{y=0}^\infty Y_F(y)\log Y_F(y)\label{eq:hy3}\\
&=H(Y_F).\label{eq:hy4}
\end{aligned}$$ We justify all of the steps above. Observe that and follow by definition. To show , note that, for fixed $y$, the function $x\mapsto Y_x(y)$ is a bounded, continuous function of $x$. Therefore, by the properties of the weak-\* topology, it follows that $$Y_F(y)=\int_0^\infty Y_x(y)dF(x)$$ is a continuous function of $F$ for each $y$. Since $x\mapsto x\log x$ is continuous for $x\geq 0$ as well, holds.
It remains to prove . It suffices to show that we are in a condition to apply the dominated convergence theorem. More specifically, we need to prove that $$|Y_F(y)\log Y_F(y)|\leq g(y)$$ for all $F\in\Omega_\mu$ and $y\in\mathbb{N}$, where $g$ satisfies $\sum_{y=0}^\infty g(y)<\infty$. Fix $F\in\Omega_\mu$, and note that $$\begin{aligned}
Y_F(y)&=\int_0^\infty e^{-x}\frac{x^y}{y!}dF(x)\nonumber\\
&=\int_0^{y-y^{0.99}} e^{-x}\frac{x^y}{y!}dF(x)+\int_{y-y^{0.99}}^{y+y^{0.99}} e^{-x}\frac{x^y}{y!}dF(x)+\int_{y+y^{0.99}}^\infty e^{-x}\frac{x^y}{y!}dF(x).\label{eq:yfsep}
\end{aligned}$$ We analyze the three terms. First, since $x\mapsto Y_x(y)$ is increasing for $x<y$ and decreasing for $x>y$, we have $$\begin{aligned}
&\int_0^{y-y^{0.99}} e^{-x}\frac{x^y}{y!}dF(x)\leq Y_{y-y^{0.99}}(y),\label{eq:lower1}\\
&\int_{y+y^{0.99}}^\infty e^{-x}\frac{x^y}{y!}dF(x)\leq Y_{y+y^{0.99}}(y)\label{eq:upper3},
\end{aligned}$$ and both $Y_{y-y^{0.99}}(y)$ and $Y_{y+y^{0.99}}(y)$ converge to $0$ faster than $y^{-3/2}$ when $y\to\infty$.
For fixed $y$, it can be seen that $Y_x(y)$ is maximized when $x=y$. Furthermore, we have $Y_y(y)=O(1/\sqrt{y})$. Since $F\in\Omega_\mu$, we have $1-F(x)\leq \mu/x$, and so $$\label{eq:middle2}
\int_{y-y^{0.99}}^{y+y^{0.99}} e^{-x}\frac{x^y}{y!}dF(x)\leq \frac{\mu Y_y(y)}{y-y^{0.99}}=O(y^{-3/2}).$$ Combining with , , and yields $$\label{eq:asympyf}
Y_F(y)=O(y^{-3/2})$$ when $y\to \infty$ for all $F\in\Omega_\mu$, where the hidden constant is independent of $F$. To conclude, observe that, due to , for every ${\epsilon}$ there is a constant $y_{\epsilon}$ (possibly depending on $\mu$) such that $Y_F(y)\leq {\epsilon}$ for all $y\geq y_{\epsilon}$ and $F\in\Omega_\mu$. Therefore, $$\begin{aligned}
|Y_F(y)\log Y_F(y)|=O(Y_F(y)^{0.7})=O(y^{-1.05})
\end{aligned}$$ for all $F\in\Omega_\mu$, where we used in the last equality, the hidden constant is independent of $F$. Consequently, follows by noting that $\sum_{y=0}^\infty y^{-1.05}<\infty$. This shows that $H(Y_F)$ is weak-\* continuous, and hence $I(F)$ is weak-\* continuous too, as desired.
Finally, Lemma \[lem:max\] implies that for every $\mu\geq 0$ there exists $F^\star\in\Omega_\mu$ such that $$C(\mu)=I(F^\star).$$ We can show more: If $F^\star\in\Omega_\mu$ is capacity-achieving, then $F^\star\in\Omega^=_\mu$ necessarily. In fact, suppose not, and let $\mu'=\mathds{E}_{F_0}[X]$. We have $\mu'<\mu$ by hypothesis. Then, it is clear that $C(\mu'')=I(F^\star)$ for all $\mu''\in[\mu',\mu]$. It is easy to see that $C(\mu)$ is concave in $\mu$. As a result, we have $C(\mu'')=I(F^\star)$ for all $\mu''\geq \mu'$. However, it can be shown that $C(\mu)$ is unbounded when $\mu\to \infty$. This is a contradiction, and so $F^\star\in\Omega^=_\mu$ necessarily.
Proof of Theorem \[thm:dual\] {#sec:proof1}
=============================
In this section, we prove Theorem \[thm:dual\] for “well-behaved" channels. The technical meaning of “well-behaved" will be made clear later on.
Since our input alphabet is continuous, we have to deal with input distributions that do not have associated probability density/mass functions. In fact, the input distribution may be a mixture of discrete and continuous distributions. Because of this, we are forced to work solely with cumulative distribution functions, which we may call just “distributions". Our output alphabet is discrete, and so we may identify distributions with the corresponding probability mass functions. Overall, this leads to a more technical proof, although the methods used are still standard. Our approach mimics in part those of [@Smi71; @Sha90; @AFTS01]. Additionally, we present proofs of standard results whose proofs we could not find in the literature.
We note that if we deal with discrete inputs only, then the proof of the analogous result in this case is shorter [@Che17], but leads to the exact same conclusions.
Before we proceed with the proof of Theorem \[thm:dualtech\], we need some auxiliary definitions and results. Given a functional $f\colon \Omega\to\mathbb{R}$, where $\Omega$ is a convex subset of a linear vector space, the *weak derivative of $f$ at $F\in\Omega$ in the direction of $Q\in\Omega$*, denoted by $f'_F(Q)$, is defined as $$f'_F(Q)=\lim_{\theta\to 0^+}\frac{f((1-\theta)F+\theta Q)-f(F)}{\theta}.$$ The functional $f$ is said to be *weakly differentiable in $\Omega$ at $F$* if $f'_F(Q)$ exists for all $Q\in\Omega$. If $f$ is weakly differentiable in $\Omega$ at $F$ for all $F\in\Omega$, then we simply say $f$ is *weakly differentiable in $\Omega$*. We have the following result.
\[lem:weakder\] Fix a concave $f\colon\Omega\to\mathbb{R}$ in a convex space $\Omega$, and suppose that $f$ achieves a maximum in $\Omega$. If $F^\star\in\Omega$ is a maximizer of $f$ in $\Omega$ and $f'_{F^\star}(Q)$ exists, then $$f'_{F^\star}(Q)\leq 0.$$ Moreover, if $f$ is weakly differentiable in $\Omega$ at $F$ and $f'_{F}(Q)\leq 0$ for all $Q\in\Omega$, then $F$ maximizes $f$ in $\Omega$.
Fix $f$ satisfying the conditions of the lemma statement, and let $F^\star\in\Omega$ be a maximizer of $f$ over $\Omega$. Therefore, $$\frac{f((1-\theta)F^\star+\theta Q)-f(F^\star)}{\theta}\leq 0$$ for every $\theta\in(0,1]$, since $(1-\theta)F^\star+\theta Q\in\Omega$ by the convexity of $\Omega$ and $f(F^\star)\geq f(F)$ for every $F\in\Omega$ by hypothesis. As a result, if $f'_{F^\star}(Q)$ exists, then we must have $f'_{F^\star}(Q)\leq 0$.
For the second statement, suppose that $F$ is not a maximizer. Then, there exists $Q\in\Omega$ such that $f(Q)>f(F)$. For every $\theta\in (0,1]$, we have $$\frac{f((1-\theta)F+\theta Q)-f(F)}{\theta}\geq \frac{(1-\theta)f(F)+\theta f(Q)-f(F)}{\theta}=f(Q)-f(F)>0,$$ where the first inequality follows from the concavity of $f$. Since this result holds for every $\theta\in(0,1]$, we conclude that $f'_F(Q)\geq f(Q)-f(F)>0$. Therefore, if $f'_F(Q)\leq 0$ for all $Q$, then $F$ must be a maximizer of $f$ in $\Omega$.
The following lemma states a generalized form of Lagrange duality. Informally, this result transforms a constrained convex optimization problem (such as determining the capacity of a channel under some average-power constraint) into an unconstrained optimization problem. This is accomplished by moving the constraint into the objective function to be optimized. The necessary and sufficient conditions for optimality of a candidate solution to the unconstrained problem have a more useful form, as we shall see later in this section.
\[lem:lagdual\] Let $f\colon\Omega\to\mathbb{R}$ be convex, where $\Omega$ is a convex subset of a vector space $X$, and let $G:\Omega\to\mathbb{R}$ be a convex map. Suppose there exists an $x\in\Omega$ such that $G(x)<0$, and that $\inf\{f(x):G(x)\leq 0,x\in\Omega\}$ is finite. Then, $$\inf\{f(x):G(x)\leq 0,x\in\Omega\}=\max\{\varphi(z):z\geq 0\},$$ where $\varphi(z)=\inf\{f(x)+zG(x):x\in\Omega\}$, and the maximum on the right hand side is achieved by some $z^\star$.
Moreover, if the infimum on the left hand side is achieved by some $x^\star$, then $$z^\star G(x^\star)=0,$$ and $x^\star$ minimizes $f(x)+z^\star G(x)$ over $\Omega$.
Given some channel ${\mathsf{Ch}}$ with input alphabet $\mathcal{X}$ and output alphabet $\mathcal{Y}$, we can define the associated mutual information functional $I(\cdot)$. Suppose that the output distribution of ${\mathsf{Ch}}$ given input $x\in\mathcal{X}$ has an associated probability density function $Y_x(\cdot)$. Given a distribution $F$ on $\mathcal{X}$, we define $I(F)=I(X_F;Y_F)$, where $X_F$ is an input distribution to ${\mathsf{Ch}}$ distributed according to $F$ and $Y_F$ is the corresponding output distribution satisfying $$Y_F(y)=\int_\mathcal{X}Y_x(y)dF(x),\quad\forall y\in\mathcal{Y}.$$
The following result characterizes the weak derivative of $I(\cdot)$, conditioned on a certain quantity being finite. This characterization, combined with Lemma \[lem:weakder\], is the key to determining the conditions under which an input distribution is capacity-achieving.
\[lem:weakderexp\] Let $I(\cdot)$ denote the mutual information functional of some channel ${\mathsf{Ch}}$ with input alphabet $\mathcal{X}\subseteq \mathbb{R}^{\geq 0}$ and output alphabet $\mathcal{Y}\subseteq\mathbb{N}$. Fix an input distribution $F$ on $\mathcal{X}$ such that $I(F)<\infty$ with corresponding output distribution $Y_F$. Suppose that $$\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_F)dQ(x)<\infty$$ for a distribution $Q$ on $\mathcal{X}$. Then, $I'_F(Q)$ exists and is given by $$\label{eq:weakderexp}
I'_F(Q)=\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_F)dQ(x) - I(F).$$
Let $F_\theta=(1-\theta)F+\theta Q$ for $\theta\in[0,1]$. Denote the output distribution associated to $F_\theta$ by $Y_\theta=(1-\theta)Y_0+\theta Y_1$, where $Y_0$ and $Y_1$ denote the output distributions of $F$ and $Q$, respectively. We have $$\begin{aligned}
\frac{I(F_\theta)-I(F)}{\theta}&=\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_\theta)dQ(x)-I(F)\nonumber\\&+\frac{1-\theta}{\theta}{\mathopen{}\mathclose\bgroup\originalleft}(\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_\theta)dF(x)-\int_0^\infty{D_\mathsf{KL}}(Y_x||Y_0)dF(x){\aftergroup\egroup\originalright}).\label{eq:expandweakder}
\end{aligned}$$ We deal with the limit of each term on the right hand side of separately. First, we show that $$\label{eq:limterm1}
\lim_{\theta\to 0^+}\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_\theta)dQ(x)=\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_0)dQ(x).$$ Observe that $$\begin{aligned}
{D_\mathsf{KL}}(Y_x||Y_\theta)&=-H(Y_x)-\sum_{y=0}^\infty Y_x(y)\log((1-\theta)Y_0(y)+\theta Y_1(y))\nonumber\\
&\leq -H(Y_x)-\sum_{y=0}^\infty Y_x(y)\log((1-\theta)Y_0(y))\nonumber\\
&={D_\mathsf{KL}}(Y_x||Y_0)-\log(1-\theta)\label{eq:ineqKL1}
\end{aligned}$$ for all $\theta\in [0,1)$. Since $\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_0)dQ(x)<\infty$ by hypothesis, we conclude from that $$\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_\theta)dQ(x)<\infty$$ for all $\theta\in [0,1)$. Hence, by Fubini’s theorem, $$\begin{aligned}
\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_\theta)dQ(x)&=-\int_0^\infty {\mathopen{}\mathclose\bgroup\originalleft}(H(Y_x)+\sum_{y=0}^\infty Y_x(y)\log Y_\theta(y){\aftergroup\egroup\originalright})dQ(x)\nonumber\\
&=-H(Y|X_Q)-\sum_{y=0}^\infty \int_0^\infty Y_x(y)\log Y_\theta(y) dQ(x)\nonumber\\
&=-H(Y|X_Q)-\sum_{y=0}^\infty Y_1(y)\log Y_\theta(y).\label{eq:term1}
\end{aligned}$$ Therefore, $$\label{eq:simp1}
\lim_{\theta\to 0^+}\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_\theta)dQ(x)=-H(Y|X_Q)-\lim_{\theta\to 0^+}\sum_{y=0}^\infty Y_1(y)\log Y_\theta(y).$$ We now show that we can swap the limit and infinite sum on the right hand side of . Observe that $$-\log Y_\theta(y)\leq -\log Y_0(y)-\log(1-\theta)\leq -\log Y_0(y)+2,$$ provided that $\theta$ is small enough. Since $-\sum_{y=0}^\infty Y_1(y)\log Y_0(y)<\infty$ by hypothesis (recall we assume $\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_0)dQ(x)<\infty$), it follows by the dominated convergence theorem that $$-\lim_{\theta\to 0^+}\sum_{y=0}^\infty Y_1(y)\log Y_\theta(y)=-\sum_{y=0}^\infty Y_1(y)\lim_{\theta\to 0^+}\log Y_\theta(y)=-\sum_{y=0}^\infty Y_1(y)\log Y_0(y). \label{eq:domconv}$$ Combining with yields , as desired.
We now show that $$\label{eq:rightder}
\lim_{\theta\to 0^+}\frac{1-\theta}{\theta}\int_0^\infty ({D_\mathsf{KL}}(Y_x||Y_\theta)-{D_\mathsf{KL}}(Y_x||Y_0))dF(x)=0.$$ The limit on the left hand side of equals the right derivative of $\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_\theta)dF(x)$ with respect to $\theta$ at $\theta=0$. We show that this limit is zero by a reasoning similar to the proof of Leibniz’s integral rule. First, from it follows that $$\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_\theta)dF(x)=-H(Y|X_F)+\int_0^\infty\sum_{y=0}^\infty Y_x(y)\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{Y_\theta(y)}{\aftergroup\egroup\originalright})dF(x)\leq I(F)-\log(1-\theta)<\infty$$ for $\theta\in[0,1)$. As a result, by Fubini’s theorem we have $$\label{eq:simp3}
\int_0^\infty\sum_{y=0}^\infty Y_x(y)\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{Y_\theta(y)}{\aftergroup\egroup\originalright})dF(x)=\sum_{y=0}^\infty\int_0^\infty Y_x(y)\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{Y_\theta(y)}{\aftergroup\egroup\originalright})dF(x)=\sum_{y=0}^\infty Y_0(y)\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{Y_\theta(y)}{\aftergroup\egroup\originalright})$$ for $\theta\in[0,1)$, with the convention that $0\log x=0$ for all $x\geq 0$.
Let $h(\theta,y)=\log{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{Y_\theta(y)}{\aftergroup\egroup\originalright})$. Since $Y_\theta(y)=(1-\theta)Y_0(y)+\theta Y_1(y)$, we have $$\frac{\partial h}{\partial\theta} (\theta,y)=\frac{Y_0(y)-Y_1(y)}{(1-\theta)Y_0(y)+\theta Y_1(y)}$$ for all $\theta\in (0,1)$ and $y$. In particular, $h(\cdot,y)$ is, say, continuous in $[0,1/2]$ and differentiable in $(0,1/2)$ for every $y\in\mathsf{supp}(Y_0)$. Moreover, we have the general bound $$\label{eq:derbound}
{\mathopen{}\mathclose\bgroup\originalleft}|\frac{\partial h}{\partial\theta} (\theta,y){\aftergroup\egroup\originalright}|\leq \frac{1}{1-\theta}{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{Y_1(y)}{Y_0(y)}{\aftergroup\egroup\originalright}|\leq 2{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{Y_1(y)}{Y_0(y)}{\aftergroup\egroup\originalright})=:g(y)$$ for all $y\in\mathsf{supp}(Y_0)$, provided that $\theta<1/2$. Observe that $$\sum_{y=0}^\infty Y_0(y)g(y)=2+2\sum_{y=0}^\infty Y_1(y)=4,$$ and so $g$ is integrable with respect to $Y_0$.
Via , we can write the left hand side of as $$\lim_{\theta\to 0^+}\sum_{y=0}^\infty Y_0(y)\frac{h(\theta,y)-h(0,y)}{\theta}.$$ Since $h(\cdot,y)$ is continuous in $[0,1/2]$ and differentiable in $(0,1/2)$ for every $y\in\mathsf{supp}(Y_0)$, then, by the mean value theorem, for every $\theta\in(0,1/2]$ and $y\in\mathsf{supp}(Y_0)$ there exists some $z\in (0,\theta)$ such that $$\frac{h(\theta,y)-h(0,y)}{\theta}=\frac{\partial h}{\partial\theta} (z,y).$$ Taking into account , it follows that $${\mathopen{}\mathclose\bgroup\originalleft}|\frac{h(\theta,y)-h(0,y)}{\theta}{\aftergroup\egroup\originalright}|\leq g(y)$$ for all $y\in\mathsf{supp}(Y_0)$ and $\theta\in(0,1/2]$. Therefore, by the dominated convergence theorem we can conclude that $$\lim_{\theta\to 0^+}\sum_{y=0}^\infty Y_0(y)\frac{h(\theta,y)-h(0,y)}{\theta}=\sum_{y=0}^\infty Y_0(y)\lim_{\theta\to 0^+}\frac{h(\theta,y)-h(0,y)}{\theta}=\sum_{y=0}^\infty Y_0(y){\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{Y_1(y)}{Y_0(y)}{\aftergroup\egroup\originalright})=1-1=0,$$ which shows that holds.
Finally, combining , , and yields the desired result.
The following is a generalization to continuous alphabets of a well-known convex duality result for discrete memoryless channels [@CK11 Chapter 2, Theorem 3.4]. We use it to derive our general capacity upper bound in an easy way.
\[lem:lapi\] Fix a channel ${\mathsf{Ch}}$ with input alphabet $\mathcal{X}\subseteq \mathbb{R}^{\geq 0}$ and output alphabet $\mathcal{Y}\subseteq\mathbb{N}$. Suppose that for every set $S\subseteq\mathbb{N}$ the map $x\mapsto Y_x(S)=\sum_{y\in S} Y_x(y)$ is Borel-measurable. Let $F$ be any distribution on $\mathcal{X}$, and $Y$ any distribution on $\mathcal{Y}$. Then, $$I(F)\leq \int_0^\infty {D_\mathsf{KL}}(Y_x||Y)dF(x).$$
Finally, we define some sets which will be relevant in the proof. $$\begin{aligned}
{\Omega_{Y,A,\mu}}&={\mathopen{}\mathclose\bgroup\originalleft}\{F\in\mathcal{F}:\mathsf{supp}(F)\subseteq [0,A],\mathds{E}[Y_F]=\int_0^\infty\mathds{E}[Y_x]dF(x)\leq \mu{\aftergroup\egroup\originalright}\},\\
{\Omega^=_{Y,A,\mu}}&={\mathopen{}\mathclose\bgroup\originalleft}\{F\in\mathcal{F}:\mathsf{supp}(F)\subseteq [0,A],\mathds{E}[Y_F]=\int_0^\infty\mathds{E}[Y_x]dF(x)= \mu{\aftergroup\egroup\originalright}\},\\
{\Omega_{Y,A,\mathsf{fin}}}&= {\mathopen{}\mathclose\bgroup\originalleft}\{F\in\mathcal{F}:\mathsf{supp}(F)\subseteq [0,A],\mathds{E}[Y_F]=\int_0^\infty\mathds{E}[Y_x]dF(x)< \infty{\aftergroup\egroup\originalright}\}.\end{aligned}$$ Note that all of these sets are convex subsets of the set $\mathcal{F}$ of all distributions on $\mathbb{R}$. We are now ready to prove the following theorem.
\[thm:dualtech\] Let ${\sf Ch}$ be a channel with input alphabet $\mathcal{X}=[0,A]$ (where we may set $A=\infty$) and output alphabet $\mathbb{N}$. Furthermore, let ${\mathsf{Ch}}_\mu$ denote ${\mathsf{Ch}}$ under an output average-power constraint $\mu$. Suppose that, for every $S\subseteq \mathbb{N}$, the map $x\mapsto Y_x(S)=\sum_{y\in S} Y_x(y)$ is Borel-measurable. Then, we have the following:
1. Assume that there exist a random variable $Y$, supported on $\mathcal{Y}$, and parameters $\nu_0\in{\mathbb{R}}$ and $\nu_1\in\mathbb{R}^{\geq 0}$ such that $$D_{{\sf KL}}(Y_x\| Y)\leq \nu_0+\nu_1{\mathds{E}}[Y_x]$$ for every $x\in\mathcal{X}$. Then, we have $$C({\sf Ch}_\mu)\leq \nu_0+\nu_1\mu$$ for every $\mu$ such that ${\Omega_{Y,A,\mu}}\neq\emptyset$.
2. Suppose that ${D_\mathsf{KL}}(Y_x||Y_F)$ exists for all $x\in\mathcal{X}$ and all output distributions $Y_F$ associated to input distributions $F$ satisfying $\mathds{E}[X_F]>0$, that the map $x\mapsto {D_\mathsf{KL}}(Y_x||Y)$ is continuous in $x$, that for each $\mu>0$ there is $F$ such that $\mathds{E}[Y_F]<\mu$, and that $x\mapsto\mathds{E}[Y_x]$ is continuous in $x$. Then, if $F^\star\in{\Omega_{Y,A,\mu}}$ with $I(F^\star)<\infty$ is capacity-achieving for ${\mathsf{Ch}}_\mu$ we must have $$\label{eq:KLline}
{D_\mathsf{KL}}(Y_x||Y_{F^\star})\leq \nu_0+\nu_1\mathds{E}[Y_x],\quad\forall x\in\mathcal{X}$$ for some $\nu_0\in\mathbb{R}$ and $\nu_1\in\mathbb{R}^{\geq 0}$, with equality for $x\in\mathsf{supp}(F^\star)$. Moreover, if $F\in{\Omega^=_{Y,A,\mu}}$ satisfies for some $\nu_0\in\mathbb{R}$ and $\nu_1\in\mathbb{R}^{\geq 0}$ with equality for $x\in\mathsf{supp}(F)$, then $F$ is capacity-achieving for ${\mathsf{Ch}}_\mu$ and the capacity in this case is exactly $$C({\mathsf{Ch}}_\mu)=\nu_0+\nu_1\mu.$$
We begin by proving the first part of the theorem statement. Fix some distribution $Y$ in $\mathcal{Y}$ such that $$\label{eq:condY}
{D_\mathsf{KL}}(Y_x||Y)\leq \nu_0+\nu_1\mathds{E}[Y_x],\quad\forall x\in\mathcal{X}.$$ Furthermore, let $F\in{\Omega_{Y,A,\mu}}$ be some input distribution. By Lemma \[lem:lapi\], we have $$I(F)\leq \int_0^\infty {D_\mathsf{KL}}(Y_x||Y)dF(x)\leq \nu_0+\nu_1\int_0^\infty \mathds{E}[Y_x]dF(x)\leq \nu_0+\nu_1\mu.$$ The first inequality follows from Lemma \[lem:lapi\]. The second inequality follows from . Finally, the third inequality holds because $F\in{\Omega_{Y,A,\mu}}$. This implies that $C({\mathsf{Ch}}_\mu)\leq \nu_0+\nu_1\mu$, as desired.
We now prove the second part of the theorem statement. First, suppose that $F^\star\in{\Omega_{Y,A,\mu}}$ is capacity-achieving for ${\mathsf{Ch}}_\mu$. Instantiate Lemma \[lem:lagdual\] with $\Omega={\Omega_{Y,A,\mathsf{fin}}}$, $f(F)=-I(F)$, and $G(F)=\mathds{E}[Y_F]-\mu$. By hypothesis, we have that $I(F^\star)<\infty$, and that there exists $F$ with $G(F)<0$ whenever $\mu>0$. Moreover, both $-I$ and $G$ are convex, and ${\Omega_{Y,A,\mathsf{fin}}}$ is a convex subspace of a vector space. As a result, there exists $z^\star\geq 0$ such that $F^\star$ minimizes $$J(\cdot)=-I(\cdot)+z^\star G(\cdot)$$ over ${\Omega_{Y,A,\mathsf{fin}}}$, and $z^\star G(F^\star)=0$. As a result, according to Lemma \[lem:weakder\] we must have $$\label{eq:condweakder}
J'_{F^\star}(Q)=I'_{F^\star}(Q)-z^\star G'_{F^\star}(Q)\leq 0$$ if $I'_{F^\star}(Q)$ and $G'_{F^\star}(Q)$ exist. For a fixed $\overline{x}\in\mathcal{X}$, define the unit step function $Q_{\overline{x}}\in{\Omega_{Y,A,\mathsf{fin}}}$ as $$Q_{\overline{x}}(x)=\begin{cases}
0,\text{ if $x<\overline{x}$}\\
1,\text{ else.}
\end{cases}$$ Since ${D_\mathsf{KL}}(Y_x||Y_{F^\star})$ is finite for every $x\in\mathcal{X}$, Lemma \[lem:weakderexp\] implies that $I'_{F^\star}(Q_{\overline{x}})$ exists and is given by $$\label{eq:weakderstep}
I'_{F^\star}(Q_{\overline{x}})=\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_{F^\star})dQ_{\overline{x}}(x)-I(F^\star)={D_\mathsf{KL}}(Y_{\overline{x}}||Y_{F^\star})-I(F^\star).$$ Furthermore, since $G$ is linear in ${\Omega_{Y,A,\mathsf{fin}}}$, we have $$\label{eq:weakderG}
G'_{F^\star}(Q_{\overline{x}})=\lim_{\theta\to 0^+}\frac{G(F_\theta)-G(F^\star)}{\theta}=G(Q_{\overline{x}})-G(F^\star)=\mathds{E}[Y_{\overline{x}}]-\mathds{E}[Y_{F^\star}].$$
Combining , , and , we must have $${D_\mathsf{KL}}(Y_{x}||Y_{F^\star})-I(F^\star)-z^\star\mathds{E}[Y_{x}]+z^\star\mathds{E}[Y_{F^\star}]\leq 0$$ for every $x\in\mathcal{X}$. Equivalently, $${D_\mathsf{KL}}(Y_{x}||Y_{F^\star})\leq I(F^\star)+z^\star(\mathds{E}[Y_{x}]-\mu)$$ must hold for every $x\in\mathcal{X}$. The inequality holds because, according to Lemma \[lem:lagdual\], if $G(F^\star)\neq 0$ (i.e., $\mathds{E}[Y_{F^\star}]<\mu$), then we must have $z^\star=0$ and the inequality would still be true in this case.
Suppose now that there is $x\in\mathsf{supp}(F)$ such that $$\label{eq:strictsupp}
{D_\mathsf{KL}}(Y_x||Y_{F^\star})<I(F^\star)+z^\star (\mathds{E}[Y_x]-\mu).$$ All terms in the inequality above are continuous in $x$ by hypothesis. As a result, actually holds in an open neighborhood $U$ of $x$. Since $x\in\mathsf{supp}(F^\star)$, by definition of support we have $\int_U dF^\star(x)=\delta>0$ for some positive $\delta$. Therefore, $$I(F^\star)=\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_{F^\star})dF^\star(x)<I(F^\star)+z^\star {\mathopen{}\mathclose\bgroup\originalleft}(\int_0^\infty\mathds{E}[Y_x]dF^\star(x)-\mu{\aftergroup\egroup\originalright})=I(F^\star),$$ a contradiction. It follows that for $\nu_1=z^\star\geq 0$ and $\nu_0=I(F^\star)-z^\star\mu$ we must have $${D_\mathsf{KL}}(Y_{x}||Y_{F^\star})\leq \nu_0+\nu_1\mathds{E}[Y_x]$$ for every $x$, with equality for $x\in\mathsf{supp}(F^\star)$, as desired.
Finally, suppose that $F\in{\Omega^=_{Y,A,\mu}}$ satisfies $${D_\mathsf{KL}}(Y_{x}||Y_F)\leq \nu_0+\nu_1\mathds{E}[Y_x]$$ for some $\nu_0\in\mathbb{R}$ and $\nu_1\in\mathbb{R}^{\geq 0}$ and for every $x\in\mathcal{X}$, with equality for $x\in\mathsf{supp}(F)$. Then, for every distribution $Q\in{\Omega_{Y,A,\mathsf{fin}}}$ we have $$\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_F)dQ(x)\leq \nu_0+\nu_1 \mathds{E}[Y_Q]<\infty.$$ The last inequality follows since $Q\in {\Omega_{Y,A,\mathsf{fin}}}$. Furthermore, since ${D_\mathsf{KL}}(Y_{x}||Y_F)=\nu_0+\nu_1\mathds{E}[Y_x]$ for all $x\in\mathsf{supp}(F)$, we have $$I(F)=\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_F)dF(x)=\nu_0+\nu_1\mu.$$ The last equality holds because $F\in{\Omega^=_{Y,A,\mu}}$. Therefore, according to Lemma \[lem:weakderexp\], $I'_F(Q)$ exists for every $Q\in {\Omega_{Y,A,\mathsf{fin}}}$ and satisfies $$\label{eq:weakdercond}
I'_F(Q)=\int_0^\infty {D_\mathsf{KL}}(Y_x||Y_F)dQ(x)-I(F)\leq \nu_0+\nu_1 \mathds{E}[Y_Q]-(\nu_0+\nu_1\mu)=\nu_1 (\mathds{E}[Y_Q]-\mu).$$ As a result, $$I'_F(Q)-\nu_1 G'_F(Q)=I'_F(Q)-\nu_1(\mathds{E}[Y_Q]-\mu)\leq 0$$ for every $Q\in{\Omega_{Y,A,\mathsf{fin}}}$. Via Lemma \[lem:weakder\], it follows that $F$ minimizes the functional $J(\cdot)=-I(\cdot)+\nu_1 G(\cdot)$ over ${\Omega_{Y,A,\mathsf{fin}}}$. Moreover, since $\mathds{E}[Y_F]=\mu$, we have $G(F)=0$. This means that $F$ minimizes $J$ with value $J(F)=-I(F)$. If $F$ is not capacity-achieving over ${\Omega_{Y,A,\mu}}$, there exists some $F^\star\in{\Omega_{Y,A,\mu}}$ such that $I(F^\star)>I(F)$. Note that $G(F^\star)\leq 0$ and $\nu_1\geq 0$. Therefore, $$J(F^\star)=-I(F^\star)+\nu_1 G(F^\star)\leq-I(F^\star)< -I(F),$$ contradicting the fact that $F$ minimizes $J$ over ${\Omega_{Y,A,\mathsf{fin}}}$. Therefore, we conclude that $F$ is capacity-achieving and $C({\mathsf{Ch}}_\mu)=I(F)=\nu_0+\nu_1\mu$.
Note that, a priori, there may not exist capacity-achieving distributions for $\mathsf{Ch}_\mu$ as in the statement of Theorem \[thm:dualtech\]. Moreover, even if capacity-achieving distributions exist, they may not lie in ${\Omega^=_{Y,A,\mu}}$. It is possible to come up with stronger, but still general, assumptions about $\mathds{E}[Y_x]$ which ensure that there exist capacity-achieving distributions for ${\mathsf{Ch}}_\mu$, and that they lie in ${\Omega^=_{Y,A,\mu}}$. An example, which covers the DTP channel, is when $\mathds{E}[Y_x]$ is an increasing affine function of $x\in\mathcal{X}$. In this case, imposing an output average-power constraint is equivalent to imposing an input average-power constraint on the channel. As a result, the desired properties transfer directly from one setting to the other.
It remains to see that the DTP channel satisfies the hypotheses of Theorem \[thm:dualtech\] in order to derive Theorem \[thm:dual\]. First, the map $x\mapsto Y_x(S)$ is continuous in $x$ for all $S\subseteq \mathbb{N}$, and hence it is Borel-measurable. Second, ${D_\mathsf{KL}}(Y_x||Y)$ is finite and continuous in $x$ whenever $Y$ is an output distribution of the DTP channel with full support over $\mathbb{N}$. This happens whenever $\mathds{E}[X_F]>0$. Third, the results of Appendix \[sec:existoptimal\] imply that $I(F)<\infty$ for every $F\in{\Omega_{Y,A,\mathsf{fin}}}$. In particular, this means that $C({{\sf DTP}}_{\lambda,A,\mu})<\infty$ always. Fourth, observe that $\mathds{E}[Y_x]=\lambda+x$. Therefore, $x\mapsto \mathds{E}[Y_x]$ is continuous. Finally, note that we have $\mathds{E}[Y_F]=\mu$ if and only if $\mathds{E}[X_F]=\mu-\lambda$. It follows that ${\Omega_{Y,A,\mu}}\neq\emptyset$ and that there exists $F$ with $\mathds{E}[Y_F]<\mu$ whenever $\mu> \lambda$. Furthermore, because of this property, the results of Appendix \[sec:existoptimal\] imply that capacity-achieving distributions for the DTP channel over ${\Omega_{Y,A,\mu}}$ exist and are contained in ${\Omega^=_{Y,A,\mu}}$. Combining all of these observations with Theorem \[thm:dualtech\] leads to Theorem \[thm:dual\].
[^1]: Department of Computing, Imperial College London, UK. Emails: {m.cheraghchi, j.lourenco-ribeiro17}@imperial.ac.uk.
[^2]: We note that the duality framework of [@Che17] uses standard techniques and the dual-feasibility of the digamma distribution also has a simple proof.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
I. Nikitin[^1]\
\
bibliography:
- 'mm.bib'
title: Bivariate systems of polynomial equations with roots of high multiplicity
---
Let $P=c_{1}t^{a_{1}}+c_{2}t^{a_{2}}+\ldots+c_{k}t^{a_{k}}$ be a sparse polynomial of $k$ monomials in a single variable $t$ over the field of complex numbers. Consider $P(t)$ as a function from the complex torus $\mathbb{C}^{*}$ to $\mathbb{C}$. There is a well known result that for some choice of coefficients $P(t)$ has a nonzero root of multiplicity $k-1$. We formulate and proof the two-dimensional generalisation of that fact. We also study some properties of monomial maps.
Introduction
============
In the beginning we discuss the following classical result$\colon$
\[ts\] Let $P(t)$ be a sparse polynomial in a single complex variable $t$ such that $P(t)$ has exactly $k$ monomials. Then for some set of coefficients $P(t)$ has root of any given multiplicity up to $k-1$.
The proof is based on a simple linear-algebraic argument. We compute the rank of a special [*Wronski matrix*]{} that will be introduced latter. We will use the same idea to obtain the main result of the paper so let us exhibit the proof.
Let $P(t)=c_{1}t^{a_{1}}+c_{2}t^{a_{2}}+\ldots+c_{k}t^{a_{k}}$. We show that there exists a set of coefficients such that $t=1$ is a root of desirable multiplicity $k-1$. Since $P(t)$ is a polynomial in one variable the condition {$t=1$ is a root of multiplicity $k-1$} is equivalent to existence of a non-zero solution of the following system of linear of equations. $$P(1)=P'(1)=P''(1)=\ldots=P^{(i)}(1)=\ldots=P^{(k-2)}(1)=0$$ or in matrix notation $$\begin{bmatrix}
1 & 1 & 1 & \dots & 1 \\
a_{1} & a_{2} & a_{3} & \dots & a_{k-1} \\
\dots & \dots & \dots & \dots & \dots \\
a_{1}!/(a_{1}-k+2)! & a_{2}!/(a_{2}-k+2)! & a_{3}!/(a_{3}-k+2)! & \dots & a_{k-1}!/(a_{k-1}-k+2)!
\end{bmatrix}
\begin{bmatrix}
c_{1} \\ c_{2} \\ \dots \\ c_{k-1}
\end{bmatrix}
=
c_{k}\begin{bmatrix}
1 \\ a_{k} \\ \dots \\ a_{k}!/(a_{k}-k+2)!
\end{bmatrix}$$
Applying elementary transformations to the system, we get the following$\colon$
$$\begin{bmatrix}
1 & 1 & 1 & \dots & 1 \\
a_{1} & a_{2} & a_{3} & \dots & a_{k-1} \\
a^2_{1} & a^2_{2} & a^2_{3} & \dots & a^2_{k-1} \\
\dots & \dots & \dots & \dots & \dots \\
a^{k-2}_{1} & a^{k-2}_{2} & a^{k-2}_{3} & \dots & a^{k-2}_{k-1}
\end{bmatrix}
\begin{bmatrix}
c_{1} \\ c_{2} \\ \dots\\ \dots \\ c_{k-1}
\end{bmatrix}
=
c_{k}\begin{bmatrix}
1 \\ a_{k} \\ a^{2}_{k} \\ \dots\\ a^{k-2}_{k}
\end{bmatrix}$$ Now, by Cramer’s rule we get a nontrivial solution. Indeed, each determinant of that system is Vandermonde determinant and all $a_{k}$ are distinct.
The goal of this article is, in particular, to generalise that classical theorem to the case $n=2$.
For further discussions, let us introduce some notation. For each $a=(a_{1}, a_{2})\in\mathbb{Z}^{2}$ we use the expression $z^{a}$ to denote the monomial $z_{1}^{a_{1}}z_{2}^{a_{2}}$, where $z_{i}\in\mathbb{C}^{*}$ and for each $A\subset\mathbb{Z}^{2}$ we denote by $\mathbb{C}^{A}$ the space of Laurant polynomials supported at $A$, that is $f\in\mathbb{C}^{A}\iff f=\sum c_{a}z^{a}$, $c_{a}$ – are complex coefficients. For a polynomial $f\in\mathbb{C}^{A}$, the convex hull of its support is called the Newton polytope of $f$. For example $A=\{(1, 2), (3, 5), (0, 1)\}$ then $f=a xy^{2}+bx^{3}y^{5}+cy\in \mathbb{C}^{A}\ \forall a, b, c$. For a system of two polynomial equations supported at $(A_{1}, A_{2})\subset\mathbb{Z}^{2}$, $\mathbb{C}^{A}=\mathbb{C}^{A_{1}}\oplus\mathbb{C}^{A_{2}}$, $(f_{1}, f_{2})\in\mathbb{C}^{A}\iff f_{i}\in\mathbb{C}^{A_{i}}$. We identify each $f\in\mathbb{C}^{A}$ with the corresponding system of polynomial equations $f_{1}=f_{2}=0$. For any two subsets $A, B$ of $\mathbb{Z}^{2}$ their Minkowski sum is denoted by $A+B$ and defined as follows $A+B=\{a+b|a\in A, b\in B\}\subset\mathbb{Z}^{2}$. We also define $A\ominus B$ by the formula $A\ominus B=\{c|c+B\subset A\}\subset\mathbb{Z}^{2}$ i.e. the set of shift vectors such that any of them takes $B$ to a subset of $A$. All definitions may easily be generalised for higher dimensions. $\mathcal{Z}(\ldots)$ - the zero set of an ideal or a set of polynomials.
1. A tuple of finite sets $A_{1}, \ldots , A_{k}\subset\mathbb{Z}^n$ is said to be reduced, if they cannot be shifted to the same proper sublattice of $\mathbb{Z}^n$.
2. A tuple of finite sets $A_{1}, \ldots , A_{k}\subset\mathbb{Z}^n$ is said to be irreducible if it is impossible to shift all but $m$ of them to the same codimension $m$ sublattice for $m>0$.
In the paper [@AE] it was shown that under these conditions on $A$, the discriminant $\triangle_{A}$ has the following property$\colon$for generic tuple $(f_{1},\ldots, f_{n})\in\triangle_{A}$ there exists an isolated root of multiplicity exactly $2$. But there is no information about roots of higher multiplicity in this paper. If the maximal possible value of multiplicity is denoted by $I_{A}$ then the mentioned theorem states that $I_{A}\geq 2$.
In this work we are considering the following problem[\[q1\]]{}$\colon$what conditions should a tuple of Newton bodies satisfy to make the lower bound of $I_{A}$ be greater than 2. Finally, the main result to be proved in this paper is
\[t1\] Let $(A, B)$ be a reduced, irreducible tuple of convex subsets of $\mathbb{Z}^{2}$, satisfying the conditions$\colon A\ominus B=\varnothing$ or equivalently $B$ cannot be shifted to any subset of $A$. Then for an irreducible curve $\mathcal{Z}(f)$, that is given as a zero locus of a sparse polynomial $f\in\mathbb{C}^{B}$ and for almost all smooth points $p\in\mathcal{Z}(f)$ there exists a polynomial $g$ supported at $A$ with the property$\colon\mathcal{Z}(g)$ and $\mathcal{Z}(f)$ intersect at $p$ with multiplicity $|A|-1$.
The proof is given in Section $\ref{thr12}$. Despite this theorem makes non-trivial assumptions on $(A,B)$, we shall now see that actually theorems \[ts\] and \[t1\] together allow to find roots of nontrivial multiplicity for every pair $(A,B)$.
1. Theorem $\ref{t1}$ is obviously not applicable to the case when $A$ equals $B$ up to a shift. However, in this case the theorem remains valid for multiplicity $|A|-2$ instead of $|A|-1$. Let $(A, B)$ be a pair of convex bodies such that $B=A+b$, where $b$ is a suitable shift vector. In that case consider a sparse system of equations, supported at $(A, B)$ $$f_{A}=f_{B}=0$$ Then consider a new system that is obtained from the initial by a linear transformation $$f_{A}-c^{A}_{1}/c_{1}^{B}f_{B}=f_{B}=0$$ The support of the first equation in the system is $\bar{A}=A\setminus{a_{1}}$. Zeros of the new system are the same as zeros of the initial. Moreover, the coordinate transformation is never degenerate since its Jacobian is of maximal rank everywhere. After the change of variables, we will get a new system supported at $(\bar{A}, B)$. And the conditions of the theorem \[t1\] are satisfied.\
Replacing $|A|-2$ instead of $|A|-1$ is inevitable for the case $A=B$. See example \[ex1\].
2. If A consists of $p$ points in one line, and the projection of B along this line has $q$ points, then there exists a system of equations $f=g=0$ supported at $(A,B)$ with a multiple root of any multiplicity up to $\max(p,q)$. To prove this, make a monomial change of variables $(x,y)$ so that the first equation $f$ depends only on $x$, and apply Theorem $\ref{ts}$ to the first equation $f=0$, or to the equation $g(x_0,y)=0$ for a root $x_0$ of the first equation $f=0$.
3. Except for special pairs $(A, B)$ that were considered in the previous two remarks, every pair $(A, B)$ admits direct application of theorem \[t1\]. This is because for non-equal $A$ and $B$ either $A\setminus B=\emptyset$ or $B\setminus A=\emptyset$. Indeed, let $x+B\subset A$ and $A+y\subset B$ for some $x, y$. Then $|B|=|A|$ and $B=A+z$ for some $z$ but that is a contradiction.
4. The last example of this section shows that for some interesting pairs $(A, B)$ the theorem \[t1\] is applicable both to $(A,B)$ and $(B,A)$ and these two applications give two different estimates from which we can choose the maximal one.
Theorem \[t1\] may be considered as a generalisation of \[ts\] and as a first satisfactory answer to the mentioned question about $I_{A}$. On the contrary in higher dimensions the situation is sugnificaantly more complicated (see example \[ex5\]). We now work out some examples to examine the tightness of the estimate on the multiplicity of a root in theorem \[t1\].
\[ex1\] Let $A=B=\{(0,0), (1,0), (0,1), (1,1)\}$. The system, supported at that is as follows$\colon$ $$\begin{cases}
a_{1}x+b_{1}y+c_{1}xy+d_{1}=0\\
a_{2}x+b_{2}y+c_{2}xy+d_{2}=0
\end{cases}$$ Apply the following transformation$\colon$ $$\begin{cases}
a_{1}x+b_{1}y+c_{1}xy+d_{1}=0| \times (-c_{2})\\
a_{2}x+b_{2}y+c_{2}xy+d_{2}=0| \times c_{1}
\end{cases}$$ and get the system$\colon$ $$\begin{cases}
a_{1}x+b_{1}y+c_{1}xy+d_{1}=0\\
(a_{2}-a_{1}c_{2})x+(b_{2}-b_{1}c_{2})y+d_{2}-d_{1}c_{2}=0
\end{cases}$$ Thus we reduce a generic system of equations supported at the tuple $(A, A)$ to the system supported at $(A, A\setminus(1,1))$.
\[ex2\] Consider two plane curves which are given explicitly by the polynomials $$\gamma_{l}\colon y-p(x)=0$$ $$\gamma_{m}\colon y-q(x)=0$$ where $p(x), q(x)$ are polynomials in one variable $x$ of degrees $n$ and $m$. Let us denote their Newton polytopes by $A_{n}, A_{m}$ respectively. Without loss of generality we may assume that the point $p=(0, 0)$ is their common root. Choose the following local parametrization $x=t,\ y=p(t)$. Then the value of multiplicity is defined by the least non-zero term in the expression$\colon$ $$p(t)-q(t)$$ It is easy to understand, that the maximal possible value of multiplicity is $\max(m,n)$. Indeed, set $p(t)$ to be generic and $q(t)=p(t)+x^{m}(-p_{n}+q_{m})$. That tuple is not optimal for $m>n+1$ since $S_{A}=\{n\}$, but there exists a system with root of multiplicity higher than $n+1$ but for the partial case $m=n+1$ it is optimal. In addition, for each $i\in\{n+2, m\}$ there exists a system with root whose multiplicity equals $i$. The Berstein-Koushnerenko theorem tells us that the generic system has exactly $m$ isolated roots. Then in that case we can glue together all roots, as it was shown above.
\[ex3\] Let $A_{m}$ be two dimensional simplex, that is bounded from above by monomials of degree $n$, i.e. the following set $\{(a_{x}, a_{y})\in\mathbb{Z}^2|a_{x}+a_{y}\leq n, a_{x}\geq 0, a_{y}\geq 0\}\subset\mathbb{Z}^2$. According to famous Bezout theorem, a system supported at $A=(A_{n}, A_{m})$ has at most $mn$ isolated roots. Let us assume that $m>n$. Set $f=\prod h^{A}_{i}(x, y),\ g=\prod h^{B}_{i}(x, y)$, where $h_{i}^{A}$, $h_{i}^{B}$ are linear homogeneous forms of $x, y$ such that any two of them $h_{i}^{A}, h_{j}^{B}$ are linearly independent. That system has an isolated root of multiplicity $mn$ at zero.\
Let $j$ be of the form $j=nk+l<nm$ for $l\leq k$. Then there exists a system supported at $A$ such that it has a root of multiplicity $j$. To show that we take $$u=\prod_{i=1}^{n} h_{i}\in\mathbb{C}^{A_{n}}$$ $$v=\prod_{i=1}^{l} h_{i}^{a_{i}}+H^{k+1}\in\mathbb{C}^{A_{m}},\ a_{1}+a_{2}+\ldots+a_{l}=k$$ where $h_{i}$ are linear forms such that any two of them are linearly independent and $H$ is any homogeneous polynomial of degree $k+1$ such that $H$ is not identically zero on the line, defined by $h_{i}$ for all $i$. For $i\in\{1, 2, \ldots, l\}$ we have $x=t, y=-th_{i}^{x}/h_{i}^{y}$ and $v(t)=H(t, -th_{i}^{x}/h_{i}^{y})=t^{k+1}C,\ C\neq 0$. And for $i\in\{l+1,\ldots, n\}$ we have $v(t)=t^{k}C_{1}+C_{2}t^{k+1},\ C_{1}\neq 0$. Then the total multiplicity is $l(k+1)+(n-l)k=nk+l$ as desired.
\[ex4\] Consider two plane curves which are given explicitly by the polynomials$\colon$ $$\gamma_{n}\colon y-p_{n}(x)=0$$ $$\gamma_{m}\colon x-q_{m}(y)=0$$ By Berstein-Koushnerenko theorem that system may have at most $mn$ isolated roots, as in the previous example, But the difference is that we cannot get the system that has a root of multiplicity $mn$. Indeed, without loss of generality we may asuume that $p=(0, 0)$ is the root of multiplicity $mn$ then after substitution, we will have the equation$\colon$ $$x=q_{m}(p_{n}(x))$$ that has a root of multiplicity $mn$. Therefore $q_{m}(p_{n}(x))-x=a x^{mn}$, $a\in\mathbb{C}$. Let the coefficient in the term $x^{mn-1}$ be equal $c$, then $c=0$ and $c=mq_{m}p_{n}^{m-1}p_{n-1}$. But $p_{n}\neq 0$ since $a\neq 0$, therefore $p_{n-1}=0$. By the same computations for $q$ we obtain $q_{m-1}=0$, and then by induction we finally get a contradiction. Now let us show that the tuple $A=\{(0, 0), (0, 1), (1, 0), (2, 0)\},\ B=\{(0, 0), (1, 0), (0, 1), (0, 2), (0, 3)\}$ is optimal. Indeed, consider the following polynomial$\colon$ $$a(fx^2+gx+h)^3+b(fx^2+gx+h)^2+c(fx^2+gx+h)+d-x=0$$ the system has a root of multiplicity 4 if and only if the first four coefficients vanish and the fifth doesn’t$\colon$ $$\begin{cases}
d+ch+bh^2+ah^3=0\\
-1+cg+2bgh+3agh^2=0\\
cf+bg^2+2bfh+3ag^2h+3afh^2=0\\
2bfg+ag^3+6afgh=0
\end{cases}$$ Fortunately, that system is linear in $(a, b, c, d)$ and has the solution$\colon$ $$a=2f^2/g^5,\ b=-(fg^2+6f^2 h)/g^5,\ c=(g^4+2fg^2h+6f^2h^2)/g^5,\ d=-(g^4h+fg^2h^2+2f^2h^3)/g^5$$ Substituting all variables to the fourth term, we get $$5f^3/g^3$$ Then we cannot vanish the fourth coefficient without getting a contradiction. The tuple is optimal as desired.
Monomial map {#mm}
============
In this section the definition of the monomial map $m_{A}$ will be given, and we will also verify some important properties of it. This section can be considered and used as an independent material.
Let $(\mathbb{C}^{*})^{n}$ be a complex torus and $A$ be a subset of $\mathbb{Z}^{n}$. By $\mathbb{CP}^{A}=\{[\ldots\colon z_{a_{i}}\colon\ldots]\}$ we denote the projective space of homogeneous coordinates whose index set is $A$. The monomial map supported at $A$ is given by the rule
$$m_{A}\colon(\mathbb{C}^{*})^{n}\rightarrow\mathbb{CP}^{A}$$
$$z\mapsto [\ldots\colon z^{a_{i}} \colon\ldots]$$
By $m_{A}(X)$ we denoted the image of a set $X$ under the monomial map. The map is well defined since $z\neq 0$. Let $A=(A_{1}, \ldots, A_{n})$ be a tuple of subsets of $\mathbb{Z}^{n}$ and $f=(f_{1}, f_{2}, \ldots, f_{n})$ be a set of polynomials supported at $A$, i.e. $\operatorname{supp}f_{i}=A_{i}$. We now shall give the precise statement of the result mentioned in the previous section.
\[t2\] Let $B=(B_{1}, \ldots,B_{n-1})$ and $A$ be subsets of $\mathbb{Z}^{n}$ and let $m_{A}(f)$ be the image of the curve, defined as the zero set of a system of polynomials $f=(f_{1}, \ldots, f_{n-1})\in\mathbb{C}^{B}$. Then
1. If $(A, B)$ and $f$ are irreducible, then $m_{A}(f)$ is irreducible.
2. If $(A, B)$ is moreover reduced, then the map $m_{A}|_f$ has degree $1$.
3. If $(A, B)$ is reduced and irreducible, then $m_{A}$ is injective unless $(A, B)$ can be shifted to the same lattice simplex of lattice volume $1$.
The goal of this section is to formulate a criteria to determine whether $m_{A}$ is injective. We are also considering the following question$\colon$when does the image under $m_{A}$ of an algebraic curve lie in a hyperplane. For the sake of that let’s prove some technical facts. Let $g=\sum_{a\in A}c_{a}z^{a}$ be a polynomial in $\mathbb{C}^A$, by $\pi_{g}$ we denote the hyperplane in $\mathbb{CP}^{A}$ that is given by the same set of coefficients as $g$. The image of any proper subset $X$ of complex tori under $m_{A}$ is called $A$ – image of $X$. We shall say that the tuple $(A_{1}, A_{2}, \ldots, A_{k})\subset\mathbb{Z}^{n}$ is irreducible if it is impossible to shift all but $m$ of them to the same sublattice of codimension $m$. We denote the $A$ – image of the complex torus by $\mathbb{T}$.
\[p1\] Let $g\in\mathbb{C}^{A}$, $f\in\mathbb{C}^{B}$, $B=(B_{1}, \ldots, B_{n-1})$ then the following hold true$\colon$
1. $A$ – image of $\mathcal{Z}(g)$ and $\mathbb{T}\cap\pi_{g}$ coincide
2. If $m_{A}$ is injective (or at least the restriction of $m_{A}$ to $\mathcal{Z}(f)$ is injective) then the intersection multiplicity of $Z(f)$ and $Z(g)$ at $p$ and the intersection multiplicity of their $A$ – images at $m_{A}(p)$ coincide
3. If $(A, B)$ is irreducible then for generic $f\in\mathbb{C}^{B}\colon A$ – image of $\mathcal{Z}(f)\subset\pi_{g}$ $\iff\exists c_{i}\in\mathbb{C}[x_{1},\ldots, x_{n}]$ such that $g=\sum c_{i}f_{i}$
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1. Let $q\in\mathbb{T}\cap\pi_{g}\Rightarrow$ there exists $x\in(\mathbb{C}^{*})^{n}$ and $t\in\mathbb{C}^{*}$ such that $\sum_{a\in A}c_{a}tx^{a}=0$, where $c_{a}$ are coefficients of the polynomial $g$. Dividing by $t$ we have the following condition $x\in Z(g)$. In converse direction$\colon$let $\sum_{a\in A}c_{a}x^{a}=0$, then the homogeneous coordinates of the image of $x$ will be $x^{a}$ and they will obviously satisfy $\sum_{a\in A}c_{a}x^{a}=0\Rightarrow m_{A}(x)\in\pi_{g}$.
2. Let $p$ be a smooth point of the curve $f\in\mathbb{C}^{B}$ and $g\in\mathbb{C}^{A}\colon g(p)=0$, and $\phi$ be its local parametrization at $p$. Consider the restriction of $m_{A}$ to $f$ in some neighbourhood of $p$ and denote that by $m_{A, p}$. If $m_{A}$ is injective, then the composition $m_{A, p}\circ\phi$ is a parametrization of $A$ –image of $f$. Since the equations of $\pi_{g}$ and the surface $g=0$ have the same coefficients, then their Laurant expansions $g\circ\phi$ and $\pi_{g}\circ m_{A, p}\circ\phi$ will coincide$\Rightarrow$ the intersection multiplicities of $\mathcal{Z}(g)$ and $\mathcal{Z}(f)$ at $p$ and their $A$ –images will be equal.
3. Assume that $A$ – image of $\mathcal{Z}(f)$ lies in $\pi_{g}$. Take $p\in A$ – image of $\mathcal{Z}(f)$, then $\exists x\in(\mathbb{C}^{*})^{n}$ such that $p=m_{A}(x)$. $p\in\pi_{g}$ then $\pi_{g}\colon\sum_{a\in A}c_{a}tx^{a}=0, t\in\mathbb{C}^{*}$. Dividing by $t$ we have $\sum c_{a}x^{a}=0\Rightarrow x\in \mathcal{Z}(g)$ and $\mathcal{Z}(f)\subset \mathcal{Z}(g)\Rightarrow$ by Hilbert nullstellensatz $\exists c_{i}\in\mathbb{C}[x_{1}, \ldots, x_{n}], i=1\ldots n-1$ and $k\in\mathbb{Z}$ such that $g^{k}=c_{1}f_{1}+\ldots+c_{n-1}g_{n-1}$. By assumption $(A, B)$ is irreducible. Then, according to the theorem 3.21 in [@AE] for generic $f\in\mathbb{C}^{B}$, $\mathcal{Z}(f)$ is also irreducible as an algebraic subset. $(f)\subset\mathbb{C}[x_{1}, \ldots, x_{n}]$ is irreducible if and only if its vanishing set is irreducible. For the prime ideal $I$ the following implication holds true $a^k\in I\Rightarrow a\in I$. Thus $g^{k}=c_{1}f_{1}+\ldots+c_{n-1}g_{n-1}\Rightarrow\exists \bar{c}_{i}\in\mathbb{C}[x_{1},\ldots, x_{n}]$ such that $g=\sum \bar{c}_{i}f_{i}$. In converse direction let $g=c_{1}f_{1}+\ldots+c_{n-1}f_{n-1}$, then the vanishing set of $f$ vanishes $g$ as well$\Rightarrow \mathcal{Z}(f)\subset \mathcal{Z}(g)$. Therefore $A$–image $\mathcal{Z}(f)\subset\pi_{g}$ .
Let us now describe the set-theoretical properties of $m_{A}$ in terms of combinatorial properties of $A$. Note that$\colon$ $$m_{A}-\text{is injective}\iff \{p^{a_{1}}/q^{a_{1}}=\ldots=p^{a_{i}}/q^{a_{i}}\Rightarrow p=q\}$$ That condition is equivalent to the following $$\{p^{a_{1}}=\ldots=p^{a_{i}}\Rightarrow p=1\}$$ The property of $m_{A}$ to be injective and to generate $\mathbb{Z}^{n}$ are not equivalent.
Consider $A=\{a=(2, 1), b=(-3, -1), c=(3, 2)\}$ it is easy to show that $A$ generates $\mathbb{Z}^{2}$ as a $\mathbb{Z}$–module. Indeed $(1, 0)=-a-b, (0, 1)=c+b$. But the system $$\{x^{2}y^{1}=x^{-3}y^{-1}=x^{3}y^{2}\}$$ has at least two solutions$\colon(1, 1)$ and $\{(\epsilon, \epsilon^{-1})|\epsilon^{3}=1\}$ thus $m_{A}$ is not injective.
We denote $\{\sum t_{i}a_{i}|\sum {t_{i}}=0\}$ by $<A>_{0}$.
\[pr2\] $<A+(-A)>=<A>_{0}$
Let $x\in<A+(-A)>$, then there exists the decomposition $$x=\sum t_{ij}(a_{i}-a_{j})$$ each term can be rewritten as follows $t_{ij}a_{i}-t_{ij}a_{j}\Rightarrow x\in<A>_{0}$. In converse direction let $x\in<A>_{0}$. Consider the decomposition $$x=\sum t_{i}a_{i}=\sum_{+} t_{i}a_{i}+\sum_{-} t_{i}a_{i}$$ with positive terms in first summand and negative in the second. Let $(t_{\alpha}, t_{\beta})$ be a pair of terms whose absolute value is minimal. Without lose of generality it can be assumed that $|t_{\alpha}|>|t_{\beta}|$. Then the decomposition of $x$ can be rewritten in the following way$\colon$ $$x=\sum_{+}t_{i}a_{i}+\sum_{-}t_{j}a_{j}=a_{\alpha}(t_{\alpha}-t_{\beta})+\sum_{+}t_{i}a_{i}+\sum_{-}t_{j}a_{j}+(a_{\alpha}-a_{\beta})|t_{\beta}|$$ In first three terms the sum of the coefficients is still zero and the sums $\sum |t_{i}|$ and $\sum |t_{j}|$ strictly decreased. By induction on the height $\sum |t_{i}|+\sum |t_{j}|$ we proved that $x=\sum t_{ij}(a_{i}-a_{j})+ta-tb\Rightarrow x=\sum t_{ij}(a_{i}-a_{j})\Rightarrow x\in <A+(-A)>$
\[pr3\] $m_{A}$ is injective $\iff <A>_{0}=\mathbb{Z}^{n}$
We denote the left and the right statements by $L, R$ respectively. First let us prove $R\Rightarrow L$ and then $\rceil R\Rightarrow \rceil L$. The first implication is straightforward, set $p^{a_{i}}=t$, where $p\in(\mathbb{C}^{*})^n$, then we have a system of linear combinations $e_{\alpha}=\sum\lambda_{i}^{\alpha}a_{i}$, where $\{\lambda_{i}\}$ satisfy the equality $\sum \lambda_{\alpha}^{i}=0$, then $$p^{\sum\lambda_{i}^{\alpha}a_{i}}=p^{e_{\alpha}}=x_{\alpha}$$ on the other hand $$p^{\sum\lambda_{i}^{\alpha}a_{i}}=t^{\sum \lambda_{\alpha}^{i}}=1$$ thus $m_{A}$ is injective. Conversely suppose $<A>_{0}$ is the image of $B$ under the lattice embedding $i\colon\mathbb{Z}^{m}\rightarrow\mathbb{Z}^{n}$ for some $m$ – dimensional lattice $B$. Let $n$ be greater than $\dim<A>_{0}$, then choose a generating system in $<A>_{0}$ and complete it to the basis of $\mathbb{Z}^{n}$ then it is obvious that the system $p^{a_{i}}=p^{a_{j}}$ has infinitely many solutions since the system is free from at least one of the coordiantes. In the second case we assume that $\dim<A>_{0}=n$ but $<A>_{0}\neq \mathbb{Z}^{n}$ and show that each point has more than one preimage. Indeed, choose a generating set $\{l_{i}\}\subset<A>_{0}$. The system $p^{a_{i}}=p^{a_{j}}$ is equivalent to the system $p^{a_{1}-a_{j}}=1$ for all $j$. Express each vector $a_{i}-a_{j}$ as a linear combination of $l_{i}$ and set $p^{l_{i}}=q_{i}$, let $M$ be the matrix whose columns are $l_{i}$ in the standard basis $e_{\alpha}$. Since the dimension of $<A>_{0}$ equals $n$ then the cardinality of the set $\{l_{i}\}$ also equals $n$. Consider the new system$\colon$ $$q^{t^{j}}=1$$ Where $t^{j}=(t^{j}_{1},\ldots, t^{j}_{n})$ are coordinates of $a_{1}-a_{j}$ in basis $l_{i}$ and $q=(q_{1},\ldots, q_{n})$. That system has a sollution $q_{i}=1$. Then in terms of the original variables we obtain the system $p^{l_{i}}=1$ that has $|\det(M)|> 1$ sollutions. Thus $m_{A}$ is not injecitve.
$m_{A}$ is an injective map if and only if $<A+(-A)>$ generates ${Z}^{n}$ as a $\mathbb{Z}$–module.
According to proposition \[pr2\] and proposition \[pr3\], $m_{A}$ is injective $\iff \mathbb{Z}^{n}=<A>_{0}=<A+(-A)>$
Proof of the main result {#thr12}
========================
\[lemwr\] Let $\gamma$ be a projective curve that is given locally (in some neighbourhood of a point $p\in\gamma$) by a family of holomorphic functions$\colon$ $$f\colon U\rightarrow\mathbb{P}^n$$ $$z\mapsto[\ldots\colon f_{i}(z)\colon\ldots]$$ If the dimension of the projective hull of $\gamma$ equals $k$, then for all $i\in\{1\ldots k\}$ there exists an osculating hyperplane $H_{i}$ such that $\gamma$ and $H_{i}$ intersect at $p$ with multiplicity $i$.
Let the dimension of the projective hull of $\gamma$ equal $k$ then by definition there exists a set of $n-k$ linearly independent rows $\{c_{j}\}_{j=0}^{n}$ such that $\sum c_{j}f_{j}=0$ for all $z$ in some neighbourhood of $p$. Each row associated to some function $\{f_{j}\}$. The well known fact [@W] that the set of analytic functions is linearly independent if and only if the Wronskian of this set $\colon$ $$\begin{bmatrix}
f_{0} & f_{1} & f_{2} & \dots & f_{k} \\
f_{0}^{(1)} & f_{1}^{(1)} & f_{2}^{(1)} & \dots & f_{k}^{(1)} \\
\hdotsfor{5} \\
f_{0}^{(k)} & f_{1}^{(k)} & f_{2}^{(k)} & \dots & f_{k}^{(k)}
\end{bmatrix}$$ has the maximal rank i.e. there exists a non-vanishing minor of dimension $k+1$.\
Now let us provide a relation between that property and the statement of the lemma. Expand each $f_{j}$ as a Laurant power series at $p$ and substitute them to the equation $H=c_{0}x_{0}+c_{1}x_{1}+\ldots+c_{n}x_{n}$ so we can find $c_{i}$ using method of undetermined coefficients. Group terms by their exponents $z^i$ and then we have $\colon$ $$z^{0}(c_{0}f_{0}(0)+\ldots+c_{n}f_{n}(0))/0!+$$ $$z^{1}(c_{0}f_{0}^{(1)}(0)+\ldots+c_{n}f_{n}^{(1)}(0))/1!+\ldots$$ $$z^{j}(c_{0}f_{0}^{(j)}(0)+\ldots+c_{n}f_{n}^{(j)}(0))/j!+\ldots$$ $$z^{n}(c_{0}f_{0}^{(n)}(0)+\ldots+c_{n}f_{n}^{(n)}(0))/n!$$ Thus it is necessary for the matrix $$\begin{bmatrix}
f_{0}(0) & f_{1}(0) & f_{2}(0) & \dots & f_{k}(0) \\
f_{0}^{(1)}(0) & f_{1}^{(1)}(0) & f_{2}^{(1)}(0) & \dots & f_{k}^{(1)}(0) \\
\hdotsfor{5} \\
f_{0}^{(k)}(0) & f_{1}^{(k)}(0) & f_{2}^{(k)}(0) & \dots & f_{k}^{(k)}(0)
\end{bmatrix}$$ to have a principal non-vanishing minor of order at least $k$. Assume that in $\{f_{i}\}_{i=0}^{n}$ there exists a linearly independent subset $\{f_{i_{l}}\}_{l=0}^{k}$ of cardinality $k$. Its Wronskian is not identically zero on $U$, let us assume that it is not zero at a point $p$. Rearrange the homogeneous coordinates so that the minor associated to the system $\{f_{i_{l}}\}_{l=0}^{k}\}$ is principal. In that case the hyperplanes $0=c_{0}f_{0}^{(i)}+\ldots+c_{0}f_{k}^{(i)}$ intersect transversely and we may satisfy first $j$ equations without satisfying the others as desired. Since the augmented matrix of the system and the Wronski matrix coincide at $p$ we obtain the statement of lemma.
For a given tuple of polynomials $u=(u_{1}, \ldots, u_{n-1})\in\mathbb{C}^{B}$ in $n$ complex variables consider the ideal $J_{u}$ generated by $u$ in $\mathbb{C}[x_{1}, \ldots, x_{n}]$. Assume that $J_{u}$ is prime. Let $A$ be a subset of $\mathbb{Z}^{n}$. For $(B, A)$ consider the vector space $$\mathbb{V}^{u}_{A}=\mathbb{C}^{A}\cap J_{u}$$ and a linear map $$\alpha_{A}^{u}\colon\mathbb{C}[x_{1},\ldots,x_{n}]^{n-1}\rightarrow J_{u}\rightarrow \mathbb{V}^{u}_{A}$$ Where the first arrow acts by the rule $$(c_{1}, \ldots, c_{n-1})\mapsto c_{1}u_{1}+\ldots+c_{n-1}u_{n-1}$$ and the second one is a projection on $\mathbb{C}^A$ i.e $z^{a}\mapsto z^{a}$ iff $a\in A$ and $z^{a}\mapsto 0$ in other cases. Given an arbitrary subset $X\subset\mathbb{P}^{A}$, we denote its projective hull by $<X>$.
Let $(A, B)$ be a tuple of convex sets in $\mathbb{Z}^n$ and $u\in\mathbb{C}^{B}$ be an irreducible set of polynomials. Then the following holds true$\colon$
1. $\operatorname{codim}<m_{A}(\mathcal{Z}(u))>=\dim\mathbb{V}^{u}_{A}$
2. For the case $n=2$, the restriction of $\alpha_{A}^{u}$ to the subset ${\mathbb{C}^{A\ominus B}}\rightarrow\mathbb{V}^{u}_{A}$ is a surjective map.
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1. Set $\operatorname{rk}\alpha_{A}^{u}=l$, therefore we have a set $\{f_{j}\}_{j=1}^{j=l}$ of $l$ linearly independent polynomials $f_{j}\in\mathbb{C}^{A}$ such that$\colon$ $$f_{j}=c_{1}^{j}u_{1}+\ldots+c_{n-1}^{j}u_{n-1}\\$$ It follows that $\forall p\in\mathcal{Z}(u)$, $m_{A}(p)\in\pi_{f_{j}}$ where $\pi_{f_{j}}$ are linearly independent $\Rightarrow l\leq\operatorname{codim}<m_{A}(u)>$. In converse direction let $\operatorname{codim}<m_{A}(u)>$ be equal $l$, but $\operatorname{rk}\alpha_{A}^{u}\leq l-1$. Represent $<m_{A}(\mathcal{Z}(u))>$ as a complete intersection of $l$ hyperplanes $\{\pi_{j}\}_{1}^{l}$. The linear system of hyperplanes is naturally associated with linear system of polynomials $f_{j}\in\mathbb{C}^{A}$. Then by proposition \[p1\], there exist $\{c_{i}^{j}\}$ such that $f_{j}=c_{1}^{j}u_{1}+\ldots+c_{n-1}^{j}u_{n-1}\Rightarrow f_{j}\in\operatorname{Im}\alpha_{A}^{u}$ and $\operatorname{rk}\alpha_{A}^{u}\geq l$ as desired.
2. Let $f=cu$, where $f$ and $u$ are sparse polynomials supported at $A$ and $B$ respectively. Without loss of generality we may assume that $C=\operatorname{supp}(c)$ is a convex set. Vertices of a Minkowski sum of two convex bodies are sums of vertices and vice versa. Suppose that $C\not\subset A\ominus B$ then there exists at least one vertex $v$ of $C$ such that $v\not\in A\ominus B$. Take $b\in B$ such that $v+b$ is a vertex of $A$. Since $v+b\not\in A$ we conclude that the coefficient of the monomial $z^{v}$ is zero and $v\not\in C$. By induction we get the inclusion $C\subset A\ominus B$ as desired.
We are now ready to prove the main result
Take an irreducible sparse polynomial $u\in\mathbb{C}^{B}$ and consider the map $\alpha_{A}^{u}$. Let $Q_{\alpha}$ be the matrix representation of $\alpha_{A}^{u}$. The coefficients of $Q_{\alpha}$ are linear forms of $u$. Set $\operatorname{rk}Q_{\alpha}=l$ then the largest order of a non-vanishing minor in $Q_{\alpha}$ is $l$. Consider the holomorphic embedding $u\hookrightarrow\mathbb{P}^{l}$ as a restriction of the map $m_{A}|_{u}$ and apply lemma \[lemwr\]. We can see that for generic point $p$ of the curve $m_{A}(\mathcal{Z}(u))$, for all $i\in\{1\ldots l\}$ there exists an osculating hyperplane $H_{i}$ such that $H_{i}$ intersects $m_{A}(\mathcal{Z}(u))$ at $p$ with multiplicity exactly $i$. By proposition \[p1\], the set $\{H_{i}\}$ corresponds to the system of hypersurfaces $\in\mathbb{C}^{A}$ that intersect $u$ with the same multiplicity. For the partial case $A\ominus B=\emptyset$ we obviously have $l=|A|-1$.
\[ex5\] Consider a tuple of bodies from $\mathbb{Z}^3\colon$\
$A=\{(0,0,0),(1,0,0),(1,1,0),(0,1,0)\}, B=\{(0,0,0),(0,1,0),(0,1,1),(0,0,1)\}, C=\{(0,0,0),(1,0,0),(1,0,1),(0,0,1)\}$ and a system of equations supported at that$\colon$ $$\begin{cases}
a_{1}x+b_{1}y+c_{1}xy+d_{1}=0\\
a_{2}x+b_{2}z+c_{2}xz+d_{2}=0\\
a_{3}y+b_{3}z+c_{3}yz+d_{3}=0
\end{cases}$$ Choose the curve given by the system of equations, say $$\gamma\colon\begin{cases}
3x+y+xy+1=0\\
x+z+2xz+1=0\\
\end{cases}$$ It is easy to see that $A\ominus B=\emptyset, A\ominus C=\emptyset$. But the image under the monomial map of the curve $\gamma$ lies in a hyperplane. Indeed the image of $\gamma$ is given by the parameter $t\colon$ $$[1\colon-3+2/(t+1)\colon-1/2-0.5/(2t+1)\colon 3/2-0.5/(2t+1)]$$ then $-2\omega_{1}-\omega_{3}+\omega_{3}=0$, where $\omega_{i}$ are homogeneous coordinates. And the maximal possible value of multiplicity is $\dim\mathbb{V}^{u}_{A}=2\neq |A|-1=3$. That example shows that the theorem \[t1\] cannot be generalised to the higher dimensions with the same conditions as in the case $n=2$.
Acknowledgements
================
I would like to express my gratitude to Alexander Esterov for productive discussions and his great support.
[^1]: Faculty of Mathematics, National Research University Higher School of Economics, e-mail$\[email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A high-energy spin resonance mode is known to exist in many high-temperature superconductors. Motivated by recent scanning tunneling microscopy (STM) experiments in superconducting Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, we study the effects of this resonance mode on the local density of states (LDOS). The coupling between the electrons in a $d$-wave superconductor and the resonance mode produces high-energy peaks in the LDOS, which displays a two-unit-cell periodic modulation around a nonmagnetic impurity. This suggests a new means to not only detect the dynamical spin collective mode but also study its coupling to electronic excitations.'
author:
- 'Jian-Xin Zhu'
- Jun Sun
- Qimiao Si
- 'A. V. Balatsky'
title: 'Effects of a Collective Spin Resonance Mode on the STM Spectra of D-Wave Superconductors'
---
A prominent feature in the excitation spectrum of the high-$T_c$ superconductors is the “41 meV” collective spin resonance mode, seen by inelastic neutron scattering experiments in most of the cuprate families [@Rossat-Mignod91; @Mook93; @Fong96; @He01]. The physics of this resonance mode – including its microscopic origin, its connections with other physical properties, as well as its role on superconductivity itself – has been the subject of considerable debate. Given the recent developments of the atomic resolution scanning tunneling microscopy (STM) [@Hoffman02a; @Hoffman02b; @Howald02a; @Howald02b; @McElroy02], it is timely to address the possible manifestation of this resonance mode in the local density of states (LDOS). A number of theoretical works [@Polkovnikov02; @Podolsky02; @Han02; @Kivelson02] have addressed the effect of related spin fluctuations on the LDOS. These works focused on the pinning of the spin fluctuations by impurity: a dynamical spin mode centered around the wavevector ${\bf Q}$ leads to a $2{\bf Q}$ spatial modulation in the low-energy LDOS. This result is smoothly connected to what happens in the case of a static spin-density-wave ordering [@Zachar98; @Zhu02]. However, for the resonance mode – which is sharply peaked at ${\bf Q} = ({\pi,\pi})$ – such effects would not be manifested \[since $2 {\bf Q} = ({2\pi,2\pi})$ is equivalent to $(0,0)$\]. There are also works about quasiparticle scattering from nomagnetic impurities [@Wang02; @Zhang02].
In this paper, we show that the coupling of $d$-wave quasiparticles to the resonance mode does produce spatial modulations in the LDOS around an impurity. The feature is located at relatively high energies, $\approx \pm (\Delta_0 + \Omega_0)$, where $\Delta_0$ is the maximum superconducting energy gap and $\Omega_0$ the resonance energy. In addition, the wavevector of the LDOS modulation is close to ${\bf Q} =(\pi,\pi)$. Our predictions could be observable by the STM experiments. Such STM studies represent a new means to characterize the coupling between the electronic excitations and the resonance mode. The STM feature we discuss relates to the “peak-dip-hump” structure of the angle resolved photoemission spectroscopy (ARPES) [@Dessau91; @Ding96; @Abanov02; @Kee02; @Eschrig00; @Abanov00]; the inference about the electron-spin coupling from the ARPES and related spectroscopies is a topic of recent controversy [@Kee02; @Abanov02] and we hope that the STM studies we propose will shed new light on this important issue.
We start with a model Hamiltonian describing two-dimensional electrons coupled to a collective spin mode and in the presence of a single-site impurity: $\mathcal{H}=\mathcal{H}_{BCS}+
\mathcal{H}_{sp} + \mathcal{H}_{imp}$. Here the BCS-type Hamiltonian for a uniform $d$-wave superconductor is given by $
\mathcal{H}_{BCS}=\sum_{\mathbf{k},\sigma}
(\varepsilon_{\mathbf{k}}-\mu) c_{\mathbf{k}\sigma}^{\dagger}
c_{\mathbf{k}\sigma} +\sum_{\mathbf{k}}(\Delta_{\mathbf{k}}
c_{\mathbf{k}\uparrow}^{\dagger}c_{-\mathbf{k}\downarrow}^{\dagger}
+\Delta_{\mathbf{k}}^{*}
c_{-\mathbf{k}\downarrow}c_{\mathbf{k}\uparrow}) $, where $c_{\mathbf{k}\sigma}^{\dagger}$ ($c_{\mathbf{k}\sigma}$) creates (annihilates) a conduction electron of spin $\sigma$ and wavevector $\mathbf{k}$, $\varepsilon_{\mathbf{k}}$ is the normal state energy dispersion for the conduction electrons, $\mu$ the chemical potential, and $\Delta_{\mathbf{k}}=\frac{\Delta_{0}}{2}(\cos k_x -\cos k_y)$ the $d$-wave superconducting energy gap. The coupling between the electrons and the resonance mode is modeled by an interaction term $\mathcal{H}_{sp}=g\sum_{i} \mathbf{S}_{i}\cdot \mathbf{s}_{i}$, where the quantities $g$, $\mathbf{s}_{i}$, and $\mathbf{S}_{i}$ are the coupling strength, the electron spin operator at site $i$, and the operator for the collective spin degrees of freedom, respectively. The dynamics of the collective mode will be specified below. The impurity scattering is given by $\mathcal{H}_{imp}=U_{0}\sum_{\sigma} c_{0\sigma}^{\dagger}
c_{0\sigma}$, where without loss of generality we have taken a single-site impurity of strength $U_0$ located at the origin, $\mathbf{r}_{i}=0$. By introducing a two-component Nambu spinor operator, $\Psi_{\mathbf{k}}=(c_{\mathbf{k}\uparrow},
c_{-\mathbf{k}\downarrow}^{\dagger})^{T}$, the matrix Green’s function for the $d$-wave BCS Hamiltonian $\mathcal{H}_{BCS}$ is determined by $
%\begin{equation}
G_{0}^{-1}(\mathbf{k};i\omega_{n})=\left( \begin{array}{cc}
i\omega_{n}-\xi_{\mathbf{k}} & -\Delta_{\mathbf{k}} \\
-\Delta_{\mathbf{k}} & i\omega_{n}+\xi_{\mathbf{k}}
\end{array}
\right)\;,
%\end{equation}
$ where $\xi_{\mathbf{k}}=\varepsilon_{\mathbf{k}}-\mu$ and $\omega_{n}=(2n+1)\pi T$ is the fermionic Matsubra frequency. We have also assumed that the $d$-wave pair potential is real.
For a homogeneous system, where only the inelastic scattering of quasiparticles from the collective mode occurs, we calculate the self-energy to the second order in the coupling constant (see Fig. \[FIG:diagrams\]b): $$%\begin{equation}
%\begin{eqnarray*}
\hat{\Sigma}(\mathbf{k};i\omega_{n})=\frac{3g^{2}T}{4}\sum_{\mathbf{q}}
\sum_{\Omega_{l}} \chi(\mathbf{q};i\Omega_{l})
% \\
%&&\times
%\left(
%\begin{array}{cc}
%G_{0,11} & G_{0,12}
%\\
%G_{0,21} & G_{0,22}
%\end{array}
%\right)
G_{0}(\mathbf{k}-\mathbf{q};i\omega_{n}-i\Omega_{l})\;,
%\end{eqnarray*}
%\end{equation}$$ where $\chi(\mathbf{q};i\Omega_{l})$ is the dynamical spin susceptibility $\chi_{ij}(\tau)=\langle T_{\tau}
(S^{x}_{i}(\tau)S^{x}_{j}(0))\rangle$ and $\Omega_{l}=2l\pi T$ the bosonic Matsubra frequency. The dressed Green’s function is: $$\underline{{G}}_{0}^{-1}(\mathbf{k};i\omega_{n})=\left(
\begin{array}{cc}
i\omega_{n}-\xi_{\mathbf{k}}-\Sigma_{11} & -\Delta_{\mathbf{k}}-\Sigma_{12} \\
-\Delta_{\mathbf{k}}-\Sigma_{21} &
i\omega_{n}+\xi_{\mathbf{k}}-\Sigma_{22}
\end{array}
\right)\;.$$ The corresponding real-space dressed Green’s function $\underline{{G}}_{0}(i,j;i\omega_{n})$ is obtained through a Fourier transform with respect to $\mathbf{r}_{i}-\mathbf{r}_{j}$. For the $d$-wave pairing symmetry, one can show that the local Green’s function, $\underline{\hat{g}}_{0}(i\omega_{n})=
\underline{{G}}_{0}(i,i;i\omega_{n})$ is diagonal. In the presence of a single-site impurity at $\mathbf{r}_{i}=0$ with potential strength $U_0$, the site-dependent Green’s function can be written in terms of the $T$-matrix: $$\begin{aligned}
{G}(i,j;E)&=&{\underline{G}}_{0}(i,j;E)\nonumber \\
&&+\sum_{lm} {\underline{G}}_{0}(i,l;E)\hat{T}_{lm}(E)
{\underline{G}}_{0}(m,j;E)\;.
\label{gij}\end{aligned}$$ Due to the vertex corrections induced by the coupling to the collective modes (Fig. \[FIG:diagrams\]a), the $T$ matrix in general contains site-off-diagonal terms. We will first carry out the calculation without the vertex corrections, in which case $\hat{T}_{lm} = \hat{T} \delta_{l,0} \delta_{m,0}$, with $\hat{T}^{-1}=U_{0}^{-1} \sigma_{3}-\hat{\underline{g}}_0 $, where $\sigma_{3}$ is the $z$-component of the Pauli matrix. The LDOS at the $i$-th site, summed over two spin components, is $$\begin{aligned}
\rho(\mathbf{r}_{i},E)=-\frac{2}{\pi} \mbox{Im} G_{11}(i,i;E+i\gamma)\;,
\label{ldos}\end{aligned}$$ where $\gamma=0^+$.
Up to now, our discussion and formulation are quite general and can be used to study the effects of any dynamic mode once the susceptibility $\chi$ is known. We treat the susceptibility in a phenomenological form (based on the inelastic neutron scattering observations), see also [@Eschrig00]: $$\chi(\mathbf{q};i\Omega_{l})=-\frac{\delta_{\mathbf{q},\mathbf{Q}}}{2}
\left[\frac{1}{i\Omega_{l}-\Omega_{0}}-\frac{1}
{i\Omega_{l}+\Omega_{0}}\right]\;,$$ where we denote the wavevector $\mathbf{Q}=(\pi,\pi)$ and the mode energy by $\Omega_{0}$. This form is especially suitable for the optimally doped YBa$_2$Cu$_3$O$_{6+y}$ (YBCO) compounds in the superconducting phase, where the observed neutron resonance peak is almost resolution-limited in energy and fairly sharp in wavevector. The resonance peak in BSCCO is broadened in both energy and wavevector. The finite width in the wavevector space might be important for the ARPES lineshape in general and in particular the understanding of the ARPES spectra away from the $M$ points \[$\mathbf{k}=(\pi,0)$ and symmetry-related points\] of the Brillouin zone [@Eschrig00], but should not change the qualitative conclusion of our work: the LDOS effects we will discuss arise from the fact that the dominant effects of the resonance mode on the single-electron spectral functions occur near the $M$ points which is expected to remain to be the case beyond our simplified form for the susceptibility. In addition, given that the peak in BSCCO is still quite sharp in energy, we expect that the main effect of the broadening in energy of the resonance mode is to extend the bias window for the LDOS feature we will discuss. We have also neglected the incommensurate peaks seen in the inelastic neutron scattering experiments in YBCO (the part that disperses “downward” away from the resonance peak) [@Arai99; @Fong00; @Brinckmann99; @Kao00], since their spectral weight is significantly smaller than that of the resonance mode. For the normal-state energy dispersion, we use $\varepsilon_{\mathbf{k}}=-2t (\cos k_x + \cos k_y) -4t^{\prime}
\cos k_{x} \cos k_y$, where $t$ and $t^{\prime}$ are the nearest and next-nearest neighbor hopping integral. Unless specified explicitly, the energy is measured in units of $t$. We choose $t^{\prime}=-0.2$ to model the band structure of the hole-doped cuprates. Since the maximum energy gap for most of the cuprate superconductors at the optimal doping is about $30 \;\mbox{meV}$ while the resonance mode energy is in the range between $35$ and $47$ meV, we take $\Delta_0=0.1$ and $\Omega_0=0.15$ (i.e., $1.5
\Delta_{0}$). The chemical potential is tuned to give an optimal doping value 0.16. To mimic the intrinsic life time broadening, in our numerical calculation we take $\gamma$ of Eq. (\[ldos\]) to be $0.04\Delta_{0}$. A system size of $N_x\times N_y=1000 \times
1000$ is taken in the numerical calculation.
\[FIG:SPECTRUM\]
In the absence of impurities, the density of states is the summation of the spectral function, $A_{\mathbf{k}}(E)=-\frac{2}{\pi}
\mbox{Im}\underline{G}_{0,11}(\mathbf{k};E+i\gamma)$, over all wavevectors $\mathbf{k}$. Fig. \[FIG:SPECTRUM\](a) shows the spectral function at an $M$ point of the Brillouin zone. Without the electron-mode coupling, as is well known, the spectral function is peaked at the maximum gap edges $\pm \Delta_0$. As the electron-mode coupling is switched on, new peaks emerge at the energies $\pm E_{1} \approx \pm (\Delta_{0}+\Omega_{0})$. (For simplicity, we have neglected the broad “background” part of the single-electron spectral function in our consideration.) The peaks in $A_{\mathbf{k}}$ originate from the poles of the Green’s function $\underline{G}_{0,11}$. Note that the weight of the peak at $-\Delta_0$ is larger than at $\Delta_0$ because the van Hove singularity is below the Fermi energy. Since the spectral weight of the spin resonance mode \[i.e., $\mbox{Im}
\chi(\mathbf{q};\omega)$\] is peaked at $\mathbf{Q}$, the feature of the quasiparticle self-energy is the strongest around the $M$ points of the zone because they are connected by $\mathbf{Q}$. The singularity in the quasiparticle self-energy causes additional poles in the Green’s function. As the coupling constant $g$ increases, these peaks are shifted to higher energies and, in addition, their spectral weight is enhanced; simultaneously, the weight of the superconducting coherent peaks is reduced to obey the sum rule. The shift of states due to inelastic scattering is expected in DOS and is also expected for scattering off local mode [@Balatsky02]. Fig. \[FIG:BAND\_DOS\](b) plots the density of states and clearly shows that the high energy peaks still occur around $\pm E_{1}$. In other words, the contributions from near the $M$ points dominate the wavevector summation for the density of states, reflecting the flat nature of the normal state band near this point. Furthermore, the highly asymmetrical structure in the DOS at energies $-E_{1}$ and $E_{1}$ comes from the singular structure in the quasiparticle self-energy. These results, for the clean case, are consistent with earlier studies of the ARPES [@Dessau91; @Ding96; @Abanov02; @Kee02; @Eschrig00] and DOS [@Abanov00].
We are now in a position to address the LDOS in the presence of a single nonmagnetic impurity. For concreteness, we take the on-site potential $U_0=100\Delta_{0}$. Fig. \[FIG:LDOS\] shows the LDOS directly at the impurity site, as well as at its nearest neighbor. The near-zero energy resonant state triggered by the quasiparticle scattering from the impurity [@Balatsky95] is robust against the electron-mode coupling. Our key new results are two-fold. First, the impurity modifies the shape of the spectral features at $\pm E_{1}$, which can now be either a dip or a peak. Second, and more importantly, these high energy features at $\pm E_{1}$ exhibit a spatial dependence. At the impurity site, the LDOS displays a dip at $-E_{1}$ but a peak at $+E_{1}$. The behavior is reversed at the site closest to the impurity.
To explore this spatial variation of the LDOS in more detail, we have calculated the LDOS, with the energy fixed at $-E_1$, in the vicinity of the impurity with and without the mode coupling. In the absence of mode coupling, we obtain the results (not shown) which are consistent with previous studies [@Balatsky95]: the LDOS exhibits a Friedel oscillation along the diagonals of CuO$_{\rm 2}$ plane but no other non-trivial features. When the mode coupling is turned on, in addition to the Friedel oscillation along the diagonals, the LDOS displays a new type of modulation with a period of $2a$ in the wide regions along the bond direction. This new modulation can be more easily seen Fig. \[FIG:IMAGE\](a) when the pre-dominant Friedel oscillation is filtered away. It occurs in four disconnected triangles in the field of view.
Also in order to highlight the new modulation, we find it useful to perform a filtered Fourier transform, $\rho({\bf q},E) =
\sum_{i}^{\prime} {\rm e}^{i {\bf q} \cdot \mathbf{r}_{i}}
\rho(\mathbf{r}_{i},E)$, where $\sum_{i}^{\prime}$ denotes a summation over all the sites in four triangles in Fig. \[FIG:IMAGE\](a). The resulting Fourier-transformed image [@note3] is given in Fig. \[FIG:IMAGE\](b), which unambiguously shows that the new feature induced by the coupling to the spin resonance mode has a spatial modulation wavevector $(\pi,\pi)$.
This new type of LDOS modulation with a wavevector close to ${\bf Q} = (\pi,\pi)$ reflects the dominance of the collective mode effect near the $M$ points of the Brillouin zone. It follows from Eqs. (\[gij\],\[ldos\]), with the form of the $T$-matrix under consideration and when the chosen field of view has an inversion symmetry with respect to the impurity site, that the Fourier transformed LDOS for any finite ${\bf q}$ is, $$\begin{aligned}
\rho({\bf q}, E)=-\frac{2}{\pi} \mbox{Im} \int d {\bf p}
\left [ {\underline{G}}_{0}({\bf p}+\mathbf{q};E) \hat{T}(E)
{\underline{G}}_{0}({\bf p},E) \right ]_{11}\;.
\label{convolution}\end{aligned}$$ At positive energies, the poles shown in Fig. \[FIG:SPECTRUM\] for ${\underline{G}}_{0}({\bf p};E)$ are located [@Dessau91; @Ding96; @Abanov02; @Kee02; @Eschrig00] at $E_{\bf
p}$ and $E_{{\bf p}-{\bf Q}} + \Omega_0$, respectively. (Here, $E_{\bf p} \equiv \sqrt{\xi_{\bf p}^2 + \Delta_{\bf p}^2}$.) Likewise, ${\underline{G}}_{0}({\bf p}+{\bf q};E)$ is the sum of two poles, one at $E_{{\bf p}+{\bf q}}$ and the other at $E_{{\bf
p}+{\bf q}-{\bf Q}} + \Omega_0$. At $E=E_1$, Eq. (\[convolution\]) is dominated by the convolution between the poles at $E_{{\bf p}-{\bf Q}} + \Omega_0$ and $E_{{\bf p}+{\bf
q}-{\bf Q}} + \Omega_0$. This term in turn is dominated by the contributions corresponding to when both ${\bf p}-{\bf Q}$ and ${\bf p}+{\bf q}-{\bf Q}$ are near to the $M$ points, leading to a $\rho({\bf q}, -E_1)$ that is peaked near $({\pi, \pi})$.
Similar phase-phase considerations show that the vertex correction terms lead to a similar momentum dependence. In the presence of vertex corrections, the $T$-matrix satisfies an integral equation in the wavevector space. The vertex correction to the $T$-matrix to the same order ($g^2$) of our calculation for the self-energy is shown in Fig. \[FIG:diagrams\](a). It involves a wavevector convolution of a form similar to that given in Eq. (\[convolution\]). We therefore expect [@note2] an additive contribution to $\rho({\bf q}, -E_1)$ that is also peaked near $( {\pi, \pi})$.
Finally, we remark on issues which go beyond the idealized model we have considered so far: (i) In the presence of a gap inhomogeneity at the nanoscale, the mode signature would appear at $\langle\Delta_{0}\rangle+\omega_0$, with a slight smearing. Here, $\langle \Delta_{0} \rangle $ is the spatially averaged superconducting gap. The fact that a well-defined peak-dip-hump structure appears in the break-junction tunneling spectrum [@Zasadzinski01] implies that the smearing is not too large; (ii) The mode signature depends on the detailed band structure. In the cuprates, the band is flat near the antinodal points (i.e., $(\pi,0)$ [*etc.*]{}), and a mode with a momentum significantly different from $(\pi,\pi)$ will have a weaker effect compared to that of the $(\pi,\pi)$ mode we addressed; (iii) Tunneling matrix elements need to be taken into account in order to understand the detailed spatial variation of the LDOS as observed around zinc impurities in BSCCO [@Pan00; @Martin02]. Such a filtering effect, however, will not affect our conclusion on the momentum of the LDOS modulation.
To summarize, we have studied the effects of the magnetic resonance mode on the tunneling spectrum in the presence of a nonmagnetic impurity. The LDOS around the impurity displays resonant features at relatively high energies \[close to $\pm
(\Omega_0+\Delta_0)$\], which modulates in space with a wavevector close to $(\pi,\pi)$. Our prediction can be tested straightforwardly by operating the existing high resolution STM at a relatively high energy window. Such studies should shed considerable new light on the physics of the spin resonance mode, in particular its coupling to the electronic excitations.
We are especially grateful to J. C. Davis for helpful conversations at the early stage of this work. The authors have benefited considerably from discussions with Ar. Abanov, Y. Bang, A. V. Chubukov, K. Damle, and M. Norman. This work was supported by the US DOE (JXZ and AVB), by TcSAM and the NSF Grant No. DMR-0090071 (JS and QS). JXZ also acknowledges the hospitality of the Rice University, where part of this research was carried out.
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Similar procedure for the LDOS in the absence of a mode coupling yields a Fourier-transformed LDOS that is featureless near $(\pi,\pi)$.
We have numerically calculated the leading order vertex correction contribution to the LDOS, finding that it is indeed peaked near $(\pi,\pi)$.
J.F. Zasadzinski [*et al.*]{}, Phys. Rev. Lett. [**87**]{}, 067005 (2001).
S. H. Pan [*et al.*]{}, Nature [**403**]{}, 746 (2000).
I. Martin, A. V. Balatsky, and J. Zannen, Phys. Rev. Lett. [**88**]{}, 097003 (2002).
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{
"pile_set_name": "ArXiv"
}
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abstract: 'It is generally known that multi-spacecraft observations of interplanetary coronal mass ejections (ICMEs) more clearly reveal their three-dimensional structure than do observations made by a single spacecraft. The launch of the STEREO twin observatories in October 2006 has greatly increased the number of multipoint studies of ICMEs in the literature, but this field is still in its infancy. To date, most studies continue to use on flux rope models that rely on single track observations through a vast, multi-faceted structure, which oversimplifies the problem and often hinders interpretation of the large-scale geometry, especially for cases in which one spacecraft observes a flux rope, while another does not. In order to tackle these complex problems, new modeling techniques are required. We describe these new techniques and analyze two ICMEs observed at the twin STEREO spacecraft on 22-23 May 2007, when the spacecraft were separated by $\sim$8$^{\circ}$. We find a combination of non-force-free flux rope multi-spacecraft modeling, together with a new non-flux rope ICME plasma flow deflection model, better constrains the large-scale structure of these ICMEs. We also introduce a new spatial mapping technique that allows us to put multispacecraft observations and the new ICME model results in context with the convecting solar wind. What is distinctly different about this analysis is that it reveals aspects of ICME geometry and dynamics in a far more visually intuitive way than previously accomplished. In the case of the 22-23 May ICMEs, the analysis facilitates a more physical understanding of ICME large-scale structure, the location and geometry of flux rope sub-structures within these ICMEs, and their dynamic interaction with the ambient solar wind.'
title: 'Advancing In Situ Modeling of ICMEs: New Techniques for New Observations'
---
Introduction
============
Coronal mass ejections (CMEs) are eruptions from the Sun that release vast quantities of plasma into the expanding solar wind. When observed at 1 AU, the interplanetary counterpart of CMEs (ICMEs) often exhibit depressed plasma temperatures, enhanced and twisted magnetic fields, and bi-directional suprathermal electron signatures making them distinguishable from the ambient solar wind \[Burlaga et al., 1981; Burlaga and Behannon, 1982, Gosling, 1990; Neugebauer and Goldstein, 1997; Richardson and Cane, 2004a; Zurbuchen and Richardson, 2006)\]. In the interplanetary medium, the appearance of ICMEs varies greatly such that only about one-third to one-half of all ICMEs in the inner heliosphere are observed as magnetic clouds (MCs) (Gosling 1990; Wang and Richardson, 2004), though this may be a product of viewing geometry, as recent results suggest ICMEs intersected far from their axis do not exhibit MC signatures (Jian, 2006, Kilpua, 2009, 2011). In MCs, simplifying assumptions have been made about their three-dimensional structure and they have successfully been modeled as magnetic flux ropes using force-free bessel functions based on the Lundquist solution (Lundquist 1950; Goldstein, 1983; Burlaga et al., 1990, Kilpua et al., 2011 and references therein). This basic flux rope model assumes that ICMEs are in equilibrium and have circular cross-sections. However, in-situ studies with multiple spacecraft have found ICMEs are dynamic structures that undergo extensive expansion and interact with the ambient solar wind. These processes result in mild to extreme distortions in the magnetic field geometry and flux rope cross-sectional shapes (e.g., Gosling, 1990; Farrugia et al., 1995; Mulligan et al., 1999; Mulligan and Russell, 2001; Russell and Mulligan, 2002; Riley and Crooker, 2004; Lepping et al., 2006; Liu et al., 2006). Consequently, the Lundquist flux rope is now considered an approximate solution and has prompted several revisions to the basic model, such as the inclusion of non-constant alpha and expansion effects (Marubashi, 1986, 1997), uniform-twist (Farrugia et al., 1999), and non-force free kinematic effects (Mulligan and Russell, 2001; Cid et al., 2002; Hidalgo et al., 2002, Owens et al., 2006).
Although flux rope modeling has dominated ICME analyses over the past decade, there is a significant fraction of ICMEs that have no observable flux rope signature, which fuels the debate as to whether or not an ICME must contain a flux rope in its interior. In the ecliptic plane ICMEs typically span up to 60$^{\circ}$ in longitude (consistent with the angle the average CME subtends at the Sun) so there are many examples in which multiple spacecraft have observed the same ICME. However, the probability of intercepting a central flux rope structure within the ICME diminishes with increasing impact parameter (the closest approach to the center of the ICME). In the Jian et al., (2006) study, a majority of ICME observations occur with the observing spacecraft intercepting the ICME disturbance sufficiently far from its center such that a central flux rope is not identifiable. Kilpua et al., (2009, 2011) also show examples of multispacecraft observations in which one spacecraft crosses the central flux rope near its apex, while the other spacecraft traverses the ICME flank, where the flux rope structure is no longer discernible.Thus at 1 AU, the association between multiple spacecraft observations of an ICME is not always straightforward, resulting in significant differences, even when separation distances of the spacecraft are small. In addition, ICME characteristics such as size, magnetic field and plasma signatures, and the structure of the surrounding solar wind can vary significantly from event to event. Despite this difficulty, multi-spacecraft observations, separated by at least a few degrees, comprise the best tool currently available to study the large-scale properties of these vast structures. Since the majority of ICME observations either come from single spacecraft or multispacecraft encounters that are difficult to reconcile, the three dimensional structure of ICMEs is still poorly understood.
Data from the Solar TErrestrial Relation Observatory (STEREO) (Kaiser et al., 2007), launched in 2006, are ideally suited to help characterize the elusive, large-scale properties of ICMEs. STEREO consists of two functionally identical satellites, one that leads the Earth (STEREO-A), and one that lags the Earth (STEREO-B) in its orbit around the Sun with gradually increasing angular separation. The focus of this paper is to better exploit the available multi-spacecraft observations by combining measurements from STEREO with those from near-earth spacecraft like Wind and ACE (approximately halfway between the STEREO spacecraft) into a single, coherent model reconstruction of large-scale ICME strucutre. Promising methods already exist, such as those based on the Grad-Shafranov Reconstruction (GSR) technique, which physically constrain magnetic field and plasma observations into a single model inversion (Hu and Sonnerup, 2001, 2002). However, models based on this technique have only been applied to flux rope observations. Though GSR-based models have merits, in practice they also also suffer from non-uniqueness and have been shown to incorrectly determine the shape of flux rope cross-sections in complex MHD simulations of ICMEs (e.g. Riley 2004).
To advance beyond these existing reconstructions, new methods must be employed to tackle complex ICME structures, such as those having no discernible magnetic flux rope signature at one or more spacecraft locations. Ideally, such a model would include magnetic field and plasma measurements of the surrounding solar wind to constrain the inversion and to help characterize the distortion and dynamics of ICME structure on a large scale. Consequences of ICMEs are far-reaching in the heliosphere and there is a need to create a more realistic three-dimensional picture of these structures in context with the ambient interplanetary environment. Utilizing data in this way gives us clues to how ICMEs evolve in the solar wind and what relationship they share with the surrounding plasma envelope, its composition, sub-structures, and dynamics. Overall, it is this better understanding of the large-scale structure that is of key importance for solar-terrestrial research and space weather forecasting.
In the following sections we outline the development of two new techniques designed to extend the utility of the Mulligan and Russell (2001) magnetic flux rope model to simultaneously invert data from flux rope and non-flux rope multi-spacecraft ICME observations. Specifically, in Section 2 we outline a technique that uses the solar wind plasma velocity data and the ICME boundary normal to allow the inclusion of non-flux-rope observations in a new large-scale or “synoptic” ICME model inversion. In Section 3 we discuss the development of a new spatial mapping technique that exploits solar wind magnetic field, plasma, and ionic composition measurements observed at multiple spacecraft in order to create a more quantitative understanding of the ambient plasma environment and how this environment affects the dynamics and evolution of the ICME structure. Also in this section (and in the Appendix) we test the spatial mapping method on a period of solar wind, when the ACE, Wind, and twin STEREO spacecraft are in close proximity and reconstruct the time-series data from the spatial maps. In Section 4 we apply the new techniques developed in Sections 2 and 3 to multispacecraft measurements of two ICMEs on 22-23 May 2007 seen at STEREO-A, STEREO-B, ACE, and Wind. In the closing sections 5 through 7, we discuss the implications of this work and future developments that promise to usher in a new era in the analysis of multi-spacecraft observations.
Development of the Synoptic ICME model
=======================================
As noted previously, MCs are well approximated by flux rope models, but they are not always observable when an ICME is intersected far from its central axis. When this occurs, magnetic field observations do not provide enough information about the global ICME structure at this location. However, velocity flow deflections around the ICME can provide the missing insight into the large-scale orientation of the structure in these cases and by incorporating this information into an existing flux rope model we can extend its capabilities.
The baseline for the development of the synoptic ICME model is the non-cylindrically symmetric, non-force-free flux rope model described in detail in Mulligan and Russell, (2001) and Mulligan (2002). Briefly explained, this model has an axial magnetic field component that falls off with a stretched-exponential form from the axis of the rope. The azimuthal or poloidal field increases as 1 minus a stretched-exponential dependence so that it maximizes at the rope edge. The stretched exponential form allows the modeling of non-cylindrically symmetric cross sectional geometries. When used with multiple spacecraft observations the model returns the azimuthal stretching of the flux rope cross-section, the bending along the flux rope axis, and the spatial (temporal) expansion of the modeled structure. The model flux rope is fit to the data using a downhill simplex inversion technique (Nelder and Mead, 1965) that varies the fitting parameters in an orderly manner. The parameters already allow for gross measurements of cross-sectional distortion, large-scale curvature, and residual expansion forces, and by adding velocity component modeling, these quantities will be better resolved.
In general, velocity flow deflections are only an approximation of the ICME orientation, subject to the assumptions of obstacle shape, and are not sufficient alone to determine ICME structure, especially from single spacecraft observations. However, when inverting multiple observations, for example, in which one spacecraft observation does not have a clear flux rope signature, the addition of this data can provide a valuable constraint on the large-scale geometry of the entire structure and improve the overall fit dramatically.
Velocity component modeling (Lindsay technique)
-----------------------------------------------
The first step towards incorporating solar wind flow deflection around an ICME obstacle originates in the work by G. Lindsay, (1996). In her thesis she exploited the fact that as an ICME travels outward faster than the solar wind ahead, the solar wind ahead must be deflected away from and around the ICME. This implies that non-radial, solar wind velocity components will be associated with ICME passage. For clarity, it should be emphasized that it is the deflection across the ICME leading boundary that is being examined, not the deflections across the interplanetary shock associated with the ICME. This means that slow ICMEs, which do not have shocks can also be studied using this method, provided they are moving faster than the ambient solar wind. This effect has been previously studied by Gosling et al., (1987) for east-west flow deflections in ICMEs at 1 AU. By considering east-west and north-south flow deflections and making the assumption that the gross structure of an ICME is cylindrical, Lindsay et al., (1996) was able to determine the relationship between the velocity flow deflection around the ICME obstacle and the obstacle orientation. The can diagrams in Figure 1 illustrate how this determination is performed. For an ICME with its axis in the ecliptic plane (Figure 1a), the solar wind velocity will be deflected both above and below the ICME obstacle as it passes through the upstream solar wind. This means that only northward or southward velocity deflections (in GSE coordinates, deflections in $v_z$) will be observed ahead of the ICME. In the other extreme, an ICME with its axis oriented perpendicular to the ecliptic plane (Figure 1b) will produce only eastward or westward deflections (deflections in $v_y$) ahead of the ICME.
Assuming that an ICME can be approximated by a cylindrical structure, the bottom diagrams in Figure 1 illustrate the expected flow deflections of the velocity components as the spacecraft traverses the ICME via trajectory T1 or T2. As an example, for an observing trajectory following that of T1 in Figure 1b, sufficiently upstream of the ICME (where the influence of the ICME is not present) radially oriented solar wind flow will be observed. As the ICME approaches the observation location, the presence of the obstacle will begin to influence the solar wind flow and the speed of the solar wind plasma will begin to increase. Within the region of increasing plasma speed and across the ICME leading boundary (at time $t_0$ in the figure), the plasma will be observed to be deflected eastward ($v_y>0$). Inside the ICME, a westward deflection ($v_y<0$) will be observed. Through the remainder of the ICME, the solar wind speed begins to decline until, near the trailing boundary (at time $t_1$), it attains a radial orientation. If the observing spacecraft follows trajectory T2, the observed $v_y$ deflections will occur in the opposite sense (i.e. westward, then eastward). These deflections show a predictable pattern at the ICME leading edge boundary and reveal a circulation pattern within the ICME structure (Lindsay 1996).
The systematic variations in ICME related velocity flow deflections can be used to infer ICME orientation in the plane of the sky. Again, for example, the $v_y<0$ deflection ahead of the ICME shown in Figure 1b indicates that the spacecraft is intercepting the ICME in a manner similar to trajectory T2. For this same trajectory, a flow deflection $v_y<0$ and $v_z>0$ ahead of the ICME implies that the ICME is oriented as in Figure 2a. Formally, the orientation of the cylinder axis can be calculated from the observed velocity flow deflections by $$tan\theta_c = - \frac{v_z}{v_y}$$ where $\theta_c$ is defined as in Figure 2a and where $\theta_c > 0$ indicates an eastward orientation and $\theta_c < 0$ represents a westward orientation. In the extremes that $v_z=0$ or $v_y=0$, implies that the ICME is either perpendicular to or aligned with the ecliptic plane, respectively.
Lindsay (1996) performed this analysis on 23 MCs observed by the Pioneer Venus Orbiter spacecraft and found orientations in the plane of the sky (a.k.a. the clock angle, defined in the y-z plane in GSE coordinates as the angle with 0$^{\circ}$ pointing due northward and a positive angle in the positive y direction in a counterclockwise sense) that were consistent with the Lundquist flux rope model solutions performed by Lepping (1990). The clear advantage of the Lindsay method, especially with respect to ICME modeling, is that it can be used on spacecraft traversals of non-flux rope ICMEs and traversals of suspected flux rope ICMEs for which there is little or no magnetic field information available.
Velocity component modeling (Owens and Cargill technique)
---------------------------------------------------------
Because the Lindsay technique (Lindsay 1996) only determines ICME orientation in the plane of the sky (a.k.a. the clock angle), additional information must be acquired to determine the obstacle orientation out of this plane. Using a method outlined by Owens and Cargill, (2004), it is possible to expound upon the Lindsay technique and make a quantitative comparison of the complete flux rope model orientation by calculating the normal to the ICME obstacle using velocity flow deflections.
Figure 2 summarizes how the flow deflections from the Lindsay technique are used to determine the obstacle clock angle. In the work by Owens and Cargill (2004), two unit vectors in the frame of the ICME must be determined to obtain the normal to the ICME obstacle. Retaining the naming convention for these vectors as in the Owens and Cargill paper, the unit vector $\hat{a}$ can be expressed as a function of the components of the velocity flow deflection in the y-z plane $$\hat{a} = \frac{v_y}{|v|}\hat{y} - \frac{v_z}{|v|}\hat{z}$$
where $\sqrt{v_y^2 + v_z^2}$. Recall that for a cylindrical obstacle oriented with its axis perpendicular to the ecliptic plane, a vector tangent to the obstacle at the spacecraft location is the velocity flow deflection in the x-y plane. Figure 2b shows the cross section of a cylindrical obstacle oriented with its axis perpendicular to the ecliptic plane. The unit vector $\hat{c}$ is tangent to the obstacle surface and points in the direction of the velocity flow deflection in the x-y plane at point p. Note that $\hat{c}$ can be related to the impact parameter (IP) of the spacecraft trajectory through the obstacle by the angle $\alpha$, where $\alpha$ is the angle between the x-axis of the rope and the radial line between the center of the cross-section and the spacecraft point of entry, as Figure 2b shows. Thus it is possible to calculate the impact parameter using the flow deflection components in the x-y plane $$IP = sin(tan^{-1}(-\frac{v_y}{v_x})).$$ As first suggested by Owens and Cargill, by crossing vector $\hat{a}$ with $\hat{c}$, the obstacle normal $\hat{n}$ can be determined. This is illustrated in Figure 2c for two different impact parameters.
For simplicity, the diagrams in Figure 2 do not consider all possible orientations of the ICME obstacle (i.e. they ignore axial orientations out of the plane of the page), nor do they illustrate all non-cylindrical rope cross-sections (Mulligan and Russell 2001). However, the methods outlined can be easily generalized to any rope orientation and elliptical cross-section. In this fashion, both the normal obtained from the additional spacecraft data and the normal from the magnetic field model can be compared and the residuals between these normals minimized. The additional impact parameter information helps to constrain the geometry of the ICME relative to the spacecraft location. In this sense, the minimization of the normals becomes part of the free parameter space of the synoptic model and a new, higher-dimensional response surface is created. Due to the versatility of the Nelder and Mead simplex algorithm (Nelder and Mead, 1965), this new response surface is easily traversed in the optimization process.
It is worthy to note that the determination of the spacecraft impact parameter and obstacle normal (based on the flow deflection components in the x-y plane) is heavily dependent upon the choice of the reference frame, which must be co-moving with the ICME obstacle, such that the bulk flow of the solar wind (primarily in the radial direction) is minimized. One way this can be performed is by transforming into the deHoffman-Teller frame in which a velocity V$_{HT}$ is found from the reference frame to the spacecraft frame that it minimizes the residual electric field in the least-squares sense (de Hoffmann and Teller, 1950). Other methods for determining the appropriate velocity of the co-moving reference frame can also be applied and include the Constant Velocity Approach (CVA; Russell et al., 1983), the Minimum Faraday Residue (MFR; Terasawa et al., 1996; Khrabrov and Sonnerup 1998) and Minimum Mass-Flux-Residue (MMR; Sonnerup et al., 2004), which will not be described here. At the current time, we employ a simple approach of transforming into a frame co-moving with the average radial solar wind speed of the ICME.
Additional methods to determine the ICME normal such as the coplanarity theorem can be applied successfully when magnetic field data is available, the variance of the field remains low, and when the magnetic field direction is not aligned parallel or perpendicular to the obstacle normal. If no useful magnetic field data exists, the velocity coplanarity normal is an alternate method. The mixed-mode normal, which employs both the field components and the velocity components in the spacecraft frame (Abraham-Shrauner and Yun, 1976; Russell et al., 1983) is another possibility. The advantage of using these other methods for calculation of the obstacle normal is that they do not suffer from errors caused by a translation into a co-moving frame. In a future publication we plan to refine the technique by computing the ICME normal using the various methods mentioned above and average the result over an ensemble of such pairs. This process will build-up an ensemble statistical deviation and give an error estimate of the result. (e.g. Russell et al., 1983; Schwartz (1998)).
Spatial mapping technique
=========================
Spatial interpolation of data between spacecraft is not new to analyses of space plasmas. The advent of multi-spacecraft missions has resulted in the development of several techniques for multipoint data analyses (e.g. Hu and Sonnerup, 2002; Chanteur and Mottez, 1993). In particular, Chanteur (1998), outlines the use of barycentric coordinates and reciprocal vectors, a method well-known in applied mathematics and computational geometry, but not yet fully explored in space physics. Most recently this technique has been used to interpolate data within the four-spacecraft Cluster tetrahedron to infer three-dimensional spatial information about transient structures in the Earth’s magnetosphere. The technique we introduce in this section employs a similar method as that from Chanteur and Mottez, (1993) using barycentric coordinates, but in our case we limit the problem to a two-dimensional geometry. This allows us to simplify the interpolation algorithm using a Delaunay triangulation of the data on a two-dimensional spatial grid. Background and mathematical details of Delaunay triangulation using the natural neighbor interpolation method (Sibson, 1981) are found in the Appendix. The following sections discuss how we transform time-series data onto a spatial domain and apply this Delaunay-Sibson interpolation method.
Defining the Spatial Domain
---------------------------
In the case of multiple spacecraft separated by several degrees in longitude in the ecliptic plane, the relatively small latitudinal differences between the spacecraft can be ignored, reducing the system to a two-dimensional space in which temporal data can be mapped into the spatial domain. This mapping can be accomplished by a linear transformation of the time-series data using the bulk plasma velocity of the solar wind at each time step. Using the spacecraft locations at time $t_o$ as the midpoint of the spatial grid, we can track the spacecraft positions in the spatial domain at n earlier (t$_o$-n$\Delta$t) and n later (t$_o$+n$\Delta$t) times using $$(r_{n^+} - r_0) = \sum_{k=1}^{n} v_k \cdot \Delta t$$
and
$$(r_{n^-} - r_0) = \sum_{k=1}^{n} -v_k \cdot \Delta t$$
where $\Delta$t indicates the time resolution of the data, $r_o$ is the position vector at the midpoint of the spatial grid, $v_k$ is the vector measurement of the solar wind bulk velocity at time k, and 2n is the number of data samples in the time series. The mapped data is centered on time t$_o$ to minimize propagation errors from the velocity data in the spatial range calculation. Figure 3 highlights the use of this technique in two-dimensions (R, T) in which R is the radial dimension and T is the transverse dimension. Note the sampling frequency will affect the resolution of features in the map. (For more details about sampling frequency and error see the discussion in the Appendix.)
Once time-series for each spacecraft has been mapped as separate spacecraft tracks through the spatial domain, we perform a Delaunay triangulation, which computes a set of two-dimensional simplices (triangles) from the points comprising the spacecraft tracks in the spatail domain (a tessellation, e.g. Figure A5). Each point in the tessellation has a spatial location as well as a corresponding data value for the observed solar wind parameters (e.g. $Np$, $|Vp|$, $Tp$, and $B$). These values are then interpolated between the spacecraft tracks using the weighted estimators determined from the Sibson natural neighbor method (Sibson, 1981; e.g. Figure A4). By interpolating over the multi-spacecraft data set in the entire spatial domain, scalar and vector fields of the solar wind parameters can be created. Because the spatial map and the weighted estimators are solely a function of the Delaunay triangulation geometry, the spatial grid is not recalculated for each of the solar wind parameters. Thus, for example, the solar wind $Np$, $|Vp|$, and $Tp$ scalar fields shown in Figure 4 are calculated using the same underlying Delaunay triangulation and weighted estimators.
Because the spacecraft “trajectories” in the maps are created from the solar wind convecting outward past near-stationary spacecraft locations, the mapping is not analogous to a snapshot in time at all locations (e.g. at all R, and T, from Figure 3). A resultant spatial map is more akin to a running time plot, in which snapshots at a single time and location are captured, propagated forward, and pieced together. Since the solar wind moves at a finite speed, the spacecraft trajectories in the spatial domain take a finite amount of time to traverse the grid (equivalent to the time it takes to measure the 2n samples in the time series). This results in an inherent time dependence in the maps, such that any interpolated data nearest to the left side of the plot (e.g. in Figure 3) will have a $2n\Delta t$ difference in time than those at the right hand side of the plot.
Spatial mapping test case: March 25-30, 2007
---------------------------------------------
To illustrate the usefulness of this technique and perform a preliminary error analysis, we perform a Delaunay triangulation and Sibson interpolation of ACE, Wind, STEREO-A (STA), and STEREO-B (STB) plasma data during a moderately disturbed period centered on 27 March 2007, when the STEREO spacecraft separation was 3.6$^{\circ}$ (approximatley half the separation distance during the 22-23 May 2007 event series analyzed in Section 4). In this test case, 10-minute averaged data is used. Contour maps resulting from the spatial mapping and interpolation processes are shown in Figure 4 for the proton bulk speed $|Vp|$. (Spatial maps for the other bulk plasma parameters of the solar wind, $Np$, and $Tp$ are shown in the Appendix.) In each panel the time series data ha been converted to spatial coordinates, with the origin of the spatial map located at the Earth’s position in the y direction and at the right side of the figure in the x direction. The axes are given in AU in the ecliptic plane.
We chose this interval because it allows a test of how well the interpolation can reconstruct observed data between two spacecraft separated by 3.6$^{\circ}$, which is the mean separation distance between STA and ACE in Figure 9. We purposely use a time period of nearly five days, with a time $t_o$ defined at the beginning of the time series to maximize the effect of solar wind velocity error propagation. Four separate Delaunay triangulations and Sibson interpolations are performed between the various spacecraft data. In the top panel we use data from all four spacecraft (STA, ACE, Wind, and STB) against which we compare the other less constrained maps. The middle two panels use data from three spacecraft (STA, Wind, and STB followed by STA, ACE, and STB). The bottom panel uses data from only STA and STB in the mapping and interpolation processes. Grey lines in each panel show which spacecraft data tracks are used in the interpolation and how the spacecraft trajectories track through the spatial domain. As made obvious from viewing the growing separation between the ACE and Wind tracks in the y (transverse) dimension, from the right of the plot (e.g. beginning of the data series) to the left of the plot (e.g. end of the data series), there is a transverse distortion of the map due to propagation of error in the velocity data. The amount of transverse distortion between ACE and Wind is on the same order as the y-distance (transverse spread) between them. Despite this limitation, it is clear that the three-spacecraft interpolations (panels 2 and 3) and the two-spacecraft interpolation (panel 4) suffer only slight degradation from the four-spacecraft interpolation in the top panel. Upon close inspection, some fine-scale structures are either severely smoothed or completely lost in the three-spacecraft and two-spacecraft interpolations. However, fine-scale structures in the velocity exceeding 0.03 AU in radial thickness are retained even in the map using data from only the two STEREO spacecraft.
By extracting interpolated data in the three- and two-spacecraft spatial maps corresponding to the missing ACE and Wind spacecraft tracks (missing grey lines in panels 2-4 of Figure 4), it is possible to more directly compare the results of the three- and two-spacecraft interpolations to the actual time-series data. This is accomplished through a reverse transformation of the interpolated data back to the time domain using the same bulk plasma velocity components in the ecliptic plane. The purpose of this exercise is to determine to what degree the mapping and interpolation processes can recover spatial information between spacecraft data tracks and how well the observed time-series data can be reconstructed. Figure 5 shows three examples of reconstructed solar wind $|V_p|$ $N_p$, and $T_p$ data for Wind. The top panel shows the comparison of Wind data for the contour maps of $|V_p|$ shown in Figure 4. In the plot, the blue line shows the observed Wind data, while the red and black lines show the reconstructions from the three-spacecraft spatial map and two-spacecraft spatial map in Figure 4b and 4d, respectively. Note the similarity of the interpolated data to the actual observations.
The second and third panels show similar results for the reconstructed $N_p$ and $T_p$ data at Wind. The spatial maps associated with these two- and three-spacecraft time-series reconstructions are shown in Figures A5 and A6. Note that the error is largest for the more highly variable $N_p$ and $T_p$ and lowest for the smoother profile of $|V_p|$. As expected, timing errors and amplitude errors also increase as the data moves further to the left of the map, away from time t$_o$ (right side of the map). This is due to the propagation of velocity error in the estimation of spatial grid (see Figure 3) and in practice the spatial mapping technique is not recommended for use over such long time intervals.
Overall, the success of the mapping and interpolation processes to reconstruct nearly quantitatively accurate time-series in a moderately disturbed period lends confidence that the method is robust enough to apply to the 22-23 May 2007 event period. During the ICME event series, the spacecraft separation distances between the Earth positioned spacecraft (ACE and Wind) and the STEREO spacecraft will be on the same order as the separation between STA and STB in the 25-30 March 2007 exercise. Thus, spatial mapping of the 22-23 May 2007 event period should allow assessments of the bulk plasma parameters down to fine structures having thicknesses of $\sim$0.03 AU in the radial dimension.
Observations
============
On 19 May 2007, at 1324 UT, LASCO observed a halo CME originating from AR10956 and associated with a B9.5 flare. A second partial halo CME was launched less than a day later on 20 May 2007 from the same active region associated with a second B-class flare. The source region of these CME events has been well studied by Li et al., (2008) and references therein. The first CME was associated with a flux rope observed at STB and at L1, whereas clear signatures of the second CME were observed only by STA and at L1. On 21- 23 May 2007, the corresponding ICMEs arrived at ACE, Wind, STA, and STB, followed shortly thereafter by a high speed stream. STA and STB were separated by 8.5$^{\circ}$. Additional details of the interplanetary observations can be found in Kilpua et al., (2009).
Figures 6 and 7 show magnetic field and plasma time series observations from ACE, Wind, and the STEREO spacecraft during the ICME event series. In each figure, the top four panels show the magnetic field components in GSE coordinates for ACE and Wind and in an analogous coordinate system for STA and STB, in which the x direction points from the spacecraft towards the Sun, the z direction points northward, and the y direction completes the right-handed coordinate system. For consistency, we use the nomenclature and color coding from Kilpua et al., (2009) in labeling the two ICME regions MC1 and MC2 in each figure. Focusing ont Figure 6, both ACE and Wind observe the two ICMEs, MC1 and MC2 highlighted by the orange and blue regions, rspectively. Vertical dashed lines mark the leading and trailing boundaries of both ICMEs and indicate $t_o$ and $t_1$, the entry and exit velocity flow deflections (see Figure 1), which are particularly distinct in the $v_y$ and $v_z$ components. The ICME modeling results (discussed in Section 4.1) are overlaid on the magnetic field components. Figure 7 shows similar time series data for STA and STB. In this figure, the MC1 region is split into two sections because of the different observing conditions at STA and STB. The orange highlighted region in the top four panels mark the observation of MC1 at STB, with the magnetic flux rope fit overlaid on the magnetic field components. Although an ICME signature was present, no flux rope was observed at STA for this event. Thus we have not highlighted any portion of the ICME magnetic field at STA for MC1. Instead, we highlight the velocity flow defection boundaries (between $t_o$ and $t_1$) at STA in the bottom three panels of Figure 7. Note that the ICME corresponding to MC1 is observed approximately four hours earlier at STA than STB. Turning our focus to the second event in the series, MC2, which is only observed at STA, the blue region highlights both the magnetic field model fit and the $t_o$ and $t_1$ velocity flow deflection boundaries used in the model fit. Unfortunately, velocity flow deflections for MC2 are not reliable at STB because the ICME speed in this region at STB is slower than the surrounding solar wind speed.
We apply the non-force-free flux rope model to the ACE and STB magnetic field data shown in Figures 6 and 7 using the same boundaries for the flux ropes as indicated in Kilpua et al., (2009). Resulting single spacecraft flux rope inversions at ACE and STB indicate highly inclined flux ropes relative to the ecliptic plane. Details of the single spacecraft fits and the associated can diagrams for this event series are found in Reinard et al., (2010).
Synoptic ICME Model Inversion
-----------------------------
As discussed in Section 2, the velocity flow deflection around the ICME obstacle can be used to estimate an impact parameter and normal vector to the leading ICME boundary, assuming a cylindrical shape for the obstacle. Since the magnetic field observed at STA for MC1 does not have a clearly identifiable flux rope signature, we can employ velocity flow deflections around the ICME to determine its orientation at STA. As a preliminary analysis, looking at the STA velocity components for MC1 (orange region) in the bottom three panels of Figure 7, the flow deflection ahead of$t_o$ is initially westward, indicating STA is on the west side of the ICME symmetry axis, similar to the trajectory T2 shown in Figure 1b. At the leading edge of the ICME, the flow deflection clock angle given by equation (1) gives 19$^{\circ}$, which is generally consistent with the single spacecraft model clock angles at ACE of 35$^{\circ}$ and STB 435$^{\circ}$ (see Reinard et al., 2010). The smaller angle at STA may indicate a gentle distortion of the cross-section of the rope from a high-inclination (19$^{\circ}$) at STA to a less high inclination (43$^{\circ}$) at STB.
The results of the synoptic ICME inversion for MC1 using the magnetic field data from Wind, ACE, and STB along with the obstacle normal constraints from the velocity flow deflections at STA are shown in the can diagrams Figure 8a and 8b. These are 3-D representations of the flux rope structures within the ICMEs. Cuts through these structures at each spacecraft correspond to the black dashed lines in the magnetic field time series in Figures 6 and 7. Two different views are shown in Solar-Ecliptic coordinates. The orange can represents the flux rope portion of the MC1 ICME with its size, orientation, and relativity to the Sun clearly indicated. (The orbit of the Earth is shown as the green ellipse.) In Figure 8a, a dipole field line tangent to the axial field in the ecliptic plane threads the rope and serves as a reminder that these structures have magnetic footpoints rooted on the solar surface. In both 8a and 8b, tracks from all four spacecraft through the structure are shown. The flux rope axis is quasi-perpendicular to the ecliptic plane with a mean clock angle of 39$^{\circ}$ (meaning a 39$^{\circ}$ clockwise rotation from the z-axis in the plane of the sky, as viewed from the Sun). Similar to the single spacecraft results in Reinard et al., (2010), the resulting flux rope structure is highly inclined to the ecliptic plane at ACE, Wind, and STB. (Note the trajectory of STA does not intersect the flux rope.) Unlike the single spacecraft fits, the flux rope is shown to have an elliptical structure, is elongated azimuthally by a ratio of 1.86, and the cross-section of the rope is slightly distorted between STB and ACE through a clock angle of 8$^{\circ}$. (If we include data from STA containing information about the non-rope portion ICME cross section, the distortion of the cross-section of the entire ICME envelope increases to 24$^{\circ}$.)
Performing a similar fit for the MC2 ICME, requires use of magnetic field and velocity flow deflection data from ACE, Wind, and STA. Unfortunately, STB does not exhibit an identifiable flux rope magnetic field signature during this period. Equally unfortunate is that this MC2 ICME region also has the slowest solar wind speed of the entire 21-24 May 2007 interval. Because of this slow speed, the surrounding solar wind may actually be overtaking the ICME at the STB location, and thus the velocity flow deflections around the obstacle are unreliable. Using the data at Wind, ACE, and STA results in a synoptic model fit shown by the dashed lines in Figure 6 and 7 (for ACE and STA) and represented by the blue-green can in Figure 8a and 8b. In the case of MC2, the flux rope axis is also quasi-perpendicular to the ecliptic plane, but with a mean axial orientation having a clock angle of 161$^{\circ}$. This orientation indicates the rope’s axial magnetic field is pointing southward, and that the two rope axes are nearly perpendicular to one another. This flux rope is also shown to have an elliptical shape (stretched azimuthally by a ratio of 1.2) and the cross section distorted between ACE and STA by an angle of over 30$^{\circ}$. As shown in Figure 8b, the trajectory of STB does not intersect the rope.
Figure 8c, adapted from Figure 1 of Ki;pua et al., (2009), plots the Grad-Shafranov reconstructions for MC1 (orange) and MC2 (blue) in the ecliptic plane, with the colored ovals the expected elongation of the ICME boundaries (magnetic field contours from the Grad-Shafranov fits are shown interior to the ICME boundaries). Comparing Figures 8a-8c, the Grad-Shafranov and synoptic model results are generally consistent, for both MC1 and MC2. In MC1, STB passes to the east of center, while ACE and Wind pass to the west of center of the flux rope. STA just grazes the edge of the ICME structure, with an impact parameter passing beyond the edge of the flux rope in the ICME interior. Comparing elliptical scale size ratios (major to minor axes) the synoptic result gives 1.84 9 1.200 for MC1 (MC2), consistent with the GSR values of 2.25 (1.15) for MC1 given in Kilpua et al., (2009).
Although the panels in Figure 8 clearly indicate where the magnetic flux ropes, MC1 and MC2, are relative to the spacecraft, it is difficult to picure how the non-flux rope ICME signatures are incorporated into the overall ICME structure (at STA for MC1 and at STB for MC2). The elliptical ICME boundaries shown in Figure 8c were qualitatively suggested by Kilpua et al., (2009) using the observed size of the flux ropes in the radial direction, impact parameters, and axial field directions from the GSR analyses. In the next section, the spatial mapping procedure will remove much of the remaining uncertainty and reveal the impact of the solar wind environment, both inside and around the coherent magnetic flux rope structure. This will enable us to know the posistion of the flux rope relative to the ICME envelope and construct a more quantitative description of the large-scale ICME structure of the event series.
Analyses of the 21-24 May 2007 events using spatial maps
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Much of this section focuses on placing the results from the synoptic modeling of MC1 and MC2 in context with the spatial maps of the ambient solar wind. By putting these techniques together we are able to reconstruct an ecliptic plane view of the large-scale ICME geometry for each event and analyze it within the larger framework of the surrounding plasma environment. What is distinctly new about this composite technique is that it reveals aspects of the flux rope and ICME structure and dynamics in a fashion that is far more visually intuitive than individual time-series data tracks alone.
Figure 9 shows the results of these techniques applied in combination to the 2007 May 21-24 time period, including the MC1 and MC2 ICME events. As described in Section 3, these maps extend a few tenths of an AU upstream and downstream of the two ICMEs, equivalent to observations made from several days earlier to several days after ICME passage. The maps are centered on a point in time that is nearly halfway between MC1 and MC2 as observed at L1. This time (2100 UT on 22 May 2007) is defined as the time ($t_o$) which maps to position R= 0 (x=y=0) on the spatial grid, meaning the origin of the spatial map is located at the Earth’s position in the y direction and centered between the two flux ropes in the x direction. Similar to Figure 4, the Sun is off to the left, time increases to the left, and spacecraft tracks through the maps are shown in grey, with the STA and STB tracks bounding the edges of the “visible” region of the ICME structures and solar wind. In the top panel, and overlaid on the maps in panels 2-5, are black arrow vector field regions showing the GSE Bx and By components of the interpolated magnetic field (constructed from the fits to the highlighted regions MC1 and MC2 shown in Figure 6 and 7 at STB, ACE, Wind, and STA). In panels 2-5, the spatial maps for solar wind Np, Vp, Tp, and the He$^{+2}$/H$^+$ ratio are shown for the same interval. Note that for the He$^{+2}$/H$^+$ ratio map, only three spacecraft data are available for interpolation with a much reduced data resolution of 1-hour, resulting in a coarser profile.
### Magnetic Field Analysis
In order to maintain consistency with Figure 8c, adapted from the Kilpua et al., (2009) study, the top panel in Figure 9 shows the magnetic field interpolation of the two-ICME event series, with the MC1 (MC2) region shaded by the orange (blue) ellipse. The field in thes shaded regions reveal coherent, large-scale rotations consistent with the existence of flux ropes nested within larger ICME structures. Note that no attempt has been made to ensure the interpolation of the magnetic field results in a divergenceless field and, as such, the arrows are not intended to represent actual field lines, but rather indicate the location and general direction of the ICME magnetic field in context with the surrounding environment. The extension of the shaded boundaries beyond the STA and STB spacecraft tracks illustrates how far the flux ropes and the surrounding ICME envelopes extend beyond the range of our viewing window.
In the case of the ICME containing MC1, velocity flow deflections from the synoptic model fit indicate the western edge of the ICME envelope is very close to the STA location (IP$\geq$0.9). Thus it is expected that the western flank of the ICME envelope does not extend much beyond the location of STA, the range of our view. Looking closely at the magnetic field direction at the leading westward flank of MC1, the $B_y$ component, which near ACE is pointing westward, abruptly reverses direction approximatley halfway between ACE and STA. The location of this reversal in $B_y$, at which the field begins to point in the opposite sense as the field within MC1, delineates geometrically for the first time, the magnetic boundary of a flux rope region within a larger ICME envelope. In this instance, the MC1 flux rope is revealed as a sub-structure of a larger ICME envelope. The eastern edge of this envelope is not visible because the edge of our viewing window extends only down to STB, which traverses barely east of center of the MC1 flux rope.
Turning our attention to the magnetic field of MC2, the spatial map in the top panel of Figure 9 shows the westernmost boundary of MC2 extends beyond our viewing window. In the synoptic model fit, there is consistency between the impact parameter given by the velocity flow deflections at STA and the impact parameter given by the magnetic field, indicating the large-scale ICME boundary probably does not extend much beyond the flux rope itself. Unfortunately, for MC2 we do not benefit from having velocity flow deflections at the location of STB, where the magnetic field signature is non-flux rope like so we cannot project how much beyond the eastern edge of the viewing window the ICME containing MC2 extends. However, this may prove unnecessary as there are other indicators within the ambient solar wind plasma that give us clues as to where the ICME envelope for MC2 extends in this case. Though our viewing window is constrained to only a small portion of the overall structure of these ICMEs, it is still possible to discern much information about the large-scale geometry and dynamics of these events by looking at the plasma environment during this period.
### Solar Wind Plasma Analysis of the MC1 ICME
Panels 2-5 of Figure 9 allow us to concentrate on the structure of the plasma environment within and surrounding the ICMEs. Concentrating first on the unusual distortion of MC1in the radial direction, the proton speed in panel 2 reveals a higher-speed region near the ICME trailing edge at STA compared to the other spacecraft. That this portion of the ICME is embedded in a region of higher speed plasma may explain why the ICME is first observed at STA and may be why the western flank of this ICME leads its apex, looking very unlike typical kinematic flux rope model cross-sections (e.g. Owens et al. 2006). This increased speed also compresses the tail end of the ICME at STA, perhaps contributing to the difficultly of observing anything resembling a flux rope structure at this location. Along with the grazing incidence, the compression of the ICME due to the increased solar wind speed may also explain why the observation at STA is so much shorter in duration than at ACE or STB. The third panel in Figure 9 shows the proton density is depressed near the core of the MC1 flux rope, with an extended enhanced density region trailing the flux rope proper. $N_p$ is strongly enhanced at the compression region at the trailing edge of the ICME at STA, but the fine-scale structure in this trailing region is poorly resolved. In fact, returning to Figure 7, much of the fine-scale structure at STA during this period (approximately 1900 UT on 22 May 2007 to 1200 UT on 23 May 2007) has been averaged over by the interpolation process. The region shows enhancement, but the details are lost. (To the degree the interpolation fails to capture the correct density profile in this region is discussed in Section 6.) Still, the qualitative detail is enough to confirm that the bulk of the dense material is near the trailing end of the ICME. What is also interesting is that $N_p$ (and to a lesser extent the He$^{+2}$/He$^+$ ratio in panel 5) is also elevated in the ambient plasma at the trailing edge of MC1 as observed by ACE, Wind, and STB. This elevated $N_p$ may be indicative of the filamentary material often observed at the back end of CMEs and at the trailing boundary of ICMEs. Taken together, the spatial maps are indicating that a significant portion of this material is external to the associated flux rope structure, rather than co-located with the trailing edge of the rope. This suggests the trailing portion of the ICME envelope extends well beyond the coherent flux rope sub-structure MC1.
As is typical, the proton temperature, $T_p$ in panel 4, shows the coldest temperatures inside MC1, with elevated temperatures in between the two ICME events. There is also a hint of elevated $T_p$ at the trailing edge of the MC1 ICME at STA, consistent with a compression region at this location, but with a much smoother overall profile than the density.
Moving to the spatial map containing the He$^{+2}$/He$^+$ ratio in panel 5, we see some interesting features. Typically, the helium density is elevated in the ICME sheath and envelope regions and is generally depressed in the flux rope interior. However, near the leading edge of MC1 at STB, the He$^{+2}$/H$^+$ ratio is elevated over typical solar wind and these levels remain high even as STB nears the core of the flux rope. This enhancement is also observed at ACE in the latter portion of the rope. Another interesting feature at ACE is that the helium enhancement at the trailing edge of MC1 continues, unbroken into the second ICME containing MC2. This interconnection, which occurs at both STB and at ACE, may be an artifact of the interpolation process or it may mean that the solar source region never fully“’closed” after the first CME eruption, resulting in a single, extended envelope connecting both events. CME interaction or merging in the solar wind is another possible explanation for this observation (e.g. Gopalswamy 2001, 2004). However, above 20 $R_s$ one could argue that the frozen-in flux condition should result in the two ICME envelopes being distinct from one another with a highly compressed parcel of ambient solar wind separating the two independent events. In addition, CME merging requires the second CME to have a much higher speed than the first, which is not evident in the solar or interplanetary observations (Li et al., 2008; Kilpua et al., 2009).
### Solar Wind Plasma Analysis of the MC2 ICME
Examining the plasma environment surrounding the ICME containing MC2, it is clear the extent of the flux rope, and possibly the ICME envelope do not reach the location of STB. Instead, $Np$ in panel 3 shows the easternmost flank of MC2 nested within a large density enhancement that extends the entire radial dimension of the ICME. Just upstream of MC2, the magnetic field in the ecliptic plane reverses direction in $B_x$ (from anti-sunward to sunward), seeming to bend around the density structure. Although this upstream magnetic field region may not necessarily be part of the ICME envelope, the field vectors have been included in the spatial map because the reversing field signature well defines the leading-eastern boundary of the flux rope. The peak of this density structure in the spatial map at ACE and Wind is marked by the dotted line in the time-series of Figure 6 and is bracketed by the $t_o$ and $t_1$ boundaries (blue highlighted region) in Figure 7 at STB. Another reason this upstream region has been included is to highlight the role it may play in the distortion of the flux rope cross section. As the magnetic field arrows in panel 1 clearly show, the center of MC2 is located near the trajectories of Wind and ACE, but the synoptic model results indicate ACE and Wind pass on the east side, approximately one-third of the way from the rope center. This difference is best illustrated by noting the location of the center of the blue oval in panel 1 is not co-located with the center of the flux rope as indicated by the curling magnetic field. In the magnetic field spatial map it appears as though the eastern half of MC2 has been compressed. That the model indicates the flux rope cross section is distorted through an angle of over 30$^{\circ}$ (see Figure 8) suggests the density enhancement may be compressing the eastern side of the flux rope, perhaps bending its shape out of the ecliptic plane.
The proton temperature in panel 4 and the proton speed in panel 2 are also consistent with this picture. The high density region at the eastern flank of the ICME containing MC2 is also a region of enhanced temperature and slowest speed. By comparison, the coolest and fastest regions of this ICME are inside the actual flux rope MC2. Although the difference between the fastest speed inside MC2 ($\sim$500 km/s) and the slowest speed in the density enhancement ($\sim$450 km/s) is not too remarkable, it may contribute to a slight asymmetric compression of the ICME overall structure, as the faster plasma overtakes the slower dense structure ahead on the ICME’s eastern flank.
Discussion
==========
The power of this innovative approach comes from combining two separate techniques into a single in situ modeling tool. If we return to Figure 8c for comparison with Figure 9, it is obvious that more information can be derived from these spatial maps than from previous reconstruction methods. Combing the synoptic model inversions with the spatial mapping for use alongside time-series analyses provides a much more comprehensive view of a large fraction of the two ICMEs given as examples. The result is a multi-dimensional reconstruction that reveals many long-sought-after characteristics of ICMEs in a visually intuitive manner; this includes resolving flux rope sub-structure within the larger ICME structures, determination of the ICME envelope boundaries, and the distortion of these ICMEs due to interaction with plasma structures in the surrounding solar wind.
Another benefit of this approach is that it makes multispacecraft identification of ICMEs much easier than by using individual time series alone. For the highly inclined ropes as MC1 and MC2, magnetic flux rope field rotations are extremely easy to locate as are the boundaries of these ICME substructures. Kilpua et al., (2009) noted that if only observations at L1 had been available, MC2 would not have been identified at all and the identification of MC1 would have been difficult. From Figure 9, the coherent rotations in the flux rope magnetic field signature are clearly recognizable and stand-out easily among the ICME envelope magnetic field and the surrounding ambient plasma signatures. In fact, the spatial mapping of the magnetic field made possible the refinement of the ICME flux rope boundaries indicated by the vertical lines in the time-series of Figures 6 and 7. Indeed, MC2 is easily recognizable at L1 as is MC1, regardless of whether the magnetic field mapped in Figure 9 is interpolated from the synoptic model results or taken straight from the in situ measurements of the magnetic field at each spacecraft.
A consequence of this composite technique is that the reconstructed flux rope cross-sections show a more realistic distortion when compared with results using the GSR technique. As discussed by Riley et al. (2004), the GSR technique has difficulty capturing the true distortion of simulated flux ropes, presumably because the method assumes that the structure is in approximate magnetostatic equilibrium. Kilpua et al., (2009) confirmed that the GSR technique successfully fits the ropes MC1 and MC2. However, the solution did not return a cross-section consistent with the distortion observed at the different spacecraft locations. In contrast, the methods presented in this paper are essentially empirical; there are few limiting assumptions: flux rope topology at one (or more) spacecraft, an ICME geometry that is locally elliptical or cylindrical in shape, and frozen-in-flux convection of the plasma and field. Our method does not assume any equilibrium state. Thus, the cross-sections presented in Figure 9 are consistent with the size, shape, and arrival time indicated by the observations at each spacecraft location. In the case of MC1, the spatial map reveals an asymmetrical distortion causing the rope flank (and thus the edge of the ICME proper) to lead its apex. Geometries such as these are not allowed by current static, quasi-static, or kinematic models of flux ropes. Only MHD simulations show these kinds of distortions.
One of the most outstanding implications from performing this synoptic ICME analysis on the 21-24 May 2007 ICME events is that the result argues for a non-closing of the solar source between two eruptions. In the Li et al., (2008) study, the fast CME on 19 May 2007 appeared to be part of a complex eruption with a second, slower CME observed very close in time and space. The authors found these two CMEs difficult to separate in coronagraph measurements so it was unclear if they were part of the same eruption. Kilpua et al., (2009) concluded that in-situ observations by STB and Wind supported the interpretation of a complex eruption due to an interval of counter-streaming electrons and a small magnetic cloud-like region immediately following the trailing boundary of MC1. In the bottom panel of Figure 9, the He$^{+2}$/H$^+$ ratio remains enhanced between the two ICMEs. If we take this to mean that the solar wind in between these two structures is more typical of ICME envelope material than solar wind, then this argues for the two ICME structures being embedded within a single, larger envelope, consistent with being component parts of a multi-eruption composite or complex ICME event (Gopalswamy et al., 2001, 2004).
As will be shown in a companion paper (Reinard et al., ApJ submitted 2012), spatial mapping of the Fe charge states confirm the existence of ICME-like plasma (enhanced $Q_{Fe}$) connecting MC1 and MC2 at multiple spacecraft locations. Thus, these maps give us clues as to how the solar source region responds when CMEs erupt into the solar wind. The continued “leakage” of elevated helium density (and creation of high Fe charge states) may be indicative of the solar source region remaining magnetically open to the solar wind after the traditional flare signature associated with CME expulsion. It also suggests the presence of continued eruption- (or post-eruption-) related heating that serves to pre-condition the source region environment prior to the second eruption. In fact, recent numerical simulations of magnetically coupled sympathetic CME eruptions by Török et al., (2011) depict a multi-eruption scenario that almost certainly describes the CME origins for these two May ICME events. If one compares the pre-eruption potential field source surface (PFSS) extrapolation in Figure 4 of Li et al,. (2008) to the magnetic topology and evolution of the Török et al., (2011) simulations (their Figure 3), one can immediately associate the first eruption above the west neutral line (WNL) and the second eruption above the center neutral line (CNL) with the simulated flux ropes labelled FR2 and FR3, respectively, in the Török et al., (2011). In addition, the Török et al. figure panels 3(e) and 3(f) show that the flare current sheet formed during the FR2 eruption acts as the “breakout” reconnection that facilitates the sympathetic FR3 eruption. Implications of this scenario are that the elemental and ionic composition, usually associated with the dynamic topological opening of low-lying flux and eruptive flare heating could be present, not just in the interior of each of these ICME flux ropes, but between them as well, precisely what the spatial mapping results show here and in the Reinard et al., (ApJ submitted, 2012) analysis.
Future Work
===========
There are many aspects of this composite modeling technique that have yet to be explored. It is understood that the spatial mapping of low-inclination flux rope orientations relative to the ecliptic plane will be more difficult to interpret since much of the cross-section will be perpendicular to the spatial mapping plane, making interpretation more difficult. Other unknowns include errors in determining the obstacle normal since the accuracy of using flow deflections around an obstacle is not yet well quantified. Future work will include using coplanarity, mixed mode, and velocity coplanarity computations to help provide an estimate of the errors to the obstacle normal (e.g. Russell et al., 1983; Schwartz, 1998; Paschmann and Sonnerup, 2008).
As discussed in Section 3.2, propagation of error is another factor that needs to be quantified for the spatial mapping technique, as is interpolation in Cartesian space as opposed to a spherical coordinate system. From the limited analysis in Section 3.4, these errors are expected to be small compared to the length-scale of the interpolations and the ICME structures under study. However, there are obvious limitations to the mapping as exemplified in panel 3 of Figure 9. Although enhancements in $N_p$ are observed at all four spacecraft at the trailing edge of MC1, the mapping algorithm interprets these as having local maxima at each spacecraft location, making the overall density enhancement appear “clumpy.” This type of artifact results from the algorithm constraining the length-scale of interpolated features in the transverse dimension to be on the same order as the length-scale of the (observed) features in the radial dimension. Thus, instead of a single density enhancement with a much larger transverse dimension as compared to radial dimension (which is probably more representative of reality), the enhancement in Figure 9, panel 3, appears “clumpy” as if several independent enhancements occur at each spacecraft. Future work will include the quantification of such local artifacts and a modification of the algorithm to use an analytic description of flux rope magnetic fields (obeying $\nabla \cdot B = 0$ at all locations within the grid). Testing the velocity flow deflections with MHD simulations will also help quantify flow deflection errors in obstacle normal determinations.
Conclusions
===========
In this paper we have developed two new techniques that greatly aid in the modeling and analysis of multispacecraft in situ observations of ICMEs. The first of these techniques improves upon the Mulligan and Russell flux rope model by incorporating velocity flow deflections around the obstacle that provide a measurement of the large-scale ICME orientation. This additional information relaxes the constraint requiring the model to fit only flux rope structures and opens the possibility of modeling a larger class of ICMEs. The second technique involves creating spatial maps of multispacecraft data by intelligently interpolating the data in the intervening space between the known data tracks. Combining these two techniques together, we have created a powerful composite modeling tool that complements and greatly extends the utility of more traditional time-series analyses. Besides successfully reconstructing the ICME structure previously reported with existing models, it is now possible to go beyond these models to quantitatively reveal substructures, spatial distortion, and dynamics in the large-scale ICME structure, and place them in context with the ambient plasma environment. What is also distinctly new about the approach in this paper is that it reveals aspects of flux rope and ICME geometry and dynamics in a way that is far more visually intuitive than individual time-series data tracks alone. Although the assessment of numerical errors associated with this new model have only begun to be probed, studies are already underway to better understand the limitations using different spatial and temporal resolution data and MHD simulations. The results of this study will be included in a future publication.
Consequences of ICMEs are far-reaching in the heliosphere and the ability to exploit multi-spacecraft plasma and magnetic field data as demonstrated in this paper creates a deeper understanding of these structures in context with the plasma environment in which they are surrounded. Utilizing data in this way gives us clues to how CMEs are formed and ejected into the heliosphere, how ICMEs are structured and evolve in the solar wind, and what is the relationship between solar and in situ observations of these phenomena. Overall, it is this better understanding of the CME-ICME connection that is of key importance for solar-terrestrial research and space weather forecasting.
Delaunay Triangulation and Voronoi Diagrams
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In general, triangulation is a type of discretization of a set of points defining, say a geometric object, into a number of smaller spatial elements called simplices. These simplices contain a minimum number of vertices $(d+1)$ for the dimension in which they are defined $\mathbb{R}^2$. In two dimensions, a triangulation of a set of points $P \subset \mathbb{R}^2$ can be thought of as a function $T$ that operates on $P$ and returns a surface comprised entirely of triangles $T(P)$. The convex hull of $P$ is defined as the convex polygon that traces the perimeter of the points. Figure A1 shows an example of a convex hull in two dimensions. The concept is most easily illustrated by imagining an elastic band being placed around the collection of points and having it constrict until it wraps tightly around the outermost points in the set. The area bounded by the band is the convex hull.
In a triangulation, the convex hull of $P$ is subdivided into simplices such that any two simplices intersect in a common face or not at all. The set of points used as vertices of the subdividing simplices is $Q \subseteq P$. Returning to the two-dimensional case, for each triangle on the surface, a unique circle can be defined such that it passes through all the vertices of the triangle. These circles are known as “circumcircles” (because they circumscribe the vertices of their respective triangles). If the circumcircle of a triangle contains no points in its interior, irrespective of how many vertices it has on its circumference, the triangle is called “Delaunay.” This means that there can be any number of vertices on the circumcircle, but the circle itself encloses no vertex– it is empty. Not surprisingly, Delaunay triangulation $DT(P)$ is one that generates a surface comprised only of Delaunay triangles. Such a triangulation for $P \subset \mathbb{R}^2$ is shown in Figure A2. Note that the union of all simplices in the triangulation is the convex hull of the points.
The geometric dual of the Delaunay triangulation is the Voronoi tessellation. Defined mathematically, for a $T$ nonempty subset of $S$, the Voronoi face $V(T)$ is the set of points in $\mathbb{R}^d$ equidistant from all members of $T$ and closer to any member of $T$ than to any member of $S$ that is not in $T$. This means that the Voronoi face is always a nonempty, open, convex, full-dimensional subset of $\mathbb{R}^d$. In two dimensions, the Voronoi face is a polygon (see Figure A3). The Voronoi diagram of $S$ is the collection of all nonempty Voronoi faces $V(T)$ for $T \subseteq S$. The Voronoi diagram forms a cell complex that partitions the convex hull of $S$ in a similar fashion as Delaunay triangulation. In fact, these two structures can be easily constructed from one another. The vertices in a Voronoi tessellation are centers of the circumcircles of the Delaunay triangles. Figure A3 shows that by connecting the centers of the circumcircles shown as red dots in the left panel, the Voronoi diagram (right) can be produced. The resulting Voronoi diagram is an extremely powerful discretization tool and is used in many interpolation techniques. As will be discussed in the next section, Voronoi polygons are useful for determining weights in the natural neighbor interpolation method.
Natural Neighbor Interpolation Method
-------------------------------------
The natural neighbors of a point x are defined as the neighbors of x in the Delaunay triangulation of $P \bigcup {x}$. Equivalently, the natural neighbors are the points of $P$ whose Voronoi cells would be partially or wholly removed upon the insertion of $x$ into the set. More precisely, let $V_{p_i}$ be the Voronoi cell of $p_i$ in the Voronoi diagram of $P$ and let $Vx$ be the Voronoi cell of $x$ in the Voronoi diagram of $P \bigcup$ [x]{}. The natural region $NR_{ x,p_i}$ is the portion of $V_{p_i}$ stolen by x, (i.e. $V_x \bigcap V_{p_i}$.) Let $w_{p_i}(x)$ define the n-dimensional volume of $NR_{x,p_i}$ (e.g. an area in $\mathbb{R}^2$ and a volume in $\mathbb{R}^3$). Then $ NR_{x,p_i}=\emptyset $ and $w_{p_i} = 0$ if $p_i$ is not a natural neighbor of $x$. The natural coordinate associated with $p_i$ is defined by the fractional area: $$\lambda_{p_i}(x) = \frac{w_{p_i}(x)}{\sum\limits_i w_{p_i}(x)}.$$ These natural coordinates have three important properties:
1. $\lambda_{p_i}(x)$ is a continuous function of $x$, and is continuously differentiable, except at the data sites.
2. the $\lambda_{p_i}(x)$ are bounded if $x$ belongs to the convex hull of $P$. In two dimensions, this means $\lambda_{p_i}(x)$ vanishes outside the union of the circumcircles circumscribing the Delaunay triangles incident to $p_i$.
3. The $\lambda_{p_i}(x)$ satisfy the local coordinate property (LCP) identity, which states that $x$ is a convex combination of its neighbors: $$\sum\limits_{i} \lambda_{p_i}(x) p_i = x.$$
When $x$ lies outside the convex hull of $P$, $w_{p_i}(x)$ is unbounded if $p_i$ is a vertex of the convex hull. In order to keep the $w_{p_i}(x)$ bounded, the domain over which the natural coordinates are to be computed must be bounded. This can be accomplished by limiting the insertion of point $x$ to a bounding box within the convex hull. Now assume that each $p_i$ is defined along a continuously differentiable function $h_{p_i}$ in $\mathbb{R}^d$ satisfying $h_{p_i}(p_i)=0$. The natural neighbor interpolation of the $h_{p_i}$ in two dimensions is defined as: $$H(x) = \sum\limits_i^n \lambda_{p_i}(x) h_{p_i}(x).$$ As indicated above, a bounding box $B$ is needed to bound the natural coordinates of any point $x$ to lie within the convex hull of $P$. The set of data points consists of the $p_i$ plus some points $q_i$ added on $B$. Let $h_{q_i}=0$ for all points $q_i$ on the bounding box. Once the interpolation is complete, $P$ will denote the union of the sample points $p_i$ and the $q_i$. For a given $x$, $H(x)$ is easily evaluated once the $\lambda_{p_i}(x)$ have been calculated. In the simplest terms, $H(x)$ is the estimate of the function value $h_{p_i}$ at the point $x$, arrived at by summing the function value at the known $p_i$ times the weights $\lambda_{p_i}$, which have been predetermined by the area stolen from the change in the Voronoi faces when the new point $x$ has been inserted into the mix.
Figure A4 illustrates this process. The green circles, which represent the interpolating weights in each of the cells ($\lambda_{p_i}$), are generated using the ratio of the blue shaded area in each cell to that of the cell area of the surrounding points. The blue shaded area represents the new Voronoi cell created after inserting point $x$ into the set. The area comprising this new cell is stolen from existing cells by inserting the point $x$ and recalculating the Delaunay triangles (and thus the Voronoi cells) in the affected area.
Although the nearest neighbor method is not the only way to estimate the gradients, it has advantages over simpler methods of interpolation in that it provides a continuous interpolation of the underlying function except at the site of the data points ($C^1$ continuous). There exist several other methods developed recently that may provide superior results, such as the Farin’s $C^1$, and Hiyoshi’s $C^2$ methods, which are proving robust under irregular triangulations such as time series data. Such methods may be incorporated in future versions of the model.
Application to Time Series Observations
---------------------------------------
Figure A5 shows a distribution of points along four simulated spacecraft tracks transformed onto a spatial domain using the technique outlined in section 3.1 using a simulated solar wind speed. Performing a Delaunay triangulation of this spatial data results in the blue triangles connecting the data points along and between the spacecraft tracks. Triangles within the convex hull (between the two outermost spacecraft tracks) are constructed from points that are in close proximity and sequential in time. Natural neighbor interpolation is expected to work well in this region. Triangles beyond the convex hull (outside the two outermost spacecraft tracks) connect points remote from each other and non-sequential in time. In these regions there is insufficient sampling to accurately capture the surface; thus interpolations in these regions will be spurious and should be ignored. The estimators, $H(x)$, determined from the natural neighbor method are applied to the bulk solar wind parameters $N_p$, $|V_p|$, $T_p$ and $\bf{B}$, corresponding to each spacecraft track on the spatial grid. This results in scalar and vector fields interpolated over the entire spatial domain determined by the multi-spacecraft data points within a bounding box $B$. Because the Delaunay triangulation and the Voronoi diagram are based on the spacecraft observational geometry and relative trajectories, the spatial grid and the the natural neighbor coordinates, $\lambda_{p_i}$, need to be calculated only once per event. The advantage of this is the continuous functions determined from the multi-spacecraft plasma parameters $N_p$, $|V_p|$, $T_p$ and $\bf{B}$ representing the $h_{p_i}(x)$ can be changed independently, without the need for a re-triangulation. Thus, the solar wind $N_p$,$ |V_p|$, $T_p$, and $\bf{B}$ scalar and vector fields for a single event are all calculated using the same underlying Delaunay triangulation and weighting functions.
### March 25-27, 2007 solar wind contour maps
To illustrate the usefulness of this technique we perform a Delaunay triangulation and Sibson interpolation of ACE, Wind, STA, and STB plasma data during a moderately disturbed period centered on March 27, 2007, when the STA and STB spacecraft separation was 3.6 degrees, nearly half their separation distance during the May 22, 2007 event analyzed in Section 4. Contour maps resulting from the spatial mapping and interpolation processes are shown in Figure A6 for the solar wind parameters $N_p$ and $T_p$. (Contour maps for the proton bulk speed $|Vp|$ is shown in Figure 4.)
Four separate Delaunay triangulations and natural neighbor interpolations are performed between the various spacecraft data. In the top panel we use data from all four spacecraft (STA, ACE, Wind, and STB) against which we compare the other less constrained maps. The middle two panels use data from three spacecraft (STA, Wind, and STB followed by STA, ACE, and STB). The bottom panel uses data from only STA and STB in the mapping and interpolation processes. In each panel the Sun is to the left and the units are in AU in the ecliptic plane in Solar Ecliptic coordinates. Grey lines in each panel show which spacecraft data is used and how the spacecraft trajectories track through the spatial domain. As mentioned previously, these contour maps are not a single snapshot in time (i.e. the spacecraft trajectory in the spatial domain takes a finite amount of time to traverse the grid) resulting in features further to the left (nearest the Sun) being later in time than features to the right of the map. Despite this limitation, it is clear that the three- and two-spacecraft interpolations suffer only slight degradation from the four-spacecraft interpolation in the top panel.
By extracting interpolated data in the three- and two-spacecraft contour maps that correspond to the missing ACE and Wind spacecraft tracks (missing grey lines in panels 2-4 in Figures A6 and A7), it is possible to compare the results of the three- and two-spacecraft interpolations to the actual time series data. This is easily accomplished through a reverse transformation of the interpolated data back to the time domain using the same bulk plasma velocity components in the ecliptic plane.
Figure A8 shows the results of three sets of time-series reconstructions for Wind and ACE (the other three sets are shown in Figure 5). The purpose for this exercise is to determine to what degree the mapping and interpolation processes can recover spatial information between spacecraft data tracks, how well the time-series data can be reconstructed, and quantify the error. Figure A8 shows three examples of reconstructed solar wind $|Vp|$, $ Np$, $Tp$ data for Wind and ACE. The top panel shows the comparison of Wind data for the contour maps of Vp shown in Figure 4. In the plot, the blue line shows the observed Wind data, while the red line and the black lines show the reconstructed or “modeled” data from the three-spacecraft contour map and two-spacecraft contour map in Figure 4b and 4d, respectively. Note the similarity of the interpolated data to the actual observations.
The second and third panels show similar results for the reconstructed proton density and temperature at ACE. The contour maps associated with these two- and three-spacecraft time series model results are shown in Figures A5 and A6. Note that the error is largest for the more highly variable Np and Tp and lowest for the smoother profile of $|Vp|$. Note also that timing errors and amplitude errors increase as the data moves away from the center time t. This is expected due to the propagation of error in the estimation of spatial grid (see Figure 3) as the mapping moves further from the center time t, which causes increased error, not only in the spatial mapping of the spacecraft locations relative to one another, but also in the estimation of the gradients used for interpolation.
Overall, the success of the mapping and interpolation processes to reconstruct nearly quantitatively accurate time-series in a moderately disturbed period similar to that encountered during the May 22, 2007 ICME, lends confidence that the method is robust enough to apply to the May 22, 2007 event period. During the ICME event, the spacecraft separation distances between the Earth positioned spacecraft (ACE and Wind) and the STEREO spacecraft will be on the same order as the separation between STA and STB in the March 27, 2007 exercise. This successful demonstration indicates that we can make (at the very least) qualitative assessments of the bulk plasma parameters when interpreting the spatial maps on similar length-scales.
T.M. and A.A.R. acknowledge support from NASA SR&T NNX08AH54G. B.J.L.acknowledges support of AFOSR YIP FA9550-11-1-0048 and NASA HTP NNX11AJ65G. Support for the STEREO mission in-situ data processing and analysis was provided through NASA contracts to the IMPACT (NAS5-03131) and PLASTIC (NAS5-00132) teams. The authors thank the ACE MAG, SWEPAM, and SWICS teams for making their data available on the ACE Science Center Web site (http://www.srl.caltech.edu/ACE/ASC/). T.M. also acknowledges R.A. Leske for his invaluable discussions during the early stages of this manuscript.
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---
abstract: 'Semiconductor quantum dots can emit antibunched, single photons on demand with narrow linewidths [@Michler.2000; @He.2013; @Kuhlmann.2015]. However, the observed linewidths are broader than lifetime measurements predict, due to spin and charge noise in the environment [@Kuhlmann.2013; @Matthiesen.2014]. This noise randomly shifts the transition energy and destroys coherence [@Muller.2007; @Flagg.2009] and indistinguishability [@Santori.2002; @Matthiesen.2013] of the emitted photons. Fortunately, the fluctuations can be reduced by a stabilization using a suitable feedback loop [@Latta.2009; @Prechtel.2013; @Hansom.2014]. In this work we demonstrate a fast feedback loop that manifests itself in a strong hysteresis and bistability of the exciton resonance fluorescence signal. Field ionization of photogenerated quantum dot excitons leads to the formation of a charged interface layer that drags the emission line along over a frequency range of more than $30{{\,\mbox{GHz}}}$. This internal charge-driven feedback loop could be used to reduce the spectral diffusion and stabilize the emission frequency within milliseconds, presently only limited by the sample structure, but already faster than nuclear spin feedback.'
author:
- 'B. Merkel'
- 'A. Kurzmann'
- 'J.-H. Schulze'
- 'A. Strittmatter'
- 'M. Geller'
- 'A. Lorke'
title: 'A charge-driven feedback loop in the resonance fluorescence of a single quantum dot'
---
A vision in quantum information technology is a quantum network [@Bennett.2000; @Kimble.2008] where the quantum bits (qubits) are stored in nodes and connected via quantum channels. Semiconductor quantum dots (QDs) are one possible candidate for such a quantum node [@Ladd.2010]. They can host, for instance, a spin qubit [@Vamivakas.2009], and the quantum information can be transferred between different nodes via single photons [@Ylmaz.2010]. However, these photons suffer from frequency noise, induced by residual charges and spins in their environment [@Kuhlmann.2013; @Matthiesen.2014], which reduces their coherence and indistinguishability [@Santori.2002; @Matthiesen.2013].
A possibility to reduce this spectral jitter is a feedback loop that counteracts the external fluctuations. One option is an [*external*]{} feedback that measures the emission frequency and then corrects it by setting a gate voltage, using the Stark effect[@Prechtel.2013; @Hansom.2014]. For example, the resonance fluorescence (RF) of a self-assembled QD produces single photons while simultaneously the emission energy is determined from the differential transmission [@Prechtel.2013]. Such a feedback loop effectively stabilizes the QD photon emission up to a frequency of about $100{{\,\mbox{Hz}}}$. Alternatively, the feedback can be based on [*internal*]{} mechanisms, using the coupling of the electron-hole pairs to the environment. In magnetic fields, the interaction of the excitons with the nuclear spins in the dot leads to a dragging-like hysteresis of the resonance fluorescence. The hysteresis is caused by dynamic nuclear spin polarization [@Hogele.2012; @Chekhovich.2013] and can also be used to stabilize the emission line [@Latta.2009; @Yang.2013]. Such an internal feedback loop needs no further optical or electrical detection scheme. However, the bandwidth of this internal nuclear spin feedback seems to be restricted to about $10{{\,\mbox{Hz}}}$.
{width="0.9\twocolfig"}
We use here a micro-patterned device that consists of a Schottky diode with a layer of self-assembled InAs/GaAs QDs (see the schematic band structure in Fig. \[fig-1\]a and methods below) to demonstrate an inherently fast internal and charge-driven feedback loop in the resonance fluorescence of a single dot. The bottom trace in Fig. \[fig-1\]b shows the RF signal when the exciton transition is shifted upwards or downwards across the laser frequency (blue and red lines, respectively) by sweeping the voltage that is applied to the device. Compared to the commonly found Lorentzian line shape of excitons in single InAs quantum dots [@Kuhlmann.2015], the resonances observed here show several distinctly different characteristics. They are much broader and exhibit a strongly asymmetric shape with a gradual increase on the low energy side and an abrupt decrease on the high energy side. More importantly, the traces show a pronounced hysteresis over a range of $15{{\,\mbox{GHz}}}$ not only in the absence of magnetic field ($B= 0$) but also for both Zeeman-split exciton transitions at $B=4.5{{\,\mbox{T}}}$.
The width and the asymmetry of the resonance as well as the hysteresis indicate that the resonance is “dragged along” as the excitation frequency is shifted. This can be explained by a three-step process that leads to the buildup of transient charge at the GaAs/(AlGa)As interface, as indicated in Fig. \[fig-1\]a [@Hosoda.1996; @Luttjohann.2005; @Kroner.2008; @Hauck.2014]: (1) Upon resonant excitation, bound electron hole pairs are generated in the dot. (2) These excitons can be field ionized by tunnelling [@Seidl.2005]. The electron escapes to the back contact, while the hole becomes trapped in the triangular well at the GaAs-(AlGa)As interface. (3) The stored, positive charge in that well will slowly drain, e.g. by tunnelling through the (AlGa)As barrier. Process (3) depends on the total number of holes in the well, so that for a given pumping efficiency (process (2)), a specific amount of transient positive charge will be stored at the interface. This charge affects the electric field across the dot, which will slightly detune the resonance with respect to the laser frequency.
The resulting feedback loop (pumping efficiency $\rightarrow$ amount of trapped charge $\rightarrow$ resonance position $\rightarrow$ pumping efficiency) can explain the resonance width and shape as well as the bistability: When the pumping frequency is on the low energy side of the exciton resonance line (red point at zero detuning in Fig. \[fig-1\]c), the feedback will be negative, so that a stable configuration is achieved: A small increase in the number $n_h$ of stored holes will blue-shift the resonance to more positive detuning, as indicated by the arrow. This will reduce the overlap of the exciton line with the laser, reduce the pumping efficiency, and decrease $n_h$ again. Correspondingly, there will be a positive feedback on the high energy side of the resonance, which explains the abrupt drop of the fluorescence on this side. When the laser coincides with the maximum of the exciton line (red point at 30 GHz detuning in Fig. \[fig-1\]c), a small decrease in $n_h$ will red-shift the resonance. This will further reduce the pumping efficiency until the resonance has almost completely shifted out of resonance.
Just below the maximum, there will be two possible stable configurations, as depicted in Fig. \[fig-2\]a. For a more quantitative description of the observed line shape and hysteresis, we model the system by a rate equation approach. The number $n_h$ of trapped holes is given by the interplay between the processes (2) and (3), described by the rates $\gamma_{pump}$ and $\gamma_{leak}$, respectively. $\gamma_{pump}$ is determined by an ionization probability $P_{ionize}$ and the overlap between the laser line and the Lorentzian line shape ${\ensuremath{\mathrm{\mathscr{L}}}}(n_h)$ of the exciton absorption line, with a position that depends on $n_h$. The leakage rate is assumed to be proportional to $n_h$. The resulting differential equation $$\frac{d n_h}{dt}=\gamma_{pump}-\gamma_{leak}= \gamma_{abs}\;{\ensuremath{\mathrm{\mathscr{L}}}}(n_h)\;P_{ionize} -\gamma_{esc} n_h$$ is solved numerically, taking the proportionality factors $\gamma_{abs}$, $P_{ionize}$ and $\gamma_{esc}$ as experimental parameters. For more details on the modelling, see the supplementary material. The stationary solutions of Eq. 1 are shown as a black line in Fig. 2b. Our model can well account for all characteristics of the observed resonances. The hysteresis regime of the fluorescence curves is given by the fold-over region (between $20$ and $30{{\,\mbox{GHz}}}$ in Fig. \[fig-2\]b) in which three stationary solutions for a fixed detuning exist. Only two solutions are stable, corresponding to the observed fluorescence states, and switching occurs when the edge of the fold-over region is reached (solid dots). As already suggested above, the broad, triangular shape of the resonance results from the dragging-along of the resonance.
![[**Explanation of the optical bistability and hysteresis curve.**]{} [**a**]{}, The upper panel shows schematically the resonance line for two different hole gas populations. The blue absorption curve exists for a small hole gas population. The red curve is an example for a higher population, e. g. shifting the resonance to more positive detuning. As a consequence, an excitation laser with fixed frequency (black vertical line) can drive the transition on the high energy side (blue solid dot) and on the low energy side (red solid dot). The resonance florescence signal will be different in this region of bistability and a strong hysteresis is observed (see lower panel for calculated fluorescence curves). [**b**]{}, The measured resonance fluorescence for increasing (red line) and decreasing detuning (blue line) is compared with the stationary solutions of a rate equation model (black line, see supplementary information for details on the model). []{data-label="fig-2"}](Paper-Fig2.pdf){width="0.9\onecolfig"}
So far, the observed phenomena share some characteristics with the recently observed nuclear-spin-induced dragging and the resulting hysteresis [@Latta.2009; @Hogele.2012]. In the present system, however, the feedback is based on photoionization in electric fields rather than magnetic interaction with nuclear spins, which makes the response orders of magnitude more rapid and allows for fast manipulation by externally applied voltage or light pulses. This will be demonstrated in the following.
{width="0.85\twocolfig"}
Figure \[fig-3\]a shows how the resonance fluorescence in the hysteresis region is affected by an interruption of the laser excitation. The quantum dot is first initialized in the high-fluorescence “on”-state by laser irradiation and application of a suitable gate voltage. Then, the gate voltage is switched to the hysteresis region (at time $t= 400{{\,\mbox{ms}}}$ in Fig. \[fig-3\]a). At $600 {{\,\mbox{ms}}}$, the laser excitation is interrupted for a short, variable time span $\Delta t$. Subsequently, it is recorded whether the system is in the “on”-state (blue curve) or the “off”-state (red curve). In Fig. \[fig-3\]b, the “on”-state probabilities are plotted as a function of the interruption time $\Delta t$. We observe a latency time (time required for the “on”-state probability to drop below 50%) of roughly $22{{\,\mbox{ms}}}$.
Similarly, the fluorescence can be switched on by optical means, using a second laser. Laser 1 is set to the hysteresis region and the QD is initialized in the “off”-state (Fig. \[fig-3\]c). Then, for a short pumping time $\Delta t$, laser 2 is turned on, which is resonant with the initial exciton frequency. This populates the hole gas and shifts the transition to higher frequencies, closer to laser 1 in the hysteresis region. Depending on the length of the additional laser pulse, the system will afterwards be in the “off” or “on”-state with respect to laser 1 (red and blue traces in Fig. \[fig-3\]c, respectively). Again, a latency period can be observed, which is $70{{\,\mbox{ms}}}$ for the present experimental conditions.
The time dependence of the switch-off process is given by the leakage rate of the hole gas (process (3) in Fig. 1a). Note, however, that for very short interruption times $\Delta t$, the negative feedback mechanism will ensure a fast recovery of the RF signal. Only for sufficiently long $\Delta t$, the resonance position will have shifted far enough that a recovery is no longer possible. This explains the observed latency time. Correspondingly, the observed latency period of the switch-on process reflects the pumping rate of the hole gas by exciton generation and field ionization. It is worth noting that the particular characteristic times strongly depend on the excitation power and the position of the laser frequency within the bistability region.
The operation principle of our feedback loop, which is based on simple charge transport and therefore offers great tunability, opens up a broad variety of possibilities. Similar to the proposal based on the nuclear dragging effect, the charge driven feedback loop may be used for stabilization of the QD resonance position. Also, the bistability may be useful for optical memory applications, such as switching the fluorescence signal from the “off” to the “on”-state when a short optical signal is received and holding that state for later read-out.
Methods {#methods .unnumbered}
=======
Sample growth and processing {#sample-growth-and-processing .unnumbered}
----------------------------
The investigated quantum dots were grown using the now well-established Stranski-Krastanov growth mode and consist of InAs, embedded in a GaAs/(AlGa)As field-effect-transistor-like heterostructure [@Warburton.2000]. The electrically and optically active part of the sample is sketched in Fig. \[fig-1\]a. It consists (along the growth direction from right to left) of a highly doped, metallic GaAs back contact, a GaAs-(AlGa)As tunnelling barrier, the quantum dot layer, a GaAs spacer layer, and an (AlGa)As blocking layer. In a pre-growth etching process the sample was patterned into cylindrical mesas of $\sim18.6{{\,\mbox{$\mu$m}}}$ diameter [@Strittmatter.2012; @Strittmatter.2012b]. Lithographically defined contacts on top of a single mesa allow for controlling the electric field at the quantum dots by applying a gate voltage. A more detailed description of the layer sequence and the growth and patterning processes can be found in the supplementary information.
Optical measurements
--------------------
The optical response of a single quantum dot was probed by resonance fluorescence spectroscopy (see Refs. [@Flagg.2009; @Kuhlmann.2013]). Resonance between the fixed laser frequency and the excitonic transition in the dot was achieved by applying a voltage between a gate electrode on top of the heterostructure and the back contact. The resulting change in electric field across the quantum dot leads to a linear Stark shift of the exciton transition [@Warburton.2002]. This shift can be calibrated using different laser frequencies and is given in GHz. All experiments were carried out with the sample placed in a helium bath cryostat at $4.2{{\,\mbox{K}}}$. A confocal dark-field microscope was used to focus the beam of a tunable diode laser onto a single dot and detect its resonance fluorescence with an avalanche photodiode (APD). To increase the collection efficiency, a zirconia solid immersion lens was placed on top of the sample and centered above a mesa. By the use of superconducting magnetic coils inside the bath cryostat parts of the experiments were performed at a magnetic field of $4.5{{\,\mbox{T}}}$ along the sample growth direction. For time-resolved measurements, a function generator was used to apply voltage pulses either directly at the gate contact or to the driver of an acusto-optic modulator, switching a laser beam on and off. The counts of the APD were then binned by a quTAU time-to-digital converter triggered by the pulse generator.
Authors contribution
====================
B. M. and A. K. carried out the experiments. B. M. implemented the model and performed the calculations. J.-H. S. and A. S. grew the samples. B. M., A. K., M. G. and A. L. planned the experiments and wrote the manuscript.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Stein kernels are a way of comparing probability distributions, defined via integration by parts formulas. We provide two constructions of Stein kernels in free probability. One is given by an explicit formula, and the other via free Poincaré inequalities. In particular, we show that unlike in the classical setting, free Stein kernels always exist. As corollaries, we derive new bounds on the rate of convergence in the free CLT, and a strengthening of a characterization of the semicircular law due to Biane.'
author:
- 'Guillaume Cébron[^1], Max Fathi[^2] and Tobias Mai[^3]'
title: A note on existence of free Stein kernels
---
MSC: 46L54; 60F05.
Introduction
============
Let $M$ be a von Neumann algebra and $\varphi$ a faithful normal state. Let $\mathcal{P} = \mathbb{C}\langle t_1,\dots,t_n \rangle$ be the algebra of noncommutative polynomials in $n$ variables $t_1,\dots,t_n$.
We are interested in Stein kernels, which were defined in [@FN17] as follows:
A free Stein kernel for a $n$-tuple $X$, with respect to a potential $V \in \mathcal{P}$, is an element $A \in L^2(M_n(M \bar{\otimes} M^{op}, (\varphi \otimes \varphi^{op})\circ \operatorname{Tr}))$ such that for any $P \in \mathcal{P}^n$ we have $$\langle [\mathcal{D}V](X), P(X)\rangle_{\varphi} = \langle A, [\mathcal{J}P](X) \rangle_{(\varphi \otimes \varphi^{op})\circ \operatorname{Tr}}.$$ The free Stein discrepancy of $X$ relative to $V$ is defined as $$\Sigma^*(X|V) := \inf_A \|A - (1 \otimes 1^{op})\otimes I_n\|_{L^2(M_n(M \bar{\otimes} M^{op}, (\varphi \otimes \varphi^{op})\circ \operatorname{Tr}))}$$ where the infimum is taken over all admissible Stein kernels $A$ of $X$.
In this definition, $\mathcal{D}V$ is the cyclic gradient of the potential $V$, and $\mathcal{J}P$ is the Jacobian matrix of $P$. We shall recall the precise definitions of these notions in Section 2.
The motivation behind this definition is that $(1 \otimes 1^{op})\otimes I_n$ is a Stein kernel for $X$ relative to $V$ if and only if $X$ is a free Gibbs state with potential $V$. Hence these kernels and the discrepancy give an estimate of how far away $X$ is from being this Gibbs state. When $V$ is the quadratic potential $\frac{1}{2}\sum_{i=1}^n t_i^2$, this corresponds to the situation where the reference state is an $n$-dimensional semicircular law. Stein kernels have been used to establish functional inequalities, and to estimate rates of convergence in approximation theorems, such as the free central limit theorem. Their classical counterpart are one way of implementing the now widely used Stein’s method [@Ste72; @Ros11] to bound distances between probability distributions. Another way of implementing Stein’s method that has recently been adapted to the context of free probability is to use it in combination with Malliavin calculus [@KNPS].
Instead of working with polynomials, we could consider power series $\mathcal{P}^{(R)}$, as in [@FN17]. All the results proven below remain valid in that generality, by approximation with polynomials.
An immediate constraint is that for a Stein kernel to exist, a necessary condition is that $\varphi([\mathcal{D}V](X)) = (0,\dots,0)$, since otherwise we would get a contradiction by testing the above identity with constant polynomials. We shall show that this is also a sufficient condition:
Given any potential $V\in \mathcal{P}$ such that $\varphi([\mathcal{D}V](X)) = (0,\dots,0)$ and any self-adjoint $n$-tuple $X = (x_1,\dots,x_n) \in M^n$, there exists a Stein kernel for $X$ relative to $V$.
This result stands in contrast with the situation in the classical setting, where Stein kernels may not exist, and known constructions typically require some regularity condition [@Cha09; @CFP17; @NP12].
We shall give two constructions that work in full generality: one explicit construction, with an exact formula, and one implicit, via the Riesz representation theorem. Interestingly, the two constructions do *not* give the same kernels, which in particular implies that free Stein kernels are never unique. In the classical setting, Stein kernels are not generally unique in dimension higher than two, but are unique for one-dimensional measures with nice density.
An immediate consequence of our constructions will be the following rate of convergence in the free CLT, obtained via Theorem 5.2 of [@FN17]:
Let $(X^{(i)})_{i \in {\mathbb{N}}}$ be a sequence of freely independent, identically distributed $n$-tuples of self-adjoint random variables. Assume that each of them are centered, and that the covariance matrix of $X = (x_1,\dots,x_n)$ is the identity. Let $Y^k = k^{-1/2}\left(\sum_{i=1}^{k} X^{(i)}\right)$. We have $$\Sigma^*\left(Y^k\left|\frac{1}{2}\sum_{i=1}^n t_i^2 \right.\right) \leq \frac{C}{\sqrt{k}}$$ with $C=\sqrt{n \min\left(C_{opt}(X)-1,(n+nm_4-1)/2\right)}$ (where $m_4$ is the maximum fourth moment $\max_{i=1}^n\varphi(x_i^4)$ and $C_{opt}(X)$ is the Poincaré constant of $X$). If moreover the non-microstates free Fisher information of $X$ is finite, then the non-microstates free relative entropy $\chi^*$ satisfies $$\chi^*\left(Y^k\left|\frac{1}{2}\sum_{i=1}^n t_i^2 \right.\right) \leq \frac{C\log k}{k}$$ with a constant $C$ that only depends on the free Fisher information and the norm of $X$.
As an immediate corollary obtained via a comparison to the Stein discrepancy proved in [@C18 Section 2.5], we obtain the rate of convergence $W_2(Y^k,S)\leq C \cdot k^{-1/2}$ for the non-commutative $2$-Wasserstein distance of [@BV01] from $Y^k$ to a $n$-tuple $S$ of free standard semicircular variables. Interestingly, obtaining such rates in classical probability under boundedness or moment assumptions is much more difficult, since Stein kernels may not exist [@Bon16; @Zha17]. For one-dimensional free probability, a sharper rate, without the logarithmic factor, has been obtained in [@CG13]. A rate of convergence at the level of the Cauchy transforms of polynomial evaluations was obtained in [@MS13]; see [@BM18] for an improved result.
Explicit formula
================
In this section, we want to prove existence of free Stein kernels for an arbitrary potential by providing an explicit formula. This requires some notation.
The unital complex algebra $\mathcal{P} = \mathbb{C}\langle t_1,\dots,t_n\rangle$ becomes a $\ast$-algebra if it is endowed with the involution $\ast:\mathcal{P}\to \mathcal{P}$, that is the unital antilinear map uniquely determined by $(PQ)^*=Q^*P^*$ and $t_i^*=t_i$ for $i=1,\dots,n$. This map naturally induces an involution $\ast: \mathcal{P} \otimes \mathcal{P} \to \mathcal{P} \otimes \mathcal{P}$ on the algebraic tensor product $\mathcal{P}\otimes \mathcal{P}$ over $\mathbb{C}$ by $(P\otimes Q)^* = P^*\otimes Q^*$.
Furthermore, we define on $\mathcal{P} \otimes \mathcal{P}$ the binary operation $$\sharp:\ (\mathcal{P} \otimes \mathcal{P})^2 \to \mathcal{P} \otimes \mathcal{P},\qquad (P_1 \otimes P_2) \sharp (Q_1 \otimes Q_2) = (P_1Q_1) \otimes (Q_2P_2),$$ as well as the *multiplication mapping* $$m:\ \mathcal{P} \otimes \mathcal{P} \to \mathcal{P}, \qquad P \otimes Q \mapsto PQ,$$ and the *flip mapping* $$\sigma:\ \mathcal{P} \otimes \mathcal{P} \to \mathcal{P} \otimes \mathcal{P},\qquad P\otimes Q \mapsto Q\otimes P.$$
We may view $\mathcal{P}\otimes \mathcal{P}$ as an $\mathcal{P}$-$\mathcal{P}$-bimodule via the natural actions that are determined by $P_1\cdot (Q_1\otimes Q_2)\cdot P_2= (P_1 Q_1)\otimes (Q_2 P_2)$. Accordingly, we can consider derivations $d$ on $\mathcal{P}$ with values in $\mathcal{P}\otimes \mathcal{P}$, i.e., linear maps $d: \mathcal{P} \to \mathcal{P} \otimes \mathcal{P}$ that satisfy the *Leibniz rule* $$d(PQ) = d(P) \cdot Q + P \cdot d(Q) \qquad\text{for all $P,Q\in\mathcal{P}$}.$$ Based on this terminology, we may introduce the *non-commutative derivatives* $\partial_1,\dots,\partial_n$ as the unique derivations $$\partial_i:\ \mathcal{P} \to \mathcal{P} \otimes \mathcal{P},\qquad i=1,\dots,n,$$ that satisfy $\partial_i(t_j) = \delta_{ij} 1 \otimes 1$ for $j=1,\dots,n$. Further, we will work with the linear map $$\delta:\ \mathcal{P} \to \mathcal{P} \otimes \mathcal{P}, \qquad P \mapsto P\otimes 1 - 1 \otimes P,$$ which is easily checked to be a derivation as well. Note that we can write $\delta(P)=[P,1\otimes 1]$ with respect to the $\mathcal{P}$-$\mathcal{P}$-bimodule structure of $\mathcal{P}\otimes \mathcal{P}$. From their definitions, one easily infers that those derivations are related by the formula $$\label{eq:Voi-formula}
\delta(P) = \sum_{i=1}^n \partial_i(P) \sharp \delta(t_i) \qquad\text{for all $P\in\mathcal{P}$}.$$ We point out that this formula underlies the proof of Proposition \[prop:Poincare\] below.
For any given $P=(P_1,\dots,P_n) \in \mathcal{P}^n$, the *Jacobian matrix* $\mathcal{J}P$ of $P$ is defined by $$\mathcal{J}P=\left(\begin{array}{ccc}
\partial_1 P_1 & \cdots& \partial_n P_1 \\
\vdots & \ddots & \vdots \\
\partial_1 P_n & \cdots & \partial_n P_n
\end{array} \right) \in M_n(\mathcal{P} \otimes \mathcal{P}).$$
Associated to the noncommutative derivatives are the so-called *cyclic derivatives*; those are the linear maps $$\mathcal D_i:\ \mathcal{P} \to \mathcal{P},\qquad i=1,\dots,n,$$ that are defined by $\mathcal D_i := m\circ \sigma \circ \partial_i$. For any given potential $V \in \mathcal{P}$, we denote by $\mathcal{D}V$ the *cyclic gradient* of $V$ which is defined as $$\mathcal{D}V = (\mathcal{D}_1V, \ldots, \mathcal{D}_nV) \in \mathcal{P}^n$$
For every $n$-tuple $X=(x_1,\dots,x_n)$ of self-adjoint operators in the von Neumann algebra $M$, we have a canonical evaluation homomorphism ${\operatorname{ev}}_X: \mathcal{P} \to M$ that is determined by ${\operatorname{ev}}_X(1)=1$ and ${\operatorname{ev}}_X(t_i) = x_i$ for $i=1,\dots,n$; we put $P(X) := {\operatorname{ev}}_X(P)$. Analogously, we define $Q(X) := ({\operatorname{ev}}_X \otimes {\operatorname{ev}}_X)(Q)$ for every $Q\in \mathcal{P} \otimes \mathcal{P}$. Even though by definition $Q(X) \in M\otimes M$, we will often view $Q(X)$ as an element in $M \otimes M^{op}$; note that then $(Q_1 \sharp Q_2)(X) = Q_1(X) \cdot Q_2(X)$ for all $Q_1,Q_2\in \mathcal{P} \otimes \mathcal{P}$. Similarly, we can evaluate elements in $\mathcal{P}^n$ and $M_n(\mathcal{P} \otimes \mathcal{P})$.
On the complex vector spaces $M^n$ and $M_n(M \otimes M^{op})$, we may introduce the inner products $\langle \cdot,\cdot \rangle_{\varphi}$ and $\langle \cdot,\cdot\rangle_{(\varphi \otimes \varphi^{op})\circ \operatorname{Tr}}$ by $$\langle X,Y\rangle_{\varphi}=\sum_{i=1}^n\varphi(x_i y_i^*)$$ for all $X=(x_1,\ldots, x_n), Y=(y_1,\ldots, y_n) \in M^n$ and by $$\langle A,B\rangle_{(\varphi \otimes \varphi^{op})\circ \operatorname{Tr}}=\sum_{i,j=1}^n(\varphi \otimes \varphi^{op})(a_{i,j} \cdot b_{i,j}^*)$$ for all $A=(a_{i,j}),B=(b_{i,j})\in M_n(M \otimes M^{op})$, respectively. Note that $$\langle A,B\rangle_{(\varphi \otimes \varphi^{op})\circ \operatorname{Tr}}=(\varphi \otimes \varphi^{op})\circ \operatorname{Tr}(A \cdot B^*)$$ where $B^*$ is given by $B^*=(b_{j,i}^*)_{i,j=1}^n$.
Now, we are ready to formulate the announced theorem which gives a free Stein kernel by an explicit formula. This is inspired by an unpublished note of Giovanni Peccati and Roland Speicher, where $\frac{1}{2}(X^2 \otimes 1 + 1 \otimes X^2 - 2 X \otimes X)$ was given as an expression for a free Stein kernel in the single variable case $n=1$ and for the special potential $V=\frac{1}{2}t^2$.
Given any $V\in \mathcal{P}$, the matrix $$A :=\frac{1}{2}\Big(\delta(\mathcal{D}_iV)\sharp [\delta(t_j)]\Big)_{i,j=1}^n\in M_n(\mathcal{P} \otimes \mathcal{P})$$ satisfies $$\langle [\mathcal{D}V](X)-\varphi([\mathcal{D}V](X)), P(X)\rangle_{\varphi} = \langle A(X), [\mathcal{J}P](X) \rangle_{(\varphi \otimes \varphi^{op})\circ \operatorname{Tr}}.$$ In particular, if $\varphi([\mathcal{D}V](X)) = (0,\ldots,0)$, then $A(X)\in M_n(M \otimes M^{op})$ is a free Stein kernel for $X$ relative to $V$.
Moreover, if $V=\frac{1}{2}\sum_{i=1}^n t_i^2$ and $X$ is centered, then $$\Sigma^*(X|V)^2 \leq \|A - (1 \otimes 1^{op})\otimes I_n\|_{L^2(M_n(M \bar{\otimes} M^{op}, (\varphi \otimes \varphi^{op})\circ \operatorname{Tr}))}\leq (n^2+n^2 m_4-n)/2.$$
Let $P\in \mathcal{P}$ be given. Remark that, in $M_n(M \otimes M^{op})$, $$A(X)\cdot ([\mathcal{J}P](X))^*=[A\sharp (\mathcal{J}P)^*](X),$$ where $A\sharp (\mathcal{J}P)^*$ is the element of $M_n(\mathcal{P} \otimes\mathcal{P})$ whose $(i,j)$-coefficient is $$\begin{aligned}
[A\sharp (\mathcal{J}P)^*)]_{i,j}&=\sum_{k=1}^n\frac{1}{2}\Big(\delta(\mathcal{D}_iV)\sharp [\delta(t_k)]\Big)\sharp \Big((\partial_k P_j)^*\Big)\\
&=\sum_{k=1}^n\frac{1}{2}\delta(\mathcal{D}_iV)\sharp \Big( [\delta(t_k)]^*\sharp (\partial_k P_j)^*\Big) =\sum_{k=1}^n\frac{1}{2}\delta(\mathcal{D}_iV)\sharp \Big( \big( (\partial_k P_j) \sharp [\delta(t_k)]\big)^*\Big)\\
&=\frac{1}{2}\delta(\mathcal{D}_iV)\sharp \bigg(\Big( \sum_{k=1}^n (\partial_k P_j) \sharp [\delta(t_k)] \Big)^*\bigg) \stackrel{\eqref{eq:Voi-formula}}{=} \frac{1}{2}\delta(\mathcal{D}_iV)\sharp \Big( \delta(P_j)^*\Big).\end{aligned}$$ If $i=j$, we can pursue the computation of the diagonal $(i,i)$-coefficient, namely $$\begin{aligned}
[A\sharp (\mathcal{J}P)^*)]_{i,i}&=\frac{1}{2}(\mathcal{D}_iV\otimes 1-1\otimes\mathcal{D}_iV)\sharp ( P_i^*\otimes 1-1\otimes P_i^* )\\
&=\frac{1}{2}\Big((\mathcal{D}_iV)P_i^*\otimes 1-P_i^*\otimes (\mathcal{D}_iV)-(\mathcal{D}_iV)\otimes P_i^* +1\otimes (\mathcal{D}_iV)P_i^*\Big),\end{aligned}$$ from which we deduce that $$(\varphi \otimes \varphi^{op})\left([A\sharp (\mathcal{J}P)^*)]_{i,i}(X)\right)=\varphi\Big( \big([\mathcal{D}_i V](X)-\varphi([\mathcal{D}_i V](X)) \big) P_i^*(X) \Big).$$ Using this observation, we may now check that $$\begin{aligned}
\langle A(X), &[\mathcal{J}P](X) \rangle_{(\varphi \otimes \varphi^{op})\circ \operatorname{Tr}}=(\varphi \otimes \varphi^{op})\circ \operatorname{Tr}(A(X)\cdot ([\mathcal{J}P](X))^*)\\
&=\sum_{i=1}^n(\varphi \otimes \varphi^{op})\big([A\sharp (\mathcal{J}P)^*)]_{i,i}(X)\big)\\
&=\sum_{i=1}^n \varphi\Big( \big([\mathcal{D}_i V](X)-\varphi([\mathcal{D}_i V](X))\big) P_i^*(X)\Big)\\
&=\langle [\mathcal{D}V](X)-\varphi([\mathcal{D}V](X)), P(X)\rangle_{\varphi} \end{aligned}$$ which is the asserted formula. Now, if $V=\frac{1}{2}\sum_{i=1}^n t_i^2$ and $X$ is centered, we have $$A(X)=\frac{1}{2}\Big((x_i\otimes 1-1\otimes x_i)\cdot (x_j\otimes 1-1\otimes x_j)\Big)_{i,j=1}^n$$ and consequently, $$\begin{aligned}
&\|A - (1 \otimes 1^{op})\otimes I_n\|^2_{L^2(M_n(M \bar{\otimes} M^{op}, (\varphi \otimes \varphi^{op})\circ \operatorname{Tr}))}\\&=\|A\|^2_{L^2(M_n(M \bar{\otimes} M^{op}, (\varphi \otimes \varphi^{op})\circ \operatorname{Tr}))}-n\\
&=\frac{1}{4}\sum_{i,j=1}^n\Big(2\varphi(x_ix_j^2x_i)+2\varphi(x_ix_j)\varphi(x_jx_i)++2\varphi(x_j^2)\varphi(x_i^2)\Big)-n\\
&\leq \frac{1}{2}(n^2 m_4+n+n^2)-n=(n^2+n^2 m_4-n)/2,\end{aligned}$$ which concludes the proof.
Construction via a free Poincaré inequality
===========================================
In this section, we adapt the construction of [@CFP17] to the free setting. To state our results, we first define free Poincaré inequalities:
A self-adjoint $n$-tuple $X$ satisfies a free Poincaré inequality with constant $C$ is for any polynomial $P \in \mathcal{P}$ we have $$\|P(X) - \varphi(P(X))\|_2^2 \leq C\underset{i=1}{\stackrel{n}{\sum}} \hspace{1mm} \|\partial_i P(X)\|_2^2.$$ The best possible constant in this inequality will be denoted by $C_{opt}(X)$.
In [@Bia03], Biane showed that the best constant in the Poincaré inequality for a semicircular law is one, and that this property characterizes it among all $n$-tuples with covariance matrix equal to the identity. We shall later see a refinement of this result. The Poincaré constant is always finite, and satisfies the following bound:
\[prop:Poincare\] Consider a self-adjoint $n$-tuple $X=(x_1,\dots,x_n)$. For any self-adjoint $P \in \mathcal{P}$ we have $$\|P(X) - \varphi(P(X))\|_2^2 \leq 2n\|X\|^2\underset{i=1}{\stackrel{n}{\sum}} \hspace{1mm} \|\partial_i P(X)\|_2^2,$$ where $\|X\| := \max^n_{i=1} \|x_i\|$. If $P$ is not self-adjoint, we have $$\|P(X) - \varphi(P(X))\|_2^2 \leq 4n\|X\|^2\underset{i=1}{\stackrel{n}{\sum}} \hspace{1mm} \|\partial_i P(X)\|_2^2,$$ so that $C_{opt}(X) \leq 4n\|X\|^2$.
The first part is a result of Voiculescu, which can be found in [@Dab10; @MS17]. The second part is a trivial extension to non self-adjoint polynomials: if we consider a general $P$, it can be written as $P = P_1 + iP_2$. Then $$\begin{aligned}
\|P(X) - \varphi(P(X))\|_2^2 &\leq 2\|P_1(X) - \varphi(P_1(X))\|_2^2 + 2\|P_2(X) - \varphi(P_2(X))\|_2^2 \\
&\leq 4n\|X\|^2\underset{j=1}{\stackrel{n}{\sum}} \hspace{1mm} \|\partial_j P_1(X)\|_2^2 + \|\partial_j P_2(X)\|_2^2 \\
&= 4n\|X\|^2\underset{j=1}{\stackrel{n}{\sum}} \hspace{1mm} \|\partial_j P(X)\|_2^2.\end{aligned}$$
We note that if $\varphi$ is tracial, then the bound on $C_{opt}(X)$ given in Proposition \[prop:Poincare\] can be improved to $C_{opt}(X) \leq 2n\|X\|^2$, with the same proof based on as in the self-adjoint case.
The construction we shall now see will lead to the following estimates:
\[thm\_stein\_const2\] Assume that $\varphi([\mathcal{D}V](X)) = 0$. Then there exists a free Stein kernel $A$ for $X$ relative to $V$, and moreover it satisfies $$\Sigma^*(X|V)^2 \leq \|A - (1 \otimes 1^{op})\otimes I_n\|_{(\varphi \otimes \varphi^{op})\circ \operatorname{Tr}}^2 \leq n + C_{opt}(X)\|[\mathcal{D}V](X)\|_2^2 - 2\langle [\mathcal{D}V](X), X\rangle_{\varphi}.$$
In particular, if $X=(x_1,\dots,x_n)$ is centered and satisfies $\sum^n_{i=1} \varphi(x_i^2) = n$, then we have $$\Sigma^*\left(X \left|\frac{1}{2}\sum_{i=1}^n t_i^2\right.\right)^2 \leq n(C_{opt}(X)-1).$$
Before giving the actual construction, it will be convenient to introduce the free Sobolev space $H^1(\mu)$ associated to a noncommutative distribution $\mu: \mathcal{P} \to \mathbb{C}$. We can think of it as the distribution associated to a $n$-tuple of variables $\mu: \mathcal{P} \to \mathbb{C}, P \mapsto \varphi(P(X))$. We can define on $\mathcal{P}^n$ a sesqui-linear form $\langle \cdot,\cdot \rangle_{H^1(\mu)}$ by $$\langle P, Q \rangle_{H^1(\mu)} := (\mu\otimes\mu)\circ \operatorname{Tr}\big((\mathcal{J}Q)^\ast \sharp (\mathcal{J}P)\big) \qquad\text{for $P,Q\in\mathcal{P}^n$}.$$ The latter then induces an inner product on the quotient space $\mathcal{P}^n / \mathcal{N}_\mu$, which we denote again by $\langle \cdot,\cdot\rangle_{H^1(\mu)}$, where $\mathcal{N}_\mu := \{ P\in \mathcal{P}^n \mid \langle P, P\rangle_{H^1(\mu)} = 0\}$. Unlike in the classical setting, there may be nonconstant tuples of polynomials in $\mathcal{N}_\mu$, since due to relations between the $x_i$ the Jacobian matrix $[\mathcal{J}P](X)$ may be zero. The Hilbert space obtained by completing $\mathcal{P}^n / \mathcal{N}_\mu$ with respect to the norm $\|\cdot\|_{H^1(\mu)}$ induced by $\langle \cdot,\cdot \rangle_{H^1(\mu)}$ is then denoted by $H^1(\mu)$.
In view of the classical setting, it would be more natural to define the scalar product on $H^1(\mu)$ as $\mu(Q^\ast P) + (\mu\otimes\mu)\circ \operatorname{Tr}\big((\mathcal{J}Q)^\ast \sharp (\mathcal{J}P)\big)$. Because of the Poincaré inequality, the two norms would be equivalent on $\mathcal{P}^n / \mathcal{N}_\mu$, and for our purpose the definition we gave above is better adapted.
Let $\mu$ be the distribution associated to the $X_i$. The application $$f_{V,X}:\ P \mapsto \langle [\mathcal{D}V](X), P(X)\rangle_{\varphi}$$ is a linear form over $\mathcal{P}^n$. We shall show that there exists a constant $C>0$ such that $$\label{eq:functional-bound}
|f_{V,X}(P)| \leq C\|P\|_{H^1(\mu)} \qquad\text{for any $P \in \mathcal{P}^n$},$$ where $\|\cdot\|_{H^1(\mu)}$ stands for the semi-norm induced by $\langle \cdot,\cdot \rangle_{H^1(\mu)}$ on $\mathcal{P}^n$. Once we will have proved this, we may infer from the definition of $\mathcal{N}_\mu$ that $f_{V,X}$ is invariant by adding an element of $\mathcal{N}_\mu$, so that we can view it as a linear form on the quotient space $H^1(\mu)$; in fact, we see that this yields a continuous form on $H^1(\mu)$. Thus, we shall be able to apply the Riesz representation theorem: there exists a unique element $Q_0$ of $H^1(\mu)$ such that for any element $P$ of $H^1(\mu)$ we have $$f_{V,X}(P) = \langle Q_0, P \rangle_{H^1(\mu)}.$$ The map $Q\mapsto [\mathcal{J}Q](X)$ is an isometry from $\mathcal{P}^n / \mathcal{N}_\mu$ to $L^2(M_n(M \bar{\otimes} M^{op}, (\varphi \otimes \varphi^{op})\circ \operatorname{Tr}))$ which extends to $H^1(\mu)$, and denoting by $[\mathcal{J}Q_0](X)$ the image of $Q_0$ via this isomorphism, we have $$f_{V,X}(P) = \langle [\mathcal{J}Q_0](X), [\mathcal{J}P](X) \rangle_{(\varphi \otimes \varphi^{op})\circ \operatorname{Tr}}$$ for any element $P$ of $\mathcal{P}^n$ and hence $[\mathcal{J}Q_0](X)$ would be a free Stein kernel for $X$, relative to $V$.
So all that is left to show is . As pointed out in [@CFP17], this continuity can be proved when a Poincaré inequality holds. Indeed, if we consider an element $P = (P_1,\dots,P_n) \in \mathcal{P}^n$, then we have, by Proposition \[prop:Poincare\] and since $\varphi([\mathcal{D}V](X)) = (0,\dots,0)$ by assumption, $$\begin{aligned}
|f_{V,X}(P)| &= |\langle [\mathcal{D}V](X), P(X) - \varphi(P(X))\rangle_{\varphi}|\\
&\leq \|[\mathcal{D}V](X)\|_2 \sqrt{\sum_i \|P_i(X) - \varphi(P_i(X))\|_2^2} \\
&\leq \|[\mathcal{D}V](X)\|_2\sqrt{C_{opt}(X) \sum_{i,j} \|\partial_j P_i(X)\|_2^2} \\
&= \sqrt{C_{opt}(X)}\|[\mathcal{D}V](X)\|_2 \times \|P\|_{H^1(\mu)},\end{aligned}$$ which establishes with $C = \sqrt{C_{opt}(X)}\|[\mathcal{D}V](X)\|_2$.
This implies existence of a Stein kernel $A = [\mathcal{J}Q_0](X)$, and moreover it satisfies $\|A\|_2^2 \leq C_{opt}(X)\|[\mathcal{D}V](X)\|_2^2$. Expanding the square in the definition of the discrepancy then yields $$\Sigma^*(X|V)^2 \leq n + C_{opt}(X)\|[\mathcal{D}V](X)\|_2^2 - 2\langle [\mathcal{D}V](X), X\rangle_{\varphi}.$$
Note that the construction is inherently different than the one from Section 2: while this one is the Jacobian of an element of $H^1(\mu)$, the first one is not. In particular, since projecting on the $L^2$-closure of the subspace $[\mathcal{J}(\mathcal{P}^n)](X) = \{[\mathcal{J} P](X) \mid P\in \mathcal{P}^n\}$ reduces the norm, the second one has the smallest possible norm, and hence gives rise to the infimum in the definition of the Stein discrepancy. More generally, two free Stein kernels for the same potential can differ only by an element in $[\mathcal{J}(\mathcal{P}^n)](X)^\bot \subset L^2(M_n(M \bar{\otimes} M^{op}, (\varphi \otimes \varphi^{op})\circ \operatorname{Tr}))$.
An interesting immediate corollary is the following reinforcement of Biane’s characterization of the semicircular law. It is a free counterpart of a result of [@CFP17] on Gaussian distributions.
\[cor:Poincare-characterization\] Let $X=(x_1,\dots,x_n)$ be an $n$-tuple of centered self-adjoint variables, and assume it satisfies $\sum^n_{i=1} \varphi(x_i^2) = n$. Then $$C_{opt}(X) \geq 1 + \frac{\Sigma^*\left(X| \frac{1}{2}\sum_{i=1}^n t_i^2\right)^2}{n}.$$
Hence not only the semicircular law is the only isotropic distribution with Poincaré constant equal to one, but among all such distributions, if the Poincaré constant is close to one, then the variable must be close to a semicircular law. In fact, as Corollary \[cor:Poincare-characterization\] shows, this holds not only for isotropic distributions but more generally for distributions that come from $n$-tuples $X=(x_1,\dots,x_n)$ of centered self-adjoint variables satisfying $\sum^n_{i=1} \varphi(x_i^2) = n$.
: G. C. and M. F. were partly supported by the Project MESA (ANR-18-CE40-006) of the French National Research Agency (ANR). M. F. was also partly supported by Project EFI (ANR-17-CE40-0030) and ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02.
T. M. was supported by the ERC Advanced Grant NCDFP (339760) held by Roland Speicher.
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[^1]: Institut de Mathématiques de Toulouse, Université de Toulouse, [email protected]
[^2]: CNRS and Institut de Mathématiques de Toulouse, Université de Toulouse, [email protected]
[^3]: Faculty of Mathematics, Saarland University, [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Studies of nucleosynthesis in neutrino-driven winds from nascent neutron stars show that the elements from Sr through Ag with mass numbers $A\sim 88$–110 are produced by charged-particle reactions (CPR) during the $\alpha$-process in the winds. Accordingly, we have attributed all these elements in stars of low metallicities (${\rm [Fe/H]}\lesssim -1.5$) to low-mass and normal supernovae (SNe) from progenitors of $\sim 8$–$11\,M_\odot$ and $\sim 12$–$25\,M_\odot$, respectively, which leave behind neutron stars. Using this rule and attributing all Fe production to normal SNe, we previously developed a phenomenological two-component model, which predicts that ${\rm [Sr/Fe]}\geq -0.32$ for all metal-poor stars. The high-resolution data now available on Sr abundances in Galactic halo stars show that there is a great shortfall of Sr relative to Fe in many stars with ${\rm [Fe/H]}\lesssim -3$. This is in direct conflict with the above prediction. The same conflict also exists for two other CPR elements Y and Zr. The very low abundances of Sr, Y, and Zr observed in stars with ${\rm [Fe/H]}\lesssim -3$ thus require a stellar source that cannot be low-mass or normal SNe. We show that this observation requires a stellar source leaving behind black holes and that hypernovae (HNe) from progenitors of $\sim 25$–$50\,M_\odot$ are the most plausible candidates. Pair-instability SNe from very massive stars of $\sim 140$–$260\,M_\odot$ that leave behind no remnants are not suitable as they are extremely deficient in producing the elements of odd atomic numbers such as Na, Al, K, Sc, V, Mn, and Co relative to the neighboring elements of even atomic numbers, but this extreme odd-even effect is not observed in the elemental abundance patterns of metal-poor stars.
If we expand our previous phenomenological two-component model to include three components (low-mass and normal SNe and HNe) and use for example, the observed abundances of Ba, Sr, and Fe to separate the contributions from these components, we find that essentially all of the data are very well described by the new model. This model provides strong constraints on the evolution of \[Sr/Fe\] with \[Ba/Fe\] in terms of the allowed domain for these abundance ratios. This model also gives an equally good description of the data when any CPR element besides Sr (e.g., Y or Zr) or any heavy $r$-process element besides Ba (e.g., La) is used. As the stars deficient in Sr, Y, and Zr are dominated by contributions from HNe, they define the self-consistent yield pattern of that hypothecated source. This inferred HN yield pattern for the low-$A$ elements from Na through Zn ($A\sim 23$–70) including Fe is almost indistinguishable from what we had previously attributed to normal SNe. As HNe are plausible candidates for the first generation of stars and are also known to be ongoing in the present epoch, it is necessary to re-evaluate the extent to which normal SNe are substantial contributors to the Fe inventory of the Galaxy. We conclude that HNe are important contributors to the abundances of the low-$A$ elements over the history of the universe. We estimate that they contributed $\sim 24\%$ of the bulk solar Fe inventory while normal SNe contributed only $\sim 9\%$ (not the usually assumed $\sim 33\%$). This implies a greatly reduced role of normal SNe in the chemical evolution of the low-$A$ elements.
author:
- 'Y.-Z. Qian and G. J. Wasserburg'
title: 'Abundances of Sr, Y, and Zr in Metal-Poor Stars and Implications for Chemical Evolution in the Early Galaxy'
---
Introduction
============
In this paper we consider that the elements from Sr through Ag in metal-poor stars represent the products of nucleosynthesis in neutrino-driven winds from forming neutron stars. This approach allows us to obtain information on the stellar sources that contributed to the chemical enrichment of the interstellar medium (ISM) in the Galaxy and the intergalactic medium (IGM) at early and recent times. We previously proposed a phenomenological two-component model (@qw07; hereafter QW07) to account for the abundances of heavy elements in metal-poor stars. That model focused on the elements commonly considered to be produced by the generic “$r$-process”. It specifically attributed all the elements from Sr through Ag in metal-poor stars to the charged-particle reactions (CPR) in the neutrino-driven winds from nascent neutron stars and used this as a diagnostic of the sources for these CPR elements. In contrast, the true $r$-process elements (e.g., Ba and higher atomic numbers) are produced by extensive rapid neutron capture. It was assumed in the two-component model that Fe was only produced by normal supernovae (SNe) from progenitors of $\sim 12$–$25\,M_\odot$, which leave behind neutron stars, and that the heavy $r$-process elements ($r$-elements) with mass numbers $A>130$ were formed in low-mass SNe from progenitors of $\sim 8$–$11\,M_\odot$, which also leave behind neutron stars but produce no Fe. Thus the CPR elements would be produced by both low-mass and normal SNe and the corresponding yields were estimated (QW07). It follows that if these SNe were the only sources, then the presence of Fe should always be associated with that of the CPR elements. A more extensive study of the available observational data shows that some low-metallicity stars have Fe but essentially no Sr. In particular, @fulbright04 found a star with ${\rm [Fe/H]}=\log{\rm (Fe/H)}-\log{\rm (Fe/H)}_\odot=-2.88$ and $\log\epsilon({\rm Sr})=\log({\rm Sr/H})+12<-2.6$ in the dwarf galaxy Draco. From the two-component model we would have estimated $\log\epsilon({\rm Sr})=-0.28$ for this star, which is far above the observational upper limit. These results clearly indicate that if the CPR elements are always produced during the formation of neutron stars, then there must be an additional stellar source contributing Fe that does not leave behind neutron stars, or else the above model for the production of the CPR elements is in error.
Utilizing a more extensive data base than QW07 and especially treating the data on stars very deficient in Sr, Y, and Zr relative to Fe at ${\rm [Fe/H]}<-3$, the present paper will show that a third source in addition to the two sources (low-mass and normal SNe) in the model of QW07 is required to account for the elemental abundances in metal-poor stars. It will be argued that the third source producing Fe but no CPR elements is most likely associated with hypernovae (HNe) from progenitors of $\sim 25$–$50\,M_\odot$, which leave behind black holes instead of neutron stars. It is then shown that essentially all of the stellar data on elemental abundances at ${\rm [Fe/H]}\lesssim -1.5$ can be decomposed in terms of three distinct types of sources. This decomposition also identifies a yield pattern for the elements from Na through Zn including Fe that is attributable to HNe. An important conclusion is that this HN yield pattern is almost indistinguishable from what is attributed to normal SNe. Further, the discovery of extremely energetic HNe associated with gamma-ray bursters (e.g., @galama [@iwa98]) in the present universe requires that contributions from this source must be considered both in early epochs and on to the present. This leads to a reassessment of the contributions from different sources to the Galactic Fe inventory, which shows that ongoing HNe must play an important role and that the usual attribution of $\sim 1/3$ of the solar Fe inventory to normal SNe is not valid.
We aim to present a phenomenological three-component (low-mass and normal SNe and HNe) model for the chemical evolution of the early Galaxy that may provide a quantitative, self-consistent explanation for many of the results from stellar observations. We focus on three groups of elements: the low-$A$ elements from Na through Zn ($A \sim 23$–70), the CPR elements from Sr through Ag ($A \sim 88$–110), and the heavy $r$-elements ($A > 130$, Ba and higher atomic numbers). In §\[sec-2cm\] we give a brief outline of the two-component model of QW07 with low-mass and normal SNe represented by the $H$ and $L$ sources, respectively. In §\[sec-data\] we present the data on abundances of Sr and Ba as well as Y and La for a large sample of metal-poor stars, and show that the two-component model fails at ${\rm [Fe/H]}\lesssim -3$ and that an additional source producing Fe but no Sr or heavier elements is required to account for the data at such low metallicities. This source is identified with HNe. It is then shown that the extended three-component model with HNe, $H$, and $L$ sources gives a good representation of nearly all the data on the CPR elements Sr, Y, and Zr, but leads to the conclusion that the HN yield pattern is indistinguishable from that of the $L$ source for all the low-$A$ elements. Considering that HNe not only represent the first massive stars (Population III stars) but also must continue into the present epoch, we reinterpret the yields attributed to the hypothetical $L$ source as the combined contributions from normal SNe, which we designate as the $L^*$ source, and HNe. In §\[sec-3cm\] we show that the three-component model with HNe, $H$, and $L^*$ sources gives a very good representation of essentially all the data on the CPR elements Sr, Y, and Zr and further discuss the characteristics of these sources and their roles in the chemical evolution of the universe. We give our conclusions in §\[sec-con\].
The Two-Component Model with the $H$ and $L$ Sources {#sec-2cm}
====================================================
The two-component model[^1] of QW07 was based on the observations of elemental abundances in metal-poor stars and a basic understanding of stellar evolution and nucleosynthesis. It was directed toward identifying the stellar sources for the heavy $r$-elements. The following are the key assumptions and inferences of this model:
\(1) The heavy $r$-elements must be produced by an $H$ source that contributes essentially none of the low-$A$ elements including Fe. The $H$ source is most likely associated with low-mass SNe from progenitors of $\sim 8$–$11\,M_\odot$ that undergo O-Ne-Mg core collapse.
\(2) The low-$A$ elements are produced by an $L$ source associated with normal SNe from progentiors of $\sim 12$–$25\,M_\odot$ that undergo Fe core collapse. (It was assumed that this source provided $\sim 1/3$ of the bulk solar Fe inventory.)
\(3) The so-called light “$r$”-elements from Sr through Ag, especially Sr, Y, and Zr, in metal-poor stars must have been produced by CPRs in the $\alpha$-process [@wh92] that occurs as material expands away from a nascent neutron star in a neutrino-driven wind (e.g., @dsw86). Thus, the CPR elements are not directly related to the $r$-process (i.e., they are not the true $r$-elements). Instead, their production is a natural consequence of neutron star formation in low-mass and normal SNe associated with the $H$ and $L$ sources, respectively, as proposed by QW07.
The above points were incorporated in the two-component model of QW07 to account for the elemental abundances in metal-poor stars. For ${\rm [Fe/H]}\lesssim -1.5$, Type Ia SNe (SNe Ia) associated with low-mass stars (typically of several $M_\odot$) in binaries had not contributed significantly to the Fe group elements in the ISM. Similarly, there were no significant contributions to Sr and heavier elements in the ISM of this early regime from the $s$-process in asymptotic giant branch (AGB) stars. Thus, it was considered in QW07 that for ${\rm [Fe/H]}\lesssim -1.5$, the $H$ source is solely responsible for the heavy $r$-elements such as Eu and the $L$ source is solely responsible for the low-$A$ elements such as Fe while both sources produce the CPR elements. The yield pattern for the prototypical $H$ source was taken from the data on a star (CS 22892–052, @sneden03) with extremely high enrichment in the heavy $r$-elements relative to the low-$A$ elements. In contrast, the yield pattern for the prototypical $L$ source was taken from the data on a star (HD 122563, @honda06) with very little enrichment in the heavy $r$-elements relative to the low-$A$ elements and the abundances of the latter elements in this star were attributed to the $L$ source only. For this two-component model, the (number) abundance of an element E in the ISM at ${\rm [Fe/H]}\lesssim -1.5$ can be calculated as $$\left(\frac{\rm E}{\rm H}\right)=\left(\frac{\rm E}{\rm Eu}\right)_H
\left(\frac{\rm Eu}{\rm H}\right)+\left(\frac{\rm E}{\rm Fe}\right)_L
\left(\frac{\rm Fe}{\rm H}\right),
\label{eq-eh}$$ where (E/Eu)$_H$ and (E/Fe)$_L$ are the (number) yield ratios of E to Eu and Fe for the $H$ and $L$ sources, respectively. Given these yield ratios, the abundances of all the other elements (relative to hydrogen) in a star can be obtained from the above equation using only the observed abundances of Eu and Fe in that star. The results from the above model were in good agreement with the data on a large sample of metal-poor stars. We note that so long as there are no significant $s$-process contributions to the ISM (which is the case for ${\rm [Fe/H]}\lesssim -1.5$) or the star has not undergone mass transfer from an AGB companion in a binary, the element Ba can also be used as a measure of the $r$-process contributions. The yield ratios (E/Eu)$_H$ and (E/Fe)$_L$ for the heavy $r$-elements and the CPR elements, as well as the yield ratios (E/Ba)$_H$ for the CPR elements, are given in Tables \[tab-rhl\] and \[tab-yhl\].
We emphasize the phenomenological nature of the two-component model and its extension presented below. As discussed above, the yield patterns for the $H$ and $L$ sources were taken from the observed abundance patterns in two template stars. The validity of the model should be judged by its predictions for the abundances in other metal-poor stars. Insofar as the predictions agree with the data, the model can be considered to have identified some key characteristics of nucleosynthesis in the relevant stellar sources. This approach cannot replace the ab initio models of stellar nucleosynthesis, but is complimentary to the latter.
We note that the production of the low-$A$ elements including Fe in normal SNe from progentiors of $\sim 12$–$25\,M_\odot$ ($L$ source) is demonstrated by extensive modeling of SN nucleosynthesis (e.g., @ww95 [@tnh96; @cl04]), and so is the production of the CPR elements in the neutrino-driven wind associated with neutron star formation ($H$ and $L$ sources; e.g., @meyer92 [@taka94; @woosley94; @hoffman97]). However, the theoretical yields of the low-$A$ elements, especially the Fe group, are subject to the many uncertainties in modeling the evolution and explosion of massive stars. In the absence of a solid understanding of the SN mechanism, the explosion is artificially induced and the associated nucleosynthesis is parametrized by a “mass cut” (e.g., @ww95) or constrained to fit the yields of $^{56}$Ni inferred from SN light curves (e.g., @cl04). For the CPR elements, no reliable quantitative yields are yet available. QW07 concluded that the neutrino-driven wind does not play a significant role in the production of the heavy $r$-elements and suggested that another environment with rapid expansion timescales inside low-mass ($\sim 8$–$11\,M_\odot$) SNe from O-Ne-Mg core collapse ($H$ source) is responsible for making these elements. Subsequent work by @ning07 showed that the propagation of a fast shock through the surface layers of an O-Ne-Mg core can provide the conditions leading to the production of the heavy $r$-elements. However, the required shock speed is not obtained in the current SN models [@janka07] based on the pre-SN structure of a $1.38\ M_\odot$ core calculated by @nomoto84 [@nomoto87]. Clearly, more studies of the pre-SN evolution of O-Ne-Mg cores and their collapse are needed to test whether the heavy $r$-elements can indeed be produced in the shocked surface layers of such cores. In the following we assume that low-mass SNe from O-Ne-Mg core collapse are the $H$ source solely responsible for producing the heavy $r$-elements and that the CPR elements are produced by both the $H$ and $L$ sources. At the present time, stellar models cannot calculate the absolute yields from first principles for any of the sources discussed above and some ad hoc parametric treatment is required for modeling the nucleosynthesis of these sources.
Failure of the Two-Component Model and Requirement of HNe {#sec-data}
=========================================================
In Figure \[fig-esr\]a we show the data on $\log\epsilon({\rm Sr})$ from an extensive set of the available high-resolution observations over the wide range of $-5.5\lesssim {\rm [Fe/H]}\lesssim -1.5$ (squares: @johnson02; pluses: @honda04; diamonds: @aoki05; circles: @francois07; crosses: @cohen08; asterisks: @depagne02 [@aoki02; @aoki06; @aoki07]; downward arrows indicating upper limits: @christlieb04 [@fulbright04; @frebel07; @cohen07; @norris07]). All the data are for stars in the Galactic halo except for the downward arrow at ${\rm [Fe/H]}=-2.88$, which is for a star (Draco 119, @fulbright04) in the dwarf galaxy Draco.
We now use the two-component model of QW07 to analyze the data shown in Figure \[fig-esr\]a. We first apply equation (\[eq-eh\]) to calculate the $H$ and $L$ contributions to the solar Sr abundance assuming that the $H$ source provided all of the solar Eu abundance and the $L$ source provided 1/3 of the solar Fe abundance. Further assuming that the sun represents the sampling of a well-mixed ISM, we can show that such an ISM has \[Sr/Fe\]$_{\rm mix}=-0.10$ resulting from the mixing of $H$ and $L$ contributions only (see Appendix \[sec-app1\] and Table \[tab-mix\]). This Sr/Fe ratio corresponds to $$\log\epsilon({\rm Sr})={\rm [Fe/H]}+2.82,$$ shown as the solid line in Figure \[fig-esr\]a. It can be seen from this figure that the bulk of the data lie close to the solid line, but for ${\rm [Fe/H]}\lesssim -3$, almost all of the data depart greatly from this line.
There is a lack of Eu data for many stars with $[{\rm Fe/H}]<-3$. As Ba data are more readily available for such stars, we use Ba instead of Eu as the index heavy $r$-element to identify the contributions from the $H$ source (this is robust so long as there are no $s$-process contributions). Then equation (\[eq-eh\]) can be rewritten for Sr as $$\left(\frac{\rm Sr}{\rm H}\right)=\left(\frac{\rm Sr}{\rm Ba}\right)_H
\left(\frac{\rm Ba}{\rm H}\right)+\left(\frac{\rm Sr}{\rm Fe}\right)_L
\left(\frac{\rm Fe}{\rm H}\right).
\label{eq-srh}$$ The yield ratios (E/Ba)$_H$ and (E/Fe)$_L$ for Sr and other CPR elements are given in Table \[tab-yhl\]. Using these yield ratios and the above equation, we calculate the $\log\epsilon_{\rm cal}{\rm (Sr)}$ values for those stars shown in Figure \[fig-esr\]a that have observed Ba and Fe abundances. The differences $\Delta\log\epsilon({\rm Sr})\equiv\log\epsilon_{\rm cal}{\rm (Sr)}-
\log\epsilon_{\rm obs}{\rm (Sr)}$ between the calculated and observed values are shown in Figure \[fig-esr\]b. Note that for ${\rm [Fe/H]}> -2.7$, the agreement between the model predictions and the data is very good. However, for ${\rm [Fe/H]}\lesssim -2.7$, there is great discrepancy in the sense that the calculated $\log\epsilon_{\rm cal}{\rm (Sr)}$ values for many stars far exceed the observed values. It is this discrepancy that we will focus on in this paper.
The large disgreement between the model predictions and the data for ${\rm [Fe/H]}\lesssim -2.7$ shown in Figure \[fig-esr\]b is caused by assigning all the Fe to the $L$ source. If there is an additional source producing Fe but no Sr or heavier elements at such low metallicities, then equation (\[eq-srh\]) overestimates the Sr abundances. The requirement of such a source can also be seen from the Sr/Fe ratios for the stars. The yield ratio (Sr/Fe)$_L$ corresponds to \[Sr/Fe\]$_L=-0.32$ (see Table \[tab-mix\]). Of the $H$ and $L$ sources, both produce Sr but only the latter can produce Fe. Thus any mixture of the contributions from these two sources should have ${\rm [Sr/Fe]}\geq -0.32$. Figure \[fig-srfe\] shows \[Sr/Fe\] vs. \[Ba/Fe\] for those stars in Figure \[fig-esr\] that have observed Ba abundances or upper limits. It can be seen that many stars have ${\rm [Sr/Fe]}\ll -0.32$ and quite a few have ${\rm [Sr/Fe]}\lesssim -2$. These observations are in direct conflict with the two-component model and can only be accounted for if there is an additional source for Fe (and the associated elements) that produce none or very little of Sr and heavier elements. If we expand the framework of QW07 to include this third source, then a self-consistent interpretation of all the data may be possible.
Effects of the Third Source {#sec-p}
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In the extended model including the third source in addition to the $H$ and $L$ sources, only a fraction $f_{{\rm Fe},L}$ of the Fe is produced by the $L$ source. Then equation (\[eq-srh\]) becomes $$\left(\frac{\rm Sr}{\rm H}\right)=\left(\frac{\rm Sr}{\rm Ba}\right)_H
\left(\frac{\rm Ba}{\rm H}\right)+\left(\frac{\rm Sr}{\rm Fe}\right)_L
\left(\frac{\rm Fe}{\rm H}\right)f_{{\rm Fe},L}.
\label{eq-srhf}$$ For $f_{{\rm Fe},L}=1$ the extended model reduces to the two-component model. For $f_{{\rm Fe},L}=0$ only the $H$ source and the third source are relevant with the latter being the sole contributor of the low-$A$ elements such as Fe and the former being the sole contributor of Sr and heavier elements. Equation (\[eq-srhf\]) can be rewritten as $${\rm [Sr/Fe]}=\log\left(10^{{\rm [Sr/Ba]}_H+{\rm [Ba/Fe]}}+
f_{{\rm Fe},L}\times 10^{{\rm [Sr/Fe]}_L}\right).
\label{eq-srfe}$$ Note that the above equation represents very strong constraints on the evolution of \[Sr/Fe\] with \[Ba/Fe\] in the extended model. Whether a system starts with the initial composition of big bang debris or with an initial state inside the region defined by the curves representing equation (\[eq-srfe\]) for $f_{{\rm Fe},L}=0$ and 1, it cannot have \[Sr/Fe\] and \[Ba/Fe\] values outside this region upon further evolution so long as only the $H$ and $L$ sources and the third source contribute metals.
The curves representing equation (\[eq-srfe\]) for $f_{{\rm Fe},L}=0$, 0.1, 0.5, and 1 are shown along with the data on \[Sr/Fe\] vs. \[Ba/Fe\] in Figure \[fig-srfe\]. Most of the data lie between the curves for $f_{{\rm Fe},L}=0$ and 1 with a clustering of the data around the curve for $f_{{\rm Fe},L}=1$. There are also quite a few data on the curve for $f_{{\rm Fe},L}=0$. Some data lie distinctly above the curve for $f_{{\rm Fe},L}=1$. This could be partly due to observational uncertainties. We also note that all SNe associated with the $H$ and $L$ sources are assumed here to have fixed yield patterns. If there were variations in the yield ratios by a factor of several, then some “forbidden” region above the curve for $f_{{\rm Fe},L}=1$ would be accessible. In §\[sec-3cm\] we will show that with a reinterpretation of the $L$ source there is no longer a need to call upon such variabilities. In any case, we consider that the comparison between the theoretical model curves for $f_{{\rm Fe},L}=0$ to 1 and the data shown in Figure \[fig-srfe\] justifies the extended model where a third source is producing Fe but no Sr or heavier elements.
With no production of Sr or heavier elements assigned to the third source, the Sr/Ba ratio is determined exclusively by the $H$ and $L$ sources. As both these sources produce Sr but only the $H$ source can produce Ba, any mixture of the contributions from these two sources should have ${\rm [Sr/Ba]}\geq {\rm [Sr/Ba]}_H$. Figure \[fig-srba\] shows the data on \[Sr/Ba\] vs. \[Fe/H\] along with two reference lines, one corresponding to \[Sr/Ba\]$_H=-0.31$ (see Table \[tab-mix\]) and the other to \[Sr/Ba\]$_{\rm mix}=0.10$ for an ISM with well-mixed $H$ and $L$ contributions (see Appendix \[sec-app1\] and Table \[tab-mix\]). An excess of $L$ over $H$ contributions relative to the well-mixed case displaces \[Sr/Ba\] above the line for \[Sr/Ba\]$_{\rm mix}$. It can be seen from Figure \[fig-srba\] that almost all of the data are compatible with the lower bound of ${\rm [Sr/Ba]}\geq {\rm [Sr/Ba]}_H$ and that a substantial fraction of the stars did not sample a well-mixed ISM. Excluding the lower limits, we note that some data lie below the line for \[Sr/Ba\]$_H$. However, the deviation below \[Sr/Ba\]$_H$ is $\lesssim 0.4$ dex, which is comparable to the observational uncertainties[^2] and does not represent serious violation of the lower bound. It is important to note that the four stars shown as asterisks A, B, C, and D in Figures \[fig-esr\], \[fig-srfe\], and \[fig-srba\] appear to be well behaved in terms of Sr and Ba although they have very high abundances of C and O and anomalous abundance patterns of the low-$A$ elements (see §\[sec-pan\]).
To further test the robustness of the extended model including the third source, we have carried out a similar analysis of the medium-resolution data from the HERES survey of metal-poor stars [@barklem05]. This sample contains 253 stars of which eight have neither Sr nor Ba data. Five of the remaining stars were clearly recognized from their Ba/Eu ratios as having dominant $s$-process contributions [@barklem05] and are excluded. This leaves 240 stars to be analyzed here. The data on $\log\epsilon({\rm Sr})$ vs. \[Fe/H\] are shown in Figure \[fig-heres\]a analogous to Figure \[fig-esr\]a. It can be seen that the bulk of the data again cluster around the line for an ISM with well-mixed $H$ and $L$ contributions but there are again many stars with ${\rm [Fe/H]}\lesssim -3$ showing a great deficiency in Sr. The description for the evolution of \[Sr/Fe\] with \[Ba/Fe\] by the extended model is compared with the medium-resolution data in Figure \[fig-heres\]b. In general, these medium-resolution data are in accord with the results presented above for the high-resoltuion data shown in Figure \[fig-srfe\], but with some exceptions. There are a number of data that lie well above the upper bound for $f_{{\rm Fe},L}=1$ (e.g., the data at ${\rm [Ba/Fe]}=-0.87$, ${\rm [Sr/Fe]}=0.68$ and ${\rm [Ba/Fe]}=-0.62$, ${\rm [Sr/Fe]}=0.70$ representing HE 0017–4838 and HE 1252–0044, respectively). This may be partly due to observational uncertainties, but it will be shown in §\[sec-3cm\] that a reinterpretation of the $L$ source raises the upper bound above essentially all the data. There are also a number of data that lie far to the right of and below the lower bound for $f_{{\rm Fe},L}=0$. We consider that the corresponding stars most plausibly have large $s$-process contributions to the Ba (these stars are: HE 0231–4016, HE 0305–4520, HE 0430–4404, HE 1430–1123, HE 2150–0825, HE 2156–3130, HE 2227–4044, and HE 2240–0412). This explanation can be tested by high-resolution observations covering more elements heavier than Ba. While not showing as clear-cut a case as the high-resolution data, the bulk of the HERES data appear to be in broad accord with the requirement of a third source producing Fe but no Sr or heavier elements as presented above.
Requirement of a Third Source from Data on Y and La as well as Zr and Ba {#sec-yla}
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As a final test for the requirement of a third source and the robustness of the extended model including this source, we repeat the analysis using the high-resolution data on Y and La in metal-poor stars. Like Sr and Ba, these two elements represent the CPR elements and the heavy $r$-elements, respectively. The abundances of Y and La in a star are generally much lower than those of Sr and Ba, respectively. Consequently, there are much fewer data on Y and La than on Sr and Ba in metal-poor stars. On the other hand, the abundances of Y and La are less susceptible to uncertainties in the spectroscopic analysis if they can be measured and therefore, may be better indicators for the trends of chemical evolution (e.g., @simmerer). In Figure \[fig-yla\]a we show the data on $\log\epsilon({\rm Y})$ from the high-resolution observations of @johnson (squares) and @francois07 (circles) over the wide range of $-4.1\lesssim[{\rm Fe/H}]\lesssim -1.5$. The solid line in this figure represents $$\log\epsilon({\rm Y})=[{\rm Fe/H}]+1.97,$$ which corresponds to \[Y/Fe\]$_{\rm mix}=-0.24$ for an ISM with well-mixed contributions from the $H$ and $L$ sources only (see Appendix \[sec-app1\] and Table \[tab-mix\]). It can be seen from Figure \[fig-yla\]a that the bulk of the data again cluster around the solid line, but there are again many stars with \[Fe/H\] $\lesssim -3$ showing a great deficiency in Y.
The failure of the two-component model with the $H$ and $L$ sources as found for Sr can also be shown by comparing the Y abundances predicted from this model with the data on metal-poor stars. The yield patterns of the $H$ and $L$ sources given in Tables \[tab-rhl\] and \[tab-yhl\] correspond to $\log({\rm Y/La})_H=0.27$ and $\log({\rm Y/Fe})_L=-5.67$. Using these yield ratios and the La and Fe data on the stars shown in Figure \[fig-yla\]a, we calculate the Y abundances for these stars from $$\left(\frac{\rm Y}{\rm H}\right)=\left(\frac{\rm Y}{\rm La}\right)_H
\left(\frac{\rm La}{\rm H}\right)
+\left(\frac{\rm Y}{\rm Fe}\right)_L\left(\frac{\rm Fe}{\rm H}\right)$$ and show the differences $\Delta\log\epsilon({\rm Y})\equiv
\log\epsilon_{\rm cal}({\rm Y})-\log\epsilon_{\rm obs}({\rm Y})$ between the calculated and observed values in Figure \[fig-yla\]b. As many stars lack La data, their $\log\epsilon_{\rm cal}({\rm Y})$ values are calculated from the $L$ contributions only. The resulting $\Delta\log\epsilon({\rm Y})$ values represent lower limits and are shown as symbols with upward arrows \[the value of $\log({\rm Y/Fe})_L=-5.67$ should represent the minimum value of $\log({\rm Y/Fe})=\log\epsilon({\rm Y})-
\log\epsilon({\rm Fe})$ predicted for metal-poor stars based on the two-component model\]. It can be seen from Figure \[fig-yla\]b that there is again good agreement between the two-component model and the data for ${\rm [Fe/H]}>-3$ but the model tends to greatly overpredict Y abundances (by up to $\sim 1.3$ dex) for ${\rm [Fe/H]}\lesssim -3$.
As discussed using just the Sr and Ba data, the two-component model must be modified by including a third source producing Fe but no CPR or heavier elements in order to account for the observations. The effects of such a source are shown in Figure \[fig-yla\]c for the CPR element Y and the heavy $r$-element La analogous to Figures \[fig-srfe\] and \[fig-heres\]b for Sr and Ba. The data on the evolution of \[Y/Fe\] with \[La/Fe\] shown in Figure \[fig-yla\]c are for the stars shown in Figure \[fig-yla\]a except for those with only upper limits on both Y and La abundances. The distribution of the data on Y and La with respect to the curves calculated from the three-component model for $f_{{\rm Fe},L}=0$, 0.1, 0.5, and 1 is similar to those discussed in §\[sec-p\] for the evolution of \[Sr/Fe\] with \[Ba/Fe\] and shown in Figures \[fig-srfe\] and \[fig-heres\]b. There is one exceptional star, CS 22968–014, which is indicated by the downward arrow labeled as such in Figure \[fig-yla\]c. This star has an anomalously high abundance of La corresponding to $\log({\rm La/Ba})=0.7$ [@francois07], which greatly exceeds the yield ratio of $\log({\rm La/Ba})_H=-0.71$ assumed for the $H$ source (see Table \[tab-rhl\]). If the measured La/Ba ratio were correct, then CS 22968–014 must have sampled an extremely anomalous event producing the heavy $r$-elements. However, the Ba and La abundances for this star were derived from a single line for either element [@francois07], and therefore, could be in error. More observations of these two elements in this star are needed to resolve this issue. In any case, we consider that the overall comparison between the theoretical model curves for $f_{{\rm Fe},L}=0$ to 1 and the data shown in Figure \[fig-yla\]c justifies the three-component model where the third source is producing Fe but no CPR or heavier elements.
With no production of CPR or heavier elements assigned to the third source, the Y/La ratio is determined exclusively by the $H$ and $L$ sources. Any mixture of the contributions from these two sources should have ${\rm [Y/La]}$ exceeding ${\rm [Y/La]}_H=-0.81$ (see Table \[tab-mix\]). An ISM with well-mixed $H$ and $L$ contributions should have \[Y/La\]$_{\rm mix}=-0.36$ (see Appendix \[sec-app1\] and Table \[tab-mix\]). The data on \[Y/La\] for the stars shown in Figure \[fig-yla\]c are displayed in Figure \[fig-yla\]d analogous to Figure \[fig-srba\]. It can be seen from Figure \[fig-yla\]d that except for the anomalous star CS 22968–014 noted above, all other data are compatible with the lower bound of ${\rm [Y/La]}\geq {\rm [Y/La]}_H$ and that a large fraction of the stars did not sample a well-mixed ISM.
Based on the analysis of the Sr and Ba data as well as the Y and La data, we consider that a three-component model including a third source producing Fe but no CPR or heavier elements is adequately justified. Our analysis of the Zr and Ba data (not presented in detail here) is in full accord with the three-component model and leads to the same quantitative conclusion (see §\[sec-3cm\] and Figure \[fig-csrfe\]d). We will now pursue the consequences of this approach.
HNe as the Third Source {#sec-plowa}
-----------------------
Star formation in the early universe responsible for the enrichment of metal-poor stars is still not well understood. Simulations indicate that the first stars were likely to be massive, ranging from $\sim 10$ to $\sim 1000\,M_\odot$ \[see @abel02 [@bromm04] for reviews of earlier works and @yoshida06 [@oshea07; @gao07] for more recent studies\]. It is generally thought that stars form in the typical mass range of $\sim 1$–$50\,M_\odot$ subsequent to the epoch of the first stars. Below we assume this simple scenario of star formation and focus on considerations of nucleosynthesis to identify the stellar types for the third source.
The assumed third source produces the low-$A$ elements including Fe but no CPR elements such as Sr or heavier elements. As the CPR elements are here considered to be produced in the neutrino-driven wind from nascent neutron stars, there are two main candidates for the third source : (1) pair-instability SNe (PI-SNe) from very massive ($\sim 140$–$260\,M_\odot$) stars (VMSs), in which the star is completely disrupted by the explosion and no neutron star is produced, and (2) massive SNe with progenitors of $\sim 25$–$50\,M_\odot$, in which a black hole forms either directly by the core collapse or through severe fallback onto the neutron star initially produced by the core collapse. There is observational evidence that massive SNe have two branches: HNe and faint SNe with the latter thought to be much rarer. Compared with normal SNe, HNe have up to $\sim 50$ times higher explosion energies and $\sim 7$ times higher Fe yields while faint SNe have several times lower explosion energies and $\gtrsim 10$ times lower Fe yields \[see @iwa98 for interpretation of SN 1998bw as an HN, @tura for the case of SN 1997D as a faint SN, and @nomoto06 and references therein for other studies of HNe and faint SNe\]. It is important to note that HNe are ongoing events in the present universe as evidenced by the occurrences of the associated gamma-ray bursts \[see e.g., @galama for the discovery of SN 1998bw, an HN associated with a gamma-ray burst\].
In our assumed scenario of star formation, PI-SNe can only occur at zero metallicity but HNe and faint SNe can occur at all epochs. In addition, these three types of events have very different yield patterns of the low-$A$ elements. Compared with HNe and faint SNe, PI-SNe have extremely low production of those low-$A$ elements with odd atomic numbers such as Na, Al, K, Sc, V, Mn, and Co relative to their neighboring elements with even atomic numbers (see Figure 3 in @heger02). This is because unlike HNe and faint SNe that occur after all stages of core burning, PI-SNe occur immediately following core C-burning and there is not sufficient time for weak interaction to provide the required neutron excess for significant production of the low-$A$ elements with odd atomic numbers (e.g., @heger02). Further, the production of the low-$A$ elements from Na through Mg relative to those from Si through Zn differs greatly between HNe and faint SNe. This is because the former elements are produced by hydrostatic burning during the pre-explosion evolution and the latter ones by explosive burning. The extremely weak explosion of faint SNe would lead to very high yield ratios of the hydrostatic burning products relative to the explosive burning products.
The decomposition of elemental abundances in terms of three components discussed in §\[sec-p\] and §\[sec-yla\] identifies those stars in which the Fe is exclusively the product of the third source. Such stars lie on the curve for $f_{{\rm Fe},L}=0$ representing the mixture of contributions from the $H$ source and the third source in Figure \[fig-srfe\]. As the $H$ source produces none of the low-$A$ elements, these elements in the stars lying on the $f_{{\rm Fe},L}=0$ curve should be attributed to the third source. The abundance patterns of these elements in five such stars (open square: BD $-18^\circ 5550$, ${\rm [Fe/H]}=-2.98$, @johnson; open circle: CS 30325–094, ${\rm [Fe/H]}=-3.25$, open diamond: CS 22885–096, ${\rm [Fe/H]}=-3.73$, open triangle: CS 29502–042, ${\rm [Fe/H]}=-3.14$, @cayrel; plus: BS 16085–050, ${\rm [Fe/H]}=-2.85$, @honda04) are shown in Figure \[fig-p\]. It can be seen that all the abundance patterns of the low-$A$ elements assigned to the third source are quasi-uniform. By quasi-uniformity, we mean that for element E, the \[E/Fe\] values for different stars are within $\sim 0.3$ dex of some mean value. It is also clear that there are no drastic variations in the \[E/Fe\] values either between the elements with odd and even atomic numbers or between the hydrostatic and explosive burning products. We conclude that neither PI-SNe nor faint SNe can be the third source. This leaves HNe as the third source.
The abundance patterns of the low-$A$ elements in those stars that lie on the curve for $f_{{\rm Fe},L}=1$ in Figure \[fig-srfe\] should represent the yield pattern of these elements for the hypothecated $L$ source. The patterns for three such stars (filled square: BD $+4^\circ 2621$, @johnson; filled circle: HD 122563, @honda04 [@honda06]; filled diamond: CS 29491–053, @cayrel) are compared with those assigned to the third source in Figure \[fig-p\]. It can be seen that the third source (now taken to be HNe) and the $L$ source are indistinguishable in terms of their assigned contributions to the low-$A$ elements. This is also reflected by the fact that essentially all the stars in the region bounded by the curves for $f_{{\rm Fe},L}=0$ and 1 shown in Figure \[fig-srfe\] have the same quasi-uniform abundance patterns of the low-$A$ elements as established by the observations of @cayrel (see §\[sec-pan\] for discussion of the exceptional stars). As an example, we show in Figure \[fig-p\] the pattern for BD $+17^\circ 3248$ (solid curve, @cowan02) with a relatively high value of ${\rm [Fe/H]}=-2$. We are thus left with a most peculiar conundrum: the yield pattern of the low-$A$ elements attributed to the third source is the same as that attributed to the $L$ source. This is the same result that we [@qw02] found earlier in attempting to estimate the yield patterns of the stellar sources contributing in the regime of ${\rm [Fe/H]}\lesssim -3$ using the data of @mcw and @nrb. The recent more extensive and precise data of @cayrel lead to the same conclusion.
We have associated the third source with HNe and the $L$ source with normal SNe. As HNe and normal SNe are concurrent in our assumed scenario of star formation and cannot be distinguished based on their production of the low-$A$ elements, the contributions to these elements, especially Fe, that we previously assigned to the $L$ source only may well be a combination of the contributions from both HNe and normal SNe. In this case, the Sr/Fe ratio assigned to the $L$ source represents a mixture of Sr contributions from normal SNe and Fe contributions from both HNe and normal SNe. In what follows, we designate normal SNe as the $L^*$ source and consider the $L$ source as a combination of HNe and the $L^*$ source ($L\to {\rm HNe}+L^*$). The apparent near identity in the abundance patterns of the low-$A$ elements attributed to HNe and the $L$ source may mean that the dominant contributor to these elements is HNe. The stellar types and the nucleosynthetic characteristics assigned to HNe, $H$, and $L^*$ sources are summarized in Table \[tab-phl\].
In our earlier efforts to decompose the stellar sources of elemental abundances at low metallicities, we recognized that there must be a source producing Fe and other low-$A$ elements but none of the $r$-elements [@qw02]. We therefore proposed a source that only occurred in very early epochs and did not occur later. This inference, in conjunction with the rather sharp break in the observed abundances of the heavy $r$-elements at ${\rm [Fe/H]}\sim-3$, led us to propose that PI-SNe from VMSs might be the source. It was argued that VMSs were the first stars and that the very disruptive PI-SNe associated with them provided a baseline of metals to the IGM at a level of ${\rm [Fe/H]}\sim-3$. This apparent baseline was also found in damped Lyman $\alpha$ systems [@qsw]. However, in the framework of hierarchical structure formation, for halos that are not disrupted by explosions of massive stars (see §\[sec-halo\]), the initial rate of growth in metallicity is so rapid that it would be very rare to find stars with ${\rm [Fe/H]}<-3$ [@qw04]. It is thus plausible that the rarity of ultra-metal-poor stars with ${\rm [Fe/H]}<-3$ results from the initial phase of rapid metal enrichment in all bound halos and is not due to a general “prompt inventory” in the IGM. In addition, as discussed above, none of the metal-poor stars with ${\rm [Fe/H]}\lesssim-3$ exhibit the abundance patterns calculated for PI-SNe, which are extremely deficient in the elements with odd atomic numbers such as Na, Al, K, Sc, V, Mn, and Co (e.g., @heger02). Further, the search for ultra-metal-poor stars has shown that while stars with ${\rm [Fe/H]}<-3$ are rare, they do occur and show some evidence of elements heavier than the Fe group in their spectra (see @christlieb02 [@frebel05] for the discovery of the two most metal-poor stars with ${\rm [Fe/H]}<-5$). Thus, low-mass stars must be able to form from a medium with ${\rm [Fe/H]}\ll -3$. Based on all the above considerations, we now must withdraw the “prompt inventory” hypothesis and must consider an IGM with widely variable “metal” content and that ${\rm [Fe/H]}\sim -3$ represents a transition to the regime where halos are no longer disrupted by the explosions of massive stars.
The Three-Component Model with HNe, $H$, and $L^*$ Sources {#sec-3cm}
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With the revised interpretation of the $L$ source as a combination of HNe and the $L^*$ source, we can relate ${\rm [Sr/Fe]}_L=-0.32$ (see Table \[tab-mix\]) to the yield ratio of Sr to Fe for the $L^*$ source. For example, if we assume that 24% of the Fe in the $L$ mixture is from the $L^*$ source (see §\[sec-hne\]), this corresponds to ${\rm [Sr/Fe]}_{L^*}={\rm [Sr/Fe]}_L-\log0.24=0.30$. Equation (\[eq-srfe\]) now becomes $${\rm [Sr/Fe]}=\log\left(10^{{\rm [Sr/Ba]}_H+{\rm [Ba/Fe]}}+
f_{{\rm Fe},L^*}\times 10^{{\rm [Sr/Fe]}_{L^*}}\right),
\label{eq-csrfe}$$ where $f_{{\rm Fe},L^*}$ is the fraction of the Fe in a star contributed by the $L^*$ source. The curves representing the above equation for ${\rm [Sr/Fe]}_{L^*}=0.30$ and $f_{{\rm Fe},L^*}=0$, 0.1, 0.24, and 1 are shown along with the data in Figures \[fig-csrfe\]a (high-resolution data) and \[fig-csrfe\]b (medium-resolution data) analogous to Figures \[fig-srfe\] and \[fig-heres\]b. It can be seen from Figures \[fig-csrfe\]a and \[fig-csrfe\]b that essentially all the data lie inside the allowed region for the evolution of \[Sr/Fe\] with \[Ba/Fe\] bounded by the curves for $f_{{\rm Fe},L^*}=0$ and 1 (as mentioned near the end of §\[sec-p\], the exceptional data points far to the right of and below the curve for $f_{{\rm Fe},L^*}=0$ in Figure \[fig-csrfe\]b most likely represent stars that received large $s$-process contributions to Ba).
Assuming that 24% of the Fe in the $L$ mixture is from the $L^*$ source as for Figures \[fig-csrfe\]a and \[fig-csrfe\]b, we obtain ${\rm [Y/Fe]}_{L^*}={\rm [Y/Fe]}_L-\log0.24=0.19$ (see Table \[tab-mix\]). Using this yield ratio, we show the curves representing $${\rm [Y/Fe]}=\log\left(10^{{\rm [Y/La]}_H+{\rm [La/Fe]}}+
f_{{\rm Fe},L^*}\times 10^{{\rm [Y/Fe]}_{L^*}}\right)
\label{eq-cyfe}$$ for $f_{{\rm Fe},L^*}=0$, 0.1, 0.24, and 1 along with the data in Figure \[fig-csrfe\]c analogous to Figure \[fig-yla\]d. It can be seen from Figure \[fig-csrfe\]c that with the exception of the anomalous star CS 22968–014 as noted in §\[sec-yla\], all other data again lie inside the allowed region for the evolution of \[Y/Fe\] with \[La/Fe\] bounded by the curves for $f_{{\rm Fe},L^*}=0$ and 1.
For completeness, we also show the high-resolution data on the evolution of \[Zr/Fe\] with \[Ba/Fe\] (squares: @johnson; diamonds: @aoki05; circles: @francois07) in Figure \[fig-csrfe\]d along with the curves representing $${\rm [Zr/Fe]}=\log\left(10^{{\rm [Zr/Ba]}_H+{\rm [Ba/Fe]}}+
f_{{\rm Fe},L^*}\times 10^{{\rm [Zr/Fe]}_{L^*}}\right)
\label{eq-czrfe}$$ for $f_{{\rm Fe},L^*}=0$, 0.1, 0.24, and 1. In the above equation, we take \[Zr/Ba\]$_H=-0.20$ and \[Zr/Fe\]$_{L^*}=0.46$ (see Table \[tab-mix\]). The latter yield ratio again assumes that 24% of the Fe in the $L$ mixture is from the $L^*$ source as for Figures \[fig-csrfe\]a, \[fig-csrfe\]b, and \[fig-csrfe\]c. It can be seen from Figure \[fig-csrfe\]d that essentially all the data again lie inside the allowed region for the evolution of \[Zr/Fe\] with \[Ba/Fe\] bounded by the curves for $f_{{\rm Fe},L^*}=0$ and 1.
Based on the comparison of the theoretical model curves and the data on Sr, Y, and Zr shown in Figure \[fig-csrfe\], we consider that the three-component model with HNe, $H$, and $L^*$ sources provides a very good description of the elemental abundances in metal-poor stars. For an overwhelming portion of the metal-poor stars shown in this figure, their inventory of Fe and other low-$A$ elements received significant but not dominant contributions from the $L^*$ source (normal SNe) as indicated by the corresponding low values of $f_{{\rm Fe},L^*}$. We conclude that the bulk of the low-$A$ elements including Fe in metal-poor stars with ${\rm [Fe/H]}\lesssim -1.5$ was provided by HNe. This may explain why wide fluctuations in the abundance patterns of the low-$A$ elements expected from the contributions of just a few normal SNe are not actually observed. The matter remains as to what the detailed yield patterns of the $L^*$ source are for the low-$A$ elements. This is not easily addressable from the observations of metal-poor stars as the $L^*$ contributions only constitute a small fraction of the total abundances of these elements. It appears that we must rely on stellar model calculations (e.g., @ww95 [@cl04]) to estimate the $L^*$ yield patterns of the low-$A$ elements.
A straightforward application of the three-component model is to calculate the contributions from the $H$ and $L^*$ sources to the solar inventory of the CPR elements. Assuming that all of the Eu in the sun was provided by the $H$ source and a fraction $f_{{\rm Fe},L^*}^\odot=0.08$ of the solar Fe inventory was provided by the $L^*$ source ($f_{{\rm Fe},L^*}^\odot=0.24f_{{\rm Fe},L}^\odot$ with $f_{{\rm Fe},L}^\odot=1/3$ being the fraction contributed by sources other than SNe Ia as usually assumed), we calculate the $H$ and $L^*$ contributions to a CPR element E in the sun from $$\left(\frac{\rm E}{\rm H}\right)_{\odot,HL^*}=
\left(\frac{\rm E}{\rm Eu}\right)_H
\left(\frac{\rm Eu}{\rm H}\right)_\odot+
\left(\frac{\rm E}{\rm Fe}\right)_{L^*}
\left(\frac{\rm Fe}{\rm H}\right)_\odot f_{{\rm Fe},L^*}^\odot\ ,
\label{eq-cprs}$$ where the yield ratios (E/Eu)$_H$ and (E/Fe)$_{L^*}$ are given in Table \[tab-yhl\]. We present the results in terms of $\log\epsilon_{\odot,HL^*}({\rm E})$ in Table \[tab-cprs\], where the corresponding fraction $f_{{\rm E},HL^*}^\odot$ of the solar inventory contributed by the $H$ and $L^*$ sources is also given. The fraction $f_{{\rm E},HL^*}^\odot$ is in approximate agreement with the fraction $1-f_{{\rm E},s}^\odot$ attributed to non-$s$-process sources by @arlandini99 and @travaglio for the elements Mo, Ru, Rh, Pd, and Ag with small to moderate $s$-process contributions (see Table \[tab-cprs\]). For the elements Sr, Y, Zr, and Nb with large $s$-process contributions, the fraction $f_{{\rm E},HL^*}^\odot$ is a factor of $\approx 2$ larger than the fraction $1-f_{{\rm E},s}^\odot$ estimated by @arlandini99. This latter result is in agreement with what was found earlier by us [@qw01] and confirmed later by @travaglio, who carried out a detailed study of Galactic chemical evolution for the $s$-process contributions. To calculate the fraction $1-f_{{\rm E},s}^\odot$, @travaglio used as input the $s$-process yields for stars of low and intermediate masses with a wide range of metallicities, the formation history of these stars, and the mixing characteristics of their nucleosynthetic products with gas in the Galaxy. In contrast, the fraction $f_{{\rm E},HL^*}^\odot$ is calculated directly from the yield templates of the $H$ and $L^*$ sources. These templates are taken from data on metal-poor stars that formed in the regime where there cannot be major $s$-process contributions to the ISM and only massive stars can plausibly contribute. The only assumption with regard to the solar abundances used in calculating $f_{{\rm E},HL^*}^\odot$ is the assignment of a fraction $f_{{\rm Fe},L^*}^\odot=0.08$ of the solar Fe inventory to the $L^*$ source (the fraction from this source and HNe combined is 1/3). It appears that the results from this simple and self-consistent approach, and hence, the assumptions used in the three-component model, are compatible with the non-$s$-process contributions to the solar abundances of the CPR elements. This provides a further test of the model and does not challenge the assignment of major Fe production by HNe as argued here.
Below we further discuss the characteristics of HNe and the $H$ and $L^*$ sources in the three-component model and their roles in the chemical evolution of the universe.
Yields of HNe, $H$, and $L^*$ Sources {#sec-hne}
-------------------------------------
The yields of the low-$A$ elements for HNe are not known although these were estimated by parameterized calculations (e.g., @tominaga). The Fe yields for some HNe were inferred from their light curves. Comparison of the yield patterns of the low-$A$ elements from various parameterized models of HNe with the abundance patterns observed in metal-poor stars can be found in @tominaga. We here focus on the contributions from HNe to the Fe in the ISM. In the regime of ${\rm [Fe/H]}\lesssim -1.5$, only HNe and normal SNe contribute Fe. The fraction of the Fe in a well-mixed ISM contributed by HNe can be estimated as $$\frac{\int_{25}^{50}Y_{\rm Fe}^{\rm HN}m^{-2.35}dm}
{\int_{12}^{25}Y_{\rm Fe}^{L^*}m^{-2.35}dm+
\int_{25}^{50}Y_{\rm Fe}^{\rm HN}m^{-2.35}dm}\sim 0.72,
\label{eq-fe}$$ where we have assumed a Salpeter initial mass function (IMF) with $m\sim 12$–25 and 25–50 (stellar mass in units of $M_\odot$) corresponding to progenitors of normal SNe and HNe, respectively, and we have taken $Y_{\rm Fe}^{\rm HN}\sim 0.5\,M_\odot$ and $Y_{\rm Fe}^{L^*}\sim 0.07\,M_\odot$ as the (mass) yields of Fe for an HN and a normal SN, respectively (see e.g., Figure 1 in @tominaga and references therein). The fraction of the Fe contributed by normal SNe is then $\sim 0.28$. This is close to the fraction of 0.24 assumed for the $L^*$ contribution to the $L$ mixture and used in Figure \[fig-csrfe\]. As $\sim 2/3$ of the solar Fe abundance came from SNe Ia, HNe and normal SNe contributed $\sim 24\%$ and $\sim 9\%$ of the solar Fe inventory, respectively.
Using the Salpeter IMF and the progenitor mass ranges assumed in equation (\[eq-fe\]), we estimate the relative rates of HNe and low-mass ($H$) and normal ($L^*$) SNe as $$R_{\rm HN}:R_H:R_{L^*}\sim\int_{25}^{50}m^{-2.35}dm:
\int_8^{11}m^{-2.35}dm:\int_{12}^{25}m^{-2.35}dm
\sim 0.36:0.96:1,
\label{eq-rphl}$$ where we have taken the mass range for the progenitors of low-mass SNe to be $m\sim 8$–11. The rate of all core-collapse SNe in the Galaxy is estimated to be $R_{\rm SN}^G\sim 10^{-2}$ yr$^{-1}$ (e.g., @cap). This gives the Galactic rates of HNe and low-mass and normal SNe as $R_{\rm HN}^G\sim 1.6\times 10^{-3}$ yr$^{-1}$, $R_H^G\sim 4.1\times10^{-3}$ yr$^{-1}$, and $R_{L^*}^G\sim 4.3\times10^{-3}$ yr$^{-1}$, respectively. Assuming that HNe and normal SNe provided a total mass $M_{\rm gas}^G$ of gas with $\sim 1/3$ of the solar Fe abundance over the period of $t_G\sim 10^{10}$ yr prior to the formation of the solar system, we have $$M_{\rm gas}^G\sim
\frac{(Y_{\rm Fe}^{\rm HN}R_{\rm HN}^G+Y_{\rm Fe}^{L^*}R_{L^*}^G)t_G}
{X_{{\rm Fe},\odot}/3}\sim 3.3\times 10^{10}\,M_\odot,$$ which is comparable to the total stellar mass in the Galactic disk at the present time. In the above equation, $X_{{\rm Fe},\odot}\approx 10^{-3}$ is the mass fraction of Fe in the sun [@anders]. As low-mass SNe are the predominant source for Eu, we can estimate the (mass) yield of Eu for this source as $$Y_{\rm Eu}^H\sim
\frac{X_{{\rm Eu},\odot}M_{\rm gas}^G}{R_H^Gt_G}
\sim 3\times 10^{-7}\,M_\odot,$$ where $X_{{\rm Eu},\odot}\approx 3.75\times 10^{-10}$ is the mass fraction of Eu in the sun [@anders]. Using the above Eu yield and $\log{\rm (Sr/Eu)}_H=1.41$ (see Table \[tab-yhl\]), we can estimate the (mass) yield of Sr for a single low-mass SN (see also QW07) as $$Y_{\rm Sr}^H=Y_{\rm Eu}^H
\left(\frac{\rm Sr}{\rm Eu}\right)_H
\left(\frac{A_{\rm Sr}}{A_{\rm Eu}}\right)
\sim 4.5\times 10^{-6}\,M_\odot,$$ where $A_{\rm Sr}\approx 88$ and $A_{\rm Eu}\approx 152$ are the atomic weights of Sr and Eu, respectively. The above estimate is consistent with the amount of ejecta from the neutrino-driven wind (e.g., @qw96).
Using $Y_{\rm Fe}^{L^*}\sim 0.07\,M_\odot$ and \[Sr/Fe\]$_{L^*}=0.30$ \[corresponding to $\log{\rm (Sr/Fe)}_{L^*}=-4.23$, see Tables \[tab-yhl\] and \[tab-mix\]\], we can estimate the (mass) yield of Sr for a single normal SN (see also QW07) as $$Y_{\rm Sr}^{L^*}=Y_{\rm Fe}^{L^*}
\left(\frac{\rm Sr}{\rm Fe}\right)_{L^*}
\left(\frac{A_{\rm Sr}}{A_{\rm Fe}}\right)
\sim 6.5\times 10^{-6}\,M_\odot,$$ where $A_{\rm Fe}\approx 56$ is the atomic weight of Fe. The above result is very close to the Sr yield estimated for low-mass SNe and consistent with the production of the CPR elements in the neutrino-driven wind.
We emphasize that we have included the large contributions to the solar Fe inventory from HNe in estimating the Eu and Sr yields for low-mass SNe. This then requires that the Fe contributions from normal SNe be reduced by a factor of $\sim 4$ from what were assumed previously. Likewise, the Galactic rate of $\sim 10^{-2}$ yr$^{-1}$ usually assumed for normal SNe must be reduced to $R_{L^*}^G\sim 4.3\times 10^{-3}$ yr$^{-1}$.
Effects of HNe, $H$, and $L^*$ Sources on Chemical Evolution of Halos {#sec-halo}
---------------------------------------------------------------------
We now estimate the enrichment resulting from a single HN or low-mass ($H$) or normal ($L^*$) SN. In the framework of hierarchical structure formation, chemical enrichment depends on the mass of the halo hosting the stellar sources and the extent to which the gas is bound to the halo after the explosions of these sources. In the simplest case, the gas is bound to the halo so that all sources contribute to the evolution of metal abundances in the halo. For these bound halos, the amount of gas to mix with the debris from a stellar explosion can be estimated as (e.g., @thornton) $$M_{\rm mix}\sim 3\times 10^4E_{{\rm expl},51}^{6/7}\,M_\odot,
\label{eq-mmix}$$ where $E_{{\rm expl},51}$ is the explosion energy in units of $10^{51}$ erg. The explosion energy of an HN is inferred from the light curves to be $E_{\rm expl}^{\rm HN}\sim (1$–$5)\times 10^{52}$ erg (see e.g., Figure 1 in @tominaga and references therein), which corresponds to $M_{\rm mix}^{\rm HN}\sim\mbox{(2--$9)\times 10^5\,M_\odot$}$. With $Y_{\rm Fe}^{\rm HN}\sim 0.5\,M_\odot$ and $X_{{\rm Fe},\odot}\approx 10^{-3}$, this gives $${\rm [Fe/H]}_{\rm HN}\sim\log\frac{Y_{\rm Fe}^{\rm HN}}
{X_{{\rm Fe},\odot}M_{\rm mix}^{\rm HN}}
\sim\mbox{$-3.3$ to $-2.6$}
\label{eq-fehn}$$ for enrichment of the ISM by a single HN in bound halos. Similarly, using $Y_{\rm Fe}^{L^*}\sim 0.07\,M_\odot$ and $M_{\rm mix}^{L^*}\sim 3\times 10^4\,M_\odot$ corresponding to $E_{\rm expl}^{L^*}\sim 10^{51}$ erg, we find that a single normal SN would result in ${\rm [Fe/H]}_{L^*}\sim -2.6$. As the relative rates of HNe and low-mass and normal SNe are comparable \[see equation (\[eq-rphl\])\], we expect that multiple types of stellar sources would be sampled at ${\rm [Fe/H]}>-2.6$ in bound halos. This may explain why HNe and the $L^*$ source can be effectively combined into the $L$ source and the two-component model with the $H$ and $L$ sources works rather well at such relatively high metallicities (see Figures \[fig-esr\]b and \[fig-yla\]b). To illustrate the effects of low-mass SNe, we consider the enrichment of Eu. Using $Y_{\rm Eu}^H\sim3\times 10^{-7}\,M_\odot$, $X_{{\rm Eu},\odot}\approx 3.75\times 10^{-10}$, and a mixing mass of $M_{\rm mix}^H\sim 3\times 10^4\,M_\odot$, we find that a single low-mass SN would result in ${\rm [Eu/H]}_H\sim -1.6$. This is close to the Eu abundances observed in CS 22892–052 and CS 31082–001 with ${\rm [Fe/H]}\approx -3$ but with extremely high enrichments of heavy $r$-elements.
The mixing mass $M_{\rm mix}^{\rm HN}\sim\mbox{(2--$9)\times 10^5\,M_\odot$}$ for an HN exceeds the amount of gas ($\sim 1.5\times 10^5\,M_\odot$) in a halo with a total mass of $M_h\sim 10^6\,M_\odot$ (only a fraction $\approx 0.15$ in gas and the rest in dark matter), in which the first stars are considered to have formed at redshift $z\sim 20$. On the other hand, the interaction of the HN debris with the gas in such a halo is complicated by the gravitational potential of the dark matter and by the heating of the gas due to the radiation from the HN progenitor. @kitayama studied the effects of photo-heating of the gas by a $200\,M_\odot$ VMS and found that with photo-heating, an explosion with $E_{\rm expl}\gtrsim 10^{50}$ erg is sufficient to blow out all the gas from a halo of $10^6\,M_\odot$. In contrast, without photo-heating, $10^{52}<E_{\rm expl}<10^{53}$ erg is required for the same halo. The effects of the dark matter potential are also important. The gravitational binding energy of the gas in a halo at $z\gg 1$ (e.g., @barkana) increases with the halo mass as $$E_{b,{\rm gas}}\approx 2\times 10^{49}
\left(\frac{M_h}{10^6\,M_\odot}\right)^{5/3}
\left(\frac{1+z}{10}\right)\ {\rm erg}.$$ To blow out all the gas from a halo of $3\times 10^6\,M_\odot$ requires $10^{52}<E_{\rm expl}<10^{53}$ erg and $E_{\rm expl}>10^{53}$ erg with and without photo-heating, respectively. The effects of photo-heating by HN progenitors of $\sim 25$–$50\,M_\odot$ were not studied. Based on the above results of @kitayama, we consider it reasonable to assume that an HN with $E_{\rm expl}\sim (1$–$5)\times 10^{52}$ erg would blow out all the gas from a halo of $\sim 10^6\,M_\odot$ but a low-mass or normal or faint SN with $E_{\rm expl}\sim 10^{51}$ erg or less would not.
@greif showed that subsequent to the blowing-out of the gas from a halo of $\sim 10^6\,M_\odot$ by an explosion with $E_{\rm expl}=10^{52}$ erg, collecting the debris and the swept-up gas requires the assemblage of a much larger halo of $\gtrsim 10^8\,M_\odot$. It is conceivable that the debris from several or more HNe originally hosted by different halos would be mixed and then assembled into the much larger halo. Stars that formed subsequently from this material would have sampled multiple HNe and have a quasi-uniform abundance pattern of the low-$A$ elements. The debris from a single HN mixed with $\sim 1.5\times 10^7\,M_\odot$ of gas in a halo of $\sim 10^8\,M_\odot$ would give ${\rm [Fe/H]}\sim -4.5$ (cf. equation \[\[eq-fehn\]\]). This is close to the lower end of the range of \[Fe/H\] values for metal-poor stars. @kitayama showed that even with photo-heating, to blow out all the gas from a halo of $\sim 10^7\,M_\odot$ requires $E_{\rm expl}>10^{53}$ erg. Consequently, after the debris from the first HNe in halos of $\sim 10^6\,M_\odot$ were collected into halos of $\gtrsim 10^8\,M_\odot$, the debris from all subsequent stellar explosions in the larger halos would be bound to these halos and mixed therein. As estimated above, a single HN results in ${\rm [Fe/H]}\sim -3.3$ to $-2.6$ and a single normal SN results in ${\rm [Fe/H]}\sim -2.6$ for bound halos. We therefore expect that for these halos, multiple types of stellar sources would be sampled at ${\rm [Fe/H]}>-2.6$ following a transition regime at $-4.5<{\rm [Fe/H]}\lesssim -3$. Considerations of bound halos with gas infall and normal star formation rates show that a metallicity at the level of ${\rm [Fe/H]}\sim -3$ is reached shortly after the onset of star formation in these halos [@qw04]. Thus, it is reasonable that ${\rm [Fe/H]}\sim -3$ signifies the end of a transition regime for the behavior of abundance patterns.
The occurrences of HNe and low-mass and normal SNe in bound halos would result in ${\rm [Sr/Fe]}=-0.10$ and ${\rm [Ba/Fe]}=-0.20$ for a well-mixed ISM (see Appendix \[sec-app1\]). As shown in Figure \[fig-srfe\], many stars have ${\rm [Sr/Fe]}\sim -2.5$ to $-1$ and ${\rm [Ba/Fe]}\sim -2.5$ to $-1$. Such low values of \[Sr/Fe\] and \[Ba/Fe\] largely reflect the composition of the IGM immediately following the blowing-out of the gas by the first HNe in halos of $\sim 10^6\,M_\odot$. As HNe produce the low-$A$ elements including Fe but no Sr or heavier elements, this IGM would have no Sr or Ba if none of the debris from the first low-mass and normal SNe escaped from halos of $\sim 10^6\,M_\odot$. The very low values of ${\rm [Sr/Fe]}\sim -2.5$ to $-1$ and ${\rm [Ba/Fe]}\sim -2.5$ to $-1$ may indicate that $\sim 1$–10% of the debris from the first low-mass and normal SNe escaped from their hosting halos. Alternatively, such low \[Sr/Fe\] and \[Ba/Fe\] values could be explained by the mixing of the IGM that fell into the halos forming at later times with small amounts of the debris from low-mass and normal SNe therein.
Exceptional Stars and Faint SNe {#sec-pan}
-------------------------------
The three-component model with HNe, $H$, and $L^*$ sources describes the available data on nearly all the stars very well. However, there are four exceptional stars that are identified as asterisks A, B, C, and D in Figures \[fig-esr\], \[fig-srfe\], \[fig-srba\], and \[fig-csrfe\]a. These stars are not anomalous in terms of Sr and Ba as shown by the above figures. However, Figure \[fig-pan\] shows that their abundance patterns of the low-$A$ elements differ greatly from those for HNe and all the other stars (see discussion of Figure \[fig-p\] in §\[sec-plowa\]). More specifically, while all stars have indistinguishable patterns of the explosive burning products from Si through Zn, these stars have extremely high abundances of the hydrostatic burning products Na, Mg, and Al relative to the explosive burning products. Such anomalous production patterns can be accounted for by faint SNe (e.g., @iwamoto), in which fall-back coupled with a weak explosion would hinder the ejection of the explosive burning products in the inner region much more than that of the hydrostatic burning products in the outer region. Due to the weak explosion, the debris from faint SNe would always be bound to their hosting halos. Mixing with the debris from low-mass and normal SNe to some small extent would preserve the anomalous patterns of the low-$A$ elements and add small amounts of Sr and Ba to the mixture. The stars that formed from this mixture would then appear as the exceptional stars discussed above. Using $Y_{\rm Fe}\sim 4\times 10^{-3}\,M_\odot$ and $E_{\rm expl}\sim 4\times 10^{50}$ erg inferred from the light curve of SN 1997D and $Y_{\rm Fe}\sim 2\times 10^{-3}\,M_\odot$ and $E_{\rm expl}\sim 6\times 10^{50}$ erg for SN 1999br (see Figure 1 in @tominaga and references therein), we find that for the corresponding mixing mass (see equations \[\[eq-mmix\]\]) a single faint SN like these two would result in ${\rm [Fe/H]}\sim -3.5$ (SN 1997D) and $\sim -4$ (SN 1999br) (cf. equation \[\[eq-fehn\]\]). These \[Fe/H\] values are close to those of the exceptional stars B (${\rm [Fe/H]}=-3.94$) and C (${\rm [Fe/H]}=-3.70$). This is compatible with a single faint SN giving rise to the anomalous abundance pattern of the low-$A$ elements in each exceptional star.
Conclusions {#sec-con}
===========
The two-component model of QW07 with the $H$ and $L$ sources provided a good description of the elemental abundances in metal-poor stars of the Galactic halo for $-2.7<{\rm [Fe/H]}\lesssim -1.5$. A key ingredient of that model is the attribution of the elements from Sr through Ag in metal-poor stars to the charged-particle reactions in neutrino-driven winds from nascent neutron stars but not to the $r$-process. However, that model cannot explain the great shortfall in the abundances of Sr, Y, and Zr relative to Fe for stars with ${\rm [Fe/H]}\lesssim -3$. The observations on these three CPR elements require that there be an early source producing Fe but no Sr or heavier elements. It is shown that if such a third source is assumed, then the data can be well explained by an extended three-component model. From considerations of the abundance patterns of the low-$A$ elements (from Na through Zn), it is concluded that this third source is most likely associated with HNe from massive stars of $\sim 25$–$50\,M_\odot$ that do not leave behind neutron stars. We here consider the third source to be HNe.
It is shown that the available data on the evolution of \[Sr/Fe\] with \[Ba/Fe\], that of \[Y/Fe\] with \[La/Fe\], and that of \[Zr/Fe\] with \[Ba/Fe\] are well described by the extended model with HNe, $H$ and $L$ sources, which also provides clear constraints on the abundance ratios that should be seen. It is further shown that the abundance patterns of the low-$A$ elements for HNe and the $L$ sources are not distinguishable. Considering that HNe are observed to be ongoing events in the present universe, we are forced to conclude that the $L$ source, which was assumed to have provided $\sim 1/3$ of the solar Fe inventory (the rest attributed to SNe Ia), is in fact a combination of normal SNe (from progenitors of $\sim 12$–$25\,M_\odot$), which we define as the $L^*$ source, and HNe. The net Fe contributions from HNe are found to be $\sim 3$ times larger than those from normal SNe.
Using the three-component model with HNe, $H$, and $L^*$ sources, we obtain a very good quantitative description of essentially all the available data. In particular, this model provides strong constraints on the evolution of \[Sr/Fe\] with \[Ba/Fe\] in terms of the allowed domain for these abundance ratios. It gives an equally good description of the data when any CPR element besides Sr (e.g., Y or Zr) or any heavy $r$-element besides Ba (e.g., La) is used. The model is also compatible with the non-$s$-process contributions to the solar abundances of all the CPR elements. The anomalous abundance patterns of the low-$A$ elements observed in a small number of stars appear to fit the description of faint SNe (e.g., @iwamoto), which are a rarer type of events from the same progenitor mass range as HNe but with even weaker explosion energies and smaller Fe yields than normal SNe (e.g., @nomoto06). The anomalous abundance patterns observed reflect the fact that faint SNe produce very little of the Fe group elements but an abundant amount of the elements from hydrostatic burning in their outer shells. This gives rise to the extremely high abundances of Na, Mg, and Al relative to Fe observed in the anomalous stars. The quasi-uniform abundance patterns of the elements from Si through Zn in all cases (including the stars with anomalous abundances of Na, Mg, and Al) appear to reflect some robustness in the outcome of explosive burning that may arise from the limited range of conditions required for such nucleosynthesis.
In this paper we used the elemental yield patterns for three prototypical model sources to calculate the abundances of an extensive set of elements (relative to hydrogen) for metal-poor stars with ${\rm [Fe/H]}\lesssim -1.5$. As the yield patterns adopted for the assumed prototypical sources are taken from the data on two template stars, they must represent the results of stellar nucleosynthesis. The full version of the three-component model appears very successful in calculating the abundances of the elements ranging from Na through Pt in stars with ${\rm [Fe/H]}\lesssim -1.5$. In contrast to this phenomenological approach, there are extensive studies of Galactic chemical evolution (GCE) that use the various theoretical results on the absolute yields of metals for different stellar types. These theoretical yields are not calculated from first principles, but are dependent on the parametrization used in the various stellar models. In those GCE studies, the elemental abundances for an individual star are not predicted. Instead, general trends for the elemental abundances are calculated assuming different sources, the rates at which they contribute, and a model of mixing in the ISM for different regions of the Galaxy. These results give a good broad description for typical elemental abundances in the general stellar population at higher metallicities of ${\rm [Fe/H]}>-1.5$. This is a regime in which the observational data are quite convergent with only limited variability. However, as anticipated by @gilroy and supported by the considerable scatter in the abundances of heavy elements observed in stars with ${\rm [Fe/H]}\lesssim -1.5$, the chemical composition of the ISM in the early Galaxy was extremely inhomogeneous. For ${\rm [Fe/H]}<-2$ there are gross discrepancies between the observations and the smoothed model of GCE. In no case does that model give the elemental abundances for an individual star. It is our view that the simple phenomenological model used here permits a clearer distinction between the different stellar sources contributing to the ISM and the IGM at early times. This model also gives specific testable predictions, which can be used to further check its validity.
In conclusion, we consider that the general three-component model with HNe, $H$, and $L^*$ sources provides a quantitative and self-consistent description of nearly all the available data on elemental abundances in stars with ${\rm [Fe/H]}\lesssim -1.5$. Further, HNe may be not only explosions from the first massive stars (i.e., the Population III stars much sought after by many) that provided a very early and variable inventory to the IGM through ejection of enriched gas from small halos, but also are important ongoing contributors to the chemical evolution of the universe.
We thank an anonymous reviewer for criticisms and suggestions that greatly improve the paper. This work was supported in part by DOE grants DE-FG02-87ER40328 (Y. Z. Q.) and DE-FG03-88ER13851 (G. J. W.), Caltech Division Contribution 9004 (1125). G. J. W. acknowledges NASA’s Cosmochemistry Program for research support provided through J. Nuth at the Goddard Space Flight Center. He also appreciates the generosity of the Epsilon Foundation.
Abundance Ratios in a Well-Mixed ISM {#sec-app1}
====================================
Using equation (\[eq-eh\]), we calculate the $H$ and $L$ contributions (Sr/H)$_{\odot,HL}$ to the solar Sr abundance as $$\left(\frac{\rm Sr}{\rm H}\right)_{\odot,HL}=
\left(\frac{\rm Sr}{\rm Eu}\right)_H\left(\frac{\rm Eu}{\rm H}\right)_{\odot,H}+
\left(\frac{\rm Sr}{\rm Fe}\right)_L\left(\frac{\rm Fe}{\rm H}\right)_{\odot,L}.
\label{eq-srs}$$ As Eu is essentially a pure heavy $r$-element, we take the $H$ contributions to the solar Eu abundance to be ${\rm (Eu/H)}_{\odot,H}\approx {\rm (Eu/H)}_\odot$. Allowing for contributions from SNe Ia, we take the $L$ contributions to the solar Fe abundance to be ${\rm (Fe/H)}_{\odot,L}\approx {\rm (Fe/H)}_\odot/3$. Using the yield ratios (Sr/Eu)$_H$ and (Sr/Fe)$_L$ given in Table \[tab-yhl\], we obtain $${\rm [Sr/Fe]}_{\rm mix}\equiv\log{\rm (Sr/H)}_{\odot,HL}-
\log{\rm (Fe/H)}_{\odot,L}-\log{\rm (Sr/Fe)}_\odot=-0.10,
\label{eq-srfem}$$ which we assume to be characteristic of an ISM with well-mixed $H$ and $L$ contributions. Here and throughout the paper (particularly when presenting the data from different observational studies), we have consistently adopted the solar abundances given by @ags05.
The $H$ contributions (Ba/H)$_{\odot,H}$ to the solar Ba abundance can be calculated as $$\left(\frac{\rm Ba}{\rm H}\right)_{\odot,H}=
\left(\frac{\rm Ba}{\rm Eu}\right)_H\left(\frac{\rm Eu}{\rm H}\right)_{\odot,H}
\approx\left(\frac{\rm Ba}{\rm Eu}\right)_H\left(\frac{\rm Eu}{\rm H}\right)_\odot.$$ Together with equation (\[eq-srs\]), this gives $${\rm [Sr/Ba]}_{\rm mix}\equiv\log{\rm (Sr/H)}_{\odot,HL}-
\log{\rm (Ba/H)}_{\odot,H}-\log{\rm (Sr/Ba)}_\odot=0.10$$ for an ISM with well-mixed $H$ and $L$ contributions. Combining the above equation with equation (\[eq-srfem\]) gives $${\rm [Ba/Fe]}_{\rm mix}={\rm [Sr/Fe]}_{\rm mix}-
{\rm [Sr/Ba]}_{\rm mix}=-0.20.$$
Other abundance ratios such as \[Y/Fe\]$_{\rm mix}$, \[Y/La\]$_{\rm mix}$, \[Zr/Fe\]$_{\rm mix}$, and \[Zr/Ba\]$_{\rm mix}$ for an ISM with well-mixed $H$ and $L$ contributions can be calculated similarly and are given in Table \[tab-mix\].
Abel, T., Bryan, G. L., & Norman, M. L. 2002, Science, 295, 93 Anders, E., Grevesse, N. 1989, , 53, 197 Aoki, W., et al. 2005, , 632, 611 Aoki, W., et al. 2006, , 639, 897 Aoki, W., et al. 2007, , 660, 747 Aoki, W., Norris, J. E., Ryan, S. G., Beers, T. C., & Ando, H. 2002, , 576, L141 Arlandini, C., et al. 1999, , 525, 886 Asplund, M., Grevesse, N., & Sauval, A. J. 2005, in ASP Conf. Ser. 336, Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, ed. T. G. Barnes III & F. N. Bash (San Francisco: ASP), 25 Barkana, R., & Loeb, A. 2001, Phys. Rep., 349, 125 Barklem, P. S., et al. 2005, , 439, 129 Bromm, V., & Larson, R. B. 2004, , 42, 79 Cappellaro, E., Evans, R., & Turatto, M. 1999, , 351, 459 Cayrel, R., et al. 2004, , 416, 1117 Chieffi, A., & Limongi, M. 2004, , 608, 405 Christlieb, N., et al. 2002, , 419, 904 Christlieb, N., et al. 2004, , 603, 708 Cohen, J. G., et al. 2007, , 659, L161 Cohen, J. G., et al. 2008, , 672, 320 Cowan, J. J., et al. 2002, , 572, 861 Depagne, E., et al. 2002, , 390, 187 Duncan, R. C., Shapiro, S. L., & Wasserman, I. 1986, , 309, 141 François, P., et al. 2007, , 476, 935 Frebel, A., et al. 2005, , 434, 871 Frebel, A., et al. 2007, , 658, 534 Fulbright, J. P., Rich, R. M., & Castro, S. 2004, , 612, 447 Galama, T. J., et al. 1998, , 395, 670 Gao, L., et al. 2007, , 378, 449 Gilroy, K. K., Sneden, C., Pilachowski, C. A., & Cowan, J. J. 1988, , 327, 298 Greif, T. H., Johnson, J. L., Bromm, V., & Klessen, R. S. 2007, , 670, 1 Heger, A., & Woosley, S. E. 2002, , 567, 532 Hill, V., et al. 2002, , 387, 560 Hoffman, R. D., Woosley, S. E., & Qian, Y.-Z. 1997, , 482, 951 Honda, S., et al. 2004, , 607, 474 Honda, S., Aoki, W., Ishimaru, Y., Wanajo, S., & Ryan, S. G. 2006, , 643, 1180 Iwamoto, K., et al. 1998, , 395, 672 Iwamoto, N., Umeda, H., Tominaga, N., Nomoto, K., & Maeda, K. 2005, Science, 309, 451 Janka, H.-T., Müller, B., Kitaura, F. S., & Buras, R. 2007, preprint (arXiv: 0712.4237 \[astro-ph\]) Johnson, J. A. 2002, , 139, 219 Johnson, J. A., & Bolte, M. 2002, , 579, 616 Kitayama, T., & Yoshida, N. 2005, , 630, 675 McWilliam, A., Preston, G. W., Sneden, C., & Searle, L. 1995, , 109, 2757 Meyer, B. S., Mathews, G. J., Howard, W. M., Woosley, S. E., & Hoffman, R. D. 1992, , 399, 656 Ning, H., Qian, Y.-Z., & Meyer, B. S. 2007, , 667, L159 Nomoto, K. 1984, , 277, 791 Nomoto, K. 1987, , 322, 206 Nomoto, K., Tominaga, N., Umeda, H., Kobayashi, C., & Maeda, K. 2006, Nucl. Phys. A, 777, 424 Norris, J. E., Ryan, S. G., & Beers, T. C. 2001, , 561, 1034 Norris, J. E., et al. 2007, , 670, 774 O’Shea, B. W., & Norman, M. L. 2007, , 654, 66 Qian, Y.-Z., Sargent, W. L. W., & Wasserburg, G. J. 2002, , 569, L61 Qian, Y.-Z., & Wasserburg, G. J. 2001, , 559, 925 Qian, Y.-Z., & Wasserburg, G. J. 2002, , 567, 515 Qian, Y.-Z., & Wasserburg, G. J. 2004, , 612, 615 Qian, Y.-Z., & Wasserburg, G. J. 2007, Phys. Rep., 442, 237 (QW07) Qian, Y.-Z., & Woosley, S. E. 1996, , 471, 331 Simmerer, J., et al. 2004, , 617, 1091 Sneden, C., et al. 2003, , 591, 936 Takahashi, K., Witti, J., & Janka, H.-T. 1994, , 286, 857 Thielemann, F.-K., Nomoto, K., & Hashimoto, M. 1996, , 460, 408 Thornton, K., Gaudlitz, M., Janka, H.-Th., & Steinmetz, M. 1998, , 500, 95 Tominaga, N., Umeda, H., & Nomoto, K. 2007, , 660, 516 Travaglio, C., et al. 2004, , 601,864 Turatto, M., et al. 1998, , 498, L129 Wasserburg, G. J., Busso, M., & Gallino, R. 1996, , 466, L109 Woosley, S. E., & Hoffman, R. D. 1992, , 395, 202 Woosley, S. E., & Weaver, T. A. 1995, , 101, 181 Woosley, S. E., Wilson, J. R., Mathews, G. J., Hoffman, R. D., & Meyer, B. S. 1994, , 433, 229 Yoshida, N., Omukai, K., Hernquist, L., & Abel, T. 2006, , 652, 6
[crrcrr]{} Ba&0.97&$-\infty$&Tm&$-0.45$&$-\infty$\
La&0.26&$-\infty$&Yb&0.26&$-\infty$\
Ce&0.46&$-\infty$&Lu&$-0.50$&$-\infty$\
Pr&$-0.03$&$-\infty$&Hf&$-0.13$&$-\infty$\
Nd&0.58&$-\infty$&Ta&$-0.88$&$-\infty$\
Sm&0.28&$-\infty$&W&$-0.20$&$-\infty$\
Gd&0.48&$-\infty$&Re&$-0.27$&$-\infty$\
Tb&$-0.22$&$-\infty$&Os&0.82&$-\infty$\
Dy&0.56&$-\infty$&Ir&0.85&$-\infty$\
Ho&$-0.05$&$-\infty$&Pt&1.14&$-\infty$\
Er&0.35&$-\infty$&Au&0.28&$-\infty$\
[crrrr]{} Fe&$-\infty$&$-\infty$&0&0\
Eu&0&$-0.97$&$-\infty$&$-\infty$\
Ba&0.97&0&$-\infty$&$-\infty$\
Sr&1.41&0.44&$-4.85$&$-4.23$\
Y&0.53&$-0.44$&$-5.67$&$-5.05$\
Zr&1.19&0.22&$-5.02$&$-4.40$\
Nb&0.15&$-0.82$&$-6.22$&$-5.60$\
Mo&0.55&$-0.42$&$-5.61$&$-4.99$\
Ru&1.03&0.06&$-5.60$&$-4.98$\
Rh&0.40&$-0.57$&$<-5.94$&$<-5.32$\
Pd&0.66&$-0.31$&$-6.10$&$-5.48$\
Ag&0.07&$-0.90$&$-6.62$&$-6.00$\
[lrrr]{} ${\rm [E/Fe]}_{\rm mix}$&$-0.10$&$-0.24$&0.04\
${\rm [E/Ba]}_{\rm mix}$&0.10&$-0.03$&0.24\
${\rm [E/La]}_{\rm mix}$&$-0.23$&$-0.36$&$-0.09$\
${\rm [E/Ba]}_H$&$-0.31$&$-0.48$&$-0.20$\
${\rm [E/La]}_H$&$-0.64$&$-0.81$&$-0.53$\
${\rm [E/Fe]}_L$&$-0.32$&$-0.43$&$-0.16$\
${\rm [E/Fe]}_{L^*}$&0.30&0.19&0.46\
[cccc]{} stellar types&HNe from stars&low-mass SNe from&normal SNe from\
&of $\sim 25$–$50\,M_\odot$&stars of $\sim 8$–$11\,M_\odot$& stars of $\sim 12$–$25\,M_\odot$\
&&&\
remnants&black holes&neutron stars&neutron stars\
&&&\
nucleosynthetic&dominant source for&source for CPR elements& source for low-$A$\
characteristics&low-$A$ elements&from Sr through Ag& and CPR elements\
&from Na through Zn&and only source for heavy&\
&$f_{\rm Fe, HN}^\odot\sim 0.24$ &$r$-elements with $A>130$& $f_{{\rm Fe},L^*}^\odot\sim 0.09$\
[cccccc]{} Sr&2.92&2.34&0.26&0.15&0.20\
Y&2.21&1.50&0.19&0.08&0.26\
Zr&2.59&2.15&0.36&0.17&0.33\
Nb&1.42&1.01&0.39&0.15&0.31\
Mo&1.92&1.54&0.42&0.50&0.61\
Ru&1.84&1.77&0.85&0.68&0.76\
Rh&1.12&$>0.92$&$>0.63$&0.86&0.90\
Pd&1.69&1.35&0.46&0.54&0.64\
Ag&0.94&0.79&0.71&0.80&0.91\
![(a) High-resolution data on $\log\epsilon({\rm Sr})$ vs. \[Fe/H\] (squares: @johnson02, pluses: @honda04, diamonds: @aoki05, circles: @francois07, crosses: @cohen08, asterisks representing stars with very high C and O abundances and anomalous abundance patterns of the low-$A$ elements: @aoki06 (A, HE 1327–2326); @depagne02 (B, CS 22949–037); @aoki02 (C, CS 29498–043); @aoki07 (D, BS 16934–002), downward arrows indicating upper limits: @christlieb04 [@fulbright04; @frebel07; @cohen07; @norris07]). Symbols connected with a line indicate results for the same star assuming two different atmospheric models (subgiant vs. dwarf). Typical observational errors in $\log\epsilon({\rm Sr})$ are $\sim 0.2$–0.3 dex. The solid line is for an ISM with well-mixed $H$ and $L$ contributions. The data mostly cluster around this line but drastically depart to low $\log\epsilon({\rm Sr})$ values for ${\rm [Fe/H]}\lesssim -3$. (b) Comparison of the two-component model of QW07 and the observations in terms of $\Delta\log\epsilon({\rm Sr})\equiv
\log\epsilon_{\rm cal}({\rm Sr})-\log\epsilon_{\rm obs}({\rm Sr})$ as a function of \[Fe/H\] for those stars shown in (a) that have observed Ba abundances. In general, the model grossly overestimates the Sr abundance below ${\rm [Fe/H]}\sim -2.7$. However, the calculated Sr abundance for HE 1327–2326 with ${\rm [Fe/H]}=-5.45$ (asterisk A) using the upper limit on its Ba abundance appears to be in good agreement with its observed Sr abundance. Measurement of the exact Ba abundance in this star will provide an extremely important test of the model.[]{data-label="fig-esr"}](f1a.eps "fig:") ![(a) High-resolution data on $\log\epsilon({\rm Sr})$ vs. \[Fe/H\] (squares: @johnson02, pluses: @honda04, diamonds: @aoki05, circles: @francois07, crosses: @cohen08, asterisks representing stars with very high C and O abundances and anomalous abundance patterns of the low-$A$ elements: @aoki06 (A, HE 1327–2326); @depagne02 (B, CS 22949–037); @aoki02 (C, CS 29498–043); @aoki07 (D, BS 16934–002), downward arrows indicating upper limits: @christlieb04 [@fulbright04; @frebel07; @cohen07; @norris07]). Symbols connected with a line indicate results for the same star assuming two different atmospheric models (subgiant vs. dwarf). Typical observational errors in $\log\epsilon({\rm Sr})$ are $\sim 0.2$–0.3 dex. The solid line is for an ISM with well-mixed $H$ and $L$ contributions. The data mostly cluster around this line but drastically depart to low $\log\epsilon({\rm Sr})$ values for ${\rm [Fe/H]}\lesssim -3$. (b) Comparison of the two-component model of QW07 and the observations in terms of $\Delta\log\epsilon({\rm Sr})\equiv
\log\epsilon_{\rm cal}({\rm Sr})-\log\epsilon_{\rm obs}({\rm Sr})$ as a function of \[Fe/H\] for those stars shown in (a) that have observed Ba abundances. In general, the model grossly overestimates the Sr abundance below ${\rm [Fe/H]}\sim -2.7$. However, the calculated Sr abundance for HE 1327–2326 with ${\rm [Fe/H]}=-5.45$ (asterisk A) using the upper limit on its Ba abundance appears to be in good agreement with its observed Sr abundance. Measurement of the exact Ba abundance in this star will provide an extremely important test of the model.[]{data-label="fig-esr"}](f1b.eps "fig:")
![Evolution of \[Sr/Fe\] with \[Ba/Fe\]. Data symbols are the same as in Figure \[fig-esr\] except that the left-pointing arrows indicate the upper limits on \[Ba/Fe\]. Typical observational errors in \[Sr/Fe\] and \[Ba/Fe\] are $\sim 0.1$–0.25 dex. The curves show ${\rm [Sr/Fe]}=\log\left(10^{{\rm [Sr/Ba]}_H+{\rm [Ba/Fe]}}+ f_{{\rm Fe},L}
\times 10^{{\rm [Sr/Fe]}_L}\right)$ based on the three-component model with the $H$ and $L$ sources and a third source (HNe) for $f_{{\rm Fe},L}=0$ (dot-dot-dashed), 0.1 (dashed), 0.5 (dot-dashed), and 1 (solid). The parameter $f_{{\rm Fe},L}$ is the fraction of Fe contributed by the $L$ source ($f_{{\rm Fe},L}=0$ corresponds to all the Fe being from the third source). The filled circle labeled “$L$” indicates the value of \[Sr/Fe\]$_L=-0.32$ for the $L$ source. Almost all of the data lie within the allowed region of the model. Note the presence of quite a few data on the curve for $f_{{\rm Fe},L}=0$ as well as the abundant data near the curve for $f_{{\rm Fe},L}=1$.[]{data-label="fig-srfe"}](f2.eps)
![Data on \[Sr/Ba\] vs. \[Fe/H\]. Symbols are the same as in Figure \[fig-esr\] except that the upward arrows represent lower limits on \[Sr/Ba\]. Typical observational errors in \[Sr/Ba\] are $\sim 0.2$–0.3 dex. The dashed line shows the lower bound of \[Sr/Ba\]$_H=-0.31$ for pure $H$ contributions and the solid line shows the value of \[Sr/Ba\]$_{\rm mix}=0.10$ for an ISM with well-mixed $H$ and $L$ contributions. Data above the solid line represent higher proportions of $L$ contributions than in the well-mixed case. Note that considering observational uncertainties, there are no serious exceptions to the rules of Fe, Sr, and Ba production for the $H$ and $L$ sources and the third source (HNe) in the three-component model.[]{data-label="fig-srba"}](f3.eps)
![(a) Medium-resolution data on $\log\epsilon({\rm Sr})$ vs. \[Fe/H\] from the HERES survey [@barklem05]. Typical observational errors in $\log\epsilon({\rm Sr})$ are $\sim 0.3$ dex. The data in general follow the same distribution as presented in Figure \[fig-esr\]a for the high-resolution data, where the same solid line for an ISM with well-mixed $H$ and $L$ contributions is also shown. The majority of the data cluster around the solid line but there is a great dispersion below ${\rm [Fe/H]}\sim -2.5$. (b) Evolution of \[Sr/Fe\] with \[Ba/Fe\] for the HERES sample. Typical observational errors in \[Sr/Fe\] and \[Ba/Fe\] are $\sim 0.3$ dex. The data distribution is again quite similar to the case for the high-resolution data presented in Figure \[fig-srfe\], where the same curves are shown. A number of data lie far to the right of and below the curve for $f_{{\rm Fe},L}=0$. We consider that the corresponding stars (HE 0231–4016, HE 0305–4520, HE 0430–4404, HE 1430–1123, HE 2150–0825, HE 2156–3130, HE 2227–4044, and HE 2240–0412) may have received large $s$-process contributions. This can be tested by high-resolution observations covering more elements heavier than Ba.[]{data-label="fig-heres"}](f4a.eps "fig:") ![(a) Medium-resolution data on $\log\epsilon({\rm Sr})$ vs. \[Fe/H\] from the HERES survey [@barklem05]. Typical observational errors in $\log\epsilon({\rm Sr})$ are $\sim 0.3$ dex. The data in general follow the same distribution as presented in Figure \[fig-esr\]a for the high-resolution data, where the same solid line for an ISM with well-mixed $H$ and $L$ contributions is also shown. The majority of the data cluster around the solid line but there is a great dispersion below ${\rm [Fe/H]}\sim -2.5$. (b) Evolution of \[Sr/Fe\] with \[Ba/Fe\] for the HERES sample. Typical observational errors in \[Sr/Fe\] and \[Ba/Fe\] are $\sim 0.3$ dex. The data distribution is again quite similar to the case for the high-resolution data presented in Figure \[fig-srfe\], where the same curves are shown. A number of data lie far to the right of and below the curve for $f_{{\rm Fe},L}=0$. We consider that the corresponding stars (HE 0231–4016, HE 0305–4520, HE 0430–4404, HE 1430–1123, HE 2150–0825, HE 2156–3130, HE 2227–4044, and HE 2240–0412) may have received large $s$-process contributions. This can be tested by high-resolution observations covering more elements heavier than Ba.[]{data-label="fig-heres"}](f4b.eps "fig:")
![(a) High-resolution data on $\log\epsilon({\rm Y})$ vs. \[Fe/H\] (squares: @johnson; circles: @francois07). Downward arrows indicate upper limits. The solid line is for an ISM with well-mixed $H$ and $L$ contributions. Note that many stars with \[Fe/H\] $\lesssim -3$ lie below this line. (b) Comparison of the two-component model of QW07 and the observations in terms of $\Delta\log\epsilon({\rm Y})\equiv
\log\epsilon_{\rm cal}({\rm Y})-\log\epsilon_{\rm obs}({\rm Y})$ as a function of \[Fe/H\] for the stars shown in (a). For those stars with only upper limits on the La abundance, only the $L$ contributions to Y are calculated to give the lower limits on $\Delta\log\epsilon({\rm Y})$ shown as the upward arrows. The two-component model grossly overestimates the Y abundances at ${\rm [Fe/H]}\lesssim -3$ but describes the observations very well at ${\rm [Fe/H]}> -3$. (c) Evolution of \[Y/Fe\] with \[La/Fe\] for those stars shown in (a) that have observed Y abundances. Left-pointing arrows indicate upper limits on \[La/Fe\]. The curves are calculated from the three-component model with the $H$ and $L$ sources and a third source (HNe) for $f_{{\rm Fe},L}=0$ (dot-dot-dashed), 0.1 (dashed), 0.5 (dot-dashed), and 1 (solid). The filled circle labeled “$L$” indicates the value of \[Y/Fe\]$_L=-0.43$ for the $L$ source. Note that the data mostly lie between the curves for $f_{{\rm Fe},L}=0$ and 1. The anomalous star CS 22968–014 is an exception. (d) Data on \[Y/La\] vs. \[Fe/H\] for the stars shown in (c). Except for CS 22968–014, all the other stars are consistent with the lower bound of ${\rm [Y/La]}\geq{\rm [Y/La]}_H=-0.81$ from the three-component model. Typical observational errors in $\log\epsilon({\rm Y})$ \[see (a)\], \[Y/Fe\], \[La/Fe\], and \[Y/La\] are $\sim 0.1$–0.3 dex.[]{data-label="fig-yla"}](f5a.eps "fig:") ![(a) High-resolution data on $\log\epsilon({\rm Y})$ vs. \[Fe/H\] (squares: @johnson; circles: @francois07). Downward arrows indicate upper limits. The solid line is for an ISM with well-mixed $H$ and $L$ contributions. Note that many stars with \[Fe/H\] $\lesssim -3$ lie below this line. (b) Comparison of the two-component model of QW07 and the observations in terms of $\Delta\log\epsilon({\rm Y})\equiv
\log\epsilon_{\rm cal}({\rm Y})-\log\epsilon_{\rm obs}({\rm Y})$ as a function of \[Fe/H\] for the stars shown in (a). For those stars with only upper limits on the La abundance, only the $L$ contributions to Y are calculated to give the lower limits on $\Delta\log\epsilon({\rm Y})$ shown as the upward arrows. The two-component model grossly overestimates the Y abundances at ${\rm [Fe/H]}\lesssim -3$ but describes the observations very well at ${\rm [Fe/H]}> -3$. (c) Evolution of \[Y/Fe\] with \[La/Fe\] for those stars shown in (a) that have observed Y abundances. Left-pointing arrows indicate upper limits on \[La/Fe\]. The curves are calculated from the three-component model with the $H$ and $L$ sources and a third source (HNe) for $f_{{\rm Fe},L}=0$ (dot-dot-dashed), 0.1 (dashed), 0.5 (dot-dashed), and 1 (solid). The filled circle labeled “$L$” indicates the value of \[Y/Fe\]$_L=-0.43$ for the $L$ source. Note that the data mostly lie between the curves for $f_{{\rm Fe},L}=0$ and 1. The anomalous star CS 22968–014 is an exception. (d) Data on \[Y/La\] vs. \[Fe/H\] for the stars shown in (c). Except for CS 22968–014, all the other stars are consistent with the lower bound of ${\rm [Y/La]}\geq{\rm [Y/La]}_H=-0.81$ from the three-component model. Typical observational errors in $\log\epsilon({\rm Y})$ \[see (a)\], \[Y/Fe\], \[La/Fe\], and \[Y/La\] are $\sim 0.1$–0.3 dex.[]{data-label="fig-yla"}](f5b.eps "fig:") ![(a) High-resolution data on $\log\epsilon({\rm Y})$ vs. \[Fe/H\] (squares: @johnson; circles: @francois07). Downward arrows indicate upper limits. The solid line is for an ISM with well-mixed $H$ and $L$ contributions. Note that many stars with \[Fe/H\] $\lesssim -3$ lie below this line. (b) Comparison of the two-component model of QW07 and the observations in terms of $\Delta\log\epsilon({\rm Y})\equiv
\log\epsilon_{\rm cal}({\rm Y})-\log\epsilon_{\rm obs}({\rm Y})$ as a function of \[Fe/H\] for the stars shown in (a). For those stars with only upper limits on the La abundance, only the $L$ contributions to Y are calculated to give the lower limits on $\Delta\log\epsilon({\rm Y})$ shown as the upward arrows. The two-component model grossly overestimates the Y abundances at ${\rm [Fe/H]}\lesssim -3$ but describes the observations very well at ${\rm [Fe/H]}> -3$. (c) Evolution of \[Y/Fe\] with \[La/Fe\] for those stars shown in (a) that have observed Y abundances. Left-pointing arrows indicate upper limits on \[La/Fe\]. The curves are calculated from the three-component model with the $H$ and $L$ sources and a third source (HNe) for $f_{{\rm Fe},L}=0$ (dot-dot-dashed), 0.1 (dashed), 0.5 (dot-dashed), and 1 (solid). The filled circle labeled “$L$” indicates the value of \[Y/Fe\]$_L=-0.43$ for the $L$ source. Note that the data mostly lie between the curves for $f_{{\rm Fe},L}=0$ and 1. The anomalous star CS 22968–014 is an exception. (d) Data on \[Y/La\] vs. \[Fe/H\] for the stars shown in (c). Except for CS 22968–014, all the other stars are consistent with the lower bound of ${\rm [Y/La]}\geq{\rm [Y/La]}_H=-0.81$ from the three-component model. Typical observational errors in $\log\epsilon({\rm Y})$ \[see (a)\], \[Y/Fe\], \[La/Fe\], and \[Y/La\] are $\sim 0.1$–0.3 dex.[]{data-label="fig-yla"}](f5c.eps "fig:") ![(a) High-resolution data on $\log\epsilon({\rm Y})$ vs. \[Fe/H\] (squares: @johnson; circles: @francois07). Downward arrows indicate upper limits. The solid line is for an ISM with well-mixed $H$ and $L$ contributions. Note that many stars with \[Fe/H\] $\lesssim -3$ lie below this line. (b) Comparison of the two-component model of QW07 and the observations in terms of $\Delta\log\epsilon({\rm Y})\equiv
\log\epsilon_{\rm cal}({\rm Y})-\log\epsilon_{\rm obs}({\rm Y})$ as a function of \[Fe/H\] for the stars shown in (a). For those stars with only upper limits on the La abundance, only the $L$ contributions to Y are calculated to give the lower limits on $\Delta\log\epsilon({\rm Y})$ shown as the upward arrows. The two-component model grossly overestimates the Y abundances at ${\rm [Fe/H]}\lesssim -3$ but describes the observations very well at ${\rm [Fe/H]}> -3$. (c) Evolution of \[Y/Fe\] with \[La/Fe\] for those stars shown in (a) that have observed Y abundances. Left-pointing arrows indicate upper limits on \[La/Fe\]. The curves are calculated from the three-component model with the $H$ and $L$ sources and a third source (HNe) for $f_{{\rm Fe},L}=0$ (dot-dot-dashed), 0.1 (dashed), 0.5 (dot-dashed), and 1 (solid). The filled circle labeled “$L$” indicates the value of \[Y/Fe\]$_L=-0.43$ for the $L$ source. Note that the data mostly lie between the curves for $f_{{\rm Fe},L}=0$ and 1. The anomalous star CS 22968–014 is an exception. (d) Data on \[Y/La\] vs. \[Fe/H\] for the stars shown in (c). Except for CS 22968–014, all the other stars are consistent with the lower bound of ${\rm [Y/La]}\geq{\rm [Y/La]}_H=-0.81$ from the three-component model. Typical observational errors in $\log\epsilon({\rm Y})$ \[see (a)\], \[Y/Fe\], \[La/Fe\], and \[Y/La\] are $\sim 0.1$–0.3 dex.[]{data-label="fig-yla"}](f5d.eps "fig:")
![Comparison of the abundance patterns of the low-$A$ elements for the third source (HNe) and the $L$ source. The patterns for the third source are taken from five stars that lie on the curve for $f_{{\rm Fe},L}=0$ in Figure \[fig-srfe\] (open square: BD $-18^\circ 5550$, ${\rm [Fe/H]}=-2.98$, @johnson; open circle: CS 30325–094, ${\rm [Fe/H]}=-3.25$, open diamond: CS 22885–096, ${\rm [Fe/H]}=-3.73$, open triangle: CS 29502–042, ${\rm [Fe/H]}=-3.14$, @cayrel; plus: BS 16085–050, ${\rm [Fe/H]}=-2.85$, @honda04). Those for the $L$ source are from three stars that lie on the curve for $f_{{\rm Fe},L}=1$ in Figure \[fig-srfe\] (filled square: BD $+4^\circ 2621$, @johnson; filled circle: HD 122563, @honda04 [@honda06]; filled diamond: CS 29491–053, @cayrel). The solid curve represents a star (BD $+17^\circ 3248$, @cowan02) with a relatively high value of ${\rm [Fe/H]}=-2$. Typical observational errors in \[E/Fe\] are $\sim 0.1$–0.25 dex. All the patterns shown are essentially indistinguishable.[]{data-label="fig-p"}](f6.eps)
![(a) Evolution of \[Sr/Fe\] with \[Ba/Fe\] in the three-component model with HNe, $H$, and $L^*$ sources compared with the high-resolution data (analogous to Figure \[fig-srfe\]). (b) The same relationships compared with the medium-resolution data (analogous to Figure \[fig-heres\]b). (c) Evolution of \[Y/Fe\] with \[La/Fe\] compared with the high-resolution data (analogous to Figure \[fig-yla\]c). (d) Evolution of \[Zr/Fe\] with \[Ba/Fe\] compared with the high-resolution data (squares: @johnson; diamonds: @aoki05; circles: @francois07). Typical observational errors in \[Zr/Fe\] and \[Ba/Fe\] are $\sim 0.1$–0.25 dex. The parameter $f_{{\rm Fe},L^*}$ is the fraction of Fe contributed by the $L^*$ source. The filled circles labeled “$L$” indicate the (number) yield ratios of \[Sr/Fe\]$_L=-0.32$ (a) and (b), \[Y/Fe\]$_L=-0.43$ (c), and \[Zr/Fe\]$_L=-0.16$ (d) for the $L$ source, while those labeled “$L^*$” indicate the yield ratios of \[Sr/Fe\]$_{L^*}=0.30$ (a) and (b), \[Y/Fe\]$_{L^*}=0.19$ (c), and \[Zr/Fe\]$_{L^*}=0.46$ (d) for the $L^*$ source (see Table \[tab-mix\]). The increase from the $L$ to the $L^*$ yield ratio is the same for all the CPR elements. Note that except for the data points far to the right of and below the curve for $f_{{\rm Fe},L^*}=0$ in (b), which may represent stars with large $s$-process contributions, and the anomalous star CS 22968–014 noted in the text, essentially all the data lie inside the allowed region bounded by the curves for $f_{{\rm Fe},L^*}=0$ and 1.[]{data-label="fig-csrfe"}](f7a.eps "fig:") ![(a) Evolution of \[Sr/Fe\] with \[Ba/Fe\] in the three-component model with HNe, $H$, and $L^*$ sources compared with the high-resolution data (analogous to Figure \[fig-srfe\]). (b) The same relationships compared with the medium-resolution data (analogous to Figure \[fig-heres\]b). (c) Evolution of \[Y/Fe\] with \[La/Fe\] compared with the high-resolution data (analogous to Figure \[fig-yla\]c). (d) Evolution of \[Zr/Fe\] with \[Ba/Fe\] compared with the high-resolution data (squares: @johnson; diamonds: @aoki05; circles: @francois07). Typical observational errors in \[Zr/Fe\] and \[Ba/Fe\] are $\sim 0.1$–0.25 dex. The parameter $f_{{\rm Fe},L^*}$ is the fraction of Fe contributed by the $L^*$ source. The filled circles labeled “$L$” indicate the (number) yield ratios of \[Sr/Fe\]$_L=-0.32$ (a) and (b), \[Y/Fe\]$_L=-0.43$ (c), and \[Zr/Fe\]$_L=-0.16$ (d) for the $L$ source, while those labeled “$L^*$” indicate the yield ratios of \[Sr/Fe\]$_{L^*}=0.30$ (a) and (b), \[Y/Fe\]$_{L^*}=0.19$ (c), and \[Zr/Fe\]$_{L^*}=0.46$ (d) for the $L^*$ source (see Table \[tab-mix\]). The increase from the $L$ to the $L^*$ yield ratio is the same for all the CPR elements. Note that except for the data points far to the right of and below the curve for $f_{{\rm Fe},L^*}=0$ in (b), which may represent stars with large $s$-process contributions, and the anomalous star CS 22968–014 noted in the text, essentially all the data lie inside the allowed region bounded by the curves for $f_{{\rm Fe},L^*}=0$ and 1.[]{data-label="fig-csrfe"}](f7b.eps "fig:") ![(a) Evolution of \[Sr/Fe\] with \[Ba/Fe\] in the three-component model with HNe, $H$, and $L^*$ sources compared with the high-resolution data (analogous to Figure \[fig-srfe\]). (b) The same relationships compared with the medium-resolution data (analogous to Figure \[fig-heres\]b). (c) Evolution of \[Y/Fe\] with \[La/Fe\] compared with the high-resolution data (analogous to Figure \[fig-yla\]c). (d) Evolution of \[Zr/Fe\] with \[Ba/Fe\] compared with the high-resolution data (squares: @johnson; diamonds: @aoki05; circles: @francois07). Typical observational errors in \[Zr/Fe\] and \[Ba/Fe\] are $\sim 0.1$–0.25 dex. The parameter $f_{{\rm Fe},L^*}$ is the fraction of Fe contributed by the $L^*$ source. The filled circles labeled “$L$” indicate the (number) yield ratios of \[Sr/Fe\]$_L=-0.32$ (a) and (b), \[Y/Fe\]$_L=-0.43$ (c), and \[Zr/Fe\]$_L=-0.16$ (d) for the $L$ source, while those labeled “$L^*$” indicate the yield ratios of \[Sr/Fe\]$_{L^*}=0.30$ (a) and (b), \[Y/Fe\]$_{L^*}=0.19$ (c), and \[Zr/Fe\]$_{L^*}=0.46$ (d) for the $L^*$ source (see Table \[tab-mix\]). The increase from the $L$ to the $L^*$ yield ratio is the same for all the CPR elements. Note that except for the data points far to the right of and below the curve for $f_{{\rm Fe},L^*}=0$ in (b), which may represent stars with large $s$-process contributions, and the anomalous star CS 22968–014 noted in the text, essentially all the data lie inside the allowed region bounded by the curves for $f_{{\rm Fe},L^*}=0$ and 1.[]{data-label="fig-csrfe"}](f7c.eps "fig:") ![(a) Evolution of \[Sr/Fe\] with \[Ba/Fe\] in the three-component model with HNe, $H$, and $L^*$ sources compared with the high-resolution data (analogous to Figure \[fig-srfe\]). (b) The same relationships compared with the medium-resolution data (analogous to Figure \[fig-heres\]b). (c) Evolution of \[Y/Fe\] with \[La/Fe\] compared with the high-resolution data (analogous to Figure \[fig-yla\]c). (d) Evolution of \[Zr/Fe\] with \[Ba/Fe\] compared with the high-resolution data (squares: @johnson; diamonds: @aoki05; circles: @francois07). Typical observational errors in \[Zr/Fe\] and \[Ba/Fe\] are $\sim 0.1$–0.25 dex. The parameter $f_{{\rm Fe},L^*}$ is the fraction of Fe contributed by the $L^*$ source. The filled circles labeled “$L$” indicate the (number) yield ratios of \[Sr/Fe\]$_L=-0.32$ (a) and (b), \[Y/Fe\]$_L=-0.43$ (c), and \[Zr/Fe\]$_L=-0.16$ (d) for the $L$ source, while those labeled “$L^*$” indicate the yield ratios of \[Sr/Fe\]$_{L^*}=0.30$ (a) and (b), \[Y/Fe\]$_{L^*}=0.19$ (c), and \[Zr/Fe\]$_{L^*}=0.46$ (d) for the $L^*$ source (see Table \[tab-mix\]). The increase from the $L$ to the $L^*$ yield ratio is the same for all the CPR elements. Note that except for the data points far to the right of and below the curve for $f_{{\rm Fe},L^*}=0$ in (b), which may represent stars with large $s$-process contributions, and the anomalous star CS 22968–014 noted in the text, essentially all the data lie inside the allowed region bounded by the curves for $f_{{\rm Fe},L^*}=0$ and 1.[]{data-label="fig-csrfe"}](f7d.eps "fig:")
![Comparison of the abundance patterns of the low-$A$ elements for HNe and faint SNe. The patterns for HNe are taken to be the same as those for the third source shown in Figure \[fig-p\] and the data on CS 22885–096 (open diamonds connected by line segments) are shown here as a typical example. The patterns in the anomalous stars (A, B, C, and D) are assumed to represent faint SNe \[filled circle: @aoki06 (A, HE 1327–2326); filled triangle: @depagne02 (B, CS 22949–037); filled square: @aoki02 (C, CS 29498–043); filled diamond: @aoki07 (D, BS 16934–002)\]. Typical observational errors in \[E/Fe\] are $\sim 0.1$–0.25 dex. Note that the latter patterns are characterized by extremely high abundances of the hydrostatic burning products Na, Mg, and Al relative to the explosive burning products from Si through Zn. Note also that the patterns of the explosive burning products are indistinguishable for HNe and faint SNe.[]{data-label="fig-pan"}](f8.eps)
[^1]: The original two-component model was inspired by the meteoritic data on $^{129}$I and $^{182}$Hf in connection with the $r$-process. See @wbg for the requirement of two distinct types of $r$-process sources based on these data and QW07 for a review on the development of the two-component model.
[^2]: Four of the data points are repeated measurements of well-studied stars with large $r$-process enrichments: the plus at ${\rm [Fe/H]}=-2.86$, ${\rm [Sr/Ba]}=-0.53$ and the circle at ${\rm [Fe/H]}=-2.98$, ${\rm [Sr/Ba]}=-0.44$ for CS 22892–052, the plus at ${\rm [Fe/H]}=-2.75$, ${\rm [Sr/Ba]}=-0.60$ for CS 31082–001, and the circle at ${\rm [Fe/H]}=-2.02$, ${\rm [Sr/Ba]}=-0.60$ for BD $+17^\circ 3248$. Observational studies focused on these stars give ${\rm [Sr/Ba]}=-0.31$ (CS 22892–052, @sneden03), $-0.43$ (CS 31082–001, @hill02), and $-0.16$ (BD $+17^\circ 3248$, @cowan02). We note that having some $s$-process contributions to Ba would also lower \[Sr/Ba\].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Hubble Ultra Deep Field (HUDF) contains a significant number of [$B$]{}-, [$V$]{}- and [$i''$]{}-band dropout objects, many of which were recently confirmed to be young star-forming galaxies at $z\!\simeq\!4\!-\!6$. These galaxies are too faint individually to accurately measure their radial surface brightness profiles. Their average light profiles are potentially of great interest, since they may contain clues to the time since the onset of significant galaxy assembly. We separately co-add [$V$]{}, [$i''$]{}- and [$z''$]{}-band HUDF images of sets of $z\!\simeq\!4,5$ and $6$ objects, pre-selected to have nearly identical compact sizes and the roundest shapes. From these stacked images, we are able to study the average(d) radial structure of these objects at much higher signal-to-noise ratio than possible for an individual faint object. Here we explore the reliability and usefulness of a stacking technique of compact objects at $z\!\simeq\!4\!-\!6$ in the HUDF. Our results are: (1) image stacking provides reliable and reproducible average surface brightness profiles; (2) the shape of the average surface brightness profile shows that even the faintest $z\!\simeq\!4\!-\!6$ objects are *resolved*; and (3) if late-type galaxies dominate the population of galaxies at $z\!\simeq\!4\!-\!6$, as previous *HST* studies have shown for $z\!\lesssim\!4$, then limits to dynamical age estimates for these galaxies from their profile shapes are comparable with the SED ages obtained from the broadband colors. We also present accurate measurements of the sky-background in the HUDF and its associated 1$\sigma$ uncertainties.'
author:
- |
N. P. Hathi, R. A. Jansen, R. A. Windhorst, S. H. Cohen,\
W. C. Keel, M. R. Corbin and R. E. Ryan Jr.
title: |
Surface Brightness Profiles of Composite Images of Compact\
Galaxies at in the HUDF
---
Introduction
============
In the last decade, ground and space based observations of high redshift galaxies have begun to outline the process of galaxy assembly. The details of that process at high redshifts, however, remain poorly constrained. There is increasing support for the model of galaxy formation, in which the most massive galaxies assemble earlier than their less massive counterparts [e.g. @cowi96; @guzm97; @koda04; @mcca04]. A detailed analysis of the ‘fossil record’ of the current stellar populations in nearby galaxies selected from the *Sloan Digital Sky Survey* [SDSS; @york00] provides strong evidence for this downsizing picture [@heav04; @pant07]. The increasing number of luminous galaxies spectroscopically confirmed to be at $z\!\simeq\!6.5$ [e.g. @hu02; @kodi03; @kurk04; @rhoa04; @ster05; @tani05], or [$\lesssim$]{}0.9 Gyr after the Big Bang, also supports this general picture. In an alternate hierarchical scenario, arguments have been made that significant number of low luminosity dwarf galaxies were present at these times, and were the main contributor to finish the process of reionization of the intergalactic medium [@yan04a; @yan04b]. However, there is presently little information on the dynamical structure of these or other galaxies at $z\!\simeq\!6$. It is not clear whether these objects represent isolated disk systems, or collapsing spheroids, mergers or other dynamically young objects.
@ravi06 used deep, multi-wavelength images obtained with the *Hubble Space Telescope* (*HST*) Advanced Camera for Surveys (ACS) as part of the Great Observatories Origins Deep Survey (GOODS) to analyze 2-D surface brightness distributions of the brightest Lyman Break Galaxies (LBGs) at $2.5\!<\!z\!<\!5$. They distinguish various morphologies based on the Sérsic index $n$, which measures the shape of the azimuthally averaged surface brightness profile (where $n$=1 for exponential disks and $n$=4 for a de Vaucouleurs law). @ravi06 find that 40% of the LBGs have light profiles close to exponential, as seen for disk galaxies, and only $\sim$30% have high $n$, as seen in nearby spheroids. They also find a significant fraction ($\sim$30%) of galaxies with light profiles *shallower* than exponential, which appear to have multiple cores or disturbed morphologies, suggestive of close pairs or on-going galaxy mergers. Distinction between these possible morphologies and, therefore, a better estimate of the formation redshifts of the systems observed at $z\!\simeq\!4\!-\!6$ in particular, is important for testing the galaxy assembly picture, and for the refinement of galaxy formation models.
One possible technique involves the radial surface brightness profiles of the most massive objects — those that will likely evolve to become the massive elliptical galaxies, which we see in place at redshifts $z\!\lesssim\!2$ [@driv98; @vand03; @vand04]. This can be analytically understood in the context of the @lynd67 relaxation formalism and the numerical galaxy formation simulations of @vana82, which describe collisionless collapse and violent relaxation as the formation mechanism for elliptical galaxies. As the time-scale for relaxation is shorter in the inner than in the outer parts of a galaxy, convergence toward a $r^{1/n}$-profile will proceed from the inside to progressively larger radii at later times. Moreover, @korm77 has shown that tidal perturbations due to neighbors can cause the radial surface brightness profile to deviate from a pure de Vaucouleurs profile in the outer parts of a galaxy. This implies that the radius where surface brightness profiles start to deviate significantly from an $r^{1/n}$ profile *might* serve as a “*virial clock*” that traces the time since the onset of the last major merger, accretion events or global starburst in these objects.
Image stacking methods have been used extensively on X-ray [@nand02; @bran01] and radio [@geor03; @whit07] data to study the mean properties (e.g. flux, luminosity) of well-defined samples of sources that are otherwise too faint to be detected individually. @pasc96 applied such a stacking method to a large number of optically very faint, compact objects at $z\!=\!2.39$ to trace their “average” structure. This approach was also applied by @zibe04 to detect the presence of faint stellar halos around disk galaxies selected from the SDSS. An attempt to apply this technique to high redshift galaxies in the Hubble Deep Field [HDF; @will96] was not conclusive (H. Ferguson; private communication) due to the poorer spatial sampling and shallower depth of the HDF compared to the Hubble Ultra Deep Field [HUDF; @beck06].
In this paper, we use the exceptional depth and fine spatial sampling of the HUDF to study the potential of this image stacking technique, and will estimate limits to dynamical ages of faint, young galaxies at $z\!\simeq\!4\!-\!6$. The HUDF reaches $\sim$1.5 mag deeper than the equivalent HDF exposure in the [$i'$]{}-band, and has better spatial sampling than the HDF. The HUDF depth also allows us to characterize the sky background very accurately, which is critical for successfully using a stacking method to measure the mean surface-brightness profiles for these faint young galaxies.
This paper is organized as follows: In [§ \[observations\]]{} we summarize the HUDF observations, and in [§ \[sample\]]{} we discuss the selection of our $z\!\simeq\!4,5$ and $6$ samples. In [§ \[analysis\]]{} we describe our data analysis, which includes accurately measuring the 1$\sigma$ sky-subtraction error, the image stacking method to generate mean surface-brightness profiles, and our test of its reliability. In [§ \[results\]]{} we present and discuss our results in terms of the average surface-brightness profiles of $z\!\simeq\!4\!-\!6$ galaxies, and in [§ \[conclusion\]]{} we conclude with a summary of our results.
Throughout this paper we refer to the *HST*/ACS F435W, F606W, F775W, and F850LP filters as the [$B$]{}-, [$V$]{}-, [$i'$]{}-, and [$z'$]{}-bands, respectively. We assume a *Wilkinson Microwave Anisotropy Probe* (WMAP) cosmology of $\Omega_m$=0.24, $\Omega_{\Lambda}$=0.76 and [$H_{0}$]{}=73 km s$^{-1}$ Mpc$^{-1}$, in accord with the most recent 3-year WMAP results of @sper07. This implies a current age for the Universe of 13.65 Gyr. All magnitudes are given in the AB system [@oke83].
Observations
============
The HUDF contains [$\gtrsim$]{}100 objects that are [$i'$]{}-band dropouts, making them candidates for galaxies at $z\!\simeq\!6$ [@bouw04; @bouw06; @bunk04; @yan04b]. Similarly, there are larger numbers of objects in the HUDF that are [$B$]{}-band dropouts (415 in total) or [$V$]{}-band dropouts (265 in total), and are candidates for galaxies at $z\!\simeq\!4$ and $z\!\simeq\!5$, respectively. @beck06 and @bouw07 find similar number of [$B$]{}- and [$V$]{}-band dropouts in the HUDF. A significant fraction of these objects to AB[$\lesssim$]{}27 mag have recently been spectroscopically confirmed to have redshifts $z\!\simeq\!4\!-\!6$ through the detection of Ly$\alpha$ emission or identifying their Lyman break [@malh05; @dow07]. We discuss our detailed drop-out selection criteria below. Despite the depth (AB[$\lesssim$]{}29.5 mag) of the HUDF images, however, these objects appear very faint and with little, if any, discernible structural detail. Visual inspection of all these objects shows their morphologies to divide into four broad categories: symmetric, compact, elongated, and amorphous.
Sample Selection {#sample}
================
We construct three separate catalogs for these $z\!\simeq\!4,5,6$ galaxy candidates, selecting only the *isolated*, *compact* and *symmetric* galaxies. We exclude objects with obvious nearby neighbors, to avoid a bias due to dynamically disturbed objects and complications due to chance superpositions. [Figure \[fig1\]]{} demonstrates that our completeness limit for $z\!\simeq\!4$ and $z\!\simeq\!5$ objects is AB[$\lesssim$]{}29.3 mag, and for $z\!\simeq\!6$ objects it is AB[$\lesssim$]{}29.0 mag. Therefore, all three catalogs are complete to AB[$\lesssim$]{}29.0 mag, which is equivalent to at least a 10$\sigma$ detection for objects that are nearly point sources. For each object in our $z\!\simeq\!4,5,6$ samples, we extracted 51$\times$51 pixel postage stamps (which at 003 pix$^{-1}$ span $1\farcs53$ on a side) from the HUDF [$V$]{}, [$i'$]{}and [$z'$]{}-band images, respectively. Each postage stamp was extracted from the full HUDF, such that the centroid of an object (usually coincident with the brightest pixel) was at the center of that stamp.
We used the [$i'$]{}-band selected $BVi'z'$ HUDF catalog [@beck06] to select the $z\!\simeq\!4$ and $z\!\simeq\!5$ objects. With the `HyperZ` code [@bolz00], we computed photometric redshift estimates, using the magnitudes and associated uncertainties tabulated in the HUDF catalog. All objects with 3.5$\leq\!z_{\rm
phot}\!\leq$4.5 were assigned to the bin of $z\!\simeq\!4$ candidates, and all objects with 4.5$\leq\!z_{\rm phot}\!\leq$5.5 to the bin of $z\!\simeq\!5$ candidates.
We then applied color criteria, similar to those adopted by @giav04, to select the [$B$]{}($z\!\simeq\!4$) and [$V$]{}($z\!\simeq\!5$) dropout samples. For [$B$]{}-band dropouts, we require: $$\left\{ \begin{array} {ll}
(B-V) \ge 1.2 + 1.4 \times (V-z') \quad \hbox{mag} \\
\hbox{and}\quad (B-V) \ge 1.2 \quad \hbox{mag} \\
\hbox{and}\quad (V-z') \le 1.2 \quad \hbox{mag}
\end{array} \right.$$ For [$V$]{}-band dropouts, the following color selection was applied: $$\left\{ \begin{array} {ll}
(V-i') > 1.5 + 0.9 \times (i'-z') \; \; \hbox{or} \; \; (V-i') > 2.0 \quad \hbox{mag} \\
\hbox{and}\quad (V-i') \ge 1.2 \quad \hbox{mag} \\
\hbox{and}\quad (i'-z') \le 1.3 \quad \hbox{mag}
\end{array} \right.$$ We note, that only objects satisfying *both* color *and* photometric redshift criteria were selected in our samples. @vanz06 using VLT/FORS2 observed $\sim$100 [$B$]{}-, [$V$]{}- and [$i'$]{}-band dropout objects in the Chandra Deep Field South (CDFS) selected based on above mentioned color criteria [@giav04]. They have spectroscopically confirmed $>$90% of their high redshift galaxy candidates. Therefore, we expect only a small number ($<$10%) of contaminants in our sample of dropouts. One or two objects in our final sample could be such contaminants, but because we have 3 different realizations of 10 objects (3$\times$10), each showing similar profiles, they do not appear to affect our results.
The $z\!\simeq\!4$ sample has 415 objects, while the $z\!\simeq\!5$ sample has 265 objects. In [Figure \[fig2\]]{}ab, we show the distribution of the FWHM and ellipticity, $\epsilon=(1-b/a)$, measured in each of the two samples using `SExtractor` [@bert96]. We further constrained our samples by imposing limits on compactness and on roundness of FWHM $\le 0\farcs3$ and $\epsilon \le 0.3$. Again, this is to minimize the probability that the $z\!\simeq\!4\!-\!5$ candidates are significantly dynamically disturbed, and to maximize the probability of selecting physically similar objects. Our goal is to find the visibly most symmetric, least disturbed systems for the current study. This sub-selection leaves 204 objects in the $z\!\simeq\!4$ sample and 102 objects in the $z\!\simeq\!5$ sample. Most of these objects are faint, and are only a few pixels across in size, and, hence, have larger uncertainties in their measurements of FWHM and ellipticity. Therefore, we also checked our objects visually to eliminate any possibility of our selected objects being contaminated by unrelated nearby objects, being clearly extended, or objects with complex morphologies.
@yan04b found 108 possible 5.5$\leq\!z\!\leq$6.5 candidates in the HUDF to $m_{AB}$($z_{850}$)=30.0 mag. @bunk04 independently found the brightest 54 of these 108 $z\!\simeq\!6$ candidates to AB=28.5 mag. Similarly, deep *HST*/ACS grism spectra of the HUDF [$i'$]{}-band dropouts confirm [$\gtrsim$]{}90% of these objects at AB[$\lesssim$]{}27.5 mag to be at $z\!\simeq\!6$ [@malh05; @hath07]. Using the catalog of @yan04b, we extracted 108 postage stamps, each 51$\times$51 pixels in size, from the HUDF [$z'$]{}-band image.
Like for the $z\!\simeq\!4$ and $z\!\simeq\!5$ objects, for each $z\!\simeq\!6$ object we measured its [$z'$]{}-band FWHM and ellipticity using `SExtractor`. [Figure \[fig2\]]{}c shows the measured ellipticity versus FWHM for all 108 $z\!\simeq\!6$ candidates. A smaller sample of 67 objects satisfies our constraints on the FWHM and ellipticity. Further visual inspection, to make sure that our sample has only isolated, compact and round objects, leaves 30 objects in our $z\!\simeq\!6$ sample. We therefore imposed a sample size of 30 objects also on the two lower redshift bins after visual inspection.
The results in this paper are therefore based on approximately (30/415)$\sim$7%, (30/265)$\sim$11%, and (30/108)$\sim$28% of the total $z\!\simeq\!4,5$ and $6$ galaxy populations.
Results {#analysis}
=======
The HUDF Sky Surface-Brightness Level and its rms Variation {#skyerror}
-----------------------------------------------------------
For the present work, it is *critical* that we accurately characterize the sky-background, and correctly propagate the true 1$\sigma$ errors due to the subtraction of this sky-background. In the following, we will pursue two complimentary approaches to determine the sky surface-brightness, and compare the results. Here, we discuss the [$z'$]{}-band measurements in detail.
We first measured the sky-background in each of the 415 $z\!\simeq\!4$ object stamps (‘local’ sky measurements). The Interactive Data Language (IDL[^1]) procedure `SKY/MMM.pro`[^2] was used to measure the sky-background. This procedure is adapted from the `DAOPHOT` [@stet87] routine of the same name and works as follows. First, the average and sigma are obtained from the sky pixels. Second, these values are used to eliminate outliers with a low probability. Third, the values are then recomputed and the process is repeated up to 20 iterations. If there is a contamination due to an object, then the contamination is estimated by comparing the mean and median of the remaining sky pixels to get the true sky value. The output of this procedure is the modal sky-level in the image.
[Figure \[fig3\]]{}c shows a histogram of the [$z'$]{}-band modal sky values obtained from all 415 object stamps extracted from the drizzled HUDF images. The 1$\sigma$ uncertainty in the sky, $\sigma_{\rm sky}$, determined from a Gaussian fit to the histogram, is $2.19\times10^{-5}$ electrons sec$^{-1}$ in the [$z'$]{}-band. The sky-background level within the HUDF was obtained from the original flat-fielded ACS images, because the final co-added HUDF data products are sky-subtracted. The header parameters MDRIZSKY and EXPTIME were used to obtain the actually observed sky-value. MDRIZSKY is the sky value in electrons ($e^-$) computed by the MultiDrizzle code [@koek02], while EXPTIME is the total exposure time for the image in seconds, so that the average sky-value in the HUDF has the units of $e^-$ sec$^{-1}$. [Figure \[fig4\]]{}d shows the histogram of the sky-values obtained from 288 HUDF [$z'$]{}-band flat-fielded exposures. The average value of the sky background, I$_{\rm sky}$, is 0.02051 $e^-$ sec$^{-1}$ pix$^{-1}$. That sky-value is measured from the flat-fielded individual ACS images with pixel sizes of 005 pix$^{-1}$ and hence, in the following calculations, the average sky-value is multiplied by a factor of (0.030/0.05)$^2$=0.60$^2$ to obtain the corresponding average sky-value for the HUDF drizzled pixel size of 0030 pix$^{-1}$. Using these values, we estimate the relative rms random sky-subtraction error as follows: $$\Sigma_{\rm ss,ran} = \frac{\sigma_{\rm sky,ran}}{{\rm I}_{\rm sky}} = \frac{2.19\times10^{-5}}{2.05\times10^{-2}\ \cdot\ 0.60^2} = 2.97\times10^{-3}$$
The measured average sky background level can then be expressed as the [$z'$]{}-band sky surface brightness as follows: $$\begin{aligned}
\mu_{z'} &=& 24.862 - 2.5 \cdot \log\left(\frac{0.0205\ \cdot\ 0.60^2}{0.030^2} \right) \\
&=& 22.577 \pm 0.003 \ \hbox{mag arcsec}^{-2} \qquad\qquad\qquad\qquad\quad\null \end{aligned}$$ where 24.862 is the ACS/WFC [$z'$]{}-band AB zero-point, and 0030 pixel$^{-1}$ is the drizzled pixel scale. This is consistent with the values obtained by extrapolating the on-orbit $BVI$ sky surface brightness of @wind94 [@wind98] to [$z'$]{}, with the sky-background estimates from the ACS Instrument Handbook [@gonz05], and with the colors obtained by convolving the filter transmission curves with the solar spectrum. [Table \[table1\]]{} gives the measured electron detection rate, surface brightness and colors of the sky background with their corresponding errors for the HUDF $BVi'z'$ bands as calculated from [Figure \[fig3\]]{} and [Figure \[fig4\]]{}. The contribution of the zodiacal background dominates the total sky-background, which we find to be only $\sim$10% redder in ([$V$]{}–[$i'$]{}) and ([$i'$]{}–[$z'$]{}) than the Sun. The [$z'$]{}-band surface brightness corresponding to the 1$\sigma$ sky-subtraction uncertainty is therefore: $$\begin{aligned}
\mu_{z'} - 2.5\cdot \log (\Sigma_{\rm ss,ran}) &=& 22.577 - 2.5\cdot \log (2.97\times10^{-3}) \\
&=& 28.895\ \hbox{mag arcsec}^{-2}\end{aligned}$$
Next, we measure the sky-background from 415 ‘*blank*’ sky stamps (51$\times$51 pixel) distributed throughout the HUDF (‘global’ sky measurements). We measure the sky background using the same IDL algorithm as used above.
[Figure \[fig5\]]{}c shows the histogram of the measured [$z'$]{}-band modal sky values. A Gaussian distribution was fit to this histogram, giving a sky-sigma of $2.00\times10^{-5}$ $e^-$ sec$^{-1}$. The average value of the sky remains 0.02051 $e^-$ sec$^{-1}$ ([Figure \[fig4\]]{}d). Using these values, we can estimate a relative rms systematic sky-subtraction error as follows: $$\Sigma_{\rm ss,sys} = \frac{\sigma_{\rm sky,sys}}{{\rm I}_{\rm sky}} = \frac{2.00\times10^{-5}}{2.05\times10^{-2}\ \cdot\ 0.60^2} = 2.71\times10^{-3}$$ Since the [$z'$]{}-band sky surface brightness remains 22.577 mag arcsec$^{-2}$, this gives us for the surface brightness corresponding to the 1$\sigma$ sky subtraction uncertainty: $$\begin{aligned}
\mu_{z'} - 2.5\cdot \log (\Sigma_{\rm ss,sys}) &=& 22.577 - 2.5\cdot \log (2.71\times10^{-3}) \\
&=& 28.995\ \hbox{mag arcsec}^{-2}\end{aligned}$$
From these two complementary approaches, we can conclude that all surface brightness measurements become unreliable for surface-brightness levels fainter than 28.95$\pm$0.05 mag arcsec$^{-2}$ in the [$z'$]{}-band. We have also experimented with slightly larger cutouts (75$\times$75 pixels instead of 51$\times$51 pixels) to estimate the sky-subtraction error. We find that with the larger cutouts, the surface brightness corresponding to the 1$\sigma$ sky-subtraction error is $\sim$0.1–0.2 mag arcsec$^{-2}$ fainter. For larger cutouts we expect this surface brightness to be $\sim$0.4 mag fainter but we find about 0.1–0.2 mag fainter. This might be because of residual systematic errors in the HUDF images. Therefore, we are at the limit of accurately measuring this surface brightness and hence, we will here quote the conservative brighter limit of the surface brightness corresponding to this 1$\sigma$ sky-subtraction error. Expected contributions to this surface brightness due to uncertainties in the bias level determinations, which correspond to $\sim$0.001 counts sec$^{-1}$ for typical HUDF exposures (A. M. Koekemoer; private communication), are less than 1%.
[Figure \[fig5\]]{} clearly shows that the distribution of the modal sky-values is not as symmetric around zero as in [Figure \[fig3\]]{}, and hence, the use of a ‘global’ sky value for the HUDF is not as reliable as ‘local’ sky measurements. Therefore, for the surface brightness profiles and the following discussion, we will adopt the *local* 1$\sigma$ random sky-subtraction error for all objects in our study.
The average modal sky values and their 1$\sigma$ errors in the [$V$]{}- and [$i'$]{}-bands were calculated in exactly the same way as for the [$z'$]{}-band, as shown in [Figure \[fig3\]]{}, 4 and 5. The resulting $BVi'z'$ sky values and the sky surface-brightness levels are all given in [Table \[table1\]]{}.
[cccccc]{}
$B$ & 112 & 0.015909 $\pm$ 0.000065 & 23.664 $\pm$ 0.003 & ($B-V$)$_{\rm sky}$=0.800 & 29.85 $\pm$ 0.05\
$V$ & 112 & 0.070276 $\pm$ 0.000297 & 22.864 $\pm$ 0.002 & ($V-i'$)$_{\rm sky}$=0.222 & 30.15 $\pm$ 0.15\
$i'$ & 288 & 0.040075 $\pm$ 0.000088 & 22.642 $\pm$ 0.002 & ($i'-z'$)$_{\rm sky}$=0.065 & 29.77 $\pm$ 0.20\
$z'$ & 288 & 0.020511 $\pm$ 0.000047 & 22.577 $\pm$ 0.003 & ($V-z'$)$_{\rm sky}$=0.287 & 28.95 $\pm$ 0.05\
Composite Images and Surface Brightness Profiles
------------------------------------------------
For each redshift bin ($z\!\simeq\!4,5,6$), we generated three “stacked” composite images from subsets of 10 postage stamps that were selected as follows. After placing all 30 image stamps per redshift bin into a $30\times$ (51$\times$51) pixel IDL array, 10 stamps were randomly drawn without selecting any object more than once. An output image was generated, in which the values at each pixel are the average of the corresponding pixels in the 10 selected input stamps. From the remaining 20 stamps, we again randomly select 10, from which we generated a second composite image, after which the final 10 images were averaged into the third composite image. The three composite images per redshift bin are therefore independent of each other. In none of our realizations did we produce composite images that were essentially unresolved. Even the faintest $z\!\simeq\!4\!-\!6$ galaxies are clearly resolved. The $z\!\simeq\!4,5,6$ objects used to generate the composite images have an apparent magnitude range of approximately 27.5$\pm$1.0 AB mag. Because the magnitude range is relatively small and the S/N per pixel is low even in their central pixel, we have given all objects equal weight. To test whether this range in magnitude will affect our stacks and hence, our profiles, we created 3 stacks depending on the apparent magnitude, i.e. one stack of the 10 brightest objects in the sample, a second stack of the 10 next brightest objects in the sample and a third stack of the 10 faintest objects in the sample. This is summarized in [Figure \[fig6\]]{}d. We found that the profiles were very similar except that the profiles of the fainter stacks fall-off more quickly at larger radius compared to the profile of the brightest stack, but the inner profile and the deviation in the profiles are clearly visible in all 3 stacks. Therefore, we conclude that for our range in apparent magnitudes, our stacks/profiles are not affected. Perhaps most surprisingly, [Figure \[fig6\]]{}d shows that $r_e$ value of all 3 flux ranges ($\sim$26.0–27.0, $\sim$27.0–28.0 & $\sim$28.0–29.0 mag) are all about the same over $\sim$3-4 mag in flux, so the primary parameter that distinguishes the brighter from the fainter $z\!\simeq\!6$ dropouts is their central surface brightness (which thus also varies by $\sim$3–4 mag).
We used the IRAF[^3] procedure `ELLIPSE` to fit surface brightness profiles shown in [Figure \[fig6\]]{} to each of the three independent composite images per redshift bin. We also computed a mean surface-brightness profile from the three composite surface brightness profiles generated from the three independent composite images for each redshift bin. [Figure \[fig7\]]{} shows composite images for $z\!\simeq\!4,5,6$ objects. Here each composite image is a stack of 30 objects. [Figure \[fig8\]]{} shows the average surface brightness profiles for each of the redshift intervals $z\!\simeq\!4,5,6$. The thin solid curves in [Figure \[fig6\]]{} and the dot-dash curves in [Figure \[fig8\]]{} represent the observed ACS [$V$]{}, [$i'$]{}and [$z'$]{}-band Point Spread Functions (PSFs), while the horizontal dashed lines indicate the surface brightness level corresponding to the 1$\sigma$ sky–subtraction error in each of the HUDF images as discussed in [§ \[skyerror\]]{}. It is important to note that we scaled the ACS PSFs to match the surface brightness of the central data point in our mean surface-brightness profile, to determine how extended the mean surface-brightness profile is with respect to the PSFs.
In [Figure \[fig8\]]{}, we fitted all possible combinations of the Sersíc profiles (convolved with the ACS PSF) to the observed profiles and using $\chi^2$ minimization, found the best fits for galaxies at $z\!\simeq\!4,5,6$. The best fit Sérsic index ($n$) for all three profiles ($z\!\simeq\!4,5,6$) is $n\!<\!2$, meaning these galaxies follow mostly exponential disk-type profiles in their central regions. We find that the observed profiles start to deviate from the best-fit profiles at $r\!\gtrsim$0[[$\buildrel{\prime\prime}\over .$]{}]{}27, somewhat depending on the redshift. From [Figure \[fig8\]]{}, we also see that in each of $V$ ($z\!\simeq\!4$), $i'$ ($z\!\simeq\!5$) and $z'$ ($z\!\simeq\!6$), the PSF declines more rapidly with radius than the composite radial surface brightness profile for $r\!\gtrsim$0[[$\buildrel{\prime\prime}\over .$]{}]{}27. It is therefore unlikely that the observed ‘breaks’ result from the halos and structure of the ACS PSFs. Specifically, at $z\!\simeq\!6$ the most significant deviations in the light-profiles are seen at levels 1.5–2.0 mag above the 1$\sigma$ sky-subtraction error, and well above the PSF wings. Each of the mean surface brightness profiles display a well-defined break, the radius of which appears to change somewhat with redshift. The vertical dotted lines (in [Figure \[fig6\]]{} and [Figure \[fig8\]]{}) mark the radius at which the mean surface brightness profiles start to deviate significantly from the extrapolation of the $r^{1/n}$ profile observed at smaller radii.
Test of the Stacking Technique on Nearby Galaxies
-------------------------------------------------
To test the general validity of the stacking technique itself on a local galaxy sample, we used surface photometry from the Nearby Field Galaxy Survey [NFGS: @jans00a; @jans00b]. The NFGS sample contains 196 nearby galaxies, that were objectively selected from the CfA redshift catalog [CfAI; @davi83; @huch83] to span the full range in absolute $B$ magnitude present in the CfAI ($-14.7\lesssim
\!M_B\!\lesssim -22.7$ mag). The absolute magnitude distribution in the NFGS sample approximates the local galaxy luminosity function [e.g., @marz94], while the distribution over Hubble type follows the changing mix of morphological types as a function of luminosity in the local galaxy population. The NFGS sample [as detailed in @jans00a] minimizes biases, and yields a sample that, with very few caveats, is representative of the local galaxy population. As part of the NFGS, $UBR$ surface photometry, both integrated (global) and nuclear spectrophotometry, as well as internal kinematics were obtained [see @jans01]. Here, we will concentrate on the $U$-band surface photometry, since it is closest in wavelength to the rest-frame wavelengths observed at $z\!\simeq\!4\!-\!6$. Although, ideally, we would want a filter further into the UV, @tayl07 and @wind02 show that for the majority of late-type nearby galaxies, the apparent structure of galaxies does not change dramatically once one observes shortward of the Balmer break. Early-type galaxies, however, are a clear exception to this, but these are not believed to dominate the galaxy population at $z\!\simeq\!4\!-\!6$, as discussed before.
[Figure \[fig9\]]{} shows stacked profiles for relatively luminous early-, spiral-, and late-type galaxies drawn from the NFGS. Vertical dotted lines indicate the half-light radii and their intersection with the profiles, the surface brightness at that radius. Dashed lines indicate exponential fits to the outer portion of each profile. [Figure \[fig9\]]{} also shows that co-adding profiles for disparate morphological types and for mid-type spiral galaxies with a range in bulge-to-disk ratios can produce breaks in the composite profile. No such breaks are seen when the profiles of either early-type galaxies (E, S0) or late-type galaxies (Sd–Irr) are co-added. This figure shows that, *if* galaxies at $z\!\simeq\!4\!-\!6$ had similar morphological types as local galaxies, then it would be possible to produce a break in the profiles (as shown in [Figure \[fig6\]]{} and [Figure \[fig8\]]{}), merely by mixing different types of galaxies. We do not believe that the galaxy populations at $z\!\simeq\!4\!-\!6$ morphologically resemble those at low redshift. Hence, for primarily late-type galaxies, which dominate the faint blue galaxy population at AB$\ge$24 mag [@driv98], and which likely dominate the fainter end of the luminosity function at $z\!\simeq\!4\!-\!6$ that we sample here [@yan04a; @yan04b], the image stacking is likely a valid exercise.
The primary goal of this section was to show that the profile stacking technique is valid and can be used to get meaningful surface brightness profiles. We are not comparing our nearby sample with galaxies at $z\!\simeq\!4\!-\!6$. These nearby galaxies are unlikely to be local analogues of high redshift galaxies. If we apply surface brightness dimming to UV light-profiles of these nearby galaxies, they would be mostly invisible to *HST*, and in some cases visible to *JWST* in long integration [see e.g. @wind06]. This is another way of saying that the $z\!\simeq\!4\!-\!6$ objects are truly different from $z\!\simeq\!0$ objects.
Discussions {#results}
===========
[Figure \[fig8\]]{} shows that the mean surface brightness profiles deviate significantly from an inner $r^{1/n}$ profile at radii $r\!\gtrsim$0[[$\buildrel{\prime\prime}\over .$]{}]{}27–0[[$\buildrel{\prime\prime}\over .$]{}]{}35, depending somewhat on the redshift bin. These deviations appear real, with the break/point of departure located [$\gtrsim$]{}1.5–2 mag above the 1$\sigma$ sky-subtraction error and above the PSF-wings. In the following, we discuss several possible explanations for the observed shapes of our composite surface brightness profiles.
Galaxies with Different Morphologies
------------------------------------
Our test on nearby galaxies ([Figure \[fig9\]]{}) shows that, if we stack many galaxies with different morphologies (early-type, late-type or spiral galaxies), it is possible to get a slope-change (‘break’) in the average surface brightness profile. @ravi06 find that 40% of the brighter LBGs at $2.5\!<\!z\!<\!5$ have light profiles close to exponential, as seen for disk galaxies, and only $\sim$30% have high $n$, as seen in nearby spheroids. They also find a significant fraction ($\sim$30%) of galaxies with light profiles *shallower* than exponential, which appear to have multiple cores or disturbed morphologies, suggestive of close pairs or on-going galaxy mergers. Therefore, if galaxies at $z\!\simeq\!4\!-\!6$ have a variety of morphological types, then the shape of the average surface brightness profile that we see may be due to the stacking of different types of galaxies. Therefore, we find that the exponential and the flatter profiles found by @ravi06 for galaxies at $2.5\!<\!z\!<\!5$ also apply to higher redshifts ($z\!\ge\!5$).
Also, we believe that it is more likely that the high redshift, faint galaxy population consists primarily of small galaxies with late-type morphologies and with sub-L$^{*}$ luminosities, as seen at $z\!\simeq\!2\!-\!3$ [@driv95; @driv98]. So if the $z\!\simeq\!4\!-\!6$ population consists of such a late-type galaxy population, then the slope-change in the light profiles is likely not the result of co-adding images of objects with disparate morphological types.
Central Star Formation/Starburst
--------------------------------
*HST* optical images of galaxies at $z\!\simeq\!4\!-\!6$ sample their rest-frame UV ($\sim$1200 [Å]{}), where the contribution from the actively star-forming regions (very young, massive stars) dominates the UV-light. @hath07 have shown that galaxies at $z\!\simeq\!5\!-\!6$ are high redshift starbursts and these galaxies have similar starburst intensity limit as local starbursting galaxies. Therefore, it is possible that galaxies at $z\!\simeq\!4\!-\!6$ have centrally concentrated star formation or starburst. This possibility is based on three key assumptions: (1) most of the galaxies at $z\!\simeq\!4,5,6$ are intrinsically later-type galaxies [@driv98; @stei99]; (2) the Spectral Energy Distribution (SED) of these galaxies at $z\!\simeq\!4,5,6$ are dominated by early A- to late O-type stars, respectively; and (3) there are no old stars with ages at $z\!\simeq\!4\!-\!6$ greater than 2-1 Gyr in WMAP cosmology, respectively.
@hunt06 studied azimuthally averaged surface photometry profiles for large sample of nearby irregular galaxies. They find some galaxies have double exponentials that are steeper (and bluer) in the inner parts compared to outer parts of the galaxy. @hunt06 discuss that this type of behavior is expected in galaxies where the centrally concentrated star formation or starburst steepens the surface brightness profiles in the center. If that is the case, then one might expect a better correlation between the break in the surface brightness profiles and changes in color profiles. Unfortunately, for our sample of galaxies at $z\!\simeq\!4\!-\!6$, we don’t have high- resolution restframe [$U$]{}[$B$]{}[$V$]{}color information. The objects are generally too faint for *Spitzer Space Telescope*, and hence we cannot confirm or reject this possibility for the shape of our composite surface brightness profiles.
{#ages}
The average compact $z\!\simeq\!4\!-\!6$ galaxy is clearly extended with respect to the ACS PSFs ([Figure \[fig8\]]{}), and is best fit by an exponential profile ($n\!<\!2$) out to a radius of about $r\!\simeq$0[[$\buildrel{\prime\prime}\over .$]{}]{}35, 0[[$\buildrel{\prime\prime}\over .$]{}]{}31, and 0[[$\buildrel{\prime\prime}\over .$]{}]{}27 at $z\!\simeq\!4,5$ and $6$, respectively. The apparent progression with redshift is noteworthy. The radius at which the profile starts to deviate from $r^{1/n}$ (in this case at radius $r$[$\gtrsim$]{}0[[$\buildrel{\prime\prime}\over .$]{}]{}35–0[[$\buildrel{\prime\prime}\over .$]{}]{}27) may be an important constraint to the dynamical time scale of the system, as discussed in [§ \[introduction\]]{}. If this argument is valid, then we can estimate limits to the dynamical ages of $z\!\simeq\!4,5,6$ galaxies as follows.
In WMAP cosmology, a radius of $r$[$\gtrsim$]{}0[[$\buildrel{\prime\prime}\over .$]{}]{}35 at $z\!\simeq\!4$ corresponds to $r$[$\gtrsim$]{}2.5 kpc. The dynamical time scale [e.g., @binn87], $\tau_{dyn}$, goes as $\tau_{dyn}$ = $C
r^{3/2} \!/\!\sqrt{G\,M}$, where the constant $C=\pi/2$. For a typical dwarf galaxy mass range of $\sim10^9\!-\!10^8$[[ $M_{\odot}$]{} ]{}inside $r$=2.5 kpc, we infer that the limits to the dynamical age would be $\tau_{dyn}\simeq$ 90–290 Myr, which is the lifespan expected for a late-type B-star. This means that the last major merger that affected this surface brightness profile and that triggered its associated starburst may have occurred $\sim$0.20 Gyr before $z\!\simeq\!4$, —assuming that the star-formation wasn’t spontaneous, but associated with some accretion or a merging event.
Table 2 shows the break-radius and inferred limits to dynamical ages for the $z\!\simeq\!4\!-\!6$ objects. At $z\!\simeq\!5$, we find that the limits to dynamical age at the break radius would be $\tau_{dyn}\simeq$ 70–210 Myr, which is the lifespan expected for a mid B-star, while at $z\!\simeq\!6$, $\tau_{dyn}\simeq$ 50–150 Myr, which is the lifespan expected for a late O–early B-star. This means that the last major merger that affected these surface brightness profiles at $z\!\simeq\!5$ and $6$ and that triggered its associated starburst may have occurred $\sim$0.14 and $\sim$0.10 Gyr before $z\!\simeq\!5$ and $6$, respectively.
[cccc]{}
4 & 0.35 & 2.5 & 0.09–0.29 Gyr\
5 & 0.31 & 2.0 & 0.07–0.21 Gyr\
6 & 0.27 & 1.6 & 0.05–0.15 Gyr\
The dynamical time is a lower limit to the actual time available, since it assumes matter starts from rest. Any angular momentum at start will increase the available time. The best-fit SED age from the GOODS *HST* and *Spitzer* photometry on some of the brighter of these objects — using @bruz03 templates — is in the range of about $\sim$150–650 Myr [@yan05; @eyle05; @eyle07], the lower end of which is consistent with our limits to their dynamical age estimates, while the somewhat larger SED ages could also be affected by the onset of the AGB in the stellar population increasing the observed *Spitzer* fluxes and hence possibly overestimating ages [@mara05]. Our age estimates for $z\!\simeq\!4\!-\!6$ are consistent with the trend of SED ages suggested for $z\!\simeq\!7$ [@labb06]. It is noteworthy that, given the uncertainties, the two independent age estimates are consistent. If our limits to dynamical age estimates for the image *stacks* are thus valid, they are consistent with the SED ages, and point to a consistent young age for these objects.
Furthermore, the presence of young, massive late O–early B-stars at $z\!\simeq\!6$ has implications for the reionization of the universe. From observations of the appearance of complete Gunn-Peterson troughs in the spectra of $z\!\ga\!5.8$ quasars [@fan06], we know that the epoch of reionization had ended by $z\!\simeq\!6$. From the steep ($\alpha$=–1.8) faint-end slope of the luminosity function of $z\!\simeq\!6$ galaxies, @yan04a [@yan04b] concluded that dwarf galaxies, and not quasars, likely finished reionization by $z\!\simeq\!6$. Should the present interpretation of their light profiles be correct, then it would appear to add support to this picture, in the sense that such objects are dominated by B-stars and did not start their most recent major starburst long before $z\!\simeq\!6$.
Summary {#conclusion}
=======
We used the stacked HUDF images to analyze the average surface brightness profiles of $z\!\simeq\!4\!-\!6$ galaxies. Our analysis shows that even the faintest galaxies at $z\!\simeq\!4\!-\!6$ are resolved. This may have implications on the stellar density and its relation to the stellar density in present-day galaxies. We also find that the average surface brightness profiles display breaks at a radius that progresses toward lower redshift from $r\simeq$0[[$\buildrel{\prime\prime}\over .$]{}]{}27 (1.6 kpc) at $z\!\simeq\!6$ to $r\simeq$0[[$\buildrel{\prime\prime}\over .$]{}]{}35 (2.5 kpc) at $z\!\simeq\!4$.
The shape of the radial surface brightness profile that we observe could result from a mixture of different morphological types of galaxies, if they exist at $z\!\simeq\!4\!-\!6$, because we can produce similar breaks in the surface brightness profiles when we mix different types of nearby galaxies. Alternatively, if these galaxies are dominated by a central starburst then they could show such double exponential-type profiles, as discussed by @hunt06. In a third scenario, if the galaxies at $z\!\simeq\!4-6$ are truly young and mostly late-type, the outer profiles seen in our mean radial surface brightness profiles at $z\!\simeq\!4-6$ bear the imprint of the hierarchical build-up process, and are still dominated by infalling material, which is *not* detectable in the individual HUDF images of these faint objects. We have estimated limits to dynamical ages from the break radius at $z\!\simeq\!4, 5, 6$, very roughly as $\sim$0.20, 0.14 and 0.10 Gyr, respectively, and those ages are similar to the SED ages inferred at $z\!\simeq\!4\!-\!6$ [@yan05; @eyle05; @eyle07], and consistent with SED ages suggested for $z\!\simeq\!7$ [@labb06]. Hence, at $z\!\simeq\!4, 5, 6$, the last major merger that affected the surface brightness profiles that we observe, and that triggered the observed star-burst, may have occurred respectively $\sim$0.20, 0.14 and 0.10 Gyr earlier, or very approximately at $z\!\simeq\!4.5, 5.5, 6.5$. This would be consistent with the hierarchical assembly of galaxies and with the end of reionization, since it would imply that from $z\!\simeq\!4$ to $z\!\simeq\!6$, the SEDs become progressively more dominated by late-B–late-O stars. This implies that the sub-$L^*$ (i.e. dwarf) galaxies may have produced sufficient numbers of energetic UV photons to complete the reionization process by $z\!\simeq\!6$, as @yan04a [@yan04b] suggested. It will be imperative to study with future instruments like *HST*/WFC3 and *JWST* [@wind06; @wind07] whether the dominant stellar population indeed changes from late-O–early-B at $z\!\simeq\!6$ (i.e. capable of reionizing) to mid- to late-B at $z\!\simeq\!4-5$ (i.e. capable of maintaining reionization), and to what extent the intrinsic sizes of these faint objects will ultimately limit deep *JWST* surveys.
This work was partially supported by HST grants AR 10298 and GO 9780 from the Space Telescope Science Institute, which is operated by AURA under NASA contract NAS 5-26555. The authors thank Deidre Hunter, Alan Dressler, Henry Ferguson, Anton Koekemoer, Robert Morgan for their helpful discussions. RAW acknowledges support from NASA JWST Interdisciplinary Scientist grant NAG5-12460 from GSFC, that supported an investigation of the implications of this work for JWST. We specially thank our referee, Dr. Patrick McCarthy, for his helpful comments that have improved this paper.
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[^3]: IRAF (http://iraf.net) is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the boundary value problem for the stationary rotating black hole solutions to the five-dimensional vacuum Einstein equation. Assuming the two commuting rotational symmetry and the sphericity of the horizon topology, we show that the black hole is uniquely characterized by the mass, and a pair of the angular momenta.'
author:
- Yoshiyuki Morisawa
- Daisuke Ida
date: 'January 22, 2004'
title: 'A boundary value problem for the five-dimensional stationary rotating black holes'
---
Introduction
============
In recent years there has been renewed interest in higher dimensional black holes in the context of both string theory and brane world scenario. In particular, the possibility of black hole production in linear collider is suggested [@'tHooft:rb; @Banks:1999gd; @Giddings:2001bu; @Dimopoulos:2001hw]. Such phenomena play a key role to get insight into the structure of space-time; we might be able to prove the existence of the extra dimensions and have some information about the quantum gravity. Since the primary signature of the black hole production in the collider will be Hawking emission from the stationary black hole, the classical equilibrium problem of black holes is an important subject. The black holes produced in colliders will be small enough compared with the size of the extra dimensions and generically have angular momenta, they will be well approximated by higher dimensional rotating black hole solutions found by Myers and Perry [@Myers:un]. The Myers-Perry black hole which has the event horizon with spherical topology can be regarded as the higher-dimensional generalization of the Kerr black hole. One might expect that such a black hole solution describes the classical equilibrium state continued from the black hole production event, if it equips stability and uniqueness like the Kerr black hole in four-dimensions. The purpose of this paper is to consider the uniqueness and nonuniqueness of the rotating black holes in higher dimensions.
The uniqueness theorem states that a four-dimensional black hole with regular event horizon is characterized only by mass, angular momentum and electric charge [@Carter:hk; @Heusler:book]. Recently, uniqueness and nonuniqueness properties of five or higher-dimensional black holes are also studied. Emparan and Reall have found a black ring solution of the five-dimensional vacuum Einstein equation, which describes a stationary rotating black hole with the event horizon homeomorphic to $S^2 \times S^1$ [@Emparan:2001wn]. In a certain parameter region, a black ring and a (Myers-Perry) black hole can carry the same mass and angular momentum. This might suggest the nonuniqueness of higher-dimensional stationary black hole solutions. For example, Reall [@Reall:2002bh] conjectured the existence of stationary, asymptotically flat higher-dimensional vacuum black hole admitting exactly two commuting Killing vector fields although all known higher dimensional black hole solutions have three or more Killing vector fields. In six or higher dimensions, Myers-Perry black hole can have an arbitrarily large angular momentum for a fixed mass. The horizon of such black hole highly spreads out in the plane of rotation and looks like a black brane in the limit where the angular momentum goes to infinity. Hence, Emparan and Myers [@Emparan:2003sy] argued that rapidly rotating black holes are unstable due to the Gregory-Laflamme instability [@Gregory:vy] and decay to the stationary black holes with rippled horizons implying the existence of black holes with less geometric symmetry compared with the Myers-Perry black holes. For supersymmetric black holes and black rings, string theoretical interpretation are given by Elvang and Emparan [@Elvang:2003mj]. They showed that the black hole and the black ring with same asymptotic charges correspond to the different configurations of branes, giving a partial resolution of the nonuniqueness of supersymmetric black holes in five dimensions. On the other hand, we have uniqueness theorems for black holes at least in the static case [@Hwang:1998; @Gibbons:2002bh; @Gibbons:2002av; @Gibbons:2002ju; @Rogatko:2002bt; @Rogatko:2003kj]. Furthermore, the uniqueness of the stationary black holes is supported by the argument based on linear perturbation of higher dimensional static black holes [@Kodama:2003jz; @Ishibashi:2003ap]. There exist regular stationary perturbations that fall off at asymptotic region only for vector perturbation, and then the number of the independent modes corresponds to the rank of the rotation group, namely the number of angular momenta carried by the Myers-Perry black holes [@Kodama:private]. This suggests that the higher-dimensional stationary black holes have uniqueness property in some sense, but some amendments will be required. Here we consider the possibility of restricted black hole uniqueness which is consistent with any argument about uniqueness or nonuniqueness. Though the existence of the black ring solution explicitly violates the black hole uniqueness, there still be a possibility of black hole uniqueness for fixed horizon topology [@Kol:2002dr]. Hence we restrict ourselves to the stationary black holes with spherical topology.
In this paper, we consider the asymptotically flat, black hole solution to the five-dimensional vacuum Einstein equation with the regular event horizon homeomorphic to $S^3$, admitting two commuting spacelike Killing vector fields and stationary (timelike) Killing vector field. The two spacelike Killing vector fields correspond to the rotations in the ($X^1$-$X^2$)-plane and ($X^3$-$X^4$)-plane in the asymptotic region ($\{X^{\mu}\}$ are the asymptotic Cartesian coordinates), respectively, which are commuting with each other. Along with the argument by Carter [@Carter:1970], it is possible to construct a timelike Killing vector field tangent to the fixed points (namely, axis) of the axi-symmetric Killing vector field from the given timelike Killing vector field. Repeating this procedure for each commuting spacelike Killing vector field, the obtained timelike Killing vector field is also commuting with both spacelike Killing vector fields. Hence, it is natural to assume all the three Killing vector fields are commuting with each other. The five-dimensional vacuum space-time admitting three commuting Killing vector fields is described by the nonlinear $\sigma$-model [@Maison:kx]. Then the Mazur identity [@Mazur:1984wz] for this system is derived. We show that the five-dimensional black hole solution with regular event horizon of spherical topology is determined by three parameters under the appropriate boundary conditions.
The remainder of the paper is organized as follows. In Section \[sec:metric\], we give the field equations for the five-dimensional vacuum space-time admitting three commuting Killing vector fields. In Section \[sec:matrix\], we introduce the matrix form of field equations to clarify the hidden symmetry of this system following Maison [@Maison:kx]. Then the Mazur identity which is useful to show the coincidence of two solutions is derived in Section \[sec:identity\]. In Section \[sec:conditions\], we determine the boundary conditions. We summarize this paper and make discussions on related matters in Section \[sec:summary\].
Five-dimensional vacuum space-time admitting three commuting Killing vector fields
==================================================================================
Assuming the symmetry of space-time, the Einstein equations reduce to the equations for the scalar fields defined on three-dimensional space. Then, we show that the system of the scalar fields is described by a nonlinear $\sigma$-model.
Weyl-Papapetrou metrics {#sec:metric}
-----------------------
We consider the five-dimensional space-time admitting two commuting Killing vector fields $\xi_I=\partial_I,\,(I=4,5)$. The metric can be written in the form $$\begin{aligned}
g &=&
f^{-1}\gamma_{ij} dx^i dx^j
+f_{IJ}(dx^I+w^I_i dx^i)(dx^J+w^J_j dx^j),\end{aligned}$$ where $i,j=1,2,3$, $f=\det(f_{IJ})$. The three-dimensional metric $\gamma_{ij}$, the functions $w^I_i$ and $f_{IJ}$ are independent on the coordinates $x^I$ ($x^4=\phi$, $x^5=\psi$, and we will later identify $\xi_4$ and $\xi_5$ as Killing vector fields corresponding to two independent rotations in the case of asymptotically flat space-time). We define the twist potential $\omega_I$ by $$\begin{aligned}
\omega_{I,\mu} = f\,f_{IJ}\sqrt{|\gamma|}\epsilon_{ij\mu}
\gamma^{im}\gamma^{jn}\partial_m w^J_n,
\label{eq:def-twist}\end{aligned}$$ where $\mu=1,\cdots,5$, $\gamma=\det(\gamma_{ij})$, $\gamma^{ij}$ is the inverse metric of $\gamma_{ij}$, and $\epsilon_{\lambda\mu\nu}$ denotes the totally skew-symmetric symbol such that $\epsilon_{123}=1,\epsilon_{I\mu\nu}=0$. Then the vacuum Einstein equation reduces to the field equations for the five scalar fields $f_{IJ}$ and $\omega_{I}$ defined on the three-dimensional space: $$\begin{aligned}
D^2 f_{IJ} &=&
f^{KL}Df_{IK}\cdot Df_{JL} -f^{-1}D\omega_I \cdot D\omega_J,
\label{eq:EOM-f}
\\
D^2 \omega_I &=&
f^{-1}Df\cdot D\omega_I +f^{JK}Df_{IJ}\cdot D\omega_K,
\label{eq:EOM-omega}\end{aligned}$$ and the Einstein equations on the three-dimensional space: $$\begin{aligned}
{}^{(\gamma)}R_{ij}
&=&
{1 \over 4}f^{-2}f_{,i}f_{,j}
+{1 \over 4}f^{IJ}f^{KL}f_{IK,i}f_{JL,j}
+{1 \over 2}f^{-1}f^{IJ}\omega_{Ii}\omega_{Jj},\end{aligned}$$ where $D$ is the covariant derivative with respect to the three-metric $\gamma_{ij}$ and the dot denotes the inner product determined by $\gamma_{ij}$.
Here we assume the existence of another Killing vector field $\xi_3=\partial_3$ which commutes with the other Killing vectors as $[\xi_3,\xi_I]=0$ (we will later identify the $\xi_3$ as the stationary Killing vector field in the case of asymptotically flat space-time). Then the metric can be written in the Weyl-Papapetrou–type form [@Ida:2003wv] $$\begin{aligned}
g &=&
f^{-1}e^{2\sigma}(d\rho^2+dz^2)-f^{-1}\rho^2 dt^2
+f_{IJ}(dx^I+w^I dt)(dx^J+w^J dt),
\label{eq:weyl-metric}\end{aligned}$$ where we denote $x^3=t$, and all the metric functions depend only on $\rho$ and $z$. Once the five scalar fields $f_{IJ}, \omega_I$ are determined, the other metric functions $\sigma$ and $w^I$ are obtained by solving the following partial derivative equations: $$\begin{aligned}
{2 \over \rho}\sigma_{,\rho} &=&
{1 \over 4} f^{-2}[(f_{,\rho})^2-(f_{,z})^2]
+{1 \over 4} f^{IJ}f^{MN}(f_{IM,\rho}f_{JN,\rho}-f_{IM,z}f_{JN,z})
\nonumber\\&&
+{1 \over 2} f^{-1}f^{IJ}
(\omega_{I,\rho}\omega_{J,\rho}-\omega_{I,z}\omega_{J,z}),
\\
{1 \over \rho}\sigma_{,z} &=&
{1 \over 4} f^{-2} f_{,\rho} f_{,z}
+{1 \over 4} f^{IJ}f^{MN} f_{IM,\rho} f_{JN,z}
+{1 \over 2} f^{-1}f^{IJ} \omega_{I,\rho} \omega_{J,z},
\\
w^I_{,\rho} &=&
\rho f^{-1} f^{IJ} \omega_{J,z},
\\
w^I_{,z} &=&
-\rho f^{-1} f^{IJ} \omega_{J,\rho}.\end{aligned}$$ The $f_{IJ}$ and $\omega_I$ are given by axi-symmetric solution of the field equations (\[eq:EOM-f\]) and (\[eq:EOM-omega\]) on the abstract flat three-space with the metric $$\begin{aligned}
\gamma &=& d\rho^2+dz^2+\rho^2 d\varphi^2.
\label{eq:flat-metric}\end{aligned}$$ Thus the system is described by the action $$\begin{aligned}
S&=&\int d\rho dz \,\rho
\left[{1 \over 4}f^{-2}(\partial f)^2
+{1 \over 4}f^{IJ}f^{KL}\partial f_{IK} \cdot \partial f_{JL}
+{1 \over 2}f^{-1}f^{IJ}\partial\omega_I \cdot \partial\omega_J \right].
\label{eq:eff-action}\end{aligned}$$
Matrix representation {#sec:matrix}
---------------------
The action (\[eq:eff-action\]) is invariant under the global $SL(3,{\bf R})$ transformations as shown by Maison [@Maison:kx]. Instead of the nonlinear representation by the scalar fields $f_{IJ}$ and $\omega_I$, we introduce the $SL(3,{\bf R})$ matrix field $\Phi$ as $$\begin{aligned}
\Phi &=&
\left(
\begin{array}{rrr}
f^{-1} & -f^{-1}\omega_{\phi} & -f^{-1}\omega_{\psi}
\\
-f^{-1}\omega_{\phi} & f_{\phi\phi}+f^{-1}\omega_{\phi}\omega_{\phi} & f_{\phi\psi}+f^{-1}\omega_{\phi}\omega_{\psi}
\\
-f^{-1}\omega_{\psi} & f_{\phi\psi}+f^{-1}\omega_{\phi}\omega_{\psi} & f_{\psi\psi}+f^{-1}\omega_{\psi}\omega_{\psi}
\\
\end{array}
\right),
\label{eq:def-Phi}\end{aligned}$$ which is symmetric (${}^{t}\Phi=\Phi$) and unimodular ($\det\Phi=1$). $\Phi$ transforms as a covariant, symmetric, second-rank tensor fields under global $SL(3,{\bf R})$ transformations. When the Killing vector fields $\xi_{\phi}$ and $\xi_{\psi}$ are spacelike, all the eigenvalues of $\Phi$ are real and positive. Therefore, there is an $SL(3,{\bf R})$ matrix field $g$ which is a square root of the matrix field $\Phi$, namely $$\begin{aligned}
\Phi = g \, {}^{t}g.
\label{eq:squareroot}\end{aligned}$$ This square root matrix $g$ is determined upto global $SO(3)$ rotation because the rotation $g \mapsto g\Lambda$ for any $\Lambda \in SO(3)$ is canceled by $\Lambda^{-1}={}^{t}\Lambda$. Since any $SL(3,{\bf R})$ matrix field $g$ conversely defines a symmetric and unimodular matrix field by $\Phi=g \, {}^{t}g$, the matrix $\Phi$ defines a map from two-dimensional $\rho$-$z$-half plane (base space) to the coset space $SL(3,{\bf R})/SO(3)$.
The inverse matrix of $\Phi$ is explicitly given by $$\begin{aligned}
\Phi^{-1} &=&
\left(
\begin{array}{ccc}
f+f^{IJ}\omega_I \omega_J & f^{\phi J}\omega_J & f^{\psi J}\omega_J
\\
f^{\phi J}\omega_J & f^{\phi\phi} & f^{\phi\psi}
\\
f^{\psi J}\omega_J & f^{\phi\psi} & f^{\psi\psi}
\end{array}
\right),\end{aligned}$$ and transforms as a second rank contravariant tensor field on the base space.
The current matrix defined by $$\begin{aligned}
J_i = \Phi^{-1}\partial_i \Phi\end{aligned}$$ linearly transforms according to the adjoint representation of $SL(3,{\bf R})$. This current is conserved, namely every element of $D_{i}J^{i}$ independently vanishes due to the field equations (\[eq:EOM-f\]) and (\[eq:EOM-omega\]).
The action (\[eq:eff-action\]) can be expressed in terms of $J_i$ or $\Phi$ as $$\begin{aligned}
S&=& {1 \over 4}\int d\rho dz \,\rho{\rm tr}(J_i J^i),
\\
&=& {1 \over 4}\int d\rho dz \,\rho
{\rm tr}(\Phi^{-1}\partial_i \Phi \Phi^{-1}\partial^i \Phi).\end{aligned}$$ This action takes a nonlinear $\sigma$-model form.
Mazur identity {#sec:identity}
==============
Let us consider two different sets of the field configurations $\Phi_{[0]}$ and $\Phi_{[1]}$ satisfying the field equations (\[eq:EOM-f\]) and (\[eq:EOM-omega\]). To show the coincidence of the two solutions, we will derive the Mazur identity for the nonlinear $\sigma$-model on the symmetric space $SL(3,{\bf R})/SO(3)$
A bull’s eye ${}^{\odot}$ denotes the difference between the value of functional obtained from the field configuration $\Phi_{[1]}$ and value obtained from $\Phi_{[0]}$, [*e.g.*]{}, $$\begin{aligned}
\stackrel{\odot}{J}\!{}^i = {J}^i_{[1]}-{J}^i_{[0]}
=\Phi^{-1}_{[1]}\partial^{i}\Phi_{[1]}-\Phi^{-1}_{[0]}\partial^{i}\Phi_{[0]}.\end{aligned}$$ The deviation matrix $\Psi$ is defined by $$\begin{aligned}
\Psi = \stackrel{\odot}{\Phi}\Phi^{-1}_{[0]}
=\Phi_{[1]}\Phi^{-1}_{[0]}-{\bf 1},\end{aligned}$$ where ${\bf 1}$ is the unit matrix. The deviation $\Psi$ vanishes if and only if the two sets of field configurations ($[1]$ and $[0]$) coincide with each other. Differentiating $\Psi$, $$\begin{aligned}
D^{i}\Psi = \Phi_{[1]} \stackrel{\odot}{J}\!{}^{i}\, \Phi^{-1}_{[0]},\end{aligned}$$ and taking divergence, we obtain $$\begin{aligned}
D_{i}(D^{i}\Psi) =
\Phi_{[1]} D_{i}\stackrel{\odot}{J}\!{}^{i}\, \Phi^{-1}_{[0]}
+\Phi_{[1]} \left\{
J_{[1]i}J_{[1]}^{i}-2J_{[1]i}J_{[0]}^{i}+J_{[0]i}J_{[0]}^{i}
\right\}\Phi^{-1}_{[0]}.
\label{eq:Psiii}\end{aligned}$$ Due to the current conservation $D_{i}J^{i}=0$, the first term of the right hand side of Eq. (\[eq:Psiii\]) vanishes. Since ${}^{t}\!J^i = \Phi J^i \Phi^{-1}$, the second term of the right hand side can be rewritten as $$\begin{aligned}
\Phi_{[1]} \left\{
J_{[1]i}J_{[1]}^{i}-2J_{[1]i}J_{[0]}^{i}+J_{[0]i}J_{[0]}^{i}
\right\}\Phi^{-1}_{[0]}
&=&
\Phi_{[1]} \left(
J_{[1]}^{i} \stackrel{\odot}{J}\!{}_{i}\,
-\stackrel{\odot}{J}\!{}_{i}\,J_{[0]}^{i}
\right)\Phi^{-1}_{[0]}
\\
&=&
{}^{t}\!J_{[1]}^{i} \Phi_{[1]} \stackrel{\odot}{J}\!{}_{i}\, \Phi^{-1}_{[0]}
-\Phi_{[1]} \stackrel{\odot}{J}\!{}_{i}\,\Phi^{-1}_{[0]}\,{}^{t}\!J_{[0]}^{i}.\end{aligned}$$ Then taking trace, we obtain the identity $$\begin{aligned}
(D_{i}D^{i}{\rm tr}\Psi) =
{\rm tr}\left\{
{}^{t}\!\stackrel{\odot}{J}\!{}^{i}\, \Phi_{[1]}
\stackrel{\odot}{J}\!{}_{i}\, \Phi^{-1}_{[0]}
\right\}.
\label{eq:Mazure1}\end{aligned}$$ Since $D$ is covariant derivative with respect to the abstract flat three-metric (\[eq:flat-metric\]) and all quantities are independent on $\varphi$, the above identity (\[eq:Mazure1\]) is $$\begin{aligned}
\partial_{a}(\rho \partial^{a}{\rm tr}\Psi) =
\rho h_{ab}{\rm tr} \left\{
{}^{t}\!\stackrel{\odot}{J}\!{}^{a}\, \Phi_{[1]}
\stackrel{\odot}{J}\!{}^{b}\, \Phi^{-1}_{[0]}
\right\},
\label{eq:Mazur2}\end{aligned}$$ where $h_{ab}$ is the flat two-dimensional metric $$\begin{aligned}
h = d\rho^2 + dz^2.\end{aligned}$$ Integrating Eq. (\[eq:Mazur2\]) over the relevant region $\Sigma=\{(\rho,z)|\rho \ge 0\}$ in $\rho$-$z$ plane, and using Green’s theorem, we find $$\begin{aligned}
\oint_{\partial\Sigma} \rho \partial^{a} {\rm tr}\Psi dS_{a}
= \int_{\Sigma} \rho h_{ab}{\rm tr} \left\{
{}^{t}\!\stackrel{\odot}{J}\!{}^{a}\, \Phi_{[1]}
\stackrel{\odot}{J}\!{}^{b}\, \Phi^{-1}_{[0]}
\right\}d\rho dz,
\label{eq:Mazur3}\end{aligned}$$ where the boundary $\partial\Sigma$ is corresponding to the horizon, the two planes of rotation and infinity. Since the matrix $\Phi$ has the square root matrix $g$ as Eq. (\[eq:squareroot\]), the integrand of the right hand side of Eq. (\[eq:Mazur3\]) is written by $$\begin{aligned}
\rho h_{ab}{\rm tr} \left\{
{}^{t}\!\stackrel{\odot}{J}\!{}^{a}\, \Phi_{[1]}
\stackrel{\odot}{J}\!{}^{b}\, \Phi^{-1}_{[0]}
\right\}
=
\rho h_{ab}{\rm tr} \left\{
g^{-1}_{[0]}\, {}^{t}\! \stackrel{\odot}{J}\!{}^{a}\, g_{[1]}
\,{}^{t}\! g_{[1]} \stackrel{\odot}{J}\!{}^{b}\, {}^{t}\!g^{-1}_{[0]}
\right\}\end{aligned}$$ Thus, we obtain the Mazur identity $$\begin{aligned}
\oint_{\partial\Sigma} \rho \partial^{a} {\rm tr} \Psi dS_{a}
= \int_{\Sigma} \rho h_{ab}
{\rm tr} \left\{ {\cal M}^{a}\, {}^{t}\!{\cal M}^{b} \right\}d\rho dz,
\label{eq:Mazur4}\end{aligned}$$ where the matrix ${\cal M}$ is defined by $$\begin{aligned}
{\cal M}^{a} =
g^{-1}_{[0]}\, {}^{t}\!\stackrel{\odot}{J}\!{}^{a}\, g_{[1]}.\end{aligned}$$ When the current difference $\stackrel{\odot}{J}\!{}^{a}$ is not zero, the right hand side of the identity (\[eq:Mazur4\]) is positive. Hence we must have $\stackrel{\odot}{J}\!{}^{a}=0$ if the boundary conditions under which the left hand side of Eq. (\[eq:Mazur4\]) vanishes are imposed at $\partial\Sigma$. Then the difference $\Psi$ is a constant matrix over the region $\Sigma$. The limiting value of $\Psi$ is zero on at least one part of the boundary $\partial\Sigma$ is sufficient to obtain the coincidence of two solutions $\Phi_{[0]}$ and $\Phi_{[1]}$.
Boundary conditions and coincidence of solutions {#sec:conditions}
================================================
When one use the Mazur identity, the boundary conditions for the fields $\Phi$ ([*i.e.*]{}, $f_{IJ}$ and $\omega_I$) are needed at the infinity, the two planes of rotation and the horizon. We will require asymptotic flatness, regularity at the two planes of rotation, and regularity at the spherical horizon. Under these conditions, the Mazur identity shows that the coincidence of the solutions.
An asymptotically flat space-time with mass $M={3\pi m / 8G}$, angular momenta $J_{\phi}={\pi ma / 4G}$ and $J_{\psi}={\pi mb / 4G}$ (where we restrict ourselves to the case in which $m>a^2+b^2+2|ab|$) has metric as the following form: $$\begin{aligned}
g &=& -\left[1-{m \over r^2}+O(r^{-3})\right] dt^2
-\left[{2ma \over r^{4}}+O(r^{-5})\right] dt(ydx-xdy)
\nonumber\\&&
-\left[{2mb \over r^{4}}+O(r^{-5})\right] dt(wdz-zdw)
\nonumber\\
&&
+\left[1+{m \over 2 r^2}+O(r^{-3})\right] [dx^2+dy^2+dz^2+dw^2].\end{aligned}$$ Here introducing the coordinates $$\begin{aligned}
x &=& \sqrt{r^2+a^2} \sin\theta \cos[\bar{\phi}-\tan^{-1}(a/r)],\\
y &=& \sqrt{r^2+a^2} \sin\theta \sin[\bar{\phi}-\tan^{-1}(a/r)],\\
z &=& \sqrt{r^2+b^2} \cos\theta \cos[\bar{\psi}-\tan^{-1}(b/r)],\\
w &=& \sqrt{r^2+b^2} \cos\theta \sin[\bar{\psi}-\tan^{-1}(b/r)],\end{aligned}$$ and proceeding further coordinate transformations $$\begin{aligned}
d\bar{\phi} &=& d\phi-{a \over r^2+a^2} dr,\\
d\bar{\psi} &=& d\psi-{b \over r^2+b^2} dr,\end{aligned}$$ then one obtains $$\begin{aligned}
g &=& -\left[1-{m \over r^2}+O(r^{-3})\right] dt^2
+\left[{2ma(r^2+a^2) \over r^{4}}\sin^2\theta+O(r^{-3})\right] dt d\phi
\nonumber\\&&
+\left[{2mb(r^2+b^2) \over r^{4}}\cos^2\theta+O(r^{-3})\right] dt d\psi
\nonumber\\
&&
+\left[1+{m \over 2 r^2}+O(r^{-3})\right]\times
\biggl[
{r^2+a^2\cos^2\theta+b^2\sin^2\theta \over (r^2+a^2)(r^2+b^2)} r^2 dr^2
\nonumber\\
&& \quad
+(r^2+a^2\cos^2\theta+b^2\sin^2\theta) d\theta^2
+(r^2+a^2)\sin^2\theta d\phi^2+(r^2+b^2)\cos^2\theta d\psi^2
\biggr].
\label{eq:asympt-metric}\end{aligned}$$ Here the metric (\[eq:asympt-metric\]) admits two orthogonal planes of rotation $\theta=\pi/2$ and $\theta=0$, which are specified by the azimuthal angles $\phi$ and $\psi$, respectively. The planes $\theta=0$ and $\theta=\pi/2$ are invariant under the rotation with respect to the Killing vector fields $\partial_\phi$ and $\partial_\psi$, respectively. Both angles $\phi$ and $\psi$ have period $2\pi$. Comparing the asymptotic form (\[eq:asympt-metric\]) with the Weyl-Papapetrou form (\[eq:weyl-metric\]), we derive boundary conditions.
The regularity on invariant planes requires $$\begin{aligned}
g_{\phi\phi} = f_{\phi\phi} &=& \sin^2\theta \tilde{f}_{\phi\phi},\\
g_{\psi\psi} = f_{\psi\psi} &=& \cos^2\theta \tilde{f}_{\psi\psi},\\
g_{\phi\psi} = f_{\phi\psi} &=& \sin^2\theta \cos^2\theta \tilde{f}_{\phi\psi},\end{aligned}$$ where the quantities with tilde are regular at both the invariant planes and the black hole horizon.
The asymptotic behavior of $\tilde{f}_{\phi\phi}$ and $\tilde{f}_{\psi\psi}$ are derived from Eq. (\[eq:asympt-metric\]), and $\tilde{f}_{\phi\psi}$ is at most $O(r^{-1})$ since Killing vectors $\partial_{\phi}$ and $\partial_{\psi}$ are asymptotically orthogonal. $$\begin{aligned}
\tilde{f}_{\phi\phi} &=& r^2+a^2+{m \over 2}+O(r^{-1}),\\
\tilde{f}_{\psi\psi} &=& r^2+b^2+{m \over 2}+O(r^{-1}),\\
\tilde{f}_{\phi\psi} &=& O(r^{-1}).\end{aligned}$$
Since $f_{\phi\psi}$ is negligible as compared with $f_{\phi\phi}$ and $f_{\psi\psi}$ in the asymptotic region, the leading terms of $g_{t\phi}$ and $g_{t\psi}$ are $f_{\phi\phi}w^{\phi}$ and $f_{\psi\psi}w^{\psi}$, respectively. Then, we have $$\begin{aligned}
f_{\phi\phi}w^{\phi}
&=& {ma \sin^2\theta \over r^2} + O(r^{-3}),
\\
f_{\psi\psi}w^{\psi}
&=& {mb \cos^2\theta \over r^2} + O(r^{-3}).\end{aligned}$$ Thus we obtain $$\begin{aligned}
w^{\phi} &=& {ma \over r^4} + O(r^{-5}),
\\
w^{\psi} &=& {mb \over r^4} + O(r^{-5}).\end{aligned}$$
Similarly, we have $$\begin{aligned}
g_{tt}
&=& -f^{-1}\rho^2
+f_{\phi\phi}w^{\phi}w^{\phi} +2f_{\phi\psi}w^{\phi}w^{\psi}
+f_{\psi\psi}w^{\psi}w^{\psi}
\\
&=&
-1 + {m \over r^2} + O(r^{-3}).\end{aligned}$$ Here $O(r^{-2})$ term must come from $-f^{-1}\rho^2$ term since the $w^{I}$ are $O(r^{-4})$. Therefore $\rho$ behaves as $$\begin{aligned}
\rho^2 = \left[r^4+(a^2+b^2)r^2+O(r)\right] \sin^2\theta \cos^2\theta.\end{aligned}$$
$\rho^2$ does not only vanish at $\phi$-invariant plane ($\sin\theta=0$) and $\psi$-invariant plane ($\cos\theta=0$), but also vanishes at the horizon due to the form of the metric (\[eq:weyl-metric\]). Since the horizon has topology of $S^3$, let us introduce the spheroidal coordinates on $\Sigma$ as $$\begin{aligned}
z &=& \lambda\mu,\\
\rho^2 &=& (\lambda^2-c^2)(1-\mu^2),
\label{eq:def-lambda}\end{aligned}$$ where $\mu = \cos2\theta$. Then the relevant region is $\Sigma=\{(\lambda,\mu)|\lambda \ge c, -1\le \mu \le 1\}$. The boundaries $\lambda=c$, $\lambda=+\infty$, $\mu=1$ and $\mu=-1$ correspond to the horizon, the infinity, the $\phi$-invariant plane and the $\psi$-invariant plane, respectively. In these coordinates, the two-dimensional metric on $\Sigma$ is given by $$\begin{aligned}
h = d\rho^2+dz^2 = (\lambda^2-c^2\mu^2)
\left({d\lambda^2 \over \lambda^2-c^2}+{d\mu^2 \over 1-\mu^2}\right).\end{aligned}$$ The boundary integral in the left hand side of the Mazur identity (\[eq:Mazur4\]) is explicitly written as $$\begin{aligned}
\oint_{\partial\Sigma} \rho \partial^{a}{\rm tr}\Psi dS_{a}
&=&
\int_{c}^{\infty} d\lambda \left.\left(
\sqrt{h_{\lambda\lambda} \over h_{\mu\mu}} \rho
{\partial{\rm tr}\Psi \over \partial\mu}
\right)\right|_{\mu=-1}
+
\int_{-1}^{+1} d\mu \left.\left(
\sqrt{h_{\mu\mu} \over h_{\lambda\lambda}} \rho
{\partial{\rm tr}\Psi \over \partial\lambda}
\right)\right|_{\lambda=\infty}
\nonumber\\&&
+
\int_{\infty}^{c} d\lambda \left.\left(
\sqrt{h_{\lambda\lambda} \over h_{\mu\mu}} \rho
{\partial{\rm tr}\Psi \over \partial\mu}
\right)\right|_{\mu=+1}
+
\int_{+1}^{-1} d\mu \left.\left(
\sqrt{h_{\mu\mu} \over h_{\lambda\lambda}} \rho
{\partial{\rm tr}\Psi \over \partial\lambda}
\right)\right|_{\lambda=c},
\label{eq:boundary_integral}\end{aligned}$$ where $$\begin{aligned}
{\partial{\rm tr}\Psi \over \partial x^{a}} =
{\partial \over \partial x^{a}}\left[
f^{-1}_{[1]}\left(-\stackrel{\odot}{f}
+f^{IJ}_{[0]}\stackrel{\odot}{\omega}{}_{I}\stackrel{\odot}{\omega}{}_{J}
\right)
+f^{IJ}_{[0]}\stackrel{\odot}{f}{}_{IJ}
\right],
\quad \mbox{for $x^a=\lambda,\mu$.}\end{aligned}$$ Here the relation between $\lambda$ and $r$ is given by $$\begin{aligned}
\lambda = {r^2 \over 2}+{a^2+b^2 \over 4}+O(r^{-1}),\end{aligned}$$ or $$\begin{aligned}
r = \sqrt{2}\lambda^{1/2}
\left[1-{a^2+b^2 \over 8\lambda}+O(\lambda^{-3/2})\right].\end{aligned}$$
The boundary conditions for $f_{IJ}$ are summarized as follows:
------------------------ ------------------------ ------------------------ ----------------- -------------------------------------------- --
$\phi$-invariant plane $\psi$-invariant plane horizon infinity
$\mu \to +1$ $\mu \to -1$ $\lambda \to c$ $\lambda \to +\infty$
$\tilde{f}_{\phi\phi}$ $O(1)$ $O(1)$ $O(1)$ $2\lambda+(a^2-b^2+m)/2+O(\lambda^{-1/2})$
$\tilde{f}_{\phi\psi}$ $O(1)$ $O(1)$ $O(1)$ $O(\lambda^{-1/2})$
$\tilde{f}_{\psi\psi}$ $O(1)$ $O(1)$ $O(1)$ $2\lambda+(b^2-a^2+m)/2+O(\lambda^{-1/2})$
------------------------ ------------------------ ------------------------ ----------------- -------------------------------------------- --
where $$\begin{aligned}
f_{\phi\phi} &=& {(1-\mu) \over 2}\tilde{f}_{\phi\phi},\\
f_{\phi\psi} &=& {(1-\mu)(1+\mu) \over 4}\tilde{f}_{\phi\psi},\\
f_{\psi\psi} &=& {(1+\mu) \over 2}\tilde{f}_{\psi\psi}.\end{aligned}$$
Next, let us derive the boundary conditions for the twist potentials. By the definition of twist potentials, Eq. (\[eq:def-twist\]), $$\begin{aligned}
{\partial\omega_{\phi} \over \partial\lambda}
= -{f\,f_{\phi J} \over \lambda^2-c^2}{\partial w^{J} \over \partial\mu},
&\quad&
{\partial\omega_{\phi} \over \partial\mu}
= {f\,f_{\phi J} \over 1-\mu^2}{\partial w^{J} \over \partial\lambda},
\\
{\partial\omega_{\psi} \over \partial\lambda}
= -{f\,f_{\psi J} \over \lambda^2-c^2}{\partial w^{J} \over \partial\mu},
&\quad&
{\partial\omega_{\psi} \over \partial\mu}
= {f\,f_{\psi J} \over 1-\mu^2}{\partial w^{J} \over \partial\lambda}.\end{aligned}$$ From the $\mu$ dependence of $f_{IJ}$, the $\mu$ dependence of the derivatives of the twist potentials are given as follows: $$\begin{aligned}
{\partial\omega_{\phi} \over \partial\lambda}=
{\partial\omega_{\phi} \over \partial\mu}=
{\partial\omega_{\psi} \over \partial\lambda}=0
\quad\mbox{at $\mu=+1$},&&
{\partial\omega_{\psi} \over \partial\mu}
\mbox{ does not have } (1-\mu) \mbox{ as a factor},
\\
{\partial\omega_{\psi} \over \partial\lambda}=
{\partial\omega_{\psi} \over \partial\mu}=
{\partial\omega_{\phi} \over \partial\lambda}=0
\quad\mbox{at $\mu=-1$},&&
{\partial\omega_{\phi} \over \partial\mu}
\mbox{ does not have } (1+\mu) \mbox{ as a factor}.\end{aligned}$$
In the asymptotic region ($\lambda \to +\infty$), the derivatives of the twist potentials behave as $$\begin{aligned}
{\partial\omega_{\phi} \over \partial\lambda}
&=& O(\lambda^{-3/2}),
\\
{\partial\omega_{\phi} \over \partial\mu}
&=& -{ma \over 2}(1-\mu) +O(\lambda^{-1/2}).\end{aligned}$$ Thus we obtain $$\begin{aligned}
\omega_{\phi} = -{ma\over4}\mu(2-\mu) + (1-\mu)^2(1+\mu) O(\lambda^{-1/2}),\end{aligned}$$ and similarly $$\begin{aligned}
\omega_{\psi} = -{mb\over4}\mu(2+\mu) + (1-\mu)(1+\mu)^2 O(\lambda^{-1/2}).\end{aligned}$$
Then, of course, the condition that $\omega_{I}$ are regular on the horizon is required.
The boundary conditions for $\omega_{I}$ are summarized as follows:
------------------------- --------------------------- --------------------------- ----------------- ----------------------- --
$\phi$-invariant plane $\psi$-invariant plane horizon infinity
$\mu \to +1$ $\mu \to -1$ $\lambda \to c$ $\lambda \to +\infty$
$\tilde{\omega}_{\phi}$ $O\left((1-\mu)^2\right)$ $O(1+\mu)$ $O(1)$ $O(\lambda^{-1/2})$
$\tilde{\omega}_{\psi}$ $O(1-\mu)$ $O\left((1+\mu)^2\right)$ $O(1)$ $O(\lambda^{-1/2})$
------------------------- --------------------------- --------------------------- ----------------- ----------------------- --
where $$\begin{aligned}
\omega_{\phi} &=& -{ma \over 4}\mu(2-\mu)+\tilde{\omega}_{\phi},
\\
\omega_{\psi} &=& -{mb \over 4}\mu(2+\mu)+\tilde{\omega}_{\psi}.\end{aligned}$$
The behavior of the following quantities which appear in the boundary integral (\[eq:boundary\_integral\]) are easily calculated as follows.
------------------------------------------ ------------------------ ------------------------ ----------------------- ----------------------- --
$\phi$-invariant plane $\psi$-invariant plane horizon infinity
$\mu \to +1$ $\mu \to -1$ $\lambda \to c$ $\lambda \to +\infty$
$\partial{\rm tr}\Psi / \partial\lambda$ — — O(1) $O(\lambda^{-5/2})$
$\partial{\rm tr}\Psi / \partial\mu$ $O(1)$ $O(1)$ — —
$\rho$ $O(\sqrt{1-\mu})$ $O(\sqrt{1+\mu})$ $O(\sqrt{\lambda-c})$ $O(\lambda)$
$\sqrt{h_{\mu\mu} / h_{\lambda\lambda}}$ — — $O(\sqrt{\lambda-c})$ $O(\lambda)$
$\sqrt{h_{\lambda\lambda} / h_{\mu\mu}}$ $O(\sqrt{1-\mu})$ $O(\sqrt{1+\mu})$ — —
------------------------------------------ ------------------------ ------------------------ ----------------------- ----------------------- --
Then, the boundary integral (\[eq:boundary\_integral\]) vanishes. The difference matrix $\Psi$ is constant and has asymptotic behavior as $$\begin{aligned}
\Psi \to \left(
\begin{array}{ccc}
O(\lambda^{-3/2}) & O(\lambda^{-7/2}) & O(\lambda^{-7/2})\\
O(\lambda^{-1/2}) & O(\lambda^{-3/2}) & O(\lambda^{-3/2})\\
O(\lambda^{-1/2}) & O(\lambda^{-3/2}) & O(\lambda^{-3/2})\\
\end{array}
\right),
\quad (\lambda\to +\infty).\end{aligned}$$ $\Psi$ vanishes at the infinity, and then $\Psi$ is zero over $\Sigma$. Thus, the two configurations $\Phi_{[0]}$ and $\Phi_{[1]}$ coincide with each other.
Summary and Discussion {#sec:summary}
======================
We show uniqueness of the asymptotically flat, black hole solution to the five-dimensional vacuum Einstein equation with the regular event horizon homeomorphic to $S^3$, admitting two commuting spacelike Killing vector fields and stationary Killing vector field. The solution of this system is determined by only three asymptotic charges, the mass $M={3\pi m / 8G}$ and the two angular momenta $J_{\phi}={\pi m a / 4G}$ and $J_{\psi}={\pi m b / 4G}$. The five-dimensional Myers-Perry black hole solution is unique in this class.
The vacuum black ring solution fulfills above conditions other than that on the topology of the horizon. There exist two black ring solutions which have same mass and angular momentum, which means uniqueness property fails for the $S^2 \times S^1$ event horizon. It is intriguing to investigate how this nonuniqueness occurs.
It will be impossible to extend our argument using the Mazur identity to the six or higher dimensional Myers-Perry black hole solutions. An $n$-dimensional space-time admitting $(n-3)$ commuting Killing vector fields is always described by nonlinear $\sigma$-model as shown by Maison [@Maison:kx]. To derive the Mazur identity for this nonlinear $\sigma$-model, all the $(n-3)$ Killing vector fields have to be spacelike. However, the $n$-dimensional Myers-Perry black hole space-time has only $[(n-1)/2]$ commuting spacelike Killing vector fields. Thus our method cannot be used except for the five-dimensional Myers-Perry black hole.
The rigidity theorem in four dimensions claims that the asymptotically flat, stationary analytic space-time is also axi-symmetric [@Hawking:1971vc]. However the existence of additional space-time Killing vector fields is not justified in the case of five-dimensional black holes any longer. Therefore uniqueness shown in the present work does not exclude the possibility of existence of the black hole solutions with less symmetry as suggested by Reall [@Reall:2002bh].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank H. Kodama for valuable discussions and comments. D.I. was supported by JSPS Research, and this research was supported in part by the Grant-in-Aid for Scientific Research Fund (No. 6499).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The conductance of metallic graphene nanoribbons (GNRs) with single defects and weak disorder at their edges is investigated in a tight-binding model. We find that a single edge defect will induce quasi-localized states and consequently cause zero-conductance dips. The center energies and breadths of such dips are strongly dependent on the geometry of GNRs. Armchair GNRs are much more sensitive to a vacancy than zigzag GNRs, but are less sensitive to a weak scatter. More importantly, we find that with a weak disorder, zigzag GNRs will change from metallic to semiconducting due to Anderson localization. But a weak disorder only slightly affects the conductance of armchair GNRs. The influence of edge defects on the conductance will decrease when the widths of GNRs increase.'
author:
- 'T. C. Li'
- 'Shao-Ping Lu'
bibliography:
- 'GrapheneRibbons.bib'
title: Quantum conductance of graphene nanoribbons with edge defects
---
\[sec1\] Introduction
=====================
Recently, graphene (a single atomic layer of graphite) sheets were successfully isolated for the first time and demonstrated to be stable under ambient conditions by Novoselov et al.[@novoselov04; @novoselov05] Due to their unique two dimensional(2D) honeycomb structures, their mobile electrons behave as massless Dirac fermions,[@wallace47; @novoselov05; @zhang05] making graphene an important system for fundamental physics[@gusynin05; @peres06; @aleiner06]. Moreover, graphene sheets have the potential to be sliced or lithographed to a lot of patterned graphene nanoribbons (GNRs)[@banerjee05] to make large-scale integrated circuits[@wilson06].
The electronic property of GNRs has attracted increasing attention. Recent studies using tight-binding models[@nakada96; @ezawa06] and the Dirac equation[@brey06] have shown that GNRs can be either metallic or semiconducting, depending on their shapes. This allows GNRs to be used as both connections and functional elements[@wakabayashi00; @obradovic06] in nanodevices, which is similar to carbon nanotubes(CNTs).[@yao99; @chen06]
However, GNRs are substantially different from CNTs by having two open edges at both sides (see Fig.\[fig1\]). These edges not only remove the periodic boundary condition along the circumference of CNTs, but also make GNRs more vulnerable to defects than CNTs[@chico96B; @anantram98]. In fact, nearly all observed graphene edges[@kobayashi06; @banerjee05; @niimi06] contain local defects or extended disorders, while few defects are found in the bulk of graphene sheets. These edge defects can significantly affect the electronic properties of GNRs. Recent theoretical studies of perfect GNRs have also considered some edge corrections.[@ezawa06; @fujita97; @miyamoto99] But all edge atoms (see Fig. \[fig1\]) are assumed to be identical and the GNRs still have translational symmetry along their axis. There have been no studies of changes in the conductance caused by local defects or extended disorder on the edges, which break the translational symmetry of GNRs. We address this issue by calculating the conductance of metallic GNRs with such edge defects using a tight-binding model.
In our calculation, external electrodes and the central part (sample) are assumed to be made of GNRs. And the edge defects are modeled by appropriate on-site (diagonal) energy in the Hamiltonian of the sample. We utilize a quick iterative scheme[@li05; @sancho84] to calculate the surface Green’s functions of electrodes and an efficient recursive algorithm[@li05; @krompiewski02] to calculate the total Green’s function of the whole system. Finally, the conductance is calculated by the Landauer formula.[@imry99; @datta95] The calculation time of this method is only linearly dependent on the length of the sample and a GNR with disorder distributed over a length of 1 $\mu$m is tractable.
(8,6.5) (0,6.4)[(a) Zigzag ribbon]{} (0.8,3.6)[![\[fig1\] (Color online) Geometry of graphene ribbons. (a): A zigzag ribbon (N=4); (b): An armchair ribbon (N=7). A black circle denotes an edge carbon and a gray circle denotes a bulk carbon. A unit cell contains $2N$ atoms. From the top down, atoms in a unit cell are labeled as $1$A, $1$B, $2$A, $2$B, ..., $N$A, $N$B. Atoms close to 1B are 1A and 2A, and so on. ](Fig1a.eps "fig:")]{} (0,2.9)[(b) Armchair ribbon]{} (1,0.0)[![\[fig1\] (Color online) Geometry of graphene ribbons. (a): A zigzag ribbon (N=4); (b): An armchair ribbon (N=7). A black circle denotes an edge carbon and a gray circle denotes a bulk carbon. A unit cell contains $2N$ atoms. From the top down, atoms in a unit cell are labeled as $1$A, $1$B, $2$A, $2$B, ..., $N$A, $N$B. Atoms close to 1B are 1A and 2A, and so on. ](Fig1b.eps "fig:")]{}
We first study the conductance of zigzag and armchair GNRs with the simplest possible edge defects, a single vacancy or a weak scatter. Then we use a simple 1D model to explain the zero-conductance dips caused by edge defects. Finally, we study some more realistic structures, GNRs with weak scatters randomly distributed on their edges. We find that a weak disorder can change zigzag ribbons from metallic to semiconducting, but only changes the conductance of armchair ribbons slightly. The paper is organized as follows: in section \[sec2\], we introduce the model and method employed in this paper. Results and discussion are presented in section \[sec3\]. We conclude our findings in section \[sec4\].
\[sec2\] Model and method
=========================
The geometry of GNRs is shown in Fig.\[fig1\]. A graphene ribbon contains two unequal sublattices, denoted by A and B in this paper. We use $N$, the number of A(B)-site atoms in a unit cell, to denote GNRs with different widths. [@nakada96] Then the widths of ribbons with zigzag edges and armchair edges are $W_z=\frac{N}{2}\sqrt{3}\; a_0 $ and $W_a=\frac{N}{2} a_0$ respectively, where $a_0=2.49$ is the graphene lattice constant. From the top down, atoms in a unit cell will be labeled as $1$A, $1$B, ..., $N$A, $N$B. As shown in Fig. \[fig1\], an edge atom is a carbon atom at the edge of GNRs that is connected by only two other carbon atoms. In this paper, we consider defects that locate at the sites of these edge atoms only.
The system under consideration is composed of two electrodes and a central part(sample). The sample(unit cells 1, ... , $M$) is a finite GNR, which may contain edge defects, while the left and right electrodes are assumed to be semi-infinite perfect GNRs. We describe the GNR by a tight-binding model with one $\pi$-electron per atom. The tight-binding Hamiltonian of the system is $$\label{eq1}
H=\sum_{i}\varepsilon_i\;a_i^{\dag}\;a_i-V_{pp\pi}\sum_{<i,
j>}a_i^{\dag}\;a_j+c.c.$$ where $\varepsilon_i$ is the on-site energy and $V_{pp\pi}$ is the hopping parameter. The sum in $<i, j>$ is restricted to the nearest-neighbor atoms. In the absence of defects, $\varepsilon_i$ is taken to be zero and $V_{pp\pi}=2.66$ eV. [@chico96B] In the presence of defects, both the on-site energy and the hopping parameter can change. Here, we only consider the variation in the on-site energy. A vacancy is simulated by setting its on-site energy to infinity[@chico96B; @wakabayashi02]. A weak scatter caused by impurity or distortion will be simulated by setting $\varepsilon_i$ to a small value $V_i$. In the case of a weak disorder, $V_i$ is randomly selected from the interval $\pm |V_{random}|$ for every edge atom.
In what follows we show how to calculate the conductance of the GNRs:
First, the surface retarded Green’s functions of the left and right leads ($g^L_{0,0}$, $g^R_{M+1,M+1}$) are calculated by:[@li05; @jiang03; @sancho84] $$\begin{aligned}
\label{eq2}
&&g^L_{0,0}=[E^+I-H_{0,0}-H_{-1,0}^{\dag}\tilde\Lambda]^{-1},\\
\label{eq3} &&g^R_{M+1,M+1}=[E^+I-H_{0,0}-H_{-1,0}\Lambda]^{-1}\end{aligned}$$ where $E^+$=$E+i\eta$ ($\eta \to 0^+$)[@eta] and $I$ is a unit matrix. $H_{0,0}$ is the Hamiltonian of a unit cell in the lead, and $H_{-1,0}$ is the coupling matrix between two neighbor unit cells in the lead. Here $\Lambda$ and $\tilde\Lambda$ are the appropriate transfer matrices, which can be calculated from the Hamiltonian matrix elements via an iterative procedure:[@sancho84; @nardelli99] $$\begin{aligned}
\label{eq4}
\Lambda=&&t_0+\tilde{t}_0t_1+\tilde{t}_0\tilde{t}_1t_2+\ldots+\tilde{t}_0\tilde{t}_1\tilde{t}_{2}\cdots t_n,\\
\label{eq5}
\tilde\Lambda=&&t_0+t_0\tilde{t}_1+t_0t_1\tilde{t}_2+\ldots+t_0t_1t_{2}\cdots\tilde{t}_n,\end{aligned}$$ where $t_i$ and $\tilde{t_i}$ are defined via the recursion formulas $$\begin{aligned}
\label{eq6}
&&t_i=(I-t_{i-1}\tilde{t}_{i-1}-\tilde{t}_{i-1}t_{i-1})^{-1} t_{i-1}^2,\\
\label{eq7}
&&\tilde{t}_i=(I-t_{i-1}\tilde{t}_{i-1}-\tilde{t}_{i-1}t_{i-1})^{-1}
\tilde{t}_{i-1}^2,\end{aligned}$$ and $$\begin{aligned}
\label{eq8}
&&t_0=(E^+I-H_{0,0})^{-1}H_{-1,0}^{\dag},\\
\label{eq9} &&\tilde{t}_0=(E^+I-H_{0,0})^{-1}H_{-1,0}.\end{aligned}$$ The process is repeated until $t_n$,$\tilde{t}_n\leq\delta$ with $\delta$ arbitrarily small.
Second, including the sample as a part of the right lead layer by layer (from $l=M$ to $l=2$), the new surface Green’s functions are found by:[@li05; @krompiewski02] $$\label{eq10}
g^R_{l,l}=[E^+I-H_{l,l}-H_{l,l+1}\,g^R_{l+1,l+1}H_{l,l+1}^{\dag}]^{-1}.$$
Third, the total Green’s function $G_{1,1}$ can then be calculated by $$\label{eq11} G_{11}=[E^+I-H_{1,1}-\Sigma^L-\Sigma^R]^{-1},$$ where $$\begin{aligned}
\label{eq12}
&&\Sigma^L=H_{0,1}^{\dag}g^L_{0,0}H_{0,1}\\
\label{eq13} &&\Sigma^R=H_{1,2}g^R_{2,2}H_{1,2}^{\dag}\end{aligned}$$ are the self energy functions due to the interaction with the left and right sides of the structure. From Green’s function, the local density of states (LDOS) at site $j$ can be found $$\begin{aligned}
\label{eq14} &&n_j=-\frac{1}{\pi} \text{Im}[G_{(j,j)}],\end{aligned}$$ where $G_{(j,j)}$ is the matrix element of Green’s function at site $j$.
Finally, the conductance $G_{(E)}$ of the graphene ribbon can be calculated using the Landauer formula[@imry99; @datta95] $$\begin{aligned}
\label{eq15} &&G_{(E)}=\frac{2e^2}{h}T_{(E)}.\end{aligned}$$ Here $T_{(E)}$ is the transmission coefficient, which can be expressed as:[@fisher81; @meir92] $$\begin{aligned}
\label{eq16}
&&T_{(E)}=\text{Tr}[\Gamma^LG_{11}\Gamma^RG_{11}^{\dag}]\end{aligned}$$ where $$\begin{aligned}
\label{eq17} &&\Gamma^{L,R}=i[\Sigma^{L,R}-(\Sigma^{L,R})^{\dag}].\end{aligned}$$
In this calculation, no matrix larger than $N \times N$ is involved. And its cost is only linearly dependent on the length of the GNRs. We have used this method to study the effects of dangling ends on the conductance of side-contacted CNTs.[@li05] We have also calculated the band structures of perfect GNRs by diagonalizing the Hamiltonian. The conductances of perfect GNRs agree with the band structures.
\[sec3\] Results and discussion
===============================
The electronic properties of GNRs are strongly dependent on their geometry. There are two basic shapes of regular graphene edges, namely zigzag and armchair edges, depending on the cutting direction of the graphene sheet (see Fig. \[fig1\]). All ribbons with zigzag edges (zigzag ribbons) are metallic; however, two thirds of ribbons with armchair edges (armchair ribbons) are semiconducting.[@nakada96; @ezawa06]. The bands of zigzag GNRs are partially flat around Fermi energy ($E_F=0$ eV),[@nakada96] which means the group velocity of mobile electrons is close to zero. On the other hand, the bands of metallic armchair GNRs are linear around Fermi energy.[@nakada96; @kobayashi06] So the group velocity of their mobile electrons around Fermi energy should be a constant value, which is measured to be about $10^6 \text{m/s}$.[@novoselov05] Since zigzag GNRs and armchair GNRs have so different electronic properties, the effects of edge defects on their conductance should also be very different.
\[sec3:sec1\] Single defects
----------------------------
In this section, we will study the conductance and local density of states(LDOS) of GNRs with some single edge defects. A study of the effects of a single defect is not only realistic( e.g., a single two-atom vacancy at an armchair edge has been observed[@kobayashi06]), but also can serve as a guide for us to understand the effects of more complex edge defects.
### \[sec3:sec1:sec1\] Single vacancies
![\[fig2\] (Color online) (a) Conductance of a zigzag ribbon (N=8, $W=17.3$ ) with (dashed line) and without a vacancy at its edge (solid line). (b) LDOS of a 1B atom when there is no vacancy (solid line) and when there is a vacancy on its nearest 1A site at the edge (dashed line). ](Fig2.EPS)
![\[fig3\] (Color online) (a) Conductance of an armchair ribbon ($N=14$, $W=17.4$ ) without vacancy (solid line), with a two-atom vacancy (dashed line) and with a one-atom vacancy (dash dot line) at its edge . (b) LDOS of a 2B atom when there is no vacancy (solid line) and when its nearest (1A, 1B) pair atoms are removed (dashed line). (c) LDOS of a 1B atom when there is no vacancy (solid line) and when its nearest 1A atom is removed (dashed line).](Fig3.EPS)
One of the simplest defects in a zigzag GNR is a single vacancy caused by the loss of one edge atom (one-atom vacancy). In Fig.\[fig2\](a), we plot the conductance of a zigzag GNR (N=8) with a single 1A vacancy (dashed line) as a function of the energy. The solid line is for the perfect GNR. The defect almost does not affect the conductance around the Fermi energy. There are two conductance dips close to the first band edges and simultaneously two peaks appear in the LDOS of the 1B atom near the vacancy (dashed line in Fig.\[fig2\](b)). These two peaks have energies different from Van Hove singularities of a perfect GNR, which are extreme points of the 1D energy bands. So they are quasi-localized states caused by the vacancy. And the conductance dips are due to the antiresonance of these quasi-localized states. The relation between the conductance and quasi-localized states will be discussed further in Sec. \[sec3:sec1:sec2\].
The conductance and LDOS of armchair GNRs are displayed in Fig.\[fig3\]. In an armchair GNR, edge atoms appear in pairs. So a “single vacancy" can be formed by the loss of one edge atom (one-atom vacancy) or a pair of nearest edge atoms (two-atom vacancy). These two types of vacancies have very different properties especially around the fermi energy. For the two-atom vacancy, the conductance is similar to that of the single vacancy situation of the zigzag ribbon, as well as the LDOS. However, when there is a one-atom vacancy, a large LDOS will be formed and consequently a large conductance dip will appear at the fermi energy. In fact, it is expected that the effect of a one-atom vacancy is much larger than a two-atom vacancy because a one-atom vacancy breaks the symmetry between the two sublattices, while a two-atom vacancy keeps such symmetry. This is the same as in CNTs.
![\[fig4\] (Color online) The decreasing rate of the average conductance (between $\pm 0.5$eV) due to a single vacancy at the edge. As the width of a zigzag ribbon increases, its conductance becomes immune from a vacancy at its edge. ](Fig4.EPS)
The conductance around the Fermi energy is a very important parameter for the application of GNRs. It is affected by edge defects shown above. Also it depends on the width of the ribbon. For example, a one-atom vacancy at the edge of an armchair GNR will always cause a zero-conductance dip at the Fermi energy. But the breadth of the dip will change when the width of GNR changes. In order to describe the effect of a defect on the conductance quantitatively, we introduce the decreasing rate of the average conductance, which is $$\begin{aligned}
\label{eq18} {\overline{\Delta G}/\, \overline{G_0}}={\int^{\Delta
E}_{-\Delta E}[G_0(E)-G(E)] \text{d}E \over \int^{\Delta E}_{-\Delta
E}G_0(E) \text{d}E},\end{aligned}$$ where $G_0(E)$ is the conductance of a GNR without defects, and $G(E)$ is the conductance of the GNR with a defect. The decreasing rate of the average conductance (between $\pm 0.5$ eV) as a function of the ribbon width is plotted in Fig.\[fig4\]. From the figure, we know that edge vacancies affect armchair GNRs much more strongly than zigzag GNRs. And the effect of edge vacancies decreases when the width of GNRs increases. This is because there are more atoms in the cross section of a wider GNR, so electrons are easier to go around the defect. Thus we can use wide GNRs as connections in a nanodevice to avoid the change of conductance due to edge vacancies. There is a small bump in $\overline{\Delta G}/\, \overline{G_0}$ of armchair GNRs with a two-atom vacancy at $40$. It is because the conductance dips at band edges(Fig. \[fig3\]) enter into the energy range of $\pm 0.5$ eV as the width of GNRs increases . This does not change the overall decreasing tendency of $\overline{\Delta
G}/\, \overline{G_0}$ when the width of GNRs increases.
### \[sec3:sec1:sec2\] Single weak scatters
Another kind of single defect is a weak scatter, which can be caused by a local lattice distortion, an absorption of an impurity atom at the edge, or a substitution of a carbon atom by an impurity atom. Such a single weak scatter will be simulated by changing the on-site energy of an edge atom to a small defect potential $V$.
![\[fig5\] (Color online) (a) Conductance of a zigzag ribbon ($N=8$) for various strengths of defect potential when the ribbon has a weak scatter at its edge. (b) The LDOS of an edge atom for $V=0$ eV (no defect) and $V=2.0$ eV at that site. ](Fig5.EPS)
![\[fig6\] (Color online) (a) Conductance of an armchair ribbon ($N=14$) for various strengths of defect potential when the ribbon has a weak scatter at its edge. (b) The LDOS of an edge atom for $V=0$ eV (no defect) and $V=2.0$ eV at that site. ](Fig6.EPS)
The conductances and LDOS of zigzag GNRs under the influences of single weak scatters with different strengths are presented in Fig.\[fig5\]. It can be seen that even a very weak scatter($V=0.5$ eV) can produce a quasi-localized state around Fermi energy and cause a zero-conductance dip. And the energy level and breadth of the dip increase when the defect potential increases. This is because the kinetic energy of mobile electrons in a zigzag GNR is nearly zero around Fermi energy. And these mobile electrons are localized to ribbon edges (edge states).[@nakada96] So they can be easily reflected by a weak scatter at the edge. On the other hand, the group velocity of mobile electrons in armchair GNRs around Fermi energy is in the order of $10^6 \text{m/s}$, which gives a large kinetic energy. So these mobile electrons will not as sensitive to a weak scatter as those in zigzag GNRs. In fact, there is no conductance dips near Fermi energy for armchair GNRs with a weak scatter. And the conductance dip at the band edge only becomes visible when the defect potential is larger than $2.0 $ eV (see Fig.\[fig6\]).
### \[sec3:sec1:sec2\] A simple one dimensional model
There are some common characters in the conductance curves and LDOS curves shown above (Figs. \[fig2\], \[fig3\], \[fig5\], \[fig6\]). First, there are sharp peaks in LDOS curves of perfect GNRs, which are Van Hove singularities(VHS) corresponding to extreme points in the energy bands. VHS are characteristic of the dimension of a system. In 3D systems, VHS are kinks due to the change in the degeneracy of the available phase space, while in 2D systems, the VHS appear as stepwise discontinuities with increasing energy.[@odom00] Unique to 1D systems, the VHS display as peaks. So GNRs are expected to exhibit sharp peaks in the LDOS due to the 1D nature of their band structures. Second, besides these VHS, there are new peaks in the LDOS of GNRs with an edge defect. And zero-conductance dips occur at the same energy of these new peaks simultaneously. These new peaks in LDOS only occur near the defect, but have effects on the conductance of GNRs. So they correspond to quasi-localized states. And the zero-conductance dips are due to the anti-resonance of these quasi-localized states. The relation between quasi-localized states and zero-conductance dips can be understood by a simple 1D model.
![\[fig7\] A one-dimensional model of a system including conducting bands and a quasi-localized state (QLS). The bulk of the system is represented by a quantum wire (QW), which has one conducting band. ](Fig7.EPS)
A GNR with an edge defect which induces a quasi-localized state(QLS) can be modeled as a 1D quantum wire (QW) with a side quantum dot (see Fig.\[fig7\]). The quantum wire has one conducting band with dispersion relation $E=2\, \nu \, \cos(k d)$, where $E$ the energy of electrons, $\nu$ the hopping coefficient in the QW and $d$ the lattice spacing. The energy level of the quasi-localized state (side quantum dot) is $\varepsilon_{L}$. And the coupling between the quasi-localized state and the QW is $t_{LC}$. If $t_{LC}=0$, the state is completely localized and has no effect on the conductance of the QW. When $t_{LC}\neq 0$, the electrons not only can transport in the QW, but also can transport through “QW$\rightarrow$QLS$
\rightarrow$QW", “QW$\rightarrow$QLS$\rightarrow$QW$\rightarrow$ QLS$\rightarrow$QW", and so on. These different channels will interfere with each other and can cause resonance or antiresonance. It is easy to show that they will always cause antiresonance:[@orel03]
To calculate the conductance of this simple system, we assume that the electrons are described by a plane wave incident from the far left with unity amplitude and a reflection amplitude $r$ and at the far right by a transmission amplitude $t$. So the probability amplitude to find the electron in the site $j$ of the QW in the state $k$ can be written as $$\begin{aligned}
\label{eq19} &&a^k_j=e^{ikdj}+r\, e^{-ikdj}, \; j<0, \\
\label{eq20} &&a^k_j=t\, e^{ikdj}, \; j>0.\end{aligned}$$ Then the transmission amplitude $t$ and thus the conductance of the system can be easily calculated by its tight-binding Hamiltonian.[@orel03] The conductance is $$\begin{aligned}
\label{eq21} &&G_{(E)}=\frac{2e^2}{h}\frac{1}{1+\displaystyle
\frac{t^4_{LC}}{ 4\nu^2\sin^2(kd)\, (E-\varepsilon_{L})^2}}.\end{aligned}$$
From Eq.\[eq21\], we can see that when $E=\varepsilon_L$, the conductance $G$ will be zero and a dip will appear in the conductance curve. In other words, the incident electrons will be totally reflected when their energy is equal to the energy level of the quasi-localized state. So the quasi-localized state causes an antiresonance. This analytical result agrees with our numerical results of GNRs. This relation between the conductance dips and localized states is very useful in experiments. It’s not easy to measure the conductance of GNRs directly because of their small size. But the quasi-localized states can be find in the STS (Scanning Tunneling Spectroscopy) images or low bias STM images.[@kobayashi06] Then with a STS image or a low bias STM image, the conductance dips can be predicted.
\[sec3:sec2\] Weak disorders
----------------------------
Experimental observed graphene edges[@kobayashi06; @banerjee05; @niimi06] have a lot of randomly distributed defects. Most of these defects are likely to be avoided in future with improvements in the processing of GNRs. However, as all materials have defects, real GNRs will always have some uncontrollable defects at their edges due to lattice distortion or impurity. In this section, we will consider the properties of GNRs under the influence of weak uniform disorders at their edges. An edge disorder distributed over a length $L$, will be simulated by setting the on-site energies of all edge atoms within a length $L$ to energies randomly selected from the interval $\pm
|V_{random}|$, where $ |V_{random}|$ is the disorder strength.
![\[fig8\] (Color online) Conductance versus energy for a zigzag ribbon ($N=8$) with disorder distributed at both edges over a length of $100$ and $1000$ . With a weak disorder, zigzag GNRs change from metallic to semiconducting.](Fig8.EPS)
The conductances of zigzag GNRs with different disorder strengths and distribution lengths are displayed in Fig.\[fig8\]. The most important feature of the conductance curves is that there are gaps around the Fermi energy. For a $N=8$ zigzag GNR with a very weak disorder ($V_{random}=0.25$ eV) distributed over a length of 1000, the conductance have a $0.25$ eV gap, within which its maximum is less than $10^{-3}\cdot (2e^2/h)$. If the disorder strength is $1.0$ eV, the conductance gap is $1.04$ eV. This is enormous, because a semiconducting perfect GNR with a similar width only has a gap less than $0.7$ eV.[@ezawa06; @brey06]
![\[fig9\] (Color online) Conductance versus ribbon length for two zigzag ribbons ($N=8$ and $N=12$) with different disorder strengths ($V_{random}=0.25$ eV and $V_{random}=1.0$ eV). The straight lines are exponential fits to the simulated data with the ribbon length larger than $50$ . ](Fig9.EPS)
The conductance gaps around the Fermi energy come from the Anderson localization of electrons.[@anderson58; @biel05; @gornyi05] In a perfect GNR or a GNR with periodic defects, the constructive interference of tunneling allows that electrons within certain energy bands can propagate through an infinite GNR (Bloch tunneling). However, the disorder can disturb the constructive interference sufficiently to localize electrons. In an infinite 1D system, even weak disorder localizes all states, yielding zero conductance. If the disorder is only distributed within a finite length $L$, the conductance is expected to decrease exponentially with length, $G=G_0 \exp(-L/L_0)$, when $L$ is much larger than localization length $L_0$.[@lee85; @beenakker97] We observe this is true in our simulations (see Fig. \[fig9\]). Each point in Fig. \[fig9\] is an average over several thousand disorder configurations. The localization length of electrons with energy close to zero is very small in the zigzag ribbons. From the top down, the fitted localization lengths are $L_0=59$, $L_0=53$ and $L_0=45$ for curves in Fig.\[fig9\], respectively. So the wider the ribbon, the longer the localization length. And the stronger the disorder strength, the shorter the localization length.
![\[fig10\] (Color online) Conductance versus energy for an armchair ribbon ($N=14$) with disorder distributed at its both edges over a length of 1000 . ](Fig10.EPS)
The conductance of the armchair GNRs with weak disorders is plotted in Fig.\[fig10\]. Unlike the conductance of zigzag ribbons, there is no gap around the Fermi energy. We also calculate the conductance versus disorder length, which is shown in Fig.\[fig11\]. The conductance also decreases exponentially but much slower. The $N=14$ armchair GNRs ($W=17.4$) have nearly the same width of $N=8$ zigzag GNRs ($W=17.3$). But their localization lengths are very different. When the disorder strength is $V_{random}=0.25$ eV, the localization length of $N=14$ armchair GNRs is larger than $2$ $ \mu$m, while $L_0=59$ in $N=8$ zigzag GNRs. So the localization is much weaker in armchair GNRs than in zigzag GNRs. As discussed in Sec. \[sec3:sec1\], this is because the kinetic energy of mobile electrons around Fermi energy in armchair GNRs are larger than in zigzag GNRs, and also because these mobile electrons in zigzag GNRs are localized to edges, while they distribute in the whole cross section of armchair GNRs. So when compared to zigzag GNRs, armchair GNRs are more like 2D systems where electrons are easier to travel around defects. There is no such difference between zigzag and armchair CNTs. The conductances of both zigzag CNTs and armchair CNTs are not significantly affected by disorder.[@anantram98]
![\[fig11\] (Color online) Conductance versus ribbon length for two armchair ribbons ($N=8$ and $N=14$) with different disorder strengths.](Fig11.EPS)
Recent studies of perfect GNRs have found that, with edge corrections which keep the translational symmetry of GNRs, all zigzag GNRs will be still metallic.[@ezawa06] Here we shown that with a weak disorder at edges, zigzag GNRs will change from metallic to semiconducting due to Anderson localization. So narrow zigzag GNRs with a weak disorder can be used as functional elements in a nanodevice. This result is important because nearly all realistic GNRs contain some edge disorder.
\[sec4\] Conclusion
===================
Using a tight-binding model, we have investigated the conductance of the zigzag and armchair graphene nanoribbons with single defects or weak disorder at their edges. We first study the simplest possible edge defects, a single vacancy or a weak scatter. We find that even these simplest defects have highly non-trivial effects. A single edge defect will induce quasi-localized states and consequently cause zero-conductance dips. And the center energies and breadths of such dips are strongly dependent on the geometry of GNRs. A one-atom edge vacancy will completely reflect electrons at Fermi energy in an armchair GNR, while only slightly affecting the transport of electrons in a zigzag GNR. The effect of a two-atom vacancy in an armchair GNR is similar to the effect of a one-atom vacancy in a zigzag GNR. A weak scatter can cause a quasi-localized state and consequently a zero-conductance dip near Fermi energy in a zigzag GNR. But its effect on the conductance of armchair ribbons near Fermi energy is negligible. The influence of edge defects on the conductance will decrease when the widths of GNRs increase. Then we use a simple one dimensional model to discuss the relation between quasi-localized states and zero-conductance dips of GNRs. We find that a quasi-localized state caused by a defect will cause antiresonance and corresponds to a zero-conductance dip.
Finally, we study some more realistic structures, GNRs with weak scatters randomly distributed on their edges. We find that with a weak disorder distributed in a finite length, zigzag GNRs will change from metallic to semiconducting due to Anderson localization. But a weak disorder only slightly affects the conductance of armchair GNRs. The effect of edge disorder decreases as the width of GNRs increases. So narrow zigzag GNRs with a weak disorder can be used as functional elements in a nanodevice. And GNRs used as connections should be wider than GNRs used as functional elements. These results are useful for better understanding the property of realistic graphene nanoribbons, and will be helpful for designing nanodevices based on graphene.
The authors would like to thank N. M. R. Peres for helpful discussions.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'A finite group $G$ is said to be a [*POS-group*]{} if the number of elements of every order occurring in $G$ divides $|G|$. We answer two questions by Finch and Jones in [@finch2002curious] by providing an infinite family of nonabelian POS-groups with orders not divisible by $3$.'
address: ' Department of Mathematics, College of Saint Benedict and Saint John’s University, 37 College Avenue South, Saint Joseph, MN 56374-5011, USA '
author:
- Bret Benesh
bibliography:
- 'MasterBibliography.bib'
title: On two questions by Finch and Jones about Perfect Order Subset Groups
---
Let $G$ be a finite group, and define the [*order subset of an element $x \in G$*]{} to be $\{g \in G \mid o(g)=o(x)\}$, where $o(x)$ denotes the order of $x$. We say that $G$ has [*perfect order subsets*]{} if the number of elements in every order subset divides $|G|$; in this case, we say that $G$ is a POS-group. It is easy to see that ${\mathbb{Z}}_2$, ${\mathbb{Z}}_4$, and the symmetric group $S_3$ are POS-groups, whereas ${\mathbb{Z}}_3$, ${\mathbb{Z}}_5$, and $S_4$ are not.
This definition is due to Finch and Jones, who worked with abelian groups [@finch2002curious] and direct products of abelian groups with $S_3$ [@finch2003nonabelian]. They provided the following open questions at the end of [@finch2002curious].
1. Are there nonabelian POS-groups other than the symmetric group $S_3$?
2. If the order of a POS-group is not a power of $2$, is the order necessarily divisible by $3$? This is also Conjecture 5.1 from [@finch2003nonabelian], also by Finch and Jones.
The answers are “yes" and “no," respectively. The first question was answered by Finch and Jones in [@finch2003nonabelian], although all groups were direct products of $S_3$ with an abelian group. Das [@das2009finite] answered both questions by proving that there exists an action $\theta$ such that the semidirect product ${\mathbb{Z}}_{p^{k}} \rtimes_{\theta} {\mathbb{Z}}_{2^{l}}$ is a POS-group, where $p$ is a Fermat prime, $k \geq 1$, and $2^{l} \geq p-1$. Feit also answered both questions by indicating that a Frobenius group of order $p(p-1)$ for a prime $p>3$ is a POS-group [@finch2003nonabelian].
We now provide an infinite family of groups that simultaneously answers both questions. Let $n \geq 1$, and consider the group ${\mathbb{Z}}_4 \rtimes {\mathbb{Z}}_{2 \cdot 5^{n}}$ with the inversion action. The order of this group is $2^3 \cdot 5^{n}$, which is not divisible by $3$. Consequently, $S_3$ cannot appear as a subgroup or quotient of any of these groups, as $3$ divides the order of $S_3$.
Table \[tab:OrderTable\] summarizes the easy calculations required to find the size of each order subset of ${\mathbb{Z}}_4 \rtimes {\mathbb{Z}}_{2 \cdot 5^{n}}$ (one can use geometric sums to verify that all elements of the group are accounted for). Note that the number of elements of each order divides the order of the group, thereby proving that the groups are POS-groups. This confirms that ${\mathbb{Z}}_4 \rtimes {\mathbb{Z}}_{2 \cdot 5^{n}}$ is a POS-group.
[lllllll]{}\
Order: & $1$ & $2$ & $4$ & $5^{m}$ & $2 \cdot 5^{m}$ & $4 \cdot 5^{m}$\
Elements: & $1$ & $5$ & $2$ & $4 \cdot 5^{m-1}$ &$4 \cdot 5^{m}$ &$8 \cdot 5^{m-1}$\
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is geodesically complete and has everywhere negative sectional curvature. An important consequence of this negative curvature for applications is that the Fréchet mean of a set of Dirichlet distributions is uniquely defined in this geometry.'
address:
- 'SAMM 4543, Université Paris 1 Panthéon Sorbonne, Centre PMF, Paris, France.'
- 'Department of Mathematics, Brooklyn College and CUNY Graduate Center, New York, USA.'
- 'Ecole Nationale de l’Aviation Civile, Université de Toulouse, Toulouse, France.'
author:
- Alice Le Brigant
- 'Stephen C. Preston'
- Stéphane Puechmorel
bibliography:
- 'bibliography.bib'
title: 'Fisher-Rao geometry of Dirichlet distributions'
---
Introduction
============
The differential geometric approach to probability theory and statistics has met increasing interest in the past years, from the theoretical point of view as well as in applications. In this approach, probability distributions are seen as elements of a differentiable manifold, on which a metric structure is defined through the choice of a Riemannian metric. Two very important ones are the Wasserstein metric, central in optimal transport, and the Fisher-Rao metric (also called Fisher information metric), essential in information geometry. Unlike optimal transport, information geometry is foremost concerned with parametric families of probability distributions, and defines a Riemannian structure on the parameter space using the Fisher information matrix [@fisher1922]. It was Rao who showed in 1945 [@rao1945] that the Fisher information could be used to locally define a scalar product on the space of parameters, and interpreted as a Riemannian metric. Later on, Cencov [@cencov1982] proved that it was the only metric invariant with respect to sufficient statistics, for families with finite sample spaces. This result has been extended more recently to non parametric distributions with infinite support [@ay2015; @bauer2016].
Information geometry has been used to obtain new results in statistical inference as well as gain insight on existing ones. In parameter estimation for example, Amari [@amari2016information] shows that conditions for consistency and efficiency of estimators can be expressed in terms of geometric conditions; in the presence of hidden variables, the famous Expectation-Maximisation (EM) algorithm can be described in an entirely geometric manner; and in order to insure invariance to diffeomorphic change of parametrization, the so-called natural gradient [@amari1998natural] can be used to define accurate parameter estimation algorithms [@ollivier2017].
Another important use of information geometry is for the effective comparison and analysis of families of probability distributions. The geometric tools provided by the Riemannian framework, such as the geodesics, geodesic distance and intrinsic mean, have proved useful to interpolate, compare, average or perform segmentation between objects modeled by probability densities, in applications such as signal processing [@arnaudon2013riemannian], image [@schwander2012model; @angulo2014morphological] or shape analysis [@peter2006; @srivastava2007], to name a few. These applications rely on the specific study of the geometries of usual parametric families of distributions, which has started in the early work of Atkinson and Mitchell. In [@atkinson1981], the authors study the trivial geometries of one-parameter families of distributions, the hyperbolic geometry of the univariate normal model as well as special cases of the multivariate normal model, a work that is continued by Skovgaard in [@skovgaard1984]. The family of gamma distributions has been studied by Lauritzen in [@lauritzen1987statistical], and more recently by Arwini and Dodson in [@arwini2008], who also focus on the log-normal, log-gamma, and families of bivariate distributions. Power inverse Gaussian distributions [@zhang2007], location-scale models and in particular the von Mises distribution [@said2019], and the generalized gamma distributions [@rebbah2019] have also received attention.
In this work, we are interested in Dirichlet distributions, a family of probability densities defined on the $(n-1)$-dimensional probability simplex, that is the set of vectors of ${\mathbb R}^n$ with non-negative components that sum up to one. The Dirichlet distribution models a random probability distribution on a finite set of size $n$. It generalizes the beta distribution, a two-parameter probability measure on $[0,1]$ used to model random variables defined on a compact interval. Beta and Dirichlet distributions are often used in Bayesian inference as conjugate priors for several discrete probability laws [@o1999bayesian; @griffiths2002gibbs; @briggs2003probabilistic], but also come up in a wide variety of other applications, e.g. to model percentages and proportions in genomic studies [@yang2017beta], distribution of words in text documents [@madsen2005modeling], or for mixture models [@bouguila2004unsupervised]. Up to our knowledge, the information geometry of Dirichlet distributions has not yet received much attention. In [@calin2014], the authors give the expression of the Fisher-Rao metric for the family of beta distributions, but nothing is said about the geodesics or the curvature.
In this paper, we give new results and properties for the geometry of Dirichlet distributions, and its sectional curvature in particular. The derived expressions depend on the trigamma function, the second derivative of the logarithm of the gamma function, however we will avoid using its properties when possible to obtain our results. Instead, we consider a more general metric written using a function $f$, for which we only make the strictly necessary assumptions. Section \[sec:dirichlet\] gives the setup for our problem by considering the Fisher-Rao metric on the space of parameters of Dirichlet distributions. In Section \[sec:general\], we consider the more general metric where $f$ replaces the trigamma function, and show that it induces the geometry of a submanifold in a flat Lorentzian space. This allows us to show geodesic completeness, and that the sectional curvature is everywhere negative. Section \[sec:beta\] focuses on the two-dimensional case, i.e. beta distributions.
Fisher-Rao metric on the manifold of Dirichlet distributions {#sec:dirichlet}
============================================================
Let $\Delta_n$ denote the $(n-1)$-dimensional probabilty simplex, i.e. the set of vectors in ${\mathbb R}^n$ with non-negative components that sum up to one $$\Delta_n=\{q=(q_1,\hdots,q_n)\in{\mathbb R}^n, \,\sum_{i=1}^nq_i=1,\, q_i\geq 0, i=1,\hdots,n\}.$$ The family of Dirichlet distributions is a family of probability distributions on $\Delta_n$ parametrized by $n$ positive scalars $x_1,\hdots, x_n >0$, that admits the following probability density function with respect to the Lebesgue measure $$f_n(q|x_1,\hdots,x_n) = \frac{\Gamma(x_1+\hdots+x_n)}{\Gamma(x_1)\hdots\Gamma(x_n)} q_1^{x_1-1}\hdots q_n^{x_n-1}.$$ As an open subset of ${\mathbb R}^n$, the space of parameters $M = ({\mathbb R}_+^*)^n$ is a differentiable manifold and can be equipped with a Riemannian metric defined in its matrix form by the Fisher information matrix $$g_{ij}(x_1,\hdots,x_n) = -{\mathbb E}\left[ \frac{\partial^2}{\partial x_i\partial x_j} \log f_n(Q|x_1,\hdots,x_n)\right], \quad i,j=1,\hdots,n,$$ where $\mathbb E$ denotes the expectation taken with respect to $Q$, a random variable with density $f_n(\cdot|x_1,\hdots,x_n)$. The Dirichlet distributions form an exponential family and so the Fisher-Rao metric is the hessian of the log-partition function [@amari2016information], namely $$g_{ij}(\alpha,\beta) = \frac{\partial^2}{\partial x_i\partial x_j} \varphi(x_1,\hdots, x_n), \quad i,j=1,\hdots,n,$$ where $\varphi$ is the logarithm of the normalizing factor $$\varphi (x_1,\hdots, x_n) = \sum_{i=1}^n \log \Gamma(x_i) - \log\Gamma(x_1+\hdots+x_n).$$ We obtain the following metric tensor. $$\label{fisherraodirichlet}
g_{ij}(x_1,\hdots,x_n)= \psi'(x_i)\delta_{ij} - \psi'(x_1+\hdots+x_n),$$ where $\delta_{ij}$ is the Kronecker delta function, and $\psi$ denotes the digamma function, that is the first derivative of the logarithm of the gamma function, i.e. $$\psi(x) = \frac{d}{dx}\log\Gamma(x).$$ Its derivative $\psi'$ is called the trigamma function. As noted below, the trigamma function is a function whose reciprocal is increasing, convex, and sublinear on $\mathbb{R}^+$. For slightly greater generality, and to emphasize what properties of this function are needed for our results, we will work in the sequel with a more general function $f$ on which we make only the necessary assumptions; in our special case we have $f = 1/\psi'$.
The general framework {#sec:general}
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The metric
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In this section we consider a more general geometry, that admits the Fisher-Rao geometry of Dirichlet distributions as a special case. The goal is to avoid using the properties of the trigamma function when possible. For this, we consider the quadrant $M=({\mathbb R}_+^*)^n$ equipped with a metric of the form $$\label{basicmetric}
ds^2 = \frac{dx_1^2}{f(x_1)} + \cdots + \frac{dx_n^2}{f(x_n)} - \frac{(dx_1 + \cdots + dx_n)^2}{f(x_1+\cdots + x_n)},$$ where $f:{\mathbb R}_+\rightarrow {\mathbb R}$ is a function on which we make the following assumptions: $$\label{fassumptions}
f(x)\underset{x\to 0}{=}O(x^2), \quad f'(x) \underset{x\to 0}{=} O(x),\quad f(x) \underset{x\to \infty}{=} O(x^2),\quad f''>0 \text{ and } \frac{d^2}{dx^2}\left( \frac{f}{f'}\right) > 0.$$ We retrieve the Fisher-Rao metric when $$f(x)=\frac{1}{\psi'(x)}.$$ Notice that this choice for $f$ satisfies the conditions of . Indeed, that $f(0)=f'(0)=0$ comes from the asymptotic formula $\psi'(x) \approx x^{-2}$ valid near $x=0$, since $$f'(0) = \lim_{x\to 0} \frac{-\psi''(x)}{\psi'(x)^2} = \lim_{x\to 0} \frac{2x^{-3}}{x^{-4}} = 0.$$ The fact that the reciprocal of the trigamma function $f(x) = \frac{1}{\psi'(x)}$ is convex comes from an argument of Trimble-Wells-Wright [@trimble1989superadditive], based on an inequality later proved in Alzer-Wells [@alzer1998inequalities]. The fact that $f/f'$ is convex comes from Yang [@yang2017].
Another example of a function satisfying the conditions is $$\label{rationalapprox}
\tilde{f}(x) = \frac{(2x+1)x^2}{2x^2+2x+1},$$ a simple rational function which approximates the reciprocal of the trigamma function well, in both the small-$x$ and large-$x$ regions.
Some useful consequences of our assumptions are given in the following lemma. These results are well-known, but we include the simple proofs for completeness.
\[fconsequences\] If $f$ satisfies , then we have $f'(x)>0$ and $f(x)>0$ for all $x>0$. In addition $f$ and $f/f'$ are superadditive: $$\begin{aligned}
f(x_1 + \cdots + x_n) &> f(x_1) + \cdots + f(x_n), \label{fsuperadditive} \\
\frac{f(x_1+\cdots + x_n)}{f'(x_1+\cdots x_n)} &> \frac{f(x_1)}{f'(x_1)} + \cdots + \frac{f(x_n)}{f'(x_n)}, \label{ffprimesuperadditive}\end{aligned}$$ for all $x_1,\ldots, x_n>0$.
That $f''>0$ implies $f'>0$ and thus $f>0$ for all $x>0$ is obvious. It has been known since Petrovich [@petrovich1932fonctionnelle] that a convex function $f$ with $f(0)=0$ is superadditive: an easy argument in the differentiable case is that $$f(x+y) - f(x) - f(y) = \int_0^x \int_0^y f''(s+t) \, dt \, ds \ge 0.$$ By induction the general case follows. Since $\lim_{x\to 0} f(x)/f'(x) = 0$, the same argument applies to $f/f'$ to give .
Lorentzian submanifold geometry
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We now show that after a change of coordinates, $M$ can be seen as a codimension 1 submanifold of the $(n+1)$-dimensional flat Minkowski space $L^{n+1}=({\mathbb R}^{n+1},ds_L^2)$, where $$\label{minkowskimetric}
ds_L^2=dy_1^2+\hdots+dy_n^2-dy_{n+1}^2.$$ In the sequel, we will denote by $\langle\cdot,\cdot\rangle$ the scalar product induced by this metric.
\[flatteningprop\] The mapping $$\begin{aligned}
\Phi&:M\rightarrow L^{n+1},\\
&(x_1,\hdots,x_n)\mapsto (\eta(x_1),\hdots,\eta(x_n),\eta(x_1+\hdots+x_n))\end{aligned}$$ where $\eta\colon \mathbb{R}_+\to \mathbb{R}$ is defined by $$\eta(x) = \int_1^x \frac{dr}{\sqrt{f(r)}},$$ is an isometric embedding.
Since $f(x) \approx \tfrac{1}{2} f''(0) x^2$ for $x\approx 0$, we see that $$\int_0^1 \frac{dr}{\sqrt{f(r)}} = \infty,$$ so that the image of $\eta$ must include all negative reals. Therefore $\eta$ maps $\mathbb{R}_+$ bijectively to $(-\infty, N)$ for some $N\in (0,\infty]$. The behavior of $f$ at infinity assumed in implies that $f(x) \le C x^2$ for all $x\ge K$, for some $K>0$, $C>0$, which in turns leads to $$\int_K^{\infty} \frac{dx}{\sqrt{f(x)}} = \infty.$$ Therefore $\eta$ maps bijectively $\mathbb{R}_+$ to $\mathbb{R}$, and $\Phi$ is a homeomorphism onto its image. Since $\eta'(x)>0$ for all $x$, it is also an immersion. Finally, if $(y_1,\hdots,y_{n+1})=\Phi(x_1,\hdots,x_n)$, $$dy_i^2 = \eta'(x_i)^2 dx_i^2 = \frac{dx_i^2}{f(x_i)}, i=1,\hdots,n, \quad dy_{n+1}^2 = \frac{(dx_1+\hdots+dx_n)^2}{f(x_1+\hdots+x_n)},$$ and $\Phi$ is isometric.
\[positivemetric\] $S=\Phi(M)$ is a codimension $1$ submanifold of $L^{n+1}$ given by the graph of $$\label{submanifoldeq}
y_{n+1}=\eta(\xi(y_1)+\hdots+\xi(y_n)), \quad y_i>0,$$ where $\xi=\eta^{-1}$. On this submanifold the metric is positive-definite and thus Riemannian. A basis of tangent vectors of $T_yS$ is defined by $$\label{tangentvecs}
e_i = \frac{\partial}{\partial y_i} + \sqrt{\frac{f\circ\xi(y_i)}{f\circ\xi(y_{n+1})}}\frac{\partial}{\partial y_{n+1}}, \quad i=1,\hdots,n,$$
Let $\gamma(u)=(y_1(u),\hdots,y_{n+1}(u))$ be a parametrized curve in $S$. Then its coordinates verifiy the following relations $$\begin{aligned}
y_{n+1}&=\eta(\xi(y_1)+\hdots+\xi(y_n)),\\
y_{n+1}' &=\eta'(\xi(y_1)+\hdots+\xi(y_n))(\xi'(y_1)y_1'+\hdots+\xi'(y_n)y_n'),
\end{aligned}$$ and so, since $\xi'(x)=\sqrt{f(\xi(x))}$, $$\begin{aligned}
\gamma'(u)&=\sum_{i=1}^ny_i'(u)\frac{\partial}{\partial y_i}+\eta'(\xi(y_{n+1}(u)))(\xi'(y_1(u))y_1'(u)+\hdots+\xi'(y_n(u))y_n'(u))\frac{\partial}{\partial y_{n+1}}\\
&= \sum_{i=1}^ny_i'(u)\left(\frac{\partial}{\partial y_i} + \frac{\sqrt{f\circ\xi(y_i(u))}}{\sqrt{f\circ\xi(y_{n+1}(u))}} \frac{\partial}{\partial y_{n+1}}\right),
\end{aligned}$$ yielding as basis tangent vectors. The metric components on $S$ take the form $$\label{metrictrans}
g_{ij} = \langle e_i,e_j\rangle = \delta_{ij} - W_iW_j, \qquad \text{or}\quad g = I - WW^T,$$ where $\langle\cdot,\cdot\rangle$ denotes the flat Minkowskian metric and $W_i = \sqrt{f(\xi(y_i))/f(\xi(y_{n+1}))}$ for $i=1,\hdots,n$. Applying Lemma \[positivetransposelemma\] of the appendix gives the result upon computing $$\label{Wnorm}
W^TW = \frac{\sum_{i=1}^n f\big(\xi(y_i)\big)}{f\big(\sum_{i=1}^n \xi(y_i)\big)} < 1,$$ by superadditivity of $f$, as in .
In other words, the metric is the restriction of the flat Lorentzian metric $$ds^2 = \frac{dx_1^2}{f(x_1)} + \cdots + \frac{dx_n^2}{f(x_n)} - \frac{dt^2}{f(t)}$$ to the hyperplane $t = x_1 + \cdots + x_n$. In the sequel, we will use this Lorentzian submanifold geometry to study the sectional curvature and geodesic completeness of $M$. We state the results in the original coordinate system of $M$ when possible, using the following notations for any $y=(y_1,\hdots,y_{n+1})\in S$: $$\label{xcoord}
x_i=\xi(y_i), \,\, i=1,\hdots,n, \quad t=x_1+\hdots+x_n=\xi(y_{n+1}).$$
Negative sectional curvature
----------------------------
The goal of this section is to prove that the sectional curvature of $M$ is everywhere negative. This has important implications in information geometry, as it guarantees the uniqueness of the Fréchet mean of a set of points in this geometry. The Fréchet mean, also called intrinsic mean, is a popular choice to extend the notion of barycenter to a Riemannian manifold. It is defined for a set of points as the minimizer of the sum of the squared distances to the points of the set. It is in general not unique and refers itself to a set. Uniqueness holds however when the curvature of the Riemannian manifold is negative [@karcher1977], which is the case here. This implies that the notion of barycenter of Dirichlet distributions is well defined in the Fisher-Rao geometry.
We start by computing the shape operator.
\[secondff\] The shape operator of $S=\Phi(M)$ has the following components in the basis of tangent vectors $$\langle \Sigma(e_i),e_j\rangle = -\frac{1}{2\sqrt{f(t)-\sum_{\ell=1}^nf(x_\ell)}}\left( f'(x_i)\delta_{ij} - \frac{f'(t)}{f(t)}\sqrt{f(x_i)f(x_j)}\right).$$
We first observe that the basis vectors can be expressed in coordinates as $$\label{tangentvecsx}
e_i = \frac{\partial}{\partial y_i} + \sqrt{\frac{f(x_i)}{f(t)}}\frac{\partial}{\partial y_{n+1}}, \quad i=1,\hdots,n.$$ Since $S$ can be obtained as the graph of $F(y_1,\hdots,y_n)=\eta(\xi(y_1),\hdots,\xi(y_n))$, a normal vector field to $S$ at $y$ is given by $$\label{normalfield}
N=\sum_{i=1}^n\frac{\partial F}{\partial y_i}\frac{\partial}{\partial y_i}+\frac{\partial}{\partial y_{n+1}}=\sum_{i=1}^n\sqrt{\frac{f(x_i)}{f(t)}}\frac{\partial}{\partial y_i}+\frac{\partial}{\partial y_{n+1}},$$ which yields a timelike vector since $$\label{normalnorm}
\langle N,N \rangle = \frac{1}{f(t)}\left(f(x_1)+\hdots+f(x_n) - f(t)\right)<0,$$ by superadditivity of $f$. Since $\langle N, e_i\rangle=0$, the shape operator is then given by $$\label{shapeop}
\langle \Sigma(e_i),e_j\rangle =-\langle \nabla_{e_i}\left(\frac{N}{\sqrt{-\langle N,N\rangle}}\right), e_j\rangle=-\frac{\langle \nabla_{e_i} N, e_j\rangle}{\sqrt{-\langle N,N \rangle}},$$ $\nabla$ is the flat connection of the Minkowski space. Denoting $\partial_i=\partial/\partial y_i$, we get from , and the flatness of $\nabla$, $$\nabla_{e_i}N = \nabla_{\partial_i }N +\frac{f(x_i)}{f(t)}\nabla_{\partial_{n+1}}N = \sum_{j=1}^n \partial_i\partial_j F \partial_j.$$ Inserting this last equation along with into yields $$\langle \Sigma(e_i),e_j\rangle =-\sqrt{\frac{f(t)}{f(t)-\sum_{\ell=1}^nf(x_\ell)}}\partial_i\partial_j F.$$ Straightforward computations give $$\begin{aligned}
\partial_iF&=\eta'(t)\xi'(y_i)=\sqrt{f(x_i)/f(t)},\\
\partial_i\partial_jF &= \eta''(t)\xi'(y_i)\xi'(y_j) + \eta'(t)\xi''(y_i)\delta_{ij} = \frac{1}{2\sqrt{f(t)}}\left(\frac{-f'(t)}{f(t)}\sqrt{f(x_i)f(x_j)} + f'(x_i)\delta_{ij}\right),\end{aligned}$$ and the result follows after simplification.
\[posdefsecondff\] The second fundamental form given by Proposition \[secondff\] is positive-definite.
This follows from Lemma \[positivetransposelemma\] and the decomposition of the matrix $\Sigma$ with components $\Sigma_{ij}=\langle \Sigma(e_i),e_j\rangle$ as $$\Sigma=-\frac{1}{2}k(D-cVV^T),$$ where $D=\mathrm{diag}(d_1,\hdots,d_n)$ is a diagonal matrix, $V=(v_i)_{1\leq i\leq n}$ is a column vector and $c$ and $k$ are constants, defined for $i=1,\hdots,n$ by$$\label{decompsecondff}
d_i=f'(x_i), \quad v_i=\sqrt{f(x_i)}, \quad k = \frac{1}{\sqrt{f(t)-\sum_{\ell=1}^nf(x_\ell)}}, \quad c = \frac{f'(t)}{f(t)}.$$ Recalling that $f>0$ and $f'>0$ by Lemma \[fconsequences\], we see that the matrix $D$ and constant $c$ are positive. There remains to verify that $$cV^TD^{-1}V = \frac{f'(t)}{f(t)}\sum_{i=1}^n\frac{f(x_i)}{f'(x_i)}<1,$$ by the superadditivity property .
We can now show our main result.
The sectional curvature of the Riemannian metric is negative on $M$.
We use a result from O’Neill [@o1983semi Chapter 4, Corollary 20], which states that if the normal vector field $N$ of a hypersurface $M$ in a flat Lorentzian manifold $L$ is timelike, then the sectional curvature of the submanifold is given by $$\label{gausscodazzi}
K(U,V) = - \frac{\langle \Sigma(U), U\rangle \langle \Sigma(V), V\rangle - \langle \Sigma(U),V\rangle^2}{\langle U,U\rangle \langle V,V\rangle - \langle U,V\rangle^2},$$ where $U$ and $V$ are tangent to the submanifold and $\Sigma$ is the shape operator. The result now follows by the Cauchy-Schwarz inequality: since $\Sigma$ is a positive-definite symmetric matrix, we know that $\langle \Sigma(U),U\rangle \langle \Sigma(V),V\rangle \ge \langle \Sigma(U),V\rangle^2$ with equality iff $V$ is a multiple of $U$, but in that case the denominator vanishes as well. So the sectional curvature must be strictly negative.
We know give more specifically the formula of the sectional curvature of the planes generated by the basis tangent vectors .
\[seccurv\] The sectional curvature along the axes defined by is given by $$K(e_i,e_j)=\frac{f(x_i)f'(x_j)f'(t)+f'(x_i)f(x_j)f'(t)-f'(x_i)f'(x_j)f(t)}{4(f(t)-\sum_{\ell=1}^nf(x_\ell))(f(t)-f(x_i)-f(x_j))}.$$
This follows from applying formula for the sectional curvature of a hypersurface in a flat Lorentzian manifold, with $$\langle e_i,e_i\rangle=1-\frac{f(x_i)}{f(t)}, \quad \langle e_i, e_j\rangle=\sqrt{\frac{f(x_i)f(x_j)}{f(t)^2}}, \quad i\neq j,$$ and $\langle \Sigma(e_i),e_j\rangle$ given by Proposition \[secondff\].
Finally, we state a result about the eigenvalues of the shape operator, which will be useful to show geodesic completeness in the next section.
\[boundedprincurv\] The principal curvatures at any point in $S=\Phi(M)$ are bounded.
The principal curvatures at a given point $(y_1,\hdots,y_{n+1})=\Phi(x_1,\hdots,x_n)\in S$ are the eigenvalues of the shape operator $$\Sigma=-\frac{1}{2}k(D-cVV^T),$$ where $D$, $V$, $c$ and $k$ are defined by . Without loss of generality, we assume that the $n$-tuple $(x_1,\hdots,x_n)$ is ordered. Let us first show that when at least $n-1$ variables go to zero, i.e. $x_i\to 0$ for $i=1,\hdots,n-1$ with the previous assumption, the principal curvatures go to zero. Let $\tau=x_1+\hdots+x_{n-1}$, then $t=x_n+\tau$ and $$f(t)-f(x_1)-\hdots -f(x_n)\underset{\tau\to 0}{\sim} \tau f'(x_n)$$ since $f$ has limit zero in zero, and so $$k \underset{\tau\to 0}{\sim} \frac{1}{\sqrt{\tau f'(x_n)}}.$$ Using the fact that when $\tau\to 0$, recalling assumptions , $$f(x_i)=O(x_i^2), \quad f'(x_i) = O(x_i), \quad x_i = O(\tau), \quad i=1,\hdots,n-1,$$ we see that the diagonal terms of $D-cVV^T$ behave as $$\begin{aligned}
&f'(x_i) -\frac{f'(t)}{f(t)}f(x_i) = O(\tau), \quad i=1,\hdots,n-1,\\
&f'(x_n) -\frac{f'(t)}{f(t)}f(x_n) =\frac{f'(x_n)f(x_n+\tau) -f'(x_n+\tau)f(x_n)}{f(x_n+\tau)} \underset{\tau\to 0}{\sim} \tau\left(f''(x_n) + \frac{f'(x_n)^2}{f(x_n)}\right),\end{aligned}$$ while the antidiagonal terms verify $$\begin{aligned}
&-\frac{f'(t)}{f(t)}\sqrt{f(x_i)f(x_j)} = O(\tau^2), \quad 1\leq i,j\leq n-1,\\
&-\frac{f'(t)}{f(t)}\sqrt{f(x_i)f(x_n)} = O(\tau).\end{aligned}$$ Finally, we obtain that $$\Sigma_{ij}=\langle \Sigma(e_i),e_j\rangle \underset{\tau\to 0}{=} O(\sqrt{\tau}), \quad 1\leq i,j \leq n,$$ and so the principal curvatures go to zero when $\tau\to 0$. Therefore there exists $\delta>0$ such that, at any point $(x_1,\hdots,x_n)$ belonging to the set $$\mathcal D_\delta=\{(x_1,\hdots,x_n)\in ({\mathbb R}_+^*)^n, x_i<\delta \text{ for at least } n-1 \text{ indices }i\in\{1,\hdots,n\}\},$$ the principal curvatures are upper bounded by, say, $1$. Now let us consider an $n$-tuple $(x_1,\hdots,x_n)\notin\mathcal D_\delta$, ordered as before. Then the diagonal elements of $D$ are ordered as well since $f'$ is increasing, and the ordered eigenvalues of $k(D-cVV^T)$ verify $$0\leq \lambda_1 \leq \hdots \leq \lambda_n \leq kd_{n},$$ where the lower bound comes from the the positive-definiteness shown in Corollary \[posdefsecondff\], and the upper bound comes from [@golub1973some]. Since $d_n=f'(x_n)$ and $f'$ is increasing and upper bounded by $\underset{x\to\infty}{\lim}f'(x)=1$, we have that $d_n\leq 1$. Since the function $$(x_1,\hdots,x_n)\mapsto f(x_1+\hdots+x_n)-f(x_1)-\hdots-f(x_n)$$ is increasing in all of its variables, it is larger than its limit as the first $n-2$ variables go to zero, and since at least $x_{n-1}>\delta$ and $x_n>\delta$, we obtain $$k =\frac{1}{\sqrt{f(t)-f(x_1)-\hdots -f(x_n)}}\leq \frac{1}{\sqrt{f(2\delta)-2f(\delta)}},$$ and the principal curvatures are again bounded.
Geodesics and geodesic completeness
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The geodesics of $M$ for the metric are parametrized curves $u\mapsto(x_1(u),\hdots,x_n(u))$ solution of the standard second-order ODEs $$\ddot x_k + \sum_{1\leq i,j\leq n}\Gamma_{ij}^k\dot x_i\dot x_j=0, \quad k=1,\hdots,n,$$ whose coefficients can be computed using the following result.
\[prop:christoffel\] The Christoffel symbols for metric are given by $$\begin{aligned}
\Gamma_{ij}^k = \frac{1}{2}\left[\frac{f(x_k)}{f(t)-\sum_{\ell=1}^n f(x_\ell)}\left(g(t)-g(x_j)\delta_{ij}\right) - g(x_k)\delta_{ij}\delta_{jk}\right],\end{aligned}$$ where $t=x_1+\hdots+x_n$ and $g(x) = f'(x)/f(x)$, while $\delta$ denotes the Kronecker delta function.
The Christoffel symbols of the second kind $\Gamma_{ij}^k$ can be obtained from the Christoffel symbols of the first kind $\Gamma_{ijk}$ and the coefficients $g^{ij}$ of the inverse of the metric matrix using the formula $$\Gamma_{ij}^k = \Gamma_{ijl} g^{kl},$$ where we have used the Einstein summation convention. It is easy to see that the Christoffel symbols of the first kind are given by $$\Gamma_{ijk} = \frac{1}{2}\left(\frac{f'(t)}{f(t)^2}-\frac{f'(x_k)}{f(x_k)^2}\delta_{ik}\delta_{jk}\right).$$ Applying the Sherman-Morrison formula, we obtain that the inverse of the metric matrix $$g(x_1,\hdots,x_n) = \text{diag}\left(\frac{1}{f(x_1)},\hdots,\frac{1}{f(x_n)}\right) - \frac{1}{f(t)} J,$$ where $J$ denotes the $n$-by-$n$ matrix with all entries equal to one, is given by $$\label{invmetric}
g(x_1,\hdots,x_n)^{-1}= \text{diag}(f(x_1),\hdots,f(x_n)^{-1}) + \frac{1}{f(t)- \sum_{\ell=1}^nf(x_\ell)}[f(x_i)f(x_j)]_{1\leq i,j\leq n}.$$ Noticing that the sum of all the elements of the $k^{th}$ line (or column) of the inverse of the metric matrix is given by $$\label{sumline}
\sum_{\ell=1}^ng^{k\ell}= f(x_\ell)+\frac{\sum_{\ell=1}^nf(x_k)f(x_\ell)}{f(t)- \sum_{\ell=1}^nf(x_\ell)}=\frac{f(x_k)f(t)}{f(t)- \sum_{\ell=1}^nf(x_\ell)},$$ we obtain $$\Gamma_{ij}^k = \frac{1}{2}\sum_{\ell=1}^n\left(\frac{f'(t)}{f(t)^2}-\frac{f'(x_\ell)}{f(x_\ell)^2}\delta_{ij}\delta_{j\ell}\right)g^{k\ell}=\frac{1}{2}\frac{f'(t)}{f(t)^2}\sum_{\ell=1}^ng^{k\ell}-\frac{1}{2}\delta_{ij}\frac{f'(x_j)}{f(x_j)^2}g^{kj}.$$ Inserting and the general term of the inverse matrix in the above yields $$\Gamma_{ij}^k=\frac{1}{2}\frac{f'(t)}{f(t)^2}\frac{f(x_k)f(t)}{f(t)- \sum_{\ell=1}^nf(x_\ell)} -\frac{1}{2}\delta_{ij}\frac{f'(x_j)}{f(x_j)^2}\left(f(x_k)\delta_{kj} + \frac{f(x_k)f(x_j)}{f(t)-\sum_{\ell=1}^nf(x_\ell)}\right)$$ and the result follows.
Now, using the result of Proposition \[boundedprincurv\] and a theorem from [@harris1988closed], we can show that $M$ is geodesically complete.
\[geodcompletethm\] $M$ equipped with the Riemannian metric is geodesically complete.
The image of $M$ by $\Phi$ is a hypersurface of the $(n+1)$-Minkowski space $L^{n+1}$. Moreover, $\Phi$ is an embedding and it is closed since $\Phi(M)$ is a closed subset of $L^{n+1}$ as preimage of the singleton $\{0\}$ by the continuous map $(y_1,\hdots,y_{n+1})\mapsto\xi(y_1)+\hdots+\xi(y_n)-\xi(y_{n+1})$. Therefore $\Phi$ is proper [@harris1988closed Theorem 1]. Then, [@harris1988closed Theorem 6] allows us to conclude that since $\Phi$ has bounded principal curvatures by Proposition \[boundedprincurv\], $M$ equipped with the pullback of the Minkowski metric by $\Phi$ is complete.
The two-dimensional case of beta distributions {#sec:beta}
==============================================
The simplest case is obviously when $n=2$, and even in this case the formulas are nontrivial. When $f=\frac{1}{\psi'}$, the metric comes from the well-known two-parametric family of beta distributions defined on the compact interval $[0,1]$, which is important in statistics and useful in many applications.
The geodesic equations are given by $$\label{geodeq}
\begin{aligned}
a(x,y)\ddot x + b(x,y) \dot x^2 + c(x,y)\dot x\dot y + d(x,y)\dot y^2 = 0,\\
a(y,x)\ddot y + b(y,x) \dot y^2 + c(y,x)\dot x\dot y + d(y,x)\dot x^2 = 0,
\end{aligned}$$ where $$\begin{aligned}
a(x,y) &= 2\big[ f(x+y) - f(x)-f(y)\big] \\
b(x,y) &= f(y) g(x) + f(x) g(x+y) - f(x+y)g(x) \\
c(x,y) &= 2f(x) g(x+y) \\
d(x,y) &= f(x)g(x+y) - g(y)f(x),\end{aligned}$$ with the shorthand $g(x) = f'(x)/f(x)$.
The geodesic equations can be expressed in terms of the Christoffel symbols as $$\begin{aligned}
\ddot x + \Gamma_{11}^1 \dot x^2 + 2\Gamma_{12}^1\dot x\dot y + \Gamma_{22}^1\dot y^2 = 0,\\
\ddot y + \Gamma_{22}^2 \dot y^2 + 2\Gamma_{12}^2\dot x\dot y + \Gamma_{11}^2\dot x^2 = 0,\end{aligned}$$ and the coefficients can be computed using Proposition \[prop:christoffel\].
![On the left, geodesic balls and and on the right, sectional curvature of the manifold of beta distributions (n=2).[]{data-label="fig:dim2"}](beta_manifold.png "fig:"){width="35.00000%"} ![On the left, geodesic balls and and on the right, sectional curvature of the manifold of beta distributions (n=2).[]{data-label="fig:dim2"}](curvature_beta.png "fig:"){width="35.00000%"}
![On the left, geodesic between the beta distributions of parameters $(2,5)$ and $(2,2)$ and on the right, Fréchet mean (full red line) compared to the Euclidean mean (dashed red line) of the beta distributions of parameters $(2,5)$, $(2,2)$ and $(5,1)$, shown in terms of probability density function.[]{data-label="fig:dim2dens"}](beta_geod.png "fig:"){width="35.00000%"} ![On the left, geodesic between the beta distributions of parameters $(2,5)$ and $(2,2)$ and on the right, Fréchet mean (full red line) compared to the Euclidean mean (dashed red line) of the beta distributions of parameters $(2,5)$, $(2,2)$ and $(5,1)$, shown in terms of probability density function.[]{data-label="fig:dim2dens"}](beta_mean.png "fig:"){width="35.00000%"}
No closed form is known for the geodesics, but they can be computed numerically by solving , see the left-hand side of Figure \[fig:dim2\]. Nonetheless we can notice that, due to the symmetry of the metric with respect to parameters $x$ and $y$, both equations in yield a unique ordinary differential equation when $x=y$.
Solutions of the geodesic equation with $x(0)=y(0)$ and $\dot{x}(0)=\dot{y}(0)$ satisfy $$\label{diagconserv}
\sqrt{q(x(t))} \dot{x}(t) = \text{constant},$$ where $q(x) = \frac{1}{f(x)} - \frac{2}{f(2x)}$, and thus can be found by quadratures.
If at some time $t_0$ we have $x=y$ and $\dot{x}=\dot{y}$, then the equations imply that $\ddot{x}=\ddot{y}$ at $t_0$. Differentiating repeatedly in time shows that all higher derivatives must also be equal at $t_0$, and we conclude by analyticity of the solutions that $x(t)=y(t)$ on some interval. The usual extension arguments for ODEs then imply that $x(t)=y(t)$ on the entire domain of the solution, which by Theorem \[geodcompletethm\] is $\mathbb{R}$.
When $x=y$ equation reduces to $$2 \big[ f(2x) - 2f(x)\big] \ddot{x} + \left(\frac{4f(x)f'(2x)}{f(2x)} - \frac{f(2x) f'(x)}{f(x)} \right) \dot{x}^2 = 0,$$ which is equivalent to $$2 q(x) \ddot{x} + q'(x) \dot{x}^2 = 0.$$ This clearly implies the conservation law . The differential equation can then be solved by writing $$t = \frac{1}{\dot{x}_0 \sqrt{q(x_0)}} \, \int_{x_0}^x \sqrt{q(s)}\, ds$$ and inverting the resulting function.
For example, if $f(x)=1/\psi'(x)$, then the duplication formula for the trigamma function implies $$q(x) = \psi'(x) - 2\psi'(2x) = \tfrac{1}{2} [\psi'(x) - \psi'(x+\tfrac{1}{2})].$$ Asymptotically this looks like $q(x)\approx \frac{1}{2x^2}$ for $x\approx 0$ and $q(x)\approx \frac{1}{4x^2}$ as $x\to \infty$. We conclude that it takes infinite time for a geodesic along the diagonal to either reach “diagonal infinity” or the origin, as Theorem \[geodcompletethm\] of course implies.
From an aplications point of view, the geodesics for the Fisher-Rao geometry allow us to define a notion of optimal interpolation between beta and more generally Dirichlet distributions. An example of such an optimal interpolation is shown on the left-hand side of Figure \[fig:dim2dens\]$,$ in terms of probability density function.
Now we give the formula for the sectional curvature in two dimensions.
\[gaussiancurvprop\] If $n=2$, the sectional curvature is given by $$\label{gaussian}
K(x,y) = -\frac{1}{4} \, \frac{f(t)f'(x)f'(y) - f(x)f'(t)f'(y)-f(y)f'(t)f'(x)}{\big[ f(t)-f(x)-f(y)\big]^2},$$ where $t=x+y$.
This is just a particular case of Proposition \[seccurv\].
Notice that in two dimensions, the negativity of the sectional curvature is straightforward, as there is only one Gaussian curvature to consider, which is given by , in which one can easily see that the numerator is positive by factorizing by $f'(x)f'(y)f'(t)>0$ and using the superadditivity property of $f/f'$.
As previously mentioned, the negative curvature of the Fisher-Rao geometry also has interesting implications for applications: it entails that the Fréchet mean of a set of beta, or more generally Dirichlet distributions is well defined. An example of Fréchet mean of beta distributions is shown in terms of probability density function on the right-hand side of Figure \[fig:dim2dens\].
Numerically we observe that when $f=1/\psi'$, the function $K(x,y)$ given by is decreasing in both the $x$ and $y$ variables – see the right-hand side of Figure \[fig:dim2\] – but we do not yet have a proof of this fact. However we may analyze the asymptotics of the function relatively easily.
![The difference between the sectional curvatures of the plane generated by $e_1$ and $e_2$ in two and three dimensions changes sign for $z=0.01$.[]{data-label="fig:dim2dim3"}](diff_z10.png "fig:"){width="0.3\linewidth"} ![The difference between the sectional curvatures of the plane generated by $e_1$ and $e_2$ in two and three dimensions changes sign for $z=0.01$.[]{data-label="fig:dim2dim3"}](diff_z01.png "fig:"){width="0.3\linewidth"} ![The difference between the sectional curvatures of the plane generated by $e_1$ and $e_2$ in two and three dimensions changes sign for $z=0.01$.[]{data-label="fig:dim2dim3"}](diff_z001.png "fig:"){width="0.3\linewidth"}
\[prop:asymptotic\] If $f=1/\psi'$, then the asymptotic behavior of the sectional curvature given by approaching the boundary square is given by $$\begin{aligned}
\lim_{y\to 0}K(x, y) &= \lim_{y\to 0}K(y, x) = \frac{3}{4} - \frac{\psi'(x)\psi'''(x)}{2\, \psi''(x)^2}, \label{zeroasymptote} \\
\lim_{y\to \infty} K(x, y) &= \lim_{y\to \infty} K(y, x) = \frac{x\, \psi''(x) + \psi'(x)}{4(x\,\psi'(x) - 1)^2}.\label{infiniteasymptote}\end{aligned}$$ Moreover, we have the following limits at the asymptotic corners: $$\begin{aligned}
\lim_{x,y\to 0} K(x,y) = 0, \quad \lim_{x,y\to \infty}K(x,y) = -\frac{1}{2},\quad
\lim_{x\to 0, y\to \infty} K(x, y)=\lim_{x\to \infty, y\to 0} K(x, y) = -\frac{1}{4}.\end{aligned}$$
Writing $K(x,y) = -\frac{A(x,y)}{4B(x,y)^2}$, with $$\begin{aligned}
A(x,y) &= f(x+y)f'(x)f'(y) - f(x)f'(x+y)f'(y) - f(y)f'(x+y)f'(x), \\
B(x,y) &= f(x+y)-f(x)-f(y),\end{aligned}$$ we note that $A(x,0)=M_y(x,0)=0$ and $N(x,0)=0$, so that $$\displaystyle \lim_{y\to 0} K(x,y) = \frac{A_{yy}(x,0)}{8B_y(x,0)^2},$$ which gives after rewriting in terms of $\psi'$.
For the infinite limits, we use the facts that $\displaystyle \lim_{y\to\infty} f(y)-y = -\tfrac{1}{2}$ and $\lim_{y\to\infty} f'(y)=1$, and that $\lim_{y\to\infty} y(f'(y)-1) = 0$, to obtain limits of $A(x,y)$ and $B(x,y)$ separately with elementary computations.
These limits and strong numerical evidence allow us to conjecture that the sectional curvature in two dimensions is lower bounded by $-1/2$. Comparing the two-dimensional sectional curvature $K_2(x,y)=K(x,y)$ with the sectional curvature of the plane generated by $e_1$ and $e_2$ in three dimensions, that we denote by $K_3(x,y,z)$, we observe numerically that for a given $z>0$, the function $$\label{diffseccurv}
(x,y)\mapsto K_3(x,y,z)-K_2(x,y)$$ does not have a fixed sign in general, as can be observed on Figure \[fig:dim2dim3\] for small values of $x$, $y$ and $z$.
Acknowledgments {#acknowledgments .unnumbered}
===============
S. C. Preston was partially supported by Simons Foundation, Collaboration Grant for Mathematicians, no. 318969. A. Le Brigant and S. Puechmorel would like to thank Fabrice Gamboa and Thierry Klein for bringing this problem to their attention and for fruitful discussions.
Appendix {#appendix .unnumbered}
========
Here we give a well-known principle to establish positivity of matrices.
\[positivetransposelemma\] Suppose $A$ is a positive-definite symmetric matrix, $V$ is a vector, and $c$ is a positive real number. Then $B = A - c VV^T$ is positive-definite if and only if $$c V^TA^{-1}V < 1.$$
Since $A$ is positive-definite and symmetric, we may write $A = P^2$ for some positive-definite symmetric matrix $P$. Let $X = P^{-1}V$; then we may write $$B = P^2 - cVV^T = P\big(I - c(P^{-1}V) (P^{-1}V)^T \big)P = P (I - cXX^T) P.$$ Denoting by $\langle U|U \rangle=U^TU$ the usual scalar product on ${\mathbb R}^n$, we have for any vector $U$, $$\begin{aligned}
\langle U|BU\rangle &= \langle PU|PU\rangle - c \langle PU| X\rangle^2 = \lvert Y\rvert^2 - c\langle Y|X\rangle^2 \\
&\ge \lvert Y\rvert^2 - c\lvert X\rvert^2 \lvert Y\rvert^2 = \lvert Y\rvert^2 (1 - c\lvert X\rvert^2),\end{aligned}$$ where $Y = PU$, using the Cauchy-Schwarz inequality. This is positive for all $U$ if and only if the right side is positive for all $Y$, which translates into $c \lvert X\rvert^2 < 1$. Since $\lvert X\rvert^2 = \langle P^{-1}V | P^{-1}V\rangle = \langle V| A^{-1}V\rangle$, we obtain the claimed result.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We present photometric and spectroscopic observations of a luminous type IIP Supernova (SN) 2009kf discovered by the Pan-STARRS 1 (PS1) survey and detected also by *GALEX*. The SN shows a plateau in its optical and bolometric light curves, lasting approximately 70 days in the rest frame, with absolute magnitude of M$_V = -18.4$ mag. The P-Cygni profiles of hydrogen indicate expansion velocities of 9000[$\mathrm{km\;s^{-1}}$ ]{}at 61 days after discovery which is extremely high for a type IIP SN. SN 2009kf is also remarkably bright in the near-ultraviolet (NUV) and shows a slow evolution 10–20 days after optical discovery. The NUV and optical luminosity at these epochs can be modelled with a black-body with a hot effective temperature ($T\sim
16,000$K) and a large radius ($R\sim 1\times 10^{15}$cm). The bright bolometric and NUV luminosity, the lightcurve peak and plateau duration, the high velocities and temperatures suggest that 2009kf is a type IIP SN powered by a larger than normal explosion energy. Recently discovered high-z SNe ($0.7 < z < 2.3$) have been assumed to be IIn SNe, with the bright UV luminosities due to the interaction of SN ejecta with a dense circumstellar medium (CSM). UV bright SNe similar to SN 2009kf could also account for these high-z events, and its absolute magnitude $M_{\rm NUV} = -21.5 \pm0.5$mag suggests such SNe could be discovered out to $z\sim2.5$ in the PS1 survey.
author:
- 'M. T. Botticella, C. Trundle, A. Pastorello , S. Rodney, A. Rest, S. Gezari, S. J. Smartt, G. Narayan, M. E. Huber, J. L. Tonry, D.Young, K. Smith, F. Bresolin, S. Valenti, R. Kotak, S. Mattila, E. Kankare, W. M. Wood-Vasey, A. Riess, J. D. Neill, K. Forster, D. C. Martin, C. W. Stubbs, W. S. Burgett, K. C. Chambers, T. Dombeck, H. Flewelling, T. Grav, J. N. Heasley, K. W. Hodapp, N. Kaiser, R. Kudritzki, G. Luppino, R. H. Lupton, E. A. Magnier, D. G. Monet , J. S. Morgan, P. M. Onaka, P. A. Price, P. H. Rhoads, W. A. Siegmund, W. E. Sweeney, R. J. Wainscoat, C. Waters, M. F. Waterson, C. G. Wynn-Williams'
title: 'SN 2009kf : a UV bright type IIP supernova discovered with Pan-STARRS 1 and GALEX'
---
Introduction {#sec:intro}
============
Type II SNe are hydrogen rich explosions and fall into three main sub-classes. Type IIP events have plateaus in their optical and near-infrared (NIR) lightcurves, type IIL events show a linear decay after peak, type IIn events present strong signatures of the presence of dense CSM and are characterized by narrow hyrdogen emission lines superimposed on broad wings. The relative fractions of these SNe are now well measured in the nearby Universe [see @Smartt2009A for a review of relative rates]. The majority of these (around 60%) are IIP with typical mid-plateau magnitudes of $M_{V}\sim-17$mag [@Richardson2002]. However, they are heterogeneous and span a factor of 100 both in luminosity and in mass of $^{56}$Ni created explosively [@Hamuy2002; @Pastorello2003]. The progenitor stars of several of the nearest IIP SNe have been discovered [@Li2006; @Smartt2009] and are the red supergiants (RSG) that both stellar evolutionary theory and lightcurve modelling have predicted.
Type IIL and type IIn SNe are significantly less frequent by volume making up about 3% and 4% respectively of the total core-collapse (CC) SNe [@Smartt2009]. The lack of an extended plateau suggests that type IIL SNe have more massive progenitor stars that shed a considerable amount of their hydrogen envelope before explosion but there are not progenitor detections that confirm this scenario. The progenitors of type IIn SNe have likely undergone large mass ejection just before their explosion [@GalYam2007; @GalYam2009].
Type II SNe are now being searched for in medium and high redshift surveys for a variety of reasons. The IIP SNe appear to be reasonably standard candles and have produced precise distance estimates with different methods [@2004ApJ...616L..91B; @Dessart2008], but the empirical correlation of plateau-luminosity and expansion velocity, the standardised candle method (SCM), seems a promising method of measuring the distances to large numbers of type IIP SNe. The dispersion of order 0.2–0.3mag [@Hamuy2003; @Nugent2006; @Poznanski2009], 0.1–0.15mag in NIR range[@Maguire2009b], is potentially similar to type Ia SNe but they are significantly fainter, restricting their use with current surveys to $z<0.3$.
Although this is CC SNe of all varieties have been used to estimate the star formation rate (SFR) out to $z\sim0.2$ [@Botticella2008; @Bazin2009] and to $z\sim0.7$ [@Dahlen2004]. Recently the highest redshift SNe ($0.7 < z < 2.3$) have been found, and proposed to be UV bright IIn SNe [@Cooke2009]. @Cooke2008 suggested that this type of SNe could be used to probe the SFR of the Universe out to $z\sim2$ in upcoming surveys.
The PS1 survey has the potential to discover thousands of SNe between $0 < z\le 1$ [@Young2008]. The seven square degree camera and 1.8m aperture could allow IIP SNe to be used as cosmological probes at $z\sim0.2$ and the brightest events to be found out to $z\sim2$. One of the first discoveries of PS1, SN 2009kf [@2009CBET.1988....1Y], is a very bright SN that shares some characteristics with IIP SNe with its luminous plateau and broad P-Cygni features. Simultaneous *GALEX* images show that it is also remarkably bright in the NUV. We discuss the implication of this rare SN for understanding the explosions and the use of type IIP events for probing cosmology and SFR at high redshifts. We adopt the cosmological parameters $H_0=70\,{\rm km}\,{\rm s}^{-1} {\rm Mpc}^{-1}$, $\Omega_M=0.3$, $\Omega_\Lambda=0.7$.
Discovery and observational data {#data}
================================
SN 2009kf ($\rm{\alpha_{J2000}=16^{h}12^{m}54^{s}.05, \delta_{J2000}=+55\degr38\arcmin13\arcsec.7}$) was discovered on 2009 Jun 10.9 UT by PS1 during the course of the Medium Deep Survey (10 extragalactic fields observed nightly using 25% of the telescope time) in the sky-field MD08 (Fig. \[figdisc\]). The last non detection was on 2009 Jun 04.9 UT so we adopt an explosion date ${\rm JD}=2,454,989.5\pm 3$. The reported phases are with respect to the explosion date and in the SN rest frame. The PS1 MD08 coverage provided $g,r,i,z$ photometry, and we supplemented this with images from the Liverpool Telescope (LT), William Herschel Telescope (WHT) and Gemini-North Telescope (GN). The PS1 images were reduced with the custom built Image Processing Pipeline while LT, WHT and GN images were reduced with IRAF[^1] tasks. The instrumental magnitudes were derived from host galaxy template subtracted images using PSF fitting techniques [as in @Botticella2009]. A local sequence of SDSS stars was used to measure the relative magnitude for each observation. The uncertainties in the measurements are estimated by combining in quadrature the error of the photometric calibration and the error in the PSF fitting. Apparent and absolute magnitudes reported in this letter are in the AB and Vega system, respectively.
The *GALEX* Time Domain Survey (TDS) detected SN 2009kf in monitoring observations of the MD08 field. The host galaxy was detected in the pre-explosion TDS observations between ${\rm JD}=2,454,960.7$ to 2,454,968.9 with ${\rm NUV }= 21.58 \pm 0.15$mag. This is corrected for the flux enclosed in a 6radius aperture [@Morrissey2007] and the error is measured from the dispersion of measurements between the observations. The SN was detected in 5 observations with exposure times of $1.0-1.5$ks from ${\rm JD}=2,455,003.7$ to 2,455,013.8, with a peak magnitude of ${\rm NUV }=22.34 \pm 0.38$mag. The SN magnitude was measured by subtracting the flux of the host galaxy from the total flux enclosed in a 6radius aperture centered on the host galaxy NUV centroid as determined in the pre-discovery images. The errors in the SN magnitude include both the error in the host galaxy magnitude as well as the error in the observed magnitude, which is measured empirically as a function of magnitude from the dispersion of 4275 matched sources between observations in bins of 0.5mag.
Spectroscopic follow-up was obtained with Gemini Multi-Object Spectrographs (GMOS) at GN on ${\rm JD}=2,455,061.7$ and 2,455,094.7 and with Andalucia Faint Object Spectrograph and Camera (ALFOSC) at the Nordic Optical Telescope (NOT) on ${\rm JD}=2,455,117.6$. Spectra were reduced using the Gemini pipeline and standard routines within IRAF for the NOT spectrum. The spectra of the host galaxy SDSS $J161254.19+553814.4$ provided a redshift measurement of $z=0.182 \pm 0.002$ from the nebular emission lines, in agreement with the SDSS photometric redshift of $0.185 \pm 0.065$[^2]. A reddening coefficient of C(${\rm H}\beta)=1.3\pm 0.25$ was derived from the ${\rm H}\alpha/{\rm H}\beta$ emission line ratio, which corresponds to an host galaxy extinction of $E(B-V)=0.9$mag, while the Galactic extinction is negligible ($E(B-V)=0.009$mag @Schlegel1998). The N[ii]{} and O[iii]{}N[ii]{} line ratio methods of estimating metallicity were employed [@Pettini2004], giving $12+\log ({\rm O/H})$ abundance of $8.50 \pm 0.1$ dex.
Photometric and spectroscopic evolution {#analysis}
=======================================
The $r,i,z$ lightcurves of SN 2009kf (Fig. \[fig1\]) display a plateau which is similar to that observed in type IIP SNe [@Hamuy2003]. However there are differences in that SN 2009kf shows a relatively slow rise to peak, and a clear maximum, which occurs progressively earlier from the red to the blue bands. In the last observation at about 280 days after explosion we estimated an upper limit of $i > 24$mag.
To meaningfully compare the lightcurves of SN 2009kf with those of nearby type IIP SNe we estimated extinction and redshift corrections, the latter requiring time dilation correction and K-correction. We adopted a redshift of $0.182\pm0.002$ ($\mu=39.76$mag) and determined the K-correction from observed $riz$ AB magnitudes to rest frame $VRI$ Vega magnitudes using the spectra of SN 2009kf, a sample of spectra of different type II SNe and employing the [*IRAF*]{} package [*synphot*]{}.
The comparison of colour evolution with that of similar SNe for which the colour excess has been previously determined can be used to estimate the extinction correction. Fig. \[abscomp\] shows the evolution of the $V-R$ and $V-I$ colours not corrected for internal extinction for SN 2009kf and of the intrinsic colours for SNe 1992H and 1992am [@Clocchiatti1996; @Schmidt1994; @Hamuy2003]. The intrinsic colour $V-I$ of type IIP SNe in the plateau phase appears fairly homogeneous, due to the photospheric temperature being close to that of hydrogen recombination. @Olivares2010 suggest that this temperature leads to an intrinsic $(V-I)_0 = 0.66\pm0.05$mag at the end of the plateu phase and that $E(V-I)$ can be used to determine the extinction correction. SN 2009kf has $V-I= 1.3\pm0.4$mag at this epoch which implies a value of $A_V = 1.6\pm1$mag. We also compared the $V-R$ colour of SN 2009kf with that of SN 1992H [@Clocchiatti1996] since these SNe are similar both in the photometric and spectroscopic evolution. We measured, at the same epoch, $V-R=0.55\pm 0.4$mag for SN 2009kf and $V-R~0.25$mag for SN 1992H, which implies a value of $A_V=1\pm 1.4$mag. The absorption component of the Na[i]{} doublet ($\lambda\lambda
5890,5896$) from the host galaxy in the spectra of SN 2009kf is not detected, and we set a upper limit of EW(Na[i]{} D) $< 1.4$Å. The relation by [@Munari1997] then gives an upper limit on extinction of $E(B-V) \lesssim 0.3$mag. The integrated host galaxy extinction, as measured from the $c({\rm H}\beta$) index is $E(B-V)=0.9$mag. While this is not directly applicable to the the line of sight of SN 2009kf, it does suggest areas of high extinction are plausible. While the uncertainties prevent a definitive and consistent determination of reddening, the $VRI$ colors of 2009kf suggest a reddening of $A_V \simeq 1$ mag ($E(B-V)=0.32\pm 0.5$mag assuming $R_V=3.1$) is applicable. The de-reddened $V$ and $R$ band absolute light curves are illustrated in Fig. \[abscomp\]. In Fig. \[bolUV\] the pseudo-bolometric and absolute NUV lightcurves of SN 2009kf are compared with those of other type IIP SNe. The pseudo-bolometric lightcurve was obtained by first converting $,g,r,i,z$ magnitudes into monochromatic fluxes, then correcting these fluxes for the adopted extinction according to the law from @Cardelli1989, and finally integrating the resulting SED over wavelength, assuming zero flux at the integration limits. The pseudo-bolometric lightcurve clearly displays a plateau, from day 20 to day 90, which is shorter than the 100–120 days typical of IIP SNe. However, it is significantly more luminous than all other IIP SNe for which a good estimate of bolometric luminosity is available. The upper limit luminosity estimated at about day 280 suggests that the mass of radioactive $^{56}$Ni deposited in the ejecta is $M{\rm (^{56}Ni)}< 0.4$ [M$_{\odot}$]{}. The absolute NUV lightcurve is obtained correcting the *GALEX* data for the redshift and extinction discussed above. The absolute magnitude is incredibly bright at 10–20 days after explosion in comparison with other well studied IIP SNe [@Dessart2008]. The NUV flux and optical emission can be reproduced satisfactorily with a blackbody fit with a temperature dropping from $\sim$16,000K to $\sim$13,000K from 10 to 20 days (as illustrated in Fig\[bolUV\]). The radius of the blackbody fit implies an expansion velocity of about 10000[$\mathrm{km\;s^{-1}}$ ]{}, which is similar to the measured velocities from absorption lines. The evolution of the temperature, radius and luminosity of the blackbody fit is also different from normal type IIP SNe that are characterized by smaller values and different declines. We note that it would also be possible to reproduce the SED with low extinction and a temperature of $T_{\rm eff} \sim 9600$K, but this would then require an expansion velocity of about 12,000[$\mathrm{km\;s^{-1}}$ ]{}maintained to 20days, unusually high for a normal IIP.
The spectra of SN 2009kf show strong ${\rm H}\alpha$ and ${\rm H}\beta$ P-Cygni profiles, and a P-Cygni feature in the region around 5800Å that may be attributed to He[i]{} ($\lambda$5876), Na[i]{} ($\lambda$5889,5896) or their blend. However, the peak of the emission is blue-shifted from the rest wavelength of He[i]{} by $100\pm 300$[$\mathrm{km\;s^{-1}}$ ]{}and from that of Na[i]{} by $1000 \pm 300$[$\mathrm{km\;s^{-1}}$ ]{}. We suggest that this is He[i]{} since there are no prominent metal lines in the spectrum. The presence of He[i]{} is consistent with the high energy and temperature inferred by the NUV/optical lightcurves assuming an extinction of about $A_V=1$mag. Between 61 and 89 days after explosion, the continuum becomes redder (Fig. \[speccomp\]) and there is little evolution in the Balmer features. Fitting Gaussian profiles to these features we determine expansion velocities of $9000 \pm 1000$[$\mathrm{km\;s^{-1}}$ ]{}from ${\rm H}\alpha$ on day 61, and $7800 \pm 1000$[$\mathrm{km\;s^{-1}}$ ]{}from ${\rm H}\alpha$ and ${\rm H}\beta$ on day 89 after discovery. There is more evolution in He[i]{}, which becomes stronger by day 89 due to an increasing contribution of Na[i]{}. The expansion velocity from this line decreases from $7200 \pm 1000$[$\mathrm{km\;s^{-1}}$ ]{}on day 61 to $6500 \pm 1000$[$\mathrm{km\;s^{-1}}$ ]{}on day 89. The spectra of SN 2009kf show similarities to SNe 1992H and 1992am [@Clocchiatti1996; @Schmidt1994; @Hamuy2003](see Fig \[speccomp\]).
Discussion
==========
The high luminosity, both in the optical and UV, short plateau duration and large expansion velocity of SN 2009kf have important implications for understanding the origins of type IIP SNe, using them as cosmological distance indicators and detecting CC SNe at high redshift.
The bolometric lightcurve of SN 2009kf is similar to normal IIP SN but with three striking differences: it has a slow rising peak in the first 20 days, has a short plateau phase and is significantly more luminous. The NUV luminosity is also remarkably higher and exhibits a slower evolution with respect to normal type IIP SNe [@2008ApJ...685L.117G]. The source of the observed luminosity at 10 days after the explosion can be modelled with a hot photosphere of $T_{eff} \sim 16,000$K and a radius of around $1\times10^{15}$cm. The photospheric temperature remains high for an unusually long period, as we see He[i]{} ($\lambda$ 5876) up to about 89 days and the normal metal lines are weak. The expansion velocity as measured from the ${\rm H}\beta$ absorption is similar to SN 1992am and extreme for a IIP type event, 7800[$\mathrm{km\;s^{-1}}$ ]{}at 89 days after discovery, compared to 4500[$\mathrm{km\;s^{-1}}$ ]{}for a typical IIP. In conclusion 2009kf is both luminous and extremely blue hence extends the luminosity and energy range of IIP SNe.
This peculiarity raises the question of whether the standard model for type IIP SNe with a RSG progenitor of $8-20$[M$_{\odot}$]{} and an explosion energy of $0.1-2 \times 10^{51}$ergs is valid for this event. The luminosity, plateau duration and expansion velocity mainly depend on the explosion energy, envelope mass and radius of the progenitor star at the moment of the explosion and the characteristics of SN 2009kf could be explained with a large explosion energy or very large progenitor radius, but a lower than usual hydrogen envelope mass. The analytical models of @Kasen2009 and numerical simulations of @2004ApJ...617.1233Y would require explosion energies in excess of $10 \times 10^{51}$ergs, or progenitor radii of greater than 1000[R$_{\odot}$]{}. Alternative explanations could be a different mass distribution of H and He in the envelope, or possibly interaction of the ejecta with a surrounding shell. A detailed model of SN 2009kf has the potential to provide insight into the diversity of type IIP SNe, in particular as concern their use as standard candles.
The bright visual magnitude of SN 2009kf and its apparent plateau means that such events may be preferentially selected in cosmology surveys. However the distance modulus obtained from the $I$-band relation by @Nugent2006 is $\mu=40.63 \pm 0.5$, compared to the distance modulus of $\mu=39.76$ for a $\Lambda$CDM cosmology. Further work on these high luminosity and fast expansion velocity events are required to understand if this discrepancy for one event is statistical, or systematic and if they can be reliably used for the SCM method applied to IIP SNe.
The discovery of SN 2009kf demonstrates the exciting potential of the *GALEX* TDS observing campaign which is coordinated with PS1 to both probe shock breakout [@2008ApJ...683L.131G; @2008Sci...321..223S] and the nature of UV-bright type II SNe. During the first PS1 phases, three confirmed SNe discoveries had simultaneous *GALEX* imaging within $\pm$ 10 days of the PS1 discovery, but only SN 2009kf was detected. The transients discovered at high-z by [@Cooke2009] were interpreted as type IIn SNe, as up until now the only SNe which had such high UV luminosities were interacting events. SNe 2008es and 2009kf are the brightest SN in the UV known so far, ($M_{NUV}= -22.2$ and $ -21.5$mag respectively) but SN 2008es [@Gezari2009; @Miller2009] is a type IIL SN and did not have obvious signs of CSM interaction. SN 2009kf is of similar brightness to the inferred NUV fluxes of @Cooke2009 high-z SNe. The NUV lightcurves of all types of CC SNe are not well quantified, hence it is quite possible that some fraction of the @Cooke2009 high-z SNe are UV bright type II SNe similar to 2009kf. The seasonal stacked PS1 MDS images will allow 2009kf-like UV bright SNe to be detected beyond $z\sim2$ where the NUV band would be redshifted to the $r$-band and use these events to probe the SF history of the high-z Universe. The rate of these SNe at low redshift, their progenitor scenarios, and their UV evolution are key areas we need to understand before we can confidently use them to probe the SFR at high redshift.
The PS1 Surveys have been made possible through contributions of the Institute for Astronomy at the University of HawaiÕi in Manoa, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, the University of Durham, the University of Edinburgh, the QueenÕs University Belfast, the Harvard- Smithsonian Center for Astrophysics, and the Los Cumbres Observatory Global Telescope Network, Incorporated. This work is also based on observations collected at LT, WHT and NOT (La Palma) and Gemini (HawaiÔi). This work, conducted as part of the award “Understanding the lives of massive stars from birth to supernovae” (S.J. Smartt) made under the European Heads of Research Councils and European Science Foundation EURYI Awards scheme, see www.esf.org/euryi. M.T.B. would like to thank E. Cappellaro, L. Zampieri and S. Benetti for helpful discussions. SM and EK acknowledge support from the Academy of Finland (project:8120503).
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[^1]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^2]: Information on the host galaxy is available at http://cas.sdss.org/astrodr7
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We derive and assess the sharpness of analytic upper bounds for the instantaneous growth rate and finite-time amplification of palinstrophy in solutions of the two-dimensional incompressible Navier-Stokes equations. A family of optimal solenoidal fields parametrized by initial values for the Reynolds number ${\textrm{Re}}$ and palinstrophy $\P$ which maximize $d\P/dt$ is constructed by numerically solving suitable optimization problems for a wide range of ${\textrm{Re}}$ and $\P$, providing numerical evidence for the sharpness of the analytic estimate $d\P/dt \leq \left(a + b\sqrt{\ln{\textrm{Re}}+c} \, \right) \P^{3/2}$ with respect to both ${\textrm{Re}}$ and $\P$. This family of instantaneously optimal fields is then used as initial data in fully resolved direct numerical simulations and the time evolution of different relevant norms is carefully monitored as the palinstrophy is transiently amplified before decaying. The peak values of the palinstrophy produced by these initial data, i.e., $\sup_{t > 0} \P (t)$, are observed to scale with the magnitude of the initial palinstrophy $\P(0)$ in accord with the corresponding *a priori* estimate. Implications of these findings for the question of finite-time singularity formation in the three-dimensional incompressible Navier-Stokes equation are discussed.'
author:
- Diego Ayala
- 'Charles R. Doering'
- 'Thilo M. Simon'
title: 'Maximum palinstrophy amplification in the two-dimensional Navier-Stokes equations'
---
Introduction {#sec:intro}
============
Energy methods are among the most popular techniques used in the mathematical analysis of evolutionary partial differential equations. At their core these methods rely on the existence of bounds for a norm $\Q$—an “energy” that is not necessarily related to a physical energy—that provides relevant information about the magnitude and regularity (smoothness) of solutions. The idea is to derive bounds on the growth rate $d\Q/dt$ from the equations of motion utilizing rigorous functional estimates and, given initial data with finite norm $\Q(0)$, to subsequently control $\Q(t)$ for $t>0$ via methods of ordinary differential (in)equations. In some cases this kind of energy analysis can establish that $\Q(t)$ remains finite—sometimes even uniformly bounded—for all $t>0$ while in others it can only show that that $\Q(t)$ is finite for all time if $\Q(0)$ is sufficiently small. And in some situations all that can be proved is that $\Q(t) < \infty$ during a finite time interval whose length depends on $\Q(0)$. Among the many well-known examples amenable to such analysis are the incompressible Navier-Stokes equations.
In the case of unforced flows on the d-dimensional torus ($\mathbb{T}^d$), for example, the $L^2$ norm of the velocity vector field—proportional to the square root of the kinetic energy—decays monotonically due to viscous dissipation so it is bounded uniformly in time by its initial value. Other norms may grow, however, and their amplification reflects aspects of the cascade processes that characterize much of the complexity of nonlinear fluid mechanics.
In spatial dimension $d = 3$ the enstrophy—the square of the $H^1$ semi-norm of the velocity, which is the same as the $L^2$ norm squared of the vorticity—can be amplified by vortex stretching. Analysis establishes that the growth rate of enstrophy is a bounded function of the enstrophy, but the resulting differential (in)equations only ensure that the enstrophy remains bounded forever if the initial data is sufficiently small—in particular if the product of the initial kinetic energy and enstrophy is sufficiently small; see, for example, [@d09]. That is, the possibility of finite-time singularities is not ruled out by energy analysis. The enstrophy corresponds to a particularly relevant norm in this problem because limits on the $H^1$ semi-norm of the velocity can be “bootstrapped” to bounds on norms of higher derivatives establishing infinite differentiability of the solutions while the enstrophy remains finite. On the other hand divergent upper bounds on enstrophy do not guarantee divergent solutions and no “blow-up” has been demonstrated to date. Hence the question of long-time existence of smooth solutions of the three-dimensional (3-D) incompressible Navier-Stokes equations remains one of the grand challenges for mathematical physics.
Much more is known about solutions of the two-dimensional (2-D) Navier-Stokes equations. In the case of unforced flows on $\mathbb{T}^2$ both the $L^2$ norm and the $H^1$ semi-norm of solutions (i.e., both the kinetic energy and the enstrophy) decay monotonically in time while the $H^2$ semi-norm of the velocity field—which is the same $H^1$ semi-norm of the pseudo-scalar vorticity field, the square of which is known as the [*palinstrophy*]{}—can be amplified by a vorticity gradient stretching mechanism. Energy methods can be used to show that the palinstrophy of solutions of 2-D incompressible Navier-Stokes equations remains finite for all time and there are no potential finite-time singularities. Moreover, energy methods can be used to derive rigorous upper limits on the peak palinstrophy as a function of the norms of the initial data.
In this paper we investigate the quantitative accuracy of palinstrophy amplification bounds in order to evaluate the practical predictive power of energy method analysis. We consider the incompressible 2-D Navier-Stokes equation, written here as an evolution equation for the vorticity $\omega$, on spatial domain $\Omega = \mathbb{T}^2$ (the $L \times L$ square with periodic boundary conditions): $$\label{eq:2DNSE}
\omega_t + {\mathbf{u}}\cdot\nabla\omega = \nu{\Delta}\omega, \quad
\nabla\cdot{\mathbf{u}}= 0, \quad \text{and} \quad \omega(\cdot,0) = \omega_0$$ where ${\mathbf{u}}$ is the velocity field, $\nu$ is the kinematic viscosity, and the velocity and the vorticity are related via $$\omega = {\hat {\bf e}}_3 \cdot \nabla \times {\mathbf{u}}\ \ \Leftrightarrow \ \ -{\Delta}{\mathbf{u}}= {\nabla^{\perp}}\omega$$ with ${\nabla^{\perp}}= \left[\partial_{x_2},-\partial_{x_1}\right]$. We are interested in the time evolution of the the energy, enstrophy and palinstrophy defined here, respectively, as
\[eq:Q\_defn\] $$\begin{aligned}
\K\{{\mathbf{u}}(\cdot,t)\} & = \frac{1}{2}\int_\Omega | {\mathbf{u}}(\cdot,t) |^2 \;d\Omega, \\
\E\{{\mathbf{u}}(\cdot,t)\} & = \frac{1}{2}\int_\Omega | \nabla{\mathbf{u}}(\cdot,t) |^2 \;d\Omega = \frac{1}{2}\int_\Omega | \omega(\cdot,t) |^2 \;d\Omega,\\
\P\{{\mathbf{u}}(\cdot,t)\} & = \frac{1}{2}\int_\Omega | {\Delta}{\mathbf{u}}(\cdot,t) |^2\;d\Omega = \frac{1}{2}\int_\Omega | {\nabla^{\perp}}\omega(\cdot,t) |^2 \;d\Omega = \frac{1}{2}\int_\Omega | \nabla\omega(\cdot,t) |^2 \;d\Omega.\end{aligned}$$
The temporal evolution of of $\K$, $\E$ and $\P$ are given by
\[eq:dQdt\_defn\] $$\begin{aligned}
\frac{d\K}{dt} & = -\nu\int_\Omega | \omega(\cdot,t) |^2 \;d\Omega = -2\nu\E\{{\mathbf{u}}(\cdot,t)\}, \label{eq:dKdt_defn} \\
\frac{d\E}{dt} & = -\nu\int_\Omega | \nabla\omega(\cdot,t) |^2 \;d\Omega = -2\nu\P\{{\mathbf{u}}(\cdot,t)\}, \label{eq:dEdt_defn}\\
\frac{d\P}{dt} & = -\nu\int_\Omega | {\Delta}\omega(\cdot,t) |^2\;d\Omega -
\int_{\Omega}\nabla\omega\cdot\nabla{\mathbf{u}}\cdot\nabla\omega \;d\Omega =: \R\{{\mathbf{u}}(\cdot,t)\}, \label{eq:dPdt_defn}\end{aligned}$$
defining, in the last line, the palinstrophy generation rate functional $\R\{{\mathbf{u}}(\cdot,t)\}$.
We are focused on the question of the sharpness of rigorous analytic bounds on the instantaneous and finite-time growth of palinstrophy, the only quantity from with nonmonotonic temporal dynamics. This is of interest because the functional analysis techniques used to derive the estimates rely only on the structure of the palinstrophy generation rate functional and fundamental relations between norms and [*not*]{} specifically on the physics associated to the problem at hand, making this study relevant to other partial differential equations.
Dimensional analysis requires that bounds on the instantaneous rate of palinstrophy generation, $\R\{{\mathbf{u}}(\cdot,t)\} = d\P/dt$ as a function of the instantaneous energy, enstrophy, palinstrophy, viscosity $\nu$ and domain length scale $L$—if such bounds exist at all—must be of the form $$\label{eq:gen_Estimate}
\R\{{\mathbf{u}}(\cdot,t)\} \leq \Gamma(\K,\E,\P,\nu,L) \ \P^{3/2}$$ where the prefactor $\Gamma$ is a dimensionless function of dimensionless combinations of the energy, enstrophy, palinstrophy, viscosity and system size. An estimate of this form will be declared sharp if and only if $$\max_{{\mathbf{u}}\in \mathcal{S}} \P\{{\mathbf{u}}\}^{-3/2} \R\{{\mathbf{u}}\} \sim \Gamma(\K,\E,\P,\nu,L)$$ where the maximum is over the set $\mathcal{S}$ of all spatially periodic divergence-free vector fields with energy, enstrophy and palinstrophy values $\K$, $\E$ and $\P$ on the $L \times L$ square. Thus the sharpness of such estimates can be addressed by solving the constrained optimization problem $${\widetilde{\mathbf{u}}}_{\mathcal{S}} = \mathop{\arg\max}_{{\mathbf{u}}\in \mathcal{S}} \; \R\{{\mathbf{u}}\}$$ where the constraint manifold $\mathcal{S}$ can be interpreted as the intersection (in infinite-dimensional space) of spheres of given radius measured in different norms.
Given the anisotropic nature of the constraint manifold it is desirable to seek bounds and test estimates with the fewest possible number of parameters in the prefactor $\Gamma$. Moreover, if we are interested in estimates that could conceivably make sense in the infinite volume limit, i.e., if we seek optimizing flow structures and bounds that are [*independent*]{} of the domain scale $L$, then the prefactor can only be a function of the dimensionless combinations $\K^{1/2}/\nu$—a ratio that is naturally referred to as the Reynolds number ${\textrm{Re}}$—and $(\K \P)^{1/2} / \E$. If we further conjecture an $L$-independent estimate for $\R\{{\mathbf{u}}(\cdot,t)\}$ whose large palinstrophy behavior is dominated non-trivally by the leading $\P^{3/2}$ in (\[eq:gen\_Estimate\]) then the asymptotic prefactor can be a function of ${\textrm{Re}}$ alone. That is, the prefactor would be of the form $\Gamma(\K,\E,\P,\nu,L) = \gamma({\textrm{Re}})$. In fact there are rigorous upper bounds on the palinstrophy generation rate of the form $ \R \le \gamma({\textrm{Re}}) \, \P^{3/2}$. [@ap13a] proved that $\R \lesssim {\textrm{Re}}\, \P^{3/2}$ and computed optimal vector fields indicating that the $3/2$ exponent for $\P$ is sharp but that the ${\cal O}({\textrm{Re}})$ prefactor is [*not*]{} sharp.
In this work we assess the sharpness of the improved estimate $$\label{eq:dPdt_LogEstimate}
\begin{aligned}
\frac{d\P}{dt} & = \R\{{\mathbf{u}}\} \le \left(a + b\sqrt{\ln{\textrm{Re}}+c} \right) \P^{3/2} \\
\text{with} \quad & a = 0, \quad b = \sqrt{2\pi}, \quad c = -\ln\left( \frac{2}{\sqrt{\pi}} \right)
\end{aligned}$$ for sufficiently large ${\textrm{Re}}$ and $\P$. It is derived in Appendix \[sec:dPdt\_Estimate\]. We demonstrate (1) the sharpness of the upper bound (\[eq:dPdt\_LogEstimate\]) and (2) the extent to which the Navier-Stokes flow starting from instantaneously optimal fields ${\widetilde{\mathbf{u}}}_{\mathcal{S}}$ saturates the corresponding finite-time estimates for palinstrophy amplification $$\P^{1/2}(t) - \P^{1/2}(0) \leq \phi({\textrm{Re}}_0) \left[\E(0) - \E(t)\right]$$ and $$\mathop{\max}_{t > 0} \P(t) \le \psi({\textrm{Re}}_0) \, \P(0).$$ That the prefactors $\phi$ and $\psi$ depend on ${\mathbf{u}}$ and $\nu$ only via the (initial) Reynolds number ${\textrm{Re}}_0 = \K^{1/2}(0)/\nu$ implies in particular that the peak palinstrophy amplification factor depends on the viscosity only through its appearance in the explicit function $\psi({\textrm{Re}}_0)$. We note, however, that our study does [*not*]{} indicate that our estimate for $\psi({\textrm{Re}})$ is sharp with respect to its Reynolds number dependence.
The remainder of this manuscript is as follows. The instantaneous growth of palinstrophy and structure of the optimal vector fields is reviewed in section \[sec:2D\_InstOpt\]. Finite-time growth and maximal amplification of palinstrophy is studied in section \[sec:TimeEvol\]. Section \[sec:Conclusion\] contains a discussion of the results along with conclusion and closing remarks. Detailed derivation (from elementary first principles) of the improved analytic estimate on the (\[eq:dPdt\_LogEstimate\]) can be found in the appendix.
Instantaneously Optimal Growth of Palinstrophy {#sec:2D_InstOpt}
==============================================
Given the dependence of analytic estimate on palinstrophy $\P$ and Reynolds number $\textrm{Re} = \K^{1/2}/\nu$, the sharpness of the estimate is addressed by numerically solving the constrained optimization problem for the objective functional $\R$ defined in : $$\label{eq:OptProb_K0P0}
\qquad\mathop{\max}_{{\mathbf{u}}\in\mathcal{S}_{\K_0,\P_0}} \;\; \R({\mathbf{u}})$$ where $$\mathcal{S}_{\K_0,\P_0} = \{ {\mathbf{u}}\in H^3(\Omega) : \nabla\cdot{\mathbf{u}}= 0, \; \K({\mathbf{u}}) = \K_0, \; \P({\mathbf{u}}) = \P_0 \}.$$ In order to investigate the asymptotic behavior of $\R$ as $\P_0\to\infty$, is solved numerically for a wide range of values of the energy $\K_0 \in [1,100]$ and some choices of $\P_0 \gg \K_0/C_P^2$, where $C_P = 1/(2\pi)^2$ is the Poincaré constant for the unit two dimensional torus. We compute with the numerical value $\nu = 10^{-3}$ for the kinematic viscosity, kept constant in all computations, allowing us to probe the dependence of the optimal rate of growth of palinstrophy for values of the Reynolds number in the range ${\textrm{Re}}_0 \in [10^3,10^4]$. For a given value of ${\textrm{Re}}_0$, the value of $\P_0$ defining the constraint manifold $\mathcal{S}_{\K_0,\P_0}$ is chosen so that the optimal vorticity field ${\widetilde{\omega}_{{\textrm{Re}}_0,\P_0}}$ of the optimal vector field ${\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}}= \arg\max \R({\mathbf{u}})$ is sufficiently localized in the computational domain, allowing for the effect of boundaries to be neglected. A family of optimal fields parametrized by their palinstrophy is then constructed by rescaling the optimal field ${\widetilde{\omega}_{{\textrm{Re}}_0,\P_0}}$ using the self-similar approach described in Appendix \[sec:SelfSimilarAnalysis\]. Details of the numerical methods for the variational problem can be found in the work of [@ap13a].
Figures \[fig:OptimVort\_vsK\_P07\](a) and \[fig:OptimVort\_vsK\_P07\](b) show the optimal vorticity fields ${\widetilde{\omega}_{{\textrm{Re}}_0,\P_0}}$ corresponding to values of Reynolds number ${\textrm{Re}}_0 = 10^3$ and ${\textrm{Re}}_0 = 10^4$, respectively, for a fixed value of palinstrophy $\P_0 = 10^8$. In each case, the optimal vorticity field consists of a vortex quadrupole of finite area, and a localized region of strong vorticity responsible for the extreme growth of palinstrophy. The size of the optimal vortical structure, relative to the size of the domain $\Omega$, has a positive correlation with the Reynolds number. A thorough discussion of the properties of the optimal vorticity fields, and their corresponding time evolution under 2-D Navier-Stokes dynamics can be found in [@ap13a] and [@ap13b].
\
To verify the power-law behavior of the optimal instantaneous rate of production of palinstrophy and assess the sharpness of estimate \[eq:dPdt\_LogEstimate\] with respect to the exponent $\alpha = 3/2$, figure \[fig:maxdP\_vsP\] shows the dependence on $\P_0$ of the compensated optimal rate of growth of palinstrophy $$\widetilde{\R}_{{\textrm{Re}}_0,\P_0} = \P^{-3/2}_0\;\R({\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}}),$$ for different values of Reynolds number in the interval ${\textrm{Re}}_0\in [10^3,10^4]$. The data shown here is an extension of the results reported by [@ap13a]. To streamline our discussion, we only include the portion of the data where a clear power-law behavior for $\R({\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}})$ is observed, corresponding to values of palinstrophy much larger than the Poincaré limit $\P_0 \to (\nu{\textrm{Re}}_0/C_P)^2$. As expected from the fact that estimate is sharp with respect to the exponent $\alpha = 3/2$, figure \[fig:maxdP\_vsP\] shows that the compensated optimal rate of growth of palinstrophy $\widetilde{\R}_{{\textrm{Re}}_0,\P_0}$, which corresponds to the prefactor $\gamma({\textrm{Re}})$, is indeed independent of palinstrophy in the limit $\P_0\to\infty$.
On the other hand, to assess the sharpness of estimate with respect to the prefactor $$\gamma({\textrm{Re}}) = a + b\sqrt{\ln{\textrm{Re}}+ c},$$ figures \[fig:maxdP\_vsRe\](a) and \[fig:maxdP\_vsRe\](b) show the dependence of the optimal rate of growth of palinstrophy $\R({\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}})$ on $\ln {\textrm{Re}}_0$ and of the compensated optimal rate of growth of palinstrophy $\widetilde{\R}_{{\textrm{Re}}_0,\P_0}$ on $\ln {\textrm{Re}}_0$, respectively, for three different values of palinstrophy $\P_0 = 4.6\times10^7$, $\P_0 = 6.8\times 10^7$ and $\P_0 = 10^8$. It can be observed in figure \[fig:maxdP\_vsRe\](b) that all data points collapse into a single curve of the form $$\label{eq:Prefactor_Fitted}
\gamma_{{\textrm{Re}}_0} = \tilde{a} + \tilde{b}\sqrt{ \ln{\textrm{Re}}_0 + \tilde{c} }$$ with fitted parameters $$\quad \tilde{a} = -0.093, \quad \tilde{b} = 0.128, \quad \tilde{c} = -4.38$$ shown in the figure as a red dashed curve. The values of $\tilde{a}$, $\tilde{b}$ and $\tilde{c}$ are obtained by averaging over $\P_0$ the values of $a_{\P_0}$, $b_{\P_0}$ and $c_{\P_0}$ corresponding to the parameters that provide the least-squares fit of the data to a model of the same form as estimate . Although the values of the fitted parameters $\tilde{a}$, $\tilde{b}$ and $\tilde{c}$ differ from the corresponding values in estimate , the fundamental dependence of $\gamma({\textrm{Re}})$ on ${\textrm{Re}}$ is correctly captured by the behavior of $\gamma({\textrm{Re}}_0) = \widetilde{\R}_{{\textrm{Re}}_0,\P_0}$ providing positive evidence for the sharpness of estimate with respect to the prefactor $\gamma({\textrm{Re}})$. To summarize, the information presented in Figure \[fig:maxdP\_vsP\] and Figures \[fig:maxdP\_vsRe\](a)-(b) indicates that the estimate $$\frac{d\P}{dt} \leq \left( a + b\sqrt{ \ln{\textrm{Re}}+c} \right)\P^{3/2} =
\gamma({\textrm{Re}})\,\P^{3/2}$$ is indeed sharp with respect to *both* the exponent $\alpha = 3/2$ and the functional form of the prefactor $\gamma({\textrm{Re}}) = a + b\sqrt{\ln {\textrm{Re}}+ c}$.
![ Compensated optimal instantaneous rate of growth of palinstrophy $\widetilde{\R}_{{\textrm{Re}}_0,\P_0} = \P^{-3/2}_0\R({\widetilde{\mathbf{u}}_{\K_0,\P_0}})$ as a function of $\P_0$, for values of Reynolds number ${\textrm{Re}}_0\in[10^3,10^4]$. The arrow indicates the direction of increasing ${\textrm{Re}}_0$. []{data-label="fig:maxdP_vsP"}](maxdPcomp_vsP_AllK){width="75.00000%"}
\
To complete our analysis of the optimal instantaneous growth of palinstrophy we now look at the structure of the spectrum of the optimal fields ${\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}}$. A key step in the derivation of estimate (see Appendix \[sec:dPdt\_Estimate\]) is the choice of cut-off wave numbers $\Lambda_1$ and $\Lambda_2$ that define the sets $\{ |{\mathbf{k}}| \leq \Lambda_1 \}$, $\{\Lambda_1 < |{\mathbf{k}}| \leq \Lambda_2 \}$ and $\{ |{\mathbf{k}}| > \Lambda_2 \}$ in wavenumber space where $\sum | {\mathbf{k}}| | {\widehat{u}}({\mathbf{k}}) |$ displays different behavior. As discussed in the appendix, the cut-off wave numbers depend on $\K$, $\P$ and $\nu$ as $$\Lambda_1^2 = c_1\frac{\P^{1/2}}{\K^{1/2}}\quad\mbox{and}\quad
\Lambda_2^2 = \frac{1}{c_2}\frac{\P^{1/2}}{\nu},$$ where $c_1$ and $c_2$ are dimensionless parameters. Figure \[fig:Sk\_vsP\] shows the rescaled compensated spectral density $(|{\mathbf{k}}|/\lambda_0)^2 \, S(|{\mathbf{k}}|/\lambda_0)$ corresponding to the optimal fields ${\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}}$ for ${\textrm{Re}}= 10^3$ and palinstrophy values $\P_0 = 1.71\times 10^6$, $\P_0 = 1.71\times 10^7$ and $\P_0 = 1.71\times 10^8$. The spectral density $S( |{\mathbf{k}}| )$ is computed as $$S(|{\mathbf{k}}|) = \sum_{2\pi k \leq | {\mathbf{k}}| \leq 2\pi(k+1)} |{\mathbf{k}}| |\widehat{{\mathbf{u}}}({\mathbf{k}})|^2,$$ and the scaling factor $\lambda_0$ is given by $$\label{eq:Lambda_0}
\lambda_0 = \frac{\int_0^\infty |{\mathbf{k}}|^2 \; S( |{\mathbf{k}}| ) \; d |{\mathbf{k}}| }
{\int_0^\infty S(|{\mathbf{k}}|)\;d|{\mathbf{k}}|}.$$ The wave number $\lambda_1$, computed as $$\label{eq:Lambda_1}
\lambda_1 = \frac{\int_0^\infty |{\mathbf{k}}| \; S( |{\mathbf{k}}| ) \; d |{\mathbf{k}}| }
{\int_0^\infty S(|{\mathbf{k}}|)\;d|{\mathbf{k}}|}$$ and the wave number $\lambda_2$ defined as the solution to the equation $$\label{eq:Lambda_2}
\int_0^{\lambda_2} |{\mathbf{k}}|^2 \; S(|{\mathbf{k}}|) \; d|{\mathbf{k}}| = \frac{1}{2}\int_0^{\infty} |{\mathbf{k}}|^2 \; S(|{\mathbf{k}}|) \; d|{\mathbf{k}}|$$ are shown in figure \[fig:Sk\_vsP\] as vertical lines. These wave numbers correspond to the location of local maxima of the compensated spectral density $|{\mathbf{k}}|^2 \, S(|{\mathbf{k}}|)$. As expected from the self-similar construction of the optimal fields ${\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}}$, the rescaled compensated spectral densities collapse into a single “universal” spectral density, confirming the scale-invariant nature of ${\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}}$.
![Rescaled compensated spectral density $\left(|{\mathbf{k}}| / \lambda_0 \right)^2 \, S (|{\mathbf{k}}| / \lambda_0)$, for $\lambda_0$ defined in equation , corresponding to ${\textrm{Re}}= 10^3$ and palinstrophy values $\P_0 = 1.71\times 10^6$, $\P_0 = 1.71\times 10^7$ and $\P_0 = 1.71\times 10^8$. All curves collapse onto a “universal" spectral density. []{data-label="fig:Sk_vsP"}](k2Sk_L0Scale_K01_AllP){width="75.00000%"}
Finite-time Growth of Palinstrophy {#sec:TimeEvol}
==================================
We now focus our attention on the growth of palinstrophy $\P$ over finite time. Time integration of the sharp instantaneous estimate and neglect of the time dependence of ${\textrm{Re}}$ leads to the finite-time estimates $$\label{eq:maxP_timeDep}
\P^{1/2}(t) - \P^{1/2}(0) \leq \left( a + b\sqrt{\ln{\textrm{Re}}_0 + c} \right)\left(\frac{\E(0) - \E(t)}{4\nu}\right)$$ and $$\label{eq:maxP_aPriori}
\mathop{\max}_{t>0} \P(t) \leq \psi({\textrm{Re}}_0)\P(0), \quad\mbox{with}\quad
\psi({\textrm{Re}}_0) = \left(1+ \frac{a + b\sqrt{\ln{\textrm{Re}}_0 + c}}{4} \,{\textrm{Re}}_0\right)^2,$$ where the prefactor $\psi({\textrm{Re}}_0)$ depends exclusively on the initial Reynolds number ${\textrm{Re}}_0 = \K^{1/2}(0)/\nu$, and the values of $a$, $b$ and $c$ are given in and obtained by the fitting procedure described in section \[sec:2D\_InstOpt\]. Note that although estimate is time-dependent and not in the form power law, it can still be used to determine to what extent the fields saturating a *sharp* instantaneous estimate produce time-dependent flows which also saturate a time-dependent estimate. Also note that while is an *a priori* estimate in the form of a power law which has been obtained from a *sharp* instantaneous estimate, there is no guarantee that it will be sharp with respect to either the exponent or the prefactor in the power law. The derivation of estimates and can be found in Appendix \[sec:maxP\_Estimate\].
In order to assess the sharpness of the finite-time estimates and , we numerically solve the Cauchy problem using the instantaneously optimal vorticity fields ${\widetilde{\omega}_{{\textrm{Re}}_0,\P_0}}$ as initial condition, and carefully monitor the time evolution of different diagnostics, e.g. $\K(t)$, $\E(t)$ and $\P(t)$, for sufficiently long times. Time integration is performed using an adaptive Runge-Kutta scheme of order 4, and spatial discretization is performed using a pseudo-spectral Fourier-Galerkin method, with the standard “2/3” dealiasing rule. The resolution is increased accordingly, ranging from $512^2$ for low-${\textrm{Re}}_0$, low-$\P_0$ simulations, to $4096^2$ for high-${\textrm{Re}}_0$, high-$\P_0$ simulations. We refer the reader to the work by [@ap13b] for a thorough discussion of the time evolution of the instantaneously optimal vorticity ${\widetilde{\omega}_{{\textrm{Re}}_0,\P_0}}$.
The sharpness of the point-wise estimate can be studied by considering the functions: $$\varphi (t) = \P^{1/2}(t) - \P^{1/2}(0), \quad \mbox{and} \quad
\mu (t) = \left(\frac{a + b \sqrt{\ln {\textrm{Re}}_0+c}}{4\nu} \right)\left( \E(0) - \E(t) \right)$$ and comparing their behavior as functions of time. For this, consider the characteristic time scale $t_{\max}$ and the characteristic palinstrophy scale $\rho$ defined as: $$t_{\max} = \mathop{\arg\max}_{t > 0}\, \varphi (t) \quad \mbox{and} \quad
\rho = \mathop{\max}_{t \geq 0}\, \varphi (t),$$ and the rescaled diagnostics: $$f(\tau) = \varphi(t_{\max} \, \tau)/\rho$$ and $$g(\tau) = \mu(t_{\max} \, \tau)/\rho.$$ With these definitions, the point-wise estimate simply reads $f(\tau) \leq g(\tau)$. Figure \[fig:maxP\_FandG\] shows the dependence of $f$ and $g$ on the rescaled time $\tau = t/t_{\max}$, corresponding to different values of ${\textrm{Re}}\in[10^3,10^4]$ and $\P_0 \in [1.7\times 10^6, 1.7\times10^9]$. As expected from the fact that the initial condition is constructed in a self-similar manner, all data collapses onto single “universal” curves for $f$ and $g$. The data indicates that estimate is saturated only over a short interval, as expected from the fact that the fields are optimal only in an instantaneous sense, and the optimal growth of palinstrophy can be sustained only over this short interval.
On the other hand, it follows from estimate that $$\label{eq:Qmax}
Q_{\max} := 4\nu\frac{\P^{1/2}(t_{\max}) - \P^{1/2}(0)}{\E(0) - \E(t_{\max})} \leq
a + b\sqrt{\ln {\textrm{Re}}_0 + c}.$$ Figure \[fig:maxP\_FT\](a) shows the dependence of $Q_{\max}$ on ${\textrm{Re}}_0$, for values of palinstrophy in the range $\P_0 \in [10^6,10^9]$. The curve $\gamma({\textrm{Re}}_0) = \tilde{a} +\tilde{b}\sqrt{\ln{\textrm{Re}}_0 + \tilde{c}}$, with $\tilde{a}$, $\tilde{b}$ and $\tilde{c}$ given in , is included for reference as a red dashed curve. It can observed from the figure that although estimate is saturated only over a short time interval, the dependence of the finite-time growth of palinstrophy on ${\textrm{Re}}_0$, when compensated by the enstrophy dissipation occurring at the maximum palinstrophy time, has a behavior similar to the one predicted by estimate .
Figure \[fig:maxP\_FT\](b) shows the dependence on $\P(0)$ of the compensated maximum palinstrophy $$\P_{\max} = \P(0)^{-1}\,\mathop{\max}_{t > 0} \, \P(t)$$ obtained from the time evolution of the optimal vorticity ${\widetilde{\omega}_{{\textrm{Re}}_0,\P_0}}$, for fixed ${\textrm{Re}}_0$. As expected from estimate , the data indicates that $\P_{\max}$ is independent of $\P(0)$, with its value depending only on the initial Reynolds number ${\textrm{Re}}_0$. These results provide compelling evidence supporting the sharpness of the *a priori* finite-time estimate with respect to $\P(0)$.
![ Rescaled diagnostics $f(\tau)$ (black solid lines) and $g(\tau)$ (blue dashed lines) for different values of $\P_0$ and ${\textrm{Re}}_0$. All data collapses onto a “universal” pair of curves.[]{data-label="fig:maxP_FandG"}](FG_vsTau){width="75.00000%"}
\
Discussion and Conclusion {#sec:Conclusion}
=========================
We have presented numerical confirmation that the rigorous analytic estimate $$\frac{d\P}{dt} \leq \left(a + b\sqrt{\ln {\textrm{Re}}+ c}\right)\P^{3/2}$$ is sharp in its behavior with respect to both the palinstrophy $\P$ and the Reynolds number ${\textrm{Re}}$. The power-law dependence on $\P$ is predicted by the self-similar analysis from Appendix \[sec:SelfSimilarAnalysis\] and confirmed by the data shown in Figure \[fig:maxdP\_vsP\], where the compensated optimal rate of growth of palinstrophy $\P_0^{-3/2}\R({\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}})$ is plotted against $\P_0$. Similarly, the dependence of $\R({\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}})$ on ${\textrm{Re}}_0$ is correctly captured by the estimate although the constants $\tilde{a}, \tilde{b}$ and $\tilde{c}$ that fit the optimal rate of growth in the least-squares sense differ from the analytic values given in appendix \[sec:dPdt\_Estimate\]. More careful analysis might give better constants but it is important to note that the approach used to construct the optimal fields only ensures that solutions to are [*local*]{} maximizers: there is no guarantee that they are global maximizers of $\R$. However, the best one can hope for in the search of any other maximizers is to improve the value of the constants $\tilde{a}$, $\tilde{b}$ and $\tilde{c}$ so that optimal instantaneous production of palinstrophy matches that given by the analytic estimate.
Regarding the finite-time growth of palinstrophy, we have provided evidence supporting the sharpness of the *a priori* estimate $$\mathop{\max}_{t > 0} \, \P(t) \leq \psi({\textrm{Re}}_0) \P(0)$$ with respect to the initial palinstrophy $\P(0)$. For this, we have used the instantaneously optimal fields ${\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}}$ as initial condition in the 2-D incompressible Navier-Stokes equation, and carefully monitored the time evolution of palinstrophy. The sharpness with respect to the prefactor $\psi({\textrm{Re}}_0)$, on the other hand, is a more subtle issue and instead we looked at the point-wise estimate $$\P^{1/2}(t) - \P^{1/2}(0) \leq \frac{\gamma({\textrm{Re}}_0)}{4\nu} \left( \E(t) - \E(0) \right).$$ This time-dependent estimate was found to be saturated by the instantaneously optimal fields ${\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}}$ only over a short time window, resulting in a sub-optimal dependence of the compensated maximum growth of palinstrophy on ${\textrm{Re}}_0$. This does not mean, however, that the estimate is not sharp, as the fields ${\widetilde{\mathbf{u}}_{{\textrm{Re}}_0,\P_0}}$ are optimal only at time $t=0$. A better assessment of the sharpness of the point-wise estimate could be performed by solving the finite-time optimization problem: $$\label{eq:OptProb_FT}
\begin{aligned}
& \qquad\qquad\qquad\mathop{\max}_{{\mathbf{u}}\in\mathcal{S}} \;\; \P({\mathbf{u}}(\cdot,T)),
\quad {\textrm{with}}\\
\mathcal{S} = & \{ {\mathbf{u}}\in H^2(\Omega) : \nabla\cdot{\mathbf{u}}= 0, \; \K({\mathbf{u}}(\cdot,0)) =\K_0, \; \P({\mathbf{u}}(\cdot,0)) = \P_0, {\mathbf{u}}\mbox{ solves (1.1)} \},
\end{aligned}$$ with the constraint manifold $\mathcal{S}$ including only the fields which are solutions to the 2-D incompressible Navier-Stokes equation defined on the interval $(0,T)$. This study is, however, outside of the scope of this manuscript and it is left as an open question for subsequent work.
A possible interpretation of the results presented in this manuscript is that it is possible to saturate *a priori* finite-time estimates using fields that saturate instantaneous estimates. Time-dependent point-wise estimates, on the other hand, are saturated only over short time intervals, rendering the finite time growth of palinstrophy obtained from instantaneously optimal fields suboptimal. This implies that in the context of 3-D Navier-Stokes equation, for which no *a priori* finite-time estimates for the growth of enstrophy are available and only time-dependent point-wise estimates exist, the search of solenoidal fields that are optimal for the growth of enstrophy, the key quantity controlling the regularity of solutions in 3-D, must be performed following a finite-time optimization approach where the Navier-Stokes equation is included as part of the constraint, similar to the approach described in equation . Although computationally intensive, this task is within reach with current available resources and it is also left as an open question for subsequent work.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are extremely grateful to Felix Otto for the useful suggestions and discussions leading to the results presented in this manuscript. This research was supported in part by a NSF Award PHY-1205219, a 2014 Simons Foundation Fellowship in Theoretical Physics, NSF Award DMS-1515161, and a 2016 John Simon Guggenheim Foundation Fellowship in Applied Mathematics.
Estimates for the growth of palinstrophy {#sec:Analytic_Estimates}
========================================
Instantaneous growth of palinstrophy {#sec:dPdt_Estimate}
------------------------------------
From equation , the instantaneous rate of growth of palinstrophy $d\P/dt$ is defined as: $$\frac{d\P}{dt}({\mathbf{u}}) = -\nu\|{\Delta}\omega \|_2^2 - \int_\Omega \nabla\omega\cdot\nabla{\mathbf{u}}\cdot\nabla\omega \;d\Omega,$$ with ${\mathbf{u}}:\Omega\to\mathbb{R}^2$ such that $\nabla\cdot{\mathbf{u}}= 0$, and $\omega = \partial_1 u_2 - \partial_2 u_1$. As the quadratic form induced by $\nabla {\mathbf{u}}$ only depends on its symmetric part $(\nabla {\mathbf{u}})_{S} := \frac{1}{2}\left(\nabla {\mathbf{u}}+ \nabla {\mathbf{u}}^T\right)$ and is bounded by its spectral norm $ | (\nabla {\mathbf{u}})_{S} |_{\sigma} := \max\{|\lambda_1|,|\lambda_2|\}$, with $\lambda_1$ and $\lambda_2$ being the two real eigenvalues of $(\nabla {\mathbf{u}})_{S}$, it follows that: $$\label{eq:R_estim_S1}
\begin{aligned}
\frac{d\P}{dt}({\mathbf{u}}) & \leq -\nu\| {\Delta}\omega \|^2_2 +
\|(\nabla{\mathbf{u}})_{S}\|_{\sigma,\infty} \int_{\Omega} |\nabla\omega |^2 \; d\Omega \\
& = -\nu\| {\Delta}\omega \|^2_2 + 2\|(\nabla{\mathbf{u}})_{S}\|_{\sigma,\infty}\P.
\end{aligned}$$ For $\Omega$ a square of side $L$ endowed with periodic boundary conditions, it follows that a function $u:\Omega\to\mathbb{R}$ and its gradient $\nabla u$ admit a Fourier representation of the form: $$u({\mathbf{x}}) = \sum_{{\mathbf{k}}\in \mathbb{Z}_0} {\widehat{u}}({\mathbf{k}})\,e^{ i {\mathbf{k}}\cdot{\mathbf{x}}}, \quad
\nabla u({\mathbf{x}}) = \sum_{{\mathbf{k}}\in\mathbb{Z}_0} i \, {\mathbf{k}}\, {\widehat{u}}({\mathbf{k}}) \, e^{ i {\mathbf{k}}\cdot{\mathbf{x}}},$$ where $\mathbb{Z}_0 = \left(\frac{2\pi}{L}\right)\left(\mathbb{Z}\times\mathbb{Z}\right)$. In Fourier space, the incompressibility condition $\nabla\cdot{\mathbf{u}}= 0$ takes the form ${\mathbf{k}}\cdot \widehat{\mathbf{u}}({\mathbf{k}})=0$ for each ${\mathbf{k}}\in \mathbb{Z}_0 \setminus \{0\}$, which implies that $\widehat{\mathbf{u}}({\mathbf{k}}) \in \mathbb{C}{\mathbf{k}}^\perp$ with ${\mathbf{k}}^\perp= (-k_2, k_1)$. Together with the fact that the matrix $\frac{1}{2} ({\mathbf{k}}\otimes {\mathbf{k}}^\perp + {\mathbf{k}}^\perp \otimes {\mathbf{k}})$ has eigenvalues $-\frac{1}{2}|{\mathbf{k}}|^2$ and $\frac{1}{2}|{\mathbf{k}}|^2$ we get the somewhat crude estimate $$| (\nabla {\mathbf{u}})_{S}({\mathbf{x}}) |_{\sigma} \leq \sum_{{\mathbf{k}}\in\mathbb{Z}_0 \setminus \{0\}} \frac{|{\mathbf{k}}||\hat {\mathbf{u}}({\mathbf{k}})|}{|{\mathbf{k}}|^2} \left| \frac{1}{2} ({\mathbf{k}}\otimes {\mathbf{k}}^\perp + {\mathbf{k}}^\perp \otimes {\mathbf{k}}) \right|_{\sigma} \leq \frac{1}{2}\sum_{{\mathbf{k}}\in\mathbb{Z}_0} | {\mathbf{k}}| | \hat{\mathbf{u}}({\mathbf{k}}) |$$ Grouping the wave numbers into small, intermediate and high frequencies gives: $$| (\nabla {\mathbf{u}})_{S}({\mathbf{x}}) |_{\sigma} \leq \, \frac{1}{2}\left(
\sum_{|{\mathbf{k}}| \leq \Lambda_1} | {\mathbf{k}}| | \hat{\mathbf{u}}({\mathbf{k}}) | \quad +
\sum_{\Lambda_1 \leq |{\mathbf{k}}| \leq \Lambda_2} | {\mathbf{k}}| | \hat{\mathbf{u}}({\mathbf{k}}) | \quad +
\sum_{|{\mathbf{k}}| \geq \Lambda_2} | {\mathbf{k}}| | \hat{\mathbf{u}}({\mathbf{k}}) |\right) ,$$ where the cut-off wave numbers $\Lambda_1$ and $\Lambda_2$ are yet to be determined. Each term in the right-hand side of the last inequality can be upper-bounded using the Cauchy-Schwarz inequality as: $$\begin{aligned}
\sum_{|{\mathbf{k}}| \leq \Lambda_1} | {\mathbf{k}}| | \hat{\mathbf{u}}({\mathbf{k}}) |
& \leq \left( \sum_{|{\mathbf{k}}| \leq \Lambda_1} | {\mathbf{k}}|^2 \right)^{1/2} \left( \sum_{|{\mathbf{k}}| \leq \Lambda_1} |\hat{\mathbf{u}}( {\mathbf{k}}) |^2 \right)^{1/2}\\
& \leq \left( 2\pi \int_0^{\Lambda_1}k^3dk \right)^{1/2}\left(2\K\right)^{1/2} \; = \; \sqrt{\pi}\;\Lambda^2_1\;\K^{1/2},
\end{aligned}$$ $$\begin{aligned}
\sum_{\Lambda_1 \leq |{\mathbf{k}}| \leq \Lambda_2} | {\mathbf{k}}| | \hat{\mathbf{u}}({\mathbf{k}}) |
& \leq \left( \sum_{\Lambda_1 \leq |{\mathbf{k}}| \leq \Lambda_2} \frac{1}{ | {\mathbf{k}}|^2 } \right)^{1/2}
\left( \sum_{\Lambda_1 \leq |{\mathbf{k}}| \leq \Lambda_2} | {\mathbf{k}}| ^4 | \hat{\mathbf{u}}({\mathbf{k}}) |^2 \right)^{1/2} \\
& \leq \left(2\pi \int^{\Lambda_2}_{\Lambda_1} \frac{dk}{k} \right)^{1/2} \left(2\P\right)^{1/2} \; = \; 2\sqrt{\pi}\sqrt{\ln\left(\frac{\Lambda_2}{\Lambda_1} \right)}\; \P^{1/2},
\end{aligned}$$ and $$\begin{aligned}
\sum_{|{\mathbf{k}}| \geq \Lambda_2} | {\mathbf{k}}| | \hat{\mathbf{u}}({\mathbf{k}}) |
& \leq \left( \sum_{ |{\mathbf{k}}| \geq \Lambda_2} \frac{1}{ | {\mathbf{k}}|^4 } \right)^{1/2}
\left( \sum_{ |{\mathbf{k}}| \geq \Lambda_2} | {\mathbf{k}}| ^6 | \hat{\mathbf{u}}({\mathbf{k}}) |^2 \right)^{1/2} \\
& \leq \left( 2\pi \int^{\infty}_{\Lambda_2} \frac{dk}{k^3}\right)^{1/2} \| {\Delta}\omega \|_2 \; = \;
\frac{\sqrt{\pi}}{\Lambda_2} \| {\Delta}\omega \|_2.
\end{aligned}$$ Therefore, the estimate for $\| (\nabla{\mathbf{u}})_{S} \|_{\sigma,\infty}$ reads: $$\| (\nabla{\mathbf{u}})_{S} \|_{\sigma,\infty} \leq \frac{1}{2}\left(\sqrt{\pi}\;\Lambda^2_1\;\K^{1/2} + 2\sqrt{\pi}\sqrt{\ln\left(\frac{\Lambda_2}{\Lambda_1} \right)}\; \P^{1/2} + \frac{\sqrt{\pi}}{\Lambda_2} \| {\Delta}\omega \|_2\right),$$ with the estimate in leading to: $$\frac{d\P}{dt}({\mathbf{u}}) \leq -\nu\| {\Delta}\omega \|^2_2 + \sqrt{\pi}\;\Lambda^2_1\;\K^{1/2}\P + 2\sqrt{\pi}\sqrt{\ln\left(\frac{\Lambda_2}{\Lambda_1} \right)}\; \P^{3/2} + \frac{\sqrt{\pi}}{\Lambda_2} \| {\Delta}\omega \|_2\P.$$ The sum $-\nu\| {\Delta}\omega \|_2^2 + 2\sqrt{\pi}\| {\Delta}\omega\|_2\P/\Lambda_2$ of the first and last terms in the last inequality is maximal when $\| {\Delta}\omega \|_2 = \sqrt{\pi}\P/(2\nu\Lambda_2)$, attaining a maximum value of $\pi\P^2/(4\nu\Lambda^2_2)$. Thus, the estimate for $d\P/dt$ reads: $$\begin{aligned}
\frac{d\P}{dt}({\mathbf{u}}) & \leq \frac{\pi}{4}\frac{\P^2}{\nu\Lambda^2_2} + \sqrt{\pi}\;\Lambda^2_1\;\K^{1/2}\P + 2\sqrt{\pi}\sqrt{\ln\left(\frac{\Lambda_2}{\Lambda_1} \right)}\; \P^{3/2} \\
& \leq \left( \frac{\pi}{4}\frac{\P^{1/2}}{\nu\Lambda^2_2} +
\sqrt{\pi}\;\frac{\Lambda^2_1\K^{1/2}}{\P^{1/2}} +
2\sqrt{\pi}\sqrt{\ln\left(\frac{\Lambda_2}{\Lambda_1} \right)}\right) \P^{3/2}.
\end{aligned}$$ The cut-off wave numbers $\Lambda_1$ and $\Lambda_2$ can be chosen so that: $$\Lambda_1^2 = a\frac{\P^{1/2}}{\K^{1/2}},\quad\mbox{and}\quad
\Lambda_2^2 = \frac{1}{b}\frac{\P^{1/2}}{\nu},$$ for $a$ and $b$ positive dimensionless numbers. For this choice of $\Lambda_1$ and $\Lambda_2$, and the introduction of the dimensionless parameter ${\textrm{Re}}= \K^{1/2}/\nu$, the estimate becomes: $$\label{eq:dPdt_Est_gP32}
\frac{d\P}{dt} \leq \left( \sqrt{\pi} \; a + \frac{\pi}{4} \; b +
\sqrt{2\pi}\sqrt{\ln\left(\frac{\mbox{Re}}{ab} \right)} \right)\P^{3/2} = g(a,b)\;\P^{3/2}.$$ For sufficiently large ${\textrm{Re}}$, the prefactor $g(a,b)$ in the power-law estimate from is minimized when $\partial g /\partial a = 0$ and $\partial g/\partial b = 0$, i.e. the optimal values $(\tilde{a},\tilde{b})$ satisfy: $$\tilde{a} = \frac{1}{\sqrt{2}}\left[\ln\left(\frac{{\textrm{Re}}}{\tilde{a}\tilde{b}} \right)\right]^{-1/2} \quad\textrm{and}\qquad
\tilde{b} = \frac{2\sqrt{2}}{\sqrt{\pi}}\left[\ln\left(\frac{{\textrm{Re}}}{\tilde{a}\tilde{b}} \right)\right]^{-1/2}.$$ It follows that $\tilde{a} = (\sqrt{\pi}/4)\;\tilde{b}$, with the minimum value $g(\tilde{a},\tilde{b})$ given by: $$\label{eq:Optim_Prefactor}
\mathop{\min}_{(a,b)\in \textrm{QI}} \; g(a,b) = g(\tilde{a},\tilde{b}) = \sqrt{2\pi}\left(\sqrt{|z_1|} + \frac{1}{\sqrt{|z_1|}}\right),$$ where QI denotes the first quadrant in the $(a,b)$-plane, and $z_1 = - 1/(2\tilde{a}^2)$ is the smallest of the two solutions of the transcendental equation $$\label{eq:Transcend_Re}
ze^{z} = - \frac{2}{\sqrt{\pi}}{\textrm{Re}}^{-1}.$$ For values of ${\textrm{Re}}$ satisfying ${\textrm{Re}}> 2e/\sqrt{\pi}$, the two solutions to equation are given by $$z_0 = W_0\left(\frac{-2}{\sqrt{\pi}{\textrm{Re}}}\right)\qquad\textrm{and}\qquad
z_1 = W_{-1}\left(\frac{-2}{\sqrt{\pi}{\textrm{Re}}}\right),$$ where $W_k$ is the $k$-branch of the Lambert $W$ function. The asymptotic expansion for $W_{-1}$ is given by [@Corless1996]: $$W_{-1}(x) \sim \ln(-x) - \ln(-\ln(-x))\quad\textrm{as}\quad x\to 0^-,$$ from which it follows that, for sufficiently large ${\textrm{Re}}\geq 2/\sqrt{\pi}$, $$z_1 \sim - \left(\ln {\textrm{Re}}- \ln\left( \frac{2}{\sqrt{\pi}}\right) \right)
- \ln\left( \ln {\textrm{Re}}- \ln\left( \frac{2}{\sqrt{\pi}}\right) \right).$$ Hence, to leading order in the variable $\ln{\textrm{Re}}- \ln(2/\sqrt{\pi})$, the optimal prefactor $g(\tilde{a},\tilde{b})$ from equation has the form: $$g(\tilde{a},\tilde{b}) = \sqrt{2\pi} \sqrt{\ln{\textrm{Re}}- \ln\left(\frac{2}{\sqrt{\pi}}\right) },$$ giving the estimate for $d\P/dt$ as: $$\label{eq:dPdt_appendEstimate}
\frac{d\P}{dt} \leq \left(a + b \sqrt{ \ln{\textrm{Re}}+ c } \right) \P^{3/2},$$ $$a = 0, \quad b = \sqrt{2\pi},\quad c = - \ln\left(\frac{2}{\sqrt{\pi}}\right).$$
Finite-time growth of palinstrophy {#sec:maxP_Estimate}
----------------------------------
To obtain an *a priori* estimate for the finite-time growth of palinstrophy, we use the energy dissipation equation , leading to $\K(t) \leq \K(0)$ for all $t > 0$ and, since ${\textrm{Re}}= \K^{1/2}/\nu$, ${\textrm{Re}}(t) \leq {\textrm{Re}}(0) = {\textrm{Re}}_0$. It should be noted that estimate holds only for values of Reynolds ${\textrm{Re}}\geq 2/\sqrt{\pi}$, thus we expect the estimate to be valid only on the time interval where this constraint is satisfied. Using time integration of gives: $$\begin{aligned}
\frac{d\P}{dt} & \leq \left(a + b\sqrt{\ln{\textrm{Re}}_0 + c}\right)\P^{3/2} \quad \Rightarrow \nonumber \\
\int_{\P(0)}^{\P(t)} \P^{-1/2} d\P & \leq \left( a + b\sqrt{\ln{\textrm{Re}}_0+c} \right)\int_{0}^t\P(s) \,ds \quad \Rightarrow \nonumber\\
\P^{1/2}(t) - \P^{1/2}(0) & \leq \left( a + b\sqrt{\ln{\textrm{Re}}_0+c} \right)\left(\frac{\E(0) - \E(t)}{4\nu}\right) \label{eq:maxP_timeDepAppendix}\end{aligned}$$ where time integration of the enstrophy dissipation equation has been used. Although is not an *a priori* estimate, i.e. given exclusively in terms of the initial data, it can still be used to determine to what extent sharp instantaneous estimates lead to sharp finite-time estimates. To obtain an *a priori* estimate for the maximum finite-time growth of palinstrophy one can use the fact that $\E(t) \leq \E(0)$ for all $t>0$, which follows from the enstrophy dissipation equation , along with the estimate $\E \leq \K^{1/2}\P^{1/2}$ leading to $$\P^{1/2}(t) \leq \left( 1 + \frac{a + b\sqrt{\ln{\textrm{Re}}_0+c}}{4} \,{\textrm{Re}}_0 \right) \, \P^{1/2}(0),$$ which gives the *a priori* finite-time estimate $$\label{eq:finiteTime_appendEstimate}
\mathop{\max}_{t\geq0} \, \P(t) \leq C_{{\textrm{Re}}_0} \P(0) \quad\mbox{with}\quad
C_{{\textrm{Re}}_0} = \left(1+ \frac{a + b\sqrt{\ln{\textrm{Re}}_0+c}}{4} \,{\textrm{Re}}_0\right)^2,$$ where the prefactor $C_{{\textrm{Re}}_0}$ depends exclusively on the initial Reynolds number ${\textrm{Re}}_0 = \K^{1/2}(0)/\nu$, and $$a = 0,\quad b = \sqrt{2\pi},\quad c = -\ln\left(\frac{2}{\sqrt{\pi}}\right).$$
Self-similar Optimal Fields {#sec:SelfSimilarAnalysis}
===========================
As discussed in section \[sec:2D\_InstOpt\], we are interested in finding incompressible fields maximizing the instantaneous production of palinstrophy $d\P/dt$. That is, we are solving the optimization problem $$\label{eq:OptimProb_Vort}
\begin{aligned}
& \qquad\mathop{\max}_{\omega\in\mathcal{S}_{\K_0,\P_0}} \;\; \R(\omega),
\quad {\textrm{with}}\\
\mathcal{S}_{\K_0,\P_0} = & \{ \omega\in H^2(\Omega) : \; \K(\omega) = \K_0, \; \P(\omega) = \P_0 \}.
\end{aligned}$$ Optimization problem is equivalent to problem , except that it has been written in terms of the vorticity $\omega$ instead of the velocity field ${\mathbf{u}}$. The energy, palinstrophy and objective functionals are given in terms of $\omega$ as:
$$\begin{aligned}
\K(\omega) & = \frac{1}{2}\int_\Omega | {\nabla^{\perp}}\psi |^2 \, d\Omega, \\
\P(\omega) & = \frac{1}{2}\int_\Omega | {\nabla^{\perp}}\omega |^2 \, d\Omega,\\
\R(\omega) & = \int_{\Omega}\J(\omega,\psi){\Delta}\omega \, d\Omega -
\nu \int_{\Omega}\left({\Delta}\omega \right)^2 \, d\Omega,\end{aligned}$$
where ${\nabla^{\perp}}= [\partial_{x_2}, -\partial_{x_1}]^T$, $\J(f,g) = (\partial_{x_1}f)( \partial_{x_2} g) - (\partial_{x_2} f)( \partial_{x_1} g)$ is the Jacobian determinant, and the streamfunction $\psi$ and the vorticity $\omega$ satisfy the state equation: $$\label{eq:vort2Stream}
-{\Delta}\psi = \omega\qquad\mbox{in } \Omega.$$
The first-order optimality condition, i.e. the Euler-Lagrange equation for problem reads: $$\label{eq:KKT_Vort}
\frac{\delta\R}{\delta\omega} - \lambda_{\K}\frac{\delta\K}{\delta\omega} - \lambda_{\P}\frac{\delta\P}{\delta\omega} = 0,$$ where $\lambda_{\K}$ and $\lambda_{\P}$ are the Lagrange multipliers associated with the constraints defining the manifold $\mathcal{S}_{\K_0,\P_0}$, and the corresponding variations of the functionals $\R$, $\K$ and $\P$ with respect to variations in $\omega$ are given by: $$\frac{\delta\R}{\delta\omega} = \J(\psi,{\Delta}\omega) + {\Delta}\J(\omega,\psi) + \psi^* - 2\nu{\Delta}^2\omega,$$ $$\frac{\delta\K}{\delta\omega} = -\psi, \quad\mbox{and}\quad
\frac{\delta\P}{\delta\omega} = {\Delta}\omega,$$ with $\psi^*$ representing the Lagrange multiplier associated with the state equation and obtained as the solution to the elliptic problem: $$\label{eq:vort2Stream_Adj}
-{\Delta}\psi^* = \J({\Delta}\omega,\omega).$$ Evidence supporting the existence of optimal vorticity fields $\twKP$ which, for fixed energy $\K_0$, vary in a self-similar manner with $\P_0$ was presented by [@ap13a]. With this numerical evidence in mind, we look for fields $\omega_{\P_0}$, $\psi_{\P_0}$ and $\psi^*_{\P_0}$, satisfying equation and subject to the constraints $\K(\omega_{\P_0}) = \K_0$ and $\P(\omega_{\P_0}) = \P_0$, of the form: $$\begin{aligned}
\omega_{\P_0}({\mathbf{x}}) & = \P_0^\alpha\Phi(\P_0^q {\mathbf{x}}), \\
\psi_{\P_0}({\mathbf{x}}) & = \P_0^\beta\Psi(\P_0^q {\mathbf{x}}), \\
\psi^*_{\P_0}({\mathbf{x}}) & = \P_0^\gamma\Psi^*(\P_0^q {\mathbf{x}}),
\end{aligned}$$ for some real parameters $\alpha,\beta,\gamma$ and $q$, and some functions $\Phi$, $\Psi$ and $\Psi^*$ independent of $\P_0$. Using these ansatz and the rescaled spatial variables ${\mathbf{y}}= \P_0^q{\mathbf{x}}$, it follows that:
$$\begin{aligned}
\K(\omega_{\P_0}) & = \P_0^{2\beta}\left( \frac{1}{2}\int | {\nabla^{\perp}}\Psi |^2 \, d{\mathbf{y}}\right) \quad\mbox{and}\\
\P(\omega_{\P_0}) & = \P_0^{2\alpha}\left( \frac{1}{2}\int | {\nabla^{\perp}}\Phi |^2 \, d{\mathbf{y}}\right).\end{aligned}$$
From the energy and palinstrophy constraints it follows that $\alpha = 1/2$ and $\beta = 0$, and from state equations and we obtain $q = 1/4$ and $\gamma = 3/2$. Finally, the Euler-Lagrange equation reads: $$\P_0^{3/2} \left( \J(\Psi,{\Delta}\Phi) + {\Delta}\J(\Phi,\Psi) + \Psi^* -
2\nu{\Delta}^2\Phi + \lambda_1 \Psi - \lambda_2 {\Delta}\Phi \right) = 0,$$ where $\lambda_1$ and $\lambda_2$ are the Lagrange multipliers corresponding to the constraints $$\frac{1}{2}\int | {\nabla^{\perp}}\Psi |^2 \, d{\mathbf{y}}= \K_0 \quad\mbox{and}\quad
\frac{1}{2}\int | {\nabla^{\perp}}\Phi |^2 \, d{\mathbf{y}}= 1.$$ The objective functional can be thus evaluated for $\omega_{\P_0}$, yielding: $$\R(\omega_{\P_0}) = \left( -\nu \int \left( {\Delta}\Phi\right)^2\,d{\mathbf{y}}+
\int \J(\Phi,\Psi){\Delta}\Phi \,d{\mathbf{y}}\right)\P_0^{3/2},$$ in agreement with the observed optimal instantaneous growth, and in line with the power-law behavior predicted by estimate . The dependence of the prefactor $$C_{\K_0,\nu} = -\nu \int \left( {\Delta}\Phi\right)^2\,d{\mathbf{y}}+
\int \J(\Phi,\Psi){\Delta}\Phi \,d{\mathbf{y}}$$ on the ratio $\K_0^{1/2}/\nu$ is the main subject of study of the present work.
[9]{}
D. Ayala and B. Protas, *Maximum Palinstrophy Growth in [2D]{} Incompressible Flows*, Journal of Fluid Mechanics (2014), **742** : 340-367.
D. Ayala and B. Protas, *Vortices, maximum growth and the problem of finite-time singularity formation*, Fluid Dynamics Research (2014), **46** (3) 031404.
C. R. Doering, *The [3D Navier-Stokes]{} Problem*, Annual Review of Fluid Mechanics (2009), 109-128.
Corless, R. M. and Gonnet, G. H. and Hare, D. E. G. and Jeffrey, D. J. and Knuth, D. E., *On the LambertW function*, Advances in Computational Mathematics (1996), **5** (1) 329–359.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a new model of “Stealth Dark Matter”: a composite baryonic scalar of an $SU(N_D)$ strongly-coupled theory with even $N_D \geq 4$. All mass scales are technically natural, and dark matter stability is automatic without imposing an additional discrete or global symmetry. Constituent fermions transform in vector-like representations of the electroweak group that permit both electroweak-breaking and electroweak-preserving mass terms. This gives a tunable coupling of stealth dark matter to the Higgs boson independent of the dark matter mass itself. We specialize to $SU(4)$, and investigate the constraints on the model from dark meson decay, electroweak precision measurements, basic collider limits, and spin-independent direct detection scattering through Higgs exchange. We exploit our earlier lattice simulations that determined the composite spectrum as well as the effective Higgs coupling of stealth dark matter in order to place bounds from direct detection, excluding constituent fermions with dominantly electroweak-breaking masses. A lower bound on the dark baryon mass $m_B {\mathrel{\raisebox{-.6ex}{$\stackrel{\textstyle>}{\sim}$}}}300$ GeV is obtained from the indirect requirement that the lightest dark meson not be observable at LEP II. We briefly survey some intriguing properties of stealth dark matter that are worthy of future study, including: collider studies of dark meson production and decay; indirect detection signals from annihilation; relic abundance estimates for both symmetric and asymmetric mechanisms; and direct detection through electromagnetic polarizability, a detailed study of which will appear in a companion paper.'
author:
- 'T. Appelquist'
- 'R. C. Brower'
- 'M. I. Buchoff'
- 'G. T. Fleming'
- 'X.-Y. Jin'
- 'J. Kiskis'
- 'G. D. Kribs'
- 'E. T. Neil'
- 'J. C. Osborn'
- 'C. Rebbi'
- 'E. Rinaldi'
- 'D. Schaich'
- 'C. Schroeder'
- 'S. Syritsyn'
- 'P. Vranas'
- 'E. Weinberg'
- 'O. Witzel'
bibliography:
- 'SU4.bib'
title: 'Stealth Dark Matter: Dark scalar baryons through the Higgs portal'
---
**Introduction**
================
Composite dark matter, made up of electroweak-charged constituents provides a straightforward mechanism for obtaining viable electrically-neutral particle dark matter that can yield the correct cosmological abundance while surviving direct and indirect detection search limits, e.g., [@Aprile:2012nq; @Agnese:2013rvf; @Akerib:2013tjd]. In this paradigm, the dark sector consists of fermions that transform under the electroweak group and a new, strongly-coupled non-Abelian dark force. This was considered long ago in the context of technicolor theories, where the strong dynamics was doing double-duty to both break electroweak symmetry and provide a dark matter candidate [@Nussinov:1985xr; @Chivukula:1989qb; @Barr:1990ca; @Kaplan:1991ah].
In this paper, electroweak symmetry breaking is accomplished through the weakly-coupled Standard Model Higgs mechanism, while the new strongly-coupled sector is reserved solely for providing a viable dark matter candidate. This dark sector is not easy to detect in dark matter detection experiments or in collider experiments, and so we give it the name “Stealth Dark Matter”. Earlier work in this direction includes [@Nussinov:1985xr; @Chivukula:1989qb; @Barr:1990ca; @Barr:1991qn; @Kaplan:1991ah; @Chivukula:1992pn; @Bagnasco:1993st; @Pospelov:2000bq; @Dietrich:2006cm; @Foadi:2008qv; @Khlopov:2008ty; @Mardon:2009gw; @Kribs:2009fy; @Lisanti:2009am; @Alves:2010dd; @Khlopov:2010pq; @Buckley:2012ky; @Cline:2013zca; @Heikinheimo:2013fta; @Heikinheimo:2014xza; @Yamanaka:2014pva; @Hochberg:2014dra; @Boddy:2014yra; @Boddy:2014qxa; @Hochberg:2014kqa], and except for [@Lewis:2011zb; @Hietanen:2012sz; @Appelquist:2013ms; @Hietanen:2013fya; @Appelquist:2014jch; @Detmold:2014qqa; @Detmold:2014kba; @Fodor:2015eea], was often limited by the inability to perturbatively calculate the spectrum and form factors due to strong coupling.
The proposed dark matter candidate is a scalar baryon of $SU(N_D)$, and hence $N_D$ must be even.[^1] We take the dark fermions to be in vector-like representations of the electroweak group. Hence, the constituent dark fermions can acquire bare mass terms (fermion masses that do not require electroweak symmetry breaking) while also permitting Yukawa interactions that marry dark fermion electroweak doublets with singlets. This yields a theory in which dark matter couples to the Higgs boson in a tunable way that is essentially independent of the dark matter mass itself. This is somewhat analogous to dark sector models with a dark U(1) portal (e.g., [@Pospelov:2007mp; @ArkaniHamed:2008qn; @Pospelov:2008zw; @Cheung:2009qd; @Morrissey:2009ur]), where the coupling to the Standard Model is tunable through an otherwise arbitrary parameter – the kinetic mixing between the dark U(1) and hypercharge.
The existence of both electroweak-preserving and electroweak-breaking masses for the dark fermions provides two main benefits. First, given that the Higgs boson couples electroweak doublets with singlets, the global flavor symmetries of the dark fermions can be completely broken to just dark baryon number. All mesons can decay through an electroweak process (e.g., electrically charged mesons through $W$ exchange) or through the usual chiral anomaly (e.g., the lightest neutral meson). Ensuring that these particles decay before big-bang nucleosynthesis sets a weak lower bound on the Higgs interaction strength. (This is in contrast with [@Kilic:2009mi; @Buckley:2012ky] where additional interactions were required to ensure mesons decay, e.g., through higher dimensional operators). The second reason is related to the orientation of the chiral condensate after the dark force confines. Large vector-like masses for the dark fermions ensure that the condensate can be aligned toward the electroweak-preserving direction, and thus the dark sector leads to only small corrections to electroweak precision measurements. We estimate the size of these corrections in this paper.
There are many appealing features of an electroweak-neutral composite dark matter candidate made up from fermions transforming under the electroweak group, including:
- All of the dimensionful scales are technically natural, since they arise from fermion masses (vector-like and electroweak breaking) and the confinement of a strong-coupled dark force.
- Dark matter stability is an automatic consequence of dark baryon number conservation. No additional global discrete or continuous symmetries are required. For $N_D \ge 3$, operators involving dark baryon decay are necessarily dimension-6 or higher, and thus safe from GUT-scale or Planck-scale suppressed violations of dark baryon number.
- There are no dimension-4 interactions of the composite dark matter particle with the Standard Model except with the Higgs boson. The direct detection scattering cross section is thus automatically suppressed compared with a generic elementary WIMP candidate.
- Higher dimensional interactions of the dark matter with the Standard Model are suppressed, in the nonrelativistic limit, by several powers of the dark matter mass. For a composite scalar, the leading operators are charge radius (at dimension-6) and polarizability (at dimension-7). The impact of these (and other) operators on the dark matter scattering cross section in direct detection experiments has been studied in [@Chivukula:1992pn; @Bagnasco:1993st; @Pospelov:2000bq; @Sigurdson:2004zp; @Gudnason:2006ug; @Alves:2009nf; @Kribs:2009fy; @Barbieri:2010mn; @Banks:2010eh; @Chang:2010en; @Barger:2010gv; @Weiner:2012cb].
- Interactions of the dark baryon through the neutral weak current, the charge radius interaction, as well as the contributions to the electroweak precision $T$ parameter, are simultaneously eliminated if the fermion interactions obey a global custodial $SU(2)$ symmetry. Additionally, as we will see, dark matter electric neutrality also follows from custodial $SU(2)$. To simplify our analysis, here we will primarily study the subset of stealth dark matter parameter space in which the custodial $SU(2)$ is preserved. (This simplification is very familiar from composite Higgs theories, e.g. [@Agashe:2003zs]).
- The abundance of a strongly-coupled dark scalar baryon could arise through several mechanisms: an asymmetric abundance (such as through electroweak sphalerons [@Barr:1990ca; @Barr:1991qn] or other mechanisms [@Zurek:2013wia]), when the mass is not too large ${\mathrel{\raisebox{-.6ex}{$\stackrel{\textstyle<}{\sim}$}}}$ few TeV, or a symmetric abundance, when the mass is large (perhaps $\sim \mathcal{O}(100)$ TeV) [@Griest:1989wd; @Buckley:2012ky; @Nussinov:2014qva].
We focus mainly on a confining $SU(4)$ gauge theory dark sector with dark fermions transforming non-trivially under the electroweak group. We apply our recent results [@Appelquist:2014jch] using lattice simulations for the spectrum and effective Higgs interaction for $SU(4)$. As emphasized in [@Appelquist:2013ms; @Appelquist:2014jch], this theory is well suited for lattice calculations since we are not interested in the chiral limit of vanishing dark fermion masses. Indeed, lattice simulations can efficiently simulate the parameter region where the dark fermion masses are comparable to the confinement scale, exactly where the perturbative estimates are least useful.
The organization of the paper is as follows. In Sec. \[sec:constructing\] we discuss the assumptions and requirements to construct our stealth dark matter model. In Sec. \[sec:darkfermioninteractions\] we detail the dark fermion interactions and masses. In addition, we write the electroweak currents in terms of the dark fermion mass eigenstates of the theory, detailed in Appendix \[sec:weakcurrents\]. Until this point, the discussion of the model is general. In Sec. \[sec:simplifications\], we simplify the parameter space for phenomenological and calculational purposes, applying a global custodial $SU(2)$ symmetry and taking the approximately symmetric dark fermion mass matrix limit. Then in Sec. \[sec:meson\] we discuss the light non-singlet mesons in the theory, in particular their decay rates and constraints from non-observation at LEP II. In Sec. \[sec:pew\] we discuss the stealth dark matter contributions to the $S$ parameter, and demonstrate the parametric suppression that happens in several regimes. In Sec. \[sec:fermionhiggs\] we obtain the Higgs boson coupling to the dark fermions. Then in Sec. \[sec:boundshiggs\] we apply our previous model-independent results on the $SU(4)$ spectrum and effective Higgs coupling to stealth dark matter. We obtain the bounds on the parameter space from the non-observation of a spin-independent direct detection signal at LUX. We briefly discuss the relic abundance of stealth dark matter in Sec. \[sec:abundance\]. Finally we conclude with a discussion in Sec. \[sec:discussion\].
**Constructing a viable model** {#sec:constructing}
===============================
Basic assumptions
-----------------
We assume that the dark matter candidate is a composite particle of a non-Abelian, confining gauge theory based on the group $SU(N_{D})$ with $N_f$ flavors of fermions transforming in the fundamental representation. The number $N_f$ is restricted by only the condition of confinement. For reasons outlined in the introduction (abundance, detectability), the dark fermions carry electroweak charges. Our model includes a tunable Higgs “portal” coupling between the dark sector and the Standard Model via dimension-4 Higgs couplings.[^2] We do not consider QCD-colored dark fermions since with $N_D \neq 3$, dark baryons would not generally be color singlets.[^3]
Requirements
------------
We require dark matter stability to be automatic, arising from a global symmetry. This motivates considering the dark baryon of the non-Abelian dark sector to be the dark matter [@Nussinov:1985xr; @Chivukula:1989qb; @Barr:1990ca]. In the presence of GUT-scale or Planck-scale suppressed operators, the stability of the dark baryon should be sufficient to avoid cosmological constraints.
The requirement of a sufficiently preserved accidental baryon number disfavors a dark $SU(2)$ group. First, there is no automatic baryon number in $SU(2)$ because there is no fundamental distinction between mesons and baryons. Imposing a global $U(1)$ baryon number is possible (e.g. see [@Kribs:2009fy]) but in addition baryon number violating dimension-5 Planck-suppressed operators such as $f_{\rm dark} f_{\rm dark} H^{\dagger} H/M_{\rm Pl}$ must be absent, where $f_{\rm dark}$ is the dark fermion. (Otherwise, the dark $SU(2)$ baryon would decay on a timescale much shorter than the age of the Universe.)
For $N_D \ge 3$, operators involving dark baryon decay are necessarily dimension-6 or higher, and thus safe from GUT-scale or Planck-scale suppressed violations of dark baryon number. $SU(N_D)$ with odd $N_D$ is a perfectly interesting theory, having been studied before for $N_D = 3$ by our collaboration [@Appelquist:2013ms]. There it was found that a fermionic dark baryon has a magnetic dipole interaction that leads to a significant contribution to spin-independent scattering. Constraints from the XENON100 experiment were satisfied only when the dark matter mass $M {\mathrel{\raisebox{-.6ex}{$\stackrel{\textstyle>}{\sim}$}}}10$ TeV [@Appelquist:2013ms]. This strong constraint on the mass scale implies the model is difficult to test at near-future colliders.
The magnetic dipole interaction (and other higher dimensional operators that require spin) are absent when the dark baryon is a scalar. We are thus naturally led to $SU(N_D)$ with even $N_D \ge 4$, for which the otherwise strong constraints from direct detection are weakened, lowering the scales of interest into a regime that can be probed by colliders and other detection strategies.
We assume the dark fermions have masses $M_f$ on the order of the $SU(N_D)$ confinement scale $\Lambda_D$. If the masses were much smaller, the dark sector would contain light pseudo-Goldstone pions that transform under the electroweak group, which are strongly constrained by collider experiments. A dark sector with purely vector-like fermion masses has approximately stable electrically-charged mesons due to dark flavor symmetries. Conversely, a dark sector with purely electroweak breaking fermion masses has a dark matter candidate that is ruled out by spin-independent direct detection through single Higgs exchange. (For example, quirky dark matter [@Kribs:2009fy] is now completely ruled out by Higgs exchange, given the direct detection bounds from LUX [@Akerib:2013tjd] combined with the relatively light Higgs mass [@Aad:2012tfa; @Chatrchyan:2012ufa].) Fermions with both vector-like and (small) electroweak breaking contributions to their masses can avoid both problems.
We require the lightest dark baryon to be electrically neutral. We also require Higgs couplings at dimension-4 to pairs of dark fermions. These two requirements impose restrictions on the electroweak charges of the dark fermions.
One solution is familiar from old technicolor theories (e.g. [@Peskin:1980gc; @Preskill:1980mz]): requiring the dark fermion charges to roughly satisfy $|Y| {\mathrel{\raisebox{-.6ex}{$\stackrel{\textstyle<}{\sim}$}}}|T_3|$ where $T_3$ is the $SU(2)_L$ isospin. Choosing doublets ($|T_3| = 1/2$ under $SU(2)_L$) then gives a finite number of discrete possibilities.
A simple model that satisfies all of these requirements is shown in Table \[tab:particles\]. The electric charges of the dark fermions in the broken electroweak phase are $Q = \pm 1/2$, ensuring all hadrons have integer electric charges. So long as the lightest $Q = 1/2$ and $Q = -1/2$ dark fermions are close in mass, the lightest baryon will be a scalar and electrically neutral. Finally, with the assignments shown in Table \[tab:particles\], all gauge (and global) anomalies vanish, which is automatic with fermions that transform under vector-like representations of the $SU(N_D)$ and electroweak groups.
**Dark fermion interactions and masses** {#sec:darkfermioninteractions}
========================================
Field $SU(N_D)$ ($SU(2)_{L}$, $Y$) $Q$
----------------------------------------------------------------------- ------------------------- ---------------------- ---------------------------------------------------------------
$F_1 = {\left(\!\begin{array}{c}F^u_1 \\ F^d_1\end{array} \!\right)}$ $\mathbf{N}$ $(\mathbf{2}, 0)$ ${\left(\!\begin{array}{c}+1/2 \\ -1/2\end{array} \!\right)}$
$F_2 = {\left(\!\begin{array}{c}F^u_2 \\ F^d_2\end{array} \!\right)}$ $\overline{\mathbf{N}}$ $(\mathbf{2}, 0)$ ${\left(\!\begin{array}{c}+1/2 \\ -1/2\end{array} \!\right)}$
$F_3^u$ $\mathbf{N}$ $(\mathbf{1}, +1/2)$ $+1/2$
$F_3^d$ $\mathbf{N}$ $(\mathbf{1}, -1/2)$ $-1/2$
$F_4^u$ $\overline{\mathbf{N}}$ $(\mathbf{1}, +1/2)$ $+1/2$
$F_4^d$ $\overline{\mathbf{N}}$ $(\mathbf{1}, -1/2)$ $-1/2$
: Dark fermion particle content of the stealth dark matter model. All fields are two-component (Weyl) spinors. $SU(2)_L$ refers to the Standard Model electroweak gauge group, and $Y$ is the hypercharge. In the broken phase of the electroweak theory, the dark fermions have the electric charge $Q = T_3 + Y$ as shown.[]{data-label="tab:particles"}
The fermions $F_i^{u,d}$ transform under a global $U(4) \times U(4)$ flavor symmetry with \[$SU(2) \times U(1)$\]$^4$ surviving after the weak gauging of the electroweak symmetry. From this large global symmetry, one $SU(2)$ (diagonal) subgroup will be identified with $SU(2)_L$, one $U(1)$ subgroup will be identified with $U(1)_Y$, and one $U(1)$ will be identified with dark baryon number. The total fermionic content of the model is therefore 8 Weyl fermions that pair up to become 4 Dirac fermions in the fundamental or anti-fundamental representation of $SU(N_D)$ with electric charges of $Q \equiv T_{3,L} + Y = \pm 1/2$. We use the notation where the superscript $u$ or $d$ (as in $F^u$, $F^d$ and later $\psi^u$, $\psi^d$, $\Psi^u$, $\Psi^d$) denotes a fermion with electric charge of $Q=1/2$ or $Q=-1/2$ respectively.
The fermion kinetic terms in the Lagrangian are given by $$\mathcal{L} \; \supset \;
\sum_{i=1,2} i F_i^\dagger \bar{\sigma}^\mu D_{i,\mu} F_i
+ \sum_{i=3,4; j=u,d} i {F_i^j}^\dagger \bar{\sigma}^\mu D_{i,\mu}^j F_i^j
\, ,$$ where the covariant derivatives are $$\begin{aligned}
D_{1,\mu} &\equiv& \partial_\mu - i g W^a_\mu \sigma^a /2
- i g_D G^b_\mu t^b \\
D_{2,\mu} &\equiv& \partial_\mu - i g W^a_\mu \sigma^a /2
+ i g_D G^b_\mu {t^b}^* \\
D_{3,\mu}^j &\equiv& \partial_\mu - i g' Y^j B_\mu
- i g_D G^b_\mu t^b \\
D_{4,\mu}^j &\equiv& \partial_\mu - i g' Y^j B_\mu
+ i g_D G^b_\mu {t^b}^* \end{aligned}$$ with the interactions among the electroweak group and the new $SU(N_D)$. Here $Y^u = 1/2$, $Y^d = -1/2$ and $t^b$ are the representation matrices for the fundamental of $SU(N_D)$.
The vector-like mass terms allowed by the gauge symmetries are $$\mathcal{L} \supset M_{12} \epsilon_{ij} F_1^i F_2^j
- M_{34}^u F_3^u F_4^d + M_{34}^d F_3^d F_4^u + h.c.,$$ where $\epsilon_{12} \equiv \epsilon_{ud} = -1 = -\epsilon^{12}$ and the relative minus signs between the mass terms have been chosen for later convenience. The mass term $M_{12}$ explicitly breaks an \[$SU(2) \times U(1)$\]$^2$ global symmetry down to the diagonal $SU(2)_{\rm diag} \times U(1)$ where the $SU(2)_{\rm diag}$ is identified with $SU(2)_L$. The mass terms $M_{34}^{u,d}$ explicitly break the remaining \[$SU(2) \times U(1)$\]$^2$ down to $U(1) \times U(1)$ where one of the $U(1)$’s is identified with $U(1)_Y$. (In the special case when $M_{34}^{u} = M_{34}^{d}$, the global symmetry is enhanced to $SU(2) \times U(1)$, where the global $SU(2)$ acts as a custodial symmetry.) Thus, after weakly gauging the electroweak symmetry and writing arbitrary vector-like mass terms, the unbroken flavor symmetry is $U(1) \times U(1)$.
Electroweak symmetry breaking mass terms arise from coupling to the Higgs field $H$ that we take to be in the $(\mathbf{2}, +1/2)$ representation. They are given by $$\begin{aligned}
\mathcal{L} &\supset& y_{14}^u \epsilon_{ij} F_1^i H^j F_4^d
+ y_{14}^d F_1 \cdot H^\dagger F_4^u \nonumber \\
& &{} - y_{23}^d \epsilon_{ij} F_2^i H^j F_3^d
- y_{23}^u F_2 \cdot H^\dagger F_3^u
+ h.c. \, ,
\label{eq:higgsterms}\end{aligned}$$ where again the relative minus signs are chosen for later convenience. After electroweak symmetry breaking, $H = ( 0 \;\; v/\sqrt{2} )^T$, with $v \simeq 246$ GeV. Replacing the Higgs field by its VEV in Eq. (\[eq:higgsterms\]), we obtain mass terms for the fermions, in 2-component notation, $$\begin{aligned}
\mathcal{L} &\supset& - (F_1^u \;\; F_3^u) M^u {\left(\!\begin{array}{c}F_2^d \\ F_4^d\end{array} \!\right)}
- (F_1^d \;\; F_3^d) M^d {\left(\!\begin{array}{c}F_2^u \\ F_4^u\end{array} \!\right)} \nonumber \\
& &{} + h.c. \, ,
\label{eq:2compmassmatrix}\end{aligned}$$ with the mass matrices given by $$\begin{aligned}
M^u &\equiv&
{\left(\!\begin{array}{cc}M_{12}&y_{14}^u v/\sqrt{2}\\#3&M_{34}^u\end{array} \!\right)}
\label{eq:upmassmatrix} \\
M^d &\equiv& -
{\left(\!\begin{array}{cc}M_{12}&y_{14}^d v/\sqrt{2}\\#3&M_{34}^d\end{array} \!\right)} \, .
\label{eq:downmassmatrix} \end{aligned}$$
These Yukawa couplings break the remaining $U(1) \times U(1)$ flavor symmetry to $U(1)_D$ dark baryon number. The mass matrices $M^u$ and $M^d$ correspond to the masses of two sets of fermions with electric charge $Q = +1/2$ and $Q = -1/2$ respectively, in the fundamental representation of $SU(N_D)$. The two biunitary mass matrices can be diagonalized by four independent rotation angles $$\begin{aligned}
\left( \begin{array}{cc} M^u_1 & 0 \\ 0 & M^u_2 \end{array} \right)
&=& R(\theta^u_1)^{-1} M^u R(\theta^u_2)
\label{eq:diagu} \\
\left( \begin{array}{cc} M^d_1 & 0 \\ 0 & M^d_2 \end{array} \right)
&=& R(\theta^d_1)^{-1} M^d R(\theta^d_2) \, ,
\label{eq:diagd}\end{aligned}$$ where the rotation matrices are defined by $$\begin{aligned}
R(\theta_i^j) &\equiv&
{\left(\!\begin{array}{cc}\cos\theta_i^j&-\sin\theta_i^j\\#3&\cos\theta_i^j\end{array} \!\right)}
\, .\end{aligned}$$ The 2-component mass eigenstate spinors are thus $$\begin{aligned}
{\left(\!\begin{array}{c}\psi_1^u \\ \psi_2^u\end{array} \!\right)} &=& R(\theta^u_1) {\left(\!\begin{array}{c}F_1^u \\ F_3^u\end{array} \!\right)}
\label{eq:vec1} \\
{\left(\!\begin{array}{c}\psi_1^d \\ \psi_2^d\end{array} \!\right)} &=& R(\theta^u_2) {\left(\!\begin{array}{c}F_2^d \\ F_4^d\end{array} \!\right)}
\label{eq:vec2} \\
{\left(\!\begin{array}{c}\chi_1^d \\ \chi_2^d\end{array} \!\right)} &=& i R(\theta^d_1) {\left(\!\begin{array}{c}F_1^d \\ F_3^d\end{array} \!\right)}
\label{eq:vec3} \\
{\left(\!\begin{array}{c}\chi_1^u \\ \chi_2^u\end{array} \!\right)} &=& i R(\theta^d_2) {\left(\!\begin{array}{c}F_2^u \\ F_4^u\end{array} \!\right)} \, ,
\label{eq:vec4} \end{aligned}$$ where the extra phase in Eqs. (\[eq:vec3\]),(\[eq:vec4\]) ensures the $Q=-1/2$ fermions will have positive mass eigenvalues.
The Lagrangian for the fermion mass eigenstates becomes $$\begin{aligned}
\mathcal{L} &\supset& - \sum_{i=1}^2 \left( M_i^u \psi_i^u \psi_i^d
+ M_i^d \chi_{i}^d \chi_{i}^u + h.c. \right) \end{aligned}$$ where the mass eigenvalues are $M_{1,2}^u$ for $Q=1/2$, and the distinction between fermions $\psi$ and $\chi$ allows us to write the $Q=-1/2$ fermion masses as $M_{1,2}^d$. The Dirac spinor mass eigenstates are constructed from the 2-component Weyl spinor mass eigenstates in the usual way, $$\begin{aligned}
\Psi_i^u &\equiv& {\left(\!\begin{array}{c}\psi^u_i \\ {\psi^d_i}^\dagger\end{array} \!\right)} \qquad i=1,2
\label{eq:diracup} \\
\Psi_i^d &\equiv& {\left(\!\begin{array}{c}\chi^d_{i} \\ {\chi^u_{i}}^\dagger\end{array} \!\right)} \qquad i=1,2
\label{eq:diracdown} \end{aligned}$$ giving the Dirac fermion masses $$\begin{aligned}
\mathcal{L} &\supset&
- \sum_{i=1}^2 \left( M_i^u \overline{\Psi}_i^u \Psi_i^u
+ M_i^d \overline{\Psi}_i^d \Psi_i^d \right) \, .\end{aligned}$$
The fermion masses themselves are obtained from a straightforward diagonalization of the mass matrices, $$M^u_{1,2} = \frac{M_{12} + M_{34}^u}{2} \mp
\left[ \left( \frac{M_{12} - M_{34}^u}{2} \right)^2
+ \frac{y^u_{14} y^u_{23} v^2}{2} \right]^{1/2} \, ,$$ with mixing angles $$\tan 2\theta_1^u =
\frac{2 \sqrt{2} v (M_{12} y^u_{23} + M^u_{34} y^u_{14} )}{
2 M_{12}^2 - 2 (M_{34}^u)^2 + (y^u_{14} v)^2 - (y^u_{23} v)^2}$$ $$\tan 2\theta_2^u =
\frac{2 \sqrt{2} v (M_{12} y^u_{14} + M^u_{34} y^u_{23} )}{
2 M_{12}^2 - 2 (M_{34}^u)^2 -(y^u_{14} v)^2 + (y^u_{23} v)^2}
\, ,$$ with identical expressions for $M^d_{1,2}$ and $\tan 2\theta^d_{1,2}$ with the replacement $u \leftrightarrow d$ everywhere.
It is important to note that the electroweak currents ($j^\mu_{+}$, $j^\mu_{-}$, $j^\mu_{3}$, $j^\mu_{Y}$) play an important role in the upcoming phenomenological discussions. Due to the extended expressions for these quantities in terms of our Dirac spinors, we have relegated a detailed derivation of the electroweak currents to Appendix \[sec:weakcurrents\].
**Simplifications** {#sec:simplifications}
===================
Our main interest is the more specialized case where the lightest $Q = +1/2$ and $Q = -1/2$ fermions are degenerate in mass to a very good approximation. This leads to a neutral scalar baryon with a vanishing charge radius. While there are several ways this could be accomplished, we can simply impose a custodial $SU(2)$ global symmetry on the Lagrangian. In order to simplify notation, we define $c_i^j \equiv \cos\theta_i^j$, $s_i^j \equiv \sin\theta_i^j$ and $P_{L,R} = (1 \mp \gamma_5)/2$. In the custodial $SU(2)$ symmetric theory, $c_i^u=c_i^d$ and $s_i^u=s_i^d$.
Custodial SU(2)
---------------
An exact custodial $SU(2)$ symmetry implies the masses and interactions are symmetric with respect to the interchange $u \leftrightarrow d$. This means the Lagrangian parameters satisfy $$\begin{aligned}
y_{14}^u = y_{14}^d \equiv y_{14}, & \qquad &
y_{23}^u = y_{23}^d \equiv y_{23}, \label{eq:custodiallimit} \\
M_{34}^u & = M_{34}^d & \equiv M_{34} \, . \nonumber\end{aligned}$$ Defining the overall vector-like mass scale $M$ and difference $\Delta$ to be [^4] $$\begin{aligned}
M \equiv \frac{M_{12} + M_{34}}{2} \qquad
\Delta \equiv \left| \frac{M_{12} - M_{34}}{2} \right| \, ,\end{aligned}$$ the dark fermion mass eigenvalues are $$\begin{aligned}
M_{1,2} = M \mp \sqrt{\Delta^2 + \frac{y_{14} y_{23} v^2}{2}} \, .\end{aligned}$$ No $u$ or $d$ labels are necessary, since custodial $SU(2)$ symmetry implies that there is one pair of Dirac fermions with electric charge $Q = (+1/2,-1/2)$ with mass $M_1$ (the lightest pair), as well as a second pair of Dirac fermions with electric charge $Q = (+1/2,-1/2)$ with mass $M_2$ (the heavier pair). The spectrum is illustrated in Fig. \[fig:su4\_spectrum\_figure\].
In the limit $y_{14},y_{23} \rightarrow 0$, the fermions acquire purely vector-like masses, and thus the chiral condensate of the dark force is aligned to a purely electroweak-preserving direction. In order that the chiral condensate’s electroweak-preserving orientation is not significantly disrupted, we consider small electroweak breaking masses, $y_{14} v, y_{23} v \ll M$.
This leaves two distinct regimes for the spectrum, depending on the relative sizes of $\sqrt{y_{14} y_{23}} v$ and $\Delta$.
![Illustration of the fermion mass spectra considered in the paper. Four Dirac fermions ($\Psi^u_1$, $\Psi^d_1$, $\Psi^u_2$, $\Psi^d_2$) have masses ($M_1^u$, $M_1^d$, $M_2^u$, $M_2^d$). The $u$ ($d$) fermions have electric charge $Q = +1/2$ ($Q = -1/2$); we assume an exact custodial $SU(2)$ global symmetry that ensures each $Q = +1/2$ fermion is accompanied by a $Q = -1/2$ fermion with equal mass as shown in the figure. If $\Delta \ll \sqrt{y_{14} y_{23}} v$ ($\Delta \gg \sqrt{y_{14} y_{23}} v$) the mass splitting is dominated by electroweak breaking (preserving) masses that we call the Linear (Quadratic) Case. See the text for details.[]{data-label="fig:su4_spectrum_figure"}](su4_spectrum_figure.pdf){width="49.00000%"}
Approximately symmetric mass matrices
-------------------------------------
A second simplification, useful to analytically and numerically evaluate our results, is to take $y_{14} \simeq y_{23}$. The mass matrices Eqs. (\[eq:upmassmatrix\],\[eq:downmassmatrix\]) are approximately symmetric. Specifically, we can write $$\label{eq:approxsymmetric}
y_{14} = y + {\epsilon}_y \, , \quad y_{23} = y - {\epsilon}_y \, , \quad
|{\epsilon}_y| \ll |y| \, .$$ and expand in powers of $\epsilon_y$. For example, the dark fermion masses become simply $$\begin{aligned}
M_{1,2} = M \mp \sqrt{\Delta^2 + \frac{y^2 v^2}{2}} \, .\end{aligned}$$ to leading order in $O(\epsilon_y)$.
The distinct regimes are thus $y v \gg \Delta$ and $y v \ll \Delta$. In the Linear Case $y v \gg \Delta$, electroweak symmetry breaking is (dominantly) responsible for the mass *splitting* between $\Psi^{u,d}_1$ and $\Psi^{u,d}_2$. In the Quadratic Case $y v \ll \Delta$, the splitting is dominantly attributed to the vector-like mass splitting $\Delta$. As we shall see, the primary distinction between these two cases is in the Higgs coupling to the fermion mass eigenstates: proportional to $y$ for the Linear Case and $y^2$ for the Quadratic Case, hence the case names. A similar observation was also found in Ref. [@Hill:2014yka].
From this point forward unless noted otherwise, we assume the fermion mass parameters satisfy an exact custodial $SU(2)$ and the mass matrices are approximately symmetric.
Light Non-Singlet Meson Phenomenology {#sec:meson}
=====================================
Theories with new fermions that transform under vector-like representations of the electroweak group generically have enlarged global flavor symmetries that can prevent decay of the lightest non-singlet mesons and baryons. In the case of dark baryons, this is a feature, providing the rationale for the stability of the lightest dark baryon of the theory.
In the case of the lightest non-singlet mesons, this can be problematic, since some of these mesons carry electric charge.[^5] Stable integer charged mesons are strongly constrained from collider searches as well as cosmology. One solution is to postulate additional higher dimensional operators that connect a dark fermion pair with a Standard Model fermion pair [@Kilic:2009mi; @Buckley:2012ky]. This must be carefully done to avoid also writing operators that violate the approximate global symmetries protecting the stability of the dark matter. In the stealth dark matter model, however, electroweak symmetry breaking can provide the source of global flavor symmetry breaking, leading to the decay of the lightest charged mesons. (We will not discuss the lightest neutral mesons, but they are generically more difficult to produce in colliders, and they will decay through essentially the same mechanism as we describe for the charged mesons.)
The lightest electrically charged mesons are composed dominantly of the dark fermion pairs $\Pi^+ = (\overline{\Psi^d_1} \Psi_1^u)$ and $\Pi^- = (\overline{\Psi^u_1} \Psi_1^d)$. We can estimate the lightest meson lifetime by generalizing pion decay of QCD to our model. The relevant matrix element is (see, e.g., [@Donoghue:1992dd]) $$\begin{aligned}
\langle 0 | j^\mu_{\pm,{\rm axial}} | \Pi^{\pm} \rangle &=& i f_\Pi p^\mu \, ,\end{aligned}$$ where $f_\Pi$ is the “pion decay constant” associated with the dark force in this paper. The axial part of the electroweak current can be read off from the electroweak currents given in Eqs. (\[eq:jplus\]),(\[eq:jminus\]) $$\begin{aligned}
j^\mu_{+,{\rm axial}} &\supset&
c_{\rm axial}
\overline{\Psi^u_1} \gamma^\mu \gamma_5 \Psi^d_1\end{aligned}$$ where $$\begin{aligned}
c_{\rm axial} &=& \frac{c_1^u c_1^d - c_2^u c_2^d}{\sqrt{2}} \end{aligned}$$ and $j^\mu_{-,{\rm axial}}$ is identical upon $u \leftrightarrow d$. In the custodial limit, Eq. (\[eq:custodiallimit\]), the axial coefficient is $$c_{\rm axial} = \frac{(y_{14}^2 - y_{23}^2) v^2}{\sqrt{2
(8 M^2 + (y_{14} - y_{23})^2 v^2) (8 \Delta^2 + (y_{14} + y_{23})^2 v^2)}} \, .$$ Some insight can be gained using approximately symmetric mass matrices, Eq. (\[eq:approxsymmetric\]). We then obtain $$\begin{aligned}
c_{\rm axial} &=& \frac{{\epsilon}_y y v^2}{2 M \sqrt{2 \Delta^2 + y^2 v^2}}
\\ &\simeq& \frac{{\epsilon}_y v}{2 M} \times \left\{
\begin{array}{ll}
1 & \quad \mbox{Linear Case} \\
y v/(\sqrt{2} \Delta) & \quad \mbox{Quadratic Case.}
\end{array}
\right. \nonumber\end{aligned}$$ The decay width can be obtained from pion decay of QCD by replacing $V_{ud}$ in the Standard Model with $c_{\rm axial}$ for the dark mesons. Since the charged dark mesons of this model are much heavier than the QCD pions, there are many possible decay modes. For a general decay into a Standard Model doublet $(f \, f')$, assuming $m_f \gg m_{f'}$, the decay width is $$\begin{aligned}
\Gamma(\Pi^+ {\rightarrow}f \overline{f}') &=&
\frac{G_F^2}{4 \pi} f_\Pi^2 m_f^2 m_{\Pi} c_{\rm axial}^2
\left( 1 - \frac{m_f^2}{m_{\Pi}^2} \right) \, .\end{aligned}$$ If $m_{\Pi} > m_t + m_b$, the dominant decay mode is expected to be $\Pi^+ {\rightarrow}t\overline{b}$, otherwise $\Pi^+ {\rightarrow}\tau^+ \nu_\tau$ and $\Pi^+ {\rightarrow}\bar{s} c$, with branching ratios of roughly 70% and 30% respectively. Note that the decay width has several enhancement factors relative to the QCD pion decay width $$\frac{\Gamma(\Pi^+ {\rightarrow}f \overline{f}')}{\Gamma(\pi {\rightarrow}\mu^+ \nu_\mu)}
\simeq \frac{c_{\rm axial}^2}{|V_{ud}|^2} \left( \frac{f_\Pi}{f_\pi} \right)^2
\left( \frac{m_f}{m_\mu} \right)^2
\left( \frac{m_\Pi}{m_\pi} \right)$$ where for simplicity we have neglected kinematic suppression. As an example, if $f_\Pi \simeq m_\Pi \simeq v$, we find the lightest charged dark mesons decay faster than QCD charged pions so long as $c_{\rm axial} {\mathrel{\raisebox{-.6ex}{$\stackrel{\textstyle>}{\sim}$}}}10^{-8}$. This is easy to satisfy with small Yukawa couplings and dark fermion masses at or beyond the electroweak scale.
We can now make some comments about existing collider constraints on non-singlet mesons. The lightest charged mesons $\Pi^{\pm}$ can be pair produced in particle colliders through the Drell-Yan process, and will decay through annihilation of the constituent fermions into a $W$ boson. Because the Drell-Yan production is mediated by a photon and the mesons have unit electric charge, the production cross-section is substantial, leading to robust bounds from LEP-II. For charged states near the LEP-II energy threshold, the dominant decay mode is expected to be $\Pi^+ {\rightarrow}\tau^+ \nu_\tau$ as noted above. Reinterpreting the LEP-II bound from the pair production of supersymmetric partners to the tau (with the stau decaying into a tau and a nearly massless gravitino), we find $m_{\Pi} {\mathrel{\raisebox{-.6ex}{$\stackrel{\textstyle>}{\sim}$}}}86.6$ GeV [@Heister:2001nk; @Heister:2003zk; @Abdallah:2003xe; @Achard:2003ge; @Abbiendi:2004gf]. Stronger bounds from the LHC may be possible, although existing searches do not yet give any significant constraints on the charged mesons [@Buckley:2012ky]; we briefly highlight the signals in the discussion.
Using our lattice results from Ref. [@Appelquist:2014jch], we can translate the experimental bound on the mass of the pseudoscalar meson into a bound on the baryon mass, $m_B > 245, 265, 320$ GeV when the ratio of the pseudoscalar mass to the vector meson mass is $m_{\Pi}/m_V = 0.77, 0.70, 0.55$.
Contributions to Electroweak Precision Observables {#sec:pew}
==================================================
Stealth dark matter contains dark fermions that acquire electroweak symmetry breaking contributions to their masses. Consequently, there are contributions to the electroweak precision observables of the Standard Model, generally characterized by $S$ and $T$ [@Peskin:1990zt; @Peskin:1991sw]. In the custodial $SU(2)$ limit, Eq. (\[eq:custodiallimit\]), the contribution to $T$ vanishes. There is a contribution to $S$, controllable through the relative size of the electroweak breaking and electroweak preserving masses of the dark fermions. The $S$ parameter is defined in terms of momentum derivatives of current-current correlators [@Peskin:1990zt; @Peskin:1991sw], $$\begin{aligned}
S &\equiv& 16\pi \Pi^\prime_{3Y}(0) \\
&=& \frac{d}{dq^2} \left[ \frac{16\pi}{3}
\left(g^{\mu\nu}-\frac{q^\mu q^\nu}{q^2} \right)
X^{\mu\nu}(q^2)\right]_{q^2=0} \nonumber \\
X^{\mu\nu}(q^2)
&\equiv& \int d^4x\ e^{-i q x} \langle j^\mu_3(x) j^\nu_Y(0) \rangle,\end{aligned}$$ where the currents $ j^\mu_3(x)$ and $ j^\nu_Y(x)$ for the stealth dark matter model are defined in Eqs. (\[eq:j3\]) and (\[eq:jY\]). After some algebra and identifications of symmetric contractions, these definitions of the currents in terms of 4-component fermion fields lead to the current-current correlator. In the custodial limit, we obtain $$\begin{aligned}
2\langle & & j^\mu_3(x) j^\nu_Y(0) \rangle
= c_1^2 s_1^2 \left( {}^{11}G^{\mu\nu}_{LL}
+ {}^{22}G^{\mu\nu}_{LL}
- {}^{12}G^{\mu\nu}_{LL}
- {}^{21}G^{\mu\nu}_{LL} \right) \nonumber \\
& &{} + c_2^2 s_2^2 \left( {}^{11}G^{\mu\nu}_{RR}
+ {}^{22}G^{\mu\nu}_{RR}
- {}^{12}G^{\mu\nu}_{RR}
- {}^{21}G^{\mu\nu}_{RR} \right) \nonumber \\
& &{} + c_1^2s_2^2 \left( {}^{11}G^{\mu\nu}_{LR}
+ {}^{22}G^{\mu\nu}_{RL} \right)
+ c_2^2s_1^2 \left( {}^{11}G^{\mu\nu}_{RL}
+ {}^{22}G^{\mu\nu}_{LR} \right) \nonumber \\
& &{} - c_1c_2s_1s_2 \left( {}^{12}G^{\mu\nu}_{LR}
+ {}^{12}G^{\mu\nu}_{RL}
+ {}^{21}G^{\mu\nu}_{LR}
+ {}^{21}G^{\mu\nu}_{RL} \right),\end{aligned}$$ where the connected contributions to the correlation functions are given by $${}^{ij} G^{\mu\nu}_{AB}
\equiv \left. \langle \bar{\Psi}^u_i \gamma^\mu P_A
\Psi^u_j \bar{\Psi}^u_j \gamma^\nu P_B \Psi^u_i
\rangle \right|_{\text{connected}} \, . \label{eq:curr_corr}$$ Here, $A,B = L,R$ and the flavor indices $i,j = 1,2$, where it is understood that the flavors labeled $2$ have larger fermion masses than the flavors labeled $1$. Since the $u,d$ flavors have the same mass, the $u$ and $d$ labels are interchangeable (i.e. everything is written in terms of the $u$ flavors).
We can obtain expressions for the mixing angle coefficients. Like the case of light meson decay, if we consider an approximately symmetric mass matrix, with Yukawa couplings given by Eq. (\[eq:approxsymmetric\]), all of the mixing angle coefficients are approximately equal to each other, differing only at first order in $\epsilon_y$, i.e., $$\begin{aligned}
c_1^2 s_1^2 &\simeq& c_2^2 s_2^2 \simeq
c_1^2 s_2^2 \simeq c_2^2 s_1^2 \simeq c_1 c_2 s_1 s_2
\nonumber \\
&=& \frac{1}{4} \frac{y^2 v^2}{y^2 v^2 + 2 \Delta^2}
\left[ 1 + O(\epsilon_y) \ldots \right] \nonumber \\
&\simeq& \frac{1}{4} \times \left\{
\begin{array}{ll}
1 & \quad \mbox{Linear Case} \\
y^2 v^2/(2 \Delta^2) & \quad \mbox{Quadratic Case.}
\end{array}
\right. \end{aligned}$$ In the Linear Case, the mixing angles are approximately equal $c_1 \simeq s_1 \simeq c_2 \simeq s_2 \simeq 1/\sqrt{2}$. In the Quadratic Case, all of the contributions to the $S$ parameter are suppressed by $(y v/\Delta)^2$. To calculate the $S$ parameter in general requires lattice methods, paying close attention to the heavy-light splitting of the fermions, $M_2 - M_1$. To a first approximation we expect that in the limit of small mass splitting, $M_2 - M_1 \ll M$, $$G_{AB}^{\mu\nu} \equiv {}^{11}G_{AB}^{\mu\nu} \simeq
{}^{22}G_{AB}^{\mu\nu} \simeq {}^{12}G_{AB}^{\mu\nu} \simeq
{}^{21}G_{AB}^{\mu\nu}.$$ This gives for the current–current correlator $$\begin{aligned}
2 \langle j_3^{\mu}(x) j_Y^{\nu}(0)\rangle & \simeq \left[c_1^2 s_2^2
+ c_2^2 s_1^2 - 2c_1 c_2 s_1 s_2\right] G_{LR}^{\mu\nu} \nonumber \\
& \simeq \frac{\epsilon_y^2 v^2}{2M^2} G_{LR}^{\mu\nu},\end{aligned}$$ where all of the $G_{LL}$ and $G_{RR}$ contributions self-cancel. Hence, we see that the contribution to the $S$ parameter is suppressed as $M \gg v$ or $\epsilon_y \ll 1$, as expected.
Fermion Couplings to the Higgs Boson {#sec:fermionhiggs}
====================================
In terms of the gauge-eigenstate fields, the interactions of the Higgs boson with the dark-sector fermions are, in matrix notation, $$\begin{aligned}
\mathcal{L} &\supset& - \frac{h}{\sqrt{2}} (F_1^u \;\; F_3^u)
\left( \begin{array}{cc}
0 & y_{14}^u \\ y_{23}^u & 0
\end{array} \right)
{\left(\!\begin{array}{c}F_2^d \\ F_4^d\end{array} \!\right)} \nonumber \\
& &{} + \frac{h}{\sqrt{2}} (F_1^d \;\; F_3^d)
\left( \begin{array}{cc}
0 & y_{14}^d \\ y_{23}^d & 0
\end{array} \right)
{\left(\!\begin{array}{c}F_2^u \\ F_4^u\end{array} \!\right)} \nonumber \\
& &{} + h.c. \, .\end{aligned}$$ These matrices are not simultaneously diagonalizable with the mass matrices, Eqs. (\[eq:upmassmatrix\]),(\[eq:downmassmatrix\]). This means that the Higgs boson in general has off-diagonal, “dark flavor-changing” interactions with the mass eigenstate fields. Explicitly, we find in terms of the mixing angles
$$\begin{aligned}
\mathcal{L} \supset
\frac{h}{\sqrt{2}}
\left( \begin{array}{cc} \overline{\Psi}_1^u & \overline{\Psi}_2^u
\end{array} \right)
\left( \begin{array}{cc}
c_1^u s_2^u \, y_{14}^u + s_1^u c_2^u \, y_{23}^u
&~~ c_1^u c_2^u \, y_{14}^u - s_1^u s_2^u \, y_{23}^u \\
c_1^u c_2^u \, y_{23}^u - s_1^u s_2^u \, y_{14}^u
&~~ - s_1^u c_2^u \, y_{14}^u - c_1^u s_2^u \, y_{23}^u
\end{array} \right)
{\left(\!\begin{array}{c}\Psi_1^u \\ \Psi_2^u\end{array} \!\right)}
+ (u \leftrightarrow d) \, .\end{aligned}$$
In the custodial $SU(2)$ limit, we can drop the $u$ and $d$ labels since the Higgs coupling matrix is identical for both sets of fields. If we further take the limit of an approximately symmetric mass matrix, Eq. (\[eq:approxsymmetric\]), the Higgs couplings simplify to $$\mathcal{L} \supset
\frac{y h}{M_2 - M_1}
\left( \begin{array}{cc} \overline{\Psi}_1 & \overline{\Psi}_2
\end{array} \right)
\left[ \left( \begin{array}{cc}
y v & -\sqrt{2} \Delta \\
-\sqrt{2} \Delta & - y v
\end{array} \right) + O(\epsilon_y) \right]
{\left(\!\begin{array}{c}\Psi_1 \\ \Psi_2\end{array} \!\right)} \, .$$
We observe both diagonal and off-diagonal Higgs couplings to the fermions. The off-diagonal dark flavor-changing interactions vanish in the limit $\Delta \rightarrow 0$ and $\epsilon_y \rightarrow 0$. In this limit an enhanced flavor symmetry among the fermions is restored, and the analogue of the GIM mechanism forbids such interactions at tree-level. The off-diagonal Higgs couplings lead to an inelastic scattering cross section when a single Higgs is exchanged. This is highly suppressed unless the mass difference $M_2 - M_1$ is near the (non-relativistic) kinetic energy of the dark matter in galaxy. Two off-diagonal Higgs couplings can be combined in a loop involving one heavier dark fermion and double Higgs exchange, but this is suppressed by the square of the Higgs couplings times a loop factor, as well as by the mass of the heavier fermions.
The single Higgs coupling to the lightest fermions is finally $$\begin{aligned}
\mathcal{L} &\supset& y_\Psi h \overline{\Psi}_1 \Psi_1 \end{aligned}$$ where $$\begin{aligned}
y_\Psi &=& \frac{y^2 v}{M_2 - M_1} + O(\epsilon_y) \nonumber \\
&\simeq& \begin{dcases*}
\frac{y}{\sqrt{2}} & \mbox{Linear Case} \\
\frac{y^2 v}{2 \Delta} & \mbox{Quadratic Case.}
\end{dcases*}\end{aligned}$$ (Note also that the single Higgs coupling to the heaviest fermions $\Psi_2$ is identical up to an overall sign.) Depending on the relative size of $y v$ and $\Delta$, the Higgs boson couples linearly or quadratically proportional to the Yukawa coupling $y$. The additional suppression of $y v/\Delta$ in the Quadratic Case will imply that spin independent scattering through single Higgs exchange can be significantly weaker when the mass difference between the lightest and heaviest fermions is dominated by the electroweak preserving mass $\Delta$.
Direct detection bounds from Higgs exchange {#sec:boundshiggs}
===========================================
In a previous paper [@Appelquist:2014jch], we determined the model-independent bounds on direct detection from Higgs exchange for a scalar baryon of $SU(4)$. The model-independent result was expressed in terms of the effective Higgs coupling to the baryon $$g_B = \frac{m_B}{v} \alpha f^{(B)}_{f} \, . \label{gB-eq}$$ The first factor, the baryon mass $m_B$ (divided by the electroweak VEV), as well as the third factor $$f_f^{(B)} = \frac{\langle B| M_1 \overline{\Psi}_1 \Psi_1 | B \rangle}{m_B}
= \frac{M_1}{m_B} \frac{\partial m_B}{\partial M_1}
\nonumber$$ are extracted from our lattice results [@Appelquist:2014jch]. The second factor $$\alpha \equiv \frac{v}{M_1}
\frac{\partial \, M_1(h)}{\partial \, h}\bigg|_{h=v}
\simeq \begin{dcases*}
\frac{y v}{\sqrt{2} M_1} & \text{Linear Case} \\
\frac{(yv)^2}{2 M_1 \Delta} & \text{Quadratic Case}
\end{dcases*}
\label{eq:alpha}$$ provides the effective coupling of the Higgs boson to the fermions (multiplied by $v/M_1$), and we have evaluated the derivative for the two cases in our model.
Unfortunately, we cannot directly apply our previous results on constraints in $\alpha$-$m_B$ space to the parameters of the stealth dark matter model. This is because we do not know the dark fermion mass, $M_1$, independent of the lattice regularization scheme. We can, however, construct a regularization-independent parameter, the effective Yukawa coupling $y_{\rm eff}$, that is closely related to the model parameters: $$y_{\text{eff}} \equiv
\begin{dcases*}
y \frac{m_B}{\sqrt{2} M_1} & \text{Linear Case} \\
y \frac{m_B}{\sqrt{2 \Delta M_1}} & \text{Quadratic Case.}
\end{dcases*}
\label{eq:yeff}$$ The $\alpha$ parameter is therefore $$\alpha \simeq
\begin{dcases*}
y_{\rm eff} \frac{v}{m_B} & \text{Linear Case} \\
y_{\rm eff}^2 \frac{v^2}{m_B^2} & \text{Quadratic Case.}
\end{dcases*}
\label{eq:alpha2}$$ Recasting our previous constraints in $\alpha$-$m_B$ space into $y_{\rm eff}$-$m_B$ space, we can identify the region of parameter space that remains viable. The constraints for the Linear Case are shown in Fig. \[fig:lin\] and the Quadratic Case in Fig. \[fig:quad\]. In the top two plots for the respective figures, the region above the LUX bounds represents the excluded parameter space for the model at a given dark matter mass ($m_B$) and effective Yukawa coupling ($y_{\text{eff}}$). The figures show a clear qualitative trend in how the predictions change as a function of dark matter mass. In particular, the cross-section is independent of $m_B$ for the Linear Case and inversely proportional to $m_B$ in the Quadratic Case. The bottom plots in Figs. \[fig:lin\],\[fig:quad\] shows the maximum $y_{\rm eff}$ allowed for a given dark matter mass. By increasing the splitting $\Delta$ between the vector-like mass terms, significantly more $y_{\text{eff}}$ parameter space becomes available.
![Constraints on the stealth dark matter model in the Linear Case of the model. The top and middle figures show the predicted values for the smallest and largest fermion mass explored in our simulations (corresponding to the pseudoscalar to vector mass ratio $m_{\Pi}/m_V = 0.55, 0.77$) as well as LUX bounds. Various $y_{\text{eff}}$ values are plotted on the figure, where $y_{\text{eff}} \approx y m_B/M_1$ in this case. The dark grey region is excluded by the LEP constraints on charged dark mesons. The bottom figure displays the maximum $y_{\text{eff}}$ allowed for a given dark matter mass. Each of the green curves represents a different fermion mass in the lattice calculation, $m_{\Pi} / m_V = 0.55, 0.7, 0.77$ from top to bottom, and the bottom red curve is the result in the heavy fermion limit.[]{data-label="fig:lin"}](LUX-exclusion-ylin-k01572-label.png "fig:"){width="45.00000%"} ![Constraints on the stealth dark matter model in the Linear Case of the model. The top and middle figures show the predicted values for the smallest and largest fermion mass explored in our simulations (corresponding to the pseudoscalar to vector mass ratio $m_{\Pi}/m_V = 0.55, 0.77$) as well as LUX bounds. Various $y_{\text{eff}}$ values are plotted on the figure, where $y_{\text{eff}} \approx y m_B/M_1$ in this case. The dark grey region is excluded by the LEP constraints on charged dark mesons. The bottom figure displays the maximum $y_{\text{eff}}$ allowed for a given dark matter mass. Each of the green curves represents a different fermion mass in the lattice calculation, $m_{\Pi} / m_V = 0.55, 0.7, 0.77$ from top to bottom, and the bottom red curve is the result in the heavy fermion limit.[]{data-label="fig:lin"}](LUX-exclusion-ylin-k01554-label.png "fig:"){width="45.00000%"} ![Constraints on the stealth dark matter model in the Linear Case of the model. The top and middle figures show the predicted values for the smallest and largest fermion mass explored in our simulations (corresponding to the pseudoscalar to vector mass ratio $m_{\Pi}/m_V = 0.55, 0.77$) as well as LUX bounds. Various $y_{\text{eff}}$ values are plotted on the figure, where $y_{\text{eff}} \approx y m_B/M_1$ in this case. The dark grey region is excluded by the LEP constraints on charged dark mesons. The bottom figure displays the maximum $y_{\text{eff}}$ allowed for a given dark matter mass. Each of the green curves represents a different fermion mass in the lattice calculation, $m_{\Pi} / m_V = 0.55, 0.7, 0.77$ from top to bottom, and the bottom red curve is the result in the heavy fermion limit.[]{data-label="fig:lin"}](yeff-constraint-linear "fig:"){width="45.00000%"}
![Same as Fig. \[fig:lin\] but for the Quadratic Case of the model. In this case, $y_{\text{eff}} \approx y m_B/\sqrt{M_1 \Delta}$.[]{data-label="fig:quad"}](LUX-exclusion-yquad-k01572-label.png "fig:"){width="45.00000%"} ![Same as Fig. \[fig:lin\] but for the Quadratic Case of the model. In this case, $y_{\text{eff}} \approx y m_B/\sqrt{M_1 \Delta}$.[]{data-label="fig:quad"}](LUX-exclusion-yquad-k01554-label.png "fig:"){width="45.00000%"} ![Same as Fig. \[fig:lin\] but for the Quadratic Case of the model. In this case, $y_{\text{eff}} \approx y m_B/\sqrt{M_1 \Delta}$.[]{data-label="fig:quad"}](yeff-constraint-quad "fig:"){width="45.00000%"}
Abundance {#sec:abundance}
=========
We now provide a brief discussion of the relic abundance of stealth dark matter. In the regime where the dark fermions have masses comparable to the confinement scale of the dark force, calculating the relic abundance is an intrinsically strongly-coupled calculation. Unfortunately, this calculational difficulty is not easily overcome with lattice simulations, due to the different initial and final states. Nevertheless, it is straightforward to see that the relic abundance *can* match the cosmological abundance through at least two distinct mechanisms that lead to two different mass scales for stealth dark matter. In this section we discuss obtaining the abundance of stealth dark matter through thermal freezeout, leading to a symmetric abundance of dark baryons and anti-baryons. Separately, we consider the possibility of an asymmetric abundance generated through electroweak sphalerons.
Symmetric Abundance
-------------------
In the early universe at temperatures well above the confinement scale of the $SU(4)$ dark gauge force, the dark fermions are in thermal equilibrium with the thermal bath through their electroweak interactions. As the universe cools to temperatures below the confinement scale, the degrees of freedom change from dark fermions and gluons into the dark baryons and mesons of the low energy description. Some of the dark mesons carry electric charge, and so the dark mesons remain in thermal equilibrium with the Standard Model quarks, leptons, and gauge fields. Since the dark baryons are strongly coupled to the dark mesons, they also are kept in thermal equilibrium. As the temperature of the universe falls well below the mass of the dark baryons, they annihilate into dark mesons that subsequently thermalize and decay (or decay then thermalize) into Standard Model particles. The symmetric abundance of dark baryons is therefore determined by the annihilation rate of dark baryons into dark mesons.
The annihilation of dark baryons to dark mesons is a strongly coupled process. We expect $B^* B \rightarrow \Pi \, \Pi$, $B^* B \rightarrow 3 \, \Pi$, and $B^* B \rightarrow 4 \, \Pi$, (and to possibly more mesons if kinematically allowed) to occur, but we do not know the dominant annihilation channel. If the 2-to-2 process $B^* B \rightarrow \Pi \, \Pi$ dominates, one approach is to use partial wave unitarity to estimate the thermally averaged annihilation rate [@Griest:1989wd; @Blum:2014dca], $$\begin{aligned}
\langle \sigma v \rangle
&\sim& \frac{4 \pi \langle v^{-1} \rangle}{m_B^2} \, ,\end{aligned}$$ where $\langle v^{-1} \rangle \simeq 2.5$ at freezeout [@Blum:2014dca]. Matching this cross section to the required thermal relic abundance yields $m_B \sim 100$ TeV. An alternative approach is to use naive dimensional analysis [@Manohar:1983md; @Luty:1997fk; @Cohen:1997rt], which appears to lead to a larger dark matter mass.
If the 2-to-3 or 2-to-4 processes dominate instead, the additional phase space and kinematic suppression lowers the annihilation rate, and therefore lowers the scalar baryon mass needed to obtain the cosmological abundance. For recent work that has considered the thermal relic abundance in multibody processes, see [@Hochberg:2014dra; @Hochberg:2014kqa]. Suffice it to say a symmetric thermal abundance of dark baryons will match the cosmological abundance for a relatively large baryon mass that is of order tens to hundreds of TeV.
Asymmetric Abundance
--------------------
Early work on technibaryons demonstrated that strongly-coupled dark matter could arise from an asymmetric abundance [@Nussinov:1985xr; @Chivukula:1989qb; @Barr:1990ca; @Barr:1991qn; @Kaplan:1991ah]. The main ingredient to obtain the correct cosmological abundance involved the electroweak sphaleron – the non-perturbative solution at finite temperature that allows for transitions between vacua with different[^6] $B+L$ numbers.[^7] In the early universe, at temperatures much larger than the electroweak scale, electroweak sphalerons are expected to violate one accidental global symmetry, $B+L+D$ number, leaving $B-L$ and $B-D$ numbers unaffected [@Barr:1991qn; @Kaplan:1991ah; @Kribs:2009fy]. Here $D$ number is proportional to the dark baryon number, with some appropriate normalization (for examples, see [@Kaplan:1991ah; @Kribs:2009fy]).
Given a baryogenesis mechanism, the electroweak sphalerons redistribute baryon number into lepton number and dark baryon number. As the universe cools, the mass of the technibaryon becomes larger than the temperature of the Universe. Eventually, the universe cools to the point where electroweak sphalerons “freeze out” and can no longer continue exchanging $B$, $D$, and $L$ numbers. The residual abundance of dark baryons is $\rho \sim m_B n_B$ where the number density is proportional to $\exp[-m_{\rm B}/T_{\rm sph}]$, where $T_{\rm sph}$ is the temperature at which sphaleron interactions shut off.
If the baryon and dark baryon number densities are comparable, the would-be overabundance of dark matter (from $m_B \gg m_{\rm nucleon}$) is compensated by the Boltzmann suppression. Very roughly, $m_B \sim 1$-$2$ TeV is the natural mass scale that matches the cosmological abundance of dark matter [@Barr:1990ca]. A crucial component of the early technibaryon papers [@Nussinov:1985xr; @Chivukula:1989qb; @Barr:1990ca] is that the technifermions were in a purely chiral representation of the electroweak group, like the fermions of the Standard Model.
In stealth dark matter, given an early baryogenesis mechanism (or other analogous mechanism to generate an asymmetry in a globally conserved quantity [@Barr:1991qn; @Shelton:2010ta; @Davoudiasl:2010am; @Haba:2010bm; @Buckley:2010ui; @McDonald:2011zza; @Blennow:2010qp; @Falkowski:2011xh]), it is possible that electroweak sphalerons could also lead to the correct relic abundance of dark baryons consistent with cosmology.
There is one critical difference from the early technicolor models (as well as the quirky dark matter model): The dark fermions in stealth dark matter have both vector-like and electroweak symmetry breaking masses. This leads to a suppression of the effectiveness of the electroweak sphalerons by a factor of $\alpha$, c.f. Eq. (\[eq:alpha\]), leading to a somewhat *smaller* stealth baryon mass to obtain the correct relic abundance compared with a technicolor model (all other parameters equal). A more quantitative estimate is complicated by several factors:
- Determining how the electroweak sphaleron redistributes the conserved global charges in the presence of fermions that acquire both electroweak preserving and electroweak breaking masses. To the best of our knowledge, this calculation has never been done.
- Determining the precise temperature at which electroweak sphalerons shut off, in the presence of both the Standard Model and stealth dark matter degrees of freedom contributing to the thermal bath.
- The baryogenesis mechanism itself, that determines the initial $B-L$ and $B-D$ numbers.
Given the exponential suppression of the asymmetric abundance as the dark baryon mass is increased, it is clear that the upper bound on the dark baryon mass is nearly the same as the technibaryon calculation (updated to the current cosmological parameters), when stealth dark fermions have vector-like masses comparable to electroweak symmetry breaking masses. (This case is, however, constrained by the $S$ parameter, see Sec. \[sec:pew\]). We can therefore anticipate that a range of stealth dark matter masses will be viable, up to about a TeV. More precise predictions require further detailed investigation that is beyond the scope of this paper.
Discussion {#sec:discussion}
==========
We have presented a concrete model, “stealth dark matter", that is a composite baryonic scalar of a new $SU(N_D)$ strongly-coupled confining gauge theory with dark fermions transforming under the electroweak group. Though the stealth dark matter model has a wide parameter space, we focused on dark fermion masses that respect an exact custodial $SU(2)$. Custodial $SU(2)$ implies the lightest bosonic baryonic composite is an electrically neutral scalar (and not a vector or spin-2) of the $SU(N_D)$ dark spectrum, and in addition does not have a charge radius. This yields an exceptionally “stealthy” dark matter candidate, with spin-independent direct detection scattering proceeding only through Higgs exchange (studied in this paper) and the polarizability interaction (studied in our companion paper [@Appelquist:2015zfa]). Custodial $SU(2)$ also allows for stealth dark matter to completely avoid the constraints from the $T$ parameter. While contributions to the $S$ parameter are present, they are suppressed by the ratio of the electroweak symmetry breaking mass-squared divided by a vector-like mass squared of the dark fermions. We also verified the lightest non-singlet mesons decay rapidly (so long as $\epsilon_y \not= 0$), avoiding any cosmological issues with stable electrically-charged dark mesons.
Specializing to the case of $N_D = 4$, we then applied our earlier model-independent lattice results [@Appelquist:2014jch] to the parameters of stealth dark matter, and obtained constraints on the effective Higgs interaction. We find that the present LUX bound is able only to mildly constrain the Higgs coupling to stealth dark matter for relatively light dark baryons. Even weaker constraints arise when the effective Higgs interaction is quadratic in the Yukawa coupling, which is a natural possibility when the two pairs of dark fermions are split dominantly by vector-like masses, i.e., $y v \ll \Delta$.
While we have considered many aspects of stealth dark matter, several avenues warrant further investigation:
- Chiral symmetry forbids additive renormalization of the fermion masses; we have focused on the regime where the constituent fermion mass is comparable to the confinement scale $M_f \sim \Lambda_D$, since this is best-suited for lattice simulations, exactly where analytic estimates are least useful. It would be interesting to consider a broader range of fermion masses relative the confinement scale, to understand the relative scaling of the Higgs interactions.
- A more precise calculation of the $S$ parameter is possible using lattice simulations for the relevant correlators. This would allow us to place numerical bounds on the parameters of the theory, that could be stronger than the bounds from the non-observation through direct detection.
- We would like to unpack $y_{\rm eff}$ \[c.f. Eq. (\[eq:yeff\])\] and obtain constraints on the Yukawa couplings of the model. However, this requires translating the fermion masses from the lattice regularization into a continuum regularization.
- Dark meson production and decay at the LHC is ripe for exploration. Dark meson pair production would proceed through off-shell EW gauge bosons, $q \bar{q} {\rightarrow}\Pi^+ \Pi^-$, $q \bar{q} {\rightarrow}\Pi^0 \Pi^0$, and $q \bar{q}' {\rightarrow}\Pi^\pm \Pi^0$. These could have spectacular signals at the LHC. Neutral mesons decay into fermion pairs and dibosons (explored in other related models in [@Kilic:2009mi; @Kilic:2010et; @Fok:2011yc; @Buckley:2012ky]). For charged dark mesons, with masses in the range $m_{\Pi^\pm} \sim 90-180$ GeV, the decay $\Pi^+ {\rightarrow}\tau^+ \nu_\tau$ dominates, while for masses above this, $\Pi^+ {\rightarrow}t\bar{b}$ is dominant. Charged pion pair production could therefore lead to $t \bar{b} b \bar{t}$ signals with the $t \bar{b}$ and $b \bar{t}$ pairs reconstructing to the same mass. To the best of our knowledge, this type of resonance search is not being performed at the LHC.
- More insight into the thermal abundance of stealth dark matter, perhaps using lattice simulations, would help narrow the interesting mass range that matches cosmological data.
- Asymmetric production of stealth dark matter seems very promising, but has several calculational obstacles to overcome to arrive at a quantitative relationship between the abundance and the other parameters of the theory.
- If stealth dark matter has an asymmetric abundance, there are potential limits from neutron star lifetimes [@McDermott:2011jp; @Bramante:2013hn; @Bertoni:2013bsa] though the precise bounds depend sensitively on the equation of state of the neutron stars.
- There are tantalizing signals of a $\gamma$-ray excess between about $1$-$10$ GeV in the galactic center (see for example [@Hooper:2010mq; @Hooper:2011ti; @Abazajian:2012pn; @Bringmann:2012ez; @Daylan:2014rsa]). A recent analysis [@Agrawal:2014oha] suggests that this could arise from dark matter up to $300$ GeV. It is intriguing to consider the $\gamma$-ray signal spectrum that could arise from a symmetric abundance of stealth dark matter with annihilation into a multibody final state [@Elor:2015tva] with mixtures of four or more heavy fermions and multi-gauge bosons (from $B B^* {\rightarrow}\Pi \, \Pi \, \ldots {\rightarrow}$ SM states).
Finally there are broader model-building questions to consider. One is the choice of scales $M_f \sim \Lambda_D$ that has been the focus of this work. This could arise dynamically. For example, if there are sufficient flavors in the $SU(N_D)$ gauge theory such that it is approximately conformal at high energies, then as the theory is run down through the dark fermion mass scale $M_f$, the dark fermions integrate out, and confinement sets in at $\Lambda_D \sim M_f$. This is well known to occur for supersymmetric $SU(N)$ theories in the conformal window that flow to confining theories once the number of flavors drops below $N_f < 3 N/2$ [@Seiberg:1994pq]. The origin of the vector-like masses of the fermions is also an interesting model-building puzzle. However, just as SM fermion masses are vector-like below the electroweak breaking scale, we can imagine dark fermion vector-like masses could be revealed as arising from dynamics that breaks the flavor symmetries of our dark fermions at some higher scale.
Acknowledgments
===============
We thank S. Chang, O. DeWolfe, and D. B. Kaplan for many valuable discussions during the course of this work.
We thank the Lawrence Livermore National Laboratory (LLNL) Multiprogrammatic and Institutional Computing program for Grand Challenge allocations and time on the LLNL BlueGene/Q (rzuseq and vulcan) supercomputer. We thank LLNL for funding from LDRD 13-ERD-023 “Illuminating the Dark Universe with PetaFlops Supercomputing”. Computing support for this work comes from the LLNL Institutional Computing Grand Challenge program.
This work has been supported by the U. S. Department of Energy under Grant Nos. DE-SC0008669 and DE-SC0009998 (D.S.), DE-SC0010025 (R.C.B., C.R., E.W.), DE-FG02-92ER40704 (T.A.), DE-SC0011640 (G.D.K.), DE-FG02-00ER41132 (M.I.B.), and Contracts DE-AC52-07NA27344 (LLNL), DE-AC02- 06CH11357 (Argonne Leadership Computing Facility), and by the National Science Foundation under Grant Nos. NSF PHY11-00905 (G.F.), OCI-0749300 (O.W.). Brookhaven National Laboratory is supported by the U. S. Department of Energy under contract DE-SC0012704. S.N.S was supported by the Office of Nuclear Physics in the U. S. Department of Energy’s Office of Science under Contract DE-AC02-05CH11231.
**Weak Currents** {#sec:weakcurrents}
=================
We examine the dark fermion contributions to the electroweak currents. In the gauge eigenstate basis, the currents are $$\begin{aligned}
j^\mu_{+} &=& -\frac{1}{\sqrt{2}}
\left( {F_1^u}^\dagger \bar{\sigma}^\mu F_1^d
+ {F_2^u}^\dagger \bar{\sigma}^\mu F_2^d \right) \\
j^\mu_{-} &=& -\frac{1}{\sqrt{2}}
\left( {F_1^d}^\dagger \bar{\sigma}^\mu F_1^u
+ {F_2^d}^\dagger \bar{\sigma}^\mu F_2^u \right) \\
j^\mu_{3} &=& -\frac{i}{2}
\sum_{i=1,2} \left( {F_i^u}^\dagger \bar{\sigma}^\mu F_i^u
- {F_i^d}^\dagger \bar{\sigma}^\mu F_i^d \right) \\
j^\mu_{Y} &=& -\frac{i}{2}
\sum_{i=3,4} \left( {F_i^u}^\dagger \bar{\sigma}^\mu F_i^u
- {F_i^d}^\dagger \bar{\sigma}^\mu F_i^d \right) \, .\end{aligned}$$ In the mass eigenstate basis given by Eqs. (\[eq:vec1\])-(\[eq:vec4\]), the currents can be rewritten in terms of the 4-component Dirac fermions defined by Eqs. (\[eq:diracup\]),(\[eq:diracdown\]). After some algebra, one obtains
$$\begin{aligned}
\label{eq:currents}
j^\mu_{+} &=& -\frac{1}{\sqrt{2}}
\Big[
\overline{\Psi_1^u} \gamma^\mu
\left( c_1^u c_1^d P_L + c_2^u c_2^d P_R \right) \Psi_1^d +
\overline{\Psi_2^u} \gamma^\mu
\left( s_1^u s_1^d P_L + s_2^u s_2^d P_R \right) \Psi_2^d
\nonumber \\ & &{} \qquad\quad +
\overline{\Psi_1^u} \gamma^\mu
\left( c_1^u s_1^d P_L + c_2^u s_2^d P_R \right) \Psi_2^d +
\overline{\Psi_2^u} \gamma^\mu
\left( s_1^u c_1^d P_L + s_2^u c_2^d P_R \right) \Psi_1^d
\Big] \label{eq:jplus} \\
j^\mu_{-} &=& -\frac{1}{\sqrt{2}}
\Big[
\overline{\Psi_1^d} \gamma^\mu
\left( c_1^d c_1^u P_L + c_2^d c_2^u P_R \right) \Psi_1^u +
\overline{\Psi_2^d} \gamma^\mu
\left( s_1^d s_1^u P_L + s_2^d s_2^u P_R \right) \Psi_2^u
\nonumber \\ & &{} \qquad\quad +
\overline{\Psi_1^d} \gamma^\mu
\left( c_1^d s_1^u P_L + c_2^d s_2^u P_R \right) \Psi_2^u +
\overline{\Psi_2^d} \gamma^\mu
\left( s_1^d c_1^u P_L + s_2^d c_2^u P_R \right) \Psi_1^u
\Big] \label{eq:jminus} \\
j^\mu_{3} &=& \frac{1}{2}
\Big[
\overline{\Psi_1^u} \gamma^\mu
\left( (c_1^u)^2 P_L + (c_2^u)^2 P_R \right) \Psi_1^u +
\overline{\Psi_2^u} \gamma^\mu
\left( (s_1^u)^2 P_L + (s_2^u)^2 P_R \right) \Psi_2^u
\nonumber \\ & &{} \qquad -
\overline{\Psi_1^d} \gamma^\mu
\left( (c_1^d)^2 P_L + (c_2^d)^2 P_R \right) \Psi_1^d -
\overline{\Psi_2^d} \gamma^\mu
\left( (s_1^d)^2 P_L + (s_2^d)^2 P_R \right) \Psi_2^d
\nonumber \\ & &{} \qquad +
\overline{\Psi_1^u} \gamma^\mu
\left( c_1^u s_1^u P_L + c_2^u s_2^u P_R \right) \Psi_2^u +
\overline{\Psi_2^u} \gamma^\mu
\left( s_1^u c_1^u P_L + s_2^u c_2^u P_R \right) \Psi_1^u
\nonumber \\ & &{} \qquad -
\overline{\Psi_1^d} \gamma^\mu
\left( c_1^d s_1^d P_L + c_2^d s_2^d P_R \right) \Psi_2^d -
\overline{\Psi_2^d} \gamma^\mu
\left( s_1^d c_1^d P_L + s_2^d c_2^d P_R \right) \Psi_1^d
\Big] \label{eq:j3} \\
j^\mu_{Y} &=& \frac{1}{2}
\Big[
\overline{\Psi_1^u} \gamma^\mu
\left( (s_1^u)^2 P_L + (s_2^u)^2 P_R \right) \Psi_1^u +
\overline{\Psi_2^u} \gamma^\mu
\left( (c_1^u)^2 P_L + (c_2^u)^2 P_R \right) \Psi_2^u
\nonumber \\ & &{} \qquad -
\overline{\Psi_1^d} \gamma^\mu
\left( (s_1^d)^2 P_L + (s_2^d)^2 P_R \right) \Psi_1^d -
\overline{\Psi_2^d} \gamma^\mu
\left( (c_1^d)^2 P_L + (c_2^d)^2 P_R \right) \Psi_2^d
\nonumber \\ & &{} \qquad -
\overline{\Psi_1^u} \gamma^\mu
\left( c_1^u s_1^u P_L + c_2^u s_2^u P_R \right) \Psi_2^u -
\overline{\Psi_2^u} \gamma^\mu
\left( s_1^u c_1^u P_L + s_2^u c_2^u P_R \right) \Psi_1^u
\nonumber \\ & &{} \qquad +
\overline{\Psi_1^d} \gamma^\mu
\left( c_1^d s_1^d P_L + c_2^d s_2^d P_R \right) \Psi_2^d +
\overline{\Psi_2^d} \gamma^\mu
\left( s_1^d c_1^d P_L + s_2^d c_2^d P_R \right) \Psi_1^d
\Big] \, , \label{eq:jY} \end{aligned}$$
where $c_i^j \equiv \cos\theta_i^j$, $s_i^j \equiv \sin\theta_i^j$ and $P_{L,R} = (1 \mp \gamma_5)/2$ are the left- and right-handed projectors. In general, the dark fermions contribute to both the vector and axial currents with strengths given by the mixing angles. It is easy to verify that the electromagnetic current, $$\begin{aligned}
j_{\rm em}^\mu &=& j_{3}^\mu + j_{Y}^\mu \nonumber \\
&=& \sum_{i=1,2} \Big[
Q_u \overline{\Psi^u_i} \gamma^\mu \Psi^u_i
+ Q_d \overline{\Psi^d_i} \gamma^\mu \Psi^d_i
\Big] \, ,\end{aligned}$$ with $Q_{u,d} = \pm 1/2$, is consistent with a pure vector coupling of the dark fermions to the photon independent of mass mixing angles.
Interestingly, if the mass matrices Eqs.(\[eq:upmassmatrix\]),(\[eq:downmassmatrix\]) are symmetric, i.e., $y_{14}^u = y_{23}^u$ and $y_{14}^d = y_{23}^d$, then just two mixing angles are required, i.e., $\theta_1^u = \theta_2^u$ and $\theta_1^d = \theta_2^d$. In this case, the mixing angles factor out of the left-right gamma matrix structure, leaving all of the electroweak currents to be purely vector (with vanishing axial current). This is unlike the Standard Model, where the $SU(2)_L$ currents are purely $V-A$. The difference between this model and the Standard Model is the structure of the dark fermion mass matrices that include both vector-like and electroweak symmetry breaking masses.
It is also interesting to calculate the neutral current $$\begin{aligned}
j_{Z}^\mu &=& j_{3}^\mu - \sin^2\theta_W j_{\rm em}^\mu \, .\end{aligned}$$ For the neutral baryon state, $$\begin{aligned}
\langle B | j_Z^\mu | B \rangle &\simeq& \\
& &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+ \frac{1}{4} \left( (c_1^u)^2 + (c_2^u)^2 - (c_1^d)^2 - (c_2^d)^2 \right)
\langle B | \overline{\Psi_1} \gamma^\mu \Psi_1 | B \rangle \nonumber \\
& &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+ \frac{1}{4} \left( - (c_1^u)^2 + (c_2^u)^2 + (c_1^d)^2 - (c_2^d)^2 \right)
\langle B | \overline{\Psi_1} \gamma^\mu \gamma^5 \Psi_1 | B \rangle \, .
\nonumber \end{aligned}$$ In the limit of zero momentum exchange ($Q^2 = 0$), the vector form factor $\langle B | \overline{\Psi_1} \gamma^\mu \Psi_1 | B \rangle$ evaluates to $1$, while the axial-vector form factor $\langle B | \overline{\Psi_1} \gamma^\mu \gamma^5 \Psi_1 | B \rangle$ for a scalar baryon vanishes. In the presence of an exact custodial SU(2) symmetry, which is the focus of this paper, we have $c_i^u = c_i^d$ and the $Z$ coupling vanishes identically at any momentum exchange.
On the other hand, if custodial symmetry is broken, then the lightest neutral baryon acquires tree-level couplings to the $Z$ boson. To illustrate the size of these couplings, consider taking the dark fermion mass matrices to be exactly symmetric ($y_{23} = y_{14}$) but allowing for a small, custodial symmetry-violating difference in the Yukawas, $y_u = y + \xi$ and $y_d = y - \xi$ where $\xi/y \ll 1$. The coefficient of the weak neutral vector current becomes
$$\begin{aligned}
(c_1^u)^2 + (c_2^u)^2 - (c_1^d)^2 - (c_2^d)^2 &\simeq&
\begin{dcases*}
2 \sqrt{2} \frac{\xi}{y} \frac{\Delta}{y v} & \quad \mbox{Linear Case} \\
\frac{\xi}{y} \frac{(y v)^2}{\Delta^2} & \quad \mbox{Quadratic Case.}
\end{dcases*} \end{aligned}$$
Custodial symmetry violation is therefore restricted by requiring the coupling of the lightest neutral baryon to the $Z$ boson be small enough to have evaded direct detection. There are several limits in which this can occur: $\xi/y \ll 1$ (any scenario), $\Delta/(y v) \ll 1$ (Linear Case), or $(y v)/\Delta \ll 1$ (Quadratic Case). This suggests that modest custodial symmetry violation is possible but rather constrained.
[^1]: Fermionic baryons arising from odd $N_D$ were considered in Ref. [@Appelquist:2013ms] where the limit $M {\mathrel{\raisebox{-.6ex}{$\stackrel{\textstyle>}{\sim}$}}}10$ TeV was found to avoid the direct detection constraints from the magnetic dipole interaction.
[^2]: Other portals, such as a dark gauged U(1) group that kinetically mixes with hypercharge, are neither present nor required here.
[^3]: The obvious exception, when $N_D = N_c = 3$, is discussed in [@Bai:2013xga; @Appelquist:2013ms], which is not a focus for us due to the baryons being fermions. Construction of QCD-singlet dark baryons with $N_D=6,12,18,...$ may be possible, but we do not study this possibility further here.
[^4]: We assume $\Delta < M$, such that fermion masses remain positive, to avoid further fermion field rephasings.
[^5]: We use the term “lightest mesons” and not “pions” since the would-be global symmetry that protects pion masses is completely broken by the dark fermion vector-like masses. Nevertheless, we use the symbol $\Pi$ to denote the corresponding fields.
[^6]: In this section, $B$ refers to baryon number and is to not be confused with the field defined earlier
[^7]: In addition, an asymmetric abundance could be generated through other mechanisms, see Ref. [@Kaplan:2009ag], in which case the mass scales and parameters depend on the details of the particular mechanism.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Ryan Wails
- Yixin Sun
- Aaron Johnson
- Mung Chiang
- Prateek Mittal
bibliography:
- 'references.bib'
title: '[Tempest]{}: Temporal Dynamics in Anonymity Systems'
---
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional initial space or, more generally, for coefficients satisfying a finite localisability condition. Necessary and sufficient conditions are obtained for a conjugate pair of quantum stochastic cocycles on a finite-dimensional operator space to strongly satisfy such quantum stochastic differential equations. This gives an alternative approach to quantum stochastic convolution cocycles on a coalgebra.'
address:
- 'Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF'
- 'School of Mathematical Sciences, University of Nottingham NG7 2RD'
author:
- 'J. Martin Lindsay'
- 'Adam G. Skalski'
title: On quantum stochastic differential equations
---
[^1]
Introduction {#introduction .unnumbered}
============
The investigation of quantum stochastic differential equations (QSDE) for processes acting on symmetric Fock spaces dates back to Hudson and Parthasarathy’s founding paper of quantum stochastic calculus ([@RobinPartha]). As usual in stochastic analysis, these equations are understood as integral equations. By a weak solution is meant a process, consisting of operators (or mappings), whose matrix elements satisfy certain ordinary integral equations. Quantum stochastic analysis also harbours a notion of strong solution. The first existence and uniqueness theorems ([@RobinPartha]) dealt with the constant-coefficient operator QSDE with finite-dimensional noise space; these were soon extended to the mapping QSDE by Evans and Hudson ([@Evans]). Further extensions to the case of infinite-dimensional noise were obtained in [@RobinPartha2], [@MohariSinha] and [@Fagnola], and clarified in [@Meyer] and [@LWptrf]. Solutions of such QSDE’s yield quantum stochastic, or Markovian, cocycles ([@Accardi]). The converse is also true under various hypotheses ([@HuL], [@Bradshaw]); in [@LWjfa] it was proved that any sufficiently regular cocycle on a $C^*$-algebra satisfies some QSDE weakly, and moreover if the cocycle is also completely positive and contractive, then it satisfies the equation strongly. In [@LWblms] complete boundedness of the ‘columns’ of the coefficient was identified as a sufficient condition for the solution to be strong. (When the noise dimension space is finite dimensional boundedness suffices.) In all the above cases the initial condition for the QSDE was given by an identity map ampliated to the Fock space.
Parallel to the theory of quantum stochastic cocycles, Schürmann developed a theory of quantum Lévy processes on quantum groups, or more generally $*$-bialgebras, (see [@schu] and references therein). He showed that each quantum Lévy process satisfies a QSDE of a certain type, with initial condition given by the counit of the underlying $*$-bialgebra (see below). The notion of quantum Lévy process was recently generalised to quantum stochastic *convolution* cocycle on a coalgebra in [@LSaihp] where it was shown that such objects arise as solutions of coalgebraic quantum stochastic differential equations. Extension of the results of that paper to the context of compact quantum groups, or more generally operator space coalgebras ([@LSqscc2]), was our motivation for analysing quantum stochastic differential equations on an operator space with nontrivial initial conditions. Results obtained here have also enabled the development of a dilation theory for completely positive convolution cocycles on a $C^*$-bialgebra ([@Sdilations]).
The aim of this paper is to provide existence and uniqueness results for a class of quantum stochastic differential equations, under natural conditions, together with cocycle characterisation of solutions, The crucial role played by complete boundedness ([@LWblms]) suggests that the main object for consideration as initial space should be an operator space. In general operator space theory is very useful for describing properties of coefficients, initial conditions and solutions of our equations (cf. [@LWCBonOS]). The main existence theorem is proved for coefficients with ${\mathsf{k}}$-bounded columns and initial condition given by a ${\mathsf{k}}$-bounded map, where ${\mathsf{k}}$ is the ‘noise dimension space’. (The term ${\mathsf{k}}$-*bounded* means simply bounded if ${\mathsf{k}}$ is finite dimensional and completely bounded otherwise). Solutions are expressed in terms of iterated quantum stochastic integrals (cf. [@LWjlms]) and have ${\mathsf{k}}$-bounded columns themselves (completely bounded columns if the coefficient has cb-columns and the initial condition is completely bounded). Due to our choice of test vectors (exponentials of step-functions with values in a given dense subspace of the noise dimension space) the results are explicitly basis-independent. As solutions of equations of the type considered are quantum stochastic cocycles, one may ask which cocycles satisfy a QSDE. Sufficient conditions for the cocycle to satisfy a QSDE weakly, established for the case of $C^*$-algebras in [@LWjfa], remain valid in the coordinate-free, operator space context of this paper. A new result here, informed by our recent theorem on convolution cocycles ([@LSaihp]), is the characterisation of cocycles on finite dimensional operator spaces which, together with a conjugate process, satisfy a QSDE strongly — namely, they are the locally Hölder-continuous processes with exponent $1/2$ whose conjugate process enjoys the same continuity.
The plan of the paper is as follows. In Section \[preliminaries sec\] the notation is established and basic operator-space theoretic and quantum stochastic notions are introduced. There also a concept of finite localisability is discussed. Weak regularity is shown to be sufficient for uniqueness of weak solutions in Section \[regularity sec\] (cf. [@LWptrf]). Section \[existence sec\] contains the main result on the existence of strong solutions of equations on operator spaces and elucidates their dependence on initial conditions. Although in the case of (algebraic) quantum Lévy processes the initial object is a vector space $V$, rather than an operator space, the Fundamental Theorem on Coalgebras allows us to effectively work with finite-dimensional subspaces and thereby to circumvent the lack of analytic structure on $V$ (cf. [@schu]). For this purpose, the version of the existence theorem for finitely localisable maps relevant for coalgebraic quantum stochastic differential equations is given in Section \[localisable sec\]. Section \[cocycles sec\] begins by recalling known facts on relations between quantum stochastic cocycles and quantum stochastic differential equations whose initial condition is given by the identity map on a (concrete) operator space. It then gives new necessary and sufficient conditions for a conjugate pair of cocycles on a finite-dimensional operator space to satisfy a QSDE strongly and ends with an application of this result to the infinitesimal generation of quantum stochastic convolution cocycles.
Notation
--------
For dense subspaces $E$ and $E'$ of Hilbert spaces ${\mathsf{H}}$ and ${\mathsf{H}}'$, ${\mathcal{O}}(E;{\mathsf{H}}')$ denotes the space of operators ${\mathsf{H}}\to{\mathsf{H}}'$ with domain $E$ and ${\mathcal{O}^\ddagger}(E,E'):= \{ T\in{\mathcal{O}}(E;{\mathsf{H}}'): \operatorname{Dom}T^*\supset E' \}$. Thus ${\mathcal{O}^\ddagger}(E',E)$ is the conjugate space of ${\mathcal{O}^\ddagger}(E,E')$ with conjugation $T\mapsto T^{\dagger}:=T^*|_{E'}$. When ${\mathsf{H}}'={\mathsf{H}}$ we write ${\mathcal{O}}(E)$ for ${\mathcal{O}}(E;{\mathsf{H}})$. We view $B({\mathsf{H}};{\mathsf{H}}')$ as a subspace of ${\mathcal{O}^\ddagger}(E,E')$ (via restriction/continuous linear extension). For vectors $\zeta\in E$ and $\zeta'\in {\mathsf{H}}'$, $\omega_{\zeta',\zeta}$ denotes the linear functional on ${\mathcal{O}}(E;{\mathsf{H}}')$ given by $T\mapsto {\langle}\zeta', T\zeta{\rangle}$, extending a standard notation. We also use the Dirac-inspired notations $|E{\rangle}:= \{|\zeta{\rangle}: \zeta \in E\}$ and ${\langle}E| := \{{\langle}\zeta | : \zeta \in E\}$ where $|\zeta{\rangle}\in |{\mathsf{h}}{\rangle}:= B({\mathbb{C}};{\mathsf{h}})$ and ${\langle}\zeta |\in {\langle}{\mathsf{h}}| := B({\mathsf{h}};{\mathbb{C}})$ are defined by $\lambda \mapsto \lambda\zeta$ and $\zeta'\mapsto{\langle}\zeta, \zeta'{\rangle}$ respectively — inner products (and all sesquilinear maps) here being linear in their *second* argument.
Tensor products of vector spaces, such as dense subspaces of Hilbert spaces, are denoted by ${\odot}$; minimal/spatial tensor products of operator spaces by ${{\otimes}_{\text{{\textup}{sp}}}}$; and ultraweak tensor products of ultraweakly closed spaces of bounded operators by ${{\overline{\otimes}}}$. The symbol ${\otimes}$ is used for Hilbert space tensor products and tensor products of completely bounded maps between operator spaces; the symbol ${\odot}$ is also used for the tensor product of unbounded operators, thus if $S\in{\mathcal{O}}(E;{\mathsf{H}}')$ and $T\in{\mathcal{O}}(F;{\mathsf{K}}')$ then $S{\odot}T\in{\mathcal{O}}(E{\odot}F;{\mathsf{H}}'{\otimes}{\mathsf{K}}')$. We also need ampliations of bra’s and kets: for $\zeta\in{\mathsf{h}}$ define $$\label{E-notation}
E^{\zeta}:= I_{{\mathsf{H}}}{\otimes}{\langle}\zeta | \in B({\mathsf{H}}{\otimes}{\mathsf{h}};{\mathsf{H}}) \text{ and }
E_{\zeta}:= I_{{\mathsf{H}}}{\otimes}|\zeta{\rangle}\in B({\mathsf{H}};{\mathsf{H}}{\otimes}{\mathsf{h}}),$$ where the Hilbert space ${\mathsf{H}}$ is determined by context.
For a vector-valued function $f$ on ${{\mathbb{R}}_+}$ and subinterval $I$ of ${{\mathbb{R}}_+}$ $f_I$ denotes the function on ${{\mathbb{R}}_+}$ which agrees with $f$ on $I$ and vanishes outside $I$. Similarly, for a vector $\xi$, $\xi_I$ is defined by viewing $\xi$ as a constant function. This extends the standard indicator function notation. The symmetric measure space over the Lebesgue measure space ${{\mathbb{R}}_+}$ ([@Guichardet]) is denoted $\Gamma$, with integration denoted $\int_{\Gamma} \cdots d\sigma$, thus $\Gamma = \{\sigma\subset{{\mathbb{R}}_+}: \# \sigma < \infty\} =
\bigcup_{n\geq 0} \Gamma^n$ where $\Gamma^n = \{\sigma\subset{{\mathbb{R}}_+}: \# \sigma =n \}$ and $\emptyset$ is an atom having unit measure. If ${{\mathbb{R}}_+}$ is replaced by a subinterval $I$ then we write $\Gamma_I$ and $\Gamma^n_I$, thus the measure of $\Gamma^n_I$ is $|I|^n/n!$ where $|I|$ is the Lebesgue measure of $I$. Finally, we write $X \subset \subset Y$ to mean that $X$ is a finite subset of $Y$.
Preliminaries {#preliminaries sec}
=============
Quantum stochastics ([@Partha][,]{} [@Meyer][; we follow]{} [@Llnm])
---------------------------------------------------------------------
*Fix now, and for the rest of the paper*, a complex Hilbert space ${\mathsf{k}}$ which we refer to as the noise dimension space, and let ${{\widehat}{{\mathsf{k}}}}$ denote the orthogonal sum ${\mathbb{C}}\oplus{\mathsf{k}}$. Whenever $c\in {\mathsf{k}}$, ${{\widehat}{c}}:=\binom{1}{c}\in {{\widehat}{{\mathsf{k}}}}$; for $E\subset{\mathsf{k}}$, $\widehat{E}:= \operatorname{Lin}\{{{\widehat}{c}}:c \in E\}$ and when $g$ is a function with values in ${\mathsf{k}}$, ${{\widehat}{g}}$ denotes the corresponding function with values in ${{\widehat}{{\mathsf{k}}}}$ defined by ${{\widehat}{g}}(s):= \widehat{g(s)}$. Let ${\mathcal{F}}$ denote the symmetric Fock space over $L^2({{\mathbb{R}}_+};{\mathsf{k}})$. For any dense subspace $D$ of ${\mathsf{k}}$ let ${\mathbb{S}}_D$ denote the linear span of $\{d_{[0,t[}: d\in D, t\in{{\mathbb{R}}_+}\}$ in $L^2({{\mathbb{R}}_+};{\mathsf{k}})$ (we always take these right-continuous versions) and let ${\mathcal{E}_D}$ denote the linear span of $\{{\varepsilon}(g): g\in{\mathbb{S}}_D\}$ in ${\mathcal{F}}$, where ${\varepsilon}(g)$ denotes the exponential vector $\big((n!)^{-\frac{1}{2}}g^{{\otimes}n}\big)_{n\geq 0}$. The subscript $D$ is dropped when $D={\mathsf{k}}$. An *exponential domain* is a dense subspace of ${\mathfrak{h}}{\otimes}{\mathcal{F}}$, for a Hilbert space ${\mathfrak{h}}$, of the form ${\mathfrak{D}}{\odot}{\mathcal{E}_D}$. We usually drop the tensor symbol and denote simple tensors such as $v{\otimes}{\varepsilon}(f)$ by $v{\varepsilon}(f)$.
For an exponential domain ${\mathcal{D}}=
{\mathfrak{D}}{\odot}{\mathcal{E}_D}\subset{\mathfrak{h}}{\otimes}{\mathcal{F}}$ and Hilbert space ${\mathfrak{h}}'$, ${\mathbb{P}}({\mathcal{D}};{\mathfrak{h}}{\otimes}{\mathcal{F}})$ denotes the space of (equivalence classes of) weakly measurable and adapted functions $X: {{\mathbb{R}}_+}\to {\mathcal{O}}({\mathcal{D}};{\mathfrak{h}}'{\otimes}{\mathcal{F}})$: $$t\mapsto {\langle}\xi',X_t\xi{\rangle}\text{ is measurable }
\quad
(\xi'\in{\mathfrak{h}}{\otimes}{\mathcal{F}}, \xi\in{\mathcal{D}});$$ $${\langle}u'{\varepsilon}(g'),X_t u{\varepsilon}(g){\rangle}=
{\langle}u'{\varepsilon}(g'_{[0,t[}),X_t u{\varepsilon}(g_{[0,t[}){\rangle}{\langle}u'{\varepsilon}(g'_{[t,\infty[}), u{\varepsilon}(g_{[t,\infty[}){\rangle}$$ ($u\in{\mathfrak{D}}, g\in{\mathbb{S}_D}, u'\in{\mathfrak{h}}', g'\in{\mathbb{S}}, t\in{{\mathbb{R}}_+}$), with processes $X$ and $X'$ being identified if, for all $\xi\in{\mathcal{D}}$, $X_t\xi = X'_t\xi$ for almost all $t\in{{\mathbb{R}}_+}$. If ${\mathcal{D}'}$ is an exponential domain in ${\mathfrak{h}}'{\otimes}{\mathcal{F}}$ then ${\mathbb{P}^\ddagger}({\mathcal{D}}, {\mathcal{D}'})$ denotes the space of ${\mathcal{O}^\ddagger}({\mathcal{D}}, {\mathcal{D}'})$-valued processes. Thus ${\mathbb{P}^\ddagger}({\mathcal{D}'}, {\mathcal{D}})$ is the conjugate space of ${\mathbb{P}^\ddagger}({\mathcal{D}}, {\mathcal{D}'})$ with conjugation defined pointwise: $X_t^{\dagger} = (X_t)^*|_{{\mathcal{D}'}}$.
Let $F\in{\mathbb{P}}({\mathfrak{D}}{\odot}{{\widehat}{D}}{\odot}{\mathcal{E}_D};{\mathfrak{h}}'{\otimes}{{\widehat}{{\mathsf{k}}}}{\otimes}{\mathcal{F}})$ be quantum stochastically integrable ([@Llnm]). Then the process $(X_t = \int_0^tF_s d\Lambda_s)_{t\geq0}\in
{\mathbb{P}}({\mathfrak{D}}{\odot}{\mathcal{E}_D};{\mathfrak{h}}'{\otimes}{\mathcal{F}})$ satisfies $$\label{FF1}
{\langle}v'{\varepsilon}(g'), X_t v{\varepsilon}(g){\rangle}=
\int_0^t ds\
{\langle}v' {\widehat}{g'}(s){\varepsilon}(g'), F_s v {\widehat}{g}(s){\varepsilon}(g){\rangle}$$ $$\label{FE}
\| X_t v{\varepsilon}(g)\|^2 \leq
C(g,t)^2 \int_0^t ds\ \|F_s v {\widehat}{g}(s){\varepsilon}(g)\|^2$$ ($v\in{\mathfrak{D}}, g\in{\mathbb{S}}, v'\in{\mathfrak{h}}', g'\in{\mathbb{S}}, t\in{{\mathbb{R}}_+}$) for a constant $C(g,t)$ which is independent of $F$ and $v$. These are known as the Fundamental Formula and Fundamental Estimate of quantum stochastic calculus. We also need basic estimates for sums of iterated integrals. Thus let $L=\big(L_n\in{\mathcal{O}}({\mathfrak{D}}{\odot}{{\widehat}{D}}^{{\odot}n};
{\mathfrak{h}}'{\otimes}{{\widehat}{{\mathsf{k}}}}^{{\otimes}n})\big)_{n\geq 0}$ satisfy the growth condition $$\forall_{\gamma\in{{\mathbb{R}}_+}}\forall_{v\in{\mathfrak{D}}}\forall_{F\subset\subset{{\widehat}{D}}}
\ \sum_{n\geq 0} \frac{\gamma^n}{\sqrt{n!}}
\max\{\|L_nv{\otimes}\zeta_1{\otimes}\cdots{\otimes}\zeta_n\|: \zeta_1,\ldots,\zeta_n\in F\}
< \infty.$$ Then the iterated quantum stochastic integrals of the $L_n$ sum to a process $\big(\Lambda(L)\big)_{t\geq 0}$ satisfying ( for all $v\in{\mathfrak{D}}, g\in{\mathbb{S}}, v'\in{\mathfrak{h}}', g'\in{\mathbb{S}}$) $$\label{FF1full}
{\langle}v'{\varepsilon}(g'), \Lambda_t(L) v{\varepsilon}(g){\rangle}=
e^{{\langle}g,g'{\rangle}}
\int_{\Gamma_{[0,t]}}d\sigma\
{\langle}v'\pi_{{\widehat}{g'}}(\sigma),
L_{\#\sigma} v\pi_{{\widehat}{g}}(\sigma){\rangle}$$
$$\label{FEfull}
\|\Lambda_t(L) v{\varepsilon}(g)\| \leq
\|{\varepsilon}(g)\|
\sum_{n\geq 0} C(g,T)^n
\Big\{
\int_{\Gamma_{[0,t]}^n}d\sigma\
\|L_n v\pi_{{\widehat}{g}}(\sigma)\|^2\Big\}^{1/2}$$
$$\label{FDEfull}
\big\|\big[ \Lambda_t(L) - \Lambda_r(L)\big] v{\varepsilon}(g)\big\| \leq
\|{\varepsilon}(g)\|
\sum_{n\geq 0} C(g,T)^{n+1}
\Big\{
\int_r^t ds\
\int_{\Gamma_{[0,s]}^n}d\omega\
\|L_n v\pi_{{\widehat}{g}}(\omega)\|^2\Big\}^{1/2},$$
for $0\leq r\leq t\leq T$, where $$\pi_{{\widehat}{g}}(\sigma) :=
{\widehat}{g}(s_n){\otimes}\cdots {\otimes}{\widehat}{g}(s_1)
\text{ for } \sigma = \{s_1 < \cdots < s_n\}\in\Gamma,$$ with $\pi_{{{\widehat}{g}}}(\emptyset):=1$.
Forms and maps
--------------
Let $V$ and $V'$ be vector spaces and let $E$ and $E'$ be dense subspaces of Hilbert spaces ${\mathsf{H}}$ and ${\mathsf{H}}'$. For any sesquilinear map $\phi$ defined on $E'\times E$ and vectors $\zeta'\in E'$ and $\zeta \in E$ we write $\phi^{\zeta'}_{\zeta}$ for the value of $\phi$ at $(\zeta',\zeta)$. We shall be invoking the following natural relations: $$\begin{aligned}
SL\big( E',E;L(V;V')\big) &\supset
L\big( E;L(V;V'{\odot}|{\mathsf{H}}'{\rangle})\big) \label{inc 1} \\
&\supset
L\big( V;V'{\odot}{\mathcal{O}}(E;{\mathsf{H}}') \big). \label{inc 2}\end{aligned}$$ In case ${\mathsf{H}}$ is finite dimensional the inclusion is an equality. In case $V'$ is finite dimensional the inclusion is an equality. More generally the following observation is relevant here.
\[on inc 2 lem\] Let $\chi\in L\big( E;L(V;V'{\odot}|{\mathsf{H}}'{\rangle})\big)$ satisfy the localising property[:]{} $$\forall_{x\in V}\
\exists_{V'_1 \textup{ finite dimensional subspace of } V'}\
\forall_{\zeta\in E}\ \
\chi_{|\zeta{\rangle}}(x)\in V'_1{\odot}|{\mathsf{H}}'{\rangle}.$$ Then $\chi\in L\big( V;V'{\odot}{\mathcal{O}}(E;{\mathsf{H}}')\big)$.
Straightforward.
Let $\chi\in L\big(E;L(V;V{\odot}|{\mathsf{H}}'{\rangle})\big)$ for a vector space $V$, pre-Hilbert space $E$ and Hilbert space ${\mathsf{H}}'$. A subspace $V_1$ of $V$ *localises* $\chi$ if it satisfies $$\chi_{|\zeta{\rangle}}(V_1) \subset V_1{\odot}|{\mathsf{H}}{\rangle}\quad (\zeta\in E);$$ $\chi$ is *finitely localisable* if $$V = \bigcup
\{ V_1: V_1 \text{ localises } \chi \text{ and } \dim V_1 < \infty \}.$$
By Lemma \[on inc 2 lem\], if $\chi$ is finitely localisable then it belongs to $L\big( V;V{\odot}{\mathcal{O}}(E;{\mathsf{H}}')\big)$, and localisation by $V_1$ translates to $$\chi (V_1)\subset V_1{\odot}{\mathcal{O}}(E;{\mathsf{H}}').$$
Apart from the case of finite dimensional $V$, the example we have in mind is that of a coalgebra ${\mathcal{C}}$ with coproduct $\Delta$. In this context all maps of the form $\chi = (\operatorname{id}_{{\mathcal{C}}} {\otimes}\varphi ) \circ \Delta$, where $\varphi \in L\big( {\mathcal{C}}; {\mathcal{O}}(E) \big)$, are finitely localisable. This follows from the Fundamental Theorem on Coalgebras.
Matrix spaces
-------------
For the general theory of operator spaces and completely bounded maps we refer to [@ERuan] and [@ospaces]. For an operator space ${\mathsf{Y}}$ in $B({\mathsf{H}};{\mathsf{H}}')$ and Hilbert spaces ${\mathsf{h}}$ and ${\mathsf{h}}'$ define $$\label{matrix space defined}
{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}B({\mathsf{h}};{\mathsf{h}}') :=
\{ T\in B({\mathsf{H}}{\otimes}{\mathsf{h}}; {\mathsf{H}}'{\otimes}{\mathsf{h}}') = B({\mathsf{H}};{\mathsf{H}}') {{\overline{\otimes}}}B({\mathsf{h}};{\mathsf{h}}'):
\Omega_{\zeta',\zeta}(T)\in{\mathsf{Y}}\}$$ where $\Omega_{\zeta',\zeta}$ denotes the slice map $\operatorname{id}{\overline{\otimes}}\omega_{\zeta',\zeta}$: $T\mapsto E^{\zeta'}TE_{\zeta}$. For us the relevant cases are ${\mathsf{Y}}{\otimes_{{\mathrm{M}}}}B({\mathsf{h}})$ and ${\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{\mathsf{h}}{\rangle}$, referred to respectively as the ${\mathsf{h}}$-matrix space over ${\mathsf{Y}}$ and the ${\mathsf{h}}$-column space over ${\mathsf{Y}}$. (Previous notations: ${\mathrm{M}}({\mathsf{h}};{\mathsf{Y}})_{{{\text{{\textup}{b}}}}}$ and ${\mathrm{C}}({\mathsf{h}};{\mathsf{Y}})_{{{\text{{\textup}{b}}}}}$.) Matrix spaces are operator spaces which lie between the spatial tensor product ${\mathsf{Y}}{{\otimes}_{\text{{\textup}{sp}}}}B({\mathsf{h}};{\mathsf{h}}')$ and the ultraweak tensor product ${\overline}{{\mathsf{Y}}}{{\overline{\otimes}}}B({\mathsf{h}};{\mathsf{h}}')$ (${\overline}{{\mathsf{Y}}}$ denoting the ultraweak closure of ${\mathsf{Y}}$). They arise naturally in quantum stochastic analysis where a topological state space is to be coupled with the measure-theoretic noise — if ${\mathsf{Y}}$ is a $C^*$-algebra then typically the inclusion ${\mathsf{Y}}{{\otimes}_{\text{{\textup}{sp}}}}B({\mathsf{h}})\subset {\mathsf{Y}}{\otimes_{{\mathrm{M}}}}B({\mathsf{h}})$ is proper and ${\mathsf{Y}}{\otimes_{{\mathrm{M}}}}B({\mathsf{h}})$ is *not* a $C^*$-algebra. Completely bounded maps between concrete operator spaces lift to completely bounded maps between corresponding matrix spaces: for $\phi\in CB({\mathsf{Y}};{\mathsf{Y}}')$ there is a unique map $\Phi : {\mathsf{Y}}{\otimes_{{\mathrm{M}}}}B({\mathsf{h}};{\mathsf{h}}')\to{\mathsf{Y}}'{\otimes_{{\mathrm{M}}}}B({\mathsf{h}};{\mathsf{h}}')$ satisfying $$\Omega_{\zeta',\zeta}\circ \Phi = \phi\circ\Omega_{\zeta',\zeta}
\quad
(\zeta\in{\mathsf{h}}, \zeta\in{\mathsf{h}}').$$ This map is completely bounded and is denoted $\phi{\otimes_{{\mathrm{M}}}}\operatorname{id}_{B({\mathsf{h}};{\mathsf{h}}')}$. A variant on this arises when ${\mathsf{Y}}'$ has the form ${\mathsf{X}}{\otimes_{{\mathrm{M}}}}B({\mathsf{K}};{\mathsf{K}}')$: $$\label{phi hil}
\phi^{{\mathsf{h}};{\mathsf{h}}'} :=
\tau\circ (\phi{\otimes_{{\mathrm{M}}}}\operatorname{id}_{B({\mathsf{h}};{\mathsf{h}}')})$$ where $\tau$ is the flip on the second and third tensor components, so that $$\phi^{{\mathsf{h}};{\mathsf{h}}'}({\mathsf{Y}}{\otimes_{{\mathrm{M}}}}B({\mathsf{h}};{\mathsf{h}}'))\subset
{\mathsf{X}}{\otimes_{{\mathrm{M}}}}B({\mathsf{h}};{\mathsf{h}}'){\otimes_{{\mathrm{M}}}}B({\mathsf{K}};{\mathsf{K}}').$$ When ${\mathsf{h}}' = {\mathsf{h}}$ we write $\phi^{{\mathsf{h}}}$.
Tensor-extended composition
---------------------------
We develop a short-hand notation which will be useful here. Let ${\mathsf{U}}, {\mathsf{V}}$ and ${\mathsf{W}}$ be operator spaces and $V$ a vector space. If $\phi \in L(V ; {\mathsf{U}}{{\otimes}_{\text{{\textup}{sp}}}}{\mathsf{V}}{{\otimes}_{\text{{\textup}{sp}}}}{\mathsf{W}})$ and $\psi \in CB({\mathsf{V}}; {\mathsf{V}}')$ then we compose in the obvious way: $$\label{fullcomp}
\psi {\bullet}\phi :=
(\operatorname{id}_{{\mathsf{U}}} {\otimes}\psi {\otimes}\operatorname{id}_{{\mathsf{W}}} ) \circ \phi
\in L(V;{\mathsf{U}}{{\otimes}_{\text{{\textup}{sp}}}}{\mathsf{V}}' {{\otimes}_{\text{{\textup}{sp}}}}{\mathsf{W}}).$$ Ambiguity is avoided provided that the context dictates which tensor component the second-to-be-applied map $\psi$ should act on. This works nicely for matrix-spaces too. Thus if $\phi \in L(V ; {\mathsf{Y}}{\otimes_{{\mathrm{M}}}}B({\mathsf{h}};{\mathsf{h}}'))$ and $\psi \in CB({\mathsf{Y}}; {\mathsf{Y}}')$ (or $\psi \in B({\mathsf{Y}}; {\mathsf{Y}}')$ if both ${\mathsf{h}}$, ${\mathsf{h}}'$ are finite-dimensional), where ${\mathsf{Y}}$ and ${\mathsf{Y}}'$ are concrete operator spaces, then $$\psi {\bullet}\phi := (\psi {\otimes_{{\mathrm{M}}}}\operatorname{id}_{B({\mathsf{h}};{\mathsf{h}}')}) \circ
\phi \in L(V ; {\mathsf{Y}}' {\otimes_{{\mathrm{M}}}}B({\mathsf{h}};{\mathsf{h}}')) .$$
The following elementary inequality will be needed in Section 3.
\[psi dot phi\] Let $\psi \in B({\mathsf{X}}; {\mathsf{Y}})$ and $\phi_1 , \ldots , \phi_n \in B\big( {\mathsf{X}}; {\mathsf{X}}{\otimes_{{\mathrm{M}}}}|{\mathsf{H}}{\rangle}\big)$ for concrete operator spaces ${\mathsf{X}}$ and ${\mathsf{Y}}$ and finite dimensional Hilbert space ${\mathsf{H}}$. Then $$\| \psi {\bullet}\phi_1 {\bullet}\cdots {\bullet}\phi_n \|
\leq (\dim {\mathsf{H}})^{n/2} \| \psi \| \, \| \phi_1 \| \cdots \| \phi_n \| .$$
Let $(e_i)$ be an orthonormal basis for ${\mathsf{H}}$ and, for a multi-index ${\mbox{\boldmath{$i$}}} = (i_1, \ldots , i_n)$ let $e({\mbox{\boldmath{$i$}}})$ denote $e_{i_1} {\otimes}\cdots {\otimes}e_{i_n}$. Then, by a ‘partial Parseval relation’ (recall the ‘$E$ notation’ introduced in ) $$\| \psi {\bullet}\phi_1 {\bullet}\cdots {\bullet}\phi_n (x) u \|^2
= \sum_{{\mbox{\boldmath{$i$}}}} \big\| E^{e({\mbox{\boldmath{$i$}}})} ( \psi {\bullet}\phi_1 {\bullet}\cdots {\bullet}\phi_n)
(x) u \big\|^2
\quad (x\in {\mathsf{X}}, u\in{\mathsf{h}})$$ where ${\mathsf{h}}$ is the Hilbert space on which the operators of ${\mathsf{Y}}$ act. The result therefore follows since, for any unit vectors $d_1, \ldots , d_n \in {\mathsf{H}}$, $$\begin{aligned}
\| E^{d_1 {\otimes}\cdots {\otimes}d_n} \psi {\bullet}\phi_1 {\bullet}\cdots {\bullet}\phi_n \|
&= \| \psi \circ E^{d_1} \phi_1 \circ \cdots \circ E^{d_n} \phi_n \| \\
& \leq \| \psi \| \, \| \phi_1 \| \cdots \| \phi_n \| .\end{aligned}$$
The following variant on tensor-extended composition will also be useful. For $\psi \in L\big(V ; {\mathcal{O}}(E {\odot}E' ; {\mathsf{K}}{\otimes}{\mathsf{K}}') \big)$ where $V$ is a linear space, $E$ and $E'$ are dense subspaces of Hilbert spaces ${\mathsf{H}}$ and ${\mathsf{H}}'$ and ${\mathsf{K}}$ and ${\mathsf{K}}'$ are further Hilbert spaces, $$\label{omega dot}
\omega_{\zeta ,\eta} {\bullet}\psi :=
E^{\zeta} \psi ( \cdot )E_{\eta}, \quad \zeta \in {\mathsf{K}}', \eta \in E' .$$ Thus $\omega_{\zeta ,\eta}{\bullet}\psi\in L\big(V;{\mathcal{O}}(E ; {\mathsf{K}})\big)$.
Regularity and uniqueness {#regularity sec}
=========================
For this section fix a complex vector space $V$ and exponential domains ${\mathcal{D}}= {\mathfrak{D}}{\odot}{\mathcal{E}_D}$ and ${\mathcal{D}'}= {\mathfrak{D}}' {\odot}{\mathcal{E}}_{D'}$ in ${\mathfrak{h}}{\otimes}{\mathcal{F}}$ and ${\mathfrak{h}}' {\otimes}{\mathcal{F}}$ respectively. A map $V \to {\mathbb{P}}({\mathcal{D}}; {\mathfrak{h}}' {\otimes}{\mathcal{F}})$ is called a *process on* $V$. We are interested in such processes which are *linear* and denote the collection of these by ${\mathbb{P}}(V : {\mathcal{D}}; {\mathfrak{h}}' {\otimes}{\mathcal{F}})$. Also define $${\mathbb{P}}^{\ddagger} (V : {\mathcal{D}}, {\mathcal{D}'}) :=
\big\{k\in {\mathbb{P}}(V : {\mathcal{D}}; {\mathfrak{h}}' {\otimes}{\mathcal{F}}): k(V)\subset
{\mathbb{P}}^{\ddagger} ({\mathcal{D}}; {\mathcal{D}'})\big\},$$ and for such a process $k$ its *conjugate process* $k^{\dagger}\in{\mathbb{P}}^{\ddagger} (V^{\dagger} : {\mathcal{D}'}, {\mathcal{D}})$ is defined by $k^{\dagger}_t(x^{\dagger}) = k_t(x)^{\dagger}$.
A process $k$ on $V$ is $({\mathcal{D}'}, {\mathcal{D}})$-*pointwise weakly continuous* if $s \mapsto ( \omega_{\xi' , \xi} \circ k_s) (x)$ is continuous for all $\xi' \in {\mathcal{D}}'$, $\xi \in {\mathcal{D}}$ and $x \in V$; it is $({\mathcal{D}'}, {\mathcal{D}})$-*weakly regular* if, for some norm on $V$, the following set is bounded $$\big\{ \| x \|^{-1} ( \omega_{\xi' ,\xi} \circ k_s ) (x) : x \in V \setminus \{ 0 \} ,
s\in [0,t] \big\}$$ ($\xi' \in {\mathcal{D}'}, \xi \in {\mathcal{D}}, t \in {{\mathbb{R}}_+}$). In case $$\label{Dstar}
{\mathcal{D}}={\mathcal{D}_*}:=({\mathfrak{h}}{\odot}{\mathcal{E}})
\text{ and }
{\mathcal{D}}'={\mathcal{D}_*}':=({\mathfrak{h}}' {\otimes}{\mathcal{E}})$$ we drop the $({\mathcal{D}'},
{\mathcal{D}})$ and refer simply to weakly continuous and weakly regular processes. If $V$ already has a norm then weak regularity refers to that norm. We denote the spaces of such processes which are also linear by ${\mathbb{P}_{\mathrm{wc}}}(V : {\mathcal{D}}, {\mathcal{D}'})$ and ${\mathbb{P}_{\mathrm{wr}}}(V : {\mathcal{D}}, {\mathcal{D}'})$ respectively.
A weaker notion of regularity tailored to the coefficient of a quantum stochastic differential equation is also relevant to the uniqueness question. Thus let $\phi \in SL\big({\widehat}{D'} , {{\widehat}{D}}; L(V) \big)$ (sesquilinear maps). For each $R \subset \subset V$, $F \subset \subset D$ and $F' \subset \subset D'$ define the following subspace of $V$ $$V^\phi_{F', R,F} :=
\operatorname{Lin}\big\{
(\phi^{\zeta'_1}_{\zeta_1} \circ\cdots\circ \phi^{\zeta'_n}_{\zeta_n})(z):
n\in{\mathbb{Z}_+}, z \in R, \zeta'_1,\ldots, \zeta'_n\in {\widehat}{F'},
\zeta_1,\ldots , \zeta_n \in {\widehat}{F}
\big\}$$ (with the convention that an empty product in $L(V)$ equals $\operatorname{id}_V$), and for $f, f' \in {\mathbb{S}}$ write $F'_t $ and $F_t$ for $\operatorname{Ran}f|_{[0,t[}$ and $\operatorname{Ran}f'|_{[0,t[}$ respectively.
A process $k:V \to {\mathbb{P}}( {\mathcal{D}}; {\mathfrak{h}}' {\otimes}{\mathcal{F}})$ is $({\mathcal{D}'}, {\mathcal{D}})$-*weakly regular locally with respect to* $\phi$ if $V^\phi_{F'_t , R,F_t}$ has a norm for which the following is finite: $$\label{Cphi}
C^{k,\phi,t}_{\xi' , R, \xi} = \sup \Big\{ \| z\|^{-1} \big| \omega_{\xi',\xi} \circ
k_s (z) \big|
:
z \in V^\phi_{F'_t , R,F_t} \setminus \{ 0 \}, s \in [0,t[ \Big\}$$ ($R\subset\subset V,
\xi = v{\varepsilon}(f)\in{\mathcal{D}}, \xi' = v'{\varepsilon}(f') \in {\mathcal{D}}', t\in{{\mathbb{R}}_+}$).
We shall refer to such norms as *regularity norms* and let ${\mathbb{P}}_{\phi \mathrm{wr}} (V:{\mathcal{D}}, {\mathcal{D}'})$ denote the space of such processes which are linear.
Let $k \in {\mathbb{P}_{\mathrm{wc}}}(V: {\mathcal{D}}, {\mathcal{D}'})$.
Let $\phi \in SL\big({\widehat}{D'} , {{\widehat}{D}}; L(V)\big)$ and suppose that $\phi$ satisfies $$\dim V^\phi_{F'_t, R, F_t} < \infty
\quad
(R\subset\subset V, f \in {\mathbb{S}}_{D}, f'\in{\mathbb{S}}_{D'}, t\in{{\mathbb{R}}_+}).$$ Then $k \in {\mathbb{P}}_{\phi \mathrm{wr}} (V : {\mathcal{D}}, {\mathcal{D}'})$.
Suppose that $V$ is a Banach space and $\omega_{\xi' , \xi} \circ k_t$ is bounded for each $\xi' \in {\mathcal{D}'}, \xi \in {\mathcal{D}}, t \in {{\mathbb{R}}_+}$. Then $k \in {\mathbb{P}_{\mathrm{wr}}}(V: {\mathcal{D}}, {\mathcal{D}'})$.
Let $\xi = u {\varepsilon}(f) \in {\mathcal{D}}, \xi' = u' {\varepsilon}(f') \in {\mathcal{D}'}$ and $t \in {{\mathbb{R}}_+}$.
\(a) In this case let $R \subset \subset V$ and consider the $l^1$-norm on $V^\phi_{F'_t,R,F_t}$ determined by a choice of basis: $\| \sum^d_{i=1} \lambda _i e_i \| := \sum^d_{i=1} | \lambda_i |$. By linearity $$C^{k, \phi,t}_{\xi' , R, \xi} \leq \sup \Big\{ \big| {\langle}\xi' ,k_s (e_i) \xi {\rangle}\big|
: = 0 \leq s \leq t, \ i=1, \ldots , d \Big\} ,$$ which is finite by weak continuity.
\(b) In this case the family of bounded linear functionals $\{ \omega_{\xi' ,\xi} \circ k_s : 0 \leq s \leq t \}$ is pointwise bounded, by weak continuity, and so the Banach-Steinhaus Theorem applies.
In particular, if $V$ is finite dimensional then, once equipped with a norm, Part (b) applies.
If $V$ is finite dimensional then $${\mathbb{P}_{\mathrm{wc}}}(V:{\mathcal{D}}, {\mathcal{D}'}) \subset {\mathbb{P}_{\mathrm{wr}}}(V: {\mathcal{D}}, {\mathcal{D}'}) .$$
Quantum stochastic differential equations
-----------------------------------------
Now let $\phi \in SL\big({\widehat}{D'}, {{\widehat}{D}}; L(V) \big)$ and $\kappa \in L(V;W)$ where $W$ is a subspace of ${\mathcal{O}}({\mathfrak{D}}; {\mathfrak{h}}')$, for example $B({\mathfrak{h}};{\mathfrak{h}}')$. A process $k :V \to {\mathbb{P}}({\mathcal{D}}; {\mathfrak{h}}' {\otimes}{\mathcal{F}})$ is a $({\mathcal{D}'}, {\mathcal{D}})$-*weak solution* of the quantum stochastic differential equation $$\label{QSDE}
dk_t = k_t {\bullet}d\Lambda_\phi (t) , \quad k_0 = \iota \circ \kappa$$ (where $\iota$ denotes ampliation ${\mathcal{O}}({\mathfrak{D}}; {\mathfrak{h}}') \to {\mathcal{O}}({\mathcal{D}}; {\mathfrak{h}}' {\otimes}{\mathcal{F}})$), if $k$ is $({\mathcal{D}'}, {\mathcal{D}})$-pointwise weakly continuous and $$\begin{aligned}
\label{w soln}
{\langle}\xi', k_t (x) \xi {\rangle}&- {\langle}v' , \kappa (x) v {\rangle}{\langle}{\varepsilon}(g') ,{\varepsilon}(g) {\rangle}\notag \\
& \qquad\qquad\qquad\qquad = \int^t_0 ds \Big{\langle}\xi' , k_s
\big(\phi^{{\widehat}{g'}(s)}_{{\widehat}{g}(s)} (x) \big) \xi \Big{\rangle}\end{aligned}$$ ($\xi = v {\varepsilon}(g) \in {\mathcal{D}}, \xi' = v' {\varepsilon}(g') \in {\mathcal{D}'}, x \in V, t \in {{\mathbb{R}}_+}$).
Suppose that $W$ is a subspace of ${\mathcal{O}}^{\ddagger}({\mathfrak{D}},{\mathfrak{D}}')$ and ${\mathcal{D}'}= {\mathfrak{D}}'{\odot}{\mathcal{E}}_{D'}$. If a $({\mathcal{D}'}, {\mathcal{D}})$-weak solution $k$ of the equation is ${\mathbb{P}}^{\ddag} ({\mathcal{D}}, {\mathcal{D}}')$-valued then the conjugate process $k^{\dagger}:V^{\dagger}\to{\mathbb{P}}^{\ddagger}({\mathcal{D}'},{\mathcal{D}})$ is a $( {\mathcal{D}}, {\mathcal{D}'})$-weak solution of the quantum stochastic differential equation with $\phi$ and $\kappa$ replaced by $\phi^{\dagger} \in SL\big({\widehat}{D}, {\widehat}{D'} ; L(V^{\dagger}) \big)$ and $\kappa^{\dagger} \in L(V^{\dagger};W^{\dagger})$ respectively.
A process $k \in {\mathbb{P}}(V:{\mathcal{D}}; {\mathfrak{h}}' {\otimes}{\mathcal{F}})$ is a ${\mathcal{D}}$-*strong solution* of the quantum stochastic differential equation if there is a process $K \in
{\mathbb{P}}(V: {\mathfrak{D}}{\odot}{{\widehat}{D}}{\odot}{\mathcal{E}_D}; {\mathfrak{h}}'{\otimes}{{\widehat}{{\mathsf{k}}}}{\otimes}{\mathcal{F}})$ which is pointwise quantum stochastically integrable and satisfies $$\label{strong}
\omega_{\zeta' , \zeta} {\bullet}K_t = k_t \circ \phi^{\zeta'}_{\zeta}
\quad
(\zeta' \in {\widehat}{D'} , \zeta \in {{\widehat}{D}}, t \in {{\mathbb{R}}_+}),$$ and $$\label{integral eqn}
k_t (x) = \kappa (x) {\odot}I + \int^t_0 K_s (x) \, d \Lambda_s
\quad
(x \in V, t \in {{\mathbb{R}}_+}).$$ In particular strong solutions are (pointwise strongly) continuous. In view of the First Fundamental Formula , any ${\mathcal{D}}$-strong solution is a $({\mathcal{D}'}_* , {\mathcal{D}})$-weak solution. Conversely, if $k$ is a $({\mathcal{D}'}, {\mathcal{D}})$-weak solution, with ${\mathcal{D}'}$ of the form ${\mathfrak{D}}' {\odot}{\mathcal{E}}_{D'}$, and $K$ is a pointwise quantum stochastically integrable process satisfying then necessarily holds.
Strong solutions will be considered in subsequent sections. For now let $W= {\mathcal{O}}({\mathfrak{D}}; {\mathfrak{h}}')$.
\[linear-unique\] Let $\phi \in SL\big({\widehat}{D'}, {{\widehat}{D}}; L(V) \big)$ and $\kappa \in L(V;W)$ and let $k$ be a $({\mathcal{D}'},{\mathcal{D}})$-weak solution of the quantum stochastic differential equation . If $k$ is weakly regular locally with respect to $\phi$ and is such that, for each $R \subset \subset V, v {\varepsilon}(f) \in {\mathcal{D}}, v' {\varepsilon}(f') \in {\mathcal{D}'},$ $t \in {{\mathbb{R}}_+}$ and $s \in [0,t[$, the map $\phi^{{\widehat}{f'}(s)}_{{{\widehat}{f}}(s)}$ is bounded on $V^\phi_{F'_t ,R,F_t}$ with respect to a corresponding regularity norm, then
\[l-u a\] $k$ is linear, so that $k\in{\mathbb{P}}_{\phi{\textup}{wr}}(V:{\mathcal{D}},{\mathcal{D}'})$, and
\[l-u b\] the equation has no other such solutions.
Fix $\xi' = u' {\varepsilon}(f') \in {\mathcal{D}}' , \xi = u {\varepsilon}(f) \in {\mathcal{D}}$ and $t \in {{\mathbb{R}}_+}$.
\(a) Let $x,y\in V$ and $\lambda\in{\mathbb{C}}$; set $R=\{ x, y, x+\lambda y\}$, $U=V^\phi_{F'_t , R,F_t}$ with a regularity norm $\|\cdot\|$ and $C=2C^{k,\phi,t}_{\xi' ,R,\xi}$; and define $$\gamma^\lambda_s (z',z)
= \Big{\langle}\xi' , \big[ k_s (z') + \lambda k_s (z) - k_s (z' +\lambda z) \big] \xi
\Big{\rangle}\text{ for } z,z' \in U, s \in [0,t] .$$ By the regularity assumption this satisfies $$\big| \gamma^\lambda_s (z',z)\big| \leq C
\big( \| z' \| + | \lambda | \, \| z\| \big).$$ The linearity of $\kappa$ and each $\phi^{\zeta}_{\zeta'}$ yields the identity $$\gamma^\lambda_s (z',z) = \int^s_0 dr \,
\gamma^\lambda_r \Big( \phi^{{\widehat}{f'}(r)}_{{{\widehat}{f}}(r)} (z'),
\phi^{{\widehat}{f'}(r)}_{{{\widehat}{f}}(r)} (z) \Big) .$$ Iterating this and using the boundedness assumption gives $$\big| \gamma^\lambda_t (x,y) \big| \leq \frac{t^n}{n!} CM^n
\big( \| x\| + | \lambda | \, \| y\| \big), \quad n \in {\mathbb{N}},$$ where $M = \max \big\{ \| \phi^{{\widehat}{c'}}_{{{\widehat}{c}}} (z) \| : z \in U, \| z \| \leq 1, c' \in F'_t ,c\in F_t \big\}$. Thus $\gamma^\lambda_t (x,y)=0$. It follows that $k$ is linear.
\(b) Let ${\widetilde}{k}$ be another such solution. For $x\in V$ and $t\in{{\mathbb{R}}_+}$ define $$\gamma_s (z) = \big{\langle}\xi', [k_s (z) - {\widetilde}{k}_s (z) \big] \xi \big{\rangle}\qquad \big(z \in V^\phi_{F'_t ,\{ x\} ,F_t} , s \in [0,t]\big).$$ Then $$\big| \gamma_s (z)\big| \leq C
\big( \max\{\|z\|, \| z \|_{\sim} \}\big),$$ where $C=C^{k,\phi,t}_{\xi' ,\{x\},\xi} + C^{{\widetilde}{k},\phi,t}_{\xi' ,\{x\},\xi}$ and $\|\cdot\|$ and $\|\cdot\|_{\sim}$ denote the corresponding regularity norms. Arguing as in (\[l-u a\]) yields (\[l-u b\])
The following two special cases are relevant for the case of coalgebraic ([@LSaihp]) and operator space (Section \[existence sec\] of this paper) quantum stochastic differential equations respectively. The first applies in particular when $V$ is finite dimensional.
\[uniqueness cor\] Suppose that $\phi$ satisfies $$\dim V^\phi_{F' ,\{ x\} ,F} < \infty \quad
(F' \subset \subset D', x \in V, F \subset \subset D).$$ Then the quantum stochastic differential equation has at most one $({\mathcal{D}'},{\mathcal{D}})$-weak solution. Moreover any such solution is necessarily linear.
\[uniqueness cor2\] Suppose that $V$ is a Banach space and the sesquilinear map $\phi$ is $B(V)$-valued. Then the quantum stochastic differential equation has at most one linear $({\mathcal{D}'},{\mathcal{D}})$-weak solution $k$ for which each $ \omega_{\xi' ,\xi} \circ k_t$ is bounded [(]{}$\xi' \in {\mathcal{D}'}, \xi \in {\mathcal{D}}, t \in {{\mathbb{R}}_+}$[)]{}.
Existence and dependence on initial conditions {#existence sec}
==============================================
For this section let ${\mathsf{V}}$ be an operator space (with conjugate operator space ${\mathsf{V}}^{\dagger}$ and conjugation $x\mapsto x^{\dagger}$), let ${\mathsf{Y}}$ be an operator space in $B({\mathfrak{h}}; {\mathfrak{h}}')$, let ${\mathcal{D}}= {\mathfrak{h}}{\odot}{\mathcal{E}_D}$ and ${\mathcal{D}'}= {\mathfrak{h}}' {\odot}{\mathcal{E}}_{D'}$ for dense subspaces $D$ and $D'$ of ${\mathsf{k}}$ and recall the notation . Then ${\mathbb{P}}({\mathsf{V}}\to {\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$ denotes the following class of processes on ${\mathsf{V}}$: $$\big\{ k \in {\mathbb{P}}( {\mathsf{V}}: {\mathcal{D}}; {\mathfrak{h}}' {\otimes}{\mathcal{F}}) : \omega_{{\varepsilon}' , {\varepsilon}} {\bullet}k_t ({\mathsf{V}}) \subset {\mathsf{Y}}\text{ for all } {\varepsilon}' \in {\mathcal{E}}_{D'} , {\varepsilon}\in {\mathcal{E}_D}, t \in {{\mathbb{R}}_+}\big\} .$$ Recall that ${\mathsf{k}}$-bounded means bounded if the noise dimension space ${\mathsf{k}}$ is finite dimensional and completely bounded otherwise. For operator spaces ${\mathsf{V}}$ and ${\mathsf{W}}$, we write ${{\mathsf{k}}{\text{-}B}}({\mathsf{V}}; {\mathsf{W}})$ for the space of all linear ${\mathsf{k}}$-bounded maps acting from ${\mathsf{V}}$ to ${\mathsf{W}}$, and give it the operator norm if ${\mathsf{k}}$ is finite-dimensional and the cb-norm otherwise.
We consider the quantum stochastic differential equation $$dk_t = k_t {\bullet}d \Lambda_\phi (t), \quad k_0 = \iota \circ \kappa$$ where $\phi\in L\big( {{\widehat}{D}};{{\mathsf{k}}{\text{-}B}}\big({\mathsf{V}}; CB({\langle}{{\widehat}{{\mathsf{k}}}}| ;{\mathsf{V}})\big)\big)
\subset
SL({{\widehat}{{\mathsf{k}}}},{{\widehat}{D}}; B({\mathsf{V}}))$ and $\kappa \in {{\mathsf{k}}{\text{-}B}}({\mathsf{V}}; {\mathsf{Y}})$. Now ampliation is of bounded operators, so $\iota({\mathsf{Y}}) \subset {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}B({\mathcal{F}})}$. We say that $\phi$ has ‘${\mathsf{k}}$-bounded columns’ (cf. [@LWblms]). Note that $CB({\langle}{{\widehat}{{\mathsf{k}}}}|;{\mathsf{V}})={\mathsf{k}}$-$B({\langle}{{\widehat}{{\mathsf{k}}}}|;{\mathsf{V}})$ (topological isomorphism).
\[existence\] Let $\phi \in L \big( {{\widehat}{D}}; {{\mathsf{k}}{\text{-}B}}\big( {\mathsf{V}}; CB( {\langle}{{\widehat}{{\mathsf{k}}}}| ; {\mathsf{V}}) \big) \big)$ and $\kappa \in {{\mathsf{k}}{\text{-}B}}({\mathsf{V}}; {\mathsf{Y}})$. Then the quantum stochastic differential equation has a ${\mathcal{D}}$-strong solution $k \in {\mathbb{P}}({\mathsf{V}}\to {\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'}_*)$, enjoying the following properties
\[ex a\] $k$ has ${\mathsf{k}}$-bounded columns[:]{} $$k_{t,|{\varepsilon}{\rangle}} \in
{{\mathsf{k}}{\text{-}B}}( {\mathsf{V}}; {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{\mathcal{F}}{\rangle}}) \quad
(t \in {{\mathbb{R}}_+}, {\varepsilon}\in{\mathcal{E}_D}) .$$
\[ex b\] For each ${\varepsilon}\in{\mathcal{E}_D}$ the map $${{\mathbb{R}}_+}\to {{\mathsf{k}}{\text{-}B}}\big( {\mathsf{V}}; {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{\mathcal{F}}{\rangle}}\big), \quad
s \mapsto k_{s,| {\varepsilon}{\rangle}}$$ is locally Hölder-continuous with exponent $\frac12$.
\[ex c\] If ${\widetilde}{k}$ is a linear $({\mathcal{D}'}_1 , {\mathcal{D}}_1)$-weak solution of , for exponential domains ${\mathcal{D}'}_1$ and ${\mathcal{D}}_1$ contained in ${\mathcal{D}'}$ and ${\mathcal{D}}$ respectively, then ${\widetilde}{k}$ is a restriction of $k$: ${\widetilde}{k}_t (x) = k_t (x) \big|_{{\mathcal{D}}_1}$ $(x \in {\mathsf{V}}, t \in {{\mathbb{R}}_+})$.
\[ex d\] If $\phi$ has cb-columns and $\kappa$ is completely bounded then $k$ has cb-columns and [(]{}\[ex b\][)]{} holds with $CB\big( {\mathsf{V}}; {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{\mathcal{F}}{\rangle}}\big)$ in place of ${{\mathsf{k}}{\text{-}B}}\big( {\mathsf{V}}; {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{\mathcal{F}}{\rangle}}\big)$.
Define a process $k \in {\mathbb{P}}({\mathsf{V}}\to {\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$ as follows: $k_t = \Lambda_t \circ {\upsilon}$ where $${\upsilon}^n \in L \big( {{\widehat}{D}}^{{\odot}n} ; {{\mathsf{k}}{\text{-}B}}({\mathsf{V}}; {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{{\widehat}{{\mathsf{k}}}}^{{\otimes}n}{\rangle}}\big)
\subset
L \big( {\mathsf{V}}; {\mathcal{O}}({\mathfrak{h}}{\odot}{{\widehat}{D}}^{{\odot}n} ;
{\mathfrak{h}}' {\otimes}{{\widehat}{{\mathsf{k}}}}^{{\otimes}n} ) \big)
\quad
(n \in {\mathbb{Z}})$$ is defined by $$\label{upsilon}
E^{\zeta'_1 {\otimes}\cdots {\otimes}\zeta'_n} {\upsilon}^n_{| \zeta_1 {\otimes}\cdots {\otimes}\zeta_n {\rangle}}
= \kappa \circ \phi^{\zeta'_n}_{\zeta_n} \circ \cdots \circ \phi^{\zeta'_1}_{\zeta_1}
\quad
(\zeta_1 , \ldots ,\zeta_n \in {{\widehat}{D}}, \zeta'_1, \ldots , \zeta'_n \in {{\widehat}{{\mathsf{k}}}}).$$ Thus, in terms of any concrete realisation of ${\mathsf{V}}$ in $B({\mathsf{H}})$ for a Hilbert space ${\mathsf{H}}$, $${\upsilon}^n_{| \zeta_1 {\otimes}\cdots {\otimes}\zeta_n {\rangle}} = \tau \circ \left(\kappa {\bullet}\phi_{| \zeta_n{\rangle}} {\bullet}\cdots {\bullet}\phi_{| \zeta_1 {\rangle}} \right) ,$$ where $\tau: {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{{\widehat}{{\mathsf{k}}}}^{{\otimes}n}{\rangle}}\to {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{{\widehat}{{\mathsf{k}}}}^{{\otimes}n}{\rangle}}$ denotes the tensor flip reversing the order of $n$ copies of ${{\widehat}{{\mathsf{k}}}}$. Therefore, if ${\mathsf{k}}$ is finite dimensional then Lemma \[psi dot phi\] implies that $$\| {\upsilon}^n_{| \zeta_1 {\otimes}\cdots {\otimes}\zeta_n {\rangle}} \| \leq \| \kappa \|
\left( \sqrt{\dim {{\widehat}{{\mathsf{k}}}}} \, \max_i \| \phi_{| \zeta_i {\rangle}} \| \right)^n ,$$ whereas if $\kappa$ is completely bounded and $\phi$ has cb-columns then $$\| {\upsilon}^n_{| \zeta_1 {\otimes}\cdots {\otimes}\zeta_n {\rangle}} \|_{{{\text{{\textup}{cb}}}}} \leq \| \kappa \|_{{{\text{{\textup}{cb}}}}}
\left( \ \max_i \| \phi_{| \zeta_i {\rangle}} \|_{{{\text{{\textup}{cb}}}}} \right)^n .$$ It follows from and that $k_{t,| {\varepsilon}{\rangle}}({\mathsf{V}})\subset{{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{\mathcal{F}}{\rangle}}$ and $k_{t,| {\varepsilon}{\rangle}}$ is bounded ${\mathsf{V}}\to {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{\mathcal{F}}{\rangle}}$ (${\varepsilon}= {\varepsilon}(g) \in {\mathcal{E}_D}, t \in {{\mathbb{R}}_+}$), with $$\begin{aligned}
&\| k_{t, | {\varepsilon}{\rangle}} \| \leq
\| \kappa \|' \| {\varepsilon}\|
\sum_{n \geq 0} \frac{C^n}{\sqrt{n!}} , \quad \text{ and } \\
& \| k_{t, | {\varepsilon}{\rangle}} - k_{s, | {\varepsilon}{\rangle}} \|
\leq \sqrt{t-s} \| \kappa \|' \| {\varepsilon}\| C(g,T) \sum_{n \geq 0}
\frac{C^n}{\sqrt{n!}}
\quad
(0\leq s\leq t\leq T),\end{aligned}$$ where $C= C(g,T)\sqrt{C'}
\max \big\{ \| \phi_{| \zeta {\rangle}} \|' :
\zeta \in \operatorname{Ran}{{\widehat}{g}}\big|_{[0,T]} \big\}$, with $\|\cdot \|'$ and $C'$ meaning $\|\cdot\|$ and $\dim{{\widehat}{{\mathsf{k}}}}$ respectively, when ${\mathsf{k}}$ is finite-dimensional, but $\|\cdot\|_{{{\text{{\textup}{cb}}}}}$ and $1$ otherwise. We have therefore shown that $k$ satisfies (\[ex a\]) and (\[ex b\]) when ${\mathsf{k}}$ is finite dimensional.
Now suppose that $\kappa$ is completely bounded and $\phi$ has cb-columns. Then, identifying $M_N\big( {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{{\widehat}{{\mathsf{k}}}}{\rangle}}\big)= {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{{\widehat}{{\mathsf{k}}}}{\rangle}}{\otimes_{{\mathrm{M}}}}M_N$ with $M_N({\mathsf{Y}}) {\otimes_{{\mathrm{M}}}}|{{\widehat}{{\mathsf{k}}}}{\rangle}= {\mathsf{Y}}{\otimes_{{\mathrm{M}}}}M_N {\otimes_{{\mathrm{M}}}}|{{\widehat}{{\mathsf{k}}}}{\rangle}$ gives $$\label{matrix lifting}
(k_{t, | {\varepsilon}{\rangle}} )^{(N)} =
{\widetilde}{k}_{t, | {\varepsilon}{\rangle}} \quad
(N \in {\mathbb{N}}, t \in {{\mathbb{R}}_+}, {\varepsilon}\in {\mathcal{E}_D}),$$ where ${\widetilde}{k}$ is the process arising from the above construction when $\kappa$ and $\phi$ are replaced by $\kappa^{(N)}$ and $\phi^N$, $\phi^N$ being given by $(\phi^N)_{| \zeta {\rangle}} = (\phi_{| \zeta {\rangle}})^{(N)}$. It follows that the above estimates apply with cb-norms on the left-hand side (as well as the right). This completes the proof of (\[ex a\]), (\[ex b\]) and (\[ex d\]).
Recalling we next note that $k$ enjoys the following useful ‘form representation’: for ${\varepsilon}= {\varepsilon}(g) \in {\mathcal{E}_D}$, ${\varepsilon}' = {\varepsilon}(g') \in {\mathcal{E}}$ and $t \in {{\mathbb{R}}_+}$, $$\label{form repn}
e^{-{\langle}g',g{\rangle}}
\omega_{{\varepsilon}' , {\varepsilon}} {\bullet}k_t
= \int_{\Gamma_{[0,t]}} d \sigma \, {\upsilon}^{g' ,g}_\sigma
\quad (t\in{{\mathbb{R}}_+})$$ in $B({\mathsf{V}};{\mathsf{Y}})$ where $$\label{upsilon g' g}
{\upsilon}^{g',g}_\sigma
=
\kappa \circ \phi^{{\widehat}{g'}(s_1)}_{{{\widehat}{g}}(s_1)} \circ \cdots \circ
\phi^{{\widehat}{g'}(s_n)}_{{{\widehat}{g}}(s_n)}
\text{ for }
\sigma = \{ s_1 < \cdots < s_n \}\in\Gamma.$$ Therefore $$\begin{aligned}
\omega_{{\varepsilon}',{\varepsilon}}{\bullet}k_t - {\langle}{\varepsilon}',{\varepsilon}{\rangle}\kappa
&
=
{\langle}{\varepsilon}',{\varepsilon}{\rangle}\int_{\Gamma_{[0,t]}}d\sigma \, (1- \delta_{\emptyset} (\sigma ))
{\upsilon}^{g',g}_\sigma \\
&
=
{\langle}{\varepsilon}',{\varepsilon}{\rangle}\int^t_0 ds \int_{\Gamma_{[0,s]}} d \rho\
{\upsilon}^{g',g}_{\rho\cup \{s\}} \\
&
=
{\langle}{\varepsilon}',{\varepsilon}{\rangle}\int^t_0 ds \int_{\Gamma_{[0,s]}} d \rho\
{\upsilon}^{g',g}_{\rho} \circ
\phi^{{\widehat}{g'}(s)}_{{{\widehat}{g}}(s)} \\
&
=
\int^t_0 ds\
\omega_{{\varepsilon}',{\varepsilon}}{\bullet}\big(k_s \circ \phi^{{\widehat}{g'}(s)}_{{{\widehat}{g}}(s)}\big),\end{aligned}$$ so $k_s$ is a (${\mathcal{D}},{\mathcal{D}'}$)-weak solution of .
Now define a process $K \in {\mathbb{P}}\big( {\mathsf{V}}\to {{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{{\widehat}{{\mathsf{k}}}}{\rangle}}: {\mathfrak{h}}{\odot}{{\widehat}{D}}{\odot}{\mathcal{E}_D},{\mathfrak{h}}' {\odot}{\widehat}{D'} {\odot}{\mathcal{E}}\big)$ by $$K_{t, | \zeta {\otimes}{\varepsilon}{\rangle}} = k_{t,| {\varepsilon}{\rangle}} {\bullet}\phi_{| \zeta {\rangle}}
\quad
(t \in {{\mathbb{R}}_+}, \zeta \in {{\widehat}{D}}, {\varepsilon}\in {\mathcal{E}_D}).$$ Since it is (pointwise strongly) continuous, by part (b), $K$ is quantum stochastically integrable. Moreover, since $$E^{\zeta'} K_{t, | \zeta {\otimes}{\varepsilon}{\rangle}} =
E^{\zeta'} k_{t, | {\varepsilon}{\rangle}}
{\bullet}\phi_{| \zeta {\rangle}} =
k_{t, | {\varepsilon}{\rangle}} \circ \phi^{\zeta'}_{\zeta} ,$$ $K$ also satisfies . Therefore $k$ is a ${\mathcal{D}}$-strong solution of . Part (c) follows from the uniqueness result Corollary \[uniqueness cor2\]. This completes the proof.
*Notation*. The process uniquely determined by $\kappa$ and $\phi$ in this theorem will be denoted $k^{\kappa,\phi}$, extending the established notation $k^\phi$ for the case ${\mathsf{Y}}= {\mathsf{V}}$ and $\kappa = \operatorname{id}_{{\mathsf{V}}}$.
Let $\phi \in {{\mathsf{k}}{\text{-}B}}\big({\mathsf{V}}; CB(T({{\widehat}{{\mathsf{k}}}});{\mathsf{V}})\big)$ and $\kappa \in {{\mathsf{k}}{\text{-}B}}({\mathsf{V}};{\mathsf{Y}})$. Then [(]{}for any exponential domains ${\mathcal{D}}$ and ${\mathcal{D}'}$[)]{} the quantum stochastic differential equation has a unique ${\mathcal{D}},{\mathcal{D}'}$-weakly regular weak solution $k \in {\mathbb{P}}({\mathsf{V}}\to {\mathsf{Y}}:{\mathcal{D}},{\mathcal{D}'})$; it is also a ${\mathcal{D}}$-strong solution.
Here $T({{\widehat}{{\mathsf{k}}}})$ denotes the operator space of trace-class operators on ${{\widehat}{{\mathsf{k}}}}$ and we are invoking the natural complete isometry $CB\big(T({{\widehat}{{\mathsf{k}}}});{\mathsf{V}}\big) =
CB\big(|{{\widehat}{{\mathsf{k}}}}{\rangle};CB({\langle}{{\widehat}{{\mathsf{k}}}}|;{\mathsf{V}})\big)$. If ${\mathsf{V}}$ is a concrete operator space then there is a natural completely isometric isomorphism between $CB\big(T({{\widehat}{{\mathsf{k}}}});{\mathsf{V}}\big)$ and ${{\mathsf{V}}{\otimes_{{\mathrm{M}}}}B({{\widehat}{{\mathsf{k}}}})}$, so that $\phi$ above may be viewed as a map in ${{\mathsf{k}}{\text{-}B}}\big( {\mathsf{V}}; {{\mathsf{V}}{\otimes_{{\mathrm{M}}}}B({{\widehat}{{\mathsf{k}}}})}\big)$.
\[conjugate of k phi cor\] Suppose that $\phi$ has a conjugate $\phi^{\dagger}$ in $L\big( {\widehat}{D'};
{{\mathsf{k}}{\text{-}B}}({\mathsf{V}}^{\dagger};CB({\langle}{{\widehat}{{\mathsf{k}}}}|;{\mathsf{V}}^{\dagger})\big)$. Then $k^{\kappa,\phi}\in{\mathbb{P}}^{\ddagger}({\mathsf{V}}\to {\mathsf{Y}}:{\mathcal{D}},{\mathcal{D}'})$ and $(k^{\kappa,\phi})^{\dagger} =
k^{\kappa^{\dagger},\phi^{\dagger}}$.
In view of the identity $${\widetilde}{{\upsilon}}^{g,g'}_\sigma
= \big({\upsilon}^{g',g}_\sigma\big)^{\dagger}
\quad
(g\in{\mathbb{S}_D},
g'\in{\mathbb{S}}_{D'}, \sigma\in\Gamma),$$ where ${\widetilde}{{\upsilon}}$ is defined by with $\kappa^{\dagger}$ and $\phi^{\dagger}$ in place of $\kappa$ and $\phi$, this follows from the form representations for $k^{\kappa^{\dagger},\phi^{\dagger}}$ and $k^{\kappa,\phi}$.
\(i) If $U$ is a subspace of ${\mathsf{V}}$ invariant under each of the maps $\phi^{\zeta'}_\zeta$ ($\zeta'\in{{\widehat}{{\mathsf{k}}}}, \zeta\in{{\widehat}{D}}$) then $\omega_{{\varepsilon}', {\varepsilon}} {\bullet}k_{t}(U)\subset {\overline}{\kappa(U)}$ for all ${\varepsilon}'\in{\mathcal{E}}, {\varepsilon}\in{\mathcal{E}_D}$.
\(ii) The identification extends as follows. If $\phi$ has cb-columns and $\kappa$ is completely bounded then ${\mathsf{h}}$-matrix space liftings, of coefficient, initial condition and solution, are compatible: $$\label{kappaphihil}
(k^{\kappa,\phi}_t)^{{\mathsf{h}}} = k^{\kappa',\phi'}_t$$ where $\kappa' = \kappa {\otimes_{{\mathrm{M}}}}\operatorname{id}_{B({\mathsf{h}})}$ and $\phi'$ is determined by $\phi'_{|\zeta{\rangle}} = (\phi_{|\zeta{\rangle}})^{{\mathsf{h}}}$. This follows easily from the equality $$\left(\kappa {\bullet}\phi_{| \zeta_1{\rangle}} {\bullet}\cdots {\bullet}\phi_{| \zeta_n {\rangle}}\right)^{{\mathsf{h}}} =
\kappa' {\bullet}\phi'_{| \zeta_1{\rangle}} {\bullet}\cdots {\bullet}\phi'_{| \zeta_n {\rangle}}$$ (in the notation and the identity $$\Lambda^n(T{\otimes}L) = T{\otimes}\Lambda^n_t(L)
\quad
(T\in B({\mathsf{h}}), n\in{\mathbb{Z}_+},
L\in B({\mathfrak{h}};{\mathfrak{h}}'){{\overline{\otimes}}}B({{\widehat}{{\mathsf{k}}}}^{{\otimes}n})).$$
In the next result we consider the case where the operator space ${\mathsf{V}}$ is concrete itself, and so the process $k^{\kappa, \phi}$ may be compared to the process $k^{\phi}$.
\[phi to kappa phi propn\] Let $\kappa$ and $\phi$ be as in Theorem \[existence\] and suppose that the operator space ${\mathsf{V}}$ is concrete. Then the following hold.
$$\omega_{{\varepsilon}' , {\varepsilon}} {\bullet}k^{\kappa , \phi}_t =
\kappa \circ \big(\omega_{{\varepsilon}' , {\varepsilon}} {\bullet}k^\phi_t\big)
\quad
({\varepsilon}\in {\mathcal{E}_D}, {\varepsilon}' \in {\mathcal{E}}, t \in {{\mathbb{R}}_+}).$$
If $\kappa$ is completely bounded then $$k^{\kappa , \phi}_{t , | {\varepsilon}{\rangle}} = \kappa {\bullet}k^\phi_{t, | {\varepsilon}{\rangle}} \quad
(t \in {{\mathbb{R}}_+}, {\varepsilon}\in {\mathcal{E}_D}) .$$
If $\kappa$ is completely bounded and the process $k^\phi$ is completely bounded then $k^{\kappa , \phi}$ is the completely bounded process given by $$k^{\kappa , \phi}_t = \kappa {\bullet}k^\phi_t \quad (t \in {{\mathbb{R}}_+}).$$
\(a) follows easily from ; (b) and (c) are simple consequences of (a).
Since the process $k^{\kappa,\phi}$ depends linearly on $\kappa$, the proposition implies that it also depends continuously on its initial condition — in various senses, depending on the regularity of the initial condition and process $k^\phi$.
If ${\mathsf{V}}= {\mathsf{Y}}$ and the initial condition commutes with the coefficient operator, in the sense that $\kappa{\bullet}\phi_{|\zeta{\rangle}}
= \phi_{|\zeta{\rangle}}\circ\kappa$ ($\zeta\in{{\widehat}{D}}$), then $\kappa{\bullet}\phi^{{\bullet}n}_{|\eta{\rangle}}
= \phi^{{\bullet}n}_{|\eta{\rangle}}\circ\kappa$ ($n\in{\mathbb{Z}_+}, \eta\in{{\widehat}{D}}^{{\odot}n}$) and so $$k_t^{\kappa,\phi} = k_t^{\phi}\circ\kappa
\quad (t\in{{\mathbb{R}}_+}).$$
Injectivity of the quantum stochastic operation $\Lambda$ ([@LWjlms], Proposition 2.3) implies that $$k^{\kappa,\phi}= k^{\kappa',\phi'}
\text{ if and only if }
\kappa = \kappa' \text{ and }
\kappa{\bullet}\phi_{|\zeta{\rangle}}
= \kappa'{\bullet}\phi'_{|\zeta{\rangle}}
\quad
(\zeta\in{{\widehat}{D}}).$$
Localisable equations {#localisable sec}
=====================
In this section we consider the case where the source space is a vector space on which the coefficient map of the quantum stochastic differential equation is finitely localisable. Thus let $V$ be a complex vector space, let $D$ be a dense subspace of the noise dimensions space ${\mathsf{k}}$ and consider our quantum stochastic differential equation $$dk_t = k_t {\bullet}d \Lambda_\phi (t) , \quad k_0 = \iota \circ \kappa ,$$ where $\phi \in L \big( {{\widehat}{D}}; L(V ; V {\odot}| {{\widehat}{{\mathsf{k}}}}{\rangle}) \big) $. We consider two cases. Recall that if $\phi$ is finitely localisable then it necessarily belongs to $L(V;V{\odot}{\mathcal{O}}({{\widehat}{D}}))$; also recall the notation .
\[local existence\] Let $\phi \in L(V;V{\odot}{\mathcal{O}}({{\widehat}{D}}))$ be finitely localisable and let $\kappa \in L(V;{\mathsf{Y}})$, where ${\mathsf{Y}}$ is an operator space in $B({\mathfrak{h}}; {\mathfrak{h}}')$. Set ${\mathcal{D}}= {\mathfrak{h}}{\odot}{\mathcal{E}_D}$. Then there is a process $k \in {\mathbb{P}}(V \to {\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'}_*)$, which is a ${\mathcal{D}}$-strong solution of and enjoys the following further properties:
\[lex a\] $k$ is $L(V;{\mathsf{Y}}{\odot}{\mathcal{O}}({\mathcal{E}_D}))$-valued.
\[lex b\] The map $s \mapsto k_{s, | {\varepsilon}{\rangle}} (x)$ is locally Hölder-continuous ${{\mathbb{R}}_+}\to {\mathsf{Y}}{{\otimes}_{\text{{\textup}{sp}}}}| {\mathcal{F}}{\rangle}$ with exponent $\frac12$ [(]{}$x \in V, {\varepsilon}\in{\mathcal{E}_D}$[)]{}.
\[lex c\] If ${\widetilde}{k}$ is a $({\mathcal{D}'}_1 , {\mathcal{D}}_1)$-weak solution of , where ${\mathcal{D}}_1$ and ${\mathcal{D}'}_1$ are exponential domains contained in ${\mathcal{D}}$ and ${\mathcal{D}'}_*$ respectively, then ${\widetilde}{k}$ is a restriction of $k$: ${\widetilde}{k}_t (x) = k_t (x) |_{{\mathcal{D}}_1}$.
\[lex d\] For any subspace $V_1$ localising $\phi$, $k_{t} (V_1) \subset \kappa(V_1) {\odot}{\mathcal{O}}({\mathcal{E}_D})$ $(t\in{{\mathbb{R}}_+})$.
Consider a finite dimensional subspace $V_1$ of $V$ which localises $\phi$ and let $\kappa_1$ and $\phi_1$ be the restrictions of $\kappa$ and $\phi$ to $V_1$. By endowing $V_1$ with operator space structure $\kappa_1$ becomes completely bounded and $\phi_1$ enjoys completely bounded columns. Theorem \[existence\] therefore permits us to define a process $k^1 \in {\mathbb{P}}(V_1 \to {\mathsf{Y}}:{\mathcal{D}}, {\mathcal{D}'})$ by $k^1 =
k^{\kappa_1 , \phi_1}$. Now suppose that $k^2 \in {\mathbb{P}}(V_2 \to {\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$ is the process arising in this way from another finite dimensional subspace $V_2$ localising $\phi$. Then the finite dimensional subspace $V_3 := V_1 \cap V_2$ also localises $\phi$ and so gives rise to a third process $k^3 \in {\mathbb{P}}(V_3 \to {\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$. By the uniqueness part of Theorem \[existence\] it follows that $k^3$ agrees with both $k^1$ and $k^2$ on $V_3$. The following prescription therefore gives a consistent definition of a process $k \in {\mathbb{P}}(V \to {\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$: let $k_t (x) =
k^{\kappa_1, \phi_1}_t (x) $ where $\kappa_1$ and $\phi_1$ are the restrictions of $\kappa$ and $\phi$ to any finite dimensional subspace of $V$ containing $x$ which localises $\phi$. That $k$ is a ${\mathcal{D}}$-strong solution of satisfying properties (\[lex a\])-(\[lex d\]) now follows easily from Theorem \[existence\] and the subsequent remark. Observe that (\[lex d\]) implies that for each $s \geq 0$ and ${\varepsilon}\in {\mathcal{E}_D}$ the map $k_{s,|{\varepsilon}{\rangle}}$ takes values in ${\mathsf{Y}}{\odot}|{\mathcal{F}}{\rangle}$.
Clearly the following weaker localisable property suffices: for all $x \in
V$ and $F \subset \subset D$ there is a finite dimensional subspace $V_1$ of $V$ containing $x$ such that $\phi_{| \zeta {\rangle}} (V_1) \subset V_1 {\odot}| {{\widehat}{{\mathsf{k}}}}{\rangle}$ for all $\zeta
\in {\widehat}{F}$; conclusion (\[lex d\]) is then modified accordingly.
*Notation*. We again use the notation $k^{\kappa,\phi}$ for the process obtained in the above theorem.
As before, $$k^{\kappa,\phi}= k^{\kappa',\phi'}
\text{ if and only if }
\kappa = \kappa' \text{ and }
\kappa{\bullet}\phi
= \kappa'{\bullet}\phi'.$$
\[algebraic conjugate\] Suppose that $\phi\in L(V;V{\odot}{\mathcal{O}}^{\ddagger}({{\widehat}{D}},{\widehat}{D'}))$ for some dense subspace $D'$ of ${\mathsf{k}}$. Then $k^{\kappa,\phi}\in{\mathbb{P}}^{\ddagger}(V:{\mathcal{D}}, {\mathcal{D}'})$ where ${\mathcal{D}'}= {\mathfrak{h}}'{\odot}{\mathcal{E}}_{D'}$ and $(k^{\kappa,\phi})^{\dagger} =
k^{\kappa^{\dagger},\phi^{\dagger}}$.
We next give a variant of the above existence theorem. Note that the definition of ${\mathbb{P}}(V \to {\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$ extends in an obvious way if ${\mathsf{Y}}$ is replaced by $W={\mathcal{O}}({\mathfrak{D}}; {\mathfrak{h}}')$ and ${\mathcal{D}}$ by ${\mathfrak{D}}{\odot}{\mathcal{E}_D}$.
\[more local existence\] Let $\phi \in L(V;V{\odot}{\mathcal{O}}({{\widehat}{D}}))$ be finitely localisable, let $\kappa \in L(V; W)$ and set ${\mathcal{D}}= {\mathfrak{D}}{\odot}{\mathcal{E}_D}$. Then the conclusions of Theorem \[local existence\] hold with ${\mathsf{Y}}$ replaced by $W$ and [(]{}\[lex a\][)]{}, [(]{}\[lex b\][)]{} and [(]{}\[lex d\][)]{} replaced by
\[mle a\] $s \mapsto k^{\kappa , \phi}_s (x) \xi$ is locally Hölder-continuous ${{\mathbb{R}}_+}\to {\mathfrak{h}}' {\otimes}{\mathcal{F}}$ with exponent $\frac12$, for all $x \in V$ and $\xi \in {\mathcal{D}}$.
For $u \in {\mathfrak{D}}$, Theorem \[local existence\] applies, with ${\mathsf{Y}}= | {\mathfrak{h}}' {\rangle}$, to the quantum stochastic differential equation $$dk_t =
k_t {\bullet}d \Lambda_\phi (t), \quad k_0 = \iota \circ \kappa_{| u{\rangle}};$$ Let $l^u\in{\mathbb{P}}\big(V\to |{\mathfrak{h}}'{\rangle}:{\mathcal{E}}_D,{\mathfrak{h}}'{\odot}{\mathcal{E}}\big)$ be its ${\mathcal{E}_D}$-strong solution. For $u,v \in {\mathfrak{D}}$ and $\lambda \in {\mathbb{C}}$, if $g \in {\mathcal{E}_D}$ and $\xi' = v'{\varepsilon}(g') \in {\mathcal{D}'}$ then the maps $\gamma_s : V \to {\mathbb{C}}$ $(s \in {{\mathbb{R}}_+})$ given by $$\gamma_s (x) = \Big{\langle}\xi' , \big[l^u_s (x) + \lambda l^v_s (x) -
l^{(u+\lambda v)}_s (x) \big]
{\varepsilon}(g) \Big{\rangle}$$ satisfy $$\gamma_t (x) =
\int^t_0 ds \, \gamma_s \big( \phi^{{\widehat}{g'}(s)}_{{{\widehat}{g}}(s)} (x) \big)
\quad (x \in V , t \in {{\mathbb{R}}_+}) .$$ In view of finite localisability, iteration shows that $\gamma$ is identically zero. If follows that $$k^{\kappa , \phi}_t (x) u {\varepsilon}(g) := l^u_t (x) {\varepsilon}(g) \quad
(x \in V, u \in {\mathfrak{D}}, g \in {\mathbb{S}}_D , t \in {{\mathbb{R}}_+}) ,$$ defines a process $k^{\kappa ,\phi} \in {\mathbb{P}}(V \to W : {\mathcal{D}}, {\mathcal{D}'})$ which is a ${\mathcal{D}}$-strong solution of ; it is clear that it satisfies (a)$'$ and (\[lex c\]) too.
Quantum stochastic cocycles {#cocycles sec}
===========================
In this section we give a new result on the infinitesimal generation of quantum stochastic cocycles (cf. [@LWjfa]). At the end we describe how the result may be applied to quantum stochastic convolution cocycles on a coalgebra ([@LSaihp]). Fix an operator space ${\mathsf{Y}}$ in $B({\mathfrak{h}}; {\mathfrak{h}}')$ and exponential domains ${\mathcal{D}}= {{\mathfrak{h}}{\odot}{\mathcal{E}_D}}$ and ${\mathcal{D}'}= {{\mathfrak{h}}'{\odot}{\mathcal{E}}_{D'}}$.
The following notations for a process $k \in {\mathbb{P}}({\mathsf{Y}}\to {\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$ prove useful: $$\label{kfg}
k^{g',g}_t :=
e^{-{\langle}g'_{[0,t[} ,g_{[0,t[} {\rangle}}
\omega_{{\varepsilon}(g'_{[0,t[}), {\varepsilon}(g_{[0,t[})} {\bullet}k_t$$ ($g' \in {\mathbb{S}_{D'}}, g \in {\mathbb{S}_D}, t \in {{\mathbb{R}}_+}$) and $$\label{kcd}
k^{c',c}_t :=
k^{c'_{[0,t[},c_{[0,t[}} \quad (c'\in D', c\in D).$$ Thus $k^{g',g}_t\in L({\mathsf{Y}})$ and the process is called *initial space bounded* if each map $k^{g',g}_t$ is bounded (cf.the condition of having bounded columns).
A process $k \in {\mathbb{P}}({\mathsf{Y}}\to {\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$ is a $({\mathcal{D}'}, {\mathcal{D}})$-*weak quantum stochastic cocycle* on ${\mathsf{Y}}$ if it satisfies $$\label{cocycle reln}
k^{g',g}_{r+t} = k^{g',g}_r \circ k^{S^*_r g', S^*_r g}_t$$ for all $g' \in {\mathbb{S}_{D'}}$, $r,t\in{{\mathbb{R}}_+}$ and $g \in {\mathbb{S}_D}$, where $(S_t)_{t\geq0}$ is the (isometric) right-shift semigroup on $L^2({{\mathbb{R}}_+};{\mathsf{k}})$.
Let ${\mathbb{QSC}}({\mathsf{Y}}:{\mathcal{D}},{\mathcal{D}'})$ denote the collection of these. Also define $${\mathbb{QSC}}^\ddagger({\mathsf{Y}}:{\mathcal{D}},{\mathcal{D}'}) =
{\mathbb{QSC}}({\mathsf{Y}}:{\mathcal{D}},{\mathcal{D}'}) \cap
{\mathbb{P}}^\ddagger({\mathsf{Y}}\to {\mathsf{Y}}:{\mathcal{D}},{\mathcal{D}'});$$ if $k$ is in this class then $k^{\dagger\,g,g'}_t = (k_t^{g',g})^\dagger$ and it is easily seen that the conjugate process $k^\dagger$ is a cocycle on ${\mathsf{Y}}^\dagger$.
In case the process has cb-columns (each map $x\mapsto k_{t, |{\varepsilon}{\rangle}}(x)$ is completely bounded ${\mathsf{Y}}\to{{\mathsf{V}}{\otimes_{{\mathrm{M}}}}|{\mathcal{F}}{\rangle}}$) the cocycle relation is equivalent to $$k_{r+t, | {\varepsilon}(g_{[0,r+t[}) {\rangle}} = k_{r,|{\varepsilon}(g_{[0,t[}){\rangle}}
{\bullet}k_{t, | {\varepsilon}(S^*_r g_{[r,r+t[}){\rangle}} ;$$ in case the process itself is completely bounded it simplifies further, to the more recognisable cocycle property: $$k_{r+t} = k_r {\bullet}\sigma_r {\bullet}k_t$$ were $(\sigma_r)_{r \geq 0}$ is the CCR flow of index ${\mathsf{k}}$ ([@Arveson]).
\[sgp decomp lem\] Let $k \in {\mathbb{P}}({\mathsf{Y}}\to {\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$ and define $P^{c',c} := (k^{c',c}_t)_{t\geq 0}$ [(]{}$c',c\in{\mathsf{k}}$[)]{}. Then the following are equivalent[:]{}
$k\in{\mathbb{QSC}}({\mathsf{Y}}:{\mathcal{D}},{\mathcal{D}'})$.
For all $c'\in D'$ and $c\in D$, $P^{c',c}$ is a one-parameter semigroup in $L({\mathsf{Y}})$ and, for all $g' \in {\mathbb{S}_{D'}}, g \in {\mathbb{S}_D}$ and $t \in {{\mathbb{R}}_+}$, $k^{g',g}_t = l^{g',g}_t$ where $$\label{lfg}
l^{g',g}_t =
P^{g'(t_0),g(t_0)}_{t_1 -t_0} \cdots P^{g'(t_n),g(t_n)}_{t_{n+1} -t_{n}}$$ with $n\in{\mathbb{Z}_+}$, $t_0=0$, $t_{n+1}=t$ and $\{t_1 < \cdots < t_n\}$ being precisely the [(]{}possibly empty[)]{} union of the sets of points of discontinuity of $g'$ and $g$ in $]0,t[$.
For all $g' \in {\mathbb{S}_{D'}}, g \in {\mathbb{S}_D}$ and $t \in {{\mathbb{R}}_+}$, $$\label{sgp decomp}
k^{g',g}_t =
P^{g'(t_0),g(t_0)}_{t_1 -t_0} \cdots P^{g'(t_n),g(t_n)}_{t_{n+1} -t_{n}}$$ whenever $n\in{\mathbb{Z}_+}$ and $\{ 0=t_0 \leq \cdots \leq t_{n+1} =t\}$ includes all the discontinuities of $g'_{[0,t[}$ and $g_{[0,t[}$.
Straightforward, see [@LWjfa].
The one-parameter semigroups $\{ P^{c',c} : c'\in D', c\in D\}$ in $L({\mathsf{Y}})$ are referred to as the *associated semigroups* of $k$, $P^{0,0}$ as its *Markov semigroup* and as its *semigroup decomposition*. If $k$ is initial space bounded and each semigroup is norm continuous ${{\mathbb{R}}_+}\to B({\mathsf{Y}})$ then the cocycle is called *Markov-regular*. When the cocycle is contractive, norm continuity of any of the associated semigroups (such as its Markov semigroup) implies Markov-regularity ([@LWjfa], Proposition 5.4). In view of the semigroup decomposition, Markov-regular cocycles are necessarily both weakly regular and weakly continuous processes.
Now consider the quantum stochastic differential equation where $\kappa = \operatorname{id}_{{\mathsf{Y}}}$:
$$\label{id QSDE}
dk_t = k_t {\bullet}d\Lambda_\phi(t), \quad k_0 = \iota.$$
The following result is a coordinate-free counterpart to Proposition 5.2 of [@LWjfa] in the operator space setting.
\[H thm\] Let $\phi \in SL\big( {\widehat}{D'},{{\widehat}{D}}; B({\mathsf{Y}})\big)$ and let $k\in{\mathbb{P}_{\phi\mathrm{wr}}}({\mathsf{Y}}\to{\mathsf{Y}}:{\mathcal{D}'}, {\mathcal{D}})$ be a $({\mathcal{D}}, {\mathcal{D}'})$-weak solution of the quantum stochastic differential equation . Then $k$ is a Markov-regular quantum stochastic cocycle and the generators of its associated semigroups are given by $$\label{psi and phi}
\psi_{c',c}=\phi^{{\widehat}{c'}}_{{\widehat}{c}} \qquad (c'\in D', c\in D).$$
Let $\xi' = v' {\varepsilon}(g') \in {\mathcal{D}'}, \xi \in v {\varepsilon}(g) \in {\mathcal{D}}$ and $t \in {{\mathbb{R}}_+}$. Define $l^{g',g}_t \in B({\mathsf{Y}})$ by where $P^{c',c}$ is the norm continuous semigroup in $B({\mathsf{Y}})$ with generator $\phi^{{\widehat}{c'}}_{{{\widehat}{c}}}$. Then $m^{g',g}_t := k^{g',g}_t - l^{g',g}_t$ satisfies $$\big{\langle}v' , m^{g',g}_t (x) v \big{\rangle}= \int^t_0 ds \,
\big{\langle}v' ,m^{g',g}_s (\phi^{{\widehat}{g'}(s)}_{{\widehat}{g} (s)} (x) ) v \big{\rangle}.$$ Iterating this gives $$\begin{aligned}
\lefteqn{ \big{\langle}v' , m^{g',g}_t (x) v \big{\rangle}} \\
& \quad = \int^t_0 ds_n \cdots \int^{s_2}_0 ds_1 (\omega_{\xi',\xi} \circ k_{s_1}
- \omega_{v',v} \circ l^{g',g}_{s_1} )
( \phi^{{\widehat}{g'} (s_1)}_{{{\widehat}{g}}(s_1)} \circ \cdots \circ
\phi^{{\widehat}{g'}(s_n)}_{{{\widehat}{g}}(s_n)} ) (x) .\end{aligned}$$ By $\phi$-weak regularity of $k$ and norm continuity of $l^{g',g}$, the integrand has a bound of the form $C \| x\| M^n$ where the constants $C$ and $M$ are independent of $n$. The identity $k^{g',g}_t = l^{g',g}_t$ follows and so, by Lemma \[sgp decomp lem\], $k$ is a quantum stochastic cocycle with associated semigroups $\{ P^{c',c} : c' \in D' , c \in D\}$. This completes the proof.
It follows from that the associated semigroups are cb-norm continuous if and only if the sesquilinear map $\phi$ is $CB ({\mathsf{Y}})$-valued.
Note that, in this case, the ‘form representation’ of $k$ is given by: $$k^{g',g}_s =
\int_{\Gamma_{[0,s]}} \, d \sigma \, {\upsilon}^{g',g}_\sigma$$ where ${\upsilon}^{g',g}_\sigma = \operatorname{id}_{{\mathsf{Y}}}$ when $\sigma = \emptyset$ and $${\upsilon}^{g',g}_\sigma =
\phi^{{\widehat}{g'}(s_1)}_{{\widehat}{g}(s_1)}\circ \cdots \circ
\phi^{{\widehat}{g'}(s_n)}_{{\widehat}{g}(s_n)}
\text{ for } \sigma = \{s_1 < \cdots < s_n\}.$$ In particular, if $k=k^\phi$ where $\phi\in L\big({{\widehat}{D}}; {{\mathsf{k}}{\text{-}B}}({\mathsf{Y}};{{\mathsf{V}}{\otimes_{{\mathrm{M}}}}|{{\widehat}{{\mathsf{k}}}}{\rangle}}) \big)$ then $${\upsilon}^{g',g}_\sigma =
\omega_{\pi_{{\widehat}{g'}}(\sigma),\pi_{{{\widehat}{g}}}(\sigma)}
{\bullet}{\upsilon}_{\# \sigma},$$ where ${\upsilon}= {\upsilon}^\phi$ is defined by with $\kappa = \operatorname{id}_V$, and the cocycle relation may be expressed as follows: $$\int_{\Gamma_{[0,r+t]}}\! d\sigma \, {\upsilon}^{g',g}_\sigma =
\int_{\Gamma_{[0,r]}}\! d\rho
\int_{\Gamma_{[0,t]}}\! d\tau \, {\upsilon}^{g',g}_\rho \circ {\upsilon}^{S^*_r g',S^*_r g}_\tau .$$ In this case the associated semigroup generators are given by $$\label{psi omega phi}
\psi_{c',c} = \omega_{{\widehat}{c'},{\widehat}{c}}{\bullet}\phi.$$
\[H cor\] Let $\phi \in L\big( {\mathsf{Y}};{\mathsf{Y}}{\odot}{\mathcal{O}}({{\widehat}{D}}) \big)$ and suppose that ${\mathsf{Y}}$ is finite dimensional. Then $k^\phi$ is an $L\big( {\mathsf{Y}};{\mathsf{Y}}{\odot}{\mathcal{O}}({\mathcal{E}_D})\big)$-valued Markov-regular quantum stochastic cocycle.
This follows from the theorem above and Theorem \[existence\] since, for finite dimensional ${\mathsf{Y}}$, there are natural linear identifications $$L\big( {\mathsf{Y}};{\mathsf{Y}}{\odot}{\mathcal{O}}(E)\big) =
L\big( E ; L({\mathsf{Y}};{\mathsf{Y}}{\odot}| {\mathsf{H}}{\rangle}) \big) =
L \big( E ; CB({\mathsf{Y}};{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{\mathsf{H}}{\rangle}) \big) ,$$ for $(E,{\mathsf{H}})$ equal in turn to $({{\widehat}{D}},{{\widehat}{{\mathsf{k}}}})$ and $({\mathcal{E}_D},{\mathcal{F}})$.
We now begin to develop converse results. The first is a coordinate-free counterpart to Theorem 5.6 of [@LWjfa] in the operator space setting.
\[cocycle thm\] Let $k \in {{\mathbb{QSC}}^\ddagger}({\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$ and suppose that $k$ is Markov-regular and the maps $t \mapsto k_t (x) \xi$ and $t \mapsto k_t (x)^* \xi'$ [(]{}$x \in {\mathsf{Y}}, \xi \in {\mathcal{D}}, \xi' \in {\mathcal{D}'}$[)]{} are all continuous at $0$. Then $k$ is a $({\mathcal{D}'},{\mathcal{D}})$-weak solution of the quantum stochastic differential equation for some $\phi \in SL\big({\widehat}{D'},{\widehat}{D}; B({\mathsf{Y}})\big)$.
Define a map as follows $$\phi : {\widehat}{D'} \times {{\widehat}{D}}\to B({\mathsf{Y}}),
\left( \binom{z'}{c'} , \binom{z}{c} \right) \mapsto
\begin{bmatrix} {\overline}{z'-1} & 1 \end{bmatrix}
\begin{bmatrix} \psi_{0,0} & \psi_{0,c} \\ \psi_{c',0} & \psi_{c',c}
\end{bmatrix}
\begin{bmatrix} z-1 \\ 1 \end{bmatrix}$$ where {$\psi_{c',c}:c'\in D', c\in D$} are the generators of $k$’s associated semigroups and, for $x \in {\mathsf{Y}}$, let $\phi (x)$ denote the corresponding map ${\widehat}{D'} \times {\widehat}{D} \to
{\mathsf{Y}}$. Markov-regularity implies that $l^{g',g}$, given by , satisfies $$l^{g',g}_t = \operatorname{id}_{{\mathsf{Y}}} + \int_0^t ds \, l^{g',g}_s \circ \psi_{c',c},$$ where $c'=g'(t_-)$ and $c=g(t_-)$. But, by the semigroup decomposition, $l^{g',g}=k^{g',g}$; since $\phi^{{\widehat}{c'}}_{{{\widehat}{c}}} = \psi_{c',c}$ it therefore suffices only to prove that $\phi$ is sesquilinear.
Accordingly, fix $v' \in {\mathfrak{h}}' , v \in {\mathfrak{h}}$ and $ x \in {\mathsf{Y}}$ and note the identity $$\big{\langle}v', \phi^{\zeta'}_{\zeta}(x) v\big{\rangle}=
\lim_{t \to 0^+} t^{-1}
\big{\langle}\alpha (t) , \beta (t) \big{\rangle}$$ where $\zeta' = \binom{z'}{c'}\in{\widehat}{D'}$, $\zeta = \binom zc \in{{\widehat}{D}}$, $$\begin{aligned}
\alpha (t) &= \big( k^\dagger_t (x^*) - x^* {\otimes}1 \big)
\Big( v' {\otimes}\big\{ (z'-1) {\varepsilon}(0) + {\varepsilon}(c'_{[0,t[}) \big\} \Big)
\text{ and} \\
\beta (t) &= v {\otimes}\big( z,c_{[0,t[} , (2!)^{-1/2} (c_{[0,t[})^{ {\otimes}2} ,
\ldots \big) ,\end{aligned}$$ Thus if $\zeta = \zeta_1 +\lambda \zeta_2$ for $\zeta_i = \binom{z_i}{c_i} \in {{\widehat}{D}}$ $(i=1,2)$ and $\lambda \in
{\mathbb{C}}$ then $$\big{\langle}v', \big(
\phi^{\zeta'}_\zeta(x) -
\phi^{\zeta'}_{\zeta_1}(x)
- \lambda
\phi^{\zeta'}_{\zeta_2}(x)
\big) v \big{\rangle}= \lim_{t \to 0^+} \big{\langle}\alpha (t) , \gamma (t) \big{\rangle}$$ where $$\gamma (t) =
t^{-1} v{\otimes}\big( (n!)^{-1/2} \big\{ c^{{\otimes}n} -(c_1)^{{\otimes}n}
- (\lambda c_2)^{{\otimes}n} \big\}
{\otimes}1_{[0,t[^n} \big)_{n \geq 2} .$$ Since $\gamma$ is locally bounded and $\alpha (t) \to 0$ as $t \to 0$, by the continuity of the process $k^\dagger$, this shows that $\phi(x)$ is linear in its second argument. A very similar argument, in which the roles of $k$ and $k^\dagger$ are exchanged, shows that $\phi(x)$ is conjugate linear in its first argument. The result follows.
In view of Corollary \[uniqueness cor\], $k$ is the *unique* linear $({\mathcal{D}'}, {\mathcal{D}})$-weak solution of . In particular, if either
$\phi\in L\big({{\widehat}{D}}; {{\mathsf{k}}{\text{-}B}}({\mathsf{Y}};{{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{{\widehat}{{\mathsf{k}}}}{\rangle}}\big)$, or
${\mathsf{Y}}$ is finite dimensional and $\phi\in L\big( {\mathsf{Y}};{\mathsf{Y}}{\odot}{\mathcal{O}}({{\widehat}{D}}) \big)$,
then $k=k^\phi$ and so satisfies the equation *strongly*. If ${\mathsf{Y}}$ is a $C^*$-algebra and $k$ is completely positive and contractive then (a) holds (by [@LWjfa], Theorem 5.4 and [@LWblms], Theorem 2.4); it also holds if ${\mathsf{k}}$ is finite dimensional.
We next identify a necessary and sufficient condition for (b) to hold. To this end let ${{\mathbb{QSC}}_{\mathrm{Hc}}}({\mathsf{Y}}: {\mathcal{D}},{\mathcal{D}'})$ denote the collection of cocycles $k\in{\mathbb{QSC}}({\mathsf{Y}}: {\mathcal{D}},{\mathcal{D}'})$ for which $$\label{Holder}
k_{t,|{\varepsilon}{\rangle}}(x) \text{ is bounded and }
s\mapsto k_{s,|{\varepsilon}{\rangle}}(x)\in{{\mathsf{V}}{\otimes_{{\mathrm{M}}}}|{\mathcal{F}}{\rangle}}\text{ is H\"older } \tfrac{1}{2}\text{-continuous at } 0$$ $(t\in{{\mathbb{R}}_+}, {\varepsilon}\in{\mathcal{E}_D}, x\in{\mathsf{Y}})$. Let ${{\mathbb{QSC}}^\ddagger}({\mathsf{Y}}: {\mathcal{D}},{\mathcal{D}'})$ denote the set of processes $k\in{\mathbb{P}^\ddagger}(V:{\mathcal{D}},{\mathcal{D}'})$ such that both $k$ and $k^{\dagger}$ satisfy .
\[B lem\] Let $k\in{{{\mathbb{QSC}}^\ddagger}_{\mathrm{Hc}}}({\mathsf{Y}}:{\mathcal{D}},{\mathcal{D}'})$ be Markov-regular, with resulting $\phi$ [(]{}from Theorem \[cocycle thm\][)]{} viewed as a linear map ${\mathsf{Y}}\to SL({\widehat}{D'},{\widehat}{D};{\mathsf{Y}})$. Then, for all $x\in{\mathsf{Y}}$, $\phi(x)$ is separately continuous in each argument.
Fix $x \in {\mathsf{Y}}$ and let $\zeta' = \binom{z'}{c'} \in {\widehat}{D'}$ and $\zeta = \binom zc \in {{\widehat}{D}}$. Then, in terms of the generators of the associated semigroups, $\phi^{\zeta'}_\zeta(x)$ equals $${\overline}{z'} \big\{ (z-1) \psi_{0,0} (x) + \psi_{0,c} (x) \big\} +
(z-1)\big\{ \psi_{c',0} (x) - \psi_{0,0} (x) \big\} +
\big\{ \psi_{c',c} (x) - \psi_{0,c} (x) \big\}$$ and, for each $v' \in {\mathfrak{h}}' , e \in D$ and $v \in {\mathfrak{h}}$, setting $C(x,e)
=
\sup \big\{ t^{-1/2} \| k_{t,| {\varepsilon}{\rangle}} (x) -
x {\otimes}| {\varepsilon}{\rangle}\| :t \in ]0,1[ \big\}$ where ${\varepsilon}= {\varepsilon}(e_{[0,1[})$, $$\begin{aligned}
\lefteqn{ \big|
\big{\langle}v', (\psi_{c',e} (x) - \psi_{0,e} (x) ) v \big{\rangle}\big| } \\
& \quad = \lim_{t \to 0^+} t^{-1} e^{-t {\langle}c',e {\rangle}}
\Big| \big{\langle}v' {\otimes}\{ {\varepsilon}(c'_{[0,t[}) - {\varepsilon}(0) \big\} ,
\big( k_t (x) - x {\otimes}1 \big) v {\otimes}{\varepsilon}(e_{[0,1[}) \big{\rangle}\Big| \\
& \quad \leq \| v' \| \, \| c' \| C(x,e) \| v \| .\end{aligned}$$ Thus $\| \psi_{c',e} (x) - \psi_{0,e} (x) \| \leq \| c' \| C(x,e)$. It follows that $$\begin{aligned}
\lefteqn{ \| \phi^{\zeta'}_\zeta(x) \|} \\
& \quad \leq |z'| \big\| (z-1) \psi_{0,0} (x) + \psi_{0,c} (x) \big\|
+ | z-1 | \, \| c'\| C(x,0) + \| c'\| C(x,c) \\
& \quad \leq \| \zeta' \| M(\zeta , x) ,\end{aligned}$$ where $M(\zeta ,x)$ is a constant independent of $\zeta'$. Thus the sesquilinear map $\phi(x)$ is continuous in its first argument. Again applying the above argument to $k^\dagger$ yields continuity in the second argument.
If ${\mathsf{Y}}$ is finite dimensional then the continuity assumption introduced in is equivalent to Hölder-continuity at $0$ of the map $$s\mapsto k_{s,|{\varepsilon}{\rangle}}\in B({\mathsf{Y}};{{\mathsf{Y}}{\otimes_{{\mathrm{M}}}}|{\mathcal{F}}{\rangle}})
\quad
({\varepsilon}\in{\mathcal{E}_D}).$$ If ${\mathfrak{h}}$ is finite dimensional then this further reduces to the pointwise strong continuity condition $$s\mapsto k_s(x) \xi \in {\mathfrak{h}}' {\otimes}{\mathcal{F}}\text{ is H\"older } \tfrac{1}{2}\text{-continuous at } 0
\quad
(x \in {\mathsf{Y}}, \xi \in {\mathcal{D}}).$$ We alert the reader to the fact that not all finite dimensional operator spaces can be concretely realised in $B({\mathsf{H}})$, in the sense of a completely isometric embedding, for a finite dimensional Hilbert space ${\mathsf{H}}$. For more on this point, and for details of an example given by the operator space spanned by the canonical unitary generators of the universal $C^*$-algebra of a free group $\mathbb{F}_n$ ($n \geq 3$), we refer to [@Pisier].
\[f.d. cocycle propn\] Let $k \in {{{\mathbb{QSC}}^\ddagger}_{\mathrm{Hc}}}({\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$ and suppose that ${\mathsf{Y}}$ is finite dimensional. Then there is $\phi \in L \big( {\mathsf{Y}};{\mathsf{Y}}{\odot}{\mathcal{O}^\ddagger}({{\widehat}{D}}, {\widehat}{D'}) \big)$ such that $k = k^\phi$.
Note first that, since ${\mathsf{Y}}$ is finite dimensional, the continuity assumption implies that $k$ is Markov-regular. Let $\phi\in L({\mathsf{Y}}; SL({\widehat}{D'},{\widehat}{D};{\mathsf{Y}}))$ be the map resulting from Theorem \[cocycle thm\]. Choose an ordered basis $\{ x_1, \ldots , x_n \}$ of ${\mathsf{Y}}$ and for $x \in {\mathsf{Y}}$, $\zeta' \in {\widehat}{D'}$ and $\zeta \in {{\widehat}{D}}$, let $\phi^{\zeta'}_\zeta(x)^i$, $i = 1, \ldots , n$, denote the components of $\phi^{\zeta'}_\zeta(x)$, with respect to this basis. By Lemma \[B lem\] each functional $\phi(x)^i : {\widehat}{D'} \times {{\widehat}{D}}\to {\mathbb{C}}$ is sesquilinear and continuous in each argument; it is therefore given by an operator $\phi^{(i)}(x) \in {\mathcal{O}^\ddagger}({{\widehat}{D}}, {\widehat}{D'})$: $$\phi^{\zeta'}_\zeta(x)^i = {\langle}\zeta' , \phi^{(i)}(x) \zeta {\rangle}\qquad (\zeta'\in{\widehat}{D'}, \zeta\in{{\widehat}{D}}).$$ Moreover, each map $x \mapsto \phi^{(i)}(x)$ is clearly linear. Thus, setting $$\phi (x) = \sum^n_{i=1} x_i {\otimes}\phi^{(i)}(x)$$ defines a linear map $\phi : {\mathsf{Y}}\to {\mathsf{Y}}{\odot}{\mathcal{O}^\ddagger}({{\widehat}{D}}, {\widehat}{D'})$. Therefore, by Corollary \[H cor\], $\phi$ generates a stochastic cocycle. In view of the identity $$(\omega_{{\widehat}{c'} , {{\widehat}{c}}} {\bullet}\phi) (x) =
\sum^n_{i=1} \phi^{{\widehat}{c'}}_{{{\widehat}{c}}}(x)^i x_i =
\phi^{{\widehat}{c'}}_{{{\widehat}{c}}}(x) = \psi_{c',c}(x)$$ and Theorem \[H thm\], $k$ has the same associated semigroups as the cocycle $k^\phi$. Thus $k = k^\phi$ and the proof is complete.
By *finite localisability* for a process $k\in{\mathbb{P}}({\mathsf{Y}}\to{\mathsf{Y}}: {\mathcal{D}},{\mathcal{D}'})$ we mean finite localisability for each $k_t$. Combining the above result with Corollary \[uniqueness cor\] and Theorem \[more local existence\], straightforward localisation arguments allow us to summarize the new results of this section as follows.
\[QGC a\] Let $\phi \in L\big( {\mathsf{Y}}; {\mathsf{Y}}{\odot}{\mathcal{O}}({{\widehat}{D}})\big)$ be finitely localisable. Then $k^\phi \in \mathbb{QSC}_{\mathrm{Hc}}({\mathsf{Y}}: {\mathcal{D}}, {\mathcal{D}'})$ and is finitely localisable, moreover if $\phi\in L\big( {\mathsf{Y}}; {\mathsf{Y}}{\odot}{\mathcal{O}}^{\ddagger} ({{\widehat}{D}},{\widehat}{D'})\big)$ then $k^\phi \in \mathbb{QSC}^\ddagger_{\mathrm{Hc}}({\mathsf{Y}}: {\mathcal{D}},
{\mathcal{D}'})$.
\[QGC b\] Conversely, let $k \in \mathbb{QSC}^\ddagger_{\mathrm{Hc}}({\mathsf{Y}}:{\mathcal{D}}, {\mathcal{D}'})$ be finitely localisable. Then there is a finitely localisable map $\phi\in L\big( {\mathsf{Y}}; {\mathsf{Y}}{\odot}{\mathcal{O}}^{\ddagger} ({{\widehat}{D}},{\widehat}{D'})\big)$ such that $k = k^\phi$.
Application to coalgebraic cocycles
-----------------------------------
Theorem \[f.d. cocycle propn\] yields an alternative proof of the principal implication in Theorem 5.8 of [@LSaihp] which states that if ${\mathcal{C}}$ is a coalgebra with coproduct $\Delta$ and counit ${\epsilon}$, then any Hölder-continuous quantum stochastic *convolution cocycle* $l\in {\mathbb{P}^\ddagger}({\mathcal{C}}\to {\mathbb{C}};{\mathcal{E}_D},{\mathcal{E}}_{D'})$, with Hölder-continuous conjugate, satisfies a coalgebraic quantum stochastic differential equation $$\label{coalg QSDE}
dl_t = l_t\star_{\tau} d\Lambda_{\varphi}(t), \quad
l_0 = \iota\circ{\epsilon},$$ for some map $\varphi\in L({\mathcal{C}};{\mathcal{O}}^{\ddagger}({{\widehat}{D}},{\widehat}{D'}))$. We end with a sketch of a proof of this. The Fundamental Theorem on Coalgebras and localisation arguments allow us to effectively assume that ${\mathcal{C}}$ is finite dimensional. Assuming this, linearly embed ${\mathcal{C}}$ into $B({\mathfrak{h}})$, for some (finite dimensional) Hilbert space ${\mathfrak{h}}$, and observe that the process $k\in
{\mathbb{P}^\ddagger}({\mathcal{C}}\to {\mathcal{C}};{\mathfrak{h}}{\odot}{\mathcal{E}_D},{\mathfrak{h}}{\odot}{\mathcal{E}}_{D'})$, defined by the formula $$k_t = (\operatorname{id}_{{\mathcal{C}}} {\odot}l_t) \circ \Delta
\quad
(t\geq 0),
\label{conv1}$$ is a Hölder-continuous quantum stochastic cocycle on ${\mathcal{C}}$. Theorem \[f.d. cocycle propn\] then implies that $k$ satisfies the quantum stochastic differential equation for some $\phi \in L({\mathcal{C}}; {\mathcal{C}}{\odot}{\mathcal{O}}^{\ddagger}({{\widehat}{D}},{\widehat}{D'}))$. Set $$\label{conv2}
\varphi = (\epsilon {\odot}\operatorname{id}_{{\mathcal{O}}^{\ddagger}({{\widehat}{D}},{\widehat}{D'})} )\circ \phi.$$ It is then easily checked that the convolution cocycle $l$ satisfies the coalgebraic quantum stochastic differential equation .
The idea outlined here, of using correspondences such as and for moving between quantum stochastic cocycles and quantum stochastic convolution cocycles, or their respective stochastic generators, also works well in the analytic context of quantum stochastic convolution cocycles on operator space coalgebras. This enables application of known results for quantum stochastic cocycles to the development of a theory of quantum Lévy processes on compact quantum groups and the characterisation of their stochastic generators. This is done in the forthcoming paper [@LSqscc2] which also contains many examples. Dilation of completely positive convolution cocycles on a $C^*$-bialgebra to $*$-homomorphic convolution cocycles is treated in [@Sdilations]. The main results, in both the algebraic and $C^*$-algebraic cases, are summarized in [@LSbedlewo].
[SCHM]{}
L.Accardi, On the quantum Feynman-Kac formula, *Rend. Sem. Mat. Fis. Milano* **48** (1978), 135–180 (1980).
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W.S.Bradshaw, Stochastic cocycles as a characterisation of quantum flows, *Bull. Sci. Math.* (2) **116** (1992), 1–34.
E.G.Effros and Z.J.Ruan, “Operator Spaces,” Oxford University Press, Oxford 2000.
M.P.Evans, Existence of quantum diffusions, *Probab.Theory Related Fields*, **81** (1989) no.4, 473–483.
F.Fagnola, Characterization of isometric and unitary weakly differentiable cocycles in Fock space, *in* “Quantum Probability and Related Topics VIII,” *ed. L.Accardi*, World Scientific, Singapore 1993, pp.143–164.
A.Guichardet, “Symmetric Hilbert Spaces and Related Topics. Infinitely divisible positive definite functions. Continuous products and tensor products. Gaussian and Poissonian stochastic processes,” Lecture Notes in Mathematics **261**, Springer, Berlin 1972.
R.L.Hudson and J.M.Lindsay, On characterizing quantum stochastic evolutions, *Math. Proc. Cambridge Philos. Soc.* **102** (1987) no.2, 363–369.
R.L.Hudson and K.R.Parthasarathy, Quantum Itô’s formula and stochastic evolutions, *Comm.Math.Phys.* **93** (1984) no.3, 301–323.
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J.M.Lindsay and A.G.Skalski, Quantum stochastic convolution cocycles I, *Ann. Inst. H. Poincaré, Probab. Statist.* **41** (2005) no.3 (En hommage à Paul-André Meyer), 581–604.
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J.M.Lindsay and A.G.Skalski, Quantum stochastic convolution cocycles II, *in preparation*
J.M.Lindsay and S.J.Wills, Existence, positivity, and contractivity for quantum stochastic flows with infinite dimensional noise, *Probab.Theory Related Fields* **116** (2000), 505–543.
J.M.Lindsay and S.J.Wills, Markovian cocycles on operator algebras, adapted to a Fock filtration, *J.Funct.Anal.* **178** (2000) no.2, 269–305.
J.M.Lindsay and S.J.Wills, Existence of Feller cocycles on a $C^*$-algebra, *Bull.London Math.Soc.* **33** (2001) no.5, 613–621.
J.M.Lindsay and S.J.Wills, Homomorphic Feller cocycles on a $C^*$-algebra, *J.London Math.Soc.* (*2*) **68** (2003) no.1, 255–272.
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[^1]: *Permanent address of AGS*. Department of Mathematics, University of Łódź, ul. Banacha 22, 90-238 Łódź, Poland.
AGS acknowledges the support of the Polish KBN Research Grant 2P03A 03024 and EU Research Training Network HPRN-CT-2002-00279
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abstract: 'The $\Delta$-scaling method has been applied to ultra-relativistic p+p, C+C and Pb+Pb collision data simulated using a high energy Monte Carlo package, LUCIAE 3.0. The $\Delta$-scaling is found to be valid for some physical variables, such as charged particle multiplicity, strange particle multiplicity and number of binary nucleon-nucleon collisions from these simulated nucleus-nucleus collisions over an extended energy ranging from $E_{lab}$ = 20 to 200 A GeV. In addition we derived information entropy from the multiplicity distribution as a function of beam energy for these collisions.'
author:
- 'G. L. Ma'
- 'Y. G. Ma'
- 'K. Wang'
- 'B. H. Sa'
- 'W. Q. Shen'
- 'H. Z. Huang'
- 'X. Z. Cai'
- 'H. Y. Zhang'
- 'Z. H. Lu'
- 'C. Zhong'
- 'J. G. Chen'
- 'Y. B. Wei'
- 'X. F. Zhou'
title: ' $\Delta$-scaling and Information Entropy in Ultra-Relativistic Nucleus-Nucleus Collisions '
---
[^1]
[^2]
£¬
Ultra-relativistic heavy ion collisions provide a unique means to search for a new state of matter, the Quark-Gluon Plasma (QGP), where quarks and gluons over an extended volume are de-confined \[1-3\] and a phase transition between hadrons and the QGP will occur \[4-6\]. There could be a discontinuity in global features, in particular, the entropy from particle multiplicities, in nuclear collisions associated with the onset of the phase transition. In this Letter we report the first application of the $\Delta$-scaling method and information entropy calculation in ultra-relativistic heavy ion collisions.
In intermediate energy heavy ion collisions, $\Delta$-scaling was proposed by Botet and Ploszajczak \[7\]. They applied the $\Delta$-scaling law to the INDRA data in intermediate energy heavy ion collisions (Xe+Sn, 25-100 MeV/nucleon) by using $Z_{max}$ (the maximum of charge in reactions) as order parameter \[8\] and found that the distributions of $Z_{max}$ obey the $\Delta$ = 1/2 scaling law below 32 MeV/nucleon while they obey the $\Delta$ = 1 scaling law above 32 MeV/nucleon. This indicates that a transition from an order phase to the maximum fluctuation phase (disorder phase) occurs around 32 MeV/nucleon. Recently, Ma et al. analysed the multi-fragmentation data for mass around $\sim$ 36 light nuclei systems. They found a similar change of $\Delta$-scaling occurs when the excitation energy of the system around 5.6 MeV/nucleon. Combination of analysis with other probes of phase transition (eg., Zipf law \[9\] and maximum fluctuations \[10\]) reflects a transition of matter phases with the change of $\Delta$-scaling \[11\]. Since $\Delta$-scaling is a useful tool to identify the phase transition, we try to make the similar analysis in ultra-relativistic heavy-ion collisions.
Botet and Ploszajczak proposed $\Delta$-scaling to identify the transition in intermediate energy heavy ion collisions \[12,13\]. The $\Delta$-scaling law is observed when two or more probability distributions $P[m]$ of the stochastic observable $m$ collapse onto a single scaling curve $\Phi$(z) if a new scaling observable is defined by:
$$z = \frac{(m-m^*)}{\langle m\rangle^\Delta}$$
This curve is: $$\langle m\rangle^\Delta P[m] = \Phi (z) = \Phi [\frac{m-m^*}{\langle m\rangle^\Delta}]$$ where $\Delta$ is a scaling parameter, $m^*$ is the most probable value of $m$, and $\langle m\rangle$ is the mean of $m$. When $\Delta$ = 1, this kind of scaling law is called the first scaling law that is caused by self-similarity of system. This self-similarity means that, if these distributions with different $\langle m\rangle$ by a new kind of variable $z$, they entirely collapse on the same curve. In fact, the famous KNO scaling \[14\] is the special case that the $\Delta$=1 scaling law holds with a stochastic observable of multiplicity of particles. The INDRA data was explored using the $\Delta$ = 1/2 and 1 scaling laws and a phase transition was observed \[15\]. If we assume that $P[m]$ is a Gaussian distribution, we have: $$P [m] = \frac{1}{\sigma \sqrt{2\pi}} exp[-\frac{1}{2} (\frac{m-\mu}{\sigma})^2 ]$$ where $\mu = \langle m\rangle = m^*$ , $\sigma$ is the width of the Gaussian distribution, they both depend on incident energy. If this Gaussian distribution $P[m]$ obeys $\Delta$-scaling law , we should have: $$\mu^\Delta \propto \sigma$$
On the other hand, the information entropy is observable such that it was proposed to describe the fluctuation and disorder of a system by Shannon \[16\]. It is defined by: $$H = -\int P(m) lnP(m) dm$$ where $\int P(m) dm$ = 1. This indicates that the quantity of fluctuation of system and depend on the distribution $P[m]$. If $P[m]$ is the Gaussian distribution, we have : $$H = ln\sigma +(1+ln2\pi)/2 \simeq ln\sigma + 1.419$$
In this work, we investigate the $\Delta$-scaling law and the information entropy for p+p, C+C and Pb+Pb in high-energy collisions (20-200 AGeV) with help of LUCIAE 3.0 of Sa and Tai \[17\]. The head-on collisions are simulated in this work. The LUCIAE is of a Monte Carlo model and is an extension of the FRITIOF \[18\]. Here the nucleus - nucleus collision is described as the sum of nucleon-nucleon collisions. LUCIAE was improved in the following three aspects: (1) Re-scattering of final hadrons, spectator and participant nucleons are considered in the LUCIAE, because the role of re-scattering \[19\] can not be neglected in high-energy domain. (2) The LUCIAE implements the Firecracker Model that includes collective multi-gluon emission from the color fields of interacting strings in the early stage of relativistic ion collisions. (3) LUCIAE introduces the suppression of strange quarks and effective string tensor to make some parameters related to product of strange quarks and string tensor in the JETSET depend on incident energy, size of system, centrality etc. Generally the LUCIAE can explain many data very well \[20,21\].
First, we choose the particle multiplicity produced in each event as stochastic observable $m$ and simulated 10000 events for p+p at every energy point to obtain normalized multiplicity distributions correspondingly. Then we performed $\Delta$-scaling. As a result, we find that the multiplicity distributions are approximately satisfied with the $\Delta$ = 1 scaling law, as shown in Figure 1.
![$\Delta$ = 1/2 (a) and $\Delta$ = 1 (b) scalings for the multiplicity distribution of p + p reactions for the different beam energies.[]{data-label="Cv_pi"}](fig1.eps)
We calculated $\langle m\rangle = \sum m_i P(m_i)$ and $m_{RMS} = \sqrt{\sum (m_i^2 -(m^*)^2) P(m_i)}$ in terms of these normalized multiplicity distributions directly. While we fitted these distributions with Gaussian distributions to obtain $\mu$ and $\sigma$. By comparing $\mu$ with $\langle m\rangle$ and $\sigma$ with $m_{RMS}$, we found that these distributions are basically Gaussian. Table 1 shows the fitting parameters and information entropy ($H_{direct}$ or $H_{gauss}$, which was calculated directly or by Gaussian parameter, respectively (see details in Ref. \[22\] and in the following).
[lllllll]{} E(AGeV) & $H_{direct}$ & $H_{gauss}$ & $\langle m\rangle$ & $\mu$ & $\sigma$ & $m_{RMS}$\
20 & 2.237 & 2.290&7.374&7.332&2.390&2.330\
40&2.512&2.553&8.820&8.724&3.107&3.032\
60&2.634&2.687&9.767&9.638&3.555&3.470\
80&2.747&2.797&10.76&10.60&3.968&3.880\
100&2.791&2.855&11.24&11.11&4.202&4.127\
120&2.841&2.893&11.65&11.54&4.365&4.320\
140&2.912&2.921&12.18&12.25&4.489&4.617\
160&2.935&2.925&12.56&12.54&4.508&4.633\
180&2.965&2.986&12.55&12.48&4.792&4.783\
200&3.008&3.030&13.11&12.92&5.010&4.951\
In order to better investigate the $\Delta$-scaling, we defined a coefficient $$L = \frac{\langle m\rangle^\Delta}{ m_{RMS}} ,$$ which characterizes the validity of the $\Delta$-scaling. We investigat its dependence of incident energy with different $\Delta$ values. Figure 2 shows that the $L$-dependence on incident energy from $\Delta$ = 0.5 to $\Delta$ = 1.6 . As a result, we find that $L$ is nearly a constant in whole investigated range (20-200 AGeV) when $\Delta$=1.35, i.e., the system obeys the $\Delta$-scaling law most suitably with $\Delta$ = 1.35 (see Fig.3a).
![$L$ as a function of energies.[]{data-label="fig2"}](fig2.eps)
We also deal with another two systems of C+C (5000 events) and Pb+Pb (1000 events). Similarly, their multiplicity distributions basically obey Gaussian distributions. In the same way, the best $\Delta$-scaling is obtained with $\Delta$ = 1.00 for C+C and $\Delta$ = 0.80 for Pb + Pb, respectively. Figure 3b and 3c illustrate these results.
![$\Delta$-scaling for p + p with $\Delta$ = 1.35 (upper panel), for C + C with $\Delta$ = 1.00 (middle pannel) and for Pb+Pb with $\Delta$ = 0.80 (lower pannel) for the different beam energies.[]{data-label="fig3"}](fig3.eps)
Through $\Delta$-scaling for different systems, we found that the distributions of particle multiplicity obey Gaussian distributions approximately, especially for C + C and Pb + Pb systems and they obey the $\Delta$-scaling law in a wide energy range for a given system, which indicates that no phase transition and no change of reaction mechanism exist. This is expected for the simulated data because the underlying particle production dynamics in the LUCIAE is a smooth function of beam energy without a phase transition.
We calculate the respective information entropy in terms of these distributions of particle multiplicity with the method proposed by Ma in Ref. \[22\]. Figure 4 shows the dependences of the information entropy on incident energy for the p + p, C + C and Pb + Pb systems. The information entropy increases with the incident energy and with the sizes of the system monotonously. Also, the calculated values of $H_{direct}$ are consistent with the value $H_{gauss}$ obtained from Eq. (6) in the Gaussian distribution. Again, no indication of phase transition exists.
![Energy dependence of information entropy for p + p, C + C and Pb + Pb, respectively.[]{data-label="fig4"}](fig4.eps)
In addition, we use the distribution of strange particles multiplicity \[23\] and the distribution of the number of binary collisions \[24\] as two stochastic observables to make the $\Delta$-scaling plots and extract the information entropy in the same way. The similar $\Delta$-scaling was obtained and a monotonous information entropy was also observed.
In summary, we have demonstrated, for the first time to our knowledge, the $\Delta$-scaling of charged particle multiplicity, strange particle multiplicity and the number of binary collisions using simulated p + p, C + C and Pb + Pb collisions from $E_{lab}$ = 20 to $E_{lab}$ = 200 AGeV. The LUCIAE 3.0, which includes the final state interactions, was used for the simulation. The $\Delta$-scaling means that these observables obey a certain kind of universal laws, regardless of beam energy and collision system. We found that the scaling values $\Delta$ for charged particle multiplicity distributions are 1.35, 1.00 and 0.80 for p + p, C + C and Pb + Pb collisions, respectively. Moreover, the information entropy calculated from charged multiplicity distributions increases with the beam energy and with the colliding system size monotonously. Both the $\Delta$-scaling and the entropy values show no dis-continuity as a function of beam energy as expected because the LUCIAE has no change of particle production dynamics, while they are associated with a phase transition in the simulated data. We expect that the $\Delta$-scaling and the entropy variable can be a valuable tool to search for possible dis-continuities in nucleus-nucleus collisions associated with the onset of a QCD phase transition. Further checks for the models which incorporate the QGP phase transition are in progress.
Acknowledgement: YGM is grateful to Prof. J.B. Natowitz for calling his attention to the $\Delta$-scaling in intermediate energy heavy ion collision.
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Ma Y G 1999 Eur. Phys. J. A ${\bf 6}$ 367. Ma Y G 2001 J. Phys. G ${\bf 27}$ 2455. Ma Y G [*et al.*]{} 2003 arXiv:nucl-ex/0303016. Botet R and Ploszajczak M 2001 Nucl. Phys. B ${\bf 92}$ 101c Botet R and Ploszajczak M 2000 Phys. Rev. E ${\bf 62}$ 1825 Koba Z, Nielsen H B, Olesen P 1972 Nucl. Phys. A ${\bf 11}$ 554 Frankland J K and Bougault R, Contribution to International Workshop on Multifragmentation and related topics (IWM2001). C. E. Shannon C E 1948 Bell. Syst. Tech. J ${\bf 27}$ 379. Tai A and Sa B H 1999 Comp. Phys .Commu. ${\bf 116}$ 355 Sjostrand’s T Lecture Note in Lund Sa B H, Tai A and Lu Z D [*et al.*]{} 1995 Phys. Rev. C ${\bf 52}$ 2069 Sa B H [*et al.*]{} 2002 Phys. Rev. C ${\bf 66}$ 044902 Sa B H and Tai A 1997 Phys. Lett. B ${\bf 399}$ 29 Ma Y G 1999 Phys. Rev. Lett. ${\bf 83}$ 3617 Sollfrank J [*et al.*]{} 1998 Nucl. Phys. A ${\bf 638}$ 399c Matinyan S 1999 Phys. Rep. ${\bf 320}$ 261
[^1]: Supported by the Major State Basic Research Development Program under Contract No G200077400, the National Natural Science Foundation of China under Grant No 10135030, and the National Natural Science Foundation of China for Distinguished Young Scholars under Grant No 19725521
[^2]: Corresponding author. Email: [email protected]
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abstract: 'Turbulent flows in the solar wind, large scale current sheets, multiple current sheets, and shock waves lead to the formation of environments in which a dense network of current sheets is established and sustains “turbulent reconnection”. We constructed a 2D grid on which a number of randomly chosen grid points are acting as [**scatterers**]{} (i.e. magnetic clouds or current sheets). Our goal is to examine how test particles respond inside this [**large scale**]{} collection of scatterers. We study the energy gain of individual particles, the evolution of their energy distribution and their escape time distribution. We have developed a new method to estimate the transport coefficients from the dynamics of the interaction of the particles with the scatterers. Replacing the “magnetic clouds” with current sheets, we have proven that the energization processes can be more efficient depending on the strength of the effective electric fields inside the current sheets and their statistical properties. Using the estimated transport coefficients and solving the Fokker-Planck (FP) equation we can recover the energy distribution of the particles only for the sstochastic Fermi process. We have shown that the evolution of the particles inside a turbulent reconnecting volume is not a solution of the FP equation, since the interaction of the particles with the current sheets is “anomalous”, in contrast to the case of the second order Fermi process.'
author:
- 'Loukas Vlahos, Theophilos Pisokas, Heinz Isliker, Vassilis Tsiolis'
- Anastasios Anastasiadis
title: Particle Acceleration and Heating by Turbulent Reconnection
---
Introduction
============
[@Fermi49] introduced a fundamental stochastic process to solve the problem of particle energization (heating and/or acceleration) in space and astrophysical plasmas. His goal was to resolve the mystery of the stable energy distribution of Cosmic Rays (CR) [see details in @Longair11]. The core of his idea had a larger impact on non-linear processes in general and has been the driving force behind all subsequent theories on charged particle energization. He assumed that high energy particles with speed close to the speed of light collide with [magnetic clouds]{} which move in random directions with speed $V$ close to the local Alfvén speed. The reflections of the charged particles at the magnetic clouds, heat or accelerate the particles to substantial energies. The rate of the energy gain for the charged particles is proportional to the square of the ratio of the magnetic cloud speed to the speed of light $(V/c)^2$. A more realistic proposal was put forward initially by [@Kulsrud71]. The magnetic clouds were replaced by a Kolmogoroff spectrum of [**low amplitude MHD waves**]{} and the energization processes was called [**“stochastic heating and acceleration by (weak) turbulence”**]{}.
Research on reconnecting magnetic fields has undergone a dramatic evolution recently due mostly to the development of the numerical simulation techniques. Long current sheets or multiple interacting current sheets will form, on a short time scale, a turbulent environment, consisting of a collection of current sheets [@Matthaeus86; @Galsgaard96; @Drake06; @Onofri06], [see also the recent reviews @Cargill12; @Lazarian12]. On the other hand, Alfvén waves and large scale disturbances traveling along complex magnetic topologies will drive magnetic discontinuities by reinforcing existing current sheets or form new unstable current sheets [see @Biskamp89; @Lazarian99; @Dmitruk04; @Arzner04].
The goals of this article are to introduce three new and important elements in the current discussion of turbulent reconnection in [**large scale systems**]{}: (a) the study of the characteristics of the energy gain of individual particles, (b) the use of the same framework of global and statistical analysis for two types of scatterers, (i) magnetic clouds, which are representative of stochastic energy gain, (ii) [**Unstable Current Sheets (UCS)**]{}, which are representative of systematic energy gain, (c) the development of a new method to estimate the [**transport coefficients from the dynamics of the interaction of the particles with the scatterers**]{}.
Fermi type energization of particles
====================================
[@Fermi49] based his estimates for the proposed acceleration mechanism on several assumptions [see @Longair11]. The particles move with relativistic velocity $u$ and the scatterers (“magnetic clouds”) move with mean speed $V$ much smaller than the speed of light. The energy gain or loss of the particles interacting with the scatterers is $$\label{energyF}
\frac{\Delta W}{W}\approx\frac{2}{c^2}(V^2-\vec{V} \cdot \vec{u}) ,$$ where for head on collisions $\vec V\cdot \vec u < 0$ and the particles gain energy, for overtaking collisions $\vec V\cdot \vec u > 0$ and the particles lose energy. The rate of energy gain in Eq. (1) includes both, a first and a second order term. For relativistic particles the first order term dominates the energy gain. For non-relativistic particles both terms are second order.
The rate of energy gain for relativistic particles is estimated as $dW/dt=W/t_{acc},$ where $t_{acc}=(3\lambda c)/(4V^2)$ and $\lambda$ is the mean free path the particles travel between the scatterers. Assuming that the distribution of the scatterers is uniform inside the acceleration volume and their density is $n_{sc}$, the mean free path will be $\lambda \approx (\sqrt[3]{n_{sc}})^{-1}.$ The particles are not trapped inside the scatterers, [**their interaction is instantaneous**]{} and the temporal evolution of the mean energy is $$\label{Energy}
{\ensuremath{\left\langleW(t)\right\rangle}} = W_0e^{t/t_{acc}} .$$ [@Fermi49] used the FP equation in order to estimate the change of the energy distribution $n(W,t)$ of the accelerated particles. In order to simplify the diffusion equation, he assumed that spatial diffusion is not important and the particles diffuse only in energy space, $$\label{diff}
\frac{\partial n}{\partial t}+\frac{\partial }{\partial W} \left [ F n -\frac{\partial [D n] }{\partial W} \right ]=
-\frac{n}{t_{esc}}+Q ,$$ where $t_{esc}$ is the escape time from an acceleration region with characteristic length $L$, $Q$ is the injection rate, $D$ is the energy diffusion coefficient $$D(W,t) =\frac{{\ensuremath{\left\langle\left(W(t+\Delta t)- W(t)\right)^2\right\rangle}}_W}{2\Delta t},
\label{eq:DWW}$$ and $$F(W,t) =\frac{{\ensuremath{\left\langleW(t+\Delta t)- W(t)\right\rangle}}_W}{\Delta t},
\label{eq:FW}$$ is the energy convection coefficient representing the systematic acceleration, which, as mentioned, here takes the form $F(W,t)=W/t_{acc}$. With ${\ensuremath{\left\langle...\right\rangle}}_W$ we denote the conditional average that $W(t)=W$ (see e.g. [@Ragwitz2001]). Fermi reached his famous result by assuming that: (a) the particles reach a steady state before escaping from the acceleration volume and (b) the energy diffusion coefficient approaches zero asymptotically for the relativistic particles and the acceleration is mainly due to the systematic acceleration term ($F$). Based on these assumptions, the stationary solution of Eq. (\[diff\]) simply is $n(W)\sim W^{-k}$, where $k = 1+t_{acc}/t_{esc}.$ The index $k$ approaches 2 (which is close to the observed value for the CR) only if $t_{acc} \approx t_{esc}.$ In most recent theoretical studies of the second order Fermi acceleration the escape time (which is so crucial for the estimate of $k$) is difficult to estimate quantitatively.
We will expand the initial Fermi model in this article, by replacing the scatterers by randomly distributed UCS, which represents the environment present in turbulent reconnection in a fragmented large scale system. In several recent articles the 3D evolution and the fragmented UCS has been analysed (see [@Guo15; @Dahlin15]), using Particle in Cell numerical codes, and it has been found that the curvature drift competes with the electric field in the efficiency of particle acceleration inside the UCS. It will be a natural continuation of the work presented here to study also the curvature drift case, here we focus on the acceleration by the electric fields. The particle dynamics inside the UCS is complex since internally the UCS are also fragmented and the particles that interact with the fragments of the UCS can lose and gain energy on the microscopic level of description. Yet, on the average and over the entire simulation domain, the particles gain energy systematically before exiting the UCS, see Fig. 6(c) of [@Guo15] and the related discussion. The energy gain is a weak function of energy in the case of electric field acceleration and proportional to the energy in the case of curvature drift. In this article we estimate the [**macroscopic**]{} energy gain by the simple formula $$\label{e:dW_cs}
\Delta W = |q| E_\textrm{eff}\ \ell_\textrm{eff} ,$$ where $E_\textrm{eff} \approx (V/c) \; \delta B$ is the measure of the effective electric field of the UCS, and $\delta B$ is the fluctuating magnetic field encountered by the particle, which is of stochastic nature, as related to the stochastic fluctuations induced by reconnection. $\ell_\textrm{eff}$ is the characteristic length of the interaction of the particle with the UCS and should be proportional to $E_{eff}$, since small $E_{eff}$ will be related to small scale UCS. The scenario of the method used here is: particles approach the scatterers with an initial energy $W_0$ and depart with a energy $W=W_0+ \Delta W$, where $\Delta W$ [**on the macroscopic level**]{} always is positive and follows the statistical properties of the fluctuations $\delta B$.
A Fermi lattice gas model for turbulent reconnection
====================================================
We constructed a 2D grid $(N \times N),$ with linear size $L$. Each grid point is set as either *active* or *inactive*, i.e. scatterer or not. Only a small fraction R $(1-15\%)$ of the grid points are active. The mean free path of the particles moving inside the grid with minimum distance $\ell=L/(N-1)$ is $\lambda_{sc}=\ell/R.$ When a particle encounters an active grid point it is renewing its energy state depending on the physical characteristic of the scatterer (magnetic cloud or UCS).
At time, $t=0$ all particles are located at random positions on the grid. The injected distribution $n(W, t=0)$ is Maxwellian with temperature $T$. The initial direction of motion of every particle is selected randomly. The particles’ individual time $t_i$ is also adjusted between scatterings as $t_{i+1}=t_{i}+ \Delta t, \;\; \Delta t= l_i/u_i,$ with $u_i$ the particle velocity and $l_i$ the distance the particle travels between scatterings. The particles move in a random direction after interaction with the scatterers, being always confined to follow the grid-lines. It is to note that the consequent large angle scattering takes place in position space, and not in velocity space, the large angle scattering is unrelated with the particle energy, and its role is to implement a spatial random walk process on a grid that basically is influencing only the timing of the energization process. We mainly consider electrons and will just briefly comment on the energization of ions.
#### Random “scattering” by magnetic clouds
We start our analysis using the standard stochastic Fermi accelerator, Eq. (\[energyF\]), in order to validate our method for the estimate of the transport coefficients and the solution of the Fokker Planck equation, since this accelerator has been already discussed in the literature using many different approaches. The parameters used in this article are related to the plasma parameters in the low solar corona. We choose the strength of the magnetic field to be $B=100\ G,$ the density of the plasma $n_0= 10^9\ cm^{-3}$ and the ambient temperature around $10\ eV.$ The Alfvén speed is $V_A \approx 7\times 10^8\ cm/sec$, so $V_A$ is comparable with the thermal speed of the electrons. The energy increment is $(\Delta W/W) \sim (V_A/c)^2\approx 5 \times 10^{-4}$ and the length of the simulation box is $10^{10}\ cm$. We consider an open grid, so particles escape from the accelerator when they reach any boundary of the grid, at $t_i = t_{esc,i}.$ We assume in this set-up that only $R=10\%$ of the $601\times 601$ grid points are active.
The temporal evolution of the mean kinetic energy of the particles and the kinetic energy evolution of typical particles are shown in Fig. (\[f:distr\_sof:mW\]). The motion of the particles is typical for a stochastic system with random-walk like gain and loss of energy before exiting the simulation box. The mean energy increases exponentially (after a brief initial period of a few seconds), as is expected from the analysis presented by Fermi (see Eq. (\[Energy\])). The mean free path is given as $\lambda_{sc}=\ell/R \approx 1.67\times 10^8\ cm$, and, using the analytical expression derived by Fermi, we find $t_{acc_{th}}=(3\lambda_{sc} c)/(4V_A^2) \approx 8\ sec$. We can also estimate the acceleration time from our simulation (see Fig. (\[f:distr\_sof:mW\])), by fitting the asymptotic exponential form to the mean kinetic energy, as predicted by Eq. (\[Energy\]), which yields $t_{acc_{num}}\approx 10\ sec$, a value close to the analytically determined one. Fig. (\[f:distr\_sof:tdistr\]) presents the escape time, which is different for each particle, and we use the median value $(\approx 8\ sec)$ as an estimate of a characteristic escape time. In Fig. (\[f:distr\_sof:Wdistr\]) we show the energy distribution function of the particles remaining inside the box after 15 sec. The distribution is a synthesis of a hot plasma and a power law tail, which is extended to $100 MeV$, with slope $k \approx 2.3$. If we use the estimates of $t_{acc}$ and $t_{esc}$ reported, we can estimate the index of the power law tail $k=1+t_{acc}/t_{esc} \approx 2.3.$ So the slope of the accelerated particles agrees with the estimates provided by the theory of the stochastic Fermi process.
\
In Fig. (\[f:distr\_sof:DF\_W\]), the diffusion and convection coefficients at $ t = 15\ sec$, as functions of the energy, are presented. The estimate of the coefficients is based on Eqs. (\[eq:DWW\]) and (\[eq:FW\]), with $\Delta t$ small, whereto we monitor the energy of the particles at a number of regularly spaced monitoring times $t^{(M)}_k$, $k=0,1,...,K$, with $K$ typically chosen as $200$, and we use $t=t^{(M)}_{K-1}$, $\Delta t = t^{(M)}_{K} - t^{(M)}_{K-1}$ in the estimates. Also, in order to account for the conditional averaging in Eqs. (\[eq:DWW\]) and (\[eq:FW\]), we divide the energies $W\left(t^{(M)}_{K-1}\right)_i$ of the particles into a number of logarithmically equi-spaced bins and perform the requested averages separately for the particles in each bin. As Fig. (\[f:distr\_sof:DF\_W\]) shows, both transport coefficients exhibit a power-law shape, with indexes $a_D = 1.57$ and $a_F = 0.70$, for energies above $1\ keV$, $
F(W) = A W^{0.70}, \ \ \ \ \ D(W) = B W^{1.57}.$ These estimates clearly depart from the assumptions made initially by Fermi.
In order to verify the estimates of the transport coefficients, we insert them in the form of the fit into the FP equation (Eq. (\[diff\])) and solve the FP equation numerically (including the escape term, and with $Q=0$). For the integration of the FP equation on the semi-infinite energy interval $[0,\infty)$, we use the pseudospectral method, based on the expansion in terms of rational Chebyshev polynomials in energy space, combined with the implicit backward Euler method for the time-stepping (see e.g. [@Boyd2001]). The resulting energy distribution at final time is also shown in Fig. (\[f:distr\_sof:Wdistr\]), and it turns out to coincide very well with the distribution from the particle simulation in the intermediate energy range that corresponds to the heating of the population, the power-law tail can though not be reproduced by the FP solution. The differences below energies of about $10 \; eV$ are of less importance and can most likely be attributed to the fact that for simplicity we just assumed the transport coefficients to be constant at low energies.
Varying the density of the scatterers in a parametric study in the range $0.01 < R < 0.2$ and keeping the characteristic length of the acceleration volume constant, we find that the main characteristics of the distribution remain the same but the heating and the slope of the accelerated particles vary.
The ions in the asymptotic stage do not appear to have significant differences from the evolution of the electrons. We can then conclude that stochastic Fermi processes can heat and accelerate both ions and electrons in the solar corona, yet on different time scales.
#### A model for turbulent reconnection
We now use the lattice gas model to estimate the heating and acceleration of particles inside a large scale turbulent reconnection environment, where a fragmented distribution of UCS is present. The setup is $R=0.1$, $N=601$, $V=V_A$ and the simulation box has length $10^8\ cm$ and is open. The energy change of a particle that encounters an UCS is now given by Eq. (\[e:dW\_cs\]), and we assume that $\delta B$ takes random values following a power-law distribution with index $5/3$ (Kolmogorov spectrum), and $\delta B \in [10^{-5} G, 100G].$ We also assume the effective length $\ell_\textrm{eff}$ to be a linear function of $E_{eff}$, $\ell_\textrm{eff} =a E_{eff}+b$, and by restricting the size of $\ell_\textrm{eff}$ to $\ell_{eff} \in [10^3 cm, 10^5 cm]$, we determine the constants $a,b.$ Combining all the above we find that the effective electric filed lies approximately in $E_\textrm{eff} \in [10^{-7} E_D,
E_D]$, where $E_D$ is the Dreicer field, $E_D\approx 1.6\cdot
10^{-7}\ statV/cm$.
\
We initiate the simulation with a Maxwellian distribution with temperature $10\ eV$. Fig. (\[f:distr\_cs:mW\]) shows the mean energy and the energy of some typical particles as a function of time, up to final time or untill they escape from the simulation box. The rate at which the particles on the average gain energy is exponential, so $\ln\left< W \right> \approx t/t_{acc}$, and we estimate the asymptotic value of the acceleration time to be $\approx 0.3\; sec$.
The acceleration is systematic and the particles feel a rapid increase of their energy any time they cross an UCS with variable strength of the effective electric field (see the similar behaviour observed in [@Dahlin15; @Guo15]). The energy distribution reaches an asymptotic state (see Fig. (\[f:distr\_cs:Wdistr\])) in s fraction of a second. It is obvious that particles are very efficiently accelerated inside the turbulent reconnecting volume and form a power law tail with index $\approx
1.7$.
Fig. (\[f:distr\_cs:tesc\]) presents the escape time, which is different for each particle, and we use the median value $(\approx
0.5\ sec)$ as an estimate of a characteristic escape time. If we use the estimates of $t_{acc}$ and $t_{esc}$ reported, we can estimate the index of the power law tail $k=1+t_{acc}/t_{esc} \approx 1.6$, which is close to the slope of the distribution of the accelerated particles in the simulation.
In Fig. (\[f:distr\_cs:F\_W\]) the convection coefficient $F$ at $t
= 1\ sec$ is presented as function of the energy, and it exhibits a power-law shape, with index $a_F = 0.76$ for energies above $100\,$eV, an index close to the one found above in Fermi’s original scenario. For the diffusion coefficient, the estimate $D$ based on Eq. (\[eq:DWW\]) yields a power-law, applying though the finite time correction of [@Ragwitz2001], $D_{true} = D -
0.5 \Delta t F^2$, we find that $D_{true} \approx 0$, the energization process is purely convective in nature, the non-zero $D$ is an artifact resulting from the finite time contribution of $F$ to $D$ (we just note that in the Fermi case the finite time correction was negligible).
Using $F$ and $D_{true}$ in the numerical solution of the FP equation, we find only heating, on time-scales though of the order of tens of seconds, much larger than the time of 1 sec considered here. This result is in accordance with and a generalization of the result in [@Guo2014; @Guo15], who also find only heating when analytically solving the FP equation (for $D=0$ and $F\sim W$ in their case). On the other hand, the asymptotic distribution can be calculated from Eq. (\[diff\]) (assuming $\partial n/\partial
t =0$) as $n\sim W^{-0.76}$. The reason for the discrepancy between the FP solution and the asymptotic solution must be attributed to the fact that the asymptotic solution, determined as a stationary solution, cannot be reached with the initial condition being a Maxwellian (in analogy to the case in [@Guo2014] with $F\sim W$).
Concerning the difference between the FP solution and the lattice model, we find that the sample of energy differences $W_i(t+\Delta t)- W_i(t)$ in Eq. (\[eq:FW\]) (with $i$ the particle index), on which the estimate of $F$ is based, follows actually a power law distribution, and as a consequence the particles occasionally perform very large jumps in energy space (Levy flights), as illustrated in Fig. (\[f:distr\_cs:mW\]), in contrast to the second order Fermi process (see Fig. (\[f:distr\_sof:mW\])). The fact that the energy increments have a power-law distribution with the specific index has several consequences: (1) The estimate of $F$ as a mean value theoretically is finite, yet it is very noisy. (2) Both the mean (or the median, as used here) are not representative of a scale-free power-law distribution. (3) The variance of the distribution of energy-increments tends to infinity. After all, in the case at hand, the applicability of the classical random walk theory (classical Langevin and FP equation) breaks down, as it is manifested in the inability of the FP equation to reproduce the test-particles’ energy distribution, and in the practical difficulties of the expressions for $F$ and $D$ in Eqs. (\[eq:FW\]) and (\[eq:DWW\]) to yield meaningful transport coefficients. Thus, modeling tools like the Fractional FP equation become appropriate here. Similar cases of Levy flights have been observed by [@Arzner04] and [@Bian08], without further analyzing the consequences for the transport coefficients and the FP equation.
We also have explored the role of collisions and they are important for impulsive energization longer than the collision time of the system, they though play a crucial role only for the bulk of the energized plasma and just slightly modify the slope of the tail.
Summary and Discussion
======================
Turbulent reconnection is a new type of accelerator which can be modelled with the use of tools borrowed from Fermi type accelerators, namely by replacing the “magnetic clouds" with a new type of “scatterers”, the UCS. This generalization can handle large scale astrophysical systems composed from local accelerators like current sheets appearing randomly in reconnecting turbulence. We developed a 2D lattice gas model where a number of active points act as “scatterers" in order to model the new accelerator. Our main contribution in this article is the estimate of the transport coefficients from the particle dynamics and their use in solving the FP equation. Our main results from this study are: (a) Stochastic Fermi accelerators can reproduce a well known energy distribution in laboratory and astrophysical plasmas, where heating of the bulk and acceleration of the run away tail co-exist. The density of the scatterers plays a crucial role in controlling the heating and the acceleration of particles. (b) The transport coefficients show a general power-law scaling with energy. (c) The replacement of the scatterers with UCS has several effects on the energization of the particles: (i) The acceleration time is an order of magnitude faster than in the stochastic Fermi process. (ii) Estimating the transport coefficients from the dynamic particle orbits, we have shown that the final energy distribution cannot be a solution of the FP equation, since the orbits of the energetic particles in energy space depart radically from Brownian motion, showing characteristics of Levy flights. (iii) The asymptotic distribution of the accelerated particles is similar to the ones obtained in different simulations (see [@Arzner04; @Dmitruk04; @Onofri06; @Drake06; @Drake13; @Dahlin15]), where turbulent reconnection is established.
We can conclude that the stochastic Fermi acceleration and turbulent reconnection processes can play a crucial role in many astrophysical plasmas and their role depends strongly on their physical properties, such as the nature of the scatterers (e.g. large amplitude Alfvén waves or UCS), their spatio-temporal statistical properties (e.g. their spatial density), and the time evolution of the driver of the explosions.
We thank the referee, whose comments helped to improve substantily the article. The authors acknowledge support by European Union (European Social Fund -ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF) -Research Funding Program: THALES: Investing in knowledge society through the European Social Fund.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Ray optics effectively fail to detect an eleven-parameter family of deviations from a metric spacetime geometry. These ray-optically invisible deviations, however, affect quantum field theoretic scattering amplitudes and bound states. To show this, we first prove renormalizability and gauge invariance of the pertinent quantum electrodynamics to any loop order. We then calculate scattering amplitudes and radiative corrections and determine the bound states of the hydrogen atom in the presence of ray-optically undetectable deviations. In particular, we find effects on the anomalous magnetic moment of the electron as well as the ionisation energy and hyperfine transition of hydrogen, which is of direct relevance to astrophysical measurements.'
author:
- 'Simon Grosse-Holz'
- 'Frederic P. Schuller[^1]'
- Roberto Tanzi
bibliography:
- 'biblio.bib'
- 'ext.bib'
title:
- 'Quantum signatures of ray-optically invisible birefringence'
- 'Quantum signatures of ray-optically invisible geometric degrees of freedom'
- 'Quantum signatures of non-metricity invisible to ray-optics'
- 'Quantum signature of non-metricities invisible to ray-optics'
- 'Quantum signatures of ray-optically invisible non-metricity'
- 'Quantum detection of ray-optically invisible spacetime structure'
- 'Quantum detection of ray-optically invisible non-metricities'
- 'Quantum signatures of ray-optically missed non-metricity'
- 'Quantum signatures of ray-optically invisible non-metricities'
---
Introduction
============
Maxwell theory—linear electrodynamics on a metric spacetime manifold—implies that a classical light ray cannot split into two, or more, rays outside a material optical medium. But does the converse hold? Can the observational absence of such a splitting of light rays be used in order to exclude that the underlying spacetime structure deviates from the usual metric geometry, while still keeping with the linear superposition principle of classical electrodynamics? This is not an entirely academic question. For letting an assumption, which is merely sufficient for the absence of a phenomenon, pass as a necessary one, rids one of the due generality that is usually needed in order to understand other phenomena. In our case, these other phenomena are quantum effects caused by non-metricities that do not affect the paths of light rays.
In this article, we first identify all such ray-optically invisible non-metricities that can underlie any linear theory of classical electrodynamics that follows from an action [@truesdell1960classical; @postbook; @Obukhov1999; @hehlbook] on a flat background. This leads to the insight that there is an eleven-parameter family of non-metricities that are not seen to first order perturbation theory around a flat metric background. Strictly adhering to the principles of linear perturbation theory, only linear terms in the non-metricities are kept, while higher order terms are discarded entirely. We emphasize this fact since the ray-optical invisibility of the non-metricities hinges crucially on the employment of first order perturbation theory: deviations from the paths of light rays in metric spacetime are merely quadratically suppressed. This definition of invisibility, however, is in full accordance with the presence of inevitable bounds on the resolution of any experimental apparatus. For conclusions drawn from an assumption of exactly unaltered light paths, see [@Laemmerzahl] for the restriction of the background geometry and [@Klinkhamer1; @Klinkhamer2] for quantum effects. Due to our admission of a finite resolution of any experiment aimed at detecting deviations from light paths on a metric background, our results complement those drawn from the more idealized previous treatments.
Our quantization of linear electrodynamics with ray-optically invisible non-metricities, together with an explicit proof of renormalizability and gauge-invariance to any loop order, then provides the foundation for the investigation of quantum effects of experimental relevance. We calculate the pertinent corrections from these non-metricities to Bhabha scattering and to the anomalous magnetic moment of the electron to the same order as currently known in standard QED. Furthermore, we reveal a qualitatively new coupling of the electron spin and the magnetic field that necessarily obtains in the presence of ray-optically invisible deviations from a Lorentzian geometry. Our ensuing discussion of the bound states of hydrogen yields effects on the ionization energy, a scaling of the astrophysically relevant $21.1\,\textrm{cm}$ line of the hyperfine transition of hydrogen and the qualitatively new lifting of the degeneracy of the participating triplet state. Our results extend and complement existing studies of sectors of the standard model extension [@SME-flat-space; @SME-curved-spacetime; @hydrogenstudy], to whose findings we will therefore refer throughout this article.
Throughout this article, spacetime indices ranging over $0,1,2,3$ are denoted $a, b, c, \dots$, while purely spatial indices with respect to a particular observer frame, ranging over $1, 2, 3$, are denoted by $\alpha, \beta, \gamma, \dots$ . The Levi-Civita tensor densities are normalized as $\epsilon_{0123}=1$ and $\epsilon^{0123}=-1$ and the Minkowski metric $\eta$ in normal coordinates as $\eta_{00}=1$ and $\eta_{\alpha\alpha}=-1$.
Dirac-Maxwell theory with ray-optically invisible deviations {#sec:GLED}
============================================================
The most general electrodynamics that still obey a linear superposition principle have been thoroughly studied [@truesdell1960classical; @postbook; @Obukhov1999; @hehlbook]. The salient feature of these theories, as far as their ray-optical limit is concerned, is that they feature birefringence of arbitrary strength even in vacuo: an unpolarized light ray will generically split into two rays of particular polarizations and speeds. Which polarizations, and which speeds, is determined by the pertinent refined background geometry. The known results relevant for this paper are pithily reviewed in section \[sec:EDyn\]. In order to provide sources for these general linear electrodynamics, we construct first derivative order field equations on such backgrounds in section \[sec:birefringent-spacetime-fermions\]. Their formulation is achieved by refined Dirac matrices, which are defined by a quaternary algebra, rather than the usual binary one, and some subsidiary trace conditions. We will then restrict attention, from section \[sec:ROIDs\] onward, to those eleven-parameter perturbations away from a Minkowskian metric that effectively do not affect the paths of classical light rays. It is then the purpose of the remainder of this paper to show how these ray-optically invisible deviations affect scattering amplitudes of the ensuing renormalizable quantum field theory and the spectrum of atomic hydrogen.
Maximal linear extension of Maxwell theory {#sec:EDyn}
------------------------------------------
The most general electrodynamics coupled to a vector current $j$, which follow from an action and still satisfy a superposition principle, are given by $$\label{Sstrong}
S_\textrm{\tiny strong}[A] = \int{d}^4x\,\,\omega\!\left[-\frac{1}{8} G^{abcd}F_{ab}F_{cd} - j^a A_a\right]\,,$$ where $A$ is the one-form abelian gauge potential and $F = {d}A$ the associated field strength. The fourth rank contravariant tensor $G$ featuring in this action is restricted, without loss of generality, to possess the same algebraic symmetries as a curvature tensor, $$\label{area-metric-symmetries}
G^{abcd} = G^{[ab][cd]} = G^{cdab} \qquad\text{and}\qquad G^{abcd} + G^{acdb} + G^{adbc}=0\,,$$ and can physically be thought of as the constitutive tensor that relates the electromagnetic inductions to the field strengths [@postbook; @hehlbook], while the weight-one scalar density $\omega$ is a further background degree of freedom, independent of the constitutive tensor and needed to make the integral measure well-defined.
Note that we chose to build the required tensor density in the above action as the product of an individual scalar density $\omega$ and an individual tensor $G^{abcd}$, which is subtly more information than would be carried by direct use of a tensor density of the same rank. This choice, however, affords us just sufficient structure to [*gravitationally close*]{} [@SSWD] the above action, which is a procedure to constructively derive, not postulate, the most general dynamics for the geometry $(\omega,G^{abcd})$ such that the derived gravitational dynamics shares its initial value surfaces with those of the given matter dynamics. See [@SSSW], which achieves this explicitly for the weak field gravitational dynamics that will provide the non-metric geometric backgrounds to which we will soon need to restrict for phenomenological reasons. Concrete predictions—for when and where which quantum effects occur—then follow from combining the solutions of that canonical weak field gravity theory and the findings of this paper.
Returning to the pure matter theory, we find that variation of the general linear electrodynamics action with respect to the gauge potential $A$ yields the equations of motion $$\label{eq:motion}
-\frac{1}{\omega}\partial_b(\omega G^{abcd}\partial_c A_d) = j^a\,,$$ whose geometric-optical limit restricts the wave covector of light rays to satisfy a quartic dispersion relation $$P^{abcd}k_a k_b k_c k_d = 0\,,$$ rather than the familiar quadratic one that is enforced in a Lorentzian metric background geometry. The polynomial defined by the left hand side arises as the principal polynomial of the partial differential matter field equation (\[eq:motion\]). Indeed, removing the abelian gauge ambiguity (in one way or another, see [@rubilar2002generally; @itin2009light]), one obtains the explicit expression for the totally symmetric tensor field $$\label{eq:Ppolarization}
{P}^{abcd} = -\frac{1}{24}\omega^2\epsilon_{mnpq}\epsilon_{rstu}G^{mnr(a}G^{b|ps|c}G^{d)qtu}\,$$ that defines the quartic dispersion relation above. If the constitutive tensor $G$ is chosen such that the electromagnetic field equations have a well-posed initial value problem, it follows that the principal polynomial is hyperbolic [@gaarding1951linear].
The quartic structure then generically admits two different future-directed light rays in the same spatial direction, which effect is known as birefringence. Either of them corresponds to a particular wave covector $k$, which in turn is related by [@punzi2009propagation] $$G^{abcd}k_a k_c a_d = 0\,$$ to the polarization $a$ that of the light ray. As was shown in [@RRS], by employing an interplay of results from the convex analysis of hyperbolic polynomials and real algebraic geometry, the quartic dispersion relation already dictates that the action for the worldline $x(\lambda)$ of a light ray is $$\label{lightpathaction}
S_\textrm{\tiny light ray}[x,\mu] := \int d\lambda\, \mu \,P^\#_{abcd}\, \dot x^a\, \dot x^b\, \dot x^c\, \dot x^d\,,$$ where the so-called dual polynomial is given by $$P^\#_{abcd} = -\frac{1}{24}\omega^{-2}\epsilon^{mnpq}\epsilon^{rstu}G_{mnr(a}G_{b|ps|c}G_{d)qtu}\,.$$
Canonical quantizability of the above electromagnetic field equations requires that not only the principal polynomial is hyperbolic, but also this dual polynomial, as is shown in the same article. This ultimately leads to a significant restriction of the possible constitutive tensors $G^{abcd}$ to only 7 out of 23 possible algebraic classes [@SWW]. This is precisely the same mechanism as the one that restricts any metric that can underly standard Maxwell theory to the one algebraic class that is characterized by Lorentzian signature (although in the metric case, hyperbolicity of the principal polynomial $P^{ab}=g^{ab}$ already implies hyperbolicity of the dual dual polynomial given by $P^\#_{ab} = g_{ab}$).
This article is concerned with the quantum effects produced by the presence of non-metricities. This requires to also transfer the standard Dirac equation to the same refined background as underlies the general linear electrodynamics we discussed in this subsection. Remarkably, the principal polynomial defined by (\[eq:Ppolarization\]) already fully determines the relevant refined Dirac algebra in the next section.
Dirac fields on the same background geometry {#sec:birefringent-spacetime-fermions}
--------------------------------------------
In order to provide a charged vector current $j^a$ to source the general linear electrodynamics (\[Sstrong\]), we follow Dirac and consider a field $\psi$ that takes values in some finite-dimensional hermitian inner product space $V$ and satisfies a first derivative order field equation of the form $$\label{generalized-Dirac-equation}
\Big[ i \gamma^{a} \partial_{a} -m \Big] \psi (x)=0 \,,$$ where the $\gamma^a$ are four suitable endomorphisms on $V$. Suitability is here already decided by the requirement that these field equations can employ the same initial value surfaces as the general linear electrodynamics, for the simple reason that they must evolve together. This is ensured [@RiveraPhD] if the endomorphisms $\gamma^a$ satisfy the [*quaternary algebra*]{} $$\label{generalized-Dirac-algebra}
\gamma^{(a} \gamma^b \gamma^c \gamma^{d)} = P^{abcd}\; \textrm{id}_{V}$$ together with the three trace conditions $$\label{generalized-Dirac-algebra-trace}
{\ensuremath{\text{tr}}}\big[ \gamma^a \big]=0 \,, \qquad {\ensuremath{\text{tr}}}\big[ \gamma^{(a} \gamma^{b)} \big]=0 \,, \qquad {\ensuremath{\text{tr}}}\big[ \gamma^{(a} \gamma^b \gamma^{c)} \big]=0\,.$$ Any quadruplet of such matrices $\gamma^a$ thus presents a refinement of the Dirac algebra to a general linear background, and we will thus refer to them as [*refined Dirac matrices*]{}. If one can find, additionally, a hermitian endomorphism $\Gamma=\Gamma^{\dagger}$ satisfying $$(\Gamma^{\dagger})^{-1} (\gamma^a)^{\dagger}\Gamma^{\dagger}=\gamma^a\,,$$ one may also provide an action from which one obtains the refined Dirac equation (\[generalized-Dirac-equation\]) by variation with respect to $\psi$, or equivalently, $\bar{\psi} {\ensuremath{:=}}\psi^{\dagger} \Gamma$, namely $$\label{fermions-action}
S[\psi, \bar{\psi}]=\int d^{\,4}x \; \bar{\psi} (i \gamma^a \partial_a -m) \psi \,.$$
An explicit representation of this refined Dirac algebra has been provided in [@RiveraPhD], but is fortunately not needed for the purposes of this paper. This is because, starting with the next section, we will consider only such general linear backgrounds that arise from a small eleven-parameter perturbation around a flat spacetime metric. For these, the refined Dirac algebra (\[generalized-Dirac-algebra\]) effectively reduces to the standard Dirac algebra and only the electromagnetic gauge potential feels the deviation from a metric geometry, but not the Dirac field. Based on the results stated here, this fortunate reduction of the complexity of the refined Dirac field dynamics can be easily seen at the end of the following subsection.
Ray-optically invisible deviations from a metric background {#sec:ROIDs}
-----------------------------------------------------------
Splitting of light rays [*in vacuo*]{} is experimentally excluded to high precision [@Kostelecky:2002hh; @SME-constraints-gamma-ray; @SME-Cerenkov]. This section is concerned with the circumstances under which this is merely due to quadratic suppression of the effect.
We start by purposely over-interpreting the above observational finding, by taking it to mean that there is exactly zero birefringence. As ultimately follows from [@Laemmerzahl], exactly vanishing birefringence implies, in our technical implementation, that $$G_g^{abcd} := g^{ac}g^{bd} - g^{ad}g^{bc} \qquad\text{and}\qquad \omega_g := \sqrt{-\det g_{ab}}$$ in terms of a Lorentzian metric $g^{ab}$, which indeed corresponds to the dispersion relation $$P^{abcd} = g^{(ab} g^{cd)}\,.$$ This is because an additionally allowed dilaton can first be absorbed into the metric and an additionally allowed axion can be absorbed into the density and then removed by conformal invariance.
Now closing the gap—between the brutish assumption of exactly vanishing birefringence and what is supported experimentally—we admit small, but otherwise generic linear, deviations from a Lorentzian background geometry. We will find that there is an eleven-parameter family of [*linear* ]{} perturbations away from a metric for which a modification of the paths of classical light rays, as determined by the action (\[lightpathaction\]), is [*quadratically*]{} suppressed. More precisely, consider a perturbation $E^{abcd}$ of the constitutive tensor and a perturbation $e$ of the scalar density, $$\label{eq:ansatz}
G^{abcd} = g^{ac}g^{bd} - g^{ad}g^{bc} + E^{abcd} \qquad\text{and}\qquad \omega = \sqrt{-\text{det }g_{ab}} + e\,,$$ and assume that the respective entries are sufficiently small to justify dropping any terms beyond linear order. Careful calculation yields the intermediate result $$P^{abcd} = g'^{(ab} g'^{cd)} \,\,+ \,\,\textrm{second order terms in $E^{abcd}$ and $e$}$$ for $g'^{ab} := (1+e) g^{ab} + \frac{1}{2} E^{ambn} g_{mn}$. This is, of course, not quite yet the announced result of a dispersion relation that is unchanged to zeroth and first order, since the first order contributions in $E^{abcd}$ and $e$ are merely hidden in the notation, and indeed reappear as cross terms upon expansion of $g'^{ab}$. But one can easily show that these first order terms are merely an artefact that can be gauged away by a change of local frame. More precisely, employing the pointwise transformation $$e^{a'}{}_a = (1-\tfrac{1}{2}e) \delta^{a'}{}_a - \tfrac{1}{4} E^{a'mk n} g_{mn} g_{ak}\,,$$ one finds for the constitutive tensor and scalar density in the new frame that $$\label{perta}
G^{abcd} = g^{ac} g^{bd} - g^{ad} g^{bc} + E^{abcd} \qquad \textrm{ and } \qquad \omega = \sqrt{-\det g_{ab}} + e$$ for perturbations $E^{abcd}$ and $e$ that satisfy the gauge conditions $$\label{pertconditionsa}
E^{amb}{}_m = \frac{1}{4} E^{mn}{}_{mn} g^{ab} \qquad\textrm{ and } \qquad e = - \frac{1}{8} E^{mn}{}_{mn}\,.$$ This reveals two important facts. First, only eleven components of the perturbation of the constitutive tensor effect a deviation from a metric geometry. Secondly, any effective perturbation of the scalar density is entirely determined by the effective perturbation of the constitutive tensor. Most importantly, however, one now finds that the associated dispersion relation, $$\label{standardP}
P^{abcd} = g^{(ab} g^{cd)} \,\,+ \,\,\textrm{second order terms in $E$ and $e$}\,,$$ does not contain any linear correction in the effective perturbations. Deviations from light paths in the ray-optical limit are thus quadratically suppressed, as claimed. Without loss of generality, we may thus restrict attention to perturbations (\[perta\]) that satisfy the conditions (\[pertconditionsa\]) when considering ray-optically invisible perturbations.
Interacting electrodynamics with ray-optically invisible flat perturbations
---------------------------------------------------------------------------
On the scale of the hydrogen atom (whose modified spectrum we calculate in section \[sec:hydrogen\]), and below (such as for the $N$-loop correction of the anomalous magnetic moment of the electron derived in section \[sec:anomalous-magnetic-moment\]), the effects of spacetime curvature can safely be neglected. For the purpose of this article, we may thus assume spacetime to be flat, such that the components of the constitutive tensor are constant throughout the manifold and the above construction is valid globally. Moreover, we may choose to work in coordinates where $g^{ab}=\eta^{ab}$, and of course do so, for calculational convenience and because results thus are those seen from a local observer frame [@RRS].
Clearly, for ray-optically invisible deviations from metric geometry, the generically quartic Dirac algebra identified in section \[sec:birefringent-spacetime-fermions\] factorizes, and thus reduces to the familiar binary Dirac algebra. Thus these deviations are effectively invisible not only to the light rays of geometric optics, but also to Dirac fermions. In any case, coupling of Dirac fields with the electromagnetic field via minimal coupling is straightforwardly achieved by use of a charged current $j^a := Q {e_\textrm{\tiny pos}}\bar\psi \gamma^a \psi$, where ${e_\textrm{\tiny pos}}>0$. We thus have, on a ray-optically invisibly refined flat background, the total action $$\label{action-gled-fermion}
S[\psi,A]=\int d^{\,4} x\, (1+e)\! \left[ -\frac{1}{8} (\eta^{ac} \eta^{bd} - \eta^{ad} \eta^{bc} + E^{abcd}) F_{ab} F_{cd} +
\bar{\psi} (i \gamma^a \partial_a - Q {e_\textrm{\tiny pos}}\gamma^a A_a -m) \psi \right]$$ for general linear electrodynamics minimally coupled to a Dirac field of charge $Q {e_\textrm{\tiny pos}}$, where the perturbations $e$ and $E^{abcd}$ satisfy the conditions (\[pertconditionsa\]) and $\gamma^{(a} \gamma^{b)} = \eta^{ab}$. A similar contribution to $E^{abcd} F_{ab} F_{cd}$ appeared already in the action of the effective field theory known as SME, studied both in flat spacetime [@SME-flat-space] and in curved spacetime [@SME-curved-spacetime]. In this context $E$ is usually denoted by $2k_F$, which is double tracefree but does not satisfy (\[pertconditionsa\]).
We conclude that in linear perturbation theory of a flat spacetime, there is an eleven-parameter family of perturbations of the constitutive tensor that is entirely invisible to geometric optics and Dirac fields, but felt by the electromagnetic gauge potential. It is the existence of this domain that makes the results of this article relevant, since they are not at variance with the observed light paths predicted by standard theory.
Quantization and renormalization {#sec:renormalization}
================================
We show that quantum electrodynamics with ray-optically invisible deviations from a Lorentzian background is renormalizable and gauge invariant to arbitrary loop-order. To this end we prepare dimensional regularization by extending the theory to arbitrary dimensions and then quantize à la Batalin-Vilkovisky. The Feynman rules, which we identify in the course of these more formal developments, will be practically relevant for the calculations of scattering amplitudes and vertex corrections in section \[sec:scatterings\].
Dimensional regularization {#sec:weakly-birefringent-spacetime-complex-dim}
--------------------------
The action (\[action-gled-fermion\]) readily generalizes to integer dimensions $D \geq 3$, with the only difference that $A$, $F$, $E$ and $\eta$ now come as their $D$-dimensional versions, and $\gamma^a$ and $\psi$ are the $2^{\lfloor D/2 \rfloor}$/dimensional representations of the Dirac gamma matrices and the Dirac spinor, respectively. Note that there is no explicit appearance of the dimension in the Lagrangian.
In contrast, the four-dimensional perturbation conditions (\[pertconditionsa\]) do pick up an explicit dependence on the dimension $D$. Indeed, while the $D$-dimensional perturbation still takes the form $$\label{pert}
G^{abcd} = \eta^{ac} \eta^{bd} - \eta^{ad} \eta^{bc} + E^{abcd} \qquad \textrm{ and } \qquad \omega = \sqrt{-\det \eta_{ab}} + e\,,$$ the restrictions on the corresponding $D$-dimensional perturbations $E^{abcd}$ and $e$ now read $$\label{pertconditionsb}
E^{amb}{}_{m}=\frac{1}{D} E^{mn}{}_{mn}\, \eta^{ab} \qquad\textrm{ and } \qquad e = -\frac{1}{2D} E^{mn}{}_{mn} \,,$$ as one shows exactly along the same lines as before for the physical case of four dimensions. The principal polynomial, up to second order corrections, is then again simply the Minkowski metric, now in $D$ dimensions, $$P^{ab} = \eta^{ab}\,,$$ without explicit dependence on $D$.
The above distinction, between implicit and explicit dependence on the spacetime dimension, plays a role for the extrapolation of the above expressions to $D$ complex dimensions. For in the practical implementation of dimensional regularization, one may treat the implicit dimensional dependence of the various tensorial objects (in our case the electromagnetic field $A_a$, Minkowski metric $\eta^{ab}$, perturbations $E^{abcd}$ and $e$, density $\omega$, Dirac matrices $\gamma^a$ and Dirac spinors $\psi$) entirely formally. The only exception would be presented by tensors and tensor densities (such as $\epsilon^{a_1 a_2 \dots a_D}$) whose rank varies with $D$, but this case does not occur in this article. In contrast, any explicit dependence (such as the value of the trace $\eta^{ab} \eta_{ab} = D$ or the above relations (\[pertconditionsb\]) between the various perturbation components) is simply extended from the calculated expressions for integer dimensions to complex $D$. We will use the above quantities with this understanding whenever we dimensionally regularize in this article.
Batalin-Vilkovisky quantization {#sec_BVquant}
-------------------------------
We now quantize the theory in the Batalin-Vilkovisky formalism [@Batalin:1981jr; @Batalin:1984jr]. The proof of renormalizability and gauge invariance to every loop order is comparatively straightforward there. The procedure is to first parametrize the infinitesimal version of the $U(1)$ gauge symmetry $$\psi'(x)= e^{-iQ{e_\textrm{\tiny pos}}\Lambda(x)} \psi(x) \,, \quad
\bar{\psi}'(x)= e^{iQ{e_\textrm{\tiny pos}}\Lambda(x)} \bar{\psi}(x) \,, \quad
A'_n(x)=A_n(x)+\partial_n \Lambda (x) \,$$ of the classical action (\[action-gled-fermion\]) through an anti-commuting field $C(x)$, such that $\Lambda(x)=\theta C(x)$ for a Grassmann number $\theta$. The infinitesimal gauge transformations thus become $$\label{gauge-transformation-BRST}
s \psi=-iQ{e_\textrm{\tiny pos}}C(x) \psi(x) \,, \qquad s \bar{\psi}(x)=-iQ{e_\textrm{\tiny pos}}\bar{\psi}(x) C(x) \,, \qquad s A_n(x)= \partial_n C (x) \,,$$ where the generator $s$ is defined as $$\theta s \phi (x) {\ensuremath{:=}}\phi'(x)-\phi(x) \qquad \text{for} \quad \phi=\psi,\, \bar{\psi},\, A_n\,.$$ Another anti-commuting field $\bar{C}(x)$ and an auxiliary field $B(x)$ are introduced to later fix the gauge. It is convenient to collect all these fields in the sextuple $\Phi=(\psi,\bar{\psi},A,C,\bar{C},B)$. For every field $\Phi^{\alpha}$, one then introduces a BRST source $K_\alpha$, with the opposite statistics (commuting if the field is anti-commuting, and vice versa). The fields and the BRST sources are conjugate variables with respect to the anti-commuting bracket $$(F,G):=\int d^{D} \!x \; \omega \! \left( \frac{\delta_r F}{\delta \Phi^\alpha (x)} \frac{\delta_\ell G}{\delta K_\alpha (x)}
-\frac{\delta_r F}{\delta K_\alpha (x)} \frac{\delta_\ell G}{\delta \Phi^\alpha (x)} \right)$$ acting on any two functionals $F(\Phi,K)$ and $G(\Phi,K)$ of the fields and sources. The subscripts $r$ and $\ell$ on the derivatives indicate whether the differentiated variable is sorted to the right or to the left of its prefactor, which is relevant for Grassmann variables. Next one extends the classical action (\[action-gled-fermion\]) in $D$ complex dimensions to an action $S_\textrm{\tiny BV}[\Phi,K]$ that satisfies the master equation $(S_\textrm{\tiny BV},S_\textrm{\tiny BV})=0$ subject to the four boundary conditions $$S_\textrm{\tiny BV}[\psi,\bar\psi,A,0,\dots,0] = S[\psi,\bar\psi,A] \qquad \textrm{ and }\qquad \frac{\delta_r S_\textrm{\tiny BV}}{\delta K_{\tilde\alpha}}(\Phi,0) = - s \Phi^{\tilde\alpha}$$ for $\tilde\alpha$ restricted to $1,2,3$. Thus one finds the extended action of QED in a spacetime with ray-optically invisible non-metricities to be $$\label{extended-action}
\begin{aligned}
S_\textrm{\tiny BV}[\Phi^\alpha,K_\alpha]=& \int d^{D}\! x\, \omega \left[ -\frac{1}{4} F_{ab} F^{ab} -\frac{1}{8} F_{ab} F_{cd} E^{abcd}
+\bar{\psi} (i\slashed{\partial}-m-Q{e_\textrm{\tiny pos}}\slashed{A}) \psi \right]+ \\
+&\int d^{D}\! x\, \omega \left[ -\frac{\lambda}{2} B^2+ B \partial^n A_n-\bar{C} \square C \right]+ \\
+&\int d^{D}\! x\, \omega \,\Big[ K_A^n \partial_n C +iQ{e_\textrm{\tiny pos}}\bar{\psi} C K_{\bar{\psi}} +iQ{e_\textrm{\tiny pos}}K_\psi C \psi -BK_{\bar{C}} \Big] \,.
\end{aligned}$$ The first line is the classical action for birefringent electrodynamics with a minimally coupled fermion, the second line is the gauge fixing sector and the last line ensures the boundary conditions above. Since this whole construction only builds on the gauge symmetry of the theory, it is not surprising that the only non-metric term in the extended action above is the linear one inherited from the classical action.
The extended action (\[extended-action\]) becomes the bare extended action if all the fields, coupling constants and external sources are replaced by their bare counterparts, which we denote either with a subscript or a superscript $B$. After integrating the auxiliary field $B(x)$ out of the path integral, the ghosts $C(x)$ and $\bar{C}(x)$ decouple and after setting to zero also the BRST sources, the remaining action reads $$\label{qed-action}
S_B[\psi_B, A_B]\!=\!\int\!d^{D}\!x (1+e_B)\!\left[
-\frac{1}{4} F_{ab}^B F^{ab}_B -\frac{1}{8} F_{ab}^B F_{cd}^B E^{abcd}_B
+\bar \psi_B (i\slashed{\partial}\!-\!m_B\!-\!Q{e_\textrm{\tiny pos}}^B\slashed{A}_B) \psi_B
\!+\!\frac{1}{2 \lambda_B} (\partial_a A^a_B)^2
\right] \,,$$ with the usual Lorenz gauge-fixing term. Thus one directly reads off the Feynman rules. First we find the [*photon propagator*]{} $$\label{photon-propagator}
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=-\frac{i}{q^2+i\epsilon}\left[ \eta_{ab}+ (\lambda-1) \frac{q_a q_b}{q^2+i\epsilon} -E_{arbs} \frac{q^r q^s}{q^2+i\epsilon} \right]\,.$$ The photon polarizations can be found explicitly only when the specific form of the perturbation $E^{abcd}$ is known, see [@SME-scattering], where the perturbation $E$ is denoted by $2 k_F$ and does not satisfy the restrictions (\[pertconditionsb\]). However, the diagrams we consider in section \[sec:scatterings\] only contain fermions as ingoing and outgoing particles, thus we do not have to worry about photon polarizations. Secondly, we find the [*fermion propagator*]{} $$\label{fermion-propagator}
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\draw[fermion] (e2) -- (e1);
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=\frac{i(\slashed{p}+m)^{\alpha}_{\;\; \beta}}{p^2-m^2+i\epsilon}\,,$$ and finally the [*vertex*]{} $$\label{vertex}
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\draw[fermion] (e2) -- (aux);
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=-i Q {e_\textrm{\tiny pos}}\,\mu^{\varepsilon/2} (\gamma^n)^{\alpha}_{\;\; \beta}\,.$$ The factor $\mu^{\varepsilon/2}$ in the vertex, employing an arbitrary mass scale $\mu$, has been introduced in order for the charge to have correct physical dimension in complex $D = 4-\varepsilon$ spacetime dimensions. We immediately notice that both, the fermion propagator and the vertex, are unchanged compared to standard QED on flat Minkowski spacetime. This is due to the fact that the Dirac algebra remains unchanged, which also implies that the free Dirac spinors are the usual $u({\ensuremath{\boldsymbol{p}}},s)$ and $v({\ensuremath{\boldsymbol{p}}},s)$.
Gauge invariance and renormalizability at every loop {#sec:gauge-invariance-renormalization}
----------------------------------------------------
It is now straightforward to establish that the maximal linear extension of quantum electrodynamics, from the usual Lorentzian metric background to the eleven-parameter family of deviations that are invisible to ray optics, is renormalizable and thus physically meaningful.
For redefining the fields, the couplings and the external sources in the bare action as $$\label{field-coupling-redefinitions}
\begin{aligned}
(\Phi^\alpha)_B &= (Z_{\Phi^\alpha})^{1/2} \Phi^\alpha \,, \quad & (K_\alpha)_B &= Z_{K_\alpha} K_\alpha \,, \quad
& E^{abcd}_B &= Z_E E^{abcd} \,,\\
m_B &= Z_m m \,, \quad & Q {e_\textrm{\tiny pos}}^B &= Z_{{e_\textrm{\tiny pos}}} \mu^{\varepsilon/2} Q {e_\textrm{\tiny pos}}\,, \quad & \lambda_B &= Z_\lambda \lambda \,,
\end{aligned}$$ where no sum over $\alpha$ is understood, one obtains the renormalized action, whose classical sector is $$\label{renormalized-classical-action}
S= \int d^{D}\! x\, \omega \left[ -\frac{1}{4} Z_A F_{ab} F^{ab} -\frac{1}{8} Z_A Z_E F_{ab} F_{cd} E^{abcd}
+Z_{\psi} \bar{\psi} (i\slashed{\partial}-Z_m m-Z_{{e_\textrm{\tiny pos}}} Z_A^{1/2} \mu^{\varepsilon/2} Q {e_\textrm{\tiny pos}}\slashed{A}) \psi \right] \,.$$ At the end of this section, we show that every divergence can be eliminated by a suitable choice of the renormalization constants $Z$, of which we now determine the independent ones. Since the ghosts decouple and one cannot build any one-particle-irreducible diagrams with exterior legs provided by the auxiliary field $B$ or the source $K_\alpha$, the gauge fixing sector and the sources sector do not renormalize. This fact and the Ward identity (\[Ward-identity-vertex-fermion-self-energy\]), detailed later on in this subsection, imply the relations $$Z_A=Z_{{e_\textrm{\tiny pos}}}^{-2} \,, \qquad Z_\lambda=Z_{{e_\textrm{\tiny pos}}}^{-2} \,, \qquad Z_{K_\alpha}=Z_{{e_\textrm{\tiny pos}}}^{-1}(Z_{\Phi^\alpha})^{-1/2}$$ between the renormalization constants. Apart from the field and coupling redefinitions (\[field-coupling-redefinitions\]), we also allow the scalar density perturbation $e$ to be renormalized by $$e_B= Z_e\, e\,.$$ Requiring, however, that (\[pertconditionsb\]), the second restriction on the perturbation, holds after renormalization, immediately yields $Z_e=Z_E$. Thus the scalar density perturbation renormalizes exactly as the tensor perturbation. In summary, the only renormalization constants left to be determined are $Z_\psi$, $Z_m$, $Z_{{e_\textrm{\tiny pos}}}$ and $Z_E$.
Before we can prove the renormalizability of the theory at every loop, we need to discuss the possible contributions of the perturbation $E$ to the [*vacuum polarization*]{} $$\label{vacuum-polarization}
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{\ensuremath{:=}}i\Pi^{ab} (q) \,,$$ which collects the one-particle-irreducible radiative corrections to the photon propagator, to the [*fermion self-energy*]{} $$\label{fermion-self-energy}
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{\ensuremath{:=}}-i \big[\Sigma(p) \big]^\alpha_{\; \beta} \,,$$ which collects the one-particle-irreducible radiative corrections to the fermion propagator, and to the [*proper vertex*]{} $$\label{proper-vertex}
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\coordinate (aux) at (0,0);
\coordinate [label=left:$p'$,label=right:$\alpha$] (e1) at (-1,.8);
\coordinate [label=right:$p$,label=left:$\beta$] (e2) at (1,.8);
\coordinate [label=below:$n$] (e3) at (0,-.8);
\coordinate (arr1) at (.3,-.8);
\coordinate (arr2) at (.3,-.4);
\coordinate [label=right:$q$] (arrlab) at ($(arr1)!0.5!(arr2)$);
\node [circle,draw,inner sep=2.5mm,pattern=north east lines] (blob) at (aux) {};
\draw [fermion] (e2) -- (blob);
\draw [fermion] (blob) -- (e1);
\path [sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=1.5, amplitude=2mm}},postaction={sines}] (blob) -- (e3);
\draw[->] (arr1) -- (arr2);
\end{tikzpicture}
{\ensuremath{:=}}-i Q{e_\textrm{\tiny pos}}\big[\Gamma^n (p',p) \big]^\alpha_{\;\;\beta} \,,$$ which collects the one-particle-irreducible radiative corrections to the vertex. To this end, we first notice that every term already appearing in standard QED may now appear multiplied by $E^{ij}{}_{ij}$. This, in turn, implies that the renormalization constants $Z_\psi$, $Z_m$ and $Z_{{e_\textrm{\tiny pos}}}$ may now also contain a part linear in $E^{ij}{}_{ij}$. The analysis of the other possible contributions is greatly simplified if we replace $E_{arbs}$ in the photon propagator (\[photon-propagator\]) with $(E_{arbs}+E_{bras})/2$. This substitution does not change the value of the propagator, by virtue of the symmetries (\[area-metric-symmetries\]). We begin by writing all possible Lorentz-covariant terms, using as building blocks the momenta of the external legs, the gamma matrices and the components $E^{abcd}$ in the particular combination described above. We then simplify each of them by use of the symmetries (\[area-metric-symmetries\]) and the first restriction (\[pertconditionsb\]) on the perturbation.
In the vacuum polarization (\[vacuum-polarization\]), there is a new divergent term proportional to $E^{arbs} q_r q_s$, which can be eliminated through a renormalization of $E^{abcd}$ and yields the value of $Z_E$. In contrast to the other renormalization constants, $Z_E$ cannot depend on $E^{ij}{}_{ij}$ because such a contribution would be of second order in $E$. The fermion self-energy (\[fermion-self-energy\]) does not contain any other new term. The proper vertex (\[proper-vertex\]) is related with the fermion self-energy through the Ward identity $$\label{Ward-identity-vertex-fermion-self-energy}
\Gamma^n(p,p)=\gamma^n - \frac{\partial \Sigma (p)}{\partial p_n} \,,$$ which holds also in our theory, since the gauge-fixing and BRST source sectors of the extended action (\[extended-action\]) are the same as in metric QED. Thus, also the proper vertex does not contain any other new contribution.
At this point, we are ready to prove the renormalizability of the theory at every loop. For it can be shown that since the bare extended action $S_B$ satisfies the master equation , also the bare generating functional of the one-particle-irreducible diagrams $\Gamma_B$ does, so that . The proof then proceeds using induction on the number of loops in order to show that the divergences can be removed preserving the master equation, as in the standard case [@Anselmi:renormalization; @Weinberg:1996kr]. The only difference arises when considering all the possible divergent gauge-invariant local terms of dimension less than or equal to 4. Other than the standard terms $F_{ab} F^{ab}$, $\bar{\psi} \psi$ and $\bar{\psi} \slashed{D} \psi$, which can be eliminated by a suitable choice of $Z_\psi$, $Z_e$ and $Z_m$, we also need to include $E^{abcd} F_{ab} F_{cd}$, which can be eliminated by a suitable choice of $Z_E$. Thus, general linear electrodynamics can be renormalized at every loop in a gauge-invariant way. Note that the restrictions (\[pertconditionsb\]) on the perturbations were central to the background of our proof. If they are not heeded in the renormalization process, new interactions in the loop corrections of the fermion sector are generated, see [@Santos:2015koa]. Thus these corrections drive the theory outside the domain of ray-optically invisible deviations.
Renormalization constants in the on-shell scheme {#sec:one-loop-renormalization-constants}
------------------------------------------------
Using dimensional regularization in $D=4-\varepsilon$ complex dimensions, we now determine the relevant one-loop renormalization constants in the on-shell scheme, where the renormalized propagators have poles in the physical masses with a residue of 1, and the renormalized electric charge is the physical one.
The vacuum polarization in renormalized perturbation theory, i.e., with the counterterms included, takes the form $$i \Pi^{ab}(q)=i\Pi(q^2)(q^2\eta^{ab}-q^a q^b)+i\Pi^E (q^2) E^{arbs} q_r q_s\,.$$ The requirement that the photon propagator has the pole in the physical mass with a residue of 1 translates into the two conditions $\Pi(0)=0$ and $\Pi^E(0)=0$, from which one obtains the one-loop renormalization constants $$\label{renormalization-constants-photon-propagator}
Z_A^{\, \text{o.s.}}=1+\Pi_{\text{1-loop}}(0)
\qquad \text{and} \qquad
Z_E^{\, \text{o.s.}}=\frac{1+\Pi^E_{\text{1-loop}}(0)}{ Z_A^{\, \text{o.s.}}} \,.$$ On the other hand, the same requirement for the fermion propagator translates into the two conditions $$\Sigma(p)|_{\slashed{p}=m}=0
\qquad \text{and} \qquad
\left[ \frac{\partial \Sigma(p)}{\partial \slashed{p}} \right]_{\slashed{p}=m}=0$$ on the fermion self-energy $-i\Sigma(p)$ in renormalized perturbation theory, from which one obtains the one-loop renormalization constants $$\label{renormalization-constants-fermion-propagator}
Z_\psi^{\, \text{o.s.}}=1+ \left[ \frac{\partial \Sigma_{\text{1-loop}}(p)}{\partial \slashed{p}} \right]_{\slashed{p}=m}
\qquad \text{and} \qquad
Z_m^{\, \text{o.s.}}=1- \frac{ \Sigma_{\text{1-loop}}(p)|_{\slashed{p}=m}}{m} \,.$$ Moreover, one requires that the non-relativistic potential of a charged particle in a quasi-static and uniform electric field, obtained by means of the Born approximation, is the renormalized electric charge in the on-shell scheme times the electrostatic potential of the external field. This is obtained by imposing the condition $$\lim_{q \rightarrow 0} \bar{u}(p') \big[ -iQ{e_\textrm{\tiny pos}}\Gamma^n (p',p) \big] u(p) = \bar{u}(p') \big[-iQ{e_\textrm{\tiny pos}}\gamma^n \big] u(p) \,,$$ on the proper vertex in renormalized perturbation theory. By virtue of the Ward identity (\[Ward-identity-vertex-fermion-self-energy\]), the above condition is satisfied as long as $Z_e^{\, \text{o.s.}}=\big( Z_A^{\, \text{o.s.}}\big)^{-1/2}$. Thus, the proper vertex does not contain any new information for the renormalization of the theory, and we need to consider only the one-loop radiative corrections to the propagators.
The first diagram to be taken under consideration is the [*one-loop vacuum polarization*]{} $$\begin{tikzpicture}[baseline={([yshift=-2mm]current bounding box.center)}]
\coordinate[label=left:$a$] (e1) at (-1.2,0);
\coordinate[label=right:$b$] (e2) at (1.2,0);
\coordinate (aux) at (0,0);
\coordinate (aux1) at (-1,.25);
\coordinate (aux2) at (-.6,.25);
\coordinate[label=above:$q$] (label) at ($(aux1)!0.5!(aux2)$);
\coordinate (i1) at (-.5,0);
\coordinate (i2) at (+.5,0);
\draw[draw=black, postaction={decorate}, decoration={markings,mark=at position 0.55 with {\arrow[scale=.7,black]{triangle 45}}}] (i1) arc (180:0:.5cm);
\draw[draw=black, postaction={decorate}, decoration={markings,mark=at position 0.55 with {\arrow[scale=.7,black]{triangle 45}}}] (i2) arc (0:-180:.5cm);
\path [sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=2, amplitude=2mm}},postaction={sines}] (e1) -- (i1);
\path [sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=2, amplitude=2mm}},postaction={sines}] (i2) -- (e2);
\draw [->] (aux1) -- (aux2);
\end{tikzpicture}
{\ensuremath{:=}}i\Pi^{ab}_{\text{1-loop}} (q) \,.$$ Since the fermion propagator (\[fermion-propagator\]) and the vertex (\[vertex\]) of QED in a spacetime with ray-optically invisible non-metricities are the same of those of standard QED, the vacuum polarization takes the same value as that of standard QED. One thus easily obtains $$i\Pi^{ab}_{\text{1-loop}} (q)=-i(q^2 \eta^{ab} -q^a q^b) \frac{Q^2 {e_\textrm{\tiny pos}}^2}{12 \pi^2}
\left[ \frac{2}{\varepsilon}-\gamma_E+\log \left( \frac{4\pi\mu^2}{m^2} \right) +{\ensuremath{\mathcal{O}}}(q^2) \right]\,,$$ where $\gamma_E$ is the Euler-Mascheroni constant. From the above expression and the equations (\[renormalization-constants-photon-propagator\]), one finds the one-loop renormalization constants $$\label{ZA-ZH-on-shell}
Z_A^{\,\text{o.s.}}=1-\frac{Q^2 {e_\textrm{\tiny pos}}^2}{12 \pi^2}\left[ \frac{2}{\varepsilon}-\gamma_E+\log \left( \frac{4\pi\mu^2}{m^2} \right) \right]
\qquad \text{and} \qquad Z_E^{\text{o.s.}}=\big(Z_A^{\,\text{o.s.}}\big)^{-1} \,.$$ Using the relation $Z_e^{\, \text{o.s.}}=\big( Z_A^{\, \text{o.s.}}\big)^{-1/2}$, one further finds $$\label{Ze-on-shell}
Z_e^{\,\text{o.s.}}=1+\frac{Q^2 {e_\textrm{\tiny pos}}^2}{24 \pi^2}\left[ \frac{2}{\varepsilon}-\gamma_E+\log \left( \frac{4\pi\mu^2}{m^2} \right) \right] \,.$$
The second diagram to be taken under consideration is [*one-loop fermion self-energy*]{} $$\begin{tikzpicture}[baseline={([yshift=-2mm]current bounding box.center)}]
\coordinate[label=left:$\alpha$] (e1) at (-1.3,0);
\coordinate[label=right:$\beta$] (e2) at (1.3,0);
\coordinate (i1) at (-.75,0);
\coordinate (i2) at (.75,0);
\coordinate (aux) at (0,0);
\coordinate[label=above:$p$] (label) at (1.1,0);
\draw[draw=black, postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow[scale=.7,black]{triangle 45}}}] (e2) -- (i2);
\draw[draw=black, postaction={decorate}, decoration={markings,mark=at position 0.7 with {\arrow[scale=.7,black]{triangle 45}}}] (i1) -- (e1);
\draw[fermion] (i2) -- (i1);
\draw[draw=none,sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=5.5, amplitude=2mm}},postaction={sines}] (i1) arc (180:0:.75cm);
\end{tikzpicture}
{\ensuremath{:=}}-i \big[\Sigma_{\text{1-loop}}(p) \big]^\alpha_{\; \beta} \,,$$ which now depends on the perturbation $E$, due to the presence of the photon propagator. In order to avoid infrared divergences, it is useful to replace the photon propagator (\[photon-propagator\]) with $$\label{photon-propagator-IR-regulated}
\begin{tikzpicture}[baseline={([yshift=-2mm]current bounding box.center)}]
\coordinate[label=left:$a$] (e1) at (-.75,0);
\coordinate[label=right:$b$] (e2) at (.75,0);
\coordinate[label=above:$q$] (lab) at (0,0);
\path [sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=5, amplitude=2mm}},postaction={sines}] (e1) -- (e2);
\end{tikzpicture}
=-\frac{i}{q^2-\delta^2+i\epsilon}\left[ \eta_{ab}+ (\lambda-1) \frac{q_a q_b}{q^2-\lambda\delta^2+i\epsilon}
-E_{arbs} \frac{q^r q^s}{q^2-\delta^2+i\epsilon} \right]\,,$$ where $\delta$ is a small photon mass used as an infrared regulator. The Feynman prescription $i \epsilon$ will not be shown explicitly in the following equations. The one-loop fermion self-energy $$-i \Sigma_{\text{1-loop}}(p)\!=\!\big(iQ{e_\textrm{\tiny pos}}\mu^{\varepsilon/2}\, \big)^2\! \int\! \frac{d^D k}{(2\pi)^D} \gamma^a \frac{i(\slashed{p}+\slashed{k}+m)}{(p+k)^2-m^2} \gamma^b
\left\{\!-\frac{i}{k^2\!-\!\delta^2} \left[ \eta_{ab}\!+\!(\lambda\!-\!1) \frac{k_a k_b}{k^2\!-\!\lambda \delta^2}\!-\!E_{arbs} \frac{k^r k^s}{k^2\!-\!\delta^2} \right] \right\}$$ can be split into a sum of the zeroth order contribution in $E$, denoted by $-i \Sigma_{\text{1-loop}}^{0}(p)$, and the first order contribution, denoted by $-i \Sigma_{\text{1-loop}}^{E}(p)$. The former is the standard QED contribution $$\label{fermion-self-energy-divergent-QED}
\begin{aligned}
-i \Sigma_{\text{1-loop}}^{0}(p) =
& -3 \frac{iQ^2 {e_\textrm{\tiny pos}}^2}{(4\pi)^2} \left[ \frac{2}{\varepsilon}-\gamma_E+\log \left( \frac{4\pi\mu^2}{m^2} \right) +\frac{4}{3} \right]m\\
& +(\slashed{p}-m) \frac{iQ^2 {e_\textrm{\tiny pos}}^2}{(4\pi)^2}
\left\{ \lambda \left[ \frac{2}{\varepsilon}-\gamma_E+\log \left( \frac{4\pi\mu^2}{m^2} \right) +4-2 \log \left( \frac{m^2}{\delta^2} \right) \right] \right. \\
&\left.-(\lambda-1)\left[ 3+\frac{\lambda \log \lambda}{\lambda-1} -3\log \left( \frac{m^2}{\delta^2} \right) \right]\right\}
+{\ensuremath{\mathcal{O}}}\big((\slashed{p}-m)^2 \big)\,.
\end{aligned}$$ On the other hand, the contribution linear in $E$ is $$\begin{aligned}
-i \Sigma_{\text{1-loop}}^{H}(p) = &
\frac{iQ^2 {e_\textrm{\tiny pos}}^2}{4(4\pi)^2}\, E^{ij}_{\;\;\; ij} \left[ \frac{2}{\varepsilon}-\gamma_E+\log \left( \frac{4\pi\mu^2}{m^2} \right) +\frac{5}{2} \right]m\\
&+ \frac{iQ^2 {e_\textrm{\tiny pos}}^2}{4(4\pi)^2}\, E^{ij}_{\;\;\; ij} \left[ \log \left( \frac{m^2}{\delta^2} \right)-2 \right](\slashed{p}-m) +{\ensuremath{\mathcal{O}}}\big((\slashed{p}-m)^2 \big)\,.
\end{aligned}$$ Using the above result, one finds the one-loop values $$\begin{gathered}
\label{Zpsi-on-shell}
Z_\psi^{\,\text{o.s.}}=1-\frac{Q^2 {e_\textrm{\tiny pos}}^2}{(4\pi)^2}
\left\{ \lambda \left[ \frac{2}{\varepsilon}-\gamma_E+\log \left( \frac{4\pi\mu^2}{m^2} \right) +4-2 \log \left( \frac{m^2}{\delta^2} \right) \right] \right. \\
\left.-(\lambda-1)\left[ 3+\frac{\lambda \log \lambda}{\lambda-1} -3\log \left( \frac{m^2}{\delta^2} \right) \right]
-\frac{E^{ij}_{\;\;\; ij}}{4} \left[ 2-\log \left( \frac{m^2}{\delta^2} \right) \right] \right\}\end{gathered}$$ and $$\label{Zm-on-shell}
Z_m^{\,\text{o.s.}}=1-\frac{3Q^2 {e_\textrm{\tiny pos}}^2}{(4 \pi)^2 }
\left\{ \left(1-\frac{E^{ij}_{\;\;\; ij}}{12} \right)\! \left[\frac{2}{\varepsilon}-\gamma_E+\log \left( \frac{4\pi\mu^2}{m^2} \right) \right] +\frac{4}{3} -\frac{5}{24}E^{ij}_{\;\;\; ij} \right\}$$ for the other two renormalization constants.
We have thus determined the four independent renormalization constants at one-loop in the on-shell scheme. This was possible analyzing the one-loop vacuum polarization, which is the same as in standard QED, and the one-loop fermion self-energy, which contains a new term linear in the full trace of the perturbation $E^{ij}_{\;\;\; ij}$.
The one-loop on-shell renormalization of ray-optically invisible birefringent electrodynamics can find its application also in the photon sector of the SME [@SME-flat-space; @SME-curved-spacetime]. In this context, indeed, the one-loop renormalization was worked out considering the divergent part of the relevant diagrams [@SME-one-loop-renormalization], without the finite contributions necessary for the on-shell scheme.
Scattering and anomalous magnetic moment {#sec:scatterings}
========================================
We now calculate potentially measurable effects using the Feynman rules for the photon propagator (\[photon-propagator\]), fermion propagator (\[fermion-propagator\]) and vertex (\[vertex\]) for general linear quantum electrodynamics. This is pursued by analizing the effects of the perturbation (\[pert\]) in three different scattering processes: $e^+ e^-\rightarrow \bar{f}f$, Bhabha scattering and the scattering of an electron in an external magnetic field. Several scattering processes with similar backgrounds, such as SME, have been considered in the literature: see [@SME-scattering] for the study of $e^+ e^- \rightarrow \gamma\gamma$ with a perturbation close to (\[pert\]) and [@SME-Bhabha] for the study of Bhabha scattering with a modified Dirac sector, but a standard photon sector in the action.
$e^+ e^-\rightarrow \bar{f}f$ scattering
----------------------------------------
The first process taken into consideration is the scattering $e^+ e^-\rightarrow \bar{f}f$, for a fermion $f$ different from $e^\pm$, but otherwise generic. This process contains only fermions as external legs for which we can use the usual free Dirac spinors $u({\ensuremath{\boldsymbol{p}}},s)$ and $v({\ensuremath{\boldsymbol{p}}},s)$, since the Dirac sector in (\[action-gled-fermion\]) and the Dirac algebra remain unchanged by the perturbation (\[pert\]), as already mentioned in section \[sec\_BVquant\]. At tree level, the scattering consists in the diagram
\(c) at (0,0); (v1) at (0,-.5); (v2) at (0,+.5); (e1) at ($(-1.05,-.7)+(v1)$); (e2) at ($(+1.05,-.7)+(v1)$); (f1) at ($(-1.05,.7)+(v2)$); (f2) at ($(+1.05,.7)+(v2)$);
(a11); (a12) at ($(a11)+.7*(v1)-.7*(e1)$); (a1) at ($(a11)!0.5!(a12)+(+.1,-.1)$);
(a21); (a22) at ($(a21)+.7*(v1)-.7*(e2)$); (a1) at ($(a21)!0.5!(a22)+(-.1,-.1)$);
(b12); (b11) at ($(b12)+.7*(v2)-.7*(f1)$); (b1) at ($(b11)!0.5!(b12)+(+.1,+.1)$);
(b22); (b21) at ($(b22)+.7*(v2)-.7*(f2)$); (b2) at ($(b21)!0.5!(b22)+(-.1,+.1)$);
(e2) – (v1); (v1) – (e1); (f1) – (v2); (v2) – (f2); (v1) – (v2);
(a11) – (a12); (a21) – (a22); (b11) – (b12); (b21) – (b22);
.
Here, the electron has charge $-{e_\textrm{\tiny pos}}$ and mass $m$, while $f$ has charge $Q{e_\textrm{\tiny pos}}$ and mass $M$. The tree-level amplitude in the Feynman gauge ($\lambda=1$) is $$i {\ensuremath{\mathcal{M}}}= \bar{u}_f (k) \big[-iQ{e_\textrm{\tiny pos}}\gamma^b \big] v_f (k') \bar{v}_e (p') \big[i{e_\textrm{\tiny pos}}\gamma^a \big] u_e (p) \frac{-i}{q^2} \left[ \eta_{ab} -E_{ambn}\frac{q^m q^n}{q^2} \right] \,,$$ where $\varepsilon=0$, since the tree-level process is not divergent. In order to simplify the results, we move to the center of mass frame through a Lorentz transformation. One should only keep in mind that $E^{abcd}$ needs to be transformed accordingly with a Lorentzian change of frame. We further choose the orientation of the axes in such a way that $$\begin{gathered}
{\ensuremath{\boldsymbol{p}}} = -{\ensuremath{\boldsymbol{p}}}'= |{\ensuremath{\boldsymbol{p}}}| \hat{z} \quad \text{and} \\
{\ensuremath{\boldsymbol{k}}} = -{\ensuremath{\boldsymbol{k}}}'= |{\ensuremath{\boldsymbol{k}}}| \hat{r} {\ensuremath{:=}}|{\ensuremath{\boldsymbol{k}}}| (\sin \theta \cos \varphi \,\hat{x} + \sin \theta \sin \varphi \,\hat{y} + \cos \theta \,\hat{z} ) \,.\end{gathered}$$ Since spin polarizations are often not measured experimentally, we then take the average of the initial spin polarizations, the sum of the final spin polarizations and compute the differential cross section, which, with the help of the first restriction (\[pertconditionsb\]) on the perturbation, can be put in the form $$\label{cross-section-muons}
\begin{aligned}
\frac{d \sigma}{d \Omega} &= \frac{Q^2 \alpha^2}{4s} \sqrt{\frac{s-4 M^2}{s-4 m^2}} \left\{
1\!+\! \frac{4(m^2\!+\!M^2)}{s}+\! \left( 1- \frac{4 m^2}{s} \right)\!\! \left( 1- \frac{4 M^2}{s} \right)\! \cos^2 \theta + \right. \\
&-2 \sin^2\! \theta \! \left( 1-\frac{4 M^2}{s} \right)\! \big( \cos^2\! \varphi\, E^{0101}\!\! + \sin^2\! \varphi \, E^{0202}\!+2\sin \varphi \cos \varphi \,E^{0102} \big) + \\
&-2 \left[ \left( 1-\frac{4 m^2}{s} \right) + \frac{4 m^2}{s} \left( 1-\frac{4 M^2}{s} \right) \cos^2 \theta \right] E^{0303} -\frac{1}{2} E^{ij}_{\;\;\; ij}+\\
&-2 \left. \sin \theta \cos \theta \left( 1+\frac{4 m^2}{s} \right) \left( 1-\frac{4 M^2}{s} \right) \big( \cos \varphi \, E^{0103}+\sin \varphi \, E^{0203} \big) \right\} \,,
\end{aligned}$$ where $\alpha={e_\textrm{\tiny pos}}^2/(4\pi)$ is the fine structure constant and $s {\ensuremath{:=}}(p+p')^2$ is one of Mandelstam’s variables.
We notice that the factor presented by the curly brackets in (\[cross-section-muons\]) is not symmetric in $m$ and $M$. However, this does not imply a violation of time-reversal invariance when computing the inverse scattering process $\bar{f} f \rightarrow e^+ e^-$ cross section and comparing the result with the time-reversed one of the direct process. This is seen as follows. The cross section of the inverse process can be indeed inferred when switching $m$ with $M$, but is expressed in the frame where ${\ensuremath{\boldsymbol{k}}}$ is along $\hat{z}$ and ${\ensuremath{\boldsymbol{p}}}$ along $\hat{r}$. The cross-section of the time-reversed process instead is expressed in the frame where ${\ensuremath{\boldsymbol{p}}}$ is along $-\hat{z}$ and ${\ensuremath{\boldsymbol{k}}}$ along $-\hat{r}$, and does not change by flipping the sing of all the spatial momenta. The two frames are related by the rotation that interchanges ${\ensuremath{\boldsymbol{k}}}/|{\ensuremath{\boldsymbol{k}}}|$ and ${\ensuremath{\boldsymbol{p}}}/|{\ensuremath{\boldsymbol{p}}}|$. If we express the components of $E$ in the one frame in terms of its components in the other, we indeed find the differential cross section of the inverse process starting from the one of the direct process, implying that time-reversal is not violated.
Even if the detector does not resolve different angles $\varphi$, we still have an effect. Indeed, integrating out $\varphi$ in the cross section and using again the first restriction (\[pertconditionsb\]) on the perturbation, we find $$\begin{aligned}
\frac{d \sigma}{d \cos \theta} &= \frac{\pi Q^2 \alpha^2}{2s} \sqrt{\frac{s-4 M^2}{s-4 m^2}} \left\{
1+ \frac{4(m^2+M^2)}{s}+ \left( 1- \frac{4 m^2}{s} \right) \left( 1- \frac{4 M^2}{s} \right) \cos^2 \theta + \right. \\
&- \left[ \left( 1-\frac{8 m^2}{s}+\frac{4 M^2}{s} \right) + \left( 1+ \frac{8 m^2}{s} \right) \left( 1-\frac{4 M^2}{s} \right) \cos^2 \theta \right] E^{0303} + \\
&\left. - \frac{1}{4} \left[ \left(1+\frac{4M^2}{s}\right)+\left(1-\frac{4M^2}{s}\right) \cos^2\theta \right] E^{ij}_{\;\;\; ij} \right\} \,.\end{aligned}$$
Bhabha scattering
-----------------
The second process we take into consideration is the Bhabha scattering $e^+ e^- \rightarrow e^+ e^-$, which is of direct interest for the determination of luminosity [@Bhabha-colliders], both in past [@LEP-Bhabha] and in future $e^+ e^-$ colliders [@TLEP-Bhabha; @ILC-Bhabha]. As for the previous scattering, only fermions appear as external legs and we can use for them the usual free Dirac spinors $u({\ensuremath{\boldsymbol{p}}},s)$ and $v({\ensuremath{\boldsymbol{p}}},s)$, since the Dirac sector in (\[action-gled-fermion\]) and the Dirac algebra remain unchanged by the perturbation (\[pert\]). At tree level, the scattering consists in the sum of the two diagrams $$\begin{tikzpicture}[baseline={(current bounding box.center)}]
\coordinate[label=right:$q$] (c) at (0,0);
\coordinate[label=below:$a$] (v1) at (0,-.5);
\coordinate[label=above:$b$] (v2) at (0,+.5);
\coordinate[label=left:$e^-$] (e1) at ($(-1.05,-.7)+(v1)$);
\coordinate[label=right:$e^+$] (e2) at ($(+1.05,-.7)+(v1)$);
\coordinate[label=left:$e^-$] (f1) at ($(-1.05,.7)+(v2)$);
\coordinate[label=right:$e^+$] (f2) at ($(+1.05,.7)+(v2)$);
\coordinate[above=2.5mm of e1] (a11);
\coordinate (a12) at ($(a11)+.7*(v1)-.7*(e1)$);
\coordinate[label=above left:$p$] (a1) at ($(a11)!0.5!(a12)+(+.1,-.1)$);
\coordinate[above=2.5mm of e2] (a21);
\coordinate (a22) at ($(a21)+.7*(v1)-.7*(e2)$);
\coordinate[label=above right:$k$] (a1) at ($(a21)!0.5!(a22)+(-.1,-.1)$);
\coordinate[below=2.5mm of f1] (b12);
\coordinate (b11) at ($(b12)+.7*(v2)-.7*(f1)$);
\coordinate[label=below left:$p'$] (b1) at ($(b11)!0.5!(b12)+(+.1,+.1)$);
\coordinate[below=2.5mm of f2] (b22);
\coordinate (b21) at ($(b22)+.7*(v2)-.7*(f2)$);
\coordinate[label=below right:$k'$] (b2) at ($(b21)!0.5!(b22)+(-.1,+.1)$);
\draw[fermion] (v1) -- (e2);
\draw[fermion] (e1) -- (v1);
\draw[fermion] (v2) -- (f1);
\draw[fermion] (f2) -- (v2);
\path [sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=3, amplitude=2mm}},postaction={sines}] (v1) -- (v2);
\draw[->] (a11) -- (a12);
\draw[->] (a21) -- (a22);
\draw[->] (b11) -- (b12);
\draw[->] (b21) -- (b22);
\end{tikzpicture}
+
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\coordinate (c) at (0,0);
\coordinate[label=above:$q'$] (labelq) at (0,+.05);
\coordinate[label=left:$a$] (v1) at (-.6,0);
\coordinate[label=right:$b$] (v2) at (+.6,0);
\coordinate[label=left:$e^-$] (e1) at ($(-.7,-1.05)+(v1)$);
\coordinate[label=right:$e^+$] (e2) at ($(+.7,-1.05)+(v2)$);
\coordinate[label=left:$e^-$] (f1) at ($(-.7,+1.05)+(v1)$);
\coordinate[label=right:$e^+$] (f2) at ($(+.7,+1.05)+(v2)$);
\coordinate[right=2.5mm of e1] (a11);
\coordinate (a12) at ($(a11)+.7*(v1)-.7*(e1)$);
\coordinate[label=below right:$p$] (a1) at ($(a11)!0.5!(a12)+(-.1,+.1)$);
\coordinate[left=2.5mm of e2] (a21);
\coordinate (a22) at ($(a21)+.7*(v2)-.7*(e2)$);
\coordinate[label=below left:$k$] (a1) at ($(a21)!0.5!(a22)+(+.1,+.1)$);
\coordinate[right=2.5mm of f1] (b12);
\coordinate (b11) at ($(b12)+.7*(v1)-.7*(f1)$);
\coordinate[label=above right:$p'$] (b1) at ($(b11)!0.5!(b12)+(-.1,-.1)$);
\coordinate[left=2.5mm of f2] (b22);
\coordinate (b21) at ($(b22)+.7*(v2)-.7*(f2)$);
\coordinate[label=above left:$k'$] (b2) at ($(b21)!0.5!(b22)+(+.1,-.1)$);
\draw[fermion] (v1) -- (f1);
\draw[fermion] (e1) -- (v1);
\draw[fermion] (f2) -- (v2);
\draw[fermion] (v2) -- (e2);
\path [sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=3, amplitude=2mm}},postaction={sines}] (v1) -- (v2);
\draw[->] (a11) -- (a12);
\draw[->] (a21) -- (a22);
\draw[->] (b11) -- (b12);
\draw[->] (b21) -- (b22);
\end{tikzpicture} \,.$$ The charge of the electron is $-{e_\textrm{\tiny pos}}$ and its mass is $m$. The tree-level amplitude in Feynman gauge ($\lambda=1$) is $$\begin{aligned}
i {\ensuremath{\mathcal{M}}}&= \bar{v} (k) \big(i{e_\textrm{\tiny pos}}\gamma^a \big) u (p) \bar{u} (p') \big(i{e_\textrm{\tiny pos}}\gamma^b \big) v (k') \frac{-i}{q^2} \left[ \eta_{ab} -E_{ambn}\frac{q^m q^n}{q^2} \right] +\\
&-\bar{u} (p') \big(i{e_\textrm{\tiny pos}}\gamma^a \big) u (p) \bar{v} (k) \big(i{e_\textrm{\tiny pos}}\gamma^b \big) v (k') \frac{-i}{q'^2} \left[ \eta_{ab} -E_{ambn}\frac{q'^m q'^n}{q'^2} \right] \,,\end{aligned}$$ where $\varepsilon=0$, since the tree-level process is not divergent. As before, we move to the center of mass frame with the axes oriented in such a way that $$\begin{gathered}
{\ensuremath{\boldsymbol{p}}} = -{\ensuremath{\boldsymbol{k}}}= |{\ensuremath{\boldsymbol{p}}}| \hat{z} \quad \text{and} \\
{\ensuremath{\boldsymbol{p}}}' = -{\ensuremath{\boldsymbol{k}}}'= |{\ensuremath{\boldsymbol{p}}}| (\sin \theta \cos \varphi \,\hat{x} + \sin \theta \sin \varphi \,\hat{y} + \cos \theta \,\hat{z} ) \,.\end{gathered}$$ Since spin polarizations are often not measured experimentally, we can take the average of the initial spin polarizations, the sum of the final spin polarizations and compute the differential cross section. In the ultra-relativistic limit ($s \gg m^2$), using the first restriction (\[pertconditionsb\]) on the perturbation, one obtains $$\label{cross-section-Bhabha}
\begin{aligned}
\frac{d \sigma}{d \Omega} &=\frac{\alpha^2}{2 s} \left[ \frac{1}{2} (1+\cos^2 \theta)+\frac{1+\cos^4 (\theta/2)}{\sin^4 (\theta/2)} -
2\frac{\cos^4 (\theta/2)}{\sin^2 (\theta/2)} + \right. \\
&- \frac{(3+\cos^2\theta) \sin^2 \theta}{4 \sin^4(\theta/2)} \big(\cos^2\varphi \, E^{0101}\! +\sin^2\varphi \, E^{0202}\! +2\sin\varphi \cos\varphi \, E^{0102} \big) +\\
&- \frac{(7+\cos^2 \theta) \sin \theta \cos \theta}{4 \sin^4 (\theta/2)} \big(\cos \varphi \, E^{0103}+\sin \varphi \, E^{0203}\big) + \\
&- \left.\frac{3+5 \cos^2 \theta}{4 \sin^4 (\theta/2)} E^{0303} -\frac{5+3 \cos^2\theta}{16 \sin^4(\theta/2)} E^{ij}_{\;\;\; ij} \right] \,.
\end{aligned}$$ Again there is an effect, even if the detector does not resolve different angles $\varphi$. Integrating out $\varphi$ and using again the first restriction (\[pertconditionsb\]) on the perturbation, we also find $$\begin{gathered}
\frac{d \sigma}{d \cos \theta} = \frac{\pi \alpha^2}{s} \left[ \frac{1}{2} (1+\cos^2 \theta)+\frac{1+\cos^4 (\theta/2)}{\sin^4 (\theta/2)} -
2\frac{\cos^4 (\theta/2)}{\sin^2 (\theta/2)} + \right. \\
\left. - \frac{\cos^4 \theta +12 \cos^2 \theta +3}{8 \sin^4 (\theta/2)} E^{0303} - \frac{(1+\cos^2 \theta)(7+\cos^2 \theta)}{32 \sin^4 (\theta/2)}\, E^{ij}_{\;\;\; ij} \right] \,.\end{gathered}$$ We have thus shown, for two prototypical tree-level processes, how the Feynman rules derived in section \[sec\_BVquant\] can produce measurable effects in the tree-level cross-sections, both when the detector is sensitive to the angle $\varphi$ and when it is not. We now continue with the investigation of one-loop processes.
Electron in an external magnetic field {#sec:electron-external-magnetic-field}
--------------------------------------
We now study the scattering of an electron in a quasi-static and uniform external magnetic field. At tree level, this process consists of merely the vertex (\[vertex\]) and, thus, it does not contain any correction of the perturbation $E^{abcd}$. For this reason, we study one-loop radiative corrections to the vertex, which will provide us with the anomalous magnetic moment of the electron in the presence of ray-optically invisible perturbations of a Minkowskian spacetime.
The said process, up to one-loop, is the sum of two diagrams: the tree-level vertex and the one-loop contribution. We thus write $$\begin{tikzpicture}[baseline={(current bounding box.center)}]
\coordinate (aux) at (0,0);
\coordinate [label=left:$p'$] (e1) at (-1,.8);
\coordinate [label=right:$p$] (e2) at (1,.8);
\coordinate [label=below:$n$] (e3) at (0,-1);
\coordinate (arr1) at (.3,-.95);
\coordinate (arr2) at (.3,-.55);
\coordinate [label=right:$q$] (arrlab) at ($(arr1)!0.5!(arr2)$);
\node [circle,draw,inner sep=3.5mm,pattern=north east lines] (blob) at (aux) {};
\draw [fermion] (e2) -- (blob);
\draw [fermion] (blob) -- (e1);
\path [sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=1.5, amplitude=2mm}},postaction={sines}] (blob) -- (e3);
\draw[->] (arr1) -- (arr2);
\end{tikzpicture}
=
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\coordinate (aux) at (0,0);
\coordinate[label=left:$p'$] (e1) at (-1,.8);
\coordinate[label=right:$p$] (e2) at (1,.8);
\coordinate[label=below:$n$] (e3) at (0,-1);
\coordinate[above right=.2 and .25 of e3] (arr1);
\coordinate[below right=.2 and .25 of aux] (arr2);
\coordinate[label=right:$q$] (arrlab) at ($(arr1)!0.5!(arr2)$);
\draw[fermion] (aux) -- (e1);
\draw[fermion] (e2) -- (aux);
\path [sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=2.5, amplitude=2mm}},postaction={sines}] (aux) -- (e3);
\draw[->] (arr1) -- (arr2);
\end{tikzpicture}
+
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\coordinate (aux3) at (0,0);
\coordinate[label=left:$p'$] (e1) at (-1,.8);
\coordinate[label=right:$p$] (e2) at (1,.8);
\coordinate (aux1) at ($0.75*(e1)$);
\coordinate (aux2) at ($0.75*(e2)$);
\coordinate[label=below:$n$] (e3) at (0,-1);
\coordinate[above right=.2 and .25 of e3] (arr1);
\coordinate[below right=.2 and .25 of aux3] (arr2);
\coordinate[label=right:$q$] (arrlab) at ($(arr1)!0.5!(arr2)$);
\draw[-] (aux1) -- (e1);
\draw[-] (e2) -- (aux2);
\draw[fermion] (aux2) -- (aux3);
\draw[fermion] (aux3) -- (aux1);
\path [sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=2.5, amplitude=2mm}},postaction={sines}] (aux3) -- (e3);
\path [sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=4.5, amplitude=2mm}},postaction={sines}] (aux1) to[bend left] (aux2);
\draw[->] (arr1) -- (arr2);
\end{tikzpicture}
{\ensuremath{:=}}i {e_\textrm{\tiny pos}}\Gamma^n (p,p') \,,$$ where $p$ and $p'$ are the momenta of the ingoing and outgoing on-shell electrons, respectively, $q=p'-p$ is $\hbar$ times the wave covector of the oscillating external potential and $-{e_\textrm{\tiny pos}}$ is the charge of the electron. Since the external magnetic field is quasi-static and uniform, $q$ is very small. Thus it is reasonable to expand the scattering amplitude in powers of $q$ and neglect second and higher orders.
The tree-level contribution above is $$\label{tree-level-vertex}
\bar{u} (p') \big[ i{e_\textrm{\tiny pos}}\Gamma^n_{\text{tree}} (p,p') \big] u(p)=\bar{u} (p') \big[ i{e_\textrm{\tiny pos}}\mu^{\varepsilon/2}\, \gamma^n \big] u(p) \,,$$ while the one-loop contribution $I^n {\ensuremath{:=}}\bar{u}(p')\big[ i{e_\textrm{\tiny pos}}\Gamma^n_{\text{1-loop}}(p',p) \big] u(p)$ is $$\begin{gathered}
I^n =(i{e_\textrm{\tiny pos}}\mu^{\varepsilon/2})^3 \int \frac{d^{\, D} k}{(2 \pi)^D} \left[ -\frac{i}{k^2-\delta^2+i\epsilon} \left( \eta_{ab}-E_{arbs}\frac{k^r k^s}{k^2-\delta^2+i\epsilon} \right) \right] \times \\
\times \bar{u}(p') \gamma^a \frac{i(\slashed{p}' -\slashed{k} +m)}{(p'-k)^2 -m^2 +i\epsilon} \gamma^n
\frac{i(\slashed{p} -\slashed{k} +m)}{(p-k)^2 -m^2 +i\epsilon} \gamma^b u(p) \,,\end{gathered}$$ where we have chosen the Feynman gauge ($\lambda=1$) and used the photon propagator (\[photon-propagator-IR-regulated\]) with a small photon mass $\delta$ in order to avoid infrared divergences. The ultraviolet divergences, on the other hand, are treated using dimensional regularization, where integrals are performed in $D=4-\varepsilon$ complex dimensions and an arbitrary mass scale $\mu$ is introduced in order to preserve the correct physical dimension of the electric charge. The Feynman prescription $i\epsilon$ will not be shown explicitly in the equations below. The integral $I^n$ can be split into a sum consisting of a zeroth order contribution in $E$, denoted by $I^n_0$, and a first order contribution in $E$, denoted by $I^n_E$. The former gives the standard QED contribution $$\label{1-loop-vertex-qed}
I^n_0=i{e_\textrm{\tiny pos}}\mu^{\varepsilon/2}\, \frac{\alpha}{2 \pi} \bar{u}(p') \left\{ \gamma^n
\left[ \frac{1}{\varepsilon}+\frac{1}{2} \log \left( \frac{4\pi \mu^2}{m^2} \right) -\frac{\gamma_E}{2} +2 -\log \left( \frac{m^2}{\delta^2} \right) \right]+
\frac{i\sigma^{n\ell} q_{\ell}}{2m}\right\} u(p) +{\ensuremath{\mathcal{O}}}(q^2) \,,$$ where $\gamma_E$ is the Euler-Mascheroni constant, $\alpha$ the fine structure constant, $\sigma^{n \ell} {\ensuremath{:=}}\frac{i}{2}[\gamma^n,\gamma^\ell]$ and the expression has already been expanded in powers of $q$, neglecting second and higher orders.
We start the computation of $I^n_E$ by using twice the Feynman parametrization to simplify the denominators, and obtain $$\label{1-loop-integral}
I^n_E= -6 {e_\textrm{\tiny pos}}^3 \mu^{3\varepsilon/2} \int_0^1 dx \int_0^1 dy\, y(1-y)
\int \frac{d^{\, D} k}{(2\pi)^D} \frac{\bar{u}(p') E_{arbs} k^r k^s \gamma^a (\slashed{k}-\slashed{p}'-m) \gamma^n (\slashed{k}-\slashed{p}-m) \gamma^b u(p) }{\Big[ (k-y\, p_x)^2 -\big( y^2 p_x^2+(1-y) \delta^2 \big) \Big]^{4}} \,,$$ having defined $p_x {\ensuremath{:=}}x p+(1-x) p'$. We can now simplify the numerator by use of $\slashed{p} u(p)=mu(p)$, $\bar{u} (p') \slashed{p}'=\bar{u}(p') m$ and the first restriction (\[pertconditionsb\]) on the perturbation. Changing the integration variable $k'=k-y\, p_x$ and dropping all terms containing odd powers of $k'$, the numerator $N=N_A+N_B+N_C$ consists of a part $$N_A = \bar{u}(p') \left[ -\frac{E^{ij}_{\;\;\; ij}}{D} k'^2 \slashed{k}' \gamma^n \slashed{k}'+ 2 E_{a}^{\; rns} k'_r k'_s \slashed{k}' \gamma^a \slashed{k}' \right] u(p) \,,$$ which is quartic in the integrated momentum $k'$, another part $$\begin{aligned}
N_B &=\bar{u}(p') \Big\{-y^2 \frac{E^{ij}_{\;\;\; ij}}{D} \big[ k'^2 \slashed{p}_x \gamma^n \slashed{p}_x+p_x^2\, \slashed{k}' \gamma^n \slashed{k}'
+4k'\cdot p_x(k'^n \slashed{p}_x +p_x^n \slashed{k}') -4(k'\cdot p_x)^2\gamma^n \big] + \\
&+2 y^2 E_a^{\; rns} \big[ k'_r k'_s \slashed{p}_x \gamma^a \slashed{p}_x +p_{xr} p_{xs} \slashed{k}' \gamma^a \slashed{k}'+2(k'_r p_{xs}+p_{xr} k'_s) (p_x^a \slashed{k}' +k'^a \slashed{p}_x -k' \cdot p_x \gamma^a) \big] + \\
&+2y E_a^{\; rbs} \big[ k'_r k'_s (p_b \slashed{p}_x \gamma^a \gamma^n+p'_b \gamma ^n \gamma^a \slashed{p}_x) +
(k'_r p_{xs}+p_{xr} k'_s) (p_b \slashed{k}' \gamma^a \gamma^n+p'_b \gamma ^n \gamma^a \slashed{k}') \big]+ \\
& +4 E^{arbs} k'_r k'_s p_ap'_b \gamma^n \Big\} u(p) \,,\end{aligned}$$ which is quadratic in the integrated momentum $k'$, and a third part $$\begin{aligned}
N_C &=\bar{u}(p') \bigg\{ -y^4 \frac{E^{ij}_{\;\;\; ij}}{D} p_x^2\, \slashed{p}_x \gamma^n \slashed{p}_x+
2 y^4 E_a^{\; rns} p_{xr} p_{xs} \slashed{p}_x \gamma^a \slashed{p}_x+ \\
&+2 y^3 E_a^{\; rbs} p_{xr} p_{xs} (p_b \slashed{p}_x \gamma^a \gamma^n+p'_b \gamma^n \gamma^a \slashed{p}_x)
+4 y^2 E^{arbs} p_{xr} p_{xs} p_a p'_b \gamma^n \bigg\} u(p) \,,\end{aligned}$$ which does not depend on the integrated momentum $k'$ at all.
The contribution of $N_A$ to the integral (\[1-loop-integral\]) is divergent a priori, but in fact finite, yielding $$(I_E^n)_A = \frac{i \mu^{\varepsilon/2} {e_\textrm{\tiny pos}}^3}{8(4 \pi)^2} E^{ij}_{\;\;\; ij} \gamma^n \,.$$ The contributions $N_B$ and $N_C$, which we denote by $(I^n_E)_B$ and $(I^n_E)_C$ respectively, are ultraviolet finite and can thus be computed at $D=4$, yielding $$\begin{aligned}
(I_E^n)_B = \frac{i\mu^{\varepsilon/2} {e_\textrm{\tiny pos}}^3}{12(4\pi)^2} &\left\{ \frac{E^{ij}_{\;\;\; ij}}{2} \gamma^n \left[ 4-3\int_0^1 \!dx \frac{m^2}{p_x^2}
+6 \int_0^1 \!dx \frac{p\cdot p'}{p_x^2} \left( \log\!\left( \frac{p_x^2}{\delta^2} \right)\!-2 \right) \right] + \right. \\
&-\frac{E^{ij}_{\;\;\; ij}}{2}m(p+p')^n \int_0^1 \!dx \frac{1}{p_x^2} -16 E_a^{\;\;rns} \int_0^1 \!dx\, \frac{p_{xr} p_{xs}}{p_x^2} + \\
& \left. -3 (E_{arbs}+E_{abrs}) p^r p'^s (\gamma^n \gamma^a \gamma^b + \gamma^a \gamma^b \gamma^n) \int_0^1 \!dx \frac{1}{p_x^2} \right\}\end{aligned}$$ and $$\begin{aligned}
(I_E^n)_C=-\frac{i\mu^{\varepsilon/2} {e_\textrm{\tiny pos}}^3}{(4\pi)^2} &\left\{ \frac{E^{ij}_{\;\;\; ij}}{24} \gamma^n -\frac{E^{ij}_{\;\;\; ij}}{24}m (p+p')^n \int_0^1 \!\!dx \frac{1}{p_x^2}
-\frac{1}{3} E_a^{\;\;rns} \gamma^a \int_0^1 \!\!dx\, \frac{p_{xr} p_{xs}}{p_x^2}+ \right. \\
&+ E_a^{\;\;rbs} \int_0^1 \!dx\, \frac{p_{xr} p_{xs}}{p_x^2} (p_b \slashed{p}_x \gamma^a \gamma^n+ p'_b \gamma^n \gamma^a \slashed{p}_x)+ \\
& \left. +2E^{arbs} p_a p'_b \gamma^n \int_0^1 \!dx\, \frac{p_{xr} p_{xs}}{(p_x^2)^2} \left[ \log\left( \frac{p_x^2}{\delta^2} \right)-3 \right] \right\} \,.\end{aligned}$$ Expanding in powers of $q$ neglecting second and higher orders, summing all contributions and replacing $(p+p')^n$ with $2m \gamma^n-i\sigma^{n\ell} q_\ell$ by use of the Gordon identity, we then find $$\label{1-loop-vertex-h}
\begin{aligned}
I_E^n =& - i {e_\textrm{\tiny pos}}\mu^{\varepsilon/2} \frac{\alpha}{4\pi} \bar{u}(p') \left[\,\frac{1}{2} E^{ij}_{\;\;\; ij} \gamma^n -\frac{1}{4} E^{ij}_{\;\;\; ij} \gamma^n \log\left(\frac{m^2}{\delta^2} \right) \right.+\\
&+\frac{i \sigma^{ab}}{8m^3} E_{a}^{\;\; rns} (p'+p)_r (p'+p)_s q_b + 2 \frac{i \sigma^{na}}{8m^2} E_{arbs} (p'+p)^b (p'+p)^r q^s + \\
&\left. + \frac{1}{2} \frac{E_{arbs}+E_{abrs}}{4m^2} (p'+p)^r q^s (\gamma^a \gamma^b \gamma^n + \gamma^n \gamma^a \gamma^b ) \right] u(p) \,.
\end{aligned}$$ Taking the sum of the tree-level contribution (\[tree-level-vertex\]), the one-loop corrections (\[1-loop-vertex-qed\]), (\[1-loop-vertex-h\]) and the one-loop counter-term in the on-shell scheme, $$\bar{u} (p') \left[
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\coordinate (aux) at (0,0);
\coordinate[label=left:$p'$] (e1) at (-.7,.5);
\coordinate[label=right:$p$] (e2) at (.7,.5);
\coordinate[label=below:$n$] (e3) at (0,-.6);
\coordinate[above right=.1 and .25 of e3] (arr1);
\coordinate[below right=.1 and .25 of aux] (arr2);
\coordinate[label=right:$q$] (arrlab) at ($(arr1)!0.5!(arr2)$);
\node [circle,draw,inner sep=.5mm,fill] (dot) at (aux) {};
\draw[fermion] (dot) -- (e1);
\draw[fermion] (e2) -- (dot);
\path [sines/.style={
line join=round,
draw=black,
decorate,
decoration={complete sines, number of sines=1.5, amplitude=1.5mm}},postaction={sines}] (dot) -- (e3);
\draw[->] (arr1) -- (arr2);
\end{tikzpicture}
\right] u(p)=
i{e_\textrm{\tiny pos}}\mu^{\varepsilon/2} (Z_\psi^{\text{o.s.}}-1) \bar{u} (p') \gamma^n u(p) \,,$$ where the renormalization constant $Z_\psi^{\,\text{o.s.}}$ is given by (\[Zpsi-on-shell\]), after setting $\lambda=1$ and $Q=-1$, we finally obtain the relevant renormalized quantity $$\label{1-loop-vertex-complete}
\begin{aligned}
i{e_\textrm{\tiny pos}}&\bar{u} (p') \Gamma^n_{\text{o.s.}} u(p) = i{e_\textrm{\tiny pos}}\bar{u} (p') \left[ F_1\, \gamma^n + F_2\, \frac{i \sigma^{n\ell} q_\ell}{2m}+ \right. \\
&+ F_3\, \frac{i \sigma^{ab}}{8m^3} E_{a}^{\; rns} (p'+p)_r (p'+p)_s q_b + F_4\, \frac{i \sigma^{na}}{8m^3} E_{arbs} (p'+p)^b (p'+p)^r q^s + \\
& \left. +F_5\, \frac{E_{arbs}+E_{abrs}}{4m^2} (p'+p)^r q^s (\gamma^a \gamma^b \gamma^n + \gamma^n \gamma^a \gamma^b ) \right] u(p) +{\ensuremath{\mathcal{O}}}(q^2) \,,
\end{aligned}$$ where we calculated the form factors to take the values $$\label{form-factors}
F_1=1 \,, \qquad F_2=\frac{\alpha}{2\pi} \,, \qquad F_3= - \frac{\alpha}{4\pi} \,, \qquad F_4= - \frac{\alpha}{2\pi} \,, \qquad F_5= - \frac{\alpha}{8\pi} \,.$$
This concludes the computation of the one-loop proper vertex. In the next subsection, we analyze the physical implications.
Emergence of a qualitatively new spin-magnetic coupling {#sec:anomalous-magnetic-moment}
-------------------------------------------------------
We now use the renormalized one-loop vertex (\[1-loop-vertex-complete\]) in order to determine the non-relativistic potential of an electron in a quasi-static and uniform external magnetic field. This will provide us with a correction of the anomalous magnetic moment of the electron—which result we will however significantly refine in the next subsection—and a new small interaction between the spin and the magnetic field. Both are corrections to the standard QED case.
Multiplying the one-loop vertex (\[1-loop-vertex-complete\]) by the Fourier transform of the electromagnetic external potential $\tilde{A}_n(q)$, one obtains the scattering amplitude $$i \mathcal{M} = \bar{u} (p') \big[ i{e_\textrm{\tiny pos}}\Gamma_{\text{o.s.}}^n (p,p') \big] u(p) \tilde{A}_n (q) \,.$$ Then taking the non-relativistic limit $|{\ensuremath{\boldsymbol{p}}}| \ll m$, by neglecting all contributions quadratic in ${\ensuremath{\boldsymbol{p}}}$ and all those linear in both ${\ensuremath{\boldsymbol{p}}}$ and $H$, and using the Born approximation $$V({\ensuremath{\boldsymbol{x}}})=- \frac{1}{2m}\int \frac{d^{\,3} q}{(2\pi)^3} e^{-i q_\alpha x^\alpha} \mathcal{M}(q) \,,$$ one derives the non-relativistic interaction potential $$\begin{aligned}
V({\ensuremath{\boldsymbol{x}}}) = \left[ 2 (F_1+F_2)+ \frac{E^{ij}_{\;\;\; ij}}{4}(F_3-F_4+6F_5)\right] \left\langle \frac{{e_\textrm{\tiny pos}}{\ensuremath{\boldsymbol{s}}}}{2m} \right\rangle \cdot {\ensuremath{\boldsymbol{B}}}
+(F_3-F_4+6F_5) E^{0 \alpha 0 \beta} \left\langle \frac{{e_\textrm{\tiny pos}}s_\alpha}{2m} \right\rangle B_\beta \,,
\end{aligned}$$ where $\langle {\ensuremath{\boldsymbol{s}}} \rangle {\ensuremath{:=}}\xi' {\ensuremath{\boldsymbol{s}}}\, \xi$, with $\xi$ denoting the polarization of the ingoing electron and $\xi'$ that of the outgoing electron. Expanding $V({\ensuremath{\boldsymbol{x}}})$ in terms of the form factors (\[form-factors\]) and the tracefree tensor $$\label{widehatE}
\widehat{E}^{\alpha \beta}{\ensuremath{:=}}E^{0 \alpha 0 \beta}+\frac{E^{ij}_{\;\;\; ij}}{12} \delta^{\alpha \beta}$$ (which will make another appearance in section \[subsec:full\], where it will be seen to control a splitting of the triplet of the hyperfine structure of hydrogen), the non-relativistic potential for the electron in an external quasi-static and uniform magnetic field becomes $$\label{potential}
V({\ensuremath{\boldsymbol{x}}})= 2 \left[ 1+\frac{\alpha}{2\pi} \left(1 -\frac{E^{ij}_{\;\;\; ij}}{12}\right) \right] \left\langle \frac{{e_\textrm{\tiny pos}}{\ensuremath{\boldsymbol{s}}}}{2m} \right\rangle \cdot {\ensuremath{\boldsymbol{B}}}-
\frac{\alpha}{2\pi} \widehat{E}^{\alpha \beta} \left\langle \frac{{e_\textrm{\tiny pos}}s_\alpha}{2m} \right\rangle B_\beta \,.$$
The first summand in $V({\ensuremath{\boldsymbol{x}}})$ is the minimal coupling of the spin with the magnetic field, featuring a however slightly modified anomalous magnetic moment of the electron $$\label{anomalous-magnetic-moment}
a {\ensuremath{:=}}\frac{g-2}{2}= (F_1+F_2-1)+ \frac{E^{ij}_{\;\;\; ij}}{12}(F_3-F_4+6F_5) = \frac{\alpha}{2\pi}\left( 1-\frac{E^{ij}_{\;\;\; ij}}{12} \right) \,,$$ which is compatible with what is obtained by Carone *et al.* [@Carone:2006tx] for the special case of an isotropic background. On the other hand, the second summand in (\[potential\]) presents a qualitatively new interaction $$\label{potential-correction}
\delta V({\ensuremath{\boldsymbol{x}}}) = - \frac{\alpha}{2\pi}\widehat{E}^{\alpha \beta} \left\langle \frac{{e_\textrm{\tiny pos}}s_\alpha}{2m} \right\rangle B_\beta$$ between the spin and the magnetic field.
This concludes the calculation of the scattering of an electron in an external electromagnetic field in a spacetime with ray-optically invisible non-metricities. As for the tree-level scatterings, we found only very subtle differences with respect to QED, both in the modification of the Schwinger correction to the anomalous magnetic moment (\[anomalous-magnetic-moment\]) and in the appearance of a quantitatively small, but qualitatively new term in the potential (\[potential-correction\]).
Anomalous magnetic moment to any order from ray-optically invisible non-metricities {#sec:AMM-beyond-one-loop}
-----------------------------------------------------------------------------------
In section \[sec:electron-external-magnetic-field\], we have computed the one-loop radiative corrections to the vertex and used them in section \[sec:anomalous-magnetic-moment\] to derive the one-loop anomalous magnetic moment (\[anomalous-magnetic-moment\]) of the electron. Now we show how to deduce the anomalous magnetic moment of the electron at every loop order, once it is known in metric QED.
Since the renormalization constant for the vertex and the anomalous magnetic moment are scalar, they depend on the perturbation $E^{abcd}$ only through its full trace $E^{ij}_{\;\;\; ij}$. Thus when computing the anomalous magnetic moment, we can replace the perturbation $E^{abcd}$ with any other perturbation $\tilde{E}^{abcd}$, on the condition that both the perturbation have the same full trace, $E^{ij}_{\;\;\; ij}=\tilde{E}^{ij}_{\;\;\; ij}$. In particular, we can take $$\label{particular-perturbation}
\tilde{E}^{abcd}=\frac{E^{ij}_{\;\;\; ij}}{D(D-1)} (\eta^{ac}\eta^{bd}-\eta^{ad}\eta^{bc}) \,.$$ Moreover, since the physical quantities do not depend on the gauge parameter $\lambda$, we choose the *Landau gauge* to set it to zero. Then the photon propagator (\[photon-propagator\]) for the perturbation (\[particular-perturbation\]) becomes $$\label{photon-propagator-particular}
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=- \frac{i}{q^2+i\epsilon}\left( \eta_{ab} - \frac{q_a q_b}{q^2+i\epsilon}\right) \left[ 1-\frac{E^{ij}_{\;\;\; ij}}{D(D-1)} \right] \,,$$ which, up to the scalar factor in square brackets, is the usual photon propagator for standard QED in Landau gauge.
The topology of the diagrams implies that the $\ell$-loop vertex radiative correction $i{e_\textrm{\tiny pos}}\, \Gamma^n_{\ell}(p,p',\alpha)$ contains $\ell$ photon propagators and $(2\ell+1)$ vertices. Here we chose to denote the momentum of the ingoing fermion by $p$ and that of the outgoing by $p'$, to leave explicit the dependence on the fine structure constant $\alpha$ and to factor out ${e_\textrm{\tiny pos}}$. The $(2\ell+1)$ vertices carry a factor ${e_\textrm{\tiny pos}}\alpha^{\,\ell}$, while the $\ell$ photon propagators carry the factor in square brackets in (\[photon-propagator-particular\]) to the power of $\ell$. Thus, we can write $$i{e_\textrm{\tiny pos}}\, \Gamma^n_{\ell}(p,p',\alpha)=i{e_\textrm{\tiny pos}}\, \alpha^{\,\ell} \left[ 1-\frac{E^{ij}_{\;\;\; ij}}{D(D-1)} \right]^{\ell} I^n_{\ell}(p,p') \,,$$ where $I^n_{\ell}(p,p')$ is defined through the relation $\Gamma^n_{\ell,\text{\tiny{QED}}}(p,p',\alpha)=\alpha^{\,\ell} I^n_{\ell}(p,p')$. We then find, for the particular perturbation (\[particular-perturbation\]) in Landau gauge, the relation $$\label{vertex-loops-relation}
i{e_\textrm{\tiny pos}}\, \Gamma^n_{\ell}(p,p',\alpha)=i{e_\textrm{\tiny pos}}\, \Gamma^n_{\ell,\text{\tiny{QED}}}(p,p',\tilde{\alpha}) \,,
\quad\text{where}\quad
\tilde{\alpha}{\ensuremath{:=}}\alpha \left[ 1-\frac{E^{ij}_{\;\;\; ij}}{D(D-1)} \right] \,,$$ from which it follows that the $\ell$-loop anomalous magnetic moment for quantum electrodynamics, in the presence of ray-optically invisible perturbations around flat Lorentzian spacetime, is given by $$\label{AMM-loops-relation}
a_\ell(\alpha)=a_{\ell,\text{\tiny{QED}}}(\tilde{\alpha}) \,,
\quad\text{where}\quad
\tilde{\alpha}{\ensuremath{:=}}\alpha \left( 1-\frac{E^{ij}_{\;\;\; ij}}{12} \right) \,,$$ having set $D=4$, since the quantity is finite. In principle, ultraviolet and infrared regulators, as the photon mass employed in section \[sec:electron-external-magnetic-field\], could break relation (\[vertex-loops-relation\]), but this is merely an effect of the chosen regularization technique, which disappears after renormalization and does not affect relation (\[AMM-loops-relation\]) between physical quantities.
The anomalous magnetic moment of the electron is measured with high precision [@Hanneke:2008tm] and can be used, in metric QED, to determine the value of the fine structure constant [@Aoyama:2012wj]. With the same procedure, we can determine the value $$\label{alpha-tilde-value}
\alpha \left( 1-\frac{E^{ij}_{\;\;\; ij}}{12} \right)=1/137.035\, 999\, 2(2) \,,$$ where the uncertainty in brackets comes from having neglected, in the theoretical calculation, the hadronic contribution, whose order of magnitude is $a^{\text{hadron}}\simeq 2\times 10^{-12}$, and the electroweak contribution, whose order of magnitude is even smaller.
It is thus shown to be possible to determine, with high precision, the value of a particular combination of the fine structure constant $\alpha$ and the double trace of the perturbation $E^{ij}_{\;\;\; ij}$. In order to determine their individual values, it is necessary to find at least another independent measurable quantity depending on them. For the moment, we content ourself with the determination of the fine structure constant $\alpha$ in ray-optically invisible birefringent electrodynamics. Indeed, many bounds to the value of $E^{ij}{}_{ij}$ have been found in the past in the context of SME [@SME-constraints-gamma-ray; @SME-Cerenkov; @SME-bounds-laser-interferometry; @SME-bounds-1; @SME-bounds-2], with the identification $E^{ij}{}_{ij}=-12 \tilde{k}_{\ensuremath{\text{tr}}}$, and can be used to find a precise value of $\alpha$. Since the tensor $E$, together with the density $e$, provides the geometry of spacetime and vary from point to point as a consequence of its dynamics [@SSSW], it is important to compared the experimental value of the anomalous magnetic moment, which is measured on the surface of Earth, with those bounds determined on the surface of Earth. With this in mind, using the bound $|E^{ij}{}_{ij}/12|<10^{-10}$, one finds the value $$\label{alpha-value}
\alpha =1/137.035\, 999\, 2(2)$$ for the fine structure constant.
Bound states and spectrum of hydrogen {#sec:hydrogen}
=====================================
We obtain the bound states of the hydrogen atom in the presence of ray-optically invisible non-metricities. This is done by employing the Born-Oppenheimer approximation and a classical treatment of the electromagnetic field generated by the proton’s charge and magnetic moment, followed by a second order quantum mechanical perturbation around the known exact solution of the relativistic hydrogen atom. In particular, we find the effect of ray-optically invisible non-metricities on the hyperfine structure of hydrogen, which plays a significant role in astrophysics.
In the context of the SME, various studies of the hydrogen atom have been undertaken. These, however, were only concerned with modifications in the Dirac equation, while working with usual Maxwellian electromagnetism, see [@hydrogenstudy] and references therein. From this point of view, our work is complementary to the literature, for we investigate the effect of a refined electromagnetism, while Dirac’s equation remains unaltered.
The proton’s electromagnetic field {#sec:solutions}
----------------------------------
The classical electromagnetic potential produced by the proton has two constituents: one from the electric charge of the proton and one due to the proton’s magnetic moment. We calculate now both contributions from the electromagnetic field equations.
First, we obtain the refinement of the general time-independent solution of the field equations. On a flat spacetime with ray-optically invisible non-metricities, and after choosing there the Lorenz gauge $\eta^{ab}\partial_a A_b = 0$, the field equations for the gauge potential $A$ become $$\label{eq:field}
\left(\eta^{ad}\Box - E^{abcd}\partial_b\partial_c\right)A_d = j^a$$ with $\Box := \eta^{ab}\partial_a\partial_b$. Solutions to these field equations are given by convolution of the Green’s function $\Delta_{de}$ with the source term $j^a$. The Green’s function is the solution of $$\left(\eta^{ad}\Box - E^{abcd}\partial_b\partial_c\right)\Delta_{de}(x) = \delta^a_e\delta^{(4)}(x)$$ and for its Fourier transform one finds $$\left(\mathcal{F}\Delta_{de}\right)(q) = -\frac{1}{q^2}\left(\eta_{de} + E_{d\hphantom{bc}e}^{\hphantom{d}bc} \frac{q_bq_c}{q^2}\right)\,,$$ with $q^2 := \eta^{ab}q_a q_b$. For source terms $j({\ensuremath{\boldsymbol{x}}},t) = j({\ensuremath{\boldsymbol{x}}})$ without time dependence, as are provided by a resting proton’s charge and magnetic moment, we can perform an inverse Fourier transform to find the refined Biot-Savart law $$\label{eq:Biot-Savart}
A_a = \int{d}^3 x'\frac{j^b({\ensuremath{\boldsymbol{x}}}')}{4\pi|{\ensuremath{\boldsymbol{x}}}'-{\ensuremath{\boldsymbol{x}}}|}\left(\eta_{ab} + \frac{1}{2}E_{a\mu\nu b}\left(\eta^{\mu\nu}+\frac{(x'-x)^\mu(x'-x)^\nu}{|{\ensuremath{\boldsymbol{x}}}'-{\ensuremath{\boldsymbol{x}}}|^2}\right)\right)\,,$$ where $|{\ensuremath{\boldsymbol{x}}}'-{\ensuremath{\boldsymbol{x}}}| := \sqrt{-\eta_{\mu\nu}(x'-x)^\mu(x'-x)^\nu}$. With this expression at hand, it is now a simple exercise to calculate the two contributions to the proton’s potential.
The proton’s charge ${e_\textrm{\tiny pos}}$ sources the electromagnetic field equations by $$j^0({\ensuremath{\boldsymbol{x}}},t) = {e_\textrm{\tiny pos}}\delta^{(3)}({\ensuremath{\boldsymbol{x}}}) \qquad \text{and} \qquad j^\beta = 0\,,$$ which trivializes the integration in the Biot-Savart law and thus yields as the modification of the standard Coulomb potential in the presence of ray-optically invisible non-metricities $$\label{eq:pointcharge}
A^\textrm{\tiny charge}_a({\ensuremath{\boldsymbol{x}}}) = \frac{Q{e_\textrm{\tiny pos}}}{4\pi r}\left(\eta_{a0} +\frac{1}{2} E_{a\mu\nu 0}\left(\eta^{\mu\nu}+\frac{x^\mu x^\nu}{r^2}\right)\right)\,,$$ where $r^2 := -\eta_{\mu\nu}x^\mu x^\nu$. The second contribution to the electromagnetic field of the proton is due to its magnetic dipole moment. Classically, a magnetic dipole is a circular current in the limit of vanishing radius. This amounts to the electromagnetic potential of a magnetic dipole ${\ensuremath{\boldsymbol{M}}}$ being, by definition, the solution of the field equations with the source term $$j^0 = 0 \qquad \text{and} \qquad {\ensuremath{\boldsymbol{j}}} = \frac{|{\ensuremath{\boldsymbol{M}}}|}{\pi a^2} \delta(r - a)\delta({\ensuremath{\boldsymbol{M}}}\cdot{\ensuremath{\boldsymbol{x}}}){\ensuremath{\boldsymbol{M}}}\times{\ensuremath{\boldsymbol{x}}}$$ in the limit $a \rightarrow 0$. For tangibility we used the intuitive notations ${\ensuremath{\boldsymbol{M}}}\cdot{\ensuremath{\boldsymbol{x}}}$ and ${\ensuremath{\boldsymbol{M}}}\times{\ensuremath{\boldsymbol{x}}}$ which are defined as $$\label{eq:dotncross}
{\ensuremath{\boldsymbol{M}}}\cdot{\ensuremath{\boldsymbol{x}}} := -\eta_{\mu\nu}M^\mu x^\nu \qquad \text{ and }\qquad \left({\ensuremath{\boldsymbol{M}}}\times{\ensuremath{\boldsymbol{x}}}\right)^\mu := -\eta^{\mu\nu}\epsilon_{\nu\rho\sigma}M^\rho x^\sigma\,.$$ Inserting this into the Biot-Savart law , expanding in powers of $\frac{a}{r}$, calculating the occurring integrals with the help of Mathematica, and then letting $a\rightarrow 0$, one finds the electromagnetic potential of the magnetic dipole to be $$\label{eq:dipole}
A^\textrm{\tiny magnetic}_a = -\frac{M^\mu x^\nu}{4\pi r^3}\left(\epsilon_{a\mu\nu} + \frac{1}{2}\epsilon_{\beta\mu\nu} E_{a\rho\sigma}{}^\beta \left(\eta^{\rho\sigma} + 3\frac{x^\rho x^\sigma}{r^2}\right) + \epsilon_{\beta\mu\rho}\eta_{\nu\sigma} E_a{}^{(\rho\sigma)\beta}\right)\,.$$ Note that for a vanishing perturbation $E$ this reproduces the standard result ${\ensuremath{\boldsymbol{A}}} = \frac{1}{4\pi r^3}{\ensuremath{\boldsymbol{M}}}\times{\ensuremath{\boldsymbol{x}}}$.
Refined Hamiltonian of hydrogen {#subsec:setup}
-------------------------------
We consider the hydrogen atom quantum mechanically as a two-body system composed of a spin-$\frac{1}{2}$ proton of mass $m_p$ and charge ${e_\textrm{\tiny pos}}$ and a spin-$\frac{1}{2}$ electron of mass $m_e$ and charge $-{e_\textrm{\tiny pos}}$ that moves in the electromagnetic potential generated by the charge and the magnetic moment of the proton. The much larger mass of the proton allows for a Born-Oppenheimer approximation, so that the state of the entire atom can be assumed to be a simple tensor product $$\Psi^\textrm{\tiny atom} = \psi^\textrm{\tiny electron}\otimes\psi^\textrm{\tiny proton}\,,$$ rather than a linear combination of such. Additionally considering the frame where the proton is at rest, we may further describe the positive energy states of the proton simply by $\psi^\textrm{\tiny proton} \in \mathbb{C}^2$. The electron’s wave function, however, is a full Dirac spinor field $\psi^\textrm{\tiny electron}$ with reduced Dirac mass $m_\text{\tiny red} := \frac{m_e m_p}{m_e + m_p}$, whose negative energy states will be discarded later on by hand in this first quantized treatment.
With the above assumptions, the fully time-dependent quantum mechanical equation of motion for the hydrogen atom reads $$({i}\gamma^a\partial_a + {e_\textrm{\tiny pos}}\gamma^a A_a - m_\text{\tiny red})\Psi^\textrm{\tiny atom} = 0\,,$$ if all operators, with the notable exception of $A_a$, are agreed to act non-trivially only on the first factor of the tensor product $\psi^\textrm{\tiny electron}\otimes\psi^\textrm{\tiny proton}$, e.g., $\gamma^a (\psi^\textrm{\tiny electron}\otimes\psi^\textrm{\tiny proton}) := (\gamma^a\psi^\textrm{\tiny electron})\otimes\psi^\textrm{\tiny proton}$. The electromagnetic potential, in contrast, being dependent on the proton’s spin through its magnetic moment operator $${M}^\mu (\psi^\textrm{\tiny electron}\otimes\psi^\textrm{\tiny proton}) := \frac{g_p {e_\textrm{\tiny pos}}}{2m_p} \psi^\textrm{\tiny electron} \otimes (\tfrac{1}{2} \sigma^\mu \psi^\textrm{\tiny proton})\qquad\textrm{(with gyromagnetic ratio } g_p \approx 5.59\textrm{)} \,,$$ acts non-trivially on both factors. The complete electromagnetic potential generated by the proton is given by the sum of the two contributions $A^{\text{\tiny charge}}$ and $A^{\text{\tiny magnetic}}$ of \[eq:pointcharge,eq:dipole\] respectively, such that $$A_a = \frac{{e_\textrm{\tiny pos}}}{4\pi r}\left(\eta_{a0} +\frac{1}{2} E_{a\mu\nu 0}\frac{x^\mu x^\nu}{r^2}\right) - \frac{{M}^\mu x^\nu}{4\pi r^3}\left(\epsilon_{a\mu\nu} + \frac{1}{2}\epsilon_{\beta\mu\nu}E_{a\rho\sigma}{}^\beta \left(\eta^{\rho\sigma} + 3\frac{x^\rho x^\sigma}{r^2}\right) + \epsilon_{\beta\mu\rho}\eta_{\nu\sigma} E_a{}^{(\rho\sigma)\beta}\right)\,.$$ Rewriting the above equation of motion in Schrödinger form $i\partial_t\Psi^\textrm{\tiny atom} = H \Psi^\textrm{\tiny atom} $, which one obtains by acting on both sides with $\gamma^0$ and rearranging terms, one identifies the Hamiltonian $${H} := -{i}\gamma^0\gamma^\mu\partial_\mu - {e_\textrm{\tiny pos}}\gamma^0\gamma^a A_a + \gamma^0 m_\text{\tiny red}$$ of the hydrogen atom in a spacetime with ray-optically invisible non-metricities.
For stationary states $\Psi^\textrm{\tiny atom} ({\ensuremath{\boldsymbol{x}}},t) = \Psi^\textrm{\tiny atom} ({\ensuremath{\boldsymbol{x}}}){e}^{-{i}Et}$, solving the Dirac equation thus reduces to the eigenvalue problem $E\Psi^\textrm{\tiny atom} ({\ensuremath{\boldsymbol{x}}}) = {H}\Psi^\textrm{\tiny atom} ({\ensuremath{\boldsymbol{x}}})$ . With regards to the perturbative determination of the spectrum of $H$, we choose to display the Hamiltonian as a sum $$\label{hamiltonian}
{H} = {H}_0 + {H}_{\textrm{\tiny hfs}} + {H}_{\text{\tiny abs}} + {H}_{\text{\tiny rel}}$$ of individual contributions (controlled by the fine structure constant $\alpha := {e_\textrm{\tiny pos}}^2/(4\pi)$ and the hyperfine parameter $\mu := g_p/(2m_p r_\text{\tiny B})$, where $r_\text{\tiny B} := 1/(m_\text{\tiny red}\alpha)$ denotes the Bohr radius) $$\begin{aligned}
{H}_{0} :={}& -{i}\gamma^0\gamma^\mu\partial_\mu - \frac{\alpha}{r} + \gamma^0 m_\textrm{\tiny red}\,,\\
{H}_{\textrm{\tiny hfs}} :={}& \mu \frac{\alpha}{r}\frac{r_B}{r}\gamma^0\gamma^\alpha\hat{S}^\mu \frac{x^\nu}{r}\epsilon_{\alpha\mu\nu}\,,\\
{H}_{\textrm{\tiny abs}} :={}& - \frac{\alpha}{2r}\gamma^0\gamma^a E_{a\mu\nu 0}\frac{x^\mu x^\nu}{r^2}\,,\\
{H}_{\textrm{\tiny rel}} :={}& \mu \frac{\alpha}{r}\frac{r_B}{r}\gamma^0\gamma^a\hat{S}^\mu \frac{x^\nu}{r}\left(\frac{1}{2}\epsilon_{\beta\mu\nu} E_{a\rho\sigma}{}^\beta \left(\eta^{\rho\sigma} + 3\frac{x^\rho x^\sigma}{r^2}\right) + \epsilon_{\beta\mu\rho}\eta_{\nu\sigma} E_a{}^{(\rho\sigma)\beta}\right)\,.\end{aligned}$$ The part $H_0$ is known to be exactly solvable and thus provides the basis for a perturbative treatment of the other contributions. Of these, only $H_\textrm{\tiny hfs}$ appears in the calculation of the standard hyperfine structure, while the two remaining parts are entirely due to the ray-optically invisible deviations from Lorentzian geometry that we consider.
The need for a second order perturbation analysis {#subsec:secondorder}
-------------------------------------------------
Compared to the Coulomb potential $\frac{\alpha}{r}$ in ${H}_0$, the three correction operators in the full Hamiltonian (\[hamiltonian\]) are suppressed by the hyperfine parameter $\mu$, the ray-optically invisible perturbation components $E^{abcd}$, or both. More precisely, $$\label{eq:mags}
{H}_{\text{\tiny hfs}} \sim \mu H_0\,,\qquad {H}_{\text{\tiny abs}} \sim E^{abcd} H_0\,,\qquad {H}_{\text{\tiny rel}} \sim \mu E^{abcd} H_0\,.$$ With the hyperfine parameter $\mu = g_p/(2m_p r_\text{\tiny B}) \approx 1.11\times 10^{-5}$ being small compared to unity, and the eleven-parameter family of small deviations $E^{abcd}$ to be considered only to linear order in our approach, we will be able to derive the hydrogen spectrum as a twelve-parameter perturbation of the eigenvalue problem for $H_0$, whose exact solutions are known and for the conceptual discussion in this section will be denoted by $|E_n,\alpha\rangle$ with the eigenenergy $E_n$ and a degeneracy label $\alpha$.
Following usual quantum mechanical perturbation theory, the corrections to the energy levels $E_n$ are given by the eigenvalues of the transition matrix $T_n$, whose elements $\langle E_n,\alpha | T_n | E_n,\beta \rangle$ are given by the obvious first order contribution $\langle E_n,\alpha | {H}_{\textrm{\tiny hfs}} + {H}_{\text{\tiny abs}} + {H}_{\text{\tiny rel}} | E_n,\beta \rangle$ plus the second order contribution $$\sum_{m\neq n,\,\gamma}\frac{\langle E_n,\alpha | {H}_{\textrm{\tiny hfs}} + {H}_{\text{\tiny abs}} + {H}_{\text{\tiny rel}} | E_m,\gamma \rangle\langle E_m,\gamma | {H}_{\textrm{\tiny hfs}} + {H}_{\text{\tiny abs}} + {H}_{\text{\tiny rel}} | E_n,\beta \rangle}{E_m - E_n}$$ plus higher order contributions, which we will not need to consider. Bearing in mind the relative magnitudes of the single operators, it is apparent that for the first order correction due to $H_{\text{\tiny rel}}$, there is a second order term of comparable magnitude from the product of $H_\text{\tiny hfs}$ and $H_\text{\tiny abs}$. It is thus inevitable to also calculate these second order terms. However, we will not need to calculate all the second order contributions, but only those comparable in magnitude to the first order terms. Taking into account the phenomenological fact that the spectral corrections from extra geometric degrees of freedom are significantly weaker than those from the magnetic moment of the proton, these relevant terms are the second order correction to the usual hyperfine structure, $H_\text{\tiny hfs}^2$, and the above mentioned products $H_\text{\tiny hfs}H_\text{\tiny abs}$ and $H_\text{\tiny abs}H_\text{\tiny hfs}$. In the following sections, all these contributions to the transition matrix $T_n$ are calculated. After having fixed notation in a most concise review of the exact solution of the eigenproblem of $H_0$ in \[subsec:exact\], we calculate the usual hyperfine structure up to second order in \[subsec:nonbiref\]. From then on we turn to the investigation of the effects of ray-optically invisible non-metricities. While \[subsec:lambda\] is concerned with the purely non-metric corrections, in \[subsec:lambdamu\] we investigate the combined effects of non-metricity and proton spin, including the relevant second order terms identified above. The full transition matrix is then of course obtained by summing all these contributions. Because of its immediate phenomenological relevance, we investigate in \[subsec:full\] the hyperfine structure of hydrogen in the presence of ray-optically invisible non-metricities.
Exact solution without proton spin and without non-metricities {#subsec:exact}
--------------------------------------------------------------
The eigenproblem of $H_0$, the unperturbed relativistic hydrogen problem, is exactly solvable [@dirac1928quantum; @pidduck1929laguerre] and can be found in many textbooks on advanced quantum mechanics [@sakurai1967advanced; @merzbacher1998quantum]. We will thus only shortly revisit its most essential features, and thereby introduce the notation used in the sections to follow.
The fully relativistic energy spectrum of the hydrogen atom is given by $$E(n,\kappa,m,s) := E(n,|\kappa|) = \frac{m_\text{\tiny red}}{\sqrt{1+\left(\frac{\alpha}{n + \sqrt{\kappa^2 - \alpha^2}}\right)^2}}\,, $$ where $n\in\mathbb{N}$, $\kappa\in\, - \mathbb{N}$ (for $n=0$) or $\kappa\in\mathbb{Z}\backslash\lbrace 0\rbrace$ (for $n>0$), $m = -|\kappa|+\tfrac{1}{2}, \dots, |\kappa|-\tfrac{1}{2}$ and $s=-\tfrac{1}{2}, \tfrac{1}{2}$, and the corresponding eigenstates $|n\kappa ms\rangle$ explicitly read $$\langle {\ensuremath{\boldsymbol{x}}} | n\kappa ms\rangle = \mathcal{N}\left(2m_\text{\tiny red}\alpha\right)^{\frac{3}{2}}{e}^{-\frac{y}{2w}} y^{K-1} \begin{pmatrix} \left[L_n^{2K}\left(\frac{y}{w}\right) - \frac{w+\kappa}{n}L_{n-1}^{2K}\left(\frac{y}{w}\right)\right]\mathcal{Y}_\kappa^m ({\ensuremath{\boldsymbol{n}}}) \\[2ex] \frac{w-W}{{i}\alpha}\left[L_n^{2K}\left(\frac{y}{w}\right) + \frac{w+\kappa}{n}L_{n-1}^{2K}\left(\frac{y}{w}\right)\right]\mathcal{Y}_{-\kappa}^m ({\ensuremath{\boldsymbol{n}}}) \end{pmatrix} \otimes \chi^{(s)}\,,$$ where we split the coordinates ${\ensuremath{\boldsymbol{x}}}$ into radial and angular components according to $x^\mu = \frac{1}{2}r_\text{\tiny B} y n^\mu$ such that $|{\ensuremath{\boldsymbol{n}}}| = 1$ and the radial variable $y$ is dimensionless. Furthermore we abbreviated frequent combinations of quantum numbers (and the finestructure constant $\alpha$) as $$K := \sqrt{\kappa^2 - \alpha^2}\,,\qquad W := n + K\,,\qquad w := \sqrt{W^2 + \alpha^2}\,.$$ The definitions of the Laguerre functions $L_n^\lambda(z)$ and the two component spherical spinors $\mathcal{Y}^m_\kappa ({\ensuremath{\boldsymbol{n}}})$, and the precise form of the normalization constant $\mathcal{N}$ are given in the appendix. The $\chi^{(s)}$ denote the unit spinors for the proton, with $\chi^{(+\tfrac{1}{2})} = (1,\,0)$, $\chi^{(-\tfrac{1}{2})} = (0,\,1)$.
Since we wish to inspect the effects of ray-optically invisible non-metricities on the 21 cm line, we are especially interested in the level splitting of the ground state(s). These we denote by $|ms\rangle:= |0, -1, m, s\rangle$ with both the electron spin $m$ and and the proton spin $s$ taking values either $-\tfrac{1}{2}$ or $+\tfrac{1}{2}$. With all these prerequisites at hand, we can now begin to calculate all the relevant contributions to the 4-by-4 ground state transition matrix $\langle m's' | T_g | ms \rangle$. By the outline at the end of \[subsec:secondorder\], the latter is given by $$\begin{aligned}
\langle m's' | T_g | ms \rangle ={}& \langle m's' | T_\text{\tiny\labelcref{subsec:nonbiref}} |ms\rangle + \langle m's' | T_\text{\tiny\labelcref{subsec:lambda}} |ms\rangle + \langle m's' | T_\text{\tiny\labelcref{subsec:lambdamu}} | ms \rangle\,,\end{aligned}$$ where the single contributions explicitly read $$\begin{aligned}
\langle m's' | T_\text{\tiny\labelcref{subsec:nonbiref}} | ms \rangle ={}& \langle m's' | H_\text{\tiny hfs} | ms\rangle + \sum_{n'',\kappa'',m'',s''} \frac{\langle m's' | H_\text{\tiny hfs} | n''\kappa''m''s''\rangle\langle n''\kappa''m''s'' | H_\text{\tiny hfs} | ms \rangle}{E(n'',|\kappa''|)-E_g}\,, \\
\langle m's' | T_\text{\tiny\labelcref{subsec:lambda}} | ms \rangle ={}& \langle m's' | H_\text{\tiny abs} | ms\rangle\,, \phantom{\sum_{n''}}\\
\langle m's' | T_\text{\tiny\labelcref{subsec:lambdamu}} | ms \rangle ={}& \langle m's' | H_\text{\tiny rel} | ms\rangle +2 \sum_{n'',\kappa'',m'',s''} \frac{\langle m's' | H_\text{\tiny hfs} | n''\kappa''m''s''\rangle\langle n''\kappa''m''s'' | H_\text{\tiny abs} | ms \rangle}{E(n'',|\kappa''|)-E_g}\,.\end{aligned}$$ Remember that the sums in the second order terms run over only those combinations of quantum numbers for which $E(n'',|\kappa''|)$ is different from the ground state energy $E_g$, such that the denominator is non-zero. The second order term in the last line is a cross-term from the square, thus the factor of two. More precisely, this would have to be two terms, $H_\text{\tiny hfs}H_\text{\tiny abs}$ and $H_\text{\tiny abs}H_\text{\tiny hfs}$, which a priori are not equal. However, our explicit calculation below shows that this in fact is the case, which brings about the two.
Corrections from proton spin, but without non-metricities {#subsec:nonbiref}
---------------------------------------------------------
This section is dedicated to the corrections that have nothing to do with the non-metric perturbation, namely the usual hyperfine structure, to first and second order perturbation theory in $\mu$. To this end, we consider the first order correction matrix elements $\langle n'\kappa'm's'| H_\text{\tiny hfs} |n\kappa ms\rangle$, which take the explicit form $$\begin{aligned}
& \mu m_\text{\tiny red}\alpha\mathcal{N}'\mathcal{N}\notag\\
&\times \left[\left(w-W+w'-W'\right)\left({{\ensuremath{\prescript{2}{00}{I}^{n'\kappa'}_{n\kappa}}}}- {{\ensuremath{\prescript{2}{11}{I}^{n'\kappa'}_{n\kappa}}}}\right) + \left(w-W-w'+W'\right)\left({{\ensuremath{\prescript{2}{01}{I}^{n'\kappa'}_{n\kappa}}}}- {{\ensuremath{\prescript{2}{10}{I}^{n'\kappa'}_{n\kappa}}}}\right)\right]\notag\\
&\times \left[\frac{\delta_{-\kappa', \kappa - 1}}{|2\kappa - 1|}\left(\tau_{-\kappa}^\mu\right)_{m'm} + \frac{4\kappa \delta_{\kappa',\kappa}}{4\kappa^2 - 1}\left(\sigma_\kappa^\mu\right)_{m'm} - \frac{\delta_{-\kappa',\kappa+1}}{|2\kappa+1|}\left(\tau_\kappa^\mu\right)_{m'm}\right]\left(\sigma_\mu\right)_{s's}\,. \label{eq:hfs}\end{aligned}$$ The precise form of the integrals ${{\ensuremath{\prescript{a}{ij}{I}^{n'\kappa'}_{n\kappa}}}}$ and the matrices $\sigma_\kappa^\mu$ and $\tau_\kappa^\mu$ are given in the appendix. Of immediate relevance, however, is that the $\sigma_{\pm 1}^\mu$ coincide with the three standard Pauli sigma matrices $\sigma^\mu$. Using the above expression, we can now straightforwardly calculate the individual terms in $T_\text{\tiny\labelcref{subsec:nonbiref}}$ for the ground states. The first one is simply $\langle m's' | H_\text{\tiny hfs} | ms \rangle$, for which one obtains $$-\frac{2}{3}\frac{\mu m_\text{\tiny red}\alpha^3}{2-2\alpha^2-\sqrt{1-\alpha^2}}\left(\sigma^\mu\otimes\sigma_\mu\right)_{m's',ms} =: \frac{1}{4}\Delta E_\text{\tiny hfs}\begin{pmatrix} 1&&&\\&-1&2&\\&2&-1&\\&&&1 \end{pmatrix}_{m's',ms}\,.$$ This is the usual first order hyperfine transition matrix. Indeed, it has the three-fold degenerate eigenvalue $+\frac{1}{4}\Delta E_\text{\tiny hfs}$ corresponding to the hyperfine structure triplet state and the non-degenerate eigenvalue $-\frac{3}{4}\Delta E_\text{\tiny hfs}$ corresponding to the singlet state, given in terms of the hyperfine transition energy $$\Delta E_\text{\tiny hfs} \approx 9.41\times 10^{-25} \text{ J} = \frac{hc}{21.1\text{ cm}}\,.$$ Let us go on to the second term in $T_\text{\tiny\labelcref{subsec:nonbiref}}$. For generic states $|n\kappa ms\rangle$, $|n'\kappa'm's'\rangle$ of same energy, that is $n' = n$ and $|\kappa'| = |\kappa|$, we have the second order correction $$\sum_{n'',\kappa'',m'',s''}\frac{\langle n\kappa'm's'| H_\text{hfs} |n''\kappa'' m''s''\rangle \langle n''\kappa''m''s''| H_\text{hfs} |n\kappa ms\rangle}{E(n'',|\kappa''|) - E(n,|\kappa|)}\,,$$ for which one explicitly obtains $$\begin{aligned}
4m_\text{\tiny red}&\alpha^2\mu^2\delta_{\kappa',\kappa} \Bigg\lbrace \delta_{m'm}\delta_{s's}\left[ \frac{\kappa-1}{2\kappa - 1} A_{n\kappa}(-\kappa+1) + \frac{4\kappa^2}{4\kappa^2 - 1} A_{n\kappa}(\kappa) + \frac{\kappa+1}{2\kappa+1} A_{n\kappa}(-\kappa-1)\right] \notag\\
& + \left(\sigma_\kappa^\mu\otimes\sigma_\mu\right)_{m's',ms}\bigg[ \frac{\kappa-1}{(2\kappa-1)^2}A_{n\kappa}(-\kappa+1) + \frac{8\kappa^2}{(4\kappa^2-1)^2}A_{n\kappa}(\kappa) - \frac{\kappa + 1}{(2\kappa + 1)^2}A_{n\kappa}(-\kappa-1)\bigg]\Bigg\rbrace\,,\end{aligned}$$ with the function $A_{n\kappa}(\kappa'')$ defined in the appendix. For the ground states $|ms\rangle$, this becomes $$a_1 \Delta E_\text{\tiny hfs}\, \delta_{m'm}\delta_{s's} + \frac{a_2}{4}\Delta E_\text{\tiny hfs} \begin{pmatrix} 1&&& \\ &-1&2& \\ &2&-1& \\ &&&1 \end{pmatrix}_{m's',ms}$$ with $$\begin{aligned}
a_1 :&= \frac{\mu}{\alpha}(2-2\alpha^2-\sqrt{1-\alpha^2})(A_{0,-1}(2)+2A_{0,-1}(-1)) \approx 2.0\times 10^{-8}\,,\\
a_2 :&= \frac{4\mu}{3\alpha}(2-2\alpha^2-\sqrt{1-\alpha^2})(A_{0,-1}(2)-4A_{0,-1}(-1)) \approx -5.4\times 10^{-8}\,.\end{aligned}$$ The defintions are exact, but the approximate numeric values are given only as a rough estimate. They have been obtained numerically by truncating the infinite sum over $n''$ occuring in $A_{n\kappa}$ and will need to be replaced by sufficiently precise estimates before they are used to compare predictions to data.
The results of this subsection yield the contribution $T_\text{\tiny\labelcref{subsec:nonbiref}}$ to the full transition matrix. In addition to the usual first order hyperfine structure, we obtained the second order terms, which qualitatively, however, yield nothing new: the $a_1$ term is proportional to unity, $\delta_{m'm}\delta_{s's}$, and thus just shifts all the states together, while $a_2$ constitutes a small correction to the hyperfine transition energy $\Delta E_\text{\tiny hfs}$. We had to calculate them, however, in order to be consistent with the introduction of all corrections linear in the non-metricities $E$ in the next two subsections.
Corrections from non-metricities, but without proton spin {#subsec:lambda}
---------------------------------------------------------
In this section we calculate the purely non-metric contribution to the transition matrix, that is $\langle m's' | H_\text{\tiny abs} | ms\rangle$. Since we will need to use them again for $T_\text{\tiny\labelcref{subsec:lambdamu}}$, let us start by calculating all the matrix elements of $H_\text{\tiny abs}$. For $ \langle n'\kappa'm's' | H_\text{\tiny abs} | n\kappa m s\rangle$ one gets $$\begin{aligned}
& & \frac{m_\text{\tiny red}\alpha}{4}\mathcal{N}'\mathcal{N}\delta_{s's} \notag\\
& &\times\Bigg\lbrace \bigg[\left(\alpha + \frac{(w'-W')(w-W)}{\alpha}\right)\left({{\ensuremath{\prescript{1}{00}{I}^{n'\kappa'}_{n\kappa}}}}+ {{\ensuremath{\prescript{1}{11}{I}^{n'\kappa'}_{n\kappa}}}}\right) - \left(\alpha - \frac{(w'-W')(w-W)}{\alpha}\right)\left({{\ensuremath{\prescript{1}{01}{I}^{n'\kappa'}_{n\kappa}}}}+ {{\ensuremath{\prescript{1}{10}{I}^{n'\kappa'}_{n\kappa}}}}\right)\bigg] \notag\\
& &\qquad\times \bigg[\frac{\delta_{\kappa',\kappa-2}}{4(\kappa-1)^2 - 1}\left(\tau_{-\kappa+1}^\mu\tau_{-\kappa}^\nu\right)_{m'm} + \frac{\delta_{\kappa',\kappa+2}}{4(\kappa+1)^2 - 1}\left(\tau_{\kappa+1}^\mu\tau_{\kappa}^\nu\right)_{m'm} + \frac{4\delta_{-\kappa',\kappa-1}}{|2\kappa-3|(4\kappa^2-1)}\left(\tau_{-\kappa}^\mu\sigma_{-\kappa}^\nu\right)_{m'm} \notag\\
& &\qquad\qquad + \frac{4\delta_{-\kappa',\kappa+1}}{|2\kappa+3|(4\kappa^2-1)}\left(\tau_{\kappa}^\mu\sigma_{\kappa}^\nu\right)_{m'm} - \frac{2\delta_{\kappa'\kappa}}{4\kappa^2-1}\left(\sigma_\kappa^\mu\sigma_\kappa^\nu - (4\kappa^2-1)\eta^{\mu\nu}\mathbb{1}\right)_{m'm}\bigg]E_{\mu 0\nu 0}\notag\\
& &\quad +\frac{1}{2} \bigg[\left(w-W+w'-W'\right)\left({{\ensuremath{\prescript{1}{00}{I}^{n'\kappa'}_{n\kappa}}}}- {{\ensuremath{\prescript{1}{11}{I}^{n'\kappa'}_{n\kappa}}}}\right) + \left(w-W-w'+W'\right)\left({{\ensuremath{\prescript{1}{01}{I}^{n'\kappa'}_{n\kappa}}}}- {{\ensuremath{\prescript{1}{10}{I}^{n'\kappa'}_{n\kappa}}}}\right)\bigg] \notag\\
& &\qquad\quad\times \bigg[ \frac{\delta_{-\kappa',\kappa-2}}{4(\kappa-1)^2 - 1}\left(\tau_{-\kappa+1}^\mu\tau_{-\kappa}^\nu\right)_{m'm} - \frac{\delta_{-\kappa',\kappa+2}}{4(\kappa+1)^2 - 1}\left(\tau_{\kappa+1}^\mu\tau_{\kappa}^\nu\right)_{m'm} \notag\\
& &\qquad\qquad + \frac{2\delta_{\kappa',\kappa-1}}{|2\kappa+1|(2\kappa-3)}\left(\tau_{-\kappa}^\mu\sigma_{-\kappa}^\nu\right)_{m'm} + \frac{2\delta_{\kappa',\kappa+1}}{|2\kappa-1|(2\kappa+3)}\left(\tau_{\kappa}^\mu\sigma_{\kappa}^\nu\right)_{m'm} \bigg] \epsilon_{\mu\rho\sigma}E^{\rho\sigma}{}_{\nu 0} \Bigg\rbrace\,, \label{eq:abs}\end{aligned}$$ which for the ground states reduces to $$\frac{m_\textrm{\tiny red}\alpha^2 E^{ab}{}_{ab}}{12\sqrt{1-\alpha^2}}\delta_{m'm}\delta_{s's}\,.$$ Note that, just like the $a_1$ term in $T_\text{\tiny\labelcref{subsec:nonbiref}}$, this contribution is proportional to the identity $\delta_{m'm}\delta_{s's}$; it does not contribute to any splitting of the states, but merely yields an absolute shift of all the states together. Thus it is irrelevant to the hyperfine structure, but contributes to a shift in the ionization energy (\[Eionization\]) below.
Combined corrections from proton spin and non-metricities {#subsec:lambdamu}
---------------------------------------------------------
We now come to the calculation of $\langle m's' | T_\text{\tiny\labelcref{subsec:lambdamu}} | ms \rangle$, which presents the non-metric correction to the hyperfine structure. As before, for generality we will start with arbitrary states of same energy ($n' = n$, $|\kappa'| = |\kappa|$) and only then specialize to the ground states $|ms\rangle$.
The matrix elements of $H_\text{\tiny rel}$ between states of same energy, $\langle n\kappa'm's' | H_\text{\tiny rel} | n\kappa ms\rangle$, are $$\begin{aligned}
-\frac{1}{2}&\frac{m_\text{\tiny red}\alpha^2}{4\kappa^2 - 1}\mathcal{N}^2\sqrt{\frac{w-\kappa'}{w-\kappa}} (\sigma_\mu)_{s's} \notag\\
&\times\Bigg\lbrace \delta_{\kappa',-\kappa} \bigg[\left(\alpha + \frac{(w-W)^2}{\alpha}\right)\left({{\ensuremath{\prescript{2}{00}{I}^{n'\kappa'}_{n\kappa}}}}+ {{\ensuremath{\prescript{2}{11}{I}^{n'\kappa'}_{n\kappa}}}}\right) - \left(\alpha - \frac{(w-W)^2}{\alpha}\right)\left({{\ensuremath{\prescript{2}{01}{I}^{n'\kappa'}_{n\kappa}}}}+ {{\ensuremath{\prescript{2}{10}{I}^{n'\kappa'}_{n\kappa}}}}\right)\bigg] \notag\\
&\qquad\quad\,\,\times \left[\eta^{\rho\sigma}\sigma_\kappa^\mu - \frac{9}{2(4\kappa^2-9)}\left(\sigma_\kappa^{(\mu}\sigma_\kappa^\rho\sigma_\kappa^{\sigma)} + (4\kappa^2-5)\eta^{(\rho\sigma}\sigma_\kappa^{\mu)}\right)\right]_{m'm} \epsilon_{\rho\alpha\beta}E^{\alpha\beta}{}_{\sigma 0} \notag\\
&\quad\quad + \delta_{\kappa',\kappa}\frac{4\kappa(w-W)}{4\kappa^2-9}\left({{\ensuremath{\prescript{2}{00}{I}^{n'\kappa'}_{n\kappa}}}}- {{\ensuremath{\prescript{2}{11}{I}^{n'\kappa'}_{n\kappa}}}}\right) \notag\\
&\qquad\quad\,\,\times \big[4(4\kappa^2-9)\eta^{\mu\rho}\sigma_\kappa^{\sigma} + (4\kappa^2+3)\eta^{\rho\sigma}\sigma_\kappa^\mu + 3\sigma_\kappa^\mu\sigma_\kappa^\rho\sigma_\kappa^\sigma - 6{i}\epsilon^{\mu\rho}{}_\nu\sigma_\kappa^\sigma\sigma_\kappa^\nu\big]_{m'm} E_{\rho 0\sigma 0} \notag\\
&\quad\quad - \delta_{\kappa',\kappa}(w-W)\left({{\ensuremath{\prescript{2}{01}{I}^{n'\kappa'}_{n\kappa}}}}-{{\ensuremath{\prescript{2}{10}{I}^{n'\kappa'}_{n\kappa}}}}\right){i}\epsilon^{\mu\rho}{}_\nu\lbrace\sigma_\kappa^\sigma, \sigma_\kappa^\nu\rbrace_{m'm}E_{\rho 0\sigma 0} \Bigg\rbrace\,,\end{aligned}$$ which for the ground states reduces to $$\frac{1}{4}\Delta E_\text{\tiny hfs}\left(\frac{7}{10}\widehat{E}^\mu{}_\nu + \frac{1}{12}E^{ab}{}_{ab}\delta^\mu_\nu\right)(\sigma_\mu\otimes\sigma^\nu)_{m's',ms}\,,$$ and thus directly yields the first order term in $T_\text{\tiny\labelcref{subsec:lambdamu}}$. Note that the tracefree tensor (\[widehatE\]) makes another appearance here. As before, where $\widehat{E}^{\mu\nu}$ controlled a qualitatively new spin-magnetic coupling, also here the term $\widehat{E}^\mu{}_\nu(\sigma_\mu\otimes\sigma^\nu)$ produces a qualitatively new effect: it lifts the degeneracy of the triplet states in the hyperfine structure of the hydrogen atom, as we will see in detail in the next section.
For the remaining second order term in $T_\text{\tiny\labelcref{subsec:lambdamu}}$ we take recourse to the matrix elements of $H_\text{\tiny hfs}$ and $H_\text{\tiny abs}$, calculated in (\[eq:hfs\]) and (\[eq:abs\]), respectively. Between states of equal energy, we find the second order contribution $$2\sum_{n'',\kappa'',m'',s''}\frac{\langle n\kappa'm's'| H_\text{\tiny hfs} |n''\kappa'' m''s''\rangle \langle n''\kappa''m''s''| H_\text{\tiny abs} |n\kappa ms\rangle}{E(n'',|\kappa''|) - E(n,|\kappa|)}$$ to be given explicitly by $$\begin{aligned}
& - \frac{2\mu m_\text{\tiny red}\alpha^2\left(\sigma_\mu\right)_{s's}}{4\kappa^2-1} \notag\\
&\times\Bigg\lbrace \delta_{\kappa'\kappa}\bigg[\frac{B_{n\kappa}(-\kappa+1)}{4(\kappa-1)^2-1}\Big(\sigma_\kappa^\mu\sigma_\kappa^{\alpha}\sigma_\kappa^{\beta} + \left(4(\kappa - 1)^2 -1\right)\eta^{\mu\alpha}\sigma_\kappa^{\beta} + 2(2\kappa-1)\eta^{\alpha\beta}\sigma_\kappa^\mu - (2\kappa-1){i}\epsilon^{\mu\alpha}{}_\tau\sigma_\kappa^\beta\sigma_\kappa^\tau\Big)_{m'm} \notag\\
&\qquad - \frac{B_{n\kappa}(-\kappa-1)}{4(\kappa+1)^2 -1}\Big(\sigma_\kappa^\mu\sigma_\kappa^\alpha\sigma_\kappa^\beta + \left(4(\kappa+1)^2 - 1\right)\eta^{\mu\alpha}\sigma_\kappa^\beta
- 2(2\kappa+1)\eta^{\alpha\beta}\sigma_\kappa^\mu + (2\kappa+1){i}\epsilon^{\mu\alpha}{}_\tau\sigma_\kappa^\beta\sigma_\kappa^\tau\Big)_{m'm}\notag\\
&\qquad + 2\kappa B_{n\kappa}(\kappa)\Big(\sigma_\kappa^\mu\sigma_\kappa^\alpha\sigma_\kappa^\beta - (4\kappa^2-1)\eta^{\alpha\beta}\sigma_\kappa^\mu\Big)_{m'm} \bigg]E_{\alpha 0\beta 0} \notag\\
&\quad -\frac{1}{4}\delta_{\kappa',-\kappa}\bigg[\frac{C_{n\kappa}(\kappa-1)}{2\kappa-3}\Big(\sigma_\kappa^\mu\sigma_\kappa^{\alpha}\sigma_\kappa^{\beta} + \left(4(\kappa - 1)^2 -1\right)\eta^{\mu\alpha}\sigma_\kappa^{\beta} + 2(2\kappa-1)\eta^{\alpha\beta}\sigma_\kappa^\mu - (2\kappa-1){i}\epsilon^{\mu\alpha}{}_\tau\sigma_\kappa^\beta\sigma_\kappa^\tau\Big)_{m'm} \notag\\
&\qquad -\frac{C_{n\kappa}(\kappa+1)}{2\kappa+3}\Big(\sigma_\kappa^\mu\sigma_\kappa^\alpha\sigma_\kappa^\beta + \left(4(\kappa+1)^2 - 1\right)\eta^{\mu\alpha}\sigma_\kappa^\beta
- 2(2\kappa+1)\eta^{\alpha\beta}\sigma_\kappa^\mu + (2\kappa+1){i}\epsilon^{\mu\alpha}{}_\tau\sigma_\kappa^\beta\sigma_\kappa^\tau\Big)_{m'm} \bigg]\epsilon_{\alpha\rho\sigma}E^{\rho\sigma}{}_{\beta 0}\!\Bigg\rbrace\,,\end{aligned}$$ with $B_{n\kappa}(\kappa'')$ and $C_{n\kappa}(\kappa'')$ defined in the appendix. For the ground states, this reduces to $$-\frac{\lambda}{4}\Delta E_\text{\tiny hfs} (b_2\widehat{E}^\mu{}_\nu + \frac{b_1}{12} E^{ab}{}_{ab}\delta^\mu_\nu)(\sigma_\mu\otimes\sigma^\nu)_{m's',ms}\,,$$ with $$\begin{aligned}
b_1 :={}& \frac{24}{\alpha}(2-2\alpha^2-\sqrt{1-\alpha^2})B_{0,-1}(-1) \approx 1.74\,,\\
b_2 :={}& \frac{4}{5\alpha}(2-2\alpha^2-\sqrt{1-\alpha^2})B_{0,-1}(2) \approx -2.9\times 10^{-4}\,.\end{aligned}$$ Again, the second order does not bring qualitatively new features. Both terms appearing in the formula above are already present in the first order correction due to $H_\text{\tiny rel}$. The precise impact of all these terms on the energy levels will be the topic of the next section.
Hyperfine structure and ionization energy of hydrogen {#subsec:full}
-----------------------------------------------------
We find that three out of the eleven ray-optically invisible deviations affect the hyperfine structure. In particular, the triplet states are no longer degenerate in general, as illustrated in \[fig:boat1\]. Their splitting is controlled by the three cubic roots of a complex number that encodes two of the three effective degrees of freedom. The remaining relevant parameter is the full contraction $E^{ab}{}_{ab}$, which shifts both, the energy of ionization and of the hyperfine transition. The latter plays a central role for the detection of hydrogen and its state of motion in astrophysical measurements.
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Collecting all relevant terms from sections \[subsec:nonbiref\], \[subsec:lambda\] and \[subsec:lambdamu\], we obtain for the ground state transition matrix $$\langle m's' | T_g | ms \rangle = \langle m's' | T_\text{\tiny\labelcref{subsec:nonbiref}} |ms\rangle + \langle m's' | T_\text{\tiny\labelcref{subsec:lambda}} |ms\rangle + \langle m's' | T_\text{\tiny\labelcref{subsec:lambdamu}} | ms \rangle$$ the sum $$\begin{aligned}
\left(a_1\Delta E_\text{\tiny hfs} \!+\! \frac{m_\text{\tiny red}\alpha^2 E^{ab}{}_{ab}}{12\sqrt{1-\alpha^2}}\right)&\delta_{m'm}\delta_{s's}\nonumber\\
- \frac{1}{4}\Delta E_\text{\tiny hfs}&\left[\left(1\!+\!a_2\!-\!\frac{1\!-\!b_1}{12}E^{ab}{}_{ab}\right)\delta^\mu_\nu\!-\!\left(\frac{7}{10}\!-\!b_2\right)\widehat{E}^\mu{}_\nu\right]\!(\sigma_{\mu} \otimes\sigma^\nu)_{m'\!s'\,,ms}\,.\label{eq:transitionMatrix}\end{aligned}$$ In order to calculate the eigenvalues $\Delta E_k$, $k\in\lbrace 0,1,2,3 \rbrace$, we note that the transition matrix has two constituents: the first line is proportional to $\delta_{m'm}\delta_{s's}$, which is the identity on the space of ground states, diagonal in every basis, whereas the second line has non-trivial eigenvalues. Thus we may make the ansatz $$\label{eq:DE}
\Delta E_k = \left(a_1\Delta E_\text{\tiny hfs} + \frac{m_\text{\tiny red}\alpha^2 E^{ab}{}_{ab}}{12\sqrt{1-\alpha^2}}\right) + \frac{1}{4}\Delta E_\text{\tiny hfs}\, \xi_k\,.$$ Up to a factor $\frac{1}{4}\Delta E_\text{\tiny hfs}$, the $\xi_k$ controlling these eigenvalues are given by the roots of the characteristic polynomial of the second line in \[eq:transitionMatrix\]. This is a quartic polynomial, and thus algebraically solvable. One quickly finds the first root, and thus the first hyperfine energy level through (\[eq:DE\]), $$\xi_0 = -3\left(1+a_2+ \frac{b_1-1}{12}E^{ab}{}_{ab}\right)\,,\label{eq:xisa}$$ which reduces the problem of finding the remaining three hyperfine structure energy levels to the solution of a cubic polynomial. Then using the known solution formula one also finds the remaining three ($\kappa=1,2,3$) roots $$\xi_{\kappa} = 1+a_2 + \frac{b_1 - 1}{12}E^{ab}{}_{ab} + 2^{\frac{5}{3}}\left(\frac{7}{10} - b_2\right) \,\text{Re}\sqrt[3,\kappa]{\text{det }\widehat{E} + {i}\sqrt{\frac{1}{54}(\text{tr }\widehat{E}^2)^3 - (\text{det }\widehat{E})^2}}\,, \label{eq:xisb}$$ featuring the four numerical constants $a_1$, $a_2$, $b_1$, $b_2$ determined in the two subsections \[subsec:nonbiref\] and \[subsec:lambdamu\] above. Note in particular, that the last term involves a cubic root, whose three possible choices of complex phase distinguish the three $\xi_{\kappa}$. Removing all non-metricities $E_{abcd} = 0$ and second order terms ($a_{1/2} = b_{1/2} = 0$), we are left with $$\begin{aligned}
\Delta E_0 ={}& -\frac{3}{4}\Delta E_\text{\tiny hfs}& \textrm{and}&& \Delta E_{\kappa} ={}& \frac{1}{4}\Delta E_\text{\tiny hfs}\,,\end{aligned}$$ which properly recovers the standard first order hyperfine structure, see the left of fig. \[fig:boat1\]. From (\[eq:xisa\]) and (\[eq:xisb\]), we see that this standard splitting is now stretched to $$\label{stretching}
\Delta E_\text{\tiny hfs}^\textrm{\tiny ROID} = \left(1+a_2+ \frac{b_1-1}{12}E^{ab}{}_{ab}\right) \Delta E_\text{\tiny hfs}$$ for ray-optically invisible deviations from a Lorentzian spacetime. Note that the stretching factor can be both larger and smaller than unity. This reveals a modification of the hyperfine splitting $\Delta E_\text{\tiny hfs}$, but yields not a qualitatively new effect, see the middle of fig. \[fig:boat1\]. Qualitatively new, however, is the splitting of the triplet states, induced by the cubic root in \[eq:xisb\]. The real, symmetric and tracefree perturbation components $\widehat{E}^\mu{}_\nu$ feature only two invariant perturbation degrees of freedom, while the remaining three encode the spatial orientation of the eigenbasis, but do not affect the energy levels of the hydrogen atom. This can be deduced from the fact that $\text{det }\widehat{E}$ as well as $\text{tr }\widehat{E}^2$ can be written completely in terms of the eigenvalues of $\widehat{E}$. Doing so, one also quickly sees that $(\text{tr }\widehat{E}^2)^3 \geq 54(\text{det }\widehat{E})^2$, where equality holds when two eigenvalues coincide. The expression under the cubic root thus takes values in the entire complex plane, up to the fact that the sign of the imaginary part is determined by the choice of sign for the square root. Precisely this sign, however, is irrelevant, since only the real part of the three cubic roots contributes to (\[eq:xisb\]). Picturing the cubic root as an equilateral triangle in the complex plane, whose corners are projected onto the real axis, it is evident that the unweighted average of the three levels remains unaltered, since the sum of the three possible roots adds up to zero. Orientation and size of the mentioned equilateral triangle correspond to the two effective degrees of freedom of $\widehat{E}^\mu{}_\nu$, see the prototypical situation on the right of fig. \[fig:boat1\].
With the energy level of the singlet state now known, we finally collect all terms that contribute to the ionization energy $$\label{Eionization}
E^\textrm{\tiny ROID}_\textrm{\tiny ionization} = m_\textrm{\tiny red}\left(1-\sqrt{1-\alpha^2}\right) + \left(\frac{3+3a_2}{4} - a_1\right) \Delta E_\textrm{\tiny hfs} - \frac{1}{12}\left(\frac{m_\textrm{\tiny red}\alpha^2}{\sqrt{1-\alpha^2}}+\frac{3-3b_1}{4} \Delta E_\textrm{\tiny hfs}\right) E^{ab}{}_{ab}$$ of a hydrogen atom on a spacetime with ray-optically invisible deviations from a Lorentzian background. As in the scaling (\[stretching\]) of the $21.1\, \textrm{cm}$ line of hydrogen, it is again the double trace $E^{ab}{}_{ab}$ of the ray-optically invisible perturbation that single-handedly controls the shift in the ionization energy.
Conclusions
===========
In this paper, we identified an eleven-parameter deformation of Lorentzian geometry that can still carry gauge field and spinor dynamics, but is invisible to the classical ray optics of either. We showed the pertinent quantum electrodynamics to be gauge-invariantly renormalizable to any loop order and thus to provide a fundamental quantum theory for the electromagnetic interactions. This makes this non-metric geometry an interesting candidate for a hitherto not detected spacetime geometry.
Quantum effects, however, do feel these non-metricities to first order perturbation theory. In quantum field theoretic scattering, they appear already at tree level in Bhabha scattering. At one loop, we studied the scattering of an electron in an external magnetic field, where we found a qualitatively new interaction between the electron spin and the external magnetic field. Because of the special structure of ray-optically invisible deviations, we were able to compute the anomalous magnetic moment of the electron up to every loop order at which it is known in standard QED (\[AMM-loops-relation\]), yielding the particular combination (\[alpha-tilde-value\]) of the fine structure constant $\alpha$ and the double trace of the eleven-parameter perturbation $E^{abcd}$. A single further result of comparable precision, relating the same two quantities or containing either one of them as the only experimental unknown, will thus allow to determine both the fine structure constant and the double trace of the non-metric perturbation. For the moment, however, using the current known bounds on the double trace of the perturbation, it was at least possible to determine a precise value of the fine structure constant (\[alpha-value\]) when ray-optically invisible non-metricities are present.
Moreover, we investigated the bound states of atomic hydrogen in the presence of ray-optically invisible non-metricities. The pertinent result here is that only three out of the eleven perturbative degrees of freedom affect the hyperfine structure, and thus the astrophysically crucially relevant $21.1\,\textrm{cm}$ line. While two of these source a splitting of the triplet states, the remaining one, namely again the double trace $E^{ab}{}_{ab}$, simultaneously controls two effects: it shifts the ionization energy (\[Eionization\]) as well as the hyperfine transition energy (\[stretching\]). While the ionization energy yields the same functional connection of $\alpha$ and $E^{ab}{}_{ab}$ as the anomalous magnetic moment—and hence does not contribute new information—any sufficiently precise measurement of the modified hyperfine structure will, in principle, provide sufficient new information to determine the fine structure constant and the double trace of a non-metric perturbation. This would therefore amount to a direct quantum test of a non-metric spacetime structure.
While we assumed a fixed non-metric perturbation around a Lorentzian manifold in this article—and even chose to freeze them to constant values over the tiny domains on which the quantum effects studied here take place—there is indeed a way to determine the (ultimately gravitational) dynamics that such perturbations, once unfrozen, must satisfy. This has become possible by the gravitational closure mechanism [@SSWD], which allows to calculate, rather than stipulate, the dynamics that the background geometry underlying given canonically quantizable matter field dynamics must satisfy, in order to enable a common canonical evolution of the geometric degrees of freedom with those of matter. Especially in the weak field regime, which is precisely the scope of this work, the gravitational closure of the electrodynamics (\[action-gled-fermion\]) has been performed explicitly [@SSSW] by way of linear perturbation theory. Thus it is possible to predict how strongly the background geometry deviates from a Lorentzian metric in various physical situations of interest, such as in the far field of a single mass, a binary, or a rotating galactic disk. This way, gravitational closure enables one to embed the findings of this paper into a fully coherent theory: we know how these perturbations are generated, where they are to be expected, and then can make specific predictions for their actual quantitative effects on Bhabha scattering, the electron’s anomalous magnetic moment and the $21.1\,\textrm{cm}$ line in physical situations, where the weak field approximation employed here pertains.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank Nadine Stritzelberger and Marcus Werner for insightful comments and suggestions. FPS thanks the Yukawa Institute for Theoretical Physics for their generous invitation for a two-month research visit in September and October 2016, during which part of this work was completed. RT thanks the Friedrich-Alexander University Erlangen-Nuremberg, and in particular Frederic Schuller, for their kind hospitality over the course of the past year.
Appendix: Conventions and notations for special functions and integrals {#sec_specialfunctions .unnumbered}
=======================================================================
We frequently use an array of special functions and integrals in section \[sec:hydrogen\], in connection with the eigenstates $|n\kappa ms\rangle$ of the exactly diagonalizable part of the Hamiltonian of the relativistic hydrogen atom. These are collected in this appendix for the reader’s convenience.
In subsection \[subsec:exact\], the normalization constant used in the general expression for the eigenstates $|n\kappa m s\rangle$ is given by $$\mathcal{N} := \frac{1}{2w^{K+1}}\sqrt{\frac{(w-\kappa)(w+W)n!}{w\Gamma(2K + 1 + n)}}\,,$$ using the shorthands $W$, $w$ and $K$ introduced in that subsection for expressions in terms of the quantum numbers $n$, $\kappa$, $m$, $s$. Moreover, the generalized Laguerre polynomials $L_n^\lambda(z)$ used there are the solutions of Laguerre’s equation $$\left(z\frac{{d}^2}{{d}z^2} + (\lambda + 1 - z)\frac{{d}}{{d}z} + n\right)L^\lambda_n(z) = 0\,,$$ which read explicitly [@WFLaguerre] $$L^\lambda_n(z) = \sum_{k=0}^n \frac{\Gamma(\lambda + n + 1)}{\Gamma(\lambda + k + 1)}\frac{(-z)^k}{(n-k)!k!}\,.$$ The spinor spherical harmonics $\mathcal{Y}_\kappa^m({\ensuremath{\boldsymbol{n}}})$ are defined by $$\mathcal{Y}_\kappa^m ({\ensuremath{\boldsymbol{n}}}) := \frac{1}{\sqrt{2d}}\begin{pmatrix} -v\sqrt{d-vm}\,Y_{d-\frac{1}{2}}^{m-\frac{1}{2}}({\ensuremath{\boldsymbol{n}}}) \\[2ex] \sqrt{d+vm}\,Y_{d-\frac{1}{2}}^{m+\frac{1}{2}}({\ensuremath{\boldsymbol{n}}}) \end{pmatrix}\,,\qquad v := \text{sign } \kappa\,,\qquad d := \left|\kappa + \frac{1}{2}\right|$$ with the spherical harmonics [@WFSpherical] $$Y^m_l(\theta,\varphi) := \sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)!}}P^m_l(\cos \theta){e}^{{i}m\varphi}\,.$$
In subsection \[subsec:nonbiref\], we use that the spinor spherical harmonics satisfy the identity [@Ycalculus] $$\sigma^\mu\mathcal{Y}_\kappa^m = \frac{-1}{2\kappa + 1}\sum_{m' = -|\kappa|+\frac{1}{2}}^{|\kappa|+\frac{1}{2}}\mathcal{Y}_\kappa^{m'}\left(\sigma_\kappa^\mu\right)_{m'm} + \frac{1}{|2\kappa + 1|} \sum_{m' = -|\kappa'|+\frac{1}{2}}^{|\kappa'|+\frac{1}{2}}\mathcal{Y}_{\kappa'}^{m'}\left(\tau_\kappa^\mu\right)_{m'm}\,,$$ where the matrices $\sigma^\mu_\kappa$ and $\tau^\mu_\kappa$ are given by $$\begin{aligned}
\left(\sigma_\kappa^1\right)_{m'm} :={}& \sqrt{\kappa^2-\bar{m}^2}\left(\delta_{m',m-1} + \delta_{m',m+1}\right)\\
\left(\sigma_\kappa^2\right)_{m'm} :={}& \sqrt{\kappa^2-\bar{m}^2}\left({i}\delta_{m',m-1} - {i}\delta_{m',m+1}\right)\\
\left(\sigma_\kappa^3\right)_{m'm} :={}& 2\bar{m}\delta_{m'm}\\[3ex]
\left(\tau_\kappa^1\right)_{m'm} :={}& -v\sqrt{(\kappa' + \bar{m})(\bar{m} - \kappa)}\delta_{m',m-1} + v\sqrt{(\kappa + \bar{m})(\bar{m} - \kappa')}\delta_{m',m+1}\\
\left(\tau_\kappa^2\right)_{m'm} :={}& -v{i}\sqrt{(\kappa' + \bar{m})(\bar{m} - \kappa)}\delta_{m',m-1} - v{i}\sqrt{(\kappa + \bar{m})(\bar{m} - \kappa')}\delta_{m',m+1}\\
\left(\tau_\kappa^3\right)_{m'm} :={}& -2\sqrt{-\left(\kappa + \frac{1}{2}\right)\left(\kappa' + \frac{1}{2}\right) - \bar{m}^2}\delta_{m'm}\end{aligned}$$ and $\kappa' := -(\kappa + 1)$, $\bar m := (m'+m)/2$. Note that from the explicit expressions above, one can directly read off that $\sigma_{\pm 1}^\mu = \sigma^\mu$, where the $\sigma^\mu$ are the standard Pauli matrices. Furthermore, the $\sigma_\kappa$ and $\tau_\kappa$ matrices satisfy a series of identities, of which the most important for the purposes of this paper is $$\left[\sigma_\kappa^\mu, \sigma_\kappa^\nu\right] = 2{i}\varepsilon^{\mu\nu}{}_\rho\sigma_\kappa^\rho\,.$$
Throughout subsections \[subsec:nonbiref\], \[subsec:lambda\] and \[subsec:lambdamu\], we use the following shorthand symbols for repeatedly appearing radial integrals, in order to lighten the notation. $${{\ensuremath{\prescript{a}{ij}{I}^{n'\kappa'}_{n\kappa}}}} := \left(\frac{w'+\kappa'}{n'}\right)^i \left(\frac{w+\kappa}{n}\right)^j \int_0^\infty {d}y\, {e}^{-\frac{w'+w}{2w'w}y} y^{K'+K-a} L_{n'-i}^{2K'}\left(\frac{y}{w'}\right)L_{n-j}^{2K}\left(\frac{y}{w}\right)$$ Note that this is well-defined only for $n'-i \geq 0$ and $n-j \geq 0$. In all other cases ${{\ensuremath{\prescript{a}{ij}{I}^{n'\kappa'}_{n\kappa}}}} := 0$. Note also that, since $L_0^\lambda = 1$, $${{\ensuremath{\prescript{a}{n'n}{I}^{n'\kappa'}_{n\kappa}}}} = \left(\frac{w'+\kappa'}{n'}\right)^{n'} \left(\frac{w+\kappa}{n}\right)^n \left(\frac{w'+w}{2w'w}\right)^{-(K'+K-a+1)} \Gamma\left(K'+K-a+1\right)\,.$$ Furthermore, in the second order corrections, there appear the functions $A_{n\kappa}$, $B_{n\kappa}$, $C_{n\kappa}$, which are defined as $$\begin{aligned}
A_{n\kappa}(\kappa'') := \sum_{n''}&\frac{w''w}{wW''-w''W}\mathcal{N}^2\mathcal{N''}^2\notag\\
&\times \Big[\left(w-W + w'' - W''\right)\left({{\ensuremath{\prescript{2}{00}{I}^{n''\kappa''}_{n\kappa}}}} - {{\ensuremath{\prescript{2}{11}{I}^{n''\kappa''}_{n\kappa}}}}\right) \notag\\
&\hspace{2cm} + \left(w-W-w''+W''\right)\left({{\ensuremath{\prescript{2}{01}{I}^{n''\kappa''}_{n\kappa}}}}-{{\ensuremath{\prescript{2}{10}{I}^{n''\kappa''}_{n\kappa}}}}\right)\Big]^2\,,\\
B_{n\kappa}(\kappa'') := \sum_{n''}&\frac{w''w}{wW''-w''W}\mathcal{N}^2\mathcal{N''}^2 \notag\\
&\times \Big[\left(w-W + w'' - W''\right)\left({{\ensuremath{\prescript{2}{00}{I}^{n''\kappa''}_{n\kappa}}}} - {{\ensuremath{\prescript{2}{11}{I}^{n''\kappa''}_{n\kappa}}}}\right) \notag\\
&\hspace{2cm}+ \left(w-W-w''+W''\right)\left({{\ensuremath{\prescript{2}{01}{I}^{n''\kappa''}_{n\kappa}}}}-{{\ensuremath{\prescript{2}{10}{I}^{n''\kappa''}_{n\kappa}}}}\right)\Big]\notag\\
&\times \Big[\left(\alpha+\frac{(w''-W'')(w-W)}{\alpha}\right)\left({{\ensuremath{\prescript{1}{00}{I}^{n''\kappa''}_{n\kappa}}}}+{{\ensuremath{\prescript{1}{11}{I}^{n''\kappa''}_{n\kappa}}}}\right) \notag\\
&\hspace{2cm} - \left(\alpha - \frac{(w''-W'')(w-W)}{\alpha}\right)\left({{\ensuremath{\prescript{1}{01}{I}^{n''\kappa''}_{n\kappa}}}}+{{\ensuremath{\prescript{1}{10}{I}^{n''\kappa''}_{n\kappa}}}}\right)\Big]\,,\\
C_{n\kappa}(\kappa'') := \sum_{n''}&\frac{w''w}{wW''-w''W}\mathcal{N}^2\mathcal{N''}^2\sqrt{\frac{w+\kappa}{w-\kappa}} \notag\\
&\times \Big[\left(w-W + w'' - W''\right)\left({{\ensuremath{\prescript{2}{00}{I}^{n''\kappa''}_{n,\,-\kappa}}}} - {{\ensuremath{\prescript{2}{11}{I}^{n''\kappa''}_{n,\,-\kappa}}}}\right) \notag\\
&\hspace{2cm}+ \left(w-W-w''+W''\right)\left({{\ensuremath{\prescript{2}{01}{I}^{n''\kappa''}_{n,\,-\kappa}}}}-{{\ensuremath{\prescript{2}{10}{I}^{n''\kappa''}_{n,\,-\kappa}}}}\right)\Big]\notag\\
&\times \Big[\left(w-W+w''-W''\right)\left({{\ensuremath{\prescript{1}{00}{I}^{n''\kappa''}_{n\kappa}}}}-{{\ensuremath{\prescript{1}{11}{I}^{n''\kappa''}_{n\kappa}}}}\right) \notag\\
&\hspace{2cm}+ \left(w-W-w''+W''\right)\left({{\ensuremath{\prescript{1}{01}{I}^{n''\kappa''}_{n\kappa}}}}-{{\ensuremath{\prescript{1}{10}{I}^{n''\kappa''}_{n\kappa}}}}\right)\Big]\,,\end{aligned}$$ where the summation over $n''$ runs over $\mathbb{N}_0$ if $|\kappa''|\neq|\kappa|$ and over $\mathbb{N}_0\backslash\lbrace n\rbrace$ if $|\kappa''| = |\kappa|$.
[^1]: Corresponding author: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Risk spreading in bacterial populations is generally regarded as a strategy to maximize survival. Here, we study its role during range expansion of a genetically diverse population where growth and motility are two alternative traits. We find that during the initial expansion phase fast growing cells do have a selective advantage. By contrast, asymptotically, generalists balancing motility and reproduction are evolutionarily most successful. These findings are rationalized by a set of coupled Fisher equations complemented by stochastic simulations.'
author:
- Matthias Reiter
- Steffen Rulands
- Erwin Frey
title: Range Expansion of Heterogeneous Populations
---
[^1]
[^2]
Expansion of populations is a ubiquitous phenomenon in nature which includes the spreading of advantageous genes [@Fisher1937] or infectious diseases [@Mollison1977; @Grenfell2001], and dispersal of species into new territory. The latter has recently been investigated experimentally by analyzing the spreading of bacterial populations after droplet inoculation on an agar plate [@Ben-Jacob1994; @Golding1999; @Beer2010; @Hallatschek2007; @Hallatschek2008; @Korolev2010; @Datta2013]. Among others, these studies have highlighted the importance of random genetic drift in driving population differentiation along the expanding fronts of bacterial colonies [@Hallatschek2008; @Hallatschek2007; @Korolev2010; @Hallatschek2010; @Kuhr2011; @Korolev2013]. While these studies have focused on genetically uniform populations or the competition between two strains with different growth rates [@Kuhr2011; @Korolev2013] much less is known about range expansion of heterogeneous populations. Single cell studies have revealed that even genetically identical bacteria exhibit variability in phenotypic traits [@Dubnau2006]. As an example, clonal populations of *Bacillus subtilis* (in mid expontial growth phase) consist of both swarming cells, propelled by flagella, and non-motile cells [@Kearns2005]. Cells in the motile state do not divide. As a result, colonies of *B. subtilis* are heterogeneous with respect to the cells’ motility. This risk-spreading strategy allows the population to exploit nutrients at its current location and at the same time disperse to new, possibly more favorable, niches.
Motivated by these findings, we consider range expansion of a heterogeneous population. We ask what degree of risk-spreading between cell division and motility is optimal for survival during range expansion, *i.e.* whether an individual is better off by investing preferentially in growth or in motility, or by adopting a risk-spreading strategy and balance its investment in growth as well as motility. Specifically, we study range expansion dynamics on a one- and two-dimensional lattice, where each site can be occupied by an arbitrary number of individuals. We assume that each individual $i$ has a distinct genotype $A_i$, which encodes rates to migrate, $e_i$, and reproduce, $\mu_i$, *i.e.* in the language of game theory each individual plays a mixed strategy. In detail, an individual $A_i$ may reproduce with a rate in the interval $\mu_i\in(0,\mu_\text{max})$ upon consumption of resources $B$: $A_i B \xrightarrow{\mu_i} A_i A_i$; the offspring inherits the genotype and is placed on the same lattice site. In addition, individuals are able to migrate upon stochastically hopping to nearest neighbor sites with a rate $e_i$ in the range $(0,e_\text{max})$. Motivated by the behavior of bacterial populations we assume that an individual may invest its limited resources partly in motility and partly in reproduction, and model this by the constraint $e_i/e_\text{max}+\mu_i/\mu_\text{max}=1$, *i.e.* fast reproducing individuals can only move slowly and vice versa. As we will see, the implications of this biologically motivated tradeoff are more intricate than the phenomenon of front acceleration found in populations exhibiting only heterogeneous motility [@Benichou2012]. Numerical simulations of our stochastic lattice gas model were performed using Gillespie’s algorithm [@Gillespie1977] with sequential updating on square and hexagonal lattices with lattice spacing $a$. We measure time in units of the inverse maximum reproduction rate $1/\mu_\text{max}$, *i.e.* roughly speaking dimensionless time $t$ corresponds to the number of generations (of the slowest moving genotype).
We are interested in a range expansion scenario where initially a small area with a linear extension of three lattice sites (inoculum) is occupied by a genetically diverse population with $G$ different genotypes $A_i$, each site containing $\Omega$ individuals, while the remainder of the lattice sites contains $\Omega$ units of resources $B$. We assume that the local carrying capacity $\Omega$ is large: $\Omega\gg 1$ and thereby $G\gg 1$ as well. The relative hopping rates $\epsilon_i=e_i/e_\text{max}$ in the initial population are randomly drawn from a uniform distribution on $(0,1)$. Our stochastic simulations show that the inoculum quickly expands into a circular front with a concomitant loss of genetic diversity and the formation of multiple sectors composed of single genotypes; Figs. \[fig:box1\](a,b) show a typical configuration in two spatial dimensions with the ensuing spatial distribution of relative motilities and genetic diversity, respectively.
![Segregation patterns emerging from the stochastic simulation of a range expansion dynamics starting from a genetically diverse inoculum. (a) Local average motility $\langle\epsilon\rangle$ with blue (dark gray) signifying a low, yellow (light gray) a medium, and red (medium gray) a high motility. The front is dominated by genotypes with a motility close to $\epsilon =0.5$ (‘generalists’). (b) Local genetic diversity with blue (dark gray) indicating a homogeneous and red (medium gray) a heterogeneous population. Genetic diversity is rapidly lost during the range expansion process. The population remains genetically diverse only close to the inoculum and at sector boundaries. Stochastic simulations were run on a hexagonal lattice with $601\times695$ sites, with carrying capacity $\Omega=100$, initial number of genotypes $G=900$, and dimensionless time $t=490$. \[fig:box1\]](./pie_charts1_revised_steffen.pdf){width="\columnwidth"}
The rate at which genetic diversity is lost during expansion is strongly interlinked with the underlying dynamics of the range expansion processes. Of particular importance is the genetic diversity in the front region, as these individuals constitute the gene pool for the further expansion process [@Hallatschek2008]. The position of the front is defined as those lattice sites, where the fraction of resources $B$ exceeds a value of $1/2$. In polar coordinates, this yields a parametrization $r(\varphi)$ of the front, giving its distance from the origin as a function of the angle $\varphi$. Figure \[fig:box2\](a) shows the time evolution of the average number $H_f(t)$ of distinct genotypes in a region $r(\varphi)\pm\Delta r$, where $\Delta r$ is proportional to the width of the front, $\Delta r\sim\ell$ [^3]. We identify several temporal regimes characterized by different kinds of selection pressure acting on the individuals. After a short initial transient there is an intermediate regime, $10\lesssim t\lesssim 100$, where the loss of genetic diversity in the front region approximately follows a power law, $H_f(t)\propto t^{-\alpha}$ with $\alpha\approx 1.4 \pm 0.1$. This loss is significantly faster than for neutral evolution, where the neutral coalescence theory gives $\alpha=1$ [@Hinrichsen2000]. It suggests that the coalescence process is biased, meaning that some genotypes in the front region have a higher probability to go extinct than others; we will see later that this bias is related to the speed of Fisher waves for different genotypes. For $d=1$, which may for example be realized in coupled microfluidic chambers, the selection process quickly leads to the fixation of one particular genotype in the front region, as apparent from Fig. \[fig:box2\](a). As opposed to this, for $d=2$, range expansion leads to the formation of monoclonal sectors with a uniform genotype \[Fig. \[fig:box1\](a)\]. Further loss of genetic diversity is subsequently caused by annihilation of neighboring sector boundaries, and, as a result, $H_f$ decreases at a rather slow rate.
![(a) Decrease of genetic diversity $H_f(t)$ in the front region for one and two spatial dimensions. After a short transient genetic diversity decreases rapidly. While for $d=1$ genetic diversity is quickly lost, $H_f(t)=1$, it decreases only slowly in $d=2$ due to the formation of homogeneous sectors. (b) Probability to find a certain motility at a large time $t=220$. The populations is dominated by individuals with an approximately equal probability to migrate or reproduce. The histograms were averaged over $10^3$ sample runs with $\Omega=100$. \[fig:box2\]](./box21_revised.pdf){width="\columnwidth"}
This dynamics of genetic diversity leaves two key questions: First, which genotypes are selected by the expansion process and what is the asymptotic composition of the population? Second, which dynamic processes lead to the asymptotic state? To answer the first question we determined the genetic composition of the population after many generations, *i.e.* the probability $P(\epsilon)$ that an individual has a relative hopping rate $\epsilon$ and a corresponding relative reproduction rate $1-\epsilon$, \[Fig. \[fig:box2\](b)\]. We find that successful genotypes are ‘generalists’ which migrate and reproduce with approximately equal probability, while ‘specialists’, who preferentially reproduce or migrate, do not colonize. To answer the second question one needs to understand the role of evolutionary forces during range expansion. This can be achieved to a large degree within an analytical approach valid in a deterministic continuum limit where the carrying capacity, $\Omega$ is large and the local genetic diversity, *i.e.* the number of genotypes on any lattice site, $g(\vec{r},t)$, is sufficiently low: $\Omega \gg g(\vec{r},t) \gg 1$. In the spirit of a Fisher equation [@Fisher1937] one can then write down a set of coupled integro-difference-differential equations for the fraction of species with a given relative hopping rate, $n_{\epsilon_i} (\vec{r},t):=N_i (\vec{r},t)/\Omega$, and the fraction of resources, $\rho(\vec{r},t):=R (\vec{r},t)/\Omega$, where $N_i (\vec{r},t)$ and $R (\vec{r},t)$ are the local number of individuals with strategy $A_i$ and local number of resource units $B$, respectively. One obtains a set of Fisher equations for $n_{\epsilon} (\vec{r},t)$ [^4] coupled through the availability of resources $\rho (\vec{r},t)$:
\[eq:ipde\] $$\begin{aligned}
\partial_t n_\epsilon(\vec{r},t) &=D_\epsilon \Delta n_\epsilon(\vec{r},t) + (1-\epsilon) n_\epsilon(\vec{r},t) \rho(\vec{r},t) \, , \\
\partial_t\rho(\vec{r},t)&=-\rho (\vec{r},t)\int_0^1\! (1-\epsilon)\, n_\epsilon(\vec{r},t)\,\mathrm{d}\epsilon\, .\end{aligned}$$
Here $\Delta$ is the lattice Laplacian, $D_\epsilon=\epsilon/(2d\delta^2)$ with $\delta=\sqrt{\mu_\text{max}/e_\text{max}}$, and the unit of length is $\ell=a/\delta$.
Equations \[eq:ipde\](a,b) exhibit a stationary, spatially uniform state with resources only: $n(\vec{r},t)\equiv\int_0^1\! n_\epsilon(\vec{r},t)\,\mathrm{d}\epsilon=0$ and $\rho(\vec{r},t)=1$. However, a linear stability analysis shows that this state is locally unstable to small population seeds. The ensuing exponential growth is limited by the availability of resources, and saturates when resources are fully exploited, $\rho(\vec{r},t)=0$, and the population reaches the carrying capacity, $n(\vec{r},t)=1$. From classical front propagation theory we expect that a small population seed will develop into a traveling wave front [@Saarloos2003]. Indeed, in accordance with our stochastic simulations, a numerical solution of Eqs. \[eq:ipde\](a,b) shows propagating wave solutions. For a homogeneous system with growth rate $\mu_{\text{max}}$ and migration rate $e_\text{max}$ the width of such front is $\ell=a/\delta$. Hence $\delta$ measures the “coarseness” of the model: It can be read as the size of a bacterium, $a$, compared to the width of the wave front, $\ell$. Alternatively, $\delta=\sqrt{\mu_\text{max}/e_\text{max}}$ also gives the relative maximal range of growth and hopping rates. While for small values of $\delta$ diffusion is faster than growth, large values correspond to a growth-dominated expansion process.
 To investigate which genotypes are selected by the evolutionary dynamics at specific times we numerically solved the Fisher equations, Eqs. \[eq:ipde\](a,b), for various values of $\delta$ (as indicated in the graph) and computed the average motility $\langle\epsilon\rangle$ in the population. (b) The solid (dashed) line illustrates the analytical result for the genotype $\epsilon^*$ with the optimal front velocity in one (two) spatial dimensions, \[Eq. \]. Triangles indicate the average genotype $\langle\epsilon\rangle$ obtained from the numerical solution of Eqs. , evaluated at a large time $t=3000$ for $d=1$ and at $t=1100$ for $d=2$, respectively.](./box3_revised.pdf){width="\columnwidth"}
To understand the role of evolutionary forces during range expansion we computed the temporal evolution of the mean motility $\langle\epsilon\rangle$ in the whole population. Figure \[fig:box3\](a) shows $\langle\epsilon\rangle (t)$ as obtained from the numerical solution of the coupled Fisher equations, Eqs. \[eq:ipde\](a,b), for a series of values for $\delta$. During the first few generations, $t\lesssim 15$, while the population is genetically still highly heterogeneous, the population dynamics is governed by scramble competition for resources. In order to dominate the front, a potentially successful genotype must be capable of efficiently outgrowing its competitors by consumption of the majority of resources at the front. This gives a selective advantage to genotypes with a high reproduction rate. They dominate over competitors with a higher motility, which in turn leads to a decrease in the mean motility $\langle\epsilon\rangle$, cf. Fig. \[fig:box3\](a); the decrease is to a large degree independent of $\delta$. After this initial phase, for $t\gtrsim15$, macroscopic differences in the concentrations of the different genotypes have emerged which locally compete for resources. Our simulations show that the average motility reaches a minimum and starts to increase again \[Fig. \[fig:box3\](a)\]. This indicates that now the evolutionary most successful genotypes are no longer those which optimize their growth rates (specialists), but those, which balance reproduction with motility. The reason is that the decisive factor limiting the growth of a particular genotype colony is the velocity of the ensuing Fisher wave, as can be understood by analyzing the set of coupled Fisher equations, Eqs. \[eq:ipde\](a,b): Since the velocity of a Fisher wave is determined by its leading edge where resources are plentiful, we may approximately write $\rho\approx 1-n$. Following the theory of front propagation [@Brunet1997; @Saarloos2003], we assume that traveling wave solutions $n_\epsilon(r,t)=n_\epsilon(r-vt)=n_\epsilon(z)$ decay exponentially at the leading edge of the front, $n_\epsilon(z)\sim \exp{(-\gamma z)}$. Upon substituting the exponential ansatz into Eqs. \[eq:ipde\](a,b) and linearizing in the concentrations we find that the Fisher equations for different values of $\epsilon$ decouple [@Supplement]. Keeping only the highest order exponential terms, we obtain a dispersion relation $v(\gamma)=\gamma^{-1}\left\{(\epsilon/d) \left[\cosh(\delta\gamma)-1\right]+1-\epsilon\right\}$. Given a sufficiently steep initial front, the solution with minimal velocity $v(\gamma_0)$ is selected [@Bramson1983; @Saarloos2003]; for a radially expanding front in $d=2$, $v(\gamma_0)$ is approached asymptotically for $r\to\infty$ [@Murray2002]. Hence, a homogeneous subpopulation with motility $$\epsilon^*=\left[\sqrt{2/(d\delta^2)+1}\;\mathrm{arccosh}\left(1+d\delta^2\right)\right]^{-1}\label{eq:eps}$$ has the highest invasion speed. As illustrated in Fig. \[fig:box3\](b), the optimal motility is $\epsilon^*=0.5$ for $\delta\to 0$, and it decreases only slowly with increasing $\delta$. Since the fastest propagating subpopulation will take an increasingly larger fraction of the colony, this explains why the mean motility increases \[Fig. \[fig:box3\](a)\]. Concomitant with this coarsening process sectors of uniform genotypes form for $50\leq t\leq 100$; see Fig. \[fig:box1\](a,b).
For a radially expanding front in two dimensions, the subsequent population dynamics is mainly governed by annihilation of these sector boundaries. Since this is a very slow process, the mean motility $\langle\epsilon\rangle$ is only asymptotically approaching the optimal value $\epsilon^*$: The boundary $\varphi(r)$ of two adjacent sectors, propagating with velocities $v_1$ and $v_2>v_1$, forms a logarithmic spiral with $\varphi(r)=-\sqrt{v_2^2/v_1^2-1}\,\ln(r/a)$ which moves only very slowly into the direction of the slower domain [^5]. Hence any front consisting of multiple sectors will ultimately be dominated by the genotype with the fastest front velocity, given by Eq. . However, this annihilation process is far too slow to be observed within the numerically accessible time window. To heuristically account for this, we have added a constant value to $\langle\epsilon\rangle$. We find excellent agreement of $\epsilon^*$ and $\langle\epsilon\rangle$ strongly suggesting that the asymptotic genotype is determined by the optimal velocity of the corresponding homogeneous fronts.
To investigate the dependence of the asymptotic composition of the population on the strength of reaction noise, we computed $\langle\epsilon\rangle$ and $\epsilon^*$ also by employing stochastic simulations of the lattice gas model for different values of the system size $\Omega$. Demographic noise affects the evolutionary dynamics in manifold ways: The initial coarsening process leading to sector formation is an inherent stochastic process which is not well accounted for by the set of Fisher equations, Eqs. \[eq:ipde\](a,b). Figure 4 illustrates that compared to the deterministic dynamics the coarsening process for the stochastic dynamics is slightly faster.
 Comparison of $\langle\epsilon\rangle(t)$ between stochastic simulations with $\Omega=100$ and simulations of Eqs. , both for $\delta=1$ and $d=2$. The measured value $\epsilon^*$ for $\Omega=100$ from (c), and the analytic value from Eq. are indicated. (b, c) Comparison of $\langle\epsilon\rangle$ as obtained from stochastic simulations at a large time $t=220$ with $\epsilon^*$ as function of $\Omega$. These quantities differ by a small constant value $\Delta$ indicated in the graph. Stochastic simulation results in (a-c) were averaged over at least $500$ sample runs.](./box4_revised.pdf){width="\columnwidth"}
Moreover, the ensuing sector boundaries also merge faster due to the stochastic meandering motion of the sector boundaries [@Ali2010; @Hallatschek2007]. Indeed we recover that the stochastic lateral movement of domain boundaries is super-diffusive, *i.e.* its root-mean-square displacement increases as $t^\gamma$, with $\gamma>0.5$. For a sector boundary of a planar front we measured $\gamma\approx 0.63$ [@Supplement] confirming that conformation of the sector boundaries is well described by kinetic roughening [@Kardar1986; @Saito1995]. These stochastic effects become less important as the colony grows [@Ali2010; @Korolev2010; @Lavrentovich2013; @Ali2013]: Since the front is advancing uniformly, *i.e.* $t\sim r$, the front roughness $r^\gamma$ becomes small compared to deterministic drift $r\ln r$ as $r\to\infty$. Conversely, due to the absence of front inflation, sector annihilation proceeds more rapidly as a result of stochastic fluctuations for planar fronts [@Supplement; @Korolev2010; @Ali2010; @Ali2013; @Lavrentovich2013].
Finally, noise also affects the speed of propagating fronts [@Brunet1997; @Panja2004; @Brunet2006; @Hallatschek2009; @Hallatschek2011; @Hallatschek2011a]. Taken together, we find that demographic noise significantly affects the population dynamics during range expansion and leads to an asymptotic composition of the population with an increased average motility, cf. Fig. \[fig:box4\](a,b). In particular, the asymptotic value of $\langle\epsilon\rangle$ and the genotype $\epsilon^*$ with the highest velocity of homogeneous fronts both decrease with $\Omega$. In fact, $\langle\epsilon\rangle$ and $\epsilon^*$ differ only by a small constant, which can be attributed to the fact that $\langle\epsilon\rangle$ was measured at a finite time. This observation underscores our assertion that the species dominating the front will be the genotype maximizing its front speed.
In conclusion, we studied the role of risk-spreading between motility and growth during range expansion. Starting from a genetically heterogeneous population we found that during the initial phase of the expansion process scramble competition for resources favors fast growing individuals. Concomitantly, the number of distinct genotypes decreases rapidly and thereby genetically homogeneous sectors form. Therefore, the competitive advantage at larger times shifts towards those individuals with the highest front speed. We have shown that risk-spreading leads to an optimal front speed. In the deterministic limit, described by a set of coupled Fisher equations, the optimal strategy turns out to be perfect risk-spreading between motility and growth in a parameter regime dominated by diffusion (small dimensionless parameter $\delta$). Our analytical results also quantify how the optimal strategy is increasingly biased towards growth as the typical time scales for growth and diffusion become comparable. A low carrying capacity is affecting the range expansion dynamics in a twofold way: During the initial phase demographic noise may lead to an early fixation of the front and hence to a bias towards slowly migrating individuals. At later stages of range expansion, noise leads to a strong shift of the optimal value for the mean motility towards larger values.
We expect that both the spatial separation of different genotypes and the evolutionary success of generalists are generic for range expansions of heterogeneous populations. By genetically tuning the number of flagella in *E. coli* bacteria, a motility-growth tradeoff can be studied experimentally. Current experiments are investigating the implications of this tradeoff for range expansion, allowing for a test of our results [@Taute2014]. We believe that our model can also be tested using custom-build reaction-diffusion networks with synthetic nucleic acids and enzymatic reactions [@Padirac2013]. In general, our findings also pertain to other spreading processes, where motility is complementary to growth. Further work might include mutations [@Kuhr2011; @Greulich2012] or extend our findings to excitable media, systems exhibiting an Allee effect [@Taylor2005] and metastable states [@Meerson2011; @Rulands2013], and finally also to more complex reaction networks [@Reichenbach2008; @Case2010; @Dobrinevsky2012; @Knebel2013].\
We thank Madeleine Opitz and David Jahn for fruitful discussions. This research was supported by the Deutsche Forschungsgemeinschaft via contract no. FR 850/9-1 and the German Excellence Initiative via the program ‘Nanosystems Initiative Munich’. S.R. gratefully acknowledges support of the Wellcome Trust (grant number 098357/Z/12/Z).
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[^1]: M. Reiter and S. Rulands contributed equally to this work.
[^2]: M. Reiter and S. Rulands contributed equally to this work.
[^3]: For genotypes $(\mu_\text{max},\epsilon_\text{max})$ the front width is $\ell = a \sqrt{e_\text{max}/\mu_\text{max}}$ [@Saarloos2003]. In our stochastic simulations we used $\delta=\sqrt{e_\text{max}/\mu_\text{max}} = 1$, and, for simplicity, choose $\Delta r = a$ \[Fig. \[fig:box2\](a)\].
[^4]: As each individual has a distinct genotype and $G\gg 1$ we can omit the index $i$ and formally treat $\epsilon$ as a continuous variable uniquely identifying a certain genotype.
[^5]: Neglecting fluctuations the angle $\varphi$ satisfies the differential equation [@Korolev2012], $r\varphi'(r) =-\sqrt{v_2^2/v_1^2 - 1}$, which is solved by $\varphi(r)=-\sqrt{v_2^2/v_1^2-1}\ln(r/a)$ where we have taken without loss of generality $\varphi(a) =0$.
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"pile_set_name": "ArXiv"
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**Jordan algebras admitting derivations with invertible values.** [^1]
**Ivan Kaygorodov$^{a,b}$, Artem Lopatin$^{c}$ Yury Popov$^{b,d}$**
Abstract {#abstract .unnumbered}
========
The notion of derivation with invertible values as a derivation of ring with unity that only takes multiplicatively invertible or zero values appeared in a paper of Bergen, Herstein and Lanski, in which they determined the structure of associative rings that admit derivations with invertible values. Later, the results of this paper were generalized in many cases, for example, for generalized derivations, associative superalgebras, alternative algebras and many others. The present work is dedicated to description of all Jordan algebras admitting derivations with invertible values.
Introduction
============
Let $A$ be an algebra with unit element $1$ over field $F$. We denote by $U$ the set of invertible elements of $A$. Further in this article we only consider *derivations with invertible values*, by which we understand such non-zero derivations $d$ that for every $x \in A$ holds $d(x) \in U$ or $d(x)=0$.
In 1983, Bergen, Herstein and Lanski initiated the study which purpose is to relate the structure of a ring to the special behavior of one of its derivations. Namely, in their article [@Berg] they described associative rings admitting derivations with invertible values. They proved that such ring must be either a division ring, or the ring of $2 \times 2$ matrices over a division ring, or a factor of a polynomial ring over division ring of characteristic 2. They also characterized those division rings such that $2 \times 2$ matrix ring over them has an inner derivation with invertible values. Further, associative rings with derivations with invertible values (and also their generalizations) were discussed in variety of works (see, for instance, [@Car; @Chang; @CAAF; @Hongan; @Giam; @Komatsu]). So, in [@Giam] semiprime associative rings with involution, allowing a derivation with invertible values on the set of symmetric elements, were given an examination. In the work [@Car] Bergen and Carini studied associative rings admitting a derivation with invertible values on some non–central Lie ideal. Also, in the papers [@Chang] and [@Hongan] the structure of associative rings that admit $\alpha$-derivations with invertible values and their natural generalizations — $(\sigma,\tau)$-derivations with invertible values was described. In the paper [@Komatsu] Komatsu and Nakajima described associative rings that allow generalized derivations with invertible values. The case of associative superalgebras with derivations with invertible values was studied in the paper of Demir, Albas, Argac, and Fosner [@CAAF]. The description of non-associative algebras addmiting derivations with invertible values began in paper of Kaygorodov and Popov [@KP], where it was proved that every alternative (non-associative) algebra addmiting derivation with invertible values is a Cayley–Dickson over their center or a factor-algebra of polynomial algebra $C[x]/(x^2)$ over a Cayley–Dickson division algebra.
Nowadays, a great interest is shown to the studying of nonassociative algebras and superalgebras with derivations. For example, in paper [@Popov2] the structure of differentiably simple Jordan algebras is determined, and papers [@kay; @kay2; @shestakov] give the description of generalizations of derivations of Jordan algebras. Analogues of Moens’ theorem, describing nilpotent finite-dimensional algebras as those having invertible Leibniz-derivations were proved in [@kp15] for Jordan, Malcev and right alternative algebras . Nevertheless, the problem of specification of Jordan algebras admitting derivations with invertible values remains unconsidered. The present work is dedicated to the description of Jordan algebras admitting derivations with invertible values.
Preliminaries.
==============
In this article we only consider algebras over field $F$ of characteristic $\neq 2,3.$ A commutative algebra $J$ is called *Jordan* if it satisfies the *Jordan identity*: $ (x^2,y,x) = 0. $
Let $J$ be a Jordan algebra with unit element $1$. An element $x \in J$ is called invertible if there exists $y \in J$ such that $xy = 1, x^2y = x.$
During our discussion we will encounter some certain types of Jordan algebras, so, in order to make our work self-contained, we provide their definitions:
1. **Algebra of type $A^{(+)}$.** Let $(A,\cdot)$ be an algebra. Then its underlying vector space equipped with new multiplication $a \circ b = \frac{1}{2}(a\cdot b + b \cdot a)$ is again an algebra which we denote by $A^{(+)}.$ If $A$ is associative, then $A^{(+)}$ is a Jordan algebra. If a Jordan algebra $J$ can be imbedded into $A^{(+)}$ for an associative algebra $A$, it is called *special*. It is known that in a special Jordan algebra $J \subseteq A^{(+)}$ an element is invertible with inverse $y$ if and only if it is invertible in $A$ algebra with inverse $y$. Nonspecial Jordan algebras are called *exceptional*.
2. **Algebra of type $H(A,*)$.** For an algebra $A$ with involution $*$ the set $H(A,*)$ of $*$-hermitian elements (i.e., such elements $x$ that $x^* = x$) is closed under Jordan product $\circ$, so it is a subalgebra of $A^{(+)}$. Again, if $A$ is associative, $H(A,*)$ is a Jordan algebra.
3. **Algebra of symmetric bilinear vector form $J(V,f)$.** Let $F$ be a field and $V$ be a $F$-vector space with a symmetric bilinear form $f:V\otimes V \rightarrow F$. We can endow $F \oplus V$ with a structure of Jordan algebra by defining multiplication as follows: $$(\alpha + v)(\beta + u) = \alpha\beta + f(v,u) + \alpha u + \beta v, \text{ where } \alpha, \beta \in F, u, v \in V.$$ We denote this algebra by $J(V,f).$
4. **Algebras of Albert type.** Let $F$ be a field, $C$ be a Cayley-Dickson algebra over $F$, and $\gamma_1, \gamma_2, \gamma_3$ be nonzero elements of $F$. We denote the main involution in $C$ by $\bar{}$ (For more information on Cayley-Dickson algebras, we refer the reader to [@Jac; @kolca]). Consider $C_3$, the algebra of $3\times 3$ matrices with entries in $C.$ By $\bar{X}^T$ we denote the matrix obtained from $X \in C_3$ by applying the transpose and involution $\bar{}$ to every coefficient of $X,$ and by $\gamma$ we denote the matrix $diag\{\gamma_1, \gamma_2, \gamma_3\}.$ Then the mapping $*_\gamma:C\rightarrow C$ defined by $X^{*_\gamma} = \gamma^{-1} \bar{X}^T \gamma $ is an involution of $C_3$. It is known that $H(C_3,*_\gamma)$ is an exceptional simple Jordan algebra. If $\gamma_1 = \gamma_2 = \gamma_3 = 1$, and $C$ is a split Cayley-Dickson algebra over $F$, then this algebra is called the Albert algebra over $F.$
A Jordan algebra $J$ over field $F$ is called an *algebra of Albert type*, if the scalar extension $J \otimes_\Omega \Omega$ is an Albert algebra over $\Omega$, where $\Omega$ is the algebraic closure of $F.$ It is well known that all algebras of Albert type are divided in two classes: the division algebras, and the algebras $H(C_3, *_\gamma)$.
An element $A \in J = H(C_3,*_\gamma)$ has the form (the following properties of the algebra $H(C_3,*_\gamma)$ can be found in [@Jac]) $$A = \begin{pmatrix}
\alpha_1 & c & \gamma_1^{-1}\gamma_3\bar{b} \\
\gamma_2^{-1}\gamma_1\bar{c} & \alpha_2 & a \\
b & \gamma_3^{-1}\gamma_1\bar{a} & \alpha_3 \end{pmatrix},$$ where $\alpha_1, \alpha_2, \alpha_3 \in F, a, b, c \in C.$ One can see that this algebra is of degree 3, that is, it has 3 orthogonal idempotents $e_{ii}, i = 1, 2, 3$. Relative to these idempotents it has a Peirce decomposition
$$J = \sum_{i=1}^3 J_{ii} + \sum_{i<j} J_{ij},$$
where $J_{ii} = Fe_{ii},$ and $J_{ij} = \{ae_{ij} + \gamma_i\gamma_j^{-1}ae_{ji}, a \in C\}, i, j = 1, 2, 3$
For an element $A \in J$ we can define
$$n(A) = \alpha_1\alpha_2\alpha_3 - \alpha_1\gamma_3^{-1}\gamma_2n(a) - \alpha_2\gamma_1^{-1}\gamma_3n(b) - \alpha_3\gamma_2^{-1}\gamma_1n(c) + t((ca)b)$$.
It is well know that $A$ is invertible in $J$ if and only if it $n(A) \neq 0.$
We will also need the following statement which describes simple Jordan algebras:\
[**Zelmanov’s Theorem [@Zel83].**]{} Let $J$ be a simple Jordan algebra over field of characteristic $\neq 2,3$. Then one of the following holds:
1. $J$ is an algebra $A^{(+)}$, where $A$ is a simple associative algebra;
2. $J$ is an algebra $H(A, *)$, where $A$ is a simple associative algebra;
3. $J$ is an algebra of non-degenerate symmetric bilinear form $J(V,f)$ on a vector space $V$ of dimension $>1$ over field $F$;
4. $J$ is an algebra of Albert type.
Jordan algebras addmiting derivations with invertible values.
=============================================================
The purpose of this paragraph is to generalize the results of Bergen, Herstein and Lanski to the Jordan case. Further in this part, $J$ is a Jordan algebra with unit element $1$ and a derivation with invertible values $d$.
Now we shall study the ideal structure of $J$:
\[Lem1.2\] Let $I\lhd J$. Suppose that $I \not \subset \ker (d)$. Then:
1. $I$ is both minimal and maximal,
2. $I^3=0$.
**Proof.** $(a).$ Let $I_1 \subseteq I \subseteq I_2$ be proper ideals in $J$, and $d(I) \neq 0$. Since $I$ does not contain invertible elements, it is easy to see that $d(I) \cap I=0$ and $I \oplus d(I)$ is also an ideal in $J$. Since $d(I) \neq 0$, $d(I)$ contains invertible elements, hence $I \oplus d(I)=J$. Particularly, for any $j \in J$ there exist $a, b \in I$ such that $d(j)=a+d(b)$. Hence $a=d(j-b) \in I \cap d(J)=0$, therefore $J=I \oplus \ker (d)$ which implies $I \cap \ker (d) = 0$. For arbitrary $j \in I_2 $ we have $j=m+d(n)$, $m, n \in I$. Consequently, $d(n)=j-m \in I_2 \cap d(I) = 0$; thus $j=m \in I$, so $I$ is maximal. Since $I \cap ker (d) = 0$, $I_1 \not\subset ker (d)$, therefore, repeating the argument above, we conclude that $I_1 = I$, so $I$ is minimal.
$(b).$ The cube of an ideal in Jordan algebra is also an ideal. Obviously, $I^3 \subseteq I$, and $I^3$ is also an ideal of $J$, so by $(a)$ we only have two possibilities: $I^3 = I$ or $I^3 = 0$. If $I^3 = I$, then $$d(I) = d(I^3) = d(I^2I) \subseteq I^2d(I) + (d(I)I)I \subseteq I \cap d(J) = 0,$$ which contradicts the initial condition $I \not\subset ker(d)$, so we are forced to conclude that $I^3 = 0$. $\Box$
\[Lem1.3\] $I \lhd J$ implies $d(I) = 0$.
**Proof.** Let $I \lhd J$ and suppose that $d(I) \neq 0.$ Then there exists $i \in I$ such that $d(i) \in U$. Since $I^3 = 0$, $$\begin{aligned}
0=d^3(i^3)=
d^3(i^2)i + 6 d(i)^3 + 6(id^2(i))d(i) + 6(id(i))d^2(i) + i^2 d^3(i), \end{aligned}$$ and we have $6d(i)^3 \in I$. Since the power of an invertible element in Jordan algebra is also invertible, $6d(i)^3 = 0$, therefore $6 = 0.$ We have obtained a contradiction which proves the lemma. $\Box$
\[Lem1.4\] Let $I \lhd J$ and $I \subseteq \ker(d)$. Then $I^2 = 0$.
**Proof.** For any $z \in I, y \notin ker (d)$ we have $0 = d(zy) = zd(y)$. For arbitrary $x \in I$ we have $xd(y) = x^2d(y) = 0$. It follows that $(d(y), J, x^2) = 0$ (see [@Zel78]). Particulary, we have $$0 = d(y)(d(y)^{-1}x^2) = (d(y)d(y)^{-1})x^2 = x^2.$$ Linearizing this, we obtain $2ab = 0$ for $a, b \in I$, so $I^2 = 0$.$\Box$
By $M$ we denote the sum of all ideals contained in $ker (d)$. Lemma \[Lem1.3\] implies that $M$ is the largest ideal of $J$. We now prove that $\bar{J} = J/M$ is a simple Jordan algebra that admits a derivation with invertible values. Since $d(M) =0$, the map $\bar{d}: \bar{J} \rightarrow \bar{J} $ given by $\bar{d}(j + M) = d(j) + M$ is correctly defined, and it is easy to see that $\bar{d}$ is a derivation with invertible values of $\bar{J}$. Now, if $\bar{I}$ is an ideal of $\bar{J}$ such that $\bar{I} \neq \bar{J},$ from lemma \[Lem1.3\] it follows that $\bar{d}(\bar{I}) = 0$, then for the full inverse image $I$ of $\bar{I}$ we have $d(I) \subseteq M$, and since $M \cap d(J) = 0$, $d(I) = 0$ and $I \subseteq M$, which is equivalent to $\bar{I} = 0$.
We now consider simple Jordan algebras admitting derivations with invertible values.
As Zelmanov’s theorem suggests, we have to study four cases:
The case $J = A^{(+)}$, $A$ is a simple associative
---------------------------------------------------
This case is the easiest one, because by [@Herstein] every derivation of $A^{(+)}$ is a derivation of $A$. Since any element which is invertible in $A^{(+)}$ is also invertible in $A$, then $d$ is a derivation with invertible values of associative algebra $A$, so, by [@Berg] $A$ is either a division algebra $D$, or a $D_2$, the $2 \times 2$ matrix algebra over a division algebra $D$.
The case $J = H(A, *)$, $A$ is a simple associative
---------------------------------------------------
This case is dealt with in
\[Lem2.1\] Let $A$ be a simple associative algebra such that $H(A, *)$ has a derivation $d$ with invertible values. Then $A$ is isomorphic either to
1. a division algebra $D$, or
2. $D_2$, $2 \times 2$ matrix algebra over $D$, or
3. $F_4$, $4 \times 4$ matrix algebra over field $F$, $*$ is the symplectic involution, or
4. a central order in $F_4$, $*$ is an involution of symplectic kind.
**Proof.** Every derivation $d$ of $H(A, *)$ can be considered as a Jordan derivation $d:H(A, *) \rightarrow A$. By [@Lagutina] either any Jordan derivation $d:H(A, *) \rightarrow A$ can be extended to derivation $d:<H(A, *)> \rightarrow A$, or $A$ is a central order in $F_4$ and $*$ is an involution of symplectic kind. From Herstein’s work [@Herstein1] it follows that if $dim_{Z(A)} A >4$, then $<H(A, *)> = A$. Wedderburn–Artin theorem implies that if $dim_{Z(A)} A \leq 4$, then $A$ is either a division algebra over $Z(A)$ or $Z(A)_2$, which correspondingly matches the cases $1)$ and $2)$, so from now on we may assume that $d$ can be extended to a derivation of $A$. As a derivation of $A$, $d$ also has invertible values, so by [@Giam] $A$ is an algebra of type $1)$, $2)$ or $3)$. The lemma is now proved.$\Box$
The case $J = J(V,f)$
---------------------
is examined in
\[Lem2.2\] An algebra $J(V,f)$ over field $F$ admits a derivation with invertible values if and only if there exist $x, y \in V$ such that $f(x,x) \neq 0,\ f(y,y) \neq 0,\ f(x,y) = 0$ and $-\frac{f(y,y)}{f(x,x)}$ is not a square in $F$.
**Proof.** Let $d$ be any derivation of $J$. For $0 \neq v \in V$ we can write $d(v) = \beta + u$, where $\beta \in F$, $u \in V$. We have $0 = d(f(v,v)) = d(v^2) = 2vd(v) = 2\beta v + 2f(v,u)$, which yields $\beta = 0, f(v,d(v)) = 0$. Conversely, it is easy to check that for arbitrary endomorphism $\phi$ of $V$ condition $f(v, \phi(v)) = 0$ for $v \in V$ (and its linearization $f(\phi(v), u) + f(v,\phi(u)) = 0$ for $v, u \in V$) imply that $\phi$ is a derivation of $J(V,f)$. By definition of invertibility in Jordan algebras it is easy to see that $u \in V$ is invertible in $J(V,f)$ if and only if $f(u,u) \neq 0$. Since $d(F) = 0,$ the problem of describing derivations with invertible values is actually a problem of finding all $d \in End(V)$ such that $$\begin{aligned}
\label{f1}
f(v,d(v)) = 0,\end{aligned}$$ $$\begin{aligned}
\label{f2}
f(d(v),d(v)) = 0 \Rightarrow d(v) = 0\end{aligned}$$ for any $v \in V.$ Suppose that lemma condition does not hold in $V$. In this case we prove that $dim(d(J)) < 2$. Let $x, y$ be two linearly independent vectors in $d(J)$. By our hypothesis, $f(x,x) \neq 0$, thus we may substitute $y \rightarrow y - \frac{f(x,y)}{f(x,x)}x$ and assume that $f(x,y) =0$. Also, $f(y,y)$ remains nonzero. For $\alpha \in F$ we have $f(\alpha x + y, \alpha x + y) = \alpha^2 f(x,x) + f(y,y)$. But for $\alpha = \sqrt{-\frac{f(y,y)}{f(x,x)} } \in F$ this expression is equal to $0$, hence $\alpha x + y$ is not invertible in $J(V,f)$, which contradicts (\[f2\]), so we conclude that $d(J) = Fu$ for $u \in V.$ Particularly, $d(u) = \delta u,$ where $\delta \in F.$ But from (\[f1\]) it follows that $0 = f(u,d(u)) = \delta f(u,u)$, and since $f(u,u) \neq 0$, we have $\delta = 0$. Now, take $v \in V$ such that $d(v) = u$. Linearizing (\[f1\]), we have $0 = d(f(v,u))= f(d(v),u) + f(v,d(u)) = f(u,u)$, a contradiction.\
**Conversely,** suppose that $x, y \in V$ satisfy the lemma condition. By $W$ we denote subspace spanned by $x$ and $y$. Every element $v \in V$ can be considered as $v=v' + \frac{f(x,v)}{f(x,x)}x + \frac{f(y,v)}{f(y,y)}y.$ Here we have that $f(v',W)=0$. Say that $U= \{ v' \in V : f(v',W)=0 \}$ and $f(x,x) \neq 0 \neq f(y,y).$ It is easy to see that $U \cap W = 0$ and $V = U \oplus W.$
Now we are able to explicitly construct a derivation with invertible values of $J(V,f)$: define $d$ by $d(x) = y, d(y) = -\frac{f(y,y)}{f(x,x)}x$, $d(U) = 0$. It is easy to see that conditions (\[f1\]) and (\[f2\]) hold for $d$, so it is a derivation of $J(V,f)$. Also, for $\beta, \gamma \neq 0$, we have $$f(\beta x + \gamma y, \beta x + \gamma y) = \beta^2 f(x,x) + \gamma^2 f(y,y)\neq 0.$$ In the other case, $\sqrt{-\frac{f(x,x)}{f(y,y)}} \in F.$ Now, $d$ is a derivation with invertible values. $\Box$
Every simple Jordan algebra of symmetric bilinear vector form over perfect (or algebraically closed) field does not have a derivation with invertible values.
The case $J$ is an algebra of Albert type
-----------------------------------------
As we have said, the algebras of Albert type are divided in two classes: the division algebras (which are not interesting to us, because all their derivations have invertible values), and the algebras $H(C_3,*_\gamma)$. We deal with the later ones in the following
\[Lem2.3\] An algebra $H(C_3,*_\gamma)$ does not have nonzero derivations with invertible values.
**Proof.** Let $d$ be a derivation with invertible values of $J.$ Then $d(e_{11}) = d(e_{11}^2) = 2e_{11}d(e_{11}),$ which means that $d(e_{11}) \in J_{12} + J_{13}, $ that is, has the form
$$d(e_{11}) = \begin{pmatrix}
0 & c & \gamma_1^{-1}\gamma_3\bar{b} \\
\gamma_2^{-1}\gamma_1\bar{c} & 0 & 0 \\
b & 0 & 0 \end{pmatrix},$$ where $ b, c \in C.$
It is easy to see that $n(d(e_{11})) = 0,$ therefore, that element is not invertible and is 0 by the definition of invertible derivation. Analogously, $d(e_{22}) = d(e_{33}) = 0.$ Now, let $A \in J_{12}.$ Then, since $d(e_{11}) = 0,$ we have $d(A) = 2d(e_{11}A) = 2e_{11}d(A).$ Analogously, $d(A) = 2e_{22}d(A),$ therefore, $d(A) \in J_12.$ One can see that $n(d(A))$ is again zero, therefore, $d(A)$ is zero. Analogously, one can prove that $d(J_{13}) = d(J_{23}) = 0.$ Hence, $d(J_{ij}) = 0$ for all Peirce components of $J$, and $d = 0.$ The lemma is now proved.
Let $J$ be a Jordan algebra of characteristic $\neq 2, 3$ with unit element $1$, admitting derivation with invertible values $d$. Then one of the following holds:
1. $J$ is an algebra $A^{(+)}$, where $A = D$ or $D_2$, $D$ is an associative division algebra;
2. $J$ is an algebra $H(A, *)$, where $A = D$ or $D_2$, $D$ is an associative division algebra, or $A$ is either $F_4$ or a central order in $F_4$, $* = Symp$;
3. $J$ is an algebra of symmetric bilinear form $J(V,f)$;
4. $J$ is a division algebra of Albert type;
5. $J$ is an extension of cases $(1)$ – $(4)$ by $M = \mathbf{P}(J)$ – the prime radical of $J$, $M \subseteq \ker d$, $M$ is the largest ideal of $J$.
**Proof.** Follows immediately from lemmas (\[Lem1.2\]) – (\[Lem2.3\]).
[**Acknowledgements.**]{} The authors grateful to Prof. Ivan Shestakov (IME-USP, Brasil) and Prof. Alexandre Pozhidaev (Sobolev Institute of Math., Russia) for interest and constructive comments.
[20]{}
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[^1]: The first author was supported by FAPESP, Proc. 2014/24519-8, all authors were supported by RFBR 15-31-21169.
|
{
"pile_set_name": "ArXiv"
}
|
---
address:
- 'Department of Mathematics, University of California, Berkeley, CA 94720, USA'
- 'Department of Mathematics, Harvard University, Cambridge, MA 02138, USA'
author:
- Edward Frenkel
- Dennis Gaitsgory
date: 'November 2005; revised July 2006'
title: 'Fusion and convolution: applications to affine Kac-Moody algebras at the critical level'
---
Introduction {#introduction .unnumbered}
============
The goal of this article is two-fold. On the one hand, it constitutes the second in the series of papers devoted to the study of the category of representations of an affine Kac-Moody algebra at the critical level in connection with the local geometric Langlands correspondence. On the other hand, we study from the geometric point of view the concept of fusion of modules over an affine Kac-Moody algebra, or, more generally, chiral modules over a chiral algebra that admits an algebraic group as a group of its symmetries.
Let us explain the first perspective in some detail. In [@FG2] we proposed a framework for the local geometric Langlands correspondence. Namely, let $G$ be a semi-simple algebraic group (over $\BC$ or any other algebraically closed field of characteristic $0$), and let us consider $G\ppart$ as a group ind-scheme. We proposed that to a “local Langlands parameter” $\sigma$, which is a $\cG$–bundle with a connection on the punctured disc (where $\cG$ is the Langlands dual group of $G$), there should correspond a category $\C_\sigma$ equipped with an action of $G\ppart$.
Unfortunately, in general we could not characterize $\C_\sigma$ in a unique way by a universal property. However, we conjectured that this category is closely connected to the category of representations of the affine Kac-Moody algebra $\hg$ at the critical level.
To describe this connection, let us recall that, according to [@FF; @F:wak], the center $\fZ_{\fg}$ of the completed enveloping algebra of $\hg_\crit$ is isomorphic to the algebra of functions on the space $\Op(\D^\times)$ of $\cg$-opers on the punctured disc $\D^\times$ (where $\cg$ is the Langlands dual Lie algebra of $\g$). We recall that a $\cg$-oper on $\D^\times$ (or, more generally, on any curve) is by definition a $\cG$-bundle $\CF_G$ equipped with a connection $\nabla$, together with an additional datum, namely, a reduction of $\CF_G$ to a Borel subgroup of $\cG$, which is in a particular relative position with respect to $\nabla$.
For a fixed $\cg$-oper $\chi$, which we regard as a character of $\fZ_\fg$, we can consider the sub-category $\hg_\crit\mod_\chi$ of the category $\hg_\crit\mod$ of all discrete modules at the critical level, on which the center acts according to this character. This category carries a canonical action of $G\ppart$ via its adjoint action on $\hg_\crit$.
We proposed in [@FG2] that $\hg_\crit\mod_\chi$ should be equivalent to the sought-after category $\C_\sigma$, where $\sigma$ corresponds to the pair $(\CF_G,\nabla)$ underlying the oper $\chi$. This guess entails a far-reaching corollary that the categories $\hg_\crit\mod_{\chi^1}$ and $\hg_\crit\mod_{\chi^2}$ for two different central characters $\chi^1$ and $\chi^2$ are equivalent once we identify the underlying local systems $(\CF^1_G,\nabla^1)$ and $(\CF^2_G,\nabla^2)$.
In [@FG2] we discussed various consequences of this proposal, and in particular, a concrete conjecture about the structure of the full subcategory $\hg_\crit\mod_\nilp^{I,m}$ of the category of $\hg_\crit$-modules. Its objects are $I$-monodromic $\hg_\crit$-modules (where $I$ is the Iwahori subgroup of $G\ppart$), which are supported, as $\fZ_{\fg}$-modules, over a certain sub-scheme $\nOp\subset\Op(\D^\times)$.
Our conjecture was that the derived category of $\hg_\crit\mod_\nilp^{I,m}$ is equivalent to the derived category of quasi-coherent sheaves on the scheme $\tg/\cG\underset{\cg/\cG}\times
\nOp$, where $\tg\to \cg$ is Grothendieck’s alteration, and $\nOp\to
\cg/\cG$ is a canonical residue map $\Res^\nilp$, introduced in [@FG2]. In [@FG2] we proved a weaker statement that certain quotients of these two categories are equivalent.
In order to prove this conjecture in full we need to know a wide range of results about the structure of the category $\hg_\crit\mod$. One class of such results has to do with the Harish-Chandra convolution action of D-modules on $G\ppart$ on the derived category $D(\hg_\crit\mod)$.
In [@FG2] we explained, following [@BD], that the appropriately defined $I$-equivariant derived category $D^b(\on{D}(\Fl_G)_\crit\mod)^{I}$ of critically twisted D-modules on the affine flag scheme $\Fl_G=G\ppart/I$ acts on the $I$-equivariant derived category $D^b(\hg_\crit\mod)^{I}$ of $\hg_\crit$-modules.
This structure is called the Harish-Chandra convolution action, and its origin is the action of the affine Hecke algebra (whose categorification is $D^b(\on{D}(\Fl_G)_\crit\mod)^{I}$) on the space of $I$-invariant vectors in any representation of the group $G$ over a local non-archimedian field.
The category $D^b(\on{D}(\Fl_G)_\crit\mod)^{I}$ contains special objects, the so-called central sheaves $\CZ_V, V \in \on{Rep}(\cG)$, introduced in [@Ga]. They correspond to the central elements of the affine Hecke algebra.
The main result of the present paper, is that all objects of $\hg_\crit\mod^{I}$ are eigen-objects with respect to the functor of convolution with the central sheaves. This is one of the crucial steps in our project describing $\hg_\crit\mod_\nilp^{I,m}$ in terms of quasi-coherent sheaves. Let us now explain more precisely what being an eigen-object means in our set-up.
In [Sect. \[statement of Iwahori theorem\]]{}, we explain that the support over the ind-scheme $\Spec(\fZ_\fg)$ of any object from $\hg_\crit\mod^{I}$ is contained in the ind-subscheme $\Spec(\wt\fZ^{\int,\nilp}_\fg)$ that corresponds to opers which, as local systems on $\D^\times$, have regular singularities and a unipotent monodromy. We show also that to each $V\in \Rep(\cG)$ there corresponds a vector bundle $\CV_{\wt\fZ^{\int,\nilp}_\fg}$ over $\Spec(\wt\fZ^{\int,\nilp}_\fg)$. The geometric meaning of this vector bundle is the following: for a $\BC$-point $\chi\in \Spec(\wt\fZ^{\int,\nilp}_\fg)\simeq
\on{Op}_{\cg}(\D^\times)$, the fiber $\CV_\chi$ of $\CV_{\wt\fZ^{\int,\nilp}_\fg}$ is isomorphic to the fiber at the origin of the canonical (i.e., Deligne’s) extension from $\D^\times$ to $\D$ of the local system $(\CF_G,\nabla)$, underlying $\chi$.
The vector bundle $\CV_{\wt\fZ^{\int,\nilp}_\fg}$ is equipped with a nilpotent endomorphism, corresponding to the monodromy of the underlying oper.
The main result of the present paper, [Theorem \[main\]]{}, states that for any $\CM\in \hg_\crit\mod^{I}$ and $V\in \Rep(\cG)$ we have a canonical isomorphism of $\hg_\crit$-modules: $$\label{main formula}
\CZ_V \star \CM \simeq \CV_{\wt\fZ^{\int,\nilp}_\fg}
\underset{\wt\fZ^{\int,\nilp}_\fg}\otimes \CM,$$ such that the action of the monodromy on the left hand side, coming from the definition of $\CZ_V$ via the nearby cycles functor as in [@Ga], goes under this isomorphism to the nilpotent endomorphism of $\CV_{\wt\fZ^{\int,\nilp}_\fg}$ mentioned above. Moreover, the isomorphisms of for different representations $V$ are compatible, in a natural sense, with the operation of tensor product.
Thus, for example, if $\CM$ is an $I$-equivariant $\hg_\crit$–module on which the center $\fZ_{\fg}$ acts via the character corresponding to a particular $\chi\in \wt\fZ^{\int,\nilp}_\fg$, then the isomorphism becomes $$\CZ_V \star \CM \simeq \CV_\chi \underset{\CC}\otimes \CM,$$ where $\CV_\chi$ is as above.
The isomorphism stated in has an analogue for the category of $G[[t]]$-equivariant $\hg_\crit$-modules (rather than $I$-equivariant ones). In this case the role of the category $\on{D}(\Fl_G)_\crit\mod^{I}$ is played by the category $\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$ of $G[[t]]$-equivariant D-modules on the affine Grassmannian $\Gr_G = G\ppart/G[[t]]$. This category is a tensor category that is equivalent to the category $\Rep(\cG)$ of finite-dimensional representations of $\cG$, see [@MV] (it may also be thought of as a categorification of the spherical Hecke algebra).
Hence, for each $V \in \Rep(\cG)$ we have the corresponding object $\CF_V$ in the category $\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$. A spherical version of our main theorem, [Theorem \[main, spherical\]]{}, states that for every $\CM\in \hg_\crit\mod^{G[[t]]}$ there is a canonical isomorphism $$\label{main formula, spherical}
\CF_V \star \CM \simeq \CV_{\wt\fZ^{\int,\nilp}_\fg}
\underset{\wt\fZ^{\int,\nilp}_\fg}\otimes \CM.$$ Moreover, we obtain that the support of every such $\CM$ over $\Spec(\wt\fZ^{\int,\nilp}_\fg)$ is contained in the ind-subscheme $\Spec(\wt\fZ^{\on{m.f}}_\fg)$, corresponding to monodromy-free opers.
Tautologically, the isomorphism is a particular case of that of . Let us specialize further to the case when $\CM\in \hg_\crit\mod^{G[[t]]}$ is the vacuum module $\BV_\crit$. In the latter case, the corresponding isomorphism $$\label{main formula, vacuum}
\CF_V \star \BV_\crit \simeq \CV_{\fz_\fg}
\underset{\fz_\fg}\otimes \BV_\crit.$$ is known, due to [@BD].
The main idea of this paper is that one can derive results such as and from the special case when $\CM=\BV_\crit$. This is based on the operation of fusion product, and this brings us to the discussion of the second perspective in which one can view this paper.
Let $\CA$ be a chiral algebra on a curve $X$, and let $\CM_1,\CM_2$ and $\CN$ be chiral $\CA$-modules. In this case one can consider the set of [*chiral pairings*]{} $\{\CM_1,\CM_2\}\to \CN$. These notions were introduced by A. Beilinson and V. Drinfeld in [@CHA].
When $\CA$ is a chiral algebra attached to a conformal vertex algebra ${\mathsf V}$, and $\CM_1,\CM_2,\CN$ are obtained from ${\mathsf
V}$-modules, the notion of chiral pairing is similar to that of intertwining operator between modules over ${\mathsf V}$, see [@FHL].
If for fixed $\CM_1$ and $\CM_2$ the functor that assigns to $\CN$ the set of chiral pairings $\{\CM_1,\CM_2\}\to \CN$ is representable, we shall call the representing object [*the fusion product*]{} of $\CM_1$ and $\CM_2$.
Assume now that $\CA$ is the chiral algebra $\CA_{\fg,\kappa}$, attached to a semi-simple Lie algebra $\fg$ and a symmetric invariant form $\kappa:\fg\otimes \fg\to \BC$. Then the category of chiral $\CA_{\fg,\kappa}$-modules, supported at a given point $x\in X$ is naturally equivalent to the category $\hg_\kappa\mod$ of representations of the affine Kac-Moody algebra $\hg$ at the level $\kappa$. (More generally, we can consider chiral algebras that admit $G$ as a group of symmetries, see [Sect. \[general alg\]]{}).
Let us suppose now that $\kappa$ is non-positive and integral. In this case, to every $V\in \Rep(\cG)$ we can attach a chiral $\CA_{\fg,\kappa}$-module, denoted $\Gamma(\Gr_{G,X},\CF_{V,X})$, or $\CF_{V,X}\star \CA_{\fg,\kappa}$. It is constructed using the D-module $\CF_V\in \on{D}(\Gr_G)_\kappa\mod^{G[[t]]}$, which is well-defined for every integral $\kappa$.
Let $\CM_2=\CM$ be any $G[[t]]$-equivariant (resp., $I$-equivariant) $\hg_\kappa$-module. We set $\CM_1:=\Gamma(\Gr_{G,X},\CF_{V,X})$, or more generally, $\CM_1:=\Gamma(\Gr_{G,X},\CF_{V,X})\otimes \CE$, where $\CE$ is a local system on the punctured curve $X-x$ with a nilpotent monodromy around $x$.
Our key technical tool is the assertion (see [Theorem \[representability, spherical\]]{} and [Theorem \[representability, Iwahori\]]{}) that in this case fusion products exist and are given by $\CF_V\star \CM$ in the $G[[t]]$-equivariant case, and by $$\Bigl((\CZ_V\star \CM)\otimes
\Psi(\CE)\Bigr)_N$$ in the $I$-equivariant case, where $\Psi(\CE)$ denotes nearby cycles of $\CE$ at $x$, viewed as a vector space with a nilpotent operator, and the subscript $N$ stands for taking coinvariants of the monodromy.
The proofs of the main Theorems, namely, \[main, spherical\] and \[main\], are obtained by showing that the right-hand sides of the stated isomorphisms also represent the above functors of chiral pairings, and it is this last assertion that uses the result of [@BD] about the isomorphism .
Let us remark that the proofs of the main theorems do not actually use the full statements of Theorems \[representability, spherical\] and \[representability, Iwahori\], but only the existence of the corresponding maps in one direction, and their compatibility with tensor products of representations.
Let us briefly describe the way this paper is organized. It is divided into two parts.
In Part I we discuss in detail the $G[[t]]$-equivariant situation. In [Sect. \[sect 1\]]{} we state [Theorem \[main, spherical\]]{}; in [Sect. \[sect 2\]]{} we discuss the relationship between the fusion product and Harish-Chandra convolution and state [Theorem \[representability, spherical\]]{}. In [Sect. \[sect 3\]]{} we complete the proof of [Theorem \[main, spherical\]]{}, and finally in [Sect. \[proof of rep\]]{} we prove a generalization of [Theorem \[representability, spherical\]]{} in the framework of chiral algebras, endowed with a Harish-Chandra action of the group $G$.
In Part II we show how to modify the material of Part I for the $I$-equivariant situation. Thus, in [Sect. \[sect 5\]]{} we state [Theorem \[main\]]{}, and in [Sect. \[sect 6\]]{} we derive it from (a part of) [Theorem \[representability, Iwahori\]]{}. In [Sect. \[sect 7\]]{} we prove a suitable generalization of [Theorem \[representability, Iwahori\]]{}.
In the appendix, [Sect. \[app A\]]{}, we give a proof of the fact that the chiral bracket on chiral algebras such as $\CA_{\fg,\kappa}$ can be described using D-modules on the Beilinson-Drinfeld Grassmannian.
The notation in this paper follows closely that of [@FG2].
D.G. would like to thank Sasha Beilinson for patient explanations and stimulating discussions.
The research of E.F. was supported by the DARPA grant HR0011-04-1-0031 and by the NSF grant DMS-0303529.
**[Part I: The spherical case]{}**
Convolution at the critical level {#sect 1}
=================================
Recollections {#recol}
-------------
Let $\fg$ be a simple finite-dimensional Lie algebra. For an invariant inner product $\kappa$ on $\fg$ (which is unique up to a scalar) define the central extension $\hg_\kappa$ of the formal loop algebra $\fg
\otimes \BC\ppart$ which fits into the short exact sequence $$0 \to \BC {\mb 1} \to \hg_\kappa \to \fg \otimes \BC\ppart \to 0.$$ This sequence is split as a vector space, and the commutation relations read $$\label{KM rel}
[x \otimes f(t),y \otimes g(t)] = [x,y] \otimes f(t) g(t) - \kappa(x,y)\cdot
\on{Res}(f \cdot dg)\cdot {\mb 1},$$ and ${\mb 1}$ is a central element. The Lie algebra $\hg_\kappa$ is the [*affine Kac-Moody algebra*]{} associated to $\kappa$. We will denote by $\hg_\kappa\mod$ the category of [*discrete*]{} representations of $\hg_{\kappa}$ (i.e., such that any vector is annihilated by $\fg \otimes t^n\BC[[t]]$ for sufficiently large $n$), on which ${\mb 1}$ acts as the identity.
Let $\kappa_{Kil}$ be the Killing form: $$\kappa_{Kil}(x,y) = \on{Tr} (\on{ad} (x) \circ \on{ad}(y)).$$ A level $\kappa$ is called critical (resp., positive, negative, irrational) if $\kappa=c\cdot \kappa_{Kil}$ and $c=-\frac{1}{2}$ (resp., $c+\frac{1}{2}\in \BQ^{>0}$, $c+\frac{1}{2}\in \BQ^{<0}$, $c\notin \BQ$). A level $\kappa$ is called integral is it is an integral multiple of the standard inner product, normalized so that the square length of the maximal root is equal to $2$.
Next, we recall some notation and terminology from the theory of chiral algebras introduced in [@CHA]. Our chiral algebras will be defined on a smooth algebraic curve $X$. We will fix a point $x\in X$, and identify D-modules supported at $x$ with underlying vector spaces.
We will denote by $\D_x$ and $\D_x^\times$, respectively, the formal disc and the formal punctured disc around $x$. If we choose a coordinate $t$ near $x$, we obtain the identifications $\D_x\simeq
\D:=\Spec(\CC[[t]])$ and $\D_x^\times\simeq
\D^\times:=\Spec(\CC\ppart)$,
Following [@CHA] (see also [@FG2], Sect. 10), we associate to $\fg$ the Lie-\* algebra $L_\fg = \fg\otimes D_X$, and for each level $\kappa$ its central extension $L_{\fg,\kappa}$ by means of $\omega_X$. By definition, the chiral algebra $\CA_{\fg,\kappa}$ is the quotient of the chiral universal envelope of $L_{\fg,\kappa}$ obtained by identifying the two copies of $\omega_X$.
For the rest of this section we shall fix the level $\kappa$ to be critical. Let $\fz_\fg$ be the center of $\CA_{\fg,\crit}$, viewed as a commutative D-algebra on a curve $X$. Denote by $\fz_{\fg,x}$ the fiber of $\fz_\fg$ at the point $x\in X$, and let $\fZ_\fg$ be the topological commutative algebra corresponding to $\fz_\fg$ and $x$, as defined in [@CHA], Sect. 3.6.18.
There exists a canonical map from $\fZ_\fg$ to the center of the category $\hg_\crit\mod$. (The latter identifies tautologically with the center of the corresponding completed universal enveloping algebra $\wt{U}_\crit(\hg)$, see [@FG2], Sect. 5.1 for more details.) One can show that the map from $\fZ_\fg$ to $Z(\wt{U}_\crit(\hg))$ is in fact an isomorphism, but we will not use this fact.
Denote by $\check \fg$ the Langlands dual Lie algebra to $\fg$. Let $\on{Op}_{\cg,X}$ be the D-scheme of $\cg$-opers on $X$ [@BD]. The following isomorphism is proved in [@FF; @F:wak]: $$\label{isom with opers global}
\fz_{\fg} \simeq \on{Fun} (\on{Op}_{\cg,X}).$$
Let consider also the spaces of $\cg$-opers $\on{Op}_{\cg}(\D_x)$ and $\on{Op}_{\cg}(\D_x^\times)$ on the formal disc $\D_x$ and the formal punctured disc $\D_x^\times$, respectively. The former is a scheme of infinite type, and the latter is an ind-scheme. We refer the reader to Part I of [@FG2] for a detailed discussion of opers.
The isomorphism implies that: $$\label{isom with opers reg}
\fz_{\fg,x} \simeq \on{Fun} (\on{Op}_{\cg}^\reg),$$ where $\on{Op}_{\cg}^\reg:=\on{Op}_{\cg}(\D_x)$, and $$\label{isom with opers}
\fZ_\fg \simeq \on{Fun} (\on{Op}_{\cg}(\D_x^\times)).$$
Vector bundles over opers {#bundle on opers}
-------------------------
Let us denote by $\cG$ the simple algebraic group of simply-connected type, corresponding to $\cg$. Then there is a tautological $\cG$-bundle $\CP_{\cG,\on{Op}_{\cg,X}}$ over $\on{Op}_{\cg,X}$ equipped with a connection along $X$. For any finite-dimensional representation $V$ of $\cG$ we will denote by $\CV_{\on{Op}_{\cg,X}}$ the corresponding associated vector bundle over $\on{Op}_{\cg,X}$, equipped with a connection along $X$.
Via the isomorphism we can view $\CV_{\on{Op}_{\cg,X}}$ as a locally free $\fz_\fg$-module with a connection, and we shall also denote it by $\CV_{\fz_\fg}$. (We remark that as a $\fz_\fg$-module, $\CV_{\fz_\fg}$ is actually free, but there is no canonical choice of generators.) The fiber of $\CV_{\fz_\fg}$ over $x\in X$ is a locally free $\fz_{\fg,x}$-module, which we will denote by $\CV_{\fz_{\fg,x}}$.
Denote by $\wt\fz_{\fg,x}$ the topological algebra equal to the completion of $\fZ_{\fg}$ with respect to the ideal defining $\fz_{\fg,x}$, i.e., $$\wt\fz_{\fg,x}:=\underset{n}{\underset{\longleftarrow}{\lim}}\,
\fZ_{\fg}/\CJ_n,$$ where $\CJ_n$ is the closure of the $n$th power of $\on{ker}(\fZ_{\fg}\to \fz_{\fg,x})$. Let $\Spec(\wt\fz_{\fg,x})$ be the resulting ind-subscheme of $\Spec(\fZ_\fg)$.
(In what follows, for a topological algebra $A\simeq \underset{\longleftarrow}{\lim}\, A_i$, we shall denote by $\Spec(A)$ the corresponding ind-scheme, i.e., $\underset{\longrightarrow}{"\lim"}\, \Spec(A_i)$. By a vector bundle over an ind-scheme we shall mean a compatible system of vector bundles over its closed subschemes.)
\[extension of bundle over center\]
[*(1)*]{} The vector bundle $\CV_{\fz_{\fg,x}}$ naturally extends to a vector bundle $\CV_{\wt\fz_{\fg,x}}$ over $\Spec(\wt\fz_{\fg,x})$.
[*(2)*]{} For $V,W\in \Rep(\cG)$ and $U=V\otimes W$, there is a natural isomorphism $\CV_{\wt\fz_{\fg,x}}\underset{\wt\fz_{\fg,x}}\otimes
\CW_{\wt\fz_{\fg,x}}\simeq \CU _{\wt\fz_{\fg,x}}$.
[*(3)*]{} Each $\CV_{\wt\fz_{\fg,x}}$ is equipped with a (pro)-nilpotent endomorphism $N_{\CV_{\wt\fz_{\fg,x}}}$, and these endomorphisms are compatible with the identifications of (2) above.
The rest of this subsection is devoted to the proof of this proposition.
By definition, $$\wt\fz_{\fg,x}\simeq
\underset{\fz'_\fg}{\underset{\longleftarrow}{\lim}}\, \fz'_{\fg,x},$$ where $\fz'_\fg$ runs over the family of D-subalgebras of $\fz_\fg$, such that the ideal $\on{ker}(\fz'_{\fg,x}\to \fz_{\fg,x})$ is nilpotent (here $\fz'_{\fg,x}$ denotes the fiber of $\fz'_\fg$ at $x$). Consider the following general set-up:
Let $\CB$ be a commutative chiral algebra on $X$, and let $\CV_{X-x}$ be a free finite-rank $\CB$-module, equipped with a compatible connection along $X$. Assume that $\CV_{X-x}$ admits an extension to a free finite rank $\CB$-module $\CV_X$, on which the connection along $X$ has a pole of order $\leq 1$ at $x$ and its residue $\Res(\nabla,\CV_X)$, thought of as an endomorphism of the fiber $\CV_x$ of $\CV_X$ at $x$, is nilpotent.
Let $\CB'\hookrightarrow \CB$ be a chiral subalgebra, such that $\CB'|_{X-x}\simeq \CB|_{X-x}$, and such that $\on{ker}(\CB'_x\to
\CB_x)$ is nilpotent.
\[extension of chiral modules\] Under the above circumstances, $\CV_{X-x}$, viewed as a $\CB'$-module over $X-x$, admits a unique extension to a free finite rank $\CB'$-module $\CV'_X$ such that $\CV_X\simeq
\CV'_X\underset{\CB'}\otimes \CB$, and such that the connection on $\CV'_X$ has a pole of order $\leq 1$ at $x$. In this case $\Res(\nabla,\CV'_X)$ is also nilpotent.
Applying this lemma in our situation we obtain a locally free sheaf $\CV_{\fz'_\fg}$ over each $\fz'_\fg$ as above. Moreover, the formation of $\CV_{\fz'_\fg}$ is compatible with tensor products of representations, by the uniqueness statement of the lemma.
By taking the fiber of $\CV_{\fz'_\fg}$ at $x$ we obtain a vector bundle $\CV_{\fz'_\fg,x}$ over each $\Spec(\fz'_{\fg,x})$ as above, i.e., a vector bundle over $\Spec(\wt\fz_{\fg,x})$. The endomorphisms $N_{\CV_{\wt\fz_{\fg,x}}}|_{\Spec(\fz'_{\fg,x})}$ are equal to $\Res(\nabla,\CV_{\fz'_\fg})$.
Let us now prove [Lemma \[extension of chiral modules\]]{}. We have the natural morphisms $\CB\to \CB_x[[t]]$ and $\CB'\to \CB'_x[[t]]$, compatible with connections. It follows from the Beauville-Laszlo theorem [@BL] (see also [@BD]) that the problem of extension of $\CV_{X-x}$ to $X$ translates to a problem of extensions of locally free modules with connections over $\CB_x[[t]]$ and $\CB'_x[[t]]$. Thus, we have to prove the following:
\[extensions of modules\] Let $B'\twoheadrightarrow B$ be a surjection of commutative algebras with a nilpotent kernel. Let $T'$ be a free module over $B'\ppart$, endowed with a connection along $t$, and let $T$ be the corresponding $B\ppart$-module. Let $T_0\subset T$ be a $B[[t]]$-lattice, such that $t\partial_t\cdot T_0\subset T_0$, and the endomorphism induced on $T_0/t\cdot T_0$ is nilpotent. Then there exists a unique $B'[[t]]$-lattice $T'_0$ in $T'$ with the same properties, such that $T'_0\underset{B'[[t]]}\otimes B[[t]]=T_0\subset
T$.
By induction, we can assume that $I:=\on{ker}(B'\to B)$ is such that $I^2=0$. Then we have a short exact sequence $$0\to I\cdot T'\to T'\to T'/I\cdot T'\to 0,$$ where both $I\cdot
T'\simeq I\underset{B}\otimes T$ and $T'/I\cdot T'\simeq T$ are $B\ppart$-modules.
It is easy to see that $T'$ admits at least one $B'[[t]]$-lattice $'T'_0$, satisfying $$\label{cond on lattice}
'T'_0\on{mod} \, I\cdot T'=T_0 \text{ and }
'T'_0\cap \, I\cdot T'=I[[t]]\underset{B[[t]]}\otimes T_0.$$
The operator $t\partial_t$, acting on $'T_0'$ defines a $B[[t]]$-linear map $$\phi:T_0\to I[[t]]\underset{B[[t]]}\otimes T/T_0.$$
Any other lattice $T'_0$ in $T'$ that satisfies differs from $'T'_0$ by a $B[[t]]$-linear operator $$E:T_0\to I[[t]]\underset{B[[t]]}\otimes T/T_0.$$ The condition for $T'_0$ to satisfy the assumption of the lemma reads as follows: $$\phi=[E,t\partial_t].$$ This equation is uniquely solvable by induction on the order of the pole, since $\phi-k\cdot \on{Id}$ is invertible whenever $k\neq 0$.
Let $V^\mu$ be the irreducible $\fg$-module with highest weight $\mu$, and set $\BV^\mu=\on{Ind}^{\hg_\crit}_{\fg[[t]]}(V^\mu)$. Recall from [@FG2], Sect. 7.6 that to any dominant integral weight $\lambda$ we have attached a subscheme $\Spec(\fZ^{\lambda,\reg}_{\fg})\subset \Spec(\fZ_{\fg})$. The following will be proved in [Sect. \[another proof mon-free\]]{}:
\[sup of ind\] The support of $\BV^\mu$ over $\Spec(\fZ_{\fg})$ is contained in $\Spec(\fZ^{\mu,\reg}_{\fg})$.
Let $\fz^{\lambda,\reg}_\fg$ be the commutative chiral algebra on $X$, isomorphic to $\fz_\fg$ over $X-x$ whose fiber at $x$ is $\Spec(\fZ^{\lambda,\reg}_{\fg})$. According to [@FG2], Sect. 2.9, for $V\in \Rep(\cG)$, the locally free sheaf $\CV_{\fz_\fg}$ with a connection, defined on $\fz^{\lambda,\reg}_\fg|_{X-x}$, extends to a locally free sheaf with a [*regular connection*]{} over $\fz^{\lambda,\reg}_\fg$. We will denote the resulting vector bundle over $\Spec(\fZ^{\lambda,\reg}_{\fg})$ by $\CV_{\fZ^{\lambda,\reg}_{\fg}}$.
Let $\wt\fZ^{\lambda,\reg}_{\fg}$ be the formal completion of $\Spec(\fZ_{\fg})$ along $\Spec(\fZ^{\lambda,\reg}_{\fg})$. The proof of [Proposition-Construction \[extension of bundle over center\]]{} implies that $\CV_{\fZ^{\lambda,\reg}_{\fg}}$ naturally extends to a vector bundle $\CV_{\wt\fZ^{\lambda,\reg}_{\fg}}$ over $\Spec(\wt\fZ^{\lambda,\reg}_{\fg})$, equipped with a nilpotent endomorphism $N_{\CV_{\wt\fZ^{\lambda,\reg}_{\fg}}}$, in a way compatible with tensor products of objects of $\cG$.
Let us denote by $\Spec(\fZ^{\int,\reg}_{\fg})$ (resp., $\Spec(\wt\fZ^{\int,\reg}_{\fg})$) the ind-subscheme of $\Spec(\fZ_{\fg})$ equal to the disjoint union $\underset{\lambda}\sqcup\, \Spec(\fZ^{\lambda,\reg}_{\fg})$ (resp., $\underset{\lambda}\sqcup\, \Spec(\wt\fZ^{\lambda,\reg}_{\fg})$), and by $\CV_{\fZ^{\int,\reg}_{\fg}}$ (resp., $\CV_{\wt\fZ^{\int,\reg}_{\fg}}$) the vector bundle on it, corresponding to $V\in \Rep(\cG)$. We shall denote by $N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}$ the endomorphism of the latter vector bundle.
Let $\Spec(\fZ_{\fg}^{\on{m.f.}}) \subset \Spec(\fZ_{\fg})$ be the sub-functor that corresponds to opers that are monodromy-free as local systems. It is easy to see that we have the following inclusions of functors: $$\Spec(\fZ^{\int,\reg}_{\fg}) \subset \Spec(\fZ_{\fg}^{\on{m.f.}})
\subset \Spec(\wt\fZ^{\int,\reg}_{\fg}).$$ (In fact, one can show that $\Spec(\fZ_{\fg}^{\on{m.f.}})$ is the minimal ind-subscheme of $\Spec(\wt\fZ^{\int,\reg}_{\fg})$, containing $\Spec(\fZ^{\int,\reg}_{\fg})$, stable under the action of the Lie algebroid $\Omega^1(\fZ_{\fg})$, see [@FG1], Sect. 6.12.)
Moreover, from the definitions it follows that $\Spec(\fZ_{\fg}^{\on{m.f.}})$ is the closed ind-subscheme of $\Spec(\wt\fZ^{\int,\reg}_{\fg})$, equal to the locus of vanishing of $N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}$ for all $V\in \cG$.
Let $\CM$ be an object of $\fZ_{\fg}\mod$, i.e., a (discrete) vector space, endowed with a continuous action of $\fZ_\fg$. We shall say that $\CM$ is supported on $\Spec(\wt\fZ^{\lambda,\reg}_{\fg})$ if every element of $\CM$ is annihilated by some power of the ideal of $\fZ^{\lambda,\reg}_{\fg}$ in $\fZ_\fg$.
We shall say that a module $\CM$ is supported on $\Spec(\wt\fZ^{\int,\reg}_{\fg})$ if it is a union (or, automatically, a direct sum) of module $\CM^\lambda$, where each $\CM^\lambda$ is supported on $\Spec(\wt\fZ^{\lambda,\reg}_{\fg})$.[^1]
Given $V\in \Rep(\cG)$ we have a well-defined functor on the category of $\fZ_\fg$-modules, supported on $\Spec(\wt\fZ^{\int,\reg}_{\fg})$: $$\CM\mapsto
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM.$$ Moreover, these functors come equipped with nilpotent endomorphisms $N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}$, compatible with tensor products of objects of $\Rep(\cG)$.
The above constructions have a relevance for us due to the following:
\[sup of sph\] If $\CM$ is a $G[[t]]$-integrable $\hg_\crit$-module, then, viewed as a module over $\fZ_\fg$, it is supported on $\Spec(\wt\fZ^{\int,\reg}_{\fg})$.
The proof follows from [Lemma \[sup of ind\]]{}, since every object of $\hg_\crit\mod^{G[[t]]}$ admits a filtration, such that each subquotient is isomorphic to a quotient module of some $\BV^\lambda$. (For a different argument see [Sect. \[another proof mon-free\]]{}.)
Let $G$ be the group of adjoint type with the Lie algebra $\fg$, so that the Langlands dual group $\cG$ is simply-connected. We denote by $\Gr_G = G\ppart/G[[t]]$ the corresponding affine Grassmannian.[^2]
We will consider the category $\on{D}(\Gr_G)_\crit\mod$ of critically twisted right D-modules on $\Gr_G$. Since the critical level is integral, this category is equivalent to category of usual right D-modules on $\Gr_G$, via the tensor product by the corresponding line bundle. Let $\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$ be the corresponding category of $G[[t]]$-equivariant twisted D-modules.
The geometric Satake equivalence (see [@MV]) defines a functor $$V\in \Rep(\cG)\mapsto \CF_V\in \on{D}(\Gr_G)_\crit\mod^{G[[t]]}.$$
Moreover, $\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$ is endowed with a structure of tensor category via the convolution product, such that the above functor becomes an equivalence of tensor categories.
To $\CM\in \hg_\crit\mod^{G[[t]]}$ and $\CF\in \on{D}(\Gr_G)_\crit\mod$ we can associate their convolution $$\CF\star \CM\in D(\hg_\crit),$$ (see [@FG2], Sect. 22.5 for the corresponding definitions).
If $\CF$ is an object of $\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$, then $\CF\star \CM$ will be naturally an object of $D(\hg_\crit)^{G[[t]]}$.
\[main, spherical\] For $\CM\in \hg_\crit\mod^{G[[t]]}$ and $V\in \Rep(\cG)$, the convolution $\CF_V\star \CM$ is acyclic off cohomological degree $0$, and we have a functorial isomorphism $$\label{main eq}
\fs_V:\CF_V\star \CM \simeq
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM,$$ compatible with tensor products of $\cG$-representations, i.e., for $V,W\in \cG$ the diagrams $$\label{comp1}
\CD \CF_V\star (\CF_W\star \CM) @>{\fs_V}>>
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
(\CF_W\star \CM) \\ @V{\sim}VV @V{\on{id}_V\otimes \fs_W}VV \\
(\CF_V\star \CF_W)\star \CM @>{\fs_{V\otimes W}}>>
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
\CM, \endCD$$ and $$\label{comp2}
\CD \CF_V\star (\CF_W\star \CM) @>{\on{id}_V\star \fs_W}>> \CF_V\star
(\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM) \\ @V{\sim}VV @V{\fs_V}VV \\ (\CF_V\star \CF_W)\star \CM
@>{\fs_{V\otimes W}}>>
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes
\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM, \endCD$$ are commutative. The endomorphism on RHS of , given by $N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}$, is identically equal to $0$.
The last statement of [Theorem \[main, spherical\]]{} implies the following statement, conjectured by A. Beilinson (see [@FG1], Conjecture 6.13):
\[monodromy-free\] The support in $\Spec(\fZ_\fg)$ of any $G[[t]]$-integrable $\hg_\crit$-module is contained in $\Spec(\fZ_\fg^{\on{m.f.}})$.
\[thm for diff op\]
Let $\fD_{G,\crit}$ denote the chiral algebra of differential operators on $G$ at the critical level introduced in [@AG]. It comes equipped homomorphisms of chiral algebras $$\fl:\CA_{\fg,\crit}\to \fD_{G,\crit}\rightarrow \CA_{\fg,\crit}:\fr$$ whose images mutually Lie-\* commute. Let $\fD_{G,\crit,x}$ be its fiber at $x$, which we view as a $G[[t]]$-equivariant bimodule at the critical level.
Using the map $\pi:G\ppart\to \Gr_G$, starting from $\CF\in
\on{D}(\Gr_G)_\kappa\mod$ (for any level $\kappa$), we produce a chiral $\fD_{G,\crit}$-module supported at $x$ by considering $$\Gamma(G\ppart,\pi^*(\CF)).$$ We can also view it as $\hg_\crit$-bimodule, which is $G[[t]]$-equivariant with respect to the $\fr$ action.
We have: $$\label{sections on D as conv}
\Gamma(G\ppart,\pi^*(\CF))\simeq \CF\star \fD_{G,\crit,x},$$ as $\fD_{G,\crit}$-modules with respect to the action of $G$ on itself by left translations. Hence, the corresponding isomorphism holds also on the level of $\hg_\crit$-bimodules. [Theorem \[main, spherical\]]{} then immediately implies the following:
\[main, spherical, diff\] We have a canonical isomorphism of $\hg_\crit$-bimodules, $$\label{main eq diff}
\Gamma(G\ppart,\pi^*(\CF_V))\simeq
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \fD_{G,\crit,x},$$ where the tensor product is taken with respect to the $\fZ_\fg$-module structure on $\fD_{G,\crit,x}$, given by $\fl$. These isomorphisms are compatible with tensor products of objects of $\Rep(\cG)$. The endomorphism on the RHS of , given by $N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}$, is zero.
Conversely, [Theorem \[main, spherical, diff\]]{} implies the first two statements of [Theorem \[main, spherical\]]{}:
Recall (see [@FG2], Sect. 21.13) that for a $G[[t]]$-integrable $\hg_\kappa$-module $\CM$ (at any level $\kappa$) and $\CF\in
\on{D}(\Gr_G)_\kappa\mod$ there exists a canonical isomorphism of individual cohomologies $$\label{conv as semiinf}
h^i(\CF\star \CM) \simeq h^i\Bigl(
\Gamma(G\ppart,\pi^*(\CF))\underset{\fg\ppart,\fg}{\overset{\semiinf}
\otimes}\CM\Bigr),$$ where the semi-infinite Tor is taken with respect to action of $\hg_{\kappa'}$ on $\Gamma(G\ppart,\CF)$ given by $\fr$ (here $\kappa'$ is the opposite level). We refer the reader to [@FG2] for the precise definition of this functor.
Applying this to $\CF=\CF_V$, and using the isomorphism of [Theorem \[main, spherical, diff\]]{}, we obtain that the cohomologies of $\CF_V\star \CM$ are isomorphic to those of $$\label{co 1}
\Bigl(\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes \fD_{G,\crit,x}\Bigr)
\underset{\fg\ppart,\fg}{\overset{\semiinf}\otimes}\CM.$$
However, since $\CV_{\wt\fZ^{\int,\reg}_\fg}$ is locally free over $\wt\fZ^{\int,\reg}_\fg$, we obtain that the complex is quasi-isomorphic to $$\label{co 2}
\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes \Bigl(\fD_{G,\crit,x}
\underset{\fg\ppart,\fg}{\overset{\semiinf}\otimes}\CM\Bigr).$$
However, by [@FG2], Corollary 21.14, at any level $\kappa$, we have a quasi-isomorphism of $\hg_\kappa$-modules $$\label{semi-jective}
\fD_{G,\kappa,x}\underset{\fg\ppart,\fg}{\overset{\semiinf}\otimes}
\CM\simeq \CM.$$
Combining and we obtain that $\CF\star \CM$ has the same cohomologies as $\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes \CM $, as contended.
Interpretation of convolution as fusion {#sect 2}
=======================================
Our strategy of proof of [Theorem \[main, spherical\]]{} is to show that both sides of represent the same functor on the category $\hg_\crit\mod$. This functor has to do with the notion of [*chiral pairing*]{} or [*fusion*]{} of modules over a chiral algebra, which we shall presently define. Having introduced this functor, in the rest of the section we consider various properties of fusion and its connection to the convolution functors.
Let $\CA$ be a chiral algebra on a smooth curve $X$, and $\CM_1,\CM_2$ and $\CN$ be chiral $\CA$-modules. As usual, we shall denote by $\Delta$ the embedding of $X$ into $X^n$ as the main diagonal, and by $j$ the embedding of the complement of the diagonal divisor.
A chiral pairing $\{\CM_1,\CM_2\}\to \CN$ is by definition a map of D-modules on $X\times X$: $$\phi:j_*j^*(\CM_1\boxtimes \CM_2)\to \Delta_!(\CN),$$ which is compatible with the $\CA$ action in the following sense:
We need that the sum of the three morphisms $$j_*j^*(\CA\boxtimes \CM_1\boxtimes \CM_2)\to \Delta_!(\CN)$$ is zero, where the first morphism is $$j_*j^*(\CA\boxtimes \CM_1\boxtimes \CM_2)
\overset{\CA\text{-action on } \CM_1}\longrightarrow
\Delta_{x_1=x_2}{}_!(j_*j^*(\CM_1\boxtimes\CM_2))
\overset{\phi}\to \Delta_!(\CN),$$ the second morphism is the negative of $$j_*j^*(\CA\boxtimes \CM_1\boxtimes \CM_2)
\overset{\CA\text{-action on } \CM_2}\longrightarrow
\Delta_{x_1=x_3}{}_!(j_*j^*(\CM_1\boxtimes\CM_2))
\overset{\phi}\to \Delta_!(\CN),$$ and the third morphism is $$j_*j^*(\CA\boxtimes \CM_1\boxtimes \CM_2)\overset{\phi}\to
\Delta_{x_1=x_3}{}_!(j_*j^*(\CA\boxtimes \CN))
\overset{\CA\text{-action on } \CN}\longrightarrow \Delta_!(\CN).$$
Chiral pairings evidently form a functor $\CA\mod^o\times
\CA\mod^o\times \CA\mod\to \on{Vect}$.
[*Remark.*]{} The notion of chiral pairing was introduced in [@CHA]. As was mentioned in the introduction, in the case when $\CA$ is obtained from a conformal vertex algebra ${\mathsf V}$, and the modules $\CM_1,\CM_2$ and $\CN$ correspond to some ${\mathsf
V}$-modules $M_1,M_2$ and $N$, the set of chiral pairings $\{\CM_1,\CM_2\}\to \CN$ is in a natural bijection with the set of intertwining operators $M_1\otimes M_2\to N$ in the sense of [@FHL], such that in the corresponding fields all powers of the formal variable $z$ are integral.
Let us regard $\CA$ as a chiral module over itself. Then for any other chiral $\CA$-module $\CM$, we have a canonical chiral pairing $\{\CA,\CM\} \to \CM$ given by the action of $\CA$ on $\CM$. The following result is obtained directly from the definitions:
\[A is unit\] The canonical chiral pairing $\{\CA,\CM\}\to \CM$ establishes a bijection between the set of maps of chiral modules $\CM\to \CN$ and the set of chiral pairings $\{\CA,\CM\}\to \CN$.
Let $\kappa$ be any level, and let $\CA_{\fg,\kappa}$ be the corresponding chiral algebra on $X$.
Consider the relative version $\Gr_{G,X}$ of the affine Grassmannian over $X$. Let $\CF_X$ be a (right, $\kappa$-twisted) D-module on $\Gr_{G,X}$. In what follows we will assume that $\CF_X$ is torsion-free with respect to $X$. By a slight abuse of notation, we will denote by $\Gamma(\Gr_{G,X},\CF_X)$ the quasi-coherent direct image of $\CF_X$ on $X$; this is a chiral $\CA_{\fg,\kappa}$-module. By replacing the subscript “$X$” by either “$X-x$” or “$x$” we will denote the fibers of the above objects over $X-x$ and $x$ respectively.
\[const of basic pairing\] Given an object $\CM\in \hg_\kappa\mod^{G[[t]]}$ there exists a canonical chiral pairing $$\label{basic pairing for modules}
\{\Gamma(\Gr_{G,X},\CF_X),\CM\}\to h^0(\CF_x\star \CM).$$
The rest of this subsection is devoted to the proof of this proposition.
Let $\on{Jets}^{\mer}(G)_X$ be the D-ind scheme over $X$ of meromorphic jets into $G$. This is a relative version of the ind-scheme $G\ppart$. By definition, the category $\on{D}(\on{Jets}^{\mer}(G)_X)_\kappa\mod$ of (right $\kappa$-twisted) D-modules on $\on{Jets}^{\mer}(G)_X$ is equivalent to that of chiral $\fD_{G,\kappa}$-modules.
We have a natural projection $\pi:\on{Jets}^{\mer}(G)_X\to \Gr_{G,X}$ and by considering the quasi-coherent pull-back, we obtain a functor $$\pi^*:\on{D}(\Gr_{G,X})_\kappa\mod\to
\on{D}(\on{Jets}^{\mer}(G)_X)_\kappa\mod.$$ For $\CF_X\in
\on{D}(\Gr_{G,X})_\kappa\mod$, we will denote by $\Gamma(\on{Jets}^{\mer}(G)_X,\pi^*(\CF_X))$ the resulting $\fD_{G,\kappa}$-module.
One reconstructs $\Gamma(\Gr_{G,X},\CF_X)$ as a subset of $\Gamma(\on{Jets}^{\mer}(G)_X,\pi^*(\CF_X))$ as follows: this is the D-submodule consisting of sections that \*-commute with $\fr(\CA_{\fg,\kappa'})$, cf. [@AG] or [@FG1], Theorem 2.5.
For $\CF_X\in \on{D}(\Gr_{G,X})_\kappa\mod$ consider the action map $$\label{act of diff op}
j_*j^*\Bigl(\Gamma(\on{Jets}^{\mer}(G)_X,\pi^*(\CF_X))\boxtimes
\fD_{G,\kappa}\Bigr)\to
\Delta_!\Bigl(\Gamma(\on{Jets}^{\mer}(G)_X,\pi^*(\CF_X))\Bigr).$$
This is a $\fD_{G,\kappa}$-chiral pairing, and hence, in particular, a chiral pairing with respect to the action of $\CA_{\fg,\kappa}$ via $\fl$. In particular, we obtain a chiral $\CA_{\fg,\kappa}$-pairing $$\label{basic pairing univ}
j_*j^*(\Gamma(\Gr_{G,X},\CF_X)\boxtimes \fD_{G,\kappa})\to
\Delta_!(\Gamma(\on{Jets}^\mer(G)_X,\pi^*(\CF_X)).$$
This map commutes with the right $\CA_{\fg,\kappa'}$-action in the sense that the map $$\begin{aligned}
&\wt{ j}_*\wt{j}^*(\Gamma(\Gr_{G,X},\CF_X)\boxtimes
\fD_{G,\kappa}\boxtimes \CA_{\fg,\kappa'})
\to \Delta_{x_1=x_2}{}_!\Bigl(\Gamma(\on{Jets}^{\mer}(G)_X,\pi^*(\CF_X))
\boxtimes \CA_{\fg,\kappa'}\Bigr) \to \\
&\to \Delta_!\Bigl(\Gamma(\on{Jets}^{\mer}(G)_X,\pi^*(\CF_X))\Bigr)\end{aligned}$$ coincides with $$\begin{aligned}
&\wt{ j}_*\wt{j}^*(\Gamma(\Gr_{G,X},\CF_X) \boxtimes
\fD_{G,\kappa}\boxtimes \CA_{\fg,\kappa'}) \to
\Delta_{x_2=x_3}{}_!\Bigl(\Gamma(\Gr_{G,X},\CF_X)\boxtimes
\fD_{G,\kappa}\Bigr) \to \\ &\to
\Delta_!\Bigl(\Gamma(\on{Jets}^{\mer}(G)_X,\pi^*(\CF_X))\Bigr),\end{aligned}$$ where $\wt{j}$ denotes the embedding of the complement of the union of divisors corresponding to $x_1=x_2$ and $x_2=x_3$.
Let us restrict both sides of to $X\times
x\subset X\times X$. We obtain a chiral pairing $$\label{univ pairing at point}
j_x{}_*j_x^*\Bigl(\Gamma(\Gr_{G,X-x},\CF_{X-x})\Bigr)\otimes
\fD_{G,\kappa,x}\to i_x{}_! \Bigl(\Gamma(G\ppart,\pi^*(\CF_x))\Bigr),$$ where we view $\CF_x$ as a $\kappa$-twisted D-module over the affine Grassmannian, and $i_x$ (resp., $j_x$) denotes the embedding of $x$ into $X$ (resp., the embedding of this complement).
By the above, the map in commutes with the action of $\hg_{\kappa'}$ via $\fr$. (On the LHS this action affects only the $\fD_{G,\kappa,x}$ multiple.)
Let now $\CM$ be a $G[[t]]$-integrable $\hg_\kappa$-module. By considering the complex, computing the semi-infinite Tor, $$\fC^\semiinf(\fg\ppart;\fg,?\otimes \CM)$$ against the two sides of , considered as $\hg_{\kappa'}$-modules, we obtain a chiral pairing of complexes of $\CA_{\fg,\kappa}$-modules: $$\{\Gamma(\Gr_{G,X},\CF_X),
\fC^\semiinf\Bigl(\fg\ppart;\fg,\fD_{G,\kappa,x}\otimes \CM\Bigr)\}\to
\fC^\semiinf\Bigl(\fg\ppart;\fg,\Gamma(G\ppart,\pi^*(\CF_x))\otimes
\CM\Bigr).$$
By passing to the $0$th cohomology, and taking into account we obtain the chiral pairing of .
Let us now specialize to the case when $\kappa$ is integral non-positive (see [Sect. \[recol\]]{} for the definition of what this means). For $V\in \Rep(\cG)$, let $\CF_{V,X}$ be the corresponding object of $\on{D}(\Gr_{G,X})_\kappa\mod$. Then from we obtain a canonical chiral pairing $$\label{pairing with sph}
\{\Gamma(\Gr_{G,X},\CF_{V,X}),\CM\}\to h^0(\CF_V\star \CM).$$
In [Sect. \[proof of rep\]]{} we will prove the following theorem:
\[representability, spherical\]
[*(1)*]{} For any $\CM\in \hg_\kappa\mod^{G[[t]]}$ and $V\in
\Rep(\cG)$, the convolution $\CF_V\star \CM$ is acyclic away from cohomological degree $0$.
[*(2)*]{} The functor on $\hg_\kappa\mod$ that sends $\CN$ to the set of chiral parings $\{\Gamma(\Gr_{G,X},\CF_{V,X}),\CM\}\to \CN$ is representable by $\CF_V\star \CM$.
Our proof of [Theorem \[main, spherical\]]{} will be independent of [Theorem \[representability, spherical\]]{}. However, it is useful to keep [Theorem \[representability, spherical\]]{} in mind as it gives us an important insight into the connection between fusion and convolution.
\[sect pairing with sph\]
We will need the following result about the associativity property of the map given by .
For any two objects $V,W\in \Rep(\cG)$ there exists a canonical chiral pairing $$\{\Gamma(\Gr_{G,X},\CF_{V,X}),\Gamma(\Gr_{G,X},\CF_{W,X})\}\to
\Gamma(\Gr_{G,X},\CF_{V\otimes W,X}).$$ (Its construction will be recalled in the sequel.) Let $\CM$ be an object of $\CM\in
\hg_\kappa\mod^{G[[t]]}$, such that $\CF_{V'}\star \CM$ is acyclic away from cohomological degree $0$ for any $V'\in \Rep(\cG)$ (we choose not to rely here on [Theorem \[representability, spherical\]]{}(1), which says that the latter assumption is satisfied automatically).
Then there are three maps $$j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
\Gamma(\Gr_{G,X},\CF_{W,X})\boxtimes \CM\Bigr)\to
\Delta_!(\CF_{V\otimes W}\star \CM):$$
The first one is the composition $$\begin{aligned}
& j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
\Gamma(\Gr_{G,X},\CF_{W,X})\boxtimes \CM\Bigr)\to \\ &\to
\Delta_{x_2=x_3}{}_!\Bigl(j_*j^*(\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
(\CF_W\star \CM))\Bigr) \to \Delta_!(\CF_V\star (\CF_W\star
\CM))\simeq \Delta_!(\CF_{V\otimes W}\star \CM).\end{aligned}$$ The second map is the negative of a similar map with the roles of the first and the second factor swapped. The third map is the composition $$\begin{aligned}
&j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
\Gamma(\Gr_{G,X},\CF_{W,X})\boxtimes \CM\Bigr)\to \\ &\to
\Delta_{x_1=x_2}{}_!\Bigl(j_*j^*(\Gamma(\Gr_{G,X},\CF_{V\otimes W,X})
\boxtimes \CM)\Bigr)\to \Delta_!(\CF_{V\otimes W}\star \CM).\end{aligned}$$
\[comp with ten products\] The sum of the above three maps is $0$.
\[Jets n\]
By the construction of the chiral pairing , it is sufficient to consider the universal case, i.e., when $\CM\simeq
\fD_{G,\kappa}$ as a chiral $\CA_{\fg,\kappa}$-module under $\fl$.
Let us rewrite the map in more geometric terms. For any $n$, let $\Gr_{G,X^n}$ be the Beilinson-Drinfeld affine Grassmannian over $X^n$. I.e., this is the ind-scheme classifying the data of $$(x_1,...,x_n,\CP_G,\beta),$$ where $x_1,...,x_n$ is an $n$-tuple of points on $X$, $\CP_G$ is a principal $G$-bundle on $X$, and $\beta$ is a trivialization of $\CP_G$ on $X-\{x_1,...,x_n\}$.
For a decomposition $n=n_1+n_2$ there exists a natural map $${\bf 1}_{n_1,n_2}:X^{n_1}\times \Gr_{G,X^{n_2}}\to \Gr_{G,X^n},$$ which sends $((x_{1},...x_{n_1}), (x_{n_1+1},...,x_{n},\CP_G,\beta))
\mapsto (x_1,...,x_{n_1},x_{n_1+1},...,x_n,\CP_G,\beta)$.
Consider now the ind-scheme $\on{Jets}^\mer(G)_{X^n}$, fibered over $\Gr_{G,X^n}$, where in addition to the data $(x_1,...,x_n,\CP_G,\beta)$ we have that of a trivialization of $\CP_G$ on a formal neighborhood of $\underset{i}\cup\, x_i\subset X$. Let us denote by $\pi$ the corresponding projection. This ind-scheme also carries a flat connection along $X^n$.
The scheme $\on{Jets}^\mer(G)_{X^n}$ has the usual factorization pattern: $$\label{factorization}
j^*\Bigl(\on{Jets}^\mer(G)_{X^n}\Bigr)\simeq
j^*\Bigl(\on{Jets}^\mer(G)_X^{\times n}\Bigr) \text{ and }
\Delta^*\Bigl(\on{Jets}^\mer(G)_{X^n}\Bigr)\simeq
\on{Jets}^{\mer}(G)_X.$$
For two objects $V,W\in \Rep(\cG)$ we consider the D-module $j^*(\CF_{V,X}\boxtimes \CF_{W,X})$ on $\Gr_{G,X^2}|_{X^2-X}$. As is well-known, its Goresky-MacPherson extension $j_{!*}\left(j^*(\CF_{V,X}\boxtimes \CF_{W,X})\right)$ onto $\Gr_{G,X^2}$ has the property that $$\Delta^!(j_{!*}\left(j^*(\CF_{V,X}\boxtimes \CF_{W,X})\right))[1]\simeq
\CF_{V\otimes W,X}.$$
Consider the (right $\kappa$-twisted) D-module $$({\bf 1}_{1,2})_!\Bigl(\omega_X\boxtimes j_{!*}\left(j^*(\CF_{V,X}\boxtimes
\CF_{W,X})\right)\Bigr)$$ on $\Gr_{G,X^3}$. We have three maps
$$j_*j^*\Biggl(({\bf 1}_{1,2})_!\Bigl(j_{!*}\left(\omega_X\boxtimes
j^*(\CF_{V,X}\boxtimes \CF_{W,X})\right)\Bigr)\Biggr)\to
\Delta_!(\CF_{V\otimes W,X}),$$ corresponding to the three diagonals in $X^3$, whose sum is equal to $0$.
Applying the pull-back by means of $\pi$ to the two sides of the above formula to $\on{Jets}^\mer(G)_{X^3}$, followed by the direct image onto $X^3$, we obtain three maps $$\begin{aligned}
\label{triple map downstairs}
&j_*j^*\Bigl(\fD_{G,\kappa}\boxtimes
\Gamma(\on{Jets}^\mer(G)_X,\pi^*(\CF_{V,X}))\boxtimes
\Gamma(\on{Jets}^\mer(G)_X,\pi^*(\CF_{W,X}))\Bigr)\to \\ &\to
\Delta_!(\Gamma(\on{Jets}^\mer(G)_X,\pi^*(\CF_{V\otimes W})).\end{aligned}$$
Finally, note that we have a natural inclusion $$\begin{aligned}
&j_*j^*\Bigl(\fD_{G,\kappa}\boxtimes
\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
\Gamma(\Gr_{G,X},\CF_{W,X})\Bigr)\hookrightarrow \\
&j_*j^*\Bigl(\fD_{G,\kappa}\boxtimes
\Gamma(\on{Jets}^\mer(G)_X,\pi^*(\CF_{V,X}))\boxtimes
\Gamma(\on{Jets}^\mer(G)_X,\pi^*(\CF_{W,X}))\Bigr).\end{aligned}$$
By composing this inclusion with the three maps above we obtain three maps $$\label{same three maps}
j_*j^*\Bigl(\fD_{G,\kappa}\boxtimes
\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
\Gamma(\Gr_{G,X},\CF_{W,X})\Bigr)\to
\Delta_!\Bigl(\Gamma(\on{Jets}^\mer(G)_X,
\pi^*(\CF_{V\otimes W,X}))\Bigr).$$
The three maps of coincide with those that appear in the statement of [Proposition \[comp with ten products\]]{}.
Clearly, this implies the statement of the proposition. To prove the above lemma we will consider a slightly more general framework.
\[sect action as fusion\]
Let $\CF_1$ be a ($\kappa$-twisted, right) D-module on $G\ppart/K$, where $K\subset G[[t]]$ is a subgroup of finite codimension. Let $\CF'_1$ be the pull-back of $\CF_1$ to $G\ppart$.
Consider the ind-scheme $\Gr_{G,X;K}$, classifying the data $(x_1,\CP_G,\beta,\alpha)$, where $(x_1,x,\CP_G,\beta)$ is a point of $\Gr_{G,X\times x}$, and $\alpha$ is a reduction of $\CP_G|_{\D_x}$ to $K$. We have the isomorphisms $$\Gr_{G,X-x;K}\simeq \Gr_{G,X-x}\times G\ppart/K \text{ and }
\Gr_{G,x;K}\simeq G\ppart/K.$$ We also have a natural projection $\pi_K:\on{Jets}^\mer(G)_{X\times x}\to \Gr_{G,X;K}$, where $\on{Jets}^\mer(G)_{X\times x}$ denotes the preimage of $X\times
x\subset X^2$ in $\on{Jets}^\mer(G)_{X^2}$.
Consider the direct image of $\omega_X\boxtimes \CF_1$ under the tautological embedding ${\bf 1}_{1,1}:X\times G\ppart/K\to
\Gr_{G,X;K}$. It gives rise to a map of twisted D-modules on $\Gr_{G,X;K}$: $$\label{basic map down}
j_x{}_*j_x^*(\delta_{1,\Gr_{G,X}}\boxtimes \CF_1)\to i_x{}_!(\CF_1).$$
Lifting the two sides of by means of $\pi_K$ and taking the (quasi-coherent direct image onto $X\simeq X\times x$, we obtain a map $$\label{basic map up}
j_x{}_*j_x^*(\fD_{G,\kappa}) \otimes \Gamma(G\ppart,\CF'_1)\to
i_x{}_!\Bigl(\Gamma(G\ppart,\CF'_1)\Bigr).$$
The following assertion is built into the interpretation of twisted D-modules on $G(\ppart$ as chiral $\fD_{G,\kappa}$-modules:
\[action as fusion\] The map of coincides with the chiral action map for $\Gamma(G\ppart,\CF'_1)$ considered as a chiral $\fD_{G,\kappa}$-module supported at $x$.
In the appendix ([Sect. \[app A\]]{}) we shall give a proof of this result for the sake of completeness.
Assume now that the D-module $\CF_1$ on $G\ppart/K$ is $G[[t]]$-equivariant with respect to the left action of $G[[t]]$ on $G\ppart/K$. Let now $\CF_X$ be a ($\kappa$-twisted, right) D-module on $\Gr_{G,X}$. In this case we can form the convolution $\CF_X\underset{G[[t]]}\star \CF_1$, which will be an object of the derived category of ($\kappa$-twisted, right) D-modules on $\Gr_{G,X;K}$. We have: $$\CF_X\underset{G[[t]]}\star \CF_1|_{\Gr_{G,X-x;K}}\simeq
\CF_{X-x}\boxtimes \CF_1 \text{ and } \CF_x\underset{G[[t]]}\star
\CF_1|_{\Gr_{G,x;K}}\simeq \CF_{x}\underset{G[[t]]}\star \CF_1.$$
In particular, we obtain a map of D-modules on $\Gr_{G,X;K}$: $$\label{F map down}
j_x{}_*j_x^*\Bigl(\CF_{X-x}\boxtimes \CF_1\Bigr)\to
i_x{}_!\Bigl(h^0(\CF_{x}\underset{G[[t]]}\star \CF_1)\Bigr).$$ Pulling the two sides of under the map $\pi_K:\on{Jets}^\mer(G)_{X\times x}\to \Gr_{G,X;K}$, and taking the (quasi-coherent) direct image onto $X$, we obtain a map of D-modules on $X$: $$\begin{aligned}
&j_x{}_*j_x^*\Bigl(\Gamma(\on{Jets}^\mer(G)_X,\pi^*(\CF_X))\Bigr)
\otimes \Gamma(G\ppart,\CF'_1) \to
i_x{}_!\Bigl(\Gamma(G\ppart,h^0(\CF_x\star \CF'_1))\Bigr)\simeq \\
&\simeq i_x{}_!\Bigl(h^0\Bigl(\CF_x\star
\Gamma(G\ppart,\CF'_1)\Bigr)\Bigr).\end{aligned}$$
\[two pair\] The above map coincides with for $\CM=\Gamma(G\ppart,\CF'_1)$.
It is easy to see that this lemma, combined with a version of [Proposition \[action as fusion\]]{}, where the point $x$ varies on $X$, imply the required property of the three maps of .
(of [Lemma \[two pair\]]{})
Consider first the case when $K=G[[t]]$ and $\CF_1=\delta_{1,\Gr_G}$. Then we are dealing with a map $$j_x{}_*j_x^*\Bigl(\Gamma(\on{Jets}^\mer(G)_X,\pi^*(\CF_X))\Bigr)\otimes
\fD_{G,\kappa,x}\to i_x{}_!\Bigl(\Gamma(G\ppart,\pi^*(\CF_x))\Bigr),$$ and we claim that it coincides with the map, obtained by restriction to $X\times x$ from the map of . This follows from a version of [Proposition \[action as fusion\]]{} for $K=G[[t]]$, when instead of the fixed point $x$ we consider a morphism of D-modules on $X\times X$, and the map itself is described as a Cousin map corresponding to $({\bf 1}_{1,1})_!(\CF_X)$, viewed as D-module on $\Gr_{G,X^2}$.
To pass to the general case, consider the category of $\kappa$-twisted D-modules on $\on{Jets}^\mer(G)_{X,x}$ as acted on by the group ind-scheme $G\ppart$ on the right, which is of Harish-Chandra type with respect to the extension $\hg_{\kappa'}$ of $\fg\ppart$. Then the convolution action of $\CF_1$ on $$j_*{}_*j_x^*(\pi^*(\CF_{X-x})\boxtimes \pi^*(\delta_{1,\Gr_G}))
\text{ (resp., } i_x{}_!(\CF_x))$$ equals $$j_*{}_*j_x^*(\pi^*(\CF_{X-x})\boxtimes \CF'_1) \text{ and }
i_x{}_!(\CF_x\star \CF'_1),$$ respectively.
Consider also the category of D-modules on $X$, endowed with an action of $\hg_{\kappa'}$ as a category with a Harish-Chandra action of $G\ppart$. Then the convolution action of $\CF_1$ on a module of the type $j_x{}_*j_x^*(\CM)\otimes \fD_{G,\kappa,x}$ (resp., $i_x{}_!(\CN)$) (where $\CM$ is a D-module, and $\CN$ is a representation of $\hg_{\kappa'}$) equals $$j_x{}_*j_x^*(\CM)\otimes \Gamma(G\ppart,\CF'_1) \text{ and }
i_x{}_!\Bigl(\fC^\semiinf(\fg\ppart;\fg,\CN\otimes
\Gamma(G\ppart,\CF'_1))\Bigr),$$ respectively.
Note now that the (quasi-coherent) direct image is a functor between the above two categories that respects the $G\ppart$-actions. Hence, the assertion of the lemma follows from the case when $K=G[[t]]$ and $\CF_1=\delta_{1,\Gr_G}$ by the functoriality of the convolution.
Proof of [Theorem \[main, spherical\]]{} {#sect 3}
========================================
Having established some preliminary results in the previous section, we are now ready to prove [Theorem \[main, spherical\]]{}. The plan of the proof is as follows: we will first show that the chiral maps $\{\CM_1,\CM \} \to \CN$ in the special case when $\CM_1 =
\CV_{\fz_\fg}\underset{\fz_\fg}\otimes \CA_{\fg,\crit}$ are the same as ordinary homomorphisms of $\hg_\crit$-modules $\CM' \to \CN$ where $\CM'\simeq
\CV_{\wt\fZ_\fg^{\int,\reg}}\underset{\wt\fZ_\fg^{\int,\reg}}\otimes
\CM$. Roughly speaking, this means that we can “swap”, under the chiral product, the tensor product with $\CV_{\fz_\fg}$ from $\CA_{\fg,\crit}$ to $\CM$. This is the content of [Theorem \[repr of pair from ten product\]]{}.
Once we have that, we obtain, for each $V \in \Rep(\cG)$, a map in one direction in , provided that $\CF_V \star \CM$ is acyclic away from cohomological dimension $0$. But we do know this for $\CM = \fD_{G,\crit,x}$, according to [Theorem \[main, spherical, diff\]]{}. Moreover, according to [Sect. \[thm for diff op\]]{}, it is sufficient to prove the statement of [Theorem \[main, spherical\]]{} just in this special case. Hence all we need is to check that the maps we have constructed in this case are indeed isomorphisms and that they satisfy the properties listed in [Theorem \[main, spherical\]]{}. We first check that the maps in question do satisfy the required properties; then we show that this already implies that the above maps are necessarily isomorphisms, thus completing the proof.
\[crit fusion\]
In this section we fix $\kappa=\kappa_\crit$, and study the map of in this case. We will use the following crucial result of [@BD], Sects. 5.5-5.6: there is a canonical isomorphism of chiral $\CA_{\fg,\crit}$-modules, $$\label{BD isom}
\Gamma(\Gr_{G,X},\CF_{V,X})\simeq
\CV_{\fz_\fg}\underset{\fz_\fg}\otimes \CA_{\fg,\crit},$$ where the tensor product makes sense since $\fz_\fg$ is the center of $\CA_{\fg,\crit}$.
Given a chiral $\CA_{\fg,\crit}$-module $\CM$, let us study the functor on the category of chiral $\CA_{\fg,\crit}$-modules that assigns to a chiral $\CA_{\fg,\crit}$-module $\CN$ the set of chiral pairings $$\label{pairings from ten product}
\{\CV_{\fz_\fg}\underset{\fz_\fg}\otimes \CA_{\fg,\crit},\CM\}\to \CN.$$
\[repr of pair from ten product\] Let $\CM$ be concentrated at $x\in X$, and assume that, when viewed as a module over $\fZ_\fg$, it is supported on $\wt\fZ^{\int,\reg}_{\fg}$. Then chiral pairings for $\CN\in \CA_{\fg,\crit}\mod_x$ are in bijection with maps of $\hg_\crit$-modules $$\Bigl(\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
\CM\Bigr)_{N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}}\to \CN.$$
In the statement of the theorem the subscript $N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}$ stands for the coinvariants with respect to the action of this nilpotent operator. Before giving a proof of this theorem, which will occupy the next few subsections, let us note that [Theorem \[repr of pair from ten product\]]{} implies the existence of the map in one direction in [Theorem \[main, spherical\]]{}:
Indeed, the chiral pairing gives rise to a map $$\label{desired map almost}
\Bigl(\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
\CM\Bigr)_{N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}} \to h^0(\CF_V\star
\CM).$$
Note that if we assume [Theorem \[representability, spherical\]]{}, then we obtain that the map of is an isomorphism (and moreover, that $h^0(\CF_V\star \CM) = \CF_V\star \CM$). However, this would still not be enough to prove [Theorem \[main, spherical\]]{}: we would also need to show that ${N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}}$ acts trivially on the left-hand side, which requires a separate argument. The proof of [Theorem \[main, spherical\]]{} that we give below will be independent of [Theorem \[representability, spherical\]]{}.
To prove [Theorem \[repr of pair from ten product\]]{} we need to study chiral pairings between modules over commutative chiral algebras.
Recall the set-up of the proof of [Proposition-Construction \[extension of bundle over center\]]{}, i.e., let $\CB$ be a commutative chiral algebra on $X$, and let $\CV_{X-x}$ be a \*-commutative module over $\CB|_{X-x}$, extended to the entire $X$ such that the connection has a pole of order $\leq 1$ and the residue $\Res(\nabla,\CV_X)\in \End(\CV_x)$ is nilpotent.
\[commutative fusion\] Let $\CM$ be a $\CB_x$-module, viewed as \*-commutative $\CB$-module, supported at $x$. Let $\CN$ be another chiral $\CB$-module, supported at $x$. Then chiral pairings $$\label{com pair one}
\{\CV_{X-x},\CM\}\to \CN$$ are in bijection with maps of $\hat\CB_x$-modules $$\label{com pair two}
\Bigl(\CV_x\underset{\CB_x}\otimes \CM\Bigr)_{\Res(\nabla,\CV_X)}\to
\CN,$$ where $\hat\CB_x$ denotes the topological commutative algebra, corresponding to $\CB$ at $x$.
First, from the definition of chiral pairings it is easy to see that any pairing $\{\CV_{X-x},\CM\}\to \CN$ factors through $\CN'\subset
\CN$, where $\CN'$ is the maximal submodule of $\CN$ on which $\hat\CB_x$ acts via $\hat\CB_x\twoheadrightarrow \CB_x$. In other words, we can assume that $\CN$ is also \*-commutative.
Secondly, for a fixed $\CN$, both and , regarded as contravariant functors with respect to $\CM$ are both right exact and commute with direct limits. Hence, we can replace $\CM$ by $\CB_x$.
Let us denote by $B$ the fiber $\CB_x$ and by $T_0$ the module over $B[[t]]$, corresponding to $\CV_X$. Let $T$ be the localization of $T_0$ with respect to $t$.
Then chiral pairings $\{\CV_{X-x},\CB_x\}\to \CN$ are in bijection with $B[[t]]$-linear maps $$\label{all poles}
\phi:T \to \CN\underset{B}\otimes B\ppart dt/B[[t]]dt,$$ that respect that action of $\partial_t$.
Given such $\phi$, the finite rank assumption on $T_0$ implies that for some integer $k$ the composition $$t^k\cdot T_0\hookrightarrow T\overset{\phi}\to
\CN\underset{B}\otimes B\ppart/B[[t]]$$ is zero. However, the nilpotency assumption on $t\partial_t$ acting on $T_0/t\cdot T_0$ implies that for any $k>0$ the map $$\partial_t:\Bigl(t^k\cdot T_0\Bigr)\to \Bigl(t^{k-1}\cdot
T_0\Bigr)$$ is surjective.
Therefore, $\phi$ in fact annihilates $T_0$. Restricting $\phi$ to $t^{-1}\cdot T_0$ we obtain a map $$\label{pole one}
t^{-1}\cdot T_0/T_0\to (t^{-1}\BC[[t]]/\BC[[t]])\otimes \CN,$$ which is zero on the image of $\partial_t(T_0)$. Hence, we obtain a map $$\on{coker}\Bigl(t\partial_t:T_0/t\cdot T_0\to T_0/t\cdot
T_0\Bigr)\to \CN,$$ as desired.
Conversely, starting from a map as in, it is easy to see that it uniquely extends to a map as in .
Let us denote by $can_{\CV}$ the resulting canonical map $$j_*j^*(\CV_{X-x}\boxtimes \CM)\to
\Delta_!(\CV_x\underset{\CB_x}\otimes \CM)_{\Res(\nabla,\CV_X)}.$$ For the proof of [Theorem \[main, spherical\]]{}, we will need the following additional property of this map.
Let $(\CV_{X-x},\CV_X)$ and $(\CW_{X-x},\CW_X)$ be two pairs of $\CB$-modules, satisfying the assumptions of [Proposition \[commutative fusion\]]{}. Note that the tensor product $(\CV_{X-x}\underset{\CB_{X-x}}\otimes \CW_{X-x},
\CV_X\underset{\CB}\otimes \CW_X)$ also has the same property.
We have three maps $$j_*j^*(\CV_{X-x}\boxtimes \CW_{X-x}\boxtimes \CM)\to
\Delta_!\Bigl(\CV_x\underset{\CB_x}\otimes \CW_x
\underset{\CB_x}\otimes\CM
\Bigr)_{\Res(\nabla,\CV_X),\Res(\nabla,\CW_X)},$$ where the subscript refers to the fact that we are taking coinvariants of the above two endomorphisms acting on $\CV_x$ and $\CW_x$.
The first map is the composition $$\begin{aligned}
&j_*j^*(\CV_{X-x}\boxtimes \CW_{X-x}\boxtimes \CM)
\overset{\on{id}_{\CV_{X-x}}\boxtimes can_{\CW}}\longrightarrow
\Delta_{x_2=x_3}{}_!\Bigl(j_*j^*(\CV_{X-x}\boxtimes
(\CW_x\underset{\CB_x}\otimes\CM)_{\Res(\nabla,\CW_X)})\Bigr)
\overset{can_{\CV}}\longrightarrow \\ &\to
\Delta_!\Bigl(\CV_x\underset{\CB_x}\otimes
\CW_x\underset{\CB_x}\otimes \CM\Bigr)
_{\Res(\nabla,\CV_X),\Res(\nabla,\CW_X)}.\end{aligned}$$ The second map is obtained by interchanging the roles of $\CV$ and $\CW$. The third map if the composition $$\begin{aligned}
&j_*j^*(\CV_{X-x}\boxtimes \CW_{X-x}\boxtimes \CM_x) \to
\Delta_{x_1=x_2}{}_!\Bigl(j_*j^*((\CV_{X-x}\underset{\CB_{X-x}}\otimes
\CW_{X-x}) \boxtimes \CM_x)\Bigr)\overset{can_{\CV\otimes
\CW}}\longrightarrow \\ &\to
\Delta_!\Bigl(\CV_x\underset{\CB_x}\otimes
\CW_x\underset{\CB_x}\otimes \CM
\Bigr)_{\Res(\nabla,\CV_X)+\Res(\nabla,\CW_X)}\to
\Delta_!\Bigl(\CV_x\underset{\CB_x}\otimes
\CW_x\underset{\CB_x}\otimes \CM
\Bigr)_{\Res(\nabla,\CV_X),\Res(\nabla,\CW_X)}.\end{aligned}$$
\[add assoc\] The sum of the three maps above is equal to zero.
The proof follows by unfolding the construction of the bijection in the proof of [Proposition \[commutative fusion\]]{}.
Consider the following general set-up. Let $\CA$ be a chiral algebra, and let $\fz$ be its center. Let $\CV$ be a \*-commutative chiral module over $\fz$, free of finite rank as a $\fz$-module. Let $\CM,\CN$ be two chiral $\CA$-modules.
Given a chiral pairing $\{\CV\underset{\fz}\otimes \CA,\CM\}\to
\CN$ we can restrict it and obtain a chiral pairing $$j_*j^*(\CV\boxtimes \CM)\to \Delta_!(\CN)$$ of chiral $\fz$-modules. This map has the following commutation property with respect to $\CA$: the map $$\wt{j}_*\wt{j}{}^*(\CV\boxtimes \CA\boxtimes \CM)\to
\Delta_{x_2=x_3}{}_! \Bigl(j_*j^*(\CV\boxtimes \CM)\Bigr)\to
\Delta_!(\CN)$$ equals $$\wt{j}_*\wt{j}{}^*(\CV\boxtimes \CA\boxtimes \CM)\to
\Delta_{x_1=x_3}{}_! \Bigl(j_*j^*(\CA\boxtimes \CN)\Bigr)\to
\Delta_!(\CN),$$ where $\wt{j}$ denotes the embedding of the complement of the union of the divisors $x_1=x_3$ and $x_2=x_3$.
We have the following general assertion:
\[pairings over center\] The set of $\CA$-chiral pairings $\{\CV\underset{\fz}\otimes
\CA,\CM\}\to \CN$ is in a bijection with the set of $\fz$-chiral pairings $\{\CV,\CM\}\to \CN$, satisfying the additional condition above.
Together with [Proposition \[commutative fusion\]]{}, this lemma implies [Theorem \[repr of pair from ten product\]]{}.
According to [Sect. \[thm for diff op\]]{}, in order to prove [Theorem \[main, spherical\]]{}, it suffices to consider the universal case, namely, the one when $\CM=\fD_{G,\crit,x}$. We will work in a more general framework, assuming only that $\CM$ is such that $\CF_{V'}\star \CM$ is acyclic away from cohomological degree $0$ for any $V'\in \Rep(\cG)$, which is satisfied in the above case by [Theorem \[main, spherical, diff\]]{}. (Of course, [Theorem \[main, spherical\]]{} will imply that this assumption is satisfied for any $\CM$.)
Composing the map with the tautological projection $$\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes \CM \twoheadrightarrow
\Bigl(\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
\CM\Bigr)_{N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}},$$ we obtain a map $$\label{desired map}
\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes \CM \to \CF_V\star \CM.$$
We will denote this map by $\fs_V^{-1}$, as it will be the inverse of the desired map of [Theorem \[main, spherical\]]{}. We will deduce [Theorem \[main, spherical\]]{} from the following:
\[map is assoc\] The map of is compatible with tensor products of representations, in the sense that for $V,W\in \Rep(\cG)$ the 3 compositions $$\label{compo 1}
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes
\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM\overset{\fs^{-1}_V}\longrightarrow \CF_V\star
\Bigl(\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM\Bigr) \overset{\on{id}_{\CF_V}\star
\fs^{-1}_W}\longrightarrow \CF_V\star (\CF_W\star \CM),$$ $$\label{compo 2}
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes
\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM \overset{\on{id}_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}\otimes
\fs^{-1}_W}\longrightarrow
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \Bigl(\CF_W\star \CM\Bigr) \overset{\fs^{-1}_V}\longrightarrow
\CF_V\star (\CF_W\star \CM)$$ and $$\label{compo 3}
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes
\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM \overset{\fs^{-1}_{V\otimes W}}\to \CF_{V\otimes W}\star
\CM\simeq \CF_V\star (\CF_W\star \CM)$$ coincide.
Let us finish the proof of [Theorem \[main, spherical\]]{} modulo this proposition. Without restriction of generality, we can assume that the representation $V$ is finite-dimensional.
Suppose the map of is not injective, and let $\CN$ be its kernel. Let $V^*$ be representation dual to $V$. Consider the diagram $$\CD \CM & @>{\on{id}}>> & \CM \\ @AAA & & @AAA \\
\CV^*_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM @>{\on{id}_{\CV^*}\otimes \fs^{-1}_V}>>
\CV^*_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes
(\CF_V\star \CM) @>{\fs^{-1}_{V^*}}>> \CF_{V^*}\star (\CF_V\star \CM)
\\ @AAA \\
\CV^*_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes
\CN. \endCD$$ It is commutative since the maps and of [Proposition \[map is assoc\]]{} coincide. Hence, we obtain that the composed map $$\CV^*_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CN\to \CM$$ is zero. But this is a contradiction since this map is adjoint to the tautological embedding $\CN\to
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM$.
In particular, we obtain that $\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM \to
\Bigl(\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM\Bigr)_ {N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}}$ is injective, implying that the action of $N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}$ on $\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM$ is trivial.
Suppose now that the map of is not surjective, and let $\CN'$ be its kernel. Let us recall that the functors $\CF_V\star
?$ and $\CF_{V^*}\star ?$ are mutually (both left and right) adjoint on the category $D(\hg_\crit\mod)^{G[[t]]}$. Consider the diagram $$\CD & & & & \CF_{V^*}\star \CN' \\ & & & & @AAA \\
\CV^*_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM @>{\fs^{-1}_{V^*}}>> \CF_{V^*}\star
(\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM) @>{\on{id}_{\CF_{V^*}}\star \fs^{-1}_V}>> \CF_{V^*}\star
(\CF_V\star \CM) \\ @AAA & & @AAA \\ \CM & @>{\on{id}}>> & \CM.
\endCD$$ This diagram is commutative because the maps and of [Proposition \[map is assoc\]]{} coincide. Hence, we obtain that the composed map $\CM\to \CF_{V^*}\star \CN'$ is zero, which is a contradiction.
Consider the diagram $$\CD j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
(\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM)\Bigr) @>{\fs^{-1}_W}>>
j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes (\CF_W\star
\CM)\Bigr) \\ @V{can_V}VV @V{can_V}VV \\
\Delta_!\Bigl(\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
(\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM)\Bigr)_{N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}}
@>{\fs^{-1}_W}>> \Delta_!
\Bigl(\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes (\CF_W\star \CM)\Bigr)_{N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}}
\\ @V{\fs^{-1}_V}VV @V{\fs^{-1}_V}VV \\
\Delta_!\Bigl(\CF_V\star(\CW_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes \CM)\Bigr)
@>{\fs^{-1}_W}>> \Delta_!(\CF_V\star (\CF_W\star \CM)). \endCD$$
To prove that and coincide we have to show the commutativity of the lower square in this diagram.
Note that the upper square is commutative by functoriality of the map $can_V$; moreover the vertical maps in this square are surjective, by [Proposition \[commutative fusion\]]{}. Hence, it suffices to see that the outer square is commutative, but this follows from the functoriality of the map .
Consider now the composed map $$\CD j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
\Gamma(\Gr_{G,X},\CF_{W,X})\boxtimes \CM\Bigr) \\ @VVV \\
\Delta_{x_2=x_3}{}_!\Bigl(j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})
\boxtimes
(\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM)_{N_{\CW_{\wt\fZ^{\int,\reg}_{\fg}}}} \Bigr)\Bigr) \\
@VVV \\ \Delta_!\Bigl(\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM\Bigr)
_{N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}},N_{\CW_{\wt\fZ^{\int,\reg}_{\fg}}}}
\\ @V{\text{\eqref{compo 1}}}VV \\ \Delta_!(\CF_V\star \CF_W\star
\CM), \endCD$$ and a similar composition when the roles of $V$ and $W$ are interchanged. The resulting maps are equal to the first and the second maps, respectively, of [Proposition \[comp with ten products\]]{}.
Hence, their sum equals the composed map from the commutative diagram $$\CD j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
\Gamma(\Gr_{G,X},\CF_{W,X})\boxtimes \CM\Bigr) \\ @VVV \\
\Delta_{x_1=x_2}{}_!\Bigl(j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V\otimes
W,X})\boxtimes \CM \Bigr)\Bigr) @>>> \Delta_!(\CF_{V\otimes W}\star
\CM) \\ @VVV @V{\sim}VV \\ \Delta_!\Bigl(
\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
\CM\Bigr)_
{N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}+N_{\CW_{\wt\fZ^{\int,\reg}_{\fg}}}}
@>{\text{\eqref{compo 3}}}>> \Delta_!(\CF_V\star \CF_W\star \CM),
\endCD$$ by [Proposition \[comp with ten products\]]{}. Moreover, by [Lemma \[add assoc\]]{}, the sum of the three composed maps $$\begin{aligned}
&j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
\Gamma(\Gr_{G,X},\CF_{W,X})\boxtimes \CM\Bigr) \to \\ &\to
\Delta_!\Bigl(\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM\Bigr)
_{N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}},N_{\CW_{\wt\fZ^{\int,\reg}_{\fg}}}}\end{aligned}$$ is also zero.
Let $\CM_{V,W}\subset j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
\Gamma(\Gr_{G,X},\CF_{W,X})\boxtimes \CM\Bigr)$ be the kernel of the map to $$\Delta_{x_1=x_3}{}_!\Bigl(j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{W,X})
\boxtimes
(\CV_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM)_ {N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}}}\Bigr)\Bigr).$$
Then the map $$\CM_{V,W}\to \Delta_{x_2=x_3}{}_!
\Bigl(j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\boxtimes
(\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM)_ {N_{\CW_{\wt\fZ^{\int,\reg}_{\fg}}}}\Bigr)\Bigr)$$ is still surjective. Hence, the two surviving maps $$\CM_{V,W}\to \Delta_!\Bigl(\CV_{\wt\fZ^{\int,\reg}_{\fg}}
\underset{\wt\fZ^{\int,\reg}_{\fg}}\otimes
\CW_{\wt\fZ^{\int,\reg}_{\fg}}\underset{\wt\fZ^{\int,\reg}_{\fg}}
\otimes \CM\Bigr) _{N_{\CV_{\wt\fZ^{\int,\reg}_{\fg}}},
N_{\CW_{\wt\fZ^{\int,\reg}_{\fg}}}}$$ coincide and are surjective. By the above, the two surviving maps $$\CM_{V,W}\to \Delta_!(\CF_V\star \CF_W\star \CM)$$ coincide as well, and the equality of the maps and follows.
Convolution and fusion for general chiral algebras {#proof of rep}
==================================================
The goal of this section is to prove [Theorem \[representability, spherical\]]{}. In fact, we will prove a more general result, valid for any chiral algebra, endowed with a Harish-Chandra action of $\on{Jets}(G)_X$. We will make a more extensive use of the formalism developed in the appendix to [@FG2], but we should note that the main results of this paper, [Theorem \[main, spherical\]]{} and [Theorem \[main\]]{}, are independent of this section.
\[general alg\]
Recall the Lie-\* algebras $L_\fg$ and $L_{\fg,\kappa}$ introduced in [Sect. \[recol\]]{}. Let $\on{Jets}(G)_X$ denote the group-like object in the category of D-schemes on $X$ corresponding to jets into $G$. Its relative cotangent sheaf is the D-module on $X$ canonically isomorphic to the dual $L_\fg^\vee$ of $L_\fg$.
Let $\CA$ be a chiral algebra on $X$, endowed with an action of $\on{Jets}(G)_X$, such that the adjoint action of its Lie algebra is inner at the level $\kappa$. In other words, we assume being given a homomorphism of Lie-\* algebras $L_{\fg,\kappa}\to \CA$, such that the map $$\CA\to \CA\otimes L_\fg^\vee,$$ corresponding to the Lie-\* bracket equals the derivative of the action of $\on{Jets}(G)$ on $\CA$.
\[group act on cat\] The category of chiral $\CA$-modules, supported at $x\in X$ carries an action of the group-scheme $G\ppart$ of Harish-Chandra type with respect to the central extension $\hg_\kappa$ (cf. [@FG2], Sect. 22).
We will give two proofs. One, discussed below, is local in terms of the De Rham cohomology of $\CA$ on the formal punctured disc $\D_x^\times$ around $x$. Another proof will be of chiral nature.
Let us first recall the following general construction, cf. [@CHA], 3.6.9. For a chiral algebra $\CA$ let us consider the topological vector space $H_{DR}^0(\D_x^\times,\CA)$. We remind that for a D-module $\CV$ on $X$, we define $$H_{DR}^0(\D_x^\times,\CV):=
\underset{\CL}{\underset{\longrightarrow}{\lim}}\,
H_{DR}^0(\D_x^\times,\CL),$$ where $\CL$ runs over the filtered set of finitely-generated D-submodules of $\CV$, and the direct limit is taken in the category of topological vector spaces. [^3] Consider in addition the topological vector space $H_{DR}^0(\D_x^\times\times \D_x^\times-\Delta_{\D_x^\times},
\CA\boxtimes \CA)$, which is acted on by the transposition $\sigma$. We have a canonical map $$H_{DR}^0(\D_x^\times\times
\D_x^\times-\Delta_{\D_x^\times},\CA\boxtimes \CA)\to
H_{DR}^0(\D_x^\times,\CA)\arrowtimes H_{DR}^0(\D_x^\times,\CA),$$ defined in fact for any pair of D-modules on $X$, cf. [@CHA] 3.6.9.
In addition, the structure of chiral algebra defines a map $$\{\cdot,\cdot\}:H_{DR}^0(\D_x^\times\times
\D_x^\times-\Delta_{\D_x^\times},\CA\boxtimes \CA)\to
H_{DR}^0(\D_x^\times,\CA).$$
The category $\CA\mod_x$ of chiral $\CA$-modules supported at $x$ identifies with the category of vector spaces $\CM$, endowed with an action map $$\on{act}_\CM:H_{DR}^0(\D_x^\times,\CA)\arrowtimes \CM\to \CM,$$ such that the difference of $$\begin{aligned}
&H_{DR}^0(\D_x^\times\times
\D_x^\times-\Delta_{\D_x^\times},\CA\boxtimes \CA)\arrowtimes \CM\to
H_{DR}^0(\D_x^\times,\CA)\arrowtimes
H_{DR}^0(\D_x^\times,\CA)\arrowtimes \CM
\overset{\on{act}_\CM}\longrightarrow \\ &\to
H_{DR}^0(\D_x^\times,\CA)\arrowtimes \CM
\overset{\on{act}_\CM}\longrightarrow \CM\end{aligned}$$ and the map obtained by first acting by $\sigma$ on the first factor, equals $$H_{DR}^0(\D_x^\times\times
\D_x^\times-\Delta_{\D_x^\times},\CA\boxtimes \CA)\arrowtimes \CM
\overset{\{\cdot,\cdot\}\otimes \on{id}}\longrightarrow
H_{DR}^0(\D_x^\times,\CA)\arrowtimes \CM
\overset{\on{act}_\CM}\longrightarrow \CM.$$
Let $\bg$ be an $S$-point of $G\ppart$ for some base-scheme $S$. It gives rise to a map $$\CO_{\on{Jets}(G)_X}\to \CO_S\ppart,$$ compatible with the connection. Composing it with the map $\CA\to
\CO_{\on{Jets}(G)_X}\underset{\CO_X}\otimes \CA$, given by the action of $\on{Jets}(G)_X$ on $\CA$, we obtain a map $$H_{DR}^0(\D_x^\times,\CA) \to \CO_S\shriektimes
H_{DR}^0(\D_x^\times,\CA).$$ Similarly, we have a map $$H_{DR}^0(\D_x^\times\times
\D_x^\times-\Delta_{\D_x^\times},\CA\boxtimes \CA)\to
\CO_S\shriektimes H_{DR}^0(\D_x^\times\times
\D_x^\times-\Delta_{\D_x^\times},\CA\boxtimes \CA).$$
Given an object $\CM\in \CA\mod_x$, we define an action map $$H_{DR}^0(\D_x^\times,\CA) \arrowtimes (\CM\otimes \CO_S)\to
\CM\otimes \CO_S$$ as a composition $$\begin{aligned}
&H_{DR}^0(\D_x^\times,\CA) \arrowtimes (\CM\otimes \CO_S)\to
\Bigl(\CO_S\shriektimes H_{DR}^0(\D_x^\times,\CA)\Bigr)\arrowtimes
(\CM\otimes \CO_S)\to \\ &\to \CO_S\shriektimes
\Bigl(H_{DR}^0(\D_x^\times,\CA)\arrowtimes \CM)\shriektimes
\CO_S\overset{\on{act}_\CM}\to \CO_S\otimes \CM\otimes \CO_S\to
\CM\otimes \CO_S,\end{aligned}$$ where the last arrow is given by the multiplication on $\CO_S$.
By construction, it follows that the relation, that singles out representations among all vector spaces endowed with an action of $H_{DR}^0(\D_x^\times,\CA)$ (cf. above), holds. Thus, we obtain a $G\ppart$-action on $\CA\mod_x$, which is of Harish-Chandra type by construction.
Let now $\CM'$ be a torsion-free chiral $\CA$-module on $X$. Assume that $\CM'$ is weakly $\on{Jets}(G)_X$-equivariant. I.e., we have an action of $\on{Jets}(G)_X$ on $\CM'$, compatible with its action on $\CA$ in the natural sense.
Let $\CM$, $\CN$ be two chiral $\CA$-modules, both supported at the point $x\in X$. Let $\bg$ be an $S$-point of $G\ppart$ and let $\CM^{\bg}$ and $\CN^{\bg}$ the corresponding $S$-families of objects of $\CA\mod_x$, defined by [Proposition-Construction \[group act on cat\]]{}.
\[twisting pairing\] To every chiral pairing $$\label{initial pairing}
\phi:j_*j^*(\CM'\boxtimes \CM)\to \Delta_!(\CN)$$ there functorially corresponds a chiral pairing $$\phi:j_*j^*(\CM'\boxtimes \CM^\bg)\to \Delta_!(\CN^\bg).$$
The proof is largely parallel to that of [Proposition-Construction \[group act on cat\]]{} above.
First, let $\CM'$ be any torsion-free module over a chiral algebra $\CA$. Consider the topological vector spaces $H^0_{DR}(\D_x^\times,\CM')$ and $H_{DR}^0(\D_x^\times\times
\D_x^\times-\Delta_{\D_x^\times},\CA\boxtimes \CM')$; the action of $\CA$ on $\CM'$ gives rise to a map $$\on{act}_{\CM'}:
H_{DR}^0(\D_x^\times\times \D_x^\times-\Delta_{\D_x^\times},
\CA\boxtimes \CM')\to H^0_{DR}(\D_x^\times,\CM').$$
For two objects $\CM,\CN\in \CA\mod_x$, chiral pairings $\{\CM',\CM\}\to \CN$ are in bijection with maps $$\phi:H^0_{DR}(\D_x^\times,\CM')\arrowtimes \CM\to \CN,$$ such that two difference of the two compositions $$\begin{aligned}
&H_{DR}^0(\D_x^\times\times
\D_x^\times-\Delta_{\D_x^\times},\CA\boxtimes \CM')\arrowtimes \CM \to
H^0_{DR}(\D_x^\times,\CA)\arrowtimes
H^0_{DR}(\D_x^\times,\CM')\arrowtimes \CM \overset{\on{id}\CA\otimes
\phi} \longrightarrow \\ &\to H^0_{DR}(\D_x^\times,\CA)\arrowtimes \CN
\overset{\on{act}_\CN}\longrightarrow \CN\end{aligned}$$ and $$\begin{aligned}
&H_{DR}^0(\D_x^\times\times
\D_x^\times-\Delta_{\D_x^\times},\CA\boxtimes \CM')\arrowtimes \CM \to
H^0_{DR}(\D_x^\times,\CM') \arrowtimes
H^0_{DR}(\D_x^\times,\CA)\arrowtimes \CM
\overset{\on{id}_{\CM'}\otimes \on{act}_\CM}\longrightarrow \\ &\to
H^0_{DR}(\D_x^\times,\CM') \arrowtimes \CM \overset{\phi}\to \CN\end{aligned}$$ equals $$H_{DR}^0(\D_x^\times\times
\D_x^\times-\Delta_{\D_x^\times},\CA\boxtimes \CM')\arrowtimes \CM
\overset{\on{act}_{\CM'}}\longrightarrow
H^0_{DR}(\D_x^\times,\CM')\arrowtimes \CM\overset{\phi}\to \CN.$$
Suppose now that $\CM'$ is weakly $\on{Jets}(G)_X$-equivariant. Given an $S$-point $\bg$ of $G\ppart$ as above, as in the case of $\CA$, we obtain a map $$H^0_{DR}(\D_x^\times,\CM')\to \CO_S\shriektimes
H^0_{DR}(\D_x^\times,\CM').$$
For a chiral pairing $\phi$ as above, we define a chiral pairing $$H^0_{DR}(\D_x^\times,\CM')\arrowtimes (\CM\otimes \CO_S)\to
(\CN\otimes \CO_S)$$ as a composition $$\begin{aligned}
&H^0_{DR}(\D_x^\times,\CM')\arrowtimes (\CM\otimes \CO_S)\to
\Bigl(\CO_S\shriektimes H^0_{DR}(\D_x^\times,\CM')\Bigr) \arrowtimes
(\CM\otimes \CO_S) \to \\ &\to\CO_S\shriektimes
\Bigl(H^0_{DR}(\D_x^\times,\CM')\arrowtimes \CM\Bigr) \shriektimes
\CO_S\overset{\on{id}\otimes \phi\otimes \on{id}}\longrightarrow
\CO_S\otimes \CN\otimes \CO_S\to \CN\otimes \CO_S.\end{aligned}$$
It is easy to see that the relation involving the $\CA$-actions on $\CM$ and $\CN$, described above, holds.
Let us assume now that in the circumstances of the above proposition, module $\CM'$ is strongly $\on{Jets}(G)_X$-equivariant. By definition, this means that the derivative of the group-action, which is a map $$\CM'\to \CM'\otimes L_\fg^\vee,$$ coincides with the one coming from the Lie-\* action of $L_{\fg,\kappa}$ via $L_{\fg,\kappa}\to \CA$.
Let $\bg_1$ and $\bg_2$ be two $S$-points of $G\ppart$, whose ratio is a map from $S$ to the first infinitesimal neighborhood of the identity in $G\ppart$. In particular, for every choice of the splitting $\fg\ppart\to \hg_\kappa$ we have the canonical isomorphisms $\CM^{\bg_1}\simeq \CM^{\bg_2}$ and $\CN^{\bg_1}\simeq \CN^{\bg_2}$.
From the proof of [Proposition-Construction \[twisting pairing\]]{} we obtain:
\[strong twist\] Under the above circumstances, the diagram $$\CD
j_*j^*(\CM'\boxtimes \CM^{\bg_1})@>>> \Delta_!(\CN^{\bg_1}) \\
@V{\sim}VV @V{\sim}VV \\
j_*j^*(\CM'\boxtimes \CM^{\bg_2})@>>> \Delta_!(\CN^{\bg_2})
\endCD$$ is commutative.
Let now $\CF'$ be a $\kappa$-twisted D-module on $G\ppart$, which strongly $K$-equivariant on the right, where $K$ is an “open-compact” group-subscheme of $G\ppart$. By [@FG2], Sect. 22.4, given an object $\CM\in \CA\mod_x$, which is strongly $K$-equivariant, there exists a well-defined complex of objects in $\CA\mod_x$: $$\fC^\semiinf\Bigl(\fg\ppart;K_{red},\CF'\otimes \CM\Bigr),$$ where $K_{red}$ denotes the reductive quotient of $K$.
The image of this complex in the derived category is by definition the convolution $\CF\star \CM$, where $\CF$ is the twisted D-module on $G/K$, corresponding to $\CF'$.
From [Proposition-Construction \[twisting pairing\]]{} above, we obtain that given two strongly $K$-equivariant objects $\CM,\CN\in \CA\mod_x$ and a chiral pairing $\{\CM',\CM\}\to \CN$ we obtain a chiral pairing of graded objects $$\label{conv pairing}
\{\CM',\fC^\semiinf\Bigl(\fg\ppart;K_{red},\CF'\otimes \CM\Bigr)\}\to
\fC^\semiinf\Bigl(\fg\ppart;K_{red},\CF'\otimes \CN\Bigr).$$ Moreover, by [Corollary \[strong twist\]]{}, the above pairing is a map of complexes, i.e., it respects the differentials on both sides.
As an application we shall now establish the following result. Let $\on{Av}_{G[[t]]}$ denote the functor $D^+(\CA\mod_x)\to
D^+(\CA\mod_x)^{G[[t]]}$, right adjoint to the forgetful functor. In particular, it is left-exact and the functor $$\CN\to h^0\Bigl(\on{Av}_{G[[t]]}(\CN)\Bigr)$$ is the right adjoint to the forgetful functor $\CA\mod_x^{G[[t]]}\to \CA\mod_x$.
\[equivariance preserved\] Let $\{\CM',\CM\}\to \CN$ be a chiral pairing with $\CM'$ being strongly $\on{Jets}(G)_X$-equivariant, and $\CM\in \CA\mod_x$ being $G[[t]]$-equivariant. Then this pairing canonically factors through $\{\CM',\CM\}\to h^0\Bigl(\on{Av}_{G[[t]]}(\CN)\Bigr)$.
Let us recall that the functor $\on{Av}_{G[[t]]}$ is represented by the complex $$\CN\mapsto \fC^\bullet\Bigl(\fg[[t]],\CO_{G[[t]]}\otimes
\CN\Bigr).$$ Hence, as in , given a chiral pairing $\{\CM',\CM\}\to \CN$, we obtain a chiral pairing of complexes $$\{\CM',\on{Av}_{G[[t]]}(\CM)\}\to \on{Av}_{G[[t]]}(\CN),$$ compatible with the differential. Composing with the canonical map $\CM\to \on{Av}_{G[[t]]}(\CM)$, we obtain the desired chiral pairing $$\{\CM',\CM\}\to \on{Av}_{G[[t]]}(\CN).$$
\[global case\]
We shall now generalize the discussion of the previous subsections to the case of chiral $\CA$-modules, which are not necessarily supported at a single point $x\in X$.
Recall that in addition to $\on{Jets}(G)_X$, we have the group D-ind scheme $\on{Jets}^{mer}(G)_X$.
\[global\]
[*(1)*]{} Let $\CM$ be a chiral $\CA$-module on $X$, and let $\bg$ be an $S$-point of $\on{Jets}^{mer}(G)$, where $S$ is an affine D-scheme on $X$. Then the D-module $\CM^\bg:=\CM\underset{\CO_X}\otimes \CO_S$ acquires a natural structure of chiral $\CA$-module.
[*(2)*]{} If the ratio of two points $\bg_1$ and $\bg_2$ lies in the first infinitesimal neighborhood of the unit section of $\on{Jets}^{mer}(G)$, then for every choice of the splitting as a D-module $L_\fg\to L_{\fg,\kappa}$ there exists a functorial isomorphism $\CM^{\bg_1}\simeq \CM^{\bg_2}$.
[*(3)*]{} If $\CM'$ is a chiral $\CA$-module, which is weakly $\on{Jets}(G)$-equivariant, then to every chiral pairing $\{\CM',\CM\}\to \CN$ there functorially corresponds a chiral pairing $\{\CM',\CM^\bg\}\to \CN^\bg$.
[*(4)*]{} In the circumstances of points (2) and (3) above assume in addition that $\CM'$ is strongly $\on{Jets}(G)$-equivariant. Then the diagram of [Corollary \[strong twist\]]{} commutes.
Let us first recall the following general construction. Let $$j_*j^*(\CM_1^i\boxtimes \CM_2^i)\to \Delta_!(\CN^i),$$ be maps of D-modules, where $i$ runs over some finite set of indices $i$. Then we have a map $$\label{mult chiral}
j_*j^*\Bigl((\underset{i}\otimes \CM_1^i)\boxtimes
(\underset{i}\otimes \CM_2^i)\Bigr) \to \Delta_!(\underset{i}\otimes
\CN^i).$$
Let us recall also that the data of an $S$-point of $\on{Jets}^{mer}(G)$ is equivalent to that of a map $$j_*j^*(\CO_{\on{Jets}(G)_X}\boxtimes \CO_X)\to \Delta_!(\CO_S),$$ such that the diagram $$\CD j_*j^*\Bigl((\CO_{\on{Jets}(G)_X}\otimes
\CO_{\on{Jets}(G)_X})\boxtimes \CO_X\Bigr) @>>>
\Delta_!\Bigl((\CO_S\otimes \CO_S)\boxtimes \CO_X\Bigr) \\ @VVV @VVV
\\ j_*j^*(\CO_{\on{Jets}(G)_X}\boxtimes \CO_X) @>>> \Delta_!(\CO_S)
\endCD$$ commutes, where the upper horizontal arrow comes from the map , and vertical arrows are given by the algebra multiplication.
As in the proofs of [Proposition-Construction \[group act on cat\]]{} and [Proposition-Construction \[twisting pairing\]]{}, the proof follows from the next general construction. Let $$j_*j^*(\CM_1\boxtimes \CM_2)\to \Delta_!(\CN) \text{ and }
\CM_1\to \CM_1\otimes \CO_{\on{Jets}(G)_X}$$ be maps of D-modules and $\bg$ be as above. Then from we obtain a map $$j_*j^*(\CM_1\boxtimes \CM_2)\to
j_*j^*\Bigl((\CM_1\otimes \CO_{\on{Jets}(G)_X})\boxtimes
(\CM_2\boxtimes \CO_S)\Bigr)
\to \Delta_!(\CN\otimes \CO_S),$$ which, in turn, gives rise to a map $$\label{acted map}
j_*j^*\Bigl(\CM_1\boxtimes (\CM_2\otimes \CO_S)\Bigr)\to
\Delta_!(\CN\otimes \CO_S).$$
By putting first $\CM_1:=\CA$, $\CM_2:=\CM$ and $\CN:=\CM$, we arrive to the chiral action map of point (1). By putting $\CM_1:=\CM'$, $\CM_2:=\CM$ and $\CN:=\CN$, we arrive to the chiral pairing of point (3).
To prove point (2), we can assume being given a map $$j_*j^*(L^\vee_\fg\boxtimes \CO_X)\to \Delta_!(\CO_S),$$ and we have to construct a map $\varphi:\CM\to \CM\otimes \CO_S$, which fits into the commutative diagram: $$\CD j_*j^*(\CA\boxtimes \CM) @>>> j_*j^*\Bigl((\CA\otimes
L^\vee_\fg)\boxtimes \CM\Bigr) \\ @V{\on{id}\boxtimes \varphi}VV @VVV
\\ j_*j^*\Bigl(\CA\boxtimes (\CM\otimes \CO_S)\Bigr) @>>>
\Delta_!(\CM\otimes \CO_S), \endCD$$ where the right vertical arrow comes from , and the bottom horizontal arrow comes from the initial chiral action of $\CA$ on $\CM$.
The desired map $\varphi$ is constructed as follows. The chiral bracket of $L_\fg$ with $\CM$ (which is well-defined since we chose a splitting of $L_{\fg,\kappa}$) and give rise to a map $$j_*j^*\Bigl((L_\fg\otimes L^\vee_\fg)\boxtimes \CM\Bigr)\to
\Delta_!(\CM\otimes \CO_S).$$ The required map if the Lie-\* bracket induced by the above map applied to the canonical element ${\bf 1}\in
H^0_{DR}(L_\fg\otimes L^\vee_\fg)$.
The fact that the axioms are satisfied, and point (4) of the proposition follow from the construction.
Let $\CM$ be a chiral $\CA$-module as above, and let $\CF'$ be a chiral module over $\fD_{G,\kappa}$. Then [Proposition-Construction \[global\]]{} implies that proceeding as in [@FG2], Sect. 22.4, we can form a twisted product of $\CF'$ and $\CM$, denoted $\CF'\tboxtimes \CM$.
Let us write down this construction explicitly. As a D-module, this will be the usual tensor product $\CF'\otimes \CM$, but it will carry a new action of $\CA$, and a commuting action of $\CA_{\fg,2\kappa_\crit}$. (Note that $2\kappa_\crit$ equals the negative of the Killing form on $\fg$.)
Namely, the action of $\CA$ is the composition: $$j_*j^*(\CA\boxtimes (\CF'\otimes \CM))\to
j_*j^*((\CO_{\on{Jets}(G)_X}\otimes \CA)\boxtimes (\CF'\otimes
\CM))\to \Delta_!(\CF'\otimes \CM),$$ where the last arrow comes from the chiral action of $\CO_{\on{Jets}(G)_X}$ on $\CF$ and $\CA$ on $\CM$ via .
The chiral action of $L_{\fg,2\kappa_\crit}$ is the diagonal one with respect to the $\fr$-action of $\CA_{\fg,\kappa'}$ on $\CF'$ and the action of $L_{\fg,\kappa}$ on $\CM$ that comes from $L_{\fg,\kappa}\to
\CA$.
Hence, by tensoring with tensoring with the Clifford chiral algebra, we obtain a well-defined complex $\fC^\semiinf(L_\fg,\CF'\otimes \CM)$ of chiral $\CA$-modules. More generally, if $\{\CM',\CM\}\to \CN$ is a chiral pairing of $\CA$-modules, with $\CM'$ being $\on{Jets}(G)_X$-equivariant, as in point (3) of the above proposition, we obtain a chiral pairing of complexes $$j_*j^*\Bigl(\CM'\boxtimes \fC^\semiinf(L_\fg,\CF'\otimes \CM)\Bigr)\to
\Delta_!\Bigl(\fC^\semiinf(L_\fg,\CF'\otimes \CN)\Bigr).$$
In particular, if $\CF'$ is strongly equivariant with respect to the right action of $\on{Jets}(G)_X$ and $\CM$ is also $\on{Jets}(G)_X$-equivariant, by considering the corresponding subcomplex of chains relative to $\fg\in \Gamma(X,{{\mathcal L}}_\fg)$, we obtain a map $$\label{global pairing}
j_*j^*\Bigl(\CM'\boxtimes \fC^\semiinf(L_\fg;\fg,\CF'\otimes \CM)\Bigr)\to
\Delta_!\Bigl(\fC^\semiinf(L_\fg;\fg,\CF'\otimes \CN)\Bigr),$$ and on the level of individual cohomologies the chiral pairings $$\label{global, cohomologies}
j_*j^*(\CM'\boxtimes h^i(\CF\star \CM))\to \Delta_!(h^i(\CF\star \CN)),$$ where $\CF$ denotes the corresponding twisted D-module on $\Gr_{G,X}$.
Let us take as an example the case when $\CM=\CA$, and the canonical chiral pairing $\{\CM',\CA\}\to \CM'$. We obtain the chiral pairings $$\label{general basic pairing}
\{\CM',h^i(\CF\star \CA)\}\to h^i(\CF\star \CM').$$
Consider the case $\CA\simeq \CA_{\fg,\kappa}$. Then, by construction, $$\CF\star \CA_{\fg,\kappa}\simeq \Gamma(\Gr_{G,X},\CF).$$
\[two def\] For $\CA\simeq \CA_{\fg,\kappa}$, the chiral pairings of coincide with those of .
By the construction of the pairings in , it is sufficient to consider the case when $\CM'\simeq
\fD_{G,\kappa}$. The latter reduces to the case of the chiral algebra $\fD_{G,\kappa}$ rather than $\CA_{\fg,\kappa}$.
Now the assertion of the lemma follows from the next general observation: if $\CM'=\CA$, then the map of [Proposition-Construction \[global\]]{}(3) coincides with the chiral action of $\CA$ on $\CM^\bg$. In particular, the maps of are also given by the chiral action of $\CA$ on $h^i(\CF\star \CM)$.
Let us now establish some further compatibilities, satisfied by the maps of [Proposition-Construction \[global\]]{}.
Let $\CM_1,\CM_2,\CM_3$ be chiral modules over $\CA$, and let $\{\CM_1,\CM_2\}\to \CM_{1,2}$, $\{\CM_2,\CM_3\}\to \CM_{2,3}$, $\{\CM_1,\CM_3\}\to \CM_{1,3}$ be chiral pairings. In addition, let $\{\CM_{1,2},\CM_3\}\to \CN$, $\{\CM_{2,3},\CM_1\}\to \CN$, $\{\CM_{1,3},\CM_2\}\to \CN$ be chiral pairings such that the sum of the three maps $$j_*j^*(\CM_1\boxtimes \CM_2\boxtimes \CM_3)\to \Delta_!(\CN)$$ equals zero.
Assume also that all of the above modules are $\on{Jets}(G)_X$-equivariant, and let $\CF$ be a twisted D-module on $\Gr_{G,X}$.
\[comp with triple\] Under the above circumstances the sum of the three induced maps $$j_*j^*\Bigl(h^i(\CF\star \CM_1)\boxtimes \CM_2\boxtimes \CM_3\Bigr)\to
\Delta_!(h^i(\CF\star \CN)\Bigr)$$ is zero.
By the construction of convolution, it suffices to note the following. Let $\CM_1,\CM_2,\CM_3$, $\CM_{1,2}, \CM_{2,3}, \CM_{1,3},
\CN$ be as above, but let us only assume that $\CM_2,\CM_3$ and $\CM_{2,3}$ are $\on{Jets}(G)_X$-equivariant. Let $\bg$ be an $S$-point of $\on{Jets}^{mer}(G)_X$ for some D-scheme $S$.
Then we have the chiral pairings, $$\begin{aligned}
&\{\CM_1^\bg,\CM_2\}\to \CM_{1,2}^\bg,\,\, \{\CM_1,\CM_3\}\to
\CM_{1,3}^\bg,\,\, \\ &\{\CM^\bg_{1,2},\CM_3\}\to \CN^\bg, \,\,
\{\CM^\bg_{1,3},\CM_2\}\to \CN^\bg, \,\, \{\CM_1^\bg,\CM_{2,3}\}\to
\CN^\bg,\end{aligned}$$ and the sum of the resulting three maps $$j_*j^*(\CM_1^\bg\boxtimes \CM_2\boxtimes \CM_3)\to \Delta_!(\CN^\bg)$$ is zero.
Let now $\CF'_1$, $\CF'_2$ be two chiral $\fD_{G,\kappa}$-modules with $\CF'_1$ being strongly $\on{Jets}(G)_X$-equivariant on the right and $\CF'_2$ being strongly $\on{Jets}(G)_X$-equivariant on the left. Let $\CF'_1\star \CF'_2$ be their convolution, which is by definition represented by the complex $$\fC^\semiinf(L_\fg;\fg,\CF'_1\otimes \CF'_2).$$
Given a chiral pairing $\{\CM',\CM\}\to \CN$ with $\CM'$ being strongly $\on{Jets}(G)$-equivariant, we on the one hand, obtain a chiral pairing of complexes $$\{\CM', \CF'_2\star \CM\}\to \CF'_2\star \CN,$$ from which we further obtain a chiral pairing of bi-complexes $$\{\CM', \CF'_1\star (\CF'_2\star \CM)\}\to
\CF'_1\star (\CF'_2\star \CN).$$ On the other hand, from the original pairing we obtain another pairing of bi-complexes $$\{\CM', (\CF'_1\star \CF'_2)\star \CM\}\to
(\CF'_1\star \CF'_2)\star \CN.$$
However, by [@FG2], Sect. 22.9.1, the complexes associated to $$(\CF'_1\star \CF'_2)\star \CM'' \text{ and }
\CF'_1\star (\CF'_2\star \CM'')$$ for $\CM''=\CM$ or $\CM''=\CN$ are isomorphic. The next assertion follows from the construction:
\[assoc of fusion\] Under the above circumstances, the diagram of complexes $$\CD j_*j^*\Bigl(\CM'\boxtimes \left(\CF'_1\star (\CF'_2\star
\CM)\right)\Bigr) @>>> \Delta_!\Bigl(\CF'_1\star (\CF'_2\star
\CN)\Bigr) \\ @V{\sim}VV @V{\sim}VV \\ j_*j^*\Bigl(\CM'\boxtimes
\left((\CF'_1\star \CF'_2)\star \CM)\right)\Bigr) @>>>
\Delta_!\Bigl((\CF'_1\star \CF'_2)\star \CN)\Bigr) \endCD$$ commutes.
We shall now prove the following generalization of [Theorem \[representability, spherical\]]{}. Let $\CA$ be a chiral algebra as in [Sect. \[general alg\]]{}. Let us assume that the level $\kappa$ is integral, i.e., the spherical D-modules $\CF_{V,X}$ on $\Gr_{G,X}$ for $V\in \Rep(\cG)$ make sense.
Assume that for any $V$ as above the convolution $\CF_{V,X}\star \CA$ is acyclic away from degree $0$.
\[repr, general\] Let $\CM$ be a strongly $\on{Jets}(G)_X$-equivariant chiral module.
[*(1)*]{} The convolution $\CF_{V,X}\star \CM$ is acyclic away from cohomological degree $0$.
[*(2)*]{} Chiral pairings $\{\CF_{V,X}\star \CA,\CM\}\to
\CN$, where $\CN$ is any other chiral $\CA$-module, are in bijection with maps of chiral $\CA$-modules $\CF_{V,X}\star \CM\to \CN$.
In this subsection we will prove the first point of the theorem.
The functor $\CM\mapsto \CF_{V,X}\star \CM$ on the derived category of strongly $\on{Jets}(G)_X$-equivariant chiral modules over $\CA$ is both left and right adjoint to $\CN\mapsto \CF_{V^*,X}\star
\CN$. Hence, it is enough to show that it is right-exact. Suppose not, and let $k$ be the maximal integer, for which $h^k(\CF_{V,X}\star
\CM)$ is non-zero for some $\CM\in \CA\mod^{\on{Jets}(G)_X}$. Then $k$ is also the maximal integer, for which $h^{-k}(\CF_{V^*,X}\star
\CN)\neq 0$ for $\CN\in \CA\mod^{\on{Jets}(G)_X}$.
Let us choose an object $\CM$ as above that saturates this bound, and let us denote by $\CN$ the $k$-th cohomology of $\CF_{V,X}\star
\CM$. By adjunction we have a non-zero map $\CM\to
h^{-k}(\CF_{V^*,X}\star \CN)$.
We can represent $\tau^{\geq 0}(\CF_{V,X}\star \CM)$ by a complex $\CM_1^\bullet$, supported in degrees $\geq 0$, such that the map of gives rise to a chiral pairing $\{\CF_{V,X}\star \CA,\CM\}\to \CM_1^\bullet$. Moreover, we can represent $\CN$ by a complex $\CN^\bullet$, also supported in degrees $\geq 0$, such that the map $\CF_{V,X}\star \CM\to \CN[-k]$ is represented by a map of complexes $\CM_1^\bullet\to \CN^\bullet[-k]$.
Consider the diagram of complexes: $$\label{one comp}
\CD & & \Delta_!\Bigl(h^0(\CF_{V^*,X}\star \CN^\bullet[-k])\Bigr) \\ &
& @AAA \\ j_*j^*\Bigl(h^0(\CF_{V^*,X}\star (\CF_{V,X}\star
\CA))\boxtimes \CM\Bigr) @>>> \Delta_!\Bigl(h^0(\CF_{V^*,X}\star
\CM^\bullet_1)\Bigr) \\ @AAA @AAA \\ j_*j^*(\CA\boxtimes \CM) @>>>
\Delta_!(\CM), \endCD$$ where the left vertical arrow comes from the map $\delta_{1,\Gr_G}\to
\CF_{V^*,X}\star \CF_{V,X}$, the lower right vertical arrow comes from $$\CM\to \CF_{V^*,X}\star \CF_{V,X}\star \CM\to \CF_{V^*,X}\star \CM_1,$$ and the upper horizontal arrow comes by functoriality from [Proposition-Construction \[global\]]{}. This diagram is commutative by [Lemma \[assoc of fusion\]]{}.
The composed arrow from the lower left corner to the upper right corner of the above diagram vanishes, since the composition $$j_*j^*\Bigl((\CF_{V,X}\star \CA)\boxtimes \CM\Bigr)\to
\Delta_!(\CM^\bullet_1)\to \CN^\bullet[-k]$$ is evidently equal to $0$.
This is a contradiction, since the lower horizontal arrow is surjective, and the composition $$\Delta_!(\CM)\to \Delta_!\Bigl(h^0(\CF_{V^*,X}\star \CM_1)\Bigr)\to
\Delta_!\Bigl(h^0(\CF_{V^*,X}\star \CN^\bullet[-k])\Bigr)$$ equals the map $\CM\to h^{-k}(\CF_{V^*,X}\star \CN)$ above.
The canonical map of assigns to every map of chiral modules $\CF_{V,X}\star \CM\to \CN$ a chiral pairing $\phi:\{\CF_{V,X}\star \CA,\CM\}\to \CF_{V,X}\star \CM$.
Let us construct a map in the opposite direction. By [Proposition \[equivariance preserved\]]{}, it is sufficient to consider the case when $\CN$ is also $\on{Jets}(G)_X$-equivariant. For $\phi$ as above, consider the chiral pairing $$j_*j^*\Bigl((\CF_{V^*,X}\star \CF_{V,X}\star \CA)\boxtimes
\CM\Bigr)\to \Delta_!(\CF_{V^*,X}\star \CN),$$ obtained from [Proposition-Construction \[global\]]{}. Using the canonical map $\CA\to
\CF_{V^*,X}\star \CF_{V,X}\star \CA$ we thus obtain a chiral pairing $\{\CA,\CM\}\to \CF_{V^*,X}\star \CN$. By [Lemma \[A is unit\]]{}, the latter gives rise to a map of chiral modules $\CM\to \CF_{V^*,X}\star
\CN$. By adjunction, we obtain a map $\psi:\CF_{V,X}\star \CM\to
\CN$, as required.
The fact that, if the initial chiral pairing $\phi$ came from a map $\CF_{V,X}\star \CM\to \CN$, then the resulting map $\psi$ equals the initial one, follows from .
Let us start with a map $\phi$, and show that the pairing $\phi':\{\CF_{V,X}\star \CA,\CM\}\to \CM$ obtained from the corresponding $\psi$, equals the initial $\phi$.
Consider the three maps $$j_*j^*\Bigl(\CA\boxtimes (\CF_{V,X}\star \CA)\boxtimes \CM\Bigr)\to
\Delta_!(\CN),$$ obtained from the map $\phi$. By the definition of chiral pairings, their sum equals $0$. Using [Proposition-Construction \[global\]]{}, we obtain three maps $$j_*j^*\Bigl(\CA\boxtimes (\CF_{V^*,X}\star \CF_{V,X}\star
\CA)\boxtimes \CM\Bigr)\to \Delta_!(\CF_{V^*,X}\star \CN),$$ whose sum is still equal to $0$, by [Lemma \[comp with triple\]]{}.
Furthermore, using [Proposition-Construction \[global\]]{} again, we obtain three maps $$j_*j^*\Bigl((\CF_{V,X}\star \CA)\boxtimes (\CF_{V^*,X}\star
\CF_{V,X}\star \CA) \boxtimes \CM\Bigr)\to \Delta_!(\CF_{V,X}\star
\CF_{V^*,X}\star \CN).$$ Composing these maps with $\CA\to
\CF_{V^*,X}\star \CF_{V,X}\star \CA$ and $\CF_{V,X}\star
\CF_{V^*,X}\star \CN\to \CN$ we obtain three maps $$\label{second comp}
j_*j^*\Bigl((\CF_{V,X}\star \CA)\boxtimes \CA\boxtimes \CM\Bigr)\to
\Delta_!(\CN),$$ that sum up to zero.
Let us calculate the resulting maps explicitly. It is easy to see that the first of these maps, namely, the one obtained by first fusing the $x_1$ and $x_2$ coordinates equals $$j_*j^*\Bigl((\CF_{V,X}\star \CA)\boxtimes \CA\boxtimes \CM\Bigr) \to
\Delta_{x_1=x_2}{}_!\Bigl(j_*j^*((\CF_{V,X}\star \CA)\boxtimes
\CM)\Bigr)\overset{\phi}\to \Delta_!(\CN),$$ where the first arrow is given by the chiral action of $\CA$ on $\CF_{V,X}\star \CA$.
The second map, namely, the one obtained by first fusing the $x_2$ and $x_3$ coordinates equals $$j_*j^*\Bigl((\CF_{V,X}\star \CA)\boxtimes \CA\boxtimes \CM\Bigr) \to
\Delta_{x_2=x_3}{}_! \Bigl(j_*j^*((\CF_{V,X}\star \CA)\boxtimes
\CM)\Bigr)\overset{\phi'}\to \Delta_!(\CN),$$ where the first arrow is the chiral action of $\CA$ on $\CM$.
The third arrow, by construction, factors through a D-module supported on the diagonal $x_1=x_3$.
Let us now compare the three maps of with the three maps between the same objects that correspond to the initial chiral pairing $\phi$, and subtract one from another.
We have two equal maps $$j_*j^*\Bigl((\CF_{V,X}\star \CA)\boxtimes \CA\boxtimes \CM\Bigr) \to
\Delta_!(\CN),$$ such that one factors as $$j_*j^*\Bigl((\CF_{V,X}\star \CA)\boxtimes \CA\boxtimes \CM\Bigr)
\twoheadrightarrow \Delta_{x_2=x_3}{}_! \Bigl(j_*j^*((\CF_{V,X}\star
\CA)\boxtimes \CM)\Bigr) \overset{\phi'-\phi}\longrightarrow
\Delta_!(\CN),$$ and the other through a D-module, supported on the diagonal $x_1=x_3$. this implies that both maps are in fact $0$, and in particular, $\phi=\phi'$.
**[Part II: The Iwahori case]{}**
In this part of the paper we consider convolutions of the central sheaves on the affine flag variety with objects of the category of $I$-equivariant $\hg_\crit$-modules.
Convolution with central sheaves {#sect 5}
================================
From now on we shall fix a point $x\in X$. Let $\lambda$ be an integral weight such that $\lambda+\rho$ is dominant. Recall that to such $\lambda$ in [@FG2], Sect. 7.6, we have attached a subscheme $$\Spec(\fZ_\fg^{\lambda,\nilp})\subset \Spec(\fZ_\fg).$$ In terms of the isomorphism between $\Spec(\fZ_\fg)$ and the ind-scheme of $\check\fg$-opers on $\D_x^\times$, the subscheme $\Spec(\fZ_\fg^{\lambda,\nilp})$ corresponds to opers with a regular singularity and residue $\varpi(-\lambda-\rho)$, see [@FG2], Sect. 2.9.
According to [@FG2], Sect. 7.6, if a weight $\mu$ is of the form $w(\lambda+\rho)-\rho$ for some $w\in W$, then the support of the $\hg_\crit$-module $\BM^\mu:=\on{Ind}^{\hg_\crit}_{\fg[[t]]}(M^\mu)$, where $M^\mu$ denotes the Verma module of highest weight $\mu$ over $\fg$, is contained (and in fact equal to) $\Spec(\fZ_\fg^{\lambda,\nilp})$.
Let $\fz_\fg^{\lambda,\nilp}$ be the modification of the D-algebra $\fz_\fg$ at $x$, corresponding to $\fZ_\fg^{\lambda,\nilp}$. From [@FG2], Sect. 2.9, we obtain that for every $V\in
\Rep(\cG)$, the module $\CV_{X-x}$ extends naturally to a free $\fZ_\fg^{\lambda,\nilp}$-module, such that the connection has a pole of order $\leq 1$ at $x$ and a nilpotent monodromy.
Let $\wt\fZ_\fg^{\lambda,\nilp}$ be the completion of $\fZ_\fg$ with respect to the ideal of $\fZ_\fg^{\lambda,\nilp}$. Let $\Spec(\fZ_\fg^{\int,\nilp})$ (resp., $\Spec(\wt\fZ_\fg^{\int,\nilp})$) be the sub-ind scheme of $\Spec(\fZ_\fg)$ defined to the disjoint union $\underset{\lambda}\sqcup\, \Spec(\fZ_\fg^{\lambda,\nilp})$ (resp., $\underset{\lambda}\sqcup\,
\Spec(\wt\fZ_\fg^{\lambda,\nilp})$).
Applying [Lemma \[extension of chiral modules\]]{}, we obtain that for each $V\in \Rep(\cG)$, the module $\CV_{X-x}$ gives rise to a vector bundle $\CV_{\wt\fZ_\fg^{\int,\nilp}}$ over $\Spec(\wt\fZ_\fg^{\int,\nilp})$, equipped with a nilpotent endomorphism, which we will denote by $N_{\CV_{\wt\fZ_\fg^{\int,\nilp}}}$. Moreover, for $U\simeq
V\otimes W$ we have an isomorphism $$\CU_{\wt\fZ_\fg^{\int,\nilp}} \simeq
\CV_{\wt\fZ_\fg^{\int,\nilp}}
\underset{\wt\fZ_\fg^{\int,\nilp}} \otimes
\CW_{\wt\fZ_\fg^{\int,\nilp}},$$ so that $N_{\CU_{\wt\fZ_\fg^{\int,\nilp}}} =
N_{\CV_{\wt\fZ_\fg^{\int,\nilp}}}+
N_{\CW_{\wt\fZ_\fg^{\int,\nilp}}}$.
\[statement of Iwahori theorem\]
Let $I\subset G\ppart$ be the Iwahori subgroup, i.e., the preimage of $B\subset G$ under the evaluation map $G[[t]]\to G$. Following the notation of [@FG2], we shall denote by $\hg_\kappa\mod^I$ the category of $I$-integrable representations of $\hg$ at the level $\kappa$.
The above estimate on the support of the modules $\BM^\mu$ implies:
The support of every object $\CM\in \hg_\crit\mod^I$ over $\Spec(\fZ_\fg)$ is contained in $\Spec(\wt\fZ_\fg^{\int,\nilp})$.
Thus, for every $\CM\in \hg_\crit\mod^I$ and $V\in \Rep(\cG)$ we can functorially attach another object of $\hg_\crit\mod^I$: $$\CV_{\wt\fZ_\fg^{\int,\nilp}}
\underset{\wt\fZ_\fg^{\int,\nilp}}\otimes \CM,$$ which carries a nilpotent endomorphism $N_{\CV_{\wt\fZ_\fg^{\int,\nilp}}}$.
Let $\Fl_G=G\ppart/I$ be the affine flag scheme of $G$. For a level $\kappa$ we shall denote by $\on{D}(\Fl_G)_\kappa\mod$ the category of $\kappa$-twisted right D-modules on $\Fl_G$. By $\on{D}(\Fl_G)_\kappa\mod^I$ we shall denote the corresponding category of $I$-equivariant objects in $\on{D}(\Fl_G)_\kappa\mod$. When $\kappa$ is integral (e.g., critical) we will sometimes identify $\on{D}(\Fl_G)_\kappa\mod$ with the usual category of D-modules by means of the tensor product with the corresponding line bundle.
Given an object $\CF\in \on{D}(\Fl_G)_\kappa\mod$ and $\CM\in
\fg_\kappa\mod^I$ we can form their convolution, denoted $\CF\underset{I}\star \CM$, which is an object of $D(\fg_\kappa\mod)$. When no confusion is likely to occur, we will omit the subscript $I$ from $\underset{I}\star$.
Let us recall from [@Ga] that to every object $V\in \Rep(\cG)$ there corresponds an object $\CZ_V\in \on{D}(\Fl_G)\mod^I$ called a central sheaf. Each central sheaf is endowed with a functorial endomorphism $N_V$. The construction of D-modules $\CZ_V$ will be reviewed in some detail in the sequel. Slightly abusing the notation, we shall denote by the same symbol $\CZ_V$ the corresponding $I$-equivariant object of $\on{D}(\Fl_G)_\crit\mod^I$.
Our main result is the following:
\[main\] For every $\CM\in \hg_\crit\mod^I$ and $V\in \Rep(\cG)$ the convolution $\CZ_V\star \CM$ is acyclic away from cohomological degree $0$, and we have a canonical isomorphism $$\fs_V:\CZ_V\underset{I}\star \CM \simeq
\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}
\otimes \CM,$$ such that the endomorphism induced by $N_V$ on the LHS goes over to the endomorphism, induced by $N_{\CV_{\wt\fZ_\fg^{\int,\nilp}}}$ on the RHS. This system of isomorphisms is compatible with tensor products of $\cG$-representations in the same sense as in [Theorem \[main, spherical\]]{}
\[another proof mon-free\]
Let us note that [Theorem \[main\]]{}, whose prove is parallel to, but independent of, the proof of [Theorem \[main, spherical\]]{}, implies the latter theorem. Indeed, let $\fp$ denote the natural projection $\Fl_G\to \Gr_G$; then by [@Ga], there is a canonical isomorphism $$\fp_!(\CZ_V)\simeq \CF_V.$$ For an object $\CM\in \hg_\crit\mod^{G[[t]]}$ we have: $$\CZ_V\underset{I}\star \CM\simeq \CF_V\underset{G[[t]]}\star \CM,$$ and the assertion follows from the fact that the restriction of $\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}$ to $\Spec(\wt\fZ^{\int,\reg}_{\fg})\subset
\Spec(\wt\fZ^{\int,\nilp}_{\fg,x})$ identifies canonically with $\CV_{\wt\fz^{\int,\reg}_{\fg,x}}$.
In particular, since the nilpotent endomorphism that $N_V$ induces on $\CF_V$ is zero, we obtain the assertion of [Corollary \[monodromy-free\]]{}. This, in turn, gives an alternative proof of [Lemma \[sup of sph\]]{}, as was promised earlier. Let us now prove [Lemma \[sup of ind\]]{}:
One the one hand, by [@FG2], Corollary 7.6.2, the support of $\BV^\mu$ over $\Spec(\fZ_\fg)$ is contained in the subscheme $\Spec(\fZ_\fg^{\mu,\nilp})$. On the other hand, by [Corollary \[monodromy-free\]]{}, which was proved independently, the support of $\BV^\mu$ is contained in the ind-subscheme $\Spec(\fZ_{\fg}^{\on{m.f.}})$. The assertion of the lemma follows now from the fact that $$\Spec(\fZ_\fg^{\mu,\nilp})\cap \Spec(\fZ_{\fg}^{\on{m.f.}})=\Spec(\fZ_\fg^{\mu,\reg}),$$ (see [@FG2], Sect. 2.9).
Parallel to the spherical situation, let us consider a particular case of the above theorem, corresponding to differential operators. Consider the object of $\delta_{I,G\ppart}\in
\fD_{G,\crit}\mod_x$, corresponding to distributions on $I$.
In other words, if $\pi_{\Fl}$ denotes the projection $G\ppart\to
\Fl_G$, then $$\delta_{I,G\ppart}\simeq
\Gamma(G\ppart,\pi_{\Fl}^*(\delta_{1,\Fl_G})),$$ where $\delta_{1,\Fl_G}$ is the $\delta$-function at $1\in \Fl_G$. As a $\hg_\crit$-module, $\delta_{I,G\ppart}$ can be described as $\on{Ind}^{\hg_\crit}_{\fg[[t]]}(\CO_I)$. We have:
\[main for diff op\] We have a canonical isomorphism of $\hg_\crit$-bimodules $$\Gamma(G\ppart,\pi_{\Fl_G}^*(\CZ_V))\simeq
\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}
\otimes \delta_{I,G\ppart},$$ which intertwines the endomorphism $N_V$ on the LHS with $N_{\CV_{\wt\fZ_\fg^{\int,\nilp}}}$ on the RHS. This system of isomorphisms is compatible with tensor products of $\cG$-representations.
The same argument as in the spherical case shows that [Theorem \[main for diff op\]]{} actually implies [Theorem \[main\]]{}.
Fusion in the Iwahori case {#sect 6}
==========================
Our strategy of proof of [Theorem \[main\]]{} will be parallel to that of [Theorem \[main, spherical\]]{}. Namely, we will construct a functor on the category of $\hg_\crit$-modules, and show that it is represented by both sides of the isomorphism stated in the theorem.
An additional ingredient in the present situation is that we will have to twist the chiral $\CA_{\fg,\kappa}$-modules $\Gamma(\Gr_{G,X-x},\CF_{V,X-x})$ by certain local systems on the punctured curve.
Let us fix a family of local systems $\CE_n$, $n\in \BN$, defined on a punctured Zariski neighborhood of $x\in X$, each $\CE_n$ being an $n$-fold extension of the trivial local system (i.e., $\CE_1$ is [*the*]{} trivial local system), endowed with a nilpotent endomorphism $N_{\CE}$ of order $n-1$, and such that we are given a compatible system of surjections $$\label{change of n}
\CE_n\twoheadrightarrow \CE_m,$$ defined for every $m\leq n$ and compatible with the action of $N_{\CE}$. Then the action of $N_\CE$ gives rise to a system of exact sequences: $$0\to \CE_m\to \CE_n\overset{N^m_{\CE}}\to \CE_n\to \CE_{n-m}\to 0.$$
For example, if we choose a local coordinate $t$ near $x$, such a system can be obtained as follows. Let $E_n$ be the standard $n$-dimensional Jordan block, i.e., the vector space $\BC[s]/s^{n-1}$ with the nilpotent operator $N_E$ equal to the multiplication by $s$. Set $\CE_n:=E_n\otimes \CO_{X-x}$, with $t\nabla_t$ acting as $N_E+t\partial_t$.
Let us denote by $\CE_{n,x}$ the fiber at $x$ of the extension of $\CE_n$, given by [Lemma \[extension of chiral modules\]]{}. This is an $n$-dimensional vector space, endowed with a nilpotent operator of order $n-1$, which coincides with the one induced by $N_\CE$. We shall fix a system of identifications $\CE_{n,x} \simeq E_n$, such that $N_\CE$ goes over to $N_E$, and which is compatible with the morphisms .
Recall now that if $\CA$ is any chiral algebra and $\CM$ is a chiral module over it, and $\CE$ is any D-module on $X$, then the D-module $\CM\otimes \CE$ is naturally a chiral module over $\CA$.
\[can pairing with monod\] Let $\kappa$ be any integral level. Then for any $\CM\in
\hg_\kappa\mod^I$ and $n\in \BN$ there exists a canonical chiral pairing of $\fD_{G,\crit}$-modules: $$j_x{}_*j_x^*\biggl(\Gamma\Bigl(G\ppart,\pi^*(\CF_V) \Bigr)\otimes
\CE_n\biggr) \otimes \delta_{I,G\ppart}\to
i_x{}_!\biggl(\Gamma\Bigl(G\ppart,\pi_{\Fl}^*(\CZ_V)
\Bigr)\biggr)_{N_V^n},$$ such that for $n\geq m$ the diagram $$\CD j_x{}_*j_x^*\biggl(\Gamma \Bigl(G\ppart,\pi^*(\CF_V)\Bigr)
\otimes \CE_n\biggr)\otimes \delta_{I,G\ppart} @>>>
i_x{}_!(\Gamma\Bigl(G\ppart,\pi_{\Fl}^*(\CZ_V)\Bigr)\biggr)_{N_V^n} \\
@VVV @VVV \\
j_x{}_*j_x^*\biggl(\Gamma\Bigl(G\ppart,\pi^*(\CF_V)\Bigr)\otimes
\CE_m\biggr)\otimes \delta_{I,G\ppart} @>>>
i_x{}_!\biggl(\Gamma\Bigl(G\ppart,\pi_{\Fl}^*(\CZ_V) \Bigr)
\biggr)_{N_V^m} \endCD$$ is commutative.
Before giving the proof we need to review the construction of the D-modules $\CZ_V$. Consider the ind-scheme $\Fl_{G,X}$ over $X$, whose fiber over $x_1\in X$ is the set of triples $(\CP_G,\beta, \alpha)$, where $\beta$ is a trivialization of $\CP_G$ on $X-\{x,x_1\}$ and $\alpha$ is a reduction of the fiber $\CP_{G,x}$ at $x$ to $B$. Note that $\Fl_{G,X}$ is the same as $\Gr_{G,X;K}$ for $K=I$.
The preimage of $X-x$ in $\Fl_{G,X}$, denoted $\Fl_{G,X-x}$, is isomorphic to $\Gr_{G,X-x}\times \Fl_G$, whereas the preimage of $x\in
X$ is isomorphic to $\Fl_G$. More generally, we will consider the ind-scheme $\Fl_{G,X^n}$ over $X^n$, whose fiber over $(x_1,...,x_n)\in X^n$ is the set of triples $(\CP_G,\beta, \alpha)$, where $\beta$ now is a trivialization defined away from $x_1\cup...\cup x_n\cup x$.
Recall the ind-scheme $\on{Jets}^\mer(G)_{X^n}$ from [Sect. \[Jets n\]]{}, and let $\on{Jets}^\mer(G)_{X^{n-1}\times x}$ be the closed ind-subscheme defined by the condition that the last point in the $n$-tuple $x_1,...,x_n$ is fixed to be $x$.
Note that we have a natural projection $\pi_{\Fl}:\on{Jets}^\mer(G)_{X^n\times x}\to \Fl_{G,X^n}$, which corresponds to remembering the reduction to $B$ at $x$ out of the trivialization of $\CP_G$ on the formal neighborhood of $x_1\cup...\cup x_n\cup x$.
Given an object $V\in \Rep(\cG)$ and $n\in \BN$, we consider $(\CF_{V,X-x}\otimes \CE_n)\boxtimes \delta_{1,\Fl_G}$ as a D-module on $\Fl_{G,X-x}$. For $n\in \BN$ consider the fiber over $x$ of its intermediate extension onto the entire $\Fl_{G,X}$, i.e., $$\CZ_{V,n}:=i_x^! j_x{}_{!*}\Bigl((\CF_{V,X-x}\otimes \CE_n)\boxtimes
\delta_{1,\Fl_G}\Bigr).$$
This is a D-module on $\Fl_G$, endowed with an action of $N_\CE$, by the transport of structure. The following summarizes the main construction of [@Ga]:
- For $n$ large enough and $n'\geq n$ the maps $\CZ_{V,n'}\to \CZ_{V,n}$ are isomorphisms.
- The D-module $\CZ_V$ is isomorphic to $\CZ_{V,n}$ for all $n$ large enough. Under this identification, the endomorphism $N_V$ equals the one induced by $N_{\CE}$.
- For any other $n''$, $\CZ_{V,n''}\simeq (\CZ_V)_{N^{n''}_V}$.
In fact, the above is a paraphrase of the fact that $\CZ_V$ is obtained by applying the functor of unipotent nearby cycles $\Psi^{un}$ to the D-module $(\CF_{V,X-x}\boxtimes \delta_{1,\Fl_G})$ on $\Fl_{G,X-x}$.
Note that if $\CE'$ is any local system on $X-x$ with a nilpotent monodromy around $x$, then the formalism of nearby cycles, developed in [@Be] implies that $$\label{extension against local system}
i_x^! j_x{}_{!*}\Bigl((\CF_{V,X-x}\otimes \CE')\boxtimes
\delta_{1,\Fl_G}\Bigr)[1]\simeq \Bigl(\CZ_V\otimes
\CE'_x\Bigr)_{N_V+N_{\CE'}},$$ where $\CE'_x$ is the fiber of the extension of $\CE'$ across $x$, given by [Lemma \[extension of chiral modules\]]{}, and $N_{\CE'}$ is its canonical nilpotent endomorphism.
Consider the canonical map of twisted D-modules on $\Fl_{G,X}$ $$\label{map downstairs, Iwahori}
j_x{}_*\Bigl((\CF_{V,X-x}\otimes \CE_n)\boxtimes
\delta_{1,\Fl_G}\Bigr)\to i_x{}_!(\CZ_{V,n}).$$
By taking the pull-back of under $\pi_{\Fl}:\on{Jets}^\mer(G)_{X\times x}\to \Fl_{G,X}$, we obtain a map of D-modules on $X$: $$\label{map upstairs, Iwahori}
j_x{}_*j_x^*\biggl(\Gamma\Bigl(\on{Jets}^\mer(G)_X,\pi^*(\CF_{V,X})\Bigr)
\otimes \CE_n\biggr) \otimes \delta_{I,G\ppart} \to
i_x{}_!\biggl(\Gamma\Bigl(G\ppart,\pi_{\Fl}^*(\CZ_V)\Bigr)\biggr)_{N_V^n},$$ as was stated in [Proposition-Construction \[can pairing with monod\]]{}. Thus, in order to finish the proof, we need to show that the above map respects the action of $\fD_{G,\kappa}$, i.e., that it is indeed a chiral pairing.
Consider the ind-scheme $\Fl_{G,X^2}$ over $X^2$; let $\Fl_{G,(X-x)^2-\Delta_{X-x}}$ denote its open subscheme equal to the preimage of the corresponding open subscheme in $X^2$; let $j$ denote the corresponding open embedding. Let $\Delta$ be the embedding of $\Fl_{G,x\times x}\simeq \Fl_G$.
We have an isomorphism: $$\Fl_{G,(X-x)^2-\Delta_{X-x}}\simeq \Bigl((\Gr_{G,X-x}\times
\Gr_{G,X-x})\underset{(X-x)^2}\times ((X-x)^2-\Delta_{X-x})\Bigr)
\times \Fl_G.$$
As in the case of $\Gr_{G,X^n}$, we have a map ${\bf 1}_{1,1}:X\times
\Fl_{G,X}\to \Fl_{G,X^2}$, and consider the twisted D-module on $\Fl_{G,X^2}$ equal to $$({\bf 1}_{1,1})_!\biggl(\omega_X\boxtimes
\Bigl(j_x{}_{!*}(\CF_{V,X-x}\otimes \CE_n)\Bigr)\biggr).$$ It gives rise to three maps $$\label{three maps down flags}
j_*j^*\Bigl(\delta_{1,\Gr_{G,X}}\boxtimes (\CF_{V,X-x}\otimes
\CE_n)\boxtimes \delta_{1,\Fl_G}\Bigr)\to \Delta_!(\CZ_V)_{N_V^n},$$ which sum up to zero.
Pulling back to the two sides of to $\on{Jets}^\mer(G)_{X^2\times x}$ and taking the (quasi-coherent) direct image onto $X^2$ we obtain three maps $$j_*j^*\biggl(\fD_{G,\kappa}\boxtimes
\Bigl(\Gamma(\on{Jets}^\mer(G)_X,\pi^*(\CF_{V,X})) \otimes
\CE_n\Bigr)\boxtimes \delta_{I,G\ppart}\biggr)\to
\Delta_!\biggl(\Gamma\Bigl(G\ppart,\pi_{\Fl}^*(\CZ_V)\Bigr)\biggr)_{N_V^n}.$$
From [Proposition \[action as fusion\]]{}, we obtain that these three maps are equal to those that appear in the definition of chiral pairings.
As in the spherical situation, from the chiral pairing given by [Proposition-Construction \[can pairing with monod\]]{}, we obtain a chiral pairing over $\CA_{\fg,\kappa}$: $$\label{another can pairing, Iwahori}
j_x{}_*j_x^*\biggl(\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X} \Bigr)\otimes
\CE_n\biggr)\otimes \delta_{I,G\ppart}\to
i_x{}_!\biggl(\Gamma\Bigl(G\ppart,\pi_{\Fl}^*(\CZ_V)
\Bigr)\biggr)_{N_V^n},$$ which commutes with the action of $\hg_{\kappa'}$ on $\delta_{I,G\ppart}$ and $\Gamma\Bigl(G\ppart,\pi_{\Fl}^*(\CZ_V)\Bigr)$, given by $\fr$.
Given an $I$-integrable $\fg_\kappa$-module $\CM$ we obtain a chiral pairing of complexes of $\CA_{\fg,\kappa}$-modules: $$\{\Gamma(\Gr_{G,X},\CF_{V,X})\otimes \CE_n,
\fC^\semiinf(\fg\ppart;\fh,\delta_{I,G\ppart}\otimes \CM)\} \to
\fC^\semiinf\Bigl(\fg\ppart;\fh,\Gamma(G\ppart,\pi_{\Fl}^*(\CZ_V))\otimes
\CM\Bigr)_{N_V^n}.$$ In particular, by taking $n>>0$ so that $N_V^n=0$, we obtain a chiral pairing $$\label{can pairing, Iwahori}
\{\Gamma(\Gr_{G,X-x},\CF_{V,X-x})\otimes \CE_n,\CM\}\to
h^0(\CZ_V\underset{I}\star \CM),$$ such that the action of $N_{\CE}$ on LHS corresponds to the action of $N_V$ on the RHS.
More generally, for an arbitrary local system $\CE'$ with a nilpotent monodromy around $x$, we obtain a chiral pairing $$\label{can pairing against loc sys}
\{\Gamma(\Gr_{G,X-x},\CF_{V,X-x})\otimes \CE',\CM\}\to
\Bigl(h^0(\CZ_V\underset{I}\star \CM)\otimes \CE'_x\Bigr)_{N_V+N_{\CE'}}.$$
Parallel to [Theorem \[representability, spherical\]]{}, we will prove:
\[representability, Iwahori\] Assume that $\kappa$ is non-positive, Then:
[*(1)*]{} For every $\CM\in \hg_\kappa\mod^I$, the convolution $\CZ_V\underset{I}\star \CM$ is acyclic away from cohomological degree $0$.
[*(2)*]{} The functor on $\hg_\kappa\mod$ that sends $\CN$ to the set of chiral pairings $$\{\Gamma(\Gr_{G,X},\CF_{V,X})\otimes \CE',\CM\}\to\CN,$$ is representable by $\Bigl((\CZ_V\underset{I}\star \CM)\otimes
\CE'_x\Bigr)_{N_V+N_{\CE'}}$.
We shall now proceed with the proof of [Theorem \[main\]]{}, which does not rely on [Theorem \[representability, Iwahori\]]{}. We will need to establish the following generalization of [Proposition \[comp with ten products\]]{}:
For $V,W\in \Rep(\cG)$, recall from [@Ga] that there exists a natural isomorphism $$\label{tensor center}
\CZ_V\underset{I}\star \CZ_W\simeq \CZ_{V\otimes W},$$ such that the endomorphism $N_V+N_W$ on the LHS goes over to the endomorphism $N_{V\otimes W}$.
Let $n,m\in \BN$ be such that $\CZ_V\to (\CZ_V)_{N_V^n}$ and $\CZ_W\to
(\CZ_W)_{N_V^n}$ are isomorphisms. Let $\CM$ be an object of $\hg_\crit\mod^I$, such that all convolutions $\CZ_{V'}\star \CM$, $V'\in \Rep(\cG)$ are acyclic away from cohomological degree $0$.
In this case, as in [Proposition \[comp with ten products\]]{}, we have three maps $$\label{three maps, Iw}
j_*j^*\Bigl((\Gamma(\Gr_{G,X},\CF_{V,X})\otimes \CE_n)\boxtimes
(\Gamma(\Gr_{G,X},\CF_{W,X})\otimes \CE_m)\boxtimes \CM\Bigr)\to
i_x{}_!(\CZ_{V\otimes W}\underset{I}\star \CM),$$ defined as follows.
The first map is the composition $$\begin{aligned}
&j_*j^*\biggl(\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\otimes
\CE_n\Bigr)\boxtimes \Bigl(\Gamma(\Gr_{G,X},\CF_{W,X})\otimes
\CE_m\Bigr)\boxtimes \CM\biggr)\to \\ &
\Delta_{x_2=x_3}{}_!\biggl(j_*j^*\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\otimes
\CE_n) \boxtimes (\CZ_W\star \CM)\Bigr)\biggr)\to
\Delta_!\Bigl(\CZ_V\underset{I}\star \CZ_W\underset{I}\star \CM\Bigr).\end{aligned}$$
The second map is the negative of the one defined by interchanging the roles of $V$ and $W$. To define the third map note that from we obtain a map of D-modules on $\Fl_{G,X}$: $$\label{n & m}
j_x{}_*j_x^*\biggl(\Bigl(\Gamma(\Gr_{G,X},\CF_{V\otimes W})\otimes
\CE_n\otimes \CE_m\Bigr) \boxtimes \delta_{1,\Fl_G}\biggr)\to
\Delta_!(\CZ_V\underset{I}\star \CZ_W),$$ which comes from the projection $$\begin{aligned}
&\Bigl(\CZ_{V\otimes W}\otimes E_n\otimes E_m\Bigr) _{N_{V\otimes
W}+N_\CE|_{E_n}+N_\CE|_{E_m}}\simeq \Bigl((\CZ_V\otimes E_n)\star
(\CZ_W\otimes E_m)\Bigr) _{N_{V}+N_{W}+N_\CE|_{E_n}+N_\CE|_{E_m}} \\
&\twoheadrightarrow (\CZ_V\otimes E_n)_{N_V+N_\CE|_{E_n}}\star
(\CZ_W\otimes E_m)_{N_W+N_\CE|_{E_m}}\simeq \CZ_V\underset{I}\star
\CZ_W.\end{aligned}$$
Corresponding to it there is a chiral pairing, defined for every $\CM$ as above: $$\label{n & m, modules}
j_x{}_*j_x^*\biggl(\Bigl(\Gamma(\Gr_{G,X},\CF_{V\otimes W})\otimes
\CE_n\otimes \CE_m\Bigr) \boxtimes \CM\biggr)\biggr)\to
\Delta_!(\CZ_V\underset{I}\star \CZ_W\underset{I}\star \CM).$$
The third map in equals the composition $$\begin{aligned}
&j_*j^*\biggl(\Bigl(\Gamma(\Gr_{G,X},\CF_{V,X})\otimes
\CE_n\Bigr)\boxtimes \Bigl(\Gamma(\Gr_{G,X},\CF_{W,X})\otimes
\CE_m\Bigr)\boxtimes \CM\biggr)\to \\ &\to \Delta_{x_1=x_2}{}_!
\biggl(j_x{}_*j_x^*\biggl(\Bigl(\Gamma(\Gr_{G,X},\CF_{V\otimes
W})\otimes \CE_n\otimes \CE_m\Bigr) \boxtimes
\CM\biggr)\biggr)
\overset{\text{\eqref{n & m, modules}}}\longrightarrow
\Delta_!(\CZ_V\underset{I}\star \CZ_W\underset{I}\star \CM).\end{aligned}$$
\[assoc, Iwahori\] The sum of the three maps above is $0$.
Recall that $$\Fl_{G,(X-x)^2-\Delta_{X-x}}\simeq
\Bigl((\Gr_{G,X-x}\times \Gr_{G,X-x})\underset{(X-x)^2}\times
((X-x)^2-\Delta_{X-x})\Bigr) \times \Fl_G,$$ and consider the twisted D-module $$\label{on square}
j_{!*}\Bigl((\CF_{V,X-x}\otimes \CE_n)\boxtimes
(\CF_{W,X-x}\otimes \CE_m)\boxtimes \delta_{1,\Fl_G}\Bigr)$$ on $\Fl_{G,X^2}$.
Let $\Fl_{G,X\times x}$ (resp., $\Fl_{G,x\times X}$, $\Fl_{G,\Delta_X}$) denote the preimage in $\Fl_{G,X^2}$ of the corresponding subvariety in $X^2$. By considering the iterated version of $\Fl_{G,X^2}$, the following description of the D-module was obtained in [@Ga1]:
- The restriction of the D-module to $\Fl_{G,X\times x}\simeq \Fl_{G,X}$ identifies with $$\on{ker}
\biggl(j_x{}_*j_x^*\Bigl((\CF_{V,X}\otimes \CE_n)\boxtimes \CZ_W\Bigr)\to
i_x{}_!(\CZ_V\underset{I}\star \CZ_W)\biggr).$$
- The restriction of to $\Fl_{G,x\times X}\simeq
\Fl_{G,X}$ identifies with $$\on{ker}
\biggl(j_x{}_*j_x^*\Bigl((\CF_{W,X}\otimes \CE_m)\boxtimes \CZ_V\Bigr)\to
i_x{}_!(\CZ_W\underset{I}\star \CZ_V)\biggr).$$
- The restriction of to $\Fl_{G,\Delta_X}\simeq
\Fl_{G,X}$ identifies with $$\on{ker} \biggl(j_x{}_*j_x^*\Bigl((\CF_{V\otimes W,X}\otimes
\CE_n\otimes \CE_m)\boxtimes \delta_{1,\Fl_G}\Bigr)
\overset{\text{\eqref{n & m}}}\longrightarrow i_x{}_!
(\CZ_V\underset{I}\star \CZ_W)\biggr).$$
Hence, we obtain three maps $$\label{three maps downstairs, Iwahori}
j_*j^*\Bigl((\CF_{V,X}\otimes \CE_n)\boxtimes (\CF_{W,X}\otimes
\CE_m)\boxtimes \delta_{1,\Fl_G}\Bigr)\to
\Delta_!(\CZ_V\underset{I}\star \CZ_W),$$ whose some equals to zero. By lifting the terms of and the corresponding maps by means of $\pi_{\Fl}:\on{Jets}^\mer(G)_{X^2\times x}\to \Fl_{G,X^2}$, we obtain three maps $$j_*j^*\Bigl((\Gamma(\Gr_{G,X},\CF_{V,X})\otimes \CE_n)\boxtimes
(\Gamma(\Gr_{G,X},\CF_{W,X})\otimes \CE_m)\boxtimes
\delta_{I,G\ppart}\Bigr)\to \Delta_!\Bigl(\CZ_{V\otimes W}\underset{I}\star
\delta_{I,G\ppart}\Bigr).$$ We claim that these maps coincide with those of of [Proposition \[assoc, Iwahori\]]{} for $\CM=\delta_{I,G\ppart}$. This follows from [Lemma \[action as fusion\]]{} in the same way as in the proof of [Proposition \[comp with ten products\]]{}.
The case of a general $\CM$ follows from that of $\delta_{I,G\ppart}$ by the construction of the maps .
The proof will be parallel to that of [Theorem \[main, spherical\]]{}. It suffices to construct the isomorphisms $$\label{req iso}
\fs_{V}^{-1}:
\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}
\otimes \CM\simeq \CZ_V\underset{I}\star \CM,$$ for $\CM=\delta_{I,G\ppart}$, which commute with the right action of $\hg_\crit$, which intertwine the actions of $N_{\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}}$ and $N_V$, and which are compatible with tensor products of representations. Given $V\in
\Rep(\cG)$ and $\CM\in \hg_\crit\mod^I$ will construct the map $\fs_V^{-1}$ functorially, provided that $\CZ_V\underset{I}\star \CM$ is acyclic away from cohomological degree $0$ for all $V\in
\Rep(\cG)$.
Let $\CN_1$ and $\CN_2$ be two $\hg_\crit$-modules, such that the support of $\CN_1$ over $\Spec(\fZ_\fg)$ is in $\Spec(\wt\fZ^{\int,\nilp}_{\fg,x})$. The next assertion is deduced from [Lemma \[pairings over center\]]{} and [Proposition \[commutative fusion\]]{} as was the case of [Theorem \[repr of pair from ten product\]]{}:
\[com pairing, Iwahori\] Chiral pairings $$\{(\CV_{\fz_\fg}\underset{\fz_\fg}\otimes
\CA_{\fg,\crit})\underset{\CO_X}\otimes \CE_n,\CN_1\} \to \CN_2$$ are in bijection with maps of $\hg_\crit$-modules $$\Bigl(\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}
\underset{\wt\fZ^{\int,\nilp}_{\fg,x}} \otimes \CN_1\Bigr)
_{N_{\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}}^n}\to \CN_2.$$
More generally, for a local system $\CE'$ on $X-x$ with a nilpotent monodromy around $x$, chiral pairings $$\{(\CV_{\fz_\fg}\underset{\fz_\fg}\otimes
\CA_{\fg,\crit})\underset{\CO_X}\otimes \CE',\CN_1\} \to \CN_2$$ are in bijection with maps of $\hg_\crit$-modules $\Bigl(\Bigl(\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}
\underset{\wt\fZ^{\int,\nilp}_{\fg,x}} \otimes \CN_1\Bigr) \otimes
\CE'_x\Bigr)_{N_{\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}}+N_{\CE'}}\to
\CN_2$.
Setting $\CN_1=\CM$ and $\CN_2=\CZ_V\underset{I}\star \CM$ and taking $n>>0$ so that $N_{\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}}^n=0$, from the map , we produce the desired map $$\fs^{-1}_V:\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}
\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}\otimes \CM \to
\CZ_V\underset{I}\star \CM.$$
We claim that this map is compatible with tensor products of objects of $\Rep(\cG)$, i.e., that the following three maps coincide: $$\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}
\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}\otimes
\CW_{\wt\fZ^{\int,\nilp}_{\fg,x}}
\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}\otimes \CM
\overset{\fs^{-1}_V}\to \CZ_V \underset{I}\star
(\CW_{\wt\fZ^{\int,\nilp}_{\fg,x}}
\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}\otimes \CM)
\overset{\on{id}_{\CZ_V} \underset{I}\star \fs^{-1}_W}\to
\CZ_V\underset{I}\star \CZ_W \underset{I}\star \CM,$$ $$\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}
\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}\otimes
\CW_{\wt\fZ^{\int,\nilp}_{\fg,x}}\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}
\otimes
\CM \overset{\on{id}_{\CV}\otimes \fs^{-1}_W}\to
\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}
\otimes
(\CZ_W\underset{I}\star \CM)\overset{\fs^{-1}_V}\longrightarrow
\CZ_V\underset{I}\star \CZ_W\underset{I}\star \CM$$ and $$\CV_{\wt\fZ^{\int,\nilp}_{\fg,x}}\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}
\otimes
\CW_{\wt\fZ^{\int,\nilp}_{\fg,x}}\underset{\wt\fZ^{\int,\nilp}_{\fg,x}}
\otimes \CM \overset{\fs^{-1}_{V\otimes W}}\to \CZ_{V\otimes
W}\underset{I}\star \CM \simeq \CZ_V\underset{I}\star
\CZ_W\underset{I}\star \CM.$$ This follows from [Proposition \[assoc, Iwahori\]]{} and [Lemma \[add assoc\]]{} as in the proof of [Proposition \[map is assoc\]]{}.
Finally, the fact that the maps $\fs^{-1}_V$ are isomorphisms follows from the above compatibility with tensor products as in the proof of [Theorem \[main, spherical\]]{}.
[*Remark.*]{} Let us notice that the assertion of [Theorem \[representability, Iwahori\]]{} is stronger than that of its spherical counterpart, [Theorem \[representability, spherical\]]{}. Indeed, the former theorem, combined with [Lemma \[com pairing, Iwahori\]]{}, implies the existence of an isomorphism directly. In other words, the additional argument, involving duailization is not necessary.
Convolution and twisting for general chiral algebras, continued {#sect 7}
===============================================================
The goal of this section is to prove [Theorem \[representability, Iwahori\]]{} and give another interpretation of the map , parallel to what was done in [Sect. \[proof of rep\]]{} in the spherical case.
We shall first discuss a generalization of the construction of [Proposition \[global\]]{}. Let $\CA$ be a chiral algebra as in [Sect. \[general alg\]]{}, and let $\{\CM_1,\CM_2\}\to \CM_3$ be a chiral pairing between $\CA$-modules.
Let also $\CF'_1,\CF'_2,\CF'_3$ be chiral modules over $\fD_{\kappa,G}$, and $\{\CF'_1,\CF'_2\}\to \CF'_3$ be a chiral pairing.
\[very general\] In the above situation there exists a naturally defined chiral pairing $$\{\CF'_1\tboxtimes \CM_1,\CF'_2\tboxtimes \CM_2\} \to
\CF'_3\tboxtimes \CM_3,$$ compatible with the actions of $\CA$ and $\CA_{\fg,2\kappa_\crit}$.
Each $\CF'_i\tboxtimes \CM_i$ is isomorphic to $\CF'_i\otimes \CM_i$ for $i=1,2,3$ as a D-module, and we define the desired pairing as the one coming from . The fact that this pairing is compatible with the action of chiral algebras follows from the construction of $\CF'_i\tboxtimes \CM_i$ in [Sect. \[global case\]]{}.
By multiplying the terms of the pairing of [Proposition-Construction \[very general\]]{} by the chiral Clifford algebra we obtain a chiral pairing of complexes of $\CA$-modules: $$\label{fund pairing}
\{\fC^\semiinf(L_\fg,\CF'_1\otimes
\CM_1),\fC^\semiinf(L_\fg,\CF'_2\otimes \CM_2)\} \to
\fC^\semiinf(L_\fg,\CF'_3\otimes \CM_3).$$
Let us consider a particular case when $\CF'_1\simeq \fD_{G,\kappa}$, $\CF'_2\simeq \CF'_3=:\CF'$ and the pairing is given by the chiral action. Assume also that $\CM_1$ is $\on{Jets}(G)_X$-equivariant. In this case the map $\CM_1\to \CO_{\on{Jets}(G)_X}\otimes \CM_1$ induces a map of complexes of $\CA$-modules $$\CM_1\to \fC^\semiinf(L_\fg,\CF'_1\otimes \CM_1).$$
The resulting chiral pairing $$\{\CM_1, \fC^\semiinf(L_\fg,\CF'\otimes \CM_2)\}
\to \fC^\semiinf(L_\fg,\CF'\otimes \CM_3)$$ coincides with that of .
Let us assume now that in the previous set-up the chiral $\fD_{G,\kappa}$-module $\CF'_1$ is $\on{Jets}(G)_X$-equivariant on the right. Let $\CF'_2$ and $\CF'_3$ be both supported at $x\in X$, and assume that as twisted D-modules on $G\ppart$ they are both $I$-equivariant on the right.
Let $\CM_1$ be $\on{Jets}(G)_X$-equivariant, and let $\CM_2,\CM_3$ be supported at $x\in X$ and $I$-equivariant. In this case the pairing gives rise to a chiral pairing $$\label{fund pairing, Iwahori}
\{\fC^\semiinf\Bigl(L_\fg;\fg,\CF'_1\otimes \CM_1\Bigr),
\fC^\semiinf\Bigl(L_\fg;\fh,\CF'_2\otimes \CM_2\Bigr)\}
\to \fC^\semiinf\Bigl(L_\fg;\fh,\CF'_3\otimes \CM_3\Bigr).$$
Let us specialize further the case when $\CM_1\simeq \CA$, $\CM_2\simeq \CM_3:=\CM$, with the chiral pairing being given by the action. Let the terms of $\{\CF'_1,\CF'_2\}\to \CF'_3$ be those of for $n>>0$. We obtain a chiral pairing $$\{\fC^\semiinf\Bigl(L_\fg;\fg,\pi^*(\CF_{V,X-x})\otimes
\CA\Bigr)\otimes \CE_n,
\fC^\semiinf\Bigl(L_\fg;\fh,\delta_{I,G\ppart}\otimes \CM\Bigr)\}\to
\fC^\semiinf\Bigl(L_\fg;\fh,\pi_{\Fl}^*(\CZ_V)\otimes \CM\Bigr)$$
Let us assume now that $\kappa$ is such that $\CF_{V,X}\star \CA$ is acyclic away from cohomological degree $0$. Passing to the $0$-th cohomology in the previous expression, we obtain a functorial chiral pairing $$\label{main pairing, Iwahori}
\{\Bigl(\CF_{V,X-x}\star \CA)\otimes \CE_n\Bigr),\CM\}\to
h^0(\CZ_V\underset{I}\star \CM),$$ such that the action of $N_{\CE}$ on the LHS goes over to the action of $N_V$ on the RHS.
Parallel to the spherical case, we have the following generalization of [Theorem \[representability, Iwahori\]]{}:
\[gen repr, Iwahori\]
[*(1)*]{} For any $\CM\in \CA\mod_x^I$ the convolution $\CZ_V\underset{I}\star \CM$ is acyclic away from cohomological degree $0$.
[*(2)*]{} For all $n$ that are large enough, the covariant functor on $\CA\mod_x$ that sends an object $\CN$ to the set of chiral pairings $$\{\Bigl((\CF_{V,X-x}\star \CA)\otimes \CE_n\Bigr),\CM\}\to \CN$$ is representable by $\CZ_V\underset{I}\star \CM$.
The proof of this theorem is based on the following construction. Let $$\{\Bigl((\CF_{V,X-x}\star \CA)\otimes \CE_n\Bigr),\CM\}\to
\CN^\bullet$$ be a chiral pairing of complexes of $\CA$-modules, where $\CN^\bullet$ is a strongly $\on{Jets}(G)_X$-equivariant complex.
Then for $m>>0$, from we obtain a chiral pairing of complexes $$\{\fC^\semiinf\Bigl(L_\fg;\fg, (\pi^*(\CF_{V^*,X})\otimes \CE_m)
\otimes ((\CF_{V,X}\star \CA)\otimes \CE_n)\Bigr), \CM\}\to
\fC^\semiinf(L_\fg;\fh,\pi^*_{\Fl}(\CZ_{V^*})\otimes \CN^\bullet),$$ and by passing to the $0$-th cohomology and composing with $\delta_{1,\Gr_{G,X}}\to \CF_{V,X}\star \CF_{V^*,X}$ and $\CE_{m+n}\to
\CE_m\otimes \CE_n$, we obtain a chiral pairing $$\{(\CA\otimes \CE_{m+n}),\CM\}\to h^0(\CZ_{V^*}\underset{I}\star
\CN^\bullet).$$ Any such pairing factors through $\CE_{m+n}\twoheadrightarrow \CO_X$, and therefore corresponds to a map $\CM\to h^0(\CF_{V^*}\underset{I}\star \CN^\bullet)$. The rest of the argument repeats that of the the proof of [Theorem \[repr, general\]]{}.
Appendix: proof of [Proposition \[action as fusion\]]{} {#app A}
=======================================================
Let $\CF_X$ be a $\kappa$-twisted D-module on $\Gr_{G,X}$ and let $\CF'_X$ denote its pull-back to $\on{Jets}^\mer(G)_X$. Consider the D-module $({\bf 1}_{1,1})_!(\omega_X\boxtimes \CF_X)$ on $\Gr_{G,X^2}$. Corresponding to it there is a map $$j_*j^*(\delta_{1,\Gr_{G,X}}\boxtimes \CF_X)\to \Delta_!(\CF_X)$$ of D-modules on $\Gr_{G,X^2}$. Lifting this map by means of $\pi$ we obtain a map $$\label{eq act as fusion}
j_*j^*\Bigl(\fD_{G,\kappa}\boxtimes
\Gamma(\on{Jets}^\mer(G)_X,\CF'_X)\Bigr)\to
\Delta_!\Bigl(\Gamma(\on{Jets}^\mer(G)_X,\CF'_X)\Bigr).$$
Along with [Proposition \[action as fusion\]]{} we will prove the following:
\[action as fusion moving\] The map of equals the map corresponding to the chiral action of $\fD_{G,\kappa}$ on $\Gamma(\on{Jets}^\mer(G)_X,\CF'_X)$.
Applying this to $\CF_X=\delta_{1,\Gr_{G,X}}$, we obtain a description of the chiral bracket on $\fD_{G,\kappa}$ in terms of distributions on $\on{Jets}^\mer(G)_{X^2}$.
Since the chiral algebra $\fD_{G,\kappa}$ is generated by $\CO_{\on{Jets}(G)_X}$ and $L_{\fg,\kappa}$, to prove both [Proposition \[action as fusion\]]{} and [Proposition \[action as fusion moving\]]{}, it suffices to show that the two maps in question coincide when instead of $\fD_{G,\kappa}$, as one of the multiples in the LHS, we take $\CO_{\on{Jets}(G)_X}$ or $L_{\fg,\kappa}$.
The assertion concerning $\CO_{\on{Jets}(G)_X}$ follows tautologically from the definition of the group ind-scheme $\on{Jets}^\mer(G)_{X^2}$. In the case of $L_{\fg,\kappa}$ we shall discuss the set-up of [Proposition \[action as fusion\]]{}, while the case of [Proposition \[action as fusion moving\]]{} is similar. We need to establish the following:
Let $\bg$ be any point of $G\ppart/K$ and let ${\bf 1}_{1,1}(\bg)$ be the corresponding section $X\to \Gr_{G,X;K}$. The normal to this section, considered as a right D-module on $X-x$, has the property that $$\label{ident normal}
\CN_{{\bf 1}_{1,1}(\bg)}|_{X-x}\simeq L_\fg|_{X-x}\oplus
T_\bg(G\ppart/K)\otimes \omega_{X-x} \text{ and } \CN_{{\bf
1}_{1,1}(\bg)}|_x\simeq T_\bg(G\ppart/K).$$ In particular, we obtain a map $$j_x{}_*j_x^*(L_\fg)\to i_x{}_!\Bigl(T_\bg(G\ppart/K)\Bigr),$$ which is equivalent to a map $$\fg\ppart\simeq H^0_{DR}(\D^\times_x,L_\fg)\to T_\bg(G\ppart/K).$$
We need to show that the latter map equals the natural projection $$\fg\ppart\twoheadrightarrow \fg\ppart/T_\bg(G\ppart/K),$$ corresponding to the left action of $G\ppart$ on $G\ppart/K$. Using the above $G\ppart$-action, we reduce the assertion to the case when $\bg$ is the unit point of $G\ppart$, in which case $T_\bg(G\ppart/K)\simeq \fg\ppart/\sk$, where $\sk$ is the Lie algebra of $K$.
Thus, we need to show that the map $L_\fg|_{X-x}\to \CN_{{\bf
1}_{1,1}(\bg)}|_{X-x}$ extends to a map of D-modules $L_{\fg.\sk}\to
\CN_{{\bf 1}_{1,1}(\bg)}$, where $$L_{\fg.\sk}:=\on{ker}\Bigl(j_x{}_*j_x^*(L_\fg)\to
i_x{}_!(\fg\ppart/\sk)\Bigr).$$ This is a direct calculation performed below.
Without restriction of generality, we can assume that $X$ is affine, and let $\xi(x_1,x_2)$ be a $\fg$-valued map on $X\times X-(\Delta_X\sqcup X\times x)$. We will show that $\xi(x_1,x_2)$ gives rise to a vertical vector field on $\Gr_{G,X;K}$, relative to its projection onto $X$, and that the arising normal vector field to the unit section vanishes if and only if $\xi(x_1,x_2)$ has no poles at the diagonal, and for any fixed $x_1$ the Laurent expansion of $\xi(x_1,\cdot)$ at $x_2=x$, viewed as an element of $\fg\ppart$, belongs to $\sk$. This will imply the required assertion.
It will be more convenient to use a group-theoretic notation. I.e., we will think of $\xi(x_1,x_2)$ as a map $X\times X-(\Delta_X\sqcup
X\times x)\to G$, parameterized by the scheme of dual numbers. More generally, we will work with a map $\bg(x_1,x_2):\bigl(X\times
X-(\Delta_X\sqcup X\times x)\bigr)\times S\to G$ for an arbitrary scheme $S$.
To $\bg(x_1,x_2)$ as above we attach an $S$-valued automorphism of $\Gr_{G,X;K}$ as follows. Given a point $(x_1,\CP_G,\beta,\alpha)$ of $\Gr_{G,X;K}$ we define a new point by leaving $(x_1,\CP_G,\alpha)$ the same, but multiplying $\beta$ by $\bg(x_1,\cdot)$, thought of as a $G$-valued function on $X-(x_1\sqcup x)$.
Applying this automorphism to the unit section of $\Gr_{G,X;K}$, the resulting new $S\times X$-valued point of $\Gr_{G,X;K}$ will be isomorphic to the initial one if and only if there exists an $S\times
X$-valued automorphism of $\CP^0_G$, preserving $\alpha$, and whose value at any $x_1\in X$ equals that of $\bg(x_1,\cdot)$. But this precisely means that $\bg(x_1,x_2)$ extends regularly to the diagonal and $X\times x$, and the Taylor expansion of $\bg(x_1,\cdot)$ around $x$ belongs to $K$.
[199]{}
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[^1]: In fact, this notion makes sense for any pair of affine ind-schemes, one being a closed ind-subscheme of the other.
[^2]: Note that $\Gr_G$ has connected components labeled by elements of the fundamental group of $G$
[^3]: In other words, we first take the limit in the category of vector spaces, and then complete it in the natural topology.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'By detailed Molecular Dynamics and Monte Carlo simulations we show that granular materials at rest can be described as thermodynamics systems. First we show that granular packs can be characterized by few parameters, as much as fluids or solids. Then, in a second independent step, we demonstrate that these states can be described in terms of equilibrium distributions which coincide with the Statistical Mechanics of powders first proposed by Edwards. We also derive the system equation of state as a function of the “configurational temperature”, its new intensive thermodynamic parameter.'
author:
- Massimo Pica Ciamarra
- Antonio Coniglio
- Mario Nicodemi
title: Thermodynamics and Statistical Mechanics of dense granular media
---
Granular materials, such as sand or powders, for many respect are similar to fluids or solids [@Jaeger], even though in absence of external drive they rapidly come rest, due to strong dissipation and negligible thermal energy scales (they are non-thermal systems), in disordered states very similar to glasses [@Nicodemi97; @Liu1; @Liu2; @Silbert; @KurchanMakse; @Coniglio]. As standard thermodynamics is not applicable to describe them, it is natural to ask whether we can even refer to granular packs as “states". The problem of finding the correct theoretical framework where to describe granular media is in fact of deep relevance to civil engineering, geophysics and physics [@Jaeger; @Behringer; @Nedderman; @Siegfried].
Edwards proposed [@Edwards] a thermodynamic description for static granular media, which was partially investigated by recent experiments [@Knight95; @Nowak97; @Swinney05]. These experiments have established that a granular system subject to a tapping dynamics, such as subsequent mechanical oscillations of the container, may loose memory of its initial state and reach a stationary state of volume fraction only dependent on the tapping intensity, a precondition for a statistical mechanics description of static granular material to be possible. The study of out-of equilibrium (aging) slowly sheared granular assemblies is also useful for the validation of the statistical mechanics of granular media [@KurchanMakse], but it is inherently restricted to a small range of very high volume fractions, where the system is jammed [@potiguar].
Here we give strong evidences supporting the existence of a thermodynamical and statistical mechanical description of granular media. First we demonstrate, via Molecular Dynamics (MD) simulations, that granular packs at rest are genuine thermodynamic states, as they are characterized by a small set of parameteres regardless of the procedure with which they are generated. Then we show, via Monte Carlo (MC) simulations, that these states can be described in terms of the equilibrium distribution proposed by Edwards. The coincidence between time averages (MD) and ensemble averages (MC) is a strong evidence in favour of the statistical mechanics approach to granular media. For details of materials and methods, see the supplementary materials available online [@epaps].
[*MD simulations: time averages*]{} –
We run Molecular Dynamics simulations of $N=1600$ monodisperse spherical grains of diameter $d = 1$cm and mass $m=1$g. Grains, under gravity, are confined in a box with a square basis of length $L = 10$cm (see Fig. \[fig1\]), with periodic boundary conditions in the horizontal directions. The bottom of the box is made of other immobile, randomly displaced, grains (to prevent crystallization). Two grains in contact interact via a normal and a tangential force. The former is given by the spring-dashpot model, while the latter is implemented by keeping track of the elastic shear displacement throughout the lifetime of a contact [@Silbert1; @modello]. The coefficient of restitution is constant, $e= 0.8$.
![\[fig2\] (color online) Compaction of systems subject to a tapping dynamics with $\tau_0 = 0.03$s and different values of the fluid velocity as indicated, averaged over $32$ runs. Stars refers to simulations with $4$ times more particles, outlining the absence of finite size effects. Dashed lines are fits to a stretched exponential law, Eq. \[eq-se\].](fig2.eps)
As in a recent experiment [@Swinney05] the system is immersed in a fluid and, starting from a random configuration, it is subject to a dynamics made of a sequence of flow pulses where the fluid flows through the grains (see Fig. \[fig1\]). In a single pulse the flow velocity, directed against gravity, is $V>0$ for a time $\tau_0$; then the fluid comes to rest. We model the fluid-grain interaction [@King; @Crowe] via a viscous force proportional to the fluid grain relative velocity: ${\bf F}_{fg}= -A({\bf v} - {\bf V})$ where ${\bf v}$ is the grain and ${\bf V}$ is the fluid velocity. The prefactor $A={\gamma}{(1-\Phi_l)^{-3.65}}$ is dependent on the local packing fraction, $\Phi_l$, in a cube of side length $3d$ around the grain, and the constant [@Crowe] is $\gamma = 1$ Ns/cm.
During each pulse, grains are fluidized and then come to rest under the effect of gravity. The tapping dynamics, therefore, allows for the exploration of the phase space of the mechanically stable granular packs. When the system is subject to such a tap dynamics, it compactifies until it reaches a stationary state where its properties do not depend on the dynamics history. Fig. \[fig2\] shows that the volume fraction of our system increases by following a stretched exponential low, $$\label{eq-se}
\Phi(t) = \Phi_\infty - (\Phi_\infty - \Phi_0) \exp\left(-(t/\tau)^c\right),$$ in agreement with the experiment by P. Philippe [*et al.*]{} [@Philippe]. The relaxation time diverges as the tapping intensity decreases, indicating the presence of a glassy like behavior which will be discussed elsewhere [@futuro]. As the thermodynamics approach to granular media aims to describe stationary states, all of measures shown below (averaged over $32$ runs) are recorded after the application of a long sequence of flow pulses, when the system is at stationarity.
We plot in Fig. \[fig3\] the stationary values of the volume fraction, $\phi(V,\tau_0)$ (measured in the bulk of the system), and its fluctuations, $\Delta \phi(V,\tau_0)$, recorded after a sequence of such flow pulses of duration $\tau_0$ and velocity $V$. $\Delta \phi$ is by definition the standard deviation of $\phi$ around its average value at stationarity. Actually, the volume fraction probability distributions is Gaussian [@Swinney05; @epaps]. $\phi$ decreases with $V$ and with $\tau_0$: the stronger the pulse, i.e., the larger $V$ or $\tau_0$, the fluffier the pack settled after it. Similarly, $\Delta \phi$ increases with $V$ and $\tau_0$.
 [**Upper panel**]{} The $\Delta \phi$ data of Fig. \[fig3\] (same symbols are used) are here plotted as a function of $\phi$: they collapse on a single master curve, showing that $\Delta \phi$ is in a one-to-one correspondence with $\phi$, irrespectively of the dynamical protocol used to arrive at $\phi$. The dotted line is a linear best fit. A similarly good collapse is also found ([**lower panel**]{}) when we plot the radial distribution function $g(r)$ for packs having the same volume fraction (the same packs of Fig. \[fig3\] are shown with the same symbols). In particular, we show $g(r)$ for $\phi=0.615$ and $\phi=0.575$ (shifted for clarity). The empty circles in both panels are the corresponding ensemble averages independently calculated from Eq. \[pr\]: within numerical errors, they scale on the same curves, pointing out this Statistical Mechanics measure is an excellent approximation of the time averages over the pulse dynamics. ](fig4.eps)
Even though, $\phi$ depends on both the parameters of the dynamics, $V$ and $\tau_0$, we show now that such stationary states are indeed genuine “thermodynamic states”, i.e., they can be described, in this system, by one macroscopic parameter. Actually, the upper panel of Fig. \[fig4\] shows that when $\Delta \phi$ is parametrically plotted as function of $\phi$, the scattered data of Fig. \[fig3\] collapse, within numerical approximation, onto a single master function. This is a clear indication that $\Delta \phi$ and $\phi$ are in a one-to-one correspondence, no matter how the state with packing fraction $\phi$ is attained. Our claim is that such a property should be found for any macroscopic observable of the system: we checked some of them, including the energy and its fluctuations and the coordination number of grains. Actually, in Fig. \[fig4\] lower panel we show that the whole radial distribution function $g(r)$ of a pack is characterized only by its corresponding value of $\phi$, i.e., states attained with different dynamical protocols ($V,\tau_0$), but having the same $\phi$, have the same $g(r)$. From these results we derive our first conclusion: at stationarity, we can describe the pack with only one parameter, e.g., $\phi$, independently of the dynamical protocol. Such a parameter characterizes, thus, the “thermodynamics state” of the system.
[*MC simulations: ensamble averages*]{} – Our second important step is to identify the correct Statistical Mechanics distribution for these states. Under a very strong assumption (discussed for instance in [@Coniglio; @Brey00; @Edwards; @Dean01; @Coniglio01; @Barrat00; @Fierro02; @Ono02; @DeSmedt03; @Tarjus04; @Richard; @epaps]), Edwards proposed to use for the grains of a powder the standard machinery of Statistical Mechanics. He suggested, however, to consider a reduced configurational space: the system at rest (i.e., not in its “fluidized” regime) is described by a flat ensemble average restricted to its blocked configurations (i.e., its mechanically stable microstates). Under these hypotheses [@Coniglio01; @Fierro02; @epaps] the canonical ensemble probability, $P_r$, to find the blocked microstate $r$, of energy $E_r$, is: $$\begin{aligned}
P_r\propto e^{-\beta_{conf} E_r},
\label{pr}\end{aligned}$$ where the inverse of $\beta_{conf}$ is the conjugate parameter of the energy, called [configurational temperature]{}, $T_{conf}$. In order to check whether such a Statistical Mechanics scenario applies, we compared ensemble averages over the distribution of Eq. \[pr\] with those over the flow tap dynamics. For instance, the average value of $\phi$ over the distribution of Eq. \[pr\] is $$\langle \phi\rangle(T_{conf})=
\frac{\sum_r\phi_r\exp(- E_r/T_{conf})}{\sum_r\exp(- E_r/T_{conf})},$$ where the sum runs over all blocked microstates, and $\phi_r$ is the volume fraction of microstate $r$. We evaluated these ensemble averages by use of a Monte Carlo method which is an extension of that introduced in Ref. [@KurchanMakse] to the frictional case [@epaps] (frictional forces are essential to assure the stability of granular packs with small volume fraction). Fig. \[fig4\] shows, as empty circles, the functions $\langle \Delta \phi\rangle(\langle \phi\rangle)$ (resp. $\langle g(r) \rangle(\langle\phi\rangle)$) in the upper (resp. lower) panel. These ensemble averages collapse, to a very good approximation, on the same master function of the time averaged data from the flow-pulse dynamics discussed before (notice that there are no adjustable parameters). Thus, the present Statistical Mechanics description appears to hold, up to the current numerical accuracy, at least as a first very good approximation. Interestingly, the off-equilibrium dynamical effective temperature defined at high volume fractions from dynamical fluctuation-dissipation relations [@KurchanMakse] appears to coincide with the configurational temperature derived here at stationarity [@epaps].
![\[eq\_state\] (color online) The equation of state of granular materials at rest, showing the volume fraction, $\phi$, as a function of the configurational temperature, $T_{conf}$. Ensamble averages are obtained via the Monte Carlo procedure, while time averages are obtained from the data of Fig. \[fig3\] via the use of the static fluctuation dissipation relation [@epaps]. ](fig5.eps)
The function $\phi(T_{conf})$, derived from the above ensemble averages, is the system [equation of state]{}. We plot $\phi(T_{conf})$ in Fig. \[eq\_state\], where the data from MD simulations are included too by using the data collapse from Fig. \[fig4\] and integration of the static fluctuation-dissipation relations [@Knight95; @Coniglio01; @Fierro02; @Swinney05; @epaps].
[*Conclusions*]{} – Summarizing, in the present MD simulations of a non-thermal monodisperse granular system under flow pulses, we find that the stationary configurations of the system can be fully described by only one parameter, e.g., $\phi$, and can be, thus, considered genuine “thermodynamic states”. Within our numerical accuracy, we also showed that a Statistical Mechanics based on the distribution of Eq. \[pr\] is grounded to describe these “states”. We could derive as well the equation of state, $\phi(T_{conf})$, of the system.
These evidences strongly support the existence of a fundamental theory of dense granular media and address, thus, a variety of important issues in the next future, such as response functions in a granular system, mixing/segregation phenomena, the nature of their jamming transition and phase diagram [@Coniglio; @Liu1; @Richard; @Corwin].
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\
Contents:\
1. The Statistical Mechanics approach to granular media.
2. Configurational temperature and dynamical temperature.
3. Monte Carlo method.
[**Volume fraction fluctuations in the stationary state**]{}
[**Figure A**]{} Probability distributions of the volume fraction $\Phi$ in the steady state reached by the system subject to the flow pulses dynamics. The data, obtained by averaging over the stationary dynamics in $32$ different runs starting from random initial conditions, are Gaussian distributed (plain lines). The standard deviations of these distributions are the volume fraction fluctuations, $\Delta \Phi$, plotted in Fig. 3.
[**Statistical Mechanics approach to granular media**]{}\
In the Statistical Mechanics of powders introduced by S. Edwards (S.F. Edwards and R.S.B. Oakeshott, Physica A [**157**]{}, 1080, 1989) it is postulated that the system at rest can be described by suitable ensemble averages over its “mechanically stable” states. Edwards proposed a method to individuate the probability, $P_r$, to find the system in its mechanically stable state $r$, under the assumption that these mechanically stable states have the same a priori probability to occur. This is the simplest assumption one can imagine: it is the assumption of standard Statistical Mechanics (equiprobability of microstates) with the additional constraint of mechanical stability. A possible approach to find $P_r$ is as follow (A. Fierro [*et al.*]{}, Europhys. Lett. [**59**]{}, 642, 2002; Phys. Rev. E [**66**]{}, 061301, 2002; Europhys. Lett. [**60**]{} 684, 2002). $P_r$ is obtained as the maximum of the entropy, $$S = -\sum_r P_r \log P_r
\nonumber$$ with the macroscopic constraint, in the case of the canonical ensemble, of fixed system energy $E = \sum P_r E_r$ (here $E_r$ is the energy of the mechanically stable microstate $r$). For a granular medium at rest in the gravity field, $E_r$ is the sum of the gravitational energy, and of the interaction energy between grains: $$E_r = \sum_i mgz_i + \sum_{i\neq j} V_{ij},
\label{eq-energy}$$ where $z_i$ is the height of grain $i$ (with mass $m$) and $V_{ij}$ is the interaction (elastic energy) between grain $i$ and $j$ in the microstate $r$: $V_{ij} = 0$ if grains $i$ and $j$ are not in contact, otherwise $$V_{ij} = \frac{1}{2} k_n |\vec \delta_{ij}|^2 + \frac{1}{2} k_t |\vec u_{ij}|^2,$$ where $\vec \delta_{ij}$ is the overlap between grain $i$ and $j$, and $\vec u_{ij}$ is their shear displacement. The interaction energy $V_{ij}$ is derived by the normal and the tangential forces acting on the contacting grains (we use the model L3 described in L.E. Silbert [*et al.*]{}, Phys Rev E [**64**]{}, 051302, 2001): $\vec f^n = k_n \vec \delta_{ij}, \vec f^t = k_t\vec u_{ij}$.
From Edwards’ hypothesis, in analogy to the Gibbs result, you derive that: $$P_r = Z^{-1} \exp\left(-\beta_{conf}E_r\right)
\nonumber$$ where $\beta_{conf}$ is a Lagrange multiplier, called inverse configurational temperature, enforcing the above constraint on the energy: $$\beta_{conf} = \frac{\partial S_{conf}}{\partial E}; \;\;\; S_{conf} = \ln \Omega_{\rm stable}(E).
\nonumber$$ The configurational temperature is $T_{conf} = \beta_{conf}^{-1}$. Here $\Omega_{\rm stable}(E)$ is the number of mechanically stable states with energy $E$. $Z$ is fixed by the normalization condition $\sum_r P_r = 1$, where the sum is restricted to $\Omega_{\rm stable}$, i.e., to the mechanically stable states.\
\
We note here that for stiff grains the elastic energy $\sum_{i\neq j} V_{ij}$ is much smaller than the gravitational energy $\sum_i mgz_i$ (the elastic energy is strictly zero in hard sphere systems). Accordingly $E_r$ is to a good approximation proportional to the gravitational energy, i.e. to the volume of the system, as originally suggested by Edwards.
[**Configurational temperature and dynamical temperature**]{}\
We describe here the two different definitions of temperature for granular systems mentioned in our paper.\
\
[**Configurational temperature**]{}\
As discussed above, the configurational temperature is the inverse of the derivative of the configurational entropy with respect to the energy. It is therefore an ‘equilibrium’ temperature, defined for a granular system at stationarity under a given dynamics allowing the exploration of mechanically stable states.\
\
In our main text we have validated Statistical Mechanics approaches to powders by showing that, within numerical errors, the granular packs we consider are characterized, at stationarity, by a single parameter regardless of the dynamical procedure with which they were prepared (they are in a thermodynamic ‘state’), and that time averages coincide with ensemble averages over the distribution detailed above. This result justifies the use of Edwards ‘equilibrium’ partition function to make analytical calculations, and particularly to derive the following equilibrium fluctuation-dissipation relation (Coniglio and Nicodemi Physica A [**296**]{}, 451 (2001)), $$\beta_{conf}(E) = \beta_{conf}(E_0) - \int_{E_0}^{E} \frac{dE}{\Delta E^2}$$ which relates the energy and its fluctuations. This relation can be exploited to experimentally measure the configurational temperature (Nowak [*et al.*]{}, Powder. Tech. [**94**]{}, 79, 1997; Knight [*et al.*]{}, Phys. Rev. E [**57**]{}, 1971, 1998; Schröeter and Swinney, Phys. Rev. E [**71**]{}, 030301(R), 2005).\
\
[**Dynamical temperature**]{}\
In thermal glassy systems far from stationarity, dynamical off-equilibrium fluctuation-dissipation relations hold. Particularly, in the aging dynamics of mean-field glassy models in contact with a very small bath temperature, $T_{\rm bath}$, generalized out-of equilibrium fluctuation-dissipation relations were discovered where the role of the external temperature is played by a ‘dynamical temperature’ $T_{\rm dyn} \neq T_{\rm
bath}$, equal for all slow modes (L.F. Cugliandolo and J. Kurchan, Phys. Rev. Lett. [**71**]{}, 173, 1993). This scenario was later extended to aging granular materials (for a review see P. Richard [*et al.*]{}, Nat. Mat. [**4**]{}, 121, 2005, and references therein), were the dynamical temperature is defined and measured via a dynamical, off-equilibrium fluctuation dissipation relation, $$\nonumber
\langle (r(t)-r(t_w))^2 \rangle = T_{\rm dyn}\frac{\delta\langle r(t)-r(t_w) \rangle}{\delta f}$$ where $r$ is the position of a grain, $f$ a constant perturbing field, and $t_w$ the ‘waiting’ time.\
As the dynamical temperature is defined far from stationarity and, conversely, the configurational temperature at stationarity, Makse and Kurchan have shown that in granular packs, at very high density, the two are equal within numerical errors (Nature [**415**]{}, 614, 2002). At low volume fraction, on the contrary, the dynamical temperature appears to be no longer defined, as the granular system is no longer jammed and flows as soon as an external perturbation is applied, as shown by Potiguar and Makse (European Physical Journal E [**19**]{}, 171, 2006).
[**Monte Carlo method to test the Statistical Mechanics approach to granular materials at rest**]{} In order to test the Statistical Mechanics approach to granular media one has to compare time averaged data, obtained as explained in the text, with ensemble averaged data obtained by sampling Edwards distribution, eq.(2) in the text, $$P_r = Z^{-1}\exp(-\beta_{conf}E_r).
\label{eq-edwards}$$ where $E_r$ is the energy of the state $r$, $$E_r = \sum_i mgz_i + \sum_{i\neq j} V_{ij}.
\label{eq-energy1}$$
As the Statistical Mechanics approach to granular media deals with mechanically stable states, the phase space of interest (over which eq. (\[eq-edwards\]) must be sampled) is not the usual phase space, $\Omega_{\rm tot}\{\vec r, \vec v, \vec \omega \}$ (here $\vec r$ is the $3N$ vector of grains c.o.m. positions, $\vec v$ their velocities, and $\vec \omega$ their angular velocities). Instead it is the subset $\Omega_{\rm stable} \subset \Omega_{\rm tot}\{\vec r, 0, 0 \}$ of all states $r$ where the forces and the torques acting on each single grain sum to zero, and grains neither translate nor rotate. Since the states to be considered are so highly constrained it is difficult to sample the distribution of eq. (\[eq-edwards\]) via a standard Monte Carlo procedure. For instance, if a state $r$ is stable, then there is little chance to transform it in a new mechanically stable state $r'$ via the displacement of a single particle. Introducing many particles Monte Carlo moves is also useless as the probability of selecting a collective move that transform a mechanically stable state into a new mechanically stable state is practically zero. A Monte Carlo (MC) method to explore $\Omega_{\rm stable}$ was proposed by H.A. Makse and J. Kurchan, Nature [**415**]{} 614 (2002), which uses the following computational trick. The MC algorithm explores the usual phase space, $\Omega_{\rm tot}$, but in this phase space one introduces an auxiliary energy $E_{aux}$ which measures the degree of ‘mechanical instability’ of a pack in a microstate $r$. In the present case, we have defined: $$E_{aux} = \sum_{i=1}^N |m\vec{g}d + \sum_{j\neq i} (\vec f_{ij}^n + \vec f_{ij}^t)| + \sum_{i=1}^N |\vec{T}_i|,
\label{eq-eaux}$$ where $\vec{T}_i$ is the total torque acting on grain $i$. The first term of the above equation enforce the balance of forces on each single grain, and the second the balance of torques. Different expressions could be used for $E_{aux}$, the important point being that $E_{aux}(r) \ge 0$ $\forall r$, and that $E_{aux}(r) = 0$ if and only if the state $r$ is mechanically stable.
Then one samples via a standard Monte Carlo procedure the distribution $$\overline{P}_r = \overline{Z}^{-1}\exp(-\beta_{conf}E_r-\beta_{aux}E_{aux}(r)),
\label{eq-prob-aux}$$ where $T_{aux}=\beta_{aux}^{-1}$ is the so-called ‘auxiliary temperature’ which controls the equilibrium value of the auxiliary energy. By definition of $E_{aux}$, in the limit $T_{aux}\to 0$ we have: $$P_r = \lim_{T_{aux}\to 0} \overline{P}_r,\;\;\;
\lim_{T_{aux}\to 0} E_{aux} = 0.$$ Therefore in the limit $T_{aux} \to 0$ we sample the distribution of mechanically stable states ($E_{aux} = 0$) with probability $P_r$, as desired. This is precisely the limit which is considered in by Makse and Kurchan method.
In our simulations we start by sampling eq. (\[eq-prob-aux\]) in the phase space of all granular packs at the desired value of $T_{conf}$ and at $T_{aux} > 0$ via a Monte Carlo procedure (see below). Then we slowly decrease the auxiliary temperature (carefully checking for thermalization) until $T_{aux} = 0$. By repeating this procedure severals times we generate several packs (a total of $172$) at the desired configurational temperature. These packs are then used to compute ensemble averages relative to the chosen value of $T_{conf}$.\
\
[**Implementation of the Monte Carlo method**]{}\
In our definition of mechanical stability (eq. (\[eq-eaux\])) we also include tangential forces (neglected in H.A. Makse and J. Kurchan, Nature [**415**]{} 614, 2002). Tangential forces model friction, which is essential for our purposes since, under gravity, frictionless stable packs are only found for high volume fractions.
In MD simulations friction has important effects on the dynamics of the system: the frictional force between two grains depends on their shear displacement. In the Monte Carlo algorithm, however, it is convenient to consider the frictional force between two grains (that is their shear displacement) as an independent variable. This leads to the definition of two kind of MC moves: standard single particle displacement, and variation of the tangential shear displacement. The idea of separating geometrical properties of the pack (particle displacement) from the tangential forces (tangential shear variation) has been exploited previously in the literature, even though in a different contest and with simpler models (as for example in T.Unger, J.Kertész, and D. E. Wolf, Phys. Rev. Lett. [**94**]{}, 178001 (2005); S. McNamara, R. Garcma-Rojo, and H. Herrmann, Phys. Rev. E [**72**]{}, 021304 (2005)). Below we discuss briefly the two moves.\
\
[**Single-particle displacement**]{}\
One selects a particle $n$ in position $\vec r_n$ and a random displacement vector $\vec \Delta$, with $|\vec \Delta| < \lambda$ and $\lambda$ dynamically varied in order to obtain an acceptance probability $p_{\rm acc} = 0.5$. The displacement of particle $n$ from position $\vec r_n$ to position $\vec r_n + \vec \Delta$ induces a variation $\Delta E_r$ of the energy (eq. (\[eq-energy\])), and of the auxiliary energy, $\Delta E_{aux}$ (eq. (\[eq-eaux\])), of the system. $\Delta E_{aux}$ is due to:
1. changes of the overlaps $\vec \delta_{ni}$ between the displaced particle $n$ and a particle $i$, and therefore of the normal force $\vec f_{ni}^{n}$. In particular contacts may disappear (after the displacement $| \vec \delta_{ni} < 0|)$, or appear (before the displacement $|\vec \delta_{ni} < 0|$, after the displacement $|\vec \delta_{ni} > 0|)$.
2. variations of the shear displacements $\vec u_{ni}$, and therefore of the tangential force $\vec f_{ni}^{t}$. If particles $i$ and $n$ are in contact both before and after the displacement of particles $n$, we consider particle $n$ as sliding over particle $i$, inducing a variation of the shear displacement $\vec u_{ni}$. If the displacement of particle $n$ creates (destroys) a contact, then the shear displacement $\vec u_{ni}$ is created (destroyed) accordingly.
The shear displacement is rescaled by a factor $\mu |\vec f^n_{ij}| / k_t |\vec u_{ni}|$ if the Coulomb condition is violated. A move of this kind is accepted with probability $\exp(-\beta_{conf}\Delta E_r - \beta_{aux}\Delta E_{aux})$.\
\
[**Shear-displacements variation**]{}\
In this MC move one varies the tangential force between two contacting grains, respecting the Coulomb criterion. Two particles $n$ and $m$ are selected. If they are in contact one varies their shear displacement $\vec u_{nm}$ of a random amount $\vec \Delta_t$, with $|\vec \Delta_t| < \sigma$ and $\sigma$ dynamically varied in order to obtain an acceptance probability $p_{\rm acc} = 0.5$. $\Delta_t$ is actually chosen in such a way that $\vec u_{nm}+\Delta_t$ (and therefore the tangential force) lies in the plane tangent to both grains in their point of contact. Before varying $\vec u_{nm}$ by $\vec \Delta_t$ this latter is rescaled in order to satisfy the Coulomb criterion, if necessary.
The tangential move leads to a variation of both the energy of the system, $E_r$ (as the elastic energy in tangential interaction between the two particles varies), and of its auxiliary energy, $E_{aux}$. As before, the move is accepted with probability $\exp(-\beta_{conf}\Delta E_r - \beta_{aux}\Delta E_{aux})$.\
\
[**Generation of mechanically stable states with the MC algorithm**]{}
\
Here we show how the volume fraction of the system evolves during the MC procedure. First we thermalize the system at the desired value of $T_{conf}$ and at $T_{aux} = 0.04$ mgd (upper panel). Different values of $T_{aux}$ may be convenient in different runs. Then we slowly decrease the auxiliary temperature until the system reaches a mechanically stable state (lower panel). We check that the same results are obtained with slower cooling rates.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a general numerical approach to construct local Kohn-Sham potentials from orbital-dependent functionals within the all-electron full-potential linearized augmented-plane-wave (FLAPW) method, in which core and valence electrons are treated on an equal footing. As a practical example, we present a treatment of the orbital-dependent exact-exchange (EXX) energy and potential. A formulation in terms of a mixed product basis, which is constructed from products of LAPW basis functions, enables a solution of the optimized-effective-potential (OEP) equation with standard numerical algebraic tools and without shape approximations for the resulting potential. We find that the mixed product and LAPW basis sets must be properly balanced to obtain smooth and converged EXX potentials without spurious oscillations. The construction and convergence of the exchange potential is analyzed in detail for diamond. Our all-electron results for C, Si, SiC, Ge, GaAs semiconductors as well as Ne and Ar noble-gas solids are in very favorable agreement with plane-wave pseudopotential calculations. This confirms the adequacy of the pseudopotential approximation in the context of the EXX-OEP formalism and clarifies a previous contradiction between FLAPW and pseudopotential results.'
author:
- Markus Betzinger
- Christoph Friedrich
- Stefan Blügel
- Andreas Görling
bibliography:
- 'biblio.bib'
title: 'Local exact exchange potentials within the all-electron FLAPW method and a comparison with pseudopotential results'
---
Introduction
============
Its wide applicability, accuracy, and computational efficiency have made density-functional theory (DFT)[@Hohenberg-Kohn; @DFT-review] the standard method for describing the ground state of many-electron systems. The vast majority of practical calculations employ the Kohn-Sham (KS) formalism,[@Kohn-Sham] where the interacting many-electron system is mapped onto an auxiliary noninteracting system. In this KS system the noninteracting electrons move in a local effective potential that is defined in such a way that the electron densities of the real and auxiliary systems coincide. This potential is the sum of the external, Hartree, and the exchange-correlation (xc) potentials. The latter two contributions take into account the electron-electron interactions including, in an indirect way, all many-body effects. The form of the density-functional for the xc potential, which can further be divided into an exchange and a correlation term, is unknown and must be approximated in practice.
Fortunately, already simple approximations, like the local-density approximation (LDA)[@LDA-Ceperley/Adler; @LDA-VWN] or generalized gradient approximation (GGA),[@GGA-PBE; @GGA-PW] give reliable results for a wide range of materials and properties. Nevertheless, the LDA and GGA suffer from several shortcomings. First, the electrostatic interaction of the electron with the total electron charge, described by the Hartree potential, contains an unphysical interaction of the electron with itself, commonly referred to as Coulomb self-interaction. This extra term should be compensated exactly by an identical term with opposite sign in the exchange potential, in the same manner as in Hartree-Fock theory. However, as the LDA and GGA exchange potentials are only approximate, this cancellation is incomplete and part of the self-interaction remains. This error leads, in particular, to an improper description of localized states, which appear too high in energy and tend to delocalize. Second, the LDA and GGA functionals do not give rise to a discontinuity of the xc potential with respect to changes in the particle number. This discontinuity should in general be finite (and positive), as it corresponds to the difference of the real and the KS band gap.[@KS-GAP1; @KS-GAP2] The latter is well known to underestimate experimental gaps by typically 50% or even more. This is often called the band-gap problem of LDA and GGA. The significance of the discontinuity for a meaningful prediction of the fundamental band gap is discussed in Refs. and .
Functionals that depend explicitly on the electron orbitals and thus only implicitly on the electron density form a new generation of xc functionals.[@Review:O-functionals; @Review-Goerling] Already the simplest variant, the EXX functional,[@EXX1; @EXX2; @EXX3] which treats electron exchange exactly but neglects correlation altogether, remedies the aforementioned shortcomings of the more conventional local and semilocal functionals: the Coulomb self-interaction term is exactly canceled, and the local EXX potential exhibits a nonzero discontinuity at integral particle numbers because of the orbital dependence.
After applications to atoms[@OEP-Talman-Shadwick; @OEP-atoms; @OEP-atoms-1] the first implementation for periodic systems was published in 1994 by Kotani[@EXX-LMTO-Kotani-I] who employed the atomic sphere approximation within the linearized muffin-tin orbital method. In this approximation only the spherical part of the potential around each atom is taken into account. The KS band gap turned out to be closer to experiment than in LDA or GGA.[@EXX-LMTO-Kotani-I; @EXX-LMTO-Kotani-II] This indicates that the xc discontinuity and the effect of neglecting electron correlation for these systems are roughly of the same magnitude but of different sign and thus tend to cancel each other. Later, results even closer to experimental values were obtained from plane-wave calculations[@EXX-PP-Staedele-1; @EXX-PP-Staedele-2; @EXX-PP-Fleszar] employing the pseudopotential (PP) approximation, which allows accurate treatment of the warped shape of the EXX potential except for the regions close to the atomic nuclei where the potential is smoothed.
However, when the first all-electron (AE), full-potential results were reported,[@Sharma] they deviated substantially from the PP values and were also in considerably worse agreement with experiment. In Ref. , Sharma *et al.*, who implemented the EXX functional within the FLAPW method, argue that the success of EXX in the earlier PP calculations is only an artifact of the neglect of the core-valence exchange interaction. They conclude that treatment of core and valence electrons on the same footing is imperative for a proper EXX calculation. This work started a controversy about the adequacy of the PP approximation in the EXX formalism. Recently, Engel[@Engel] showed that, on the contrary, AE and PP results for lithium and diamond differ only marginally. The AE calculations were performed with a plane-wave basis by pushing the PP and plane-wave cutoffs to the AE limit. Also recently, Makmal *et al.* reported a similarly good agreement between AE and PP calculations for the diatomic molecules BeO and CO using real-space grid approaches.[@Makmal]
A final comparison between PP results and results obtained from a genuine AE approach for periodic systems, such as the FLAPW method, is still missing, though. With this work we want to fill this gap. We present an alternative implementation of the EXX approach within the FLAPW method, which uses a numerical approach different from the one reported in Ref. . It employs a specifically designed basis, the mixed product basis, in which the optimized-effective-potential (OEP) equation for the EXX potential is solved. The mixed product functions form an all-electron basis for the products of single-particle wave functions occurring in the OEP equation. Within this approach, both spherical and nonspherical as well as warped interstitial contributions to the EXX potential are fully taken into account. For the example of diamond, we discuss in detail the convergence of the local EXX potential and the resulting KS band gaps with respect to the quality of the mixed product and LAPW basis sets. We demonstrate that a smooth potential and a direct KS band gap very close to the result by Engel[@Engel] are obtained if the two basis sets are properly balanced. A similar behavior has been reported for Gaussian and plane-wave basis sets.[@Balance-Gaussian; @Hesselmann; @Comparison-OEP-KS] To validate our findings, we also report results for Si, SiC, Ge, GaAs, and crystalline Ne and Ar. For all materials, we find a very good agreement between our AE and previously published plane-wave PP values. We conclude that the large discrepancies found in Ref. cannot be attributed to the core-valence interaction.
The paper is organized as follows. Sections \[sec: Theory\] and \[sec: FLAPW-Method\] give brief introductions into the theory and the FLAPW method. Our implementation of the EXX functional within the FLAPW program [<span style="font-variant:small-caps;">Fleur</span>]{}[@Fleur] is described in Section \[sec: Implementation\]. Section \[sec: Results-&-Discussion\] discusses the convergence of the effective potential for the example of diamond and compares AE EXX KS eigenvalue differences commonly interpreted as transition energies for C, Si, SiC, Ge, GaAs, crystalline Ne, and Ar with theoretical plane-wave and experimental values from the literature. Finally, we draw our conclusions in Section \[sec: Conclusions\].
Theory\[sec: Theory\]
=====================
The KS formalism[@Kohn-Sham] of DFT[@Hohenberg-Kohn] relies on an auxiliary system of noninteracting electrons, which move in the spin-dependent local effective potential $$V_{\mathrm{eff}}^{\sigma}({\mathbf{r}})=V_{\mathrm{ext}}({\mathbf{r}})+V_{\mathrm{H}}({\mathbf{r}})+V_{\mathrm{xc}}^{\sigma}({\mathbf{r}})$$ with the external, Hartree, and xc potential, respectively. The latter is defined in such a way that the electron spin densities coincide with those of the real interacting system. It is given by the functional derivative of the xc energy functional $E_{\mathrm{xc}}[n^{\uparrow},n^{\downarrow}]$ with respect to the electron spin density $n^{\sigma}({\mathbf{r}})$ $(\sigma=\uparrow,\downarrow)$,$$V_{\mathrm{xc}}^{\sigma}({\mathbf{r}})=\frac{\delta E_{\mathrm{xc}}}{\delta n^{\sigma}({\mathbf{r}})}\,.\label{eq: definition V_xc}$$ The orbitals describing the electrons in the auxiliary system obey the KS equations$$\left[-\frac{1}{2}\nabla^{2}+V_{\mathrm{eff}}^{\sigma}({\mathbf{r}})\right]\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}})=\epsilon_{n{\mathbf{k}}}^{\sigma}\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}})\,,\label{eq: KS Hamiltonian}$$ where $\varphi_{n{\mathbf{k}}}^{\sigma}$ denotes the KS orbital of spin $\sigma$, band index $n$ and Bloch vector ${\mathbf{k}}$. Hartree atomic units are used except where explicitly noted. The electron spin density is given by a sum over the occupied states $$\begin{aligned}
n^{\sigma}({\mathbf{r}}) & = & \sum_{{\mathbf{k}}}\sum_{n}^{\mathrm{occ.}}|\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}})|^{2}\,.\label{eq: density}\end{aligned}$$ By a summation over Bloch vectors we mean an integration over the Brillouin zone, which is sampled by a finite set of mesh points.
For the conventional LDA and GGA functionals, the xc energy functional depends locally on the spin densities and, in the case of GGA, on the their gradients, and the functional derivative in Eq. translates to a derivative of a function and is evaluated in a straightforward way. However, for orbital-dependent functionals, $E_{\mathrm{xc}}$ depends only indirectly on the electron spin densities: the KS orbitals, which define $E_{\mathrm{xc}}\left[\varphi^{\uparrow},\varphi^{\downarrow}\right]$, are functionals of the effective potential $V_{\mathrm{eff}}^{\sigma}({\mathbf{r}})$ through Eq. , and $V_{\mathrm{eff}}^{\sigma}({\mathbf{r}})$ is a functional of $n^{\sigma}$. Therefore, one must apply the chain rule to calculate the functional derivative in Eq. $$\begin{aligned}
\lefteqn{V_{\mathrm{xc}}^{\mathrm{\sigma}}({\mathbf{r}})}\label{eq: V_xc chain rule}\\
& = & \sum_{n,{\mathbf{k}}}\iint\left[\frac{\delta E_{\mathrm{xc}}}{\delta\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}}')}\frac{\delta\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}}')}{\delta V_{\mathrm{eff}}^{\sigma}({\mathbf{r}}'')}+\mathrm{c.c.}\right]\frac{\delta V_{\mathrm{eff}}^{\sigma}({\mathbf{r}}'')}{\delta n^{\sigma}({\mathbf{r}})}d^{3}r'\, d^{3}r''\nonumber \end{aligned}$$ where the sum runs over all KS states present in $E_{\mathrm{xc}}$. Then, multiplication with the single-particle spin-density response function$$\chi_{\mathrm{s}}^{\sigma}({\mathbf{r}},{\mathbf{r}}')=\frac{\delta n^{\sigma}({\mathbf{r}})}{\delta V_{\mathrm{eff}}^{\sigma}({\mathbf{r}}')}\,,\label{eq: response function}$$ integration, and use of $\chi_{\mathrm{s}}^{\sigma}({\mathbf{r}},{\mathbf{r}}')=\chi_{\mathrm{s}}^{\sigma}({\mathbf{r}}',{\mathbf{r}})$ yields an integral equation for the xc potential$$\begin{aligned}
\lefteqn{\int\chi_{\mathrm{s}}^{\sigma}({\mathbf{r}},{\mathbf{r}}')V_{\mathrm{xc}}^{\sigma}({\mathbf{r}}')d^{3}r'}\nonumber \\
& = & \sum_{{\mathbf{k}}}\sum_{n}\int\left[\frac{\delta E_{\mathrm{xc}}}{\delta\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}}')}\frac{\delta\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}}')}{\delta V_{\mathrm{eff}}^{\sigma}({\mathbf{r}})}+\mathrm{c.c.}\right]d^{3}r'\,.\label{eq: general OEP equation}\end{aligned}$$ In this work we employ, as a practical example, the orbital-dependent EXX functional
$$\begin{aligned}
E_{\mathrm{x}} & = & -\frac{1}{2}\sum_{\sigma}\sum_{{\mathbf{k}},{\mathbf{q}}}\sum_{n,n'}^{\mathrm{occ.}}\iint d^{3}r\, d^{3}r'\nonumber \\
& & \times\frac{\varphi_{n{\mathbf{k}}}^{\sigma*}({\mathbf{r}})\varphi_{n'{\mathbf{q}}}^{\sigma}({\mathbf{r}})\varphi_{n'{\mathbf{q}}}^{\sigma*}({\mathbf{r}}')\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}}')}{|{\mathbf{r}}-{\mathbf{r}}'|}\label{eq: EXX functional}\end{aligned}$$
whose functional derivative with respect to the KS wave functions is given by the well-known Hartree-Fock expression$$\begin{aligned}
\frac{\delta E_{\mathrm{x}}}{\delta\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}}')} & = & \int\varphi_{n{\mathbf{k}}}^{\sigma*}({\mathbf{r}}'')V_{\mathrm{x},\mathrm{NL}}^{\sigma}({\mathbf{r}}'',{\mathbf{r}}')d^{3}r''\label{eq: HF term}\end{aligned}$$ with$$V_{\mathrm{x,NL}}^{\sigma}({\mathbf{r}}'',{\mathbf{r}}')=-\sum_{{\mathbf{q}}}\sum_{n'}^{\mathrm{occ.}}\frac{\varphi_{n'{\mathbf{q}}}^{\sigma}({\mathbf{r}}'')\varphi_{n'{\mathbf{q}}}^{\sigma*}({\mathbf{r}}')}{|{\mathbf{r}}'-{\mathbf{r}}''|}\,.\label{eq: nonlocal HF kernel}$$ First-order perturbation theory yields the wave-function response $$\frac{\delta\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}})}{\delta V_{\mathrm{eff}}^{\sigma}({\mathbf{r}}')}=\sum_{n'(\ne n)}\frac{\varphi_{n'{\mathbf{k}}}^{\sigma*}({\mathbf{r}}')\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}}')}{\epsilon_{n{\mathbf{k}}}^{\sigma}-\epsilon_{n'{\mathbf{k}}}^{\sigma}}\varphi_{n'{\mathbf{k}}}^{\sigma}({\mathbf{r}})\label{eq: wave-function response}$$ and together with Eq. the spin-density response function $$\chi_{s}^{\sigma}({\mathbf{r}},{\mathbf{r}}')=2\sum_{{\mathbf{k}}}\sum_{n}^{\mathrm{occ.}}\sum_{n'}^{\mathrm{unocc.}}\frac{\varphi_{n{\mathbf{k}}}^{\sigma*}({\mathbf{r}})\varphi_{n'{\mathbf{k}}}^{\sigma}({\mathbf{r}})\varphi_{n'{\mathbf{k}}}^{\sigma*}({\mathbf{r}}')\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}}')}{\epsilon_{n{\mathbf{k}}}^{\sigma}-\epsilon_{n'{\mathbf{k}}}^{\sigma}}\label{eq: response explicit}$$ where time-reversal symmetry has been used. Using Eqs. , , and the integral equation [\[]{}Eq. \] turns into$$\begin{aligned}
\int\chi_{\mathrm{s}}^{\sigma}({\mathbf{r}},{\mathbf{r}}')V_{\mathrm{x}}^{\sigma}({\mathbf{r}}')d^{3}r' & = & t^{\sigma}({\mathbf{r}})\label{eq: x OEP equation}\end{aligned}$$ with$$\begin{aligned}
t^{\sigma}({\mathbf{r}}) & =\frac{\delta E_{\mathrm{x}}}{\delta V_{\mathrm{eff}}^{\sigma}({\mathbf{r}})}\nonumber \\
& =2\sum_{{\mathbf{k}}}\sum_{n}^{\mathrm{occ.}}\sum_{n'}^{\mathrm{unocc.}}\left[\langle\varphi_{n{\mathbf{k}}}^{\sigma}|V_{\mathrm{x,\mathrm{NL}}}^{\sigma}|\varphi_{n'{\mathbf{k}}}^{\sigma}\rangle\frac{\varphi_{n'{\mathbf{k}}}^{\sigma*}({\mathbf{r}})\varphi_{n{\mathbf{k}}}^{\sigma}({\mathbf{r}})}{\epsilon_{n{\mathbf{k}}}^{\sigma}-\epsilon_{n'{\mathbf{k}}}^{\sigma}}\right]\label{eq: rhs t explicit}\end{aligned}$$ and$$\begin{aligned}
\lefteqn{\langle\varphi_{n{\mathbf{k}}}^{\sigma}|V_{\mathrm{x,\mathrm{NL}}}^{\sigma}|\varphi_{n'{\mathbf{k}}}^{\sigma}\rangle}\label{eq: matrix elements of V_x_NL}\\
& = & \iint\varphi_{n{\mathbf{k}}}^{\sigma*}({\mathbf{r}})V_{\mathrm{x,NL}}^{\sigma}({\mathbf{r}},{\mathbf{r}}')\varphi_{n'{\mathbf{k}}}^{\sigma}({\mathbf{r}}')d^{3}r\, d^{3}r'\,.\nonumber \end{aligned}$$ In this form the integral equation is called OEP equation and goes back to Sharp and Horton,[@OEP/OPM] who derived Eq. as a result of a variational minimization of the Hartree-Fock total energy under the additional constraint that the orbitals experience a local rather than a nonlocal potential. Sahni *et al.* finally realized that the OEP approach is equivalent to the construction of a local EXX potential within the KS formalism.
FLAPW Method\[sec: FLAPW-Method\]
=================================
The LAPW basis[@FLAPW1; @FLAPW2; @FLAPW3] is constructed from piecewise defined functions to deal, at the same time, with the atomic-like potential close to the nuclei and the smooth potential in the region far away from the nuclei. For this purpose, space is partitioned into nonoverlapping atom-centered muffin-tin (MT) spheres and the remaining interstitial region (IR), where the smoothness of the potential allows to employ plane waves as basis functions. At the MT sphere boundaries, these plane waves are matched in value and first radial derivative to linear combinations of spin-dependent MT solutions $u_{l0}^{a\sigma}(r)Y_{lm}(\hat{{\mathbf{r}}})$ of the radial scalar-relativistic Dirac equation and their energy derivatives $u_{l1}^{a\sigma}(r)Y_{lm}(\hat{{\mathbf{r}}})$ using the spherical average of the effective potential and predefined energy parameters that lie in the energy range of the occupied states. Here, $Y_{lm}(\hat{{\mathbf{r}}})$ denote the spherical harmonics, ${\mathbf{r}}$ is measured from the MT center of atom $a$ and $\hat{{\mathbf{r}}}={\mathbf{r}}/r$ is a unit vector. This gives the LAPW basis functions
$$\phi_{{\mathbf{k}}{\mathbf{G}}}^{\sigma}({\mathbf{r}})=\left\{ \begin{array}{cl}
\frac{1}{\sqrt{\Omega}}\exp\left[i({\mathbf{k}}+{\mathbf{G}})\cdot{\mathbf{r}}\right] & \mathrm{if\,}{\mathbf{r}}\in\mathrm{IR}\\
\sum_{l=0}^{l_{\mathrm{max}}}\sum_{m=-l}^{l}\sum_{p=0}^{1}A_{lmp}^{\sigma}({\mathbf{k}},{\mathbf{G}})u_{lp}^{a\sigma}(|{\mathbf{r}}-{\mathbf{R}}_{a}|)Y_{lm}(\widehat{{\mathbf{r}}-{\mathbf{R}}_{a}}) & \mathrm{if\,}{\mathbf{r}}\in\mathrm{MT}(a)\end{array}\right.\label{eq: APWs}$$
for the valence electrons with the unit-cell volume $\Omega$ and reciprocal lattice vectors ${\mathbf{G}}$. For a practical calculation cutoff values for the reciprocal lattice vectors $|{\mathbf{k}}+{\mathbf{G}}|\le G_{\mathrm{max}}$ and the angular momentum $l\le l_{\mathrm{max}}$ are employed. The core states are obtained by solving the fully relativistic Dirac equation with the spherical average of the effective potential.
The basis functions, defined in Eq. , can represent only those wave functions accurately whose energies are sufficiently close to the energy parameters, which are usually located in the valence-band region. For a precise description of semicore and high-lying unoccupied states that are far away from the energy parameters the basis must be augmented and local orbitals (lo)[@Local_Orbitals1; @Local_Orbitals2; @Local_Orbitals3] are currently the best developed technique. Let us assume that we want to improve the basis for states with an angular momentum $l$ around an energy $\epsilon^{\mathrm{lo}}$. Then we construct an additional radial function $u_{l}^{a\sigma}(r,\epsilon^{\mathrm{lo}})$ from the radial scalar-relativistic Dirac equation with the energy parameter $\epsilon^{\mathrm{lo}}$ and form a linear combination $u_{lp}^{a\sigma}(r)Y_{lm}(\hat{{\mathbf{r}}})$ ($p\ge2$, the index $p$ is a label numbering the basis functions for a given $l$, $a$, and $\sigma$) from $u_{l}^{a\sigma}(r,\epsilon^{\mathrm{lo}})$ and the radial functions $u_{l0}^{a\sigma}(r)$ and $u_{l1}^{a\sigma}(r)$, already defined above, such that $u_{lp}^{a\sigma}(r)$ is normalized and its value and radial derivative vanish at the MT boundary. In this way, the local orbital $u_{lp}^{a\sigma}(r)Y_{lm}(\hat{{\mathbf{r}}})$ is completely confined to the MT sphere and need not be matched to a plane wave outside. For semicore states, which are nearly dispersion-less, the energy parameter $\epsilon^{\mathrm{lo}}$ is fixed at the semicore energy level. For the unoccupied states we use energy parameters chosen such that the solutions of the radial scalar-relativistic Dirac equation fulfill$$\left.\frac{d}{dr}\mathrm{ln}[u_{l}^{a\sigma}(r,\epsilon^{\mathrm{lo}})]\right|_{r=S}=-(l+1)$$ at the MT sphere boundary $r=S$, following a procedure proposed in Ref. . This condition yields for each $l$ quantum number a series of orthogonal solutions of increasing energies. We use the resulting local orbitals to converge the LAPW basis in a systematic way.
Implementation\[sec: Implementation\]
=====================================
To solve the integral equation [\[]{}Eq. \], we introduce a basis $\{ M_{I}({\mathbf{r}})\}$ that reformulates the equation as a linear-algebra problem $$\sum_{J}\chi_{\mathrm{s},IJ}^{\sigma}V_{\mathrm{x},J}^{\sigma}=t_{I}^{\sigma}\,,\label{eq: algebraic OEP}$$ which can be solved for the exchange potential $V_{\mathrm{x}}^{\sigma}$ by matrix inversion of $\chi_{\mathrm{s}}^{\sigma}$ applying standard numerical techniques. As all quantities appearing in Eq. are defined in terms of wave-function products, the basis should be constructed foremost of products of LAPW basis functions. In recent publications we have already used such a mixed product basis (MPB), which was first proposed by Kotani and van Schilfgaarde,[@Kotani-MixedBasis] to implement hybrid functionals[@PBE0-NonLocalExactExchangePotential] and the *GW* approximation[@GW-MixedBasis] as well as calculate EELS spectra.[@CoulombMatrix-MixedBasis] However, we will introduce a slightly modified version for the present purpose: (1) since the potential is strictly periodic, the MPB may be restricted to ${\mathbf{k}}={\mathbf{0}}$, (2) we add the atomic exact exchange potential as a basis function, and (3) we make the functions $M_{I}({\mathbf{r}})$ continuous over the whole space.
The construction of the MPB and the implementation of the spin-density response function $\chi_{s,IJ}^{\sigma}$ and $t_{I}^{\sigma}$ are described in Sec. \[sub: Auxiliary-basis-set\] and \[sub: Single-particle-response\], respectively. Numerical tests of the implementation are shown in Sec. \[sub: Tests\].
Mixed product basis\[sub: Auxiliary-basis-set\]
-----------------------------------------------
The MPB consists of plane waves in the IR and MT functions $M_{LP}^{a}(r)Y_{LM}(\hat{{\mathbf{r}}})$ in the spheres that derive from products of the functions $u_{lp}^{a\sigma}(r)Y_{lm}(\hat{{\mathbf{r}}})$. As in the LAPW basis, cutoff values $G'_{\mathrm{max}}$ for the interstitial plane waves and $L_{\mathrm{max}}$ for the angular momentum quantum numbers $L$ are employed. For mathematical details of the construction of the MPB we refer the reader to our previous publications Refs. . Here, we lay emphasis on the modifications for the present implementation of the EXX-OEP method.
From EXX-OEP calculations of atoms it is known that the local exchange potential shows pronounced humps which reflect the atomic shell structure.[@OEP-atoms] As the electron orbitals contract spatially for atoms with larger atomic numbers, these humps move closer and closer to the atomic nucleus. Near the nucleus, the exchange potential of a periodic crystal resembles that of the corresponding atom, because the long-range exchange interactions with electrons on neighboring atoms contribute only a slowly varying potential there. Therefore, we augment the MPB with the spherical exchange potential from an atomic EXX-OEP calculation performed with the relativistic atomic structure program RELKS.[@RELKS1; @RELKS2; @RELKS3] It is added to the set of spherical MT functions $(L=0)$. The rest of the basis must then only describe the difference between the atomic and the crystal exchange potential. In this way all nonlocal exchange contributions are fully taken into account.
To avoid discontinuities of the resulting potential at the MT sphere boundaries, we form linear combinations of the MT functions and interstitial plane waves that are continuous in value and first derivative there. In analogy to the construction of the LAPW basis (s. Sec. \[sec: FLAPW-Method\]), two radial functions per $lm$ channel are used to augment the interstitial plane waves in the MT spheres, while the remaining functions are combined to form local orbitals. We note that there are usually far more than two radial functions per $lm$ channel in the MPB. We also note that such a construction was not needed in our earlier implementations.
Spin-density response function and $t_{I}^{\sigma}({\mathbf{r}})$\[sub: Single-particle-response\]
--------------------------------------------------------------------------------------------------
In the MPB the spin-density response function in Eq. and $t_{I}^{\sigma}({\mathbf{r}})$ in Eq. become$$\chi_{\mathrm{s},IJ}^{\sigma}=2\sum_{{\mathbf{k}}}\sum_{n}^{\mathrm{occ.}}\sum_{n'}^{\mathrm{unocc.}}\frac{\langle M_{I}\varphi_{n'{\mathbf{k}}}^{\sigma}|\varphi_{n{\mathbf{k}}}^{\sigma}\rangle\langle\varphi_{n{\mathbf{k}}}^{\sigma}|\varphi_{n'{\mathbf{k}}}^{\sigma}M_{J}\rangle}{\epsilon_{n{\mathbf{k}}}^{\sigma}-\epsilon_{n{\mathbf{'k}}}^{\sigma}}\label{eq: response matrix}$$ and
$$t_{I}^{\sigma}=2\sum_{{\mathbf{k}}}\sum_{n}^{\mathrm{occ.}}\sum_{n'}^{\mathrm{unocc.}}\langle\varphi_{n{\mathbf{k}}}^{\sigma}|V_{\mathrm{x,\mathrm{NL}}}^{\sigma}|\varphi_{n'{\mathbf{k}}}^{\sigma}\rangle\frac{\langle M_{I}\varphi_{n'{\mathbf{k}}}^{\sigma}|\varphi_{n{\mathbf{k}}}^{\sigma}\rangle}{\epsilon_{n{\mathbf{k}}}^{\sigma}-\epsilon_{n'{\mathbf{k}}}^{\sigma}}\,.\label{eq: represenatation rhs t}$$
Both core and valence states are taken into account in the sums over the occupied states in Eqs. and .
As said before, solving Eq. for the exchange potential involves the matrix inversion of $\chi_{s,IJ}^{\sigma}$. The Hohenberg and Kohn theorem guarantees that the response function is invertible except for variations of the potential given by an addition of a constant. The latter restriction gives rise to a constant eigenfunction of $\chi_{s,IJ}^{\sigma}$ with eigenvalue $0$, which we eliminate from the outset by orthogonalizing all MPB functions to a constant function such that variations in the potential by a constant are excluded.
Equation contains the matrix elements of the nonlocal exchange potential [\[]{}Eq. \] between occupied (core and valence) and unoccupied states. In a recent publication, we described an efficient scheme to calculate the valence-valence and valence-conduction matrix elements within the FLAPW method.[@PBE0-NonLocalExactExchangePotential] For the present implementation, this scheme has been extended to the core-conduction matrix elements. Furthermore, spatial and time-reversal symmetries are exploited to restrict the ${\mathbf{k}}$-point sums to the irreducible wedge of the Brillouin zone in Eqs. and .[@PBE0-NonLocalExactExchangePotential; @GW-MixedBasis]
Numerical tests\[sub: Tests\]
-----------------------------
In this section, we present numerical tests of the spin-density response function, the function $t_{I}^{\sigma}({\mathbf{r}})$, and the resulting exchange potential. According to the Eqs. , , and all three quantities are functional derivatives of the form $\frac{\delta A({\mathbf{r}})}{\delta B({\mathbf{r}}')}$. Thus, they describe the linear response of a quantity $A$ with respect to changes of a quantity $B$.
For the case of diamond, we calculate the changes $\Delta n({\mathbf{r}})$ and $\Delta E_{\mathrm{x}}$ that result from an explicit perturbation $V_{\mathrm{per}}({\mathbf{r}})=\alpha\sum_{I}V_{\mathrm{per},I}M_{I}({\mathbf{r}})$, where $V_{\mathrm{per,}I}$ are random numbers, by exact diagonalization of the perturbed Hamiltonian and check whether they correspond to their linear counterpart $\Delta n^{\mathrm{lin}}({\mathbf{r}})=\int\chi_{\mathrm{s}}({\mathbf{r}},{\mathbf{r}}')V_{\mathrm{per}}({\mathbf{r}}')d^{3}r'$, $\Delta E_{\mathrm{x}}^{\mathrm{lin}}=\int t({\mathbf{r}}')V_{\mathrm{per}}({\mathbf{r}}')d^{3}r'$, and $\Delta E_{\mathrm{x}}^{\mathrm{lin}}=\int V_{\mathrm{x}}({\mathbf{r}}')\Delta n({\mathbf{r}})d^{3}r$ up to linear order in $\alpha$.
Tables \[tab: Response function test\], \[tab: t test\], and \[tab: vx test\] show that, indeed, the differences $|\Delta n^{\mathrm{lin}}-\Delta n|$ and $|\Delta E_{\mathrm{x}}^{\mathrm{lin}}-\Delta E_{\mathrm{x}}|$ depend quadratically on the perturbation strength $\alpha$ which confirms the validity of our implementation.
$\alpha$ 0.01 0.001 0.0001
---------------------------------------- ------------------------ ------------------------ ------------------------
$|\Delta n^{\mathrm{lin}}$-$\Delta n|$ $2.16710\times10^{-4}$ $2.17018\times10^{-6}$ $2.17035\times10^{-8}$
: Numerical test for the response function $\chi_{\mathrm{s}}({\mathbf{r}},{\mathbf{r}}')=\delta n({\mathbf{r}})/\delta V_{\mathrm{s}}({\mathbf{r}}')$ of diamond. The exact response of the density $\Delta n({\mathbf{r}})$ is compared with its linear approximation $\Delta n^{\mathrm{lin}}({\mathbf{r}})=\sum n_{I}^{\mathrm{lin}}M_{I}({\mathbf{r}})$ with $n_{I}^{\mathrm{lin}}=\alpha\sum_{J}\chi_{\mathrm{s},IJ}V_{\mathrm{per,}J}$. The L2 norm $[\int|\Delta n^{\mathrm{lin}}({\mathbf{r}})-\Delta n({\mathbf{r}})|^{2}d^{3}r]^{1/2}$clearly shows a quadratic dependence on the perturbation strength $\alpha$. \[tab: Response function test\]
$\alpha$ 0.01 0.001 0.0001
------------------------------------------------------------------ ------------------------ ------------------------ --------------------------
$\Delta E_{\mathrm{x}}$ $1.69455$ $1.74639\times10^{-1}$ $1.75171\times10{}^{-2}$
$\Delta E_{\mathrm{x}}^{\mathrm{lin}}$ $1.75230$ $1.75230\times10^{-1}$ $1.75230\times10^{-2}$
$|\Delta E_{\mathrm{x}}^{\mathrm{lin}}$-$\Delta E_{\mathrm{x}}|$ $5.77479\times10^{-2}$ $5.90882\times10^{-4}$ $5.92211\times10^{-6}$
: Same as Table \[tab: Response function test\] for $t({\mathbf{r}})=\delta E_{\mathrm{x}}/\delta V_{\mathrm{s}}({\mathbf{r}})$. The difference of the exact response $\Delta E_{\mathrm{x}}$ of the exchange energy and its linear approximation $\Delta E_{\mathrm{x}}^{\mathrm{lin}}=\alpha\sum_{I}t_{I}V_{\mathrm{per},I}$ depends quadratically on $\alpha$.\[tab: t test\]
$\alpha$ 0.01 0.001 0.0001
------------------------------------------------------------------ ------------------------ ------------------------ ------------------------
$\Delta E_{\mathrm{x}}$ $1.69455$ $1.74639\times10^{-1}$ $1.75171\times10^{-2}$
$\Delta E_{\mathrm{x}}^{\mathrm{lin}}$ $1.67924$ $1.74479\times10^{-1}$ $1.75155\times10^{-2}$
$|\Delta E_{\mathrm{x}}^{\mathrm{lin}}$-$\Delta E_{\mathrm{x}}|$ $1.53093\times10^{-2}$ $1.59919\times10^{-4}$ $1.60625\times10^{-6}$
: Same as Table \[tab: t test\], for $V_{\mathrm{x}}({\mathbf{r}})=\delta E_{\mathrm{x}}/\delta n({\mathbf{r}})$ with $\Delta E_{\mathrm{x}}^{\mathrm{lin}}=\sum_{I}V_{\mathrm{x},I}\int M_{I}^{*}({\mathbf{r}})\Delta n({\mathbf{r}})d^{3}r$.\[tab: vx test\]
Results & Discussion\[sec: Results-&-Discussion\]
=================================================
In this section, we present results for a variety of semiconductors and insulators obtained with our implementation of the EXX-OEP approach within the FLAPW method. In particular, we demonstrate for the case of diamond that a smooth and physical EXX potential requires a balance of the basis sets: the LAPW basis for the wave functions must be converged with respect to a given MPB until the EXX potential does not change anymore. This is somewhat counterintuitive and in contrast to our implementation of the hybrid functionals where, conversely, the MPB must be converged for a given LAPW basis. A similar behavior has been found in implementations employing plane-wave and Gaussian basis sets.[@Balance-Gaussian; @Hesselmann] We will analyze and explain this point later in this section.
Figure \[fig: C EXX potential (L=3D4)\] shows the local EXX potential on lines connecting two neighboring carbon atoms along the $[111]$ [\[]{}Fig. \[fig: C EXX potential (L=3D4)\](a)\] and the $[100]$ [\[]{}Fig. \[fig: C EXX potential (L=3D4)\](b)\] directions, see Fig. \[fig: C unit-cell\]. A $4$$\times$$4$$\times$$4$ ${\mathbf{k}}$-point sampling is employed, and the MPB parameters are $G'_{\mathrm{max}}=3.4\, a_{0}^{-1}$ ($a_{0}$ is the Bohr radius) and $L_{\mathrm{max}}=4$, giving rise to five $s$, four $p$, four $d$, and three $f$, and two $g$-type radial functions per atom. These cutoff values are well below those of the LAPW basis, $l_{\mathrm{max}}=6$ and $G_{\mathrm{max}}=4.2\, a_{0}^{-1}$, which reflects the relative smoothness of the potential compared with the shape of the wave functions. However, if we only use the conventional basis of augmented plane waves, defined in Eq. , the potential (dashed lines) shows an overpronounced intershell hump and tends to an unphysical positive value close to the atomic nucleus ($r=0$). This is a case where the basis sets are *unbalanced*. In particular, the LAPW basis lacks flexibility in the MT spheres as becomes obvious when we add local orbitals, which are nonzero only in the MT spheres. We find that six local orbitals for each $lm$ channel with $l=0,\dots,5$ and $|m|\le l$, placed at higher energies according to the prescription described in Sec. \[sec: FLAPW-Method\], are needed to converge the local EXX potential. This is reasonable since, with the cutoff $L_{\mathrm{max}}=4$, the occupied $2s$ and $2p$ states of diamond couple maximally to the $l=5$ contribution of the unoccupied states. With so many local orbitals the number of basis functions is increased roughly by a factor of five: there are about $100$ augmented plane waves (the exact number depends on the ${\mathbf{k}}$ point) and $432$ additional local orbitals. All resulting KS bands, about $530$, are taken into account in the sums of Eqs. and . The resulting potential is shown as solid lines in Fig. \[fig: C EXX potential (L=3D4)\] and looks smooth and physical. It is remarkable that, even for diamond, it takes so much effort to converge the EXX potential since, in conventional LDA or GGA calculations, diamond is treated readily with a very modest LAPW basis without any local orbitals.
(a)
(b)
![Cubic unit cell of diamond with the ($01\overline{1}$) plane. The lines connecting two carbon atoms along the $[111]$ and $[100]$ directions are indicated as solid lines. \[fig: C unit-cell\]](fig2)
Before analyzing this point in more detail, we want to identify the MPB functions that contribute most to the MT part of the potential. Not surprisingly, the function that corresponds to the atomic EXX potential gets the largest weight. In fact, close to the atomic nuclei this function (dotted lines in Fig. \[fig: C EXX potential (L=3D4)\]) and its bulk counterpart are indistinguishable. They deviate more and more towards the MT sphere boundary ($S\mathrm{=1.42\,}a_{0}$), where the atomic EXX potential already enters the typical $1/r$ behavior, while the crystal EXX potential is periodic. The second largest contribution comes from the constant MT function, which helps to align the MT potential to the interstitial one. We note that there is no ambiguity with respect to adding a constant to the potential over the whole space, since the constant function has been eliminated explicitly from the MPB (see Sec. \[sub: Single-particle-response\]) giving rise to the condition $\int V_{\mathrm{x}}({\mathbf{r}})d^{3}r=0$.
So far, we have only discussed the MT potential, whose proper convergence requires additional local orbitals in the spheres. We find an analogous behavior for the interstitial potential. As is seen in Fig. \[fig: IR balance\], the cutoff radius of the reciprocal lattice vectors included in the LAPW basis set must be converged with respect to that of the MPB. To show this effect clearly, the latter was chosen much larger than necessary, $G'_{\mathrm{max}}=5.8\, a_{0}^{-1}$. Similarly to the MT potential, the interstitial potential exhibits spurious oscillations in the underconverged cases. Only if $G_{\mathrm{max}}$ is large enough, $G_{\mathrm{max}}\gtrsim5.0\, a_{0}^{-1}$, the oscillations are suppressed and a smooth potential is obtained. Furthermore, Fig. \[fig: IR balance\] shows that, in the underconverged cases, the potential is not continuous at the MT sphere boundary because the oscillatory potentials possess large-${\mathbf{G}}$ Fourier coefficients and require spherical harmonics beyond $L_{\mathrm{max}}=4$ in the spheres for a proper matching. Fortunately, the converged EXX potential is a smooth function and already moderate reciprocal cutoff radii are sufficient, typically 75% of the usual LAPW cutoff. For diamond, for example, the combination of $G'_{\mathrm{max}}=3.4\, a_{0}^{-1}$ and $G_{\mathrm{max}}=4.2\, a_{0}^{-1}$ leads to stable results.
An overall view of the MT and the interstitial exchange potential on the diamond ($01\overline{1}$) plane is shown in Fig. \[fig: contour plot\](a) as a contour plot. The plane is displayed in Fig. \[fig: C unit-cell\]. It contains the connecting lines along [\[]{}111\] and [\[]{}100\] that correspond to Fig. \[fig: C EXX potential (L=3D4)\]. We see that in the regions close to the atomic nuclei the potential is predominantly spherical. However, towards the MT sphere boundaries the potential becomes strongly anisotropic and matches continuously to the warped interstitial potential, which is far from constant also. In fact, the nonsphericity of the EXX potential is considerably more pronounced than in the LDA potential, Fig. \[fig: contour plot\](b). The latter is similar in shape to the electron density distribution [\[]{}cf. Fig. \[fig: density-contour plot\](a)\], of which it is a direct function $V_{\mathrm{x}}^{\mathrm{LDA}}({\mathbf{r}})\propto n({\mathbf{r}})^{1/3}$. The EXX potential, in contrast, incorporates the full nonlocality of the EXX functional, which makes it much more corrugated than the LDA one, in particular in the MT spheres, where the KS orbitals are highly oscillatory. All this stresses the importance of a full-potential treatment within the EXX-OEP approach.
The LDA potential corresponds in each point ${\mathbf{r}}$ to the exchange potential of the homogeneous electron gas with an electron density that equals the local electron density $n({\mathbf{r}})$ of the real system. Thus, by construction it is exact for the homogeneous electron gas but misses the effects of density variations. The EXX potential, in contrast, takes all density variations exactly into account. Thus, the differences between Figs. \[fig: contour plot\](a) and (b) must be attributed to the influence of the density inhomogeneities on the exchange potential. This influence is particularly large in regions where the density varies a lot, that is, close to the atomic nuclei, while in the interstitial region the two potentials are more similar.
The differences in the exchange potentials naturally affect the electron density distribution. The EXX electron density, Fig. \[fig: density-contour plot\](a), clearly shows a pronounced contraction of the electron distribution compared with the LDA one. This is a direct consequence of the self-interaction error, which is eliminated in the EXX approach, while it gives rise to an unphysical delocalization in the case of the LDA potential. This becomes clearer in Fig. \[fig: density-contour plot\](b) where we plot the difference between the EXX and the LDA densities. The exactly compensated self-interaction allows the charge to accumulate in the atomic cores, but also in the covalent bonds between the atoms.
(a)![Two-dimensional plot of (a) the local EXX and (b) the LDA exchange potential on the $(01\overline{1})$ plane of diamond. Both potentials are shifted such that $\int V_{\mathrm{x}}({\mathbf{r}})d^{3}r=0$. The contour lines start at $V_{\mathrm{x}}=-1.00\,\mathrm{htr}$ and have an interval of $0.05\,\mathrm{htr}$. The contour line with the maximal value is found (a) at $V_{\mathrm{x}}=0.30\,\mathrm{htr}$ and (b) at $V_{\mathrm{x}}=0.20\,\mathrm{htr}$. The dotted line corresponds to $V_{\mathrm{x}}=0$. The MT sphere boundaries and the lines corresponding to Fig. \[fig: C EXX potential (L=3D4)\] are indicated.\[fig: contour plot\]](fig4a "fig:")
(b)![Two-dimensional plot of (a) the local EXX and (b) the LDA exchange potential on the $(01\overline{1})$ plane of diamond. Both potentials are shifted such that $\int V_{\mathrm{x}}({\mathbf{r}})d^{3}r=0$. The contour lines start at $V_{\mathrm{x}}=-1.00\,\mathrm{htr}$ and have an interval of $0.05\,\mathrm{htr}$. The contour line with the maximal value is found (a) at $V_{\mathrm{x}}=0.30\,\mathrm{htr}$ and (b) at $V_{\mathrm{x}}=0.20\,\mathrm{htr}$. The dotted line corresponds to $V_{\mathrm{x}}=0$. The MT sphere boundaries and the lines corresponding to Fig. \[fig: C EXX potential (L=3D4)\] are indicated.\[fig: contour plot\]](fig4b "fig:")
(a)![(a) Total electron densities obtained from the EXX (solid lines) and LDA potential (dotted lines) on the diamond $(01\overline{1})$ plane. Contour lines for the values $0.03\, a_{0}^{-3},\,0.06\, a_{0}^{-3},\dots,\,0.99\, a_{0}^{-3}$ are shown. (b) Plot of the density difference with contour lines between $-0.039\, a_{0}^{-3}$ and $0.390\, a_{0}^{-3}$ at intervals of $0.003\, a_{0}^{-3}$. The dotted line corresponds to $\Delta n({\mathbf{r}})=0$. \[fig: density-contour plot\]](fig5a "fig:")
(b)![(a) Total electron densities obtained from the EXX (solid lines) and LDA potential (dotted lines) on the diamond $(01\overline{1})$ plane. Contour lines for the values $0.03\, a_{0}^{-3},\,0.06\, a_{0}^{-3},\dots,\,0.99\, a_{0}^{-3}$ are shown. (b) Plot of the density difference with contour lines between $-0.039\, a_{0}^{-3}$ and $0.390\, a_{0}^{-3}$ at intervals of $0.003\, a_{0}^{-3}$. The dotted line corresponds to $\Delta n({\mathbf{r}})=0$. \[fig: density-contour plot\]](fig5b "fig:")
To understand the requirement of the basis-set balance in more detail, we go back to the OEP Eq. , whose solution involves the inversion of the response function, Eq. . The response function describes the linear response of the electron density with respect to changes of the effective potential. For the latter we employ the MPB, while the former is given by the orbital densities of the occupied states, Eq. , and, hence, ultimately by the LAPW basis set. Thus, the LAPW basis must provide enough flexibility for the density to enable it to respond adequately to the changes of the effective potential.
This explains the observed behavior and becomes evident in the convergence of the response function with respect to the LAPW basis. In Fig. \[fig: Response convergence\] we show the changes of the eigenvalues of $\chi_{\mathrm{s}}$, ordered according to increasing moduli, when we add more and more local orbitals, $2l+1$ additional local orbitals per $l$ quantum number ($l=0,\dots,5$) in each step. We again use $L_{\mathrm{max}}=4$ and $G'_{\mathrm{max}}=3.4\, a_{0}^{-1}$ as MPB cutoff values. In the case of the maximal number of local orbitals per $lm$ channel, $n_{\mathrm{lo}}=6$, the basis is increased by $432$ functions relative to the conventional LAPW basis. Clearly, the eigenvalues can be systematically converged. The relative changes are in the order of 0.1% to 1.0% between $n_{\mathrm{lo}}=5$ and $n_{\mathrm{lo}}=6$. Especially the small eigenvalues converge well, which are particularly important in the inversion of the response function. We note that a straightforward elimination of the small eigenvalues by singular value decomposition is not advisable and leads to an ill-defined response function.
Up to now, we have discussed the importance of the quality of the LAPW basis for the shape of the local EXX potential. Only for well-balanced basis sets, the potential is smooth and physical (cf. Fig. \[fig: C EXX potential (L=3D4)\]). We now address the question to what extent this effect influences the KS one-particle energies that result from the self-consistent solution of Eq. with the local EXX potential obtained from the OEP Eq. . Table \[tab: convergence of transition energies\] gives the transition energies, that is, the KS eigenvalue differences, from the valence-band maximum at the $\Gamma$ point to the $\mathrm{L}$ and $\mathrm{X}$ point of the lowest conduction band of diamond for the basis sets with $n_{\mathrm{lo}}=0,\dots,6$ local orbitals per $lm$ channel. Obviously, at least three local orbitals are necessary to converge these transition energies to within $0.01\,\mathrm{eV}$. Between the unbalanced and balanced basis sets ($n_{\mathrm{lo}}=0$ and $n_{\mathrm{lo}}=6$, respectively) the values change by about 0.2 eV.
$n_{\mathrm{lo}}$ $\Gamma\rightarrow\Gamma$ $\Gamma\rightarrow\mathrm{L}$ $\Gamma\rightarrow\mathrm{X}$
------------------- --------------------------- ------------------------------- -------------------------------
0 6.351 9.243 5.307
1 6.196 9.086 5.125
2 6.186 9.069 5.144
3 6.180 9.063 5.138
4 6.178 9.059 5.139
5 6.177 9.057 5.136
6 6.176 9.055 5.136
: KS transition energies (in eV) for diamond obtained from the self-consistent solution of the KS equation with the local EXX potential for LAPW basis sets including zero to six local orbitals per $lm$ channel ($l=0,\dots,5$, $|m|\le l$). As the LAPW basis is made more flexible, the transition energies converge. \[tab: convergence of transition energies\]
Another balance condition we find for the $1s$ core state that goes into both the left- and the right-hand sides of Eq. . In particular, we now distinguish between four cases: (a) core state considered in $\chi_{\mathrm{s},IJ}$ and $t_{I}$, (b) core state only considered in $\chi_{\mathrm{s},IJ}$, (c) core state only considered in $t_{I}$, and (d) core state neglected in both. Case (a) corresponds to the full calculations presented so far. Figure \[fig: balance left/right\] shows that the resulting potentials look very different for the different cases. Surprisingly, potential (d) is closest to the full potential, while the potentials (b) and (c) are much too shallow and too strongly varying, respectively. Obviously, the inclusion of the core state only on one side of the OEP equation gives rise to an equation that is *out of balance* and that yields an unphysical $V_{\mathrm{x}}({\mathbf{r}})$. This also influences the resulting KS transition energies. In the balanced case (d), the results $6.19\,\mathrm{eV}$, $9.08\,\mathrm{eV}$, and $5.28\,\mathrm{eV}$ for $\Gamma\rightarrow\Gamma$, $\Gamma\rightarrow\mathrm{L}$, and $\Gamma\rightarrow\mathrm{X}$, respectively, are surprisingly close to the full calculation (a) (cf. Table \[tab: convergence of transition energies\]), while the energies for the unbalanced case (b) deviate more strongly, especially for the $\Gamma\rightarrow\mathrm{X}$ transition, $6.22\,\mathrm{eV}$, $9.16\,\mathrm{eV}$, and $6.03\,\mathrm{eV}$. The energetic position of the $1s$ state with respect to the Fermi energy, however, is only realistic for the full calculation (a), $\epsilon(1s)=-265.7\,\mathrm{eV}$ (cf. $-262.9\,\mathrm{eV}$ for LDA), whereas (b) and (d) give binding energies, that are smaller by $31.6\,\mathrm{eV}$ and $10.5\,\mathrm{eV}$, respectively. Calculation (c) is unstable and does not converge. These results indicate that the PP approximation is, indeed, suitable for the EXX-OEP approach, a conjecture that will be confirmed by our reference calculations later-on.
As outlined in Sec. \[sec: Theory\], the construction of a local EXX potential within the KS formalism of DFT is equivalent to the OEP approach of Sharp and Horton,[@OEP/OPM] where the Hartree-Fock total energy is minimized under the constraint that the wave functions feel a local multiplicative potential. This constraint reduces the Hilbert space for the wave functions and, thus, increases the total energy due to the variational principle. In fact, we find that the total energy for diamond obtained with the nonlocal Hartree-Fock potential, Eq. , and an $8$$\times$$8$$\times$$8$ ${\mathbf{k}}$-point set is $0.24\,\mathrm{eV}$ per unit cell lower than that of the EXX-OEP approach.
As reference, we now report fully converged KS transition energies for a variety of semiconductors and insulators in Table \[tab: EXX+EXX-VWN results\]. All calculations are performed with an $8$$\times$$8$$\times$$8$ Brillouin zone sampling and at the experimental lattice constants (C: $6.746\, a_{0}$, Si: $10.26\, a_{0}$, SiC: $8.24\, a_{0}$, Ge: $10.67\, a_{0}$, GaAs: $10.68\, a_{0}$, Ne: $8.44\, a_{0}$, Ar: $9.93\, a_{0}$). Apart from the EXX-only calculations, we also show transition energies obtained with the EXX+VWNc functional in which the LDA correlation functional from Ref. was added to the EXX functional. Table \[tab: EXX+EXX-VWN results\] shows that both the EXX and EXX+VWNc functionals yield KS transition energies much closer to experiment (last column) than the LDA functional (first column). While for semiconductors the resulting energies are even quantitatively in very good agreement with experiment, there are larger discrepancies for the insulators diamond and, in particular, crystalline Ne and Ar. There is only little difference between the EXX and EXX+VWNc values. The inclusion of the LDA correlation functional does not lead to a definite improvement. The direct band gap $\Gamma\rightarrow\Gamma$ for diamond of $6.21\,\mathrm{eV}$ agrees very well with the value $6.18\,\mathrm{eV}$ recently reported by Engel,[@Engel] who used a plane-wave PP approach with cutoff values pushed to the AE limit. It is even identical to the value obtained with a standard valence-only plane-wave PP approach.[@Hesselmann-privat] In contrast to that, Sharma *et al.* calculated a much larger value of $6.67\,\mathrm{eV}$ with their FLAPW-EXX implementation. For neon, there is a somewhat larger discrepancy with the calculation by Magyar *et al.*,[@EXX-Ar/Ne] though. In conclusion, with our EXX-OEP implementation within the AE FLAPW method, we obtain results in very good agreement with previous plane-wave PP calculations, provided that the basis sets for the wave functions and the potential are properly balanced. This shows that the PP approximation is adequate for the EXX-OEP approach at least for the systems examined here, which is at variance with the findings of Ref. .
------ ------------------------------- --------- --------- ---------- --------------------- ------------- -------------
LDA EXX EXX+VWNc EXX EXX+VWNc Expt.
C $\Gamma\rightarrow\Gamma$ $5.56$ $6.21$ $6.26$ $6.19^{a},6.21^{b}$ $6.28^{c}$ $7.3^{e}$
$\Gamma\rightarrow\mathrm{L}$ $8.43$ $9.09$ $9.16$ $9.15^{b}$ $9.18^{c}$
$\Gamma\rightarrow\mathrm{X}$ $4.71$ $5.20$ $5.33$ $5.34^{b}$ $5.43^{c}$
Si $\Gamma\rightarrow\Gamma$ $2.53$ $3.13$ $3.21$ $3.12^{b}$ $3.26^{c}$ $3.4^{e}$
$\Gamma\rightarrow\mathrm{L}$ $1.42$ $2.21$ $2.28$ $2.21^{b}$ $2.35^{c}$ $2.4^{e}$
$\Gamma\rightarrow\mathrm{X}$ $0.61$ $1.30$ $1.44$ $1.25^{b}$ $1.50^{c}$
SiC $\Gamma\rightarrow\Gamma$ $6.27$ $7.18$ $7.24$ $7.37^{c}$
$\Gamma\rightarrow\mathrm{L}$ $5.38$ $6.14$ $6.21$ $6.30^{c}$
$\Gamma\rightarrow\mathrm{X}$ $1.32$ $2.29$ $2.44$ $2.52^{c}$ $2.42^{e}$
Ge $\Gamma\rightarrow\Gamma$ $-0.14$ $1.24$ $1.21$ $1.28^{c}$ $1.0^{e}$
$\Gamma\rightarrow\mathrm{L}$ $0.06$ $0.89$ $0.94$ $1.01^{c}$ $0.7^{e}$
$\Gamma\rightarrow\mathrm{X}$ $0.66$ $1.15$ $1.28$ $1.34^{c}$ $1.3^{e}$
GaAs $\Gamma\rightarrow\Gamma$ $0.29$ $1.72$ $1.74$ $1.82^{c}$ $1.63^{e}$
$\Gamma\rightarrow\mathrm{L}$ $0.85$ $1.79$ $1.86$ $1.93^{c}$
$\Gamma\rightarrow\mathrm{X}$ $1.35$ $1.95$ $2.12$ $2.15^{c}$ $2.18^{e}$
Ne $\Gamma\rightarrow\Gamma$ $11.43$ $14.79$ $15.46$ $14.15^{d}$ $14.76^{d}$ $21.51^{f}$
$\Gamma\rightarrow\mathrm{L}$ $16.97$ $20.49$ $21.16$
$\Gamma\rightarrow\mathrm{X}$ $18.27$ $21.85$ $22.56$
Ar $\Gamma\rightarrow\Gamma$ $8.19$ $9.65$ $10.09$ $9.61^{d}$ $9.95^{d}$ $14.15^{f}$
$\Gamma\rightarrow\mathrm{L}$ $11.06$ $12.22$ $12.60$
$\Gamma\rightarrow\mathrm{X}$ $10.86$ $12.08$ $12.49$
------ ------------------------------- --------- --------- ---------- --------------------- ------------- -------------
----------------- -- ----------------- -- ----------------- --
$^{a}$Reference $^{b}$Reference $^{c}$Reference
$^{d}$Reference $^{e}$Reference $^{f}$Reference
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Conclusions\[sec: Conclusions\]
===============================
We have developed an all-electron full-potential implementation of the EXX-OEP approach to DFT within the FLAPW method. We analysed the conditions and requirements on the basis sets and numerical cutoff parameters to obtain reliable and numerically stable results. Based on this knowledge we presented as proof of principle results on KS transition energies for some typical semiconductors, insulators and noble-gas solids that are in very good agreement with pseudopotential results.
The OEP equation is formulated utilizing the mixed product basis (MPB),[@PBE0-NonLocalExactExchangePotential; @GW-MixedBasis; @CoulombMatrix-MixedBasis] which has been adjusted for the present purpose: it is augmented with the atomic EXX potential, the constant function is eliminated, and the basis functions are made continuous all over the space. In this basis, the OEP equation becomes an algebraic equation, which is solved for the local EXX potential with standard numerical tools.
For the case of diamond, we have demonstrated that the local EXX potentials are spatially strongly corrugated, which makes a full-potential treatment even more important than in conventional LDA or GGA calculations. Furthermore, the two basis sets, LAPW and MPB are not independent. They must be properly balanced to obtain a smooth and physical EXX potential over the whole space. In the *unbalanced* case, the potential shows spurious oscillations, which we have traced back to an insufficiently converged response function, a function that gives the response of the electron density with respect to changes of the effective potential. If the LAPW basis, which parametrizes the electron density, is not flexible enough, the electron density cannot follow the changes of the effective potential that are described by the MPB leading to a corrupted response function. As a result, the LAPW basis must be converged with respect to a given MPB. Already in the simple case of diamond, we must add six local orbitals at different energies for each $lm$ channel from $l=0$ to $l=5$ in order to obtain a smooth potential in the spheres. This shows that the LAPW basis must be converged to an accuracy that is far beyond that of conventional LDA or GGA calculations. Similarly, also the LAPW reciprocal cutoff radii must be chosen large enough.
Not surprisingly, the shape of the EXX potential – oscillatory or smooth – has an impact on the resulting KS transition energies. We find that with properly balanced basis sets, the transition energies for a variety of semiconductors and insulators obtained with the EXX and the EXX+VWNc functionals are in very good agreement with plane-wave pseudopotential results from the literature (crystalline neon is an exception). This confirms that the pseudopotential approximation works reliably within the EXX-OEP approach. Our finding is in contradiction to a previously published implementation (Ref. ) based on the FLAPW method, where large discrepancies with pseudopotential results were reported.
Currently, reliable all-electron full-potential EXX-OEP calculations are computationally very demanding, because of the need for large orbital basis sets, which we attribute partly to the fact that the LAPW basis functions depend explicitly on the effective potential. To refine our full-potential implementation of the EXX-OEP approach, we suggest as an task for the future the investigation of schemes that treat the response of the LAPW basis with respect to changes of the potential more efficiently than employing local orbitals.
Financial support from the Deutsche Forschungsgemeinschaft through the Priority Program 1145 is gratefully acknowledged.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Clustering analysis, a classical issue in data mining, is widely used in various research areas. This article aims at proposing a self-adaption grey DBSCAN clustering (SAG-DBSCAN) algorithm. First, the grey relational matrix is used to obtain the grey local density indicator, and then this indicator is applied to make self-adapting noise identification for obtaining a dense subset of clustering dataset, finally, the DBSCAN which automatically selects parameters is utilized to cluster the dense subset. Several frequently-used datasets were used to demonstrate the performance and effectiveness of the proposed clustering algorithm and to compare the results with those of other state-of-the-art algorithms. The comprehensive comparisons indicate that our method has advantages over other compared methods.'
address: ' $^1$College of Economics and Management, Nanjing University of Science and Technology, Nanjing, 210094, China (e-mail: [email protected])\'
author:
- 'Shizhan Lu$^{1*}$'
title: 'Self-adaption grey DBSCAN clustering'
---
Clustering analysis, density-based, B-style grey relationship, SAG-DBSCAN.
[2]{}
Introduction
============
Cluster analysis, which focuses on the grouping and categorization of similar elements, is widely used in different research areas, such as climate predictions [@KCP], gene expression [@WTX], bioinformatics [@PEA], finance and economics [@HJD; @LG], and neuroscience [@GS; @AHK].
In general, different clustering methods can be basically classified as follows: density-based (DP [@RA], DP-HD [@MR], DBSCAN [@EM], NQ-DBSCAN [@CHEN] and CSSub [@ZHY]); grid-based (CLIQUE [@AGR], Gridwave [@DEC] and WaveCluster [@SG]); model-based (Gaussian parsimonious [@MUK], Gaussian mixture models [@OHA] and Latent tree models[@CT]); partitioning (K-means [@MJ; @ZHR; @ZHA], K-partitioning [@CDW] and TLBO [@LK]); graph-based (SEGC [@WAJ], ProClust [@PPI] and MCSSGC [@VIE]); and hierarchical (BIRCH [@ZHT], K-d tree and Quadtree [@DAJ] and CHAMELEON [@GAK]) approaches.
DBSCAN [@EM] is a representative density-based algorithm which clusters data by defining the density criterion with two parameters, Eps-distance and MinPts. NQ-DBSCAN [@CHEN], AA-DBSCAN [@KIM], RNN-DBSCAN [@BRY], ReCon-DBSCAN [@ZHTM] and ReScale-DBSCAN [@ZHTM] are some up-to-date developments of DBSCAN. As a disadvantage, DBSCAN and its extensions are difficult for their parameters to be set, which are ruleless on account of the different densities for variant datasets.
Grey relational analysis is a significant tool for data mining [@WUD; @YHH] and clustering analysis[@CKC; @LCH; @LIX]. In this article, grey relational analysis is applied to obtain grey local density indicator for every object in clustering dataset. And then, grey local density indicator is applied to make noise identification for obtaining a dense subset. The dense subset which the border points of $i$th cluster are far away from the border points of $j$th cluster is easily to set parameters for DBSCAN. This method overcomes the disadvantage of DBSCAN which is difficult for its parameters to be set for variant datasets.
The remainder of this article is organized as follows: Section 2 presents a grey local density indicator, proposes a method for self-adaption noise identification and proposes a self-adaption grey-DBSCAN clustering method. Section 3 demonstrates our algorithms by some numerical experiments of both simulated and real datasets and make comparisons with the state-of-the-art clustering algorithms. Section 4 gives a conclusion.
Proposed methods
================
The main framework of the SAG-DBSCAN algorithm can be described as follows. Step 1 obtains the matrix of the B-style grey relationship degree and grey local density indicator $\rho$. Step 2 utilizes linear regression to obtain dense subset $C$ (as shown in Fig. 1 (b)). Step 3 applies DBSCAN algorithm to cluster dense subset (as shown in Fig. 1 (c)). Step 4 assigns the object in $X-C$ to its nearest cluster (as shown in Fig. 1 (d)).
Grey local density indicator
----------------------------
For all $x_i=(x_i(1),x_i(2),\cdots,x_i(N))$ and $x_j=(x_j(1),x_j(2),\cdots,x_j(N))$ in N-dimensional dataset $X$,
$$\gamma(x_i,x_j)=\frac{1}{1+d^{(0)}_{ij}/N+d^{(1)}_{ij}/(N-1)+d^{(2)}_{ij}/(N-2)}$$
is the B-style grey relationship degree of $x_i$ and $x_j$ [@DJL; @FEW], where $d^{(0)}_{ij}=\sum\limits_{k=1}^N|x_i(k)-x_j(k)|$, $d^{(1)}_{ij}=\sum\limits_{k=1}^{N-1}|x_i(k+1)-x_j(k+1)-x_i(k)+x_j(k)|$ and $d^{(2)}_{ij}=\sum\limits_{k=2}^{N-1}|x_i(k+1)-x_j(k+1)-2(x_i(k)-x_j(k))+x_i(k-1)-x_j(k-1)|$. $G=[\gamma(x_i,x_j)]_{n\times n}$ is denoted as B-style grey relationship degree matrix.
B-style grey relationship degree is superior to other grey relationship degrees and Euclidean distance function for describing the objects’ relationships about object displacement, such as the DrivFace dataset in our experiments. Meanwhile, it can work well for other simulated data (Euclidean space points) like Euclidean distance function. B-style grey relationship degree has strong applicability and generality for various datasets, hence, it is selected to make following analysis.
KNN-density is a frequently-used indicator to describe the local density indicator $\rho_i$ [@CYW; @DUM]. A relatively straightforward and useful grey KNN-density indicator is projected as equation (2), where $|GKNN(x_i)|=k$ and $G(i,j)\geqslant G(i,t)$ for all $x_j\in GKNN(x_i)$ and $x_t\in X-GKNN(x_i)$.
$$\rho_i=\sum\limits_{x_j\in GKNN(x_i)} G(i,j),$$
In general, the dense family of a cluster is composed of many objects with large grey relationship degree between each other, on the contrary, the border objects has small grey relationship degree with its neighbors. As shown in Fig. 1 (c), the dense subset $C$ is composed of many dense families $C_i$ ($1\leqslant i \leqslant t$). The object $x_j$ of dense family has a large value of $\rho_j$ via calculating by equation (2) if we set $k\leqslant min\{|C_i|: i=1,2,\cdots, t\}$, where $C_i$ is the dense family of $i$th cluster.
Self-adaption method for noise identification
---------------------------------------------
Data preprocessing is necessary for the grey local density indicator $\rho$ before the noise identification. Let $\rho'=\{\rho_i': 1\leqslant i\leqslant n\}$ be the descending sequence of $\rho=\{\rho_j: 1\leqslant j\leqslant n\}$ and $V=\{v_i: 5\leqslant i\leqslant n\}$ be the mean smoothing sequence of $\rho'$, where $v_i=(\rho_i'+\rho_{i-1}'+\rho_{i-2}'+\rho_{i-3}'+\rho_{i-4}')/5$.\
As shown in Fig. 2, the elements of $V$ have two distinctly different distribution trends. The elements in $V$ are divided into two parts $V_{p}^-=\{v_i:5\leqslant i\leqslant p\}$ and $V_{p}^+=\{v_i:p<i\leqslant n\}$ for linear regression, and $f_1$ and $f_2$ are the regression equations for $V_{p}^-$ and $V_{p}^+$, respectively.
$$\left\{
\begin{aligned}
e_1(i)=f_1(i)-v_i, 5\leqslant i\leqslant p, \\
e_2(i)=f_2(i)-v_i, p<i\leqslant n.
\end{aligned}
\right.$$
$R=\sum\limits_{5\leqslant i\leqslant p} |e_1(i)|+\sum\limits_{p<i\leqslant n}|e_2(i)|$ is the regression residual.
As shown in Fig. 2, we pick $p_1$, $p_2$ and $p_3$ in different positions for linear regressions and obtain three regression residuals $R_1$, $R_2$ and $R_3$, where $f_1$ and $f_2$ are the regression straight line with respect to $V_{p_2}^-$ and $V_{p_2}^+$, respectively. The results show that $R_2<R_1$ and $R_2<R_3$.
After investigating the characteristic of the grey local density indicator $\rho$ and the smoothing sequence $V$, we can make a sequence of linear regressions to obtain a residual sequence $R=\{R_i:5+5\leqslant i\leqslant n-5\}$ (5 points for regression). If $R_p=min(R)$, the object $x_j$ is considered as a member of dense subset for the case $\rho_j\geqslant \rho'_p$. Then, the dense subset (as shown in Fig. 1 (c)) is obtained for clustering by DBSCAN method.
Self-adaption grey DBSCAN clustering method (SAG-DBSCAN)
--------------------------------------------------------
The detailed processes of the SAG-DBSCAN algorithm are shown as Algorithm 1.
------------------------------------------------------------------------
[**Algorithm 1:**]{} SAG-DBSCAN algorithm.
------------------------------------------------------------------------
[**Input:**]{} Dataset, parameters $k\in N^+$ and $m\in N^+$.
[**Output:**]{} The clustering result.
1\. Use equation (1) to obtain B-style grey relationship degree matrix $G$.
2\. Apply equation (2) to obtain grey local density indicator $\rho$ with respect to $k=|GKNN(x_i)|$.
3\. Sort $\rho$ to obtain $\rho'$, smooth $\rho'$ to obtain $V$.
4\. Obtain dense subset $C\subset X$ by linear regression of $V$.
5\. Set MinPts=$m$ and Eps-distance=$max\{d(x_i,x_i^{(m)}): x_i\in C\}$, where $x_i^{(m)}$ is the $m$th neighbor of $x_i$.
6\. Apply DBSCAN to cluster dense subset $C$ with MinPts=$m$ and Eps-distance.
7\. Assign the objects of $X-C$ to their nearest clusters.
------------------------------------------------------------------------
The objects of $X-C$ can be assigned as follows: let $A\subset X$ be the subset that contains the already classified points and $U\subset X$ be the subset of unclassified points. If $f(x_i',x_j')=min\{f(x_i,x_j):x_i\in A, x_j\in U\}$, then $x_j'$ is assigned to the category that contains $x_i'$.
The time complexities are $\mathscr{O}(n^2)$ and $\mathscr{O}(n)$ for obtaining B-style grey relationship degree matrix $G$ and dense subset $C$, respectively. The time complexity of DBSCAN is $\mathscr{O}(n^2)$ at the worst case. Moreover, if $|C|=n'$, then $|X-C|=n-n'$, and the time complexity of assigning border points is hence $\mathscr{O}(n-n')$.
Experiments
===========
In this section, we evaluate the performance and effectiveness of the proposed method on both simulation data and real data, and then compare it with some up to date methods: the affinity propagation algorithm (AP) [@FBJ], automatic find of density peaks (ADPC) [@TLI], Neighbor Query DBSCAB (NQ-DBSCAN) [@CHEN] and NK hybrid genetic algorithm (NKGA) [@TR].
Descriptions of Experiment data
-------------------------------
\[tab:parametervalues\]
Dataset Instances Features Clusters Dataset Instances Features Clusters Detail
--------- ----------- ---------- ---------- ----------- ----------- ---------- ---------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
D31 3100 2 31 Iris 150 4 3 three kinds of irises: Setosa, Versicolour and Virginica. Each kind has 50 samples
S1 5000 2 15 Wifi 2000 7 4 2000 times of signal records in 4 rooms, 500 records in each room
R15 600 2 15 Vertebral 310 6 2 310 orthopaedic samples, 210 abnormal samples and 100 normal samples
Dim2 1350 2 9 TumTyp 801 20531 5 gene expressions of patients having different types of tumor: BRCA, KIRC, COAD, LUAD and PRAD.
ShapeT 10000 2 3 DrivFace 606 6400 4 dataset contains images sequences of subjects while driving in real scenarios. It is composed of 606 samples and acquired over different days from 4 drivers with several facial features.
First, some frequently-used datasets obtained from different references are used to test the algorithms, such as R15 [@VCJ], D31 [@VCJ], S1 [@FP] and Dim2 [@FP1] etc. And then a dataset, ShapeT (Fig. 3), is constructed for the supplementary tests. All the simulation data are points of two-dimensional Euclidean space.
Several real-world datasets are used to test the performance of the proposed method, including a plant dataset: Iris [@FRA; @HF]; a wireless signal dataset: Wifi [@RJG]; a human vertebral column dataset: Vertebral [@BEE]; a gene dataset: TumTyp [@WJN]; and a face image dataset: DrivFace [@KDC]. Datasets were taken from the UCI[^1] repository. Simple descriptions of these real datasets are provided in Table 1.
Results and Comparisons
-----------------------
\[tab:parametervalues\]
--------- ------------ ------------- ------------ ------------ ------------ ----------- ---- ----------- ----------- ----------- ------------
Dataset Dataset
AP ADPC NKGA NQ-DBSCAN SAG-DBSCAN AP ADPC NKGA NQ-DBSCAN SAG-DBSCAN
D31 8 [**31** ]{} 19 [**31**]{} [**31**]{} Iris 2 2 11 [**3**]{} [**3**]{}
S1 [**15**]{} [**15**]{} 14 [**15**]{} [**15**]{} Wifi 5 [**4**]{} 1 [**4**]{} [**4**]{}
R15 5 [**15**]{} [**15**]{} [**15**]{} [**15**]{} Vertebral 1 1 [**2**]{} [**2**]{} [**2**]{}
Dim2 7 [**9**]{} [**9** ]{} [**9** ]{} [**9** ]{} TumTyp 3 [**5**]{} 6 [**5**]{} [**5**]{}
ShapeT 27 5 11 [**3**]{} [**3**]{} DrivFace 5 6 3 5 [**4**]{}
--------- ------------ ------------- ------------ ------------ ------------ ----------- ---- ----------- ----------- ----------- ------------
\[tab:parametervalues\]
Dataset Measures AP ADPC NKGA NQ-DBSCAN SAG-DBSCAN Dataset Measures AP ADPC NKGA NQ-DBSCAN SAG-DBSCAN
--------- ---------- -------- ----------------- -------- ----------------- ---------------- ----------- ---------- -------- ---------------- -------- ---------------- -----------------
D31 Accuracy 0.2210 [**0.9677**]{} 0.3539 0.5416 [**0.9677**]{} Iris Accuracy 0.5333 0.6667 0.4533 0.7867 [**0.9067**]{}
F-Score 0.3466 [**0.9679**]{} 0.4537 0.6937 [**0.9679**]{} F-Score 0.4329 0.5714 0.5883 0.8697 [**0.9168**]{}
ARI 0.1704 [**0.9352**]{} 0.3290 0.1240 [**0.9352**]{} ARI 0.4120 0.5681 0.2681 0.6789 [**0.7592**]{}
NMI 0.4929 [**0.9573**]{} 0.6498 0.2994 [**0.9573**]{} NMI 0.4509 0.7337 0.0138 0.7603 [**0.8057**]{}
S1 Accuracy 0.7642 0.9262 0.6992 0.9614 [**0.9932**]{} Wifi Accuracy 0.1405 0.8625 0.2500 0.7545 [**0.9355** ]{}
F-Score 0.7907 0.9332 0.7315 0.9647 [**0.9934**]{} F-Score 0.1671 0.8859 0.1000 0.8561 [**0.9402**]{}
ARI 0.6518 0.8915 0.5685 0.9378 [**0.9858**]{} ARI 0.1948 0.8103 0.0000 0.6868 [**0.8470**]{}
NMI 0.8382 0.9450 0.7878 0.9695 [**0.9895**]{} NMI 0.2646 0.8309 0.0078 0.6531 [**0.8635**]{}
R15 Accuracy 0.2217 0.9917 0.8983 0.8200 [**0.9933**]{} Vertebral Accuracy 0.6774 0.6774 0.6645 0.1516 [**0.7710** ]{}
F-Score 0.3416 0.9918 0.9035 0.9011 [**0.9935**]{} F-Score 0.4038 0.4038 0.3992 0.1437 [**0.7976** ]{}
ARI 0.2574 0.9817 0.7968 0.7667 [**0.9857**]{} ARI 0.0335 0.0304 0.0166 0.2381 [**0.2916**]{}
NMI 0.5460 0.9864 0.8705 0.3609 [**0.9893**]{} NMI 0.0000 0.0000 0.0145 0.1635 [**0.3129**]{}
Dim2 Accuracy 0.8259 [**1.0000** ]{} 0.9289 [**1.0000**]{} [**1.0000**]{} TumTyp Accuracy 0.3483 [**0.9975**]{} 0.3059 [**0.9975**]{} [**0.9975**]{}
F-Score 0.8482 [**1.0000**]{} 0.9384 [**1.0000**]{} [**1.0000**]{} F-Score 0.5092 [**0.9976**]{} 0.2666 [**0.9976**]{} [**0.9976**]{}
ARI 0.7549 [**1.0000**]{} 0.8714 [**1.0000**]{} [**1.0000**]{} ARI 0.1445 [**0.9938**]{} 0.0025 [**0.9938**]{} [**0.9938**]{}
NMI 0.8782 [**1.0000**]{} 0.9337 [**1.0000** ]{} [**1.0000**]{} NMI 0.2577 [**0.9898**]{} 0.0042 [**0.9898**]{} [**0.9898**]{}
ShapeT Accuracy 0.2667 0.8808 0.2799 [**1.0000**]{} [**1.0000**]{} DrivFace Accuracy 0.3053 0.6436 0.2871 0.5924 [**0.6485**]{}
F-Score 0.3600 0.9586 0.3732 [**1.0000**]{} [**1.0000**]{} F-Score 0.3685 [**0.7489**]{} 0.3303 0.5614 0.6853
ARI 0.0910 0.7819 0.0938 [**1.0000**]{} [**1.0000**]{} ARI 0.2394 0.4684 0.0076 [**0.4701**]{} 0.3663
NMI 0.0930 0.6104 0.0882 [**1.0000**]{} [**1.0000**]{} NMI 0.2727 0.4285 0.0138 [**0.5018**]{} 0.4604
### Results presentation
Table 2 presents the number of clusters estimated by different methods. Table 3 shows the clustering results when compared with other methods.
To evaluate and compare the performance of the clustering methods, we apply the evaluation metrics: Accuracy, F-Score, Adjusted Rand Index (ARI) [@DUL] and Normalized Mutual Information (NMI) [@ABB] in our experiments to do a comprehensive evaluation. The higher the value, the better the clustering performance for all these measures. Compared with the best results of other algorithms, our method has relative advantages of 0.0936,0.3938, 0.0535 and 0.1494 (TABLE 3) with respect to Accuracy, F-Score, ARI and NMI for the Vertebral dataset, respectively.
We conduct the Friedman test with the post-hoc Nemenyi test [@ZHY] to examine whether the difference between any two clustering algorithms is significant in terms of their average ranks. The difference between two algorithms is significant if the gap between their ranks is larger than CD. There is a line between two algorithms if the rank gap between them is smaller than CD. This test shows that SAG-DBSCAN, ADPC and NQ-DBSCAN are significantly better than NKGA and AP. SAG-DBSCAN is the best performer of these algorithms, followed by ADPC (as shown in Fig. 4).
In summary, our method obtains better results with respect to the estimation of cluster number, Accuracy, F-Score, ARI and NMI, compared comprehensively with other methods.
### Parameter analysis
The parameters of the proposed method can be set with a reference of the number of objects in clustering dataset $X$. For these datasets, we set $m=3$ when $n<500$, $m=4$ when $500\leqslant n<1000$, $m=5$ when $1000\leqslant n<5000$ and $m=10$ when $n\geqslant 5000$. We set $k=ceil(2\%n)$ when $n<1000$, $k=ceil(1\%n)$ when $1000\leqslant n<2000$ and $k=20$ when $n\geqslant2000$.
The parameters of NQ-DBSCAN are shown in Table 4, for example Eps-distance=0.6 (left) and MinPts=23 (right) for D31 dataset. The parameter settings of NQ-DBSCAN are random and ruleless for the datasets. It is thus difficult to guess the right parameters for NQ-DBSCAN if the results are unknown before clustering occurs.\
--------- ---------- ----------- ----------- ------------
D31 S1 R15 Dim2 ShapeT
0.6, 23 5000, 19 0.3, 6 5000, 10 0.2746, 10
Iris Wifi Vertebral TumTyp DrivFace
0.42, 5 6, 20 16, 7 166.05, 5 10.26, 5
--------- ---------- ----------- ----------- ------------
[ ]{}
The proposed method is robust. It can obtain the same clustering result when we select values for parameters $k$ and $m$ at wide intervals. For example, SAG-DBSCAN can obtain the same results for dataset Iris with $k\in[5,11]\cap N^+$ when $m=3$. For dataset Iris, NQ-DBSCAN cannot obtain the same results for three cases; Eps=0.41, Eps=0.42 and Eps=0.43, when MinPts=5. NQ-DBSCAN is not robust in their parameters.
The parameter of ADPC [@TLI] is set with $d_c=0.02$ for these datasets. $d_c=0.02$ means that the parameter of ADPC takes the value at the position of first 2% of all distances [@TLI].\
### Comparisons and discussions
AP [@FBJ] is an unsupervised algorithm without any parameters. The parameters of NKGA [@TR] are recommended by the publication [@TR]. The algorithms of parameter-free or fixed parameter value may not be adaptive to various kinds of datasets.
ADPC can obtain good results in most instances. However, as a centroid-based method, ADPC and its variants cannot cluster objects correctly when a category has more than one center, such as the ShapeT dataset which has no single point can be considered as the geometrical centroid of the T shape cluster.
The NQ-DBSCAN produces good results for two-dimensional data after it tunes the parameters many times with reference to two-dimensional figures. However, it does not work well for high-dimensional data, because these data cannot show well in two-dimensional figures. It is hampered by the ruleless parameters when it deals with multidimensional data. SAG-DBSCAN sets its parameters according to the number of objects, it is easier to set parameters than NQ-DBSCAN. SAG-DBSCAN obtains better results more easily than NQ-DBSCAN do, when faced with a new high-dimensional dataset that has no references to known clustering results.
Conclusion
==========
In this article, the SAG-DBSCAN algorithm is proposed, and then some simulation and real data are used to test the performance and effectiveness of the proposed method. Moreover, our proposed algorithm is also compared with several frequently-used clustering algorithms, including the intelligent algorithm NKGA, the centroid-based algorithm ADPC, the parameter-free self-adaption algorithm AP and an improved DBSCAN algorithm NQ-DBSCAN. The experiments indicate that our method obtains better results, in terms of the evaluation metrics (TABLE 3) and the estimated number of clusters (TABLE 2), than the other methods under comparison. Based on this work, it will be interesting to extend our method into a fully adaptive method in the future.
[00]{}
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[^1]: http://archive.ics.uci.edu/ml/datasets.php
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have performed a metrological characterization of the quantum Hall resistance in a 1 $\mu$m wide graphene Hall-bar. The longitudinal resistivity in the center of the $\nu=\pm 2$ quantum Hall plateaus vanishes within the measurement noise of 20 m$\Omega$ upto 2 $\mu$A. Our results show that the quantization of these plateaus is within the experimental uncertainty (15 ppm for 1.5 $\mu$A current) equal to that in conventional semiconductors. The principal limitation of the present experiments are the relatively high contact resistances in the quantum Hall regime, leading to a significantly increased noise across the voltage contacts and a heating of the sample when a high current is applied.'
author:
- 'A.J.M. Giesbers'
- 'G. Rietveld'
- 'E. Houtzager'
- 'U. Zeitler'
- 'R. Yang'
- 'K.S. Novoselov'
- 'A.K. Geim'
- 'J.C. Maan'
title: Quantum resistance metrology in graphene
---
The Hall resistance in two-dimensional electron systems (2DESs) is quantized in terms of natural constants only, $R_H = h/ie^2$ with $i$ an integer number [@Klitzing]. Due to its high accuracy and reproducibility this quantized Hall resistance in conventional 2DESs is nowadays used as a universal resistance standard [@metrology].
Recently a new type of half-integer quantum Hall effect [@NovoselovNature; @ZhangNature] was found in graphene, the purely two-dimensional form of carbon [@ReviewGeim]. Its unique electronic properties [@ReviewCastroNeto] (mimicking the behavior of charged chiral Dirac fermions [@Semenoff; @Haldane]) allow the observation of a quantized Hall resistance up to room-temperature [@NovoselovScience2; @Giesbers], making graphene a promising candidate for a high-temperature quantum resistance standard. Although the quantized resistance in graphene around the $\nu = 2$ plateau is generally believed to be equal to $h/2e^{2}$, up to now it has not been shown to meet a metrological standard. In this Letter we present results of the first metrological characterization of the quantum Hall resistance in graphene. In particular, we will address the present accuracy of quantization (15 ppm) and the experimental conditions limiting this accuracy.
![(Color online) Longitudinal resistivity $\rho_{xx}$ (blue, measured across contacts 3 and 5) and Hall resistance $\rho_{xy}$ (red, measured across 5 and 6) at $B=14$ T and $T=0.35$ K as a function of gate voltage (top x-axis) and the corresponding carrier concentration (bottom x-axis). A bias current $I=100$ nA was applied between the contacts 7 and 8. The inset shows a false color scanning electron micrograph of the graphene Hall-bar with the contact configuration of the device. []{data-label="Figure1"}](Fig1.eps){width="6cm"}
Our sample consists of a graphene Hall-bar on a Si/SiO$_{2}$ substrate forming a charge-tunable ambipolar field-effect transistor (A-FET), where the carrier concentration can be tuned with a back-gate voltage $V_g$ [@NovoselovScience]. In order to remove most of the surface dopants that make graphene generally strongly hole doped and limit its mobility, we have annealed the sample [*in-situ*]{} for several hours at 380 K prior to cooling it down slowly ($\Delta T / \Delta t < 3$ K/min) to the base temperature (0.35 K) of a top-loading $^{3}$He-system equipped with a 15 T superconducting magnet. After annealing, the charge neutrality point in the A-FET was situated at 5 V and the sample displayed a (low-temperature) mobility $\mu = 0.8$ m$^2$(Vs)$^{-1}$.
We have performed standard DC resistance measurements using a Keithley 263 current source and two HP3458a multimeters or, for the most sensitive longitudinal resistance measurements, an EM N11 battery-operated nanovolt meter. A low-pass [*LC*]{}-filter at the current-source output protects the sample from large voltage peaks during current reversal. Special care was taken to achieve high leakage resistance of the wiring in the insert ($R_{leak} > 10^{13}$ $ \Omega$). The high precision measurements were performed with a Cryogenic Current Comparator (CCC) [@NPLCCC] using a 100 $\Omega$ transfer resistor, where special attention was devoted to measuring at low currents ($I_{sd}=1.5$ $\mu$A).
Figure \[Figure1\] shows a typical quantum Hall measurement at $B=14$ T and $T=0.35$ K with the Hall resistance $\rho_{xy}$ and the longitudinal resistivity $\rho_{xx}$ plotted as a function of the carrier concentration $n$. Around filling factors $\nu= \pm2$ the device displays well defined flat plateaus in $\rho_{xy}$ accompanied by zero longitudinal resistivity minima in $\rho_{xx}$.
![(Color online) (a) Detailed sweep of $\rho_{xx}$ for holes on both sides of the sample, $\rho_{3,\, 5}$ (red) and $\rho_{4,\, 6}$ (blue), with $I_{sd}=0.5$ $\mu$A at $B=14$ T and $T=0.35$ K. The curves were taken for two different cooldowns (solid and dotted lines). (b) Detailed sweep of $\rho_{xx;\, 4,\, 6}$ for electrons at different source-drain currents, $I_{sd}=0.5,\, 1.5,\, 2.5$ $\mu$A, respectively solid black, dashed red and dotted blue at $B=14$ T and $T=0.35$ K.[]{data-label="Figure2"}](Fig2.eps){width="3.5cm"}
In a next step we characterize the sample following the metrological guidelines [@Dalahaye] for DC-measurements of the quantum Hall resistance, especially making sure that the longitudinal resistivity $\rho_{xx}$ is well enough zero in order to provide a perfect quantization of $\rho_{xy}$ [@metrology]. Qualitatively, the absolute error in the quantization of $\rho_{xy}$ due to a finite $\rho_{xx}$, can be estimated as $\Delta \rho_{xy} = -s \rho_{xx}$, where $s$ is in the order of unity [@Furlan].
In order to address the quantization conditions in some detail, we have investigated the longitudinal resistivities in the $\nu=\pm 2$ minima along both sides of the sample under different conditions. Figure \[Figure2\](a) shows that the $\nu=-2$ resistivity minima for holes are indeed robustly developed on both sides of the sample for two different cooldowns. A similar robustness of the resistivity minima is also observed for electrons around the $\nu=2$ minimum.
Figure \[Figure2\](b) displays the behaviour of $\rho_{xx}$ around $\nu= 2$ for increasing source-drain currents. All minima remain robust and symmetric, the position of the middle of the minimum does not change for neither the holes nor the electrons when the bias current is increased.
![(Color online) Precise measurement of the zero longitudinal resistance for (a) holes ($n=-7.68 \cdot 10^{15}$ m$^{-2}$), and (b) electrons ($n=+7.89 \cdot 10^{15}$ m$^{-2}$) at $B=14$ T and $T=0.35$ K. Current densities of 2.5 A/m for holes and 3.5 A/m for electrons are achievable in graphene before the quantum Hall effect starts to breakdown (gray arrow). The inset shows the same hole measurements for a poorly annealed sample. []{data-label="Figure3"}](Fig3.eps){width="4.5cm"}
A more detailed investigation of the longitudinal resistance in its zero-minima is shown in Figure 3. On the hole side of the sample (Fig. \[Figure3\](a)) the resistivity in the $\nu = -2$ minimum remains zero for bias currents up to 2.5 $\mu$A within the measurement noise (20 m$\Omega$ for the highest current). At higher currents the resistance starts to rise significantly above zero, indicating current breakdown of the quantum Hall effect.
For electrons (Fig. \[Figure3\](b)), even higher currents are attainable; no breakdown is observed for currents as high as 3.5 $\mu$A, corresponding to a current density of 3.5 A/m. For a 1 $\mu$m wide Hall bar, this is a very promising result indeed, as wider samples might therefore easily sustain currents up to several tens of microamperes before breakdown of the quantum Hall effect starts [@current].
As a reference we also investigated a poorly annealed sample (charge neutrality point at 9 V, mobility $\mu=0.5$ m$^2$(Vs)$^{-1}$ at 0.35 K). Here the quantum-Hall minimum breaks down for considerably smaller currents (see insert in (Fig. \[Figure3\](a)) and already reaches 30 $\Omega$ at a current of 1 $\mu$A, making it unsuitable for high precision measurements of the QHE.
These characterization measurements presented so far are a promising starting point to anticipate that the Hall resistance in graphene is indeed quantized accurately. From the fact that $\rho_{xx}$ remains below 20 m$\Omega$ for currents up to 2.5 $\mu$A one may expect an accuracy as good as 1 ppm for the quantum Hall plateaus in this well annealed sample.
![(Color online) Deviations from quantization in ppm measured with the CCC ($I_{sd}=1.5$ $\mu$A) for different contact configurations and the average of them (blue circles). The red square ($R_{p.a.}$) represents the deviation for a poorly annealed sample at a source-drain current of 0.5 $\mu$A. []{data-label="Figure4"}](Fig4.eps){width="4.5cm"}
In order to check this expectation we performed high precision measurements on the quantum Hall plateaus using a CCC with a source-drain current of 1.5 $\mu$A (see Fig. \[Figure4\]). Variations measured in the quantum Hall resistance in a many hour CCC measurement (Fig. \[Figure4\]) were more than one order of magnitude larger than the 1 to 2 parts in $10^6$ noise attained in a single five minute CCC measurement run. The fluctuations in the precision measurement are considerably reduced when better voltage contacts are chosen. Still, the variations were two orders of magnitude larger than in a measurement at the same current of an AlGaAs heterostructure.
Combining several measurement runs using different contacts, we achieved an average resistance value of the $\nu=\pm 2$ quantum Hall plateaus in graphene of $R_{H}=(12,906.34 \pm 0.20)$ $\Omega$, showing no indication of a different quantization in graphene with respect to conventional two-dimensional electron systems at the level of ($-5\pm15$) parts in $10^6$.
For comparison we also determined the quantization of the poorly annealed sample at a source-drain current of 0.5 $\mu$A. The deviation of ($85\pm 20$) ppm is consistent with an $s$-factor of $-0.41$ due to the finite longitudinal resistance, $\rho_{xx}=2.3\;\Omega$.
------------- ------------------------- ----------------------------- --
Contact \# $R_{holes}$ (k$\Omega$) $R_{electrons}$ (k$\Omega$)
\[0.5ex\] 1 5.6 1.25
3 0.95 6.3
4 0.03 2.7
5 1.4 4.8
6 0.3 1.1
7 1.0 5.5
8 0.3 0.8
\[1ex\]
------------- ------------------------- ----------------------------- --
: Contact resistances of the graphene sample, measured in the quantum Hall regime where $\rho_{xx} \simeq 0$ $\Omega$ (all values for the voltage contacts (1-6) were measured at 0.1 $\mu$A, whereas the current contacts 7 and 8 where measured at 3 $\mu$A).
\[table1\]
The main limitation in the CCC measurements appeared to be the contact resistance of the voltage contacts [@Dalahaye]. The rather high resistances induce additional measurement noise and fluctuations in the voltage contacts thereby limiting the attainable accuracy of quantum-Hall precision experiments. Table \[table1\] shows the contact resistances for our specific sample in the center of the $\rho_{xx}$ minima around $\nu= \pm 2$ in a three terminal setup. They reveal large variations for the different contacts, and, furthermore a significant difference between holes ($n<0$) and electrons ($n>0$). The latter might be explained by doping effects of the contacts [@contacts1] and the high contact resistance of the contacts could be accounted for by non-ideal coupling between the gold contacts and the graphene sheet. [@contacts2]. Besides noise on the voltage contacs, high contact resistances also lead to local heating at the current contacts thereby limiting the maximum breakdown current.
In conclusion, we have presented the first metrological characterization of the quantum Hall effect in graphene. We have shown that the quantum Hall resistance in a only 1 $\mu$m wide graphene sample is already within ($-5\pm15$) ppm equal to that in conventional AlGaAs and Si-MOSFET samples. A proper annealing of the sample ensuring well pronounced zeroes in $\rho_{xx}$ and sufficiently high breakdown currents were shown to be crucial to obtain such an accuracy. The main limitation for high accuracy measurements in our experiments are the relatively high contact resistances of the sample used, inducing measurement noise and local heating. Extrapolating our results to samples with lower resistance contacts for both electrons and holes and using wider samples with high breakdown currents, would most probably allow precision measurements of the quantum Hall effect in graphene with an accuracy in the ppb range.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Spare representation of signals has received significant attention in recent years. Based on these developments, a sparse representation-based classification (SRC) has been proposed for a variety of classification and related tasks, including face recognition. Recently, a class dependent variant of SRC was proposed to overcome the limitations of SRC for remote sensing image classification. Traditionally, greedy pursuit based method such as orthogonal matching pursuit (OMP) are used for sparse coefficient recovery due to their simplicity as well as low time-complexity. However, orthogonal least square (OLS) has not yet been widely used in classifiers that exploit the sparse representation properties of data. Since OLS produces lower signal reconstruction error than OMP under similar conditions, we hypothesize that more accurate signal estimation will further improve the classification performance of classifiers that exploiting the sparsity of data. In this paper, we present a classification method based on OLS, which implements OLS in a classwise manner to perform the classification. We also develop and present its *kernelized* variant to handle nonlinearly separable data. Based on two real-world benchmarking hyperspectral datasets, we demonstrate that class dependent OLS based methods outperform several baseline methods including traditional SRC and the support vector machine classifier.'
address: |
Department of Electrical and Computer Engineering, University of Houston\
Contact Author: Saurabh Prasad, Email: [email protected]
author:
- Minshan Cui
- Saurabh Prasad
title: 'Sparse Representation-Based Classification: Orthogonal Least Squares or Orthogonal Matching Pursuit?'
---
Orthogonal least square ,orthogonal matching pursuit ,sparse representation-based classification ,hyperspectral image classification.
Introduction
============
In recent years, sparse representation of signals has drawn considerable interest and has shown to be powerful in many applications — particularly in compression and denoising. It is based on the observation that most natural signals can be sparsely represented in an appropriate representation. Applications of sparse signal representations can be found in various fields such as image denoising [@elad2006image; @dabov2007image], restoration [@mairal2008sparse], visual tracking [@bai2012robust; @lu2013robust], detection [@zhu2014sparse; @cong2013abnormal], and classification [@WYG2009; @chen2015vehicle; @zhou2013kernel; @wang2012supervised]. Recent work in [@WYG2009], Wright *et al.* proposed a sparse representation-based classification (SRC) for face recognition. The basic idea of SRC is to learn a sparse representation for a test sample as a (sparse) linear combination of all training samples (over-complete dictionary), wherein the class-specific dictionary yielding the lowest reconstruction error determines the class label for the test sample. SRC has also been actively applied in various classification problems including vehicle classification [@mei2011robust], multimodal biometrics [@shekhar2014joint], digit recognition [@labusch2008simple], speech recognition [@gemmeke2011exemplar], hyperspectral image classification [@CNT2011; @MStgars2013].
Finding the sparsest solution in SRC is a combinatorial problem as it involves searching through every combination of $S$ atoms in a dictionary, where $S$ denotes the optimal sparsity level. There are two major approaches to approximate this problem. One is to relax this non-convex combinatorial problem into an $\ell_1$ convex optimization problem — also known as basis pursuit. Several methods have been proposed to solve this $\ell_1$-norm problem including interior-point method [@KKLG2007], gradient projection [@FNW2007] etc. The other major category is based on iterative greedy pursuit algorithms such as matching pursuit, orthogonal matching pursuit (OMP) and orthogonal least square (OLS). These greedy approaches have been widely used due to their computational simplicity and easy implementation. They find an atom at a time based on different criterion and update the sparse solution iteratively. Among these approaches, the OMP algorithm is by far the most popular approach and is used in a wide range of applications. The main difference between OMP and MP is that OMP uses an orthogonal dictionary while MP does not. Making the dictionary orthogonal will reduce the redundancy of the dictionary when estimating the signal. OLS is similar to OMP except for the atom selection process. A major difference between OMP and OLS relies on their atom selection procedure in that OMP selects an atom that best correlates with the current residual, while OLS selects an atom giving the smallest residual after orthogonalization. The time complexity of OMP is $O(dnS)$ where $d$ is number of features, $n$ is the dictionary size and $S$ is the sparsity level. The time complexity of OLS is slightly higher than OMP which is caused by the difference in the atom selection process. Note that the first atom selected by OMP is identical to OLS. For more detailed information about the differences between these two algorithms, readers can refer to [@blumensath2007difference; @rebollo2002optimized] and a $k$-step analysis of OMP and OLS can be found in [@soussen2011joint].
OLS has been widely used in many applications [@chen2006local; @chen2009orthogonal; @huang2005determining; @huang2012orthogonal; @zhang2014two], but it has not gained much attention for classification problems. In [@MStgars2013], the authors implement SRC in a classwise manner to improve the classification accuracy, in which the sparse coefficient is recovered by OMP. In this work, we implement A class-dependent version of OLS to perform classification. [Since OLS produces lower signal reconstruction error compared to OMP under similar condition [@blumensath2007difference] (such as the same sparsity level, same dictionary etc.) — an observation that will be further analyzed and explained in the next section, we hypothesize that more accurate signal estimation will further improve the classification performance of SRC. Compared with convex optimization based techniques such as interior point and gradient projection methods [@sjk; @mat], greedy pursuit-based approaches are more efficient and appropriate to recover the sparse coefficient in SRC due to their low time-complexity. By using the kernel trick, we extend the proposed cdOLS into its kernel variant to handle nonlinearly separable data as well.]{}
The remainder of this paper is organized as follows. In Sec. \[sec:sparse\], we briefly introduce the basic concept of SRC and illustrate the recovery performance of OMP and OLS using an illustrative case study. The proposed cdOLS as well as its kernel variant are also described in Sec. \[sec:sparse\]. Experimental hyperspectral datasets and comparative classification results are presented in Sec. \[sec:experiment\]. We provide concluding remarks in Sec. \[sec:conclusion\].
Sparse representation {#sec:sparse}
=====================
Sparse representation-based classification
------------------------------------------
Assume $\boldsymbol{a}_{ij} \in \mathbb{R}^{d}$ represent the $j$-th training sample from class $i$, $\boldsymbol{A} = [\boldsymbol{A}_{1},\boldsymbol{A}_{2}, \hdots, \boldsymbol{A}_{c}]$, where $\boldsymbol{A}_{i} = [\boldsymbol{a}_{i1},\boldsymbol{a}_{i2}, \hdots, \boldsymbol{a}_{i n_{i}}] \in \mathbb{R}^{d \times n_{i}}$ is the $i$-th class training sample set, $c$ is the number of classes, $n_{i}$ represents the number of training samples from class $i$, and $n$ is the total number of training samples, $n = \sum_{i=1}^{c}{n_{i}}$. Based on the assumption of SRC, a test sample $\boldsymbol{x} \in \mathbb{R}^{d}$ from class $i$ approximately lies in the linear span of training samples from class $i$ which can be described as $$\begin{aligned}
\boldsymbol{x} & \approx & \beta_{i1}\boldsymbol{a}_{i1} + \beta_{i2}\boldsymbol{a}_{i2} + \hdots + \beta_{in_{i}}\boldsymbol{a}_{in_{i}} \nonumber \\
& = & [\boldsymbol{a}_{i1},\boldsymbol{a}_{i2}, \hdots, \boldsymbol{a}_{in_{i}}] [\beta_{i1},\beta_{i2}, \hdots, \beta_{in_{i}}]^{\top} \nonumber \\
& = & \boldsymbol{A}_{i} \boldsymbol{\beta}_{i} \end{aligned}$$ where $\boldsymbol{\beta}_{i}$ is a coefficient vector whose entries are the weights of the corresponding training samples in $\boldsymbol{A}_{i}$.
In real-world classification problems, the true label of the test sample is unknown. Thus $\boldsymbol{x}$ needs to be represented as a linear combination of all training samples in $\boldsymbol{A}$ as described below $$\boldsymbol{x} = \boldsymbol{A}\boldsymbol{\beta}
\label{eq:linear}$$ where $\boldsymbol{\beta} = [\beta_{11},\beta_{12}, \hdots, \beta_{cn_{c}}]$ is a coefficient vector corresponding to $\boldsymbol{A}$.
Ideally, the entries of $\boldsymbol{\beta}$ are all zeros except those related to the training samples from the same class as the test sample. The residual of each class can be calculated via $$\mathbf{r}_i(\boldsymbol{x}) = \|\boldsymbol{a} - \boldsymbol{A}_{i}\hat{\boldsymbol{\beta}}_{i}\|_{2}, \quad \quad i = 1,2, \ldots, c$$ where $\hat{\boldsymbol{\beta}}_{i}$ denotes the entries of the coefficient vector $\boldsymbol{\beta}$ associated with the training samples from the $i$-th class.
Finally, $\boldsymbol{x}$ is assigned a class label $i$ corresponding to a class that resulted in the minimal residual.
Sparse solution via OMP and OLS {#sec:omp_ols}
-------------------------------
The sparsest solution of $\boldsymbol{x}$ in can be obtained by solving $$\hat{\boldsymbol{\beta}} = \operatorname{argmin}\|\boldsymbol{\beta}\|_{0}, \quad \operatorname{s.t.} \quad \boldsymbol{A}\boldsymbol{\beta} = \mathbf{x},
\label{eq:l0}$$ where the l0-norm $\|\cdot\|_{0}$ simply counts the number of nonzero entries in $\boldsymbol{\beta}$.
The problem in is NP-hard, and it cannot be solved in polynomial time. There are several different approaches [@CDS1998; @KKLG2007; @wipf2004sparse] to solving this sparse approximation problem in , in this letter, we focus on the two greedy pursuit based approaches — OMP and OLS. Both OMP and OLS can be used to approximate the sparsest solution in . In each iteration, the atom selected by OMP is not designed to minimize the residual norm after projecting the target signal onto the selected elements, while OLS selects the atom that minimizes the residual based on the previously selected atoms. Thus the final residual norm generated by OLS is always smaller than OMP under similar conditions. However, OLS does not always give the sparsest solution. To find an optimal $S$-term representation of an signal $\boldsymbol{x}$ in , a simple approach to finding the sparsest solution then is to search over all possible linear combinations of $S$ atoms in $\boldsymbol{A}$. Let us denote this exhaustive searching algorithm as combinatorial orthogonal least square (COLS). The first atom selected by OLS and OMP is the same and fixed. However, COLS iteratively select each of the atom as the first atom and remaining atoms are selected based on OLS. Specifically, it first selects the first atom and then select the remaining ($S - 1$) atoms based on OLS. After selecting $S$ atoms, it uses them to estimate the signal and calculates the residual (least square error) between the signal and the estimated signal. Following this, it selects the next atoms as the first set of atoms and repeats the above process. After calculating all $n$ ($n$ is the dictionary size) residuals using each atom as the first atom, it chooses the minimal residual as the final output. This is further explained graphically in fig. 1 next.
We use an intuitive example to illustrate the differences of OMP, OLS and COLS algorithms. [In [@blumensath2007difference], the authors use a graphical interpretation to show the difference between OMP and OLS in terms of atom selection procedure. In this example, we will further illustrate that the norm of residual generated by OLS is smaller than OMP but they are both not optimal. We will demonstrate later that the signal reconstruction performance of OLS is close to optimal.]{} Assume the true sparsity level in is $S$. Let $\boldsymbol{z}_1$, $\boldsymbol{z}_2$ and $\boldsymbol{z}_3$ be the axes in a 3-dimensional space, and $\boldsymbol{a}_1$, $\boldsymbol{a}_2$, $\boldsymbol{a}_3$ be the atoms in a dictionary $D$. Without loss of generality, assume $\boldsymbol{a}_1$ and $\boldsymbol{z}_1$ are overlapped with each other, and $\boldsymbol{a}_2$ and $\boldsymbol{a}_3$ are in the $\boldsymbol{z}_1\boldsymbol{z}_2$-plane and $\boldsymbol{z}_1\boldsymbol{z}_3$-plane respectively. Let $\boldsymbol{x}$ be a target signal, and assume that $\boldsymbol{a}_1$ is the most correlated with $\boldsymbol{x}$ than $\boldsymbol{a}_2$ and $\boldsymbol{a}_3$. Let $\vec{OF} = \vec{AD}$. Let $\phi_1$ and $\phi_2$ be the angles between $\boldsymbol{a}_2$ and $\vec{OF}$, and $\boldsymbol{a}_3$ and $\vec{OF}$ respectively. Under this scenario, we will analyze the optimal sparse $S$-term representation using OMP, OLS and COLS, where $S$ equals to 2. 1) OMP first selects the most correlated atom which is $\boldsymbol{a}_1$, and produces the residual $\vec{AD}$ by projecting $\boldsymbol{x}$ onto it. Next, OMP selects an atom that is mostly correlated with $\vec{AD}$. Since $\vec{OF}=\vec{AD}$ and $\phi_1 < \phi_2$, OMP selects $\boldsymbol{a}_2$. Therefore, the final residual norm produced by OMP is $\|\vec{AB}\|_2$, which is obtained by projecting $\boldsymbol{x}$ onto $\boldsymbol{a}_1\boldsymbol{a}_2$-plane. 2) For OLS, the first atom selected is $\boldsymbol{a}_1$, since OMP and OLS are the same in the first iteration. Next, OLS calculates the residual norms of $\|\vec{AC}\|_2$ and $\|\vec{AB}\|_2$ obtained by projecting $\boldsymbol{x}$ onto $\boldsymbol{a}_1\boldsymbol{a}_3$-plane and $\boldsymbol{a}_1\boldsymbol{a}_2$-plane respectively, and selects $\boldsymbol{a}_3$, since $\|\vec{AC}\|_2 < \|\vec{AB}\|_2$. Thus, the final residual norm of OLS is $\|\vec{AC}\|_2$ obtained by projecting $\boldsymbol{x}$ onto $\boldsymbol{z}_1\boldsymbol{z}_3$-plane. 3) COLS calculates all residuals by projecting $\boldsymbol{x}$ onto planes formed by every combination of two atoms. Since $\|\vec{AE}\|_2 < \|\vec{AC}\|_2 < \|\vec{AB}\|_2$, COLS selects $\boldsymbol{a}_2$ and $\boldsymbol{a}_3$. The final residual norm is $\|\vec{AE}\|_2$. For the special case when $D$ is an orthonormal dictionary, all of the above three methods will find an optimal $S$-term representation [@tropp2004greed]. Overall, the performance of these methods with regard to the reconstruction error are COLS $\ge$ OLS $\ge$ OMP.
![Graphically illustrating OMP, OLS and COLS. []{data-label="fig:omp_ols"}](figure1_eps-eps-converted-to.pdf "fig:"){width="7cm"}\
The proposed OLS-based classification {#sec:classification}
-------------------------------------
The recent work in [@MStgars2013] demonstrates that operating SRC in a class-wise manner can significantly improve the classification performance of SRC. As is explained in the previous section, the recovery ability of OLS is always better than OMP in terms of the least square error under the same condition (i.e. the same sparsity level). Therefore, it is expected that the classification performance can be significantly enhanced by replacing OMP with OLS under this framework. We name this algorithm class-dependent OLS (cdOLS). Note that the stopping criterion in cdOLS is based on the sparsity level. This is because the signal estimation error monotonically decreases as the sparsity level increases. [Hence, we use the same sparsity level for each class to circumvent this bias.]{} We also extend cdOLS to a “kernel” cdOLS (KcdOLS). The cdOLS and KcdOLS algorithms are described in Algorithm \[alg:cdols\] and Algorithm \[alg:kcdols\]. For a faster implementation of OLS, readers can refer to [@blumensath2007difference].
**Input:** A training dataset $\boldsymbol{A} \in \{\boldsymbol{A}_l\}_{l=1}^{c} \in \mathbb{R}^{d \times n}$, test sample $\boldsymbol{x} \in \mathbb{R}^{d}$ and sparsity level $S$.\
Set $\Lambda^{0} = \emptyset$, $\boldsymbol{r}^{0} = \boldsymbol{y}$, and iteration counter $m = 1$. Update the support set $\Lambda^{m} = \Lambda^{m-1} \cup \lambda^{m}$ by solving $$\lambda^{m} = {\operatornamewithlimits{argmin}}_{j=1,2,\ldots, n}\|\boldsymbol{x} - (\boldsymbol{A}_l)_{:,\Lambda^{m\!-\!1}\cup j}\tilde{\boldsymbol{\beta}}\|_2, \nonumber$$ where $\tilde{\boldsymbol{\beta}} = (\boldsymbol{A}_l^{\dagger})_{:,\Lambda^{m\!-\!1}\cup j}\boldsymbol{x}.$ Calculate the residual $\boldsymbol{r}^{m}$ by solving $$\boldsymbol{r}^{m} = \boldsymbol{x} - \boldsymbol{A}_{:,\Lambda^{m}}\hat{\boldsymbol{\beta}}, \nonumber$$ where $\hat{\boldsymbol{\beta}} = (\boldsymbol{A}^{\dagger}_l)_{:,\Lambda^{m}}\boldsymbol{x}$. $m \leftarrow m + 1$ Calculate the $l$-th class residual norm $\displaystyle{\nu_l = \|\boldsymbol{r}^{m-1}\|_2}$. Class label of $\boldsymbol{x}$: $\displaystyle{\omega = {\operatornamewithlimits{argmin}}_{l = 1,2,\ldots, c}\nu_l}$.\
**Output:** A class label $\omega$.
Experimental Validation
=======================
We validate the proposed cdOLS and KcdOLS and compare with various baselines using two benchmark hyperspectral datasets. The first dataset is acquired using an ITRES-CASI (Compact Airborne Spectrographic Imager) 1500 hyperspectral imager over the University of Houston campus and the neighboring urban area in 2012. This image has [a spatial dimension of $1905 \times 349$ with a]{} spatial resolution [of]{} $2.5m$. There are 15 number of classes and 144 spectral bands over [the]{} $380 - 1050nm$ wavelength range. Fig. \[data\_uh\] shows the true color image of University of Houston dataset inset with the ground truth.
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The second hyperspectral data is acquired using ProSpecTIR instrument [in]{} May 2010 over an agriculture area in Indiana, USA. This image covering agriculture fields has $1342 \times 1287$ spatial dimension with 2$m$ spatial resolution. It has 360 spectral bands over $400 - 2500nm$ wavelength range with approximately $5nm$ spectral resolution. The 19 classes consist of agriculture fields with different residue cover. Fig. \[data\_i\] shows the true color image of the Indian Pines dataset with corresponding ground truth.
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![ (a) True color image and (b) ground-truth of the Indian Pines Data[]{data-label="data_i"}](figure29_eps-eps-converted-to.pdf "fig:"){width="4cm"} ![ (a) True color image and (b) ground-truth of the Indian Pines Data[]{data-label="data_i"}](figure30_eps-eps-converted-to.pdf "fig:"){width="3.45cm"}
(a) (b)
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![ (a) True color image and (b) ground-truth of the Indian Pines Data[]{data-label="data_i"}](figure31_eps-eps-converted-to.pdf){width="7.5cm" height="1.7cm"}
**Input:** A training dataset $\boldsymbol{A} = \{\boldsymbol{A}_{l}\}_{l=1}^{c} \in \mathbb{R}^{d \times n}$, where $\boldsymbol{A}_{l}=\{\boldsymbol{a}_{li}\}_{i=1}^{n_{l}} \in \mathbb{R}^{d \times n_l}$, test sample $\boldsymbol{x} \in \mathbb{R}^{d}$, kernel function $\kappa$, sparsity level $S$.\
Calculate $l$-th class kernel matrix $\boldsymbol{K}_{l} \in \mathbb{R}^{n_{l} \times n_{l}}$ whose $(i,j)$-th entry is $\kappa(\boldsymbol{a}_{li},\boldsymbol{a}_{lj})$ and $\boldsymbol{k}_{l} \in \mathbb{R}^{n_{l}}$ whose $i$-th entry is $\kappa(\boldsymbol{x}, \boldsymbol{a}_{li})$. [Set index set $\Lambda^{1}$ to be the index corresponding to the largest entry in $\boldsymbol{k}_{l}$ and iteration counter $m = 2$]{}. Update the support set $\Lambda^{m} = \Lambda^{m-1} \cup \lambda^{m}$ by solving $$\begin{aligned}
\nonumber
\lambda^{m} &= {\operatornamewithlimits{argmin}}_{j \in {1,2,\ldots, n}}(\kappa(\boldsymbol{x},\boldsymbol{x}) \ - \ 2(\boldsymbol{k}_l^{\top})_{\Lambda^{m\!-\!1} \cup j}\tilde{\boldsymbol{\beta}} \ + \ \\
& \qquad \tilde{\boldsymbol{\beta}}^{\top}(\boldsymbol{K}_l)_{\Lambda^{m\!-\!1} \cup j,\Lambda^{m\!-\!1} \cup j}\tilde{\boldsymbol{\beta}}), \nonumber
\end{aligned}$$ where [$\tilde{\boldsymbol{\beta}}=((\boldsymbol{K}_l)_{\Lambda^{m\!-\!1} \cup j,\Lambda^{m\!-\!1} \cup j})^{-1}(\boldsymbol{k}_l)_{\Lambda^{m\!-\!1} \cup j}$]{}. $m \leftarrow m + 1$ The $l$-th class residual norm can be calculated via $$\nu_l = \sqrt{\kappa(\boldsymbol{y},\boldsymbol{y})-2(\hat{\boldsymbol{\beta}})^{\top}(\boldsymbol{k}_l)_{\Lambda^{m\!-\!1}}+(\hat{\boldsymbol{\beta}})^{\top}(\boldsymbol{K}_l)_{\Lambda^{m\!-\!1},\Lambda^{m\!-\!1}}\hat{\boldsymbol{\beta}}}, \nonumber$$ where $\hat{\boldsymbol{\beta}}=\big((\boldsymbol{K}_l)_{\Lambda^{m},\Lambda^{m}}\big)^{\!-1}(\boldsymbol{k}_l)_{\Lambda^{m}}.$ Class label of $\boldsymbol{x}$: $\displaystyle{\omega = {\operatornamewithlimits{argmin}}_{l = 1,2,\ldots, c}\nu_l}$.\
**Output:** A class label $\omega$.
Results and analysis {#sec:experiment}
--------------------
To evaluate the classification performance of cdOLS and KcdOLS, several baseline approaches including SRC, kernel SRC (KSRC), class-dependent OMP (cdOMP), kernel cdOMP (KcdOMP), and linear and nonlinear support vector machine (SVM) are compared. For SRC (KSRC), we use OMP (KOMP) as the recovery method for fair comparison, although convex optimization-based approaches generally outperform greedy-based approaches. Additionally, we also implement the COLS in a class-wise manner (cdCOLS) as well as its kernel version KcdCOLS — these COLS based variants can be considered as upper bounds in performance of OLS based methods. We also include cdSRC-l1 as a baseline method. It is a class-dependent version of SRC with l1-norm as the constraint, analogous to a class-dependent basis pursuit problem. The kernel functions used in these kernel-based methods was the radial basis function (RBF). The optimal parameters including sparsity level and kernel parameter in RBF are determined via cross-validation.
The classification results for these two datasets are presented in Table \[tab:result\_h\] and Table \[tab:result\_i\] respectively. As expected, we observe that the higher the reconstruction accuracy, the better the classification result. Since COLS is a combinatorial searching method, it is practically unfeasible, particularly when the dictionary size is large. We add it as a comparative method in this work in order to compare the performance gap between cdOLS and cdCOLS. We note that cdCOLS may be feasible in scenarios where the dictionary size is small, and so is the underlying sparsity level for the representations. The overall performance of cdCOLS and cdOLS are similar with a slightly better performance for cdCOLS (as expected). The average performance of cdOLS is generally better than cdOMP.
\[tab:result\_h\]
\[tab:result\_i\]
To analyze the effect of sparsity level, we evaluate the performance of cdCOLS, cdOLS and cdOMP under the different sparsity levels. Fig. \[fig:spa\_h\] show the classification accuracy as a function of sparsity level for University of Houston data respectively. The number of samples per class in this experiment is set to 30. Hence we test the sparsity level starting from 1 to the highest possible number 30. From these two figures, we notice that the optimal sparsity level for these methods are generally very low. This is due to the fact that the within-class hyperspectral data samples are very correlated with each other, and a low residual norm can be derived using a small number of atoms.
Next, we analyze the class-specific residuals obtained for cdCOLS, cdOLS and cdOMP. In this experiment, we select a test sample from class-1 and calculate the residual of the test sample using the training samples from class-1 for both datasets. This experiment is repeated 100 times and the average residuals are reported. Fig. \[fig:res\_h\] show the residual plots for University of Houston data. As can be seen from the figures, the residual obtained from cdOLS in each iteration is smaller than the residual obtained from cdOMP. Also, the residual obtained from cdOLS is close to the optimal one obtained from cdCOLS in each iteration.
Finally, in order to validate the generalization capabilities of these classifiers, we plot for the University of Houston dataset in Fig. \[fig:map\_h\] respectively. In this experiment, 30 training samples per class are used. As can be seen from these maps, cdCOLS and cdOLS generally gives much more accurate classification maps compared with cdOMP, especially in the areas of clouds.
![Overall classification accuracy (%) versus sparsity level $S$ for the University of Houston data.[]{data-label="fig:spa_h"}](figure8_eps-eps-converted-to.pdf "fig:"){width="9cm"}\
![Norm of residual versus iteration number for the University of Houston data.[]{data-label="fig:res_u"}](figure9_eps-eps-converted-to.pdf "fig:"){width="9cm"}\
![Norm of residual versus iteration number for the Indian Pines data.[]{data-label="fig:res_h"}](figure10_eps-eps-converted-to.pdf "fig:"){width="9cm"}\
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Conclusion {#sec:conclusion}
==========
In this paper, we present a class-dependent OLS-based classification method named cdOLS for the problem of hyperspectral image classification. We also extend cdOLS into its kernel variant. Through two real-world hyperspectral datasets, we demonstrate that our proposed methods outperform cdOMP, KcdOMP as well as SVM. We also demonstrate that the classification performance of the proposed methods are close to that of cdCOLS and KcdCOLS. Our proposed developments are based on the observation that OLS is generally better suited for sparse coefficient recovery. We also present an *combinatorial* OLS based classifier - COLS, that acts as an upper bound on the performance of such classifiers, and can itself be used as well when the training dictionary is small. For scenarios where training dictionaries are not small, the more feasible cdOLS method has very similar performance to cdCOLS (in both the input and kernel induced space).
Acknowledgement {#acknowledgement .unnumbered}
===============
[This work was funded in part by NASA grant NNX14AI47G]{}.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We calculate the character of the Weil representation using previous results which express the Weyl symbol of metaplectic operators in terms of the symplectic Cayley transform and the Conley–Zehnder index.'
author:
- 'Maurice de Gosson[^1]'
- 'Franz Luef[^2]'
title: 'The Pseudo-Character of the Weil Representation and its Relation with the Conley–Zehnder Index'
---
Introduction
============
Let $\operatorname*{Sp}(2n,\mathbb{R})$ be the standard symplectic group: it consists of all linear automorphisms of $\mathbb{R}^{2n}=T^{\ast}\mathbb{R}^{n}$ preserving the standard symplectic form $\sigma=\sum_{j=1}^{n}dp_{j}\wedge dx_{j}$ (the generic element of $\mathbb{R}^{2n}$ is $z=(x,p)$). It is well-known that $\operatorname*{Sp}(2n,\mathbb{R})$ is a connected Lie group and $\pi_{1}[\operatorname*{Sp}(2n,\mathbb{R})]$ is isomorphic to the integer group $(\mathbb{Z},+)$ hence $\operatorname*{Sp}(2n,\mathbb{R})$ has covering groups $\operatorname*{Sp}_{q}(2n,\mathbb{R})$ of all orders $q=2,3,...,\infty$. It turns out that the double cover $\operatorname*{Sp}_{2}(2n,\mathbb{R})$ can be faithfully represented by a group of unitary operators on $L^{2}(\mathbb{R}^{n})$. This group is denoted by $\operatorname*{Mp}(2n,\mathbb{R})$ and is called the Weil (or metaplectic) representation of $\operatorname*{Sp}(2n,\mathbb{R})$. Its elements are called metaplectic operators. The covering projection is denoted by $\Pi
:\operatorname*{Mp}(2n,\mathbb{R})\longrightarrow\operatorname*{Sp}(2n,\mathbb{R})$.
Trace formulas for diverse Weil representations have been recently obtained (see for instance [@gurhad; @tho], also see [@luma]); such formulas are important in many contexts, for instance in the theory of theta functions. In this Note we study the analogue of trace formulas for the continuous case, that is, for the full metaplectic representation. Of course, for metaplectic operators the notion of trace does not make sense since such operators are not of trace class. It is however possible to define what we call a pseudo character by the formula $$\operatorname*{Tr}(S)=\int_{\mathbb{R}^{n}}K_{S}(x,x)dx$$ provided that $s=\Pi(S)$ has no eigenvalue equal to one; here $K_{S}$ is the kernel of $S\in\operatorname*{Mp}(2n,\mathbb{R})$. We will see that the phase of the pseudo-trace in the right-hand side is obtained in terms of the Conley–Zehnder index of symplectic paths, familiar from the theory of periodic orbits of Hamiltonian systems (see [@CZ; @GGP; @HWZ; @Long] and the references in these works). This index has been expressed in terms of the Leray–Maslov index (see [@Leray; @JMPA1]) in de Gosson [@JMP; @Birk; @RMP; @JMPA2] and in de Gosson et al [@GGP]. The Conley–Zehnder index also plays a key role in the semiclassical quantization of chaotic Hamiltonian systems (the physicist’s Gutzwiller formula) as has been recognized by Meinrenken [@minibis; @miniter].
Metaplectic operators as Weyl operators
=======================================
Recall that if $a\in\mathcal{S}^{\prime}(\mathbb{R}^{2n})$ the Weyl operator with symbol $a$ is the operator $A:\mathcal{S}(\mathbb{R}^{n})\longrightarrow
\mathcal{S}^{\prime}(\mathbb{R}^{n})$ defined by$$A=(2\pi)^{-n}\int_{\mathbb{R}^{2n}}a_{\sigma}(z)T(z)dz$$ where $a_{\sigma}$ is the symplectic Fourier transform of $a$,$$a_{\sigma}(z)=(2\pi)^{-n}\int_{\mathbb{R}^{2n}}e^{-i\sigma(z,z^{\prime})}a(z^{\prime})dz^{\prime}$$ and $T(z)$ is the Heisenberg operator:$$T(z)f(x^{\prime})=e^{-i(px^{\prime}-\frac{1}{2}px)}f(x^{\prime}-x).$$ The distribution $a$ is the Weyl symbol of $A$.
In de Gosson [@Mp; @Birk; @JMP] metaplectic operators are studied from the point of view of Weyl pseudo-differential calculus. The main results are summarized in the following Theorem:
\[th1\]Let $S\in\operatorname*{Mp}(2n,\mathbb{R})$ have projection $\Pi(S)=s$ on $\operatorname*{Sp}(2n,\mathbb{R})$ such that $\det(s-I)\neq0$. Then the symplectic Fourier transform of the Weyl symbol $a^{S}$ of $S$ is given by the formula $$a_{\sigma}^{S}(z)=\left( \frac{1}{2\pi}\right) ^{n}\frac{i^{\nu(S)}}{\sqrt{|\det(s-I)|}}e^{\frac{i}{2}Mz\cdot z} \label{1}$$ where $$M=\tfrac{1}{2}J(s+I)(s-I)^{-1}\text{ \ , \ }J=\begin{pmatrix}
0 & I\\
-I & 0
\end{pmatrix}
\label{2}$$ The number $\nu(S)$, defined modulo $4$, is the Conley–Zehnder index of a path joining the identity to $s$ in $\operatorname*{Sp}(2n,\mathbb{R})$ and whose homotopy class depends on the choice of $S$ .
It is easily verified that $(s+I)(s-I)^{-1}\in\mathfrak{sp}(2n,\mathbb{R})$ (the Lie algebra of $\operatorname*{Sp}(2n,\mathbb{R})$), hence $M=M^{T}$ (the mapping $s\longmapsto(s+I)(s-I)^{-1}$ is sometimes called the symplectic Cayley transform). The index $\nu(S)$ corresponds to a choice of the argument of $\det(s-I)$: $$\arg\det(s-I)\equiv(\nu(S)-n)\pi\text{ \ }\operatorname{mod}2\pi\text{.}
\label{Maslov2}$$
For a detailed study of this relationship see de Gosson [@JMPA2], where the Conley–Zehnder index is expressed in terms of the Leray–Maslov index [@JMPA1] on the symplectic space $(\mathbb{R}^{2n}\oplus\mathbb{R}^{2n},\sigma\oplus(-\sigma))$.
Recall that the symbol $a$ of a Weyl operator $A$ is related to the kernel $K_{A}$ of $A$ by the formula$$K_{A}(x,y)=\left( \frac{1}{2\pi}\right) ^{n}\int_{\mathbb{R}^{n}}e^{ip\cdot(x-y)}a(\tfrac{1}{2}(x+y),p)dp \label{kax}$$ (interpreted in the sense of distributions).
Pseudo-Trace Formulas
=====================
An immediate consequence of Theorem \[th1\] is the following formula:
Let $S\in\operatorname*{Mp}(2n,\mathbb{R})$ be as above. We have$$\operatorname*{Tr}(S)=\left( \frac{1}{2\pi}\right) ^{n}\frac{i^{\nu(S)}}{\sqrt{|\det(s-I)|}}. \label{3}$$
In view of formula (\[kax\]) we can write $$\begin{aligned}
\operatorname*{Tr}(S) & =\int_{\mathbb{R}^{n}}K_{S}(x,x)dx\\
& =\left( \frac{1}{2\pi}\right) ^{n}\int_{\mathbb{R}^{2n}}a(z)dz\\
& =a_{\sigma}^{S}(0)\end{aligned}$$ hence (\[3\]) in view of (\[1\]).
Assume now that $s\ell_{P}\cap\ell_{P}=\{0\}$ where $\ell_{P}=\{0\}\times
\mathbb{R}^{n}$; in the canonical symplectic basis of $(\mathbb{R}^{2n},\sigma)$ we may identify $s$ with a block matrix $\begin{pmatrix}
A & B\\
C & D
\end{pmatrix}
$ with $\det B\neq0$, and $S\in\operatorname*{Mp}(2n,\mathbb{R})$ has projection $\Pi(S)=s$ if and only if $$Sf(x)=\left( \frac{1}{2\pi i}\right) ^{n}i^{m}\sqrt{|\det B^{-1}|}\int_{\mathbb{R}^{n}}e^{iW(x,x^{\prime})}f(x^{\prime})dx^{\prime}$$ for $f\in\mathcal{S}(\mathbb{R}^{n})$ (Leray [@Leray], de Gosson [@Birk]); here $$W(x,x^{\prime})=\tfrac{1}{2}DB^{-1}x\cdot x-B^{-1}x\cdot x^{\prime}+\tfrac
{1}{2}B^{-1}Ax^{\prime}\cdot x^{\prime}$$ is the generating function of $s$ and $m$ is the Maslov index: $$\arg\det B^{-1}=m\pi\text{ \ }\operatorname{mod}2\pi.$$ We will write from now on $s=s_{W}$ and $S=S_{W,m}$. It is proven in de Gosson ... that if $\det(s_{W}-I)\neq0$ then$$\nu(S_{W,m})=m-\operatorname*{Inert}W_{xx}^{\prime\prime} \label{Morse}$$ where $\operatorname*{Inert}W_{xx}^{\prime\prime}$ (the Morse index) is the signature of the Hessian matrix of the mapping $x\longmapsto W(x,x)$. Thus:
When $s=s_{W}$ and $\det(s_{W}-I)\neq0$ then$$\operatorname*{Tr}(S_{W,m})=\frac{i^{m-\operatorname*{Inert}W_{xx}^{\prime\prime}}}{\sqrt{|\det(s_{W}-I)|}}.$$
Note that we have, explicitly,$$\det(s_{W}-I)=(-1)^{n}\det B\det(B^{-1}A+DB^{-1}-B^{-1}-(B^{T})^{-1})$$ (see de Gosson [@Mp], Lemma 4).
It turns out that we have the following factorization result (de Gosson [@Mp]):
Every $S\in\operatorname*{Mp}(2n,\mathbb{R})$ can be written (in infinitely many ways) as a product $S=S_{W,m}S_{W^{\prime},m^{\prime}}$ such that $\det(s_{W}-I)\neq0$ and $\det(s_{W^{\prime}}-I)\neq0$.
Using formula (\[Morse\]) together with the product formula $$\nu(SS^{\prime})=\nu(S)+\nu(S^{\prime})+\tfrac{1}{2}\operatorname*{sign}(M+M^{\prime})$$ proven in [@GGP; @JMPA2] the result above allows the calculation of the Conley–Zehnder index in the general case. The constructions in [@GGP] are certainly useful in this context.
The first author has been financed by the Austrian Science Foundation FWF (Projektnummer P20442-N13). The second author has been financed by the Marie Curie Outgoing Fellowship PIOF 220464.
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de Gosson, M.: The Weyl Representation of Metaplectic operators. Letters in Mathematical Physics 72 129–142 (2005)
de Gosson, M.: Symplectic Geometry and Quantum Mechanics. Birkhäuser, Basel, series Operator Theory: Advances and Applications (subseries: Advances in Partial Differential Equations), Vol. 166 (2006)
de Gosson, M., de Gosson, S.: An extension of the Conley–Zehnder Index, a product formula and an application to the Weyl representation of metaplectic operators. J. Math. Phys., 47(12) (2006)
de Gosson, M., de Gosson, S., Piccione, P.: On a product formula for the Conley–Zehnder Index of symplectic paths and its applications. To appear in Ann. Global Analysis and Geom. 34, 167–183 (2008). Preprint 2006 (arXiv math.SG/0607024)
de Gosson, M.: Metaplectic Representation, Conley–Zehnder Index, and Weyl Calculus on Phase Space. Rev. Math. Physics, 19(8), 1149–1188 (2007)
de Gosson, M.: On the usefulness of an index due to Leray for studying the intersections of Lagrangian and symplectic paths. Journal de Mathématiques Pures et Appliqués 91 (2009) 598–613 \[Preprint MPIM2007-119, Max Planck Institute for Mathematics preprint server: http://www.mpim-bonn.mpg.de/ preprints/retrieve (2008)\]
Gurevich, S. and Hadani, R.: The geometric Weil representation. Selecta Math.13(3) (2007) 465–481
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Leray, J.: Lagrangian Analysis and Quantum Mechanics, a mathematical structure related to asymptotic expansions and the Maslov index. The MIT Press, Cambridge, Mass. (1981)
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Long, Y.: Index iteration theory for symplectic paths and multiple periodic solution orbits. Frontiers of Math. 8, 341–353 (2006)
Meinrenken, E.: Trace formulas and the Conley–Zehnder index. J. Geom. Phys. 13, 1–15 (1994)
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[^1]: [email protected]
[^2]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection $\nabla$ on a two dimensional torus is complete and, up to a finite cover, homogeneous.
Let $\nabla$ be a unimodular real analytic affine connection on a real analytic compact connected surface $M$. If $\nabla$ is locally homogeneous on a nontrivial open set in $M$, we prove that $\nabla$ is locally homogeneous on all of $M$.\
address: '${}^\star$ Département de Mathématiques d’Orsay, Bat. 425, U.M.R. 8628 C.N.R.S., Univ. Paris-Sud (11), 91405 Orsay Cedex, France'
author:
- 'Sorin DUMITRESCU$^\star$'
title: Locally homogeneous rigid geometric structures on surfaces
---
[^1]
Introduction
============
Riemannian metrics are the most commun (rigid) geometric structures. A locally homogeneous Riemannian metric on a surface has constant sectional curvature and it is locally isometric either to the standard metric on the two-sphere, or to the flat metric on $\RR^2$, or to the hyperbolic metric on the Poincaré’s upper-half plane (hyperbolic plane). Obviously the flat metric is translation invariant on $\RR^2$. Recall also that the isometry group of the hyperbolic plane is isomorphic to $PSL(2,\RR)$ and contains copies of the affine group of the real line (preserving orientation) $Aff(\RR)$, which act locally free (they are the stabilizers of points on the boundary). Consequently, the hyperbolic plane is locally isometric to a translation invariant Riemannian metric on $Aff(\RR)$.
We generalize here this phenomena to all rigid geometric structures in Gromov’s sense (see the definition in the following section).
\[surfaces\] Let $\phi$ be a locally homogeneous rigid geometric structure on a surface. Then $\phi$ is locally isomorphic to a rigid geometric structure which is either rotation invariant on the two-sphere, or translation invariant on $\RR^2$, or translation invariant on the affine group of the real line preserving orientation $Aff(\RR)$.
If $\phi$ is an affine connection this was first proved by B. Opozda [@Opozda] for the case torsion-free (see also the group-theoretical approach in [@KOV]), and then by T. Arias-Marco and O. Kowalski in the case of arbitrary torsion [@AM-K].
Theorem \[surfaces\] stands, in particular, for projective connections. In this case the theorem can also be deduced from the results obtained in [@BMM]. Recall that a well-known class of locally homogeneous projective connections $\phi$ on surfaces are those which are [*flat*]{}, i.e. locally isomorphic to the standard projective connection of the projective plane $P^2(\RR)$. The automorphism group of $P^2(\RR)$ is the projective group $PGL(3,\RR)$ which contains $\RR^2$ acting freely (by translation) on the affine plane: it is the subgroup which fixes each point in the line at the infinity. Consequently, $\phi$ is locally isomorphic to a translation invariant projective connection on $\RR^2$.
We prove also the following global results dealing with projective and affine connections on surfaces:
\[projective\] A locally homogeneous projective connection on a compact surface is flat.
\[affine connection1\] Let $M$ be a compact connected real analytic surface endowed with a unimodular real analytic affine connection $\nabla$. If $\nabla$ is locally homogeneous on a nontrivial open set in $M$, then $\nabla$ is locally homogeneous on $M$.
The main result of the article is Theorem \[affine connection1\]. It is motivated by the celebrated open-dense orbit theorem of M. Gromov [@DG; @Gro] (see also [@Benoist2; @CQ; @Feres]). Gromov’s result asserts that a rigid geometric structure admitting an automorphism group which acts with a dense orbit is locally homogeneous on an open dense set. This maximal locally homogeneous open (dense) set appears to be mysterious and it might very well happen that it coincides with all of the (connected) manifold in many interesting geometric backgrounds. This was proved, for instance, for Anosov flows with differentiable stable and instable foliations and transverse contact structure [@BFL] and for three dimensional compact Lorentz manifolds admitting a nonproper one parameter group acting by automorphisms [@Zeghib]. In [@BF], the authors deal with this question and their results indicate ways in which the rigid geometric structure cannot degenerate off the open dense set.
Surprisingly, the extension of a locally homogeneous open dense subset to all of the (connected) manifold might stand [*even without assuming the existence of a big automorphism group*]{}. This is known to be true in the Riemannian setting [@Tri], as a consequence of the fact that all scalar invariants are constant (see also Corollary \[metric\] in our section \[section4\]). This was also recently proved in the frame of three dimensional real analytic Lorentz metrics [@Dumitrescu] and for complete real analytic pseudo-Riemannian metrics [@Melnick].
Theorem \[affine connection1\] proves the extension phenomenon in the setting of affine connections on surfaces. We don’t know if the result is still true when $\nabla$ is not unimodular analytic, or in higher dimension.
As a by-product of the proof we get the following:
\[sur tore\] A locally homogeneous unimodular affine connection on a two dimensional torus is complete and, up to a finite cover, homogeneous.
The composition of the article is the following. In section \[section2\] we introduce the basic facts about rigid geometric structures and prove theorem \[surfaces\]. Section \[section3\] deals with global phenomena and proves theorem \[projective\]. Theorems \[affine connection1\] and \[sur tore\] will be proved in sections \[section4\] and \[section 5\] respectively.
Locally homogeneous rigid geometric structures {#section2}
==============================================
In the sequel all manifolds will be supposed to be smooth and connected. The geometric structures will be also assumed to be smooth.
Consider a $n$-manifold $M$ and, for all integers $r \geq 1$, consider the associated bundle $R^r(M)$ of $r$-frames, which is a $D^r(\RR^n)$-principal bundle over $M$, with $D^r(\RR^n)$ the real algebraic group of $r$-jets at the origin of local diffeomorphisms of $\RR^n$ fixing $0$ (see [@AVL]).
Let us consider, as in [@DG; @Gro], the following
A [*geometric structure*]{} (of order $r$) $\phi$ on a $M$ is a $D^r(\RR^n)$-equivariant smooth map from $R^r(M)$ to a real algebraic variety $Z$ endowed with an algebraic action of $D^r(\RR^n)$.
Riemannian and pseudo-Riemannian metrics, affine and projective connections and the most encountered geometric objects in differential geometry are known to verify the previous definition [@DG; @Gro; @Benoist2; @CQ; @Feres]. For instance, if the image of $\phi$ in $Z$ is exactly one orbit, this orbit identifies with a homogeneous space $D^r(\RR^n)/G$, where $G$ is the stabilizer of a chosen point in the image of $\phi$. We get then a reduction of the structure group of $R^r(M)$ to the subgroup $G$. This is exactly the classical definition of a $G$-structure (of order $r$): the case $r=1$ and $G=O(n,\RR)$ corresponds to a Riemannian metric and that of $r=2$ and $G=GL(n,\RR)$ gives a torsion free affine connection [@Kobayashi; @AVL].
A (local) Killing field of $\phi$ is a (local) vector field on $M$ whose canonical lift to $R^r(M)$ preserves $\phi$.
Following Gromov [@Gro; @DG] we define rigidity as:
A geometric structure $\phi$ is rigid at order $k \in \NN$, if local Killing fields are determined by their $k$-order jet at any chosen point in $M$.
Consequently, in the neighborhood of any point of $M$, the algebra of Killing fields of a rigid geometric structure is finite dimensional.
Recall that (pseudo)-Riemannian metrics, as well as affine and projective connections, or conformal structures in dimension $\geq 3$ are known to be rigid [@DG; @Gro; @Benoist2; @CQ; @Feres; @Kobayashi].
The geometric structure $\phi$ is said to be locally homogeneous on the open subset $U \subset M$ if for any tangent vector $V \in T_{u}U$ there exists a local Killing field $X$ of $\phi$ such that $X(u)=V$.
The Lie algebra of Killing fields is the same at the neighborhood of any point of a locally homogeneous geometric structure $\phi$. In this case it will be simply called [*the Killing algebra of $\phi$.*]{}
Let $G$ be a connected Lie group and $I$ a closed subgroup of $G$. Recall that $M$ is said to be [*locally modelled*]{} on the homogeneous space $G/I$ if it admits an atlas with open sets diffeomorphic to open sets in $G/I$ such that the transition maps are given by restrictions of elements in $G$.
In this situation any $G$-invariant geometric structure $\tilde{\phi}$ on $G/I$ uniquely defines a locally homogeneous geometric structure $\phi$ on $M$ which is locally isomorphic to $\tilde{\phi}$.
We recall that there exists locally homogeneous Riemannian metrics on $5$-dimensional manifolds which are not locally isometric to a invariant Riemannian metric on a homogeneous space [@K; @LT]. However this phenomenon cannot happen in lower dimension:
\[model\] Let $M$ be a manifold of dimension $\leq 4$ bearing a locally homogeneous rigid geometric structure $\phi$ with Killing algebra $\mathfrak{g}$. Then $M$ is locally modelled on a homogeneous space $G/I$, where $G$ is a connected Lie group with Lie algebra $\mathfrak{g}$ and $I$ is a closed subgroup of $G$. Moreover, $(M, \phi)$ is locally isomorphic to a $G$-invariant geometric structure on $G/I$.
Let $\mathfrak{g}$ be the Killing algebra of $\phi$. Denote by $\mathfrak{I}$ the (isotropy) subalgebra of $\mathfrak{g}$ composed by Killing fields vanishing at a given point in $M$. Let $G$ be the unique connected simply connected Lie group with Lie algebra $\mathfrak{g}$. Since $\mathfrak{I}$ is of codimension $\leq 4$ in $\mathfrak{g}$, a result of Mostow [@Mos] (chapter 5, page 614) shows that the Lie subgroup $I$ in $G$ associated to $\mathfrak{I}$ is [*closed*]{}. Then $\phi$ induces a $G$-invariant geometric structure $\tilde{\phi}$ on $G/I$ locally isomorphic to it. Moreover, $M$ is locally modelled on $G/I$.
\[model simply connected\] By the previous construction, $G$ is simply connected and $I$ is connected (and closed), which implies that $G/I$ is simply connected (see [@Mos], page 617, Corollary 1). In general, the $G$-action on $G/I$ admits a nontrivial discrete kernel. We can assume that this action is effective considering the quotient of $G$ and $I$ by the maximal normal subgroup of $G$ contained in $I$ (see proposition 3.1 in [@Sharpe]).
Theorem \[surfaces\] is a direct consequence of theorem \[model\] and of the following:
A two dimensional homogeneous space $G/I$ of a connected simply connected Lie group $G$ is either the two sphere with $G$ being $S^3$, or it bears an action of a two dimensional subgroup of $G$ (isomorphic either to $\RR^2$, or to the affine group of the real line preserving orientation) which admits an open orbit.
Here the $3$-sphere $S^3$ is endowed with its standard structure of Lie group [@Kir; @Olver].
This follows directly from Lie’s classification of the two dimensional homogeneous space (see the list in [@Olver] or [@Mos]). More precisely, any (finite dimensional) Lie algebra acting transitively on a surface either admits a two dimensional subalgebra acting simply transitively (or equivalently, which trivially intersects the isotropy subalgebra), or it is isomorphic to the Lie algebra of $S^3$ acting on the real sphere $S^2$ by the standard action.
Global rigidity results {#section3}
=======================
Recall that a manifold $M$ locally modelled on a homogeneous space $G/I$ gives rise to a [*developing map*]{} defined on its universal cover $\tilde M$ with values in $G/I$ and to a [*holonomy morphism*]{} $\rho : \pi_{1}(M) \to G$ (well defined up to conjugacy in $G$) [@Sharpe]. The developing map is a local diffeomorphism which is equivariant with respect to the action of the fundamental group $\pi_{1}(M)$ on $\tilde M$ (by deck transformations) and on $G$ (by its image through $\rho$).
The manifold $M$ is said to be [*complete*]{} if the developing map is a global diffeomorphism. In this case $M$ is diffeomorphic to a quotient of $G/I$ by a discrete subgroup of $G$ acting properly and without fixed points.
We give now a last definition:
A geometric structure $\phi$ on $M$ is said to be of [*Riemannian type*]{} if there exists a Riemannian metric on $M$ preserved by all Killing fields of $\phi$.
Roughly speaking a locally homogeneous geometric structure is of Riemannian type if it is constructed by putting together a Riemannian metric and any other geometric structure (e.g. a vector field). Since Riemannian metrics are rigid, a geometric structure of Riemannian type it is automaticaly rigid.
With this terminology we have the following corollary of theorem \[model\].
\[isotropie compacte\] Let $M$ be a compact manifold of dimension $\leq 4$ equipped with a locally homogeneous geometric structure $\phi$ of Riemannian type. Then $M$ is isomorphic to a quotient of a homogeneous space $G/I$, endowed with a $G$-invariant geometric structure, by a lattice in $G$.
By theorem \[model\], $M$ is locally modelled on a homogeneous space $G/I$. Since $\phi$ is of Riemannian type, $G/I$ admits a $G$-invariant Riemannian metric. This implies that the isotropy $I$ is compact.
On the other hand, compact manifolds locally modelled on homogeneous space $G/I$ with compact isotropy group $I$ are classically known to be complete: this is a consequence of the Hopf-Rinow’s geodesical completeness [@Sharpe].
A $G$-invariant geometric structure on $G/I$ is of Riemannian type if and only if $I$ is compact.
Recall that a homogeneous space $G/I$ is said to be [*imprimitive*]{} if the canonical $G$-action preserves a non trivial foliation.
\[imprimitive\] If $M$ is a compact surface locally modelled on an imprimitive homogeneous space, then $M$ is a torus.
The $G$-invariant one dimensional foliation on $G/H$ descends on $M$ to a non singular foliation. Hopf-Poincaré’s theorem implies then that the genus of $M$ equals one: $M$ is a torus.
Note that the results of [@KOV; @AM-K] imply in particular:
\[imprimitive connections\] A locally homogeneous affine connection on a surface which is neither torsion free and flat, nor of Riemannian type, is locally modelled on an imprimitive homogeneous space.
Indeed, T. Arias-Marco and O. Kowalski study in [@AM-K] all possible local normal forms for locally homogeneous affine connections on surfaces with the corresponding Killing algebra. Their results are summarized in a nice table (see [@AM-K], pages 3-5). In all cases, except for the Killing algebra of the (standard) torsion free affine connection and for Levi-Civita connections of Riemannian metrics of constant sectional curvature, there exists at least one Killing field non contained in the isotropy algebra which is normalized by the Killing algebra. Its direction defines then a $G$-invariant line field on $G/I$.
The previous result combined with proposition \[imprimitive\] imply the main result in [@Opozda]:
(Opozda) \[Opozda\] A compact surface $M$ bearing a locally homogeneous affine connection of non Riemannian type is a torus.
Recall first that a well known result of J. Milnor shows that a compact surface bearing a flat affine connection is a torus [@Milnor] (see also [@Benzecri]).
In the case of a non flat connection, theorem \[imprimitive connections\] shows that $M$ is locally modelled on an imprimitive homogeneous space. Then proposition \[imprimitive\] finishes the proof.
We give now the proof of theorem \[projective\].
The starting point of the proof is the classification obtained in [@BMM] of all possible Killing algebras of a two dimensional locally homogeneous projective connection. Indeed, Lemma 3 and Lemma 4 in [@BMM] prove that eiher $\phi$ is flat, or the Killing algebra of $\phi$ is the Lie algebra of one of the following Lie groups: $Aff(\RR)$ or $SL(2,\RR)$. Moreover, in the last case the isotropy is generated by a one parameter unipotent subgroup.
Assume, by contradiction, that the Killing algebra of $\phi$ is that of $Aff(\RR)$. Then, by theorem \[model\], $M$ is locally modelled on $Aff(\RR)$ and, by theorem \[isotropie compacte\], $M$ has to be a quotient of $Aff(\RR)$ by a uniform lattice. Or, $Aff(\RR)$ is not unimodular and, consequently, doesn’t admit lattices: a contradicition.
Assume, by contradiction, that the Killing algebra is that of $SL(2,\RR)$. Then, by theorem \[model\], $M$ is locally modelled on $SL(2,\RR)/I$, with $I$ a one parameter unipotent subgroup in $SL(2, \RR)$.
Equivalently, $I$ is conjugated to $\left( \begin{array}{cc}
1 & b \\
0 & 1 \\
\end{array} \right)$, with $b \in \RR$. The homogeneous space $SL(2,\RR)/I$ is difffeomorphic to $\RR^2 \setminus \{0 \}$ endowed with the linear action of $SL(2, \RR)$.
Notice that the action of $SL(2,\RR)$ on $SL(2,\RR)/I$ preserves a nontrivial vector field. The expression of this vector field in linear coordinates $(x_{1}, x_{2})$ on $\RR^2 \setminus \{ 0 \}$ is $\displaystyle x_{1} \frac{\partial}{\partial x_{1} } + x_{2} \frac{\partial}{\partial x_{2}}$, which is the fundamental generator of the one parameter group of homotheties. The flow of this vector field doesn’t preserve the standard volume form and has a nonzero constant divergence: $\lambda=2$.
Let $X$ be the corresponding vector field induced on $M$ and $div (X)$ the divergence of $X$ with respect to the volume form $vol$ induced on $M$ by the standard $SL(2,\RR)$-invariant volume form of $\RR^2 \setminus \{ 0 \}$. Recall that, by definition, $L_{X} vol =div(X) \cdot vol,$ where $L_{X}$ is the Lie derivative. Here $div(X)$ is the constant function $\lambda$.
Denote by $\Psi^t$ the time $t$ of the flow generated by $X$. We get $(\Psi^t)^*vol=exp( \lambda t) \cdot vol,$ for all $t \in \RR$. But the flow of $X$ has to preserve the global volume $\int_{M}vol$. This implies $\lambda =0$: a contradiction.
It remains that $\phi$ is flat.
We terminate the section with the following.
If $M$ is a compact surface endowed with a locally homogeneous rigid geometric structure admitting a semi-simple Killing algebra of dimension $3$, then either $M$ is globally isomorphic to a rotation invariant geometric structure on the two-sphere (up to a double cover), or the Killing algebra of $\phi$ preserves a hyperbolic metric on $M$ (and $M$ is of genus $g \geq 2$).
By theorem \[model\], $M$ is locally modelled on $G/I$, with $G$ a $3$-dimensional connected simply connected semi-simple Lie group and $I$ a closed one parameter subgroup in $G$.
Up to isogeny, there are only two such $G$: $S^3$ and $SL(2,\RR)$ [@Kir]. If $G=S^3$ then $I$ is compact and coincides with the stabilizer of a point under the standard $S^3$-action on $S^2$. Consequently, $G/I$ identifies with $S^2$ and the $G$-action on $G/I$ preserves the canonical metric of the two-sphere.
The developing map from $\tilde M$ to $G/I$ has to be a diffeomorphism (see theorem \[isotropie compacte\]). Consequently, $M$ is a quotient of the sphere $S^2$ by a discrete subgroup of $G$ acting by deck transformations. Since a nontrivial isometry of $S^2$ which is not $-Id$ always admits fixed points, this discrete subgroup has to be of order two. Up tp a double cover, $(M, \phi)$ is isomorphic to $S^2$ endowed with a rotation invariant geometric structure.
Consider now the case where $G=SL(2,\RR)$. Then $M$ is locally modelled on $G/I$, where $I$ is a closed one parameter subgroup in $SL(2, \RR)$. We showed in the proof of theorem \[projective\] that $I$ is not conjugated to a unipotent subgroup.
Assume now that $I$ is conjugated to a one parameter semi-simple subgroup in $SL(2, \RR)$. We prove that this assumption yields a contradiction. In order to describe the geometry of $SL(2,\RR)/I$, consider the adjoint representation of $SL(2, \RR)$ into its Lie algebra $sl(2, \RR)$. This $SL(2, \RR)$-action preserves the Killing quadratic form $q$, which is a non degenerate Lorentz quadratic form. Choose $x \in sl(2, \RR)$ a vector of unitary $q$-norm and consider its orbit under the adjoint representation. This orbit identifies with our homogeneous space $SL(2, \RR)/I$, on which the restriction of the Killing form induces a two dimensional $SL(2, \RR)$-invariant complete Lorentz metric $g$ of constant nonzero sectional curvature [@Wolf].
We prove now that there is no [*compact*]{} surface locally modelled on the previous homogeneous space $SL(2, \RR)/I$.
Observe that $x$ induces on $SL(2,\RR)/I$ a $SL(2, \RR)$-invariant vector field $X$ and $g(X, \cdot)$ induces on $SL(2, \RR)/I$ an invariant one form $\omega$. Remark that $d \omega$ is a volume form. Indeed, $d\omega(Y,Z)=-g(X, \lbrack Y, Z \rbrack)$, for all $SL(2, \RR)$-invariant vector fields $Y,Z$ tangents to $SL(2, \RR)/I$.
Assume, by contradiction, that $M$ is locally modelled on $SL(2, \RR)/I$. Then $M$ inherits the one form $\omega$ whose differential is a volume form. This is in contradiction with Stokes’ theorem. Indeed $\int_{M}d \omega = \int vol \neq 0$, but $d \omega$ is exact.
It remains that $I$ is conjugated to a one parameter subgroup of rotations in $SL(2,\RR)$ and then the $SL(2, \RR)$-action on $SL(2,\RR)/ I$ identifies with the action by homographies on the Poincaré’s upper-half plane. This action preserves the hyperbolic metric. Therefore $M$ inherits a hyperbolic metric and, consequently, its genus is $\geq 2$.
Dynamics of local Killing algebra {#section4}
=================================
A manifold $M$ bearing a geometric structure $\phi$ admits a natural partition given by the orbits of the action of the Killing algebra of $\phi$. Precisely, two points $m_{1}, m_{2} \in M$ are in the same subset of the partition if $m_{1}$ can be reached from $m_{2}$ by flowing along a finite sequence of local Killing fields. A connected open set in $M$ where $\phi$ is locally homogeneous lies in the same subset of this partition.
The Gromov’s celebrated stratification theorem [@DG; @Gro] which was used and studied by many authors [@DG; @Gro; @Benoist2; @CQ; @Feres] roughly states that, if $\phi$ is rigid, the subsets of this partition are locally closed in $M$. We adapt here Gromov’s proof and get a more precise result in the particular case, where $\phi$ is of Riemannian type.
Let $M$ be a connected manifold endowed with a geometric structure of Riemannian kind $\phi$. Then the orbits of the Killing algebra of $\phi$ are closed.
\[metric\] If $\phi$ is locally homogeneous on an open dense set, then $\phi$ is locally homogeneous on $M$.
Let $g$ be a Riemannian metric preserved by all Killing fields of $\phi$. Consider also the $g$-orthonormal frame bundle $\pi : B \to M$. Then $B$ is a principal sub-bundle of the frame bundle $R^1(M)$ with structure group $O(n, \RR)$.
Gromov’s proof shows that (for any rigid geometric structure) there exists an integer $s \in \NN$ such that two points of $M$ where the $s$-jet of $\phi$ is the same are in the same orbit of the Killing algebra [@DG; @Gro; @Benoist2; @CQ; @Feres].
We consider exponential local coordinates with respect to $g$. For each element of $(m, b) \in B$ we get local exponential coordinates around the point $m \in M$ in which we take the $k$-jet $ \phi^k$ of $\phi$. This gives a map $$\phi^k : B \to Z^k$$
with values in the variety $Z^k$ of $k$-jets of $\phi$.
By Gromov’s proof, orbits of the Killing algebra are the connected components of the projections on $M$ (through $\pi$) of the pull-back through $\phi^k$ of $O(n,\RR)$-orbits of $Z^k$ (see the arguments in [@DG], section 3.5). The $O(n,\RR)$-orbits of $Z^k$ being compact, their pull-back through $\phi^k$ in $B$ are saturated closed sets. By compactness of the fibers, the projection to $M$ of a saturated closed subset in $B$ is a closed set. Since connected components in $M$ of closed sets are also closed, we get that the orbits of the Killing algebra are closed.
The corollary was known for Riemannian metrics. Indeed, in [@Tri] the authors proved that Riemannian metrics whose all scalar invariants are constant are locally homogeneous (this is known to fail in the pseudo-Riemannian setting [@BV]).
In the real analytic realm this implies the following more precise:
\[extension\] If $M$ and $\phi$ are real analytic and $\phi$ is locally homogeneous on a nontrivial open set, then $\phi$ is locally homogeneous on $M$.
In the real analytic setting, Gromov’s proof shows that away from a nowhere dense analytic subset $S$ in $M$, the orbits of the Killing algebra are connected components of fibers of an analytic map of constant rank [@Gro] (section 3.2). With our hypothesis, the orbits of the Killing algebra are exactly connected components of $M \setminus S$. Consequently, they are open sets. One apply now corollary \[metric\] and get that the orbits are also closed. Since $M$ is connected, the Killing algebra admits exactly one orbit.
In the previous results the compactness of the orthogonal group was essential.
We prove now the following result for affine unimodular connections which are not (necessarily) of Riemannian kind:
\[affine connection\] Let $M$ be a compact connected real analytic surface endowed with a real analytic unimodular affine connection $\nabla$. If $\nabla$ is locally homogeneous on a nontrivial open set in $M$, then $\nabla$ is locally homogeneous on $M$.
Recall that $\nabla$ is said [*unimodular*]{} if there exists a volume form on $M$ which is invariant by the parallel transport [@AVL]. This volume form is automaticaly preserved by any local Killing field of $\nabla$. We prove first the following useful:
\[unimodularlemma\] Let $\nabla$ be a unimodular analytic affine connection on an analytic surface $M$. Then the dimension of the isotropy algebra at a point of $M$ is $\neq 2$.
Assume by contradiction that the isotropy algebra $\mathcal{I}$ at a point $m \in M$ has dimension two. Consider a system of local exponential coordinates at $m$ with respect to $\nabla$ and, for all $k \in \NN$ take the $k$-jet of $\nabla$ in these coordinates. Any volume preserving linear isomorphism of $T_{m}M$ gives another system of local exponential coordinates at $m$, with respect to which we consider the $k$-jet of the connection. This gives an algebraic $SL(2,\RR)$-action on the vector space $Z^k$ of $k$-jets of affine connections on $\RR^2$ admitting a trivial underlying $0$-jet [@DG; @Gro].
Elements of $\mathcal{I}$ linearize in exponential coordinates at $m$. Since they preserve $\nabla$, they preserve in particular the $k$-jet of $\nabla$ at $m$, for all $k \in \NN$. This gives an embedding of $\mathcal{I}$ in the Lie algebra of $SL(2,\RR)$ such that the corresponding (two dimensional) connected subgroup of $SL(2,\RR)$ preserves the $k$-jet of $\nabla$ at $m$ for all $k \in \NN$.
Now we use the fact that [*the stabilizers of a linear algebraic $SL(2,\RR)$-action are of dimension $\neq 2$*]{}. Indeed, it suffices to check this statement for irreducible linear representations of $SL(2, \RR)$ for which it is well-known that the stabilizer in $SL(2,\RR)$ of a nonzero element is one dimensional [@Kir].
It follows that the stabilizer of the $k$-jet of $\nabla$ at $m$ is of dimension three and contains the connected component of identity in $SL(2, \RR)$. Consequently, in exponential coordinates at $m$, each element of the connected component of the identity in $SL(2, \RR)$ gives rise to a local linear vector field which preserves $\nabla$ (for it preserves all $k$-jets of $\nabla$). The isotropy algebra $\mathcal{I}$ contains a copy of the Lie algebra of $SL(2, \RR)$: a contradiction, since $\mathcal{I}$ is of dimension two.
Our proof of theorem \[affine connection\] will need analyticity in another essential way. We will make use of an extendibility result for local Killing fields proved first for Nomizu in the Riemannian setting [@Nomizu] and generalized then for rigid geometric structures by Amores et Gromov [@Amores; @Gro] (see also [@CQ; @DG; @Feres]). This phenomena states roughly that a local Killing field of a [*rigid analytic*]{} geometric structure can be extended along any curve in $M$. We then get a multivalued Killing field defined on all of $M$ or, equivalently, a global Killing field defined on the universal cover. In particular, the Killing algebra in the neighborhood of any point is the same (as long as $M$ is connected).
As an application of the Nomizu’s phenomena we give (compare with theorem \[model\]):
Let $M$ be a compact simply connected real analytic manifold admitting a real analytic locally homogeneous rigid geometric structure. Then $M$ is isomorphic to a homogeneous space $G/I$ endowed with a $G$-invariant geometric structure.
Since $\phi$ is locally homogeneous and $M$ is simply connected and compact, the local transitive action of the Killing algebra extends to a global action of the associated simply connected Lie group $G$ (we need compactness to insure that vector fields on $M$ are complete). All orbits have to be open, so there is only one orbit: the action is transitive and $M$ is a homogeneous space.
Let’s go back now to the proof of theorem \[affine connection\]. As before, in the real analytic setting Gromov’s stratification theorem shows that the locally homogeneous open dense set has to be dense [@Gro; @DG]. Note also that Nomizu’s extension phenomena doesn’t imply that the extension of a family of linearly independent Killing fields, stays linearly independent. In general, the extension of a localy transitive Killing algebra, fails to be transitive on a nowhere dense analytic subset $S$ in $M$. The unimodular affine connection is locally homogeneous on each connected component of $M \setminus S$.\
[ *We prove now that $S$ is empty.*]{}\
Assume by contradiction that $S$ is not empty. Then we have the following crucial:
\[dim isotropy\] (i) The Killing algebra $\mathfrak{g}$ of $\nabla$ has dimension two and the isotropy algebra at a point of $S$ is one dimensional.
\(ii) $\mathfrak{g}$ is isomorphic to the Lie algebra of the affine group of the line.
\(i) Since the Killing algebra admits a nontrivial open orbit in $M$, its dimension is $\geq 2$. Pick up a point $s \in S$ and consider the linear morphism $ev(s): \mathfrak{g} \to T_{s}M$ which associates to an element $K \in \mathfrak{g}$ its value $K(s)$. The kernel of this morphism is the isotropy $\mathcal{I}$ at $s$. Since the $\mathfrak{g}$-action is nontransitive in the neighborhood of $s$, the range of $ev(s)$ is $\leq 1$. This implies that the isotropy at $s$ is of dimension at least dim $\mathfrak{g}-1$.
Assume, by contradiction, that the Killing algebra has dimension at least three. Then the isotropy at $s$ is of dimension at least two. By lemma \[unimodularlemma\], this dimension never equals two. Consequently, the isotropy algebra at $s \in S$ is three dimensional. The isotropy algebra contains then a copy of the Lie algebra of $SL(2,\RR)$ (see the proof of lemma \[unimodularlemma\]).
The local action of $SL(2,\RR)$ in the neighborhood of $s$ is conjugated to the the standard linear action of $SL(2,\RR)$ on $\RR^2$. This action has two orbits: the point $s$ and $\RR^2 \setminus \{s \} $. The open orbit $\RR^2 \setminus \{s \} $ identifies with a homogeneous space $SL(2,\RR)/I$. Precisely, the stabilizer $I$ in $G=SL(2, \RR)$ of a nonzero vector $x \in T_{s}M$ is conjugated to the following one parameter unipotent subgroup of $SL(2, \RR)$: $\left( \begin{array}{cc}
1 & b \\
0 & 1 \\
\end{array} \right)$, with $b \in \RR$. The action of $SL(2,\RR)$ on $SL(2,\RR)/I$ preserves the induced flat torsion free affine connection coming from $\RR^2$.
By proposition 8 in [@AM-K], the only $SL(2,\RR)$-invariant affine connection on $SL(2,\RR)/I$ is the previous flat torsion free connection. Another way to prove this result is to consider the difference of a $SL(2,\RR)$-invariant connection with the standard one. We get a $(2,1)$-tensor on $SL(2,\RR)/I$ which is $SL(2,\RR)$-invariant. Equivalently, we get a $ad(I)$-invariant $(2,1)$-tensor on the quotient of the Lie algebra $sl(2,\RR)$ by the infinitesimal generator of $I$ [@AVL]. A straightforward computation shows that the tensor has to be trivial. This gives a different proof of the unicity of a $SL(2,\RR)$-invariant connection.
By analyticity, $\nabla$ is torsion free and flat on all of $M$. In particular, $\nabla$ is locally homogeneous on all of $M$. This is in contradiction with our assumption.
\(ii) The Killing algebra is two dimensional. Thus it coincides with $\RR^2$ or with the Lie algebra of the affine group. Consider $K_{1}, K_{2}$ a basis of the local Killing algebra and extend $K_{1}, K_{2}$ along a topological disk reaching a point $s$ in $S$.
Recall that $\nabla$ is unimodular and let $vol$ be the volume form associated to $\nabla$.
In the case where the Lie algebra is $\RR^2$, $vol(K_{1}, K_{2})$ is a nonzero constant (for being invariant by the Killing algebra and thus constant on the locally homogenous open set). Hence the Lie algebra acts transitively in the neighborhood of $s \in S$. This is a contradiction: $S$ is then empty and $\phi$ is locally homogeneous on all of $M$.
For the sequel, let $K_{1}, K_{2}$ be two local Killing fields at $s \in S$ which span the Killing algebra. We assume, without loss of generality, that $K_{1}$ and $K_{2}$ verify the Lie bracket relation $\lbrack K_{1}, K_{2} \rbrack =K_{1}$.
Recall that $K_{1}, K_{2}$ don’t vanish both at a point $s \in S$. Indeed, if not the isotropy at $s$ has dimension $2$, which is impossible by lemma \[unimodularlemma\].
Notice that $vol(K_{1}, K_{2})$ is not invariant by the action of the Killing algebra (since the adjoint representation of $Aff(\RR)$ is nontrivial). But we still have:
\[local volume\] The local function $vol(K_{1},K_{2})$ is constant on the orbits of the flow generated by $K_{1}$.
The adjoint action $ad(K_{1})$ on $\mathfrak{g}$ is nilpotent. In particular, the adjoint representation of the one parameter subgroup generated by $K_{1}$ is unimodular. Consequenlty, the local flow generated by $K_{1}$ preserves $vol(K_{1},K_{2})$.
\[dim1\] $S$ is a smooth $1$-dimensional manifold (diffeomorphic to a finite union of circles).
By lemma \[dim isotropy\], the isotropy $\mathcal{I}$ at a chosen point $s \in S$ is one dimensional and the range of the map $ev(s)$ equals one. In particular, the orbit of $s$ under the action of $\mathfrak{g}$ is one dimensional. The image of $ev(s)$ coincides with $T_{s}S$. Consequently, the $\mathcal{I}$-action on $T_{s}M$ preserves the line $T_{s}S$.
This shows that $\mathfrak{g}$ acts transitively on each connected component of $S$.
In particular, each connected component of $S$ is a one dimensional smooth submanifold in $M$ (recall that $S$ is nowhere dense in $M$).
Since $M$ is compact, each connected component of $S$ is diffeomorphic to a circle.
We also have:
\[cadre\] (i) $M \setminus S$ is locally modelled on the affine group of the line endowed with a translation invariant connection.
\(ii) $M \setminus S$ admits a (nonsingular) $\mathfrak{g}$-invariant foliation $\mathcal{F}_{1}$ by lines.
\(iii) $\mathcal{F}_{1}$ coincides with the kernel of a (nonsingular) closed $\mathfrak{g}$-invariant one form $\omega$.
\(iv) The leafs of $\mathcal{F}_{1}$ are closed (in $M \setminus S$). They are endowed with a $\mathfrak{g}$-invariant translation structure.
\(v) The space of leafs of $\mathcal{F}_{1}$ is Hausdorff.
\(i) This comes from the fact that the $\mathfrak{g}$-action on $M \setminus S$ is simply transitive.
\(ii) The Lie bracket relation $\lbrack K_{1}, K_{2} \rbrack =K_{1}$ implies that the flow of $K_{2}$ normalizes the flow of $K_{1}$ and thus the foliation spaned by $K_{1}$ is $K_{2}$-invariant. Consequently, the foliation spaned by $K_{1}$ is $\mathfrak{g}$-invariant and it defines a foliation $\mathcal{F}_{1}$ well defined on $M \setminus S$.
\(iii) We locally define the one form $\omega$ such that $\omega(K_{1})=0$ and $\omega(K_{2})=1$. Since the basis $K_{1}, K_{2}$ is well defined, up to a Lie algebra isomorphism (which necessarily preserves the derivative algebra $\RR K_{1}$ and sends $K_{2}$ on $K_{2} + \beta K_{1}$, for $\beta \in \RR$), $\omega$ is globally defined on $M \setminus S$. Also by Lie-Cartan’s formula [@AVL] $d \omega(K_{1},K_{2})=-\omega(\lbrack K_{1}, K_{2} \rbrack)=0$.
Thus the foliation is transversaly Riemannian in the sense of [@Molino]. Another way to see it, is to observe that the projection of the local Killing field $K_{2} + \beta K_{1}$ on $TM/ T\mathcal{F}_{1}$ is well defined (it doesn’t depend on $\beta$). It defines a transverse vector field $\tilde K_{2}$ and, consequently, $\mathcal{F}_{1}$ is transversaly Riemannian.
\(iv) By proposition \[local volume\] and by the previous point, $vol(K_{1},K_{2})$ is a well define (nonconstant) function on $M \setminus S$ which is constant on the leafs of $\mathcal{F}_{1}$. Therefore the leafs are closed in $M \setminus S$.
Moreover, the action of the derivative algebra $\RR K_{1}$ preserves each leaf of $\mathcal{F}_{1}$. Hence each leaf inherits a $\mathfrak{g}$-invariant translation structure.
\(v) This is a consequence of (iii) and (iv) (see [@Molino], chapter 3, Proposition 3.7). The connected components of the space of leafs are parametrized by the flow of $\tilde K_{2}$.
Denote by $Y \in \mathfrak{g}$ a generator of $\mathcal{I}$ and by $X \in \mathfrak{g}$ a Killing field such that $X(s)$ span $T_{s}S$. By applying an automorphism of the Lie algebra of the affine group we can assume that either $Y=K_{1}$ and $X=K_{2}$, or $Y=K_{2}$ and $X=K_{1}$.\
[**Case I: unipotent isotropy**]{}\
Assume first that $Y=K_{1}$ and $X=K_{2}$.\
We study the local situation in the neighborhood of $s \in S$. For this local analysis we will also denote by $S$ the connected component of $s$ in $S$.
\[unipotent isotropy\] The isotropy at any point of $S$ is unipotent.
Because of the Lie bracket relation, the isotropy $\mathcal{I}$ at $s$ acts trivially on $T_{s}S$. This implies that its generator $Y$ acts trivially on the unique geodesic passing through $s$ and tangent to the direction $T_{s}S$. Consequently $Y$ vanishes on this geodesic which has to coincide locally to $S$ (for $S$ is the subset of $M$ where $\mathfrak{g}$ doesn’t act freely).
Since the $Y$-action on $T_{s}M$ is trivial on $T_{s}S$ and volume preserving, it is unipotent. In local exponential coordinates $(x,y)$ at $s$, $Y$ is linearized and so conjugated to the linear vector field $y \frac{\partial}{\partial x}$. In these coordinates $S$ identifies locally with $y=0$ and the time $t$ of the flow generated by $Y$ is $(x,y) \to (x+ty,y)$.
Notice that the flow of $Y$ preserves a unique foliation in the neighborhood of $s$, which is given by $dy=0$. We will see later that this rules out the case of a connected components of $S$ with isotropy $Y=K_{2}$ (see Step 2 in the proof of Case II, below).
\[extension\] $\mathcal{F}_{1}$ extends to a nonsingular foliation with closed leafs defined on all of $M$.
The foliation $\mathcal{F}_{1}$ spaned by $Y$ extends on $M$ by adding the leafs $S$. Moreover, all the leafs of the foliation $\mathcal{F}_{1}$ are closed on $M$. Indeed, the first integral $vol(K_{1},K_{2})$ on $M \setminus S$ vanishes exactly on the extra leafs $S$. Therefore, $vol(K_{1},K_{2})$ defines a first integral on all of $M$.
\[both extensions\] (i) Each connected component of $M \setminus S$ is diffeomorphic to a cylinder $\RR \times S^1$.
\(ii) The elements of $\mathfrak{g}$ extend to global Killing fields on each connected component of $M \setminus S$.
By proposition \[extension\], the leafs of $\mathcal{F}_{1}$ are circles.
\(i) The flow of the transverse vector field $\tilde K_{2}$ (see the proof of proposition \[cadre\]) acts transitively on each connected component of the space of leafs of $\mathcal{F}_{1}$. Let $\rbrack a, b \lbrack$ be the maximal domain of definition of a integral curve of $\tilde K_{2}$. Then the corresponding connected component of $M \setminus S$ is diffeomorphic to $\rbrack a, b \lbrack \times S^1$ (we will se further that the maximal domain of definition is $\RR$).
\(ii) Choose a local determination of $K_{1}$ and extend it along a leaf of $\mathcal{F}_{1}$. Our local determination changes into $\alpha K_{1}$, with $\alpha \in \RR$. Also extending $K_{2}$ along the same leaf, we will find $K_{2}+ \beta K_{1}$, with $\beta \in \RR$. We get then two multivalued vector fields defined on a small cylinder containing the chosen leaf.
But $vol(\alpha K_{1}, K_{2}+ \beta K_{1})=vol(K_{1}, K_{2} )$ is a nonzero constant function on our leaf (for it is locally constant by proposition \[local volume\]). This implies $\alpha=1$, so $ K_{1}$ is globally defined (and univalued) on the small cylinder containing the leaf. Since the fundamental group of the corresponding connected component of $M \setminus S$ is generated by our leaf (as a simple closed curve), $K_{1}$ extends in a global Killing field on all of the corresponding connected component of $M \setminus S$.
This also proves that the projection of $K_{2}$ on $T \mathcal{F}_{1}$ extends in a global vector field (which is not Killing but preserves each leaf of $\mathcal{F}_{1}$ and its translation structure: its local expression is $f K_{1}$, with $f$ a function defined on the space of leafs). Since the kernel of this projection is also well defined globally, $K_{2}$ extends on the corresponding connected component of $M \setminus S$ as well.
We have now that $X$ and $Y$ extend to globally defined vector fields on any connected component of $M \setminus S$. We have seen that the flow of $X$ parametrized the space of leafs of $\mathcal{F}_{1}$. In particular, the positive and the negative orbit of any point in $M \setminus S$ acccumulate on $S$.
A contradiction is obtained in the following;
\[negative orbits\] The negative orbit of points in $M \setminus S$ under the flow of $X$ (equivalently the positive orbit under the flow of $-X$) cannot accumulate on $S$.
Recall that the Lie bracket relation implies that the flow of $X$ contracts $Y$ exponentialy. Thus the action of the negative flow of $X$ expands $Y$. Therefore, a negative orbit under the flow of $X$ of a point $m$ in $M \setminus S$ never accumulates on $S$ (for $Y$ vanishes on $S$).
Another way to get a contradiction is to prove the following:
\[both complete\] $X$ and $Y$ are complete on $M \setminus S$.
By proposition \[negative orbits\], the negative orbits of $X$ are complete since they stay in a compact subset of $M \setminus S$. The positive orbits of $X$ are complete because they stay in a compact subset of the maximal domain of definition of $X$ (for $X$ can be extended to an open neighborhood of $S$ as in the proof of point (ii) in proposition \[both extensions\]).
As for $Y$, its orbits lie in the (compact) leafs of $\mathcal{F}_{1}$, so they are complete.
Proposition \[both complete\] implies then that there is a locally free transitive action of $Aff(\RR)$ on each connected component of $M \setminus S$. Consequently, each connected component of $M \setminus S$ is diffeomorphic to a homogeneous space $Aff(\RR) / \Gamma$, where $\Gamma$ is a discrete subgroup of $Aff(\RR)$.
Since the $Aff(\RR)$-action preserves the finite volume of $M \setminus S$ given by $vol$, $\Gamma$ has to be a lattice in $Aff(\RR)$: a contradiction.\
[**Case II: semi-simple isotropy**]{}\
We assume now that $Y=K_{2}$ and $X=K_{1}$.\
[*We will show that the Killing algebra $\mathfrak{g}$ preserves two transverse foliations by lines $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ on $M$, with $\mathcal{F}_{2}$ geodesic.*]{}\
[**Step 1: Construction of $\mathcal{F}_{1}$**]{}. We prove that the foliation $\mathcal{F}_{1}$ constructed in proposition \[cadre\] extends to all of $M$.
Assume first that $S$ is connected. In the neighborhood of $s \in S$ the local foliation spaned by the nonsingular vector field $K_{1}$ agrees with $\mathcal{F}_{1}$ on $M \setminus S$. Since this stands in the neigborhood of each point of $S$, the foliation $\mathcal{F}_{1}$ extends to a nonsingular foliation on $M$ by adding the leaf $S$.
If $S$ admits several connected components, the previous argument applies in the neighborhood of each component of $S$.
Notice that Poincaré-Hopf’s theorem shows that $M$ is a torus.\
[**Step 2: Construction of $\mathcal {F}_{2}$.**]{} Since the action of $\mathcal{I}$ on $T_{s}M$ is unimodular (for it preserves $vol(s)$) and contracts the direction $T_{s}S$ (for $\lbrack Y, X \rbrack=-X$), this action (preserves and) expands a unique line $\mathcal {F}_{2}(s) \in T_{s}M$. This constructs a smooth $\mathfrak{g}$-invariant line field along $S$: the unique line field (preserved and) expanded by the local isotropy.
In the neighborhood of $s$, consider the unique geodesic foliation $\mathcal {F}_{2}$ which extends the previous line field. It is the image of the line field through the exponential map, with respect to $\nabla$, along $S$. Since $\mathcal{F}_{2}$ is $\mathfrak{g}$-invariant, it extends to a foliation (which we still denote by $\mathcal{F}_{2}$) on $M$ transverse to $\mathcal {F}_{1}$. We obtain then:
\[a Lorentz metric\]$M$ admits a flat $\mathfrak{g}$-invariant lorentz metric.
Let $v \in T_{m}M$ and consider $q(v)=vol(v_{1} ,v_{2}),$ where $v_{1},v_{2}$ are the components of $v$ under the splitting $$T_{m}M=T\mathcal{F}_{1}(m)\oplus T\mathcal {F}_{2}(m).$$ This constructs a $\mathfrak{g}$-invariant lorentz metric on $M$. In particular, $q$ is locally homogeneous and hence of (lorentzian) constant sectional curvature [@Wolf].
Since $M$ is a torus, Gauss-Bonnet’s theorem (see its Lorentzian version in [@Wolf]) implies that $q$ is flat (locally isomorphic to $\RR^2$ endowed with $dxdy$) [@Wolf].
[**Step 3: Construction of a $\mathfrak{g}$-invariant vector field $T$ on $M$ which vanishes exactly on $S$.**]{}
Recall that the local vector field $Y$ generates the isotropy in the neighborhood of $s\in S$. Let $T$ be the projection of $Y$ on the second factor of the decomposition $$TM=T\mathcal {F}_{1} \oplus T\mathcal {F}_{2}.$$ Then $T$ is obviously $Y$-invariant. Since $X$ spans $\mathcal F_{1}$ and $X$ and $Y$ commutes modulo $X$, it follows that $T$ is also $X$-invariant. The vector field $T$ is $\mathfrak{g}$-invariant and vanishes on $S$. As before, $T$ is defined locally in the neighborhood of $s \in S$, but since $T$ is $\mathfrak{g}$-invariant, it extends to a global vector field on $M$.
Note that $\mathfrak{g}$ being transitive on $M \setminus S$, the vector field $T$ doesn’t vanish on $M \setminus S$. We prove now:
The flow of $T$ preserves the volume form of the unimodular connection (and also the volume form of the flat lorentz metric $q$).
Since both $T$ and the volume form $vol$ are $\mathfrak{g}$-invariant, the divergence $div(T)$ is also $\mathfrak{g}$-invariant. Consequently, $div(T)$ is constant on each connected component of $M \setminus S$. The time $t$ of the $T$-flow acts then on $vol$ by multiplication with $exp(\lambda t)$, where $div(T)=\lambda \in \RR$.
But this $T$-action has to preserve the global (finite) volume of each connected component of $M \setminus S$ . This implies $\lambda=0$ and, consequently, $vol$ is $T$-invariant.
Remark that the underlying volume form of the lorentz metric being also $\mathfrak{g}$-invariant, it is a constant multiple of $vol$. Consequently, it is also $T$-invariant.
We will get a contradiction by showing the following
The divergence of $T$ with respect to the volume form of the flat lorentz metric equals $1$.
Choose a point $s$ in $S$. In exponential coordinates $(x,y)$, at $s$, with respect to the flat lorentz metric $q$, the foliations $\mathcal F_{1}$ and $\mathcal F_{2}$ are generated by the isotropic directions $\frac{\partial}{\partial x}$ and $\frac{\partial}{ \partial y}$ of the standard lorentz metric $dxdy$.
The isotropy $\mathcal I$ at $s$ is generated by the linear vector field $Y=-x \frac{\partial}{\partial x} + y \frac{\partial}{ \partial y}$.
It follows, by construction, that $T$ equals $y \frac{\partial}{ \partial y}$. The divergence of $T$ with respect to $dx \wedge dy$ equals $1$.
Locally homogeneous affine connections {#section 5}
======================================
Recall that T. Nagano and K. Yagi completely classified torsion free flat affine connections (affine structures) on the real two torus [@Nagano]. They proved that, except a well described family of incomplete affine structures (see the nice description given in [@Benoist1]), all the other are homogeneous constructed from faithful affine actions of $\RR^2$ on the affine plane $GL(2, \RR) \ltimes \RR^2 / GL(2, \RR)$ admitting an open orbit. This induces a translation invariant affine structure on $\RR^2$. The quotient of $\RR^2$ by a lattice is a homogeneous affine two dimensional tori.
Notice that the previous homogeneous affine structures are complete if and only if the corresponding $\RR^2$-actions on the affine plane are transitive.
In particular, Nagano-Yagi’s result prove that a real two dimensional torus locally modelled on the affine space $GL(2, \RR) \ltimes \RR^2 / GL(2, \RR)$ such that the linear part of the holonomy morphism lies in $SL(2, \RR)$ is always complete and homogeneous (the corresponding torsion free flat affine connection on the torus is translation invariant).
Here we prove:
\[complete and homogeneous\] Let $M$ be a compact surface locally modelled on a homogeneous space $G/I$ such that $G/I$ admits a $G$-invariant affine connection $\nabla$ and a volume form. Then:
\(i) the corresponding affine connection on $M$ is homogeneous (up to a finite cover), except if $\nabla$ is the Levi-Civita connection of a hyperbolic metric (and the genus of $M$ is $\geq 2$).
\(ii) the corresponding affine connection on $M$ is complete.
In particular, a unimodular affine connection on the two torus has constant Christoffel symbols with respect to global translation invariant coordinates.
Note first that, in the case where $I$ is compact, theorem \[isotropie compacte\] proves that $M$ is complete. Moreover, if $M$ is of genus $0$ then the underlying locally homogeneous Riemannian metric has to be of positive sectional curvature (by Gauss-Bonnet’s theorem) and $G/I$ coincides with the standard sphere $S^2$ seen as a homogeneous space of $S^3$. Since $M$ is simply connected, the developing map gives a global isomorphism with $S^2$.
Also if the genus of $M$ is one, the underlying locally homogeneous metric on $M$ has to be flat. It follows that $G/I$ coincides with the only homogeneous flat Riemannian space $O(2, \RR) \ltimes \RR^2 / O(2, \RR)$. Bieberbach’s theorem (see, for instance, [@Wolf]) implies then that $M$ is homogeneous (up to a finite cover).
Surfaces of genus $g \geq 2$ cannot admit a homogeneous geometric structure locally modelled on $G/I$, with $I$ compact. Indeed, here the Riemannian metric induced on $M$ is of constant negative curvature. It is well know that the isometry group of a hyperbolic metric on $M$ is finite (hence it cannot act transitively) [@Kobayashi].\
[*Consider now the case where $I$ is noncompact*]{}.\
Denote also by $\nabla$ the locally homogeneous affine connection induced on $M$. The Killing algebra $\mathfrak{g}$ acts locally on $M$ preserving $\nabla$ and a volume form $vol$. The proof of theorem \[affine connection\] implies that $\mathfrak{g}$ is of dimension at most $3$. Therefore $G$ is of dimension $3$ and $I$ is a one parameter subgroup in $G$. Since $I$ is supposed noncompact, its (faithful) isotropy action on $T_{o}G/I$, where $o$ is the origin point of $G/I$, identifies $I$ either with a semi-simple, or with a unipotent one parameter subgroup in $SL(2,\RR)$.\
[**Case I: semi-simple isotropy**]{}\
The isotropy action on $T_{o}G/I$ preserves two line fields. Consequently, there exists on $G/I$ two $G$-invariant line fields. Since $G/I$ admits also a $G$-invariant volume form, there exists on $G/I$ a $G$-invariant Lorentz metric $q$ (see the construction in proposition \[a Lorentz metric\]).
By Poincaré-Hopf’s theorem $M$ is a torus and $q$ has to be flat (by Gauss-Bonnet’s theorem). It follows that $G$ is the automorphism group $SOL$ of the flat Lorentz metric $dxdy$ on $\RR^2$. By the Lorentzian version of Bieberbach’s theorem, this structure is known to be complete and homogeneous (up to a finite cover of $M$) [@CD].\
[**Case II: unipotent isotropy**]{}\
Here the isotropy action preserves a vector field in $T_{o}G/I$. This yields a $G$-invariant vector field $\tilde X$ on $G/I$ and a $G$-invariant one form $\tilde{\omega}=vol(\tilde X, \cdot)$. We then have:
$M$ admits a $\mathfrak{g}$-invariant vector field $X$ and a $\mathfrak{g}$-invariant closed one form $\omega$ vanishing on $X$. Thus the foliation $\mathcal{F}$ generated by $X$ is transversally Riemannian.
Since $d \tilde{ \omega}$ and $vol$ are $G$-invariant, there exists $\lambda \in \RR$ such that $d \tilde{ \omega}= \lambda vol$.
Denote by $\omega$ the one form on $M$ associated to $\omega$ and by $X$ the vector field induced on $M$ by $\tilde X$. It follows that $\omega$ vanishes on $X$. Also $\int_{M} d \omega= \lambda vol(M)$, where the volume of $M$ is calculated with respect to the volume form induced by $vol$. Stokes’ theorem yield $\lambda =0$ and, consequently, $\omega$ is closed.
The normal subgroup $H$ of $G$ which preserves each leaf of the foliation $\tilde{\mathcal{F}}$ generated by $\tilde X$ on $G/I$ is two dimensional and abelian.
The group $G$ preserves the foliation $\tilde{\mathcal{F}}$ and its transverse Riemannian structure. Therefore an element of $G$ fixing one leaf of $\tilde{\mathcal{F}}$, will fixe all the leafs. The subgroup $H$ is nontrivial, since it contains the isotropy. The action of $G/H$ on the transversal of the foliation preserves a Riemannian structure, so it is of dimension at most one. Since this action has to be transitive, the dimension of $G/H$ is exactly one and the dimension of $H$ is two.
Since $H$ preserves $\tilde X$, the elements of $H$ commutes in restriction to each leaf of $\tilde{\mathcal{F}}$. Hence $H$ is abelian.
We will make use of the following result proved in [@Dumitrescu] (pages 17-19):
$\tilde X$ is a central element in $\mathfrak{g}$. Consequently, $X$ is a global Killing field on $M$ preserved by $\mathfrak{g}.$
In order to prove that $M$ is homogeneous, we will construct a second global Killing field on $M$ which is not tangent to the foliation $\mathcal F$ generated by $X$. For this we will study the holonomy group.
Once again $M$ is of genus one, since it admits a nonsingular vector field. Since the fundamental group of the torus is abelian, its image $\Gamma$ by the holonomy morphism is an abelian subgroup of $G$.
We prove first that $\Gamma$ is nontrivial. Assume by contradiction that $\Gamma$ is trivial. Then the developing map is well defined on $M$ and we get a local diffeomorphism $dev : M \to G/I$. The image has to be open and closed (for $M$ is compact). Hence, $dev$ is surjective. By Ehresmann’s submersion theorem, $dev$ is a covering map. Remark \[model simply connected\] shows that $G/I$ can be considered simply connected, which implies $dev$ is a diffeomorphism: a contradiction, since $M$ is not simply connected.
Considering finite covers of $M$, the previous proof rules out the case where $\Gamma$ is finite. Consequently, $\Gamma$ is an infinite subgroup of $G$.
Consider its (real) Zariski closure $\overline{\Gamma}$ in $G$, which is an abelian subgroup of positive dimension. This is possible, since by Lie’s classification [@Mos; @Olver], $G$ is locally isomorphic to a real algebraic group acting (transitively) on $G/I$. Therefore we can assume that $G$ is algebraic.
Up to a finite cover of $M$, we can assume $\overline{\Gamma}$ connected (algebraic groups admit at most finitely many connected components).
\[holonomie et champs de Killing\] Any element $a$ of the Lie algebra of $\overline{\Gamma}$ defines a global Killing field on $M$.
Moreover, if the one parameter subgroup of $G$ generated by $a$ intersects $\Gamma$ nontrivially, then the orbits of the corresponding Killig field on $M$ are closed.
The action of $\Gamma$ on $a$ (by adjoint representation) being trivial, $a$ defines a $\Gamma$-invariant vector field on $G/I$ which descends on $M$.
Assume now that $\Gamma$ intersects nontrivially the one parameter subgroup generated by $a$. One orbit of the corresponding Killing field on $M$ develops in the model $G/I$ as one orbit of the Killing field $a$. Since vector fields on compact manifolds are complete, the image of the developing map contains all of the orbit of $a$. In particular, the corresponding orbit of $a$ contains distinct points which are in the same $\Gamma$-orbit. Therefore, the corresponding orbit on $M$ is closed.
We prove now:
$\overline{\Gamma}$ is not a subgroup of $H$.
Consider a one parameter subgroup in $G$, generated by an element in the Lie algebra of $\overline{\Gamma}$, which nontrivially intersects $\Gamma$. By proposition \[holonomie et champs de Killing\], $a$ defines a global Killing field $K$ with closed orbits on $M$.
Assume by contradiction that $\overline{\Gamma}$ is a subgroup of $H$. It follows that the orbits of $K$ coincide with those of $X$. Consequently, the orbits of $X$ are closed and the space of leafs of $\mathcal F$ is a one dimensional manifold (see proposition \[cadre\], point (v)). Since $M$ is compact, the space of leafs is diffeomorphic to a circle $S^1$.
Consider the developing map $dev: \tilde{M} \to G/I$ of the $G/I$-structure. In particular, this is also the developing map of the transverse structure of the foliation $\mathcal{F}$. Since the holonomy group acts trivially on the transversal $\tilde T$ of $\tilde{\mathcal F}$, $dev$ descends to a local diffeomorphism from the space of leafs of $\mathcal{F}$ (parametrized by $S^1$) to $\tilde T$.
Since $G/I$ is simply connected, the closed one form $\tilde{\omega}$ admits a primitive $\tilde f: G/I \to \RR$. Consequently, $\tilde f$ is a first integral for $\tilde{\mathcal F}$ and $\tilde f \circ dev$ descends to a local diffeomorphism from $S^1$ to $\RR$. This map has to be onto since the image is open and closed. We get a topological contradiction.
By proposition \[holonomie et champs de Killing\], any one parameter subgroup in $G$ generated by an element of the Lie algebra of $\overline{\Gamma}$ non contained in the Lie algebra of $H$ provides a global Killing field $K$ on $M$ such that the abelian group generated by the flows of $X$ and of $K$ acts transitively on $M$. Therefore the $G/I$-structure on $M$ is homogeneous.
\(ii) Consider $\tilde X, \tilde K$ the corresponding Killing vector fields on $G/I$. They generate a two dimensional abelian subgroup $A$ of $G$ acting with an open orbit on $G/I$. Recall that $vol(\tilde X, \tilde K)$ is an $A$-invariant function on $G/I$. Hence, $vol(\tilde X, \tilde K)$ is a nonzero constant in restriction to the open orbit. By continuity, $vol(\tilde X, \tilde K)$ equals the same nonzero constant on the closure of the open orbit. This implies that $\tilde X, \tilde K$ remain linearly independent on the closure of the open orbit, which implies that the open orbit is also closed. Since $G/I$ is connected, the open orbit is all of $G/I$.
The previous proof combined with a result of [@Dumitrescu] leads to the following classification result:
\[classification\] Let $M$ be a compact surface locally modelled on a homogeneous space $G/I$ such that $G/I$ admits a $G$-invariant affine connection $\nabla$ and a volume form.
If $I$ is noncompact, then $G$ is three dimensional and:
\(i) either $G$ is the unimodular $SOL$ group and its action preserves a flat (two dimensional) Lorentz metric;
\(ii) or $G$ is the Heisenberg group and its action preserves the standard flat torsion free affine connection on $\RR^2$ together with a nonsingular closed one form $\tilde{\omega}$ and with a nontrivial parallel vector field $\tilde X$ such that $\tilde{\omega}(\tilde X)=0$;
\(iii) or $G$ is isomorphic to the product $\RR \times Aff(\RR)$ and its action preserves the canonical bi-invariant torsion free and complete connection of $Aff(\RR)$ (for which the full automorphism group is $Aff(\RR) \times Aff(\RR)$) together with a geodesic Killing field.
The model (i) was obtained in the previous proof in the case where the isotropy is semi-simple. The models (ii) and (iii) correspond to the case where the isotropy is unipotent. This classification follows from [@Dumitrescu] (see Proposition 3.5 and pages 15-19).
Theorem \[complete and homogeneous\] can be deduced from theorem \[classification\]. Indeed, in the case (ii) $G$ preserves a flat torsion free affine connection and Nagano-Yagi’s result applies. The completeness of compact surfaces locally modelled on the homogeneous space (iii) was proved in Proposition 9.3 of [@Zeghib] (the homogeneity follows from the proof of Proposition 10.1 in [@Zeghib]).
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[^1]: This work was partially supported by the ANR Grant Symplexe BLAN 06-3-137237
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Focused electron beam induced deposition (FEBID) is a direct-write method for the fabrication of nanostructures whose lateral resolution rivals that of advanced electron lithography but is in addition capable of creating complex three-dimensional nano-architectures. Over the last decade several new developments in FEBID and focused electron beam induced processing (FEBIP) have led to a growing number of scientific contributions in solid state physics and materials science based on FEBID-specific materials and particular shapes and arrangements of the employed nanostructures. In this review an attempt is made to give a broad overview of these developments and the resulting contributions in various research fields encompassing mesoscopic physics with nanostructured metals at low temperatures, direct-write of superconductors and nano-granular alloys or intermetallic compounds and their applications, the contributions of FEBID to the field of metamaterials, and the application of FEBID structures for sensing of force or strain, dielectric changes or magnetic stray fields. The very recent development of FEBID towards simulation-assisted growth of complex three-dimensional nano-architectures is also covered. In the review particular emphasis is laid on conceptual clarity in the description of the different developments, which is reflected in the mostly schematic nature of the presented figures, as well as in the recurring final sub-sections for each of the main topics discussing the respective ”challenges and perspectives”.'
address: 'Institute of Physics, Goethe University, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany'
author:
- 'M. Huth'
- 'F. Porrati'
- 'O. V. Dobrovolskiy'
bibliography:
- 'mee\_2016\_rev1.bib'
title: Focused electron beam induced deposition meets materials science
---
focused electron beam induced deposition ,focused electron beam induced processing ,materials science ,superconductivity ,nanomagnetism ,sensors ,metamaterials ,three-dimensional nanostructures
Introduction
============
Focused electron beam induced deposition (FEBID) is a direct-write approach for the fabrication of 2D- and 3D-nanostructures [@Randolph2006_febid_review; @Utke2008_febid_review; @Huth2012_febid_review]. Over the last decade FEBID or, more generally, focused electron beam induced processing (FEBIP) has developed from a rather exotic technique employed by a small number of specialist groups for a rather limited but important selection of applications, such as mask repair [@Edinger2014_mask_repair], into a highly versatile technology for various materials research areas. These comprise amorphous and polycrystalline superconductors [@Makise2014_ebid_superconductivity; @Sengupta2015_W_ebid_superconductivity; @Winhold2014_Pb_superconductor], magnetic materials [@DeTeresa2016_ebid_magnetic_review], alloys and intermetallic compounds [@Che2005_FEBID_FePt_holography; @Winhold2011_PtSi_alloy; @Porrati2012_CoPt_alloy; @Porrati2013_CoSi_alloy; @Shawrav2014_AuFe_alloy; @Porrati2015_CoFe_precursor; @Porrati2016_FeSi_alloy], multilayer structures [@Porrati2016_FeSi_alloy; @Porrati2017_FeCoSi_multilayer] and metamaterials in which suitable materials combinations result in a desired functionality [@Dobrovolskiy2015_CoPt_treatment_H2_O2; @DeTeresa2016_ebid_magnetic_review]. The latter approach, in particular, has opened a new pathway to the realization of different sensor applications [@Schwalb2010_strain_sensing; @Huth2014_diel_sensing_theory; @Huth2014_sensor_ttfca; @Dukic2016_afm_sensor; @Moczala2017_ebid_sensor].
In this review an attempt is made to give an overview of important milestones along this development and, in parallel, keep a critical eye on the future perspective of this field. Important issues to be addressed will be material purity and reproducibility in the targeted material properties. Following this goal setting, this review will give a brief introduction into the basics of FEBID (see [@Utke2008_febid_review; @Utke2012_book] for a comprehensive overview) and will then, in particular, address recent developments in the fabrication of all-metal FEBID structures, discuss FEBID approaches to direct-write superconductors and provide several examples of how alloys and intermetallic compounds can be obtained by FEBIP. In two additional sections, the potential of FEBID materials for different sensor application areas will be addressed and examples will be given for the use of FEBID materials towards the realization of metamaterials. A very recent development in FEBID is complex, high-resolution 3D-structure fabrication [@Fowlkes2016_febid_3D_simulation], e.g. for plasmonics [@Winkler2017_3D_plasmonic]. This development holds great potential which is why it will also be covered in this review. Many aspects relating to the fabrication and characterization of magnetic FEBID materials have been discussed in depth in a very recent, excellent review by De Teresa and collaborators [@DeTeresa2016_ebid_magnetic_review], therefore here magnetic FEBID structures are only covered with regard to 3D variants, their combination with superconductors and their application as sensor elements.
Basics of focused electron beam induced deposition
==================================================
The basic principle of focused electron beam induced deposition is simple. Provided by a gas injection system or an environmental chamber inside of an electron microscope, a precursor gas adsorbed on a surface is dissociated in the focus of an electron beam (see Fig.\[fig\_intro\_febip\_scheme\] for illustration). This brief description shows an apparent conceptual similarity to 3D printing, in particular if one considers the 3D writing capabilities of FEBID. Even the use of ”liquid ink” precursors has been pioneered by the Hastings group and holds great potential for metallic nanostructure fabrication with very decent writing speeds [@Donev2009_Pt_liquid_precursor; @Bresin2013_bimetallic_liquid_precursor]. A closer look, however, reveals the intrinsic complexity of the FEBID process. The electron-induced dissociation process is mostly triggered by low-energy electrons, i.e. the secondary electrons generated by the primary electrons (SEI) and also by the backscattered electrons (SEII). For the dissociation process several channels are available with strongly energy-dependent and precursor-specific cross-sections [@Thorman2015_dissociation_case_studies]. Which precursor to choose for a given application has to be carefully considered, as the growth process depends on several precursor-specific aspects, such as its vapor pressure at around room temperature, the adsorption characteristics of the precursor molecules and their stability under adsorption. By now several examples are documented of the intrinsic instability of several precursors under adsorption in dependence of the surface state of the underlying substrate [@Muthukumar2011_wco6_adsorption_dft; @Muthukumar2012_dissociation_co2co8]. This is not to say that this instability cannot be turned into a profit. By proper pre-conditioning with the electron beam, a site-selective auto-dissociation of some precursor species, such as Fe(CO)$_5$ and Co$_2$(CO)$_8$ can be induced resulting in metallic magnetic nanostructures. Pioneered by the Marbach group this process has been termed EBISA (electron beam induced surface activation) [@Walz2010_ebisa]. Additional contributions to the growth characteristics relate to the surface mobility of the precursor molecules and their average residence time. As the last important criterion in precursor selection we mention the final composition of the deposits. The composition may depend strongly on the writing parameters (beam energy and current, precursor flux, writing strategy), as has often been observed for metal-carbonyl precursors, or else may be quite insensitive to variations of the writing process, such as in metal-organic complexes with cyclopentadienyl ligands. Most often the aim is to obtain deposits with high metal content which will be covered in more detail in the next section.
![Schematic of a SEM adapted for FEBIP with additional process equipment: (A) Steady-state or pulsed IR-laser heat supply for enhanced desorption of undesired organic dissociation products. (B) In-situ electrical characterization of the deposits by using substrate materials with pre-patterned electrodes. (C) Multi-component deposits by employing several precursor gas injection channels. Illustration of the FEBID process with two different precursor species. Precursor molecules (here: organometallic complex; blue or red: metal, gray: organic ligands) are supplied by the gas-injection system and physisorb (1) on the surface. Surface diffusion (2), thermally induced desorption (3’) and electron-stimulated desorption (3) take place. Within the focus of the electron beam, adsorbed precursor molecules are dissociated followed by desorption of volatile organic ligands (4). Upper right: For pattern definition the electron beam is moved in a raster fashion (here: serpentine) over the surface and settles on each dwell point for a specified dwell time $t_D$. After one raster sequence is completed the process is repeated until a predefined number of repeated loops is reached. Neighboring dwell points have distances of $p_x$ and $p_y$ in $x$- and $y$-direction, respectively.[]{data-label="fig_intro_febip_scheme"}](fig_intro_febip_scheme.pdf){width="70.00000%"}
The practical adaption of the typically employed scanning electron microscopes (SEM) for more advanced electron beam induced processing has significantly progressed in the last couple of years, as also indicated schematically in Fig.\[fig\_intro\_febip\_scheme\]. The optional parallel use of more than one precursor gas injection channel has become routine in several laboratories. This will be discussed in more detail in a separate section on alloys and intermetallic compounds. Also, steady-state heating of the substrate [@Botman2009_purification_first_review] or, even more effective, pulse-heating of the substrate surface with an IR-laser synchronized to the electron beam dwell events [@Roberts2012_purification_pulsed_laser] have become valuable additions to the process arsenal in FEBIP with a goal to improve the metal content. This will be taken up again in the section on all-metal FEBID structures. Recently, the group of Fedorov at Georgia Tech has demonstrated that the use of a supersonic precursor gas jet can yield very high metal content deposits [@Henry2016_gasjet_pure_tungsten]. With a view to metallic deposits, the option for in-situ electrical characterization during or after growth has been shown to be an important added value. This is especially true with regard to semi-automatic deposition optimization approaches pioneered by us. They use the time-dependent conductance increase during growth as a parameter for feedback control of the writing process, viz. by suitably changing the dwell time and pitches employing a genetic algorithm [@Weirich2013_ga_first_pub; @Winhold2014_ga_modeling].
Metallic FEBID structures
=========================
For a wide range of application fields, such as nano-contact fabrication or mesoscopic structure preparation, the ability to obtain fully metallic deposits is in fact a condition to which FEBIP has to live up to. This is not an easy task to fulfill. In the vast majority of cases, precursor materials used in FEBID are adopted from chemical vapor deposition (CVD) and are not specifically designed to be efficiently and completely dissociated under electron irradiation. As a consequence, the elemental composition of FEBID materials very often shows substantial contributions from carbon, oxygen and other undesired dissociation products that interfere with good metallic conductivity; at least at low temperatures. Several developments over the last decade or so have brought significant progress into advancing FEBID to a state that it is now possible to get fully metallic structures of the elements Au, Pt, Fe, Co, Pb, and Co$_3$Fe alloy. These developments are (1) the optimization of the deposition processes for carbonyl precursors of Fe, Co and a hetero-nuclear precursor of Fe and Co, (2) the development of several post-growth purification protocols for Au, Pt and Co, (3) the combination of FEBID with selective-area atomic layer deposition of Pt and (4) the establishment of dual gas channel deposition techniques in which the precursor is combined with a reactive gas to directly yield metallic Au and Pt. These developments are reviewed in compact form in the following sub-sections.
As-grown metal structures
-------------------------
Without doubt, the most convenient FEBID process for metal structure definition would be a single-step process. In addition, such a process should be quite forgiving of slight variations in the process parameters, such as, e.g. beam current and energy or precursor gas flow. This is, however, not the case and it is also not to be expected. The dissociation process is precursor-specific and strongly dependent on the beam energy. The residual gas contribution of the process chamber and the associated partial pressures have a strong influence on the final deposit composition. The ratio of electron and precursor flux, in conjunction with precursor residence time and diffusion coefficient determine the precursor density on the growing surface, which, in turn, determines the growth regime. Depending on the growth regime and the character of the dissociation products, the latter may be able to desorb before they are incorporated in the growing deposit or not. Because of this complex interplay of factors that influence the final composition of the deposit, it is mandatory to monitor and control as many parameters as possible in a FEBID process and to pay special attention to the reported process conditions, if an attempt is made to reproduce experiments reported in the literature.
In Tab.\[tab\_precursors\_metal\_contents\] an overview is given of the small subset of precursors so far used in FEBID for which metallic deposits can be obtained in a single-step process. By metallic we mean materials whose room temperature resistivity is clearly below the critical value set by the Mott-Ioffe-Regel criterion [@Hussey2004_mott_ioffe_regel]. In this case one can expect good conductivity also at low temperatures with, ideally, increasing values as the temperature is lowered. This latter point is particularly important with a view to the use of FEBID structure in mesoscopic physics for which sufficiently long dephasing times are mandatory.
Precursor Max. metal content Impurities Microstructure Ref
------------------------------ --------------------------- ------------ ----------------- -------------------------------------------
Me$_3$CpMePt(IV) + O$_2$ Pt, $\approx 100\,$at$\%$ C polycrystalline [@Villamor2015_Pt_purification_direct_O2]
Me$_2$(tfac)Au(III) + H$_2$O Au, $>90\,$at$\%$ C, O polycrystalline [@Shawrav2016_Au_with_H2O]
W(CO)$_6$, gas jet W, $95\,$at$\%$ C, O unknown [@Henry2016_gasjet_pure_tungsten]
W(CO)$_6$ W, $50\,$at$\%$ C, O amorphous [@Sengupta2015_W_ebid_superconductivity]
Et$_4$Pb Pb, $46\,$at$\%$ C, O granular [@Winhold2014_Pb_superconductor]
Fe(CO)$_5$ Fe, $76\,$at$\%$ C, O amorphous [@Lavrijsen2011_Fe_febid_pure]
Fe$_2$(CO)$_9$ Fe, $75\dots 80\,$at$\%$ C, O amorphous [@Cordoba2016_Fe_nonacarbonyl]
Co$_2$(CO)$_8$ Co, $93\,$at$\%$ C, O polycrystalline [@Ramon2011_Co_febid_clean_highres]
HCo$_3$Fe(CO)$_{12}$ Co$_3$Fe, $84\,$at$\%$ C, O granular [@Porrati2015_CoFe_precursor]
AgO$_2$Me$_2$Bu Ag, $75\,$at$\%$ C, O polycrystalline [@Hoeflich2017_Ag_metal]
: Precursor materials for FEBID that result in metallic deposits, if process conditions for maximum metal content are used. See listed references for details.[]{data-label="tab_precursors_metal_contents"}
The single-step deposition of metallic Pt by FEBID using Me$_3$CpMePt(IV) was pioneered by the gas chemistry group around Hans Mulders and Piet Trompenaars of FEI company [@Villamor2015_Pt_purification_direct_O2], which was also the pacemaker for the post-growth purification activities of Pt and Au [@Mulders2014_review_purification; @Mehendale2015_Au_purification_O2], as is briefly reviewed in the next sub-section. By the parallel injection of molecular O$_2$ during deposition with the precursor Me$_3$CpMePt(IV), Villamor and collaborators could obtain void-free Pt deposits with resistivity values only about a factor of six larger than the bulk value of Pt [@Villamor2015_Pt_purification_direct_O2]. It was found that the O$_2$-to-precursor flux ratio had to be as large as $3.5\times 10^4$ to obtain the highest Pt metal content. This is in accordance with the results of other work which found that below a threshold value of the flux ratio pure Pt deposition does not occur [@Mehendale2015_Au_purification_O2]. These extreme flux ratios can only be realized by reducing the precursor flux to very small values while in parallel increasing the O$_2$ flow. As a result, the deposition yield is reduced to about $1.2\times 10^{-5}\,\mu$m$^3/$nC which renders this process most suitable for small deposit volumes. As is generally the case when O$_2$ is used together with Pt-deposits, the necessary dissociative chemisorption of O$_2$ and the delayed desorption of CO formed by the catalytically supported oxidation of the carbon has to be finely balanced to get the most efficient removal of unwanted carbon in the deposit (see also next sub-section). Inspired by this single-step approach, the group of Heinz Wanzenböck developed a similar process for Au by employing the precursor Me$_2$(tfac)Au(III) together with water as oxidative enhancer [@Shawrav2016_Au_with_H2O]. In this case the water-to-precursor flux ratio was estimated to be $10$. Interestingly, the authors reported resistivity values as low as $8.8\,\mu\Omega$cm, i.e. only a factor of $4$ larger than the bulk value of Au.
A different approach was followed by Andrey Fedorov’s group. By employing an inert carrier gas jet together with the precursor W(CO)$_6$ Henry and collaborators obtained in a single-step process FEBID structures with up to $95\,$at$\%$ of W [@Henry2016_gasjet_pure_tungsten]. So far resistivity values for these deposits have not been reported, but they are expected to be clearly in the metallic regime. Interestingly, Sengupta and co-workers found that an optimized standard FEBID process using W(CO)$_6$ can yield deposits with resistivity values just on the metallic side of the Mott-Ioffe-Regel criterion [@Sengupta2015_W_ebid_superconductivity]. These deposits become superconducting below $2\,$K (see section on superconductivity). In this case the metallic behavior is percolative in nature. Similar results were obtained by us using the precursor Et$_4$Pb in a single-step standard FEBID process. At metal contents up to $46\,$at$\%$ we found percolating metallic behavior with room temperature resistivity values of about $160\,\mu\Omega$cm that dropped by more than an order of magnitude under cooling, followed by a superconducting transition between $5$ and $7\,$K [@Winhold2014_Pb_superconductor].
Another effective method was developed by the group of Philip Rack [@Roberts2012_purification_pulsed_laser]. Synchronizing laser pulses with the FEBID dwell events and optimizing the pulse duration and laser power, Roberts and collaborators obtained deposits from the precursor Me$_3$CpMePt(IV) with strongly reduced resistivity values of about $1000\,\mu\Omega$cm. The same method was also applied with the precursor W(CO)$_6$ [@Roberts2013_laebid_tungsten]. In both cases the improved metal content in the deposits was attributed to the enhanced by-product desorption caused by the short-time heating of the substrate surface induced by the laser pulses.
A particularly promising precursor group for metallic deposits in a single-step FEBID process are the carbonyls of Fe and Co. Apparently, an intrinsic instability of these precursor materials after loss of one or several of the carbonyl groups under electron irradiation provides favorable conditions for a rather complete dissociation and desorption of the carbonyl ligands [@Thorman2015_dissociation_case_studies]. In particular, the group of José María De Teresa at Zaragoza developed optimized deposition protocols for the precursors Co$_2$(CO)$_8$, Fe(CO)$_5$ and Fe$_2$(CO)$_9$ resulting in metal contents well above $90\,$at$\%$ [@Ramon2011_Co_febid_clean_highres; @Lavrijsen2011_Fe_febid_pure; @Cordoba2016_Fe_nonacarbonyl]. Importantly, these protocols are suitable for high-resolution work and established a standard for FEBID nanostructure fabrication of Fe and Co for micromagnetic studies. In our work on the hetero-nuclear precursor HCo$_3$Fe(CO)$_{12}$ we found similar results, as metal contents above $80\,$at$\%$ can be obtained under beam conditions suitable for high-resolution work with resistivity values as low as $43\,\mu\Omega$cm [@Porrati2015_CoFe_precursor].
As a last, very recent example with significant potential for plasmonics, we refer to work by Katja Höflich and collaborators [@Hoeflich2017_Ag_metal]. By using the precursor AgO$_2$Me$_2$Bu, which must be heated to $150^\circ$C and for which the substrate has to be kept at elevated temperature ($120^\circ$C) during the FEBID process, Ag metal contents as high as $75\,$at$\%$ could be obtained under optimized beam conditions.
Post-growth purification and surface-activated growth {#sec_post_growth_purification}
-----------------------------------------------------
Deposits with improved metal content up to the level of complete removal of undesired dissociation products can also be obtained by post-growth purification. With a focus on Pt and Au, but also Fe and Co, different approaches have been developed which are schematically indicated in Fig.\[fig\_pgp\_approaches\]. The figure also depicts an electron-induced surface activation (EBISA) approach that has been developed by Hubertus Marbach and collaborators [@Walz2010_ebisa; @Porrati2011_Fe_nanowires_ebisa; @Vollnhals2013_ebisa_Fe] and will be discussed first.
![Schematic of different post-growth purification approaches. (a) SiO$_2$ (native) is electron irradiated resulting in SiO$_2$-activation. Fe(CO)$_5$ dissociates spontaneously on top of the activated region, followed by auto-catalytic Fe growth. (b) Standard FEBID is used to obtain M = Au, Pt or Co nanostructures within a carbonaceous matrix. Electron irradiation in the presence of O$_2$ (M = Au or Pt) or H$_2$ (M = Co) leads to removal of matrix material. For M = Co in the presence of H$_2$ heating to about $250$–$300^\circ$C is necessary. (c) After a standard FEBID process to obtain M = Au or Pt, electron irradiation in an H$_2$O atmosphere inside an environmental SEM is used to purify the deposit. (d) Following standard FEBID of Pt within a carbonaceous matrix, pulsed oxygen exposure at a substrate temperature of about $150^\circ$C is used to purify the deposit without the need of electron irradiation.[]{data-label="fig_pgp_approaches"}](fig_pgp_approaches.pdf){width="70.00000%"}
EBISA is a two-step process to grow clean metallic nanostructures, in particular Fe when using the precursor Fe(CO)$_5$ [@Walz2010_ebisa]. In the first step, the surface of a Si$(001)$ substrate with $300\,$nm thermally grown SiO$_x$ is locally activated by a highly focused electron beam whose raster pattern defines a latent image for the second step. According to Walz *et al.* the surface site activation is a consequence of the formation of oxygen vacancies caused by electron-induced desorption of oxygen by a Knotek-Feibelman mechanism [@Walz2010_ebisa]. Following the activation, the precursor Fe(CO)$_5$ is introduced close to the substrate surface where it spontaneously decomposes at the activated sites forming Fe nano-clusters on which the growth proceeds further by autocatalytic processes. An important prerequisite for EBISA to work is a very low background pressure in the SEM chamber requiring an ultra-high vacuum setup. If this condition is met, EBISA was shown to be also applicable for the growth of clean Fe nanostructures on TiO$_2$ (rutile phase) and porphyrin-layers on Ag$(111)$ [@Vollnhals2013_ebisa_Fe].
We now turn to a brief description of different post-growth purification approaches for Pt-, Au- and Co-containing FEBID structures which have been developed over the last few years. As briefly reviewed by Hans Mulders in [@Mulders2014_review_purification], first electron-beam stimulated purification experiments on nano-granular Pt(C) deposits from the precursor Me$_3$CpMePt(IV) have been inspired by two observations. First, it was shown by Botman and collaborators that annealing of Pt(C) deposits at temperatures above about $200^\circ$C is effective in removing the carbon matrix and resulting in structures with metal contents of up to $70\,$at$\%$ [@Botman2009_purification_first_review]. Second, in later work it was demonstrated that electron irradiation of Pt(C) is very effective in increasing the conductivity by up to four orders of magnitude as compared to as-grown deposits [@Porrati2011_PtC_irradiation; @Plank2011_PtC_irradiation; @Sachser2011_PtC_universal_conductance]. This was attributed to modest grain-size increases and the partial removal of carbon leading to strongly enhanced tunnel couplings between the Pt nano-crystallites. By using O$_2$, as reactive gas species, in conjunction with electron irradiation of as-grown Pt(C) FEBID structures, a complete purification to metallic Pt was demonstrated by several groups [@Mehendale2013_PtC_purification_O2; @Plank2014_PtC_purification_O2; @Lewis2015_purification_pt_oxygen_modeling], as schematically depicted in Fig.\[fig\_pgp\_approaches\](b). The same oxygen-based approach was shown to be effective for Au-containing FEBID deposits [@Mehendale2015_Au_purification_O2]. Geier, Winkler and collaborators could show that using H$_2$O as reactive gas in an environmental SEM is very efficient in purifying Pt(C) and Au(C) FEBID structures [@Geier2014_Pt_purification_H2O; @Winkler2017_3D_plasmonic] (see Fig.\[fig\_pgp\_approaches\](c)). A particular advantage of this approach is the shape-fidelity that can be reached by careful selection of suitable process conditions leading to pore-free and compact Pt and Au nanostructures. By employing H$_2$ as reactive gas we could show that oxidation-sensitive deposits with carbon and oxygen impurities, such as Co-C-O, can also be purified under post-growth irradiation [@Begun2015_Co_purification_H2]. For this process to work the substrate temperature has to be elevated to about $250^\circ$C. We could also demonstrate that the post-growth purification of Pt and Co can be combined to fabricate magnetic heterostructures with controlled magnetic anisotropy [@Dobrovolskiy2015_CoPt_treatment_H2_O2]. If a substrate heater is an available option for a SEM stage, Pt(C) purification can also be accomplished without electron irradiation. Roland Sachser and colleagues could show that the catalytic efficacy of Pt is sufficient to remove the carbon matrix in a pulsed-oxygen process starting to operate sufficiently fast at about $150^\circ$C [@Sachser2014_PtC_purification_pulsed_O2]. Dissociative chemisorption of O$_2$ on the Pt nano-grains facilitates the oxidation of carbon to CO. If the O$_2$ supply is periodically stopped, the formed CO can desorb from the surface. After $8$ to 12 pulse cycles with a duration of several minutes each, a complete removal of the carbon matrix is accomplished. As the pulsed-oxygen approach does not need electron-assisted activation, it can work over larger substrate surface areas. Tab.\[tab\_postgrowth\_purification\] gives an overview of post-growth purification results obtained so far for a range of selected precursors.
Precursor \[Metal\] (at$\%$) Purification method $\rho(300\,\mathrm{K})$ ($\mu\Omega$cm) Ref
--------------------- -------------------- ------------------------------ ----------------------------------------- ------------------------------------------------------------------------------------------------------------------
Me$_3$CpMePt(IV) $96$, –, – e-irradiation in O$_2$ $70\pm 8$, $<350$, – [@Mehendale2013_PtC_purification_O2; @Plank2014_PtC_purification_O2; @Lewis2015_purification_pt_oxygen_modeling]
Me$_3$CpMePt(IV) – e-irradiation in H$_2$O – [@Geier2014_Pt_purification_H2O]
Me$_3$CpMePt(IV) – pulsed O$_2$ at $150^\circ$C $79.5$ [@Sachser2014_PtC_purification_pulsed_O2]
Me$_2$(tfac)Au(III) – e-irradiation in O$_2$ $17\pm 2$ [@Mehendale2015_Au_purification_O2]
Me$_2$(acac)Au(III) $> 99$ e-irradiation in H$_2$O – [@Winkler2017_3D_plasmonic]
Co$_2$(CO)$_8$ $85$, $92$ e-irradiation in H$_2$ $22.4$, – [@Begun2015_Co_purification_H2; @Dobrovolskiy2015_CoPt_treatment_H2_O2]
: Selection of metallic FEBID materials obtained after applying different post-growth purification protocols. \[Metal\] denotes the metal content in the deposit after purification. $\rho(300\,\mathrm{K})$ refers to the room-temperature resistivity. A dash indicates that the respective values were not stated in the respective references.[]{data-label="tab_postgrowth_purification"}
Area-selective atomic layer deposition
--------------------------------------
In 2010 Mackus and collaborators introduced a highly versatile new methodology for pure Pt thin film growth on the lateral nanoscale [@Mackus2010_Pt_asald_first; @Mackus2012_Pt_asad_resolution; @Mackus2013_Pt_asald_highres]. In area-selective atomic layer deposition (AS-ALD) the preparation of a Pt-containing seed layer with FEBID, that defines the desired lateral Pt thin film shape, is followed by the well established Pt ALD process that uses Me$_3$CpMePt(IV) as precursor and O$_2$ as reactive gas. The principle of AS-ALD is depicted in Fig.\[fig\_as\_ald\_principle\].
![Schematic of area-selective atomic layer deposition of Pt. By FEBID a Pt-containing seed layer is deposited (top), thus defining the lateral shape of the metallic Pt layer to be grown with ALD (bottom). (Pt) denotes the precursor Me$_3$CpMePt(IV).[]{data-label="fig_as_ald_principle"}](fig_as_ald_principle.pdf){width="70.00000%"}
In contradistinction to most other AS-ALD methods, the combinatorial FEBID/ALD approach relies on locally stimulating catalytic Pt ALD growth instead of using masking techniques for specific area deactivation. In addition, the FEBID seed layers is effective in overcoming the commonly observed nucleation problem of ALD metal layer growth. Thus, the combination of FEBID and AS-ALD merges the patterning capability of FEBID with the ability of ALD to deposit high purity and low resistive materials with good thickness control [@Altonen2003_Pt_ald] and room temperature resistivity values as low as $11\,\mu\Omega$cm [@Mackus2012_Pt_asad_resolution]. The pulsed-oxygen Pt(C) purification process described in the last sub-section is in fact closely related to the microscopic working principle of AS-ALD.
*In-situ* monitoring is a highly useful technique to follow the evolution of the electrical conductance of FEBID layers during or after growth, as well as during post-growth electron irradiation [@Porrati2011_PtC_irradiation]. In addition, conductance monitoring during growth has been shown to allow for semi-automatic growth optimization by using the conductance monitor signal as input to a genetic algorithm (GA) that adapts a set of deposition parameters, such as dwell time and pitches, until the rate of conductance increase per writing loop is maximized [@Weirich2013_ga_first_pub; @Winhold2014_ga_modeling]. The same approach is also applicable to AS-ALD and the conductance monitor signal can be used to optimize the cycling times for the oxygen and precursor flow, as well as the duration of the pumping times between the gas-flow periods [@Diprima2017_asald_monitoring]. In Fig.\[fig\_as\_ald\_conductance\_monitor\] we show an example for the conductance monitor signal during several ALD cycles showing a step-like behavior from which, e.g., the height increase per cycle can be deduced.
![Conductance of Pt layer as monitored during several cycles of ALD-growth of Pt (left). The ALD-growth proceeds on top of a Pt-containing FEBID seed layer in six-probe geometry. The inset shows an SEM image of the final Pt structure after about $100$ ALD cycles. A detailed view of the conductance progression during the different cycle phases reveals a mostly linear growth over time during the Pt-precursor feed and subsequent pump period (green), whereas the conductance first drops and then increases again during oxygen flow. See [@Diprima2017_asald_monitoring] for details.[]{data-label="fig_as_ald_conductance_monitor"}](fig_as_ald_conductance_monitor.pdf){width="70.00000%"}
For FEBID-induced AS-ALD growth monitoring, so far established monitoring approaches, such as ellipsometry, X-ray photoelectron or UV spectroscopy and X-ray reflectivity, are not suitable, as they rely on the availability of sufficiently large lateral growth areas. Conductance monitoring is therefore particularly suited for AS-ALD on the lateral nanoscale. Considering the sub-$10\,$nm resolution structure obtained by Mackus [@Mackus2013_Pt_asald_highres], the fabrication of point-contact structures on the nm-scale with controlled coupling strength appears feasible, if conductance monitoring during growth is employed.
Challenges and perspectives
---------------------------
A metal content of, ideally, $100\,$at$\%$ is certainly the most important factor that has guided the development of the different FEBID-based approaches briefly reviewed in this section. However, additional factors have to be taken into account depending on the particular application which is addressed. These factors are, among others, the void-free nature of the deposits, the electronic properties of possible residual contaminants (e.g. magnetic), the compatibility of the approach with high-resolution writing conditions in FEBID, and the degree of co-deposit formation, as well as the electronic nature of the co-deposit, either directly after growth or after the respective processing steps, such as purification. In particular with a view to the future use of (3D) metallic FEBID structures in plasmonics, further efforts are necessary in developing processes that result in functional, metallic nanostructures with good shape-fidelity [@Winkler2017_3D_plasmonic]. Nevertheless, large progress has been made and this is expected to continue in the future, as more and more (pure) metals become accessible by the FEBID approach.
Superconducting FEBID materials
===============================
By application of a magnetic field $H$ or by increasing the temperature $T$ the superconducting state of a material is suppressed. For bulk samples, it is the material itself that determines the superconductor’s phase boundary, as represented in the $H-T$ phase diagram. Because of the very small surface-to-volume ratio in bulk materials, the sample topology can be mostly neglected. For mesoscopic samples, on the other hand, this ratio is large and the spatial development of the superconducting state depends strongly on the sample shape which imposes specific boundary conditions. Moreover, the Cooper pair state can be substantially modified by the properties of materials in close contact with the mesoscopic superconductor. Broadly speaking, on the mesoscopic scale one has the freedom to design confinement patterns for the Cooper pair condensate and its magnetic excitations (vortices) by using their quantum nature. Sophisticated nanostructuring is needed to fabricate such mesoscopic patterns and it is quite obvious that the development of a FEBID-based direct write approach for superconducting nanostructures would be of high interest.
As-grown FEBID superconductors
------------------------------
First research on the direct-write fabrication of superconducting nanostructures was done employing a focused ion beam (FIBID – focused ion beam induced deposition) and not with FEBID. In 2004 Sadki and collaborators showed that type-II superconducting nanostructures of the approximate composition W$_{0.4}$Ga$_{0.2}$C$_{0.4}$ can be directly written using Ga-FIBID and W(CO)$_6$ as precursor [@Sadki2004_SC_WGa_FIB_first]. With a critical temperature $T_c$ of about $5.2\,$K, the exact value depending on the preparation conditions [@Li2008_SC_WGa_tuning], this superconducting material turned out to be quite attractive for several follow up works. Its weak-coupling BCS-like behavior was confirmed in tunneling spectroscopy [@Guillamon2008_tunneling_WGa] and it was used in studies relating to vortex matter [@Guillamon2014_vortex_matter_WGa], in spin-polarized Andreev reflection experiments with ferromagnetic counter electrodes prepared by FEBID [@Sangiao2011_SC_WGa_andreev_reflection] and in studies on induced odd-frequency triplet pairing in superconductor-ferromagnet proximity junctions [@Wang2010_odd_freq_sc_proximity_nanowire_first; @Kompaniiets2014_proximity_triplet_sc1]. By using different precursors other superconducting materials prepared by Ga-FIBID were discovered, such as Mo-Ga-C-O with the precursor Mo(CO)$_6$ [@Weirich2014_SC_MoGaCO], showing a maximum $T_c$ of $3.8\,$K for Mo$_{0.41}$Ga$_{0.26}$C$_{0.26}$O$_{0.07}$, and in Ga-C-O employing the purely organic aromatic precursor phenanthrene (C$_{14}$H$_{10}$) with a maximum $T_c$ of $7\,$K for Ga$_{0.27}$C$_{0.35}$O$_{0.38}$ [@Dhakal2010_SC_CGaO].
Evidently, the W-based FIBID superconductor, which closely follows BCS theory and exhibits homogeneous superconducting properties under optimized growth conditions, is an attractive candidate for various areas of research in which the direct-write approach provides a unique advantage. However, there are limitations which have to be considered. First, the high degree of disorder in this system puts this material close to a disorder-induced metal-insulator transition, thus limiting its usability, as statistical fluctuations of the superconducting order parameter lead to spatial variations of the critical temperature [@Sadovskii1997_sc_in_disordered_systems]. Second, the application of a Ga focused ion beam for the deposition is always accompanied by ion beam induced etching. Simultaneous etching does not necessarily have to be regarded as a drawback for the preparation of FIBID superconductors, but it will limit the application of this technique with regard to the fabrication of, e.g., superconducting multilayer structures. This is why a direct-write approach of superconducting structures with a focused electron beam is even more attractive. It is also more challenging, since the integral metal content in Ga-FIBID structures tends to be higher than in FEBID structures employing the same precursor.
Nevertheless, within less than one year three independents works were published with clear evidence for superconductivity in FEBID structures employing the precursors Mo(CO)$_6$ in conjunction with H$_2$O leading to an amorphous Mo-C-O phase that showed a broad onset to superconductivity at about $10\,$K in resistance measurements [@Makise2014_ebid_superconductivity], W(CO)$_6$ resulting in superconducting nanowires with an onset of superconductivity at $2\,$K under carefully optimized deposition conditions [@Sengupta2015_W_ebid_superconductivity] and tetraethyllead which resulted in metallic and superconducting Pb-C-O deposits with a maximum $T_c$ of $7.3\,$K for the composition Pb$_{0.44}$C$_{0.31}$O$_{0.25}$, a value which corresponds to the critical temperature of bulk Pb [@Winhold2014_Pb_superconductor]. So far no follow up works using these recent FEBID-based superconductors have been published which indicates that their preparation is quite delicate and more effort is needed to establish growth protocols that result in homogenous superconducting materials with reproducible properties.
Superconductivity in doped FEBID structures
-------------------------------------------
So far, the phases are not yet clearly identified which are the substrate of the superconducting condensate in the amorphous materials W-Ga-C-(O), Mo-Ga-C-O and C-Ga-O (FIBID-based), as well as W-C-O and Mo-C-O (FEBID-based). Amorphous Ga, W and Mo, as well as the carbides of W and Mo can show superconductivity in the temperature range reported for the Ga-FIBID and FEBID-based materials reported so far. In this regard an interesting question arises: may doping of charge carriers into an amorphous material, such as W-C-O prepared by FEBID, be sufficient to induce superconductivity by way of driving the system towards a superconductor-insulator transition, see, e.g. [@Sadovskii1997_sc_in_disordered_systems; @Feigelman2007_pseudogap_sc_ins_transition; @Gantmakher2010_sc_ins_quantum_phase_transition], on the interplay of Anderson localization and superconductivity in strongly disordered systems and the insulator-superconductor quantum phase transition . Two recent studies indicate that this may be the case.
Magnetization measurements performed on amorphous W-C-O FEBID structures which had been exposed to a sulphur atmosphere at $250^\circ$C for $24\,$h showed indications of superconductivity in both, field- and zero-field cooled curves setting in at temperatures as high as $38\,$K [@Felner2012_SC_WCO_S_doped]. The authors estimated a very small shielding fraction of only $0.013\,\%$ but argued that their results prove the effectiveness of sulfur in inducing superconductivity in amorphous carbon. They even suggested that this approach may open new pathways to achieve high-temperature superconductivity in amorphous carbon based materials.
In a very recent study, Porrati and collaborators showed that W-C-O prepared by FEBID employing the precursor W(CO)$_6$ can be driven towards a superconductor-insulator transition by means of Ga doping employing low-dose irradiation with a Ga focused ion beam at $30\,$keV [@Porrati2017_SC_W_FEBID_Ga_doping]. By increasing the irradiation dose, a pronounced reduction of the resistivity by more than one order of magnitude was observed in parallel with the occurrence of local superconductivity on the insulating side of the superconductor-insulator transition, indicated by a negative magnetoresistance. On the superconducting side of the superconductor-insulator transition, phase-coherent superconductivity was observed, as indicated by a positive sign of the magnetoresistance.
Challenges and perspectives
---------------------------
Direct-write fabrication of nanostructured superconductors by FEBID can open up a large range of possible application fields, such as for the study of (quantum) phase slips in nanowires [@Belkin2015_qps_sc_nanowires], as inducers for odd-frequency triplet states in proximity junctions with ferromagnets [@Bergeret2005_odd_freq_triplet_pairing], as sensor elements in highly sensitive nanowire bolometers [@Natarajan2012_nanowire_sc_bolometers], or for fundamental studies on the nature of the superconductor-insulator transition in disordered or granular systems [@Gantmakher2010_sc_ins_quantum_phase_transition], to indicate just a few. FEBID-based superconducting materials reported so far represent promising examples. However, with regard to just using them outside the immediate research field of the superconductor-insulator transition, several properties of these materials need to be significantly improved. A reproducible FEBID process to obtain a homogeneous materials that shows a sharp transition into the superconducting state, ideally well above $4.2\,$K, is still sought for.
Alloys and intermetallic compounds
==================================
The fabrication of multi-component polycrystalline or granular metals by FEBID represents a challenging research approach for the design of novel materials. Multi-component granular metals, i.e. materials in which metallic nano-crystallites are embedded in an insulating matrix, are particularly interesting, due to their fine tunability in composition and electronic inter-granular tunnel coupling strength. This flexibility is of great advantage for the investigation of binary and ternary alloy systems and to target specific intermetallic compounds.
Currently, there are three different approaches for the fabrication of multi-component FEBID nanostructures, as will be discussed below. These comprise: 1. co-deposition using two different precursors; 2. deposition with one single multi-component (heteronuclear) precursor; 3. intermixing of multilayer nanostructures fabricated with different precursors by low-energy electron irradiation.
Co-deposition using two different precursors
--------------------------------------------
The co-deposition using two different precursors represents the first and up to now most used approach for the fabrication of multi-component FEBID nanostructures. Technically, two different methods are available. The first consists in employing one single capillary to simultaneously inject two different gases into the SEM [@Che2005_FEBID_FePt_holography]. The second consists in using two independent gas injection systems, one for each precursor, see Fig.\[fig\_febid\_alloys\_multilayers\](a). The latter method is the most present in the literature [@Winhold2011_PtSi_alloy; @Porrati2012_CoPt_alloy; @Porrati2013_CoSi_alloy; @Shawrav2014_AuFe_alloy; @Porrati2017_FeCoSi_multilayer]. In our group a self-made two-channel gas injection system [@Keller2014_master_thesis] allows either the alternating or the simultaneous use of two gases.
![(a) Co-deposition with two precursors for the fabrication of a Co-Si alloy. The two precursors (”Co”: Co$_2$(CO)$_8$, ”Si”: neopentasilane) are injected into the SEM by two independent capillaries. (b) Direct deposition of Co$_3$Fe by the use of the heteronuclear precursor HFeCo$_3$(CO)$_{12}$ (”Co$_3$Fe”). (c) Deposition of a \[Fe/Si\]$_n$ multilayer by the successive injection of Fe(CO)$_5$ (”Fe”) and neopentasilane as precursors. (d) Low-energy electron irradiation treatment of the multilayer obtained from (c) to obtain a granular FeSi compound by atomic species intermixing.[]{data-label="fig_febid_alloys_multilayers"}](fig_febid_alloys_multilayers.pdf){width="80.00000%"}
Historically, the first material investigated was FePt [@Che2005_FEBID_FePt_holography], which was obtained in an UHV-SEM by mixing Me$_3$CpPt(IV) and Fe(CO)$_5$. An annealing process carried out at $600^\circ$C for two hours was employed to transform the as-grown FePt nano-rods into polycrystalline samples of the L1$_0$ phase. The magnetic characterization of these samples was performed by electron holography. In our group, the PtSi binary system was fabricated by using Me$_3$CpMePt(IV) and neopentasilane, Si$_5$H$_{12}$ [@Winhold2011_PtSi_alloy]. In this work, deposits with different composition were characterized by energy dispersive x-ray analysis (EDX), transmission electron microscopy (TEM) and electrical measurements. In particular, for a \[Pt\]/\[Si\] ratio of $2/3$, we found evidence for the formation of the Pt$_2$Si$_3$ metastable phase. CoPt was prepared by co-injection of the precursors Co$_2$(CO)$_8$ and Me$_3$CpMePt(IV) [@Porrati2012_CoPt_alloy]. Here it was shown that the as-grown amorphous CoPt phase transforms into the L1$_0$ face-centered tetragonal phase by means of a room temperature low-energy electron irradiation treatment. In ref. [@Porrati2013_CoSi_alloy] we discussed the fabrication of CoSi alloy nanostructures by using Co$_2$(CO)$_8$ and Si$_5$H$_{12}$. In that work, the electrical transport properties of the system were investigated by focusing on the metal-insulator transition, which was reached by tuning the composition of the deposits. In the group of Heinz Wanzenböck, AuFe nano-alloys were prepared by using Me$_2$(tfac)Au(III) and Fe(CO)$_5$ [@Shawrav2014_AuFe_alloy]. In this work, the influence of changes of the precursor pressure on the alloy composition was studied. Finally, we note that some work on bimetallic nanostructures, namely AuAg and AuPt, has been carried out by using liquid phase-electron beam induced deposition in the group of Todd Hastings [@Bresin2013_bimetallic_liquid_precursor].
Summarizing the results presented in the aforementioned papers, one can state that the composition of Pt-Si, Co-Pt and Au-Fe binary alloys are characterized by a large amount of carbon, $40\dots 60\,$at.$\%$, and oxygen, $10\dots 20\,$at.$\%$. As a consequence, the electrical properties of these carbon-rich alloys are dominated by electron tunneling between neighboring alloy nano-grains [@Winhold2011_PtSi_alloy; @Porrati2012_CoPt_alloy]. In Co-Si, the presence of carbon and oxygen is less dominant, since the sum of the concentrations of these two elements is always less than $50\,$at$\%$, so that an assumed metallic Co-Si phase would be expected to lead to a percolating metallic path [@Porrati2013_CoSi_alloy]. Therefore, in Co-Si the electrical properties mainly depend on the ratio between \[Si\] and \[Co\].
We conclude this paragraph remarking that, given the high carbon and oxygen concentrations present in these alloys, post-growth treatments, like electron irradiation or annealing, are required to obtain pure nano-alloy grains. On the other hand, the presence of carbon in the as-grown deposits may be seen as a benefit, since it allows to tune the microstructure, and thus, the electrical and the magneto-transport properties of the alloys by means of post-growth treatments [@Che2005_FEBID_FePt_holography; @Porrati2012_CoPt_alloy]. The main advantage of the co-deposition process is its flexibility, which allows to obtain alloys of the form A$_{1-x}$B$_x$ for any stoichiometric ratio. However, this fabrication approach fails if one of the precursor species dominates adsorption during the deposition process, as in the case of Fe(CO)$_5$ and Si$_5$H$_{12}$. In that case, alternative fabrication strategies have to be employed, as discussed next.
Heteronuclear precursors
------------------------
The use of heteronuclear precursors represents the most direct way to fabricate FEBID multi-component nanostructures, see Fig.\[fig\_febid\_alloys\_multilayers\](b). Usually, FEBID employs homonuclear precursors, as for example Fe(CO)$_5$, Co$_2$(CO)$_8$ or Si$_5$H$_{12}$, whose molecules contain one single metal (or other) atomic species, beyond the presence of hydrogen, carbon and oxygen. Recently, heteronuclear precursors, with two different metal atomic species, started to be investigated [@Porrati2015_CoFe_precursor; @Kumar2017_FeRu_dissociation_study]. The first precursor used, HFeCo$_3$(CO)$_{12}$, led to Co-Fe magnetic alloys with a metal content of about $80\,$at$\%$ [@Porrati2015_CoFe_precursor]. The \[Co\]/\[Fe\] ratio was found to be about $3$, which corresponds to the metal stoichiometry of the precursor. In TEM studies the microstructure of the samples was found to be a mixture dominated by bcc Co-Fe with some contributions of an FeCo$_2$O$_4$ spinel oxide phase which was later found to be localized at the surface of Co$_3$Fe nanostructures and is likely a consequence of oxidation under ambient conditions [@Keller2017_magnetic_3D]. Micro-Hall magnetometric measurements show that the nanostructures are ferromagnetic up to the highest measured temperature of $250\,$K [@Porrati2015_CoFe_precursor]. Recently, another heteronuclear carbonyl precursor, H$_2$FeRu$_3$(CO)$_{13}$, was investigated in a large collaborative effort with regard to its electron dissociation characteristics (gas phase and surface science studies) and its FEBID performance [@Kumar2017_FeRu_dissociation_study]. The metal content found in the deposits obtained under optimized FEBID conditions did not exceed $26\,$at$\%$, a much lower value than the one obtained for the above-mentioned Co$_3$Fe nanostructures. This large difference in the metal content of the deposits for the structurally similar precursors is attributed to their different dissociation characteristics, as found in the gas phase and surface science studies [@Kumar2017_FeRu_dissociation_study].
Clearly, the main advantage of the deposition from heteronuclear precursors is the possibility to target alloys and compounds of interest in a direct way. Furthermore, the composition of the deposits can be tuned, in a limited range, by varying the electron beam deposition parameters. In general, on the one hand the deposition from heteronuclear precursors is very attractive because of its simplicity, on the other hand the precursors currently available are few, greatly limiting the number of target materials.
Intermixing of multilayers
--------------------------
The third method used for the fabrication of FEBID multi-component deposits consists in the growth of multilayer nanostructures and their subsequent intermixing by a low-energy electron irradiation treatment [@Porrati2016_FeSi_alloy; @Porrati2017_FeCoSi_multilayer], see Fig.\[fig\_febid\_alloys\_multilayers\](c,d). This two-step fabrication method circumvents the limits of the co-deposition process for the case in which the two precursors cannot be used simultaneously. As described in Ref. [@Porrati2016_FeSi_alloy], we found that FeSi nano-alloys cannot be fabricated by mixing the precursors Fe(CO)$_5$ and Si$_5$H$_{12}$, probably as a consequence of the specifics of the competition for adsorption sites when these two precursors are simultaneously used during growth. Similarly, in Ref. [@Porrati2017_FeCoSi_multilayer] the Heusler compound Co$_2$FeSi could not be prepared by mixing HFeCo$_3$(CO)$_{12}$, Fe(CO)$_5$ and Si$_5$H$_{12}$. To overcome these limits, we deposited \[Fe/Si\]$_2$, \[Fe$_3$/Si\]$_2$ and \[Co$_2$Fe/Si\]$_n$ multilayers and, subsequently, subjected them to a low-energy electron irradiation treatment in order to induce atomic species intermixing. As a result, the compounds FeSi, Fe$_3$Si and Co$_2$FeSi were obtained [@Porrati2016_FeSi_alloy; @Porrati2017_FeCoSi_multilayer].
The main advantage of this fabrication method is to provide an alternative approach in those cases for which the co-deposition approach and the fabrication process based on the use of heteronuclear precursors cannot be applied. Moreover, this fabrication method explicitly shows that high-quality nanopatterned multilayers can be fabricated by FEBID [@Porrati2017_FeCoSi_multilayer], which is of potential interest for the fabrication of plasmonic metamaterials and prototype devices in spintronics.
Challenges and perspectives
---------------------------
The possibility to fabricate multi-component materials by FEBID opens the way for the fabrication of a large number of nanostructured alloys and intermetallic compounds. The investigation of entire material classes, as for example Heusler compounds, which for itself encompasses more than $1000$ members, becomes possible at the nanoscale. Two main characteristics distinguish FEBID materials from bulk or thin film samples prepared with other techniques. First, FEBID allows to fabricate one-, two- and three-dimensional nanostructures. Second, the granular microstructure of the material can be tuned with a high degree of precision. This combination makes FEBID multi-component nanostructured materials very attractive for fundamental studies, as for example the investigation of finite-size effects, and, potentially, for applications in spintronics and thermoelectrics.
We conclude remarking that the main part of the precursors potentially interesting for the fabrication of FEBID alloys and intermetallic compounds are not commercially available. Therefore, the synthesis of the precursor, the study of its dissociation behavior, the growth of the deposits and their successive characterization, is a challenging interdisciplinary work involving know-how from the fields of chemistry, physics, surface and materials science. It remains a task for the future to further expand the reservoir of high-potential FEBID precursors.
Metamaterials
=============
In the narrow sense, research on metamaterials has been largely concerned with negative refraction index materials in optics. However, the idea of rationally designing a unit from selected materials and (periodically) arrange it into an artificial solid is much broader. The properties of that solid are then determined by the structure of the artificial solid and the coupling between its units and can be tailored towards a desired functionality or a particular emerging property. In this regard, several important benefits can be gained from employing FEBID. One can directly take advantage of the particular microstructure of FEBID materials, as exemplified by nano-granular metals which are often obtained when organometallic precursors are used. Else, one can take advantage of the high-resolution writing capability of FEBID to follow the rational design approach towards metamaterials or combine nanostructured FEBID materials with other materials. Finally, these approaches can be combined, as will be exemplified below. It should be noted that the emerging simulation-guided approach for 3D nano-manufacturing, as discussed later, will likely be of large relevance for the future development of optical and magnetic metamaterials.
Ordered and disordered nano-granular metals {#sec_nanogranular_metals}
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Nano-granular metals, in the ordered form also denoted as nano-dot lattices, are model systems for the study of the interplay of electronic correlation effects, finite size induced quantization of the electronic level structure, and disorder [@Beloborodov2007_granular_electronic_systems]. The inter-granular tunnel coupling $g$ (normalized to the conductance quantum $G_0 = 2e^2/h$, including spin degeneracy) is the most important control parameter that governs the electronic properties of this material class. The coupling strength in conjunction with the metallic grain size, that determines the Coulomb charging energy $E_C$ of the grains, and the properties of the insulating matrix, in which the grains are embedded, determine the electronic ground state of a granular metal. In both phases, the weak-coupling or insulating and the strong-coupling or metallic, length scales exist which exceed by far the geometric length scales of grain diameter and grain-to-grain distance. As a result, many-body properties emerge which cause characteristic electronic properties in the respective phases.
On the weak-coupling side, higher-order tunneling events (sequential inelastic and elastic co-tunneling) start to dominate, as the temperature is reduced leading to a stretched exponential dependence of the conductivity on temperature, the correlated variable range hopping (c-VRH) [@Efetov2003_coulomb_effects_granular_metals; @Beloborodov2007_granular_electronic_systems]. $$\sigma(T) = \sigma_0 \exp{\left\{ -\left( \frac{T_0}{T} \right)^{1/2} \right\}} \,.
\label{eq_cVRH_sigma_vs_T}$$ This is indicated schematically in Fig.\[fig\_co\_tunneling\].
![Schematic of charge transport in a nano-granular metal in the c-VRH regime. Electron tunneling between metallic grains can occur over larger distances by means of co-tunneling, as indicated in the upper image. A strong reduction of the effective charging energy $E_C$ occurs, as there is no charge accumulation in the grains which take part in the co-tunneling sequence. Over the co-tunneling range $\xi$, that determines the activation temperature $T_0$, the Coulomb interaction between the initial (now positively charged) and final grain (now negatively charged) after the co-tunneling event has occurred is reduced over the distance $\xi$ and by screening effected by the surrounding material. In the lower part of the figure inelastic co-tunneling is indicated, in which the incoming and outgoing charge are on different energy levels. The energy difference is exchanged with the surrounding material via phononic excitations.[]{data-label="fig_co_tunneling"}](fig_co_tunneling.pdf){width="0.6\linewidth"}
In the strong-coupling limit a granular Fermi liquid was theoretically predicted [@Beloborodov2003_granular_metals_strong_coupling; @Beloborodov2004_granular_fermi_liquid] and experimentally observed in nano-granular Pt prepared by FEBID with the precursor Me$_3$CpMePt(IV) [@Sachser2011_PtC_universal_conductance] (see Fig.\[fig\_phase\_diagram\_granular\_metals\] for a schematic phase diagram of nano-granular metals). Furthermore, there is growing evidence for the emergence of collective states, indicated by super-ferro- or super-antiferromagnetic behavior in nano-granular Pt in a certain range of the coupling strength $g$ for temperatures below about $20$ to $30\,$K [@Porrati2014_PtC_magnetoresistance; @Porrati2014_PtC_coupling_Co_nanopillars]. These observations gain additional relevance because of the apparent similarities in the electronic structure of nano-granular metals and electronically correlated organic charge transfer systems of the $\kappa$-\[BEDT-TTF\]$_2$-X-type (BEDT-TTF: bis-ethylenedithiotetrathiafulvalene, X: halogenide or polymeric anion) close to a superconductor-insulator transition [@Diehl2015_pseudogap_kappaET; @Guterding2016_sc_order_parameter_kET]. It should also be noted that underdoped high-$T_c$ superconductors show evidence for spontaneous granularity in their electron density at low temperature [@Pan2001_STS_underdoped_htsc; @Dubi2007_sit_htsc_disordered]. In addition, microstructurally homogeneous, amorphous TiN thin films also show evidence for spontaneous electronic granularity [@Baturina2007_localized_sc_TiN].
![Schematic phase diagram of transport regimes of nano-granular metals. Adapted from [@Beloborodov2004_granular_fermi_liquid].[]{data-label="fig_phase_diagram_granular_metals"}](fig_phase_diagram_granular_metals.pdf){width="0.6\linewidth"}
Nano-granular Pt, as prepared by FEBID, has been established as a particularly fruitful model system for studies of the coupling strength dependent phase diagram of disordered granular metals (see Fig.\[fig\_phase\_diagram\_granular\_metals\]). This is due to the fact that the inter-granular coupling strength can be extremely finely tuned by a simple post-growth electron irradiation treatment combined with *in-situ* conductance monitoring [@Porrati2011_PtC_irradiation]. The research that has been done on nano-granular Pt may easily be extended to other nano-granular FEBID structures provided that their insulating matrix shows similar tunability. For as-grown nano-granular Pt the matrix consists mainly of amorphous carbon which tends to graphitize under electron irradiation, thus improving the tunnel-coupling strength between the Pt nano-grains [@Porrati2011_PtC_irradiation]. In addition, FEBID has more to offer with regard to ordered nano-granular metals or nano-dot lattices. Employing the precursor W(CO)$_6$ we fabricated ordered 2D square lattices of roughly semi-spherical and metallic W-C-O nano-islands of approximately $20\,$nm diameter and showed that a metal-insulator transition occurs as a function of lattice constant [@Sachser2009_2D_nanodot_lattice_transport; @Porrati2010_2D_nanodot_lattice_fabrication]. In this case, the metal-poor co-deposit around the metallic nano-islands served as tunneling matrix. It is to be expected that future work will expand the class of tunable nano-granular metals prepared by FEBID.
Proximity-induced superconductivity
-----------------------------------
The variety of available FEBID materials with tunable structural and magnetic properties in conjunction with its high resolution makes FEBID valuable for the fabrication of hybrid nanostructures and the investigation of emerging effects at interfaces on the meso- and nanoscale. A prominent exemplary phenomenon is the superconducting proximity effect [@Buzdin2005_review_fm_proximity] and, more generally, proximity-induced superconductivity [@Bergeret2005_odd_freq_triplet_pairing]. In the conventional superconducting proximity effect at a superconductor/ferromagnet (SF) interface, the wave function of the Cooper pairs is singlet as it is formed by two electrons with opposite spins. The exchange field of F tends to align both spins in the same direction which results in a strong pair-breaking effect and causes a rapid exponential oscillatory decay of the superconducting order parameter in F over a distance of the order of $1$nm. However, under some circumstances the presence of ferromagnetism may lead to triplet superconducting pairing [@Buzdin2005_review_fm_proximity; @Bergeret2005_odd_freq_triplet_pairing; @Eschrig2008_triplet_supercurrents; @Eschrig2015_spin_polarized_supercurrents]. In this case spin-triplet superconducting correlations are extending into F over a proximity length of several microns and this is why the spin-triplet proximity effect at a S/F interface is long-ranged and ”unconventional” [@Wang2010_odd_freq_sc_proximity_nanowire_first]. As theoretically shown by Bergeret *et al.* [@Bergeret2001_long_range_proximity_fm], local inhomogeneity of the magnetization in the vicinity of the S/F interface is necessary for spin-triplet pairing in S/F structures. These inhomogeneities can be either intrinsic to F (domains) and thus modifiable by an external field, or be caused by a material inhomogeneity as a result of experimental manipulations, such as contacting procedures or post-growth processing. The versatility of FEBID for both of these procedures makes it an especially useful experimental technique in this regard.
Sangio *et al.* [@Sangiao2011_SC_WGa_andreev_reflection; @Sangiao2011_andreev_magnetic_field_sc_fm_febid] demonstrated for the first time functional devices combining magnetic nanostructures grown by FEBID and superconducting nanostructures fabricated by Ga-FIBID. They studied the magnetic-field dependence of the conductance in planar S/F nano-contacts which allowed them to extract the magnetic field dependences of the superconducting gap of the W-based electrode. The value of the superconducting gap extracted from those experiments is in agreement with direct scanning tunneling spectroscopy experiments, emphasizing the possibility of preparing clean nano-contacts by FEBID/FIBID with a single or very few conduction channels [@Guillamon2008_tunneling_WGa]. Subsequent experiments on nano-contacts with worse definition or cleanliness gave rise to multi-channel transport [@Sharma2014_multi_channel_andreev].
Kompaniets *et al.* investigated superconducting proximity effects at S/F interfaces in nanowire structures [@Kompaniiets2014_proximity_triplet_sc1; @Kompaniiets2014_Cu_Co_proximity]. Whereas superconducting correlations induced by a W-FIBID electrode led to up to $30\%$ resistance drops of a $7\,\mu$m-long section of a polycrystalline Co nanowire [@Kompaniiets2014_proximity_triplet_sc1], proximity-induced superconductivity did not become apparent in Co-FEBID nanowire structures [@Kompaniiets2014_Cu_Co_proximity]. This may indicate that C ($15\,$at$\%$) and O ($14\,$at$\%$), residual impurities from the FEBID process with Co$_2$(CO)$_8$, are impurities responsible for suppressing the superconducting proximity effect, as these elements may be effective pair-breaking scatterers for triplet Cooper pairs. As possible routes to stimulating long-range spin-triplet superconductivity in Co-based FEBID structures one should mention annealing in a hydrogen atmosphere at elevated substrate temperatures in conjunction with electron irradiation in order to remove undesired C and O impurities [@Begun2015_Co_purification_H2], as well as enhancing the magnetic inhomogeneity on the lateral mesoscale, e.g., by formation of a Co/Pt phase with different magnetic properties [@Dobrovolskiy2015_CoPt_treatment_H2_O2].
Fluxonics
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Almost all technologically relevant superconductors are superconductors of type II. In a magnetic field $\mathbf{B}$, whose magnitude is between the lower and upper critical field, they are in the mixed state, being penetrated by a flux-line array of Abrikosov vortices, or *fluxons* [@Abrikosov1957_flux_lattice; @Brandt1995_flux_line_lattice; @Moshchalkov2010_sc_nanoscience; @Dobrovolskiy2017_fluxonics_review; @Woerdenweber2017_sc_at_nanoscale]. Each fluxon carries one magnetic flux quantum, $\Phi_0=2.07\times10^{-15}$ Wb, and the repulsive interaction between vortices makes them to arrange in a triangular lattice with the vortex lattice parameter $a_\bigtriangleup = (2\Phi_0/B\sqrt{3})^{1/2}$, where $B=|\mathbf{B}|$. Each vortex experiences the action of the driving (Lorentz) force induced by the transport current and the pinning force related to the spatial variation of the vortex energy at different locations inside the superconductor. Thus, when the pinning force dominates the Lorentz force, the vortex lattice is locally anchored (pinned) while in the opposite case the vortex lattice moves and the superconducting state is destroyed. For this reason, artificial nanostructures inducing a spatial modulation of the vortex energy are widely used for the manipulation of Abrikosov vortices [@Velez2008_vortex_pinning; @Dobrovolskiy2017_fluxonics_review].
![(a) Atomic force microscope image of Co stripes deposited by FEBID on the surface of a superconducting Nb film. (b) Polar diagram of the total resistance of the Nb films with the Co stripes measured in the oriented-current geometry. The Co stripes are arranged along the $y$-axis and induce a washboard pinning potential periodic in the $x$-direction. The currents $I_x$ and $I_y$ allow one to pass the transport current $I=\sqrt{I_x^2+I_y^2}$ at any arbitrary angle $\alpha$ with respect to the pinning ”channels”. The measured voltage components $V_x$ and $V_y$ allow one to obtain the polar diagram of the total resistance $R(\alpha)$ of the film for a series of temperatures.[]{data-label="fig_Co_stripes_vortex_dynamics"}](fig_Co_stripes_vortex_dynamics.pdf){width="0.8\linewidth"}
Our group demonstrated for the first time that nanostructures fabricated by FEBID can be used for the extension of the dissipation-free state of superconductors to higher temperatures and larger currents [@Dobrovolskiy2010_guiding_Co_decoration; @Dobrovolskiy2011_washboard_pinning; @Dobrovolskiy2011_washboard_fabrication]. Fig.\[fig\_Co\_stripes\_vortex\_dynamics\](a) depicts an atomic force microscopy (AFM) image of an array of Co nanostripes (precursor Co$_2$(CO)$_8$) deposited by FEBID on the surface of a superconducting epitaxial Nb film. The film was patterned into an $8$-contact geometry for the oriented-current setup shown in the lower inset of Fig.\[fig\_Co\_stripes\_vortex\_dynamics\](b). The Co-FEBID stripes are aligned along the $y$-axis and induce a pinning potential of the washboard type for Abrikosov vortices, which is periodic in the $x$-direction. In the mixed state, near the superconducting transition temperature $T_c\approx 8.1\,$K, this results in a several orders of magnitude smaller resistance of the superconducting film when the transport current is applied along the stripes (vortices move in the $x$-direction) than perpendicular to the stripes (vortices move in the $y$-direction). Electrical resistance measurements clearly revealed an anisotropic behavior induced by the uniaxial pinning [@Dobrovolskiy2010_guiding_Co_decoration] as well as resistance steps ascribed to matching effects between the Co-FEBID stripe periodicity and the vortex lattice periodicity [@Dobrovolskiy2011_washboard_pinning].
Challenges and perspectives
---------------------------
Although the nano-granular nature of many of the materials prepared by FEBID has long been known, the directed use of the tunability of the tunnel coupling strength in conjunction with *in-situ* monitoring is a recent development [@Porrati2011_PtC_irradiation]. Several important questions relating to the electronic properties of nano-granular metals might thus become experimentally addressable in future work. Among them are: What is the dependence of the Coulomb charging energy $E_C$ on the coupling strength $g$ as the critical region around the metal-insulator transition is approached and how does it depend on the dimensionality of the nano-granular metal? In which part of the low-temperature transport regimes of nano-granular metals does universal behavior arise and what are the consequences of beginning coherent electron motion over several metal particles in the 1D, 2D and 3D case? Also, tunable nano-granular metals with magnetic nano-grains might help to provide insight into the formation of super-ferromagnetic or super-antiferromagnetic collective states in competition with super-spinglasses, depending on the degree of disorder [@Bedanta2007_superferromagnetism_evidence; @Morup2010_review_interacting_magnetic_nanoparticles]. Extension of previous work on ordered 2D nano-island lattices to magnetic structures is also highly promising with regard to the controlled formation of such collective phases, as well as in coupling these to other collective phases, like the superconducting. As such, the capability of fabrication of ferromagnetic FEBID structures with high metal content [@Begun2015_Co_purification_H2; @Dobrovolskiy2015_CoPt_treatment_H2_O2] is of particular advantage for studying the magnetization dynamics of nanomagnets [@Lara2014_Co_discs_half_antivortex] and helps to advance investigations of spin-triplet proximity-induced superconductivity in S/F nanostructures. At a later stage, when S/F structures with a large spatial extend of spin-triplet superconducting correlations will be available, this will open a route to studying vortex matter in low-dimensional systems with proximity-induced superconductivity — an intriguing research line [@Kopnin2013_vortex_low_dim_proximity] which lacks experimental investigation so far. While new insights into vortex matter are also expected from studying it in more sophisticated 3D nano-architectures [@Thurmer2010_nanomembrane_sc_junctions; @Fomin2012_correlated_vortex_tuning], one should expect that the application of FEBID in the next years will cover an even longer list of sample preparation tasks ranging from the deposition of nanostructures with complex topology to the fabrication of out-of-plane contact electrodes.
With respect to thin-film fluxonic applications, the following advantage of FEBID should be emphasized. A particular feature of structures prepared by conventional lithographic techniques is that they are planar, while FEBID is also suitable for the fabrication of 3D pinning structures. This is especially relevant for the fabrication of asymmetric (ratchet) pinning landscapes for Abrikosov vortices, whose peculiar feature consists in the appearance of a rectified net vortex motion and its reversal [@Plourde2009_pinning_anisotropic; @Shklovskij2014_vortex_ratchet_asymmetric]. While the functionality of superconductors with linearly-extended asymmetric pinning structures fabricated by FIB milling has been demonstrated for microwave filters [@Dobrovolskiy2015_dual_cutoff_filter] and fluxonic metamaterials [@Dobrovolskiy2015_ac_microwave_loss_modulation], we foresee next-generation electronic applications of the fluxonic type realized using re-programmable vortex pinning landscapes fabricated by FEBID. Taking into account the possibility to fabricate 3D magnetic structures by FEBID [@Cordoba2016_Fe_Co_3D_nanopillars; @Keller2017_magnetic_3D], one should expect even higher tunability of the strength of vortex pinning induced by them.
Sensor applications
===================
As has been discussed in section \[sec\_nanogranular\_metals\], charge transport in nano-granular metals depends sensitively on the inter-grain distance, the grain size and the dielectric properties. The corresponding electronic parameters are the inter-granular tunnel coupling $g$, the charging energy $E_C$ and the dielectric function of the matrix material $\epsilon(T, \omega)$. Here $\omega$ denotes the frequency if the granular metal is subject to a harmonic electric field. The nano-granular microstructure of many FEBID materials suggest their use for different sensing tasks, in particular strain and modifications in the dielectric environment, as will be discussed below. In the case of nano-granular magnetic materials, the strongly enhanced anomalous Hall effect in proximity to the metal-insulator transition can be exploited for the realization of magnetic stray field sensors, on which we will also briefly dwell in this section. The relevance of the FEBID approach for theses different sensor application fields mainly derives from the high degree of miniaturization that can be achieved, but also the possibility of fine-tuning the sensor response function by simple modifications of the FEBID process.
Strain sensing
--------------
Charge transport in nano-granular metals is dominated by tunneling, as has been pointed out in section \[sec\_nanogranular\_metals\]. From this it is apparent that nano-granular metals might be suitable materials for strain-sensing, since the tunnel coupling has an exponential dependence on the inter-grain distance which is altered under strain (see also lower right inset of Fig.\[fig\_strain\_sensing\]). A key parameter that characterizes a strain sensor is its gauge factor $\kappa$ given by $$\kappa = \frac{\Delta R}{R} \left/ \frac{\Delta\ell}{\ell} \right. \,,
\label{eq_gauge_factor}$$ where $\Delta R$ denotes the resistance change under the relative length change $\Delta \ell/\ell$ or strain. In 2010 one of us (MH) could show in a theoretical analysis, based on recent advances in the understanding of the charge transport regimes in nano-granular metals [@Beloborodov2007_granular_electronic_systems], that nano-granular FEBID materials are indeed promising with regard to strain sensing, in particular for use in highly-miniaturized micro-electromechanical (MEMS) sensors [@Huth2010_theory_strain_sensing]. In the same year we could experimentally demonstrate that nano-granular Pt FEBID structures are indeed highly linear strain sensing elements that can be quite easily integrated in cantilever sensors [@Schwalb2010_strain_sensing]. Fig.\[fig\_strain\_sensing\] shows the relative resistance change under strain for a soft SiN$_x$ cantilver equipped with a nano-granular Pt sensor.
![Exemplary resistance-strain characteristics of nano-granular Pt sensor element with gauge factor of about $8$ at the bending edge of a soft SiN$_x$ cantilever structure (see SEM image in the upper left). Lower right inset: Schematic of the change in the inter-granular distance and coupling strength $g$ under tensile strain when a force $F$ is applied to a bridge, membrane or cantilever equipped with a nano-granular sensing layer. The sensing layer is deposited at the positions of largest strain.[]{data-label="fig_strain_sensing"}](fig_strain_sensing.pdf){width="0.8\linewidth"}
The possibility to finely tune the inter-granular coupling strength of nano-granular Pt by post-growth electron irradiation (see section \[sec\_nanogranular\_metals\]) allows for the optimization of the strain sensor’s signal-to-noise ratio which is mandatory for its use in such demanding applications as AFM with self-sensing cantilevers, see [@Dukic2016_afm_sensor] for details. In very recent work Moczała and collaborators used the same nano-granular Pt FEBID structures as deflection sensing elements and applied them to read out the resonance frequency of micromechanical SiN$_x$ bridges [@Moczala2017_ebid_sensor]. By stress distribution modification employing FIB milling they could increase the deflection detection sensitivity even further.
Gas and dielectric sensing
--------------------------
Charge transport in nano-granular metals depends very sensitively on the dielectric environment in which the metallic grains are embedded. In order to see this, we briefly enlarge upon what has already been said concerning nano-granular metals in section \[sec\_nanogranular\_metals\].
The effects of granularity in the charge transport properties are most pronounced at temperatures $T$ above $\Gamma/k_{B}$, where $\Gamma$ denotes the life-time broadening of the energy levels at the Fermi level inside a grain as a consequence of the tunnel-coupling to the neighboring grains. Electron correlations come into play, if one considers that, as an electron is tunneling from one neutral grain to a neighboring grain, a charging energy $E_C$ accrues which amounts approximately to $E_{C}=e^{2}/2C$, where $e$ is the electron charge and $C$ stands for the capacitance of the grain in its dielectric environment. For a spherical grain, the capacitance is $C=4\pi\epsilon_{0}\epsilon D$ with $\epsilon_{0}$ and $\epsilon$ the dielectric constant of the vacuum and the effective dielectric constant of the granular material, respectively. The energy $E_{C}$ hinders charge transport and eventually leads to a hard or soft energy gap (depending on the degree of disorder) in the density of states at the Fermi level [@Beloborodov2004_dos_weak_coupling; @Beloborodov2005_dos_close_MIT]. The influence of $E_C$ is strongly visible only in the weak inter-grain coupling $g<0.1$ regime, in which charge transport occurs via c-VRH. In this regime the temperature-dependent conductivity $\sigma(T, \omega=0)$ follows [@Beloborodov2007_granular_electronic_systems] $$\sigma(T) = \sigma_{0}\exp\left\{ -\left(\frac{T_{0}}{T}\right)^{1/2}\right\} \quad\textnormal{with}\quad k_{B}T{}_{0}\approx e^{2}/4\pi\epsilon_{0}\epsilon\xi(T) \,,
\label{eq_cVRH}$$ in which the activation temperature $T_0$ ($k_B$: Boltzmann temperature) depends on the relevant type of sequential co-tunneling governing this transport regime via the wave function attenuation length or co-tunneling range $\xi$. $\xi$ takes on different functional forms in the elastic and inelastic co-tunneling channel, see [@Beloborodov2007_granular_electronic_systems] for details. The dielectric sensing effect is based on the fact that changes of the effective dielectric properties of the environment of the metallic grains are directly reflected in changes of the charging energy $E_{C}$ which depends on $\epsilon(T,\omega)$.
We consider a bilayer-system consisting of a nano-granular metal layer of typical thickness $5$ to $20\,$nm on which a non-conducting top layer to be dielectrically characterized is deposited (see schematic in Fig.\[fig\_dielectric\_sensing\]). The activation temperature $T_0$, characterizing the nano-granular metal in the weak-coupling regime, depends on the wave function attenuation length $\xi(T)$ which itself is governed by the charging energy $E_{C}$ of the metallic grains in the following form (inelastic co-tunneling regime) [@Beloborodov2007_granular_electronic_systems] $$\xi(T)=2D/\ln\left[E_{C}^{2}/16\pi g\left(k_{B}T\right)^{2}\right]\,.
\label{eq_attenuation_length_ineleastic_co_tunneling}$$ Apparently, the activation temperature $T_{0}$ depends on two accounts on the dielectric environment of the metallic grains. First, it is inversely proportional to the effective dielectric constant $\epsilon$ of the surrounding medium. Second, changes of the charging energy $E_{C}$ caused by the surrounding medium will directly modify the attenuation length and, thus, the activation temperature. In recent work one of us (MH) developed a model to account for the conductance changes induced in the nano-granular metal by the dielectric properties of the top layer. On the mean field level, the model describes the change of the effective dielectric constant experienced by the metallic grains at various distances to the interface of a nano-granular metal / dielectric layer heterostructure [@Huth2014_diel_sensing_theory].
The first experimental evidence for the dielectric sensing effects was demonstrated by Kolb and collaborators [@Kolb2013_H2O_sensing] from the group of Harald Plank. In this work it was shown that strong conductance changes in $5$ to $20\,$ nm thick nano-granular Pt FEBID layers occur under adsorption and desorption of water with sub-monolayer sensitivity [@Kolb2013_H2O_sensing]. These observations eventually lead to the development of the mean-field theory of the sensing mechanism mentioned above [@Huth2014_diel_sensing_theory]. This could then also be applied to understand the observed conductance modulation in nano-granular Pt FEBID sensing layers on top of which a thin film of the insulating, organic ferroelectric TTF-p-chloranil (TTF: tetrathiafulvalene) was deposited [@Huth2014_sensor_ttfca]. TTF-p-chloranil layers (under tensile strain) show a paraelectric-to-ferroelectric phase transition at about $56\,$K, at which the real part of the dielectric constant exhibits a pronounced maximum. As shown in Fig.\[fig\_dielectric\_sensing\], this causes a strong increase in the conductance of the nano-granular Pt sensing layer, which is due to the reduction of the charging energy of the Pt nano-grains caused by the enhanced dielectric screening effect of the ferroelectric at the phase transition, see [@Huth2014_sensor_ttfca] for details.
![Conductance vs. temperature behavior of nano-granular Pt FEBID sensing layer as part of a bilayer structure (red curve) consisting of nano-granular Pt (effective dielectric constant $\epsilon_1$, see inset) and an insulating dielectric top layer (dielectric constant $\epsilon_2$, see inset; here TTT-p-chloranil). The conductance increase is caused by $E_C$ renormalization effects induced by the ferroelectric transition of TTF-p-chloranil at about $56\,$K [@Huth2014_sensor_ttfca]. The blue curve shows the temperature-dependent conductance of a reference Pt nano-granular metal layer prepared under identical conditions but without TTF-p-chloranil top layer.[]{data-label="fig_dielectric_sensing"}](fig_dielectric_sensing.pdf){width="0.8\linewidth"}
Magnetic sensing
----------------
Several complementary techniques are available for the quantitative detection of local magnetic stray fields, each of which is characterized by specific strengths and weaknesses. The most relevant quantities for magnetic stray field sensors used in scanning probe microscopy applications are the smallest detectable field change and the lateral resolution. Established sensor materials in this regard are two-dimensional electron gases (2DEG), e.g. based on GaAs/AlGaAs heterostructures, and semimetals, typically Bi. The stray field is measured by detecting the associated Hall voltage $V_H$ induced in a miniaturized Hall cross which is scanned at a small distance (typically below $1\,\mu$m) over the magnetic structure to be mapped [@Bending1999_principles_local_hall_sensors]. The thermal voltage noise $V_{th} = \sqrt{4k_BTR\Delta f}$ of such a Hall cross sets a lower limit to the detectable field change $\delta B_{min}$. Here $\Delta f$ denotes the frequency bandwidth. Given the signal-to-noise ratio $\textnormal{SNR}$ [@Bending1999_principles_local_hall_sensors] $$\textnormal{SNR} = \frac{V_H}{V_{th}} = \frac{R_HIB}{\sqrt{4k_BTR\Delta f}} \,,
\label{eq_SNR_hall_cross}$$ $\delta B_{min}$ is obtained by setting $\textnormal{SNR}=1$ and solving for $B$ $$\delta B_{min} = \frac{\sqrt{4k_BTR\Delta f}}{R_HI} \,.
\label{eq_bmin_Bi}$$ This limit can only be reached in ac measurements, as the voltage noise spectrum typically shows a $1/f$-characteristic, so that the minimal voltage noise is only reached above a temperature- and current-dependent frequency value.
Besides 2DEGs and semimetals, granular ferromagnetic materials are a promising material class for sensitive magnetic stray field measurements. Soon after the discovery of the Hall effect it was found that the Hall coefficient is strongly enhanced in iron sheet metal as compared to non-ferromagnetic metals. The Hall resistivity $\rho_H$ was found to have two contributions. The first contribution relates to the conventional Hall effect $\rho_{OH}$ and is governed by the charge carrier density. The second contribution $\rho_{EHE}$ is denoted as extraordinary (EHE) or anomalous and relates to the ferromagnet’s magnetization $M(T, H)$ $$\rho_H = \rho_{OH} + \rho_{EHE} = \mu_0 \left[ R_0H + R_SM(T, H) \right] \,.$$ Here $H$ denotes the field component perpendicular to the sample surface and the applied current. $M$ accordingly represents the magnetization component in field direction, so that the corresponding magnetic flux density component $B$ inside the sample is $B = \mu_0 (H + M)$. $R_0$ and $R_S$ are material constants. The enhanced Hall effect in ferromagnets is a consequence of material-intrinsic spin-orbit effects which are reflected in the electronic band structure, as well as chiral scattering contributions from non-magnetic impurities (extrinsic effects). The intrinsic effects are related to the geometric phase of the charge carrying electronic states, see a recent review by Nagaosa for more details [@Nagaosa2010_review_anomalous_hall_effect].
In order to use the enhanced Hall effect in ferromagnets, two further observations are crucial. First, for granular ferromagnets, i.e. magnetic grains embedded in an insulating matrix, $\rho_{EHE}$ is found to be even larger than in bulk ferromagnets. For a material- and microstructure-specific volume fraction of the metal grains the anomalous Hall effect can be enhanced by up to three orders of magnitude, which is also denoted as giant anomalous Hall effect [@Denardin2003_review_giant_hall_effect_granular_ferromagnets]. Second, due to the granular microstructure the metallic grains tend to remain in a super-paramagnetic state for temperatures above a characteristic blocking temperature $T_B$. In this state magnetic hysteresis effects do not occur. Fig.\[fig\_schematic\_anomalous\_hall\_effect\] gives a schematic overview of the response of a Hall sensor based on a granular ferromagnet for $T > T_B$. Also shown is the typically linear dependence of the extraordinary Hall resistivity on the longitudinal resistivity, which is useful for estimating the achievable SNR for a given geometry of a Hall cross (see Eq.\[eq\_SNR\_hall\_cross\]). For magnetic flux density values $B > B_{sat}$, i.e. above magnetic saturation, the slope of the Hall voltage is only determined by the ordinary Hall effect. However, typical values for $B_{sat}$ are in the $1\,$T-range for granular ferromagnetic Hall structures prepared by FEBID, so that the useful range of magnetic stray field detection is large.
![(a) Hall voltage vs. applied magnetic field for a granular ferromagnet at $T > T_B$. (b) Relation between the extraordinary Hall resistivity and the longitudinal resistivity of a granular ferromagnet.[]{data-label="fig_schematic_anomalous_hall_effect"}](fig_schematic_anomalous_hall_effect.pdf){width="0.6\linewidth"}
Research into granular ferromagnetic Hall sensor for magnetic stray field detection based on nano-scale FEBID structures was pioneered by the group of Ivo Utke [@Boero2005_CoC_Hall_sensor; @Gabureac2010_granular_hall_sensors] and later applied to the detection of suspended superparamagnetic beads [@Gabureac2013_gfm_magnetic_bead]. In this work the precursor Co$_2$(CO)$_8$ was used. For Hall-cross sensor areas down to $100\times 100\,$nm$^2$ the smallest detectable field change was found to be $B_{min} \approx 5\,\mu$T/$\sqrt{\textnormal{Hz}}$ at room temperature [@Gabureac2010_granular_hall_sensors]. This value is about a factor of $10$ lower than what has been achieved with semimetallic Bi Hall sensors [@Sandhu2004_Bi_hall_sensors]. Frequency-dependent voltage noise measurement showed a $1/f$-characteristic running into the Johnson noise limit at about $1\,$kHz at the largest currents the sensors could sustain without being damaged [@Boero2005_CoC_Hall_sensor]. Within the magnetic bead detection and tracing experiments a lateral resolution of $230\,$nm$/\sqrt{\mathrm{Hz}}$ was achieved at a field resolution of $300\,\mu$T$/\sqrt{\mathrm{Hz}}$ [@Gabureac2013_gfm_magnetic_bead].
A very systematic analysis of the specific Hall resistivity for Fe FEBID structures was performed by Rosa Córdoba and collaborators employing the precursor Fe$_2$(CO)$_9$ [@Cordoba2012_GAHE_Fe]. By admixture of variable amounts of water during the depositon process the Fe-to-oxocarbide ratio of the deposits was carefully controlled and the development of the anomalous Hall effect was investigated in dependence of the longitudinal resistivity. An almost linear dependence was found up to anomalous Hall resistivity values of more than $100\,\mu\Omega$cm.
Challenges and perspectives
---------------------------
Strain sensing based on nano-granular FEBID is most promising if applied in very small resonating structures for which the high-resolution direct-write capabilities of FEBID are essentially without alternative [@Dukic2016_afm_sensor]. On the other hand, this requires additional development in high-speed detection electronics, in particular for AFM applications, as the resonance frequencies of sub-micron sized structures can extend into the several $10\,$MHz regime.
With regard to the novel dielectric sensing approach, an extension into ac driven systems could be promising. One may expect a universal frequency dependence of the real part of the conductivity, which is long known for a broad range of different disordered systems [@Dyre2000_ac_conductance_universality]. This universality has recently been shown for a nano-granular metal [@Bakkali2016_universality_ac_response_granular_metals] but not yet for nano-granular FEBID materials.
Nano-granular ferromagnetic Hall sensors feature a very high sensitivity for magnetic stray fields and show good down-scaling behavior. Nevertheless, they still have to be shown to work reliably in actual sensor devices.
In all cases discussed in this section, the sensing mechanism relied on a particular property of the nano-granular microstructure of the deposits. The longterm stability of the electric and magnetic properties of nano-granular FEBID materials is an issue on which not much has been worked on so far, but see [@Winhold2015_phd_thesis] with regard to the transient behavior in the electrical conductance of nano-granular Pt up to the time scale of months. Further work towards a better understanding of aging effects and the development of reliable stabilization procedures will certainly be needed, if nano-granular FEBID sensors are to be applied in real devices in the future.
3D FEBID structures
===================
Its 3D writing capability is probably one of the prime disciplines of FEBID. Nevertheless, the fabrication of even moderately complex 3D shapes is not straightforward. Over the last two decades, several works dedicated to 3D growth have been published. In early work by Hans Koops and collaborators, photonic crystals [@Koops2001_FEBID_photonic_crystals] and field emitters and electron optics structures were addressed [@Kretz1994_field_emitter_structures; @Koops1995_Pt_tips_FEBID; @Floreani2001_field_emitters_FEBID]. Following this, suspended FEBID structures were demonstrated, see e.g. [@Gazzadi2007_suspended_FEBID_structures], and plasmonic structures came more into focus [@Hoeflich2011_plasmonics_Au_3D; @Esposito2015_helix_FEBID_plasmonics], as well as simple 3D structures, such as nano-pillars, with application in nanomagnetism [@Pacheco2013_magnetics_Co_3D_spirals; @Cordoba2016_Fe_Co_3D_nanopillars; @Navarro2017_Co_3D_pillars]. Recently a nanospray liquid precursor FEBID process with strongly enhanced growth rates has been demonstrated by Andrei Fedorov’s group [@Fisher2015_nanospray_liquid_precursor_3d_FEBID]. By this process nominally pure Ag pillar structures have been fabricated. Future work will have to show whether sub-micron sized deposits are feasible.
The advent of simulation-assisted 3D growth goes back to very recent joint work of the groups of Harald Plank and Philipp Rack with leading part in the simulations by Jason Fowlkes [@Fowlkes2016_febid_3D_simulation]. This work was not yet aimed at functional 3D FEBID structures, but very soon several follow up works focused on 3D pure metal deposits by either a laser-assisted approach [@Lewis2017_purification_3D] or post-growth purification [@Winkler2017_3D_plasmonic]. Complex nanomagnetic 3D structures have also very recently been fabricated [@Keller2017_magnetic_3D].
Simulating 3D growth
--------------------
How to simulate the FEBID process at several levels of complexity? It must be stressed at the beginning that it is still a far way to go with regard to a full simulation of all sub-process which are relevant in FEBID, as this represents a multi-scale problem on both, time (femtoseconds to seconds) and length scales (fraction of nanometers to many micrometers).
On the *ab-initio* level several works have recently focused on the adsorption and stability of selected precursor materials on amorphous SiO$_2$ using density functional theory (DFT) [@Muthukumar2014_adsorption_several_dft]. Si/SiO$_2$ is by far the most often used substrate material. One important example is therefore the intrinsic instability of the precursor Co$_2$(CO)$_8$ under adsorption onto SiO$_2$ with non-hydroxylated dangling bonds [@Muthukumar2012_dissociation_co2co8]. This has been experimentally observed and also holds relevance for Fe(CO)$_5$, a precursor which is frequently used for the deposition of magnetic nanostructures with high metal content [@Muthukumar2012_dissociation_co2co8; @Vollnhals2013_ebisa_Fe]. A DFT approach was also used in a theoretical electronic structure investigation in the W-C-O phase field [@Muthukumar2012_tungsten_mit_uspex] with respect to the metal-insulator transition found in FEBID material obtained from the precursor W(CO)$_6$ [@Huth2009_WCO_deposits_MIT]. Very recent work has used a novel molecular dynamics approach to describe electron irradiation-driven transformations of molecular structures (IDMD – irradiation driven molecular dynamics) with relevance for FEBID, in particular focusing on W(CO)$_6$ [@Sushko2016_MD_simulation_WCO6]. One has to keep in mind that molecular dynamics of FEBID processes is probably feasible into the few microseconds time range, which is by far not sufficient to describe 3D growth. This is why other approaches to FEBID simulation do not address the microscopic details of the electron-induced chemical transformation processes, but rather tackle the problem of how to predict FEBID growth rates and shape evolution of FEBID structures using an effective theory. Such an effective approach requires to combine the spatial and energy distribution of the primary, backscattered and, especially, secondary electrons with the precursor coverage on the growth front. The interaction of an electron beam with a solid can be efficiently modeled by the Monte Carlo (MC) method, as has been recognized early on [@Joy1991_monte_carlo_sem]. From this the required spatial and energy distribution of the secondary electrons can be derived as input for a reaction-diffusion equation describing the spatial and temporal evolution of the precursor density on the growth front of the developing deposit. This latter part, commonly known as the FEBIP continuum model, has recently been reviewed by Milos Toth and colleagues for the 2D case [@Toth2015_review_cont_models] and is able to simulate both, deposition and electron-induced etching, can handle complex adsorption processes [@Bishop2012_activated_chemisorption], and can deal with etch processes that proceed through multiple reaction pathways with several reaction products present at the substrate surface. The combination of MC simulations of the electron distribution for a Gaussian beam shape with the reaction-diffusion equation has been pioneered by the Rack group [@Smith2007_MC_modeling_febid] and was recently refined by Jason Fowlkes and collaborators to the first simulation-guided approach for 3D nano-manufacturing with very impressive examples of experimental realization by the Plank group [@Fowlkes2016_febid_3D_simulation]. This may well prove to be the start signal for future activity towards 3D fabrication of sophisticated, functional nanostructures by FEBID. It is therefore appropriate to briefly summarize the main ideas of this development. For details we refer to the work by Fowlkes and collaborators [@Fowlkes2016_febid_3D_simulation].
![Schematic of FEBID growth process based on MC simulation of the electron trajectory and energy loss and numerical solution of the reaction-diffussion for the precursor density $n_s$. (a) Primary electrons with $20\,$keV (green cone and line) enter the deposit and experience a series of elastic and inelastic scattering events. Shown are three exemplary trajectories, for one of which the energy loss associated with the inelastic processes is shown as color-coded volume voxels. Secondary electrons are generated along the trajectories. If close enough to the deposit or substrate surface, they contribute to the electron flux $\phi$ which controls precursor dissociation. (b) Surface position dependent precursor density $n_s$ (see color code) for a pillar-like deposit created by a Gaussian beam, as indicated. The gas flux direction is shown by the blue cone. The shadowing effect created by the deposit is visible as reduced precursor coverage.[]{data-label="fig_3d_modeling"}](fig_3d_modeling.pdf){width="80.00000%"}
Let us assume that FEBID growth has progressed to a deposit geometry as indicated in Fig.\[fig\_3d\_modeling\](a). The electrons of the primary beam enter the deposit, experience many elastic and inelastic collision events, thereby generating secondary electrons, eventually leave the deposit and re-enter at another position on the deposit or proceed into the substrate. All generated secondary electrons which are within the energy-dependent escape depth to the surface of the deposit or the substrate can induce at a given surface position $(x_s, y_s, z_s)$ precursor dissociation with a rate proportional to the local precursor density (coverage) $n_s(x_s , y_s , z_s, t)$. Here we assume that the maximum coverage is one monolayer and consider $n_s$ to be normalized to the one-monolayer density $n_{ML}$, i.e. $n_s$ takes on values between $0$ and $1$. The dissociation rate is also proportional to the dissociation cross section $\sigma(E)$ which is, in general, energy-dependent and becomes large only for low energies [@Thorman2015_dissociation_case_studies]. For lack of detailed data of the energy dependence for almost all of the precursors used in FEBID, it is customary to replace $\sigma(E)$ by an energy-averaged value $\sigma$. The precursor density is furthermore governed by the interplay of the three elemental processes of precursor adsorption, diffusion and desorption. The secondary electron distribution $\phi(x_s, y_s, z_s, t)$ at all surface positions $(x_s, y_s, z_s)$ obtained from the MC simulation now enters the reaction-diffusion equation for the precursor density which reads $$\frac{\partial n_s}{\partial t} = D\left( \frac{\partial^2 n_s}{\partial x^2} + \frac{\partial^2 n_s}{\partial y^2} \right) - \sigma\phi(x_s, y_s, z_s, t)n_s + \frac{\Phi}{n_{ML}} \left( 1 - n_s \right) - \frac{n_s}{\tau}\,.
\label{eq_reaction_diffusion_febid}$$ $\tau$ is the temperature-dependent average residence time of a precursor molecule. $\Phi(x_s, y_s, z_s, t)$ represents the precursor flux provided by the gas injection system and can – for flat deposit shapes – be estimated from kinetic gas theory for a given precursor pressure at the substrate surface. However, for sufficiently accurate simulations of 3D FEBID a more sophisticated approach is necessary. Experimental studies have demonstrated that the precursor coverage depends sensitively on the direction of the gas flux vector, which is determined by the geometrical arrangement of the gas injector capillary [@Winkler2014_gas_flux_influence]. This has been simulated in good quantitative agreement with experimental results by Friedli and collaborators [@Friedli2009_gas_flux_simulation]. The gas flux distribution $\Phi(x_s, y_s, z_s, t)$ in Eq.\[eq\_reaction\_diffusion\_febid\] therefore has two components. The first component, $\Phi_1(t)$, which may depend on time but not on the surface position, is the result of precursor adsorption governed by the precursor pressure above the surface. The second component, $\Phi_2(x_s, y_s, z_s, t)$ depends on the surface position for two reasons [@Fowlkes2016_febid_3D_simulation]. For one, depending on the orientation of the surface normal at this position with regard to the gas flux vector, the flux will be higher or lower. Secondly, the flux may be reduced by shadowing effects caused by the current 3D deposit shape. This effect is exemplified in Fig.\[fig\_3d\_modeling\](b) that shows the result of the numerical solution of Eq.\[eq\_reaction\_diffusion\_febid\] for a stationary, Gauss-shaped beam and a directed precursor flux component, as indicated. The actual time-evolution of the growth front can directly by obtained from the known precursor density at a given time and position using the following relationship between the local height increase $\Delta h_\perp$ (perpendicular to the local surface) and $n_s(x_s, y_s, z_s, t)$ $$\Delta h_\perp(x_s, y_s, z_s, t) = V_D\sigma\phi(x_s, y_s, z_s, t)n_sn_{ML}\Delta t
\label{eq_relationship_growthrate_precursor_density}$$ within a time step $\Delta t$. Here $V_D$ denotes the volume deposited for each dissociated precursor molecule. In the next sub-section we will provide examples for 3D FEBID for which the simulation-guided approach was applied.
Examples for simulation-guided 3D FEBID growth
----------------------------------------------
By way of introducing a hybrid Monte Carlo / continuum simulation and comparison with experimental FEBID results using the precursor Me$_3$CpMePt(IV), Jason Fowlkes and collaborators established the foundation for the simulation-guided 3D nanofabrication with FEBID [@Fowlkes2016_febid_3D_simulation]. In follow-up work a software tool for 3D computer-aided design was presented, which generates the beam parameters necessary for FEBID by both, simulation and experiment [@Fowlkes2017_upcoming_cad_tool]. First application in functional 3D nano-architectures was demonstrated by Robert Winkler and collaborators in the field of resonant optics [@Winkler2017_3D_plasmonic]. Following direct-write fabrication of 3D structures using the precursor Me$_2$(acac)Au(III), a water-assisted purification process was applied, which left the 3D structures mostly intact, see also section \[sec\_post\_growth\_purification\]. By electron energy loss spectroscopy (EELS) on tetrapod Au structures the plasmonic activity of these structures was clearly demonstrated.
A highly interesting example of correlation microscopy employing 3D FEBID in conjunction with SEM and *in-situ* AFM characterization was recently published by the groups of Harald Plank (FEBID) and Georg Fantner (high-speed AFM) [@Yang2017_3d_FEBID_and_AFM]. In this work the evolution of the mechanical strength during the growth of Pt-based 3D FEBID structures was investigated by employing a slice-by-slice approach in which the sequence of FEBID growth, followed by AFM characterization was repeated many times over.
As the last example for application of the simulation-guided approach we briefly dwell on 3D nanomagnetic structures, a research field of rapidly growing interest [@Pacheco2017_3d_nanomagnetism]. In very recent work Lukas Keller and collaborators demonstrated the direct-write fabrication of freestanding ferromagnetic 3D Co$_3$Fe nano-architectures (precursor HCo$_3$Fe(CO)$_{12}$) with focus on the consequences of frustrated magnetic interactions [@Keller2017_magnetic_3D]. In particular, nano-cube and nano-tree structures were chosen, as these imply magnetic vertex segments in which three or four magnetic edges of roughly cylindrical shape join (see Fig.\[fig\_3d\_nanogmagnetism\](a) and (d)). Individual nano-cube and nano-tree Co$_3$Fe structures were carefully magnetically characterized by employing micro-Hall sensing in conjunction with micromagnetic and macro-spin simulations (see Fig.\[fig\_3d\_nanogmagnetism\](c)). By carefully adapting the writing strategy and with support by a software tool that generates suitable pattern files for FEBID [@Keller2017_pattern_generator], it was possible to write 3D arrays of Co$_3$Fe nano-trees forming a diamond-like lattice which is very interesting for future work on artificial 3D spin-ice systems. As an additional novel aspect in this work, Keller *et al.* demonstrated the combination of ferromagnetic 3D elements with other 3D elements of different chemical composition and intrinsic material properties, namely nano-granular Pt on the vertex segments positions in nano-trees whose edges were made from Co$_3$Fe alloy (see Fig.\[fig\_3d\_nanogmagnetism\](c)).
![(a) $2\times 2$-array of Co$_3$Fe nano-cubes on micro-Hall sensor. (b) Left: Magnetic stray field as measured at $30\,$K by micro-Hall magnetometry of nano-cubes in an applied magnetic field under $\theta=45^\circ$ with regard to the surface normal; see inset in (a). Right: Result of $T=0$ micromagnetic simulation. (c) Top view of 3D array of Co$_3$Fe nano-trees (diamond structure). (d) $2\times 2$-array of Co$_3$Fe nano-trees in which the vertex segment consists of nano-granular Pt, as indicated in the inset. See [@Keller2017_magnetic_3D] for details.[]{data-label="fig_3d_nanogmagnetism"}](fig_3d_nanogmagnetism.pdf){width="95.00000%"}
Challenges and perspectives
---------------------------
Simulation-assisted 3D FEBID growth in conjunction with the developed software tools to generate suitable pattern files will likely boost future activity in the fabrication of 3D nano-architectures for several application fields, such as resonant optics and nanomagnetism, to name just two. This may turn out to be one of the most promising recent developments in FEBID with a view to expanding both, the community of FEBID users as well as the application fields in which FEBID has a chance to bequeath a sustainable impression in the coming years. Nevertheless, there is plenty of room for future work on optimizing the 3D writing strategies, e.g. with regard to shape control of the different structural units that make a 3D element, the reliable fabrication of larger array structures, and the precision with which different FEBID materials can be combined in the 3D heterostructures.
Concluding remarks
==================
Chances have never been so good for FEBID to become the method of choice for nanofabrication in an ever broader range of research and application fields. The development of various processes for clean metal nanofabrication, the combination of FEBID with area-selective atomic layer deposition, as well as the introduction of well-defined protocols for 2D and true 3D nanofabrication of structures for resonant optics and nanomagnetism will strongly increase the awareness about the unique usefulness of this direct-write approach. Nevertheless, significant challenges have to be met in several ongoing FEBID research fields. E.g., a FEBID process still has to be developed that yields a homogeneous superconducting material positioned far on the superconducting side of the superconductor-insulator transition. Also, with regard to sensor applications based on nano-granular materials, long-term stability is an important issue which needs to be addressed.
Based on the present status of the field, possible directions for future development could be 3D FEBID nano-architectures for resonant optics and nanomagnetism, as well as a broadening of the application of FEBID for the fabrication of artificial solids and metamaterials. Due to the extremely high flexibility and resolution capabilities of FEBID, in conjunction with the ever growing reservoir of suitable precursors for a wide range of material properties of the deposits, one can expect many more research and application fields for FEBID to open up in the coming years, many of which are presently not anticipated.
Acknowledgements {#acknowledgements .unnumbered}
================
M. H. acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) under grant No. HU 752/11-1, through the priority program 1928 (Coordination Networks: Building Blocks for Functional Systems) under grant HU 752/12-1, and through the Collaborative Research Centre SFB/TR49 (Condensed Matter Systems with Variable Many-Body Interactions). O. D. acknowledges financial support by the DFG under grant DO 1511/3-1. This work was conducted within the framework of the COST Action CM1301 (CELINA). Part of the work on hybride superconductor/ferromagnet nanostructures was conducted within the framework of the COST Action CA16218 (NANOCOHYBRI).
References {#references .unnumbered}
==========
|
{
"pile_set_name": "ArXiv"
}
|
Theory Group, Laboratoire de Physique Nucléaire et des Hautes Énergies (LPNHE)[^1], CNRS and Université Pierre et Marie Curie, Paris\
[e-mail: `[email protected] `]{}
Instituto de F' isica Téorica, Universidade Estadual Paulista,\
Rua Pamplona, 145, 01405-900, São Paulo, SP, Brazil\
[e-mail: `[email protected]`]{}
**Abstract**
We suggest that the recently discovered charm-strange meson $D_{sJ}^+(2632)$, with unusual properties, could be a cryptoexotic tetraquark baryonium state $cd\bar d\bar s$. We predict other four narrow states, as Regge recurrences of $D^+_{sJ}(2632)$, below the possible baryon-antibaryon thresholds.
The recently discovered charm-strange meson $D^+_{sJ}(2632)$ [@Evdo04] has very intriguing properties : i) it is very narrow ($\Gamma < 17$ MeV); ii) its coupling to the $D^+_s\eta$ channel is much stronger than its coupling to the $D^{0}~K^{+}$ channel.
The above two unusual properties indicate that the $D^+_{sJ}(2632)$ maybe a cryptoexotic tetraquark baryonium state $c d \bar d\bar s$. The reasons are the following:\
- The favoured decays of a baryonium state are the baryon-antibaryon channels (see Fig. 1): $\Lambda_c^+$(2285) $\bar{\Sigma}_0$(1193), $\Sigma_c^0$(2455) $\bar{\Sigma}^+$(1197) and $\Xi_c^0$ (2472) $\bar\Xi^+$(1321). However the $D_{sJ}^+$ (2632) state has a mass well below the threshold of these baryon-antibaryon channels. Therefore its decay in these channels is forbidden.
- For a baryonium state, the meson decay channels are disfavoured as compared with the baryon-antibaryon channels. This last possibility being forbidden by the mass of the $D_{sJ}^+$ state, what remains are the meson channels and the hierarchy of decays is dictated by the quark content. A $cd\bar d\bar s$ state prefers the $D_{s}^+$ $\eta$ channel as compared with the $D^0K^+$ channel (see Fig. 2).
Let us explain in some detail the selection rules governing the decay channels shown in Fig. 1 and Fig. 2. These selection rules were very much discussed in the $70^{\prime}$ s in the framework of the dual topological $1/N_C$ expansion [@Roy04]. The crucial point here is that the gauge invariance of [*QCD*]{} selects the Y-shape of the baryon as compared with the $\Delta$-shape [@Rossi] (see Fig. 3a) and Fig. 3b)). The three quarks of the baryon are joined at a junction point. The scattering is described by duality diagrams which are ordered in terms of the topological $1/N_C$ expansion. The leading terms correspond to the propagation of the junction lines (generated by the propagation of the junction point) from initial to final state (see Fig. 3c) and Fig. 4): this is the reason why the baryonium-baryon-antibaryon coupling is stronger than the baryonium-mesons coupling. This is also the reason why the diagrams of Fig. 2a) and Fig. 2b) are suppressed as compared with the diagram of Fig. 1.
Let us make some terminological precisions. In the $70^{\prime}$ s one did distinguish between *phaneroexotic* multiquark states and *cryptoexotic* states. The phaneroexotic multiquark states (from the Greek *phaneros*= manifest, visible, open to sight) are those states whose quark content can not be confused with the usual $q\bar q$ and $qqq$ content.
The much celebrated $\Theta^+(1540)$ pentaquark ($uudd\bar s$) state [@LEPS] is also a phaneroexotic state: its quantum numbers can not be obtained from a triquark combination.
However, one must remark that evidence for phaneroexotic multiquark state was given 26 years ago from the study of ”forbidden” forward peaks in the t-channel [@Nicol78]. Namely in Ref. [@Nicol78] we presented evidence for phaneroexotic t-channel $uu\bar d\bar d$ (I=2, S=0) exchanges in $pn \to \Delta^-\Delta^{++}$, $\pi^+n\to\pi^-\Delta^{++}$ and $\pi^-p \to \pi^{+} \Delta^{-}$, for $sd\bar u\bar u$ (I=3/2, S=-1) exchanges in the in $\bar pp \to \bar{Y}^{\ast +} Y^{\ast-}$, $\pi^- p \to K^{+} Y^{\ast-}$ and $K^{-}p \to \pi^{+} Y^{\ast-}$ and for $ss\bar u\bar u$, $ss\bar d\bar d$ and $ss\bar u\bar d$ (S=-2) exchanges in $K^-p \to K^+ \Xi^{\ast-}$. In the absence of baryonium exchanges, these peaks must show an energy dependence $s^{-9}-s^{-10}$ while experimentally one observes a behavior $s^{-3.5}-s^{-4.4}$.
The cryptoexotic states (from the Greek *kruptos*=hidden, secret) are those multiquark states whose quantum numbers can be also obtained from the usual $q\bar q$ or $qqq$ states. The discussion in the $70^{\prime}$ s was concentrated on the cryptoexotic tetraquark states $qq\bar q\bar q$.
In the framework of tetraquarks, a special position is played by the baryonium states: for the reason discussed above they can be very narrow, while most of the other tetraquark states are generally very broad. No clear evidence was given till now for a cryptoexotic baryonium state. The $D_{sJ}^{+}$(2632) state is possibly the first clear evidence for such a state. It is possible that the narrow charmed meson seen in the $D_s^+\pi^0$ channel at 2317 GeV [@Babar03] belongs to the same class of states [@Barn03].
If our interpretation is correct, the $D_{sJ}^{+}$(2632) is the first known member of an entire family of very narrow tetraquark baryonium states. By assuming that all these states belong to an exchange-degenerate Regge trajectory with the universal slope $\alpha^{\prime}~\simeq~0.94\pm 0.06$ GeV$^{-2}$ (as indicated by the masses of known mesons and baryons listed in *Review of Particle Physics* [@hagi02], we can compute the masses of these recurrences of the $D_{sJ}^+$(2632) state, in terms of the mass of the $D_{sJ}^+$ and $\alpha^{\prime}$: $$m_{n}^2=\frac{n}{\alpha^{\prime}}+
m^2_{D^+_{SJ}} \text{, with } n=1,2,3 ...,$$ We therefore predict four narrow states below the $\Lambda_c^+\bar\Sigma_0,\ \Sigma_c^0\bar\Sigma^+$ and $\Xi^0_c\bar\Xi^+$ thresholds: $$\begin{aligned}
& m_1 = 2.827\pm 0.011 \text{ GeV}, & m_2 = 3.010\pm 0.022 \text{ GeV},\\ \nonumber
& m_3 = 3.182 \pm 0.031\text{ GeV}, & m_4 = 3.345\pm 0.039 \text{ GeV}. \end{aligned}$$
Let us close making some considerations on a unified terminology in view of the new born spectroscopy, which concerns the sector of narrow high mass resonances involving heavy quarks in their structure. The new spectroscopy requires an unified terminology of old and new states.
In the new sector states of baryon number 1 we can define the *barypolyquarks*. $$q^{3+n}\bar q^n, \text{ with } n=0,1,2, ...$$
For $n=0$, we get the usual $q^3$ baryons, which we propose to rebaptise as *triquarks*. For $n=1$, we get the much celebrated $q^4\bar q$ *pentaquarks*. For $n=2$ we get the $q^5\bar{q}^2$ *heptaquarks*, for $n=3$ the $q^6\bar q^3$ *enneaquarks*, etc.
By annihilating a barypolyquark with the corresponding antibarypolyquark we get, *via* the annihilation process of at least one $q\bar{q}$ pair, the baryon number sector $0$, which we propose to call *mesopolyquarks*, namely the usual $q\bar q$ mesons (which we propose to rebaptise *diaquarks*) and also the already discussed *tetraquarks* $q^2\bar q^2$.
From the requirement of keeping a minimum number of quarks and antiquarks, while still keeping a junction-antijunction pair, we define also the states $$q^{3+n}~\bar{q}^{3+n}, \text{with } n=1,2,3, ...$$
For $n=1$ we get the $q^4\bar q^4$ *octoquarks*, for $n=2$ the $q^5\bar q^5$ *decaquarks*, for $n=3$ the $q^6\bar q^6$ *dodecaquarks*, etc. The octoquarks are obtained through pentaquark-antipentaquark annihilation.
The ”hexaquarks” $q^3\bar q^3$ are absent from this list because they correspond to molecular broad states. The theoretical status of polyquark states other than mesons (diaquarks), baryons (triquarks), tetraquarks and pentaquarks requires intensive theoretical and phenomenological studies.
*Note.* After the completion of this paper, we learnt that L. Maiani et al. [@maiani] made, simultaneously with us, the assumption that $D_{sJ}^+(2632)$ is a $cd\bar d\bar s$ baryonium state.
*Acknowledgments.* One of us (JPBCM) thanks the Brazilian Agency Fundação de Amparo ' a Pesquisa do Estado de São Paulo (FAPESP) for its support and the Theory Group of LPNHE Paris, where this work was performed, for its kind hospitality.
[99]{} SELEX Collaboration: A. V. Evdokimov et al., hep-ex/0406045.
Proceedings of the Meeting on Exotic Resonances, Hiroshima, September 1-2, 1978, edited by I. Endo, Y. Sumi, S. Wakaizumi and M. Yonezawa; Proceedings of the International Workshop on Baryonium and Other Unusual Hadron States, IPN Orsay, France, June 21-22, 1979, edited by B. Nicolescu, R. Vinh Mau and J.-M. Richard; L. Montanet, G. C. Rossi and G. Veneziano, Phys. Rept. **63**, 149 (1980); for a recent review see D. P. Roy, J. Physics **G30**, R113 (2004).
G. C. Rossi and G. Veneziano, Nucl. Phys. [**B123**]{}, 507 (1977).
LEPS Collaboration: T. Nakano et al., Phys. Rev. Lett. **91**, 012002 (2003); CLAS Collaboration: S. Stepanyan et al., Phys. Rev. Lett. **91**, 252001 (2003); SAPHIR Collaboration: J. Barth et al., hep-ex/0307083; DIANA Collaboration: V. V. Barmin et al., Phys. Atom. Nucl. **66**, 1715 (2003); Yad. Fiz. **66**, 1763 (2003).
B. Nicolescu, Nucl. Phys. [**B134**]{}, 495 (1978).
BABAR Collaboration: B. Aubert et al., Phys. Rev. Lett. **90**, 242001 (2003); CLEO Collaboration: D. Besson et al., AIP Conf. Proc. **698**, 497 (2004); BELLE Collaboration, K. Abe et al., hep-ex/0307041.
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*Review of Particle Physics*, K. Hagiwara et al., **66**, 010001 (2002).
L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, hep-ph/0407025.
Figure captions
**Fig. 1.** The decay of the $D_{sJ}^+(2632)$ meson, as a baryonium state, in the baryon-antibaryon channel.
**Fig. 2.**
- The coupling of $cd\bar d\bar s$ to $D_s^+\eta$.
- The coupling of $cd\bar d\bar s$ to $D^0K^+$.
**Fig. 3.**
- Baryon as a $\Delta$-shape.
- Baryon as a Y-shape.
- Baryon-antibaryon pair leading to a tetraquark baryonium state.
**Fig. 4.** Baryon-antibaryon-baryonium coupling.
(430,180)(0,0) (5,10)(203,10) (35,10)(36,10) (288,50)(289,51) (-8,6)[(0,0)\[br\][[c]{}]{}]{} (-1,-16)[(0,0)\[br\][[d ]{}]{}]{} (-7,-60)[(0,0)\[br\][[$D_{sJ}^{+}~(2632)$]{}]{}]{} (5,-12)(205,-12) (35,-12)(36,-12) (301,30)(302,31) (334,76)[(0,0)\[br\][[c]{}]{}]{} (338,44)[(0,0)\[br\][[d]{} ]{}]{} (5,-34)(200,-34)[10]{} (35,-34)(36,-34) (310,11)(311,12) (281.9999,-35.999)(335,0) (300,-60)(30,125,-125) (281.888,-83.8)(335,-110) (318,-12)(322,-9) (315,-100)(312,-98.5) (350,-8)[(0,0)\[br\][[q]{}]{}]{} (350,-115)[(0,0)\[br\][[$\bar{q}$]{}]{}]{} (201,-130)(5,-130) (200,-81)(5,-81)[10]{} (36,-81)(35,-81) (303,-111)(302,-110) (-1,-108)[(0,0)\[br\]]{} (-1,-135)[(0,0)\[br\]]{} (201,-106)(5,-106) (36,-106)(35,-106) (303,-147)(302,-146) (335,-175)[(0,0)\[br\][[$\bar{d}$]{}]{}]{} (335,-195)[(0,0)\[br\][[$\bar{s}$]{}]{}]{} (201,-130)(5,-130) (36,-130)(35,-130) (303,-175)(302,-174)
Figure 1
(430,180)(0,0) (5,10)(203,10) (35,10)(36,10) (288,50)(289,51) (-4.0,-67)[(0,0)\[br\][[$D_{sJ}^+$ (2632)]{}]{}]{} (-1,6)[(0,0)\[br\][[c]{}]{}]{} (-1,-16)[(0,0)\[br\][[d]{}]{}]{} (5,-12)(150,-12) (35,-12)(36,-12) (340,76)[(0,0)\[br\][[c]{}]{}]{} (365,20)[(0,0)\[br\][[$\bar{s}$]{}]{}]{} (430,45)[(0,0)\[br\][[$D_{s}^{+}$ (1968)]{}]{}]{} (317,-17)(315,-19) (35,-30)(36,-30) (80,-60)(30,-90,90)[10]{} (150,-130)(5,-130) (5,-30)(60,-30)[10]{} (60,-89)(5,-89)[10]{} (36,-89)(35,-89) (150,-106)(5,-106) (-1,-108)[(0,0)\[br\][[$\bar{s}$]{}]{}]{} (-1,-135)[(0,0)\[br\][[$\bar{d}$]{}]{}]{} (430,-110)[(0,0)\[br\][[$\eta$ (547)]{}]{}]{} (36,-106)(35,-106) (355,-137)[(0,0)\[br\][[$\bar{d}$]{}]{}]{} (355,-84)[(0,0)\[br\][[$d$]{}]{}]{} (318,-63.8)(319,-64.8) (340,-130)(5,-130) (36,-130)(35,-130) (310,-130)(309,-130)
a)
(430,180)(0,0) (5,10)(340,10) (35,10)(36,10) (312,10)(314,10) (-1,6)[(0,0)\[br\][[c]{}]{}]{} (-1,-16)[(0,0)\[br\][[d]{}]{}]{} (353,10)[(0,0)\[br\][[c]{}]{}]{} (430,-4)[(0,0)\[br\][[$D^0$ (1864)]{}]{}]{} (-3,-51)[(0,0)\[br\][[$D_{sJ}^+$ (2632)]{}]{}]{} (5,-12)(80,-12) (35,-12)(36,-12) (80,-42)(30,-90,90) (5,-72)(80,-72) (36,-72)(35,-72) (50,-42)(15,-90,90)[5]{} (5,-27)(50,-27)[10]{} (35,-27)(36,-27) (36,-57)(35,-57) (5,-57)(50,-57)[10]{} (280,-42)(30,90,-90) (279,-12)(340,-12) (309,-12)(308,-12) (278,-72)(340,-72) (308,-72)(309,-72) (353,-14)[(0,0)\[br\][[$\bar{u}$]{}]{}]{} (5,-90)(340,-90) (36,-90)(35,-90) (309,-90)(308,-90) (-1,-75)[(0,0)\[br\][[$\bar{d}$]{}]{}]{} (-1,-94)[(0,0)\[br\][[$\bar{s}$]{}]{}]{} (-1,-94)[(0,0)\[br\][[$\bar{s}$]{}]{}]{} (353,-74)[(0,0)\[br\][[$u$]{}]{}]{} (353,-94)[(0,0)\[br\][[$\bar{s}$]{}]{}]{} (435,-92)[(0,0)\[br\][[$K^{+}$ (493) ]{}]{}]{}
b)\
Figure 2
(430,180)(0,0) (75,150)[8]{}[1]{} (30,99)(69.5,145) (35,100)[8]{}[1]{} (43,100)(120,100) (114,100)[8]{}[1]{} (81,145)(111,108) (86,70)[(0,0)\[br\][[a)]{}]{}]{} (235,150)[8]{}[1]{} (240.5,144)(270,120) (303,150)[8]{}[1]{} (297,144)(270,120) (270,120)(270,82.5) (270,82.5)[8]{}[1]{} (280,48)[(0,0)\[br\][[b)]{}]{}]{} (33,0)[8]{}[1]{} (39.5,-4.5)(65,-28) (95,0)[8]{}[1]{} (89,-4)(65,-28) (65,-28)(65,-58) (65,-60)[8]{}[1]{} (65,-90)[8]{}[0]{} (65,-96)(65,-120) (65,-120)(36,-140) (65,-120)(90,-140) (36,-140)[8]{}[0]{} (90,-140)[8]{}[0]{} (225,0)[8]{}[1]{} (230,-7)(260,-40) (292,0)[8]{}[1]{} (285.5,-5)(260,-40) (260,-40)(260,-90) (260,-90)(218,-145) (260,-90)(295,-145) (225,-140)[8]{}[0]{} (292,-140)[8]{}[0]{} (120,-60)(200,-60) (173,-175)[(0,0)\[br\][[c)]{}]{}]{}
Figure 3
(430,180)(0,0) (5,10)(203,10) (35,10)(36,10) (288,50)(289,51) (5,-12)(205,-12) (35,-12)(36,-12) (301,30)(302,31) (5,-34)(200,-34)[10]{} (35,-34)(36,-34) (310,11)(311,12) (281.9999,-35.999)(335,0) (300,-60)(30,125,-125) (281.888,-83.8)(335,-110) (318,-12)(322,-9) (315,-100)(312,-98.5) (201,-130)(5,-130) (200,-81)(5,-81)[10]{} (36,-81)(35,-81) (303,-111)(302,-110) (201,-106)(5,-106) (36,-106)(35,-106) (303,-147)(302,-146) (201,-130)(5,-130) (36,-130)(35,-130) (303,-175)(302,-174)
Figure 4
[^1]: Unité de Recherche des Universités Paris 6 et Paris 7, Associée au CNRS.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, a new mixed finite element scheme using element-wise stabilization is introduced for the biharmonic equation with variable coefficient on Lipschitz polyhedral domains. The proposed scheme doesn’t involve any integration along mesh interfaces. The gradient of the solution is approximated by $H({\rm div})$-conforming $BDM_{k+1}$ element or vector valued Lagrange element with order $k+1$, while the solution is approximated by Lagrange element with order $k+2$ for any $k\geq 0$.This scheme can be easily implemented and produces positive definite linear system. We provide a new discrete $H^{2}$-norm stability, which is useful not only in analysis of this scheme but also in $C^{0}$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^{2}$-norm and $L^{2}$-norm are derived. This scheme with its analysis is further generalized to the von Kármán equations. Finally, numerical results verifying the theoretical estimates of the proposed algorithms are also presented.'
address:
- 'School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Fujian, 361005, China'
- 'Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai-400076, India'
- 'Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China'
author:
- Huangxin Chen
- 'Amiya K. Pani'
- Weifeng Qiu
title: A mixed finite element scheme for biharmonic equation with variable coefficient and von Kármán equations
---
[^1]
Introduction
============
In the first part of this paper, a new mixed finite element scheme is proposed and analyzed for the following biharmonic equation with variable coefficient:
\[equ\_biharmonic\] $$\begin{aligned}
\label{equ_biharmonic1}
\Delta \left( \kappa \Delta u \right) = f, & \ \ \textrm{in} \ \ \Omega, \\
\label{equ_biharmonic_dirichlet}
u = 0, & \ \ \textrm{on} \ \ \partial \Omega,\\
\label{equ_biharmonic2}
\frac{\partial u}{\partial n} = 0, & \ \ \textrm{on} \ \ \partial \Omega,\end{aligned}$$
where $\Omega \subset \mathbb{R}^d (d\in \mathbb{N})$ is a Lipschitz polygonal or polyhedral domain, the coefficient $\kappa \in W^{1,\infty}(\Omega)$ such that $0<\kappa_{0} \leq \kappa(\boldsymbol{x})
\leq \kappa_{1}$, and $f \in H^{-1}(\Omega)$. By using element-wise stabilization, our scheme doesn’t involve any integration along mesh interfaces. Our scheme uses $H(\text{div})$-conforming $BDM_{k+1}$ or vector valued Lagrange element with order $k+1$ to approximate $\bfw = \nabla u$, and approximates $u$ with Lagrange element with order $k+2$ for any $k\geq 0$. The second part of this paper is related to an application of our scheme to the von Kármán model, which can be stated as follows:
\[equ\_von\_karman\] $$\begin{aligned}
\Delta^2 \xi - [\xi,\psi]= f, & \ \ \textrm{in} \ \ \Omega, \label{vk_eq1}\\
\Delta^2 \psi + [\xi,\xi]= 0, & \ \ \textrm{in} \ \ \Omega,\label{vk_eq2} \\
\xi =\frac{\partial \xi}{\partial n} = 0, & \ \ \textrm{on} \ \ \partial \Omega, \label{vk_bc1}\\
\psi = \frac{\partial \psi}{\partial n} = 0, & \ \ \textrm{on} \ \ \partial \Omega,\label{vk_bc2}\end{aligned}$$
where $\Omega \subset \mathbb{R}^2$ is a Lipschitz polygonal domain, $f \in H^{-1}(\Omega)$, and the von Kármán bracket $[\cdot,\cdot]$ appearing in (\[vk\_eq1\]) and (\[vk\_eq2\]) is defined by $$[\eta,\phi] = \frac{\partial^2 \eta}{\partial x_1^2} \frac{\partial^2 \phi}{\partial x_2^2} + \frac{\partial^2 \eta}{\partial
x_2^2} \frac{\partial^2 \phi}{\partial x_1^2} - 2 \frac{\partial^2 \eta}{\partial x_1 \partial x_2}
\frac{\partial^2 \phi}{\partial x_1 \partial x_2} = {\rm cof}(D^2 \eta): D^2 \phi.$$ Here ${\rm cof}(D^2 \eta)$ denotes the cofactor matrix of the Hessian of $\eta$ and $A : B$ denotes the Frobenius inner product of the matrices $A$ and $B$.
In literature, there are many numerical methods available for the biharmonic equation, that is, the problem (\[equ\_biharmonic\]) with $\kappa = 1$. Some of them can be easily generalized to include biharmonic problem with variable coefficients. We provide below a brief summary of results which are relevant to our present investigation.
- [*Numerical methods approximating both $u$ and $\Delta u$*]{}. The Ciarlet and Raviart (C-R) method [@CRmethod] uses $u$ and $\Delta u$ as unknowns and thereby, gives rise to a system of Poisson problems. Then, $H^{1}$-conforming finite element spaces are used to approximate both $u$ and $\Delta u$, and it has no stabilization along mesh interfaces. Thus, the C-R method can be easily implemented. The stability of the C-R method with respect to discrete $H^{2}$-norm is shown in [@BabushkaCR]. However, the analysis in [@BabushkaCR] requires that the domain is convex (see, the proof of [@BabushkaCR Lemma $5$]). The optimal convergence to $u$ of the C-R method, which is obtained in [@Scholtz1978] though the convergence to $\Delta u$ is suboptimal. Though the corresponding linear system of the C-R method is a saddle point one, its conditional number may be of order $O(h^{-2})$ if numerical approximations to $u$ and $\Delta u$ are calculated alternatively. Optimal convergence to $\Delta u$ is obtained by the method [@Falk1978] which is similar to the C-R method. But the analysis in [@Falk1978] doesn’t provide the stability with respect to discrete $H^{2}$-norm. For $hp$-mixed DG method with penalization of the interelement boundary jump terms applied to the split system, see [@GNP2008].By approximating $\kappa \Delta u$ instead of $\Delta u$, all these methods can be easily generalized for variable coefficient $\kappa$.
- [*Numerical methods approximating both $u$ and $D^{2}u$*]{}. The Hellan-Herrmann-Johnson (HHJ) method analyzed by Johnson in [@Johnson1973] treats $u$ and $D^{2}u$ as unknowns. It uses $H^{1}$-conforming approximations to $u$ and normal-normal continuous symmetric approximations to $D^{2}u$. Optimal convergence to both $u$ and $D^{2}u$ is shown in [@Johnson1973]. Since it naturally provides the stability with respect to discrete $H^{2}$-norm, the HHJ method and its variants are suitable for solving the von Kármán model (see [@Miyoshi1976; @Brezzi1981; @Reinhart1982]). We notice that the method [@BehrensGuzman] uses the same formulation by the HHJ method, but writes the biharmonic equation as four first-order equations instead of two second-order equations. The method [@BehrensGuzman] obtains optimal convergence to $u$, $\nabla u$ and $D^{2}u$, and its global unknowns after hybridization are numerical approximations to the trace of $u$ and $\nabla u$ along the mesh interfaces. Overall, the analysis of the HHJ method and its variant (including the method [@BehrensGuzman]) can be generalized to the von Kármán equations (\[equ\_von\_karman\]). But the implementation may not be easy and the corresponding linear system (without hybridization) is a saddle point system with a large number of degrees of freedom. Furthermore, since all these methods utilize the following identity $$\begin{aligned}
\label{biharmonic_identity}
\int_{\Omega} \Big(\frac{\partial^{2}u}{\partial x_{1}^{2}} \frac{\partial^{2} v}{\partial x_{2}^{2}}
+ \frac{\partial^{2}u}{\partial x_{2}^{2}} \frac{\partial^{2} v}{\partial x_{1}^{2}}
- 2 \frac{\partial^{2}u}{\partial x_{1}\partial x_{2}} \frac{\partial^{2} v}{\partial
x_{1}\partial x_{2}}\Big)\; d\bfx = 0
\qquad \forall u, v \in H_{0}^{2}(\Omega),\end{aligned}$$ it may not be straightforward to generalize these methods for problems with a non-constant coefficient $\kappa$. One way is to split the biharmonic term $\Delta (\kappa \Delta u)$ as $$\begin{aligned}
\label{operator_split1}
\Delta (\kappa \Delta u) = \underline{\kappa} \Delta^{2} u + \Delta ((\kappa - \underline{\kappa})\Delta u), \end{aligned}$$ where $\underline{\kappa}$ is a positive constant chosen to satisfy $$\begin{aligned}
\underline{\kappa} \leq \inf_{\bfx \in \Omega} \kappa (\bfx). \end{aligned}$$ Applying (\[biharmonic\_identity\]) to the first term on the right hand side of (\[operator\_split1\]), it is easy to see that the solution $u$ of the biharmonic equation (\[equ\_biharmonic\]) satisfies $$\begin{aligned}
\label{biharmonic_identity_var}
\dfrac{\underline{\kappa}}{2} (D^{2}u, D^{2}v)_{\Omega} + ( (\kappa - \underline{\kappa})\Delta u, \Delta v)_{\Omega}
= (f ,v)_{\Omega},\qquad \forall v \in H_{0}^{2}(\Omega).\end{aligned}$$ Based on the variational formula (\[biharmonic\_identity\_var\]), all HHJ type methods can be generalized for variable coefficient $\kappa$. However, the value of $\underline{\kappa}$ may affect the stability of HHJ type methods, if $\underline{\kappa}$ is chosen to be much smaller than $\inf_{\bfx \in \Omega} \kappa (\bfx)$ which may not be easy to discover in practice.
- [*Numerical methods approximating $u$ only*]{}. There are several sub-classes of numerical methods approximating $u$ only. These methods can produce symmetric and positive definite linear system but with condition number of order $O(h^{-4})$ (the same as our mixed finite element scheme). One class uses $C^{1}$-conforming finite element spaces, which are naturally suitable for the biharmonic equation [@Argyris1968; @Bogner1965; @Douglas1979] and the von Kármán equations [@Brezzi1978; @Miyoshi1976]. The main drawback of $C^{1}$-conforming elements is the difficulty of implementation, especially in high dimensional domains with high polynomial orders. In order to simplify the implementation of $C^{1}$-conforming elements, several kinds of non $C^{1}$-conforming numerical methods have been developed and analyzed. These include Morley element methods [@Morley1968; @Wang06], $C^{0}$- interior penalty method [@Brenner2005; @GGN2013], and DG methods [@Engel02; @Mozolevski03]. All of these methods can be applied for the von Kármán equations (\[equ\_von\_karman\]) (see [@Brenner2017; @Carstensen19; @Mallik16]). Morley element methods are very popular since these schemes use only six degrees of freedom on each element in two dimensional domains and don’t need any stabilization along mesh interfaces. However, it is not straightforward to use Morley element for equations including both the biharmonic and Laplacian operator due to the fact that it is not $H^{1}$-conforming on the whole mesh. Compared with $C^{0}$ interior penalty method and DG methods, our methods might be understood and implemented relatively easier by beginners, since there is no stabilization along mesh interfaces. All these methods can be easily modified for variable coefficient $\kappa$.
In [@SZhang2018], the biharmonic equation (\[equ\_biharmonic\]) is deduced to an equivalent system on three low-regularity spaces which are connected by a regular decomposition corresponding to a decomposition of the regularity of the high order space. A numerical method based on the equivalent system of (\[equ\_biharmonic\]) is presented in [@SZhang2018], which can approximate solutions with low regularity well. But it may not be straightforward for beginners to understand and implement the method in [@SZhang2018].
Our proposed mixed finite element scheme for the biharmonic equation (\[equ\_biharmonic\]) has the following properties.
- By using element-wise stabilization, our scheme doesn’t involve any integration along mesh interfaces. Our scheme uses $H(\text{div})$-conforming $BDM_{k+1}$ or vector valued Lagrange element with order $k+1$ to approximate $\bfw = \nabla u$, and approximates $u$ with Lagrange element with order $k+2$ for any $k\geq 0$. Thus even beginners can relatively easily implement our scheme.
- The corresponding linear system is positive definite. In fact, one method of our scheme produces symmetric and positive definite linear system.
- Our scheme can be used in arbitrary Lipschitz polyhedral domains and can approximate accurately solutions in $H^{2+\delta}(\Omega)$, where $\delta > \frac{1}{2}$. In fact, the numerical solution $u_{h}$ approximates the exact solution $u$ optimally in discrete $H^{2}$ norm and $L^{2}$ norm. We refer to Theorem \[main\_err1\] and Theorem \[thm\_L2\_con\_biharmonic\] for detailed description.
- The new method and its analysis can be generalized to nonlinear problem such as the von Kármán equations (\[equ\_von\_karman\]). Our method (\[sta\_method\_vk\]) for the von Kármán equations doesn’t involve with any integration along mesh interfaces either. We refer to Theorem \[thm\_sta\_vk\] and Theorem \[thm\_conv\_von\_karman\] for the detailed description on the existence and uniqueness of the numerical solution to the von Kármán equations and the optimal convergence. For the sake of simplicity, our analysis for the von Kármán equations is based on the assumption that $\Vert f\Vert_{H^{-1}(\Omega)}$ is small enough. The success of generalization to the von Kármán equations is because the numerical solution $u_{h}$ of our scheme for the biharmonic equation satisfies the stability result: $$\begin{aligned}
\Vert u_{h}\Vert_{H^{1}(\Omega)} +\Vert u_{h}\Vert_{2,\calT_{h}} \leq C \Vert f\Vert_{H^{-1}(\Omega)}.\end{aligned}$$ Here $\Vert \cdot \Vert_{2, \calT_{h}}$ is the discrete $H^{2}$- semi norm defined in (\[discrete\_h2\]). With the above stability result, we can extend our analysis to isolated solutions of the von Kármán equations like existing works [@Brenner2017; @Brezzi1978; @Brezzi1981; @Carstensen19; @Mallik16; @Miyoshi1976; @Reinhart1982].
In this paper, we provide a discrete $H^{2}$-norm stability (\[discrete\_embedding\_ineq2\]) in Theorem \[thm\_discrete\_embedding\]: $$\begin{aligned}
\Vert v \Vert_{H^{1}(\Omega)}^{2} + \Vert v\Vert_{2, \calT_{h}}^{2}
\leq C \left( \Vert \Delta v \Vert_{\calT_{h}}^{2} + \Sigma_{F\in \calE_{h}} h_{F}^{-1} \Vert \llbracket
\nabla v \cdot \bfn \rrbracket \Vert_{0,F}^{2} \right),
\quad \forall v \in V_{h}:= H_{0}^{1}(\Omega) \cap P_{k+2}(\calT_{h}) \text{ } (k\geq 0), \end{aligned}$$ which looks similar to the discrete Miranda–Talenti inequality [@Neilan2019a ($1.3$)] (they have different boundary conditions). In Remark \[remark\_key\_inequality\_alternative\], we explain the fundamental difference between the proof of above inequality and [@Neilan2019a ($1.3$)]. This inequality can help in analysis of the $C^{0}$ interior penalty method: to find $u_{h} \in V_{h}$ such that for any $v\in V_{h}$, $$\begin{aligned}
\label{CIP1}
& (\kappa \Delta u_{h}, \Delta v)_{\calT_{h}} + \langle {\{\hspace{-4.0pt}\{}
\newcommand{\Rb}{\}\hspace{-4.0pt}\}}\kappa \Delta u_{h} \Rb,
\llbracket \nabla v \cdot \bfn \rrbracket \rangle_{\calE_{h}}
+ \langle {\{\hspace{-4.0pt}\{}
\newcommand{\Rb}{\}\hspace{-4.0pt}\}}\kappa \Delta v \Rb,
\llbracket \nabla u_{h} \cdot \bfn \rrbracket \rangle_{\calE_{h}} \\
\nonumber
& \qquad \qquad + \tau h^{-1}\langle \kappa \llbracket \nabla u_{h} \cdot \bfn \rrbracket,
\llbracket \nabla v \cdot \bfn \rrbracket \rangle_{\calE_{h}} = (f, v)_{\calT_{h}}. \end{aligned}$$ Here, ${\{\hspace{-4.0pt}\{}
\newcommand{\Rb}{\}\hspace{-4.0pt}\}}\cdot \Rb$ and $\llbracket \cdot \rrbracket$ represent, respectively, the average and jump across the inter-element boundaries. Obviously, (\[CIP1\]) is a natural generalization of the $C^{0}$ interior penalty method in [@Brenner2005]. In addition, another discrete $H^{2}$-norm stability (\[discrete\_embedding\_ineq3\]) in Theorem \[thm\_discrete\_embedding\]: $$\begin{aligned}
\Vert \nabla \tilde{v}_{h} \Vert_{\calT_{h}}^{2} + \Vert \tilde{v}_{h}\Vert_{2, \calT_{h}}^{2}
\leq C \left( \Vert \Delta \tilde{v}_{h} \Vert_{\calT_{h}}^{2}
+ \Sigma_{F\in \calE_{h}} \big( h_{F}^{-1} \Vert \llbracket \nabla \tilde{v}_h \cdot \bfn \rrbracket \Vert_{0,F}^{2}
+ h_{F}^{-3} \Vert \llbracket \tilde{v}_h \rrbracket \Vert_{0,F}^{2} \big) \right),
\forall \tilde{v}_{h} \in P_{k+2}(\calT_{h}), \end{aligned}$$ will help to prove discrete $H^{2}$-norm stability of the DG methods in [@Mozolevski03].
We conclude this section with section wise description. Section 2 deals with our new mixed type formulation. In section 3, stability estimates are proved and [*a priori*]{} error estimates are established. As an application, the section 4 focuses on the von Kármán model and related error analysis using a generalization of the proposed method. In Section \[sec:numer\_example\], we give some numerical results to verify the efficiency of the proposed new schemes. Finally we provide a conclusion in Section \[sec:conclusion\].
A new finite element method for the biharmonic equation {#sec_method_biharmonic}
=======================================================
This section deal with the formulation of our new mixed finite element scheme for the biharmonic equation (\[equ\_biharmonic\]). At the end of this section, we introduce the idea how to derive our scheme.
Let $\calT_h$ be the conforming triangulation of $\Omega$ made of shape-regular simplicial elements. We denote by $\calE_h$ the set of all faces $F$ of all elements $K\in \calT_h$, $\calE^0_h$ the set of interior faces of $\calT_h$, $ \calE^{\partial}_h$ the set of all faces $F$ on the boundary $\partial \Omega$, and set $\partial \calT_h:=\{\partial K:K\in \calT_h\}$. For scalar-valued functions $\phi$ and $\psi$, we write $$(\phi,\psi)_{\calT_h}:=\sum_{K\in\calT_h}(\phi,\psi)_K, \, \, \langle\phi,\psi\rangle_{\partial\calT_h}
:=\sum_{K\in\calT_h}\langle\phi,\psi \rangle_{\partial K}.$$ Here $(\cdot,\cdot)_D$ denotes the integral over the domain $D\subset \Real^d$, and $\langle \cdot,\cdot \rangle_D$ denotes the integral over $D\subset \Real^{d-1}$. When $D=\Omega$, we denote $(\cdot,\cdot) := (\cdot,\cdot)_\Omega$. For vector-valued functions, we write $(\bfphi,\bfpsi)_{\calT_h}:=\sum_{i=1}^{d} (\phi_i,\psi_i)_{\calT_h}$. We denote by $h_K$ the diameter of element $K \in \calT_h$ and set $h = \max_{K \in \calT_h}h_K$. For any face $F \in \calE_h$, $h_F$ stands for the diameter of $F$. For any interior face $F= \partial K \cap \partial K'$ in $\calE^0_h$, we denote by $\llbracket \psi \rrbracket
= (\psi|_K)|_F - (\psi|_{K'})|_F $ the jump of a scalar function $\psi$ across $F$, and $\llbracket \boldsymbol{\phi} \rrbracket = (\boldsymbol{\phi} _K \cdot \bfn_K)|_F -
(\boldsymbol{\phi}_{K'}\cdot \bfn_K)|_F $ the jump of a vector-valued function $\boldsymbol{\phi} $ across $F$. On a boundary face $F = \partial K \cap \partial \Omega$, we set $\llbracket \psi \rrbracket = \psi$ and $\llbracket \boldsymbol{\phi} \rrbracket = \boldsymbol{\phi}_K\cdot \bfn_K $.
Throughout the paper, we use the standard notations and definitions for Sobolev spaces (see, e.g. [@Adams1975]). To be more precise, let $\|\cdot\|_{s,D}$ be the usual norm on the Sobolev space $H^s(D)$ and $\|\cdot\|_{L^p(D)}$ be the $L^p$-norm on $L^p(D)$. If $p=2$, we let $\|\cdot\|_{0,D}$ denote the $L^2$-norm on $L^2(D)$. Further, let $H_{0}^{1}(\Omega):= \{v\in H^{1}(\Omega): v|_{\partial\Omega} = 0\}$, $H({\rm div},\Omega) := \{ \bfw \in [L^{2}(\Omega)]^{d}: \nabla\cdot \bfw \in L^{2}(\Omega) \}$ and $H_{0}({\rm div},\Omega) := \{ \bfw \in H({\rm div},\Omega): \bfw\cdot \bfn |_{\partial\Omega} = 0\}$.
The norm $\|\cdot\|_{\calT_h}$ is the discrete norm defined as $\|\cdot\|_{\calT_h}:= (\sum_{K \in \calT_h}
\|\cdot\|^2_{L^{2}(K)})^{\frac{1}{2}}$. We also define the discrete norm $\|\cdot\|_{ \calE_h}:= (\sum_{F \in
\calE_h} \|\cdot\|^2_{0,F})^{\frac{1}{2}}$. For any set $\calF_h \subseteq \partial \calT_h$, we denote $\|h^\alpha \cdot\|_{\calF_h} := ( \sum_{F \in \calF_h}h^{2\alpha}_F \|
\cdot \|^2_{0,F})^{\frac{1}{2}}$ with optional parameter $\alpha$.
We also define a semi-norm $\Vert \cdot \Vert_{2,\calT_{h}}$ on $H^{2}(\calT_{h})$ as $$\begin{aligned}
\label{discrete_h2}
\Vert v \Vert_{2,\calT_{h}}^{2} = \Vert D^{2} v \Vert_{\calT_{h}}^{2}
+ \Sigma_{F \in \calE_{h}} h_{F}^{-1}\Vert \llbracket \nabla v\cdot \bfn \rrbracket \Vert_{0,F}^{2},
\qquad \forall v \in H^{2}(\calT_{h}).\end{aligned}$$
In this paper, $C$ denotes a positive constant depending only on the property of $\Omega$, the shape regularity of the meshes and the degree of polynomial spaces. The constant $C$ can take on different values in different occurrences.
In the following we present the detailed formulation of the mixed finite element scheme for (\[equ\_biharmonic\]). For any $k\geq 0$, we define the finite element spaces $$\begin{aligned}
\label{FEM_spaces}
\bfW_h := H_0({\rm div},\Omega) \cap \bfP_{k+1}(\calT_h),\quad V_h := H^1_0(\Omega) \cap P_{k+2}(\calT_h),\end{aligned}$$ where $P_k(D)$ denotes the set of polynomials of total degree at most $k$ defined on $D$, $\bfP_k(D)$ denotes the set of vector-valued functions whose $d$ components lie in $P_k(D)$. We would like point out that an alternative choice of $\bfW_{h}$ is $$\begin{aligned}
\label{Wh_alternative}
\bfW_h := [H_{0}^{1}(\Omega) \cap P_{k+1}(\calT_h)]^{d}.\end{aligned}$$ In Remark \[remark\_Wh\], we explain why we can use $\bfW_{h}$ in (\[Wh\_alternative\]) and what is its drawback. In this paper, our scheme focuses on the finite element spaces (\[FEM\_spaces\]).
Our mixed finite element scheme is to seek an approximation $(\bfw_h,u_h)\in \bfW_h \times V_h$ such that for $\theta \in \{-1, 1\}$, $$\begin{aligned}
\label{sta_fem_biharmonic}
B_{\theta}((\bfw_h,u_h),(\bfeta,v)) =(f,v)_{\langle H^{-1}(\Omega), H_{0}^{1}(\Omega) \rangle}
\;\;\;\forall (\bfeta,v)\in \bfW_h \times V_h,\end{aligned}$$ where $$\begin{aligned}
\label{def_bilinear}
B_{\theta}((\bfw_h,u_h),(\bfeta,v)) :=&
\left(\kappa \nabla \cdot \bfw_h, \nabla \cdot \bfeta \right)_{\calT_h}
+\left(\nabla(\kappa \nabla \cdot \bfw_h), \bfeta - \nabla v \right)_{\calT_h} \\
& +\theta \left( \bfw_h-\nabla u_h, \nabla(\kappa \nabla \cdot \bfeta) \right)_{\calT_h}
+ \frac{\tau}{h^2}\left( \kappa (\bfw_h-\nabla u_h), \bfeta - \nabla v \right)_{\calT_h}.{\nonumber}\end{aligned}$$ Here, $\tau$ is a stabilization parameter to be determined later.
\[remark\_Wh\] Notice that the boundary conditions (\[equ\_biharmonic\_dirichlet\], \[equ\_biharmonic2\]) imply that $\nabla u = \boldsymbol{0}$ on $\partial\Omega$. Thus the alternative $\bfW_{h}$ in (\[Wh\_alternative\]) is also a reasonable choice of finite element space to approximate $\nabla u$. It can be easily shown that all main results (Theorem \[thm\_discrete\_embedding\], Theorem \[thm\_main\_sta\], Theorem \[main\_err1\], Theorem \[thm\_L2\_con\_biharmonic\], Theorem \[thm\_sta\_vk\], Theorem \[thm\_conv\_von\_karman\]) will still hold. The only drawback of using the alternative $\bfW_{h}$ in (\[Wh\_alternative\]) is that when the Dirichlet boundary conditions (\[equ\_biharmonic\_dirichlet\], \[equ\_biharmonic2\]) are not homogeneous, extra calculation is needed to obtain the value of $\nabla u$ on $\partial\Omega$. On the contrast, the original choice of $\bfW_h := H_0({\rm div},\Omega) \cap \bfP_{k+1}(\calT_h)$ can handle nonhomogeneous Dirichlet boundary data directly.
In the following, we always assume that $$\begin{aligned}
\bfw = \nabla u \in \bfH^{1+\delta}(\Omega), \quad \nabla \cdot \bfw = \Delta u
\in H^{\delta}(\Omega), \quad \delta > 1/2.
\label{reg_ass}\end{aligned}$$
Derivation of the finite element scheme (\[sta\_fem\_biharmonic\])
------------------------------------------------------------------
We introduce the idea how to derive the finite element scheme (\[sta\_fem\_biharmonic\]) in the following. We temporarily assume the exact solution $u$ of (\[equ\_biharmonic\]) and $\kappa$ are smooth.
For any $v \in V_{h}$, we have $$\begin{aligned}
(\Delta (\kappa \Delta u), v)_{\Omega} = (f, v)_{\Omega}. \end{aligned}$$ Doing integration by parts, we obtain $$\begin{aligned}
- (\nabla (\kappa \Delta u), \nabla v)_{\Omega} = (f, v)_{\Omega},\end{aligned}$$ since $v = 0$ on $\partial\Omega$. Obviously, if we want to do integration by parts one more time, we will encounter terms along mesh interfaces since $\nabla v$ is not $H(\text{div})$-conforming. We take $\bfeta \in \bfW_h$ arbitrarily. We would like to “replace" $\nabla v$ by $\bfeta$ such that integration by parts can be done. So we have $$\begin{aligned}
- (\nabla (\kappa \Delta u), \bfeta)_{\Omega}- (\nabla (\kappa \Delta u), \nabla v - \bfeta)_{\Omega}
= (f, v)_{\Omega}.\end{aligned}$$ Now we can do integration by parts to the first term in the left hand side of above equation. Thus we have $$\begin{aligned}
(\kappa \Delta u, \nabla \cdot \bfeta)_{\Omega} - (\nabla (\kappa \Delta u), \nabla v - \bfeta)_{\Omega}
= (f, v)_{\Omega}.\end{aligned}$$ The above equation inspire us to use $(\bfw_{h}, u_{h}) \in \bfW_{h} \times V_{h}$ to approximate $(\nabla u, u)$: $$\begin{aligned}
(\kappa\nabla\cdot \bfw_{h}, \nabla \cdot \bfeta)_{\Omega} - (\nabla (\kappa \nabla \cdot \bfw_{h}),
\nabla v - \bfeta)_{\calT_{h}} = (f, v)_{\Omega}, \quad \forall (\bfeta, v) \in \bfW_{h} \times V_{h}.\end{aligned}$$ In order to have well-posedness, we need $\bfw_{h}$ and $\nabla u_{h}$ are “close" enough to each other. So we need to add stabilization into the above equation to have: $$\begin{aligned}
(\kappa\nabla\cdot \bfw_{h}, \nabla \cdot \bfeta)_{\Omega} +(\nabla ( \kappa \nabla \cdot \bfw_{h}),
\bfeta - \nabla v)_{\calT_{h}} + \frac{\tau}{h^2}\left( \kappa (\bfw_h-\nabla u_h), \bfeta - \nabla v \right)_{\calT_h}
= (f, v)_{\Omega}, \quad \forall (\bfeta, v) \in \bfW_{h} \times V_{h}.\end{aligned}$$ Here $\tau$ is a positive constant. In order to have a symmetric or anti-symmetric method, we add the term $\theta \left( \bfw_h-\nabla u_h, \nabla(\kappa\nabla \cdot \bfeta) \right)_{\calT_h}$ to the above equation to get the finite element scheme (\[sta\_fem\_biharmonic\]).
Analysis of the mixed finite element scheme for the biharmonic equation
=======================================================================
In this section, we discuss the stability and error estimates of the mixed finite element scheme (\[sta\_fem\_biharmonic\]) for the biharmonic equation (\[equ\_biharmonic\]).
Stability estimate
------------------
\[thm\_discrete\_embedding\] There exists a positive constant $C$ such that for any $(\bfeta_{h}, v_{h}) \in \bfW_{h}\times V_{h}$,
\[discrete\_embedding\_ineqs\] $$\begin{aligned}
\label{discrete_embedding_ineq1}
& \Vert v_{h} \Vert_{H^{1}(\Omega)}^{2} + \Vert v_{h}\Vert_{2, \calT_{h}}^{2}
\leq C \left( \Vert \nabla \cdot \bfeta_{h} \Vert_{\calT_{h}}^{2}
+ \Sigma_{K \in \calT_{h}} h_{K}^{-2} \Vert \bfeta_{h} - \nabla v_{h} \Vert_{L^{2}(K)}^{2} \right),
\quad \forall (\bfeta_{h}, v_{h}) \in \bfW_{h}\times V_{h}; \\
\label{discrete_embedding_ineq2}
& \Vert v_{h} \Vert_{H^{1}(\Omega)}^{2} + \Vert v_{h}\Vert_{2, \calT_{h}}^{2}
\leq C \left( \Vert \Delta v_{h} \Vert_{\calT_{h}}^{2}
+ \Sigma_{F\in \calE_{h}} h_{F}^{-1} \Vert \llbracket \nabla v_h \cdot \bfn \rrbracket \Vert_{0,F}^{2} \right),
\quad \forall v_{h} \in V_{h}; \\
\label{discrete_embedding_ineq3}
& \Vert \nabla \tilde{v}_{h} \Vert_{\calT_{h}}^{2} + \Vert \tilde{v}_{h}\Vert_{2, \calT_{h}}^{2}
\leq C \left( \Vert \Delta \tilde{v}_{h} \Vert_{\calT_{h}}^{2}
+ \Sigma_{F\in \calE_{h}} \big( h_{F}^{-1} \Vert \llbracket \nabla \tilde{v}_h \cdot \bfn \rrbracket \Vert_{0,F}^{2}
+ h_{F}^{-3} \Vert \llbracket \tilde{v}_h \rrbracket \Vert_{0,F}^{2} \big) \right),
\forall \tilde{v}_{h} \in P_{k+2}(\calT_{h}).\end{aligned}$$
\[remark\_key\_inequality\_alternative\] Though (\[discrete\_embedding\_ineq2\]) looks similar to the discrete Miranda–Talenti inequality [@Neilan2019a ($1.3$)] (they have different boundary conditions), their proofs are fundamentally different from ours. An enriching operator [@Neilan2019a Lemma $3$] for $H^{2}$-conforming Clough–Tocher elements is used to obtain [@Neilan2019a ($1.3$)]. Since there is no Clough–Tocher elements with polynomial order greater than $3$ in three or higher dimensional domains, [@Neilan2019a ($1.3$)] is valid for polynomial order less than $3$ in three dimensional domains. If we mimic the methodology in [@Neilan2019a], our result (\[discrete\_embedding\_ineq2\]) should have the same restriction on polynomial order as [@Neilan2019a ($1.3$)]. However, our proof of (\[discrete\_embedding\_ineq2\]) treats $\nabla v_{h}$ as $1$-form and uses standard enriching operator for vector valued $H^{1}$-conforming elements. Thus in (\[discrete\_embedding\_ineq2\]), there is no restriction on the dimension of domains and order of polynomial.
By triangle inequality and discrete inverse inequality, $$\begin{aligned}
\Vert \Delta v_{h} \Vert_{\calT_{h}}^{2}
\leq & C \left( \Vert \nabla\cdot (\bfeta_{h} - \nabla v_{h}) \Vert_{\calT_{h}}^{2}
+ \Vert \nabla\cdot \bfeta_{h} \Vert_{\calT_{h}}^{2} \right) \\
\leq & C \left( \Vert \nabla \cdot \bfeta_{h} \Vert_{\calT_{h}}^{2}
+ \Sigma_{K \in \calT_{h}} h_{K}^{-2} \Vert \bfeta_{h} - \nabla v_{h} \Vert_{L^{2}(K)}^{2} \right). \end{aligned}$$ By discrete trace inequality and the fact that $\bfeta_{h} \in H_{0}(\text{div},\Omega)$, $$\begin{aligned}
\sum_{F\in \calE_{h}} h_{F}^{-1} \Vert \llbracket \nabla v_h \cdot \bfn \rrbracket \Vert_{0,F}^{2}
= \sum_{F\in \calE_{h}} h_{F}^{-1} \Vert \llbracket (\bfeta - \nabla v_h ) \cdot \bfn \rrbracket \Vert_{0,F}^{2}
\leq C \Sigma_{K \in \calT_{h}} h_{K}^{-2} \Vert \bfeta_{h} - \nabla v_{h} \Vert_{L^{2}(K)}^{2}.\end{aligned}$$ Combing the above two inequalities, we have $$\begin{aligned}
\label{thm_discrete_embedding_tmp_ineq1}
\Vert \Delta v_{h} \Vert_{\calT_{h}}^{2} + \sum_{F\in \calE_{h}} h_{F}^{-1} \Vert \llbracket \nabla v_h \cdot
\bfn \rrbracket \Vert_{0,F}^{2} \leq C \left( \Vert \nabla \cdot \bfeta_{h} \Vert_{\calT_{h}}^{2}
+ \Sigma_{K \in \calT_{h}} h_{K}^{-2} \Vert \bfeta_{h} - \nabla v_{h} \Vert_{L^{2}(K)}^{2} \right). \end{aligned}$$
Since $v_{h} \in H_{0}^{1}(\Omega)$, the tangential components of $\nabla v_{h}$ are continuous across interior mesh interfaces and vanish along the boundary $\partial\Omega$. Thus $$\begin{aligned}
\sum_{F\in \calE_{h}} h_{F}^{-1} \Vert \llbracket \nabla v_h \cdot \bfn \rrbracket \Vert_{0,F}^{2}
= \sum_{F\in \calE_{h}} h_{F}^{-1} \Vert \llbracket \nabla v_h \rrbracket \Vert_{0,F}^{2}, \end{aligned}$$ where $\llbracket \nabla v_h \rrbracket|_{F}$ is the jump of all components of $\nabla v_{h}$ across interior mesh interface $F$ and $\llbracket \nabla v_h \rrbracket|_{F} = \nabla v_{h}|_{F}$ for any face $F$ on $\partial\Omega$. According to [@KP2003 Theorem $2.2$] and the above equality, there is $\tilde{\bfeta}_{h} \in [ H_{0}^{1}(\Omega) \cap P_{k+1}(\calT_{h}) ]^{d}$ such that $$\begin{aligned}
\label{thm_discrete_embedding_tmp_ineq2}
\Vert \tilde{\bfeta}_{h} - \nabla v_{h}\Vert_{\calT_{h}}^{2} \leq C \sum_{F\in \calE_{h}} h_{F}
\Vert \llbracket \nabla v_h \rrbracket \Vert_{0,F}^{2} \leq C \sum_{F\in \calE_{h}} h_{F}
\Vert \llbracket \nabla v_h \cdot \bfn \rrbracket \Vert_{0,F}^{2} . \end{aligned}$$ We would like to point out that though it only deals with $d=2,3$, the proof of [@KP2003 Theorem $2.2$] can be easily generalized for arbitrary dimension $d \in \mathbb{N}$. Thus (\[thm\_discrete\_embedding\_tmp\_ineq2\]) is valid for any dimension $d \in \mathbb{N}$.
Obviously, $\tilde{\bfeta}_{h}$ is a $1$-form on $\Omega$ and therefore (cf. [@Arnold2006]), $$\begin{aligned}
- \Delta \tilde{\bfeta}_{h} = \boldsymbol{d}_{0} \boldsymbol{d}_{1}^{*}\tilde{\bfeta}_{h}
+ \boldsymbol{d}_{2}^{*} \boldsymbol{d}_{1} \tilde{\bfeta}_{h} \in H^{-1} \Lambda^{1}(\Omega), \end{aligned}$$ where for any $1\leq k \leq d$, $\boldsymbol{d}_{k}$ is the exterior derivative mapping from $k$-form to $(k+1)$-form and $\boldsymbol{d}_{k}^{*} = (-1)^{d(k+1)+1} *\boldsymbol{d}_{d-k} *$ maps from $k$-form to $(k-1)$-form. Here $*$ is the Hodge star operator. In fact, $\boldsymbol{d}_{0} = \nabla$ and $\boldsymbol{d}_{1}^{*} = - \nabla\cdot$. If $d=3$, $\boldsymbol{d}_{1} = \boldsymbol{d}_{2}^{*} = \nabla \times$. Then for any $\boldsymbol{\phi} \in C_{0}^{\infty}\Lambda^{1}(\Omega)$, $$\begin{aligned}
- (\Delta \tilde{\bfeta}_{h}, \boldsymbol{\phi})_{\Omega}
= (\boldsymbol{d}_{1}^{*} \tilde{\bfeta}_{h}, \boldsymbol{d}_{1}^{*}\boldsymbol{\phi})_{\Omega}
+ (\boldsymbol{d}_{1}\tilde{\bfeta}_{h}, \boldsymbol{d}_{1}\boldsymbol{\phi})_{\Omega}.\end{aligned}$$ Since $\Vert \boldsymbol{d}_{1}^{*}\boldsymbol{\phi} \Vert_{L^{2}(\Omega)} \leq
C \Vert \boldsymbol{\phi}\Vert_{H^{1}(\Omega)}$ and $\Vert \boldsymbol{d}_{1}\boldsymbol{\phi}
\Vert_{L^{2}(\Omega)} \leq C \Vert \boldsymbol{\phi} \Vert_{H^{1}(\Omega)}$, the above identity implies $$\begin{aligned}
\Vert \Delta \tilde{\bfeta}_{h} \Vert_{H^{-1}(\Omega)} \leq C \left( \Vert \boldsymbol{d}_{1}^{*}
\tilde{\bfeta}_{h}\Vert_{L^{2}(\Omega)} + \Vert \boldsymbol{d}_{1}\tilde{\bfeta}_{h}\Vert_{L^{2}(\Omega)} \right).\end{aligned}$$ Since $\tilde{{\bfeta}_{h}} = \boldsymbol{0}$ on $\partial\Omega$, the above inequality implies $$\begin{aligned}
\Vert \tilde{\bfeta}_{h} \Vert_{H^{1}(\Omega)} \leq C \left( \Vert \boldsymbol{d}_{1}^{*}
\tilde{\bfeta}_{h}\Vert_{L^{2}(\Omega)} + \Vert \boldsymbol{d}_{1}\tilde{\bfeta}_{h}\Vert_{L^{2}(\Omega)} \right).\end{aligned}$$ By triangle inequality and the fact that $\boldsymbol{d}_{0} = \nabla$ and $\boldsymbol{d}_{1}^{*} = - \nabla\cdot$, we arrive at $$\begin{aligned}
\label{thm_discrete_embedding_tmp_ineq3}
& \Vert \tilde{\bfeta}_{h} \Vert_{H^{1}(\Omega)} \leq C \left( \Vert \boldsymbol{d}_{1}^{*}
\tilde{\bfeta}_{h}\Vert_{L^{2}(\Omega)} + \Vert \boldsymbol{d}_{1}\tilde{\bfeta}_{h}\Vert_{L^{2}(\Omega)} \right) \\
\nonumber
\leq & C \left( \Vert \boldsymbol{d}_{1}^{*}
(\tilde{\bfeta}_{h} - \nabla v_{h})\Vert_{\calT_{h}} + \Vert \boldsymbol{d}_{1}(\tilde{\bfeta}_{h} - \nabla v_{h})
\Vert_{\calT_{h}} + \Vert \boldsymbol{d}_{1}^{*} (\nabla v_{h}) \Vert_{\calT_{h}}
+ \Vert \boldsymbol{d}_{1} (\nabla v_{h}) \Vert_{\calT_{h}} \right) \\
\nonumber
= & C \left( \Vert \boldsymbol{d}_{1}^{*}
(\tilde{\bfeta}_{h} - \nabla v_{h})\Vert_{\calT_{h}} + \Vert \boldsymbol{d}_{1}(\tilde{\bfeta}_{h} - \nabla v_{h})
\Vert_{\calT_{h}} + \Vert \Delta v_{h} \Vert_{\calT_{h}} \right) \\
\nonumber
\leq & C \left( \Sigma_{K \in \calT_{h}} h_{K}^{-1} \Vert \tilde{\bfeta}_{h} - \nabla v_{h}\Vert_{\calT_{h}}
+ \Vert \Delta v_{h} \Vert_{\calT_{h}} \right). \end{aligned}$$ We utilized discrete inverse inequality to obtain the above inequality.
By (\[thm\_discrete\_embedding\_tmp\_ineq2\]) and (\[thm\_discrete\_embedding\_tmp\_ineq3\]), we obtain (\[discrete\_embedding\_ineq2\]). (\[discrete\_embedding\_ineq1\]) is an immediate application of (\[discrete\_embedding\_ineq2\]) and (\[thm\_discrete\_embedding\_tmp\_ineq1\]).
Now we want to prove (\[discrete\_embedding\_ineq3\]). Like (\[thm\_discrete\_embedding\_tmp\_ineq2\]), there is $v_{h} \in
H_{0}^{1}(\Omega) \cap P_{k+2}(\calT_{h})$ such that $$\begin{aligned}
\Vert \tilde{v}_{h} - v_{h}\Vert_{\calT_{h}}^{2} \leq C \Sigma_{F \in \calE_{h}}
h_{F}\Vert \llbracket v_h \rrbracket \Vert_{0,F}^{2}. \end{aligned}$$ The above inequality with (\[discrete\_embedding\_ineq2\]) yields (\[discrete\_embedding\_ineq3\]) and this concludes the rest of the proof.
Below, we state the stability estimate for the mixed finite element scheme (\[sta\_fem\_biharmonic\]).
\[thm\_main\_sta\] For the solution $(\bfw_h,u_h)$ of the mixed finite element scheme (\[sta\_fem\_biharmonic\]) with $\theta = 1$ if the stabilization parameter $\tau$ is chosen to be large enough, then, there exists a positive constant $C$ independent of $h$ and penalty parameter $\tau$ such that $$\begin{aligned}
\label{main_sta_est}
\|\bfw_h\|_{H({\rm div},\Omega)}+ \|u_{h}\|_{H^{1}(\Omega)} + \Vert u_{h}\Vert_{2,\calT_{h}}
\leq C \|f\|_{H^{-1}(\Omega)}.\end{aligned}$$ When $\theta=-1$, the stability estimate (\[main\_sta\_est\]) holds for any $\tau>0$.
Choose $\bfeta=\bfw_h$ and $v = u_h$ in (\[sta\_fem\_biharmonic\]) to obtain for $\theta\neq -1$ $$\begin{aligned}
\label{est_sta_1}
\| \sqrt{\kappa} \nabla \cdot \bfw_h\|^2_{\calT_h} + \frac{\tau}{h^2}\;\| \sqrt{\kappa}
(\bfw_h-\nabla u_h)\|^2_{\calT_h} =
(f,u_{h})_{\langle H^{-1}(\Omega), H_{0}^{1}(\Omega) \rangle}
- (1+\theta)\;\left(\nabla(\kappa \nabla \cdot \bfw_h), \bfw_h - \nabla u_h \right)_{\calT_h}.\end{aligned}$$ For the second term on the right hand side of (\[est\_sta\_1\]), a use of the Cauchy-Schwarz inequality with the inverse inequality and the fact that $\kappa \in W^{1, \infty}(\Omega)$ yields $$\begin{aligned}
\label{est_sta_2}
(1+\theta) \left(\nabla(\kappa\nabla \cdot \bfw_h), \bfw_h - \nabla u_h \right)_{\calT_h}
&\leq C \| \nabla(\kappa \nabla \cdot \bfw_h) \|_{\calT_h} \|\bfw_h - \nabla u_h\|_{\calT_h}\\
\nonumber
&\leq C_s \| \nabla \cdot \bfw_h\|_{\calT_h} h^{-1}\|\bfw_h - \nabla u_h\|_{\calT_h}\\
\nonumber
& \leq \frac{1}{2} \| \nabla \cdot \bfw_h\|^2_{\calT_h} + C_s^2 h^{-2}\|\bfw_h - \nabla u_h\|^2_{\calT_h}. \end{aligned}$$ Substituting (\[est\_sta\_2\]) in (\[est\_sta\_1\]), choose $\tau$ large enough such that $\tau > C_s^2.$ Then, we obtain $$\begin{aligned}
\label{est_sta_3}
\| \nabla \cdot \bfw_h\|^2_{\calT_h} + h^{-2}\|\bfw_h - \nabla u_h\|^2_{\calT_h}
\leq C \|f\|_{H^{-1}(\Omega)} \| \nabla u_h\|_{\calT_h}. \end{aligned}$$ From Theorem \[thm\_discrete\_embedding\], it follows that $$\begin{aligned}
\label{est_sta_4}
\Vert u_{h} \Vert_{H^{1}(\Omega)}^{2} + \Vert u_{h}\Vert^2_{2, \calT_{h}}
& \leq C \left( h^{-2}\|\bfw_h - \nabla u_h\|^2_{\calT_h}+ \| \nabla\cdot \bfw_h \|^2_{\calT_h} \right). \end{aligned}$$ Combining (\[est\_sta\_3\], \[est\_sta\_4\]), we obtain $$\begin{aligned}
\label{est_sta_5}
\|\nabla \cdot \bfw_h\|^2_{\calT_h} + \Vert u_{h} \Vert_{H^{1}(\Omega)}^{2}
+ \Vert u_{h}\Vert^2_{2, \calT_{h}} \leq C \|f\|_{H^{-1}(\Omega)}\| \nabla u_h\|_{\calT_h}.\end{aligned}$$ Now an application of the Cauchy-Schwarz inequality with (\[est\_sta\_5\]) yields the part of estimate (\[main\_sta\_est\]). To complete the rest of the estimate for for $\theta = 1,$ we note from (\[est\_sta\_3\]) that $$\begin{aligned}
\label{w-u-1}
\|\bfw_h - \nabla u_h\|_{\calT_h} \leq C h \|f\|_{H^{-1}(\Omega)}.\end{aligned}$$ Since $$\|\bfw_h\|_{\calT_h}\leq \|\bfw_h - \nabla u_h\|_{\calT_h} + \|\nabla u_h\|_{\calT_h},$$ (\[est\_sta\_5\], \[w-u-1\]) show the following stability estimate for $\bfw_h$ in $L^2$ norm as $$\|\bfw_h\|_{\calT_h} \leq C \|f\|_{H^{-1}(\Omega)}.$$ This concludes the desired result for $\theta = 1$. When $\theta=-1$, the second term on the right hand side of (\[est\_sta\_1\]) becomes zero and the rest of the proof follows as above for any $\tau>0.$ This completes rest of the proof.
Error estimates {#sec_err_b}
---------------
In this section, we present the detailed proof of the [*a priori*]{} error estimates for the mixed finite element scheme (\[sta\_fem\_biharmonic\]).
Now define $$\bfe_{\bfw} := \bfPi^{\rm div}_h\bfw - \bfw_h, \quad e_u := \pi_h u -u_h,$$ where $\bfPi^{\rm div}_h: H_{0}({\rm div},\Omega) \rightarrow \bfW_h$ is the $H({\rm div})$-smooth projection and $\pi_h: H^1_0(\Omega)\rightarrow V_h$ is the $H^1$-smooth projection introduced in [@ChrisWinth08; @Schob08b] (the interpolations in [@DB2005] can be used also). By the regularity assumption (\[reg\_ass\]), the following approximation properties hold true for the two projections $\bfPi^{\rm div}_h$ and $\pi_h$:
\[appros\] $$\begin{aligned}
\label{appro_1}
\| \bfw - \bfPi^{\rm div}_h\bfw\|_{H({\rm div},\Omega)} & + h^{\frac{1}{2}}\| \nabla \cdot( \bfw - \bfPi^{\rm div}_h\bfw )
\|_{\partial \calT_h} \leq C h^{s} \left( \| \bfw \|_{s,\Omega} + \| \nabla\cdot \bfw \|_{s,\Omega}\right), \\
\label{appro_2}
& \| \bfw - \bfPi^{\rm div}_h\bfw\|_{\calT_h} + \| \nabla ( u - \pi_h u) \|_{\calT_h} \leq Ch^{1+s} \|\bfw \|_{1+s,\Omega}, \\
\label{appro_3}
& \Vert u - \pi_h u \Vert_{2, \calT_{h}} \leq C h^{s} \| \bfw \|_{1+s,\Omega}, \end{aligned}$$
where $s =\min\{\delta, k+1\}$, $u$ is the solution of (\[equ\_biharmonic\]) and $\bfw = \nabla u$.
We are now ready to present the error equation for our subsequent error analysis.
\[lem\_err\_bih\] Let $u$ and $(\bfw_h,u_h)$ be the solution of (\[equ\_biharmonic\]) and (\[sta\_fem\_biharmonic\]), with $\bfw = \nabla u,$ respectively. Under the regularity assumption in (\[reg\_ass\]), there holds $$\begin{aligned}
\label{equ_err}
& B_{\theta} ( (\bfe_{\bfw},e_u), (\bfeta, v)) \\
& = -(\kappa \nabla \cdot( \bfw - \bfPi^{\rm div}_h \bfw ),\nabla \cdot \bfeta )_{\calT_h}
- \tau h^{-2} ( \bfw - \bfPi^{\rm div}_h \bfw - \nabla( u-\pi_h u ), \kappa(\bfeta - \nabla v) )_{\calT_h} {\nonumber}\\
& \quad +( \kappa\nabla \cdot(\bfw -\bfPi^{\rm div}_h \bfw ) , \nabla \cdot(\bfeta - \nabla v) )_{\calT_h}
- \langle \kappa\nabla \cdot (\bfw -\bfPi^{\rm div}_h \bfw), (\bfeta - \nabla v) \cdot
\bfn \rangle_{\partial \calT_h}{\nonumber}\\
&\quad -\theta( \bfw - \bfPi^{\rm div}_h \bfw - \nabla( u-\pi_h u ),
\nabla (\kappa\nabla \cdot \bfeta))_{\calT_h},{\nonumber}\end{aligned}$$ for any $(\bfeta,v) \in \bfW_h \times V_h$.
We assume $\delta \in (\frac{1}{2}, 1]$ where $\delta$ is introduced in (\[reg\_ass\]). We choose $(\bfeta, v)\in \bfW_h \times V_h$ arbitrarily.
We define $\tilde{v}$ to be a function on $\mathbb{R}^{d}$ satisfying $$\begin{aligned}
\tilde{v}(\boldsymbol{x}) & = v(\boldsymbol{x}) \qquad & \forall \boldsymbol{x} \in \Omega, \\
\tilde{v}(\boldsymbol{x}) & = 0 \qquad & \forall \boldsymbol{x} \in \mathbb{R}^{d} \setminus \Omega.\end{aligned}$$ Obviously, $\tilde{v} \in H^{1}(\mathbb{R}^{d})$. It is easy to verify that $\nabla \tilde{v} \in [ H^{1- \delta}(\mathbb{R}^{d}) ]^{d}$. Then, according to [@Mclean1 Theorem $3.33$], we have that $v \in H^{2 - \delta}_{0}(\Omega)$ where $H^{2 - \delta}_{0}(\Omega)$ is the closure of $C_{0}^{\infty}(\Omega)$ with respect to $\Vert \cdot \Vert_{H^{2 - \delta}(\Omega)}$ (the standard norm of $H^{2-\delta}(\Omega)$). In addition, it is easy to verify that $\bfeta \in [ H^{1 - \delta}(\Omega) ]^{d}$.
Since $\kappa \in W^{1,\infty}(\Omega)$ and $\nabla\cdot \bfw \in H^{\delta}(\Omega)$, then $\kappa \nabla\cdot \bfw \in H^{\delta}(\Omega)$. Thus, $-(\kappa\nabla\cdot \bfw, \nabla \cdot \bfeta)_{\calT_{h}}
+ \langle \kappa\nabla \cdot \bfw, (\bfeta - \nabla v)\cdot \bfn \rangle_{\partial \calT_{h}}$ is well defined. By [@Mclean1 Theorem $3.40$], $H^{1 - \delta}(\Omega) = H_{0}^{1 - \delta}(\Omega)$. Since $\bfeta, \nabla v \in [ H^{1 - \delta}(\Omega) ]^{d}$, $\langle \nabla (\kappa\nabla\cdot \bfw), \bfeta - \nabla v\rangle_{\Omega}$ is well defined where $\langle \nabla (\kappa\nabla\cdot \bfw), \bfeta - \nabla v\rangle_{\Omega}$ is the coupling between $[H^{\delta - 1}(\Omega)]^{d}$ and $[H_{0}^{1 - \delta}(\Omega)]^{d}$.
By (\[equ\_biharmonic1\]) and the fact that $f \in H^{-1}(\Omega)$, there holds $$\begin{aligned}
(\Delta (\kappa\nabla\cdot \bfw), v)_{\calT_{h}} = (f, v)_{\langle H^{-1}(\Omega), H_{0}^{1}(\Omega) \rangle}.\end{aligned}$$ We notice that for any $\bar{v}\in C_{0}^{\infty}(\Omega)$, $$\begin{aligned}
(\Delta (\kappa\nabla\cdot \bfw), \bar{v})_{\calT_{h}} =
- \left(\nabla (\kappa \nabla\cdot \bfw), \nabla \bar{v}\right)_{\Omega}.\end{aligned}$$ Since $C_{0}^{\infty}(\Omega)$ is dense in $H^{2 - \delta}_{0}(\Omega)$, then for $v\in H^{2 - \delta}_{0}(\Omega)$, it follows that $$\begin{aligned}
(\Delta (\kappa\nabla\cdot \bfw), v)_{\calT_{h}} = - \left(\nabla (\kappa\nabla\cdot \bfw), \nabla v\right)_{\Omega}
= -\left(\nabla (\kappa\nabla\cdot \bfw), \bfeta \right)_{\Omega}
+ \left(\nabla (\kappa\nabla\cdot \bfw), \bfeta - \nabla v\right)_{\Omega} .\end{aligned}$$ We recall that $\kappa \nabla\cdot \bfw \in H^{\delta}(\Omega)$ since $\kappa \in W^{1,\infty}(\Omega)$. Let $\{ \sigma_{i} \}_{i=1}^{\infty} \subset D(\bar{\Omega}):= \{ \sigma: \sigma = \bar{\sigma}|_{\Omega}
\text{ where } \bar{\sigma} \in C_{0}^{\infty}(\mathbb{R}^{d}) \}$ such that $$\begin{aligned}
\Vert \sigma_{i} - \kappa\nabla\cdot \bfw \Vert_{H^{\delta}(\Omega)} \rightarrow 0 \text{ as } i\rightarrow \infty.\end{aligned}$$ Then from [@Grisvard1 Theorem $1.4.4.6$], we arrive at $$\begin{aligned}
\label{negative_appr1}
\Vert \nabla (\sigma_{i} - \kappa\nabla\cdot \bfw) \Vert_{H^{\delta-1}(\Omega)}
\rightarrow 0 \text{ as } i\rightarrow \infty.\end{aligned}$$ For any $i \geq 1$, $$\begin{aligned}
-\left(\nabla \sigma_{i}, \bfeta \right)_{\Omega} + \left(\nabla \sigma_{i}, \bfeta - \nabla v\right)_{\Omega}
= ( \sigma_{i}, \nabla \cdot \bfeta)_{\calT_{h}} - (\sigma_{i}, \nabla\cdot (\bfeta - \nabla v))_{\calT_{h}}
+ \langle \sigma_{i}, (\bfeta - \nabla v) \cdot \bfn \rangle_{\partial \calT_{h}}.\end{aligned}$$ Then, letting $i\mapsto \infty$, we apply (\[negative\_appr1\]) and integration by parts to obtain $$\begin{aligned}
(\Delta (\kappa\nabla\cdot \bfw), v)_{\calT_{h}} = (\kappa\nabla\cdot \bfw, \nabla \cdot \bfeta)_{\calT_{h}}
- (\kappa\nabla\cdot \bfw, \nabla\cdot (\bfeta - \nabla v))_{\calT_{h}}
+ \langle \kappa\nabla\cdot \bfw, (\bfeta - \nabla v) \cdot \bfn \rangle_{\partial \calT_{h}}\end{aligned}$$ Thus, we arrive at $$\begin{aligned}
\label{consistent_biharmonic}
(\kappa\nabla\cdot \bfw, \nabla \cdot \bfeta)_{\calT_{h}}
- (\kappa\nabla\cdot \bfw, \nabla\cdot (\bfeta - \nabla v))_{\calT_{h}}
+ \langle \kappa \nabla\cdot \bfw, (\bfeta - \nabla v) \cdot \bfn \rangle_{\partial \calT_{h}}
= (f, v)_{\langle H^{-1}(\Omega), H_{0}^{1}(\Omega) \rangle}.\end{aligned}$$ By subtracting (\[consistent\_biharmonic\]) from (\[sta\_fem\_biharmonic\]), we immediately obtain the error equation (\[equ\_err\]) and this concludes the proof.
Below, we present the main theorem of this section.
\[main\_err1\] Let $u$ and $(\bfw_h,u_h)$ be the solution of (\[equ\_biharmonic\]) and (\[sta\_fem\_biharmonic\]), respectively, and $\bfw = \nabla u$. Further let the regularity assumption in (\[reg\_ass\]) hold. For $\theta = 1,$ if the stabilization parameter $\tau$ is chosen to be large enough, then there holds $$\begin{aligned}
\| \nabla \cdot (\bfw - \bfw_h) \|_{\calT_h} + \Vert u - u_{h} \Vert_{H^{1}(\Omega)}
+ \Vert u - u_{h}\Vert_{2, \calT_{h}}
\leq C h^s\| \bfw \|_{1+s,\Omega},\end{aligned}$$ where $s= \min\{\delta, k+1\}$ and $\delta$ is the parameter given in (\[reg\_ass\]). When $\theta=-1,$ the above estimate holds for $\tau>0.$
Choose $(\bfeta,v) = (\bfe_{\bfw},e_u)$ in (\[equ\_err\]) to obtain $$\begin{aligned}
\|\nabla \cdot \bfe_{\bfw} \|^2_{\calT_h} + \tau h^{-2} \| \bfe_{\bfw} - \nabla e_u \|^2_{\calT_h} = \sum^5_{k=0}T_k,\end{aligned}$$ where $$\begin{aligned}
T_0 &=- (1+\theta) ( \nabla( \kappa\nabla \cdot \bfe_{\bfw}), \bfe_{\bfw} - \nabla e_u)_{\calT_h} ,\\
T_1 &= -( \kappa\nabla \cdot( \bfw - \bfPi^{\rm div}_h \bfw ),\nabla \cdot \bfe_{\bfw} )_{\calT_h}, \\
T_2 &= - \tau h^{-2} ( \bfw - \bfPi^{\rm div}_h \bfw - \nabla( u-\pi_h u ),
\kappa(\bfe_{\bfw} - \nabla e_u ) )_{\calT_h}, \\
T_3 &= ( \kappa\nabla \cdot(\bfw -\bfPi^{\rm div}_h \bfw ) , \nabla \cdot(\bfe_{\bfw} - \nabla e_u) )_{\calT_h} ,\\
T_4 &= - \langle \kappa\nabla \cdot (\bfw -\bfPi^{\rm div}_h \bfw), (\bfe_{\bfw} - \nabla e_u) \cdot
\bfn \rangle_{\partial \calT_h},\\
T_5 &= -\theta ( \bfw - \bfPi^{\rm div}_h \bfw - \nabla( u-\pi_h u ),
\nabla (\kappa\nabla \cdot \bfe_{\bfw}))_{\calT_h}.\end{aligned}$$ By the inverse inequality, the approximation properties of the two operators $\bfPi^{\rm div}_h$, $\pi_h$ in (\[appros\]), and the fact that $\kappa \in W^{1,\infty}(\Omega)$, we can derive the upper bounds of $T_0,\cdots,T_5$ as follow: $$\begin{aligned}
T_0 & \leq C\| \nabla \cdot \bfe_{\bfw} \|_{\calT_h} h^{-1} \|\bfe_{\bfw} - \nabla e_u\|_{\calT_h},\\
T_1 &\leq C h^s\| \bfw \|_{1+s,\Omega} \| \nabla \cdot \bfe_{\bfw} \|_{\calT_h},\\
T_2 & \leq C h^s\| \bfw \|_{1+s,\Omega} h^{-1}\|\bfe_{\bfw} - \nabla e_u\|_{\calT_h},\\
T_3 & \leq C h^s\| \bfw \|_{1+s,\Omega}h^{-1}\|\bfe_{\bfw} - \nabla e_u\|_{\calT_h},\\
T_4 & \leq C h^{\frac{1}{2}} \|\nabla \cdot (\bfw -\bfPi^{\rm div}_h \bfw)\|_{\partial \calT_h} h^{-1}\|\bfe_{\bfw} - \nabla e_u\|_{\calT_h} \quad (\text{by trace inequality})\\
&\leq C h^s\| \bfw \|_{1+s,\Omega}h^{-1}\|\bfe_{\bfw} - \nabla e_u\|_{\calT_h},\\
T_5 & \leq C h^s\| \bfw \|_{1+s,\Omega} \| \nabla \cdot \bfe_{\bfw} \|_{\calT_h}.\end{aligned}$$ Now combining the above estimates for $T_k,k=0,\cdots,5$, the Cauchy-Schwarz inequality, and choosing the stabilization parameter $\tau$ to be large enough, we arrive at $$\begin{aligned}
\label{err_res1}
\|\nabla \cdot \bfe_{\bfw} \|_{\calT_h} + h^{-1} \| \bfe_{\bfw} - \nabla e_u \|_{\calT_h} \leq Ch^s\| \bfw \|_{1+s,\Omega} . \end{aligned}$$ By (\[err\_res1\]) and (\[appro\_1\]), it directly follows that $$\| \nabla \cdot (\bfw - \bfw_h) \|_{\calT_h} \leq C h^s\| \bfw \|_{1+s,\Omega}.$$
By Theorem \[thm\_discrete\_embedding\], triangle inequality and trace inequality, we obtain $$\begin{aligned}
\label{err_eu_en_est1}
\Vert e_{u} \Vert_{H^{1}(\Omega)} + \Vert e_{u} \Vert_{2, \calT_{h}}
& \leq \| \nabla \cdot (\nabla e_u)\|_{\calT_h} + \|h^{-\frac{1}{2}} \llbracket \nabla e_u \cdot \bfn \rrbracket \|_{\calE_h} \\
& \leq \| \nabla \cdot (\nabla e_u -\bfe_{\bfw} )\|_{\calT_h} + \|\nabla \cdot \bfe_{\bfw}\|_{\calT_h} + \|h^{-\frac{1}{2}} \llbracket ( \bfe_{\bfw}-\nabla e_u )\cdot \bfn \rrbracket \|_{\calE_h}{\nonumber}\\
&\leq C\left( h^{-1} \| \nabla e_u -\bfe_{\bfw} \|_{\calT_h} + \|\nabla \cdot \bfe_{\bfw}\|_{\calT_h} \right).{\nonumber}\end{aligned}$$ By (\[err\_eu\_en\_est1\]), (\[err\_res1\]), (\[appro\_3\]) and triangle inequality, we obtain $$\begin{aligned}
\Vert u - u_{h} \Vert_{H^{1}(\Omega)} + \Vert u - u_{h} \Vert_{2, \calT_{h}} \leq C h^s\| \bfw \|_{1+s,\Omega}.\end{aligned}$$ For $\theta=-1$, the term $T_0$ becomes zero and the rest of the estimates hold. This complete the rest of the proof.
In order to prove the $L^2$-norm of error estimate for $u-u_h$, we apply the Aubin-Nitsche duality argument.
Now consider the dual problem (\[dual\_p\])
\[dual\_p\] $$\begin{aligned}
\Delta (\kappa \Delta \varphi)= z, & \ \ \textrm{in} \ \ \Omega, \\
\varphi = 0, & \ \ \textrm{on} \ \ \partial \Omega, \\
\frac{\partial \varphi}{\partial n} = 0, & \ \ \textrm{on} \ \ \partial \Omega.\end{aligned}$$
with the following elliptic regularity condition: $\bfpsi = \nabla \varphi \in H^{1+\alpha}(\Omega)$ with $\alpha > 1/2$ and there holds $$\begin{aligned}
\label{reg_dual}
\|\varphi\|_{2+\alpha} + \|\bfpsi\|_{1+\alpha,\Omega} \leq C \|z\|_{0,\Omega}. \end{aligned}$$
\[thm\_L2\_con\_biharmonic\] Let the conditions in Theorem \[main\_err1\] and the regularity result (\[reg\_dual\]) of the dual problem hold. Then, the following estimates hold for sufficiently large $\tau>0$, $$\begin{aligned}
\label{l2_err_u}
\|u-u_h\|_{\calT_h} & \lesssim h^{s+\sigma} \|\bfw\|_{1+s,\Omega}, \\
\label{l2_err_w}
\|\bfw-\bfw_h\|_{\calT_h} & \lesssim h^{\min\{1+s,s+\sigma/2\}} \|\bfw\|_{1+s,\Omega},\\
\label{l2_err_gradu}
\|\nabla u-\nabla u_h\|_{\calT_h}& \lesssim h^{\min\{1+s,s+\sigma/2\}} \|\bfw\|_{1+s,\Omega},\end{aligned}$$ where $s = \min\{\delta,k+1\}, \sigma = \min\{ \alpha,k+1 \}$.
For the dual problem (\[dual\_p\]), there holds $$\begin{aligned}
B_{\theta}((\bfpsi,\varphi), (\bfw - \bfw_{h},u - u_{h}))= (z, u - u_{h}).\end{aligned}$$ Since $B_{\theta}((\bfw-\bfw_h,u-u_h), (\bfeta_h,v_h)) = 0$ for all $(\bfeta_h,v_h) \in \bfW_h \times V_h$, we obtain $$\begin{aligned}
B_{\theta}((\bfpsi-\bfpsi_h, \varphi-\varphi_h), (\bfw-\bfw_h,u-u_h) ) = (z, u - u_{h}),
\label{dual_est1}\end{aligned}$$ where $(\bfpsi_h,\varphi_h)$ is the discrete solution of the dual problem (\[dual\_p\]) based on the mixed finite element scheme (\[sta\_fem\_biharmonic\]). Then, we rewrite the left hand-side of (\[dual\_est1\]) as $$B_{\theta}((\bfpsi-\bfpsi_h, \varphi-\varphi_h), (\bfw-\bfw_h,u-u_h) ) = \sum^4_{k=1} D_k,$$ where $$\begin{aligned}
&D_1 = (\kappa\nabla\cdot(\bfpsi - \bfpsi_h), \nabla \cdot(\bfw-\bfw_h))_{\calT_h}, \\
& D_2 = ( \nabla(\kappa\nabla \cdot(\bfpsi-\bfpsi_h)), \bfw-\bfw_h-\nabla(u-u_h) )_{\calT_h},\\
&D_3 = \theta ( \nabla (\kappa\nabla \cdot (\bfw-\bfw_h)) ,
\bfpsi-\bfpsi_h - \nabla(\varphi - \varphi_h))_{\calT_h}, \\
& D_4 = \tau h^{-2} ( \kappa( \bfpsi-\bfpsi_h - \nabla(\varphi - \varphi_h)), \bfw-\bfw_h-\nabla(u-u_h) )_{\calT_h}.\end{aligned}$$ By the error estimate in $H({\rm div})$-norm (cf. Theorem \[main\_err1\]), $D_1$ can be made bounded by $$D_1 \leq C h^{s + \sigma} \| \bfpsi \|_{1+\sigma,\Omega} \| \bfw \|_{1+s,\Omega}.$$ From (\[err\_res1\]), we easily find that $$\begin{aligned}
h^{-1}\| \bfw_h - \nabla u_h \|_{\calT_h} \leq C h^{s}\|\bfw\|_{1+s,\Omega}. \label{aux_err_est1}\end{aligned}$$ Similarly, we also have the following estimate for the discrete solution of the dual problem (\[dual\_p\]) $$h^{-1}\| \bfpsi_h - \nabla \varphi_h \|_{\calT_h} \leq C h^{\sigma}\|\bfpsi\|_{1+\sigma,\Omega} .$$ Thus, a use of the Cauchy-Schwarz inequality yields $$D_4 \leq C h^{s+\sigma} \| \bfpsi \|_{1+\sigma,\Omega} \| \bfw \|_{1+s,\Omega}.$$ Next we take integration by parts for $D_2$ and $D_3$. We note that $ \| \nabla\cdot(\Pi^{\rm div}_h \bfpsi -\bfpsi_h)\|_{ \calT_h} \leq C h^{\sigma} \| \bfpsi \|_{1+\sigma,\Omega} $ can be similarly derived as in (\[err\_res1\]). Now for $D_2$, we arrive at $$\begin{aligned}
D_2 &= ( \kappa\nabla\cdot(\bfpsi - \bfpsi_h), \nabla \cdot ( \bfw_h - \nabla u_h ))_{\calT_h}
-\langle \kappa\nabla\cdot(\bfpsi - \bfpsi_h), (\bfw_h - \nabla u_h )\cdot\bfn \rangle_{\partial \calT_h}{\nonumber}\\
&\leq C \|\nabla\cdot(\bfpsi - \bfpsi_h) \|_{\calT_h} h^{-1}\|\bfw_h - \nabla u_h\|_{\calT_h} \\
&\quad + Ch^{\frac{1}{2}}( \| \nabla\cdot (\bfpsi - \bfPi^{\rm div}_h \bfpsi) \|_{\partial \calT_h} + \| \nabla\cdot(\bfPi^{\rm div}_h \bfpsi -\bfpsi_h)\|_{\partial \calT_h})h^{-1} \| \bfw_h - \nabla u_h \|_{\calT_h} \\
&\leq Ch^{s+\sigma} \| \bfpsi \|_{1+\sigma,\Omega} \| \bfw \|_{1+s,\Omega}.\end{aligned}$$ For $D_3$, it follows that $$\begin{aligned}
D_3 &= \theta (\kappa \nabla\cdot(\bfw - \bfw_h), \nabla \cdot(\bfpsi_h - \nabla \varphi_h) )_{\calT_h}
- \theta\langle \kappa \nabla \cdot(\bfw-\bfw_h), (\bfpsi_h - \nabla \varphi_h)\cdot\bfn \rangle_{\partial \calT_h}\\
& \leq C \theta \|\nabla\cdot(\bfw - \bfw_h)\|_{\calT_h} h^{-1} \|\bfpsi_h - \nabla \varphi_h)\|_{\calT_h}\\
&\quad + C \theta h^{\frac{1}{2}}( \| \nabla\cdot (\bfw - \bfPi^{\rm div}_h \bfw) \|_{\partial \calT_h} + \| \nabla\cdot(\bfPi^{\rm div}_h \bfw -\bfw_h)\|_{\partial \calT_h})h^{-1} \| \bfpsi_h - \nabla \varphi_h \|_{\calT_h} \\
&\leq C \theta h^{s+\sigma} \| \bfpsi \|_{1+\sigma,\Omega} \| \bfw \|_{1+s,\Omega}.\end{aligned}$$ Combing the estimates for $D_l,l=1,\cdots,4$, we obtain $$(z, u-u_h) \leq C\;h^{s + \sigma} \| \bfpsi \|_{1+\sigma,\Omega} \| \bfw \|_{1+s,\Omega}.$$ Let $z = u-u_h$. Then, we obtain the desired estimate (\[l2\_err\_u\]) by (\[reg\_dual\]).
Next, we decompose $\|\bfe_{\bfw}\|^2_{\calT_h}$ into $$\|\bfe_{\bfw}\|^2_{\calT_h} = ( \bfPi^{\rm div}_h \bfw - \bfw + \nabla u - \nabla u_h
+\nabla u_h - \bfw_h, \bfe_{\bfw} )_{\calT_h}.$$ By (\[appro\_2\]) and (\[aux\_err\_est1\]), we arrive at $$\begin{aligned}
\label{errw_pre1}
\|\bfe_{\bfw}\|^2_{\calT_h} \leq Ch^{1+s} \|\bfw\|_{1+s,\Omega} \|\bfe_{\bfw}\|_{\calT_h}
+ (\nabla u - \nabla u_h,\bfe_{\bfw} )_{\calT_h}.\end{aligned}$$ By integration by parts and noting that $\bfe_{\bfw} \in \bfW_h$ and $u-u_h \in H^1_0(\Omega)$, it follows that $$\begin{aligned}
(\nabla u - \nabla u_h,\bfe_{\bfw} )_{\calT_h} &= -( u - u_h, \nabla \cdot \bfe_{\bfw} )_{\calT_h} \label{errw_pre2} \\
&\leq C \| u-u_h \|_{\calT_h} \|\nabla \cdot \bfe_{\bfw} \|_{ \calT_h} {\nonumber}\\
&\leq C h^{2s + \sigma} \|\bfw\|^2_{1+s,\Omega} .{\nonumber}\end{aligned}$$ Now (\[l2\_err\_w\]) can be obtained by (\[errw\_pre1\]), (\[errw\_pre2\]) and (\[appro\_2\]).
Similarly, we can decompose $\|\nabla e_u\|^2_{\calT_h}$ into $$\|\nabla e_u\|^2_{\calT_h} = ( \bfw-\bfw_h + \bfw_h - \nabla u_h, \nabla e_u)_{\calT_h}.$$ Then, the estimate (\[l2\_err\_gradu\]) follows from the above equality, (\[l2\_err\_w\]) and (\[aux\_err\_est1\]). This completes the rest of the proof.
\[remark\_conv\] When $\kappa$ is a constant and the domain is convex, the solution of (\[equ\_biharmonic\]) are smooth enough and the solution of dual problem (\[dual\_p\]) is also smooth with $\alpha=2$, then we have $$\begin{aligned}
\|u-u_h\|_{\calT_h} &\leq C h^{k+3}\|\bfw\|_{k+2,\Omega},\quad k\geq 1,\\
\|\bfw-\bfw_h\|_{\calT_h} +\|\nabla u-\nabla u_h\|_{\calT_h}& \leq C h^{k+2}\|\bfw\|_{k+2,\Omega},\quad k\geq 1,\\
\| \nabla \cdot (\bfw - \bfw_h) \|_{\calT_h} + \| \nabla (u - u_h) \|_{1,\calT_h} &\leq C h^{k+1}\|
\bfw \|_{k+2,\Omega},\quad k\geq 0.\end{aligned}$$
A new mixed finite element scheme for the von Kármán equation {#sec_method_vonkarman}
=============================================================
In this section, we extend our new formulation to the von Kármán equation. Further, under smallness condition on the data to be defined subsequently, we present the existence and uniqueness result for the discrete nonlinear system, the [*a priori*]{} bounds and the corresponding error estimates.
Through out this section, we assume that the following regularity of the solution of the von Kármán equation (\[equ\_von\_karman\]): $$\begin{aligned}
\label{ass_reg_vk}
\xi \in H^{2+\beta}(\Omega), \psi \in H^{2+\beta}(\Omega),
\quad \text{ where } 2 \geq\beta > 1/2.\end{aligned}$$ We first recall the result of in [@Brenner2017 Lemma $2.2$]. For the solution $ (\xi,\psi)$ of the von Kármán equation (\[equ\_von\_karman\]) and any $\eta \in H^1_0(\Omega)$, there holds $$\begin{aligned}
\label{lem_equ_pre_vk}
([\xi,\psi],\eta) = ({\rm cof}(D^2\xi): D^2 \psi, \eta)
=- ({\rm cof}(D^2\xi)\nabla \psi,\nabla \eta).\end{aligned}$$
We use the same finite element spaces $\bfW_h$ and $V_h$ as in Section \[sec\_method\_biharmonic\] in two dimensions. With $\bfu=\nabla \xi$ and $\bfw=\nabla \psi$, our mixed finite element scheme for the von Kármán equation (\[equ\_von\_karman\]) is to seek an approximation $(\bfu_h,\bfw_h,\xi_h,\psi_h)\in \bfW_h\times \bfW_h\times V_h \times V_h$ such that
\[sta\_method\_vk\] $$\begin{aligned}
B_{\theta}((\bfu_h,\xi_h),(\bfv,\eta))+({\rm cof}(D^2\xi_h)\nabla \psi_h,\nabla \eta)_{\calT_h} &
= (f,\eta)_{\langle H^{-1}(\Omega), H_{0}^{1}(\Omega) \rangle},\\
B_{\theta} ((\bfw_h,\psi_h),(\bfz,\phi)) -({\rm cof}(D^2\xi_h)\nabla \xi_h,\nabla \phi)_{\calT_h} &= 0,\end{aligned}$$
for any $(\bfv,\bfz,\eta,\phi) \in \bfW_h\times \bfW_h\times V_h \times V_h$, where the bilinear form $B_{\theta}((\cdot,\cdot),(\cdot,\cdot))$ is defined as in (\[def\_bilinear\]) with $\kappa = 1$.
Existence and uniqueness of the discrete nonlinear system and stability estimate
--------------------------------------------------------------------------------
Since the discrete system (\[sta\_method\_vk\]) leads to a system of nonlinear algebraic system, therefore, in this subsection, we first prove the existence and uniqueness of the discrete system based on the following one point iterative scheme, called the Picard’s method. from the mixed finite element scheme (\[sta\_method\_vk\]).
Given an initialization $\xi^{m-1}_h \in V_h$, $m\geq 1$, the Picard’s iteration for the nonlinear system (\[sta\_method\_vk\]) is to find $(\bfu^m_h,\bfw^m_h,\xi^m_h,\psi^m_h)\in \bfW_h\times \bfW_h\times V_h \times V_h$ such that
\[picard\_iter\] $$\begin{aligned}
B_{\theta}((\bfu^m_h,\xi^m_h),(\bfv,\eta))+\left({\rm cof}(D^2\xi^{m-1}_h)\nabla \psi^m_h,\nabla \eta\right)_{\calT_h}
&= (f,\eta)_{\langle H^{-1}(\Omega), H_{0}^{1}(\Omega) \rangle},\\
B_{\theta}((\bfw^m_h,\psi^m_h),(\bfz,\phi)) - \left({\rm cof}(D^2\xi^{m-1}_h)\nabla \xi^m_h,\nabla \phi\right)_{\calT_h} &= 0,\end{aligned}$$
for any $(\bfv,\bfz,\eta,\phi) \in \bfW_h\times \bfW_h\times V_h \times V_h$. Choosing $\bfv = \bfu^m_h,
\eta = \xi^m_h, \bfz = \bfw^m_h, \phi =\psi^m_h $ and noting that $$({\rm cof}(D^2\xi^{m-1}_h)\nabla \psi^m_h,\nabla \xi^m_h)_{\calT_h}
= ({\rm cof}(D^2\xi^{m-1}_h)\nabla \xi^m_h,\nabla \psi^m_h)_{\calT_h} ,$$ we now obtain $$B_{\theta}((\bfu^m_h,\xi^m_h),(\bfu^m_h,\xi^m_h)) + B((\bfw^m_h,\psi^m_h),(\bfw^m_h,\psi^m_h))
= (f,\xi^m_h)_{\langle H^{-1}(\Omega), H_{0}^{1}(\Omega) \rangle}.$$ Similar to the stability estimate (\[main\_sta\_est\]), we easily follow the proof of Theorem \[thm\_main\_sta\] to derive the estimate below, provided for $\theta\neq -1$ the stabilization parameter $\tau$ in (\[sta\_method\_vk\]) is chosen to be large enough, and for $\theta=-1$, $\tau$ is chosen to be any arbitrary positive constant. $$\begin{aligned}
\||({{\bf u}}_h^m,\bfw_h^m, \xi_h^m, \psi_h^m)\|| \leq C_{sta} \|f\|_{H^{-1}(\Omega)},\end{aligned}$$ where $$\||({{\bf u}}_h^m,\bfw_h^m, \xi_h^m, \psi_h^m)\||:= \Big(\|\bfu^m_h\|^2_{H({\rm div},\Omega)}
+ \|\bfw^m_h\|^2_{H({\rm div},\Omega)} + \| \xi^m_h\|^2_{2,\calT_h}
+ \| \psi^m_h\|^2_{2,\calT_h}\Big)^{1/2}.$$ Inspired by the above result, we define a closed subset of $\bfW_h\times \bfW_h\times V_h \times V_h$: $$\begin{aligned}
\label{def_K_h}
\calK_h:= \{ (\bfv, \bfz, \eta, \phi) \in \bfW_h\times \bfW_h\times V_h \times V_h: \
\||(\bfv,\bfz, \eta, \phi)\|| \leq C_{sta} \|f\|_{H^{-1}(\Omega)} \}.\end{aligned}$$ We also define a mapping $\calF: \calK_h \rightarrow \calK_h$ as follows: for any $(\widehat{\bfu},\widehat{\bfw},\widehat{\xi},\widehat{\psi})\in \calK_h$, $(\bfu^\ast,\bfw^\ast,\xi^\ast,\psi^\ast)= \calF(\widehat{\bfu},\widehat{\bfw},\widehat{\xi},\widehat{\psi})$ is obtained by one step of the above Picard iteration. Clearly, $(\bfu_h,\bfw_h,\xi_h,\psi_h)$ is a solution of (\[sta\_method\_vk\]) if and only if it is a fixed point of the mapping $\calF$.
We are now ready to show the existence and uniqueness result for the nonlinear system (\[sta\_method\_vk\]) and the associated stability estimate.
\[thm\_sta\_vk\] (Existence, uniqueness and stability) If $\|f\|_{H^{-1}(\Omega)}$ is small enough and the stabilization parameter $\tau$ in (\[sta\_method\_vk\]) is large enough for $\theta = 1$ and for $\theta=-1$ any $\tau>0$, the discrete nonlinear system (\[sta\_method\_vk\]) has a unique solution $(\bfu_h,\bfw_h,\xi_h,\psi_h)\in \bfW_h\times \bfW_h\times V_h \times V_h$. More over, there also holds the following estimate:: $$\begin{aligned}
\label{main_thm_sta_vk}
\||(\bfu_h,\bfw_h, \xi_h, \psi_h)\||
\leq C_{sta} \|f\|_{H^{-1}(\Omega)}. \end{aligned}$$
Clearly, the mapping $\calF$ maps $\calK_h$ into itself. In order to show the existence and uniqueness of the solution of (\[sta\_method\_vk\]), it suffices to show that $\calF$ is a contraction on $\calK_h$. Let $
(\widehat{\bfu}^1_h,\widehat{\bfw}^1_h,\widehat{\xi}^1_h,\widehat{\psi}^1_h)$, $(\widehat{\bfu}^2_h,\widehat{\bfw}^2_h,\widehat{\xi}^2_h,\widehat{\psi}^2_h) \in \calK_h
$ and $({\bfu}^1_h,{\bfw}^1_h,{\xi}^1_h,{\psi}^1_h), ({\bfu}^2_h,{\bfw}^2_h,{\xi}^2_h,{\psi}^2_h)$ be the solutions of the Picard iteration (\[picard\_iter\]) with the initializations $(\widehat{\bfu}^1_h,\widehat{\bfw}^1_h,\widehat{\xi}^1_h,\widehat{\psi}^1_h), (\widehat{\bfu}^2_h,
\widehat{\bfw}^2_h,\widehat{\xi}^2_h,\widehat{\psi}^2_h),$ respectively. Define $$\delta_\bfu := \bfu^1_h - \bfu^2_h,\quad \delta_\bfw := \bfw^1_h - \bfw^2_h, \quad \delta_\xi := \xi^1_h - \xi^2_h,\quad \delta_\psi := \psi^1_h - \psi^2_h.$$ By (\[picard\_iter\]), it follows that $$\begin{aligned}
B_{\theta}((\delta_\bfu,\delta_\xi),(\bfv,\eta))+\left({\rm cof}(D^2\widehat{\xi}^1_h)\nabla \psi^1_h
- {\rm cof}(D^2\widehat{\xi}^2_h)\nabla \psi^2_h,\nabla \eta\right)_{\calT_h} &= 0,\\
B_{\theta}((\delta_\bfw,\delta_\psi),(\bfz,\phi)) -\left({\rm cof}(D^2\widehat{\xi}^1_h)\nabla \xi^1_h
-{\rm cof}(D^2\widehat{\xi}^2_h)\nabla \xi^2_h ,\nabla \phi\right)_{\calT_h} &= 0.\end{aligned}$$ Choose $\bfv = \delta_\bfu, \bfz = \delta_\bfw, \eta = \delta_\xi, \phi =\delta_\psi $. Following the proof of stability estimate for (\[main\_sta\_est\]) again, if the stabilization parameter $\tau>0$ in (\[sta\_method\_vk\]) is large enough for $\theta\neq -1$ and for $\theta=-1$ any arbitrary $\tau>0$, we easily obtain $$\begin{aligned}
\label{vk_exi_est1}
\||(\delta_\bfu,\delta_\bfw,\delta_\xi,\delta_\psi)\||^2 \leq C^2_{sta} \left|T_\delta\right|, \end{aligned}$$ where $T_\delta = - \left({\rm cof}(D^2\widehat{\xi}^1_h)\nabla \psi^1_h - {\rm cof}(D^2\widehat{\xi}^2_h)\nabla \psi^2_h,\nabla \delta_\xi \right)_{\calT_h} + \left({\rm cof}(D^2\widehat{\xi}^1_h)\nabla \xi^1_h -{\rm cof}(D^2\widehat{\xi}^2_h)\nabla \xi^2_h ,\nabla \delta_\psi\right)_{\calT_h}.$
In order to get an upper bound of $T_\delta$, we rewrite $T_\delta$ as follows: $$\begin{aligned}
T_\delta = R_1 + R_2, \end{aligned}$$ where $$\begin{aligned}
R_1 = & - \left({\rm cof}(D^2\widehat{\xi}^1_h-D^2\widehat{\xi}^2_h)\nabla \psi^1_h,\nabla \delta_\xi \right)_{\calT_h}
- \left({\rm cof}(D^2\widehat{\xi}^2_h) \nabla \delta_\psi, \nabla \delta_\xi \right)_{\calT_h}, \\
R_2 = & \left({\rm cof}(D^2\widehat{\xi}^1_h-D^2\widehat{\xi}^2_h)\nabla \xi^1_h ,\nabla \delta_\psi\right)_{\calT_h}
+\left({\rm cof}(D^2\widehat{\xi}^2_h) \nabla \delta_\xi ,\nabla \delta_\psi\right)_{\calT_h} .\end{aligned}$$
For $R_1$, we have $$\begin{aligned}
\label{vk_exi_est2}
|R_1| &\leq \|D^2\widehat{\xi}^1_h-D^2\widehat{\xi}^2_h\|_{\calT_{h}} \|\nabla \psi^1_h\|_{L^4(\Omega)}
\|\nabla \delta_\xi \|_{L^4(\Omega)} + \|D^2\widehat{\xi}^2_h\|_{\calT_{h}}
\Vert \nabla \delta_\psi\Vert_{L^{4}(\Omega)} \Vert \nabla \delta_\xi \Vert_{L^{4}(\Omega)} \\
&\leq C_1 \left( \|\widehat{\xi}^1_h-\widehat{\xi}^2_h\|_{2, \calT_{h}}
\| \psi^1_h\|_{2,\calT_h} \| \delta_\xi\|_{2,\calT_h}
+ \|\widehat{\xi}^2_h\|_{2, \calT_{h}}
\| \delta_{\psi}\|_{2,\calT_h} \| \delta_\xi\|_{2,\calT_h} \right)
\quad \text{(by \cite[Theorem $2.1$]{Ern2010a})} {\nonumber}\\\
& \leq C_1 C_{sta} \;\|f\|_{H^{-1}(\Omega)}
\left(\|\widehat{\xi}^1_h-\widehat{\xi}^2_h\|_{2,\calT_h} + \| \delta_{\psi}\|_{2,\calT_h} \right)
\| \delta_\xi\|_{2,\calT_h} \qquad\quad \ \text{(by the property of $\calK_h$)}{\nonumber}\\
& \leq \frac{1}{2} C_1 C_{sta} \|f\|_{H^{-1}(\Omega)} \left( \|\widehat{\xi}^1_h-\widehat{\xi}^2_h\|^2_{2,\calT_h}
+ \| \delta_{\psi}\|_{2,\calT_h}^{2}+ 2 \| \delta_\xi\|^2_{2,\calT_h}\right). {\nonumber}\end{aligned}$$ Then by the property of the subspace $\calK_h$, $R_2$ can be similarly deduced as follows: $$\begin{aligned}
\label{vk_exi_est3}
|R_2| & \leq\frac{1}{2} C_1 C_{sta}\;\|f\|_{H^{-1}(\Omega)} \left(\|\widehat{\xi}^1_h-\widehat{\xi}^2_h\|^2_{2,\calT_h}
+ \| \delta_\xi\|^2_{2,\calT_h} + 2 \|\delta_\psi\|^2_{2,\calT_h}\right).\end{aligned}$$ With $\delta_{\widehat{\xi}} = \widehat{\xi}^1_h-\widehat{\xi}^2_h$, combine (\[vk\_exi\_est2\]) and (\[vk\_exi\_est3\]) to obtain $$\begin{aligned}
\label{vk_exi_est3-1}
|T_{\delta}| &\leq |R_1|+ |R_2|
\leq C_1 C_{sta}\;\|f\|_{H^{-1}(\Omega)} \left(\|\delta_{\widehat{\xi}}\|^2_{2,\calT_h}
+ \frac{3}{2} \| \delta_\xi\|^2_{2,\calT_h} + \frac{3}{2} \|\delta_\psi\|^2_{2,\calT_h}\right){\nonumber}\\
& \leq C_1 C_{sta}\;\|f\|_{H^{-1}(\Omega)} \Big(\||(\delta_{\widehat{\bfu}},\delta_{\widehat{\bfw}},
\delta_{\widehat{\xi}},\delta_{\widehat{\psi}})\||^2 + \frac{3}{2} \||(\delta_\bfu,\delta_\bfw,\delta_\xi,\delta_\psi)\||^2
\Big).\end{aligned}$$ On sustitution of (\[vk\_exi\_est3-1\]) in (\[vk\_exi\_est1\]), we find that $$\begin{aligned}
(1- \frac{3}{2} C_1 C_{sta}^3 \;\|f\|_{H^{-1}(\Omega)})\; \||(\delta_\bfu,\delta_\bfw,\delta_\xi,\delta_\psi)\||^2
\leq C_1 C_{sta}^3\;\|f\|_{H^{-1}(\Omega)} \||(\delta_{\widehat{\bfu}},\delta_{\widehat{\bfw}},
\delta_{\widehat{\xi}},\delta_{\widehat{\psi}})\||^2.
$$ Choose $\|f\|_{H^{-1}\Omega} \leq 2/(3C_1\;C_{sta}^3)$ and obtain $$\begin{aligned}
\||(\delta_\bfu,\delta_\bfw,\delta_\xi,\delta_\psi)\||^2
\leq \frac{C_1 C_{sta}^3\;\|f\|_{H^{-1}(\Omega)}}{(1- \frac{3}{2} C_1 C_{sta}^3 \;\|f\|_{H^{-1}(\Omega)})}\;
\||(\delta_{\widehat{\bfu}},\delta_{\widehat{\bfw}},
\delta_{\widehat{\xi}},\delta_{\widehat{\psi}})\||^2.\end{aligned}$$ Obviously, the above bound implies that $\calF$ is a contraction on $\calK_h$ if we further choose $\|f\|_{H^{-1}(\Omega)}<\frac{1}{3C_1\;C_{sta}^3}$ such that $\lambda: = \frac{C_1 C_{sta}^3 \|f\|_{H^{-1}(\Omega)}}{1- \frac{3}{2} C_1 C_{sta}^3 \|f\|_{H^{-1}(\Omega)})}< 1$.
By the fixed point theorem, there is a unique fixed point $(\bfu_h,\bfw_h,\xi_h,\psi_h)$ of the mapping $\calF$, and it is also the unique solution of the system (\[sta\_method\_vk\]).
Now we have proved the existence and uniqueness of (\[sta\_method\_vk\]). Notice that (\[main\_thm\_sta\_vk\]) holds due to the definition of $\calK_{h}$ in (\[def\_K\_h\]). Now we complete the proof.
Error estimates {#error-estimates}
---------------
In this section, we present [*a priori*]{} error estimates for the mixed finite element scheme (\[sta\_method\_vk\]) for the von Kármán equation.
Define $$\bfe_{\bfu} := \bfPi^{\rm div}_h\bfu - \bfu_h,\quad \bfe_{\bfw} := \bfPi^{\rm div}_h\bfw - \bfw_h, \quad e_{\xi} := \pi_h \xi -\xi_h,\quad e_\psi = \pi_h \psi - \psi_h,$$ where $\bfPi^{\rm div}_h: H_{0}({\rm div},\Omega) \rightarrow \bfW_h$ and $\pi_h: H^1_0(\Omega)\rightarrow V_h$ are the $H({\rm div})$-smooth projection and $H^1$-smooth projection as used in Section \[sec\_err\_b\]. By the regularity assumption (\[ass\_reg\_vk\]), the following approximation properties for the solution $(\bfu,\xi)$ of (\[equ\_von\_karman\]) with $\bfu = \nabla \xi$ hold true for the two operators $\bfPi^{\rm div}_h$ and $\pi_h$:
\[appros\_vk\] $$\begin{aligned}
\label{appro_1_vk}
\| \bfu - \bfPi^{\rm div}_h\bfu\|_{H({\rm div},\Omega)} + h^{\frac{1}{2}}\| \nabla \cdot( \bfu - \bfPi^{\rm div}_h\bfu )\|_{\partial \calT_h} & \leq C h^{r} \| \bfu \|_{1+r,\Omega},\\
\label{appro_2_vk}
\| \bfu - \bfPi^{\rm div}_h\bfu\|_{\calT_h} + \| \nabla ( \xi - \pi_h \xi) \|_{\calT_h} & \leq Ch^{1+r} \|\bfu \|_{1+r,\Omega},\\
\label{appro_3_vk}
\Vert \xi - \pi_{h} \xi \Vert_{2, \calT_{h}}
& \leq C h^{r} \| \bfu \|_{1+r,\Omega},\end{aligned}$$
where $r =\min\{\beta, k+1\}$. We remark that (\[appro\_1\_vk\]) and (\[appro\_2\_vk\]) also hold true for the solution $(\bfw,\psi)$ of (\[equ\_von\_karman\]) with $\bfw = \nabla \psi$.
We first present the error equation which we need for the error estimates.
\[lemma\_erq\_vk\] With $\bfu = \nabla \xi$ and $\bfw = \nabla \psi,$ let $(\xi,\psi)$ and $(\bfu_h,\bfw_h,\xi_h,\psi_h)$ be the solutions of (\[equ\_von\_karman\]) and (\[sta\_method\_vk\]),respectively. Under the regularity assumption (\[ass\_reg\_vk\]), there holds for any $(\bfv,\bfz,\eta,\phi) \in \bfW_h\times \bfW_h\times V_h \times V_h$: $$\begin{aligned}
\label{error_eq_vk}
B_{\theta}((\bfe_{\bfu},e_{\xi}),(\bfv,\eta)) + B_{\theta}((\bfe_{\bfw},e_{\psi}),(\bfz,\phi)) = S_1 + S_2 + S_3 + S_4,\end{aligned}$$ where $$\begin{aligned}
S_1 &= -( \nabla \cdot( \bfu - \bfPi^{\rm div}_h \bfu ),\nabla \cdot \bfv )_{\calT_h} - \tau h^{-2} ( \bfu - \bfPi^{\rm div}_h \bfu
- \nabla( \xi-\pi_h \xi ), \bfv- \nabla \eta )_{\calT_h} {\nonumber}\\
& \quad +( \nabla \cdot(\bfu -\bfPi^{\rm div}_h \bfu ) , \nabla \cdot(\bfv - \nabla \eta) )_{\calT_h}
- \langle \nabla \cdot (\bfu -\bfPi^{\rm div}_h \bfu), (\bfv - \nabla \eta) \cdot \bfn \rangle_{\partial \calT_h}{\nonumber}\\
&\quad -\theta ( \bfu - \bfPi^{\rm div}_h \bfu - \nabla( \xi-\pi_h \xi ), \nabla (\nabla \cdot \bfv))_{\calT_h},\end{aligned}$$ $$\begin{aligned}
S_2 & = -( \nabla \cdot( \bfw - \bfPi^{\rm div}_h \bfw ),\nabla \cdot \bfz )_{\calT_h} - \tau h^{-2} ( \bfw - \bfPi^{\rm div}_h \bfw - \nabla( \psi-\pi_h \psi ), \bfz - \nabla \phi )_{\calT_h} {\nonumber}\\
& \quad +( \nabla \cdot(\bfw -\bfPi^{\rm div}_h \bfw ) , \nabla \cdot(\bfz - \nabla \phi) )_{\calT_h} - \langle \nabla \cdot (\bfw -\bfPi^{\rm div}_h \bfw), (\bfz - \nabla \phi) \cdot \bfn \rangle_{\partial \calT_h}{\nonumber}\\
&\quad - \theta ( \bfw - \bfPi^{\rm div}_h \bfw - \nabla( \psi-\pi_h \psi ), \nabla (\nabla \cdot \bfz))_{\calT_h},\end{aligned}$$ $$\begin{aligned}
S_3 = - \left( {\rm cof}(D^2 \xi) \nabla \psi- {\rm cof}(D^2 \xi_h) \nabla \psi_h,\nabla \eta \right)_{\calT_h},\end{aligned}$$ $$\begin{aligned}
S_4 = \left( {\rm cof}(D^2 \xi) \nabla \xi -{\rm cof}(D^2 \xi_h) \nabla \xi_h,\nabla \phi \right)_{\calT_h}.\end{aligned}$$
With (\[lem\_equ\_pre\_vk\]), the desired error equation is obtained by applying similar derivation in Lemma \[lem\_err\_bih\] to (\[sta\_method\_vk\]).
Below, we discuss the main result of this section.
\[thm\_conv\_von\_karman\] Further under the conditions in Lemma \[lemma\_erq\_vk\] and smallness condition on $\|f\|_{H^{-1}(\Omega)},$ if the stabilization parameter $\tau$ in (\[sta\_method\_vk\]) is large enough when $\theta = 1$ and is arbitrary positive $\tau$ for $\theta=-1,$ then following estimate holds: $$\begin{aligned}
\Vert \xi - \xi_{h} \Vert_{2, \calT_{h}} + \Vert \psi - \psi_{h} \Vert_{2, \calT_{h}}
+\| (\bfu - \bfu_h) \|_{H({\rm div},\Omega)} + \|(\bfw - \bfw_h) \|_{H({\rm div},\Omega)}
\leq C h^r\left(\| \bfu \|_{1+r,\Omega} + \| \bfw \|_{1+r,\Omega} \right),\end{aligned}$$ where $r= \min\{\beta, k+1\}$ and $\beta$ is the parameter given in (\[ass\_reg\_vk\]).
Taking $\bfv =\bfe_{\bfu} ,\bfz =\bfe_{\bfw} ,\eta =e_{\xi} ,\phi = e_{\psi}$ in the error equation (\[error\_eq\_vk\]) and applying the similar technique to deal with the terms $B((\bfe_{\bfu},e_{\xi}),(\bfe_{\bfu},e_{\xi})),
B((\bfe_{\bfw},e_{\psi}),(\bfe_{\bfw},e_{\psi}))$ and $S_1, S_2$, we can obtain $$\begin{aligned}
\label{eq:error-1}
\| e_{\xi}\|^2_{2,\calT_h} + \| e_{\psi}\|^2_{2,\calT_h}
+\| \nabla \cdot \bfe_{\bfu} \|^2_{\calT_h} + \| \nabla \cdot \bfe_{\bfw}\|^2_{\calT_h}
\leq Ch^{2r} (\|\bfu\|^2_{1+r,\Omega} + \|\bfw\|^2_{1+r,\Omega} ) +C (|S_3| + |S_4|),\end{aligned}$$ where $S_3$ and $S_4$ are now written as follows: $$\begin{aligned}
S_3 &= - \left( {\rm cof}(D^2 \xi) \nabla \psi- {\rm cof}(D^2 \xi_h) \nabla \psi_h,\nabla e_{\xi} \right)_{\calT_h},\\
S_4 &= \left( {\rm cof}(D^2 \xi) \nabla \xi -{\rm cof}(D^2 \xi_h) \nabla \xi_h,\nabla e_{\psi} \right)_{\calT_h}.\end{aligned}$$ For $S_3$, we obtain $$\begin{aligned}
S_3 &= -\left( ({\rm cof}(D^2 \xi) - {\rm cof}(D^2 \xi_h) ) \nabla \psi,\nabla e_{\xi} \right)_{\calT_h}
- \left( {\rm cof}(D^2 \xi_h) ( \nabla \psi - \nabla\psi_h),\nabla e_{\xi} \right)_{\calT_h}\\
&\leq C \|\nabla \psi\|_{L^{4}(\Omega)}\|{\rm cof}(D^2 \xi)
- {\rm cof}(D^2 \xi_h) \|_{\calT_h}\|\nabla e_{\xi} \|_{L^{4}(\calT_h)} \\
&\quad + C \|{\rm cof}(D^2 \xi_h)\|_{\calT_h}\|\nabla \psi - \nabla\psi_h\|_{L^4(\calT_h)}
\|\nabla e_{\xi} \|_{L^{4}(\calT_h)}\\
& \leq C \|\nabla \psi\|_{L^{4}(\Omega)} \| \xi-\xi_h\|_{2,\calT_h} \| e_{\xi}\|_{2,\calT_h}\\
&\quad + C \|{\rm cof}(D^2 \xi_h)\|_{\calT_h}\| \psi - \psi_h\|_{2,\calT_h}\| e_{\xi} \|_{2,\calT_h}
\quad \text{(by \cite[Theorem $2.1$]{Ern2010a})}\\
& \leq C \|\nabla \psi\|_{L^{4}(\Omega)}( \| \xi-\pi_h \xi\|_{2,\calT_h} + \| e_{\xi}\|_{2,\calT_h} )\| e_{\xi}\|_{2,\calT_h}\\
&\quad + C \|{\rm cof}(D^2 \xi_h)\|_{\calT_h}( \| \psi - \pi_h \psi\|_{2,\calT_h} + \| e_{\psi} \|_{2,\calT_h})
\| e_{\xi} \|_{2,\calT_h} \\
& \leq C \|\psi\|^2_{H^{2}(\Omega)}\Big( h^r \|\xi\|_{2+r} \;\| e_{\xi}\|_{2,\calT_h}
+ \| e_{\xi}\|^2_{2,\calT_h} \Big)\\
&\quad +C \|{\rm cof}(D^2 \xi_h)\|_{\calT_h} \Big(
h^r \|\psi \|_{2+r,\Omega} \| e_{\xi}\|_{2,\calT_h}
+ \| e_{\psi} \|_{2,\calT_h}\| e_{\xi} \|_{2,\calT_h}\Big).\end{aligned}$$ Similarly, for $S_4$, we establish $$\begin{aligned}
S_4 &= \left( ({\rm cof}(D^2 \xi) -{\rm cof}(D^2 \xi_h) ) \nabla \xi ,\nabla e_{\psi} \right)_{\calT_h} + \left( {\rm cof}(D^2 \xi_h) (\nabla \xi - \nabla \xi_h),\nabla e_{\psi} \right)_{\calT_h}\\
&\leq C \|\nabla \xi\|_{L^{4}(\Omega)}\|{\rm cof}(D^2 \xi) -{\rm cof}(D^2 \xi_h)\|_{\calT_h}
\|\nabla e_{\psi} \|_{L^{4}(\calT_h)} \\
&\quad +C \|{\rm cof}(D^2 \xi_h)\|_{\calT_h} \|\nabla \xi - \nabla \xi_h\|_{L^4(\calT_h)}\|\nabla e_{\psi} \|_{L^4(\calT_h)}\\
& \leq C \|\nabla \xi\|_{L^{4}(\Omega)} \| \xi-\xi_h\|_{2,\calT_h} \| e_{\psi} \|_{2,\calT_h}\\
&\quad + C\|{\rm cof}(D^2 \xi_h)\|_{\calT_h} \| \xi - \xi_h\|_{2,\calT_h}\| e_{\psi} \|_{2,\calT_h}
\quad \text{(by \cite[Theorem $2.1$]{Ern2010a})}\\
&\leq C (\|\nabla \xi\|_{L^{4}(\Omega)} +\|{\rm cof}(D^2 \xi_h)\|_{\calT_h} )( \| \xi-\pi_h\xi \|_{2,\calT_h}
+ \| e_{\xi}\|_{2,\calT_h} ) \| e_{\psi} \|_{2,\calT_h}\\
&\leq C(\|\nabla \xi\|_{L^{4}(\Omega)} +\|{\rm cof}(D^2 \xi_h)\|_{\calT_h} )(h^r\|\nabla \xi\|_{1+r,\Omega}
+ \| e_{\xi}\|_{2,\calT_h} ) \| e_{\psi} \|_{2,\calT_h}.\end{aligned}$$ On substitution in (\[eq:error-1\]), and using regularity property (\[ass\_reg\_vk\]) with stability result (\[main\_thm\_sta\_vk\]), we arrive at $$\begin{aligned}
\label{eq:error-2}
(1-C \|f\|_{H^{-1}(\Omega)}) \Big(\| e_{\xi}\|^2_{2,\calT_h} + \| e_{\psi}\|^2_{2,\calT_h}\Big)
+\| \nabla \cdot \bfe_{\bfu} \|^2_{\calT_h} + \| \nabla \cdot \bfe_{\bfw}\|^2_{\calT_h}
\leq Ch^{2r} \;\Big(\|\bfu\|^2_{1+r,\Omega} + \|\bfw\|^2_{1+r,\Omega}\Big).\end{aligned}$$ Choose $\|f\|_{H^{-1}(\Omega)}$ is small enough so that $(1-C \|f\|_{H^{-1}(\Omega)})>0$ and this completes the rest of the proof.
Numerical experiments {#sec:numer_example}
=====================
In this section, we present numerical results that illustrate the efficiency and accuracy of the newly proposed mixed finite element schemes for the biharmonic equation and von Kármán model respectively. In the following we focus on the numerical experiments in two dimensional cases. The grids we use to test are always assumed to be shape regular and quasi-uniform. In the descretization, we always use the BDM finite element space as vector function space and continuous polynomial space as scalar function space. For brevity, the mixed finite element space used in the following computation is denoted by ${BDM}_k$-${P}_{k+1}$ with $k\geq 1$. The computation is performed with FEniCS [@FEniCS]. We remark that our algorithm can be straightforwardly extended to the problem with non-homogeneous boundary conditions and other types of von Kármán model such as shown in [@Brenner2017; @Carstensen19].
Firstly, we presents some numerical results for the biharmonic equation.
\[example1\] We first test our algorithm for the biharmonic equation (\[equ\_biharmonic\]) with $\kappa=1$ on a unit square domain $[0,1]^2$ with exact solution $$u = x^2 (1-x)^2 y^2 (1-y)^2.$$ The source term $f$ can be determined by the above exact solution. For the objective of flexibility in the design of our algorithm, we use an optional parameter $\theta$ in the variational formulation (\[sta\_fem\_biharmonic\]) for the biharmonic equation.
Errors $\|e_u\|_{{{\mathcal T}}_h}$ $\| \nabla e_u\|_{{{\mathcal T}}_h} $ $ \|e_{\bfw}\|_{{{\mathcal T}}_h} $ $\|\nabla \cdot e_{\bfw}\|_{{{\mathcal T}}_h} $ $\|\nabla e_u\|_{1,{{\mathcal T}}_h} $
----------------- ------------------------------ --------------------------------------- ------------------------------------- ------------------------------------------------- ----------------------------------------
$\theta = -1,1$ 8.89e-7 4.24e-6 3.15e-6 9.11e-4 1.15e-3
: (Example \[example1\]) Errors with respect to different choices of $\theta = -1,1$ and fixed $\tau =10$ on a fixed mesh with mesh size $h=0.011$ based on ${BDM}_1$-${P}_{2}$.[]{data-label="table1"}
Errors $\|e_u\|_{{{\mathcal T}}_h}$ $\| \nabla e_u\|_{{{\mathcal T}}_h} $ $ \|e_{\bfw}\|_{{{\mathcal T}}_h} $ $\|\nabla \cdot e_{\bfw}\|_{{{\mathcal T}}_h} $ $\|\nabla e_u\|_{1,{{\mathcal T}}_h} $
-------------- ------------------------------ --------------------------------------- ------------------------------------- ------------------------------------------------- ----------------------------------------
$\tau = 1$ 1.21e-5 5.41e-5 2.34e-5 9.10e-4 1.19e-3
$\tau = 10$ 8.89e-7 4.24e-6 3.15e-6 9.11e-4 1.15e-3
$\tau = 50$ 2.27e-7 2.14e-6 2.70e-6 9.32e-4 1.15e-3
$\tau = 100$ 4.45e-7 3.27e-6 3.54e-6 9.56e-4 1.15e-3
$\tau = 200$ 7.45e-7 5.37e-6 5.48e-6 9.83e-4 1.15e-3
$\tau = 300$ 1.02e-6 7.46e-6 7.52e-6 9.99e-4 1.15e-3
: (Example \[example1\]) Errors with respect to different choices of $\tau = 1,10,50,100,200,300$ and fixed $\theta =1$ on a fixed mesh with mesh size $h=0.011$ based on ${BDM}_1$-${P}_{2}$.[]{data-label="table2"}
In order to test the influence of the choice of $\theta$ and $\tau$, we test this example based on the ${BDM}_1$-${P}_{2}$ mixed finite element space on a fixed mesh ${{\mathcal T}}_h$ with mesh size $h=0.011$. We denote $e_u = u-u_h$ and $e_{\bfw} = \bfw -\bfw_h$. Firstly, we test different choices of $\theta = -1,1$ and fixed $\tau =10$, and we always have the errors as shown in Table \[table1\]. Actually, we also test other choices of $\theta = -0.5, 0, 0.5$, and we get the same errors as in Table \[table1\]. Thus, for this example with smooth solution, we find that the influence of different choices of $\theta$ is small. In order to mainly test the accuracy and efficiency of the proposed algorithm, we always use $\theta = 1$ in the following tests. Next, we test the influence of different choices of $\tau = 1,10, 50, 100, 200, 300$ and $\theta=1$. From the viewpoint of theoretical analysis, the parameter $\tau$ should be chosen to be large enough. However, from Table [\[table2\]]{} we can see that $\tau$ can only be mildly large to get the desired results. We let $\tau=10$ in the following tests for this example.
$h$ $\|e_u\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e_u\|_{{{\mathcal T}}_h} $ order $ \|e_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla e_u\|_{1,{{\mathcal T}}_h} $ order
---------- ------------------------------ ----------- --------------------------------------- ------- ------------------------------------- ------- ------------------------------------------------- ------- ---------------------------------------- -------
$0.3536$ 9.28e-4 – 4.39e-3 – 3.03e-3 – 2.82e-2 – 4.16e-2 –
$0.1768$ 2.29e-4 2.02 1.09e-3 2.01 7.89e-4 1.94 1.45e-2 0.96 1.92e-2 1.12
$0.0884$ 5.70e-5 2.01 2.72e-4 2.00 2.00e-4 1.98 7.27e-3 1.00 9.30e-3 1.05
$0.0442$ 1.42e-5 2.01 6.78e-5 2.00 5.03e-5 1.99 3.64e-3 1.00 4.61e-3 1.01
$0.0221$ 3.56e-6 2.00 1.70e-5 2.00 1.26e-5 2.00 1.82e-3 1.00 2.30e-3 1.00
$0.0110$ 8.89e-7 2.00 4.24e-6 2.00 3.15e-6 2.00 9.11e-4 1.00 1.15e-3 1.00
$0.0055$ 2.22e-7 2.00 1.05e-6 2.01 7.88e-7 2.00 4.56e-4 1.00 5.74e-4 1.00
: (Example \[example1\]) Convergence history based on ${BDM}_1$-${P}_{2}$.[]{data-label="table3"}
$h$ $\|e_u\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e_u\|_{{{\mathcal T}}_h} $ order $ \|e_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla e_u\|_{1,{{\mathcal T}}_h} $ order
---------- ------------------------------ ----------- --------------------------------------- ------- ------------------------------------- ------- ------------------------------------------------- ------- ---------------------------------------- -------
$0.3536$ 1.08e-4 – 1.32e-3 – 1.67e-3 – 7.27e-3 – 2.80e-2 –
$0.1768$ 7.16e-6 3.91 1.71e-4 2.95 2.15e-4 2.96 1.85e-3 1.97 7.13e-3 1.97
$0.0884$ 4.57e-7 3.97 2.17e-5 2.98 2.71e-5 2.99 4.67e-4 1.99 1.75e-3 2.03
$0.0442$ 2.88e-8 3.99 2.74e-6 2.99 3.39e-6 3.00 1.17e-4 2.00 4.30e-4 2.02
$0.0221$ 1.83e-9 3.98 3.44e-7 2.99 4.23e-7 3.00 2.92e-5 2.00 1.06e-4 2.02
: (Example \[example1\]) Convergence history based on ${BDM}_2$-${P}_{3}$.[]{data-label="table4"}
We further test this example based on the mixed finite element spaces ${BDM}_1$-${P}_{2}$ and ${BDM}_2$-${P}_{3}$ respectively. From Tables \[table3\]-\[table4\], we can see that the errors always achieve almost optimal orders of convergence which are consistent with the theoretical analysis.
\[example2\] In this example we test our algorithm for the biharmonic equation (\[equ\_biharmonic\]) without exact solution in two L-shape type domains $\Omega_1 = [1,2]^2\setminus [1,1.5]^2$ and $\Omega_2 = [1,2]^2\setminus ([1,4/3]\times [4/3,5/3] \cup [1,5/3]\times [1,4/3])$. Let $\kappa=1$. By an elliptic regularity result for the clamped Kirchhoff plate (cf. [@Brenner2017 Lemma 1.1],[@BR1980]), the parameter $\delta$ given in (\[reg\_ass\]) holds that $\delta\in(1/2,1)$ for the solutions in this example. Due to the low regularity property of the solutions in this example, we only consider the approximation of the solutions based on the lowest order of mixed finite element space ${BDM}_1$-${P}_{2}$ in the proposed algorithm.
![(Example \[example2\]) Left: The solution $u_h$ for the problem in $\Omega_1$ on the mesh with mesh size $h=0.00435$. Right: The solution $u_h$ for the problem in $\Omega_2$ on the mesh with mesh size $h=0.00435$.[]{data-label="ex2_fig"}](ex2_fig1.jpg "fig:"){width="7cm" height="5.4cm"} ![(Example \[example2\]) Left: The solution $u_h$ for the problem in $\Omega_1$ on the mesh with mesh size $h=0.00435$. Right: The solution $u_h$ for the problem in $\Omega_2$ on the mesh with mesh size $h=0.00435$.[]{data-label="ex2_fig"}](ex2_fig2.jpg "fig:"){width="7cm" height="5.4cm"}
$h$ $\|e^\ast_u\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e^\ast_u\|_{{{\mathcal T}}_h} $ order $ \|e^\ast_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e^\ast_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla e^\ast_u\|_{1,{{\mathcal T}}_h} $ order
---------- ----------------------------------- ----------- -------------------------------------------- ------- ------------------------------------------ ------- ------------------------------------------------------ ------- --------------------------------------------- -------
$0.2841$ 7.96e-5 – 5.68e-4 – 6.38e-4 – 9.83e-3 – 8.76e-3 –
$0.1368$ 2.17e-5 1.88 1.58e-4 1.85 1.75e-4 1.87 5.65e-3 0.80 5.34e-3 0.71
$0.0680$ 7.95e-6 1.45 5.79e-5 1.45 6.27e-5 1.48 3.93e-3 0.52 3.94e-3 0.44
$0.0345$ 3.89e-6 1.03 2.74e-5 1.08 2.95e-5 1.09 2.94e-3 0.42 2.80e-3 0.49
: (Example \[example2\]) Convergence history for the problem in $\Omega_1$.[]{data-label="ex2_table1"}
$h$ $\|e^\ast_u\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e^\ast_u\|_{{{\mathcal T}}_h} $ order $ \|e^\ast_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e^\ast_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla e^\ast_u\|_{1,{{\mathcal T}}_h} $ order
---------- ----------------------------------- ----------- -------------------------------------------- ------- ------------------------------------------ ------- ------------------------------------------------------ ------- --------------------------------------------- -------
$0.2417$ 3.83e-5 – 2.88e-4 – 3.26e-4 – 7.03e-3 – 6.48e-3 –
$0.1320$ 1.69e-5 1.18 1.32e-4 1.13 1.43e-4 1.19 5.01e-3 0.49 5.14e-3 0.33
$0.0684$ 9.11e-6 0.89 6.44e-5 1.04 6.85e-5 1.06 3.63e-3 0.46 3.53e-3 0.54
$0.0347$ 3.50e-6 1.38 2.54e-5 1.34 2.71e-5 1.34 2.64e-3 0.46 2.56e-3 0.46
$0.0172$ 1.49e-6 1.23 1.11e-5 1.19 1.15e-5 1.24 1.70e-3 0.64 1.73e-3 0.57
: (Example \[example2\]) Convergence history for the problem in $\Omega_2$.[]{data-label="ex2_table2"}
We test the case with source term $f=1$ and set the parameter $\tau=200$. Figure \[ex2\_fig\] shows the solutions of the problems in $\Omega_1$ and $\Omega_2$ respectively on the finest mesh with mesh size $h=0.00435$. Let $u^\ast$ and $\bfw^\ast$ be the approximation solutions on the finest mesh. We denote the errors $e^\ast_u = u^\ast - u_h$ and $e^\ast_\bfw = \bfw^\ast - \bfw_h$. Tables \[ex2\_table1\]-\[ex2\_table2\] indicate that the errors also achieve almost optimal orders of convergence.
\[example3\] In this example we test our algorithm for the biharmonic equation (\[equ\_biharmonic\]) with variable coefficient which can also be assumed to satisfy the non-homogeneous boundary conditions. We consider the biharmonic equation (\[equ\_biharmonic1\]) on a unit square domain $[0,1]^2$ with the exact solution as follows: $$u = sin(2\pi x) sin(2\pi y).$$ We assume $\kappa(\bfx) = x^2+y^2+1$, then the source term $f$ and the boundary conditions can be determined by the exact solution and the coefficient $\kappa(\bfx)$.
$h$ $\|e_u\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e_u\|_{{{\mathcal T}}_h} $ order $ \|e_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla e_u\|_{1,{{\mathcal T}}_h} $ order
----------- ------------------------------ ----------- --------------------------------------- ------- ------------------------------------- ------- ------------------------------------------------- ------- ---------------------------------------- -------
$0.3536$ 6.96e-1 – 6.57 – 1.51 – 19.42 – 64.27 –
$0.1768$ 1.78e-1 1.97 1.63 2.01 4.27e-1 1.82 10.19 0.93 21.18 1.60
$0.0884$ 4.46e-2 2.00 4.04e-1 2.01 1.11e-1 1.94 5.15 0.98 7.94 1.42
$0.0442$ 1.11e-2 2.01 1.01e-1 2.00 2.79e-2 1.99 2.58 1.00 3.52 1.17
$0.0221 $ 2.78e-3 2.00 2.52e-2 2.00 7.00e-3 1.99 1.29 1.00 1.70 1.05
$0.0110 $ 6.95e-4 2.00 6.29e-3 2.00 1.75e-3 2.00 6.46e-1 1.00 8.42e-1 1.01
$0.0055 $ 1.74e-4 2.00 1.57e-3 2.00 4.38e-4 2.00 3.23e-1 1.00 4.20e-1 1.00
: (Example \[example3\]) Convergence history based on ${BDM}_1$-${P}_{2}$.[]{data-label="ex3_table1"}
$h$ $\|e_u\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e_u\|_{{{\mathcal T}}_h} $ order $ \|e_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla e_u\|_{1,{{\mathcal T}}_h} $ order
---------- ------------------------------ ----------- --------------------------------------- ------- ------------------------------------- ------- ------------------------------------------------- ------- ---------------------------------------- -------
$0.3536$ 1.94e-1 – 2.64 – 2.27 – 7.06 – 61.94 –
$0.1768$ 1.44e-2 3.75 3.46e-1 2.93 3.21e-1 2.82 1.83 2.05 14.03 2.14
$0.0884$ 9.43e-4 3.93 4.38e-2 2.98 4.13e-2 2.96 4.58e-1 2.00 3.34 2.07
$0.0442$ 5.97e-5 3.98 5.49e-3 3.00 5.21e-3 2.99 1.15e-1 1.99 8.22e-1 2.02
$0.0221$ 3.74e-6 4.00 6.87e-4 3.00 6.52e-4 3.00 2.86e-2 2.01 2.05e-1 2.00
$0.0110$ 2.39e-7 3.97 8.59e-5 3.00 8.15e-5 3.00 7.16e-3 2.00 5.11e-2 2.00
: (Example \[example3\]) Convergence history based on ${BDM}_2$-${P}_{3}$.[]{data-label="ex3_table2"}
We test our algorithm with $\theta=1$ and $\tau=10$ for this example based on the mixed finite element spaces ${BDM}_1$-${P}_{2}$ and ${BDM}_2$-${P}_{3}$ respectively. From Tables \[ex3\_table1\]-\[ex3\_table2\], we can see that for this example, our algorithm always achieve almost optimal convergence.
Next, we start to test our algorithm for the solution of von Kármán equation (\[equ\_von\_karman\]). For simplicity, in the following experiments we always test the proposed algorithm based on the mixed finite element space ${BDM}_1$-${P}_{2}$. Actually, the new algorithm can be easily extended for the solution of the von Kármán model (cf. [@Brenner2017]) as follows:
\[equ\_von\_karman\_v1\] $$\begin{aligned}
\Delta^2 \xi - [\xi,\psi] + p \Delta \xi = f, & \ \ \textrm{in} \ \ \Omega, \label{vk1_eq1}\\
\Delta^2 \psi + [\xi,\xi]= g, & \ \ \textrm{in} \ \ \Omega,\label{vk1_eq2} \\
\xi =\frac{\partial \xi}{\partial n} = 0, & \ \ \textrm{on} \ \ \partial \Omega, \label{vk1_bc1}\\
\psi = \frac{\partial \psi}{\partial n} = 0, & \ \ \textrm{on} \ \ \partial \Omega,\label{vk1_bc2}\end{aligned}$$
where $p$ is a given positive constant and the boundary conditions can also be non-homogeneous. Similar to (\[sta\_method\_vk\]), we seek an approximation $(\bfu_h,\bfw_h,\xi_h,\psi_h)\in \bfW_h\times \bfW_h\times V_h \times V_h$ such that
\[sta\_method\_vk\_1\] $$\begin{aligned}
B_{\theta}((\bfu_h,\xi_h),(\bfv,\eta))+({\rm cof}(D^2\xi_h)\nabla \psi_h,\nabla \eta)_{\calT_h} - (p \nabla \xi_h, \eta) &= (f,\eta)_{\calT_h},\\
B_{\theta} ((\bfw_h,\psi_h),(\bfz,\phi)) -({\rm cof}(D^2\xi_h)\nabla \xi_h,\nabla \phi)_{\calT_h} &= (g,\phi)_{\calT_h},\end{aligned}$$
for any $(\bfv,\bfz,\eta,\phi) \in \bfW_h\times \bfW_h\times V_h \times V_h$. The well-posedness of (\[sta\_method\_vk\_1\]) and the associated error estimates can be similarly analyzed.
\[example4\] In this example we test the proposed algorithm based on the formulation (\[sta\_method\_vk\_1\]) with $\theta =1$ for the von Kármán equation (\[equ\_von\_karman\_v1\]) with $p=0$ on a unit square domain $[0,1]^2$. The exact solution $(\xi,\psi)$ is assumed to be $$\xi = x^2(1-x)^2y^2(1-y)^2,\quad \psi = \sin^2(\pi x) \sin^2(\pi y),$$ and the source term $f$ in (\[vk1\_eq1\]) and the right-hand side $g$ in (\[vk1\_eq2\]) can be determined respectively. For the solution of nonlinear system, one can use nonlinear system solver such as Picard iteration, Newton iteration. In the following experiments, we always apply Picard iteration with zero initial guess to solve the nonlinear system.
$h$ $\|e_\xi\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e_\xi\|_{{{\mathcal T}}_h} $ order $ \|e_{\bfu}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e_{\bfu}\|_{{{\mathcal T}}_h} $ order $\|\nabla e_\xi\|_{1,{{\mathcal T}}_h} $ order
---------- -------------------------------- ----------- ----------------------------------------- ------- ------------------------------------- ------- ------------------------------------------------- ------- ------------------------------------------ -------
$0.3536$ 2.06e-3 – 9.55e-3 – 3.69e-3 – 3.43e-2 – 6.14e-2 –
$0.1768$ 4.26e-4 2.27 1.95e-3 2.29 8.17e-4 2.18 1.50e-2 1.19 2.10e-2 1.55
$0.0884$ 1.01e-4 2.08 4.64e-4 2.07 1.99e-4 2.04 7.33e-3 1.03 9.51e-3 1.14
$0.0442$ 2.50e-5 2.01 1.15e-4 2.01 4.95e-5 2.01 3.65e-3 1.01 4.63e-3 1.04
$0.0221$ 6.24e-6 2.00 2.86e-5 2.01 1.24e-5 2.00 1.82e-3 1.00 2.30e-3 1.01
$0.0110$ 1.56e-6 2.00 7.14e-6 2.00 3.09e-6 2.00 9.11e-4 1.00 1.15e-3 1.00
: (Example \[example4\]) Convergence history for the approximation of $\xi$ and $\bfu$.[]{data-label="ex4_table1"}
$h$ $\|e_\psi\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e_\psi\|_{{{\mathcal T}}_h} $ order $ \|e_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla e_\psi\|_{1,{{\mathcal T}}_h} $ order
---------- --------------------------------- ----------- ------------------------------------------ ------- ------------------------------------- ------- ------------------------------------------------- ------- ------------------------------------------- -------
$0.3536$ 1.68e-1 – 1.00 – 5.70e-1 – 6.01 – 11.93 –
$0.1768$ 4.20e-2 2.00 2.45e-1 2.03 1.55e-1 1.88 3.12 0.95 4.70 1.34
$0.0884$ 1.05e-2 2.00 6.06e-2 2.02 3.97e-2 1.97 1.58 0.98 2.11 1.16
$0.0442$ 2.62e-3 2.00 1.51e-2 2.00 9.99e-3 1.99 7.92e-1 1.00 1.02 1.05
$0.0221$ 6.54e-4 2.00 3.78e-3 2.00 2.50e-3 2.00 3.96e-1 1.00 5.06e-1 1.01
$0.0110$ 1.64e-4 2.00 9.44e-4 2.00 6.26e-4 2.00 1.98e-1 1.00 2.52e-1 1.01
: (Example \[example4\]) Convergence history for the approximation of $\psi$ and $\bfw$.[]{data-label="ex4_table2"}
We denote $e_\xi = \xi-\xi_h$, $e_{\bfu} = \bfu -\bfu_h$, $e_\psi = \psi-\psi_h$, $e_{\bfw} = \bfw -\bfw_h$. We test our algorithm with $\tau=10$ for this example. In fact, the exact solutions of this example are sufficiently smooth, and Tables \[ex4\_table1\]-\[ex4\_table2\] further show that the errors always achieve almost optimal orders of convergence as the theoretical results.
\[example5\] Now we test our algorithm for the von Kármán model (\[equ\_von\_karman\_v1\]) in the L-shape type domain $\Omega = [1,2]^2\setminus ([1.5,2]\times[1,1.5])$. We assume $f=10$ and $g=10$ in (\[equ\_von\_karman\_v1\]) for this example and choose the parameter $\tau=200$ in the proposed algorithm for this example.
$h$ $\|e^\ast_\xi\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e^\ast_\xi\|_{{{\mathcal T}}_h} $ order $ \|e^\ast_{\bfu}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e^\ast_{\bfu}\|_{{{\mathcal T}}_h} $ order $\|\nabla e^\ast_\xi\|_{1,{{\mathcal T}}_h} $ order
---------- ------------------------------------- ----------- ---------------------------------------------- ------- ------------------------------------------ ------- ------------------------------------------------------ ------- ----------------------------------------------- -------
$0.2588$ 7.09e-4 – 5.08e-3 – 5.55e-3 – 8.92e-2 – 8.18e-2 –
$0.1329$ 2.49e-4 1.51 1.80e-3 1.50 1.98e-3 1.49 5.86e-2 0.61 5.44e-2 0.59
$0.0656$ 9.86e-5 1.34 6.90e-4 1.38 7.42e-4 1.42 3.88e-2 0.59 3.72e-2 0.55
$0.0329$ 4.97e-5 0.99 3.39e-4 1.03 3.54e-4 1.07 2.55e-2 0.61 2.44e-2 0.61
$0.0166$ 1.33e-5 1.90 9.73e-5 1.80 1.00e-4 1.82 1.40e-2 0.87 1.47e-2 0.73
: (Example \[example5\]) Convergence history for the approximation of $\xi$ and $\bfu$ for the case with $p=0$.[]{data-label="ex5_table1"}
$h$ $\|e^\ast_\psi\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e^\ast_\psi\|_{{{\mathcal T}}_h} $ order $ \|e^\ast_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e^\ast_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla e^\ast_\psi\|_{1,{{\mathcal T}}_h} $ order
---------- -------------------------------------- ----------- ----------------------------------------------- ------- ------------------------------------------ ------- ------------------------------------------------------ ------- ------------------------------------------------ -------
$0.2588$ 7.09e-4 – 5.08e-3 – 5.55e-3 – 8.91e-2 – 8.18e-2 –
$0.1329$ 2.49e-4 1.51 1.80e-3 1.50 1.98e-3 1.49 5.86e-2 0.60 5.43e-2 0.59
$0.0656$ 9.86e-5 1.34 6.90e-4 1.38 7.42e-4 1.42 3.88e-2 0.59 3.72e-2 0.55
$0.0329$ 4.97e-5 0.99 3.39e-4 1.03 3.54e-4 1.07 2.55e-2 0.61 2.44e-2 0.61
$0.0166$ 1.33e-5 1.90 9.72e-5 1.80 1.00e-4 1.82 1.40e-2 0.87 1.47e-2 0.73
: (Example \[example5\]) Convergence history for the approximation of $\psi$ and $\bfw$ for the case with $p=0$.[]{data-label="ex5_table2"}
$h$ $\|e^\ast_\xi\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e^\ast_\xi\|_{{{\mathcal T}}_h} $ order $ \|e^\ast_{\bfu}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e^\ast_{\bfu}\|_{{{\mathcal T}}_h} $ order $\|\nabla e^\ast_\xi\|_{1,{{\mathcal T}}_h} $ order
---------- ------------------------------------- ----------- ---------------------------------------------- ------- ------------------------------------------ ------- ------------------------------------------------------ ------- ----------------------------------------------- -------
$0.2588$ 1.34e-3 – 9.25e-3 – 9.81e-3 – 1.34e-1 – 1.22e-1 –
$0.1329$ 4.98e-4 1.43 3.40e-3 1.44 3.62e-3 1.44 8.46e-2 0.66 7.83e-2 0.64
$0.0656$ 1.94e-4 1.36 1.29e-3 1.40 1.35e-3 1.42 5.46e-2 0.63 5.24e-2 0.58
$0.0329$ 9.49e-5 1.03 6.16e-4 1.07 6.34e-4 1.09 3.57e-2 0.61 3.41e-2 0.62
$0.0166$ 2.56e-5 1.89 1.75e-4 1.82 1.79e-4 1.82 1.96e-2 0.87 2.05e-2 0.73
: (Example \[example5\]) Convergence history for the approximation of $\xi$ and $\bfu$ for the case with $p=20$.[]{data-label="ex5_table3"}
$h$ $\|e^\ast_\psi\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e^\ast_\psi\|_{{{\mathcal T}}_h} $ order $ \|e^\ast_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e^\ast_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla e^\ast_\psi\|_{1,{{\mathcal T}}_h} $ order
---------- -------------------------------------- ----------- ----------------------------------------------- ------- ------------------------------------------ ------- ------------------------------------------------------ ------- ------------------------------------------------ -------
$0.2588$ 7.08e-4 – 5.08e-3 – 5.54e-3 – 8.91e-2 – 8.18e-2 –
$0.1329$ 2.49e-4 1.51 1.80e-3 1.50 1.98e-3 1.48 5.86e-2 0.60 5.43e-2 0.59
$0.0656$ 9.86e-5 1.34 6.90e-4 1.38 7.42e-4 1.42 3.88e-2 0.59 3.72e-2 0.55
$0.0329$ 4.96e-5 0.99 3.39e-4 1.03 3.54e-4 1.07 2.55e-2 0.61 2.44e-2 0.61
$0.0166$ 1.33e-5 1.90 9.72e-5 1.80 1.00e-4 1.82 1.40e-2 0.87 1.47e-2 0.73
: (Example \[example5\]) Convergence history for the approximation of $\psi$ and $\bfw$ for the case with $p=20$.[]{data-label="ex5_table4"}
We test two cases of von Kármán model (\[equ\_von\_karman\_v1\]) with $p=0$ and $p=20$ respectively. Let $\xi^\ast,\bfu^\ast,\psi^\ast,\bfw^\ast$ be the approximation solutions on the finest mesh with mesh size $h=0.00831$. We denote the errors $e_\xi^\ast = \xi^\ast - \xi^\ast_h$, $e^\ast_{\bfu} = {\bfu}^\ast - {\bfu}_h$, $e_{\psi}^\ast = \psi^\ast - \psi^\ast_h$, $e^\ast_\bfw = \bfw^\ast - \bfw_h$.
Tables \[ex5\_table1\]-\[ex5\_table2\] show the errors for the case with $p=0$, and the errors for the case with $p=20$ are shown in Tables \[ex5\_table3\]-\[ex5\_table4\]. As the regularity result in Example \[example2\], the parameter $\beta$ in (\[ass\_reg\_vk\]) holds for $\beta \in (1/2,1)$ for the solutions in this example. We can see from Tables \[ex5\_table1\]-\[ex5\_table4\] that the errors from different cases achieve almost optimal orders of convergence.
\[example6\] We further test our algorithm for the von Kármán model (\[equ\_von\_karman\_v1\]) with $p=20$ in another L-shape type domain $\Omega = [1,2]^2\setminus ([1,4/3]\times [4/3,5/3] \cup [1,5/3]\times [1,4/3])$. We assume $f=100$ and $g=1$ in (\[equ\_von\_karman\_v1\]) for this example and also choose the parameter $\tau=200$ in the proposed algorithm for this example.
$h$ $\|e^\ast_\xi\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e^\ast_\xi\|_{{{\mathcal T}}_h} $ order $ \|e^\ast_{\bfu}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e^\ast_{\bfu}\|_{{{\mathcal T}}_h} $ order $\|\nabla e^\ast_\xi\|_{1,{{\mathcal T}}_h} $ order
----------- ------------------------------------- ----------- ---------------------------------------------- ------- ------------------------------------------ ------- ------------------------------------------------------ ------- ----------------------------------------------- -------
$0.2313$ 7.95e-3 – 5.54e-2 – 5.93e-2 – 9.83e-1 – 8.99e-1 –
$0.1270$ 3.64e-3 1.13 2.58e-2 1.10 2.70e-2 1.14 6.71e-1 0.55 6.84e-1 0.39
$0.0658 $ 1.87e-3 0.96 1.25e-2 1.05 1.29e-2 1.07 4.60e-1 0.54 4.49e-1 0.61
$0.0335 $ 7.33e-4 1.35 4.98e-3 1.33 5.12e-3 1.33 3.13e-1 0.55 3.08e-1 0.54
$0.0166 $ 2.23e-4 1.72 1.56e-3 1.67 1.59e-3 1.69 1.64e-1 0.93 1.71e-1 0.85
: (Example \[example6\]) Convergence history for the approximation of $\xi$ and $\bfu$.[]{data-label="ex6_table1"}
$h$ $\|e^\ast_\psi\|_{{{\mathcal T}}_h}$ [order]{} $\| \nabla e^\ast_\psi\|_{{{\mathcal T}}_h} $ order $ \|e^\ast_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla \cdot e^\ast_{\bfw}\|_{{{\mathcal T}}_h} $ order $\|\nabla e^\ast_\psi\|_{1,{{\mathcal T}}_h} $ order
----------- -------------------------------------- ----------- ----------------------------------------------- ------- ------------------------------------------ ------- ------------------------------------------------------ ------- ------------------------------------------------ -------
$0.2313$ 1.89e-4 – 1.43e-3 – 1.45e-3 – 1.46e-2 – 1.35e-2 –
$0.1270$ 9.70e-5 0.96 7.41e-4 0.95 7.50e-4 0.95 8.34e-3 0.81 8.05e-3 0.75
$0.0658 $ 4.94e-5 0.97 3.71e-4 1.00 3.74e-4 1.00 4.45e-3 0.91 4.48e-3 0.85
$0.0335 $ 2.07e-5 1.25 1.52e-4 1.29 1.53e-4 1.29 2.10e-3 1.08 2.31e-3 0.96
$0.0166 $ 6.77e-6 1.61 4.88e-5 1.64 4.89e-5 1.65 8.64e-4 1.28 1.06e-3 1.12
: (Example \[example6\]) Convergence history for the approximation of $\psi$ and $\bfw$.[]{data-label="ex6_table2"}
Since there are not exact solutions for this example, we also use the approximation solutions on the finest mesh with mesh size $h=0.00843$ to test the convergence of the proposed algorithm. For the regularity result of the solutions in this example, the parameter $\beta$ in (\[ass\_reg\_vk\]) holds for $\beta \in (1/2,1)$. We can see from Tables \[ex6\_table1\]-\[ex6\_table2\] that most of the convergence rates of different kinds of errors are nearly optimal.
Conclusions {#sec:conclusion}
===========
We propose a new mixed finite element scheme using element-wise stabilization for the biharmonic equation on Lipshcitz polyhedral domains in any dimension. When solving the biharmonic equation, one merit of this scheme is that it produces symmetric and positive definite linear system, and the discrete $H^{2}$-stability and optimal convergence are obtained. Moreover, we extend the new method to solve the von Kármán equations. The existence, uniqueness and stability for the nonlinear system based on the new scheme, and the $H^{2}$-optimal convergence rate are also obtained. For the numerical experiments that we have performed thus far, our new scheme has desired efficiency and convergence rates, when solving the biharmonic equation and the von Kármán equations.
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[^1]: The work of Huangxin Chen was supported by the NSF of China (Grant No. 11771363) and the Fundamental Research Funds for the Central Universities (Grant No. 20720180003). The work of Amiya K. Pani is supported by IITB Chair Professor’s fund and also partly by a MATRIX Grant No. MTR/201S/000309 (SERB, DST, Govt. India). Weifeng Qiu is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302219). The third author is the corresponding author.
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"pile_set_name": "ArXiv"
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![ [**Chemical structure of the ejected debris for a subset of the explosion models.**]{} Blue represents intermediate mass elements (i.e., silicon, sulfur, calcium), green stable iron group elements produced by electron capture, and red [$^{56}\mathrm{Ni}$]{}. The turbulent inner regions reflect Rayleigh-Taylor and other instabilities that develop during the initial deflagration phase of burning. The subsequent detonation wave enhances the [$^{56}\mathrm{Ni}$]{} production in the center by burning remaining pockets of fuel. The lower density outer layers of debris, processed only by the detonation, consist of smoothly distributed intermediate mass elements. ](kasen_f1.pdf){width="6.0in"}
![ [**Synthetic multi-color light curves and spectra of a representative explosion model compared to observations of a normal Type Ia supernova.**]{} [**a.**]{} The angle-averaged light curves of model DFD\_iso\_06\_dc2 (solid lines) show good agreement with filtered observations of SN 20003du (Stanishev et al., 2007; filled circles) in wavelength bands corresponding to the ultraviolet (U) optical (B,V,R), and near infrared (I). [**b.**]{} The synthetic spectra of the model (black lines) compare well to observations of SN2003du (red lines) taken at times marked in days relative to B light curve maximum. Over time, as the remnant expands and thins, the spectral absorption features reflect the chemical composition of progressively deeper layers of debris, providing a strong test of the predicted compositional stratification of the model. ](kasen_f2.pdf){width="6.0in"}
![ [**Correlation of the peak brightness of the models with their light curve duration and color.**]{} The sample includes 44 models each plotted for 30 different viewing angles. Solid circles denote models computed with the most likely range of detonation criteria, while open circles denote more extreme values. [**a.**]{} Relation between the peak brightness $M_B$ (measured in the logarithmic magnitude scale) and the light curve decline rate parameter [$\Delta M_{15}$]{}, defined as the decrease in B-band brightness from peak to 15 days after peak. The shaded band shows the approximate slope and spread of the observed width-luminosity relation. [**b.**]{} Relation between $M_B$ and the color parameter B-V measured at peak. The solid line shows the slope of the observed relation of Philips et al. (1999) but with the normalization shifted, as the models are systematically redder than observed SNe Ia by 7%, likely due to the approximate treatment of non-LTE effects. In observational studies, these two relations are usually fitted jointly as: $M_B = M_{B,0}
+ \alpha (s - 1) + \beta [ (B-V)_{\rm Bmax} + (B-V)_0]$, where $s$ is a stretch parameter and $(B-V)_0$ is the color of a fiducial supernova. We take $(B-V)_0 = 0$ and determine stretch using the first order relation: $s = 1 - ({\ensuremath{\Delta M_{15}}}-1.1)/1.7)$. We find for the models fitted values of $\alpha = 2.25, \beta =
4.45$ and $M_{B,0} = -19.27$ which are in reasonable agreement with those derived from the recent observational sample of Astier et al. (2006): $\alpha = 1.52, \beta = 1.57$, and $M_{B,0} = -19.31 + 5 \log_{10}(H_0/70)$, where $H_0$ is the Hubble parameter. ](kasen_f3.pdf){width="6.5in"}
![ [**Effect of the metal content of the progenitor star population on the width-luminosity relation.**]{} The models explore two extreme values of the metallicity: 3 times (red points) and 0.3 times the solar value (blue points). For clarity, each model has been averaged over all viewing angles, and black lines connect similar explosion models of differing metallicity. The colored lines are linear fits to the width-luminosity relation of of the two metallicity samples separately. The diversity introduced by metallicity variations follows the general width-luminosity trend, but the slightly different normalization and slope of the relation for different metallicity samples indicates a potential source of systematic error in distance determinations. ](kasen_f4.pdf){width="6.0in"}
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"pile_set_name": "ArXiv"
}
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---
abstract: 'In the lepton-specific version of two Higgs doublet models, a discrete symmetry is used to couple one Higgs, $\Phi_2$, to quarks and the other, $\Phi_1$, to leptons. The symmetry eliminates tree level flavor changing neutral currents (FCNC). Motivated by strong constraints on such currents in the quark sector from meson-antimeson mixing, and by hints of $h \to \mu\tau$ in the lepton sector, we study a simple three Higgs doublet model in which one doublet couples to quarks and the other two to leptons. Unlike most other studies of three Higgs doublet models, we impose no flavor symmetry and just use a $Z_2$ symmetry to constrain the Yukawa couplings. We present the model and discuss the various mixing angles. Constraining the parameters to be consistent with observations of the Higgs boson at the LHC, we study the properties of the charged Higgs boson(s) in the model, focusing on the case in which the charged Higgs is above the top threshold. It is found that one can have the branching fraction of the charged Higgs into $\tau\nu_\tau$ comparable to $t\bar{b}$ decay without needing very large values for the ratios of vevs. One can also get a large branching fraction for the much more easily observable $\mu\nu_\tau$ decay.'
author:
- Marco Merchand
- Marc Sher
title: 'A Three Doublet Lepton-Specific Model'
---
Introduction
============
In 2012, the discovery of the Higgs boson[@Aad:2012tfa; @Chatrchyan:2012xdj] was announced. To date, the properties of the Higgs have not significantly deviated from the expectations of the Standard Model. Nonetheless, many expect new physics to be found at the TeV scale. The Standard Model has a metastable vacuum, a hierarchy problem and no dark matter candidate. In most TeV scale extensions of the Standard Model, including supersymmetric models [@Martin:1997ns], composite Higgs models [@Agashe:2004rs], twin Higgs models [@Chacko:2005pe], and left-right models [@Mohapatra:1974hk], there are additional Higgs fields. Thus, considerations of extensions of the Higgs sector are well motivated.
The most studied extension of the Higgs sector is the Two Higgs Doublet Model (2HDM); see Ref. [@Branco:2011iw] for an extensive review. As soon as one extends the Higgs sector, tree level flavor-changing neutral currents (FCNC) become an issue [@Paschos:1976ay; @Glashow:1976nt]. These can be avoided by imposing a $Z_2$ discrete symmetry such that all fermions of a given charge couple to the same multiplet. In the 2HDM, there are four models generally discussed. In the type I model, all fermions couple to one Higgs (generally taken to be $\Phi_2$), and in the type II model, the Q=2/3 quarks couple to $\Phi_2$ and the $Q=-1/3$ quarks and leptons couple to $\Phi_1$. In the other two models, the lepton specific and flipped models, the leptons couple to the other Higgs. For the lepton specific model, in particular, the quarks all couple to $\Phi_2$ and the leptons all couple to $\Phi_1$.
There are hundreds of papers on these four models. However, any discovery of a tree level FCNC would invalidate them. We are intrigued by the recent hints [@Khachatryan:2015kon] of a nonzero branching ratio of the light Higgs into $\mu\tau$ with 2.4$\sigma$ significance, with a branching ratio of $0.84\pm 0.38\%$. More recent results have not seen a signal, but the errors are large, with Ref. [@CMS:2016qvi] giving a branching ratio of $0.76\pm 0.82\%$ and Ref. [@Aad:2016blu] giving a branching ratio of $0.53\pm 0.51\%$. If such a signal is confirmed, then the conventional 2HDMs will be excluded and models with a different structure in the lepton sector will be favored. In the (more likely) event that it is not confirmed, it is still the case that bounds on scalar mediated FCNC in the quark sector are much stronger (due to meson-antimeson mixing) than those in the lepton sector, and thus we wish to study extensions of the Higgs sector with FCNC in the lepton sector but not in the quark sector.
There have been some studies of alternative 2HDMs that can address lepton flavor-violation. For example, Altmannshofer, et al. [@Altmannshofer:2015esa], Ghosh, et al. [@Ghosh:2015gpa] and Botella et al. [@Botella:2016krk] consider models in which the Standard Model Higgs couples to the third generation and the other Higgs (which may be composite) to the other two, and they study higher-dimensional operators and the effect on the low energy theory.
Since leptons must transform differently than quarks in this case, the simplest model is to extend the lepton specific model by adding a third doublet, $\Phi_3$, which behaves under the $Z_2$ symmetry exactly as $\Phi_1$. There will then be only one doublet coupling to quarks and the other two to leptons. The three doublet model has two physical pairs of charged Higgs fields, two pseudoscalars and three neutral scalars.
Three Higgs doublet models (3HDMs) have been studied previously. They allow for a very rich structure of discrete symmetry groups in the Higgs and flavor sectors. A detailed analysis of these symmetries has been carried out by Ivanov and collaborators, both for discrete symmetries [@Ivanov:2012ry; @Ivanov:2014doa] and Abelian symmetries [@Ivanov:2011ae] and Keus, King and Moretti [@Keus:2013hya] focused on flavor symmetries and analyzed Higgs masses and potentials in several models. A specific model with an $S_3$ symmetry was considered in Refs. [@Das:2014fea; @Das:2015sca] and one with an $A_4$ symmetry, that also discusses $h\to\mu\tau$ is in Ref. [@Heeck:2014qea]. Moretti and Yagyu [@Moretti:2015cwa] studied perturbative unitarity in 3HDMs in which some of the doublets are inert. More phenomenological analyses were carried out by Aranda, et al. [@Aranda:2013kq; @Aranda:2014jua] in the context of a 3HDM with a $\mathbb{Z}_5$ symmetry. A model with a global $U(1)$ flavor symmetry by Crivellin, DÕAmbrosio and Heeck [@Crivellin:2015lwa] not only can account for $h\to\mu\tau$ but can also resolve anomalies in $B\to K\mu\mu$. The possibility of a light charged Higgs and constraints from B physics in 3HDMs was considered by Akeroyd, et al. [@Akeroyd:2016ssd]. These authors, and a subsequent article by Yagyu [@Yagyu:2016whx] recently considered 3HDMs with no tree level FCNC and studied Higgs boson couplings; these works include a new “Model Z" in which the three doublets couple to the up-type quarks, down-type quarks and leptons, respectively [@Akeroyd:2016ssd]. A nice review can be found in Yagyu’s recent talk [@yagyutalk].
Our model differs from these in that we do have tree level FCNC, albeit only in the lepton sector, the discrete symmetry is a simple $Z_2$ and we will focus on charged Higgs masses above the top quark threshold. In the lepton specific 2HDM, one can only have $\tau\nu$ charged Higgs decays dominate above the top quark threshold if there is a very large ratio of vacuum expectation values and we wish to see if the requirement that there be such a large ratio still holds in the 3HDM. It will also be noted that the isospin counterpart of $h\rightarrow\mu\tau$ would be $H^+\rightarrow \mu\nu_\tau$, leading to a substantial and observable enhancement of muonic decays of the charged Higgs.
In the next section, we present the model, scalar mass matrices and mixing angles. In Section 3, we study the charged Higgs bosons, including bounds on the parameters from LHC observations of the light Higgs and focusing on the $\tau\nu$ decay mode which, we will show, can dominate the decays even above the top threshold and in Section 4 we present our conclusions.
The Model
==========
The model consists of three Higgs doublets with weak hypercharge $Y=1/2$. Since FCNC in the quark sector are very small, and we hope to allow tree level FCNC in the lepton sector, quarks and leptons must transform differently under the $Z_2$ symmetry. In the conventional 2HDMs, one model does treat quarks and leptons differently. This is the lepton specific model (sometimes referred to as the Type X model). In this model, the only fields odd under the $Z_2$ are $\Phi_1$ and the right-handed leptons, $e^i_R$. This then forces $\Phi_1$ to couple only to leptons and $\Phi_2$ only to quarks. In this simple extension, we introduce a third doublet, $\Phi_3$, which is odd under the $Z_2$. Thus $\Phi_2$ only couples to quarks, whereas $\Phi_1$ and $\Phi_3$ only couple to leptons. It is the fact that two doublets couple to leptons that will lead to FCNC in the lepton sector.
Mass matrices and mixing angles
--------------------------------
The scalar potential consistent with the $Z_2$ symmetry (under which $\Phi_1, \Phi_3$ and $e_R^i$ are odd) can be written $$\begin{aligned}
V =& m_{11}^2 \Phi_1^\dagger \Phi_1+m_{22}^2 \Phi_2^\dagger \Phi_2 + m_{33}^2 \Phi_3^\dagger \Phi_3 - m_{13}^2 (\Phi_1^\dagger \Phi_3+\Phi_3^\dagger \Phi_1)+\frac{1}{2}\lambda_{11}(\Phi_1^\dagger \Phi_1)^2 \nonumber \\
&+ \frac{1}{2}\lambda_{22}(\Phi_2^\dagger \Phi_2)^2+\frac{1}{2}\lambda_{33}(\Phi_3^\dagger \Phi_3)^2+\lambda_{12}\Phi_1^\dagger \Phi_1\Phi_2^\dagger\Phi_2 + \lambda_{13}\Phi_1^\dagger \Phi_1\Phi_3^\dagger\Phi_3 \nonumber \\
&+ \lambda_{23}\Phi_2^\dagger \Phi_2\Phi_3^\dagger\Phi_3 + \frac{\beta_{12}}{2}\left[(\Phi_1^\dagger \Phi_2)^2 + (\Phi_2^\dagger \Phi_1)^2\right] + \alpha_{12}\Phi_1^\dagger \Phi_2 \Phi_2^\dagger \Phi_1 \nonumber \\
& + \frac{\beta_{13}}{2}\left[(\Phi_1^\dagger \Phi_3)^2 + (\Phi_3^\dagger \Phi_1)^2\right] + \alpha_{13}\Phi_1^\dagger \Phi_3 \Phi_3^\dagger \Phi_1 \nonumber \\
& + \frac{\beta_{23}}{2}\left[(\Phi_2^\dagger \Phi_3)^2 + (\Phi_3^\dagger \Phi_2)^2\right] + \alpha_{23}\Phi_2^\dagger \Phi_3 \Phi_3^\dagger \Phi_2. \label{potential}\end{aligned}$$ In this expression, we have omitted quartic terms odd in either $\Phi_1$ or $\Phi_3$, such as $\Phi_1^\dagger\Phi_1\Phi_1^\dagger\Phi_3$. This is entirely done for simplicity, and will avoid needing to solve cubic equations. This simplification will only affect expressions which explicitly have the $\lambda_{ij}, \alpha_{ij}$ and $\beta_{ij}$. The phenomenology only depends on the mixing angles and not on these couplings (which realistically can’t be measured for decades). As a result, nothing more will be learned by adding these extra terms.
We choose parameters such that each doublet acquires a vacuum expectation value (vev). Some models choose parameters so that one or two of the doublets get zero vev – these models are inert models and have a dark matter candidate. Expanding around the minimum we can write $$\begin{aligned}
\Phi_a &= \begin{pmatrix}
\phi_a^{+}\\
\frac{v_a+\rho_a+i \eta _a}{\sqrt{2}} \\
\end{pmatrix}, \quad a=1,2,3. \label{doublets}
\end{aligned}$$
With this potential and our choice of minimum, one can find the mass matrices. This is done in Appendix A. The charged Higgs, pseudoscalar and scalar mass matrices are all $3\times 3$. The charged and pseudoscalar mass matrices have a zero eigenvalue, corresponding to the Goldstone bosons.
Before diagonalizing the mass matrices, it is instructive to recall the diagonalization in the more well-known 2HDMs [@Branco:2011iw]. The minimum of the potential in that model lies along the ray $(v_1,v_2)$. The mass matrices for the charged scalars and pseudoscalars have a zero eigenvalue, corresponding to the Goldstone bosons. The mass matrix for the scalars does not have a zero eigenvalue, and one eigenvalue is the $125$ GeV Higgs boson. If one rotates the basis by a rotation angle given by $\beta\equiv \tan^{-1}(v_2/v_1)$ (which is equivalent to defining a new “x-axis" along the ray), then one has the Higgs basis, in which only one field gets a vev. After this rotation, the zero eigenvalues separate out and decouple. Note that there is only a single rotation angle for both the charged scalar and pseudoscalar matrices, which is not surprising. Note also that in the 2HDM, the charged scalar and pseudoscalar matrices commute, so they are simultaneously diagonalizable.
In the 3HDM, as can be seen from the mass matrices, the charged scalar and pseudoscalar matrices do not commute, and thus they will not be simultaneously diagonalizable. Thus, in general, we will have nine different angles (three for each matrix) that diagonalize the $\phi^+$, $\eta$ (pseudoscalar) and the $\rho$ (scalar) matrices. It turns out that the charged and the pseudoscalar mass matrices have two common angles which diagonalize the matrices, leaving us with a total of seven different angles. The fact that two of the angles are common is expected. Two angles are needed to take the general vector $(v_1,v_2,v_3)$ to $(v,0,0)$, and this will in both cases lead to separating out the Goldstone mode. However, there is still another angle corresponding to an additional rotation about the $(v,0,0)$ axis, and there is no reason that this angle should be the same for the charged and pseudoscalar cases.
The first rotation matrix that partially diagonalizes the $\phi^+$ and the $\eta$ matrices is the rotation matrix that takes the ray $(v_1,v_2,v_3)$ to $(v,0,0)$. The first transformation is given by (where $v_{ij}^2 \equiv v_i^2 + v_j^2$) $$R= \left(\begin{array}{ccc}
\frac{v_1}{v} & \frac{v_2}{v} & \frac{v_3}{v} \\
-\frac{v_2}{v_{12}} & \frac{v_1}{v_{12}} & 0\\
-\frac{v_1v_3}{v v_{12}} & -\frac{v_2 v_3}{v v_{12}}& \frac{v_{12}}{v}
\end{array} \right).$$ This is equivalent to rotating the vector $(v_1,v_2,v_3)$ clockwise around the z-axis by angle $\psi$ and then rotate counter-clockwise around the y-axis by angle $\pi/2-\theta$. The azimuthal and the polar angles are given in terms of the vev’s by $$\tan{\theta}=\frac{v_{12}}{v_{3}}, \quad \quad \tan{\psi}= \frac{v_2}{v_1}. \label{angles}$$ The third rotation, which is the angle of rotation around the x-axis will be called $\beta_1$ and $\beta_2$ for the $\phi^+$ and the $\eta$ matrices respectively.
For the scalar mass-squared matrix we denote the Euler angles (in the above convention) that diagonalize it by $\beta_3$, $\theta_3$ and $\psi_3$, where we have replaced the standard notation of performing a rotation by angle $\phi$ first, around the z-axis, by $\beta_3$ instead, in order to avoid confusion with the fields.
In summary for the charged and the pseudoscalar matrices we perform the following transformation on the fields $$T_{i} = \left( \begin{array}{ccc}
\cos{\psi}& -\sin{\psi}& 0 \\
\sin{\psi} & \cos{\psi} & 0\\
0 & 0 & 1
\end{array} \right)
\left(\begin{array}{ccc}
\cos{(\pi/2-\theta)} & 0& -\sin{(\pi/2-\theta)} \\
0 & 1 & 0\\
\sin{(\pi/2-\theta)} & 0 & \cos{(\pi/2-\theta)}
\end{array} \right)
\left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos{\beta_{i}} &-\sin{\beta_{i}}\\
0 &\sin{\beta_{i}}& \cos{\beta_{i}}
\end{array}\right) ,$$ where $i=1,2$ and the angles $\theta$ and $\psi$ are given in (\[angles\]).
For the scalar we write the transformation as the product of three Euler rotations, namely $$T_3 = \left( \begin{array}{ccc}
\cos{\psi_3} &-\sin{\psi_3} & 0 \\
\sin{\psi_3}& \cos{\psi_3} & 0\\
0 & 0 & 1
\end{array} \right)
\left( \begin{array}{ccc}
\cos{\theta_3} & 0 & -\sin{\theta_3} \\
0 & 1 & 0\\
\sin{\theta_3} & 0 & \cos{\theta_3}
\end{array}\right)
\left( \begin{array}{ccc}
1 & 0 & 0\\
0 &\cos{\beta_3} & -\sin{\beta_3}\\
0 & \sin{\beta_3} & \cos{\beta_3}
\end{array} \right).$$ The field redefinitions are then given by $$\begin{pmatrix}
\phi_1^{+}\\
\phi_2^{+}\\
\phi_3^{+}
\end{pmatrix} = T_1 \begin{pmatrix}
G^+ \\
H_1^+ \\
H_2^+
\end{pmatrix}, \quad \quad
\begin{pmatrix}
\eta_1\\
\eta_2\\
\eta_3
\end{pmatrix} = T_2 \begin{pmatrix}
G^0 \\
A_1\\
A_2
\end{pmatrix}, \quad \quad \begin{pmatrix}
\rho_1\\
\rho_2\\
\rho_3
\end{pmatrix} = T_3 \begin{pmatrix}
h_1 \\
h_2 \\
h_3
\end{pmatrix}. \label{rotations}$$ The spectrum of the theory consists of two charged scalars, two pseudo scalars and three neutral scalars.
Gauge boson couplings
----------------------
The kinetic term of the Higgs doublets has the form $$(D_\mu \Phi_i)^\dagger D_\mu \Phi_i,$$ where $$D_\mu=\partial_\mu-igW_\mu^a\tau^a-i\frac{g^\prime}{2}B_\mu$$ is the covariant derivative and $W_\mu^a$ and $B_\mu$ are the $SU(2)_L$ and $U(1)_Y$ gauge bosons of the electroweak sector respectively. Expanding out in terms of the mass terms given by eq (\[rotations\]) the Lagrangian contains $$\begin{aligned}
\mathcal{L} \supseteq & -\frac{i g}{2} \sec{\theta_w}\left(\sin{2\theta_w}A^\mu + \cos{2\theta_w}Z^\mu \right) \left(\partial_\mu H_{a-1}^+ H_{a-1}^- -H_{a-1}^+\partial_\mu H_{a-1}^- \right) \\
& -\frac{i g}{2}(T_1^T \cdot T_3)_{ab}\left( W_\mu^- \left( H_{a-1}^+ \partial_\mu h_b -\partial_\mu H_{a-1}^+h_b \right) + W_\mu^+ \left(\partial_\mu H_{a-1}^- h_b -H_{a-1}^- \partial_\mu h_b \right) \right)\\
& +\frac{g^2}{4} \left(2W_\mu^+W_\mu^- + \sec{\theta_w}^2 \left(Z_\mu \cos{2\theta_w}+A_\mu \sin{2\theta_w} \right)^2\right) H_{a-1}^-H_{a-1}^+ \\
& + \frac{g^2}{4}\left(2W_\mu^-W_\mu^+ + Z_\mu^2\sec{\theta_w}^2 \right)(T_3)_{ab}v_a h_b
\end{aligned}$$ where sum over $a,b=1,2,3$ is implied and $H_0^+=G^+$ is the Goldstone boson. From this expression we can read off the couplings of the Higgs particles with the gauge bosons. A list of the relevant couplings to gauge bosons are given at the end of the section.
The quark sector
-----------------
As in the lepton-specific model, the RH quarks will couple to $\Phi_2$ and the RH leptons to $\Phi_1$ and $\Phi_3$. Thus the Yukawa terms in the Lagrangian are $$-\mathcal{L}_{Yuk} = Y^u_{ij}\bar{Q}^i \tilde{\Phi}_2 u_R^j + Y^d_{ij}\bar{Q}^i\Phi_2d_R^j + \eta^1_{ij}\bar{L}^i\Phi_1e_R^j + \eta^2_{ij}\bar{L}^i\Phi_3e_R^j + h.c. \label{Yuk}$$ where $$\tilde{\Phi}_2 \equiv i\sigma_2 \Phi_2^*,$$ and $$Q^i = \left( \begin{array}{ccc}
u_L^i \\
d_L^i
\end{array} \right), \quad L_i = \left(\begin{array}{ccc}
\nu_L^i \\
e_L^i
\end{array}\right).$$ Here $i=1,2,3$ are the generation indices.
Quarks only couple to the $\Phi_2$ doublet. The Lagrangian of the quark sector is given by $$\mathcal{L}_{quark} = \bar{Q}\tilde{\Phi}_2Y^u u_R + \bar{Q}\Phi_2 Y^d d_R.$$ Using equation (\[doublets\]) and expanding out we can write this as $$\begin{aligned}
\mathcal{L}_{quark} = &\frac{\rho_2}{\sqrt{2}}\left( \bar{u}_L Y^u u_R + \bar{d}_L Y^d d_R \right) - \frac{i \eta_2}{\sqrt{2}}\left(\bar{u}_LY^uu_R - \bar{d}_L Y^d d_R \right) \\
& -\phi_2^- \bar{d}_L Y^u u_R + \phi_2^+ \bar{u}_L Y^d d_R + h.c.\end{aligned}$$
This is the same as the conventional lepton-specific model, and one can thus immediately write, as in that model, $$\begin{aligned}
\mathcal{L}_{quark} =& \frac{\rho_2}{v_2}(m^u \bar{u}u+m^d \bar{d}d) - \frac{\eta_2}{v_2}(m^u \bar{u}i\gamma_5 u -m^d \bar{d}i\gamma_5d) \nonumber \\
& + \left(- \frac{\sqrt{2}}{v_2}\phi^-_2 \bar{d}_LV_{CKM}^\dagger m^u u_R + \frac{\sqrt{2}}{v_2}\phi^+_2 \bar{u}_L V_{CKM} m^d d_R + h.c. \right),\end{aligned}$$ $m^u$ and $m^d$ are the entries of the diagonal matrices $\frac{v_2}{\sqrt{2}}M_{u}$ and $\frac{v_2}{\sqrt{2}}M_{d}$ respectively.
Expressing this in terms of the physical fields is somewhat more complicated, since there are two charged Higgs, two pseudoscalars and three scalars. In general, one can write $$\begin{aligned}
\mathcal{L}_{quark}= & - \sum_{i=1}^{3} \sum_{f=u,d}\frac{m^f}{v}\left( \xi_{h_i}^f \bar{f}fh_i - i \xi_{A_{i-1}}^f \bar{f}\gamma_5 f A_{i-1} \right) \nonumber\\
& + \left\lbrace \sum_{i=1}^{3}\frac{\sqrt{2}V_{ud}}{v}\bar{u}\left( \xi^u_{H^+_{i-1}}m^uP_L + m^d \xi_{H^+_{i-1}}^d P_R \right)dH_{i-1}^+ + h.c. \right\rbrace \label{Lquark}\end{aligned}$$ with $A_0=G^0$, $H_0^+=G^+$ being the Goldstone bosons. The couplings are given in terms of the matrix elements of the rotations given by equation (\[rotations\]).
The Higgs basis and the lepton sector
--------------------------------------
The lepton sector of the lepton specific 3HDM has the following terms, written in matrix form (with generation indices understood) $$\mathcal{L}_{Yukawa} \supseteq - \bar{L}\left( \Phi_1 \eta^1 + \Phi_3 \eta^2 \right) e_R + h.c. \label{leptonsector}$$ where $\eta^1$ and $\eta^2$ are general complex Yukawa matrices.
We would like to go to the Higgs basis where we make a rotation $(\Phi_1,\Phi_3)\rightarrow (H_1,H_3)$ such that $H_1$ has zero vev and $\langle H_3 \rangle = v_{13}/\sqrt{2}$. This can be accomplished by performing the field redefinition $$\left(\begin{array}{ccc}
\Phi_1 \\
\Phi_3
\end{array} \right) =\frac{1}{v_{13}} \left(\begin{array}{ccc}
v_3 & v_1 \\
-v_1 & v_3
\end{array} \right) \left(\begin{array}{ccc}
H_1 \\
H_3
\end{array} \right). \label{Hbasis}$$
Following the same notation as in [@Branco:2011iw], we define the matrices $$N = \frac{1}{\sqrt{2}} \left( v_3 \eta^1 - v_1 \eta^2 \right),$$ $$M=\frac{1}{\sqrt{2}} \left(v_1 \eta^1 + v_3 \eta^2 \right).$$ So the Lagrangian of the lepton sector, in the Higgs basis, reads $$\mathcal{L}_{lepton} = - \frac{\sqrt{2}}{v_{13}} \bar{L}\left( H_1 N + H_3 M \right) e_R. + h.c.$$ In the Higgs basis, only the Yukawa couplings of the doublet $H_3$ generate fermion masses, and may be bi-diagonalized so they do not lead to tree-level FCNC.
When passing to the mass basis of the leptons in which the mass matrices are diagonal we need to simultaneously rotate the left-handed and right-handed leptons: $$e_R \rightarrow U_R e_R, \quad L \rightarrow U_L L.$$ The Lagrangian transforms as $$-\mathcal{L}_{lepton} = \frac{\sqrt{2}}{v_{13}}\bar{L} \left( U_L^\dagger N U_R H_1 + U_L^\dagger M U_R H_3 \right) e_R. + h.c.$$ Naming $$U_L^\dagger N U_R=N_d,$$ $$U_L^\dagger M U_R = M_d$$ where $M_d= diag (m_e,m_\mu,m_\tau)$ is diagonal with real and positive diagonal elements.
If after bi-diagonalization, the matrix $N_d$ is not diagonal, then there are scalar tree-level flavour changing neutral interactions in the lepton sector, and the lepton FCNC coupling for those interactions are obtained from the entries of $N_d$.
Since $U_R$ is completely unkown and $N$ is arbitrary, the $N_d$ coefficients are arbitrary; in order to look at specific processes, some assumptions must be made about their magnitudes.
Motivated by stringent constraints on flavor-changing couplings involving the first two generations, Cheng and Sher [@Cheng:1987rs] showed that if one requires no fine-tuning in the Yukawa matrices, then the flavor-changing couplings should be of the order of the geometric mean of the Yukawa couplings of the two fermions. In other words, the so-called Cheng-Sher ansatz is $$(N_d)_{ij} = k_{ij}\sqrt{m_i m_j}$$ where $k_{ij}$ are of order one. Since the most severe bounds on FCNC arise from the first two generations and the Yukawa couplings of the first two generations are small, this ansatz can explain these bounds without requiring huge scalar masses.
Using this ansatz we can write the flavor changing matrix to be approximately of the form $$N_d =\left( \begin{array}{ccc}
k_{11}m_e & 0 & 0 \\
0 & k_{22}m_\mu & k_{23} \sqrt{m_\mu m_\tau} \\
0 & k_{32} \sqrt{m_\mu m_\tau} & k_{33} m_\tau
\end{array} \right).$$ Note that we have not included the $(12), (13), (21)$ and $(31)$ elements, since FCNC involving electrons are tightly constrained and the coefficients must be small, as shown in Ref. [@Diaz:2002uk]. Using the Cheng-Sher ansatz, the current bound from $\mu\to e\gamma$ requires [@Diaz:2002uk] that $k_{12},k_{13},k_{21},k_{31}$ cannot be much bigger than one. Including these coefficients would make no difference in the physics discussed in the rest of the paper, and thus for simplicity we do not include them here.
Note that we have assumed that the $(12), (13), (21)$ and $(31)$ elements are negligible, due to the fact that a nonzero value for them could lead to too large a value for $\mu\to e\gamma$. This is consistent with the Cheng-Sher ansatz since these couplings will depend on the electron mass, which is extremely small.
The Lagrangian of the lepton sector in the mass basis reads $$-\mathcal{L}_{lepton}= \frac{\sqrt{2}}{v_{13}}\bar{L}\left(H_1N_d + H_3 M_d \right)e_R + h.c.. \label{llepton}$$
We can separate out the lepton sector Lagrangian into two components. One that includes the FCNC couplings and the other is flavor diagonal $$\mathcal{L}_{lepton} =\mathcal{L}_{FCNC} + \mathcal{L}_{diag},$$ with $$\begin{aligned}
\mathcal{L}_{diag} =- \frac{\sqrt{2}m_i}{v_{13}} \left( \left( H_1k_{ii} + H_3 \right) \bar{\nu}_L^i e_R^i + \left( H_1k_{ii} + H_3 \right) \bar{e}_L^i e_R^i \right) + h.c. \label{diag}\end{aligned}$$ where the charged component for the first term and the neutral component for the second term are implied for each Higgs doublet.
Expanding out equation (\[diag\]) in terms of the mass eigenstates we can write in analogous way as eq (\[Lquark\]) $$\begin{aligned}
\mathcal{L}_{diag} = - \sum_{f=e,\mu,\tau} \sum_{b=2}^{3} &\frac{\sqrt{2}}{v}m_f \left\lbrace \left(Z_{H_{b-1}^+}\bar{\nu}_L f_R H_{b-1}^+ + h.c. \right) + \left( Z_{h_b} \bar{f}f h_b - Z_{A_{b-1}} i \bar{f} \gamma_5 f A_{b-1} \right) \right\rbrace. \label{diagonal} \end{aligned}$$ and the coupling constant $Z$ can be found on the next section.
Similarly for the FCNC component $$\mathcal{L}_{FCNC}= - \frac{\sqrt{2}k_{23}}{v_{13}}\sqrt{m_\mu m_\tau} \left( H_1 \left( \bar{\nu}_{\tau_L}\mu_R + \bar{\nu}_{\mu_L} \tau_R \right) + H_1 \left( \bar{\tau}_L \mu_R + \bar{\mu}_L \tau_R \right) \right) + h.c. \label{LFCNC}$$
For now we are only interested in couplings which lead to $h\rightarrow \mu \tau$, thus the relevant terms are $$\mathcal{L}_{FCNC} \supseteq - \frac{\sqrt{2}}{v_{13}}H_1 \sqrt{m_\mu m_\tau} \left( k_{23} \bar{\mu}_L \tau_R + k_{32}\bar{\tau}_L \mu_R \right) + h.c.$$ where the coupling to the neutral component of the doublet is understood. Expanding in terms of the scalars $h_i$ with $i=1,2,3$, (note Eq. \[rotations\]), we can write this as $$\mathcal{L}_{FCNC} = - \frac{\sqrt{m_\mu m_\tau}}{v_{13}^2}\left\lbrace v_3 (T_3)_{1i}-v_1 (T_3)_{3i} \right\rbrace h_i \left\lbrace k_{23} \bar{\mu}_L \tau_R + k_{32}\bar{\tau}_L \mu_R \right\rbrace + h.c., \quad i=1,2,3.$$ Choosing $k_{ij}=k_{ji}$ and real gives $$\mathcal{L}_{FCNC} \supseteq \frac{\sqrt{m_\mu m_\tau}}{v_{13}^2}\left\lbrace v_3 (T_3)_{1i}-v_1 (T_3)_{3i} \right\rbrace k_{23} h_i \left\lbrace \bar{\tau}\mu + \bar{\mu}\tau \right\rbrace \quad i=1,2,3.$$ which gives the relevant couplings. The flavor changing couplings of the neutral pseudoscalars are found in a similar way, the resulting expression being given by $$\mathcal{L}_{FCNC} \supseteq \frac{i\sqrt{m_\mu m_\tau}}{v_{13}^2}\left\lbrace v_3 (T_2)_{1i}-v_1 (T_2)_{3i} \right\rbrace k_{23} A_{i-1} \left\lbrace \bar{\tau}\gamma_5\mu + \bar{\mu}\gamma_5 \tau \right\rbrace \quad i=1,2,3.$$
It should be noted that the Cheng-Sher ansatz is completely consistent with the hints from CMS. In the CMS paper[@Khachatryan:2015kon], they plot their results as a function of the Yukawa couplings $Y_{\mu\tau}$ and $Y_{\tau\mu}$, and also include the line in which the product of the two is $m_\mu m_\tau / v^2$. From this, one can see that the original Cheng-Sher ansatz gives a result slightly below the ‘observed" value for $k_{\mu\tau}=1$. Thus, generally the model would predict a branching ratio of the order of a few tenths of a percent, which is consistent with current observations.
One can also discuss the decoupling limit of the model. In the conventional 2HDM, as discussed in detail by Gunion and Haber [@Gunion:2002zf], one notes that $\cos(\alpha-\beta)=0$ implies that the light Higgs couples to gauge bosons and fermions with the same couplings as in the Standard Model. This is the decoupling limit, and only the light Higgs gets a vev. The other Higgs bosons then have vanishing coupling to gauge bosons, but may still have couplings to fermions. In the decoupling limit, the other Higgs bosons are taken to be sufficiently heavy that they do not affect quark and lepton phenomenology. In this model, one can see that the limit $$\cos\theta = \sin\theta_3=\cos\psi_3=\cos\psi= 0$$ gives the coupling of the light Higgs to WW and to fermions equal to their Standard Model values and gives vanishing couplings of the heavy Higgs to $WW$ and $hW$. This is certainly sufficient to give the decoupling limit, and means increasing precise bounds on deviations of the light Higgs couplings from the Standard Model values will not eliminate the model, but just restrict the parameter-space somewhat.
The above constraints are certainly sufficient for decoupling, but are not fully necessary. In order for the light Higgs to have SM-equivalent couplings to gauge bosons and for the heavy Higgs to have no couplings to gauge bosons, one needs to have $\psi_3 = \psi$ and $\cos \theta=\sin \theta_3$. This corresponds to the first and third equalities in the above equation. In order for the light Higgs to have SM-equivalent couplings to the fermions, one must add the condition $\cot\theta=\cos\psi$, which corresponds to $v_1=v_3$. The above equation is consistent with this, of course. Note that the constraints in the above paragraph would lead to quark-phobic charged Higgs bosons, but the more general constraint here does not. Thus, tighter bounds on deviations of the light Higgs couplings from their Standard Model values will not force the charged Higgs to decay primarily leptonically.
Summary of the couplings
-------------------------
The gauge, quark and lepton couplings are summarized in the Tables below.
Gauge boson couplings
--------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --
$g_{h_1WW}$ $ \frac{g^2}{2}v \left(\cos{\theta} \sin{\theta_3}+\sin{\theta}\cos{\theta_3}\cos({\psi_3-\psi}) \right)$
$g_{h_2WW}$ $\frac{g^2}{2} v \cos{\theta} \left[ \cos{\beta_3}\sin{(\psi-\psi_3)} + \sin{\beta_3}(\cot{\theta}\cos{\theta_3}-\sin{\theta_3}\cos{(\psi_3-\psi})) \right]$
$g_{h_3WW}$ $\frac{g^2}{2}v \cos{\theta} \left[ \sin{\beta_3}\sin{(\psi_3-\psi)}+\cos{\beta_3}\left(\cot{\theta}\cos{\theta_3}-\sin{\theta_3}\cos{(\psi_3-\psi)} \right) \right]$
$g_{H_1 h W}$ $ \frac{ig}{2}\left( \cos{\theta}\cos{\theta_3}\cos{(\psi-\psi_3)}\sin{\beta_1}-\sin{\beta_1}\sin{\theta}\sin{\theta_3}+\cos{\beta_1}\cos{\theta_3}\sin{(\psi-\psi_3)}\right)$
$g_{H_2 h W}$ $ \frac{ig}{2}\left( \cos{\beta_1}\left( \cos{\theta}\cos{\theta_3}\cos{(\psi-\psi_3)}-\sin{\theta}\sin{\theta_3} \right)-\cos{\theta_3}\sin{\beta_1}\sin{(\psi-\psi_3)} \right)$
Table $1$: A list of Higgs couplings to gauge bosons. The last two expressions are multiplied by $(p-p^\prime)_\mu$.
$\times sin{\theta}\sin{\psi}$ Quark Couplings
-------------------------------- ---------------------------------------------------------------------- --
$\xi_{h_1}^{u,d}$ $\cos{\theta_3}\sin{\psi_3}$
$\xi_{h_2}^f$ $\cos{\beta_3}\cos{\psi_3}-\sin{\beta_3}\sin{\theta_3}\sin{\psi_3}$
$\xi_{h_3}^f$ $-\cos{\psi_3}\sin{\beta_3}-\cos{\beta_3}\sin{\theta_3}\sin{\psi_3}$
$\xi_{A_{1}}^u$ $-\cos{\psi}\cos{\beta_2}+\cos{\theta}\sin{\psi}\sin{\beta_2}$
$\xi_{A_{2}}^u$ $\cos{\psi}\sin{\beta_2}+ \sin{\psi}\cos{\theta}\cos{\beta_2}$
$\xi_{A_{1}}^d $ $-\xi_{A_{1}}^u$
$\xi_{A_{2}}^d$ $-\xi_{A_{2}}^u$
$\xi_{H^+_1}^u$ $\cos{\psi}\cos{\beta_1}-\cos{\theta}\sin{\psi}\sin{\beta_1}$
$\xi_{H^+_2}^u$ $-\cos{\psi}\sin{\beta_1}- \sin{\psi}\cos{\theta}\cos{\beta_1}$
$\xi_{H^+_1}^d$ $-\xi_{H^+_1}^u$
$\xi_{H^+_2}^d$ $-\xi_{H^+_2}^u$
Table $2$: A list of the couplings appearing in equation (\[Lquark\]). We do not list the couplings of the Goldstone bosons $G^{\pm}$ and $G^0$.
$\times\frac{v}{k_{23}\sqrt{m_\mu m_\tau}}(1-\sin^2{\theta}\sin^2{\psi})$ FCNC couplings
--------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------ --
$h_1 \bar{\mu}\tau$ $\cos{\theta}\cos{\theta_3}\cos{\psi_3}-\sin{\theta}\cos{\psi}\sin{\theta_3}$
$h_2 \bar{\mu}\tau$ $- \left( \sin{\theta}\cos{\psi}\cos{\theta_3}\sin{\beta_3}+\cos{\theta}(\sin{\beta_3}\cos{\psi_3}\sin{\theta_3}+\cos{\beta_3}\sin{\psi_3}) \right)$
$h_3 \bar{\mu}\tau$ $-\left[\sin{\theta}\cos{\psi}\cos{\beta_3}\cos{\theta_3} + \cos{\theta}(\cos{\beta_3}\cos{\psi_3}\sin{\theta_3}-\sin{\beta_3}\sin{\psi_3})\right]$
$i A_1 \bar{\mu}\gamma_5\tau$ $-\left( \cos{\psi}\cos{\beta_2}-\cos{\theta}\sin{\psi}\sin{\beta_2} \right)$
$i A_2 \bar{\mu}\gamma_5\tau$ $-\left(\cos{\psi}\cos{\beta_2} + \cos{\theta}\sin{\psi}\cos{\beta_2} \right)$
$\sqrt{2} H_1^+ (\bar{\nu}_\tau P_R \mu + \bar{\nu}_\mu P_R \tau)$ $-\left( \cos{\psi}\sin{\beta_1} + \cos{\beta_1}\cos{\theta}\sin{\psi} \right)$
$\sqrt{2} H_2^+ (\bar{\nu}_\tau P_R \mu + \bar{\nu}_\mu P_R \tau)$ $-\cos{\beta_1}\cos{\psi} + \cos{\theta}\sin{\beta_1}\sin{\psi} $
Table $3$: These are the lepton flavor violating couplings.
Table $4$: These are the couplings to leptons appearing in equation .
Branching ratios and constraints on the charged Higgs bosons
=============================================================
For the analysis of this section it is useful to write down the couplings of the charged Higgs to both, quarks and leptons in one single equation, see (\[Lquark\]) and (\[diag\]), $$\begin{aligned}
\mathcal{L}= \frac{g}{\sqrt{2}M_W} \sum_{H^+=H_1^+,H_2^+} \left\lbrace \xi_{H^+}^u \bar{u}_R V_{ud}m_u d_L -\xi_{H^+}^u \bar{u}_L V_{ud}m_d d_R - Z_{H^+} \bar{\nu}_L m_l l_R \right\rbrace H^+ + h.c. \label{Lfermions}\end{aligned}$$ Note that the charged Higgs interactions depend only on three mixing angles. With the two different charged Higgs masses, this gives a five-dimensional parameter-space, as opposed to the 2HDM in which the interactions depend on the single charged Higgs mass and $\tan\beta \equiv v_2/v_1$.
Since the model has a close similarity to the lepton specific model, one can look at charged Higgs interactions in that model for guidance. Logan and MacLennan [@Logan:2009uf] studied the constraints on the charged Higgs in this model, looking at direct LEP-II searches and flavor universality in tau decays. Su and Thomas [@Su:2009fz] studied constraints from $b\rightarrow s\gamma$, $B^0-\bar{B}^0$ mixing and several other processes. Aoki, Kanemura, Tsumura and Yagyu [@Aoki:2009ha], in a very comprehensive study, considered these as well as $B\rightarrow\tau\nu$. These papers and others are discussed in the review article of Ref. [@Branco:2011iw], where it is noted that the bounds from radiative B decays and $R_b$ are the same as in the type I model, and bounds from flavor universality in tau decays are not significant unless $\tan\beta > 65$. The most intriguing result is the possibility that the $\tau\nu$ decay mode of the charged Higgs can dominate the decay, [*even above the $t\bar{b}$ threshold*]{}. For this to occur, one must have $\tan\beta > 10$. Such fairly large values of $\tan\beta$ may have problems with perturbative unitarity [@Arhrib:2009hc], although there are allowed regions of parameter-space with larger values of $\tan\beta$. We will see that in the 3HDM, a dominance of the $\tau\nu$ decay mode can occur even if the ratio of vacuum expectations values is not particularly large.
Charged Higgs decays
---------------------
As noted above, the charged Higgs sector depends on three mixing angles and two masses. As a first step, these parameters must be constrained by the requirement that they do not contradict LHC results on production and decay of the SM-Higgs like 125 GeV boson. Given the number of parameters, a full $\chi$-squared analysis is unnecessary, and we will simply require that the couplings of the light Higgs ($h_1$ in Tables 1 and 2) be within $20\%$ of their Standard Model values for the $WW$, $ZZ$ and $tt$ couplings and $30\%$ for the $bb$ coupling[@Djouadi:2013qya]. The $\gamma\gamma$ coupling will not provide useful information due to unknown contributions of heavy charged Higgs bosons (with arbitrary couplings) in the loop. All of our results below will only consider regions in parameter-space that satisfy those constraints.
We consider the branching ratios of the charged Higgs bosons. In this article we are going to assume that all of the additional neutral scalars are too heavy for the charged Higgs to decay into them. Then the most relevant decay modes to consider are $H^+ \longrightarrow h W^+$, $\bar{b}t$, $\bar{\tau}\nu$ and possibly into $\mu \nu$. Focus will be placed mainly on heavy charged scalars $M_{H^\pm_i} \gg m_t$ and therefore the decay mode into a quark-antiquark pair will be dominated by the $\bar{b}t$ decay mode. The leading order expressions for the partial widths are given by $$\Gamma(H^\pm_{i}\rightarrow h W^\pm) = \frac{\sqrt{2} G_F}{16\pi}g_{H_i hW}^2 M_{H_i^+}^3 \left[ 1 + \frac{(M_W^2-m_{h}^2)^2}{M_{H_i^+}^4}- 2\frac{(M_W^2 + m_{h}^2)}{M_{H_i^+}^2} \right]^{3/2}, \label{hdecay}$$ $$\Gamma (H_i^+ \rightarrow t \bar{b}) = \frac{3 G_F(\xi^u_{H_i^+})^2}{4 \pi \sqrt{2}} M_{H_i^+} m_t^2 \left(1-\frac{m_t^2}{M_{H^+}^2} \right)^2 , \label{quarkdecay}$$ $$\Gamma(H_i^+ \rightarrow \nu_l \bar{l}) = \frac{G_F}{4 \pi \sqrt{2}} M_{H_i^+} \times \begin{cases}
m_\tau^2 Z_{H_i^+}^2, & \ \tau \nu_\tau \\
m_\mu m_\tau C_{H_i^+}^2, & \tau \nu_\mu , \ \mu \nu_\tau
\end{cases} \label{leptondecay}$$ where $h=h_1$ being the SM-like Higgs boson is implied in . The $b$ quark and $\tau$ lepton masses have been neglected in and respectively. For the leptonic decay we included the flavor changing processes induced by and the respective couplings, $C_{H_i^+}$, are given in the last two rows of table $3$.
In the lepton-specific 2HDM the couplings of the charged Higgs boson to quarks are proportional to the SM couplings multiplied by $\cot{\beta}$. Also in that model, the couplings of the charged Higgs to right handed leptons are proportional to $\tan{\beta}$. Thus in the limit $\tan{\beta}\gg 1$, the charged scalar becomes quark-phobic but leptophilic enhancing the possibility of a large decay width into $\bar{\nu} \tau$, even above the $\bar{b}t$ threshold [@Branco:2011iw].
Here we explore that possibility in the 3HDM. The couplings to the quarks in the 3HDM are given by $$\xi_{H_1^+}^u= \frac{\cos\beta_1 \cot\psi - \cos\theta \sin\beta_1}{\sin\theta}, \label{quarkcoupling1}$$ $$\xi_{H_2^+}^u = -\frac{\cos\beta_1\cos\theta + \cot\psi\sin\beta_1}{\sin\theta}.\label{quarkcoupling2}$$ We investigate quark-phobic points in the parameter space of the mixing angles $(\theta, \psi, \beta_1)$, i.e. points for which either $\xi_{H_1^+}^u$, $\xi_{H_2^+}^u$ or both are very small. Without loss of generality we do this analysis in the region of parameter space $ 0<( \theta, \psi, \beta_1) < \pi/2 $. The reason is that we want the vevs given in to be real and positive, so that the rotation angles $\theta, \psi$ can be restricted to be in the first quadrant. The reason to consider $\beta_1$ in that region is by noticing that taking $\beta_1 \rightarrow \beta_1 + \pi/2$ in yields .
By considering as a function of $\theta$ and $\psi$ and varying $\beta_1$ in that region, we find that there is always a surface for which is equal to zero and that never crosses zero in that region. Thus the 3HDM allows for either $H_1$ or $H_2$ to be quark-phobic but not both at the same time.
When we consider the decay mode to the SM Higgs and a gauge boson, we see that the decay width is proportional to the square of the couplings $g_{H_i h W}$ given in table $1$. These couplings are dependent on the mixing angles $\theta_3$ and $\psi_3$, Therefore we also investigate points in the parameter-space that are both quark- and gauge-phobic. We shall consider values of the mixing angles $\theta_3, \ \psi_3$ in the same region described above, although they could in principle be larger since they depend on the parameters of the scalar potential. By following the same procedure described above we find that it is always possible for and both gauge couplings to cross zero at the same point. Therefore the 3HDM can always have a quark- and gauge-phobic $H_1$ decaying mostly into leptons and a gauge-phobic $H_2$ decaying most of the time to $tb$.
If one chooses parameters such that the ratio of vevs is large, one must consider unitarity and perturbativity. As remarked in the 2HDM review of Ref. [@Branco:2011iw], having high ratios of vevs is only allowed for small regions of parameter space. Therefore we consider points for which the ratios in equation are not too big ($\tan{\theta} ,\tan{\psi}< 6 $).
The production cross section for charged Higgs bosons is dominated by the gluon-gluon fusion process $gg \rightarrow \bar{t}bH^+ $ and is proportional to the square of the couplings and . This mechanism will not produce charged Higgs bosons in the quark-phobic limit. In that case, they can still be produced by vector boson fusion (VBF) which is independent of quark couplings. As shown in Ref. [@Logan:2009uf], the VBF production cross section is $2-3$ orders of magnitude smaller than the gluon-gluon fusion cross section for $\tan{\beta}=1$, see figure $14$ of Ref [@Djouadi:2016eyy], we thus consider quark-couplings which are $0, 0.03, 0.12$ times the $\tan{\beta}=1$ coupling.
A novel feature of this model is the appearance of the flavor changing decay modes given in . Usually the muonic decay of the charged Higgs is negligible due to the small muon coupling, however, here the tau coupling enters if the $\nu$ is a $\nu_\tau$. The region of parameter space for which the flavor changing branching ratios given in are bigger than $10 \%$ is very small and is concentrated in the upper right corner of the plane $(\theta, \psi)$ which corresponds to large ratios of vevs. In the limit $\theta, \psi \rightarrow \pi/2$ these modes will be dominant with $H_1$ decaying $50\%$ into $\mu \nu_\tau$ and $\tau \nu_\mu$ each.
We find the branching ratios in figure $1$. We have chosen three benchmark points which give values for $\xi$ of $0, 0.03, 0.12$. These points are $(\theta, \psi, \beta_1, \theta_3, \psi_3) $=$(1, 1.37, 0.36, 0.92, 1.14)$, $(1.30, 1.36, 0.59, 0.39, 1.36)$, $(1.41,1.41,0.25,0.28,1.41)$ respectively. Changing the benchmark points, without changing the value of $\xi$ will not substantially alter these results.
For a quark Yukawa coupling of $0.12$ times the $\tan{\beta}=1$ 2HDM coupling, one sees that the branching ratio into $\tau \nu_{\tau}$ is substantial, although not dominant. As the quark Yukawa coupling gets smaller, the branching ratio into $\tau \nu_\tau$ becomes dominant for small masses.
  
The novel flavor changing mode discussed above, into $\mu \nu_\tau$ is small, but would be easier to detect. With the LHC operating with an integrated luminosity of $4000$ fb$^{-1}$ and the pair production cross section by vector boson fusion (VBF), which is model independent, is around a few fb as can be seen from figure $8$ of Ref. [@Logan:2009uf], producing approximately 300 events. Note that a discussion of the neutral heavy Higgs decay into $\mu\tau$ can be found in Ref. [@Sher:2016rhh].
Constraints on the charged Higgs from B physics
------------------------------------------------
### $B\rightarrow X_s \gamma$
In this section we investigate bounds on the charged Higgs masses coming from the inclusive radiative decay $B\rightarrow X_s \gamma$ and give the leading order (LO) results at the matching scale $M_W$ at which the full theory is matched into an effective theory with five quark flavours in order to provide insight on the effect of new physics, as is done in Ref. [@Carone:2009nu]. This is a well suited process to probe new physics. However it suffers from large theoretical uncertainties coming mainly from the choices of the renormalization and matching scales, which are not well defined at the leading order (LO) [@Ciuchini:1997xe].
The next to leading order calculations (NLO) in the SM [@Chetyrkin:1996vx], [@Greub:1996tg] and in the 2HDM [@Ciuchini:1997xe] are well known. The branching fraction of the inclusive radiative decay at NLO is presented as [@Ciuchini:1997xe] $$\begin{aligned}
\mathcal{B}(B \rightarrow X_s \gamma) = & \mathcal{B}(B \rightarrow X_c e \bar{\nu}_e)\Big|{\frac{V_{ts}^*V_{tb}}{V_{cb}}}\Big|^2 \frac{ 6 \alpha_{em}}{\pi f(z)\kappa(z)}\frac{\bar{m}_b^2(\mu_b)}{m_b^2} \nonumber \\
&\times (|D|^2+A)\left(1-\frac{\delta_{SL}^{NP}}{m_b^2}+\frac{\delta_\gamma^{NP}}{m_b^2}+\frac{\delta_c^{NP}}{m_c^2} \right),
\end{aligned}$$ where $z=m_c^2/m_b^2$ and $f$ and $\kappa$ are phase-space supression factors. The last term in parenthesis includes corrections obtained by the method of the heavy-quark effective theory (HQEFT) which relate the quark decay rate to the hadronic process [@Neubert:1996qg]. The term $A$ is the correction coming from the Bremssthralung process $b\rightarrow s\gamma g$. Finally, $|D|^2$ contains the Wilson coefficents at the renormalization scale $\mu_b$ relevant to the radiative decay and is given by equation $(26)$ on Ref. [@Ciuchini:1997xe]. As discussed in section $5$ of Ref. [@Ciuchini:1997xe] the LO 2HDM Wilson coefficients, to be added to SM ones, at the matching scale $m_W$ are given by $$\delta C_{7,8}^{(0)eff}(m_W) = \frac{A_u^2}{3}F_{7,8}^{(1)}(y)- A_u A_d F_{7,8}^{(2)}(y),$$ with $$F_7^{(2)}(y)= \frac{y(3-5y)}{12(y-1)^2}+\frac{y(3y-2)}{6(y-1)^3}\ln{y},$$ $$F_8^{(2)}(y)= \frac{y(3-y)}{4(y-1)^2}-\frac{y}{2(y-1)^3}\ln{y},$$ $$y=\frac{\bar{m}_t^2(m_W)}{M_H^2}$$ and $A_u$ and $A_d$ are the couplings of the charged Higgs to the quarks. In the type I 2HDM these are $A_u=A_d=\cot{\beta}$.
To obtain the new Wilson coefficients of the 3HDM the only different thing is that we need to add the contribution from each $H_1^+$ and $H_2^+$ in the loop and modify the corresponding quark couplings. The result is $$\delta C_{7,8, NP}^{(0)eff}(m_W) =\sum_{i=1}^2 \left( \xi_{H_i^+}^u \right)^2 \left(\frac{1}{3}F_{7,8}^{(1)}(y_i)- F_{7,8}^{(2)}(y_i)\right), \quad y_i = \frac{\bar{m}_t^2(m_W)}{M_{H_i^2}} \label{LOcorrec}.$$ Since the above equation is dependent on the three angles $(\theta, \psi, \beta_1)$ and the two charged Higgs masses we see that the branching fraction of the radiative decay will be dependent on five parameters. The Wilson coefficients at the renormalization scale $\mu_b$ and at NLO cannot be summarized in a few lines and are extracted from Ref. [@Chetyrkin:1996vx].
Since we are interested in a region of parameter-space in which one of the charged Higgs bosons has suppressed couplings to quarks (we have checked that the suppression is sufficient to make the contribution negligible), the results will only depend on the mass of the other charged Higgs, $H_2^+$. For values of $\xi^u_{H^+_2} = 0.2, 0.6, 0.8$ and $1.4$, we find the results in Figure 2. These four points correspond to mixing angles $(\theta,\psi,\beta_1)$ = $(1.41,1.41,0.25)$, $(1.00,1.37,0.36)$, $(1.20,1.02,1.05)$ and $(0.69,1.21,0.46)$ respectively. The first two of these correspond to the two values of the points listed in the previous subsection (corresponding to $\xi=0, 0.12$ (the other point in the last subsection gives results extremely similar to the $\xi=0.12$ point). The other two points give results that are somewhat more significant for $B\to X_s\gamma$.
All black lines asymptote to the SM prediction $BR(B \rightarrow X_s \gamma)=(3.6 \pm 0.36) \times 10^{-4}$ in the limit of high mass values. The experimentally allowed region is $(3.52 \pm 0.23 \pm 0.09) \times 10^{-4}$ [@Barberio:2008fa]. The line corresponding to the first point lies entirely inside the $1 \sigma$ experimental region and therefore does not yield any bound. Including NLO QCD corrections the lower bounds on the charged Higgs mass corresponding to the points considered in figure $2$ at $95 \%$ C.L. are given by $295$ GeV, $370$ GeV and $900$ GeV for the values of $\xi^u_{H^+_2} = 0.6, 0.8$ and $1.4$. Thus, the bounds will be fairly weak unless $\xi^u_{H^+_2}$ is unusually large. Note that a more recent analysis by Misiak et al. [@Misiak:2015xwa] does the NNLO calculation and finds results that are slightly lower, but well within a single standard deviation from the NLO results.
### $R_b$
In this subsection we focus on the observable $$R_b = \frac{\Gamma(Z\rightarrow b \bar{b})}{\Gamma(Z \rightarrow hadrons)}$$ which is sensitive to radiative corrections. It is shown in Ref. [@Haber:1999zh] that in non-minimal models containing only doublets, as is our case, the loop corrections due to virtual charged Higgs bosons always worsen agreement with experiment. In that same reference they introduce a parametrization for a general extended Higgs sector and calculate the contribution to $Zb\bar{b}$ from one-loop radiative corrections involving singly charged and neutral Higgs bosons. They obtained general expressions for the corrections to the left- and right-handed $Zb\bar{b}$ couplings, and then use the measurements of $R_b$ and $A_b$ (the coupling asymmetry) to constrain specific models.
We write the interaction as $$\mathcal{L} \propto
Z_\mu \bar{b}\gamma^\mu \left[ \bar{g}_b^L P_L + \bar{g}_b^R P_R \right] b,$$ the effective couplings are then given by $\bar{g}_b^{L,R}=g_{Zbb}^{L,R} + \delta g^{L,R}$, where $g_{Zbb}$ are the tree-level couplings and $\delta g$ are the radiative corrections.
The radiative corrections to SM extensions with only doublets and singlets, are given by the second term of equation $(4.5)$ of Ref. [@Haber:1999zh] $$\delta g^{L,R} = \pm \frac{1}{32 \pi^2}\frac{e}{s_W c_W} \sum_{i \neq G^+} \left( g_{H_i^+ }^{L,R} \right)^2 \times \left[ \frac{R_i}{(R_i-1)}- \frac{R_i \log{R_i}}{(R_i-1)^2} \right]$$ where $s_W=\sin{\theta_W}$, $c_W=\cos{\theta_W}$ are the weak mixing factors, $R_i^2 = m_t^2/M_{H_i^+}^2$ and $g_{H_i^+}^{L,R}$ are obtained from the Lagrangian by writing the interaction of the charged Higgs to quarks as $$\bar{t}\left( g^L P_L + g^R P_R \right) b H^+ + h.c.$$ The correction due to Goldstone boson exchange is excluded in the sum since is the same as in the SM. In the 3HDM the above coefficients are given by $$g_{H_i^+}^L = \sqrt{2} \frac{m_t}{v}\xi_{H_i^+}^u,$$ $$g_{H_i^+}^R = - \sqrt{2} \frac{m_b}{v}\xi_{H_i^+}^u.$$ Thus we can write the corrections as $$\begin{aligned}
\delta g^L = & \frac{1}{32 \pi^2}\frac{e}{s_W c_W}\left( \frac{\sqrt{2}m_t}{v} \right)^2 \sum_{i=1,2} (\xi_{H_i^+}^u)^2 \left[ \frac{R_i}{(R_i-1)}- \frac{R_1 \log{R_i}}{(R_i-1)^2} \right] \end{aligned}$$ and $$\delta g^R = -\frac{m_b^2}{m_t^2}\delta g^L.$$
The experimentally allowed range is given by $R_b = 0.21642 \pm 0.00073$ [@Haber:1999zh]. We find bounds on the charged Higgs mass for the same quark-phobic points listed in figure $2$. However the first two points give a prediction that lies well inside the $2\sigma$ experimentally allowed region, therefore we only obtain a bound coming from the last point, namely $$m_{H_2^+}> 395 \ \GeV \ \ (95 \% \ C.L.).$$ Thus we see that the radiative process $B \rightarrow X_s \gamma$ yields stronger constraints than those of $R_b$.
### Combining $B\rightarrow X_s\gamma$ and $R_b$
In the above, we illustrated the bounds with a few benchmark points. But there are only three angles and two masses, and we can scan the parameter-space to determine the allowed regions. In the figures below we have plotted the regions in which the $\chi^2$ tests for both $B \rightarrow X_s \gamma$ and $R_b$ processes are satisfied to $95 \%$ significance level. We parametrized the rotation angles $\theta$ and $\psi$ in terms of the vevs $v_1, v_2$ and imposed the constraint $\sum_i v_i^2=246 \GeV$. In Figure 3, we show a contour plot of $v_1$ and $v_2$ for a specific value of $\beta_1$ chosen to give the quark-phobic point for $H_1$.
,
We then consider different values of $\beta_1$ in Figure 4. The patterns are clear. Some regions of parameter-space, such as small $v_2$ and intermediate $v_1$, are excluded for all values of $\beta_1$. This will become relevant when considering the possibility of $B\rightarrow \mu \nu_\tau$. We now turn to the leptonic decays of the $B$
, ,
### $B^- \rightarrow \tau \bar{\nu}_{\tau}$
We study the charged current decay $B^- \rightarrow \tau \bar{\nu}$ type of modes which are just tree-level processess mediated by the electroweak gauge bosons $W^\pm$ and the charged Higgs bosons $H_i^\pm$.
Using the quark and lepton couplings of the charged Higgs mass eigenstates given by , the $W^\pm$ and $H_i^\pm$ effectively induce the four-Fermi interaction [@Hou:1992sy] $$\mathcal{L} = -2\sqrt{2}G_{F}V_{u b} \left[ (\bar{u}\gamma^\mu P_L b )(\bar{l}\gamma_\mu P_L \nu_{l}) - R_{3HDM,l} (\bar{u}P_R b)(\bar{l}P_L \nu_{l}) \right],$$ where $$R_{3HDM,l} = \sum_{i=1}^2 \frac{m_{l}m_b}{M_{H_i^+}^2}Z_{H_i^+}\xi_{H_i^+}^u, \label{R1}$$ where the first term give the SM contribution, while the second one give that of the charged scalars. In Ref. [@Grossman:1994jb] a study of multi-Higgs doublet models with natural flavor conservation was performed. In that article they assumed that all but the lightest of the charged scalars effectively decouple from fermions and carried out an analysis of phenomenological constraints on the Yukawa couplings.
We do not make those assumptions here and instead modify the 2HDM result by the appropiate couplings and find that the charged Higgs bosons modify the SM expectation by the factor $$r_H \equiv \frac{BR(B^- \rightarrow l \bar{\nu})}{BR_{SM}(B^- \rightarrow l \bar{\nu})} = \left\lvert 1 - \sum_{i=1}^2 m_B^2 \frac{\xi_{H_i^+}^u Z_{H_i^+}}{M_{H_i^+}^2} \right\rvert^2$$ which is independent of the lepton mass. We also call that factor $r_H$, following the notation in Ref. [@Hou:1992sy]. Notice that in the limit $(\theta, \beta_1 ) \rightarrow (\pi/2, 0)$, vanishes and $H_2$ becomes quark-phobic and the sum on the right-hand side collapses to $$\frac{BR(B \rightarrow l \nu)}{BR_{SM}(B \rightarrow l \nu)} = \left\lvert 1 + \frac{m_B^2}{M_{H^+}^2} \right\lvert^2$$ which is the expression for the lepton-specific 2HDM. Therefore all the $B^- \rightarrow l \bar{\nu}$ modes are always enhanced in the lepton-specific 2HDM while they could be enhanced or supressed in the 3HDM by the same factor $r_H$. The ratio of the measured value to the SM prediction is $1.37 \pm 0.39$ [@Czarnecki:1998tn]. We compare this number with the $r_H$ prediction for the 3HDM and we find that in the quark-phobic points of $H_1$ the 3HDM prediction lies completely inside the $1 \sigma$ region for all masses of $H_2$ starting from the threshold. Larger values of the correction factor arise if we take the rotation angles $\theta, \psi \ll 1$, but that would correspond having a hierarchy on the vevs values $v_3 \gg v_1 \gg v_2 $. This is no surprise since it is remarked in [@Hou:1992sy], that for models where d-type quarks and charged leptons derive mass from different doublets there is no interesting effect. This is exactly the case for lepton-specific-type models.
### Flavor changing processess
This model has another interesting possibility that does not exist in any other versions of the 2HDM. One can study $B\rightarrow \mu\nu_\tau$. The Standard Model decay $B\rightarrow \mu\nu_\mu$ is very small due to helicity suppression, and the 2HDM charged Higgs contribution is negligible due to the small Yukawa coupling of the muon to the Higgs. But in this model, the flavor-changing couplings are proportional to the geometric mean of the Yukawa couplings, and thus the $\tau$ Yukawa coupling can play an important role. Of course, experimenters can not determine the flavor of the neutrino, so this would appear as a contribution to $B\rightarrow \mu\nu$.
From and we can write down the flavor changing interaction of the charged Higgs bosons $$\mathcal{L}_{FCNC} \supseteq -\frac{\sqrt{2}}{v}\sqrt{m_\mu m_\tau} \sum_{H^+ = H_1^+, H_2^+} C_{H^+} \left[ \bar{\nu}_{\mu}P_R \tau + \bar{\nu}_\tau P_R \mu \right]H^+ +h.c.$$ where the $C_{H^+}$ are the flavor changing coupling constants given in the last two rows of table $3$. The charged Higgs induces the four-Fermi flavor-changing interaction $$\mathcal{L}_{4F} = \frac{4G_F}{\sqrt{2}} V_{ub} \sum_{H^+} R_{H^+}(\bar{u} P_R b) (\bar{\tau} P_L \nu_{\mu} + \bar{\mu} P_L \nu_{\tau}),$$ where the sum is performed over the charged Higgs mass eigenstates and $$R_{H^+} = \sqrt{m_\mu m_\tau}m_b \frac{\xi_{H^+}^u C_{H^+}}{M_{H^+}^2},$$ similar to . The branching fraction for the flavor changing processes are given by the flavor conserving SM result times a correction factor, i.e., $$BR(B\rightarrow \mu \bar{\nu}_\tau,\tau \bar{\nu}_\mu) = BR(B\rightarrow \mu \bar{\nu}_\mu,\tau \bar{\nu}_\tau)_{SM} r_{H,l},$$ where the $l$ index stands for the charged lepton and the correction factor is $$r_{H,l}= \left( \frac{\sqrt{m_\mu m_\tau}}{m_l}\sum_{H^+}\frac{m_B^2}{M_{H^+}^2}\xi_{H^+}^u C_{H^+} \right)^2.$$ The SM branching ratio is given by [@Hou:1992sy], [@Bona:2009cj], [@Dingfelder:2016twb] $$BR_{SM}(B^- \rightarrow l^- \bar{\nu}) = \frac{G_F^2 m_B m_l^2}{8\pi} \left( 1-\frac{m_l^2}{m_B^2} \right)^2 f_B^2 |V_{ub}|^2 \tau_B.$$
Using this formula the SM predictions are [@Bona:2009cj] $$BR(B \rightarrow \tau \nu_\tau)_{SM} = (0.84 \pm 0.11)\times 10^{-4},$$ $$BR(B \rightarrow \mu \nu_\mu)_{SM} = (3.8 \pm 0.5)\times 10^{-7}.$$
The Heavy Flavor Averaging Group (HFAG) found, as of July 2016, the value $BR(B \rightarrow \tau \nu_\tau) = 1.06 \times 10^{-4}$ and the upper limit $BR(B \rightarrow \mu \nu_\mu)<1.0 \times 10^{-6} $ at $90 \%$ C.L.
We investigate the region of the parameter space that allow for an enhancement of the flavor-changing decay $B \rightarrow \mu \nu_\tau$ over the SM flavor-conserving prediction. One special case is the degenerate limit $M_{H^+_1}=M_{H^+_2}$, in which the correction factor vanishes trivially since the coupling constants satisfy the relation $$\xi_{H_1^+}^u C_{H_1^+} =- \xi_{H_2^+}^u C_{H_2^+}$$ for any rotation angle $(\theta, \psi, \beta_1)$. Thus one way to maximize the value of the correction factor is to take the decoupling limit in which either charged boson becomes extremely heavy and the other one take its threshold value.
The results are presented in figure $5$, where we consider the values of parameters in which the contribution from the charged Higgs is substantial.
Figure $5$: Region of the parameter space $(\theta, \psi)$ for which the correction factor $r_{H,\mu} >(0.2,1,2)$.\
We have taken $M_{H_1^+}$ infinitely big and $M_{H_2^+}$ at its threshold value. The chosen value of $\beta_1$ maximizes the area.\
One natural question that arise is if there exist a region of parameter space that allow a substantial enhancement and is permitted by the $\chi^2$ tests of the previous section. The answer is clearly negative as can be seen from the figure below where the colored region shows the parameter space that is allowed by the radiative processes $B \rightarrow X_s \gamma$ and $R_b$ at $95 \%$ C.L. and the same values of $M_{H_2}$ and $\beta_1$ were used. That region is concentrated at the upper right corner and there is no overlapping between them.
,
Figure $6$: Allowed region of parameter space by the $\chi^2$ tests of the processes $B \rightarrow X_s \gamma$ and $R_b$. The decoupling limit of $H_1$($H_2$) has been taken in the left (right).
Conclusions
============
Motivated by hints of lepton flavor violation in Higgs decays and the strong constraints on tree level FCNC in the quark sector, we consider a 3HDM in which one doublet couples to quarks and the other two to leptons. This structure can be imposed with a simple $Z_2$ symmetry. The model has two charged Higgs pairs, two pseudoscalars and three scalars.
The mass matrices and mixing angles are then determined. In the charged Higgs sector, there are two masses and three mixing angles, two of which come from ratios of vevs. We focus on the phenomenology of the charged Higgs bosons, concentrating on the region in which the lightest of the charged Higgs pairs is above the top quark mass. The model is similar to the 2HDM lepton specific model. In that model, the charged Higgs branching ratio into $\tau\nu$ can exceed that of $t\bar{b}$ only if the ratio of vevs ($\tan\beta$ in that model) is quite large, possibly large enough to raise unitarity concerns. In this model, even without such a large ratio of vevs, the decays into $\tau\nu$, $t\bar{b}$ and $hW$ can all be comparable. We also study the constraints from $B$ decay. We find that there is a new decay, $B\rightarrow \mu\nu_\tau$, but the region of parameter-space in which this is substantial is inconsistent with other B decay bounds.
A unique feature of the model is the possibility of charged Higgs decay into $\mu\nu_\tau$. The flavor of the neutrino can of course not be measured, but flavor does feed in through the size of the coupling, expected here to be the geometric mean of the muon and tau Yukawa couplings. The decay branching fraction is typically a few percent, although for extreme regions of parameter-space can be much higher. Although the branching fraction is small, the relative ease of detecting muons makes this decay worthy of careful study.
Acknowledgments
===============
The authors would like to thank Chris Carone, Pedro Ferreira and Keith Thrasher for useful discussion. This work was supported by the NSF under Grant PHY-1519644.
Appendix A {#appendix-a .unnumbered}
==========
Minimizing the Higgs potential yields: $$m_{11}^2= m_{13}^2 \frac{v_3}{v_1}-\lambda_{11}\frac{v_{1}^2}{2}-\frac{v_{2}^2}{2}(\lambda_{12}+\beta_{12}+\alpha_{12})-\frac{v_3^2}{2}(\lambda_{13}+\beta_{13}+\alpha_{13}),$$ $$m_{33}^2= m_{13}^2 \frac{v_1}{v_3}-\lambda_{33}\frac{v_{3}^2}{2}-\frac{v_{2}^2}{2}(\lambda_{23}+\beta_{23}+\alpha_{23})-\frac{v_1^2}{2}(\lambda_{13}+\beta_{13}+\alpha_{13}),$$ $$m_{22}^2=-\lambda_{22}\frac{v_{2}^2}{2}-\frac{v_{3}^2}{2}(\lambda_{23}+\beta_{23}+\alpha_{23})-\frac{v_1^2}{2}(\lambda_{12}+\beta_{12}+\alpha_{12}).$$
The mass terms for the charged scalars are given by $$\mathcal{L}\supseteq(\phi_1^-,\phi_2^-,\phi_3^-)\phi_{matrix} \begin{pmatrix}
\phi_1^+ \\
\phi_2^+ \\
\phi_3^+
\end{pmatrix},$$ where the mass squared matrix for the charged scalars is given by
$$\phi_{matrix} = \left( \begin{array}{ccc}
m_{13}^2 \frac{v_3}{v_1}-\frac{v_2^2}{2}A_{12}-\frac{v_3^2}{2}A_{13} & \frac{v_1 v_2}{2}A_{12} & -m_{13}^2+\frac{v_1 v_3}{2}A_{13} \\
\frac{v_1 v_2}{2}A_{12} & -\frac{v_1^2}{2}A_{12}-\frac{v_3^2}{2}A_{23} & \frac{v_2 v_3}{2}A_{23} \\
-m_{13}^2+\frac{v_1 v_3}{2}A_{13} & \frac{v_2 v_3}{2}A_{23} & m_{13}^2\frac{v_1}{v_3}-\frac{v_1^2}{2}A_{13}-\frac{v_2^2}{2}A_{23} \end{array} \right),$$
and the following definitions where made $$A_{12}=\alpha_{12}+\beta_{12},\quad
A_{13}=\alpha_{13}+\beta_{13},\quad
A_{23}=\alpha_{23}+\beta_{23}.$$ There is a zero eigenvalue corresponding to the charged Goldstone boson $G^{\pm}$ which gets eaten by the $W^{\pm}$. The mass squared of the charged Higgs particles is given by $$\begin{aligned}
m_{\pm} = &\frac{1}{4 v_1 v_3}\left(2 m_{13}^2 v_{13}^2-v_1v_3 \left( v_1^2A_{13}+ v_2^2A_{23}+A_{12}v_{12}^2+(A_{13}+A_{23}) v_3^2\right) \right. \nonumber \\
&\pm \left.\surd \left(-4 v_1 v_3 v^2 \left(A_{23} v_3^2 \left(-2 m_{13}^2+A_{13} v_1 v_3\right)+A_{12} v_1 \left(-2 m_{13}^2 v_1+A_{13} v_1^2 v_3+A_{23} v_2^2 v_3\right)\right)\right.\right. \nonumber \\
& + \left.\left.\left(-2 m_{13}^2 \left(v_1^2+v_3^2\right)+v_1 v_3 \left(A_{13} v_1^2+A_{23} v_2^2+A_{12} \left(v_1^2+v_2^2\right)+(A_{13}+A_{23}) v_3^2\right)\right)^2\right)\right),\end{aligned}$$ where we defined $v^2=v_1^2+v_2^2+v_3^2$,$v_{ij}^2=v_i^2+v_j^2$, for $i,j=1,2,3$.
The mass terms for the pseudoscalars are given by $$\mathcal{L}\supseteq(\eta_1,\eta_2,\eta_3)\eta_{matrix} \begin{pmatrix}
\eta_1 \\
\eta_2 \\
\eta_3
\end{pmatrix},$$ with $$\eta_{matrix} = \left( \begin{array}{ccc}
m_{13}^2 \frac{v_3}{v_1}-v_2^2 \beta_{12}-v_3^2\beta_{13} & v_1v_2\beta_{12} & -m_{13}^2+v_1v_3\beta_{13} \\
v_1v_2\beta_{12} &-v_1^2\beta_{12}-v_3^2\beta_{23} &v_2v_3\beta_{23} \\
-m_{13}^2+v_1v_3\beta_{13} & v_2v_3\beta_{23} & m_{13}^2\frac{v_1}{v_3}-v_1^2\beta_{13}-v_2^2 \beta_{23} \end{array} \right),$$ there is a zero eigenvalue corresponding to the neutral Goldstone boson $G^0$. The mass-squared of the physical pseudoscalar is $$\begin{aligned}
m_{A} = & -\frac{1}{2 v_1 v_3}\left(-m_{13}^2 v_{13}^2 +v_1 v_3 \left(v_1^2 (\beta_{12}+\beta_{13})+v_2^2 (\beta_{12}+\beta _{23})+v_3^2 (\beta_{13}+\beta_{23})\right) \right. \nonumber\\
&\pm \left.\surd \left(-4 v_1 v_3 v^2 \left(-m_{13}^2 \left(v_1^2 \beta_{12}+v_3^2 \beta _{23}\right)+v_1 v_3 \left(v_1^2 \beta_{12} \beta _{13}+v_2^2 \beta_{12} \beta_{23}+v_3^2 \beta _{13} \beta_{23}\right)\right) \right. \right. \nonumber \\
&+ \left.\left.\left(m_{13}^2 v_{13}^2-v_1 v_3 \left(v_1^2 (\beta_{12}+\beta_{13})+v_2^2 (\beta_{ 12}+\beta_{23})+v_3^2 (\beta_ {13}+\beta_{23})\right)\right)^2\right)\right).\end{aligned}$$
Finally the mass terms for the scalars read $$\mathcal{L}\supseteq(\rho_1,\rho_2,\rho_3)\rho_{matrix} \begin{pmatrix}
\rho_1 \\
\rho_2 \\
\rho_3
\end{pmatrix},$$ where the mass matrix is given by $$\rho_{matrix} = \left( \begin{array}{ccc}
m_{13}^2 \frac{v_3}{v_1}+v_1^2\lambda_{11} & B_{12}v_1v_2 & -m_{13}^2+B_{13}v_1v_3 \\
B_{12}v_1 v_2 &v_2^2 \lambda_{22} & B_{23}v_2v_3 \\
-m_{13}^2+B_{13}v_1v_3 & B_{23}v_2v_3 & m_{13}^2 \frac{v_1}{v_3}+v_3^2\lambda_{33} \end{array} \right),$$ where we defined $$B_{12}=A_{12}+\lambda_{12}, \quad B_{13}=A_{13}+\lambda_{13} \quad B_{23}=A_{23}+\lambda_{23}.$$ There is no zero eigenvalue in this case and the scalar masses are given by the roots of the cubic characteristic equation of this matrix.
Appendix B: Vacuum stability or bounded from below conditions {#appendix-b-vacuum-stability-or-bounded-from-below-conditions .unnumbered}
=============================================================
As the field value of each of the 12 components of the Higgs doublets go to infinity only the quartic terms of the potential given by (\[potential\]) become relevant. So we call this term $$\begin{aligned}
V_4 =& \frac{1}{2}\lambda_{11}(\Phi_1^\dagger \Phi_1)^2 + \frac{1}{2}\lambda_{22}(\Phi_2^\dagger \Phi_2)^2+\frac{1}{2}\lambda_{33}(\Phi_3^\dagger \Phi_3)^2 \nonumber \\
&+\lambda_{12}\Phi_1^\dagger \Phi_1\Phi_2^\dagger\Phi_2 + \lambda_{13}\Phi_1^\dagger \Phi_1\Phi_3^\dagger\Phi_3 + \lambda_{23}\Phi_2^\dagger \Phi_2\Phi_3^\dagger\Phi_3 \nonumber \\
&+ \frac{\beta_{12}}{2}\left[(\Phi_1^\dagger \Phi_2)^2 + (\Phi_2^\dagger \Phi_1)^2\right] + \alpha_{12}\Phi_1^\dagger \Phi_2 \Phi_2^\dagger \Phi_1 \nonumber \\
& + \frac{\beta_{13}}{2}\left[(\Phi_1^\dagger \Phi_3)^2 + (\Phi_3^\dagger \Phi_1)^2\right] + \alpha_{13}\Phi_1^\dagger \Phi_3 \Phi_3^\dagger \Phi_1 \nonumber \\
& + \frac{\beta_{23}}{2}\left[(\Phi_2^\dagger \Phi_3)^2 + (\Phi_3^\dagger \Phi_2)^2\right] + \alpha_{23}\Phi_2^\dagger \Phi_3 \Phi_3^\dagger \Phi_2. \end{aligned}$$ A simple way to obtain necessary conditions on the quartic parameters of the potential is to study its behaviour along specific field directions.
Writting the three Higgs doublets as $$\Phi_1 = \left(\begin{array}{ccc}
\phi_1 + i \phi_2\\
\phi_3 + i \phi_4
\end{array} \right), \quad \Phi_2 = \left(\begin{array}{ccc}
\phi_5 + i \phi_6\\
\phi_7 + i \phi_8
\end{array} \right), \quad \Phi_3 = \left(\begin{array}{ccc}
\phi_9 + i \phi_{10}\\
\phi_{11} + i \phi_{12}
\end{array} \right).$$ We consider for example $\phi_3 \rightarrow \infty$ and $\phi_{11} \rightarrow \infty$ and we get $$V_4 = \phi_{11}^4 \left(
\frac{\lambda_{33}}{2} + (\alpha_{13} + \beta_{13} + \lambda_{13})\left(\frac{\phi_3}{\phi_{11}}\right)^2 + \frac{\lambda_{11}}{2}\left(\frac{\phi_3}{\phi_{11}}\right)^4 \right).$$ By making $\left(\frac{\phi_3}{\phi_{11}}\right)^2=x$, this can be seen as a simple polynomial of order two, which in order to be positive semi-definite, the following conditions $$\lambda_{33} \geq 0, \quad \lambda_{11} \geq 0, \quad -\sqrt{\lambda_{11}\lambda_{33}} \leq \alpha_{13} + \beta_{13}+\lambda_{13}<0,$$ should be satisfied. By doing similarly in all different $\phi_i \phi_j$ planes we get the general necessary stability conditions $$\lambda_{ii}\geq 0,$$ $$-\sqrt{\lambda_{ii}\lambda_{jj}} \leq B_{ij}<0, \quad B_{ij}=\alpha_{ij} + \beta_{ij}+\lambda_{ij}$$ $$-\sqrt{\lambda_{ii}\lambda_{jj}} \leq \bar{B}_{ij}<0, \quad \bar{B}_{ij}=\alpha_{ij} - \beta_{ij}+\lambda_{ij}$$
[99]{}
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|
{
"pile_set_name": "ArXiv"
}
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---
address: |
Laboratoire d’Astrophysique de Grenoble\
CNRS / Université Joseph Fourier (Grenoble 1)\
BP 53, F-38041 Grenoble cedex 9, France\
[email protected]
author:
- Fabien Malbet
bibliography:
- 'malbet-vira.bib'
title: 'Tomorrow optical interferometry: astrophysical prospects and instrumental issues'
---
Introduction
============
When I was asked to give an invited review on the topic of the future of optical interferometry, I was tempted to give a short answer. Everybody knows where to go (see contribution of A. Quirrenbach in this volume) and there is no needs to detail these directions of developments:
- **Higher spatial resolution** meaning going from milli-arcsecond scale to micro-arcsec ones.
- **Higher flux sensitivity** meaning going beyond the Galaxy and reaching objects brighter than $K=13$.
- **Higher astrophysical complexity** meaning going from *visibilities* to *true* images.
Therefore these advances require **many more** telescopes, **much larger** apertures, **much longer** baselines in excellent ground-based sites and eventually in space.
*Will it be that easy?*
We have to remind us that radio interferometry took more than 30 years from the first attempts in mid-1940’s [@1960MNRAS.120..220R] to the *Very Large Array* in the mid-1970’s [@1980ApJS...44..151T]. Moreover, there is a $100\,000$ ratio between the H$_{\rm I}$ wavelength at 21cm and the Bracket $\gamma$ line at $2.165{\ensuremath{\mbox{$\mu$m}}}$ and therefore a similar ratio in accuracy requirements. Interference detection in radio is done using the heterodyne technique whereas in the optical domain one has to mix first the optical signal before directly measuring it.
In addition, optical interferometry requires nanometer precision over hundreds of meters, high reliability, complex instrumental control using active control loops working at the kilo-hertz frequency, and most of all, the atmosphere is corrugating the incoming wavefront on centimeter scale on milli-second temporal scales.
Starting from the present state of the art (Sect. \[sect:today\]), I present the directions where one can reasonably think that optical interferometry can extend the parameter space in astrophysics (Sect. \[sect:astro\]) depending on which instrumental issues (Sect. \[sect:instr\]).
Where do we stand today? {#sect:today}
========================
On the hardware side, today astronomers using optical interferometry have access to baselines ranging from 10 to 350m, aperture diameters ranging from 50cm to 8-10m, detection wavelength ranging from 1 to $10\,{\ensuremath{\mbox{$\mu$m}}}$ (a few years ago even the window $0.4-0.8\,{\ensuremath{\mbox{$\mu$m}}}$ was accessible at SUSI and GI2T) and space interferometry vessels have not yet been launched (SIM should be launched around 2015).
The observables accessible to general astronomical users range from squared amplitudes of the visibilities, color-differential phases, closure phases to dual phases. These quantities allow the astronomers to perform model fitting at different levels of complexity. Imaging has been demonstrated but it is not really a routine procedure like with radio interferometers. We are experiencing the premises of the nulling technique and narrow-angle astrometry.
![Refereed articles in optical interferometry (source: OLBIN). Left: evolution of number of refereed articles with years. Right: distribution of these papers between the different types of papers.[]{data-label="fig:refpapers"}](referee-stat-total "fig:"){width="0.48\hsize"} ![Refereed articles in optical interferometry (source: OLBIN). Left: evolution of number of refereed articles with years. Right: distribution of these papers between the different types of papers.[]{data-label="fig:refpapers"}](referee-stat-type "fig:"){width="0.48\hsize"}
Even with a rather limited scope, we have been experiencing an giant leap in the progress of optical interferometry demonstrated by the huge amount of new astrophysical results obtained during the last years (see graphs of Fig. \[fig:refpapers\] excerpted from OLBIN[^1] database).
Most of the astrophysical results in optical interferometry remains in stellar physics: stellar diameters, circumstellar environments, multiple systems,... However other fields are emerging like extragalactic studies with the advent of large aperture interferometers [@2003ApJ...596L.163S; @2004Natur.429...47J]. With increased accuracy, interferometers can measured stellar diameters of even the lowest mass stars .
Astrophysical prospects {#sect:astro}
=======================
I do not detail here all the achievements obtained by optical interferometers in the astrophysical field since S. Ridgway in this volume already tackled this issue.
Although several orders of magnitude of spatial scales may reveal the same physics, very strong changes can occur within a simple factor 10 in spatial resolution. A good example is the case of IRC 10216 which has been observed at large scales by with the *Hubble Space Telescope* with a field of 2 arcminutes and at spatial resolution by with a field of 1 arcsecond and a resolution of 50-100 milli-arcseconds. At large scales, the source appears centrosymmetric with spherical wind whereas at high spatial resolution the source is clumpy and changes at the year scale.
![Parameter space for astrophysical prospects in optical interferometry. Dark colored items corresponds to fields already tackled by today interferometry. Light colored items are possible extension areas.[]{data-label="fig:interf-fields"}](interferometry-fields){width="0.6\hsize"}
Stellar physics
---------------
Observations with optical interferometry goes already beyond the measurement of stellar diameters with the observation of many classes of stars (young, evolved, multiple, main-sequence, massive, low-mass,...) and close phenomena like accretion, outflows, every sort of shells. Some studies of the stellar surfaces have already been achieved.
In the near future, stellar atmosphere climatology, i.e. the study of dark or hot spots, should be reachable and will provide new inputs to Doppler imaging techniques. Convection in stellar interiors should be within reach and will provide complementary evidences to those obtained with asteroseismology techniques.
Since the sensitivity of interferometers but also their performance should improve, then an exploration of the whole Hertzsprung-Russel diagram should be feasible from low-mass protostars to the remnants of stellar evolution. Similarly, the connection with interstellar medium can be contemplated by studying the influence of accretion of gas and dust material as well as reversely the consequences of the ejection of stellar matter.
In conclusion, we can imagine that the progress of optical interferometry both in the visible and the infrared wavelength ranges will open avenues to the understanding of the formation of stars and planets as well as to the comprehension of the fate of stars.
Planetary science
-----------------
The field of planetary science will undoubtedly benefit from the progress in optical interferometry. The objects located at the outskirts of the solar system like the Trans-Neptunian Objects (TNOs) or those within the Kuiper belt are usually too small to be observed by classical techniques. Adaptive optics have shown that some of them can be spatially resolved from Earth, yet optical interferometry should be able to improve the spatial resolution of these objects.
More than 10 years ago a new field has opened up: the study of planets in extra-solar systems, also called exoplanets. Their proximity to their stellar host and the contrast of their brightness with the one of the central star make them difficult targets of observation. However, observations of protoplanetary disks which are probably the nurseries of such planets but also of the zodiacal light of these planetary systems should be possible. The formation of extrasolar systems is the occasion to observe the migration of giant planets and the planetary gaps opened by the formation of rocky planets.
Observations could be enlarge to the different types of objects so that we can explore the zoo of extrasolar planets. Interferometry might be a tool to explore the extent of habitable zone around close stars. Imaging of exo-Earth might seem a dream now but can be contemplated using interferometry techniques. A mission like DARWIN/TPF will certainly bring us clues on the status of extra terrestrial life out side the Solar System.
Interstellar medium
-------------------
The interstellar medium has been barely examined with optical interferometry whereas it is one of the pillar of the science performed by radio interferometry. However, prospects can be imagined that will allow to observe the impacts of stellar outflows parsec away from the central jet engine traced by shocks. This is also the location of wind collision when two massive stars have strong mass loss.
By observing the dynamics of stellar clusters, either dense ones or located far away, optical interferometry can bring pieces of knowledge to the star formation and evolution. The increase in spatial resolution will allow astronomers to observe H$-{\rm II}$ region in nearby galaxies.
Extragalactic exploration
-------------------------
As written above, optical interferometry has already started to observe much more distant objects like the central cores of galaxies like active galactic nuclei (AGN). In nearby galaxies, the brightest objects should be within reach: Cepheids, supernovae, massive star formation, but also fainter ones. It opens the possibility to study galaxy dynamics. This already started with the observation of the center of our Galaxy [@2005AN....326R.569P].
Using the brightest extragalactic sources, interferometry should be able to observe the details of superluminic jets being complementary to radio Very Long Baseline Interferometry (VLBI). In addition with spectral information, very high spatial observations of super-massive black holes should also be possible. Finally one can imagine to extent the interferometry techniques to the X-ray domain.
Top level requirements
----------------------
The requirements for optical interferometry in the current fields of astrophysics in order to increase the knowledge of these objects are **access to routine imaging**, **observe in moderate and high spectral resolution**, **increasing the spectral range** and **improve the dynamic range of observations**. To enlarge the field of investigation of interferometry toward the interstellar medium, **larger fields of view** and observations in **mid-infrared to far infrared** are required. **Higher sensitivity** and **longer baselines** are necessary to observe more distant objects like galaxies and quasars, but also to observe with enhanced spatial resolution the surface of stars. In the latter case but also in high energy physics like environments of black-holes, **access to shorter wavelengths** down to X-rays should be a priority.
Instrumental issues {#sect:instr}
===================
I have listed above a list of astrophysical prospects and their translation in top level requirements. But how do these requirements have an implication into instrumental development? For example, routine imaging is necessary because modeling cannot answer all questions and imaging requires many telescopes, but how many of them? Higher spatial resolution implies long baselines but how long, or shorter wavelength but how short: ultraviolet, X-rays?
Increasing the sensitivity is a key issue, but also the contrast between the brightest object and the faintest one. Off-axis references can be used, but requires special hardware like the PRIMA facility in the VLTI. Certainly improving the capabilities a detector is also a path to follow and since the sensitivity is more or less independent of aperture size for ground-based interferometers, finding the best site is crucial. Space-based instruments should also be continued to be contemplated. However, improving single pieces of hardware is not sufficient and we must pay attention to the interferometer as a whole.
Optical interferometers are not really complicated (made of numerous elements intricately combined) but remains complex (composed of several interconnected units). For example the VLTI in the current state is made of:
- **Light collectors**: telescopes, guiding and active optics
- **Beam routing optics**: 32 motors are involved
- **Adaptive optics**: consisting in wavefront sensors, deformable mirrors, real-time controllers
- **Delay lines**: 3 translation stages, metrology, switches, control to manage carriage trajectories
- **Beam stabilizers**: variable curvature mirrors, image and pupil sensors (ARAL/IRIS), sources (LEONARDO)
- **Fringe tracker**: fringe sensor, optical path difference controller managing fringe search, group delay or phase tracking
- **Beam combination**: mainly the instruments VINCI, AMBER or MIDI which have a variety of spectral dispersion, spectral coverage, spatial filtering, detection and should control also the atmospheric dispersion and polarization
- **Control software**: 60 computers and 750,000 lines of code [@2004SPIE.5496...21W].
This is only for the combination of 2 (MIDI) or 3 (AMBER) beams! Complexity follows the power of number of apertures and therefore combining more telescopes like several tens will be a major challenge and have a high price if one does not change the type of technology used. However with increasing number of telescopes, the impact of failure is also less important than for small number of telescopes especially if the setup is redundant enough.
Calibration is an important step of present interferometry. New advances should take into account this step which becomes decisive in a good implementation plan. For examples, interferometers with very long baselines must be prepared to calibrate very low visibilities using baseline bootstrapping or other methods. Another example are spaced-based interferometers which absolutely needs to calibrate their configuration before moving to another one since it is highly improbable they can reconfigure exactly the same way. In these perspective, spectral calibration may become an interesting way of calibrating interferometric measurements on the science target itself.
In that respect, one determining question is how to combine several tens of beams: using aperture synthesis like today or should we push forward direct imaging? Aperture synthesis imaging requires less telescopes at once, that can be compensated by more observing time. This is probably the first step to imaging. Direct combination imaging is simpler to manage and more photon efficient, but requires at least a few tens of apertures and homothetic pupil combination although densified pupil technique is possible.
Other parameters should be taken into account, like the complementarity of diluted versus filled extremely large telescopes (see contribution of Quirrenbach in this volume) which is an already known story. In fact, in my opinion, this has been one the major achievements of P. Léna to succeed with his colleagues in convincing the European astronomical community to build the *Very Large Telescope* as an interferometer. We should not forget also to continue to prepare the path to space-based interferometers (see contributions of Fridlund and Ollivier in this volume).
Conclusion
==========
We can conclude that there is certainly a future for optical interferometry! However it will not be as straightforward as initially and usually thought. While it is crucial to support the astrophysical use of current interferometric facilities, it is also essential to prepare the future (post-VLTI, KI, CHARA facility, ALOA), to identify which is the most suited site, to continue developing interferometry for space and of course to increase the number of apertures, the baseline length, the size of the apertures to access to even larger astrophysical topics.
[^1]: `http://olbin.jpl.nasa.gov`
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Before stating our main result, we first clarify through classical examples the status of the laws of macroscopic physics as laws of large numbers. We next consider the mirrors model in a finite $d$-dimensional domain and connected to particles reservoirs at fixed chemical potentials. The dynamics is purely deterministic and non-ergodic but takes place in a random environment. We study the macroscopic current of particles in the stationary regime. We show first that when the size of the system goes to infinity, the behaviour of the stationary current of particles is governed by the proportion of orbits crossing the system. This allows to formulate a necessary and sufficient condition on the distribution of the set of orbits that ensures the validity of Fick’s law. Using this approach, we show that Fick’s law relating the stationary macroscopic current of particles to the concentration difference holds in three dimensions and above. The negative correlations between crossing orbits play a key role in the argument.'
address:
- 'Laboratoire de Probabilités et Modèles Aléatoires (CNRS UMR 7599), Université Paris Diderot, UFR de Mathématiques, bâtiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris CEDEX 13 France'
- 'Laboratoire de Probabilités et Modèles Aléatoires (CNRS UMR 7599), Université Paris Diderot, UFR de Mathématiques, bâtiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris CEDEX 13 France'
author:
- Yann Chiffaudel
- Raphaël Lefevere
title: 'The Mirrors Model : Macroscopic Diffusion Without Noise or Chaos '
---
Macroscopic laws as laws of large numbers
=========================================
Take a macroscopic box $\Lambda=[0,L]^d$ that contain $N$ freely moving distinguishable particles and fixed obstacles of arbitrary shapes. $N$ should be thought to be of the order of magnitude of the Avogadro number : $6\times 10^{23}$. A first experiment is performed on this box. The $N$ particles are initially located in a cube $\Lambda'\subset\Lambda$ of side length $L'<L$, see Figure \[LambdaPrime\].
\[LambdaPrime\]
The evolution of the density of the cloud of particles is monitored through a beam of light that crosses the system. We call this density $\rho: \Lambda\times [0,\infty[\to \rho({{\ensuremath{\mathbf x}} },t)\in {{\ensuremath{\mathbb R}} }^+$. The initial state is described by $\rho({{\ensuremath{\mathbf x}} },0)={\bf 1}_{\Lambda'}({{\ensuremath{\mathbf x}} })/|\Lambda'|$. The empirical fact that is observed at the macroscopic level is that the density evolves according to the laws of diffusion: $$\left\{
\begin{array}{l}
\partial_t\rho({{\ensuremath{\mathbf x}} },t)=\kappa\,\Delta\rho({{\ensuremath{\mathbf x}} },t)\\
n_{{{\ensuremath{\mathbf x}} }}\cdot\nabla\rho({{\ensuremath{\mathbf x}} },t)=0, \quad {{\ensuremath{\mathbf x}} }\in \partial\Lambda\\
\rho({{\ensuremath{\mathbf x}} },0)=\rho_0({{\ensuremath{\mathbf x}} }):={\bf 1}_{\Lambda'}({{\ensuremath{\mathbf x}} })/|\Lambda'|,
\end{array}
\right.
\label{diffusion}$$ where $n_{{{\ensuremath{\mathbf x}} }}$ is the vector normal to the boundary of the box $\partial\Lambda$ at ${{\ensuremath{\mathbf x}} }$ and $\kappa$ is a strictly positive constant. How can we explain this phenomenon from the motion of the individual atoms ? For each [*macroscopic*]{} coordinate $\mathbf{x}=(x_1,\ldots,x_d)\in\Lambda$, we define a [*microscopic*]{} coordinate ${{\ensuremath{\mathbf q}} }=\mathbf{r}/\epsilon_N$ where $\epsilon_N=\frac {1}{N^{1/d}}$. The motion of the particles is entirely determined by the law of Newtonian mechanics. When a particle makes a collision with one of the fixed obstacles or the boundaries of the boxes, its velocity is modified according to the laws of specular reflection. We assume that each particle starts with a speed equal to $1$. Since this property is preserved by the dynamics, the microscopic motion of a given particle (with label $i\in\{1,\ldots, N\}$) is described by a map $t\to ({{\ensuremath{\mathbf q}} }_i(\epsilon^{-2}_Nt),{{\ensuremath{\mathbf p}} }_i(\epsilon^{-2}_Nt))\in [0,L/\epsilon_N]^d\times S^{d-1}$ where $S^{d-1}$ is the unit sphere in $d$ dimensions. The coordinates $({{\ensuremath{\mathbf q}} }_i,{{\ensuremath{\mathbf p}} }_i)$ are the [*microscopic*]{} positions and velocities of the $i$-th particle. The microscopic time-scale is $\epsilon^{-2}_N t$. The scaling of the time variable is a priori arbitrary but is fixed here by the fact that the solution of the diffusion equation $\rho({{\ensuremath{\mathbf x}} },t)$ is invariant under the transformation $({{\ensuremath{\mathbf x}} },t)\to (\lambda {{\ensuremath{\mathbf x}} },\lambda^2 t)$, $\lambda>0$.
In the absence of any other information, we assume that the initial positions and velocities of the particles are independent and identically uniformly distributed, with density $$f({{\ensuremath{\mathbf q}} }_i,{{\ensuremath{\mathbf p}} }_i)=\frac{\epsilon^d_N}{|\Lambda'||S^{d-1}|}{\bf 1}_{\Lambda'}(\epsilon_N {{\ensuremath{\mathbf q}} }_i){\bf 1}_{S^{d-1}}({{\ensuremath{\mathbf p}} }_i)
\label{uniform}$$ for every $i=1,\ldots, N$. If the position of each particle is chosen independently of the others with that density, the number of particles in a microscopic volume of size of order $1$ follows a Poisson distribution with finite mean as $N\to\infty$. We denote by ${{\ensuremath{\mathbb P}} }$ the law of probability of the initial positions and velocities of particles. No information about the spatial location or the shape of the obstacles in $\Lambda$ is known either. We denote by ${{\ensuremath{\mathbb Q}} }$ the probability distribution on those degrees of freedom. It is chosen such that in the limit $N\to\infty$, the number of obstacles in a microscopic volume of size $1$ follows a distribution with a finite mean and such that, almost surely, the dynamics of moving particles is well-defined at all time. The dynamical system defined in this way is an instance of the [*random Lorentz gas*]{}.
We define the empirical density of particles : $$\rho_N({{\ensuremath{\mathbf x}} },t)=\frac 1 N\sum_{j=1} ^N\delta (\epsilon_N \,{{\ensuremath{\mathbf q}} }_j(\epsilon_N^{-2} t )- {{\ensuremath{\mathbf x}} })
\label{emp_density}$$ The density $\rho_N$ contain all possible information about the density. Indeed, it is easy to see that $$\rho_N(V,t):=\int_{{{\ensuremath{\mathbb R}} }^d}d{{\ensuremath{\mathbf x}} }\; {\bf 1}_V({{\ensuremath{\mathbf x}} })\rho_N({{\ensuremath{\mathbf x}} },t)$$ gives the proportion of particles that belong to any $V\subset \Lambda$ at time $t$. It is straightforward to see that the following statement holds : if $\{({{\ensuremath{\mathbf q}} }_i(0),{{\ensuremath{\mathbf p}} }_i(0)):i=1,\ldots,N\}$ is a collection of i.i.d variables with marginals given by (\[uniform\]) then for any bounded function $h$ and any $\delta>0$ : $$\lim_{N\to\infty}{{\ensuremath{\mathbb P}} }[|\left<\hat\rho_N(0),h\right>-\left<\rho_0,h\right>|>\delta]=0,
\label{large_number0}$$ where $\rho_0$ is given by (\[diffusion\]), $\left<h,g\right>=\int_{{{\ensuremath{\mathbb R}} }^d}d{{\ensuremath{\mathbf x}} }\; h({{\ensuremath{\mathbf x}} })g({{\ensuremath{\mathbf x}} })$ and we use the notation $\hat\rho_N(t):=\rho_N(\cdot,t)$. The goal is to show the
There exists a natural [^1] distribution ${{\ensuremath{\mathbb Q}} }$ such that for any $t>0$, any bounded function $h$ and any $\delta>0$ : $$\lim_{N\to\infty}{{\ensuremath{\mathbb P}} }\times {{\ensuremath{\mathbb Q}} }[|\left<\hat\rho_N(t),h\right>-\left<\rho(t),h\right>|>\delta]=0.
\label{goal1}$$ \[large\] where $\rho(t):=\rho(\cdot,t)$ is the solution of (\[diffusion\]) for some $\kappa>0$.
The law of ordinary diffusion is therefore understood as a [*law of large numbers*]{}: as $N$ becomes very large, the probability that the empirical density $\hat\rho_N(t)$ differs significantly from the solution of the diffusion equation goes to zero.
It is natural to consider first a simpler version of the problem in which the randomness of the obstacles is removed, i.e. ${{\ensuremath{\mathbb Q}} }$ is taken to be a Dirac distribution $\delta_C$ on a special configuration of obstacles giving rise to a chaotic dynamics. This is exactly the result of Bunimovich and Sinai [@bunisinai]. They consider the 2D case in which obstacles are disks located at the vertices of a regular lattice such that the induced billiard dynamics has a finite horizon [^2]. Their result implies that for any $t>0$, any bounded function $h$ and any $\delta>0$ : $$\lim_{N\to\infty}{{\ensuremath{\mathbb P}} }\times {\bf \delta}_C[|\left<\hat\rho_N(t),h\right>-\left<\rho(t),h\right>|>\delta]=0.
\label{bunisinai}$$
Let us sketch how this statement is obtained. Let $h:{{\ensuremath{\mathbb R}} }^d\to{{\ensuremath{\mathbb R}} }$ a bounded function. First, one computes : $$\begin{aligned}
\left<\hat\rho_N(t),h\right>&=&\frac 1 N \int_{{{\ensuremath{\mathbb R}} }^d}\, d{{\ensuremath{\mathbf x}} }\sum_{j=1}^N\delta(\epsilon_N {{\ensuremath{\mathbf q}} }_i(\epsilon_N^{-2} t)-{{\ensuremath{\mathbf x}} })h({{\ensuremath{\mathbf x}} })\\
&=&\frac 1 N\sum_{j=1}^Nh(\epsilon_N {{\ensuremath{\mathbf q}} }_j(\epsilon_N^{-2} t)).\end{aligned}$$ Thus, because the initial positions of the particles are identically distributed : $$\begin{aligned}
{{\ensuremath{\mathbb E}} }[\left<\hat\rho_N(t),h\right>]&=&{{\ensuremath{\mathbb E}} }[h(\epsilon_N{{\ensuremath{\mathbf q}} }_1(\epsilon_N^{-2} t))].\end{aligned}$$ Next, the theorem 2 of [@bunisinai] implies [^3] that $$\lim_{N\to\infty}{{\ensuremath{\mathbb E}} }[h(\epsilon_N{{\ensuremath{\mathbf q}} }_1(\epsilon_N^{-2} t))]=\int_{{{\ensuremath{\mathbb R}} }^d}\rho({{\ensuremath{\mathbf x}} },t) h({{\ensuremath{\mathbf x}} },t)\,d{{\ensuremath{\mathbf x}} }\label{chaos}$$ where $\rho({{\ensuremath{\mathbf x}} },t)$ is the solution of (\[diffusion\]). To derive (\[chaos\]), one has to rely on the strong chaotic properties of the billard system under study.
Next, since $\{{{\ensuremath{\mathbf q}} }_j(t): 1\leq j\leq N\}$ are [*independent*]{}, the variance of $\left<\hat\rho_N(t),h\right>$ is $${\mathrm {Var}}[\left<\hat\rho_N(t),h\right>]=\frac 1 N{\mathrm {Var}}[h(\epsilon_N{{\ensuremath{\mathbf q}} }_1(\epsilon_N^{-2} t))]=O(\frac 1 N)$$ since $h$ is bounded. Thus, Chebychev inequality allows us to conclude the proof of Conjecture \[large\] in the case where ${{\ensuremath{\mathbb Q}} }=\delta_C$. One should note that the proof is made of two steps of very different levels of complexity. The first step is basically given by (\[chaos\]) and this is where the whole difficulty is located. The second step is a concentration result of the random variable $\left<\hat\rho_N(t),h\right>$ around the expected value ${{\ensuremath{\mathbb E}} }[\left<\hat\rho_N(t),h\right>]$. This part is trivial because when ${{\ensuremath{\mathbb Q}} }$ is replaced by $\delta_C$, the positions and velocities of the particles remain independent for all time $t$. The fact that it is so trivial is probably the reason why it is hard to find a reference where this step is mentioned or even alluded to. It is however essential and when ${{\ensuremath{\mathbb Q}} }\neq \delta_C$, the statistical independence of the motions of particles is lost. Controlling the correlations between them to ensure the concentration of $\left<\hat\rho_N(t),h\right>$ around its mean does require some work. We will see below that this issue arises in the mirrors model and how it can be dealt with.
\[Courant\]
A second experiment may be performed on the box. At the two sides of the cube $\Lambda$ perpendicular to $e_1=(1,0\ldots,0)$, particles reservoirs maintain constant values of the local densities of particles $\rho_L$ and $\rho_R$, respectively on the left and right side, see Figure 2.
A device records the net flux of mass crossing a section of $\Lambda$ perpendicular to $e_1$ per unit time. This quantity is denoted by $j(x,t)$ when the section contains the point $(x,0,\ldots,0)$ for $x\in [0,L]$. After some transient time proportional to $L^2$, it is observed that the instantaneous current of particles takes the stationary value : $$j_s(x)=\frac{\kappa}{L} (\rho_L-\rho_R).
\label{ficklaw}$$
With respect to the first experiment, the coupling to external reservoirs introduce an additional probabilistic element. The law of the reservoirs and the initial conditions of the particles inside the system is denoted by ${{\ensuremath{\mathbb P}} }$. One can introduce an empirical current of particles per unit time $\hat J_N(t)$ [^4]. Again, the goal is to show the following conjecture :
There exists a natural distribution ${{\ensuremath{\mathbb Q}} }$ such that for any bounded continuous function $h$ and any $\delta>0$ : $$\lim_{N\to\infty}\lim_{t\to\infty}{{\ensuremath{\mathbb P}} }\times {{\ensuremath{\mathbb Q}} }[|\left<\hat J_N(t),h\right>-\left<j_s,h\right>|>\delta]=0.
\label{goal2}$$ \[large2\]
While this has never been done explicitly, one should expect that with the choice ${{\ensuremath{\mathbb Q}} }=\delta_C$ the result follows from the methods of [@bunisinai]. In [@Basile], the authors show that in a low density regime limit, the expectation of the stationary current (with respect to ${{\ensuremath{\mathbb P}} }\times {{\ensuremath{\mathbb Q}} }$ ) converges to $j_s$. The statement corresponding to Conjecture \[large2\] together with the exponential convergence to the stationary current has been obtained in the case of a discrete space-time dynamics in [@Lefevere2]. We now outline how the problem may be tackled in the context of the mirrors model.
The mirrors model
=================
The mirrors model was introduced by Ruijgrok and Cohen [@Ruijgrok] as a lattice version of the random Lorentz gas or the Ehrenfest wind-tree model. The latter encompasses a Iarge class of models in which obstacles do not induce a chaotic behaviour of the trajectories of the particles. A fundamental question which remains open regarding those models is whether a non-chaotic deterministic dynamics may give rise to a [*macroscopic*]{} diffusive behaviour. In the mirrors model, particles travel on the edges of the cubic lattice generated by ${{\ensuremath{\mathbb Z}} }^d$. “Mirrors" are located at the vertices of the lattice and deflect the motion of an incoming particle in a new direction, see Figure \[mirrors\_model\] for an illustration in the 2D version of the model. The precise general definition is given below.
A quick look at the structure of the orbits reveals its total lack of ergodicity. Indeed, in Figure \[crossing\_orbits\] a sample of orbits in a finite box with periodic boundary conditions in the vertical direction and reflecting boundary conditions in the horizontal direction is pictured with different colors. For almost any configuration of the mirrors, no orbit is able to visit the entire phase space.
A perhaps even more striking fact is that, in any dimension, the motion of a particle in an environment of randomly orientated mirrors is not a gaussian diffusion [^5] [@BT]. More precisely this means that (\[chaos\]) (where the expectation is taken with respect to ${{\ensuremath{\mathbb P}} }$ and ${{\ensuremath{\mathbb Q}} }$) does not hold.
Our goal is to show that in spite of these unpromising properties, the mirrors model does exhibit normal macroscopic conductive properties when $d\geq 3$ in the sense that the analogue of (\[goal2\]) holds. It turns out that quite weak conditions on the statistics of orbits are sufficient to ensure the validity of Fick’s law at the macroscopic level. It is therefore not necessary that orbits behave as a Gaussian diffusion to ensure the validity of Fick’s law. Thus, the normal [*macroscopic*]{} laws of diffusion apply to a much wider class of dynamical systems than generally expected. The dynamics of the mirrors model is reversible in the usual sense of the word in the context of Hamiltonian dynamics. Namely, under the reversal of the velocities of all particles at a given time $t>0$, the dynamics brings the system of particles to its initial condition at time $0$ (with reversed velocities), see (\[reversible\]). This reversibility property of the dynamics will allow us to show that when the system is large, the number of orbits travelling from one side of the system to the other one basically determines the value of the current in the stationary state. This will allow to formulate a condition on the distribution of orbits that is both sufficient and necessary for the validity of Fick’s law. We recall now briefly the set-up of the original mirrors model. Particles travel on the edges of ${{\ensuremath{\mathbb Z}} }^2$ with unit speed. Mirrors are located at some vertices of the lattice and take two possible angular orientations : $\{\frac \pi 4, \frac{3\pi}{4}\}$. When a particle hits a mirror, it gets deflected according to the laws of specular reflection, see Figure \[crossing\_orbits\] for sample trajectories of particles. It is convenient to think that every particle starts at time zero with a given velocity at a vertex of the lattice ${{\ensuremath{\mathcal Q}} }$ that is obtained by taking the middle point of every edge of ${{\ensuremath{\mathbb Z}} }^2$. As all particles move with unit velocity, one can simply observe the evolution of the system at discrete times $t\in{{\ensuremath{\mathbb N}} }$. At those times, the particles will be always located at one of the vertices of the new lattice ${{\ensuremath{\mathcal Q}} }$ with a well-defined velocity. In general, the orientation of the mirrors is picked randomly. It is obvious that the motion of a single particle can not be described as a Markov process. When a particle hits a mirror for the second time, no matter how far back in the past the first visit occurred, its reflection is strongly affected by the way its was reflected at the first visit. For instance in Figure \[crossing\_orbits\], the two orientations of the mirrors are picked at random, and in that case, at the second visit the reflection is always deterministic.
We come now to a more general definition of the dynamics in $d$ dimensions. We denote by ${{\ensuremath{\mathbf z}} }=(z_1,\ldots,z_d)$ a generic element of ${{\ensuremath{\mathbb Z}} }^d$. As for ${{\ensuremath{\mathbb Z}} }^2$, we consider the set of midpoints of edges of an hypercube of ${{\ensuremath{\mathbb Z}} }^d$ of side $N$ and with periodic conditions in all but the first direction. We call this set ${{\ensuremath{\mathcal Q}} }$. It may be described as follows : ${{\ensuremath{\mathcal Q}} }=\bigcup_{i=1}^d L_i$ where $L_i=\left\{{{\ensuremath{\mathbf z}} }+\frac 1 2 {{\ensuremath{\mathbf e}} }_i: \; 0\leq z_1 \leq N-1,\;(z_2,\ldots,z_{d})\in ({{\ensuremath{\mathbb Z}} }/N{{\ensuremath{\mathbb Z}} })^{d-1}\right\}$. Let $({{\ensuremath{\mathbf e}} }_1,\ldots,{{\ensuremath{\mathbf e}} }_d)$ the canonical basis of ${{\ensuremath{\mathbb R}} }^d$, the space of possible velocities is ${{\ensuremath{\mathcal P}} }=\{\pm\frac{{{\ensuremath{\mathbf e}} }_1}{2},\ldots,\pm\frac{{{\ensuremath{\mathbf e}} }_d}{2}\}$ and the phase space of the dynamics is $${{\ensuremath{\mathcal M}} }=\{({{\ensuremath{\mathbf q}} },{{\ensuremath{\mathbf p}} }): {{\ensuremath{\mathbf q}} }\in{{\ensuremath{\mathcal Q}} },{{\ensuremath{\mathbf p}} }\in{{\ensuremath{\mathcal P}} }\;{\rm s.\; t.\; if}\;{{\ensuremath{\mathbf q}} }\in L_i\;{\rm then}\; {{\ensuremath{\mathbf p}} }=\pm\frac{{{\ensuremath{\mathbf e}} }_i}{2}\}.$$ We denote a generic point of ${{\ensuremath{\mathcal M}} }$ by $({{\ensuremath{\mathbf q}} },{{\ensuremath{\mathbf p}} })$. The set of points in ${{\ensuremath{\mathcal M}} }$ whose spatial coordinate belongs to the boundaries of the system is $B=B_-\cup B_+$, with $$\begin{aligned}
B_-&=&\{x=({{\ensuremath{\mathbf q}} },{{\ensuremath{\mathbf p}} })\in{{\ensuremath{\mathcal M}} }: {{\ensuremath{\mathbf q}} }=(q_1,\ldots,q_d) \in L_1, q_1=\frac 1 2\}\nonumber\\
B_+&=&\{x=({{\ensuremath{\mathbf q}} },{{\ensuremath{\mathbf p}} })\in{{\ensuremath{\mathcal M}} }: {{\ensuremath{\mathbf q}} }=(q_1,\ldots,q_d)\in L_1, q_1=N-\frac 1 2\}.\nonumber\end{aligned}$$ See Figure \[mirrors1\].
For each ${{\ensuremath{\mathbf z}} }\in{{\ensuremath{\mathbb Z}} }^d$, we define the action of a “mirror" on the velocity of an incoming particle by $\pi({{\ensuremath{\mathbf z}} };\cdot)$ which is a bijection of ${{\ensuremath{\mathcal P}} }$ into itself. It satisfies the following conditions : $$\left\{
\begin{array}{l}
\pi({{\ensuremath{\mathbf z}} };-\pi({{\ensuremath{\mathbf z}} };{{\ensuremath{\mathbf p}} }))=-{{\ensuremath{\mathbf p}} },\quad \forall {{\ensuremath{\mathbf z}} }\in{{\ensuremath{\mathbb Z}} }^d,\forall{{\ensuremath{\mathbf p}} }\in{{\ensuremath{\mathcal P}} }\\
\pi(0,z_2,\ldots,z_{d};- \frac{{{\ensuremath{\mathbf e}} }_1}{2})= \frac{{{\ensuremath{\mathbf e}} }_1}{2}\\
\pi(N, z_2,\ldots,z_{d}; \frac{{{\ensuremath{\mathbf e}} }_1}{2})=- \frac{{{\ensuremath{\mathbf e}} }_1}{2}, \;(z_2,\ldots,z_{d})\in ({{\ensuremath{\mathbb Z}} }/N{{\ensuremath{\mathbb Z}} })^{d-1}
\end{array}
\right.
\label{conditionsx}$$ The dynamics is defined on ${{\ensuremath{\mathcal M}} }$ in the following way. For any $({{\ensuremath{\mathbf q}} },{{\ensuremath{\mathbf p}} })\in{{\ensuremath{\mathcal M}} }$ : $$F({{\ensuremath{\mathbf q}} },{{\ensuremath{\mathbf p}} })=\left({{\ensuremath{\mathbf q}} }+{{\ensuremath{\mathbf p}} }+ \pi({{\ensuremath{\mathbf q}} }+{{\ensuremath{\mathbf p}} };{{\ensuremath{\mathbf p}} }),\pi({{\ensuremath{\mathbf q}} }+{{\ensuremath{\mathbf p}} };{{\ensuremath{\mathbf p}} })\right).$$ It is easy to check that the map $F$ is a bijection on ${{\ensuremath{\mathcal M}} }$. The two last conditions in (\[conditionsx\]) are just saying that when particles hit the boundaries they are reflected backwards. We define an operator $R:{{\ensuremath{\mathcal M}} }\to{{\ensuremath{\mathcal M}} }$ which reverses the velocities by $R ({{\ensuremath{\mathbf q}} },{{\ensuremath{\mathbf p}} })=({{\ensuremath{\mathbf q}} },-{{\ensuremath{\mathbf p}} })$ and we see that the first of the conditions (\[conditionsx\]) ensures that the map $F$ is [*reversible*]{}, $$F^{-1}=R F R.
\label{reversible}$$
We define the orbit of a point $x\in{{\ensuremath{\mathcal M}} }$, ${\mathcal O}_x=\{y\in{{\ensuremath{\mathcal M}} }: \exists t\geq 0,F^t(x)=y\}$ and its period, $T(x)=\inf\{t\geq 0: F^t(x)=x\}$. From the fact that $F$ is bijective, one infers that for every $x\in{{\ensuremath{\mathcal M}} }$, ${\mathcal O}_x$ is a loop : $T(x)\leq |{{\ensuremath{\mathcal M}} }|$ and that orbits are non-intersecting : if $ y\notin {\mathcal O}_x$, then ${\mathcal O}_x\cap{\mathcal O}_y=\emptyset$. A given orbit is also non-self-intersecting : if $y\in {\mathcal O}_x$ and $y\neq x$ then $F(y)\neq F(x)$.
As we are interested in the transport of particles, we define occupation variables $\sigma({{\ensuremath{\mathbf q}} },{{\ensuremath{\mathbf p}} };t)\in\{0,1\}$ that record the absence or presence of a particle at position ${{\ensuremath{\mathbf q}} }$ with velocity ${{\ensuremath{\mathbf p}} }$ at time $t\in{{\ensuremath{\mathbb N}} }$. When connecting the system to external particles reservoirs, we obtain the following evolution rule : given $\sigma(\cdot;t-1)$, we define $\sigma(\cdot;t)$ for all $t\in{{\ensuremath{\mathbb N}} }^*$ by $$\sigma(x;t)=\left\{
\begin{array}{lll}
\sigma(F^{-1}(x);t-1)\quad {\rm if} \quad x\notin B_{-}\cup B_{+}\\
\sigma^-_{x}(t-1)\quad {\rm if} \quad x\in B_{-}\\
\sigma^+_{x}(t-1)\quad {\rm if} \quad x \in B_{+}.
\end{array}
\right.$$ The families of random variables $\{\sigma^-_{x}(t):x\in B_{-},\, t\in{{\ensuremath{\mathbb N}} }\}$ and $\{\sigma^+_{x}(t): x\in B_{+},\, t\in{{\ensuremath{\mathbb N}} }\}$ consist of independent Bernoulli variables with respective parameters $\rho_-$ and $\rho_+$. If one chooses $\{\sigma(x;0):x\in{{\ensuremath{\mathcal M}} }\}$ to be a collection of independent random variables, then it is easy to see by induction that at any $t\geq 0$, $\{\sigma(x;t):x\in{{\ensuremath{\mathcal M}} }\}$ is a collection of i.i.d Bernoulli random variables. To simplify a bit the discussion, we choose an homogeneous initial distribution, i.e. all Bernoulli random variables have a common parameter $\rho_I$. The distribution of the collection $\{\sigma(x;t):x\in{{\ensuremath{\mathcal M}} }\}$ becomes stationary after a [*finite*]{} time. More precisely, for any $t\geq |{{\ensuremath{\mathcal M}} }|$, we have the following equality in law : $$\sigma(x,t)=\left\{
\begin{array}{lll}
\sigma_I\quad {\rm if} \quad {{\ensuremath{\mathcal O}} }_x\cap B=\emptyset\\
\sigma_-\quad {\rm if} \quad F^{-t^*}(x)\in B_-\\
\sigma_+\quad {\rm if} \quad F^{-t^*}(x)\in B_+
\end{array}
\right.$$ where $t^*=\inf\{t: F^{-t}(x)\in B\}$ and $\sigma_\pm$ and $\sigma_I$ are Bernoulli random variables of parameter $\rho_{\pm}$ and $\rho_I$.
Proceeding as in [@Lefevere2], it is possible to show that when the size of the system goes to infinity, the stationary current converges in probability to the proportion of crossing orbits times the chemical potentials difference. We define the average current of particles that crosses the hyperplane ${{\ensuremath{\mathcal Q}} }^l=\{{{\ensuremath{\mathbf q}} }\in{{\ensuremath{\mathcal Q}} }: q_1=l+\frac 1 2\}$, $l\in\{1,\ldots,N-2\}$ during a diffusive time interval $N^2$ : $$J(l,t)=\frac 1 {N^{d+1}}\sum_{s=t+1}^{t+N^2}\sum_{x\in{{\ensuremath{\mathcal M}} }}\sigma(x;s)\Delta(x,l)
\label{cud}$$ where $\Delta(x,l)=2({{\ensuremath{\mathbf p}} }\cdot{{\ensuremath{\mathbf e}} }_1){\bf 1}_{{{\ensuremath{\mathbf q}} }\in {{\ensuremath{\mathcal Q}} }^l}$,with $x=({{\ensuremath{\mathbf q}} },{{\ensuremath{\mathbf p}} })$. Thus $\Delta(x,l)$ takes the value $+1$ (resp. $-1$) if $x$ crosses the slice ${{\ensuremath{\mathcal Q}} }^l$ from left to right (resp. from right to left). We denote by ${{\ensuremath{\mathcal N}} }_{\pm}$ the numbers of crossings from $B_{\pm}$ to $B_{\mp}$ induced by $F$, i.e. ${{\ensuremath{\mathcal N}} }_{\pm}=|S_\pm|$ where $S_\pm$ is given by $$\begin{aligned}
S_\pm=\{ x\in B_{\pm}:\exists s>0,\forall 0<j<s,\; F^j(x)\notin B_{\pm}, F^s(x)\in B_{\mp}\}.\nonumber\end{aligned}$$
One notes that ${{\ensuremath{\mathcal N}} }_+={{\ensuremath{\mathcal N}} }_-$. Indeed, since every orbit is closed, it must contain as many left-to-right than right-to-left crossings. Thus, we set ${{\ensuremath{\mathcal N}} }= {{\ensuremath{\mathcal N}} }_+={{\ensuremath{\mathcal N}} }_-$. Proceeding as in [@Lefevere2], we get that for any $t\geq |{{\ensuremath{\mathcal M}} }|$, ${{\ensuremath{\mathbb E}} }[J(l,t)]=\frac{{{\ensuremath{\mathcal N}} }}{N^{d-1}}(\rho_--\rho_+)$. [^6] Moreover, for every $\delta>0$, any $t\geq |{{\ensuremath{\mathcal M}} }|$ and $l\in\{1,\ldots,N-2\}$,
$${{\ensuremath{\mathbb P}} }\left[\left|J(l,t)-\frac{{\mathcal N}}{N^{d-1}}(\rho_--\rho_+)\right|\geq \delta \right]\leq 2\exp(-\delta^2N^{d+1}).
\label{JN}$$
We take now random configurations of reflectors $\{\pi({{\ensuremath{\mathbf z}} };\cdot): {{\ensuremath{\mathbf z}} }\in {{\ensuremath{\mathbb Z}} }^d\}$. The law of the reflectors is denoted by ${{\ensuremath{\mathbb Q}} }$. The map $F$ becomes now a random map.
The model satisfies [*Fick’s law*]{} if and only if there exists some $\kappa>0$ (the conductivity) such that $\forall\delta>0$, $$\lim_{N\to\infty}\lim_{t\to\infty}{{\ensuremath{\mathbb P}} }\times{{\ensuremath{\mathbb Q}} }[|N J(l,t)-\kappa(\rho_--\rho_+)|>\delta]=0.
\label{Ficks}$$ As in [@Lefevere2], it is easy to infer from (\[JN\]) that the following theorem holds.
[**Sufficient and Necessary Condition for Fick’s law**]{} : (\[Ficks\]) holds if and only if there exists $\kappa>0$ such that for any $\delta>0$, $$\lim_{N\to\infty}{{\ensuremath{\mathbb Q}} }\left[\left|\frac{{{\ensuremath{\mathcal N}} }}{N^{d-2}}-\kappa\right|>\delta\right]=0.
\label{Ficks2}$$
We see that the central object to study is the distribution of the number of crossing orbits ${{\ensuremath{\mathcal N}} }$. The expectation of this quantity is related to the probability that one orbit crosses the system, while the variance is given in terms of the joint probability that orbits with two different starting points cross the system. Indeed by periodicity, we have, using the notations $O=((\frac 1 2,0,\ldots,0), \frac{{{\ensuremath{\mathbf e}} }_1}{2})$ and $S=S_-$ : $${{\ensuremath{\mathbb E}} }\left[\frac{{{\ensuremath{\mathcal N}} }}{N^{d-2}}\right]=\frac N {N^{d-1}}\sum_{x\in B_{-}}{{\ensuremath{\mathbb E}} }[{\bf 1}_{x\in S}]=N{{\ensuremath{\mathbb Q}} }[O\in S]
\label{expectation}$$ and $$\begin{aligned}
{\rm Var }\left[\frac{{{\ensuremath{\mathcal N}} }}{N^{d-2}}\right]
=\frac{1}{N^{2d-4}}\sum_{x,y\in B_{-}}\delta(x,y)=\frac{1}{N^{d-3}}\sum_{x\in B_{-}}\delta(O,x)\nonumber\\
\label{variance}\end{aligned}$$ with $$\delta(x,y)={{\ensuremath{\mathbb Q}} }[x\in S,y\in S]-{{\ensuremath{\mathbb Q}} }[x\in S]{{\ensuremath{\mathbb Q}} }[y\in S].
\label{corel}$$
Thus if the two following
are satisfied :
1. There exist $\kappa>0$ such that the RHS of (\[expectation\]) converges to $\kappa$ as $N\to\infty$.
2. The RHS of (\[variance\]) goes to zero as $N\to\infty$.
\[conditions\]
then Fick’s law (\[Ficks\]) holds in the stationary state. We note first that when $d=2$, (\[Ficks2\]) can not hold, whatever the distribution ${{\ensuremath{\mathbb Q}} }$ is. To see this, we adapt an argument found in [@Kozma]. Indeed, the spatial part of each crossing orbit crosses any “vertical" section ${{\ensuremath{\mathcal Q}} }^l$ an odd number of times. On the other hand, the spatial part of any non-crossing orbit must cross any vertical section an even number of times, see Figure \[crossing\_orbits\]. Thus, $N$ and ${{\ensuremath{\mathcal N}} }$ must have the same parity. This implies that there can not exist $\kappa>0$ such that (\[Ficks2\]) holds when $d=2$. The origin of this issue lies in the strong correlations between crossing orbits that are present in two dimensions.
We turn now to the higher dimensional case $d\geq 3$ equipped with some natural and spatially homogeneous distribution ${{\ensuremath{\mathbb Q}} }$. Now observe that if ${{\ensuremath{\mathbb Q}} }[\pi({{\ensuremath{\mathbf z}} };\frac{e_i}{2})=-\frac{e_i}{2}]>0$ for some $i=1,\ldots,d$ then an orbit starting from $O$ will encounter this type of reflecting mirror after an exponential number of steps and therefore ${{\ensuremath{\mathbb Q}} }[0\in S]\leq e^{-cN}$ for some $c>0$. This, in turn, implies that $\lim_{N\to\infty}N{{\ensuremath{\mathbb Q}} }[0\in S]=0$ and that Fick’s law can not hold. Thus from now on, we consider maps such that $\pi({{\ensuremath{\mathbf z}} };\frac{e_i}{2})\neq-\frac{e_i}{2}$ if $0<z_1 < N$ and such that the conditions (\[conditionsx\]) are satisfied. We call the set of such maps $\Pi$. We take ${{\ensuremath{\mathbb Q}} }$ such that the collection of maps $$\{\pi({{\ensuremath{\mathbf z}} };.): 0<z_1 < N,\;(z_2,\ldots,z_{d})\in ({{\ensuremath{\mathbb Z}} }/N{{\ensuremath{\mathbb Z}} })^{d-1}\}
\nonumber$$ is independent and that each map is uniformly distributed over $\Pi$. We note first that if the law of an orbit with respect to ${{\ensuremath{\mathbb Q}} }$ was similar to the law of a simple random walk, then there would be a $\kappa>0$ such that $\lim_{N\to\infty}N{{\ensuremath{\mathbb Q}} }[0\in S]=\kappa$, this follows from the gambler’s ruin argument. Similarly, if the orbits were independent objects, then the RHS of (\[variance\]) would go to zero because the only non-zero term would be the one with $x=O$ and ${{\ensuremath{\mathbb Q}} }[O\in S]\sim\kappa/N$. We also note that the average stationary current is identified as the difference between chemical potentials times the probability that a particle crosses the system, an idea that was put forward in [@Carlos], in the context of chaotic systems. The law ${{\ensuremath{\mathbb Q}} }$ of the mirrors induces a law on the set of orbits which is a priori very far from the distribution of independent simple random walks. The set of orbits is a very interesting lattice object in itself which features some (self-)avoiding properties as we mentioned above.
Fortunately, what is needed to ensure the validity of (\[Ficks2\]) is much less than the full joint distribution of the orbits. Thanks to (\[expectation\]) and (\[variance\]), one only has to analyze the marginal of a path starting on the boundary and also the joint probability of two such paths. The distribution of a path starting at $O$ (i.e. on the boundary) is similar to the one of a “true" self-avoiding random walk [@Amit] but defined on ${{\ensuremath{\mathcal Q}} }$ rather than on ${{\ensuremath{\mathbb Z}} }^d$ and with further constraints. The diffusive behaviour of those walks for $d\geq3$ has been conjectured in [@Amit], see also the rigorous results of [@Toth]. It can be expected that as the dimensionality of the system increases, the effect of the revisits of an orbit to the same mirror decreases. In a process where the mirrors are flipped randomly after being used (i.e memory effects are killed), we computed that in $d=3$ the crossing probability is $\sim 3/2N$. Numerical simulations in $d=3$ show that this number is indeed a good approximation. The log log plot of the crossing probability ${{\ensuremath{\mathbb Q}} }[O\in S]$ is given in Figure \[3DPcrossing\] for $N$ up to $400$. The corresponding conductivity is $\kappa=1.535\pm0.005$. As the conductivity measured in simulations is slightly higher than $3/2$, it indicates that recollisions tend to push forward the orbit.
We must show now that $\sum_{x\in B_{-}}\delta(O,x)\to 0$ as $N\to\infty$. We know that this sum is positive because it is a variance, and thus it is enough to get an upper bound on the sum. We will use a numerical analysis to show that $\delta(O,x)<0$ for all but a finite (i.e. independent of $N$) number $x\in B_-$.
But before doing that and to get a better picture of the origin of the correlations $\delta(0,x)$, we split them in two parts. Given an orbit ${{\ensuremath{\mathcal O}} }$, we denote by $\gamma({{\ensuremath{\mathcal O}} })$ the set of edges of ${{\ensuremath{\mathbb Z}} }^d$ used by ${{\ensuremath{\mathcal O}} }$. For each ${{\ensuremath{\mathbf z}} }\in Z_N:=\{{{\ensuremath{\mathbf z}} }\in{{\ensuremath{\mathbb Z}} }^d: 1\leq z_1\leq N-1, (z_2,\ldots,z_d)\in ({{\ensuremath{\mathbb Z}} }/N{{\ensuremath{\mathbb Z}} })^d\}$, let also $b_{{{\ensuremath{\mathbf z}} }}({{\ensuremath{\mathcal O}} })$ be the half of the number of times that the orbit ${{\ensuremath{\mathcal O}} }$ visits the vertex ${{\ensuremath{\mathbf z}} }$. Two [*crossing orbits*]{} ${{\ensuremath{\mathcal O}} }$ and ${{\ensuremath{\mathcal O}} }'$ are incompatible if $\gamma({{\ensuremath{\mathcal O}} })\cap\gamma({{\ensuremath{\mathcal O}} }')\neq \emptyset$ and compatible otherwise. The law of a given [*crossing*]{} orbit is : $${{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} })=\prod_{{{\ensuremath{\mathbf z}} }\in Z_N}\prod_{j=1}^{b_{{{\ensuremath{\mathbf z}} }}({{\ensuremath{\mathcal O}} })}\frac{1}{2(d-j)+1}.
\label{law_single}$$ If ${{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} },{{\ensuremath{\mathcal O}} }')$ is the joint probability of two orbits ${{\ensuremath{\mathcal O}} }$ and ${{\ensuremath{\mathcal O}} }'$ then ${{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} },{{\ensuremath{\mathcal O}} }')=0$ when ${{\ensuremath{\mathcal O}} }$ and ${{\ensuremath{\mathcal O}} }'$ are incompatible. If they are compatible, the joint law of two crossing orbits is given by $${{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} },{{\ensuremath{\mathcal O}} }')=\prod_{{{\ensuremath{\mathbf z}} }\in Z_N}\prod_{j=1}^{b_{{{\ensuremath{\mathbf z}} }}({{\ensuremath{\mathcal O}} })+b_{{{\ensuremath{\mathbf z}} }}({{\ensuremath{\mathcal O}} }')}\frac{1}{2(d-j)+1}.
\label{law_joint}$$ In particular, if they do not share any mirrors, then ${{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} },{{\ensuremath{\mathcal O}} }')={{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} }){{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} }')$. From those properties, starting from (\[corel\]), we obtain that for $x\in B_-$, $$\begin{aligned}
\delta(O,x)&=&\sum_{{{\ensuremath{\mathcal O}} }_0,{{\ensuremath{\mathcal O}} }_x} ({{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} }_0,{{\ensuremath{\mathcal O}} }_x)-{{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} }_0){{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} }_x))\nonumber\\
&-&{\sum_{{{\ensuremath{\mathcal O}} }_0,{{\ensuremath{\mathcal O}} }_x}}'{{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} }_0){{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} }_x).
\label{decomp}\end{aligned}$$ Both sums run over orbits that cross the box ${{\ensuremath{\mathcal Q}} }$. The first sum runs over compatible orbits such that $\gamma({{\ensuremath{\mathcal O}} }_0)$ and $\gamma({{\ensuremath{\mathcal O}} }_x)$ share a vertex of ${{\ensuremath{\mathbb Z}} }^d$. The second (prime) sum runs over [*incompatible*]{} orbits ${{\ensuremath{\mathcal O}} }_0,{{\ensuremath{\mathcal O}} }_x$.
Thus, from (\[decomp\]), we see that correlations $\delta(O,x)$ are created from two opposite origins, corresponding to each of the two sums in (\[decomp\]). If two orbits ${{\ensuremath{\mathcal O}} }$ and ${{\ensuremath{\mathcal O}} }'$ share some mirrors, then it is easy to see from (\[law\_joint\]) that ${{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} },{{\ensuremath{\mathcal O}} }')>{{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} }){{\ensuremath{\mathbb Q}} }({{\ensuremath{\mathcal O}} }')$. This in turn implies that the first sum is strictly positive. This is a “cooperative" effect, the orbits help each other crossing the system. The second sum corresponds to a [*jamming*]{} effect : an orbit starting from $O$ and crossing the system occupies a certain number of horizontal edges. Because distinct orbits can not share the same edges, the occupied edges are no more available for an orbit starting from $x\in B_-$, this creates negative correlations.
Numerical simulations in $d=3$ show that the latter effect dominates. For all but a few points, the correlations $\delta(O,x)$ for $x\neq O$ are not only small but negative within confidence intervals, see Figure \[3Dcorrelations\]. The only exceptions are points $((1/2,1,0),\frac{{{\ensuremath{\mathbf e}} }_1}{2})$, $((1/2,0,1),\frac{{{\ensuremath{\mathbf e}} }_1}{2})$, $((1/2,N-1,0),\frac{{{\ensuremath{\mathbf e}} }_1}{2})$ and $((1/2,0,N-1),\frac{{{\ensuremath{\mathbf e}} }_1}{2})$ which give clearly positive correlations. However, we checked that for $N=70$, $\sum_{y=1}^{N-1}\delta(O,((1/2,y,0),\frac{{{\ensuremath{\mathbf e}} }_1}{2}))=-1.360\times10^{-04}\pm 1.47\times10^{-05}$, i.e. it is negative with a margin of more than $9\sigma$. $\sum_{z=1}^{N-1}\delta(O,((1/2,0,z),\frac{{{\ensuremath{\mathbf e}} }_1}{2}))$ must be equal by symmetry. Increasing values of $N$ do not modify this behaviour. In particular, the number of points with positive correlations do not increase. Since we know already that ${{\ensuremath{\mathbb Q}} }[O\in S]\sim\kappa/N$, as $N\to\infty$, we conclude with the same margin that $\sum_{x\in B_{-}}\delta(O,x)\leq \kappa/N\to 0$, as $N\to\infty$.
We expect the same behaviour in $d>3$. A rigorous proof that the [*crossing conditions*]{} introduced above are satisfied seems to be within reach in the present model. Moreover, it is possible to draw a general conclusion from the above discussion. If one is really interested in deriving macroscopic laws from microscopic dynamics, many detailed properties of the latter are irrelevant. Only weaker properties than chaoticity, ergodicity or Gaussian behaviour of the orbits are required. In the present context, the minimal properties necessary to obtain Fick’s law are encapsulated in the crossing conditions. It is of course natural to seek similar weak conditions in different contexts as for instance in the problem of the derivation of Fourier’s law.
[10]{}
G. Basile, A. Nota, F. Pezzotti, M. Pulvirenti [*Derivation of the Fick’s law for the Lorentz model in a low density regime*]{} Communications in Mathematical Physics 04/2014; 336(3)
D.J. Amit, G. Parisi, L.Peliti, Phys. Rev. B 27, (1983), 1635
L.A. Bunimovich, Ya. G. Sinai, Ya. G. [*Statistical properties of the Lorentz gas with periodic configuration of scatterers.* ]{} Commun. Math. Phys. 78, (1980) 479-497
L. A. Bunimovich and S. E. Troubetzkoy, Journal of Statistical Physics, Vol. 67, Nos. 1/2, 1992
G. Casati, C. Mejia-Monasterio and T. Prosen Phys. Rev. Lett. 101, (2008) 016601
X.P.Kong and E.G.D.Cohen Phys. Rev. B 40, (1989), 4838-4845
G. Kozma, V. Sidoravicius, arXiv:1311.7437
R. Lefevere, Arch. Rat. Mech. and Anal. : Volume 216, Issue 3 (2015), Page 983-1008
T. W. Ruijgrok, E. G. D. Cohen, Phys. Lett. A 133 (1988) 415
I. Horváth, B.Tóth, B. Veto, Prob. Th. and Rel. F. 153 (2012) 691-726
[^1]: By natural we mean as uniform as possible over the locations and shapes of obstacles.
[^2]: For instance, the center of each (sufficiently large) disk is located at a vertex of a triangular lattice.
[^3]: To be more precise Bunimovich and Sinaï consider the case $L=\infty$ but their method should apply directly to the finite $L$ case
[^4]: This quantity will be our main object of study in the next section and will be given a precise definition there.
[^5]: In spite of this, it has been observed numerically [@Kong] that in two dimensions, the mean-square displacement of a given particle is linear in time, allowing the definition of a [*microscopic*]{} diffusion coefficient. This is of course a much weaker property than the property (\[goal2\]). In particular we will see that one can not define a [*macroscopic*]{} diffusion coefficient in $2D$.
[^6]: This relation implies that the average current flows in the “right" direction and that when $\rho_-\neq\rho_+$, the average current in the stationary state is different from $0$ if and only if ${{\ensuremath{\mathcal N}} }\neq 0$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This article describes a fast iterative algorithm for image denoising and deconvolution with signal-dependent observation noise. We use an optimization strategy based on variable splitting that adapts traditional Gaussian noise-based restoration algorithms to account for the observed image being corrupted by mixed Poisson-Gaussian noise and quantization errors.'
author:
- 'Ayan Chakrabarti[$^*$]{} and Todd Zickler[^1]'
bibliography:
- 'realnoise.bib'
title: |
Image Restoration with[\
]{} Signal-dependent Camera Noise
---
=1
Image Restoration, Signal-dependent noise, Poisson Noise, Variable Splitting.
Introduction {#sec:intro}
============
restoration refers to the recovery of a clean sharp image from a noisy, and potentially blurred, observation. Most state-of-the-art restoration techniques [@portilla; @bm3d; @levin07; @ADM; @dilip] assume that observed image is corrupted by signal-independent additive white Gaussian noise (AWGN), since this makes both analysis and estimation significantly more convenient. However, observation noise in an image captured by a real digital camera is typically signal-dependent, and therefore image restoration algorithms based on AWGN models make sub-optimal use of the information available in the observed image. For example, one source of noise in recorded intensities is the uncertainty in the photon arrival process, which is Poisson distributed and has a variance that scales linearly with the signal magnitude. Therefore, using an AWGN model fails to account for the noise variance at darker pixels being lower than that at brighter ones, and can lead to over-smoothing in darker regions of the restored image.
Due to these reasons, the development of restoration algorithms based on accurate noise models has been an area of active research [@kolaczyk; @hirakawa; @florian; @foi0; @foi]. However, this task is made challenging by the fact that state-of-the-art restoration methods use sophisticated image priors that are defined in terms of coefficients of some spatial transform, and combining these priors with a non-Gaussian noise model significantly adds to complexity during estimation. Denoising methods for signal-dependent noise are either based on a (sometimes approximate) statistical characterization of this noise in transform coefficients [@hirakawa; @kolaczyk; @florian], or use a *variance-stabilization* transform that renders the noise Gaussian followed by traditional AWGN denoising techniques [@foi0; @foi]. Recently, Harmany et al. [@spiraltv] presented an iterative deconvolution algorithm that approximates a Poisson-noise likelihood cost at each iteration with a quadratic function based on the curvature of the likelihood at the current estimate. This technique was combined with various sparsity-based priors in [@spiraltv] to enable restoration in the presence of Poisson noise.
We describe an iterative image restoration framework that accounts for the statistics of real camera noise. It is also an iterative technique like the one in [@spiraltv], but uses an optimization strategy known as *variable-splitting* leading to significant benefits in computational cost. The use of this strategy for image restoration dates back to the work of Geman et al. [@geman], and has been used to develop fast deconvolution algorithms that use the so-called *total-variation* (TV) image prior model [@ADM], with extensions that consider non-quadratic penalties on the noise— including a Poisson likelihood cost [@uclatr]. In this paper, we deploy this technique to enable efficient estimation with a likelihood cost that models observation noise as a combination of Gaussian noise, signal-dependent Poisson shot-noise, and digitization errors. Moreover, we develop a general framework that is not tied to any specific choice of image prior, and this allows us to adapt any state-of-the-art AWGN restoration technique (such as BM3D [@bm3d] for denoising) for use with the proposed signal-dependent noise model. We demonstrate the efficacy of this approach with comparisons to both AWGN and signal-dependent state-of-the-art restoration methods.
Observation Model {#sec:nmodel}
=================
Let $x(n)$ be the latent noise-free sharp image of a scene corresponding to an observed image $y(n)$, where $n \in \mathbb{R}^2$ indexes pixel location. We assume that a spatially-uniform blur acts on the scene, with a known kernel $k(n)$. We let $x_k(n) = (x*k)(n)$ denote the blurred image that would have been observed in the absence of noise. The observation $y(n)$ can then be modeled as $$\label{eq:sensor}
y(n) = Q\left[\tilde{y}(n)\right],~~\tilde{y}(n) = y_k(n) + z(n),$$ where $Q(\cdot)$ is a quantization function used to digitize the analog sensor measurement $\tilde{y}(n)$, which we in turn model as the sum of a scaled Poisson random variable $y_k(n)$ with mean $x_k(n)$, and zero-mean Gaussian random noise $z(n)$ with variance $\sigma^2$. We examine the statistical properties of each component of the model in , and define a likelihood function based on these properties with the aim of enabling accurate yet efficient inference of $x(n)$ from $y(n)$.
Shot Noise {#sec:shotnoise}
----------
At each location $n$, the random variable $y_k$ captures the uncertainty in the photon arrival process, and is modeled with a scale parameter $\alpha > 0$ as $\alpha y_k \sim \mathcal{P}(\alpha x_k)$, i.e., $$\label{eq:pzn}
P\left(y_k | x_k\right) = \frac{\left(\alpha x_k\right)^{\alpha y_k}e^{-\alpha x_k}}{\left(\alpha y_k\right)!}.$$ The difference between the observed $y_k$ and its mean $x_k$ is referred to as shot noise, and has a signal-dependent variance equal to $x_k/\alpha$. The parameter $\alpha$ depends on the ISO setting of the camera, and a high value of $\alpha$ corresponds to a low ISO setting and a higher signal-to-noise ratio (SNR). We define $L_P(x_k;y_k,\alpha)$ as the corresponding negative log-likelihood (up to a constant) of the observed pixel $y_k$ being due to $x_k$: $$\label{eq:Lpz}
L_P(x_k;y_k,\alpha) = \alpha x_k - \alpha y_k\log x_k.$$ Note that $L_P$ is a convex function of $x_k$ (since $\partial^2L_P/\partial x_k^2=\alpha{}y_k/x_k^2\geq 0, \forall x_k$), with a minimum at $x_k = y_k$.
Gaussian Noise {#sec:gnoise}
--------------
In addition to shot noise, the measurement $\tilde{y}$ may be corrupted by other signal-independent noise sources, such as thermal and amplifier noise, which we model with the AWGN variable $z \sim \mathcal{N}(0,\sigma^2)$ [@foi; @florian]. Combining this with the Poisson model in , we have: $$\label{eq:pmdl}
p\left(\tilde{y}|x_k\right) \propto \sum_{r=0}^\infty \frac{(\alpha x_k)^re^{-\alpha x_k}}{r!} \exp\left(- \frac{(\tilde{y} - r/\alpha)^2}{2\sigma^2} \right).$$ Unfortunately, the above expression can not be computed in closed form, and therefore we employ an approximation to define the log-likelihood function for this case. We note that $\tilde{y}$ is a mixed Poisson-Gaussian random variable with mean $x_k$ and variance $(x_k/\alpha+\sigma^2)$, and approximate it with a shifted Poisson likelihood as $$\label{eq:LGpsz}
L_{PG}(x_k;\tilde{y},\alpha,\sigma) = L_P(x_k + \alpha\sigma^2;\tilde{y} + \alpha\sigma^2;\alpha).$$
Quantization {#sec:quant}
------------
Finally, we account for the observed intensity $\tilde{y}$ being quantized by a function $Q(\cdot)$ that maps every interval $[y_-,y_+]$ to a single value $y$. We consider the general case where quantization is possibly preceded by a non-linear map for gamma-correction: $$\label{eq:quant1}
y = \left(\left\lfloor \tilde{y}^{1/g} \right\rfloor_q \right)^g,$$ where $g$ corresponds to the gamma-correction exponent ($1$ for linear data, and typical $2.2$ for sRGB), and $\lfloor\cdot\rfloor_q$ denotes rounding off intervals of width $q$. Note that $y$ here denotes a linearized version of the camera image, i.e., one where the inverse of the gamma-correction function has been applied to the quantized observations. The interval $[y_-,y_+]$ may therefore be asymmetric around $\tilde{y}$ and have variable widths, and the mean and variance of $\tilde{y}$ given $y$ are given by $$\begin{aligned}
\label{eq:quant2}
m_q(y) \hspace{-0.5em}&=\hspace{-0.5em}& \mathbb{E}[\tilde{y}|y] = \frac{\left(y^{1/g}+q/2\right)^{g+1}-\left(y^{1/g}-q/2\right)^{g+1} }{q(g+1)},\notag\\
\sigma_q^2(y)\hspace{-0.5em}&=\hspace{-0.5em}&\mathbb{E}[(\tilde{y}-m_q(y))^2|y]
{}{=}
\frac{\left(y^{1/g}+q/2\right)^{2g+1}-\left(y^{1/g}-q/2\right)^{2g+1} }{q(2g+1)}-m^2_q(y).~~~~\end{aligned}$$ Note that this variance is only signal-dependent when $g\neq 1$, since for $g=1$, $\sigma_q^2(y)=q^2/12$ is independent of $y$. We incorporate these to obtain the overall likelihood function as $$\begin{aligned}
\label{eq:LPGQ}
L(x_k;y,\alpha,\sigma,Q)=L_{PG}(x_k;m_q(y),\alpha,\sqrt{\sigma^2+\sigma_q^2(y)}).\end{aligned}$$
MAP-based Restoration {#sec:algo}
=====================
With a suitably defined prior model $p(x)$ for natural images, we recover the maximum a-posteriori (MAP) estimate $\hat{x}(n)$ of $x(n)$ from $y(n)$ as: $$\label{eq:map}
\hat{x} = \arg \min_x \Phi(x) + \sum_n L((x*k)(n);y(n)),$$ where $\Phi(x)=-\log p(x)$, and $L(\cdot)$ is the likelihood function defined in (with the arguments $\alpha,\sigma,Q$ omitted for brevity).
The main obstacle to computing a solution for , even in the absence of blur (i.e., $k = \delta$), is the fact that natural image priors are best defined in terms of coefficients in some transform domain. Unlike the AWGN case where noise in the coefficients of the observed image is also independent and Gaussian, estimation under the noise model in is challenging because of the complexity in characterizing the statistics of noise in the transform domain.
Variable Splitting {#sec:vsplit}
------------------
We use an optimization approach similar to the one in [@uclatr], that allows us to deal with minimizing the prior and likelihood terms in their respective *natural* domains, i.e., in terms of transform coefficients and individual pixels respectively. Specifically, we recast the unconstrained minimization problem in as an equality-constrained optimization with the addition of new variables $\tau(n)$: $$\begin{aligned}
\label{eq:map2}
\hat{x} = \arg \min_x \Phi(x) + \sum_n L(\tau(n);y(n)),{}
\mbox{~~~subject to:~} \tau(n) = (x*k)(n).\end{aligned}$$ Clearly, the problems in and are equivalent. However, this modified formulation can be solved using an efficient iterative approach that allows the noise in each pixel to be treated independently.
Note that instead of choosing a specific image prior, we assume the existence of a baseline AWGN-based image restoration algorithm that (perhaps implicitly) defines $\Phi(x)$, and also provides an estimator function $G(\cdot)$: $$\label{eq:baseline}
G(y,k,\sigma) = \arg \min_x \Phi(x) + \sum_n \frac{\left[y(n)-(x*k)(n)\right]^2}{2\sigma^2}.$$ While we treat $G(\cdot)$ as a “black box” in general, the appendix describes a “parallel” optimization algorithm for a special case when the baseline algorithm is itself based on variable-splitting.
Minimization with Augmented Lagrangian Multipliers {#sec:alm}
--------------------------------------------------
We use the augmented Lagrangian multiplier method [@uclatr] to solve this constrained optimization, and define a new augmented cost function that incorporates the equality constraint: $$\begin{aligned}
\label{eq:alm}
\hspace{-1em}C(x,\tau,\lambda)\hspace{-0.6em}&=\hspace{-0.6em}&\Phi(x) + \left[ \frac{\beta}{2}\sum_n (\tau(n)-x_k(n))^2
{}
- \sum_n \lambda(n)\left(\tau(n)-x_k(n)\right) \right] + \sum_n L(\tau(n);y(n)),\end{aligned}$$ where $x_k(n) = (x*k)(n)$, $\lambda(n)$ are the Lagrange multipliers, and $\beta$ is a fixed scalar parameter. Note that the Lagrangian terms in this expression are augmented with an additional quadratic cost on the equality constraint. This additional cost is shown to speed up convergence, and since it has a derivative equal to zero when the equality constraint is satisfied, it does not affect the Karush-Kuhn-Tucker (KKT) condition at the solution.
The solution to can be reached iteratively by making the following updates to $x(n),\tau(n)$ and $\lambda(n)$ at each iteration: $$\begin{aligned}
\label{eq:update1}
x^{t+1} &\leftarrow& \arg \min_x C(x,\tau^t,\lambda^t),\\
\label{eq:update2}
\tau^{t+1} &\leftarrow& \arg \min_\tau C(x^{t+1},\tau,\lambda^t),\\
\label{eq:update3}
\lambda^{t+1}(n) &\leftarrow& \lambda^t(n) - \gamma(n) \beta (\tau^{t+1}(n)-x_k^{t+1}(n)),\end{aligned}$$ where $x^t,\tau^t,\gamma^t$ refers to the values of these variables at iteration $t$, and the step size $\gamma(n)$ lies between $0$ and $\gamma_{\mbox{\tiny max}}=(\sqrt{5}+1)/2$.
The update to $x$ in involves minimizing the sum of the prior term $\Phi(x)$ with a uniformly-weighted quadratic cost, and this can be achieved using the baseline estimator $G(\cdot)$ as $$\label{eq:xupd}
x^{t+1} = G\left(\tau^{t}(n)-\beta^{-1} \lambda^{t}(n),k,\beta^{-1}\right).$$ The update to $\tau$ in involves a minimization that is independent of the prior term, and can be done on a per-pixel basis. For each $n$, we solve for $\partial C/\partial \tau(n) = 0$ to obtain $$\begin{aligned}
\label{eq:tausol}
&
\tau^{t+1}(n) = (b+\sqrt{b^2+4c})/2,\notag\\
&\hspace{-2.6em}
b=\left(x^{t+1}_k(n) +\beta^{-1}\lambda^t(n)\right)- \alpha\beta^{-1} - \alpha\left(\sigma^2+\sigma_q^2(y(n))\right),\notag\\
&\hspace{-1.5em}
c=\alpha\left(\sigma^2+\sigma_q^2(y(n))\right)\left(x_k^{t+1}(n)+\beta^{-1}\lambda^t(n)\right) + \alpha\beta^{-1} m_q(y(n)).\hspace{-1.5em}{}\end{aligned}$$
Algorithm Details {#sec:details}
-----------------
We begin the iterations with $\tau(n) = m_q(y(n))$ and $\lambda(n) = 0$, and stop when the relative change in $x$ falls below a threshold: $$\label{eq:thresh}
\frac{\|x^{t+1}(n)-x^{t}(n)\|^2}{\|x^{t}(n)\|^2} \leq \epsilon.$$ We find that it is optimal to vary the step-size $\gamma(n)$ at each pixel (but this is kept fixed across iterations), with a higher step-size for pixels with higher observed intensities $y(n)$ that have a higher expected noise variance. Specifically, we vary the step-size linearly with respect to $y(n)$, between $\gamma_{\mbox{\tiny max}}/2$ and $\gamma_{\mbox{\tiny max}}$.
Finally, the choice of $\beta$ involves a trade off between accuracy and speed of convergence. We find that it is best to choose a value inversely proportional to the average expected noise variance in the image, i.e. $\beta=\beta_0 / \sigma_{\mbox{avg}}^2$, where $$\label{eq:betachoice}
\sigma_{\mbox{avg}}^2= \alpha^{-1}\mbox{avg}\{m_q(y(n))\} + \sigma^2 + \mbox{avg}\{\sigma_q^2(y(n))\},$$ and $\beta_0$ depends on the choice of the baseline algorithm $G(\cdot)$.
Experimental Results {#sec:exp}
====================
We use synthetically blurred and noisy images to compare the proposed approach to traditional AWGN-based methods for deconvolution, as well as Poisson-Gaussian noise-based methods for denoising [@foi; @florian]. Table \[tab:dcnv\] shows deconvolution performance on three standard images blurred with circular pill-box kernels (that correspond to typical defocus blur) of different radii $r$, in the presence of Poisson-Gaussian noise with different values of $\alpha$ and $\sigma$, as well as with quantization errors (marked as $+Q$ in the table) corresponding to 8-bit quantization post gamma-correction ($q=1/256,g=2.2$).
[|c|c|c||c|c|c|c|c|c|c|c|c|c|c|c|c|]{} & & & & &\
&&& $r=5$&$r=7$&$r=9$&$r=11$& $r=5$&$r=7$&$r=9$&$r=11$& $r=5$&$r=7$&$r=9$&$r=11$\
& & Input& 20.55& 19.46& 18.64& 17.97& 25.02& 23.62& 22.58& 21.76& 23.12& 22.05& 21.35& 20.81\
&&AWGN [@ADM]& 23.21& 22.27& 21.50& 20.97& 28.41& 26.99& 26.23& 25.40& 25.55& 24.29& 23.64& 23.00\
&&Proposed& **24.08& **23.09& **22.29& **21.69& **28.70& **27.34& **26.51& **25.70& **26.03& **24.80& **24.12& **23.42\
& & Input& 17.23& 16.69& 16.24& 15.85& 18.54& 18.19& 17.87& 17.57& 18.27& 17.89& 17.61& 17.38\
************************
&&AWGN [@ADM]& 21.05& 20.18& 19.41& 18.73& 25.57& 24.53& 23.67& 22.95& 23.16& 22.42& 21.86& 21.41\
&&Proposed& **21.55& **20.60& **19.82& **19.11& **26.11& **24.96& **24.06& **23.22& **23.57& **22.73& **22.10& **21.57\
************************
& & Input& 20.55& 19.46& 18.64& 17.97& 25.02& 23.61& 22.58& 21.76& 23.11& 22.05& 21.34& 20.81\
&&AWGN [@ADM]& 23.21& 22.26& 21.50& 20.96& 28.41& 26.99& 26.23& 25.39& 25.55& 24.29& 23.64& 23.00\
&+Q&Proposed& **24.08& **23.08& **22.28& **21.68& **28.70& **27.34& **26.50& **25.70& **26.02& **24.79& **24.11& **23.41\
& & Input& 17.38& 16.81& 16.35& 15.94& 18.58& 18.21& 17.89& 17.58& 18.33& 17.93& 17.64& 17.41\
************************
&&AWGN [@ADM]& 20.97& 20.13& 19.38& 18.70& 25.56& 24.53& 23.66& 22.94& 23.14& 22.40& 21.85& 21.40\
&+Q&Proposed& **21.50& **20.56& **19.80& **19.09& **26.10& **24.96& **24.06& **23.22& **23.56& **22.72& **22.09& **21.56\
************************
& & Input& 19.93& 18.97& 18.23& 17.62& 23.29& 22.30& 21.51& 20.86& 21.96& 21.13& 20.54& 20.10\
&&AWGN [@ADM]& 22.20& 21.26& 20.50& 19.87& 27.26& 25.90& 25.05& 24.18& 24.48& 23.40& 22.73& 22.18\
&&Proposed& **22.88& **21.93& **21.12& **20.51& **27.64& **26.28& **25.50& **24.60& **24.95& **23.76& **23.05& **22.43\
& & Input& 16.93& 16.42& 15.99& 15.62& 18.09& 17.76& 17.47& 17.20& 17.86& 17.52& 17.26& 17.04\
************************
&&AWGN [@ADM]& 20.96& 20.09& 19.32& 18.65& 25.42& 24.39& 23.55& 22.84& 23.03& 22.33& 21.79& 21.34\
&&Proposed& **21.47& **20.52& **19.75& **19.04& **25.99& **24.85& **23.96& **23.12& **23.46& **22.64& **22.01& **21.51\
************************
& & Input& 19.93& 18.97& 18.23& 17.62& 23.29& 22.29& 21.51& 20.85& 21.96& 21.12& 20.54& 20.09\
&&AWGN [@ADM]& 22.19& 21.26& 20.50& 19.86& 27.26& 25.89& 25.05& 24.18& 24.47& 23.40& 22.73& 22.18\
&+Q&Proposed& **22.88& **21.93& **21.12& **20.51& **27.64& **26.28& **25.50& **24.60& **24.95& **23.76& **23.05& **22.43\
& & Input& 17.08& 16.54& 16.10& 15.71& 18.14& 17.80& 17.50& 17.23& 17.92& 17.56& 17.30& 17.07\
************************
&&AWGN [@ADM]& 20.88& 20.04& 19.29& 18.62& 25.40& 24.38& 23.55& 22.84& 23.01& 22.31& 21.77& 21.33\
&+Q&Proposed& **21.41& **20.47& **19.72& **19.01& **25.98& **24.84& **23.95& **23.12& **23.45& **22.63& **22.00& **21.50\
************************
& & Input& 18.06& 17.43& 16.90& 16.45& 19.65& 19.19& 18.79& 18.42& 19.09& 18.64& 18.31& 18.03\
&&AWGN [@ADM]& 21.35& 20.45& 19.68& 18.99& 25.95& 24.83& 23.96& 23.18& 23.41& 22.61& 22.02& 21.56\
&&Proposed& **21.98& **20.99& **20.20& **19.58& **26.51& **25.31& **24.44& **23.56& **23.87& **22.95& **22.31& **21.77\
& & Input& 15.88& 15.49& 15.15& 14.84& 16.64& 16.40& 16.19& 15.98& 16.51& 16.26& 16.06& 15.90\
************************
&&AWGN [@ADM]& 20.63& 19.81& 19.04& 18.43& 24.88& 23.95& 23.16& 22.54& 22.64& 22.01& 21.53& 21.11\
&&Proposed& **21.19& **20.29& **19.50& **18.79& **25.55& **24.48& **23.58& **22.86& **23.11& **22.36& **21.79& **21.33\
************************
& & Input& 14.26& 13.99& 13.77& 13.55& 14.50& 14.35& 14.21& 14.08& 14.42& 14.25& 14.13& 14.02\
&&AWGN [@ADM]& 20.10& 19.31& 18.62& 18.15& 24.02& 23.20& 22.55& 21.99& 21.99& 21.50& 21.08& 20.67\
&&Proposed& **20.87& **19.97& **19.22& **18.54& **24.90& **23.90& **23.05& **22.40& **22.58& **21.93& **21.45& **21.01\
& & Input& 13.22& 13.01& 12.83& 12.66& 13.33& 13.23& 13.12& 13.02& 13.36& 13.23& 13.13& 13.04\
************************
&&AWGN [@ADM]& 19.73& 18.97& 18.41& 18.03& 23.48& 22.76& 22.18& 21.65& 21.64& 21.20& 20.81& 20.40\
&&Proposed& **20.45& **19.60& **18.84& **18.26& **24.35& **23.42& **22.67& **22.06& **22.20& **21.63& **21.20& **20.76\
************************
& & Input& 9.02& 8.97& 8.92& 8.87& 8.72& 8.69& 8.65& 8.60& 8.81& 8.77& 8.74& 8.70\
&&AWGN [@ADM]& 18.31& 17.99& 17.71& 17.41& 20.86& 20.45& 20.07& 19.67& 19.85& 19.57& 19.26& 18.99\
&&Proposed& **18.86& **18.38& **18.08& **17.74& **21.70& **21.15& **20.72& **20.25& **20.53& **20.18& **19.79& **19.43\
& & Input& 8.69& 8.64& 8.60& 8.55& 8.39& 8.35& 8.32& 8.28& 8.49& 8.45& 8.42& 8.39\
************************
&&AWGN [@ADM]& 18.23& 17.93& 17.65& 17.36& 20.64& 20.24& 19.87& 19.49& 19.70& 19.43& 19.14& 18.89\
&&Proposed& **18.62& **18.22& **17.92& **17.58& **21.34& **20.83& **20.41& **19.96& **20.26& **19.94& **19.56& **19.24\
************************
We report performance in terms of PSNR (averaged over ten instantiations of the noise) for the AWGN-based ADM-TV method [@ADM], and for the proposed method using the ADM-TV method as the baseline. We use the parallel optimization approach as described in the appendix, and use the same baseline parameters ($\beta_\nabla=200,\kappa=8,\epsilon=10^{-5}$) in both cases. We set $\beta_0=16$ for the proposed method, and use the mean noise-variance $\sigma_{\mbox{avg}}^2$ for the AWGN results. We note that the proposed approach offers a distinct advantage over the baseline method with better estimates in all cases. Figure \[fig:dcnvex\] shows an example of the observed and restored images, and we see that the proposed method is able to account for the noise variance at darker pixels being lower, yielding sharper estimates in those regions. Our approach typically requires only two to three times as many iterations as the baseline method to converge.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Deconvolution results on the Camerman image, blurred with a circular kernel of radius $9$ pixels with noise parameters $\alpha=1024, \sigma=10^{-4}, q = 1/256,$ and $g=2.2$. Note that the proposed method correctly accounts for the noise variance being lower at darker pixels, and recovers sharper estimates in those regions in comparison to the baseline AWGN method of [@ADM].[]{data-label="fig:dcnvex"}](figs/cman_inp.png "fig:"){width="18.50000%"} ![Deconvolution results on the Camerman image, blurred with a circular kernel of radius $9$ pixels with noise parameters $\alpha=1024, \sigma=10^{-4}, q = 1/256,$ and $g=2.2$. Note that the proposed method correctly accounts for the noise variance being lower at darker pixels, and recovers sharper estimates in those regions in comparison to the baseline AWGN method of [@ADM].[]{data-label="fig:dcnvex"}](figs/cman_awgn.png "fig:"){width="18.50000%"} ![Deconvolution results on the Camerman image, blurred with a circular kernel of radius $9$ pixels with noise parameters $\alpha=1024, \sigma=10^{-4}, q = 1/256,$ and $g=2.2$. Note that the proposed method correctly accounts for the noise variance being lower at darker pixels, and recovers sharper estimates in those regions in comparison to the baseline AWGN method of [@ADM].[]{data-label="fig:dcnvex"}](figs/cman_prop.png "fig:"){width="18.50000%"}
Input (PSNR=18.64dB) AWGN Method [@ADM] (21.48 dB) Prop. Method (22.29 dB)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Next, we show results in Fig. \[fig:spiralcmp\] for the two demonstration examples provided by authors of [@spiraltv], each blurred with a $5\times 5$ kernel and corrupted only by Poisson noise. We show the running times and PSNR values of the restored images, from both the proposed method (using the same parameters as above), and the SPIRAL technique [@spiraltv] with a TV prior (which yields the highest PSNR values amongst the different priors). In both cases, our approach yields estimates with higher accuracy, and offers a significant advantage in computational cost— for the *cameraman* example, our method converges in just 25 iterations, while the SPIRAL technique requires a 100 iterations, with each iteration in-turn calling an iterative baseline TV-solver.
------------------------------------------ --------------------------------------------- ------------------------------------------- ------------------------------------------ --------------------------------------------- -------------------------------------------
{width="\wdt"} {width="\wdt"} {width="\wdt"} {width="\wdt"} {width="\wdt"} {width="\wdt"}
Input SPIRAL-TV Prop.+TV Input SPIRAL-TV Prop.+TV
PSNR=20.29 dB 23.76 dB (4.2s) 24.70 dB (0.7s) 21.33 dB 25.52 dB (94.0s) 25.60 dB (1.7s)
------------------------------------------ --------------------------------------------- ------------------------------------------- ------------------------------------------ --------------------------------------------- -------------------------------------------
Finally, we report denoising performance (i.e., $k=\delta$) in Table. \[tab:dnz\], for the standard test cases used in [@foi] with Poisson-Gaussian noise. The state-of-the-art AWGN techniques for denoising tend to be more sophisticated than those for deconvolution. We use the BM3D [@bm3d] algorithm, which uses a complex adaptive image prior, as the baseline estimator for our approach in this case (with $\beta_0=2,\epsilon=10^{-3}$). In addition to PSNR values for the proposed method, we show results for the baseline AWGN method (with $\sigma_{\mbox{avg}}^2$ as the noise variance), and for the Poisson-Gaussian denoising algorithms described in [@foi] and [@florian]. Note that the method in [@foi] also uses BM3D as a baseline. We use the same notation as in [@foi] to describe the noise parameters in Table \[tab:dnz\], where noise is synthetically added by first scaling the input image to a certain peak value, instantiating Poisson random variables with these scaled intensities, and then adding Gaussian noise with variance $\sigma^2$. The reported PSNR values are again averaged over ten instantiations of the noise. Figure \[fig:dnzex\] shows an example of input and restored images for this case.
[|l||c|c|c||c||c|c||c|c||c|]{} Image & Peak & $\sigma$ & Noisy & BM3D [@bm3d] & GAT+BM3D [@foi] & PURE-LET [@florian] & Prop.+BM3D & \# Iterations & Prop.+TV\
& 1 & 0.1 & 3.20 & 18.50 & 20.23 & 20.35 &**20.71& 7.2&17.42\
& 2 & 0.2 & 6.12 & 20.95 & 21.93 & 21.60 &**22.12& 5.1&18.53\
& 5 & 0.5 & 9.83 & 23.55 & 24.09 & 23.33 &**24.10& 4.0&21.08\
& 10 & 1 & 12.45 & 25.10 & 25.52 & 24.68 &**25.57& 4.0&22.88\
& 20 & 2 & 14.76 & 26.50 & 26.77 & 25.92 &**26.81& 4.0&24.55\
& 30 & 3 & 15.91 & 27.10 &**27.30 & 26.51 &**27.30& 4.0& 25.26\
& 60 & 6 & 17.49 & 27.97 & 28.07 & 27.35 &**28.10& 3.0&26.12\
& 120 & 12 & 18.57 & 28.52 & 28.57 & 27.89 &**28.61& 3.0&26.56\
******************
& 1 & 0.1 & 7.22 & 19.64 & 24.54 & **25.13 & 24.91& 8.8&20.79\
& 2 & 0.2 & 9.99 & 22.24 & 25.87 & **26.25 & 26.16& 7.3&22.68\
& 5 & 0.5 &13.37 & 25.43 & 27.45 & 27.60 &**27.72& 5.5&25.21\
& 10 & 1 & 15.53 & 27.53 & 28.63 & 28.59 &**28.77& 5.0&27.11\
& 20 & 2 & 17.21 & 29.18 & 29.65 & 29.47 &**29.69& 5.0&28.37\
& 30 & 3 & 17.97 & 29.91 & 30.16 & 29.84 &**30.18& 4.0&28.87\
& 60 & 6 & 18.86 & 30.72 & **30.77 & 30.42 & 30.75& 4.0&29.38\
& 120 & 12 & 19.39 & **31.15 & 31.14 & 30.70 & 31.09& 4.0&29.60\
****************
& 1 & 0.1 & 2.87 & 19.87 & 22.59 & **22.83 & 22.58& 7.0&17.01\
& 2 & 0.2 & 5.82 & 22.58 &**24.34 & 24.16 & 24.14& 5.1&19.68\
& 5 & 0.5 & 9.54 & 25.42 &26.17 & 25.74 &**26.19& 4.0&23.18\
& 10 & 1 & 12.19 & 27.20 &27.72 & 27.27 &**27.80& 4.0&25.41\
& 20 & 2 & 14.53 & 28.66 &29.01 & 28.46 &**29.12& 4.0&27.05\
& 30 & 3 & 15.72 & 29.42 &29.69 & 29.12 &**29.75& 3.5&27.69\
& 60 & 6 & 17.35 & 30.37 & 30.51 & 29.91 &**30.57& 3.0&28.12\
& 120 & 12 & 18.48 & 30.98 & 31.05 & 30.51 &**31.11& 3.0&28.08\
****************
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Denoising results (with inset close-ups) on the Fluorescent cells image, with Poisson-Gaussian noise corresponding to a peak input intensity of $5$ and $\sigma=0.5$.[]{data-label="fig:dnzex"}](figs/fc_nz_inp_cu.png "fig:"){width="19.00000%"} ![Denoising results (with inset close-ups) on the Fluorescent cells image, with Poisson-Gaussian noise corresponding to a peak input intensity of $5$ and $\sigma=0.5$.[]{data-label="fig:dnzex"}](figs/fc_nz_awgn_cu.png "fig:"){width="19.00000%"} ![Denoising results (with inset close-ups) on the Fluorescent cells image, with Poisson-Gaussian noise corresponding to a peak input intensity of $5$ and $\sigma=0.5$.[]{data-label="fig:dnzex"}](figs/fc_nz_foi_cu.png "fig:"){width="19.00000%"} ![Denoising results (with inset close-ups) on the Fluorescent cells image, with Poisson-Gaussian noise corresponding to a peak input intensity of $5$ and $\sigma=0.5$.[]{data-label="fig:dnzex"}](figs/fc_nz_prop_cu.png "fig:"){width="19.00000%"}
Input (PSNR=13.40dB) BM3D [@bm3d] (25.54 dB) GAT+BM3D [@foi] (27.57 dB) Prop.+BM3D (27.79 dB)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
We find that the proposed approach is competitive with these existing methods, with the highest PSNR in a majority of the test cases. However, it is important to remember that our approach is iterative, while the algorithms in [@foi; @florian] are single-shot. Table. \[tab:dnz\] reports the mean number of iterations required in each case, and we see that convergence is usually quick, requiring roughly three to seven calls to the baseline method. We also show results for denoising using the simple TV prior (which was used for the deconvolution results in Table. \[tab:dcnv\]), and note that using the BM3D method as the baseline instead leads to a significant improvement. This highlights the importance of the flexibility that our approach offers, in allowing the use of any baseline AWGN restoration method rather than being tied to a fixed prior.
Conclusion {#sec:conc}
==========
In this paper, we introduce a framework for image restoration in the presence of signal-dependent noise. We describe an observation model that accounts for camera sensor measurements being corrupted by both Gaussian and signal-dependent Poisson noise sources, and for errors from the subsequent digitization of these measurements. Then, we use *variable-splitting* to derive a fast iterative scheme that is able to adapt existing AWGN-based restoration methods for inference with this noise model. A MATLAB implementation of the algorithm, along with scripts to generate the results presented in this paper, is available for download at <http://vision.seas.harvard.edu/PGQrestore/>.
The flexibility in being able to incorporate any AWGN-based method as a baseline means that we can draw on a considerable amount of existing research on image statistics. Our approach can therefore be easily used for restoration of any class of images (such as medical images, or astronomical data) which is corrupted by noise with similar characteristics, by using an appropriate class-specific AWGN-restoration technique as the baseline. Moreover, the optimization scheme described here can be adapted to other noise models, as long as the likelihood functions for those models are convex, or can be so approximated.
The ADM-TV method [@ADM] is a popular choice for deconvolution under AWGN, and is itself based on variable-splitting. When using this method as the baseline algorithm, it is possible to adopt an approach where its internal optimization is effectively done in parallel to that for the noise model in our framework. We describe this approach in detail in this appendix. The ADM-TV method uses an image prior that penalizes the TV-norm of image gradients: $$\label{eq:TVL1}
\Phi(x) = \kappa \sum_n \sqrt{\sum_i |(x*\nabla_i)(n)|^2},$$ where $\nabla_i$ are gradient filters, and $\kappa$ is a model parameter. The MAP estimate in this case is computed by introducing additional auxiliary variables $d_i(n)$ corresponding to the image gradients, and the estimation problem is cast as: $$\begin{aligned}
\label{eq:adm}
x = \arg \min_x \kappa \sum_{n} \sqrt{\sum_i|d_i(n)|^2} + L(\tau(n);y(n)),\notag\\
\mbox{subject to:~} d_i(n) = (x*\nabla_i)(n);~~\tau(n) = (x*k)(n).\end{aligned}$$ We define a *joint* augmented Lagrangian-based cost for this case as in [@uclatr]: $$\begin{aligned}
\label{eq:alm2}
C(x,d_i,\tau,\lambda) {&}={&}\kappa\sum_{n} \sqrt{\sum_i |d_i(n)|^2} + \left[ \frac{\beta_\nabla}{2} \sum_{n,i} (d_i(n)-x_i(n))^2
{}
- \sum_{n,i} \lambda_i(n) (d_i(n) - x_i(n)) \right]
{\notag\\&&}
+ \left[ \frac{\beta}{2} \sum_n (\tau(n)-x_k(n))^2
{}
-\sum_n \lambda(n) (\tau(n) - x_k(n)) \right] + L(\tau(n);y(n)),~\end{aligned}$$ where $x_i(n) = (x*\nabla_i)(n)$, and the second and third term above encode the gradient equality constraint in .
The iterative optimization for this case proceeds as follows: $$\begin{aligned}
\label{eq:cupdate0}
d_i^{t+1} &\leftarrow& \arg \min_{d_i} C(x^t,d_i,\tau^t,\lambda^t),\\
\label{eq:cupdate1}
x^{t+1} &\leftarrow& \arg \min_x C(x,d_i^{t+1},\tau^t,\lambda^t),\\
\label{eq:cupdate2}
\tau^{t+1} &\leftarrow& \arg \min_\tau C(x^{t+1},d_i^{t+1},\tau,\lambda^t),\\
\label{eq:cupdate3}
\lambda^{t+1}(n) &\leftarrow& \lambda^t(n) - \gamma(n) \beta (\tau^{t+1}(n)-x_k^{t+1}(n)),\\
\label{eq:cupdate4}
\lambda_i^{t+1}(n) &\leftarrow& \lambda^t_i(n) - \gamma_{\mbox{\tiny max}} \beta_\nabla (d_i^{t+1}(n)-x_i^{t+1}(n)).\end{aligned}$$ Note that this is essentially equivalent to the optimization framework in -, with , corresponding to the $x$ update step in , and being an extra update step for the additional Lagrange multipliers (using a fixed step size equal to $\gamma_{\mbox{\tiny max}}$, as in [@ADM]). The solution to the $\tau$ update step is identical to the one described in in Sec. \[sec:algo\], since the minimization in for this update depends only on the last three terms in the cost in .
The update steps to $d_i$ and $x$ in , are largely independent of the noise model, and are similar to those described in [@ADM]. Specifically, each $d_i^{t+1}(n)$ can be computed independently on a per-pixel basis: $$\begin{aligned}
\label{eq:eqdiupd}
d_i^{t+1}(n)\hspace{-0.5em}&=\hspace{-0.5em}& \max\left(0,\sqrt{\tilde{d}(n)} -\beta_\nabla^{-1}\kappa\right) \frac{\left(x_i^t(n)+\beta_\nabla^{-1}\lambda_i^t(n) \right)}{{\sqrt{\tilde{d}(n)}}},{}
~~\tilde{d}(n){}={} \sum_i |x_i(n)+\beta_\nabla^{-1}\lambda_i^t(n)|^2.\end{aligned}$$ The updated value of $x$ can then be computed efficiently in the Fourier domain as: $$\label{eq:xtvupd}
x^{t+1} = \mathcal{F}^{-1}\left[ \frac{\beta_\nabla\sum_i \mathcal{F}[\nabla_i]^{^*}\mathcal{F}[\tilde{d}_i(n)] + \beta \mathcal{F}[k]^{^*}\mathcal{F}[\tilde{\tau}(n)] }{\beta_\nabla \sum_i |\mathcal{F}[\nabla_i]|^2 + \beta |\mathcal{F}[k] |^2}\right],~~~$$ where, $$\label{eq:xxd}
\tilde{d}_i(n) = d_i^{t+1}(n) - \beta_\nabla^{-1}\lambda_i^t(n),~~
\tilde{\tau}(n) = \tau^{t}(n) - \beta^{-1}\lambda^t(n),$$ and $\mathcal{F}$ and $\mathcal{F}^{-1}$ refer to the forward and inverse discrete Fourier transforms. To account for the periodicity assumption with the Fourier transform in , we extend $\tilde{d}_i$ and $\tilde{\tau}$ in each direction by six times the size of the kernel $k$ before computing the Fourier transform, with zeros for $\tilde{d}_i$, and values that linearly blend the intensities at opposite boundaries for $\tilde{\tau}$. After computing $x^{t+1},x_k^{t+1}$ and $x_i^{t+1}$, we crop them back to their original sizes.
We begin the iterations with $\tau(n)=m_q(y(n)),~\lambda(n)=0,~x(n)=0$ and $\lambda_i(n) = 0$, and choose the gradient filters $\nabla_i$ to correspond to horizontal and vertical finite-differences (i.e., $[-1,1]$). We vary the step sizes $\gamma(n)$ as described in Sec. \[sec:algo\]. Moreover, we do not apply the updates in and to $\tau(n)$ and $\lambda(n)$ for the first few iterations (six in our implementation), during which time the algorithm proceeds identically to the AWGN ADM-TV [@ADM] method with input $m_q(y(n))$ and noise variance $\beta^{-1}$.
[^1]: The authors are with the School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138 USA (e-mail: {ayanc,zickler}@eecs.harvard.edu).
|
{
"pile_set_name": "ArXiv"
}
|
---
bibliography:
- 'paper.bib'
---
=1
=0mu plus 1mu
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the modal properties of the $r$-modes of rotating neutron stars with the core filled with neutron and proton superfluids, taking account of entrainment effects between the superfluids. The stability of the $r$-modes against gravitational radiation reaction is also examined considering viscous dissipation due to shear and a damping mechanism called mutual friction between the superfluids in the core. We find the $r$-modes in the superfluid core are split into ordinary $r$-modes and superfluid $r$-modes, which we call, respectively, $r^o$- and $r^s$-modes. The two superfluids in the core flow together for the $r^o$-modes, while they counter-move for the $r^s$-modes. For the $r^o$-modes, the coefficient $\kappa_0\equiv\lim_{\Omega\rightarrow 0}\omega/\Omega$ is equal to $2m/[l^\prime(l^\prime+1)]$, almost independent of the parameter $\eta$ that parameterizes the entrainment effects between the superfluids, where $\Omega$ is the angular frequency of rotation, $\omega$ the oscillation frequency observed in the corotating frame of the star, and $l^\prime$ and $m$ are the indices of the spherical harmonic function representing the angular dependence of the $r$-modes. For the $r^s$-modes, on the other hand, $\kappa_0$ is equal to $2m/[l^\prime(l^\prime+1)]$ at $\eta=0$ (no entrainment), and it almost linearly increases as $\eta$ is increased from $\eta=0$. The $r^o$-modes, for which $\pmb{w}^\prime\equiv\pmb{v}^\prime_p-\pmb{v}^\prime_n\propto\Omega^3$, correspond to the $r$-modes discussed by Lindblom & Mendell (2000), where $\pmb{v}^\prime_n$ and $\pmb{v}^\prime_p$ are the Eulerian velocity perturbations of the neutron and proton superfluids, respectively. The mutual friction in the superfluid core is found ineffective to stabilize the $r$-mode instability caused by the $r^o$-mode except in a few narrow regions of $\eta$. The $r$-mode instability caused by the $r^s$-modes, on the other hand, is extremely weak and easily damped by dissipative processes in the star.'
author:
- 'Umin Lee$^1$ & Shijun Yoshida$^{1,2}$'
title: '**R-modes of neutron stars with the superfluid core**'
---
\[ =1 \] \#1
Introduction
============
One of the roles expected for the $r$-mode instability to play (Andersson 1998, Friedman & Morsink 1998) is deceleration of the spin of newly born hot neutron stars by emitting gravitational waves that carry away the angular momentum of the star (e.g., Lindblom et al 1998). We know, however, that among older and colder neutron stars as found in LMXB systems there are many rapidly rotating neutron stars like a millisecond pulsar (see, e.g, Phinney & Kulkarni 1994). This fact suggests the possibility that the $r$-mode instability does not always work well to spin down the rapid rotation of the stars. For the $r$-modes in cold neutron stars with a solid crust, for example, Bildsten & Ushomirsky (2000) suggested a damping mechanism operating in the viscous boundary layer at the interface between the solid crust and the fluid core, to explain the clustering of spin frequencies around the value of 300Hz for accreting neutron stars in LMXB systems (van der Klis 2000; see also Andersson, Kokkotas, & Stergioulas 1999). For the modal properties of the $r$-modes in neutron stars with a solid crust, see Yoshida & Lee (2001), who showed that the $r$-modes in the core are largely affected by resonance with the toroidal sound modes propagating in the solid crust.
As neutron stars cool down below $T\sim 10^9{\rm K}$, neutrons and protons in the core are believed to be in superfluid states (e.g., Shapiro & Tuekolwsky 1983). In a rotating system of superfluids, it is well known that scattering between vortices in the superfluids and normal fluid particles produces dissipation called mutual friction (e.g., Khalatnikov 1965, Tilley & Tilley 1990). Therefore, for people who are interested in the $r$-modes instability, it was a serious concern whether the $r$-mode instability could survive the dissipation due to mutual friction in the superfluid core of cold neutron stars, e.g., in LMXBs. It was Lindblom & Mendell (2000) who first examined the damping effects of mutual friction in the core on the $r$-mode instability, and concluded that the mutual friction could not be strong enough to damp out the instability in most of the parameter domains which we are interested in. In their analysis of the $r$-modes in neutron stars, Lindblom & Mendell (2000) employed a perturbative method in which the spin angular frequency $\Omega$ is regarded as the infinitesimal parameter to expand the eigenfrequencies and eigenfunctions of the modes, and they looked for the $r$-modes with the scalings given by $\beta^\prime\equiv\mu_p^\prime-\mu_n^\prime+m_e\mu_e^\prime/m_p\propto \Omega^4$ and $\pmb{w}^\prime\equiv\pmb{v}_p^\prime-\pmb{v}_n^\prime\propto\Omega^3$, where $\mu_p$, $\mu_n$, and $\mu_e$ are the chemical potentials of the proton, neutron, and electron in the core, and $\pmb{v}_p$ and $\pmb{v}_n$ are the velocities of the proton and neutron superfluids, and the prime $(^\prime)$ indicates the Euler perturbation of the quantity.
Recently, Andersson & Comer (2001) discussed the dynamics of superfluid neutron star cores, and confirmed an earlier result by Lee (1995) that there are no $g$-modes propagating in the superfluid core. They also applied their argument to the $r$-modes in the core filled with neutron and proton superfluids, and suggested the existence of two distinct families of the $r$-modes in the core, i.e., $r$-modes for which the neutrons and the protons flow together, and those for which the neutrons and the protons are counter moving. Lindblom & Mendell (2000) considered the former family of the $r$-modes, which are less strongly affected by the mutual friction than the latter.
In this paper, we employ a different method of calculation to investigate the $r$-modes in rotating neutron stars, although the basic equations describing the dynamics of superfluids in the core are essentially the same as those given in Lindblom & Mendell (1994). Our method of solution is a variant of that used in Lee & Saio (1986). Because the separation of variables is not possible for perturbations in rotating stars, we expand the perturbations in terms of spherical harmonic functions $Y_l^m(\theta,\phi)$ with different $l$’s for a given $m$. We substitute the expansions into linearized basic equations to obtain a set of simultaneous linear ordinary differential equations of the expansion coefficients, which is to be solved as an eigenvalue problem of the oscillation frequency. In this method, we do not have to assume apriori a form of solutions for the $r$-modes in the lowest order of $\Omega$. In §2 we present the basic equations employed in this paper for the dynamics of superfluids in the core, and in §3 dissipation processes considered in this paper are described. §4 gives numerical results, and §5 and §6 are for discussions and conclusions.
Oscillation Equations in the Superfluid Core of Rotating Neutron Stars
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Microscopically, the superfluids in a rotating system move irrotationally everywhere except within the core of vortex lines. Averaging over many vortices in the fluids, we may define the average superfluid velocities $<\pmb{v}>$, which can satisfy the usual relation for uniform rotation $\nabla\times <\pmb{v}>=2\pmb{\Omega}$ in the equilibrium state (see, e.g., Feynman 1972). In the following we simply use $\pmb{v}$, instead of $<\pmb{v}>$, to signify the superfluid velocities. Hydrodynamic equations for a rotating superfluid based on the two-fluid model with the normal fluid and superfluid components are derived, for example, in Khalatnikov (1965).
We derive basic equations governing superfluid motions in the neutron star core in the Newtonian dynamics, assuming uniform rotation of the star. The core is assumed to be filled with neutron and proton superfluids and a normal fluid of electron. We also assume perfect charge neutrality between the protons and electrons because the plasma frequency is much higher than the oscillation frequencies considered in this paper (see, e.g., Mendell 1991a). Since the transition temperatures $T_c\sim 10^9$K to neutron and proton superfluids are much higher than the interior temperatures of old neutron stars (see, e.g., Epstein 1988), we assume that all the neutrons and protons in the core are in superfluid states and the normal fluid components of the fluids can be ignored.
The basic hydrodynamic equations employed in this paper for the neutron and proton superfluids in rotating neutron stars are essentially the same as those given in Mendell (1991a) and Lindblom & Mendell (1994). In a fluid system in which two superfluids coexist, the entrainment between the two superfluid motions occurs because a Cooper pair of one fluid particles is affected by the force field produced by the other fluid particles (Andreev & Bashkin 1975 for a system of $^3$He and $^4$He superfluids). In the system of the neutron and proton superfluids in the core the entrainment effects between them are mediated by quantum mechanical nuclear force between the neutrons and the protons (e.g., Alpar et al 1984). Here, we introduce the entrainment effects in mass conservation equations, which are given by $${\partial\rho_n\over\partial t}+\nabla\cdot\pmb{j}_n=0,$$ and $${\partial\rho_p\over\partial t}+\nabla\cdot\pmb{j}_p=0,$$ where $\rho_n$ and $\rho_p$ are the mass densities of the neutron and proton superfluids, and the mass current vectors $\pmb{j}_n$ and $\pmb{j}_p$ are defined as $$\pmb{j}_n=\rho_{nn}\pmb{v}_n+\rho_{np}\pmb{v}_p,$$ and $$\pmb{j}_p=\rho_{pp}\pmb{v}_p+\rho_{pn}\pmb{v}_n,$$ where $\pmb{v}_n$ and $\pmb{v}_p$ denote the velocities of the neutron and the proton superfluids, respectively, and the coefficients $\rho_{nn}$, $\rho_{np}$, $\rho_{pp}$, and $\rho_{pn}$ are defined to satisfy $\rho_n=\rho_{nn}+\rho_{np}$ and $\rho_p=\rho_{pp}+\rho_{pn}$ and $\rho_{np}=\rho_{pn}$ under Galilean transformations. The mass conservation equation for the electron fluid is given by $${\partial\rho_e\over\partial t}+\nabla\cdot\left(\rho_e\pmb{v}_e\right)=0,$$ where $\rho_e$ and $\pmb{v}_e$ denote, respectively, the mass density and the velocity of the electron fluid. The velocity equation of the neutron superfluid is in an inertial frame given by $${\partial \pmb{v}_n\over\partial t}+\pmb{v}_n\cdot\nabla\pmb{v}_n=-\nabla(\mu_n+\Psi)
+{\rho_{np}\over\rho_n}\left(\pmb{v}_p-\pmb{v}_n\right)\times\left(\nabla\times\pmb{v}_n\right),$$ where $\mu_n$ is the chemical potential of neutron per unit mass, and $\Psi$ is the gravitational potential. The term proportional to $\rho_{np}$ on the right hand side of equation (6) represents a drag force between neutrons and protons. If we assume perfect charge neutrality of the proton and electron plasma, we may have $$\pmb{j}_p/\rho_p=\pmb{v}_e, \quad {\rm and} \quad \rho_p/m_p=\rho_e/m_e,$$ where $m_p$ and $m_e$ are the proton and the electron masses. The velocity equation for the proton-electron fluid is then given by $${\partial\over\partial t}\left(\pmb{v}_p+{m_e\over m_p}\pmb{v}_e\right)
+\pmb{v}_p\cdot\nabla\pmb{v}_p
+{m_e\over m_p}{\pmb{v}_e}\cdot\nabla{\pmb{v}_e}
=-\nabla\left(\mu_p+{m_e\over m_p}\mu_e+\zeta\Psi\right)
-{\rho_{np}\over\rho_p}\left(\pmb{v}_p-\pmb{v}_n\right)\times\left(\nabla\times\pmb{v}_p\right),$$ where $\zeta=1+m_e/m_p$, and $\mu_p$ and $\mu_e$ are the chemical potentials per unit mass for the proton and electron, respectively. Note that we have neglected the entropy carried by the normal fluid of electron for simplicity. The Poisson equation is given by $$\nabla^2\Psi=4\pi G\rho,$$ where $\rho=\rho_n+\rho_p+\rho_e$ and $G$ is the gravitational constant.
To linearise the hydrodynamic equations for the superfluids in rotating neutron stars, we assume that the neutron and proton superfluids and the electron normal fluid in an equilibrium state are in the same rotational motion with the angular velocity $\Omega$ around the axis of rotation, which is along the z-axis. In a perturbed state, however, the neutron and the proton superfluids move differently from each other in the core, obeying their own governing equations. The mass current vectors are linearized to be $$\pmb{j}_n^\prime=\rho_n^\prime\pmb{v}_0+\tilde{\pmb{j}}_n^\prime,
\quad {\rm and} \quad
\pmb{j}_p^\prime=\rho_p^\prime\pmb{v}_0+\tilde{\pmb{j}}_p^\prime,$$ where the prime $(^\prime)$ indicates the Eulerian perturbation of the quantity, and $\pmb{v}_{n0}=\pmb{v}_{p0}=\pmb{v}_0=r\sin\theta\Omega\pmb{e}_\phi$ is the fluid velocity in the equilibrium state, and $$\tilde{\pmb{j}}_n^\prime=\rho_{nn}\pmb{v}_n^\prime+\rho_{np}\pmb{v}_p^\prime,
\quad {\rm and} \quad
\tilde{\pmb{j}}_p^\prime=\rho_{pn}\pmb{v}_n^\prime+\rho_{pp}\pmb{v}_p^\prime$$ are the perturbed mass current vectors in a corotating frame. The perturbed superfluid velocities are then given in terms of $\tilde{\pmb{j}}_n^\prime$ and $\tilde{\pmb{j}}_p^\prime$ as $$\pmb{v}_n^\prime={\rho_{11}\over\rho_n^2}\tilde{\pmb{j}}_n^\prime
+{\rho_{12}\over\rho_n\rho_p}\tilde{\pmb{j}}_p^\prime,
\quad {\rm and} \quad
\pmb{v}_p^\prime={\rho_{21}\over\rho_n\rho_p}\tilde{\pmb{j}}_n^\prime
+{\rho_{22}\over\rho_p^2}\tilde{\pmb{j}}_p^\prime,$$ where $$\rho_{11}={\rho_{pp}\rho_n^2\over\tilde{\rho}^2}, \quad
\rho_{22}={\rho_{nn}\rho_p^2\over\tilde\rho^2}, \quad
\rho_{12}=\rho_{21}=-{\rho_{np}\rho_n\rho_p\over\tilde\rho^2},$$ and $$\tilde\rho^2=\rho_{nn}\rho_{pp}-\rho_{np}\rho_{pn}.$$ Note that $\rho_{11}+\rho_{12}=\rho_n$ and $\rho_{22}+\rho_{21}=\rho_p$.
If the equilibrium structure is axisymmetric about the rotation axis, the time dependence and $\phi-$dependence of the perturbations can be given by $\exp(i\sigma t+i m\phi)$ with $\sigma$ being the oscillation frequency observed in an inertial frame, and $m$ is an integer representing the azimuthal wavenumber. Introducing the vectors $\pmb{\xi}^n$, $\pmb{\xi}^p$, and $\pmb{\xi}^e$ defined as $$\pmb{\xi}^n\equiv{\tilde{\pmb{j}}_n^\prime\over i\omega\rho_n},
\quad
\pmb{\xi}^p\equiv{\tilde{\pmb{j}}_p^\prime\over i\omega\rho_p},
\quad {\rm and} \quad
\pmb{\xi}^e\equiv {\pmb{v}_e^\prime\over i\omega},$$ where $\omega\equiv\sigma+m\Omega$ is the oscillation frequency observed in a corotating frame of the star, the mass conservation equations are linearised to be $$\rho_n^\prime+\nabla\cdot\left(\rho_n\pmb{\xi}^n\right)=0,$$ and $$\rho_p^\prime+\nabla\cdot\left(\rho_p\pmb{\xi}^p\right)=0.$$ Note that the mass conservation equation for the electron fluid becomes the same as that for the proton fluid because of the assumption of perfect charge neutrality, that is, $\pmb{\xi}^p=\pmb{\xi}^e$, and $\rho_p^\prime/m_p=\rho_e^\prime/m_e$. The velocity equations (6) and (8) are linearized as $${\rho_{11}\over\rho_n}\pmb{F}\left(\pmb{\xi}^n\right)+
{\rho_{12}\over\rho_n}\pmb{F}\left(\pmb{\xi}^p\right)=
-\nabla\left(\mu_n^\prime+\Psi^\prime\right)
+i\omega{\rho_{np}\rho_p\over\tilde\rho^2}\left(\pmb{\xi}^p-\pmb{\xi}^n\right)
\times\left(\nabla\times\pmb{v}_0\right),$$ and $${\rho_{21}\over\rho_p}\pmb{F}\left(\pmb{\xi}^n\right)+
\left({\rho_{22}\over\rho_p}+{m_e\over m_p}\right)\pmb{F}\left(\pmb{\xi}^p\right)=
-\nabla\left(\tilde\mu^\prime+\zeta\Psi^\prime\right)
-i\omega{\rho_{np}\rho_n\over\tilde\rho^2}\left(\pmb{\xi}^p-\pmb{\xi}^n\right)
\times\left(\nabla\times\pmb{v}_0\right),$$ where $\tilde\mu\equiv\mu_p+\mu_e{m_e}/m_p$, and $$\pmb{F}\left(\pmb{\xi}\right)\equiv-\omega\sigma\pmb{\xi}+i\omega\pmb{v}_0\cdot\nabla\pmb{\xi}
+i\omega\pmb{\xi}\cdot\nabla\pmb{v}_0.$$ The Poisson equation is reduced to $$\nabla^2\Psi^\prime=4\pi G\left(\rho_n^\prime+\zeta\rho_p^\prime\right).$$ Using a variant of Gibbs-Duhem relation, the pressure perturbation is given by $$p^\prime=\rho_n\mu_n^\prime+\rho_p\tilde\mu^\prime,$$ where we have again ignored the entropy carried by the electron normal fluid. Note also that the superfluids carry no entropy.
To obtain a relation between the densities $\rho_n$ and $\rho_p$ and the chemical potentials $\mu_n$ and $\tilde\mu$, we begin with writing the energy density $e$ as $$e=e\left(\rho_n,\rho_p\right),$$ with which the chemical potentials are defined as $$\mu_n\left(\rho_n,\rho_p\right)=\left(\partial e/\partial \rho_n\right)_{\rho_p}, \quad
\mu_p\left(\rho_n,\rho_p\right)=\left(\partial e/\partial \rho_p\right)_{\rho_n}.$$ If the chemical potential of the electron is given by $$\mu_e(\rho_e)=c^2\sqrt{1+(3\pi^2\hbar^3\rho_e/m_e^4c^3)^{2/3}},$$ we have, assuming $\rho_e=\rho_pm_e/m_p$, $$\left(\matrix{\mu_n^\prime \cr \tilde\mu^\prime\cr}\right)
=\left(\matrix{{\cal P}_{11} & {\cal P}_{12}\cr {\cal P}_{21} & {\cal P}_{22}\cr}\right)
\left(\matrix{\rho_n^\prime \cr \rho_p^\prime \cr}\right),$$ where $${\cal P}_{11}=\left({\partial\mu_n\over \partial\rho_n}\right)_{\rho_p}, \quad
{\cal P}_{12}=\left({\partial\mu_n\over\partial\rho_p}\right)_{\rho_n}
=\left({\partial\tilde\mu\over\partial\rho_n}\right)_{\rho_p}={\cal P}_{21}, \quad
{\cal P}_{22}=\left({\partial\tilde\mu\over\partial\rho_p}\right)_{\rho_n}.$$ We write the inverse of equation (26) as $$\left(\matrix{\rho_n^\prime \cr \rho_p^\prime \cr}\right)=
\left(\matrix{{\cal Q}_{11} & {\cal Q}_{12}\cr {\cal Q}_{21} & {\cal Q}_{22}\cr}\right)
\left(\matrix{\mu_n^\prime \cr \tilde\mu^\prime\cr}\right),$$ where $$\left(\matrix{{\cal Q}_{11} & {\cal Q}_{12}\cr {\cal Q}_{21} & {\cal Q}_{22}\cr}\right)
=\left(\matrix{{\cal P}_{11} & {\cal P}_{12}\cr {\cal P}_{21} & {\cal P}_{22}\cr}\right)^{-1}.$$ To represent the rotationally deformed equipotential surfaces of a rotating star, we employ a coordinate system $(a,\theta,\phi)$, the relation of which to spherical polar coordinates $(r,\theta,\phi)$ is given by $r=a[1+\epsilon(a,\theta)]$, where $\epsilon$ is proportional to $\Omega^2$ and represents a small deviation of the equipotential surface from the corresponding spherical equipotential surface of the non-rotating star. We apply Chandrasekhar-Milne expansion (Chandrasekhar 1933a,b, Tassoul 1978) to the hydrostatic equations to determine the function $\epsilon$ in the form $\epsilon(a,\theta)=\alpha(a)+\beta(a)P_2(\cos\theta)$ with $P_2$ being the Legendre function. See Lee (1993) for the definition of $\alpha(a)$ and $\beta(a)$. Since for uniformly rotating stars all the physical quantities in hydrostatic equilibrium are constant on the deformed equipotential surface, labeled by the coordinate $a$, we write the linearised basic equations using the coordinates $(a,\theta,\phi)$ (Saio 1981, Lee 1993). In our formulation, the terms up to of order of $\Omega^3$ in the perturbed velocity equations are retained so that the eigenfrequencies of the $r$-modes are correctly determined to the order of $\Omega^3$ (see Yoshida & Lee 2000a).
The perturbations in a uniformly rotating star are expanded in terms of spherical harmonic functions with different $l$’s for a given $m$ (e.g., Lee & Saio 1986). For example, the vector $\pmb{\xi}^n$ is given by $${\xi_a^n\over a}=\sum_{l\geq |m|} S_l^n(a)Y_l^m(\theta,\phi)e^{i\sigma t},$$ $${\xi_\theta^n\over a}=\sum_{l\geq|m|}\left[H_l^n(a){\partial\over\partial\theta}
Y_l^m(\theta,\phi)+T_{l^\prime}^n(a){1\over\sin\theta}{\partial\over\partial\phi}
Y_{l^\prime}^m(\theta,\phi)\right]e^{i\sigma t},$$ $${\xi_\phi^n\over a}=\sum_{l\geq|m|}\left[H_l^n(a){1\over\sin\theta}{\partial\over
\partial\phi}Y_l^m(\theta,\phi)-T_{l^\prime}^n(a){\partial\over\partial\theta}
Y_{l^\prime}^m(\theta,\phi)\right]e^{i\sigma t},$$ and the Euler perturbation of the gravitational potential, $\Psi^\prime$ is given as $$\Psi^\prime=\sum_{l\geq|m|}\Psi^\prime_l(a)Y_l^m(\theta,\phi)
e^{i\sigma t},$$ where $l=|m|+2(k-1)$ and $l'=l+1$ for even modes, and $l=|m|+2k-1$ and $l'=l-1$ for odd modes, and $k=1,~2,~3,~\cdots$. Substituting these expansions and the like into the linearised basic equations (16) $\sim$ (19) and (21), we obtain oscillation equations given as a set of simultaneous linear ordinary differential equations of the expansion coefficients (see Appendix), which is to be integrated in the superfluid core. The oscillation equations solved in the normal fluid envelope are the same as those given in Yoshida & Lee (2000a).
To obtain a complete solution of an oscillation mode, solutions in the superfluid core and in the normal fluid envelope are matched at the interface between the two domains by imposing jump conditions given by $$\xi_a=\xi_a^n, \quad \xi_a=\xi_a^p, \quad
[p^\prime]^+_-=0, \quad [\Psi^\prime]^+_-=0,
\quad {\rm and} \quad [d\Psi^\prime/da]^+_-=0,$$ where $[f(x)]^+_-\equiv\lim_{s\rightarrow 0}\{f(x+s)-f(x-s)\}$. The boundary conditions at the stellar center is the regularity condition of the perturbations $\xi_a^n$, $\xi_a^p$, $\mu_n^\prime/g$, $\tilde\mu^\prime /g$, and $\Psi^\prime/g$, where $g=GM_a/a^2$. The boundary conditions at the stellar surface are $\delta p=0$ and $\Psi_l^\prime\propto a^{-(l+1)}$, where $\delta p$ is the Lagrangian perturbation of the pressure.
For numerical computation, oscillation equations of a finite dimension are obtained by disregarding the terms with $l$ larger than $l_{max}=|m|+2k_{max}-1$ in the expansions of perturbations such as given by (30) to (33). For the $r$-modes with $l^\prime=|m|$ calculated in this paper, we usually use $k_{max}=6$ so that we can get reasonable convergence of the eigenfrequency and the eigenfunction. We solve the oscillation equations of a finite dimension as an eigenvalue problem of the oscillation frequency $\omega$ using a Henyey type relaxation method (see, e.g., Unno et al 1989).
Dissipations
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The stability of an oscillation mode of a star is determined by summing up all contributions from various damping and excitation mechanisms. If we consider the contributions from gravitational radiation reaction, viscous processes, and mutual friction in the superfluid core, the energy loss (or gain) rate $dE/dt$ of a normal mode in a rotating neutron star may be given by $$\begin{aligned}
{dE\over dt}=&&-\sigma\omega\sum_{l=2}^\infty N_l\sigma^{2l}
\left(\left|D_{lm}^\prime\right|^2+\left|J_{lm}^\prime\right|^2\right)
-\int d^3x \left(\sum_{ij}{\sigma^{\prime ij}
\sigma_{ij}^{\prime*}\over 2\zeta_S}+\zeta_B\left|\nabla\cdot\pmb{v}^\prime\right|^2\right)
\nonumber \\
&&-2\Omega\int d^3x\rho_nB_n\left({\tilde\rho^2\over\rho_n\rho_p}\right)^2
\left(\pmb{w}^\prime\cdot\pmb{w}^{\prime *}-w_z^\prime w_z^{\prime *}\right) \nonumber\\
=&& \left({dE\over dt}\right)_{GD}+\left({dE\over dt}\right)_{GJ}+\left({dE\over dt}\right)_{S}
+\left({dE\over dt}\right)_{B}+\left({dE\over dt}\right)_{MF},\end{aligned}$$ where the asterisk $(^*)$ indicates the complex conjugate of the quantity, and the canonical energy $E$ of oscillation observed in the corotating frame of the star is defined in the normal fluid envelope as (Friedman & Schutz 1978) $$E={1\over 2}\int d^3x \left(\rho\pmb{v}^\prime\cdot\pmb{v}^{\prime *}+
{p^\prime\over\rho}\rho^{\prime *}-{\nabla\Psi^\prime\cdot\nabla\Psi^{\prime *}\over 4\pi G}
\right),$$ and in the superfluid core as (Mendell 1991b) $$E={1\over 2}\int d^3x\left(\rho\pmb{v}^\prime\cdot\pmb{v}^{\prime *}+
{\tilde\rho^2\over\rho}\pmb{w}^\prime\cdot\pmb{w}^{\prime *}+
\sum_{ij}{\cal P}_{ij}\rho^\prime_i\rho^{\prime *}_j
-{\nabla\Psi^\prime\cdot\nabla\Psi^{\prime *}\over 4\pi G}\right),$$ where $\pmb{v}^\prime=(\rho_n\pmb{v}_n^\prime+\rho_p\pmb{v}_p^\prime)/\rho$, $\pmb{w}^\prime=\pmb{v}_p^\prime-\pmb{v}_n^\prime$, $\rho^\prime_1=\rho^\prime_n$, and $\rho^\prime_2=\rho^\prime_p$ in the core.
The terms $(dE/dt)_{GD}$ and $(dE/dt)_{GJ}$ on the right hand side of equation (35) denote the energy loss (or gain) rates due to gravitational radiation reaction associated with the mass multipole moment $D_{lm}^\prime$ and the mass current multipole moment $J_{lm}^\prime$, where $$D_{lm}^\prime=\int d^3x\rho^\prime r^l Y_l^{m*},$$ $$J_{lm}^\prime={2\over c(l+1)}
\int d^3x r^l\left(\rho\pmb{v}^\prime+\rho^\prime\pmb{v}_0\right)\cdot
\left(\pmb{r}\times\nabla Y_l^{m*}\right),$$ and $$N_l={4\pi G\over c^{2l+1}}{(l+1)(l+2)\over l(l-1)[(2l+1)!!]^2},$$ and $c$ is the velocity of light (Thorne 1980, Lindblom et al 1998).
The terms $(dE/dt)_{S}$ and $(dE/dt)_B$ are the energy dissipation rates due to shear and bulk viscosities, and $\zeta_S$ and $\zeta_B$ are the shear and the bulk viscosity coefficients, and $\sigma_{ij}^\prime$ is the traceless stress tensor for the perturbed velocity field (e.g., Landau & Lifshitz 1987). In this paper, we ignore the contribution from the bulk viscosity, which is important only for newly born hot neutron stars without superfluids in the core. The shear viscosity coefficient we use in the superfluid core is $$\zeta_S=6\times 10^{18}\left({\rho\over 10^{15}{\rm g/cm^3}}\right)^2
\left({10^9 K\over T}\right)^2 {\rm g/cm ~s}$$ (Cutler & Lindblom 1987, Sawyer 1989), and that in the normal fluid envelope is given by $$\zeta_S=2\times 10^{18}\left({\rho\over 10^{15}{\rm g/cm^3}}\right)^{9/4}
\left({10^9 K\over T}\right)^2 {\rm g/cm ~s}$$ (Cutler & Lindblom 1987, Flowers & Itoh 1979). The stress tensor $\sigma_{ij}^\prime$ is evaluated by using $\pmb{v}_e^\prime$ in the superfluid core (Lindblom & Mendell 2000).
The term $(dE/dt)_{MF}$ is the energy loss rate due to mutual friction in the superfluid core, and the dimensionless coefficient $B_n$ is given by (Mendell 1991b) $$B_n=0.011\times {\rho_p\over\rho_n}
\left({\rho_{pp}\over\rho_{p}}\right)^{1/2}
\left({\rho_{pn}\over\rho_{pp}}\right)^2
\left({\rho_p\over 10^{14}{\rm gcm^{-3}}}\right)^{1/6}.$$ Mutual friction is a dissipation mechanism inherent to a rotating system of superfluids, and it is caused by scattering of normal fluid particles off the vortices in the superfluids. Since we have assumed perfect charge neutrality between the electrons and protons, we consider scattering between the normal electrons and vortices of the neutron superfluid (Mendell 1991b, see also Alpar etal 1984).
As is indicated by the first two terms on the right hand side of equation (35), if a normal mode has an oscillation frequency that satisfies $-\sigma\omega>0$, the oscillation energy $E$ in the corotating frame, in the absence of other damping mechanisms, increases as a result of gravitational wave radiation, indicating instability of the mode (Friedman & Schutz 1978). It was Andersson (1998) and Friedman & Morsink (1998) who realized that the $r$-modes have oscillation frequencies that satisfy the instability condition.
The damping (or growth) time-scale $\tau$ of a normal mode may be given by $${1\over\tau}={1\over 2E}\left({dE\over dt}\right)={1\over\tau_{GD}}+{1\over\tau_{GJ}}+
{1\over\tau_S}+{1\over\tau_{MF}},$$ where $\tau_i=2E/(dE/dt)_i$. For the $r$-modes of $l^\prime =|m|$, it is convenient to derive an extrapolation formula of the time-scale $\tau$ given as a function of $\Omega$ and the interior temperature $T$ (e.g., Lindblom et al 1998, Lindblom & Mendell 2000): $${1\over\tau (\Omega,T)}
={1\over\tau^0_{GD}}\left({\Omega^2\over \pi G\bar\rho}\right)^{l+2}
+{1\over\tau^0_{GJ}}\left({\Omega^2\over \pi G\bar\rho}\right)^{l}
+{1\over\tau^0_S}\left({10^9 K\over T}\right)^2
+{1\over\tau^0_{MF}}\left({\Omega^2\over \pi G\bar\rho}\right)^\gamma,$$ where $\bar\rho=M/(4\pi R^3/3)$, and only the dominant term in each of the dissipation processes with $l=|m|+1=3$ has been retained for the $r$-modes. The quantities $\tau^0_{GD}$, $\tau^0_{GJ}$, $\tau^0_S$, and $\tau^0_{MF}$ are assumed to be only weakly dependent on $\Omega$ and $T$.
Numerical Results
=================
Following Lindblom & Mendell (2000), we employ a polytropic model of index $N=1$ with the mass $M=1.4M_\odot$ and the radius $R=12.57$km as a background model for modal analysis. The model is divided into a superfluid core and a normal fluid envelope, the interface of which is set at $\rho=\rho_s=2.8\times 10^{14}$g/cm$^3$. In the normal fluid envelope, we use the polytropic equation of state given by $p=K\rho^2$ both for the equilibrium structure and for the oscillation equations. The core is assumed to be filled with neutron and proton superfluids and a normal fluid of electron, for which an equation of state, labeled A18+$\delta v$+UIX (Akmal, Pandharipande, and Ravenhall 1998), is used to give the energy density (23) and the relation (26) used for the oscillation equations. For the mass density coefficients $\rho_{nn}$, $\rho_{pp}$, and $\rho_{np}$ in the core, we employ an empirical relation given by (see Lindblom & Mendell 2000) $${\rho_p/\rho}\approx 0.031+8.8\times 10^{-17}\rho,$$ and a formula given by $$\rho_{np}=-\eta\rho_n,$$ where $\eta$ is a parameter of order of $\sim 0.04$ that parametrizes the entrainment effects between the two superfluids (Borumand, Joynt, & Kluźniak 1996).
We find that the $r$-modes of $l^\prime=m$ in the superfluid core are split into ordinary $r$-modes and superfluid $r$-modes, which we call $r^o$-modes and $r^s$-modes. The toroidal components $iT_{l^\prime}$ of the $r^o$- and $r^s$-modes of $l^\prime=m=2$ are plotted versus $a/R$ for the case of $\eta=0.04$ and $\bar\Omega\equiv\Omega/\sqrt{GM/R^3}=0.01$ in Figure 1, where the solid, dotted, dashed, and dash-dotted curves are used to indicate respectively the toroidal components $iT_m^n$, $iT_{m+2}^n$, $iT_m^p$, and $iT_{m+2}^p$ in the superfluid core, and the amplitude normalization is given by $\max(|iT_m|)=1$. The figure shows that the two superfluids in the core flow together for the $r^o$-modes, while they counter-move for the $r^s$-modes. Note that the amplitudes of $iT_{m+2}^p$ for the $r^s$-mode are not necessarily negligibly small compared with those of $iT_m^p$ in the core. This splitting of the $r$-modes in the superfluid core into two distinct families has been suggested by Andersson & Comer (2001). For the $r^o$-mode, the radial dependence of the difference $iT_m^p-iT_m^n$ in the core is given in Figure 2, which shows that the difference is quite small compared with the normalization $\max(|iT_m|)=1$. It is important to note that the $l^\prime=m$ $r$-modes having basically nodeless and dominant $iT_m$ are the only $r$-modes we can find, as in the case of the $r$-modes in isentropic models (see Yoshida & Lee 2000a,b).
In Table 1, the expansion coefficients $\kappa_0$ and $\kappa_2$, and the scaled damping (or growth) timescales $\tau^0_i$ for the $l'=m=2$ $r$-modes are tabulated for the cases of $\eta=0$, $0.02$, $0.04$, and $0.06$, where the coefficients $\kappa_0$ and $\kappa_2$ are defined in the expansion: $$\omega/\Omega=\kappa_0(\eta)+\kappa_2(\eta)\bar\Omega^2+O(\bar\Omega^4).$$ Note that the exponent $\gamma$ employed to define $\tau^0_{MF}$ in equation (45) is $\gamma=2.5$ for the $r^o$-modes and $\gamma=0.5$ for the $r^s$-modes for the non-zero $\eta$’s in the table. This is because the velocity difference $\pmb{w}^\prime=\pmb{v}^\prime_p-\pmb{v}^\prime_n$ approximately scales as $\pmb{w}^\prime\propto\Omega^3$ for the $r^o$-modes and as $\pmb{w}^\prime\propto\Omega$ for the $r^s$-modes (see Lindblom & Mendell 2000). The coefficient $\kappa_0$ for the $r^o$-modes is numerically equal to $2m/[l^\prime(l^\prime+1)]$, and $\kappa_2$ is almost independent of the entrainment parameter $\eta$. The value of $\kappa_0$ is the same as the value found by Lindblom & Mendell (2000) and the value of $\kappa_2$ differs only by 2.5%, suggesting that the $r^o$-modes are the same modes found by Lindblom & Mendell (2000). (Because of different normalization conventions the value of $\kappa_2$ reported by Lindblom & Mendell must be multiplied by a factor of $4/3$ before comparing with our results.) The coefficients $\kappa_0$ and $\kappa_2$ for the $r^s$-modes, on the other hand, appreciably depends on $\eta$, and $\kappa_0$ deviates from $2m/[l^\prime(l^\prime+1)]$ as $\eta$ is increased from $\eta=0$. This kind of deviation of $\kappa_0$ from $2m/[l^\prime(l^\prime+1)]$ for the $l^\prime=m$ $r$-modes has been found for relativistic neutron stars where the relativistic factor $GM/c^2R$ is regarded as a parameter (Yoshida 2001; Yoshida & Lee 2002a). Computing the $r^s$-mode of $l^\prime=m=2$ as a function of $\eta$, we find that a linear formula given by $$\kappa_0^s(\eta)\approx0.667+9.35\eta$$ gives a good fit to the $r^s$-mode frequency except at avoided crossings with inertial modes (see below). At $\eta=0$, the coefficients $\kappa_0$ for the $r^o$- and $r^s$-modes are both equal to $2m/[l^\prime(l^\prime+1)]$, which suggests that in the lowest order of $\Omega$ the two $r$-modes are degenerate at $\eta=0$. In Figure 3, the toroidal components $iT_m$ and $iT_{m+2}$ of the $r^o$- and $r^s$-modes of $l^\prime=m=2$ at $\bar\Omega=0.01$ are given versus $a/R$ for the case of $\eta=0$. The amplitudes of $iT_{m+2}$ are much smaller than those of $iT_m$ both for the $r^o$- and $r^s$-modes. This figure also shows that $|iT_m^p-iT_m^n|\not\ll 1$ in the core for the $r^o$-mode at $\eta=0$, for which we find that $\pmb{w}^\prime\propto\Omega$.
Inertial modes in the superfluid core are also split into ordinary and superfluid inertial modes, which we call $i^o$- and $i^s$-modes. In Table 2, we have given $\kappa_0$ for $i^o$- and $i^s$-modes for $m=2$ and $\eta=0$, and see, e.g., Lockitch & Friedman (1999) and Yoshida & Lee (2000a) for the classification scheme employed here for inertial modes. For given $m$ and $l_0-|m|$, we find at $\eta=0$ pairs of $i^o$- and $i^s$-modes that have close values of $\kappa_0$, and the number of the pairs is equal to $l_0-|m|$. As an example, for the case of $m=2$ and $\eta=0$, the eigenfunctions $iT_{l^\prime}$ are shown for the $i^o$- and $i^s$-modes of $\kappa_0=0.5180$ and $\kappa_0=0.5077$ in Figure 4, for those of $\kappa_0=0.4215$ and $\kappa_0=0.4060$ in Figure 5, and for those of $\kappa_0=1.1046$ and $\kappa_0=1.1134$ in Figure 6. The inertial modes in Figure 4 belong to $l_0-|m|=3$ and those in Figures 5 and 6 to $l_0-|m|=5$. Note that the $r^o$- and $r^s$-modes belong to $l_0-|m|=1$ (e.g., Yoshida & Lee 2000a). It is generally observed for inertial modes with long radial wavelengths that the two superfluids co-move in the core for the $i^o$-modes, and they counter-move for the $i^s$-modes. The coefficient $\kappa_0$ for the $i^o$-modes only weakly depends on the entrainment parameter $\eta$ (see Figure 7). The coefficient $\kappa_0$ for the $i^s$-modes, on the other hand, increases approximately linearly as $\eta$ is increased from $\eta=0$ (see Figure 8). See Yoshida & Lee (2002b) for extended discussions on inertial modes in the superfluid core.
The $r^s$-modes ($r^o$-modes) experience mode crossings with $i^o$-modes ($i^s$-modes) as the parameter $\eta$ is increased from $\eta=0$. For the case of $m=2$ and $\bar\Omega=0.01$, Figure 7 illustrates an avoided crossing between the $r^s$-mode of $l^\prime=m$ and the $i^o$-mode that tends to $\kappa_0=1.1046$ as $\eta\rightarrow0$. In this figure, the dashed line is given by equation (49). For mode crossings between the $r^o$-mode and $i^s$-modes, on the other hand, it is quite difficult to numerically discern whether the mode crossings result in avoided crossing or degeneracy of the mode frequencies at the crossing point. Most prominent among such mode crossings of the $r^o$-mode of $l^\prime=m=2$ are those with the $i^s$-modes that tend to $\kappa_0=0.5077$ and $\kappa_0=0.4060$ as $\eta\rightarrow 0$. For the case of $m=2$ and $\bar\Omega=0.01$, the two mode crossings which occur at $\eta\approx0.0230$ and 0.0484 are shown in Figure 8, where the solid lines and the dashed line are for the $i^s$-modes and the $r^o$-mode, respectively. We note that the ratio $\omega/\Omega\approx \kappa_0$ for the $r^o$-mode is almost constant, equal to $2m/[l^\prime(l^\prime+1)]=0.6667$ for $l^\prime=m=2$, while the ratios $\omega/\Omega$ for the $i^s$-modes increase approximately linearly with increasing $\eta$. Figure 9 shows the differences $iT_m^p-iT_m^n$ versus $a/R$ for the $l^\prime=m=2$ $r^o$-modes at $\eta=0.023$ (panel a) and 0.0484 (panel b), and Figure 10 the eigenfunctions $iT_{l^\prime}$ for the $i^s$-modes of $m=2$ at $\eta=0.023$ (panel a) and at $\eta=0.0484$ (panel b), where we have assumed $\bar\Omega=0.01$ and the normalization $\max(|iT_m|)=1$. The amplitudes of the differences at the mode crossing points are much larger than amplitudes of the difference off the crossing points (see, e.g., Figure 2). The resemblance between the $i^s$-modes of $l_0-|m|=3$ at $\eta=0$ (Figure 4b) and at $\eta=0.023$ (Figure 10a) and between the $i^s$-modes of $l_0-|m|=5$ at $\eta=0$ (Figure 5b) and $\eta=0.0484$ (Figure 10b) is obvious.
>From Table 1, we find that the growth timescales $\tau^0_{GJ}$ and $\tau^0_{GD}$ for the $r^o$-modes of $l^\prime=m=2$ are almost the same as those for the $l^\prime=m=2$ $r$-modes of the $N=1$ polytropic model without superfluidity in the core (see, e.g., Yoshida & Lee 2000a). On the other hand, the growth timescales $\tau^0_{GJ}$ and $\tau^0_{GD}$ for the $r^s$-modes are much longer than those of the $r^o$-modes, and the instability caused by the $r^s$-modes is much weaker than the instability by the $r^o$-modes. This is because the amount of gravitational wave radiation emitted from the $r^s$-modes is much smaller than that from the $r^o$-modes since $(\rho_piT_m^p+\rho_niT_m^n)/\rho\sim 0$ and $\rho^\prime\sim 0$ for the $r^s$-modes. It is interesting to note that $\tau^0_{GJ}$ of the $r^s$-modes at $\eta\not=0$ is by several orders of magnitude longer than that at $\eta=0$. Figure 11 shows, as a function of $a/R$, $\sqrt{|\omega\sigma^{2l+1}|N_l/2E}J_{mm}^\prime(a)$ for the $r^s$-modes at $\eta=0$ (dashed line) and at $\eta=0.04$ (solid line), where $J_{mm}^\prime(a)=\int_0^adadJ_{mm}^\prime/da$. As found for the case of $\eta=0.04$, the negative contributions to $J_{mm}^\prime(a=R)$ in the envelope almost completely cancel out the positive contributions in the core, which leads to extremely long growth timescales $\tau^0_{GJ}$ of the $r^s$-modes at $\eta\not=0$.
Figure 12 illustrates the dependence of $-\tau^0_{MF}$ on the entrainment parameter $\eta$ for the $l^\prime=m=2$ $r^o$-mode, and we find that $-\tau^0_{MF}$ has prominent and deep minimums at $\eta\approx0.230$ and $0.484$., which is consistent with the result by Lindblom & Mendell (2000). Note that we have assumed $\rho_s=2.8\times 10^{14}$g/cm$^3$ for the interface between the core and the envelope. These prominent minimums result from the mode crossings with the long radial wavelength $i^s$-modes that tend to $\kappa_0=0.5077$ and $\kappa_0=0.4060$ as $\eta\rightarrow 0$ (Figure 8; see also Andersson & Comer 20001). Mode crossings of the $r^o$-mode with $i^s$-modes that have much shorter radial wavelengths result in narrow and shallow minimums of $-\tau^0_{MF}$, as found for $\eta\gtsim 0.05$. These minor dips were not found by Lindblom & Mendell (2000). We guess that the $r$-modes calculated by Lindblom & Mendell (2000) are less strongly affected by inertial modes associated with large $l_0-|m|$ having short radial wavelengths since they ignored terms higher than $\Omega^4$ in their perturbative method. Comparing our results with those by Lindblom & Mendell (2000), we find that the damping timescale $-\tau^0_{MF}$ at a given $\eta$ is about by an order of magnitude shorter than that they obtained, and that our values of $\eta\approx0.0230$ and 0.0484 for the local minimums of $-\tau^0_{MF}$ are slightly larger than their $critical$ values 0.02294 and 0.04817. From Table 1 we also find that the value of $\kappa_2$ found here for the $r^o$-modes is about by 2.5% larger than the value they found. We guess that these differences between the two calculations partly come from the differences in the way of evaluating the derivatives of the thermodynamical quantities that appear in the oscillation equations. For example, Lindblom & Mendell (2000) used $\rho^\prime=(\partial \rho/\partial p)_\beta p^\prime+(\partial\rho/\partial\beta)_p\beta^\prime$, for which they assumed $(\partial \rho/\partial p)_\beta=(\rho/2p)$ from the polytropic relation $p=K\rho^2$ and employed a fitting formula to $(\partial\rho/\partial\beta)_p$ obtained from Akmal et al’s equation of state (1998), where $\beta=\tilde{\mu}-\mu_n$. In this paper, on the other hand, we used $\rho^\prime=({\cal Q}_{11}+\zeta{\cal Q}_{21})\mu_n^\prime
+({\cal Q}_{12}+\zeta{\cal Q}_{22})\mu_p^\prime$, and the coefficients ${\cal Q}_{ij}$ were all calculated by using equations (23) to (29) with Akmal et al’s equation of state (1998).
The eigenfunction $\delta\beta(r,\theta,\phi)=\delta\beta_0(r,\theta,\phi)
+(4/3)\delta\beta_2(r,\theta,\phi)\bar\Omega^2+O(\bar\Omega^4)$ in Lindblom & Mendell (2000) may be given in terms of the eigenfunctions $y^{p}_{2,k}$ and $y^{n}_{2,k}$ as $$\delta \beta(r,\theta,\phi)=(M_a/M)(R/a)\bar\Omega^{-2}
\sum_{k\ge 1}\Delta y_{2,k}(a)Y_{l_km}(\theta,\phi),$$ where $r=a[1+\epsilon(a,\theta)]$, and $\Delta y_{2,k}(a)=y_{2,k}^p(a)-y_{2,k}^n(a)$ and see Appendix for definition of the functions $y_{2,k}^{p(n)}(a)$. Assuming $\delta\beta_0=0$, we obtain $$\delta \beta_2(r,\theta,\phi)\approx 0.75\times(M_r/M)(R/r)\bar\Omega^{-4}
\sum_{k\ge 1}\Delta y_{2,k}(r)f_{l_km}P_{l_k}^m(\cos\theta)e^{im\phi}
\equiv\sum_{k\ge1}\delta\beta_{2,k}P_{l_k}^m(\cos\theta)e^{im\phi},$$ where the mean radial distance $a$ have been replaced by the radial distance $r$, and the factor $f_{lm}$ is defined by the relation $Y_l^m(\theta,\phi)=f_{lm}P^m_l(\cos\theta)e^{im\phi}$, and $l_k=|m|+2k-1$. In Figure 13, we plot the functions $\delta\beta_{2,k}(r)$ for the $r^o$-mode of $l^\prime=m=2$ at $\eta=0.04$, applying amplitude normalization given by $y_{2,k=1}(R)=f_{m+1,m}\bar\Omega^2$ at the surface, where the solid, dotted, and dashed lines are for $\delta\beta_{2,1}$, $100\times\delta\beta_{2,2}$, and $100\times\delta\beta_{2,3}$, respectively. Since $|\delta\beta_{2,1}(r)|>>|\delta\beta_{2,k}(r)|$ for $k\ge2$, the $\theta$ depenence of the function $\delta\beta_2(r,\theta,\phi)$ is well represented by a single associated Legendre function $P^m_{m+1}(\cos\theta)$, and is not necessarily the same $\theta$ dependence of the function $\delta\beta_2$ found by Lindblom & Mendell (2000). The amplitude of the function $\delta\beta_{2,1}(r)$ is about by a factor of $3$ larger than the amplitude of $\delta\beta_2$ calculated by Lindblom & Mendell (2000), which is consistent with the result that $\tau_{MF}^0$ in this paper is about by an order of magnitude shorter than $\tau_{MF}^0$ they obtained.
Discussion
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If we employ a set of the dependent variables defined as (see Lindblom & Mendell 2000) $$\pmb{\xi}={\rho_n\pmb{v}_n^\prime+\rho_p\pmb{v}_p^\prime\over i\omega\hat\rho}, \quad
\pmb{\xi}^w={\pmb{w}^\prime\over i\omega}, \quad
U={p^\prime\over\hat\rho}+\Psi^\prime, \quad
\beta^\prime=\tilde{\mu}^\prime-\mu^\prime_n,$$ where $\hat\rho=\rho_n+\rho_p$, the continuity equations (16) and (17) and the velocity equations (18) and (19) are rewritten as $$\rho_n^\prime+\rho_p^\prime+\nabla\cdot\left(\hat\rho\pmb{\xi}\right)=0,$$ $${\rho_p^\prime\over\rho_p}-{\rho_n^\prime\over\rho_n}
+\pmb{\xi}\cdot\nabla\ln{\rho_p\over\rho_n}
+{\tilde\rho^2\over\rho_n\rho_p}\left(
\pmb{\xi}^w\cdot\nabla\ln{\tilde\rho^2\over\hat\rho}+\nabla\cdot\pmb{\xi}^w\right)=0,$$ $$\pmb{F}\left(\pmb{\xi}\right)
=-\nabla U
+{\rho_n\rho_p\over\hat\rho^2}\left(\nabla\ln{\rho_p\over\rho_n}\right)\beta^\prime,$$ $$\pmb{F}\left(\pmb{\xi}^w\right)+i\omega{\rho_{np}\hat\rho\over\rho_n\rho_p}
\pmb{D}\left(\pmb{\xi}^w\right)=-\nabla\beta^\prime,$$ and the linearized Poisson equation remains the same: $$\nabla^2\Psi^\prime=4\pi G\left(\rho_n^\prime+\rho_p^\prime\right),$$ where $\pmb{D}(\pmb{\xi})=\pmb{\xi}\times(\nabla\times\pmb{v}_0)$, and the terms of order of $m_e/m_p$ have been ignored for simplicity. The term proportional to $\pmb{D}\left(\pmb{\xi}^w\right)$ in equation (56) represents the drag force between the two superfluids. In a perturbative treatment of the $r$-modes, we may expand the eigenfunctions and eigenfrequencies in terms of $\Omega$ as $$\pmb{\xi}=\pmb{\xi}_0+\pmb{\xi}_2\Omega^2+O(\Omega^4), \quad
\pmb{\xi}^w=\pmb{\xi}_0^w+\pmb{\xi}_2^w\Omega^2+O(\Omega^4),$$ $$U=U_2\Omega^2+U_4\Omega^4+O(\Omega^6), \quad
\beta^\prime=\beta_2^\prime\Omega^2+\beta_4^\prime\Omega^4+O(\Omega^6),$$ and $$\omega=\kappa_0\Omega+\kappa_2\Omega^3+O(\Omega^5), \quad
\sigma=s_0\Omega+s_2\Omega^3+O(\Omega^5),$$ where $|\pmb{\xi}_0|\sim |iT_m|\sim O(1)$. Note that the quantities like $\nabla\cdot\pmb{\xi}$, $\pmb{\xi}\cdot\nabla\rho$, $\rho_n^\prime$, $\rho_p^\prime$, and $\Psi^\prime$ are of order of $\Omega^2$ in the lowest order for the $r$-modes. For the expansions given above, we have $$\pmb{F}(\pmb{\xi})=\pmb{F}_2(\pmb{\xi}_0,\kappa_0,s_0)\Omega^2
+\pmb{F}_4(\pmb{\xi}_0,\pmb{\xi}_2,\kappa_0,\kappa_2,s_0,s_2)\Omega^4
+O(\Omega^6),$$ and we may not need to give the full expressions of $\pmb{F}_2$ and $\pmb{F}_4$ here. Equations (55) and (56) suggest the existence of two families of $r$-modes, that is, $r$-modes, which are governed by equation (55) and have $\kappa_0=2m/[l^\prime(l^\prime+1)]$, and $r$-modes, which are governed by equation (56) and have $\kappa_0\not=2m/[l^\prime(l^\prime+1)]$. If we assume $\pmb{F}_2(\pmb{\xi}_0)=-\nabla U_2$ in equation (55), we obtain the lowest order $r$-mode solution given by $iT_m\propto a^{|m|-1}$ and $\kappa_0=2m/[l^\prime(l^\prime+1)]$, which leads to $\beta_2^\prime=0$ from equation (55), and to $\pmb{\xi}^w_0=0$ from equation (56). This suggests the existence of the $r$-mode solutions, for which $\kappa_0=2m/[l^\prime(l^\prime+1)]$, and $\pmb{\xi}_0\not=0$ and $U_2\not=0$, but $\pmb{\xi}^w_0=0$ and $\beta^\prime_2=0$. The solutions of this kind correspond to the $r$-modes Lindblom & Mendell (2000) obtained in their perturbative treatment, and to the $r^o$-modes found in this paper. On the other hand, because of the drag force, i.e., the term proportional to $\pmb{D}(\pmb{\xi}^w)$, $\kappa_0$ of the $r$-modes governed by equation (56) is not necessarily equal to $2m/[l^\prime(l^\prime+1)]$. For these solutions we may expect $\pmb{\xi}^w_0\not=0$ and $\beta_2^\prime\not=0$ from equation (56), and $\pmb{\xi}_0\not=0$ and $U_2\not=0$ from equation (55). The solutions of this kind correspond to the $r^s$-modes found in this paper (see also Andersson & Comer 2001).
To show the importance of the drag force for the existence of the $r$-mode solutions with the scaling $\pmb{w}^\prime\propto\Omega^3$, we calculate the $r^o$- and $r^s$-modes by ignoring the drag force terms proportional to $\pmb{D}(\pmb{\xi})$ on the right hand side of the velocity equations (18) and (19). The results are summarized in Table 3, in which the coefficients $\kappa_0$, $\kappa_2$, and $\tau^0_i$’s are tabulated for the $r^o$- and $r^s$-modes of $l^\prime=m=2$ for the case of $\eta=0.04$. When we ignore the drag force terms, we can not find the $r^o$-modes with the scaling of $\pmb{w}^\prime\propto\Omega^3$ and the $r$-modes with the scaling $\pmb{w}^\prime\propto\Omega$ are the only $r$-mode solutions we can find, and hence the exponent $\gamma$ to define $\tau^0_{MF}$ in the table is equal to $0.5$ for both the $r^o$- and $r^s$-modes. In Figure 14, the toroidal components $iT_m$ and $iT_{m+2}$ of the two $r$-modes are given for $\eta=0.04$, where we have assumed $\bar\Omega=0.01$ and the normalization $\max(|iT_m|)=1$. The amplitudes of $iT_{m+2}$ are completely negligible compared to those of $iT_m$. In the absence of the drag force, $\kappa_0$’s for the two $r$-modes are both equal to $2m/[l^\prime(l^\prime+1)]$, independent of $\eta$, which means that the frequencies of the two $r$-modes are degenerate in the lowest order of $\Omega$. These results suggest that the $r$-mode solutions with $\pmb{w}^\prime\propto\Omega^3$ can be found only when the frequencies of the $r$-modes are not degenerate in the lowest order in $\Omega$. In this sense, it is reasonable to find $\pmb{w}^\prime\propto\Omega$ for the $r^o$-modes at $\eta=0$ since $\kappa_0$’s of the $r^o$- and $r^s$-modes are equal to each other at $\eta=0$, as shown by Table 1.
Conclusion
==========
In this paper we have discussed the modal properties of the $r$-modes of neutron stars with the core filled with neutron and proton superfluids. We numerically find that the $r$-modes of rotating neutron stars with the superfluid core are split into ordinary $r^o$-modes and superfluid $r^s$-modes, and that the two superfluids in the core flow together for the $r^o$-modes and they counter move for the $r^s$-modes. These findings are consistent with earlier suggestions made analytically by Andersoon & Comer (2001). We also find that, although $\kappa_0$ of the $r^o$-modes is numerically equal to $2m/[l^\prime(l^\prime+1)]$, almost independent of the entrainment parameter $\eta$, $\kappa_0$ of the $r^s$-modes approximately linearly increases from $2m/[l^\prime(l^\prime+1)]$ as $\eta$ is increased from $\eta=0$. The $r^o$-modes have the scaling of $\pmb{w}^\prime\propto\Omega^3$ and are the same $r$-modes discussed by Lindblom & Mendell (2000), while the $r^s$-modes have the scaling of $\pmb{w}^\prime\propto\Omega$. The instability caused by the $r^s$-modes is found much weaker than the instability by the $r^o$-mode and will be easily stabilized by various dissipation processes in the star.
We have confirmed that, except in a few narrow regions of $\eta$ around the prominent local minimums of $\tau^0_{MF}$, the dissipation due to mutual friction in the superfluid core is ineffective to stabilize the instability by the $r^o$-modes, as first shown by Lindblom & Mendell (2000). We have shown that these prominent local minimums of $-\tau^0_{MF}$ are caused by mode crossings between the $r^o$-mode and the superfluid inertial $i^s$-modes with long radial wavelengths comparable to those of the $r^o$-mode (see also Andersson & Comer 2001). Since mutual friction is almost always very strong for the $r^s$-modes and the instability by the $r^s$-modes itself is quite weak, only the instability by the $r^o$-modes will be of direct observational importance, e.g., as mechanisms that generate gravitational waves from normal modes of rotating neutron stars and/or cause the clustering of spin frequencies around 300Hz for neutron stars in LMXB’s, unless there exist other strong damping mechanisms for the $r$-modes (see Jones 2001a,b; Lindblom & Owen 2002; Haensel, Levenfish & Yakovlev 2002).
In this paper, we have made a brief report on a numerical result of inertial modes in the superfluid core. We find that inertial modes in superfluid neutron stars are also split into ordinary inertial $i^o$-modes and superfluid inertial $i^s$-modes. It is generally observed for inertial modes with long radial wavelengths that the two superfluids co-move in the core for the $i^o$-modes, and they counter-move for the $i^s$-modes (see Yoshida & Lee 2002b for more complete discussions on inertial modes in the superfluid core). Inertial modes are expected to work as a mechanism limiting the amplitude growth of the $r$-modes (Morsink 2002).
There are many interesting and challenging problems we have to deal with before we can conclude definitely about the neutron star oscillations. Confronting their cooling calculations of neutron stars with observational data for the surface temperature of several neutron stars, Kaminker, Haensel, & Yakovlev (2001), and Kaminker, Yakovlev, & Gnedin (2002) suggested that neutron superfluidity in the core of middle-aged neutron stars should be weak in the sense that the critical temperature $T_c$ is less than $10^8$K. If this is the case, there exist no neutron superfluids in the core of temperatures $T>10^8$K. If neutrons in the core are in normal state, modal properties of low frequency oscillations propagating in the core will be different from those for the core filled with neutron and proton superfluids, since buoyant force in the core, produced by thermal and/or chemical stratification (Reisenegger & Goldreich 1992), comes into play. The normal neutron fluid core with stratification will support $g$-mode propagation, and the buoyant force in it will affect inertial modes in slowly rotating neutron stars. The nodeless $r$-modes of $l^\prime=m$, however, will remain almost the same even in the presence of buoyant force (see Yoshida & Lee 2000b), and damping due to mutual friction in the core will remain weak for the $r^o$-modes, for which neutron and proton fluids co-move. Let us point out a possibility of using neutron star binaries as a probe to investigate existence or non-existence of neutron superfludity in the core, since tidally excited low frequency modes will be $g$-modes in the normal fluid core but inertial modes in the superfluid core, and the different tidal responses will result in different binary evolutions. The presence of a solid crust and a magnetic field in neutron star interior is another factor that makes complicated the problems of global oscillations of the stars. The solid crust is not a rigid body and supports its own normal modes (see, e.g., McDermott et al 1988, Lee & Strohmayer 1996). For example, there exist torsional modes propagating in the crust, which can be resonantly coupled with the $r$-modes in the normal fluid core (Yoshida & Lee 2001; Levin & Ushomirsky 2001). Quite recently, using a local analysis of perturbations, Kinney & Mendell (2002) have suggested that no $r$-mode solution to the magnetohydrodynamic equations (e.g., Mendell 1998) exists in the superfluid core when both the neutron and proton vortices are pinned to the solid crust. This may suggest that a careful formulation of boundary conditions at the core-crust interfaces will be necessary to obtain reliable solutions to global oscillations of neutron stars, particularly when a magnetic field is essential for the oscillations.
Oscillation Equations in the Superfluid Core
============================================
For the oscillation equations in the superfluid core, we employ vectors $\pmb{y}_1^n$, $\pmb{y}_2^n$, $\pmb{y}_1^p$, $\pmb{y}_2^p$, $\pmb{y}_3$, and $\pmb{y}_4$, whose components are given by $$y_{1,k}^n=S_l^n, \quad y_{2,k}^n={\mu_{n,l}^\prime+\Psi^\prime_l\over ga}, \quad
y_{1,k}^p=S_l^p, \quad y_{2,k}^p={\tilde\mu_{l}^\prime+\zeta\Psi^\prime_l\over ga}, \quad
y_{3,k}={\Psi^\prime_l\over ga}, \quad y_{4,k}={1\over g}{d\Psi^\prime_l\over da},$$ where $l=|m|+2(k-1)$ for even modes and $l=|m|+2k-1$ for odd modes, and $k=1,~2,~3, \cdots$. We also introduce vectors $\pmb{h}^n$, $\pmb{h}^p$, $i\pmb{t}^n$, and $i\pmb{t}^p$, the components of which are $$h_k^n=H_l^n, \quad it_k^n=iT_{l^\prime}^n, \quad
h_k^p=H_l^p, \quad it_k^p=iT_{l^\prime}^p,$$ where $l^\prime=l+1$ for even modes and $l^\prime=l-1$ for odd modes. Using these vectors, the perturbed continuity equations (16) and (17) are written as $$\begin{aligned}
a{d\pmb{y}_1^n\over da}&=&\left(-{d\ln\rho_n\over d\ln a}-3-a{d\chi_3(\alpha)\over da}
-a{d\chi_3(\beta)\over da}A_0\right)\pmb{y}_1^n-{ga\over\rho_n}{\cal Q}_{11}\pmb{y}_2^n
-{ga\over\rho_n}{\cal Q}_{12}\pmb{y}_2^p
+{ga\over\rho_n}\left({\cal Q}_{11}+\zeta {\cal Q}_{12}\right)\pmb{y}_3 \nonumber \\
&+& \left(\Lambda_0+3\chi_3(\beta)B_0\right)\pmb{h}^n
+3m\chi_3(\beta)Q_0i\pmb{t}^n, \end{aligned}$$ $$\begin{aligned}
a{d\pmb{y}_1^p\over da}&=&\left(-{d\ln\rho_p\over d\ln a}-3-a{d\chi_3(\alpha)\over da}
-a{d\chi_3(\beta)\over da}A_0\right)\pmb{y}_1^p-{ga\over\rho_p}{\cal Q}_{21}\pmb{y}_2^n
-{ga\over\rho_p}{\cal Q}_{22}\pmb{y}_2^p
+{ga\over\rho_p}\left({\cal Q}_{21}+\zeta {\cal Q}_{22}\right)\pmb{y}_3 \nonumber \\
&+& \left(\Lambda_0+3\chi_3(\beta)B_0\right)\pmb{h}^p
+3m\chi_3(\beta)Q_0i\pmb{t}^p.\end{aligned}$$ The radial components of equations (18) and (19) are reduced to $$\begin{aligned}
a{d\pmb{y}_2^n\over da}&=&c_1\bar\omega^2{\rho_{11}\over\rho_n}E_0\pmb{y}_1^n+(1-U)\pmb{y}_2^n
+c_1\bar\omega^2{\rho_{12}\over\rho_n}E_0\pmb{y}_1^p \nonumber \\
&-&c_1\bar\omega^2\left({\rho_{11}\over\rho_n}3\beta B_0+m\nu E_1\right)\pmb{h}^n
-c_1\bar\omega^2{\rho_{12}\over\rho_n}3\beta B_0\pmb{h}^p \nonumber \\
&-&c_1\bar\omega^2\left({\rho_{11}\over\rho_n}3m\beta Q_0+\nu E_1C_0\right)i\pmb{t}^n
-c_1\bar\omega^2{\rho_{12}\over\rho_n}3m\beta Q_0i\pmb{t}^p, \end{aligned}$$ $$\begin{aligned}
a{d\pmb{y}_2^p\over da}&=&c_1\bar\omega^2\left({\rho_{22}\over\rho_p}+{m_e\over m_p}\right)
E_0\pmb{y}_1^p+(1-U)\pmb{y}_2^p
+c_1\bar\omega^2{\rho_{21}\over\rho_p}E_0\pmb{y}_1^n \nonumber \\
&-&c_1\bar\omega^2\left[\left({\rho_{22}\over\rho_p}+{m_e\over m_p}\right)3\beta B_0
+m\nu\zeta E_1\right]\pmb{h}^p
-c_1\bar\omega^2{\rho_{21}\over\rho_p}3\beta B_0\pmb{h}^n \nonumber \\
&-&c_1\bar\omega^2\left[\left({\rho_{22}\over\rho_p}+{m_e\over m_p}\right)3m\beta Q_0
+\nu\zeta E_1C_0\right]i\pmb{t}^p
-c_1\bar\omega^2{\rho_{21}\over\rho_p}3m\beta Q_0i\pmb{t}^n.\end{aligned}$$ The perturbed Poisson equation (21) is reduced to $$a{d\pmb{y}_3\over da}=\left(1-U\right)\pmb{y}_3+\pmb{y}_4,$$ $$\begin{aligned}
a{d\pmb{y}_4\over da}&= & 4\pi G a^2\left({\cal Q}_{11}+\zeta{\cal Q}_{21}\right)I^{-1}\pmb{y}_2^n
+4\pi G a^2\left({\cal Q}_{12}+\zeta{\cal Q}_{22}\right)I^{-1}\pmb{y}_2^p \nonumber \\
&+&I^{-1}\left[\Lambda_0-4\pi G a^2\left({\cal Q}_{11}
+\zeta{\cal Q}_{21}+\zeta({\cal Q}_{12}+\zeta{\cal Q}_{22})\right)\pmb{1}
-2\left(\alpha\pmb{1}+\beta A_0\right)\Lambda_0\right]\pmb{y}_3 \nonumber \\
&+& I^{-1}\left[(-U+\chi_0(\alpha))I
+\chi_0(\beta)A_0-6\beta (A_0+B_0)\right]\pmb{y}_4.\end{aligned}$$ Algebraic equations that determine the relations between the variables $\pmb{h}$, $i\pmb{t}$, $\pmb{y}_1$, and $\pmb{y}_2$ are derived by making use of the horizontal components of the velocity equations (18) and (19), and they are $$\tilde L_0\pmb{h}^n-\tilde M_1i\pmb{t}^n
-f_nmF_0(\pmb{h}^p-\pmb{h}^n)-f_n\tilde M_1^+(i\pmb{t}^p-i\pmb{t}^n)
= \tilde J\pmb{y}_1^n +f_n \tilde J^+(\pmb{y}_1^p-\pmb{y}_1^n)
+b_{11}{\pmb{y}_2^n\over c_1\bar\omega^2}+b_{12}{\pmb{y}_2^p\over c_1\bar\omega^2},$$ $$\tilde L_0\pmb{h}^p-\tilde M_1i\pmb{t}^p
+ f_pmF_0(\pmb{h}^p-\pmb{h}^n)+f_p\tilde M_1^+(i\pmb{t}^p-i\pmb{t}^n)
= \tilde J\pmb{y}_1^p-f_p \tilde J^+(\pmb{y}_1^p-\pmb{y}_1^n)
+b_{21}{\pmb{y}_2^n\over c_1\bar\omega^2}+b_{22}{\pmb{y}_2^p\over c_1\bar\omega^2},$$ $$-\tilde M_0\pmb{h}^n+\tilde L_1i\pmb{t}^n
-f_n\tilde M_0^+(\pmb{h}^p-\pmb{h}^n)
-f_nmF_1(i\pmb{t}^p-i\pmb{t}^n)
=-\tilde K\pmb{y}_1^n-f_n\tilde K^+(\pmb{y}_1^p-\pmb{y}_1^n),$$ $$-\tilde M_0\pmb{h}^p+\tilde L_1i\pmb{t}^p
+f_p\tilde M_0^+ (\pmb{h}^p-\pmb{h}^n)
+f_pmF_1(i\pmb{t}^p-i\pmb{t}^n)
=-\tilde K\pmb{y}_1^p+f_p\tilde K^+(\pmb{y}_1^p-\pmb{y}_1^n),$$ where $$f_n={\rho_{np}\over\rho_n}{\zeta\over\tilde\zeta}, \quad
f_p={\rho_{np}\over\rho_p}{1\over\tilde\zeta},$$ $$\zeta=1+{m_e\over m_p}, \quad
\tilde\zeta=1+{m_e\over m_p}{\rho_{pp}\over\rho_p},$$ and $$\left(\matrix{b_{11}&b_{12}\cr b_{21}&b_{22}\cr}\right)=
\left(\matrix{\rho_{11}/\rho_n&\rho_{12}/\rho_n\cr
\rho_{21}/\rho_p&(\rho_{22}/\rho_p)+(m_e/m_p)\cr}
\right)^{-1}.$$ The functions $\chi_1(\alpha)$, $\chi_2(\alpha)$, $\chi_3(\alpha)$, and $\chi_0(\alpha)$ are defined as $$\chi_1(\alpha)=\alpha+a{d\alpha\over da}, \quad
\chi_2(\alpha)=2\alpha+a{d\alpha\over da}, \quad
\chi_3(\alpha)=3\alpha+a{d\alpha\over da},$$ and $$\chi_0(\alpha)=-a{d\alpha\over da}+a{d\over da}\left(a{d\alpha\over da}\right).$$ The matrices $E_0$, $E_1$, $F_0$, $F_1$, $I$, $\tilde J$, $\tilde J^+$, $\tilde K$, $\tilde K^+$, $\tilde L_0$, $\tilde L_0^+$, $\tilde L_1$, $\tilde L_1^+$, $\tilde M_0$, $\tilde M_0^+$, $\tilde M_1$, and $\tilde M_1^+$ are defined as $$E_0=\left(1+2\chi_1(\alpha)\right)\pmb{1}+2\chi_1(\beta)A_0, \quad
E_1=(1+\chi_2(\alpha))\pmb{1}+\chi_2(\beta)A_0,$$ $$F_0=\nu\Lambda_0^{-1}\left[(1+2\alpha+2\beta)\pmb{1}+12\beta A_0\right], \quad
F_1=\nu\Lambda_1^{-1}\left[(1+2\alpha+2\beta)\pmb{1}+12\beta A_1\right],$$ $$I=\pmb{1}-2\chi_1(\alpha)\pmb{1}-2\chi_1(\beta)A_0,$$ $$\tilde J=\tilde J^+-3\beta\Lambda_0^{-1}(2A_0+B_0),$$ $$\tilde J^+=m\nu\Lambda_0^{-1}\left[(1+\chi_2(\alpha))\pmb{1}+\chi_2(\beta)A_0\right],$$ $$\tilde K =\tilde K^+-3m\beta\Lambda_1^{-1}Q_1,$$ $$\tilde K^+
=\nu\Lambda_1^{-1}\left[((1+\chi_2(\alpha))\pmb{1}+\chi_2(\beta)A_1)C_1
+2((1+\chi_2(\alpha)-\chi_2(\beta))\pmb{1}+2\chi_2(\beta)A_1)Q_1\right],$$ $$\tilde L_0=(1+2\alpha)\pmb{1}-mF_0+\beta\Lambda_0^{-1}(2A_0\Lambda_0+6B_0),$$ $$\tilde L_1=(1+2\alpha)\pmb{1}-mF_1+\beta\Lambda_1^{-1}(2A_1\Lambda_1+6B_1),$$ $$\tilde M_0=\tilde M_0^+-6m\beta\Lambda_1^{-1}Q_1,$$ $$\tilde M_0^+=\nu\Lambda_1^{-1}
[(1+2\alpha-2\beta)\pmb{1}+4\beta A_1]Q_1\Lambda_0+F_1C_1, \quad$$ $$\tilde M_1=\tilde M_1^+-6m\beta\Lambda_0^{-1}Q_0,$$ $$\tilde M_1^+=\nu\Lambda_0^{-1}
[(1+2\alpha-2\beta)\pmb{1}+4\beta A_0]Q_0\Lambda_1+F_0C_0,$$ where $\pmb{1}$ denotes the unit matrix, and the matrices $A_0$, $A_1$, $B_0$, $B_1$, $C_0$, $C_1$, $Q_0$, $Q_1$, $\Lambda_0$, and $\Lambda_1$ as well as the quantities $U$ and $c_1$ are defined in Yoshida & Lee (2000a).
The oscillation equations in the superfluid core are given as a set of linear ordinary differential equations for the variables $\pmb{y}_j$ with $j=1$ to 4, which are obtained by eliminating the vectors $\pmb{h}$ and $i\pmb{t}$ in equations (A3) to (A8) using the algebraic equations (A9) to (A12).
[cccccccc]{} $r^o$& 0 & 0.6667 & 0.408 & $3.31$ & $3.83\times 10^2$ & $-7.55\times10^7$ & $-\infty$\
$r^s$& & 0.6667 & 0.511 & $2.12\times10^4$ & $2.18\times 10^6$ & $-2.99 \times 10^6$ & $-\infty$\
$r^o$& 0.02 & 0.6667 & 0.408 & $3.31$ & $3.83\times 10^2$ & $-6.76\times10^7$ & $-2.52\times10^2$\
$r^s$ & & 0.8532 & 0.652 & $4.76\times10^{8}$ & $1.21\times 10^6$ & $-1.99 \times 10^6$ & $-2.94\times 10^{-1}$\
$r^o$&0.04 & 0.6667 & 0.408 & $3.31$ & $3.83\times 10^2$ & $-6.76\times10^7$ & $-1.70\times10^3$\
$r^s$& & 1.0408 & 0.787 & $7.89\times10^{8}$ & $2.55\times 10^6$ & $-1.14 \times 10^6$ & $-7.99\times 10^{-2}$\
$r^o$&0.06 & 0.6667 & 0.408 & $3.31$ & $3.83\times 10^2$ & $-6.76\times10^7$ & $-2.62\times10^3$\
$r^s$ && 1.2296 & 0.821 & $3.64\times10^{9}$ & $1.03\times 10^7$ & $-4.67 \times 10^4$ & $-5.20\times 10^{-2}$\
\[coefficients0\]
[ccccc]{} 2 & 1.0978 & 1.1190\
& -0.5683 & -0.5109\
3 & 1.3564 & 1.3869\
& 0.5180 & 0.5077\
& -1.0293 & -0.9723\
4 & 1.5191 & 1.5526\
& 0.8639 & 0.8612\
& -0.2738 & -0.2417\
& -1.2738 & -1.2238\
5 & 1.6273 & 1.6600\
& 1.1046 & 1.1134\
& 0.4215 & 0.4060\
& -0.7028 & -0.6524\
& -1.4340 & -1.3968\
\[coefficients0\]
[ccccccc]{} $r^o$ & 0.6667 & 0.408 & $3.31$ & $3.83\times 10^2$ & $-7.49\times10^7$ & $-9.10\times10^2$\
$r^s$ & 0.6667 & 0.529 & $4.33\times10^4$ & $2.63\times 10^6$ & $-1.13 \times 10^6$ & $-7.30\times10^{-2}$\
\[coefficients0\]
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|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'In our previous work [@Morita:2017oev], we pointed out that various multi-cut solutions exist in the Chern-Simons (CS) matrix models at large-$N$ due to a curious structure of the saddle point equations. In the ABJM matrix model, these multi-cut solutions might be regarded as the condensations of the D2-brane instantons. However many of these multi-cut solutions including the ones corresponding to the condensations of the D2-brane instantons were obtained numerically only. In the current work, we propose an ansatz for the multi-cut solutions which may allow us to derive the analytic expressions for all these solutions. As a demonstration, we derive several novel analytic solutions in the pure CS matrix model and the ABJM matrix model. We also develop the argument for the connection to the instantons.'
bibliography:
- 'CS2.bib'
---
[Toward the construction of the general multi-cut solutions in Chern-Simons Matrix Models ]{}
Takeshi [Morita]{}$^{a,b}$[^1], Kento S[ugiyama]{}$^{b}$[^2]
[*a. Department of Physics, Shizuoka University\
836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan\
b. Graduate School of Science and Technology, Shizuoka University\
836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan* ]{}
Introduction
============
The $1/N$ expansion [@Brezin1978] is a quite powerful technique in matrix models, and it makes us possible to analyze the models in the non-perturbative regime. Not only that, in string theories, this expansion may correspond to the perturbative expansion of the string coupling [@Brezin:1990rb; @Douglas:1989ve; @Gross:1989vs; @Gross:1989aw; @Banks:1996vh; @Ishibashi:1996xs; @Dijkgraaf:1997vv; @Berenstein:2002jq], and it might play important role to reveal quantum gravity. Particularly, in the last decade, the analysis of the large-$N$ Chern-Simons (CS) matrix models has been developed quite remarkably. (See [@Hatsuda:2015gca; @Marino:2016new] for reviews.) The CS matrix models are obtained via the localization of the three dimensional supersymmetric CS matter theories on a sphere [@Pestun:2007rz; @Kapustin:2009kz; @Kapustin:2010xq; @Jafferis:2010un; @Hama:2010av], which describe the low energy dynamics of the superstring theories and M-theory, and, through these developments, various non-perturbative aspects of the string theories have been revealed including the derivation of the $N^{3/2}$ factor [@Drukker:2010nc] of the free energy in the $N$ M2-brane theory [@Klebanov:1996un; @Morita:2013wla; @Morita:2014ypa]. These results provide us quite strong evidences for the AdS/CFT correspondence [@Maldacena:1997re; @Itzhaki:1998dd; @Aharony:2008ug].
In this article, we mainly investigate the $U(N)$ pure CS matrix model [@Kapustin:2009kz; @Marino:2002fk; @Aganagic:2002wv] among the various CS matrix models, since other models can be regarded as the variations of this model and we can expect that the application to these other models might be straightforward.
The partition function of the pure CS matrix model is given by $$Z(k,N)
=\frac{1}{N!} \int \prod_{i=1}^N \frac{du_i}{2\pi} e^{-\frac{N}{4\pi i \lambda} \sum_i u^2_i} \prod_{i<j}^N \Bigl[ 2 \sinh{\frac{u_i-u_j}{2}} \Bigr]^2.
\label{partition-CS}$$ Here $k$ is the CS level and $\lambda := N/k$ is the ’t Hooft coupling, and we will consider the ’t Hooft limit ($N \rightarrow \infty$, $\lambda$: fixed) of this model. This partition function resembles the Gaussian Hermitian matrix model. The difference appears only in the Vandermonde determinant, but this simple difference provides quite rich structures in the CS matrix models.
It is known that we can compute this partition function exactly at arbitrary $\lambda$ and $N$ [@Kapustin:2009kz; @Tierz:2002jj]. However, the ’t Hooft expansion of this model shows non-trivial properties and it is still valuable to investigate them [@Pasquetti:2009jg; @Hatsuda:2015owa; @Honda:2016vmv; @Honda:2017qdb; @Chattopadhyay:2017ckc]. This is similar to the situations of the Gaussian matrix model and the Gross-Witten-Wadia model [@PhysRevD.21.446; @Wadia:2012fr] which show non-trivial behaviors at large-$N$ [@Buividovich:2015oju; @Alvarez:2016rmo; @Okuyama:2017pil], although we can calculate the partition functions exactly.
When we take the ’t Hooft limit, we can employ the saddle point approximation. The saddle point equation of the partition function (\[partition-CS\]) with respect to $u_i$ is given by $$u_i=\frac{2\pi i \lambda}{N} \sum_{j \neq i}^N \coth{\frac{u_i-u_j}{2}}, \qquad (i=1,\cdots, N).
\label{eom-CS}$$ The exact solution of this equation at finite $\lambda$ which is characterized by a single cut of the eigenvalue distribution is known [@Aganagic:2002wv; @Pasquetti:2009jg; @Halmagyi:2003ze]. This solution would be thermodynamically stable, since the free energy agrees with that of the $N \to \infty$ limit of the exact finite $N$ result [@Kapustin:2009kz; @Tierz:2002jj].
[cc]{}
![ Eigenvalue distributions of the pure CS matrix model. We numerically solve the saddle point equation (\[eom-CS\]) via the Newton method. []{data-label="fig-numerical-CS"}](one-cut.eps "fig:")\
One-cut solution
![ Eigenvalue distributions of the pure CS matrix model. We numerically solve the saddle point equation (\[eom-CS\]) via the Newton method. []{data-label="fig-numerical-CS"}](two-cut.eps "fig:")\
Stepwise two-cut solution
\
![ Eigenvalue distributions of the pure CS matrix model. We numerically solve the saddle point equation (\[eom-CS\]) via the Newton method. []{data-label="fig-numerical-CS"}](multi-cut.eps "fig:")\
Stepwise multi-cut solution
![ Eigenvalue distributions of the pure CS matrix model. We numerically solve the saddle point equation (\[eom-CS\]) via the Newton method. []{data-label="fig-numerical-CS"}](one+two.eps "fig:")\
Composition of the one-cut and stepwise two-cut solution
Then a question is whether this one-cut solution is unique or not. Surprisingly it turned out that an infinite number of solutions are allowed in the saddle point equation (\[eom-CS\]), which are characterized by the various multi-cuts [@Morita:2017oev; @Morita:2011cs][^3]. See Figure \[fig-numerical-CS\]. These solutions were first found by solving the saddle point equations numerically through the Newton method [@Morita:2017oev; @Morita:2011cs], which have been employed in [@Herzog:2010hf; @Niarchos:2011sn; @Minwalla:2011ma]. Later analytic expressions for some of the multi-cut solutions have been found by Ref. [@Morita:2017oev].
The purpose of this article is to develop the studies of Ref. [@Morita:2017oev], and provides analytic methods to treat all of these multi-cut solutions. We will show that an integral formula (\[composite-CS\]) for the resolvent related to the method of Migdal [@Migdal:1984gj] is quite useful. By solving this integral formula either analytically or numerically, we will demonstrate that the eigenvalue distributions of the multi-cut solutions obtained through the Newton method shown in Figure \[fig-numerical-CS\] can be reproduced. All types of the Newton method solutions as far as we find may be explained by our formula, and we presume that our method might be applicable to derive all possible solutions of the saddle point equation (\[eom-CS\]). (Hence the formula (\[composite-CS\]) is the main result of this article.)
The obtained analytic results tell us curious properties of the multi-cut solutions. We will see that there are two types of the multi-cuts in the pure CS matrix model. One is the cuts which are separated by a multiple of $2 \pi i$. We refer to such cuts as “stepwise multi-cuts" in this article. (See Figure \[fig-numerical-CS\].) Another type of the multi-cuts is the composition of the stepwise multi-cut and one-cut (or another stepwise multi-cuts). We refer to them as “composite type". We will show that, as the number of the composite type cuts increases, the genus of the resolvent increases similar to the multi-cut solutions in the ordinary matrix models. (Hence we need the higher genus generalizations of elliptic functions to describe the composite type multi-cuts.) On the other hand, the stepwise multi-cuts do not change the genus[^4], while they cause additional logarithmic singularities at the end points of each step in the resolvent. These properties might capture the geometrical natures of the multi-cut solutions.
We also discuss that our methods will work in other CS matrix models, and we propose a similar integral formula (\[general-ABJM\]) for the resolvent of the ABJM matrix model as an example. By using this formula, we will derive novel analytic solutions of the saddle point equation of the ABJM matrix model.
Generally the various multi-cut solutions in a matrix model may describe the different vacua of the system, and these vacua would affect the perturbative vacuum through the instanton effects [@David:1990sk; @David:1992za]. Indeed the connection between the multi-cut solutions and the D2-brane instantons in the ABJM matrix model [@Drukker:2011zy; @Grassi:2014cla] was conjectured in Ref. [@Morita:2017oev]. We will develop this discussion and show a quantitative evidence for this connection. Besides we comment on the relation to the membrane instanton in the pure CS matrix model [@Pasquetti:2009jg; @Hatsuda:2015owa].\
The organization of this article is as follows. In section \[sec-pCS\], we show the derivation of the multi-cut solutions in the pure CS matrix model. In section \[sec-ABJM\], we argue the multi-cut solutions in the ABJM matrix model. We also consider the connection to the D2-brane instantons. We conclude in section \[sec-discussion\] with some future directions. In appendix \[app-holomorphy\], we introduce the derivation of the multi-cut solution via holomorphy in the pure CS matrix model. This derivation is more powerful than the integral formula (\[composite-CS\]) in certain situations. In appendix \[app-negative-n\], we discuss the issue of “negative steps".
Multi-cut solutions in the pure CS matrix model {#sec-pCS}
===============================================
In this section, we will propose the integral formula (\[composite-CS\]) which provides us a method to derive possible general solutions of the saddle point equation (\[eom-CS\]) including the various multi-cut solutions shown in Figure \[fig-numerical-CS\]. Since the general solution will be characterized by a bit complicated multi-cuts sketched in Figure \[fig-image-CSgeneral\], we will first explain the derivations of several simpler multi-cut solutions which will give us insights about the general solution.
In order to derive the multi-cut solutions, we will employ the resolvent. It is convenient to introduce new variables $U_i := \exp{\left( u_i \right)}$ and rewrite the saddle point equation (\[eom-CS\]) as[^5] $$\log{U_i}=\frac{2\pi i \lambda}{N} \sum_{j \neq i}^N \frac{U_i+U_j}{U_i-U_j}, \qquad (i=1,\cdots, N).
\label{eom-CS2}$$ Following [@Marino:2011nm; @Suyama:2016nap], we define the eigenvalue density $\rho(Z)$ and resolvent $v(Z)$ $$\rho(Z) := \frac{1}{N} \sum_{i=1}^N \delta (Z-U_i),
\qquad
v(Z) := \int_\mathcal{C} dW \rho(W) \frac{Z+W}{Z-W} ,
\label{resolvent-CS}$$ where $\mathcal{C}$ is the support of $\rho(Z)$. Then the saddle point equation (\[eom-CS2\]) becomes $$V'(Z)=\lim_{\epsilon \rightarrow 0} \left[ v(Z+i\epsilon)+v(Z-i\epsilon) \right],
\qquad
(Z \in \mathcal{C}),
\qquad
V'(Z):=\frac{1}{\pi i\lambda} \log{Z}.
\label{eom-CS3}$$ Besides, the resolvent satisfies the boundary conditions $$\lim_{Z \rightarrow \infty} v(Z) = 1, \qquad \lim_{Z \rightarrow 0}v(Z) = -1,
\label{boundary-CS}$$ through the definition (\[resolvent-CS\]). By using the resolvent, the eigenvalue density is described as $$\rho(Z)=-\frac{1}{4\pi iZ} \lim_{\epsilon \rightarrow 0} \left[ v(Z+i\epsilon)-v(Z-i\epsilon) \right],
\qquad
(Z \in \mathcal{C}).
\label{rho-CS}$$ In the following subsections, we will explore the solutions of the equation (\[eom-CS3\]) which obey the boundary conditions (\[boundary-CS\]).
One-cut solution
----------------
We review the derivation of the resolvent describing the one-cut solution shown in Figure \[fig-numerical-CS\] (top-left) [@Aganagic:2002wv; @Pasquetti:2009jg; @Halmagyi:2003ze]. There are various derivations of this solution, and we employ the integral method of Migdal [@Migdal:1984gj] which is useful for finding the general solution later.
The potential $V'(Z)$ (\[eom-CS3\]) has the unique extreme at $Z=1$ (or $z=0$ where $z:=\log Z$), and the eigenvalues tend to be around there. Hence we assume that $\rho(Z)$ has a single support on the interval $[A,B]$ near $Z=1$, where $A$ and $B$ ($|A| <|B|$) will be fixed soon. We apply the ansatz [@Migdal:1984gj] for the solution of the saddle point equation (\[eom-CS3\]) [@Halmagyi:2003ze], $$v(Z)=\oint_{C_1} \frac{dW}{4\pi i} \frac{V'(W)}{Z-W} \sqrt{\frac{(Z-A)(Z-B)}{(W-A)(W-B)}}.
\label{Migdal-CS}$$ Here the contour $C_1$ encircles the support $[A,B]$ counterclockwise[^6]. By performing this integral[^7], we obtain $$\begin{aligned}
&v(Z)=\frac{1}{\pi i\lambda} \log{\left( \frac{f(Z)-\sqrt{f^2(Z)-4Z} }{2} \right)}, \nonumber \\
&f(Z)=f_0+f_1Z, \quad
f_0=\frac{2\sqrt{AB}}{\sqrt{A}+\sqrt{B}},
\quad
f_1=\frac{2}{\sqrt{A}+\sqrt{B}}.
\label{one-cut-CS}\end{aligned}$$ Then, through the boundary conditions (\[boundary-CS\]), $A$ and $B$ are determined as $$A= \exp \left(-2\, {\rm arccosh} \left( e^{\pi i\lambda} \right) \right),
\qquad
B= \exp \left( 2\, {\rm arccosh} \left( e^{\pi i\lambda} \right) \right)=1/A.
\label{CS-AB}$$ The eigenvalue density is obtained through (\[rho-CS\]), $$\rho(Z)=\frac{1}{4\pi^2 \lambda Z} \log{\left( \frac{Z+\sqrt{AB}-i \sqrt{(Z-A)(Z-B)}}{Z+\sqrt{AB}+i \sqrt{(Z-A)(Z-B)}} \right)},
\qquad
(Z \in [A,B]).
\label{density-CS}$$ We sketch the profile of this density in Figure \[fig:CS1\]. In order to compare the obtained result with the numerical result shown in Figure \[fig-numerical-CS\], we rewrite our results by using the variable $z=\log Z$ which corresponds to $u_i$ in (\[eom-CS\]). Correspondingly, $A$ and $B$ are mapped to $$b = \log{B}=2\, {\rm arccosh} \left( e^{\pi i\lambda} \right),
\qquad
a= \log{A}=-2\, {\rm arccosh} \left( e^{\pi i\lambda} \right),
\label{endpt-CS}$$ and they satisfy $a=-b$. (This is expected, since the system is symmetric under $z \to -z$.) See Figure \[fig:CS1\]. This solution describes the numerically obtained one-cut solution shown in Figure \[fig-numerical-CS\].
[cc]{}
![Schematic plot of the the eigenvalue density $\rho(Z)$ of the one-cut solution (\[density-CS\]) and the eigenvalue distribution on the $z$-plane. $A,B$ and $a,b$ are given in (\[endpt-CS\]). []{data-label="fig:CS1"}](density-one-cut.eps)
![Schematic plot of the the eigenvalue density $\rho(Z)$ of the one-cut solution (\[density-CS\]) and the eigenvalue distribution on the $z$-plane. $A,B$ and $a,b$ are given in (\[endpt-CS\]). []{data-label="fig:CS1"}](image-one-cut.eps)
Note that the resolvent in the $z$ variable has the branch cuts on $z \in [a+2\pi in , b+2\pi in]$, $(n \in \mathbb{Z})$, although the eigenvalues are distributed on $z \in [a , b]$ only. These additional infinite number of the cuts are related to the periodicity $u_i \to u_i + 2\pi i$ of the right hand side of the saddle point equation (\[eom-CS\]), and the equation of motion (\[eom-CS3\]) is not satisfied there. In this article, we refer to the solutions in which $k$ mobs of the eigenvalues exist in the $z$ plane as “$k$-cut solution", and do not count these additional cuts as “cuts”.
Stepwise two-cut solution {#sec-2-cut-CS}
-------------------------
We consider the derivation of the stepwise two-cut solution [@Morita:2017oev] plotted in Figure \[fig-numerical-CS\] (top-right).
Since the potential $V'(Z)$ (\[eom-CS3\]) has only the single extreme at $Z=1$, it might be difficult to imagine that the saddle point equation (\[eom-CS3\]) allows such a two-cut solution. The key is the periodicity $u_i \to u_i+2 \pi i$ of the right hand side of the saddle point equation (\[eom-CS\]). Thanks to this periodicity, strong interactions between the eigenvalues arise if they are separated by $2\pi i$, and these interactions make the various solutions shown in Figure \[fig-numerical-CS\] possible.
Before considering the $N=\infty$ case, we study the $N=2$ case as an example [@Morita:2017oev; @Morita:2011cs]. In this case, the saddle point equation (\[eom-CS\]) become $$\frac{u_1}{2\pi i \lambda}=\frac{1}{2} \coth{\frac{u_1-u_2}{2}}, \qquad
\frac{u_2}{2\pi i \lambda}=-\frac{1}{2} \coth{\frac{u_1-u_2}{2}}.
\label{N=2}$$ By summing these two equations, we find $u_1=-u_2$, and the equations reduce to $$\frac{u_1}{2\pi i \lambda}=\frac{1}{2} \coth u_1.
\label{N=2'}$$ This equation indeed allows infinite number of solutions. At weak coupling $|\lambda| \ll 1$, we can perturbatively obtain the solutions, $$\begin{aligned}
u_1= \pm \sqrt{\pi i \lambda} + \cdots ,\qquad
u_1= \pi i n + \frac{\lambda}{n} + \cdots ,
\label{N=2-multi}\end{aligned}$$ where $n$ is a non-zero integer. The first solution would correspond to the one-cut solution (\[density-CS\]) at large-$N$, while the second one indicates the existence of a new class of the solutions. Particularly the second solution satisfies $u_1-u_2= 2 \pi in + O(\lambda)$, and they are separated by $2\pi in$. Thus, the periodicity of the right hand side of the saddle point equation (\[eom-CS\]) causes the various solutions as we expected.
[cc]{}
![(Left) Sketch of the stepwise two-cut solution in the pure CS matrix model. The red lines describe the eigenvalue distributions. (Right) Integral contours in the integral of the resolvent. The doted lines denote the branch cuts of the integrand. The blue lines are the integral contour $C_1$ and $C_2$ in (\[Migdal-CS2\]). The green line denotes the contour $C$ in (\[Migdal-CS2”\]). []{data-label="fig-two-CS"}](image-two-cut.eps)
![(Left) Sketch of the stepwise two-cut solution in the pure CS matrix model. The red lines describe the eigenvalue distributions. (Right) Integral contours in the integral of the resolvent. The doted lines denote the branch cuts of the integrand. The blue lines are the integral contour $C_1$ and $C_2$ in (\[Migdal-CS2\]). The green line denotes the contour $C$ in (\[Migdal-CS2”\]). []{data-label="fig-two-CS"}](cycle.eps)
Let us move on the $N=\infty$ case. To find the two-cut solution corresponding to the numerical result shown in Figure \[fig-numerical-CS\], we assume the two branch cuts $[a_i,b_i]$ where ${\rm Re}(a_i) \le {\rm Re} (b_i) $ and ${\rm Im}(a_i) \le {\rm Im} (b_i) $ $(i=1,2)$ on the $z$-plane ($z= \log Z$) satisfying[^8] $$a_2=b_1+2\pi in.
\label{CS2ansatz}$$ Here $n$ is a positive integer. (We will argue why we restrict $n$ positive in Appendix \[app-negative-n\].) We also assume that the first cut and the second cut consist of $N_1$ and $N_2(=N-N_1)$ eigenvalues, respectively. See Figure \[fig-two-CS\]. Since the branch cuts are always separated by the fixed number $2\pi i n$, we call this solution “stepwise two-cut solution".
On the $Z$-plane, these cuts are mapped to $A_i=e^{a_i}$ and $B_i=e^{b_i}$ and they satisfy $$\begin{aligned}
A_2=e^{2\pi in} B_1,
\label{CS-step}\end{aligned}$$ through (\[CS2ansatz\]). We assign a new symbol $D_1:=B_1$ for this point, since the properties of this point are different from $A_1$ and $B_2$ as we will see soon. Note that, because of the branch cut of $\log Z$ in $V'(Z)$ (\[eom-CS3\]), $A_2$ and $B_1$ stand different points on the Riemann surface. See the right sketch of Figure \[fig-two-CS\].
By regarding the locations of these branch cuts, we propose the ansatz for the resolvent of the stepwise two-cut solution $$v(Z)=\oint_{C_1 \cup \, C_2} \frac{dW}{4\pi i} \frac{V'(W)}{Z-W} \sqrt{\frac{(Z-A_1)(Z-B_2)}{(W-A_1)(W-B_2)}},
\qquad
V'(Z)=\frac{1}{\pi i\lambda} \log{Z}.
\label{Migdal-CS2}$$ Here the integral contour $C_1$ and $C_2$ encircle the branch cut $[A_1,B_1]$ and $[A_2,B_2]$ counterclockwise, respectively, and they are on the different sheets as shown in Figure \[fig-two-CS\]. It is not difficult to show that this ansatz satisfies the saddle point equation (\[eom-CS3\]) on the cut $[A_1,B_1]$ and $[A_2,B_2]$ [^9].
Now we evaluate the integral in (\[Migdal-CS2\]). Since the integrand involves $\log W$ in $V'(W)$, we need to take care of the branch cut. We assume that $C_1$ is on the $n_0$-th sheet[^10]. (It implies $C_2$ is on the $n+n_0$-th sheet through the ansatz (\[CS2ansatz\]).) Then we can evaluate the integral (\[Migdal-CS2\]) as $$\begin{aligned}
v(Z)=&\oint_{C} \frac{dW}{4\pi i} \frac{1}{\pi i \lambda} \frac{\log W}{Z-W} \sqrt{\frac{(Z-A_1)(Z-B_2)}{(W-A_1)(W-B_2)}} +
\oint_{C_1} \frac{dW}{4\pi i} \frac{2n_0}{ \lambda} \frac{1}{Z-W} \sqrt{\frac{(Z-A_1)(Z-B_2)}{(W-A_1)(W-B_2)}}
\nonumber \\
&
+
\oint_{C_2} \frac{dW}{4\pi i} \frac{2(n+n_0)}{ \lambda} \frac{1}{Z-W} \sqrt{\frac{(Z-A_1)(Z-B_2)}{(W-A_1)(W-B_2)}}
\label{Migdal-CS2''}.\end{aligned}$$ Here the contour $C$ encircles the branch cut $[A_1, B_2]$ on the 0-th sheet. See Figure \[fig-two-CS\]. The first integral is identical to (\[Migdal-CS\]) and the second and third integrals have been done in [@Morita:2017oev], and we obtain [^11] $$\begin{aligned}
v(Z)=&\frac{1}{\pi i\lambda} \log{\left( \frac{f(Z)-\sqrt{f^2(Z)-4Z}}{2} \right)}
+\frac{n}{\pi i\lambda} \log{\left( \frac{q(Z)+\sqrt{q^2(Z)-4}}{2} \right)}+\frac{n_0}{\lambda},
\nonumber \\
&f(Z)=f_0+f_1 Z, \qquad\quad
f_0=\frac{2\sqrt{A_1B_2}}{\sqrt{A_1}+\sqrt{B_2}},
\quad
f_1=\frac{2}{\sqrt{A_1}+\sqrt{B_2}}, \nonumber
\\
&q(Z)=\frac{q_1Z-q_0D_1}{Z-D_1},
\qquad
q_1=\frac{2(2D_1-A_1-B_2)}{B_2-A_1},
\quad
q_0=\frac{2(D_1B_2+D_1A_1-2A_1B_2)}{D_1(B_2-A_1)}.
\label{two-cut-CS}\end{aligned}$$ This result agrees with that of Ref. [@Morita:2017oev] which employs a different method[^12]. (In Appendix \[app-holomorphy\], we show how the resolvent (\[two-cut-CS\]) satisfies the saddle point equation (\[eom-CS3\]). There, we also argue another derivation of this solution via holomorphy.) From (\[rho-CS\]), the eigenvalue density becomes $$\begin{aligned}
\rho(Z) = & \frac{1}{4\pi^2 \lambda Z} \log{\left( \frac{Z+\sqrt{A_1B_2}-i \sqrt{(Z-A_1)(Z-B_2)}}{Z+\sqrt{A_1B_2}+i \sqrt{(Z-A_1)(Z-B_2)}} \right)} \\
&+ \frac{n}{\pi^2 \lambda Z}
\left \{
\begin{array}{l}
{\rm arctanh} \left( \sqrt{\dfrac{Z-A_1}{B_2-Z}} \sqrt{\dfrac{B_2-D_1}{D_1-A_1}} \right),
\qquad
(Z \in [A_1,B_1]),
\\
-{\rm arctanh} \left( \sqrt{\dfrac{B_2-Z}{Z-A_1}} \sqrt{\dfrac{D_1-A_1}{B_2-D_1}} \right),
\qquad
(Z \in [A_2,B_2]).
\end{array}
\right.
\label{rho-CS2}\end{aligned}$$ The profile of this density at a small $\lambda$ is shown in Figure \[fig-density-two-CS\]. Particularly a logarithmic divergence at $Z=D_1$ arises from the second term due to the pole of $q(Z)$, although the integral of $\rho(Z)$ is finite [@Morita:2017oev]. The existence of the divergence is quite contrast to the one-cut solution shown in Figure \[fig:CS1\].
[cc]{}
![(Left) Schematic plot of the eigenvalue density for the stepwise two-cut solution (\[rho-CS2\]) at a weak coupling. Here we have projected the cut $[A_2,B_2]$ to the same sheet to $[A_1,B_1]$. A logarithmic singularity appears at $Z=D_1(=B_1)$. (Right) Comparison of the Newton method (blue dots) and the result through (\[two-cut-CS\]). For the Newton method, we take $N_1=70$, $N_2=30$, $n=1$ and $\lambda=0.5$. These two results agree very well. []{data-label="fig-density-two-CS"}](density-two-cut.eps "fig:")\
![(Left) Schematic plot of the eigenvalue density for the stepwise two-cut solution (\[rho-CS2\]) at a weak coupling. Here we have projected the cut $[A_2,B_2]$ to the same sheet to $[A_1,B_1]$. A logarithmic singularity appears at $Z=D_1(=B_1)$. (Right) Comparison of the Newton method (blue dots) and the result through (\[two-cut-CS\]). For the Newton method, we take $N_1=70$, $N_2=30$, $n=1$ and $\lambda=0.5$. These two results agree very well. []{data-label="fig-density-two-CS"}](endpt-analysis-two-cut.eps "fig:")\
Finally we have to fix the values of the undetermined constant $A_1,B_1(=D_1)$ and $B_2$. We impose the two boundary conditions at $Z=0$ and $Z=\infty$ (\[boundary-CS\]) and the additional condition[^13] $$\frac{N_1}{N}=\int^{B_1}_{A_1} \rho(Z) dZ
=\frac{1}{4\pi i} \oint_{C_1} \frac{v(Z)}{Z} dZ,
\label{cycle-CS2}$$ which demands that the $N_1$ eigenvalues are on the first cut $[A_1,B_1]$. Thus $A_1,B_1$ and $B_2$ should be determined as the solution of these three equations and they are given as functions of the input parameters: $\lambda$, $n$ and $N_1/N$. ($n_0$ is determined when $A_1$ is fixed.)
However finding the solution for the general input parameters is difficult. Also it is hard to answer whether the solution exists or not for the given parameters, and, even if it exists, whether it is unique or not. In addition, even if we found a solution, if the eigenvalue density $\rho(Z)$ is not positive, the solution is not allowed. For example, the one-cut solution (\[one-cut-CS\]) is allowed only when $-1 \le \lambda \le 1$, if $\lambda$ is real [@Morita:2011cs].
Some solvable cases were explored in [@Morita:2017oev]. For example, at weak coupling $|\lambda| \ll 1$, the solution is uniquely given by $$\begin{aligned}
a_1 = &~ b_1-\frac{2\pi \lambda}{n} \tan{\left( \frac{\pi}{2} \frac{N_1}{N} \right)}+O(\lambda^2), \qquad
b_2 = a_2+\frac{2\pi \lambda}{n} \tan{\left( \frac{\pi}{2} \frac{N_2}{N} \right)}
+O(\lambda^2),
\nonumber \\
b_1 = &~ d_1=a_2 -2\pi i n= - \frac{2\pi i n N_2}{N}
+\frac{\lambda \pi }{2n}
\left( \tan{\left( \frac{\pi}{2} \frac{N_1}{N} \right)}-\tan{\left( \frac{\pi}{2} \frac{N_2}{N} \right)} \right) +O(\lambda^2).
\label{weak-ab}\end{aligned}$$ Here $d_1:=\log D_1$. In this case, the cuts are parallel to the real axis if $\lambda$ is real.
In the case of a finite $\lambda$, we can find the solution if $N_1=N_2$ and $n=1,2,3$ and 4. In the case of $n=1$, the solution is given by $$\begin{aligned}
B_2=1/A_1= e^{\pi i}\left( -i (e^{\pi i \lambda}-1)+\sqrt{2 e^{\pi i \lambda}-e^{2\pi i \lambda}}\right)^2, \qquad A_2=e^{2\pi i }B_1= e^{\pi i }.\end{aligned}$$
Also we can find the solutions by solving (\[boundary-CS\]) and (\[cycle-CS2\]) numerically. For example, when we take $\{\lambda,n,N_1/N \}=\{0.5,1,0.7\}$, we obtain a solution as shown in Figure \[fig-density-two-CS\][^14]. The result agrees with the numerical result derived through the Newton method in which we solve the saddle point equation (\[eom-CS\]) at finite $N$ directly [@Herzog:2010hf; @Niarchos:2011sn; @Minwalla:2011ma][^15].
Stepwise multi-cut solution
---------------------------
[cc]{}
![Sketch of the eigenvalue distribution and the eigenvalue density of the stepwise multi-cut solution. In the eigenvalue density, we project the cuts on the different sheet to the same sheet and consider a small $\lambda$.[]{data-label="fig-g-CS"}](image-multi-cut.eps)
![Sketch of the eigenvalue distribution and the eigenvalue density of the stepwise multi-cut solution. In the eigenvalue density, we project the cuts on the different sheet to the same sheet and consider a small $\lambda$.[]{data-label="fig-g-CS"}](density-multi-cut.eps)
We develop the derivation of the stepwise two-cut solution in the previous section and consider the stepwise multi-cut solution in Figure \[fig-numerical-CS\] (bottom-left). For a stepwise $l$-cut solution, there would be cuts $[a_j, b_j]$ on the $z$-plane which satisfy ${\rm Re}(a_j) \le {\rm Re} (b_j) $ and ${\rm Im}(a_j) \le {\rm Im} (b_j) $, $(j=1, \cdots, l)$. We assume that the $j$-th cut consists of $N_j$ eigenvalues ($\sum_{j=1}^l N_j=N$). Similar to the stepwise two-cut solution, we impose that the end points of these cuts satisfy $$a_{j+1}=b_j+2\pi i n_j,
\qquad
(j=1, \cdots, l-1),
\label{CSgansatz}$$ where $\{ n_j \}$ are positive integers. See Figure \[fig-g-CS\]. In terms of the $Z$ variable, this assumption implies the cut $[A_j, B_j]$ satisfying $A_{j+1}=e^{2\pi i n_j} B_j$. Then, by generalizing (\[Migdal-CS2\]) in the stepwise two-cut solution, we use the following ansatz for the resolvent $$v(Z)= \sum_{j=1}^l \oint_{ C_j} \frac{dW}{4\pi i} \frac{V'(W)}{Z-W} \sqrt{\frac{(Z-A_1)(Z-B_l)}{(W-A_1)(W-B_l)}},
\qquad
V'(Z)=\frac{1}{\pi i\lambda} \log{Z}.
\label{Migdal-CSg}$$ Here the integral contour $C_j$ encircle the branch cut $[A_j,B_j]$ counterclockwise. Again these contours are on the different sheets of $\log Z$, and we assume that the first contour $C_1$ is on the $n_0$-th sheet.
We perform the integral in (\[Migdal-CSg\]) through the similar calculations to the stepwise two-cut solution (\[two-cut-CS\]) and obtain the resolvent of the stepwise $l$-cut solution $$\begin{aligned}
&v(Z)=\frac{1}{\pi i\lambda} \log{\left( \frac{f(Z)-\sqrt{f^2(Z)-4Z}}{2} \right)}
+\sum_{j=1}^{l-1} \frac{n_j}{\pi i\lambda} \log{\left( \frac{q^{(j)}(Z)+\sqrt{\left( q^{(j)}(Z) \right)^2-4}}{2} \right)}+\frac{n_0}{\lambda}, \nonumber \\
&f(Z)=f_0+f_1 Z, \qquad\quad
f_0=\frac{2\sqrt{A_1B_l}}{\sqrt{A_1}+\sqrt{B_l}},
\quad
f_1=\frac{2}{\sqrt{A_1}+\sqrt{B_l}}, \nonumber
\\
&q^{(j)}(Z)=\frac{q_1^{(j)}Z-q_0^{(j)}D_j}{Z-D_j},
\quad
q_1^{(j)}=\frac{2(2D_j-A_1-B_l)}{B_l-A_1},
\quad
q_0^{(j)}=\frac{2(D_jB_l+D_jA_1-2A_1B_l)}{D_j(B_l-A_1)}.
\label{multi-cut-CS}\end{aligned}$$ Here we have defined $D_j:=B_j$ in order to emphasize the distinction between $B_l$ and other $B_j$’s. Again $q^{(j)}(Z)$ has a pole at $Z=D_j$. This pole causes a logarithmic singularity and we take the branch cut as sketched in Figure \[fig-endpt-analysis-three-cut\] so that the equation (\[eom-CS3\]) is satisfied correctly.
Then we obtain the eigenvalue density $$\begin{aligned}
\rho(Z) = & \frac{1}{4\pi^2 \lambda Z} \log{\left( \frac{Z+\sqrt{A_1B_l}-i \sqrt{(Z-A_1)(Z-B_l)}}{Z+\sqrt{A_1B_l}+i \sqrt{(Z-A_1)(Z-B_l)}} \right)} + \sum_{j=1}^{l-1} \rho_s^{(j)}(Z), \nonumber \\
&
\rho_{s}^{(j)}(Z)= \frac{n_j}{\pi^2 \lambda Z}
\left \{
\begin{array}{l}
{\rm arctanh} \left( \sqrt{\dfrac{Z-A_1}{B_l-Z}} \sqrt{\dfrac{B_l-D_j}{D_j-A_1}} \right),
\quad
\left( Z \in \underset{(k=1, \cdots, j)}{[A_k,B_k]} \right),
\\
-{\rm arctanh} \left( \sqrt{\dfrac{B_l-Z}{Z-A_1}} \sqrt{\dfrac{D_j-A_1}{B_l-D_j}} \right),
\quad
\left( \underset{(k=j+1, \cdots, l-1)}{Z \in [A_k,B_k]} \right).
\end{array}
\right.
\label{rho-general}\end{aligned}$$ Again it shows the logarithmic divergence at each $D_j$, ($j=1,\cdots, l-1$). We sketch the profile in Figure \[fig-g-CS\].
[cc]{}
![ (Left) Branch cuts of the resolvent of the stepwise three-cut solution (\[multi-cut-CS\]) on the $Z$-plane. The solid lines denote the branch cuts of the square root: $[A_1,B_1]$, $[A_2,B_2]$ and $[A_3,B_3]$. The broken lines are the branch cuts of $\log$ which lie on the second sheet of the square root. The logarithmic branch cut starting from $B_1(=D_1)$ on the first sheet immediately goes to the second sheet and terminates at $B_1$ on the second sheet. (Right) Eigenvalue distributions through the Newton method (blue dots) and our method (red dots) for the stepwise three-cut solution at $N_1/N=N_2/N=N_3/N=1$, $n_1=n_2=1$, $\lambda=0.3$. We take $N=60$ in the Newton method. []{data-label="fig-endpt-analysis-three-cut"}](fig-3cut.eps "fig:")\
![ (Left) Branch cuts of the resolvent of the stepwise three-cut solution (\[multi-cut-CS\]) on the $Z$-plane. The solid lines denote the branch cuts of the square root: $[A_1,B_1]$, $[A_2,B_2]$ and $[A_3,B_3]$. The broken lines are the branch cuts of $\log$ which lie on the second sheet of the square root. The logarithmic branch cut starting from $B_1(=D_1)$ on the first sheet immediately goes to the second sheet and terminates at $B_1$ on the second sheet. (Right) Eigenvalue distributions through the Newton method (blue dots) and our method (red dots) for the stepwise three-cut solution at $N_1/N=N_2/N=N_3/N=1$, $n_1=n_2=1$, $\lambda=0.3$. We take $N=60$ in the Newton method. []{data-label="fig-endpt-analysis-three-cut"}](endpt-analysis-three-cut.eps "fig:")\
Lastly we have to fix the $l+1$ constant $A_1$, $B_l$ and $D_j$ ($j=1,\cdots, l-1$). These are determined through the two boundary conditions at $Z=0$ and $Z=\infty$ (\[boundary-CS\]) and $l-1$ normalization condition $$\frac{N_i}{N}=\int^{B_i}_{A_i} \rho(Z) dZ
=\frac{1}{4\pi i} \oint_{C_i} \frac{v(Z)}{Z} dZ
, \qquad
(i=1, \cdots, l).
\label{normalization-CS}$$ (One of the normalization condition is not independent of the other conditions.) We can solve these equations numerically. The result for $\{\lambda, n_1,n_2,N_1/N,N_2/N\}=\{0.3,1,1,1/3,1/3 \}$ is shown in Figure \[fig-endpt-analysis-three-cut\]. (In this case, the solution has a symmetry $z \to -z$ which fixes $b_3=-a_1$, $b_2=-a_2$, $b_1=-a_3$.) This agrees with the result obtained from the Newton method.
Composition of the stepwise multi-cut solutions
-----------------------------------------------
We explore the analytic solution for the last plot in Figure \[fig-numerical-CS\] (bottom-right). There, three cuts appear, and two of them are separated by $2\pi i$. Thus they may be regarded as a composition of the one-cut solution (\[one-cut-CS\]) and the stepwise two-cut solution (\[two-cut-CS\]). Hence we assume the three cuts as $[A^{(1)},B^{(1)}]$, $[A^{(2)}_1,B^{(2)}_1]$ and $[A^{(2)}_2,B^{(2)}_2]$. Here $[A^{(1)},B^{(1)}]$ corresponds to the one-cut around the $z=0$ and $[A^{(2)}_i,B^{(2)}_i]$ ($i=1,2$) describe the stepwise two-cuts. Hence we impose $$\begin{aligned}
A^{(2)}_2=e^{2\pi in} B_1^{(2)},\end{aligned}$$ where $n$ is a positive integer. We also assume that the numbers of the eigenvalues on each cuts are $N^{(1)}$ and $N^{(2)}_i$ ($i=1,2$), respectively. Then the resolvent may be given as $$v(Z)=\oint_{C^{(1)} \cup \, C^{(2)}_1 \cup \, C^{(2)}_2} \frac{dW}{4\pi i} \frac{V'(W)}{Z-W} \sqrt{\frac{(Z-A^{(1)})(Z-B^{(1)})(Z-A^{(2)}_1)(Z-B^{(2)}_2)}{(W-A^{(1)})(W-B^{(1)})(W-A^{(2)}_1)(W-B^{(2)}_2)}},
\label{1+2-CS}$$ where the contour $C^{(1)}$ encircles the cut $[A^{(1)},B^{(1)}]$ and $C_i^{(2)}$ encircles the cut $[A^{(2)}_i,B^{(2)}_i]$ ($i=1,2$). See Figure \[fig-1+2-cut\]. Note that we have the 5 constants: $A^{(1)},B^{(1)},A^{(2)}_1,B^{(2)}_1$ and $B^{(2)}_2$, and these constants can be fixed by the 3 normalization conditions similar to (\[normalization-CS\]) and the 3 boundary conditions (\[boundary-CS\])[^16]. (There are 6 conditions but only 5 of them are independent). Therefore the consistent solution would exist.
[cc]{}
![ (Left) Integral contours of the resolvent for the composite solution (one-cut + stepwise two-cut) in (\[1+2-CS\]). (Right) Eigenvalue distributions through the Newton method (blue dots) and our method (red dots) for the composite solution. We take $N^{(1)}/N=N_1^{(2)}/N=N_2^{(2)}/N=1/3$, $n=1$, $\lambda=0.2$. $N=120$ is taken in the Newton method. []{data-label="fig-1+2-cut"}](cycle-3cut.eps "fig:")\
![ (Left) Integral contours of the resolvent for the composite solution (one-cut + stepwise two-cut) in (\[1+2-CS\]). (Right) Eigenvalue distributions through the Newton method (blue dots) and our method (red dots) for the composite solution. We take $N^{(1)}/N=N_1^{(2)}/N=N_2^{(2)}/N=1/3$, $n=1$, $\lambda=0.2$. $N=120$ is taken in the Newton method. []{data-label="fig-1+2-cut"}](endpt-analysis-composite.eps "fig:")\
#### Symmetric solution
Performing the integral of the resolvent (\[1+2-CS\]) is generally difficult. However, if the solution is symmetric under $z \to -z$ ($Z \to 1/Z$), we can compute it as follows. This symmetry requires the following conditions on the ansatz, $$\begin{aligned}
A^{(1)}=1/B^{(1)}, \qquad A_1^{(2)}=1/B_2^{(2)}, \qquad
A_2^{(2)}=1/B_1^{(2)}=e^{\pi in}, \qquad N_1^{(2)}= N_2^{(2)}.\end{aligned}$$ Besides, the cut $[1/B^{(1)},B^{(1)}]$ should pass $Z=1$, and it demands $n$ to be odd so that the other two cuts do not hit this cut. In this case, the cut $[e^{\pi in}, B_2^{(2)}]$ and $[ 1/B_2^{(2)}, e^{-\pi in}]$ are on the $(n+1)/2$-th sheet and $-(n+1)/2$-th sheet[^17], respectively, and the integral (\[1+2-CS\]) can be written as $$\begin{aligned}
v(Z)=& \oint_{C^{(1)} \cup \, C^{(2)}} \frac{dW}{4\pi i} \frac{1}{\pi i\lambda} \frac{\log{W}}{Z-W} \sqrt{\frac{(Z-1/B^{(1)})(Z-B^{(1)})(Z-1/B^{(2)}_2)(Z-B^{(2)}_2)}{(W-1/B^{(1)})(W-B^{(1)})(W-1/B^{(2)}_2)(W-B^{(2)}_2)}}
\nonumber
\\
&+\oint_{C^{(2)}_1} \frac{dW}{4\pi i} \frac{-(n+1)}{\lambda} \frac{1}{Z-W} \sqrt{\frac{(Z-1/B^{(1)})(Z-B^{(1)})(Z-1/B^{(2)}_2)(Z-B^{(2)}_2)}{(W-1/B^{(1)})(W-B^{(1)})(W-1/B^{(2)}_2)(W-B^{(2)}_2)}}
\nonumber
\\
&+\oint_{C^{(2)}_2} \frac{dW}{4\pi i} \frac{n+1}{\lambda} \frac{1}{Z-W} \sqrt{\frac{(Z-1/B^{(1)})(Z-B^{(1)})(Z-1/B^{(2)}_2)(Z-B^{(2)}_2)}{(W-1/B^{(1)})(W-B^{(1)})(W-1/B^{(2)}_2)(W-B^{(2)}_2)}}.
\label{v-1+2-int}\end{aligned}$$ We can compute this integral by using the technique developed in Ref. [@Suyama:2010hr] and obtain $$\begin{aligned}
v(Z)=&\frac{1}{2\pi i\lambda} \log{\left( \frac{f(Z)-\sqrt{f^2(Z)-4Z^2}}{2} \right)}-\frac{n}{2\pi i\lambda} \log{\left( \frac{q(Z)+\sqrt{q^2(Z)-4}}{2} \right)},
\nonumber
\\
&f(Z)=f_0+f_1Z+f_0Z^2,
\qquad
q(Z)=\frac{q_0+q_1Z+q_0Z^2}{(Z+1)^2},
\label{v-1+2}\end{aligned}$$ where the constants are given as $$\begin{aligned}
&f_0=\frac{4}{c^{(1)}-c^{(2)} },
\quad
f_1=-2 \frac{c^{(1)}+c^{(2)} }{c^{(1)}-c^{(2)}}, \quad
c^{(1)}=B^{(1)}+1/B^{(1)}, \quad c^{(2)}=B_2^{(2)}+1/B_2^{(2)},
\nonumber
\\
&q_0=2\frac{c^{(1)}+c^{(2)}+4}{c^{(1)}-c^{(2)}},
\quad
q_1=-\frac{4\left( c^{(1)}+c^{(2)} \right) +4 c^{(1)} c^{(2)} }{c^{(1)}-c^{(2)}}.\end{aligned}$$ Remarkably, the first term of (\[v-1+2\]) is similar to the resolvent of the $S^3/{\mathbf Z}^2$ Lens space matrix model [@Aganagic:2002wv; @Halmagyi:2003ze] which is related to the ABJM matrix model (\[(1+1)-ABJM\]). The second term provides the logarithmic divergence at $Z= e^{\pm \pi i n}$ akin to the previous solutions (\[two-cut-CS\]) and (\[multi-cut-CS\]). The appearance of the resolvent of the Lens space matrix model indicates that some geometrical interpretations of our multi-cut solutions might be possible. We will consider it in a future research.
By numerically solving $B^{(1)}$ and $B^{(2)}_2$, we compare our solution with the one obtained via the Newton method. We can see a good agreement as shown in Figure \[fig-1+2-cut\]. Note that we attempt to solve the equations (\[boundary-CS\]) and the normalization condition like (\[normalization-CS\]) directly by Mathematica and obtain a consistent result. (Here we use FindRoot and NIntegral in (\[1+2-CS\]).) This is a good news. Although it would be difficult to perform the integral such as (\[1+2-CS\]) and obtain analytic expressions in general, this result indicates that we do not need the analytic expressions in order to evaluate the physical quantities.
Proposal for general solution in the pure CS matrix model {#sec-CS-general}
---------------------------------------------------------
![Eigenvalue distribution of the general solution (\[composite-CS\]) in the pure CS matrix model on the $z$-plane.[]{data-label="fig-image-CSgeneral"}](image-CSgeneral.eps "fig:")\
The generalization of the composite solution in the previous section is straightforward. We can consider $p$-stepwise $l_q$-cuts: $[A^{(q)}_j,B^{(q)}_j]$ ($q=1,\cdots,p $ and $j=1,\cdots,l_q $) satisfying $$\begin{aligned}
A^{(q)}_j=e^{2\pi in^{(q)}_j} B_{j+1}^{(q)},\end{aligned}$$ where $n^{(q)}_j$ are positive integers. We also assign the numbers of the eigenvalues on the cut $[A^{(q)}_j,B^{(q)}_j]$ as $N^{(q)}_j$. Then the resolvent may be given by $$v(Z)= \sum_{ r=1}^p\sum_{ j=1}^{l_r} \oint_{C^{(r)}_j} \frac{dW}{4\pi i} \frac{V'(W)}{Z-W} \prod_{q=1}^{p} \sqrt{\frac{(Z-A^{(q)}_1)(Z-B^{(q)}_{l_q})}{(W-A^{(q)}_1)(W-B^{(q)}_{l_q})}},
\label{composite-CS}$$ where the contour $C^{(r)}_j$ encircles the cut $[A^{(r)}_j,B^{(r)}_j]$ ($r=1,\cdots,p$ and $j=1,\cdots,l_r$). This integral may be performed by using the genus $p-1$ generalizations of elliptic functions. In this expression, we have the $p+\sum_{q=1}^{p} l_q $ undetermined constant $ \{ A^{(q)}_1, B^{(q)}_1, \cdots, B^{(q)}_{l_q} \} $, and these will be fixed by the $\sum_{q=1}^{p} l_q$ normalization condition (\[normalization-CS\]) and $p$ boundary condition at $Z=\infty$ and one boundary condition at $Z=0$ (\[boundary-CS\]). (Again one of these conditions is not independent.) As an example, we derive the solution shown in Figure \[fig-endpt-analysis-4cut\] in Appendix \[app-negative-n\] by using this ansatz.
In this way, our resolvent (\[composite-CS\]) may describe all the solutions in Figure \[fig-numerical-CS\] obtained through the Newton method. Then one important question is whether any other solutions of the saddle point equation (\[eom-CS\]) exist or not. We explore the solutions through the Newton method, and it seems that all the solutions might be explained by our resolvent (\[composite-CS\]). Although it is hard to exclude the possibility of the existence of the other solutions, we presume that our solution (\[composite-CS\]) may be the general solution of the saddle point equation of the pure CS matrix model.
Our method would be applicable to other CS matrix models. As a demonstration, we consider the ABJM matrix model in the next section.
Multi-cut solutions in the ABJM matrix model {#sec-ABJM}
============================================
We will apply the technique for finding the multi-cut solutions developed in the previous section to the ABJM matrix model [@Kapustin:2009kz]. The partition function of this model is given by $$Z (k,N)
= \frac{1}{(N!)^2} \int \prod_{i=1}^N \frac{d \mu_i}{2\pi}
e^{-\frac{N}{4\pi i \lambda} \mu_i^2 }
\prod_{j=1}^N \frac{d \nu_j}{2\pi}
e^{\frac{N}{4\pi i \lambda} \nu_j^2 }
\frac{\prod_{i<j}^{N} \left[ 2\sinh{\frac{\mu_i-\mu_j}{2}} \right]^2 \prod_{i<j}^{N} \left[ 2\sinh{\frac{\nu_i-\nu_j}{2}} \right]^2 }{\prod_{i,j=1}^{N} \left[ 2\cosh{\frac{\mu_i-\nu_j}{2}} \right]^2}.
\label{partition-ABJM}$$ Here $k$ is the CS level and $\lambda := N/k$. The saddle point equations of this model are $$\begin{aligned}
\mu_i &=& \frac{2\pi i \lambda}{N} \left[ \sum_{j \neq i}^{N} \coth{\frac{\mu_i-\mu_j}{2}}- \sum_{j=1}^{N} \tanh{\frac{\mu_i-\nu_j}{2}} \right],
\qquad
(i=1, \cdots, N), \nonumber
\\
-\nu_i &=& \frac{2 \pi i \lambda}{N} \left[ \sum_{j\neq i}^{N} \coth{\frac{\nu_i-\nu_j}{2}}-\sum_{j=1}^{N} \tanh{\frac{\nu_i-\mu_j}{2}} \right] ,
\qquad
(i=1, \cdots, N).
\label{eom-ABJM}\end{aligned}$$
[cc]{}
![Eigenvalue distributions of the numerical solutions of the saddle point equation (\[eom-ABJM\]) in the ABJM matrix model. We take $N=100$ and $\lambda=10$. The blue and red dots denote the eigenvalues of $\mu$ and $\nu$, respectively. The top-left plot corresponds to the DMP solution [@Drukker:2010nc], and other various multi-cut solutions exist in this model. []{data-label="fig-numerical-ABJM"}](DMP.eps "fig:")\
DMP solution
![Eigenvalue distributions of the numerical solutions of the saddle point equation (\[eom-ABJM\]) in the ABJM matrix model. We take $N=100$ and $\lambda=10$. The blue and red dots denote the eigenvalues of $\mu$ and $\nu$, respectively. The top-left plot corresponds to the DMP solution [@Drukker:2010nc], and other various multi-cut solutions exist in this model. []{data-label="fig-numerical-ABJM"}](3+1.eps "fig:")\
(Stepwise three + one)-cut solution
\
![Eigenvalue distributions of the numerical solutions of the saddle point equation (\[eom-ABJM\]) in the ABJM matrix model. We take $N=100$ and $\lambda=10$. The blue and red dots denote the eigenvalues of $\mu$ and $\nu$, respectively. The top-left plot corresponds to the DMP solution [@Drukker:2010nc], and other various multi-cut solutions exist in this model. []{data-label="fig-numerical-ABJM"}](2+1.eps "fig:")\
(Stepwise two+one)-cut solution
![Eigenvalue distributions of the numerical solutions of the saddle point equation (\[eom-ABJM\]) in the ABJM matrix model. We take $N=100$ and $\lambda=10$. The blue and red dots denote the eigenvalues of $\mu$ and $\nu$, respectively. The top-left plot corresponds to the DMP solution [@Drukker:2010nc], and other various multi-cut solutions exist in this model. []{data-label="fig-numerical-ABJM"}](3+2.eps "fig:")\
(Stepwise three+stepwise two)-cut solution
We explore the solutions of these equations in the ’t Hooft limit $N \rightarrow \infty$ at finite $ \lambda$. Again the resolvent is a convenient tool for solving these equations. We define new variable $M_i := \exp{(\mu_i)}$ and $N_j := \exp{( \nu_j )}$, and rewrite the saddle point equations (\[eom-ABJM\]) as $$\begin{aligned}
\log{M_i} &=& \frac{2\pi i \lambda}{N} \left[ \sum_{j \neq i}^{N} \frac{M_i+M_j}{M_i-M_j}- \sum_{j=1}^{N} \frac{M_i-N_j}{M_i+N_j} \right],
\qquad
(i=1, \cdots, N), \nonumber
\\
-\log{N_i} &=& \frac{2 \pi i \lambda}{N} \left[ \sum_{j\neq i}^{N} \frac{N_i+N_j}{N_i-N_j}-\sum_{j=1}^{N} \frac{N_i-M_j}{N_i+M_j} \right] ,
\qquad
(i=1, \cdots, N).
\label{eom-ABJM2}\end{aligned}$$ We introduce the eigenvalue densities of $M_i$ and $N_j$ and the resolvent as $$\begin{aligned}
&{}&
\rho_M(Z) := \frac{1}{N} \sum_{i=1}^N \delta (Z-M_i),
\qquad
\rho_N(Z) := \frac{1}{N} \sum_{i=1}^N \delta (Z-N_i),
\nonumber
\\
&{}&
w(Z) := \int_{\mathcal{C}_M} \rho_M (W) \frac{Z+W}{Z-W} dW-\int_{\mathcal{C}_N} \rho_N(W) \frac{Z-W}{Z+W} dW,
\label{resolvent-ABJM}\end{aligned}$$ where $\mathcal{C}_{M}$ and $\mathcal{C}_{N}$ are the supports of $\rho_{M}(Z)$ and $\rho_{N}(Z)$, respectively. Then the saddle point equations (\[eom-ABJM2\]) become $$\begin{aligned}
\frac{1}{\pi i\lambda} \log{Z}
&=&
\lim_{\epsilon \rightarrow 0} \left[ w(Z+i\epsilon)+w(Z-i\epsilon) \right],
\qquad
(Z \in \mathcal{C}_M),
\nonumber
\\
\frac{1}{\pi i\lambda} \log{Z}
&=&
\lim_{\epsilon \rightarrow 0} \left[ w(-Z+i\epsilon)+w(-Z-i\epsilon) \right],
\qquad
(Z \in \mathcal{C}_N),
\label{eom-ABJM3}\end{aligned}$$ and the eigenvalue densities are described by $$\begin{aligned}
\rho_M(Z) &=& -\frac{1}{4\pi iZ} \lim_{\epsilon \rightarrow 0} \left[ w(Z+i \epsilon)-w(Z-i \epsilon) \right],
\qquad
(Z \in \mathcal{C}_M),
\nonumber
\\
\rho_N(Z) &=& +\frac{1}{4\pi iZ} \lim_{\epsilon \rightarrow 0} \left[ w(-Z+i \epsilon)-w(-Z-i \epsilon) \right],
\qquad
(Z \in \mathcal{C}_N).
\label{rho-ABJM}\end{aligned}$$ Besides, the resolvent satisfies the boundary conditions $$\lim_{Z \rightarrow \infty} w(Z)=0,
\qquad
\lim_{Z \rightarrow 0} w(Z)=0.
\label{boundary-ABJM}$$ Therefore what we should do is finding the resolvent which satisfies the saddle point equations (\[eom-ABJM3\]) and the boundary conditions (\[boundary-ABJM\]).
Before considering the analytic solution, we attempt the numerical computations via the Newton method in order to gain some insight. Some of the obtained results are shown in Figure \[fig-numerical-ABJM\]. The top-left panel corresponds to the well-known solution obtained by Drukker, Marino and Putrov [@Drukker:2010nc]. We call this solution “DMP" solution. The top-right panel corresponds to the solution found in our previous study [@Morita:2017oev]. In addition, various multi-cut solutions exist. These results indicate that the dynamics of the ABJM matrix model is similar to the pure CS matrix model. While the eigenvalues tend to be around $z=0$, the strong interactions arise when the eigenvalues are separated by $2\pi i $, and they may cause various solutions[^18]. Therefore the technique in the pure CS matrix model would be useful in the ABJM matrix model too.
Derivation of the DMP solution
------------------------------
Before considering the multi-cut solutions, we first review the derivation of the DMP solution (Figure \[fig-numerical-ABJM\] top-left) by using the technique in the previous section [@Suyama:2010hr]. We assume the cut $[1/A,A]$ for $M_i$ and $[1/B,B]$ for $N_i$. (Here these cuts respect the symmetry $Z \to 1/Z$.) We also assume $|A|,|B| \geq 1$. See Figure \[fig-image-DMP\]. Then the resolvent which satisfies the saddle point equations (\[eom-ABJM3\]) is given as $$\begin{aligned}
w(Z)=&\oint_{C^{(M)}} \frac{dW}{4\pi i} \frac{V_M'(W)}{Z-W} \sqrt{\frac{(Z-A)(Z-1/A)(Z+1/B)(Z+B)}{(W-A)(W-1/A)(W+1/B)(W+B)}}
\nonumber
\\
&+\oint_{C^{(N)}} \frac{dW}{4\pi i} \frac{V_N'(W)}{Z-W} \sqrt{\frac{(Z-A)(Z-1/A)(Z+1/B)(Z+B)}{(W-A)(W-1/A)(W+1/B)(W+B)}}, \nonumber \\
&V_M'(Z):= \frac{1}{\pi i\lambda}\log{Z}, \qquad
V_N'(Z):= \frac{1}{\pi i\lambda}\log{\left( e^{\pi i}Z \right)}.
\label{Migdal-ABJM} \end{aligned}$$ Note that the cuts of this resolvent are on $[1/A,A]$ and $[-B,-1/B]$ rather than $[1/B,B]$. This is because the saddle point equations (\[eom-ABJM2\]) are singular when $M_i=-N_j$. Correspondingly the contour $C^{(M)}$ and $C^{(N)}$ encircle the cut $[1/A,A]$ and $[-B,-1/B]$, respectively.
We can perform this integral and obtain [@Suyama:2010hr] $$\begin{aligned}
w(Z)=&\frac{1}{2\pi i\lambda} \log{\left( \frac{f(Z)-\sqrt{f^2(Z)-4Z^2}}{2} \right)},
\qquad
f(Z)=f_0+f_1Z+f_0Z^2,
\nonumber
\\
&f_0=\frac{4}{A+1/A+B+1/B},
\quad
f_1=\frac{2 \left( -A-1/A+B+1/B \right)}{A+1/A+B+1/B}.
\label{(1+1)-ABJM}\end{aligned}$$ The parameter $A$ and $B$ are determined through the boundary conditions (\[boundary-ABJM\]) and the normalization condition $$1=\int_{1/A}^A \rho_M(Z) dZ=\frac{1}{4\pi i} \oint_{C^{(M)}} \frac{v(Z)}{Z} dZ.
\label{normalization-ABJM}$$ Then we obtain the relations [@Drukker:2010nc] $$A+\frac{1}{A}=2+i\kappa,
\qquad
B+\frac{1}{B}=2-i\kappa.
\label{endpt-ABJM}$$ Here $\kappa$ is related to the ’t Hooft coupling $\lambda$ through $$\lambda(\kappa) = \frac{\kappa}{8 \pi} {}_3F_2 \left( \frac12, \frac12, \frac12; 1, \frac32; - \frac{\kappa^2}{16} \right).
\label{kappa}$$ Particularly, at the strong coupling $|\lambda| \gg 1$, we obtain $$\begin{aligned}
&{}&A=e^{\alpha},
\qquad
\alpha=\pi \sqrt{2 \hat{\lambda} }+ \frac{\pi}{2} i -2i e^{-\pi \sqrt{2 \hat{\lambda}}}+\cdots,
\nonumber
\\
&{}&B=e^{\beta},
\qquad
\beta=\pi \sqrt{2 \hat{\lambda} }- \frac{\pi}{2} i +2i e^{-\pi \sqrt{2 \hat{\lambda}}}+\cdots,
\label{endpt-ABJM'}\end{aligned}$$ where $\hat{\lambda} := \lambda-\frac{1}{24}$.
[cc]{}
![(Left) Sketch of the cuts of the DMP solution. Here $\alpha=\log{A}$ and $\beta=\log{B}$. (Right) Integral contours of (\[Migdal-ABJM\]). The contour $C^{(N)}$ encircles $[-B,-1/B]$ rather than $[1/B,B]$. []{data-label="fig-image-DMP"}](image-DMP.eps)
![(Left) Sketch of the cuts of the DMP solution. Here $\alpha=\log{A}$ and $\beta=\log{B}$. (Right) Integral contours of (\[Migdal-ABJM\]). The contour $C^{(N)}$ encircles $[-B,-1/B]$ rather than $[1/B,B]$. []{data-label="fig-image-DMP"}](cycle-ABJM.eps)
Proposal for general solution in the ABJM matrix model {#sec-general-ABJM}
------------------------------------------------------
We will apply the technique developed in the pure CS matrix model to the ABJM matrix model, and propose the general solution. As the numerical computations shown in Figure \[fig-numerical-ABJM\] suggest, there are various multi-cut solutions in which the eigenvalues of the same matrix are separated by $2\pi i$. Thus each matrix can compose the stepwise multi-cuts. In addition, a composition of these stepwise multi-cuts would be a solution too as in the pure CS matrix model case. (Indeed we find these complicated solutions numerically, although we omit to show them in this article.)
By regarding these numerical results, we consider the following ansatz. Suppose the eigenvalue $\{ M_i \}$ compose $p$ stepwise $l_r$-cuts ($r=1,\cdots,p $) and the eigenvalue $\{ N_i \}$ compose $q$ stepwise $m_r$-cuts ($r=1,\cdots,q $), and we define that the cut $[A^{(M,r)}_j,B^{(M,r)}_j]$ for $\{ M_j \}$ and $[A^{(N,r)}_j,B^{(N,r)}_j]$ for $\{ N_j \}$. We assume that these cuts satisfy $$\begin{aligned}
A^{(M,r)}_j=&e^{2\pi in^{(M,r)}_j} B_{j+1}^{(M,r)}, \qquad (j=1,\cdots,l_r, \quad r=1,\cdots,p) ,\nonumber \\
A^{(N,r)}_j=&e^{-2\pi in^{(N,r)}_j} B_{j+1}^{(N,r)}, \qquad (j=1,\cdots, m_r, \quad r=1,\cdots,q) ,\end{aligned}$$ where $n^{(M,r)}_j$ and $n^{(N,r)}_j$ are positive integers. We also assign the numbers of the eigenvalues on the cut $[A^{(M,r)}_j,B^{(M,r)}_j]$ and $[A^{(N,r)}_j,B^{(N,r)}_j]$ as $N^{(M,r)}_j$ and $N^{(N,r)}_j$, respectively. Then the resolvent may be given as $$\begin{aligned}
w(Z)=&\sum_{ t=1}^p \sum_{ j=1}^{l_t} \oint_{C^{(M,t)}_j} \frac{dW}{4\pi i} \frac{V_M'(W)}{Z-W}
\prod_{r=1}^{p} \prod_{s=1}^{q} \sqrt{\frac{(Z-A^{(M,r)}_1)(Z-B^{(M,r)}_{l_r})(Z+A^{(N,s)}_1)(Z+B^{(N,s)}_{m_s})}{(W-A^{(M,r)}_1)(W-B^{(M,r)}_{l_r})(W+A^{(N,s)}_1)(W+B^{(N,s)}_{m_s})}}
\nonumber
\\
&+\sum_{ t=1}^q \sum_{ j=1}^{m_t} \oint_{C^{(N,t)}_j} \frac{dW}{4\pi i} \frac{V_N'(W)}{Z-W}
\prod_{r=1}^{p} \prod_{s=1}^{q} \sqrt{\frac{(Z-A^{(M,r)}_1)(Z-B^{(M,r)}_{l_r})(Z+A^{(N,s)}_1)(Z+B^{(N,s)}_{m_s})}{(W-A^{(M,r)}_1)(W-B^{(M,r)}_{l_r})(W+A^{(N,s)}_1)(W+B^{(N,s)}_{m_s})}}
,
\label{general-ABJM}\end{aligned}$$ where the contour $C^{(M,r)}_i$ and $C^{(N,s)}_j$ encircle the cut $[A^{(M,r)}_i,B^{(M,r)}_i]$ ($i=1,\cdots,l_r$ and $r=1,\cdots,p$) and $[-B^{(N,s)}_j,-A^{(N,s)}_j]$ ($j=1,\cdots,m_s$ and $s=1,\cdots,q$), respectively. The end points of the cuts may be determined through the boundary conditions (\[boundary-ABJM\]) and the normalization conditions akin to (\[normalization-ABJM\]).
Symmetric stepwise multi-cut solution
-------------------------------------
![Eigenvalue distribution of the symmetric stepwise $((2l+1)+(2m+1))$-cut solution (\[general-sym-ABJM\]).[]{data-label="fig-ABJMg"}](image-ABJMg.eps)
Although the general solution (\[general-ABJM\]) looks very complicated, if $p=q=1$ and the solution is symmetric under $Z \to 1/Z$, we will obtain a simple expression. To see it, we consider a stepwise $2l+1$-cuts of $\mu_i$ and stepwise $2m+1$-cuts of $\nu_i$ configuration as sketched in Figure \[fig-ABJMg\]. As we will see soon, the result depends on whether the number of each cut is odd or even, and we consider the both odd case first. We assume that $\{ \mu_i \}$ are distributed between $[-b^{(M)}_0, b^{(M)}_0 ]$, $[a^{(M)}_j, b^{(M)}_j ]$ and $[ -b^{(M)}_j, -a^{(M)}_j ]$, $(j=1, \cdots,l )$ and the number of the eigenvalues on each interval is $N^{(M)}_0$, $N^{(M)}_j$ and $N^{(M)}_j$, respectively, so that the system is symmetric under $Z \to 1/Z$. Here $N^{(M)}_0+2 \sum_{j=1}^{l}N_j^{(M)}=N$ is imposed. Similarly, for $\{\nu_i \}$, we take $[-b^{(N)}_0, b^{(N)}_0 ]$, $[a^{(N)}_j, b^{(N)}_j ]$ and $[ -b^{(N)}_j, -a^{(N)}_j ]$, $(j=1, \cdots,m )$ and $N^{(N)}_j$ which satisfies $N^{(N)}_0+2 \sum_{j=1}^{m}N_j^{(N)}=N$. Through the stepwise assumption, we impose the condition $$\begin{aligned}
a_{j}^{(M)}&=b_{j-1}^{(M)}+ 2\pi i n_{j}^{(M)}, \qquad (j=1, \cdots, l ) ,\nonumber \\
a_{j}^{(N)}&=b_{j-1}^{(N)}- 2\pi i n_{j}^{(N)}, \qquad (j=1, \cdots, m ) ,\end{aligned}$$ where $\{ n_{j}^{(M)} \}$ and $\{ n_{j}^{(N)} \}$ are positive integers. On this set up, the resolvent (\[general-ABJM\]) becomes $$\begin{aligned}
w(Z)=&\sum_{ j=1}^{2l+1} \oint_{C^{(M)}_j} \frac{dW}{4\pi i} \frac{V_M'(W)}{Z-W}
\sqrt{\frac{(Z-1/B^{(M)}_l)(Z-B^{(M)}_{l})(Z+1/B^{(N)}_m)(Z+B^{(N)}_{m})}{(W-1/B^{(M)}_l)(W-B^{(M)}_{l})(W+1/B^{(N)}_m)(W+B^{(N)}_{m})}}
\nonumber
\\
&+\sum_{ j=1}^{2m+1} \oint_{C^{(N)}_j} \frac{dW}{4\pi i} \frac{V_N'(W)}{Z-W}
\sqrt{\frac{(Z-1/B^{(M)}_l)(Z-B^{(M)}_{l})(Z+1/B^{(N)}_m)(Z+B^{(N)}_{m})}{(W-1/B^{(M)}_l)(W-B^{(M)}_{l})(W+1/B^{(N)}_m)(W+B^{(N)}_{m})}}
,
\label{general-sym-ABJM}\end{aligned}$$ where $A^{(M)}_j= \exp\left(a^{(M)}_j\right) $, $B^{(M)}_j= \exp\left(b^{(M)}_j\right) $, $A^{(N)}_j= \exp\left(a^{(N)}_j\right) $ and $B^{(N)}_j= \exp\left(b^{(N)}_j\right) $, and $C^{(M)}_j$ and $C^{(N)}_j$ are the contours which encircle the cuts as in (\[general-ABJM\]). We will use $D_j^{(M)} := A_j^{(M)} $ and $D_j^{(N)} := A_j^{(N)} $ when we emphasize the points of the steps. Through calculations similar to section \[sec-2-cut-CS\], we can perform this integral and obtain $$\begin{aligned}
&w(Z)=
\frac{1}{2\pi i\lambda} \log{\left( \frac{f(Z)-\sqrt{f^2(Z)-4Z^2}}{2} \right)}
\nonumber
\\
& \qquad \quad +\sum_{i=1}^l \frac{n_i^{(M)}}{\pi i\lambda} \log{\left( \frac{p^{(i)}(Z)+\sqrt{\left( p^{(i)}(Z) \right)^2 -4}}{2} \right)}
-\sum_{j=1}^m \frac{n_j^{(N)}}{\pi i\lambda} \log{\left( \frac{q^{(j)}(Z)+\sqrt{\left( q^{(j)}(Z) \right)^2 -4}}{2} \right)} .
\label{ABJMg}\end{aligned}$$ Here $f(Z)$, $p^{(i)}(Z)$ and $q^{(j)}(Z)$ are rational functions $$\begin{aligned}
&f(Z)=f_0+f_1Z+f_0Z^2,
\quad
p^{(i)}(Z)=\frac{p_0Z^2+p_1Z+p_0}{(Z-D_i^{(M)})(Z-1/D_i^{(M)})},
\quad
q^{(j)}(Z)=\frac{q_0Z^2+q_1Z+q_0}{(Z+D_j^{(N)})(Z+1/D_j^{(N)})},
\nonumber\end{aligned}$$ where the coefficients are given by $$\begin{aligned}
&f_0=\frac{4}{c^{(M)}+c^{(N)}},
\quad
f_1=\frac{2 \left( -c^{(M)}+c^{(N)} \right)}{c^{(M)}+c^{(N)}},
\nonumber
\\
&p^{(i)}_0=\frac{4\left( D_i^{(M)}+1/D_i^{(M)} \right)+2c^{(M)}-2c^{(N)} }{c^{(M)}+c^{(N)}},
\quad
p^{(i)}_1=\frac{2 \left( D_i^{(M)}+1/D_i^{(M)} \right) \left( c^{(M)} -c^{(N)} \right) -4c^{(M)} c^{(N)}}{c^{(M)}+c^{(N)}},
\nonumber
\\
&q^{(j)}_0=\frac{-4\left( D_j^{(N)}+1/D_j^{(N)} \right)+2c^{(M)}-2c^{(N)} }{c^{(M)}+c^{(N)}},
\quad
q^{(j)}_1=\frac{-2 \left( D_j^{(N)}+1/D_j^{(N)} \right) \left( c^{(M)} -c^{(N)} \right) -4c^{(M)} c^{(N)}}{c^{(M)}+c^{(N)}},
\nonumber
\\
&c^{(M)}=B^{(M)}_l+1/B^{(M)}_l, \qquad c^{(N)}=B^{(N)}_m+1/B^{(N)}_m.\end{aligned}$$ In (\[ABJMg\]), the first term is identical to the DMP solution (\[(1+1)-ABJM\]) and the rest of the terms resemble the terms in the stepwise multi-cut solutions in the pure CS matrix model (\[multi-cut-CS\]) and (\[v-1+2\]). Particularly, the resolvent shows the logarithmic singularities at $Z=D_i^{(M)}$, $1/D_i^{(M)}$, $-D_j^{(N)}$ and $-1/D_j^{(N)}$ ($i=1,\cdots, l $ and $j=1,\cdots, m $ ).
If the number of the cuts is even, the result should be modified, since the cut at the origin disappears. Suppose the number of the cuts of $\mu_i$ is even, we should remove the cut $[1/B_0^{(M)}, B_0^{(M)}]$ and fix $D_1^{(M)}=\exp \left( \pi i n^{M}_1 \right)$. Similarly, if the number of the cuts of $\nu_i$ is even, the cut $[1/B_0^{(N)}, B_0^{(N)}]$ is removed and $D_1^{(N)}=\exp \left( -\pi i n^{N}_1 \right)$. With these modifications, the expression (\[ABJMg\]) works in these cases.
Connection to the large-$N$ instantons {#sec-instanton}
--------------------------------------
Once we obtain the multi-cut solutions, we may obtain the large-$N$ instantons which are the “tunneling" of the eigenvalues between two solutions [@David:1990sk; @David:1992za]. Particularly the instantons in the DMP solution which corresponds to the AdS${}_4\times$CP${}^3$ vacuum of the string theory might be related to non-perturbative objects of strings. In this section, we argue that some of the instantons may be related to the so-called D2-brane instantons [@Drukker:2011zy; @Grassi:2014cla].
We consider the stepwise two+one-cut solution plotted in Figure \[fig-numerical-ABJM\] (bottom-left). If we take $N_2^{(M)} \to 0$ limit, this solution reduces to the DMP solution. Thus $N_2^{(M)} \to 1$ limit of this solution may correspond to the instanton of the single eigenvalue tunneling in the DMP solution. We can rudely estimate the instanton action of this instanton as follows [@Morita:2017oev]. We consider the effective potential for the $N$-th eigenvalue, say $\mu_N$, in the DMP solution. From (\[partition-ABJM\]), the effective potential for $\mu_N$ is given by $$\begin{aligned}
V_{\rm eff} (\mu_N)& = \frac{N}{4\pi i \lambda} \mu^2_N + V_{\rm int} (\mu_N),
\nonumber
\\
V_{\rm int} (\mu_N) & :=
-\sum_{j =1 }^{N-1}
\log \left[ 2\sinh{\frac{\mu_N-\mu_j}{2}} \right]^2+\sum_{j=1}^{N} \log \left[ 2\cosh{\frac{\mu_N-\nu_j}{2}} \right]^2 .
\label{eff-ABJM}\end{aligned}$$ Here we fix $\{ \mu_i \}$ ($i \neq N$) and $\{ \nu_j \}$ to be the DMP solution and we ignore the back-reaction of $\mu_N$ to the other eigenvalues. (We will soon see that ignoring the back-reaction is too rude.) If $\mu_N=\alpha$ where $\alpha$ is the location of the right end point of the cut defined in (\[endpt-ABJM’\]), it corresponds to the DMP solution. Then the instanton action is estimated as the difference of the values of the effective potentials $$\begin{aligned}
S_{\rm inst}(\mu) = V_{\rm eff}(\mu) -V_{\rm eff}(\alpha).\end{aligned}$$ By using this equation, we can estimate the instanton action of the $N_2^{(M)} \to 1$ limit of the stepwise two-cut solution (Figure \[fig-numerical-ABJM\]) by taking $\mu=\alpha+ 2\pi i $, $$\begin{aligned}
S_{\rm inst} (\alpha+ 2\pi i )& = \frac{N}{4\pi i \lambda} (\alpha+ 2\pi i )^2 + V_{\rm int} (\alpha+ 2\pi i ) - \left( \frac{N}{4\pi i \lambda} \alpha ^2 + V_{\rm int} (\alpha ) \right) \nonumber \\
&= \frac{ N \alpha}{\lambda} + i \frac{N \pi }{\lambda} \nonumber \\
&= N \pi \sqrt{2 /\lambda}+ \cdots, \qquad (|\lambda| \gg 1).
\label{S-inst}\end{aligned}$$ Here we have used the periodicity of the interaction $ V_{\rm int} (\alpha+ 2\pi i n)=V_{\rm int} (\alpha)$, and equation (\[endpt-ABJM’\]). Remarkably, the obtained value at strong coupling $(|\lambda| \gg 1)$ agrees with the D2-brane instanton which was obtained through a sophisticated cycle integral of the spectral curve [@Drukker:2011zy; @Grassi:2014cla] $$\begin{aligned}
S^{\rm D2}_{\rm inst}=\pi N \sqrt{2/\lambda} , \qquad (|\lambda| \gg 1).
\label{F1D2}\end{aligned}$$ This quantitative agreement indicates that our multi-cut solutions might be interpreted as the condensations of the D2-brane instantons[^19].
However our evaluation of the instanton action (\[S-inst\]) is too rude, since $\mu= \alpha+ 2\pi i $ does not satisfy the equation of motion $0=V'_{\rm eff}(\mu)=N \mu/2 \pi i \lambda +V'_{\rm int}(\mu) $. We can see it as follows. Since we have assumed that $\mu= \alpha$ is the DMP solution, it should satisfy $0=N \alpha/2 \pi i \lambda +V'_{\rm int}(\alpha) $. However it immediately means that $\mu=\alpha+2\pi i $ is not a solution due to the periodicity $V'_{\rm int}(\mu +2\pi i n )=V'_{\rm int}(\mu )$. It implies that the back-reaction to the other eigenvalues is crucial to construct the instanton solution[^20].
[cc]{}
![(Left) $N$ and $\lambda$ dependence of the real part of the instanton action through the Newton method. We compute the classical action of the DMP solution and the instanton solution and evaluate their differences $\Delta S$ at various $N$ and $\lambda$ (the red dots). Then we fit these data at each fixed $\lambda$ (solid lines) and extrapolate $\Delta S(\lambda) |_{N \to \infty} $. (Right) Plot of $\Delta S(\lambda) |_{N \to \infty}/N $ (the red dots). The solid line is analytic prediction of the D2-brane instanton action $\pi \sqrt{2/\hat{\lambda}}$ (\[F1D2\]). We can see a good agreement between them. []{data-label="fig-instanton"}](Finst-N.eps "fig:")\
${\rm Re} (\Delta S)/N $ vs. $1/N$
![(Left) $N$ and $\lambda$ dependence of the real part of the instanton action through the Newton method. We compute the classical action of the DMP solution and the instanton solution and evaluate their differences $\Delta S$ at various $N$ and $\lambda$ (the red dots). Then we fit these data at each fixed $\lambda$ (solid lines) and extrapolate $\Delta S(\lambda) |_{N \to \infty} $. (Right) Plot of $\Delta S(\lambda) |_{N \to \infty}/N $ (the red dots). The solid line is analytic prediction of the D2-brane instanton action $\pi \sqrt{2/\hat{\lambda}}$ (\[F1D2\]). We can see a good agreement between them. []{data-label="fig-instanton"}](Finst-l.eps "fig:")\
$ {\rm Re} (\Delta S)/N $ vs. $\lambda$
In principle, we can evaluate the back-reaction by using the stepwise two+one-cut solution (\[ABJMg\]). Starting from this solution, by taking $N_2^{(M)} \to 1$ in the free energy, we would obtain the instanton action including the back-reaction. However the computation of the free energy of the stepwise two+one-cut solution is technically difficult, and we instead evaluate the instanton action numerically by employing the Newton method. The result is summarized in Figure \[fig-instanton\]. It indicates that somehow the contributions of the back-reaction to the instanton action is suppressed and the rude estimation (\[S-inst\]) works well[^21]. This result supports our conjecture that the stepwise multi-cut solutions are related to the D2-brane instantons in the ABJM theory.
#### Large-$N$ instantons in the pure CS matrix model
We can apply the estimation of the instanton action in the ABJM matrix model (\[S-inst\]) to other CS matrix models if the model allows the stepwise multi-cut solutions. For example, in the case of the pure CS matrix model (\[partition-CS\]), we can estimate the instanton action as $$\begin{aligned}
S_{\rm inst} (b+ 2\pi i )& = V_{\rm eff}(b+2\pi i) -V_{\rm eff}(b) \nonumber \\
&= \frac{N}{4\pi i \lambda} (b+ 2\pi i )^2 + V_{\rm int} (b+ 2\pi i ) -\left(\frac{N}{4\pi i \lambda} b^2 + V_{\rm int} (b )\right) \nonumber \\
&= \frac{ N b}{\lambda} + i \frac{N \pi }{\lambda} = 2 \pi i N + \cdots, \qquad (|\lambda| \gg 1). \end{aligned}$$ Here $b$ is the end point of the one-cut solution (\[endpt-CS\]), and $V_{\rm eff}$ and $V_{\rm int}$ are defined similar to . Again we have ignored the back-reaction in this estimation without any justification. However, the obtained value of the instanton action agrees with the membrane instanton of the pure CS matrix model argued in [@Pasquetti:2009jg; @Hatsuda:2015owa], $$\begin{aligned}
&{}&\text{membrane instanton:}
\quad
S^{\rm M2}_{\rm inst}= \frac{2\pi t}{g_s}=2\pi iN,
\qquad (|t| \gg 1),
\label{CSinstanton}\end{aligned}$$ where $t:=ig_sN=2\pi i \lambda$. This agreement suggests that the stepwise multi-cut solutions might be regarded as the condensations of the membrane instantons.
#### Other instantons?
So far we have discussed the instanton limit of the stepwise multi-cut solutions. As we have seen in section \[sec-CS-general\] and \[sec-general-ABJM\], the composite solutions also exist in the CS matrix models. However, they are composed of at least three cuts, and we cannot take the ordinary instanton limit, namely taking the configuration of the stable solution plus single tunnelling eigenvalue. At least two tunnelling eigenvalues are required, and, in this sense, the composite solution might provide a novel type of large-$N$ instantons. (The interaction between the tunneling eigenvalues is crucial similar to the $N=2$ analysis in .) However, we have not found any simple estimation of the instanton action for these solutions so far, and the quantitative comparison to D-branes and the known non-perturbative effects in the CS matrix models [@Pasquetti:2009jg; @Hatsuda:2015owa; @Honda:2016vmv; @Honda:2017qdb; @Drukker:2011zy; @Grassi:2014cla] have not been done. We leave this issue for future work.
Conclusions and Discussions {#sec-discussion}
===========================
In this article, we proposed the ansatz (\[composite-CS\]) and (\[general-ABJM\]) for the general solutions of the pure CS matrix model and ABJM matrix model, respectively. By solving these ansatz, we obtained the multi-cut solutions which quantitatively agree with the Newton method. Besides, these solutions exhibit the various curious properties: the two types of the multi-cuts (the composite and stepwise), the logarithmic divergences of the eigenvalue densities and the instanton limit. Since the multi-cut solutions may describe the various vacua of the systems, these solutions may be crucial to reveal the non-perturbative structures of the CS matrix models. Indeed we have found the quantitatively evidences that our multi-cut solutions are related to the membrane instantons [@Pasquetti:2009jg; @Hatsuda:2015owa] and the D2-brane instantons [@Drukker:2011zy; @Grassi:2014cla].
One important future direction is the analytic computations of the integral (\[composite-CS\]) and (\[general-ABJM\]) in the general situations. They might provide us further curious structures of the CS matrix models. The holomorphy might also help us to find the general solutions as discussed in Appendix \[app-holomorphy\].
Another interesting future direction is exploring the gravity duals of our multi-cut solutions. Since we have considered the ’t Hooft limit of the CS gauge theories, the dual gravity description in superstring theory may work. Particularly the existence of the infinite number of the solutions in the gauge theories reminds us the story of the bubbling geometries [@Lin:2004nb]. If the corresponding infinite number of the gravity solutions were found, it would be very important in the supergravities. The researches on the Lens space matrix models [@Aganagic:2002wv; @Halmagyi:2003ze; @Halmagyi:2003mm; @Okuda:2004mb; @Halmagyi:2007rw] may give us some insight about the connection between the geometries and the eigenvalue distributions of the CS matrix models.
#### Acknowledgements
The authors would like to thank Tomoki Nosaka, Kazumi Okuyama, Takao Suyama and Asato Tsuchiya for valuable discussions and comments. The authors would also like to thank the referee for her/his careful reading of this manuscript and helpful comments. The authors would also like to thank YITP for financially supporting [*“Chube summer school 2017”*]{}, where they had the opportunity to present and develop this work. The work of T. M. is supported in part by Grant-in-Aid for Scientific Research (No. 15K17643) from JSPS.
Derivation of the stepwise multi-cut solution via holomorphy {#app-holomorphy}
============================================================
We will show that the stepwise multi-cut solution can be derived by using holomorphy[^22] too which have been employed in the CS matrix models [@Drukker:2010nc; @Aganagic:2002wv; @Halmagyi:2003ze; @Drukker:2011zy].
One-cut solution
----------------
We review the derivation of the one-cut solution (\[one-cut-CS\]) via holomorphy [@Aganagic:2002wv; @Halmagyi:2003ze]. We assume that the resolvent $v(Z)$ has the branch cuts on $\mathcal{C}:$ $[A,B]$. On this cut, the resolvent should satisfy the saddle point equation (\[eom-CS3\]). Then we can define a holomorphic function $$\begin{aligned}
f(Z)=e^{\pi i\lambda v (Z)}+Z e^{-\pi i\lambda v(Z)}.
\label{f(Z)}\end{aligned}$$ From the boundary conditions (\[boundary-CS\]), $f(Z)$ satisfies $f(Z) \to e^{-\pi i\lambda } Z$ ($Z\to \infty$) and $f(Z) \to e^{-\pi i\lambda } $ ($Z\to 0$). Then such a holomorphic function is uniquely determined as $$\begin{aligned}
f(Z)=f_0+f_1Z,
\qquad
f_0= e^{-\pi i \lambda},
\qquad
f_1= e^{-\pi i \lambda}.\end{aligned}$$ On the other hand, by solving (\[f(Z)\]) with respect to $v(Z)$, we obtain $$\begin{aligned}
v(Z)=\frac{1}{\pi i\lambda} \log{\left( \frac{f(Z)-\sqrt{f^2(Z)-4Z}}{2} \right)}.
\label{v_1-soln}\end{aligned}$$ This result agrees with (\[one-cut-CS\]). One can confirm that this resolvent correctly satisfies the saddle point equation (\[eom-CS3\]) $$\begin{aligned}
\lim_{\epsilon \rightarrow 0} \left[ v(Z+i\epsilon)+v(Z-i\epsilon) \right]
&=
\frac{1}{\pi i\lambda} \left[ \log{\left( \frac{f(Z)-i \sqrt{4Z-f^2(Z)}}{2} \right)}+\log{\left( \frac{f(Z)+i \sqrt{4Z-f^2(Z)}}{2} \right)} \right]
\nonumber
\\
&=
\frac{1}{\pi i\lambda} \log{Z},
\quad
\left( Z \in \mathcal{C} \right).
\label{check-eom-v0}\end{aligned}$$
Note that the end points of the cut $A$ and $B$ are determined through the relation $\sqrt{f^2(Z)-4Z} \propto \sqrt{(Z-A)(Z-B)}$, and they are given as the solution of $$\begin{aligned}
\frac{2f_0f_1-4}{f_1^2}=\left( A+B \right) ,
\qquad
\frac{f_0^2}{f_1^2}=AB
\label{f_0f_1-AB}.\end{aligned}$$
Stepwise two-cut solution {#stepwise-two-cut-solution}
-------------------------
We consider the derivation of the stepwise two-cut solution (\[two-cut-CS\]) by developing the argument in the previous section. We assume that the resolvent $v(Z)$ has the branch cuts on $\mathcal{C}_1$: $[A_1,B_1]$ and $\mathcal{C}_2$: $[A_2,B_2]$ where $A_2= e^{2\pi i n}B_1$ with a positive integer $n$ as in (\[CS-step\]). On these cuts, the resolvent should satisfy the saddle point equation (\[eom-CS3\]). As sketched in Figure \[fig-two-CS\], we assume that $\mathcal{C}_1$ locates on the $n_0$-th sheet and $\mathcal{C}_2$ locates on the $n_0+n$-th sheet.
We will see that the resolvent of the one-cut solution (\[v\_1-soln\]) plays a key role in this problem. As a trial, let us rotate $Z \to e^{2\pi i n_0} Z$ around $Z=0$ in the saddle point equation (\[check-eom-v0\]) of the one-cut solution (\[v\_1-soln\]) and see what happens. On the left hand side of (\[check-eom-v0\]), since $v(Z)$ is non-singular at $Z=0$, the rotation does not change the value[^23]. (We rotate $Z $ so that it avoids the branch cut of the square root of $v(Z)$.) On the right hand side, since $\log Z$ has the branch cut, the additional constant $2n_0/\lambda$ appears. Thus, the resolvent of the one-cut solution almost satisfies the saddle point equation (\[eom-CS3\]) on $\mathcal{C}_1$ except the constant term $ 2n_0/
\lambda$. Similarly, on $\mathcal{C}_2$, $ 2(n_0+n)/ \lambda$ arises.
Therefore, if we find a function $v_1(Z)$ which satisfies[^24] $$\begin{aligned}
\lim_{\epsilon \rightarrow 0} \left[ v_1(Z+i\epsilon)+v_1(Z-i\epsilon) \right]=&
\frac{2n_0}{\lambda},
\qquad
(Z \in \mathcal{C}_1), \nonumber
\nonumber
\\
\lim_{\epsilon \rightarrow 0} \left[ v_1(Z+i\epsilon)+v_1(Z-i\epsilon) \right]=&
\frac{2(n+n_0)}{\lambda} ,
\qquad
(Z \in \mathcal{C}_2 ),
\label{eom-v1}\end{aligned}$$ the resolvent of the stepwise two-cut solution may be given as $$\begin{aligned}
v(Z)=v_0(Z)+v_1(Z),
\label{v0-v1}\end{aligned}$$ where $v_0(Z)$ denotes the one-cut solution (\[v\_1-soln\]) which satisfies $$\begin{aligned}
\lim_{\epsilon \rightarrow 0} \left[ v_0(Z+i\epsilon)+v_0(Z-i\epsilon) \right]=&
\frac{1}{\pi i\lambda} \log Z,
\qquad
(Z \in \mathcal{C}).
\label{eom-v0}\end{aligned}$$ Here we have defined the cut $\mathcal{C} =\mathcal{C}_1 \cup \mathcal{C}_2$ on the 0-th sheet of the branch cut of $\log Z$.
However $v_0(Z)$ is not exactly identical to (\[v\_1-soln\]). This is because, through the boundary conditions (\[boundary-CS\]), $v_0(Z)$ and $v_1(Z)$ should satisfy $$\begin{aligned}
&\lim_{Z \rightarrow \infty} v_0(Z) =s,
\qquad
\lim_{Z \rightarrow \infty} v_1(Z)=1-s, \nonumber
\\
&\lim_{Z \rightarrow 0} v_0(Z) =-\tilde{s},
\qquad
\lim_{Z \rightarrow 0} v_1(Z)=-(1-\tilde{s}),
\label{decomposition-boundary}\end{aligned}$$ where $s$ and $\tilde{s}$ are constants. (We have assumed that $v_0(Z)$ and $v_1(Z)$ are finite at $Z=0$ and $Z=\infty$.) Hence $f(Z)$ is modified, $$\begin{aligned}
f(Z)=f_0+f_1Z, \qquad
f_0=e^{-\pi i\lambda \tilde{s}},
\qquad
f_1=e^{-\pi i\lambda s}.
\label{f(Z)-boundary-2cut}\end{aligned}$$ Thus $v_0(Z)$ is given by (\[v\_1-soln\]) with this $f(Z)$.
Next we consider $v_1(Z)$. Similar to the $v_0(Z)$, we define a function, $$\begin{aligned}
q(Z)=e^{-\frac{n_0 \pi i}{n}} e^{\frac{\pi i\lambda}{n} v_1(Z)}+ e^{\frac{n_0 \pi i}{n}} e^{-\frac{\pi i \lambda}{n} v_1(Z)}.
\label{q(Z)}\end{aligned}$$ We can see that $q(Z)$ is smooth on the cut $\mathcal{C}_1$ and $\mathcal{C}_2$ through (\[eom-v1\])[^25]. (Note that we will soon see that $q(Z)$ has to have a pole.) By solving (\[q(Z)\]) with respect to $v_1(Z)$, we obtain $$\begin{aligned}
v_1(Z)=\frac{n}{\pi i\lambda} \log{\left( \frac{q(Z)+\sqrt{q^2(Z)-4}}{2} \right)} +\frac{n_0}{\lambda}.
\label{v_2-soln}\end{aligned}$$
Now we determine the function $q(Z)$. Through the boundary conditions (\[decomposition-boundary\]), $q(Z)$ satisfies $$\begin{aligned}
q_1:=&\lim_{Z \rightarrow \infty} q(Z)=e^{-\frac{n_0 \pi i}{n}} e^{\frac{\pi i\lambda}{n} (1-s)}+e^{\frac{n_0 \pi i}{n}}e^{-\frac{\pi i\lambda}{n} (1-s)} ,
\nonumber \\
q_0:=&\lim_{Z \rightarrow 0} q(Z)=e^{-\frac{n_0 \pi i}{n}}e^{-\frac{\pi i\lambda}{n} (1-\tilde{s})}+e^{\frac{n_0 \pi i}{n}}e^{\frac{\pi i\lambda}{n} (1-\tilde{s})} .
\label{q(Z)-boundary}\end{aligned}$$ Besides, since $v_1(Z)$ should have the branch cut between $A_1$ and $B_2$, we demand $$\begin{aligned}
\sqrt{q^2(Z)-4} \propto \sqrt{(Z-A_1)(Z-B_2)}.
\label{q(Z)-cut}\end{aligned}$$ However we can easily see that this condition and the boundary conditions (\[q(Z)-boundary\]) are inconsistent if $q(Z)$ is a holomorphic function on the entire complex plane. Hence we relax holomorphy and allow $q(Z)$ to have poles. A natural candidate of the location of the pole is $Z=D_1:=B_1$ where the value of the right hand side of (\[eom-v1\]) changes. Then the conditions (\[q(Z)-boundary\]) and (\[q(Z)-cut\]) are satisfied, if $$q(Z)=\frac{q_1Z-q_0D_1}{Z-D_1},
\label{q(Z)-laurent}$$ where $q_0$ and $q_1$ are related to $A_1$, $D_1$ and $B_2$ via $$\begin{aligned}
\frac{2(4-q_0q_1)D_1}{q_1^2-4}=- \left( A_1+B_2 \right) , \qquad \frac{(q_0^2-4)D_1^2}{q_1^2-4}=A_1B_2.
\label{q_0q_1-AB}\end{aligned}$$
It will be instructive to see how the resolvent $v_1(Z)$ (\[v\_2-soln\]) satisfies the equation (\[eom-v1\]). On $Z \in \mathcal{C}_1$, (\[eom-v1\]) is satisfied because $$\begin{aligned}
&\lim_{\epsilon \rightarrow 0} \left[ v_1(Z+i\epsilon)+v_1(Z-i\epsilon) \right]
\nonumber
\\
=&
\frac{2n_0}{\lambda}+\frac{n}{\pi i\lambda} \left[ \log{\left( \frac{q(Z)-i \sqrt{4-q^2(Z)}}{2} \right)}+\log{\left( \frac{q(Z)+i \sqrt{4-q^2(Z)}}{2} \right)} \right]
\nonumber
\\
=&
\frac{2n_0}{\lambda},
\quad (Z \in \mathcal{C}_1).
\label{check-eom-v1}\end{aligned}$$ At $Z=B_1$, the imaginary part of $v_1(Z)$ diverges logarithmically and the real part of $v_1(Z)$ (\[v\_2-soln\]) changes by $n/\lambda$. Thus the right hand side of (\[check-eom-v1\]) becomes $2(n_0+n)/\lambda$ on $ \mathcal{C}_2$, and it satisfies (\[eom-v1\]) correctly. The configuration of the branch cut corresponding to this divergence can be seen in Figure \[fig-endpt-analysis-three-cut\]. Note that, although the cuts $[A_1,B_2]$ exist on the every sheet of $\log Z$, $v(Z)$ (\[v0-v1\]) satisfies the equation (\[eom-CS3\]) only on $ \mathcal{C}_1$ on the $n_0$-th sheet and on $ \mathcal{C}_2$ on the $n_0+n$-th sheet.
By using the obtained $v_0(Z)$ and $v_1(Z)$, the stepwise two-cut solution is given by $$\begin{aligned}
v(Z)=&\frac{1}{\pi i\lambda} \log{\left( \frac{f(Z)-\sqrt{f^2(Z)-4Z}}{2} \right)}
+\frac{n}{\pi i\lambda} \log{\left( \frac{q(Z)+\sqrt{q^2(Z)-4}}{2} \right)}-\frac{n_0}{\lambda}, \nonumber \\
&f(Z)=f_0+f_1 Z, \qquad q(Z)=\frac{q_1Z-q_0D_1}{Z-D_1} .
\label{CS-2cut-sol}\end{aligned}$$ This expression involves five constants: $f_0$, $f_1$, $q_0$, $q_1$ and $D_1$, and we can rewrite $f_0$, $f_1$, $q_0$ and $q_1$ by $A_1$, $B_2$ and $D_1$ through (\[f\_0f\_1-AB\]) and (\[q\_0q\_1-AB\]). Also we can fix $A_1$, $B_2$ and $D_1$ by imposing the normalization condition (\[cycle-CS2\]) and the boundary conditions (\[decomposition-boundary\]), (\[f(Z)-boundary-2cut\]) and (\[q(Z)-boundary\]), and will obtain the solution consistently. This agrees with the stepwise two-cut solution via the integral formula (\[two-cut-CS\]).
The generalization of such a derivation via holomorphy to the stepwise multi-cut solution (\[multi-cut-CS\]) is straightforward. However the generalization to the composite type solution (\[composite-CS\]) would be difficult. As we can see in (\[v0soln\]), the holomorphic function $f(Z)$ has a pole at $Z=-1$, which is not the location of any step. Such additional poles may appear in the composite solution generally and, we have not understood the correct rule for the assumptions on $f(Z)$ and $q(Z)$ in these cases yet.
Comments on the positivity of $\{n_i \}$. {#app-negative-n}
==========================================
[cc]{}
![ (Left) Symmetric four-cut solution through the Newton method ($\lambda=0.5, N=100$). The question is whether the red arrow interval is “a negative step" or “a small gap". (Right) Sketch of the composite (two+two)-cut solution. This type of the solution may describe the symmetric four-cut solution if $a$ is sufficiently small.[]{data-label="fig-four"}](negative-n.eps "fig:")\
Symmetric four-cut solution
![ (Left) Symmetric four-cut solution through the Newton method ($\lambda=0.5, N=100$). The question is whether the red arrow interval is “a negative step" or “a small gap". (Right) Sketch of the composite (two+two)-cut solution. This type of the solution may describe the symmetric four-cut solution if $a$ is sufficiently small.[]{data-label="fig-four"}](image-sym-four-cut.eps "fig:")\
Composite (two+two)-cut solution
When we considered the stepwise multi-cut solutions, we assumed that $\{ n_i \}$ in (\[CSgansatz\]) are positive integers. In this appendix, we discuss why we imposed this assumption.
Actually we can find a numerical solution of the saddle point equation (\[eom-CS\]) plotted in Figure \[fig-four\] (left) through the Newton method. This solution seems to have “a negative step” against our assumption. (Here “a negative step” means a negative $n_j$ in (\[CSgansatz\]).) However we cannot distinguish a negative step and a small gap through the numerical calculation. (Here “a gap” means $a_{j+1} \neq b_{j}+2 \pi i n_j$ in (\[CSgansatz\]).)
If it was a negative step, we would naively expect that this solution may be described by our stepwise multi-cut solution (\[multi-cut-CS\]) with a negative $n$. However, if we set $n$ negative, the eigenvalue density (\[rho-general\]) near the negative step may become negative[^26] as $\rho(Z) \sim n \log (Z-D)$. Since negative eigenvalue densities are not allowed physically, our stepwise multi-cut solution (\[multi-cut-CS\]) may not be applied to the solution in Figure \[fig-four\]. This is one reason that we restrict $\{ n_i \}$ to be positive.
In addition, we can indeed find a solution which has a gap rather than the negative step by composing two stepwise two-cut solutions. See the sketch in Figure \[fig-four\] (right). In the rest of this appendix, we will derive this solution and show another evidence that the negative step solution may not be allowed. Besides we will see that this solution itself has several interesting properties.
We assume that the solution is symmetric under $z \to -z$ and the four cuts locate on $[-b-\pi in, -d-\pi in]$, $[-d+\pi in, -a+\pi in]$, $[a-\pi in, d-\pi in]$ and $[d+\pi in, b+\pi in]$ as in Figure \[fig-four\] (right). We take $n$ positive even for simplicity[^27].
Through the formula for the general multi-cut solution (\[composite-CS\]), we obtain the resolvent $$\begin{aligned}
v(Z)=\oint_{C_1^{(1)} \cup \, C_2^{(1)} \cup \, C_1^{(2)} \cup \, C_2^{(2)}} \frac{dW}{4\pi i} \frac{1}{\pi i\lambda} \frac{\log{W}}{Z-W} \sqrt{\frac{(Z-A)(Z-1/A)(Z-B)(Z-1/B)}{(W-A)(W-1/A)(W-B)(W-1/B)} },
\label{Migdal-4cut}\end{aligned}$$ where the contour $C_1^{(1)}$, $C_2^{(1)}$, $C_1^{(2)}$ and $C_2^{(2)}$ encircle $[1/B, 1/D]$, $[1/D, 1/A]$, $[A, D]$ and $[D,B]$ respectively as shown in Figure \[fig-cycle-4cut\]. By regarding the value of $\log W$ on the cuts, this integral becomes $$\begin{aligned}
v(Z)=&v_0(Z)+v_1(Z),
\label{v0+v1}
\\
v_0(Z)=&\oint_{C^{(1)} \cup \, C^{(2)}} \frac{dW}{4\pi i} \frac{1}{\pi i\lambda} \frac{\log{W}}{Z-W} \sqrt{\frac{(Z-A)(Z-1/A)(Z-B)(Z-1/B)}{(W-A)(W-1/A)(W-B)(W-1/B)} },
\label{v0int}
\\
v_1(Z)=&\oint_{C_1^{(1)} \cup \, C_1^{(2)}} \frac{dW}{4\pi i} \frac{-n/\lambda}{Z-W} \sqrt{\frac{(Z-A)(Z-1/A)(Z-B)(Z-1/B)}{(W-A)(W-1/A)(W-B)(W-1/B)} },
\nonumber
\\
&+\oint_{C_2^{(1)} \cup \, C_2^{(2)}} \frac{dW}{4\pi i} \frac{+n/\lambda}{Z-W} \sqrt{\frac{(Z-A)(Z-1/A)(Z-B)(Z-1/B)}{(W-A)(W-1/A)(W-B)(W-1/B)} },
\label{v1int}\end{aligned}$$ where the contour $C^{(1)}$ and $C^{(2)}$ encircle the cut $[1/B,1/A]$ and $ [A,B]$ respectively as shown in Figure \[fig-cycle-4cut\].
[cc]{}
![ (Left) Sketch of the integral contours of the composite (two+two)-cut solution (\[Migdal-4cut\]). (Right) Branch cuts of the resolvent of the (two+two)-cut solution (\[v0soln\]) plus (\[v1-soln\]) on the $Z$-plane. The solid lines denote the branch cuts on the first sheet of the square root. The broken lines denote the branch cuts on the second sheet of the square root. []{data-label="fig-cycle-4cut"}](cycle-4cut.eps)
![ (Left) Sketch of the integral contours of the composite (two+two)-cut solution (\[Migdal-4cut\]). (Right) Branch cuts of the resolvent of the (two+two)-cut solution (\[v0soln\]) plus (\[v1-soln\]) on the $Z$-plane. The solid lines denote the branch cuts on the first sheet of the square root. The broken lines denote the branch cuts on the second sheet of the square root. []{data-label="fig-cycle-4cut"}](fig-2-2cut.eps)
First we evaluate $v_0(Z)$. Through a similar computation to (\[v-1+2-int\]) and (\[Migdal-ABJM\]), we obtain $$\begin{aligned}
v_0(Z)=& \frac{1}{\pi i\lambda} \log{\left( \frac{f(Z)-\sqrt{f^2(Z)-4Z}}{2} \right)},
\qquad
f(Z)=\frac{f_0+f_1Z+f_0Z^2}{Z+1},
\nonumber
\\
f_0=&\frac{2 \sqrt{AB}}{\left( \sqrt{A}+\sqrt{B} \right) \left( 1+\sqrt{AB} \right)},
\quad
f_1=\frac{2\left( \sqrt{B} (1+A)(B+1/B)- \sqrt{A} (1+B)(A+1/A) \right)}{\sqrt{AB} (B+1/B-A-1/A)}.
\label{v0soln}\end{aligned}$$ Here we can show that $\sqrt{f^2-4Z} \propto \sqrt{(Z-A)(Z-1/A)(Z-B)(Z-1/B)}$. This term resembles (\[v-1+2\]) and the DMP solution, while it shows a logarithmic divergence at $Z=-1$. The branch cut from $Z=-1$ may terminate at $Z=-1$ on the second sheet of the square root. See Figure \[fig-cycle-4cut\].
Next we consider $v_1(Z)$. Since this integral is complicated, we employ holomorphy discussed in Appendix \[app-holomorphy\] to derive $v_1(Z)$. Similar to (\[eom-v1\]), $v_1(Z)$ satisfies $$\begin{aligned}
-\frac{n}{\lambda}
&=
\lim_{\epsilon \rightarrow 0} \left[ v_1(Z+i\epsilon)+v_1(Z-i\epsilon) \right] ,
\qquad
(Z \in [1/B,1/D], [A,D]),
\nonumber
\\
+\frac{n}{\lambda}
&=
\lim_{\epsilon \rightarrow 0} \left[ v_1(Z+i\epsilon)+v_1(Z-i\epsilon) \right] ,
\qquad
(Z \in [1/D, 1/A], [D,B]).
\label{eom-v1-4cut}\end{aligned}$$ Besides $v_1(Z)$ is symmetric under $$\begin{aligned}
v_1(1/Z)=-v_1(Z),
\label{v-sym}\end{aligned}$$ which can be seen from the definition of $v_1(Z)$ (\[v1int\]). We also assume that $v_1(Z)$ satisfies the boundary conditions $$\begin{aligned}
\lim_{Z \rightarrow \infty} v_1(Z)=s,
\qquad
\lim_{Z \rightarrow 0} v_1(Z)=-s,
\label{v1-boundary}\end{aligned}$$ where $s$ is a constant and the symmetry (\[v-sym\]) has been taken into account.
From (\[eom-v1-4cut\]), we can find a function which is holomorphic on the cuts as $$\begin{aligned}
g(Z)=e^{\frac{\pi i \lambda}{n} v_1(Z)}-e^{-\frac{\pi i \lambda}{n} v_1(Z)}.
\label{g(Z)}\end{aligned}$$ Besides, through the boundary conditions (\[v1-boundary\]), $g(Z)$ satisfies $$\begin{aligned}
g_2:=\lim_{Z \rightarrow \infty} g(Z)=e^{\frac{\pi i \lambda}{n} s}-e^{-\frac{\pi i \lambda}{n} s},
\quad
\lim_{Z \rightarrow 0} g(Z)=e^{-\frac{\pi i \lambda}{n} s}-e^{\frac{\pi i \lambda}{n} s} = -g_2.
\label{g(Z)-boundary}\end{aligned}$$ By solving (\[g(Z)\]), we obtain $$\begin{aligned}
v_1(Z)=\frac{n}{\pi i\lambda} \log{\left( \frac{g(Z)+\sqrt{g^2(Z)+4} }{2} \right)}.
\label{v1-soln}\end{aligned}$$ Here we impose the following relation $$\begin{aligned}
\sqrt{g^2(Z)+4} \propto \sqrt{(Z-A)(Z-1/A)(Z-B)(Z-1/B)},
\label{g(Z)-cut}\end{aligned}$$ so that $v_1(Z)$ has the suitable cuts. Similar to $q(Z)$ in appendix \[app-holomorphy\], $g(Z)$ has to have some singularities in order to satisfy both this relation and the boundary conditions (\[g(Z)-boundary\]). Since the left hand side of the relations (\[eom-v1-4cut\]) is discontinuous at $Z=D$ and $1/D$, we assume that $g(Z)$ has poles there. Also $g(Z)$ satisfies $g(1/Z)=-g(Z)$ through (\[v-sym\]) and (\[g(Z)\]). Then we can find $g(Z)$ which satisfies (\[g(Z)-boundary\]) and (\[g(Z)-cut\]) as $$\begin{aligned}
g(Z)=\frac{g_2 (Z^2-1)}{(Z-D)(Z-1/D)},
\label{g(Z)-laurent}\end{aligned}$$ where the following relations have been imposed on the constants $$\begin{aligned}
-8 \frac{D+1/D}{g_2^2+4}=-\left( A+\frac{1}{A}+B+\frac{1}{B} \right) ,
\quad
\frac{-2g_2^2+4D^2+4/D^2+16}{g_2^2+4}=2+\left( A+\frac{1}{A} \right) \left( B+\frac{1}{B} \right).
\label{g2-AB}\end{aligned}$$ These relations can be written as $$\begin{aligned}
&g_2=2 \sqrt{\frac{2(D+1/D)-(A+1/A+B+1/B)}{A+1/A+B+1/B}},
\label{g2-ABZ0} \\
&\left( D+1/D \right)^2-4 \kappa \left( D+1/D \right)+4=0,
\quad
\kappa := \frac{4+(A+1/A)(B+1/B)}{2(A+1/A+B+1/B)},
\label{Z0-AB}\end{aligned}$$ and the second equation leads to $$\begin{aligned}
D=\exp{\left[ {\rm arccosh} \left( \kappa+\sqrt{\kappa^2-1} \right) \right]}.
\label{Z0-soln}\end{aligned}$$ In this way, $g_2$ and $D$ are determined by $A$ and $B$.
We can confirm that the obtained $v_1(Z)$ is consistent with the integral formula (\[v1int\]) by comparing them numerically[^28]. See Figure \[fig-v1\].
[cc]{}
![ Plots of $v_1(Z)$ via holomorphy (\[v1-soln\]) (red curves) and the integral formula (\[v1int\]) (blue curves). We take $A=1.1$ and $B=1.5$, and $D$ is fixed via (\[Z0-soln\]). (See footnote \[ftnt-Z0\] about $D$ in (\[v1int\]).) This agreement indicates that holomorphy provides the answer of the integral (\[v1int\]). Note that the plateaus in the real part correspond to the left hand side of (\[eom-v1-4cut\]). Besides, the imaginary parts of $v_1(Z)$ may provide the eigenvalue densities. []{data-label="fig-v1"}](real-v1.eps "fig:")\
Real part of $v_1(Z)$
![ Plots of $v_1(Z)$ via holomorphy (\[v1-soln\]) (red curves) and the integral formula (\[v1int\]) (blue curves). We take $A=1.1$ and $B=1.5$, and $D$ is fixed via (\[Z0-soln\]). (See footnote \[ftnt-Z0\] about $D$ in (\[v1int\]).) This agreement indicates that holomorphy provides the answer of the integral (\[v1int\]). Note that the plateaus in the real part correspond to the left hand side of (\[eom-v1-4cut\]). Besides, the imaginary parts of $v_1(Z)$ may provide the eigenvalue densities. []{data-label="fig-v1"}](imaginary-v1.eps "fig:")\
Imaginary part of $v_1(Z)$
Now we obtain the resolvent $v=v_0+v_1$ via (\[v0soln\]) and (\[v1-soln\]). It involves the undetermined constant $A$ and $B$. They can be fixed by the boundary condition (\[boundary-CS\]) and the normalization condition $$\begin{aligned}
\frac{N_1}{N} =\int_{A}^{D} \rho(Z) dZ=\frac{1}{4\pi i} \oint_{C_1^{(2)}} \frac{w(Z)}{Z} dZ
\label{cycle-4cut}.\end{aligned}$$ We can numerically solve these conditions for given $n,\lambda,N_1/N$. See Figure \[fig-endpt-analysis-4cut\] for the result at a weak coupling. It correctly reproduces the solution obtained through the Newton method.
Lastly we discuss the properties of the solution. One question is whether it continues to a negative step solution. To answer it, we regard $A$ as the input parameter of the solution instead of $N_1/N$. As we take $A \to 1$ ($a \to 0$), if the solution continues to a negative step solution, the cut $[a,d]$ should remains finite. However the relation (\[Z0-soln\]) tells us that $$\begin{aligned}
\lim_{a \rightarrow 0} d= \sqrt{\frac{2a(B-1)}{B+1}}+O(a^{3/2}).
\label{Z0-lim}\end{aligned}$$ Thus the cut shrinks as $a \to 0$, and the solution rather continues to the stepwise two-cut solution with the cuts $[-b,0]$ and $[0,b]$. This result may indicate that the negative step is dynamically not allowed, and the numerical result plotted in Figure \[fig-four\] is our composite type solution[^29].
![Composite (two+two)-cut solution via the Newton method (blue) and our result (\[v1-soln\]) (red). We take $n=2$, $\lambda=0.25$, $N_1=15$ and $N_2=35$ in the Newton method. In our method, we ignore $v_0$ by regarding small $\lambda$, and consider the contribution of $v_1$ only. We solve the conditions (\[cycle-4cut\]) and (\[boundary-CS\]) numerically, and find $A$ and $B$. These two results agree very well. []{data-label="fig-endpt-analysis-4cut"}](endpt-analysis-4cut.eps "fig:")\
[^1]: E-mail address: [email protected]
[^2]: E-mail address: [email protected]
[^3]: One important question is whether these multi-cut solutions contribute to the path-integral. We do not consider this issue in this article.
[^4]: Although the resolvent of the stepwise multi-cut solution is not described by the higher genus generalizations of elliptic functions, the free energy is suppressed by $1/N^2$ as usual [@Morita:2017oev].
[^5]: If we use a new variable $\tilde{U}_i:=U_i e^{-2\pi i \lambda}$, becomes the saddle point equation of the Stieltjes-Wigert matrix model [@Tierz:2002jj]: $ \frac{1}{\tilde{U}_i} \log \tilde{U}_i = \frac{4\pi i \lambda}{N} \sum_{j \neq i}^N \frac{1}{\tilde{U}_i-\tilde{U}_j} $. The advantage of the $U_i$ variable [@Suyama:2016nap] is that it makes equations symmetric under $U \to 1/U$ corresponding to the symmetry $u \to -u$ in the original variable .
[^6]: We employ the script $C$ for the closed contours and $\mathcal{C}$ for the supports in this article.
[^7]: To perform this integral, we deform the contour $C_1$ so that it encloses the pole at $W=Z$ and the branch cut $W \in [-\infty, 0]$ of $\log W$ [@Kazakov:1995ae].
[^8]: \[ftnt-tilt\]We assume the condition ${\rm Re}(a_i) \le {\rm Re} (b_i) $ and ${\rm Im}(a_i) \le {\rm Im} (b_i) $. This is because the potential $V'(z)= \frac{1}{\pi i \lambda} z$ forces the eigenvalues to compose such a configuration when $\lambda$ is real and positive. We can see it from the results of the Newton method.
[^9]: Our ansatz (\[Migdal-CS2\]) is similar to the ansatz for the $m$-cut solution of Hermitian matrix models [@Migdal:1984gj] $$w(z)=\sum_{i=1}^{m} \oint_{C_i} \frac{dw}{4\pi i} \frac{V'(w)}{z-w} \prod_{i=1}^m \sqrt{\frac{(z-a_i)(z-b_i)}{(w-a_i)(w-b_i)}},
\label{Migdal-hermite}$$ where $a_i$ and $b_i$ denote the end points of the $i$-th branch cuts ($i=1,\cdots, m$). The difference is that the end point $B_1$ and $A_2$ do not appear in the inside of the square root in our ansatz (\[Migdal-CS2\]). Since $B_1$ and $A_2$ are the same point on the different sheets, even though they do not appear in the square root, $v(Z)$ satisfies the saddle point equation (\[eom-CS3\]) on the cuts.
[^10]: If we sum up the saddle point equation (\[eom-CS\]), we obtain $\sum_{i=1}^N u_i=0$. This implies that the center-of-mass of the eigenvalues is at the origin. Thus the first cut $\mathcal{C}_1$ may be on a negative sheet while the second cut $\mathcal{C}_2$ may be on a positive sheet: $n_0 \le 0$ and $n+n_0 \ge 0$.
[^11]: The second term can be written as $$\begin{aligned}
\log{\left( \frac{q(Z)+\sqrt{q^2(Z)-4}}{2} \right)}
=2i \arctan{\left( \sqrt{\frac{Z-A_1}{Z-B_2}} \sqrt{\frac{B_2-D_1}{D_1-A_1}} \right)}.
\label{two-cut-arctan}\end{aligned}$$
[^12]: Our result (\[two-cut-CS\]) differs from the resolvent (64) of our previous work [@Morita:2017oev] by a constant term. This is because Ref. [@Morita:2017oev] used a different variable $Z$ which was defined on page 17 of [@Morita:2017oev].
[^13]: If $N_1=N_2$, the solution becomes symmetric under $Z \to 1/Z$ and it makes the calculation much simpler. We obtain $B_2=1/A_1$ and $A_2=1/B_1=e^{\pi i n}$, and we need to consider only one of the boundary condition (\[boundary-CS\]).
[^14]: We use FindRoot in Mathematica in our numerical computation. Then we find various solutions depending on the initial condition of FindRoot. We choose the solution which is consistent with the result of the Newton method. It is unclear whether the other solutions are all meaningful, since the equations involve several multivalued functions which may lead to wrong numerical results. Also some solutions might correspond to the eigenvalue density involving negative values which are not allowed physically.
[^15]: Although we can derive the end points of the eigenvalue distribution, obtaining the distribution curve on the complex plane is technically difficult unless the coupling $\lambda$ is small as in (\[weak-ab\]).
[^16]: The resolvent (\[1+2-CS\]) behaves as $v(Z) \to c_1 Z+ c_0 +O(1/Z) $, $(Z \to \infty)$ and the boundary condition (\[boundary-CS\]) requires the two conditions: $c_1=0$ and $c_0=1$.
[^17]: This is because the cuts would tilt as we can see from the numerical result. See footnote \[ftnt-tilt\] also.
[^18]: Note that strong interactions work between $\mu_i$ and $\nu_j$ too in the saddle point equations (\[eom-ABJM\]), if they are separated by $(2n+1) \pi i$. However we could not find numerical solution in which $\mu_i$ and $\nu_j$ are separated by $(2n+1) \pi i$. Since the sign of interactions (\[eom-ABJM\]) between $\mu_i$ and $\nu_j $ in this case are opposite to those between $\mu_i$ and $\mu_j$, we presume that the forces cannot balance and the solution could not exist. (It would be important to clarify this point rigorously.) For this reason, we do not consider the analytic solutions for these configurations in this article.
[^19]: Although the real part of the instanton action (\[S-inst\]) at the leading order of the strong coupling agrees with the result of [@Drukker:2011zy; @Grassi:2014cla], the additional imaginary factor $iN \pi/ \lambda= i \pi k $ in (\[S-inst\]) does not appear in [@Drukker:2011zy; @Grassi:2014cla]. This contributes to the phase factor of the instanton action. However, since we have merely considered the value of the effective action, we cannot evaluate the additional phase factor coming from the deformation of the contour of the path-integral. Hence we cannot ask the precise relation between our multi-cut solution and the D2-brane instanton of [@Drukker:2011zy; @Grassi:2014cla]. In order to evaluate this phase factor, we may need to consider the path integral including the back-reaction, and it is a challenging problem.
[^20]: Indeed if we do not consider the back reaction, the classical equation of motion derived from the effective action $V_{\rm eff}(\mu)$ is given by $y=0$ where $y$ is the spectral curve of the DMP solution [@Drukker:2011zy]. We can easily see that it allows only the trivial solutions $\mu= \pm \alpha$.
[^21]: We can confirm that the imaginary part of the instanton action in the numerical calculation also agrees with the estimation (\[S-inst\]). We can also check that the results in the $N_2^{(M)}=2$ case are consistent with the $N_2^{(M)}=1$ case. These results are omitted in this article.
[^22]: The advantage of the derivation of the resolvent via holomorphy is that we do not need to perform the integral of the ansatz if we found a suitable holomorphic function. However, in the case of the CS matrix models, we do not have a guidance principle to find such a holomorphic function and we have to do it through trial and error. We mention related issues in footnote \[ftnt-holo\] .
[^23]: $v(Z)$ has the logarithmic singularity at $Z=0$ only on the second sheet of the square root.
[^24]: If the cut $\mathcal{C}_1$ or $\mathcal{C}_2$ crosses the branch cut of $\log Z$, (\[eom-v1\]) should be modified. A simple way is rotating the branch cut so that it avoids the cuts $\mathcal{C}_1$ and $\mathcal{C}_2$.
[^25]: \[ftnt-holo\]$q_m(Z)=e^{\frac{\pi i \lambda m}{n} (v_1(Z)-n/\lambda)}+e^{-\frac{\pi i \lambda m}{n} (v_1(Z)-n/\lambda)}$ is also holomorphic on the cuts $\mathcal{C}_1$ and $\mathcal{C}_2$, if $m$ is an integer. However, the resolvent obtained through $q_m(Z)$ may involve constants which cannot be determined through the boundary conditions unless $m = \pm 1$, and we do not consider these cases. Similar ambiguity exists in $f(Z)$ of (\[f(Z)\]) and other cases too. For the composite type multi-cut solutions, we may need general $m$. For example, $q(Z)$ with $m=2$ appears in (\[v-1+2\]).
[^26]: The argument of the appearance of the negative eigenvalue density is subtle, since it is generally difficult to find how the eigenvalues are distributed between the end points $A_1$ and $\{B_i\}$ of the cuts. However for a small real $\lambda$, the eigenvalues are distributed parallel to the real axis as we can read off from (\[weak-ab\]), and we can indeed see that a negative $n$ always causes negative eigenvalue density.
[^27]: In the case of an odd $n$, the branch cuts $C_i^{(j)}$ may locate near the branch cut of the $\log{Z}$ in the saddle point equation, and it makes the analysis a bit complicated.
[^28]: \[ftnt-Z0\] If we take $Z \rightarrow \infty$ in the integral formula (\[v1int\]), $v_1(Z)$ linearly grows as $$\begin{aligned}
\lim_{Z \rightarrow \infty} v_1(Z)=& Z \left[ \oint_{C_1^{(1)} \cup \, C_1^{(2)}} \frac{dW}{4\pi i} \frac{-n/\lambda}{\sqrt{(W-A)(W-1/A)(W-B)(W-1/B)}} \right.
\nonumber
\\
& \left. +\oint_{C_2^{(1)} \cup \, C_2^{(2)}} \frac{dW}{4\pi i} \frac{+n/\lambda}{\sqrt{(W-A)(W-1/A)(W-B)(W-1/B)}} \right]+ \mathcal{O}(Z^0),\end{aligned}$$ and the boundary condition (\[boundary-CS\]) demands vanishing this term. We can numerically check that, if $D$ is given by (\[Z0-soln\]) which has been derived via holomorphy, this term becomes 0. This coincidence supports the consistency of the integral formula (\[v1int\]) and holomorphy.
[^29]: By using the assumption about the four cuts and the change of the variables $u_j= \pm \pi i n +x_j $ where $\pm$ depends on which cut $u_j$ belongs to, we can show that $\{ x_j \}$ feel an effective potential $$\begin{aligned}
V(x)=n[-x-d+2(x+d)\theta(x+d)-2x\theta(x)+2(x-d)\theta(x-d) ] \end{aligned}$$ at a weak coupling ($|\lambda| \ll 1 $) [@Morita:2017oev]. Then the non-existence of the negative step solution implies the non-existence of “one-cut solution" in this potential. We presume that the potential at $x=0$ is so sharp that the one-cut solution is not allowed. See [@Okuyama:2017feo] for a related problem.
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"pile_set_name": "ArXiv"
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---
abstract: YourStuffHere
author:
- YourStuffHere
bibliography:
- '/tmp/adsfiles.bib'
- 'bangalore.bib'
title: YourStuffHere
---
=1
Introduction {#yourname-sec:introduction}
============
YourStuffHere
![\[yourname-fig:XXX\] YourStuffHere ](\figspath/yourname-figXXX){width="\textwidth"}
Conclusion {#yourname-sec:conclusion}
==========
YourStuffHere
We thank the conference organisers for a very good meeting and the editors for excellent instructions.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The first part of our analysis uses the wavelet method to compare the Quantum Chromodynamic (QCD) prediction for the ratio of hadronic to muon cross sections in electron-positron collisions, $R$, with experimental data for $R$ over a center of mass energy range up to about 7 GeV. A direct comparison of the raw experimental data and the QCD prediction is difficult because the data have a wide range of structures and large statistical errors and the QCD description contains sharp quark-antiquark thresholds. However, a meaningful comparison can be made if a type of “smearing” procedure is used to smooth out rapid variations in both the theoretical and experimental values of $R$. A wavelet analysis (WA) can be used to achieve this smearing effect. The second part of the analysis concentrates on the 3.0 – 6.0 GeV energy region which includes the relatively wide charmonium resonances $\psi(1^-)$. We use the wavelet methodology to distinguish these resonances from experimental noise, background and from each other, allowing a reliable determination of the parameters of these states. Both analyses are examples of the usefulness of WA in extracting information in a model independent way from high energy physics data.'
author:
- 'V.K. Henner'
- 'C.L. Davis'
- 'T.S. Belozerova'
date: 'Received: date / Accepted: date'
title: 'Using wavelet analysis to compare the QCD prediction and experimental data on $R_{e^+e^-}$ and to determine parameters of the charmonium states above the $D\bar D$ threshold'
---
Wavelet Transformations
=======================
Let us start with a brief description of the continuous wavelet transformation (WT). The WT of function $f(t)$ is defined by $$\begin{aligned}
w(a,t)=\frac{1}{\sqrt{aC_{\varphi}}} \int\limits_{-\infty}^{+\infty}\varphi^{*}
\left(\frac{t'-t}{a}\right)f(t')dt',
\label{g1}\end{aligned}$$ where $C_{\varphi}$ is a normalization constant subject to the choice of wavelet. The decomposition described by equation (\[g1\]) is performed by convolution of the function $f(t)$ with a bi-parametric family of self-similar functions generated by dilatation and translation of the analyzing function $\varphi(t)$ called a wavelet, $$\begin{aligned}
\varphi_{a,b}(t)=\varphi \left(\frac{t-b}{a}\right),
\label{g2}\end{aligned}$$ where the scale parameter $a$ characterizes the dilatation, and $b$ characterizes the translation. It is a kind of “window function” with a non-constant window width. High frequency wavelets are narrow due to the factor $1/a$, while low frequency wavelets are much broader. The function $\varphi(t)$ should be well localized in both time and Fourier space and must obey the admissibility condition, $\int\limits_{-\infty}^{+\infty}\varphi(t)dt$. This condition requires $\varphi(t)$ must be an oscillatory function and, if the integral (\[g1\]) converges, the completeness of the wavelet functions provides the existence of inverse transformation, $$\begin{aligned}
f(t)=\frac{1}{\sqrt{C_{\varphi}}} \int\limits_{-\infty}^{+\infty}\int\limits_{0}^{+\infty}\varphi
\left(\frac{t-t'}{a}\right)w(a,t')\frac{dt'da}{a^{5/2}} dt.
\label{e3}\end{aligned}$$
In contrast to Fourier analysis, the WT depends both on $t$ and the frequency providing an optimal compromise with the uncertainty principal. One of the advantages of wavelet analysis is a fairly low sensitivity of the restored signal to any physically reasonable continuation of the function $f(t)$ outside the interval $\left(t_{min},t_{max}\right)$ where the data are known. To fill in gaps between the experimental points we use a linear interpolation (different interpolations lead to minimal difference in the restored signal). Note that since the average value of any wavelet is zero, the mean value of the WT is zero, so that $\left< f\right>$ must be added to the reconstructed signal to restore the mean value of the original signal.
Wavelets with good localization and a small number of oscillations are commonly used to recognize the local features of data, and to find the parameters of dominating structures (location and scale/width). In this work, we use one of the most popular wavelets of this type, the so-called “Mexican Hat”, $$\begin{aligned}
\varphi(t)= \left( 1-t^2 \right ) e^{-t^2/2}.
\label{e4}\end{aligned}$$ This wavelet is plotted in Figure 1 for three values of the scale parameter, $a = 1$, 2 and 0.5.
![Wavelet “Mexican Hat” $\varphi(t)= \left( 1-t^2 \right ) e^{-t^2/2}$. Solid line $-$ $\varphi(t)$, dashed line $-$ $\varphi(t/2)$, dotted line $-$ $\varphi(t/0.5)$.[]{data-label="fig:1"}](Fig1_N.eps)
Wavelets and the $R$ ratio
==========================
Our first goal is to use wavelet methodology to compare the QCD prediction of $R$ to experimental data in the center of mass energy range up to about 7 GeV. In zeroth order in the strong coupling constant $\alpha_s$, the ratio $R$ is given by
$$\begin{aligned}
R(Q)=\frac{\sigma\left( e^+e^- \rightarrow hadrons \right )}
{\sigma\left( e^+e^- \rightarrow \mu^+\mu^- \right )} \approx
3 \sum_{q} e^2_q \equiv R^{(0)}(Q),
\label{e5}\end{aligned}$$
where the summation extends over the quark flavors $q=u,d,s,c$ available up to the center of mass energy $Q$. The QCD $\alpha_s$ corrections to $R^{(0)}(Q)$ can be presented in different forms. To be specific we adopt the form of reference \[1\], $$\begin{aligned}
R(Q)= 3 \sum_{q} e^2_q T\left( \rm {v}_q \right ) \left [ 1+ g\left( \rm{v}_q \right ) {\cal R} \right ],
\label{e6}\end{aligned}$$ where $$\begin{aligned}
{\cal R} = \frac{\alpha_s}{\pi}
\left [ 1+ \frac{\alpha_s}{\pi} + C_2 \left ( \frac{\alpha_s}{\pi} \right )^2 + C_3 \left ( \frac{\alpha_s}{\pi} \right )^3 + ... \right ],
\nonumber\\
{\rm v}_q = \left [1 - 4m_q^2 / Q^2 \right ] ^{1/2}, \; \;
T({\rm v}) = {\rm v} \left (3-{\rm v}^2 \right ) /2,
\nonumber\\
g({\rm v}) = \frac{4\pi}{3} \left [\frac{\pi}{2{\rm v}} - \frac{3 + {\rm v}}{4} \left ( \frac{\pi}{2} - \frac{3}{4\pi} \right )\right ]
\nonumber\end{aligned}$$ and the summation in (\[e6\]) is over all quark flavors whose masses $m_q$ are less than $Q/2$. For this analysis we use the quark masses, the form of the coefficients $C_2$ and $C_3$ calculated in \[3\] and the energy dependence of $\alpha_s\left (Q^2 \right )$ from \[2\].
It is very hard to determine the role of these QCD corrections by comparison with experimental data for $R(Q)$ due to the large statistical errors and plethora of overlapping and interfering resonances in this region. An additional complicating factor is that the QCD perturbative approach exhibits sharp quark-antiquark thresholds. However, a meaningful comparison can be made by applying some type of “smearing” procedure, which has the effect of smoothing out rapid variations in both the theoretical and experimental values of $R$.
Before describing our wavelet analysis of this data it is worth considering the methodology developed in references \[1\] and \[2\] to compare experimental data and QCD predictions. The smearing procedure used in these analyses calculates a smeared ratio $R$ as follows, $$\begin{aligned}
R(s,{\rm \Delta})= \frac{{\rm \Delta}}{\pi} \int \limits_{0}^{s_{max}} \frac{R(s')}
{\left(s'-s \right)^2 + {\rm \Delta}^2 } ds',
\label{e7}\end{aligned}$$ where $\sqrt{s}=Q$ is the square of the center of mass energy and ${\rm \Delta}$ is the “smearing” parameter. In references \[1\] and \[2\] to evaluate the integral (\[e7\]), it was necessary to exclude sharp resonances, such as the $\psi(3.100)$ and the much wider $\rho$ peak. In addition, a term is added to account for the contribution from $s_{max}$ to $\infty$, assuming that $R$ remains constant above $s_{max}\approx 60 \; {\rm GeV}^2$. Originally, the smearing procedure in reference \[1\] supposes a global constant value ${\rm \Delta} = 3 \; {\rm GeV}^2$ in (\[e7\]). However it was found in reference \[2\] that for different energy regions it would be better to use different values of ${\rm \Delta}$. Note that the use of an energy dependent ${\rm \Delta}$ in (\[e7\]) reflects the necessity for different treatment of different energy scales.
The WT methodology provides an alternative, model independent, smearing method, which does not require different treatment in differing energy scales. Under WT to separate the signal from the background noise, wavelet reconstruction is performed for scales greater than a certain scale $a_{noise}$ – the boundary, or cut-off, scale \[4\]. In deciding on the appropriate boundary scale that will separate the noise-like high frequency components of the data we take a pragmatic line of reasoning. That is, the best choice for $a_{noise}$ is the smallest value which will smooth out any rapid variations in the data enabling us to reproduce stable results for low frequencies (resonance area). A similar pragmatic strategy was applied in the analyses of \[1\] and \[2\] in choosing the parameter ${\rm \Delta}$ of equation (\[e7\]). The best value of ${\rm \Delta}$ is large enough to compare the smeared $R$ with QCD models, but not so large that all the fine detail of the data is smoothed away.
![Wavelet reconstruction of $R_{e^+e^-}$. Dashed and solid bold curves correspond to wavelet reconstruction of the experimental data with $a_{noise} = 0.6$ and $a_{noise} = 1.2$, respectively. Dashed and solid thin curves correspond to wavelet reconstruction of QCD calculations (\[e6\]) with the same $a_{noise}$ values.[]{data-label="fig:2"}](Fig2_ND.eps)
Figure 2 displays wavelet reconstructed (smeared) experimental data with cut-off values $a_{noise} = 0.6$ (bold dashed curve) and $a_{noise} = 1.2$ (bold solid curve). The data is a compilation of measurements from many different experiments obtained from the Particle Data Group (PDG) \[5\]. In contrast to the analyses of \[1\] and \[2\], under the WT approach there is no need to remove sharp resonances ($\psi(3.100)$ etc.) by hand, they effectively become part of the high frequency noise and the wavelet analysis smears them out. The thin dashed line and thin solid line are wavelet reconstructed QCD curves with $a_{noise} = 0.6$ and $a_{noise} = 1.2$, respectively. As can be seen, the two curves with $a_{noise} = 1.2$, representing experiment and theory, are in good agreement. It should be noted that the contribution to all of the restored data curves in Figure 2 from the data region above 6.5 GeV is negligible.
$\psi$ states above the $D\bar D$ threshold and the wavelet procedure
======================================================================
Detailed information on the charmonium resonances $\psi(1^-)$ above the $D\bar D$ threshold at 3.73 GeV comes primarily from the measurement of $R_{e^+e^-}$. This data, provided by the Particle Physics Data Group (PDG) \[5\], comes from the work of many experimental collaborations over a period of more than 30 years. Several broad vector resonances are observed with varying degrees of clarity. The current best estimate of the masses and widths from the PDG are presented in Table 1.
--------------- ------------------- ------------------- ------------------- -------------------
PDG \[5\] BES \[6\] PDG \[5\] BES \[6\]
$\psi(3.770)$ 3.773$\pm$ 0.0003 3.772 $\pm$ 0.002 0.027 $\pm$ 0.001 0.030 $\pm$ 0.009
$\psi(4.040)$ 4.039 $\pm$ 0.001 4.040 $\pm$ 0.004 0.080 $\pm$ 0.010 0.085 $\pm$ 0.012
$\psi(4.160)$ 4.191 $\pm$ 0.005 4.192 $\pm$ 0.007 0.103 $\pm$ 0.008 0.072 $\pm$ 0.012
$\psi(4.415)$ 4.421 $\pm$ 0.004 4.415 $\pm$ 0.008 0.062 $\pm$ 0.020 0.072 $\pm$ 0.019
--------------- ------------------- ------------------- ------------------- -------------------
The main difficulty encountered by each of the collaborations in making these measurements is large statistical errors and hence the difficulty of separating resonance “signal” from noise. There is significant disagreement between the collaborations on many of the resonance parameters. Therefore, due to the possibility of systematic errors we do not combine measurements from different experiments, but choose to base this initial analysis on data from the BES collaboration which exhibits clear evidence of four broad resonances above the $D\bar D$ threshold. In addition to the PDG values, Table 1 shows the masses and widths of these resonances from the most recent BES analysis \[6\].
Before analyzing the charmonium resonances parameters, we first determine the non-resonant background contribution using a WT followed by wavelet reconstruction. This methodology, with its excellent scaling property, allowing the analysis of data with varying resolution, is ideally suited to separate resonances from noise, background and each other. A very helpful representation of the WT revealing all the features of the complete spectrum of the signal is the “time- frequency” plane (Figure 3(a)). This is a multi-resolution spectrogram, which shows the frequency (scale) contents of the signal as a function of energy. Each pixel on the spectrogram represents $w(a,t)$ for a particular $a$ (scale) and $t$ (in our case $t$ is energy, $E \equiv Q$). The location of spots on the vertical axis (scale axis, $a$) corresponds to the width of the maximum. The intensity of dark spots shows the amplitudes of maxima. The WT image (wavelet plane) of the BES charmonium data obtained with the “Mexican Hat” wavelet is shown in panel (a) of Figure 3. The WT localizes the structures in a fashion that allows us to estimate the masses of the resonances and their widths – all four $\psi$ resonances are clearly seen on the wavelet plane. The straight horizontal line corresponds to the boundary scale $a_{noise}$, which can be chosen to cut off small scale structures, which in this case corresponds to experimental noise. By choosing a larger value of $a_{noise}$ it is possible to also cut out the resonances, leaving only the background. The background curves for three substantially different choices of $a_{noise}$ (0.15, 0.25 and 0.35) are displayed in Figure 3(b). In the following analysis we show that the charmonium resonance parameters are not sensitive to the choice of $a_{noise}$.
![ (a) Wavelet plane for $a \in [0.01,1]$; (b) background for $a_{noise} = 0.15$ (solid line), $a_{noise} = 0.25$ (dashed line), $a_{noise} = 0.35$ (dotted line).[]{data-label="fig:3"}](Fig3_N.eps)
For each value of $a_{noise}$ we subtract this “wavelet background” from the raw experimental data leaving only the contribution from the broad charmonium resonances. We then perform a least squares fit of the “signal” in this energy range to the sum of four Breit-Wigner resonances of the form, $$\begin{aligned}
R_{res} = \frac{9}{\alpha_{em}^{2}} \sum \limits_{r=1}^4 B_{lr} B_{hr} \frac{M_{r}^{2} \Gamma_{r}^{2}}
{\left ( s-M_{r}^{2} \right )^2 + M_{r}^{2} \Gamma_{r}^{2}},
\label{e8}\end{aligned}$$ where $M_{r}$, $\Gamma_{r}$, $B_{lr}$ and $B_{hr}$ are, respectively, the mass, total width, leptonic branching fraction and hadronic branching fraction of the resonance. Each Breit-Wigner has three fitted parameters, $M$, $\Gamma$ and the product of $B_{lr}$ and $B_{hr}$. It is worth noting that since these resonances partially overlap, if their decay channels are specified, we can improve their resolution by using the multi-channel unitary scheme described in reference \[7\]. We defer this investigation to a future analysis.
The values of the fitted parameters for each of the four Breit-Wigner resonances for backgrounds obtained with three different values of $a_{noise}$ are presented in Tables 2-4. The chi squared per degree of freedom for the three fits are 1.11, 1.22 and 0.99 for the $a_{noise}$ values of 0.15, 0.25 and 0.35, respectively.
[cccc]{} Resonance & Mass (GeV) & Width (GeV) & $B_l B_h \cdot 10^5$\
$\psi(3.770)$ & 3.772 $\pm$ 0.0002 & 0.032 $\pm$ 0.005 & 0.983 $\pm$ 0.076\
$\psi(4.040)$ & 4.042 $\pm$ 0.001 & 0.088 $\pm$ 0.013 & 0.921 $\pm$ 0.067\
$\psi(4.160)$ & 4.161 $\pm$ 0.006 & 0.100 $\pm$ 0.019 & 0.664 $\pm$ 0.069\
$\psi(4.415)$ & 4.430 $\pm$ 0.007 & 0.098 $\pm$ 0.014 & 0.936 $\pm$ 0.071\
[cccc]{} Resonance & Mass (GeV) & Width (GeV) & $B_l B_h \cdot 10^5$\
$\psi(3.770)$ & 3.772 $\pm$ 0.0001 & 0.022 $\pm$ 0.004 & 0.888 $\pm$ 0.094\
$\psi(4.040)$ & 4.043 $\pm$ 0.003 & 0.097 $\pm$ 0.011 & 1.110 $\pm$ 0.064\
$\psi(4.160)$ & 4.185 $\pm$ 0.005 & 0.083 $\pm$ 0.016 & 0.715 $\pm$ 0.077\
$\psi(4.415)$ & 4.423 $\pm$ 0.004 & 0.090 $\pm$ 0.016 & 0.819 $\pm$ 0.076\
[cccc]{} Resonance & Mass (GeV) & Width (GeV) & $B_l B_h \cdot 10^5$\
$\psi(3.770)$ & 3.773 $\pm$ 0.0001 & 0.022 $\pm$ 0.004 & 0.888 $\pm$ 0.090\
$\psi(4.040)$ & 4.043 $\pm$ 0.003 & 0.097 $\pm$ 0.011 & 1.155 $\pm$ 0.061\
$\psi(4.160)$ & 4.165 $\pm$ 0.004 & 0.083 $\pm$ 0.013 & 0.907 $\pm$ 0.073\
$\psi(4.415)$ & 4.423 $\pm$ 0.004 & 0.090 $\pm$ 0.014 & 0.889 $\pm$ 0.071\
As can be seen in Tables 2-4, the fitted parameters of the four charmonium resonances are not significantly different for the three values of $a_{noise}$. Therefore, in Figure 4 we present only the fitted curve (\[e8\]) for $a_{noise}= 0.35$.
It should be noted that in this analysis, after subtracting the wavelet background, we exclude the four highest energy data points from the fitting procedure. Without these points the $\psi(4.415)$ resonance is “well-shaped”; if these four points are included the resonance is no longer “well-shaped” and its width is about twice that of the values presented in Table 1.
![Breit-Wigner fit of BES data with “wavelet background” subtracted, $a_{noise}= 0.35$[]{data-label="fig:4"}](Fig4_N.eps)
Justification for excluding the four highest energy points can be made as follows. If instead of the wavelet fitted background we fit the data with four Breit-Wigners and a parabolic (or linear) background, including the four highest energy data points, we see that the fourth resonance is not “well-shaped” and the last few data points appear to be associated with the background rather than the 4th resonance, see Figure 5. Furthermore, the fitted width of this fourth resonance (Table 5) is significantly larger than that of the second and third resonances and the published PDG value. In order to show the association of the highest energy points with the parabolic background we do not subtract the background in Figure 5. It should be emphasized that inclusion or exclusion of these four data points does not significantly alter the background curve obtained via the wavelet method and that the choice of a polynomial background is arbitrary, whereas the wavelet background is obtained from the data itself in a model independent manner.
[cccc]{} Resonance & Mass (GeV) & Width (GeV) & $B_l B_h \cdot 10^5$\
$\psi(3.770)$ & 3.770 $\pm$ 0.001 & 0.021 $\pm$ 0.004 & 1.202 $\pm$ 0.097\
$\psi(4.040)$ & 4.048 $\pm$ 0.003 & 0.107 $\pm$ 0.010 & 1.946 $\pm$ 0.079\
$\psi(4.160)$ & 4.166 $\pm$ 0.003 & 0.086 $\pm$ 0.013 & 1.335 $\pm$ 0.067\
$\psi(4.415)$ & 4.426 $\pm$ 0.005 & 0.148 $\pm$ 0.022 & 1.001 $\pm$ 0.085\
![Breit-Wigner and parabolic background fit of BES data (contrary to Fig.4 the background is not subtracted from the data).[]{data-label="fig:5"}](Fig5_N.eps)
To demonstrate the reliability of our WT methodology in extracting the parameters of these charmonium resonances we performed a study using simulated data as follows. Starting from the PDG values of masses and widths of the four charmonium resonances given in Table 1 we generate a spectrum of four Breit-Wigners of the basic form of equation (\[e8\]). To simulate real experimental data we discretize the spectrum from 2.0 to 5.5 GeV in 0.05 GeV intervals, then add a random vertical shift from -0.2 to +0.2 and random error bars in the range 0.15 to 0.4 to the data points. To complete the simulated data spectrum a representative background curve is added to the data points. Finally we pass this simulated data through the same WT analysis chain applied to the real data. This gives us a wavelet background curve and the masses and widths of the Breit-Wigner resonances after the wavelet background subtracted fit. In addition to applying our WT methodology to the simulated data we perform a Breit-Wigner and parabolic background fit in exactly the same way as we did for the real data.
This study was performed for several different initial background curves. In all such cases the WT methodology accurately reproduced the input background shape. Fitting the wavelet background subtracted simulated data, as described above, also accurately reproduces the masses and widths of the assigned Breit-Wigner resonances, independent of the particular input background curve. The parabolic background least squares fit of the simulated data was also able to reproduce the Breit-Wigner masses and widths, but in contrast led to fitted backgrounds very different from the input background. We believe the fact that the masses and widths were accurately reproduced, despite the fitted background being very different from the input background, is largely due to the clean nature of the simulated data. Real experimental data is clearly more complex, in which case the inability of the standard parabolic background least squares fit to extract an accurate background could lead to less reliable resonance parameters.
Conclusion
==========
Experimental measurements of the ratio, $R$, comprise a wide range of structures with large statistical errors, making direct comparison with the predictions of QCD very difficult. However, a meaningful comparison can be made provided that some kind of “smearing” procedure, similar to that described in \[1\], is used to smooth out rapid variations in $R$. A wavelet analysis can be used to achieve this smearing effect. We compare the WT of the predictions of perturbative QCD and experimental $R$ data. The wavelet reconstruction of the $R$ experimental data preserves its main features, but with damped statistical errors and threshold singularities. The WT of QCD perturbation theory is in good general agreement with the WT experimental data.
Using the wavelet methodology to obtain the background in the charmonium energy range above the naked charm threshold provides an important, model independent, alternative to other accepted methods. The masses and widths of the four vector mesons above the charm threshold, $\psi(3.700)$, $\psi(4.040)$, $\psi(4.160)$ and $\psi(4.415)$, obtained from fitting wavelet background subtracted data from the BES experiment, are found to be largely insensitive to the specific choice of the WT parameters and consistent with the BES and PDG reported values.\
The authors wish to thank Harrison Simrall and Adam Redwine for their work on the fitting algorithm and A. L.Kataev for useful advice.
[6]{}
E.C. Poggio, H.R. Quinn, S. Weinberg, Phys. Rev. D [**13**]{}, 1958 (1976)
A.C. Mattingly, P.M. Stevenson, Phys. Rev. D [**49**]{}, 437 (1994)
S.G. Gorishney, A.L. Kataev and S. A. Larin, Phys. Lett. B[**259**]{}, 144 (1991); L.R. Surguladze and M.A. Samuel, Phys. Rev. Lett. [**66**]{}, 560 (1991); Erratum ibid [**66**]{}, 2416 (1991)
V.K. Henner, P.G. Frick, T.S. Belozerova, V.G. Solovyev, Eur. Phys. J. C [**26**]{}, 3 (2002)
Review of Particle Physics by the Particle Physics Data Group, Chin. Phys. C [**38**]{}, 090001 (2014)
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Due to low dimensionality, the controlled stacking of the graphene films and their electronic properties are susceptible to environmental changes including strain. The strain-induced modification of the electronic properties such as the emergence and modulation of bandgaps crucially depends on the stacking of the graphene films. However, to date, only the impact of strain on electronic properties of Bernal and AA-stacked bilayer graphene has been extensively investigated in theoretical studies. Exploiting density functional theory and tight-binding calculation, we investigate the impacts of in-plane strain on two different class of commensurate twisted bilayer graphene (TBG) which are even/odd under sublattice exchange (SE) parity. We find that the SE odd TBG remains gapless whereas the bandgap increases for the SE even TBG when applying equibiaxial tensile strain. Moreover, we observe that for extremely large mixed strains both investigated TBG superstructures demonstrate direct-indirect bandgap transition.'
author:
- Zahra Khatibi
- 'Afshin Namiranian[^1]'
- Fariborz Parhizgar
bibliography:
- 'ref.bib'
title: 'Strain impacts on commensurate bilayer graphene superlattices: distorted trigonal warping, emergence of bandgap and direct-indirect bandgap transition'
---
Introduction {#intro}
============
The stacking of graphene films adds an intriguing class of graphene-based 2D materials, with new and exceptional properties [@Mele2010; @Zan2011; @Trambly2012; @Kim2016a; @cao2018unconventional]. This new class of materials that are recognized by the relative angle between the adjacent layers, namely moiré pattern, possess interesting angle-dependent properties which are different from that of bulk or monolayer graphene [@Shallcross2008; @LopesdosSantos2012; @McCann2013; @Wang2013; @Moon2013; @San-Jose2013; @Cosma2014; @Chen2014; @Moon2014a; @Pham2014; @Dai2016]. Added to the low-temperature superconductivity at magic twist angle [@cao2018unconventional], the rotation dependent low-energy electronic behavior of the twisted bilayer graphene (TBG) includes fractional quantum hall effect [@Moon2012; @Cao2016; @Kim2017], Van Hove singularities [@Li2009a; @Brihuega2012], the appearance of the secondary Dirac points [@Wallbank2013a; @Mucha-Kruczynski2013; @Nam2017], the emergence of flat bands at the Fermi energy [@SuarezMorell2010; @Luican2011a; @Fang2016; @Cao2018a], and the reduction of group velocity in the limit of small twist angles leading to localization of Dirac electrons [@Bistritzer2010; @Uchida2014]. The matching periodicity of the lower and upper layer of the TBGs forming the well established commensurate structures results in moiré pattern of longer periodicity than that of the Bernal and AA-stacked bilayer graphene (BG) reaching the high-wavelength of thousands of atomic distance [@LopesDosSantos2007; @Latil2007; @Xu2014; @Wang2014b; @Wong2015a; @Koshino2015]. This group of TBGs are identified by the sublattice exchange (SE) parity and can be classified into two distinctive groups, odd and even, which resemble the low-energy characteristic of the Bernal and AA-stacked BG. The SE even structures are gapped due to pseudospin-orbit coupling, whereas the SE odd commensurate moiré structures have two massive bands which intersect at the charge neutrality point [@Mele2010].
The low-energy electronic properties of stacking of graphene layers can be extensively affected by strain due to low dimentionality [@Mucha-Kruczynski2011; @Savini2011; @Frank2012; @Amorim2015; @Guisset2016]. Pseudoscalar potentials and transverse electric fields formed by different homogeneous in-plane strains, on each layer of BG, can lead to bandgap opening. For asymmetrically strained BG the bandgap is shown to undergo a transition from direct to indirect [@Choi2010a]. Out of plane strains can also lead to the formation of the bandgaps as a consequence of the enhancement of sublattice inequivalence when pulling the layers apart [@Wong2012]. Furthermore, compressive strain normal to the Bernal stacked BG results in the increment of the interlayer interactions leading to an enhancement in the Lifshitz transition [@Bhattacharyya2016]. For in-plane strains, on the other hand, results are found to be analogous to those of monolayer graphene, i.e. only expansion or compression along the zigzag direction can lead to the emergence of bandgaps [@Verberck2012]. For small angeled TBG, the energy separation of low-energy van Hove singularities is shown to decrease as the lattice deformation increases and well-defined pseudo-Landau levels emerge. Also, the joint effect of strain and out-of-plane deformation leads to valley polarization and formation of a significant gap [@Yan2013].
Strain, however, is a costly method for tunning and emergence of the bandgaps for mono and specifically bilayer graphene [@Pereira2009a; @Verberck2012] where a much higher interface shear stress is required compared to monolayer graphene for the same level of axial strain [@Frank2012] and thus the slippage of the layers on the substrate becomes inevitable [@Roldan2015]. Here, we show that the large angled commensurate TBGs are promising platform for manipulation of the electronic structure at low strain costs, especially because the fabrication of the moiré structures with controlled stacking is experimentally feasible [@Kim2016; @Yankowitz2016a]. We investigate the electronic structure of different commensurate TBG superlattices with large misalignment angle close to Bernal stacked BG under in-plane strain. We aim to cover possible features driven by real space symmetries, specifically the SE parity and compare the electronic behavior of the two distinct SE odd and even structures when applying strain. To this end, we conduct DFT and tight-binding (TB) calculations and measure the strain-induced modification of the low-energy electronic structure. We find that when applying biaxial tensile strain, SE odd TBG superlattices remain gapless whereas the gap energy for the SE even TBG superstructures increases monotonically. We then take the advantage of the reasonably comparable results of [*ab initio*]{} and TB calculation to use the TB method as a less computationally expensive method to study the gap modulation for numerous strain configurations including small asymmetric strains ($<$5%) where we observe direct-indirect bandgap crossover.
The paper is organized as follows. In Sec.\[theo\], we discuss the details of the geometrical structure of commensurate superlattices and the methods we used. In Sec.\[res\], we demonstrate our results for the band dispersion of unstrained and strained TBGs, the modulation and relocation of band spacing, the behavior of the low-energy bands near the charge neutrality point and strained-induced changes of the gap energy for 441 different strain configurations. We summarize our findings in Sec.\[concl\].
Theory {#theo}
======
TBGs consist of two graphene layers rotated by the angle $\theta$ with respect to each other around the vector perpendicular to their plane. Thus, the primitive vectors of the individual layers are related to each other as, $\vec{a}_{1(2)}^{\prime}=e^{i \theta}\vec{a}_{1(2)}$, where $\vec{a}_{1(2)}$ ($\vec{a}_{1(2)}^{\prime}$) is the primitive vector of the upper (lower) layer. Moreover, the lattice translation vectors of the upper and lower layers on the span of their primitive vectors can be written as $\vec{\mathcal{T}}^{(\prime)}_{m^{(\prime)}n^{(\prime)}}=m^{(\prime)}~\vec{a}^{(\prime)}_1+n^{(\prime)}\vec{a}^{(\prime)}_2$, in which $m^{(\prime)}$ and $n^{(\prime)}$ are integers. The periods of the individual layers generally might not coincide with each other and hence the TBG structures become incommensurate. On the other hand, when at a specific angle $\theta_{mn}$ and distance $l$, the periods of the lower and upper layer coincide with each other, the commensuration takes place. In other words, while rotating the layers around the common fixed A sublattices at the origin, commensuration occurs when the translation vectors of the upper and lower layer addressing the next A sublattice become equal, i.e $\vec{\mathcal{T}}_{mn}=\vec{\mathcal{T}}^\prime_{m^\prime n^\prime}$ [@Moon2013]. Also, it can be shown that the total number of disclosed atoms in the commensurate supercell is $4(n^2+nm+m^2)$, and the relative rotation angle $\theta_{mn}$ at which commensuration takes place is [@Mele2012], $$\theta_{mn}=\cos^{-1} \left({\frac{n^2+4nm+m^2}{2(n^2+nm+m^2)}}\right).$$ Hereafter, we will use the notation $(m,n)$ to address commensurate superlattices throughout this study.
As discussed earlier, the low-energy electronic behavior of the moiré commensurate structures is strongly dependent on SE parity. Regarding SE parity, the commensurate structures generally can be addressed in two distinct groups of SE odd and even. A commensurate lattice is SE odd when only one sublattice site of the upper layer, (A) sublattice site, coincides with that of the lower layer. On the other hand, when two sublattice sites, (A) and (B), of the neighboring layers coincide, the commensurate moiré structure is even [@Mele2012]. While SE odd structures are gapless, the ones that are even under SE parity are gaped and have curved bands. Also, the low-energy behavior of the SE symmetric and asymmetric structures is resembling of their limiting cases, i.e. AA and Bernal stacked BG at $\theta=0$ and $\theta=\pi /3$, respectively [@Mele2012].
![(Color online). Top view of schematic representation of commensurate TBG (a) (1,4) with $\theta \approx 38.21^{\circ}$, (b) (1,3) with $\theta \approx 32.20^{\circ}$. Red parallelograms are the supercell of $(1,4)$ and $(1,3)$, each of which include 28 and 52 atoms, respectively. (c) First BZ and the reciprocal lattice vectors of graphene (black). Blue and orange hexagons are sBZ of $(1,4)$ and $(1,3)$ TBG, respectively. (d) Magnified sBZ of $(1,4)$ and $(1,3)$ TBG to aid the visualization of high symmetry points and namings. \[fig1\]](fig1.pdf){width="\linewidth"}
Here, to capture possible features driven by real space symmetries including the SE parity and investigate the relative electronic structure and their changes under applied in-plane strain we’ll focus on two commensurate supercells, $(m,n)=(1,4)$ and $(1,3)$ that are even and odd under SE parity respectively (cf. Fig.\[fig1\]). The real space superlattice and supercell Brillouin zone (sBZ) of (1,4) and (1,3) TBGs are depicted in Fig.\[fig1\]. These structures have the shortest moiré pattern periodicity among the commensurate moiré structures and each of them consists of $28$ and $52$ atoms respectively. Also, the misalignment angle of the (1,4) and (1,3) superlattices, which are 38.21 Å and 32.20 Å, have the smallest deviations from the twist angle of Bernal BG.
To study the strain induced changes of the electronic properties of the (1,4) and (1,3) TBGs, we combine TB and first-principles calculations for both unstrained and strained TBGs. To this end, we perform first-principles calculations implemented in SIESTA code. We use double-$\zeta$ polarized basis (DZP) with Norm-conserving pseudopotential and the vdW exchange-correlation functional within the conjugate gradient method [@Soler2002]. Moreover, we sample the momentum space with $10\times 10 \times 1$ Monkhorst-Pack mesh grids. All DFT computations are converged over $400~{\rm Ry}$ energy mesh cutoff. The vacuum space perpendicular to the TBG layers is set to nearly $20~\AA$ to suppress the interactions between spurious images of the TBG. To obtain the band dispersion for unstrained TBG structures, we let both atomic coordinates and lattice vectors to relax until the forces on each atom become less than $0.04~{\rm eV/\AA}$.
To furthur study the low-energy physics of the TBGs performing TB method, we calculate the eigen energies and band dispersion of TBGs through the following Hamiltonian, $$\begin{aligned}
\mathcal{H} = -\sum_{\langle i,j\rangle}
t(\Vec{r}_i - \Vec{r}_j) |\Vec{r}_i\rangle\langle\Vec{r}_j| + {\rm h.c.},
\label{eq_Hamiltonian_TBG}\end{aligned}$$ where we use the tunneling integral equation [@Moon2014a], $$\begin{aligned}
-t(\vec{d})=V_{pp\pi}(\vec{d})[1-(\frac{\vec{d}\cdot \vec{e}_{z}}{\vec{d}})^2]+V_{pp\sigma}(\vec{d})(\frac{\vec{d}\cdot \vec{e}_{z}}{\vec{d}})^2,
\label{tunn. int.}\end{aligned}$$ to compute the hopping of the carriers in between single $p_z$ orbitals of carbon atoms located in graphene layers ($\Vec{r}_{i(j)}$) with relative distance $\vec{d}$. Here, we approximate the $a_0$ and $d_0$ with 1.42 Å and 3.3 Å, which are the intra and interlayer distance between carbon atoms, respectively. Moreover, the $\pi$ and $\sigma$ hybridization energy of the $p_z$ orbitals are approximated by $V_{pp\pi}(\vec{d})=V^0_{pp\pi}exp(-(d-a_0)/\delta_0)$, $V_{pp\sigma}(\vec{d})=V^0_{pp\sigma}exp(-(d-d_0)/\delta_0)$. We choose the nearest couplings as $V^0_{pp\pi}=-2.7$ eV, $V^0_{pp\sigma}=0.48$ eV and the decay length constant as $\delta_0=0.184a_0$.
Homogeneous lattice deformations which are uniform and equal in all in-plane directions, namely biaxial strains can be modeled by the change of the lattice constant. The more general lattice distortions that lead to the asymmetric deformation of the lattice structure including the uniaxial tensile strain can be modeled via changes of the lattice vectors. Here, to model the strained structures, we first modify the supercell vectors as, $\vec{R}_{i}^{\prime}=(1+\epsilon_{i})\vec{R}_{i}$, along any preferred direction $i=x,y$. Next, we optimize the TBG structure within DFT method by keeping the supercell vectors fixed at their strained values and letting the atoms to move. Within the TB approach, we use the same relation to alter the atomic coordinates to model the strained TBGs. Note that for both commensurate structures, based on our [*ab initio*]{} approach we find that the optimized positions of atoms deviate from the rigid ones used in TB method. However, for the sake of simplicity and also presenting a systematic approach applicable to any twist angle and external parameters such as strain, we use rigid atomic positions in the TB model.
Results and discussions {#res}
=======================
![Electronic band dispersion of (a,c) $(1,4)$ and (b,d) $(1,3)$ TBG along the high symmetry points of the first sBZ. While the lattice structures are unstrained in (a) and (b), the bands displayed in (c) and (d) belong to the commensurate TBGs when being exposed to 5% biaxial strain. Red dashed (Black solid) curves corresponds to DFT (TB) results. TB and DFT calculated bands are in excellent agreement in high energy interval $[-1,1]~{\rm eV}$ both for unstrained and strained TBGs. Insets in (a) and (b) are zoomed-in figures of band structures in low-energy limit near Fermi energy where the energy interval is $[-20,20]~$meV. While the low-energy bands are massive and gapped for unstrained SE symmetric $(1,4)$, the bands of unstrained SE odd $(1,3)$ are gapless and linear in $\vec{k}$. \[fig2\]](fig2.pdf){width="\linewidth"}
Fig.\[fig2\] illustrates the results for the DFT and TB computation of the band dispersion along the path of high symmetry points for TBG superstructures depicted in Fig.\[fig1\]. The DFT and TB results are in good agreement and demonstrate Dirac fermionic behavior for both superstructures close to the charge neutrality point. In the insets where we show the electronic bands within a small energy interval of $[-20,20]$ meV, close to Dirac cone conical points, the low-energy features driven by the SE parity emerge. As it is evident from the insets, contrary to the gapless band dispersion of unstrained $(1,3)$ TBG, we clearly observe gapped massive Dirac cones for unstrained $(1,4)$ TBG when we zoom in the vicinity of charge neutrality point. The low-energy bands of the $(1,3)$ are linear and degenerate in the scale of 20 meV and the massless Dirac cones intersect at the charge neutrality point. Hence, the energy interval in which the SE parity-driven low-energy behavior emerge scales inversely with the moiré period. Here our computations are consistent with the previous report of Ref [@Mele2012]. Also, our computations show that the renormalized Fermi velocity is 79 and 78 percent of the monolayer graphene Fermi velocity for the $(1,4)$ and $(1,3)$ superlattices respectively.
In panel (c) and (d), we present the band dispersion along the high symmetry points of the strained sBZ for $(1,4)$ and $(1,3)$ superlattices when applying 5% biaxial tensile strain. The geometrical changes of the lattice structure relative to the applied strain lead to the modification of the lattice vectors, their dual space counterparts, and the sBZ. Within the TB approach, these changes affect the carrier hopping to the $p_z$ orbitals of carbon atoms through the modulation of the relative distances between the atoms (See Eq.\[tunn. int.\]). The strain-induced changes of the DFT computed electronic structure of the TBGs which stem from the modification of expansion of the atomic orbitals and their hybridization, agrees well with those of the TB model. Furthermore, as it is can be seen from Fig.\[fig2\](c) and (d), despite the strong expansion of the lattice when applying 5% biaxial strain both TBG superlattices retain their unstrained electronic structure and remain linear close to the charge neutrality point due to the fact that biaxial strain preserves the real space lateral symmetries. Moreover, we find that the Fermi velocity reduces to an almost equal value of 0.71 ${\rm v_F^0}$ for both superlattices with ${\rm v_F^0}$ being the graphene Fermi velocity. As a result, the applied biaxial strain flattens the band dispersion close to the Fermi energy.
![(Color online). (a) DFT and TB (inset) calculated direct gap energy versus the applied biaxial strain on commensurate moiré structures shown in Fig.\[fig1\]. The gap energy for the SE symmetric $(1,3)$ TBG is robust and remains unaffected by the applied biaxial strain, whereas the gap for the SE even $(1,4)$ TBG increases monotonically regarding the applied strain. The ratio of changes in gap energy calculated by DFT for the $(1,4)$ TBG is approximately 1 meV per percent of applied strain. (b) The Fermi velocity versus the applied biaxial strain for $(1,4)$ and $(1,3)$ superstructures. The renormalized Fermi velocity for both superstructures scales inversely with the applied biaxial strain. The strain-induced renormalization of the Fermi velocity is approximately 0.15 $\rm{v^0_F}$ per percent of applied strain. \[fig3\]](fig3.pdf){width="\linewidth"}
Now, we compute the modification of the band spacing and Fermi velocity when applying biaxial strain. Fig.\[fig3\] shows the bandgap energy of commensurate $(1,4)$ and $(1,3)$ superlattices for tensile biaxial strains up to 10% computed by both TB and DFT. Both methods represent qualitatively the same results as they show similar trends for the gap energy modulation regarding the applied biaxial strain. Interestingly, we see that the gap energy remains almost unchanged for the $(1,3)$ TBG even in presence of strong biaxial strains. Hence, the huge distortion of the lattice structure has a minor effect on the gap energy of the $(1,3)$ TBG. Furthermore, the applied in-plane biaxial strain is more efficient in the modulation of the gap energy for the $(1,4)$ superlattice and the rate of the changes is 1meV per percent of applied strain based on DFT. The relative difference between the reported values of the gap energy of TB and DFT stems from the lack of electron-electron repulsions in the TB approach. Furthermore, since we use optimized structures when computing the band dispersion within the DFT approach, instead of the rigid strained atomic positions, we effectively start with different lattice structures. Although these differences in the lattice structures are small, yet, they lead to a different expansion of the atomic orbitals and their overlaps.
In panel (b) we present the Fermi velocity as a function of the applied biaxial tensile strain for both TBG superlattices. The Fermi velocity is almost identical for both superstructures and reduces to 0.63 ${\rm v_F^0}$. Thus, the strain-induced renormalization of the Fermi velocity is approximately 0.15 $\rm{v^0_F}$ per percent of applied strain and the Fermi velocity for both superstructures scales inversely with the applied biaxial strain.
![Electronic band dispersion of strained (a) $(1,4)$ and (b) $(1,3)$ TBG along the high symmetry points of the first sBZ. Blue solid curves correspond to $5\%$ tensile strain along the $x$ axis and green dashed lines are for $5\%$ tensile strain imposition along the $y$ direction. The electronic dispersion is enormously altered after application of tensile strain for both TBG structures. All strained structures bear a huge gap energy close to K valley, except for strained $(1,3)$ in the $y$ direction where the bands stay linear and gapless and the Fermi energy single state dislocates from $\vec{k}=$K. Also, the electronic bands of $(1,3)$ become massive after imposition of $5\%$ strain along the x-axis. \[fig4\]](fig4.pdf){width="\linewidth"}
We further compute the strain-induced modification of the gap energy regarding the biaxial strain for the next two smallest commensurate superstructures, $(1,7)$ and $(2,3)$ which are even and odd under SE parity. We find that similar to the $(1,3)$ superlattice, the band spacing is robust for the SE odd $(2,3)$ TBG and the band dispersion preserves its gapless behavior while being exposed to the biaxial strain. The gap energy modulation versus strain is shown in the Appendix. On the other hand, the band spacing increases monotonically with strain for the SE even $(1,7)$ superstructure, even though the gap is small. Moreover, the bandgap energy approaches 0 meV as the misalignment angle for SE even moiré superlattice becomes small in agreement with the previous report of Ref.[@LopesdosSantos2012]. Overall, when applying biaxial tensile strain, SE odd TBGs remain gapless and the SE even TBGs show increment of the gap energy.
Now to investigate whether the results of the biaxial straining, i.e. the SE odd $(1,3)$ TBG remains gapless and the gap energy for the SE even $(1,4)$ TBG increases with strain, can be generalized to other strain configurations, we study the non-equibiaxial and mixed strains. Fig.\[fig4\] illustrates two exemplary strain configurations in which both $(1,4)$ and $(1,3)$ superstructures are exposed to uniaxial tensile strains (5%) along the $x$ and $y$ directions. The asymmetric lattice distortion breaks the hexagonal symmetry of sBZ leading to three non-equivalent sBZ corners due to time reversal symmetry. Here we present the electronic bands for the strained TBGs along the path of high symmetry points close to $\kappa^{\prime}_1(\tilde{\kappa}^{\prime}_1)$. The SE even $(1,4)$ superstructure retains its unstrained electronic structure and stays gapped with two massive bands at the sBZ corner ($\kappa^{\prime}_1$). The gap energy, however, is enormously enhanced after applying uniaxial tensile strain in both directions. Compared to the unstrained $(1,4)$ superlattice the bands are flattened and the band velocity close to the charge neutrality point is reduced. Furthermore, strong band velocity discontinuity is observable for the uniaxially strained $(1,4)$ TBG along the $y$ axis at the $\kappa^{\prime}_1$. The $(1,3)$ bands close to the Fermi energy become massive after imposition of $5\%$ strain along the $x$ axis. All strained structures shown in Fig.\[fig4\] possess a huge gap energy close to $\kappa^{\prime}_1$ point, except for uniaxially strained $(1,3)$ along the $y$ direction where the bands are gapless and the conical point of the Dirac cone drifts away from the sBZ corner ($\tilde{\kappa}^{\prime}_1$). Here, the low-energy bands follow two distinct features. One becomes massive and flattened at the sBZ corner and the other preserves the linear behavior of the unstrained Dirac fermions. Also, a strong band velocity discontinuity is observable at the sBZ corner.
![(Color online). Surface maps of valence and conduction bands as a function of electronic wave vector for the diverse imposition of in-plane strain on $(1,4)$ TBG. Upper (lower) panels indicate conduction (valence) band. Black hexagons demonstrate the sBZ for the corresponding unstrained and strained structure. The band energies are shifted with regard to the Fermi energy so that the middle state between the highest occupied state and the lowest unoccupied state is set to 0 eV. Conduction band minimum and valence band maximum are displaced from the sBZ corners when applying strain, hence the gap energy shifts from sBZ corners. \[fig5\]](fig5.pdf){width="\linewidth"}
To get a deeper understanding of the strained electronic bands and to evaluate the band spacing, we plot the surface maps of the lowest conduction and the highest valence band over the entire sBZ for unstrained $(1,4)$ and $(1,3)$ superlattices and the strained configurations displayed in Fig.\[fig4\]. Fig.\[fig5\] is the resolution of the low-energy bands for the $(1,4)$ TBG and the map plots of Fig.\[fig6\] are those of the $(1,3)$ superlattice. All electronic bands displayed in figures \[fig5\] and \[fig6\] are shifted regarding the undoped Fermi energy state so that the middle state between the highest occupied level and the lowest unoccupied level is 0 eV. The first row of both plots shows the lowest conduction band whilst the second rows are the highest valence bands of the corresponding strain configuration. Moreover, Black hexagons depict the sBZ of the corresponding lattice structure. We clearly observe trigonal warping due to the non-orthogonal interlayer couplings [@Abergel2010], close to the sBZ corners of both valence and conduction bands for unstrained TBGs. This sublattice broken symmetry driven by interlayer interactions also results in the renormalization of the Fermi velocity of TBGs observable in Fig.\[fig2\](a,b). The threefold anisotropic behavior of the Fermi lines is strongly distorted as the uniaxial tensile strain is applied on both TBG superstructures. Therefore, the isoenergy lines at the sBZ corners for the unstrained TBGs split into two observable isoenergy pockets leading to the relocation of the conical point of the Dirac cone minibands from the sBZ corners. Thus, similar to the case of uniaxially strained monolayer graphene that the Dirac cones and hence the single Fermi state dislocate from the BZ corners [@Pereira2009a], the real bandgap energy of the uniaxially strained TBGs locates beyond the sBZ hexagon and cannot be identified along the path of the high symmetry points. Consequently, the bandgap should be evaluated with care. Furthermore, the strain-induced changes of the trigonal warping driven by modified interlayer coupling when applying uniaxial tensile strain are responsible for the reduction of the Fermi velocity and flattening of the bands in both TBG superstructures (cf. Fig.\[fig4\]). Also, the anisotropic strain-induced distortion of the Fermi lines results in the band velocity discontinuity observable at sBZ corners in Fig.\[fig4\]. Our computations reveal that the bandgap energy for the commensurate $(1,3)$ superstructure when stretched along the $x$ ($y$) axis with 5% uniaxial strain is 3 meV (4 meV) as a consequence of the broken real space symmetry. Therefore, the uniaxial tensile strains lead to bandgap opening in the SE odd $(1,3)$ superlattice.
![(Color online). Same plots as in Fig.\[fig5\] for TBG $(1,3)$. Analogous to the $(1,4)$ TBG, conduction band minimum and valence band maximum, and therefore the gap energy move away from sBZ corners after imposition of in-plane strain. \[fig6\]](fig6.pdf){width="\linewidth"}
Now, we use the TB model as a computationally less expensive method to investigate the modification of the bandgap with respect to the applied in-plane strain. Fig.\[fig7\] is the resolution of the gap energy for diverse strain configurations of the $(1,4)$ and $(1,3)$ superlattices. The gap energy is calculated for 441 strain configurations of the commensurate TBGs which are depicted as empty circles in Fig.\[fig7\]. These configurations include biaxial, uniaxial, compressive and mixed strains where the absolute value of the strain along the in-plane directions increases up to 10%. Panels (a) and (c) are the band spacing (direct and indirect) for $(1,4)$ and $(1,3)$ superlattices, respectively. Here we diagonalize the TB Hamiltonian over a dense mesh grid of wave vector ($\vec{k}$) in reciprocal space. Next we define and evaluate the least band spacing as ${\rm min(E_c}(\vec{k}))-{\rm max(E_v}(\vec{k}))$. Fig.\[fig7\](b) and (d) indicate the type of the bandgap, that is the bright areas depict the indirect and the dark blue areas show the direct bandgaps for the corresponding strain configuration. Interestingly, the bandgap for both commensurate TBG structures becomes indirect when applying strong mixed strains (cf. top left and bottom right corner of the panel (a) and (c)).
![(Color online). TB calculated surface plots of bandgap energy as a function of applied strain on (a) $(1,4)$ TBG and (c) $(1,3)$ TBG. The empty small circles correspond to real data and the background is the fitted formula. The resolution of the bandgap as direct or indirect for (b) $(1,4)$ and (d) $(1,3)$ TBGs. Bright areas in (b) and (d) demonstrate strain configurations in which the bandgap of the corresponding TBG is indirect. For highly mixed strain configurations in which the system is stretched along one in-plane direction and compressed along the other in-plane direction, the bandgap for both TBG structures becomes indirect. Non-equibiaxial compressive strains are more efficient in increasing the direct gap for both commensurate TBG structures. \[fig7\]](fig7.pdf){width="\linewidth"}
Moreover, the valence band maximum becomes energetically higher than the conduction band minima but in different valleys when both TBG structures are exposed to extremely large mixed strains (cf. the strain configuration $\epsilon_x=0.1$ and $\epsilon_y=-0.1$). Note that our DFT computations show that for the extremely large strained structures where the system is stretched along one in-plane direction and compressed along the other in-plane direction, the system remains integrated and the strained TBG is in the elastic region. As can be seen from the panel (a) and (c), the maximum value of gap energy is observable in symmetry broken highly compressed structures, generally when the compressive strain is along both directions but with different magnitude (See the bottom left corner of Fig.\[fig7\](c)). There also exist some mixed strain configurations that are efficient in increasing of the bandgap for $(1,4)$ and opening of the bandgap for $(1,3)$ TBG (cf. the strain situation $\epsilon_x=-0.05$ and $\epsilon_y=0.01$ for $(1,4)$ and the $\epsilon_x=-0.05$ and $\epsilon_y=0.02$ situation for the $(1,3)$). For both TBG structures studied here the bandgap does not exceed 15 meV. The general trend for the modification of the bandgap of the $(1,4)$ and $(1,3)$ superlattices regarding the in-plane strain is similar, except for the biaxial and some specific strain configurations. Contrary to the biaxial strain where the $(1,3)$ TBG remains gapless and unaffected by the lattice deformations, other strain configurations result in the emergence of a bandgap. Moreover, the equibiaxial compressive strain is not efficient in gap opening for $(1,3)$ TBG. In fact, for SE odd $(1,3)$ TBG when the lattice distortions are equal in all in-plane directions and the symmetries are not broken, the bandgap is 0 eV.
Conclusion {#concl}
==========
We have studied the impacts of in-plane strain on electronic properties of two exemplary SE odd and SE even commensurate TBG superlattices with large twist angle. We observed that the biaxial tensile strains leave the low-energy behavior of the SE odd TBG, i.e the gapless massless Dirac cones at the neutrality point, unchanged whilst they lead to an increment of bandgap for the SE even superlattice. Furthermore, we found that the renormalized Fermi velocity for both superstructures scales inversely with the applied biaxial strain. We took the advantage of the reasonable agreement between the TB and DFT calculated band dispersion of the unstrained and strained TBGs to tackle more than 400 strain configurations via the less computationally expensive approach of TB. There we found that for specific mixed strains where the TBGs are stretched along one in-plane direction and compressed along the other in-plane direction, both superlattices show direct-indirect bandgap transition. Consequently, the large angled commensurate TBGs are promising platform for manipulation of the electronic structure at low strain costs, specifically because the fabrication of the moiré structures with controlled stacking is experimentally feasible.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
We thank David S. L. Abergel for helpful discussions.
APPENDIX {#appendix .unnumbered}
========
Similar to the $(1,3)$ TBG superstructure the gap energy of the SE odd $(2,3)$ superstructure is unaffected by the biaxial tensile strain even when the strain is strong. On the other hand, the in-plane biaxial strain leads to the increment of the gap energy for the SE even $(1,7)$ superlattice analogous to the $(1,4)$ TBG. The rate of the changes in the gap energy is 0.1 meV/%.
![(Color online) Gap energy versus the applied biaxial strain for $(1,7)$ and $(2,3)$ moiré superstructures. The SE odd $(2,3)$ TBG remains gapless when applying biaxial tensile strain, whereas the gap for the SE even $(1,7)$ superlattice scales monotonically with the applied strain. \[a1\]](fig-app.pdf){width=".7\linewidth"}
REFERENCES {#references .unnumbered}
==========
[^1]: Author to whom any correspondence should be addressed
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We consider the effects that accretion from the interstellar medium onto the particles of an N-body system has on the rate of two-body relaxation. To this end, we derive an accretion-modified relaxation time by adapting Spitzer’s two-component model to include the damping effects of accretion. We consider several different mass-dependencies and efficiency factors for the accretion rate, as well as different mass ratios for the two components of the model.
The net effect of accretion is to accelerate mass segregation by increasing the average mass $\bar{m}$, since the relaxation time is inversely proportional to $\bar{m}$. Under the assumption that the accretion rate increases with the accretor mass, there are two additional effects that accelerate mass segregation. First, accretion acts to increase the range of any initial mass spectrum, quickly driving the heaviest members to even higher masses. Second, accretion acts to reduce the velocities of the accretors due to conservation of momentum, and it is the heaviest members that are affected the most. Using our two-component model, we quantify these effects as a function of the accretion rate, the total cluster mass, and the component masses. We conclude by discussing the implications of our results for the dynamical evolution of primordial globular clusters, primarily in the context of black holes formed from the most massive stellar progenitors.
author:
- |
Nathan Leigh$^{1}$, Alison Sills$^{2}$, Torsten Böker$^{1}$ [^1]\
$^{1}$European Space Agency, Space Science Department, Keplerlaan 1, 2200 AG Noordwijk, The Netherlands\
$^{2}$McMaster University, Department of Physics and Astronomy, 1280 Main St. W., Hamilton, Ontario, Canada, L8S 4M1
title: 'Modifying two-body relaxation in N-body systems by gas accretion'
---
\[firstpage\]
globular clusters: general – stellar dynamics – stars: formation – black hole physics.
Introduction {#intro}
============
For most of the life of a massive star cluster, two-body relaxation is the dominant physical mechanism driving its evolution [e.g. @henon60; @henon73; @spitzer87; @heggie03; @gieles11]. That is, the cumulative effects of long-range gravitational interactions between stars act to alter their orbits within the cluster. These interactions push the cluster toward a state of energy equipartition in which all objects have roughly the same kinetic energy. Consequently, the velocities of the most massive objects decrease, and they accumulate in the central regions of the cluster. Similarly, the velocities of the lowest mass objects increase, and they are subsequently dispersed to wider orbits. This mechanism, called mass segregation, also contributes to the escape of stars from their host cluster across the tidal boundary, with the probability of ejection increasing with decreasing stellar mass. Therefore, two-body relaxation acts to slowly modify the radial distribution of stellar masses within clusters, and can cause very dynamically evolved clusters to be severely depleted of their low-mass stars [e.g. @vonhippel98; @demarchi10; @leigh12].
Energy equipartition is an idealized state that should arise after the cumulative effects of many long-range interactions. In a real star cluster with a full spectrum of stellar masses, however, equipartition may not actually be achievable [e.g. @binney87; @heggie03]. As mentioned, the tendency towards energy equipartition reduces the velocities of the heaviest stars, causing them to sink in to the central cluster regions. Here, they are re-accelerated by the central cluster potential and gain kinetic energy. As this process proceeds, it leads to a contraction of the core and subsequently a shorter central relaxation time [e.g. @spitzer87; @heggie03]. A shorter relaxation time leads to a faster rate of energy transfer from heavier to lighter stars. Eventually, this makes the heaviest stars evolve away from equipartition.
This was first demonstrated by @spitzer69 using analytic techniques and a number of simplifying assumptions. @spitzer69 adopted a two-component system with masses $m_{\rm 1}$ and $m_{\rm 2}$ (where $m_{\rm 1} > m_{\rm 2}$), forming sub-systems with total masses $M_{\rm 1}$ and $M_{\rm 2}$. Provided that $M_{\rm 1} \ll M_{\rm 2}$, Spitzer derived the conditional requirement for a cluster to achieve energy equipartition in equilibrium. Based on this, Spitzer argued that energy equipartition could not be achieved in a cluster with a realistic mass spectrum, since there should always be enough mass in the heavier species for it to form a sub-system in the central cluster regions that decouples dynamically from the lighter species. This is commonly called the Spitzer instability [e.g. @spitzer87; @heggie03; @portegieszwart04].
A particularly compelling example of the Spitzer instability involves stellar-mass black holes (BHs) in globular clusters (GCs). @phinney91 first argued that BHs formed from the most massive stars should rapidly segregate into the core where they decouple dynamically from the rest of the cluster to form a distinct sub-system. Three-body scattering events then lead to the formation of BH-BH binaries, which in turn encounter other BHs and BH-BH binaries. These 3- and 4-body interactions are sufficiently energetic to eject the BHs from the cluster. In the end, most BHs are expected to be ejected, leaving only a handful behind.
This picture has recently been challenged in the literature. In particular, several authors have argued that the Spitzer instability should break down before most BHs are ejected [e.g. @moody09], and that the time-scale for all BHs to be ejected could exceed a Hubble time in some clusters [e.g. @downing10]. This view is supported by recent claims in the literature that stellar-mass BHs may be present in GCs in surprising numbers. For instance, @strader12 recently reported two flat-spectrum radio sources in M22, which appear to be accreting stellar-mass BHs. This suggests that this cluster could contain on the order of $\sim 5 - 100$ stellar-mass BHs. If BHs were indeed efficiently dynamically ejected, this, in turn, would suggest that a more substantial population of BHs once existed in M22, and likely other GCs as well.
The emerging picture for the formation of massive GCs involves multiple episodes of star formation [e.g. @piotto07; @gratton12; @conroy11; @conroy12]. In this context, @leigh13 recently considered the implications of the mass growth of BHs formed from massive progenitors belonging to the first generation due to accretion from the interstellar medium. The authors argued that, in principle, BHs could deplete a significant fraction of the available gas reservoir within $\lesssim 10^8$ years. If BHs were indeed to accrete efficiently from the ISM, they should not only grow in mass, but their velocities should also decrease due to conservation of momentum. This should preferentially accelerate the process of mass segregation for the BHs, causing them to rapidly accumulate in the central regions of the cluster if they did not form there in the first place. This could accelerate the dynamical decoupling of the BH sub-population from the rest of the system, and hence the phase of dynamical BH ejections due to the Spitzer instability.
In this paper, we consider how accretion from the interstellar medium affects the rate of mass segregation in a star cluster. We are especially interested in the implications for BHs in primordial GCs. Thus, we re-visit Spitzer’s two-component model to derive an accretion-modified relaxation time. We argue that the rate of mass segregation should be affected by accretion in the following way. First, assuming the accretion rate increases with the accretor mass, accretion acts to increase the range of any initial mass spectrum, driving the heaviest members to higher masses the fastest. Second, accretion acts to reduce the velocities of the accretors due to conservation of momentum, and it is the heaviest members whose velocities are reduced the fastest. Both of these effects exacerbate the Spitzer instability, and should accelerate the rate of mass segregation in a primordial star cluster.
In order to better quantify this qualitative picture, we present our adapted version of Spitzer’s two-component model in Section \[method\]. Specifically, we derive an accretion-modified relaxation time, as well as the critical accretion rate at which the rates of mass segregation due to both two-body relaxation and accretion are equal. We present our results in Section \[results\] for several different assumptions regarding the total cluster mass and accretion rate. In Section \[discussion\], we discuss the implications of our results for both star formation and stellar remnants in primordial globular clusters. We summarize our results in Section \[summary\].
Method
======
In this section, we present our analytic derivation of an accretion-modified relaxation time for a two-component model star cluster, and derive the critical accretion rate required for the mass segregation timescales due to two-body relaxation and accretion to be equal. We begin by summarizing briefly the relevant background related to both two-body relaxation and accretion.
Two-body relaxation {#relax}
-------------------
Consider a two-component model for a star cluster with component masses $m_1$ and $m_2$, such that $m_1 > m_2$. The populations for these two species have total masses $M_1$ and $M_2$ with $M_1 \ll M_2$. We let $v^2$ denote the initial mean square speed of *both* species, since at birth the cluster is not in a state of energy equipartition. The e-folding time for the tendency to equipartition bears a striking resemblance to the relaxation time [@heggie03]. Thus, to order of magnitude, the relaxation time can be approximated by calculating the time required for the mean square speed of the heavier species to fall from $v^2$ to a value $\sim m_2v^2/m_1$.
If the potential well of the lighter species is modelled using a parabolic profile, then equipartition will lead to the heavier species being confined to a region of size $\sim r_h\sqrt{m_2/m_1}$ [@heggie03]. The total mass of the heavier species within this region is $M_1$, whereas that for the lighter species is $M_2(m_2/m_1)^{3/2}$. At this point, however, it is not clear whether or not the lighter species remains the dominant mass component in this region. If not, the heavier species becomes increasingly affected by its own self-gravity, and can decouple dynamically from the remainder of the system. Consequently, it may only be possible to achieve equipartition provided [@spitzer69; @heggie03]: $$\label{eqn:spitzer-criterion}
M_1 \le M_2\Big( \frac{m_2}{m_1} \Big)^{3/2}.$$ This is known as Spitzer’s criterion.
In general, the half-mass relaxation time approximates the rate of two-body relaxation throughout the entire cluster. In GCs, it ranges from roughly a few hundred million years to the age of the Universe or longer, and is approximated by [@spitzer87]: $$\label{eqn:t-rh}
\tau_{\rm rh} [yr] = 1.7 \times 10^5[r_{\rm h} [pc]]^{3/2}N^{1/2}[\bar{m} [M_{\odot}]]^{-1/2},$$ where $r_{\rm h}$ is once again the half-mass radius, $N$ is the total number of stars within $r_{\rm h}$, and $\bar{m}$ is the average stellar mass. We assume that the value of $r_{\rm h}$ remains constant in time. This is reasonable since simulations have shown that $r_{\rm h}$ changes by not more than a factor of a few over the course of a typical cluster’s lifetime [@henon73; @spitzer87; @heggie03; @webb12]. The timescale for mass segregation due to two-body relaxation for an object of mass $m$ is then approximately [@vishniac78]: $$\label{eqn:tau-seg-time}
\tau_{\rm seg,2body}(m) = \frac{\bar{m}}{m}\tau_{\rm rh}.$$
Accretion {#accrete}
---------
A strict theoretical upper limit for the accretion rate is given by the Bondi-Hoyle limit [@bondi44]. In this approximation, the accretion is spherically symmetric, and the forces due to gas pressure are insignificant compared to gravitational forces. The background gas is treated as uniform and either stationary or moving with constant velocity relative to the accretor. This assumption gives reasonable accretion rates in the low-density, low-angular momentum regime. That is, provided the properties of the gas are such that the density, velocity, and total angular momentum are low, at least in the vicinity of the accretor, the Bondi-Hoyle limit approximately describes the true accretion rate [e.g. @fryxell88; @ruffert94; @ruffert97; @foglizzo99].
For large accretor masses and high gas densities, the Bondi-Hoyle rate can become extremely high. Here, pressure forces could play an important role in reducing the accretion rate. Indeed, this occurs if the outward continuum radiation force balances the inward gravitational force. This limit, called Eddington-limited accretion, gives considerably more modest accretion rates in the high gas density regime [@eddington26; @eddington30]. The Eddington rate should also be much closer to the true accretion rate if the gas contains significant angular momentum, and accretion proceeds mainly via angular momentum re-distribution within a disk [e.g. @rybicki79].
In general, theoretical studies have shown that the exact accretion rate can deviate substantially from the idealized cases described by Bondi-Hoyle and Eddington-limited accretion. For example, when an accretor is radiating at above the Eddington luminosity, significant amounts of gas can be expelled at high velocities due to the intense winds that are initiated [e.g. @king03]. This contributes to a reduction in the overall accretion rate. On the other hand, at very high accretion rates, photon-trapping can occur. This makes accretion disks radiatively inefficient and provides a means of circumventing the Eddington limit [@paczynsky80].
Numerical studies have also revealed the sensitivity of the accretion rate to small-scale gas dynamics. Recent work by @krumholz06 considered gas accretion onto point masses in a supersonically turbulent medium characterized by background density and velocity distributions that vary in both time and space. The authors show that in this regime, the accretion rate can either be described by the classical Bondi-Hoyle approximation, or a vorticity-dominated flow. In the latter case, the accretion rate can be significantly reduced relative to what is predicted by the Bondi-Hoyle limit [@krumholz04; @krumholz05]. Even more recently, @park13 studied the growth and luminosity of BHs in motion with respect to their surrounding medium using two-dimensional axis-symmetric numerical simulations. The authors show that the accretion rate can actually increase with increasing BH velocity, contrary to the naive predictions of simple analytic theory [@hoyle39].
In summary, theoretical work has shown that a wide range of accretion rates are possible. We will use the Bondi-Hoyle and Eddington limits for the accretion rate (combined with an accretion efficiency parameter) since these provide two different dependences on the accretor mass. However, the derivation presented in the subsequent section can be used to model other mass-dependences for the accretion rate as well.
Deriving the Relaxation Time {#derivation}
----------------------------
We are interested in the mass segregation timescale due to two-body relaxation for the heavier species in the two-component model described in Section \[relax\]. To first order, this is approximated by: $$\label{eqn:tau-seg-time}
\tau_{\rm seg,2body}(m_{\rm 1},t) = \frac{\bar{m}(t)}{m_{\rm 1}(t)}\tau_{\rm rh}(t),$$ where $\tau_{\rm rh}(t)$ denotes the half-mass relaxation time obtained by using the total number of objects $N = N_{\rm 1} + N_{\rm 2}$ and average object mass $\bar{m}(t) = (m_{\rm 1}(t)N_{\rm 1} + m_{\rm 2}(t)N_{\rm 2})/(N_{\rm 1} + N_{\rm 2})$ in Equation \[eqn:t-rh\].
We calculate the time-dependence for the mass of an object belonging to species $i$ as follows. First, assuming an object of initial mass $m_{\rm i}(0)$ accretes at a rate $\dot{m}_{\rm i} = dm_{\rm i}/dt$ for a total time $t$, we have: $$\label{eqn:mass-time}
m_{\rm i}(t) = m_{\rm i}(0) + \int_{0}^{t} \dot{m}_{\rm i}dt.$$ For the accretion rate, we assume a mass-dependence of the form: $$\label{eqn:acc-rate}
\dot{m}_{\rm i} = {\lambda}{\delta}m_{\rm i}^{\rm \epsilon},$$ where $\lambda$, $\delta$, and $\epsilon$ are all free parameters. The power-law exponent $\epsilon$ decides the mass-dependence for the accretion rate. The accretion coefficent $\delta$ is derived according to the physical assumptions that decide the rate of accretion. For example, adopting the Bondi-Hoyle approximation implies $\epsilon = 2$, and [e.g. @bondi44; @maccarone12; @leigh13]: $$\label{eqn:mdot-BH}
\delta = 7 \times 10^{-8} {M_{\odot}^{-1}}{\rm yr}^{-1} \Big( \frac{n}{\rm 10^6 cm
^{-3}} \Big) \Big( \frac{\sqrt{c_{\rm s}^2 + v^2}}{\rm 10^6 cm s^{-1}} \Big)^{-3},$$ where $n$ is the particle number density, $c_{\rm s}$ is the sound speed, and $v^2$ is the root-mean-square speed. Similarly, assuming Eddington-limited accretion implies $\epsilon = 1$, and [@eddington26; @eddington30]: $$\begin{gathered}
\label{eqn:mdot-Edd}
\delta = \frac{4{\pi}G}{{\eta}{\kappa}c} \\
= 2.2 \times 10^{-8} {\rm yr}^{-1},
\end{gathered}$$ where $c$ is the speed of light, ${\eta}c^2$ is the accretion yield from unit mass, and ${\kappa}$ is the electron scattering opacity. We take $\kappa = 0.34$ cm$^{2}$ g$^{-1}$, which assumes a hydrogen mass fraction of $X = 0.7$. We further adopt $\eta = 0.1$, which is reasonable if the accretors are BHs since this parameter quantifies the amount of energy radiated away during accretion. The accretion efficiency parameter $\lambda$, on the other hand, is left as a free parameter in our model.
In order to solve for the accretor mass as a function of time, Equation \[eqn:mass-time\] can be re-written such that the integral is with respect to mass. Provided $\epsilon > 1$, this gives for the mass of an object belonging to species $i$ at time $t$: $$\label{eqn:mass-time2}
m_{\rm i}(t) = \Big( m_{\rm i}(0)^{1-\epsilon} + {\lambda}{\delta}(1-\epsilon)t \Big)^{1/(1-\epsilon)}.$$ Similarly, for $\epsilon = 1$, we have: $$\label{eqn:mass-time3}
m_{\rm i}(t) = m_{\rm i}(0)e^{\rm {\lambda}{\delta}t}.$$ We arrive at the mass segregation timescale due to two-body relaxation for the heavier species by substituting either Equation \[eqn:mass-time2\] or Equation \[eqn:mass-time3\] for both species 1 and 2 into Equation \[eqn:tau-seg-time\].
In addition to two-body relaxation, accretion should also affect the stellar velocities via conservation of momentum. Provided the accretion rate increases with increasing accretor mass, this will reduce the kinetic energy of the heavier species faster than the lighter species, in rough analogy with the effects of two-body relaxation. Thus, accretion-induced changes in the stellar velocities should *accelerate* the rate of mass segregation. We calculate the time needed for accretion to change the velocities of the heavier species from a root-mean-square speed of $v^2$ to a value of $m_{\rm 2}/m_{\rm 1}v^2$. This corresponds to the time for accretion to affect the velocities of the heavier species by roughly the same amount as is done by two-body relaxation over a single relaxation time.
For a given accretion rate and initial masses, we calculate the time needed for the mean square speed of the heavier species to reach a value $m_{\rm 2}(t)v^2/m_{\rm 1}(t)$ using conservation of momentum, from an initial value $v$. The time needed for the heavier species to reach equipartition via accretion can then be written: $$\label{eqn:tau-acc}
\tau_{\rm seg,acc}(m_1,t) = \int_{\rm m_1(t)}^{\rm m_{1,f}(t)}\frac{dm_1}{\dot{m}_1},$$ where the final mass is $m_{\rm 1,f} = \sqrt{m_{\rm 1}^3/m_{\rm 2}}$ by conservation of momentum.
Substituting $m_{\rm 1,f}(t) = \sqrt{m_{\rm 1}(t)^3/m_{\rm 2}(t)}$ into Equation \[eqn:tau-acc\], we arrive at the timescale needed for accretion to push the heavier species into approximate equipartition: $$\label{eqn:tau-acc2}
\tau_{\rm seg,acc}(m_{\rm 1},t) = \frac{m_{\rm 1}(t)^{1-\epsilon}\Big( (m_{\rm 1}(t)/m_{\rm 2}(t))^{(1-\epsilon)/2} - 1 \Big)}{{\lambda}{\delta}(1 - \epsilon)},$$ for $\epsilon > 1$. Similarly, for $\epsilon = 1$ or Eddington-limited accretion, we obtain using Equation \[eqn:mass-time3\]: $$\label{eqn:tau-acc3}
\tau_{\rm seg,acc}(m_{\rm 1},t) = \frac{\ln(m_{\rm 1}(t)/m_{\rm 2}(t))}{2{\lambda}{\delta}}.$$ We consider accretion rates with a mass-dependence such that $\epsilon \ge 1$, since this includes both Bondi-Hoyle and Eddington-limited accretion, as well as intermediate and even steeper mass accretion rates.
The total rate (taken to be the inverse of the total mass segregation timescale $\tau_{\rm seg,tot}$) at which the heavier species achieves mass segregation can be written as the sum of the rate of two-body relaxation and the rate at which accretion pushes the heavier species to equipartition. Re-arranging this equation, we arrive at the total accretion-modified mass segregation timescale for the heavier species: $$\label{eqn:t-eq}
\tau_{\rm seg,tot}(m_{\rm 1},t) = \frac{\tau_{\rm seg,acc}(m_{\rm 1},t)\tau_{\rm seg,2body}(m_{\rm 1},t)}{\tau_{\rm seg,acc}(m_{\rm 1},t) + \tau_{\rm seg,2body}(m_{\rm 1},t)}.$$
Deriving the critical accretion rate {#critical}
------------------------------------
To derive the critical accretion rate $\delta_{\rm crit}$ at which two-body relaxation and accretion drive the mass segregation process at the same rate, we set $\tau_{\rm seg,acc} = \tau_{\rm seg,2body}$ and solve for $\delta$ as a function of $\epsilon$, $\lambda$, $m_{\rm 1}$, $m_{\rm 2}$, and $t_{\rm rh}$. This gives for $\epsilon > 1$: $$\label{eqn:delta2}
\delta_{\rm crit} = \frac{m_{\rm 1}(t)^{2-\epsilon}\Big( (m_{\rm 1}(t)/m_{\rm 2}(t))^{(1-\epsilon)/2} - 1 \Big)}{{\lambda}(1 - \epsilon)\bar{m}(t)t_{\rm rh}(t)}.$$
The procedure is similar for $\epsilon = 1$, except we use Equation \[eqn:tau-acc3\] instead of Equation \[eqn:tau-acc2\]. This gives for Eddington-limited accretion: $$\label{eqn:delta3}
\delta_{\rm crit} = \frac{m_{\rm 1}(t)\ln(m_{\rm 1}(t)/m_{\rm 2}(t))}{2{\lambda}\bar{m}(t)t_{\rm rh}(t)}.$$
In Section \[results\], we will use Equation \[eqn:delta2\] and Equation \[eqn:delta3\] in order to study the interplay between our assumptions regarding the gas properties, which affect the accretion rate, and our assumption for the total cluster mass, which determines the rate of two-body relaxation.
Accretion efficiency {#lambda-time}
--------------------
Given our limited understanding of the precise physics of accretion onto a BH, it is not possible to reliably define a functional form for the accretion efficiency parameter $\lambda$. In principle, any realistic accretion model should include a *time-dependence* for $\lambda(t)$.[^2] For example, fluctuations in the local gas density due to turbulence, a gradual or even sudden depletion of the available gas reservoir, or dynamical interactions between accreting objects may cause the accretion efficiency to vary over time.
Our analysis is easily modified to treat time-dependent accretion rates by substituting an appropriate choice for $\lambda(t)$ into either Equation \[eqn:tau-acc2\] or \[eqn:tau-acc3\], and then solving for $\tau_{\rm seg,acc}$. Plausible choices for $\lambda(t)$ may either oscillate or decline (steadily or abruptly) in time. The first case, i.e. an oscillating accretion efficiency, is more easily understood, because under these circumstances, our analysis can simply be interpreted as discussing the *time-averaged* accretion efficiency parameter. Thus, in the subsequent sections, we assume an oscillating (or constant) accretion efficiency parameter, and discuss only the time-averaged value.
For example, the function $\lambda(t) = (1+{\rm sin}({\pi}t/t_{\rm 0}))$ oscillates between 0 and 2 with a frequency of $2/t_{\rm 0}$. In this case, the time-averaged value for $\lambda(t)$ is equal to 1, so that the time-averaged value for $\tau_{\rm seg,acc}$ remains the same as for a constant $\lambda = 1$.
Accretion efficiency parameters that oscillate in time should be suitable to cases where the accretors have alternating “on” and “off” phases. This may well be the case with accreting BHs, since the radiation emitted due to accretion can heat the surrounding gas, which in turn decreases the accretion rate [e.g. @blaes95]. In this case, the source of energetic photons responsible for heating the gas is turned off, allowing the gas to cool and accretion to re-start in a “feedback regulated” loop [e.g. @king03; @yuan09].
Accretion efficiency parameters that decline in time should be appropriate to cases where the available gas reservoir is depleted over time. This could arise gradually if the gas is used to form stars, or if significant quantities of gas are accreted by BHs. Alternatively, the gas reservoir could be depleted suddenly, e.g. due to, energy injected from supernovae, stellar winds, or winds from accreting compact objects. In either case, Equations \[eqn:tau-acc2\] and \[eqn:tau-acc3\] should include the explicit time-dependence for the accretion efficiency parameter. This will contribute to an increase in $\tau_{\rm seg,acc}$ with time, since the decreasing gas mass should translate into a decreasing gas density, and hence accretion rate. In Section \[results\], we will assume that the amount of gas lost from the system is negligible over the calculated mass segregation timescales, and our interpretation of $\lambda$ as a time-independent quantity remains valid. This is reasonable provided the mass segregation timescales due to accretion are much less than the timescale for gas depletion. As we will show in Section \[results\], the current picture for the formation of globular clusters and their multiple populations is consistent with this scenario [e.g. @krause12; @krause13; @leigh13].
Results
=======
In this section, we present the results of our analytic two-component model for an accretion-modified two-body relaxation time. Our aim is to quantify the relative rates at which two-body relaxation and gas accretion drive a star cluster towards mass segregation, as a function of our assumptions for the gas properties, component masses, and total system mass. To this end, we present the time evolution of all three mass segregation timescales, namely $\tau_{\rm seg,acc}$, $\tau_{\rm seg,2body}$, and $\tau_{\rm seg,tot}$, and discuss the critical accretion rate required for the mass segregation timescales due to two-body relaxation and accretion to be equal as a function of the mass-dependence for the accretion rate.
Time evolution of the mass segregation timescales {#time-evolution}
-------------------------------------------------
We begin by quantifying the relative rates of mass segregation due to two-body relaxation and gas accretion for different model assumptions. Specifically, we consider several different mass ratios and total system masses for our two-component model, as well as different mass-dependences for the rate of accretion. This is meant to quantify the sensitivity of the two different mass segregation mechanisms to the cluster and gas properties that decide their rates.
First, we describe our assumptions for the two-component model star cluster, which are needed in order to calculate $\tau_{\rm seg,2body}$. We adopt $m_{\rm 2} = 1$ M$_{\odot}$ for the lighter species, but consider two different masses for the heavier species, namely $m_{\rm 1} = 10$ M$_{\odot}$ and $m_{\rm 1} = 50$ M$_{\odot}$. We assume a population size of $N_{\rm 1} = 10^2$ for the heavier species, but vary the population size of the lighter species by considering the values $N_{\rm 2} = 10^5, 10^6, 10^7$. The component masses and population sizes are chosen to represent reasonable mass ratios between the average stellar and BH masses, and to ensure that the Spitzer criterion (i.e. Equation \[eqn:spitzer-criterion\]) is initially satisfied. We adopt a half-mass radius for our model cluster of $r_{\rm h} = 10$ pc, and note that assuming a lower value for the half-mass radius would only shorten the calculated mass segregation timescales.
Next, we describe our assumptions for the properties of the accreted gas, which are needed to calculate $\tau_{\rm seg,acc}$, and therefore $\tau_{\rm seg,tot}$. We assume a uniform time-independent gas density throughout the cluster, so that the accretion rate changes only with the stellar mass. We further assume that the gas is always at rest relative to the accretor when calculating the final accretor velocity using conservation of momentum. For the accretors, we adopt a root mean-square-speed of $v = 10$ km s$^{-1}$, which is guided by the relation $v = \sqrt{2GM/5r_h}$ [@binney87] for a total cluster mass $M \sim 10^5-10^6$ M$_{\odot}$. For the gas, we assume a sound speed of $c_{\rm s} = 10$ km s$^{-1}$, and a particle number density of $n = 10^6$ cm$^{-3}$. These assumptions should be reasonable for what is expected in a massive primordial GC for the first $\sim 10^8$ years [e.g. @dercole08; @maccarone12; @conroy12; @krause12; @krause13; @leigh13].
We show our results for two different mass-dependencies for the accretion rate. The left panels in Figure \[fig:tau-seg-BH\] show our results assuming $\epsilon = 2$ in Equation \[eqn:acc-rate\], which corresponds to Bondi-Hoyle accretion. We use Equation \[eqn:mdot-BH\] for $\delta$, and $\lambda = 0.1$ for the accretion efficiency parameter. The panels to the right in Figure \[fig:tau-seg-BH\] show our results assuming Eddington-limited accretion, which means that $\epsilon = 1$ in Equation \[eqn:acc-rate\] and we use Equation \[eqn:mdot-Edd\] for $\delta$.
The main conclusion to be drawn from Figure \[fig:tau-seg-BH\] is that, for all but the least massive clusters and the lowest accretion rates considered here, the rate of mass segregation due to accretion can actually exceed the rate due to two-body relaxation. The timescale at which this occurs is on the order of $\sim 10^8$ years. Interestingly, this timescale is similar to the total time thought to be required for multiple episodes of star formation to occur in primordial GCs [e.g. @conroy11; @conroy12]. Thus, our results suggest that accretion from the ISM could significantly affect both the spatial and velocity distributions of the heaviest objects in a primordial GC *before* the gas reservoir is depleted. For a typical primordial GC, this should be the case provided the average accretion rate is greater than $\sim 5-10$% of the Eddington-limited rate, assuming the mass-dependence for the accretion rate is linear. Similarly, if the accretion rate scales with the square of the accretor mass, then accretion from the ISM is non-negligible as long as the average accretion rate is greater than 1-10% of the Bondi-Hoyle rate.
The critical accretion rate {#critical2}
---------------------------
In this section, we calculate the critical accretion rate required for the rates of mass segregation due to two-body relaxation and gas accretion to be equal. Our aim is to quantify the relative importance of the different parameters for the cluster and gas properties in establishing a balance between the competing effects of two-body relaxation and accretion.
In Figure \[fig:delta-crit\], we show the critical accretion rate $\delta_{\rm crit}$ as a function of the mass of the heavier species $m_{\rm 1}$. These results are calculated using Equations \[eqn:delta2\] and \[eqn:delta3\], which correspond to Bondi-Hoyle (blue) and Eddington-limited (red) accretion, respectively. In both cases, we assume a constant mass for the lighter species of $m_{\rm 2} = 1$ M$_{\odot}$, a constant population size for the heavier species of $N_{\rm 1} = 10^2$, and a constant accretion efficiency parameter $\lambda = 1.0$. In order to vary the rate of two-body relaxation without affecting the rate of mass segregation due to accretion, we consider three different population sizes for the lighter species, namely $N_{\rm 2} = 10^5, 10^6, 10^7$.
![The critical accretion rate $\delta_{\rm crit}$ at which the mass segregation timescales due to two-body relaxation and accretion are equal, shown as a function of the mass of the heavier species $m_{\rm 1}$. The blue curves show the results assuming Bondi-Hoyle accretion, whereas the red curves are for Eddington-limited accretion. The dashed, solid, and dash-dotted lines correspond to $N_{\rm 1} = 10^5, 10^6, 10^7$, respectively. For both sets of curves, we assume a constant mass for the lighter species of $m_{\rm 2} = 1$ M$_{\odot}$, and we set $\lambda = 1.0$ for the accretion efficiency parameter. \[fig:delta-crit\]](fig3.eps){width="\columnwidth"}
Figure \[fig:delta-crit\] shows that for the case of Eddington-limited accretion, the critical accretion rate increases with increasing accretor mass. This is because, as the accretor mass increases, the mass segregation timescale due to two-body relaxation decreases faster than the mass segregation timescale due to accretion. In the case of Bondi-Hoyle accretion, however, the critical accretion rate depends only very weakly on the accretor mass, which is due to the fact that the accretion rate scales as the square of the accretor mass. We emphasize that with the exception of the Eddington-limited rate at large accretor masses, the critical accretion rates are comparable to, or even smaller than, those observed in nearby star-forming regions [e.g. @mckee07].
Discussion
==========
In this section, we discuss the implications of our results for mass segregation in primordial globular clusters, in particular with regards to black holes.
Enhanced mass segregation {#primordial}
-------------------------
One of the key conclusions arising from our analysis is that accretion should accelerate the rate at which a star cluster becomes mass segregated compared to two-body relaxation alone. In fact, accretion can dominate over two-body relaxation in massive clusters for accretion rates that are below the Bondi-Hoyle or Eddington-limited rates by one or even two orders of magnitude. This is because the relaxation time increases with the cluster mass, whereas the mass segregation timescale due to accretion is independent of the cluster mass (assuming that the gas properties are independent of the cluster mass). Our results suggest that two-body relaxation should dominate the mass segregation process in low-mass primordial clusters with global relaxation times $\lesssim 10^7-10^8$ years and hence total cluster masses $\lesssim 10^4-10^5$ M$_{\odot}$, provided that our models assumptions are valid. In this regime, accretion should only have a small effect on the rate of mass segregation, and long-range gravitational interactions should alter the accretors’ velocities faster than they are reduced by the accretion process. In more massive clusters, however, the damping effects of accretion could play a significant role in accelerating the rate of mass segregation.
The effects of a realistic mass spectrum {#mass-spec}
----------------------------------------
Our assumption of a two-component model serves to demonstrate the effects of accretion from the ISM on a cluster’s dynamical evolution. The qualitative nature of our results should hold if a realistic mass spectrum is adopted instead. Accretion can modify the distribution of velocities on relatively short timescales in gas-embedded clusters. How exactly the velocities become modified depends on several parameters, in particular the mass spectrum, the total cluster mass, the properties of the gas, and the accretion rate.
In general, we expect accretion to amplify or exacerbate the Spitzer instability. This is due to the mass-dependence of the accretion rate, and the fact that typically $\epsilon > 0$ in Equation \[eqn:acc-rate\] (i.e. the accretion rate), which causes the more massive component to grow in mass the fastest. Thus, according to Equation \[eqn:spitzer-criterion\], Spitzer’s criterion should typically break down sooner as a result of accretion. However, accretion also acts to reduce the velocities of the accretors due to conservation of momentum, and this should most strongly impact the most massive objects due once again to the mass-dependence of the accretion rate. This can actually serve to combat the effects of the Spitzer instability by inhibiting the most massive objects from decoupling dynamically from the rest of the system once they have segregated to the central regions of the cluster. Clearly, a more sophisticated treatment will be needed in future studies in order to properly quantify these effects and their implications for the Spitzer instability, and a cluster’s ability to achieve energy equipartition.
Gas properties and the accretion rate {#gas-prop}
-------------------------------------
We stress that our results depend sensitively on our assumption for the accretion rate, which is poorly constrained, both theoretically and observationally. Indeed, the accretion efficiency parameter $\lambda$ adopted in Equation \[eqn:acc-rate\] is needed to account for the many sources of uncertainty in the gas properties, and hence the accretion rate. For example, our assumption of a uniform, time-independent gas density is an over-simplification. For one, stellar winds and supernovae could create over- and under-densities in the form of sheets and/or filaments, and the efficiency of these processes should fluctuate in time given the presence of a realistic mass function combined with stellar evolution and the cluster dynamics [e.g. @krause12; @krause13]. These effects could contribute to a reduction in the accretion rate by increasing the relative velocity between the gas and the accretors, or by reducing the gas density along the trajectories of the accretors. Realistic hydrodynamical simulations of star cluster formation will be needed in order to properly quantify these effects and their implications for the accretion rate.
Our results can be used to guide the parameter space relevant to these future studies. In particular, we have placed a lower limit on the minimum accretion rate required for accretion to significantly affect the distribution of stellar velocities on timescales shorter than the relaxation time, as a function of the cluster and gas properties. Specifically, the results of our simple model suggest that, for a typical primordial GC, the average accretion rate cannot be much less than $\sim 5-10$% of the Eddington-limited rate, assuming the mass-dependence for the accretion rate is linear. Similarly, for our model assumptions, the average accretion rate cannot be much less than $1-10$% of the Bondi-Hoyle rate if the accretion rate scales with the square of the accretor mass.
We have adopted the same root-mean-square speed for all models, independent of the total cluster mass. This is a reasonable assumption since the root-mean-square speed scales as $v \propto (M/r_{\rm h})^{0.5}$, and $r_{\rm h}$ itself depends weakly on the total cluster mass. Thus, in total, the root-mean-square speed depends only very weakly on the total cluster mass. Nevertheless, if the accretion rate scales inversely with the velocity of the accretor, as is the case with the Bondi-Hoyle approximation, then the dependence of the root-mean-square speed on the total cluster mass should contribute to a decrease in the accretion rate with increasing cluster mass. A proper treatment of this effect is beyond the scope of this paper, however, it should certainly be considered in future studies.
Black hole dynamics {#BH-dynamics}
-------------------
The results presented in this paper are especially relevant for black holes in primordial globular clusters, since they should be the most massive objects in the cluster within a few Myrs of its formation. Recent evidence suggests that there should be a substantial gas reservoir in GCs for the first $\sim 10^8$ years [e.g. @conroy11], albeit perhaps intermittently, and that nearly all BHs should form from the most massive cluster members within the first few Myrs [e.g. @maeder09]. It follows that the BHs could have on the order of $10^8$ years to accrete gas from the ISM. Additionally, since any BHs formed from progenitors more massive than $\sim 50$ M$_{\odot}$ are only slightly less massive than the progenitors themselves and do not experience natal kicks [@fryer12], these BHs should have both the shortest mass segregation timescales due to two-body relaxation and the highest accretion rates (ignoring BH winds and/or Compton heating; see below).
The key point is that accretion should act to reduce the mass segregation times of BHs in primordial GCs, and that this effect could be dramatic. Beyond this, more detailed modeling will be needed to determine the fates of the BHs. In particular, should the increased rate of mass segregation contribute to accelerating the onset of the hypothesized phase of dynamical BH ejections? Or could the damping effects of accretion be so dramatic that the BHs are driven to merge [e.g. @davies11]? If so, the formation of an intermediate-mass BH (IMBH) could be the inevitable result. Alternatively, it could be that black hole winds are sufficiently powerful to eject the bulk of the gas from the cluster. Another possibility is that the gas in the immediate vicinity of the BHs becomes very hot due to, for example, Compton heating [@blaes95; @yuan09], such that the accretion rate becomes drastically reduced and BH growth is severely limited?
A better understanding of how the presence of significant quantities of gas modifies the black hole dynamics in a primordial GC could help to constrain the initial cluster conditions. For example, if massive BHs should inevitably merge in the presence of gas but no IMBHs are observed in present-day GCs, does this necessarily imply that the BHs never formed in the first place? If so, this would suggest that stars with masses $\gtrsim 50$ M$_{\odot}$ must have been rare. This could be the case, for instance, if massive primordial GCs were assembled from the mergers of many low-mass sub-clumps, as opposed to a single monolothic collapse. This is because the mass of the most massive cluster member correlates with the total cluster mass [e.g. @kirk11; @kirk12], and hence the massive stellar progenitors of the most massive BHs are unlikely to form in low-mass clusters.
Summary
=======
In this paper, we have considered the effects of accretion from the interstellar medium on the rate of two-body relaxation in a star cluster. To do this, we derived an accretion-modified relaxation time by adapting Spitzer’s two-component model to include the effects of accretion. We considered several different mass-dependencies and efficiency factors for the accretion rate, as well as different mass ratios for the two components of the model.
We have shown that accretion acts to increase the rate of mass segregation. This is because the relaxation time is inversely proportional to the average mass, which increases due to accretion. There are two additional effects that accelerate the mass segregation process, assuming that the accretion rate increases with the accretor mass. First, accretion acts to increase the range of any initial mass spectrum, quickly driving the heaviest members to even higher masses. Second, accretion acts to reduce the velocities of the accretors due to conservation of momentum, and it is the heaviest members that are affected the most. Using our two-component model, these effects have been quantified as a function of the accretion rate, the total cluster mass, and the component masses. We have discussed our results in the context of the dynamical evolution of primordial globular clusters and their black hole sub-populations.
Acknowledgments {#acknowledgments .unnumbered}
===============
We kindly thank an anonymous referee for helpful suggestions that improved our manuscript, as well as Cole Miller for useful discussions. AS is supported by NSERC.
[99]{}
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\[lastpage\]
[^1]: E-mail: [email protected] (NL), [email protected] (AS), [email protected] (TB)
[^2]: Alternatively, the time-dependence can be absorbed directly into the parameter $\delta$ in Equations \[eqn:tau-acc2\] and \[eqn:tau-acc3\].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'An investigation is presented of the utility of semiclassical approximations for solving the quantum-impurity problems arising in the dynamical-mean-field approach to the correlated-electron models. The method is based on performing a exact numerical integral over the zero-Matsubara-frequency component of the spin part of a continuous Hubbard-Stratonovich field, along with a spin-field-dependent steepest descents treatment of the charge part. We test this method by applying it to one or two site approximations to the single band Hubbard model with different band structures, and comparing the results to quantum Monte-Carlo and simplified exact diagonalization calculations. The resulting electron self-energies, densities of states and magnetic transition temperatures show reasonable agreement with the quantum Monte-Carlo simulation over wide parameter ranges, suggesting that the semiclassical method is useful for obtaining a reasonable picture of the physics in situations where other techniques are too expensive.'
author:
- 'Satoshi Okamoto,$^1$[@Email] Andreas Fuhrmann,$^2$ Armin Comanac,$^1$ and Andrew J. Millis$^1$'
title: 'Benchmarking a semiclassical impurity solver for dynamical-mean-field theory: self-energies and magnetic transitions of the single-orbital Hubbard model'
---
Introduction
============
Correlated-electron materials, in which the interaction energy is comparable to or larger than the electron kinetic energy, are an important topic in materials science. In these compounds, standard band-theory is an inadequate representation of the physics. The discovery of high-$T_c$ superconductivity in the oxide cuprates[@Bednorz86] led to greatly increased interest in correlated-electron compounds. Many materials have been studied, and many novel properties have been discovered; including colossal magnetoresistance, variety of spin, orbital, charge orderings, and unconventional superconductivity.[@Imada98; @Tokura00] Understanding the novel phenomena and determining the correct electronic phases in these materials are challenging tasks, and are indispensable for developing the electronic devices exploiting the novel properties of the correlated-electron materials. However, the rapid increase of Hilbert space with system size limits the utility of exact-diagonalizaion (ED) methods, while the fermion “sign problem” renders Monte-Carlo (MC) approaches ineffective.
Dynamical-mean-field theory (DMFT) is a promising approach for treating correlation effects.[@Metzner89; @Georges96] The method may be combined with realistic band-structure calculations, and has been applied to variety of materials (see Refs. for example). In DMFT, the momentum dependent electron self-energy is approximated as a finite set of functions of frequency, and physical lattice problem is mapped into a one or several coupled impurity-Anderson models (IAM) with environment determined self-consistently. To solve the IAM, a variety of theoretical methods have been applied, including analytical approximations such as iterated-perturbation theory,[@Georges92; @Georges93; @Kajueter96] non-crossing approximation,[@Pruschke93] projective self-consistent method,[@Moeller95] and slave-boson [@Florens02] for example, and numerical methods including Quantum Monte-Carlo (QMC),[@Jarrell92; @Rozenberg92] exact diagonalization,[@Caffarel94] numerical-renormalization group,[@Sakai94; @Bulla99] and density-matrix-renormalization group.[@Garcia04; @Nishimoto04] Most interesting phenomena can not be addressed by analytical methods. Numerical methods, however, are computationally very expensive. In order to combine DMFT with a realistic band-structure calculation and survey a wide range of parameters, it is desirable to develop computationally-cheap and reliable methods. Methods which can reproduce the correct electronic phases and distinguish the phases near in energy at low temperature are particularly needed. Many ideas have been proposed to reduce the computational cost.[@Potthoff01; @Potthoff03; @Jeschke04; @Zhu04; @Savrasov04; @Dai04] These fall mainly into two groups: variations of the method simplified by truncating the number of orbitals,[@Potthoff01; @Potthoff03] and perturbative methods involving expansions around a certain solvable limit.[@Jeschke04; @Zhu04; @Savrasov04; @Dai04] The restricted method provides a good approximation to the $T=0$ Mott physics of the Hubbard model,[@Potthoff01; @Potthoff03] however because only a small number of states are used, it has difficulty dealing with the thermodynamics (as discussed later), and becomes impractical for multiorbital and multisite systems. Perturbative methods are not reliable at intermediate coupling. Non-perturbative methods which can deal with the thermodynamics are required.
In this paper, we investigate an alternative method to solve the IAM in DMFT formalism: a semiclassical approximation based on the continuous Hubbard-Stratonovich transformation.[@Hubbard59] This method is computationally very cheap (approximately two-orders of magnitude faster than QMC for single-impurity models, and with a better scaling with system size): it may be performed on a commercial PC. The semiclassical approximation, in the form used here, was apparently first proposed by Hasegawa,[@Hasegawa80] who used it to study a “single-site spin fluctuation theory” which may be viewed as an early, simplified version of dynamical-mean-field theory. Semiclassical methods were also used by Blawid and Millis[@Blawid00] and by Pankov, Kotliar and Motome[@Pankov02] to study models of electrons coupled to large-mass oscillators. In this paper, we present an implementation in the context of dynamical-mean-field theory, and test its reliability for a fully quantal model problem, namely, the Hubbard model on a variety of lattices by performing detailed comparisons between the semiclassical approximation and QMC. We also present a brief comparison a successful “restricted method”: the two-site DMFT.[@Potthoff01] The semiclassical approximation is found to be reasonable at high temperature and in the strong coupling regime, and gives good results for magnetic transitions. In the half-filled square lattice, the N[é]{}el temperature computed by the semiclassical approximation is very close to that found in QMC. In the metallic face-centered cubic (FCC) lattice, ferromagnetic Curie temperatures computed by the present method are within a factor of 2 compared with the QMC, and the competition between ferro- and antiferromagnetism is correctly captured, $T=0$ phase boundaries are obtained within the error of $\sim 20~\%$ to QMC, The $T=0$ phase boundaries are in the same range as found in two-site DMFT, but this latter method gives very poor transition temperatures. We also use the semiclassical approximation to compute dynamical properties, in particular, the electron self-energy and spectral functions, which are found to be in reasonable agreement with QMC results.
The rest of this paper is organized as follows: Section II formulates the semiclassical approximation. In Sec. III, we use the Gaussian fluctuation approximation to estimate the limits of validity of the semiclassical approximation. Sections IV and V compare numerical results for the semiclassical approximation to QMC, ED, and Hartree-Fock (HF) results, for the single-band Hubbard model on square lattice and face-centered cubic lattice, respectively. Section VI presents an application of the method to the two-impurity cluster dynamical-mean-field approximation for the half-filled Hubbard model. Comparison of nearest-neighbor spin correlation between the semiclassical approximation and QMC are given. In Sec. VII, we discuss the equilibration problem associated with the partitioned phase space in the semiclassical approximation and QMC. Finally, Sec. VIII gives a summary and discussion.
Formulation
===========
In this section, we derive the semiclassical approximation from the continuous Hubbard-Stratonovich (HS) transformation.[@Hubbard59] We consider a single-orbital repulsive Hubbard model for simplicity: $H=H_{band} + H_U$ with $H_{band}$ and $H_U$ being single-particle dispersion term and interaction term, respectively. Generalizations to multiorbital and long-range interaction would be also possible: see e.g., Refs. and .
We take the band-dispersion term to be $H_{band}=\sum_{k \sigma} \varepsilon_k c^\dag_{k \sigma} c_{k \sigma}$ with three choices of $\varepsilon_k$.\
[*Two dimensional square lattice:*]{} $$\varepsilon^{square}_{\vec k} = -2t (\cos k_x + \cos k_y).
\label{eq:ek2d}$$ [*Three dimensional FCC lattice:*]{} $$\begin{aligned}
\varepsilon^{FCC3}_{\vec k} \!\!\! &=& \!\!
4t (\cos k_x \cos k_y + \cos k_y \cos k_z + \cos k_z \cos k_x)
\nonumber \\
&&+2t' (\cos 2 k_x + \cos 2 k_y + \cos 2 k_z), \label{eq:FCC3d}\end{aligned}$$ where $t$ and $t'$ are the NN hopping and the second NN hopping, respectively.\
[*Infinite-dimensional FCC lattice:*]{} Non-interacting density of states is given by $$N^{FCC \infty}(\varepsilon)=\frac{\exp \{-(1+\sqrt{2}\varepsilon)/2\}}
{\sqrt{\pi(1+\sqrt{2}\varepsilon)}},
\label{eq:Ulmkedos}$$ which diverges at the bottom of the band. $N^{FCC \infty}$ was introduced by M[ü]{}ller-Hartmann[@Hartmann91] and studied in detail by Ulmke.[@Ulmke98] The FCC lattices are of interest because these display a ferromagnetic phase in a wide range of doping.
The interaction term is given by $$H_U = U \sum_i n_{i \uparrow} n_{i \downarrow}$$ with $U>0$.
In the DMFT approximation, one first needs to compute the partition function of an $N$-site impurity model as $$Z = \int {\cal D} [c^\dag, c] \exp\{-({\cal S}_0+{\cal S}_{int})\},
\label{eq:Z}$$ with $$\begin{aligned}
{\cal S}_0 = -\int_0^\beta \!\! d\tau d\tau' \psi_\sigma^\dag (\tau) {\bf a}_\sigma (\tau-\tau')
\psi_\sigma (\tau')\end{aligned}$$ where, $\psi = [c_1, \ldots, c_N]^t$ with $c_i (c_i^\dag)$ being a Grassmann number corresponding to the electron annihilation (creation) operator at site or orbital $i$. ${\bf a}_\sigma$ is the $N \times N$ matrix Weiss field (inverse of the non-interacting Green function) which will be determined self-consistently, and $\beta = 1/T$ is the inverse temperature. ${\cal S}_{int}$ represents the interaction term specified by the model one considers. For the single-band Hubbard model, ${\cal S}_{int} =
U \! \int \! d \tau \sum _in_{i \uparrow} (\tau) n_{i \downarrow} (\tau)$.
Next, one computes single-particle interacting Green function ${\bf G}_\sigma$ by a functional derivative of ${\rm ln} Z$ with respect to ${\bf a}_\sigma$ as $$\begin{aligned}
{\bf G}_\sigma = \frac{\delta {\rm ln} Z}{\delta {\bf a}_\sigma}.
\label{eq:Gimp}\end{aligned}$$ The electronic self-energy is obtained by inverting the Dyson equation as $$\mbox{\boldmath $\Sigma$}_\sigma = {\bf a}_\sigma - {\bf G}_\sigma^{-1}.
\label{eq:Dyson}$$ Finally, by connecting the impurity Green function $\bf G$ and the local part of the lattice Green function, the DMFT equation is closed. The self-consistency equation for the $ij$ component of ${\bf G}$ is $$G_{ij \sigma} (i \omega_n)=
\int \! \biggl(\frac{dk}{2\pi}\biggr)^d
\biggl[\frac{\varphi_{ij}(k)}{i \omega_n + \mu - {\bf t}_k - \mbox{\boldmath $\Sigma$}_{k \sigma}
(i \omega_n)}\biggr]_{ij},
\label{eq:Gloc}$$ where, $\omega_n$ is the fermionic Matsubara frequency, and ${\bf t}_k$ and $\mbox{\boldmath $\Sigma$}_k$ are the Fourier transforms of the hopping and the self-energy matrices, respectively. $\varphi_{ij}(k)$ is a form factor specified by the DMFT method chosen.[@Okamoto03] The chemical potential $\mu$ is fixed so that ${\bf G}$ gives the correct electron density $n$. The local Green function ${\bf G}$ is used to update the Weiss field ${\bf a}_\sigma^{new}$ as ${\bf a}_\sigma^{new} = {{\bf G}_\sigma}^{-1} + \mbox{\boldmath $\Sigma$}_\sigma$, and this process is repeated until the self-consistency ${\bf a}_\sigma={\bf a}_\sigma^{new}$ is obtained. The expensive computational task is evaluating the functional integral in Eq. (\[eq:Z\]).
A key point of the present approximation for the single-band Hubbard model is introducing the charge field as well as spin. Using the following identity: $$n_\uparrow n_\downarrow = \frac{1}{4}
\bigl(\rho^2 - m^2 \bigr),
\label{eq:identity}$$ with $\rho=n_\uparrow+n_\downarrow$ and $m=n_\uparrow-n_\downarrow$, we perform the HS transformation $\exp (-{\cal S}_{int}) = \int {\cal D} [\varphi, x] \exp (-{\cal S}'_{int})$ with the effective interaction $$\begin{aligned}
{\cal S}'_{int} \!\! &=& \!\!
\frac{1}{4U} \int_0^\beta \!\! d\tau
\biggl[ \bigl\{ \varphi^2 (\tau) + x^2 (\tau) \bigr\} \nonumber \\
&& \!\! - 2U \sum_\sigma c^\dag_\sigma (\tau) \{ \varphi(\tau) \sigma_z + i x (\tau) \} c_\sigma (\tau)
\biggr] ,
\label{eq:Sint}\end{aligned}$$ where, $\varphi$ and $x$ are the HS fields acting on spin and charge degrees of freedom, respectively, and $\sigma_z$ is the $z$ component of the Pauli matrices. Now, one can perform Grassmann integral over the fermionic field formally to obtain the partition function $$Z=\int \! {\cal D}[\varphi , x] \exp \bigl( -{\cal S}_{eff} \bigr)
\label{eq:Zfull}$$ with the effective action $$\begin{aligned}
{\cal S}_{eff} \!\! &=& \!\!
\frac{1}{4U} \int_0^\beta \! d\tau \bigl\{ \varphi^2 (\tau) + x^2 (\tau) \bigr\} \nonumber \\
&& \!\! - {\rm Tr} \, {\rm ln} \Bigl[-a_\sigma
- \frac{1}{2} \bigl(\varphi \sigma_z + i x \bigr) \Bigr] .
\label{eq:Seff}\end{aligned}$$ Here, the trace includes spin as well as Matsubara frequency. It is noted that the coupling constant for the charge field $x$ is imaginary. This originates from the different sign of quadratic terms for spin and charge when decoupling the interaction term. \[see Eq. (\[eq:identity\])\]
![Example of the effective potential for $\varphi$, $V(\varphi)$ given in Eq. (\[eq:Veff\]), in the square-lattice half-filled Hubbaed model with $U/t=6$. Solid line: $T/t=0.4$ (about 14 % above $T_N \sim 0.35t$); broken line: $T/t=0.3$ (about 14 % below $T_N$). At both temperatures, $V$ is far from a simple parabola, and regions far from the local minima have appreciable occupation probability. []{data-label="fig:Veff"}](Veff.eps){width="0.75\columnwidth"}
Exact evaluation of the partition function Eq. (\[eq:Zfull\]) is impossible. The simplest approximation is to approximate the integrals by the value computed using the static parts $\bar \varphi$ and $\bar x$ which extremize the action (solutions of the saddle-point equations $\partial {\cal S}(\bar \varphi, \bar x)/\partial \bar \varphi =
\partial {\cal S}(\bar \varphi, \bar x)/\partial \bar x = 0$). This is the Hartree-Fock (HF) approximation, which is known to give a poor estimates to transition temperatures and self-energies. One may correct the HF approximation by including the Gaussian fluctuations in which ${\cal S}_{eff}$ is expanded around the saddle point up to the quadratic order of the fluctuations $\delta \varphi(\tau)$ and $\delta x (\tau)$. In this case, $\delta \varphi (\tau)$ and $\delta x (\tau)$ decouple. However, Gaussian fluctuation theory is limited to the weak coupling regime. As an example, Fig. \[fig:Veff\] shows an effective potential $V$ for $\varphi$, equivalent to $T {\cal S}_{eff}$, calculated (as explained below) by the semiclassical approximation. It is seen that the potential is highly non-parabolic, and the variation in $V$ is on the scale of $T$ for reasonable $T$. Thus, the Gaussian fluctuation theory is inapplicable. Next, one can consider taking $\varphi$ and $x$ only at Matsubara-frequency $\nu_l =0$, i.e., static approximation, but evaluating the partition function $Z$ as a two-dimensional integral over two static fields (2-field approximation). The partition function is expressed as $Z_{static}=\int \! d\varphi \, dx \exp(-{\cal S}_{static})$ with $${\cal S}_{static} =
\frac{\beta}{4U} \bigl( \varphi^2 + x^2 \bigr)
- {\rm Tr} \, {\rm ln} \Bigl[-a_\sigma
- \frac{1}{2} \bigl(\varphi \sigma_z + i x \bigr) \Bigr].
\label{eq:Sstatic}$$ However, this approximation fails because the effective action ${\cal S}_{static}$ is complex and $\exp (- {\cal S}_{static})$ can not be regarded as a distribution function of $\varphi$ and $x$, leading to poor convergence of integrals. With some effort, apparently converged integrals can be obtained, but the interacting Green function computed by using Eq. (\[eq:Gimp\]) with $Z$ replaced by $Z_{static}$ does not behave correctly; the imaginary part of the self-energy changes sign and the spectral-function sum-rule is strongly violated. As an example, Fig. \[fig:Sigma\_bad\] compares self-energies computed by evaluating the static average integrating over two static fields $\varphi$ and $x$ with the action Eq. (\[eq:Sstatic\]) to the semiclassical approximation defined shortly. While we have obtained the converged solution along the Matsubara-axis, imaginary part of the self-energy changes sign and causality is strongly violated, so that the analytic continuation to the real axis is impossible.
![Comparison of the electron self-energy computed by evaluating the static average integrating over two static fields $\varphi$ and $x$ with the action Eq. (\[eq:Sstatic\]) (2-field) and the semiclassical approximation, which is defined in this section, for the infinite-dimensional FCC lattice Eq. (\[eq:Ulmkedos\]). Squares (circles) are real (imaginary) part of the self-energy, and filled and open symbols are obtained by 2-field approximation and the semiclassical approximation, respectively. []{data-label="fig:Sigma_bad"}](Sigma_bad.eps){width="0.8\columnwidth"}
![Comparison of the many-body density of states computed for the paramagnetic phase of the infinite-dimensional FCC lattice \[Eq. (\[eq:Ulmkedos\])\] by the “full” semiclassical approximation (solid line) to that computed using additional approximation in which $\xi$ is assumed to be independent of $\varphi$ (broken line). The three-peak structure seen in the broken line is unphysical; the solid line is in good agreement with QMC (not shown). Note that $T_C \sim 0.073$ for these parameters; we have presented data at $T=0.05 \simeq 3T_C/4$ (chosen to reveal the three-peak structure clearly), and have artificially suppressed ferromagnetism. []{data-label="fig:DOS_xi"}](DOS_xi.eps){width="0.8\columnwidth"}
![Example of the effective potential $V$ (upper panel) and charge field $\xi$ (lower panel) as functions of $\varphi$ in the metallic Hubbard model on an infinite-dimensional FCC lattice \[free DOS is given in Eq. (\[eq:Ulmkedos\])\] with $U=4$ and $n=0.5$. Solid lines: $T=0.08$ (about 14 % above $T_C \sim 0.073$); broken lines: $T=0.06$ (about 18 % below $T_C$). The Hartree contribution $(= n U)$ has been subtracted from $\xi$. At both temperatures, potential is seen to be highly deviated from simple parabola. $\varphi$-dependence of $\xi$ indicates strong coupling between the two fields.[]{data-label="fig:VandXi"}](VandXi.eps){width="0.75\columnwidth"}
In order to reduce the above mentioned problems and to take into account the fluctuation of both fields to some extent, we apply the semiclassical approximation following Hasegawa.[@Hasegawa80] In this method, we first solve the mean-field equation for the static charge field $\bar x_\varphi$ which extremizes ${\cal S}$ at a given value of $\varphi$. From $\partial {\cal S}_{eff}/ \partial \bar x |_\varphi = 0$, we obtain the following equation $(\xi_\varphi = i \bar x|_\varphi)$, $$\begin{aligned}
\xi_\varphi = - U T \, {\rm Tr} \frac{1}{a_\sigma+(\varphi \sigma_z + \xi_\varphi)/2}.
\label{eq:xi}\end{aligned}$$ By solving Eq. (\[eq:xi\]), one obtains $\xi_\varphi$ as a function of $\varphi$. In the single-impurity model, this equation has a unique solution, thus, Eq. (\[eq:xi\]) can be solved easily, for example, via bisection. (For multiimpurity models, we have no proof that there is a unique solution, but difficulties have not arisen in the case we have studied.) Fluctuations of the charge field around this saddle point are expected to be less important than those of spin field, because of the unique solution and because charge fluctuations are not soft, and the difficulties associated with the oscillatory convergence of Eq. (\[eq:Sstatic\]) are avoided.
The $\varphi$-dependence of $\xi$ is crucial. If it is neglected, i.e., if $\xi$ is taken to be the value $\xi_{av}$ which extremizes $S_{eff}$ after averaging over $\varphi$, then one obtains a model equivalent to the Jahn-Teller phonon model studied for example in Ref. . This latter model has different physics from the Hubbard model. In particular, the Jahn-Teller model has a spectral function with a characteristic three-peak structure quite unlike the spectral function of Hubbard-like models. An example of the density of states computed by the approximation, $\xi_\varphi \rightarrow \xi_{av}$ is shown in Fig. \[fig:DOS\_xi\] as the broken line. One can clearly observe a three-peak structure. The outer two peaks correspond to the occupied (lower band) and unoccupied (upper band) state at the occupied distorted site (from potential minima at $\varphi \sim \pm U$), and the middle peak corresponds to the unoccupied undistorted state (from potential minimum at $\varphi \sim 0$) in the phonon model. In the phonon model, the level separation between the lower occupied state and the middle unoccupied one has physical meaning as “polaron binding,” which does not exist in the original Hubbard model. The spectral function computed from the “full” semiclassical method (see below) is shown as the solid line in Fig. \[fig:DOS\_xi\], and as will be seen below is in much better agreement with QMC data.
Now that the saddle point of the effective action is determined with respect to one of two variables, $\xi$, an effective potential for remaining variable $\varphi$ is written as $$V(\varphi) = \frac{1}{4U} (\varphi^2 - \xi_\varphi^2)
- T \, {\rm Tr} \, {\rm ln} \Bigl[-a_\sigma
- \frac{1}{2} \bigl(\varphi \sigma_z + \xi_\varphi \bigr) \Bigr].
\label{eq:Veff}$$ With this effective potential, the partition function is approximated as $$\begin{aligned}
Z_{approx}=\int_{-\infty}^\infty d \varphi \exp\{-\beta V(\varphi)\}.
\label{eq:Zapprox}\end{aligned}$$ There remains only one variable $\varphi$, which is purely real. Numerical integrals can be performed without difficulty. Figure \[fig:VandXi\] shows the example of $V(\varphi)$ and $\xi_\varphi$ calculated by “full” semiclassical approximation for a non-integer filling. It is seen again that $V$ is highly non-parabolic. Furthermore, $\xi_\varphi$ depends on $\varphi$ indicating the strong coupling between spin- and charge-fields.
Now the approximate partition function $Z_{approx}$ is obtained, one can obtain physical quantity from the functional derivative form following the DMFT procedure. The impurity Green function is $$\begin{aligned}
G_\sigma = \frac{\delta {\rm ln} Z_{approx}}{\delta a_\sigma}
=\bigg\langle \frac{1}{a_\sigma+(\varphi \sigma_z + \xi_\varphi)/2} \bigg\rangle,
\label{eq:Gfull}\end{aligned}$$ where $\langle \ldots \rangle$ stands for $\int \! d\varphi \exp\{-\beta V(\varphi)\} \ldots / Z_{approx}$. Thus, Eqs. (\[eq:xi\])–(\[eq:Gfull\]) with the DMFT self-consistency equation construct the present semiclassical approximation.
It may be useful to mention the correspondence between the present theory and the single-site spin-fluctuation theory of Hasegawa.[@Hasegawa80] In Ref. , Hasegawa introduced the same HS transformation using spin- and charge-fields, and applied saddle-point approximation against the charge-field. Thus, up to this point, two theories are equivalent, but instead of Eq. (\[eq:Gfull\]) and DMFT self-consistency equation, Hasegawa used an ad-hoc procedure involving computation of averages of the magnetic moment and square of the magnetic moment.
We emphasize, however, that the semiclassical approximation is not exact. In weak coupling, it leads to a scattering rate $\propto T$, rather than $\propto T^2/E_F$, and does not give a mass renormalization; in strong coupling, while it gives correct Mott insulator behavior, i.e., divergence of the on-site self-energy $\propto 1/(i \omega_n)$, the self-energy at small $\omega$ remains finite at $T \rightarrow 0$ even in a paramagnetic metallic phase as long as the effective potential $V(\varphi)$ has more than one degenerate minimum, implying incorrect non-Fermi liquid behavior. This originates from the neglect of the quantum fluctuation of HS fields. Including the quantum fluctuation to recover the correct Fermi-liquid behavior is not easy.[@Attias97; @Pankov02; @Blawid03] Although, it is not exact, it will be shown below that the semiclassical method gives good estimates of the important physical quantities.
Weak Coupling Gaussian Approximation: Comparison of Classical and Exact Results
===============================================================================
This section uses the Gaussian approximation to the paramagnetic phase of the functional integral to gain insight into the limits of validity of the classical approximation. The advantage of the Gaussian approximation is that any desired quantity can be computed straightforwardly, permitting a detailed comparison of the results of the classical approximation to the fully quantal calculation. We consider the mean square amplitude of the spin Hubbard-Stratonovich field, $\left<\varphi^2\right>$, and also the Matsubara-axis electron self-energy, $\Sigma(i\omega_n)$.
In the Gaussian approximation one determines the values ${\bar \varphi}=UT \, {\rm Tr}_{\omega_n,\sigma}\sigma_z\left[a_0
+\frac{1}{2}({\bar \varphi}\sigma_z+ i \bar x)\right]^{-1}$ and $\bar x= i UT \, {\rm Tr}_{\omega_n,\sigma}\left[a_0
+\frac{1}{2}({\bar \varphi}\sigma_z+ i \bar x)\right]^{-1}$ which extremize the argument of the exponential, and then expand the argument of the exponential to second order in the deviations of the fields from their extremal values: $$\begin{aligned}
Z \rightarrow {\bar Z} \int {\cal D} [\varphi \, x]
\exp \Bigl[-S^{(2)}_{\varphi}-S^{(2)}_{x} \Bigr], \end{aligned}$$ where $\bar Z$ is contributions from the saddle points, and $$\begin{aligned}
S^{(2)}_{\lambda} \!\!\! &=& \!\!\! \int \! d\tau_1 d\tau_2 \, \lambda(\tau_1)
\chi_{\lambda}^{-1}(\tau_1-\tau_2) \lambda(\tau_2), \\
\chi_{\varphi/x}^{-1} \!\!\! &=& \!\!\! \frac{1}{4U} \, \delta(\tau_1-\tau_2) \nonumber \\
&& \hspace{-1.5em} \pm \frac{1}{8} {\rm Tr} \Bigr[\mbox{\boldmath $\alpha$}_{\varphi/x}
{\bf G}_0 (\tau_1-\tau_2)
\mbox{\boldmath $\alpha$}_{\varphi/x}
{\bf G}_0(\tau_2-\tau_1) \Bigl], \\
{\bf G}_0(\tau) \!\!\! &=& \!\!\! \Bigr[ a_0(\tau){\bf 1}+\frac{1}{2}({\bar \varphi}
\mbox{\boldmath $\sigma$}_{\!\! z}+ \bar x {\bf 1})\delta(\tau)\Bigl]^{-1},
\label{eq:g0}\end{aligned}$$ with $\mbox{\boldmath $\alpha$}_{\varphi} \! = \! \mbox{\boldmath $\sigma$}_{\!\! z}$ and $\mbox{\boldmath $\alpha$}_{x} \! = \! \mbox{\boldmath $1$}$, these are $2\times2$ matrices acting on spin space. In the mean-field approximation, a critical $U$ exists; for $U<U_c$, $\bar \varphi = 0$, and for $U>U_c$, $\bar \varphi \ne 0$. (This transition is removed by fluctuations.) For $U \agt U_c$, spin fluctuations are very soft.
We focus here on the $\varphi$ integral, and we consider the small-$U$ regime in which $\bar \varphi=0$, and in the paramagnetic phase, the $x$ and $\varphi$ integrals decouple. The mean square value of the fluctuations of $\varphi$, $\left< \varphi^2 \right>$ (obtained by performing the Gaussian integral over all Matsubara components of $\varphi$) and the classical approximation $\left< \varphi^2 \right>_{class}$ (obtained by performing the integral only over zero Matsubara components of $\varphi$) are $$\begin{aligned}
\left< \varphi^2 \right> \!\! &=& \!\! -T\sum_{l}\frac{U^2 \chi_0(i\nu_l)}{1+U \chi_0(i\nu_l)},\\
\left< \varphi^2 \right>_{class} \!\! &=& \!\! -T\frac{U^2 \chi_0(0)}{1+U \chi_0(0)}, \end{aligned}$$ where $\chi_0$ is an irreducible susceptibility given by $$\chi_0(i \nu_l)= 2 T \sum_n a_0^{-1}(i \omega_n+i\nu_l) \, a_0^{-1}(i\omega_n),$$ with $\nu_l$ being a bosonic Matsubara-frequency. Finally, the leading perturbative contribution to the electron self-energy $\Sigma$ and its classical approximation $\Sigma_{class}$ are given by $$\begin{aligned}
\Sigma(i\omega_n) \!\! &=& \!\! \frac{1}{4} \, T \sum_l \chi(i\nu_l) \,
a_0^{-1}(i\omega_n-i\nu_l), \\
\Sigma_{class} (i\omega_n) \!\! &=& \!\! \frac{1}{4} \, T \chi(0) \, a_0^{-1}(i\omega_n), \end{aligned}$$ with the susceptibility $$\chi(i\nu_l) = \frac{U^2 \chi_0(i\nu_l)}{1+ U \chi_0(i\nu_l)}.$$
We now present results obtained by applying the Gaussian approximation to the paramagnetic phase of the infinite-dimensional FCC lattice \[DOS is given by Eq. (\[eq:Ulmkedos\])\]. In this section, energy is scaled by the variance of DOS (=1). The lower band edge is at $\varepsilon_{min}=-1/\sqrt{2}\approx-0.71$ and the square root divergence of the density of states means that for small fillings the density of states is very close to the bottom of the band. In the non-interacting limit the chemical potential is $\mu_{n=0.5}\approx-0.6335$ ($\mu-\varepsilon_{min} \approx 0.078$) and $\mu_{n=0.75}=-0.5352$ ($\mu-\varepsilon_{min} \approx .175$). The square root divergences mean that the critical interaction strengths beyond which the mean field solution becomes unstable are themselves temperature dependent, and are small.
![Ratio of the classical approximation to the full (Gaussian) mean square fluctuation of spin Hubbard-Stratonovich field $R_\chi=\left<\varphi^2\right>_{class}/\left<\varphi^2\right>$ for infinite-dimensional FCC lattice at densities $n=0.5$ (upper panel) and $n=0.75$ (lower panel). The curves cross because of the temperature dependence of the critical interaction strength (evident from the $U$ at which the ratio approaches unity).[]{data-label="fig:Chi_ratio_ulm"}](Chi_ratio_ulm.eps){width="0.8\columnwidth"}
Figure \[fig:Chi\_ratio\_ulm\] gives the comparison of the full and Gaussian fluctuation approximation to the mean square fluctuation of the spin Hubbard-Stratonovich field for the infinite-dimensional FCC lattice model with $n=0.5$ and $n=0.75$. One sees that the classical approximation captures most of the fluctuations either for $U$ near the critical value or for temperatures of the order of the bandwidth. The panels of Fig. \[fig:Sigma\_ulm\_Udep\] show the full self-energy and the classical approximation for different interaction values, for $n=0.75$. For small $U$, the semiclassical approximation fails, but as $U$ approaches the critical value (here approximately $0.84$) the self-energy becomes well represented by its classical approximation
![Frequency dependence of full (Gaussian) self-energy (filled circles) and classical approximation (open circles), for infinite-dimensional FCC lattice with $n=0.75$, $T=0.04$ and $U$ values indicated.[]{data-label="fig:Sigma_ulm_Udep"}](Sigma_ulm_Udep.eps){width="0.95\columnwidth"}
The panels of Fig. \[fig:Sigma\_ulm\_Tdep\] show the dependence of the self-energy on temperature, at a fixed moderate interaction strength (about 3/4 of the critical value). We see that at all temperatures studied, the classical approximation provides a reasonable estimate of the low frequency self-energy. At temperatures less than about half of the band width, the classical approximation grossly underestimates the high frequency part of the self-energy, but by $T=0.08$ it is within about $30~\%$ of the exact value, and for higher temperatures or for interactions close to the critical value the approximation is quite good.
![Frequency dependence of imaginary part of full (Gaussian) self-energy (filled circles) and classical approximation (open circles) for $n=0.75$, $U=0.5$ and $T$ values indicated.[]{data-label="fig:Sigma_ulm_Tdep"}](Sigma_ulm_Tdep.eps){width="0.95\columnwidth"}
Comparison of semiclassical approximation to QMC: Two-dimensional square lattice
================================================================================
The Hubbard model on a bipartite lattice is known to exhibit an antiferromagnetic N[é]{}el ordering at half-filling and finite $U$. In this section, we apply the semiclassical approximation to the Hubbard model with a square lattice \[non-interacting electron dispersion is given by Eq. (\[eq:ek2d\])\], and investigate the self-energy, density of states and magnetic transition temperature. Note that the finite $T_N$ in two-dimensional square lattice is an artifact arising from the neglect of the low-lying spin-wave excitation which DMFT method can not capture.
![Self-energies for a square-lattice half-filled Hubbard model computed from the semiclassical approximation (filled circles) and QMC (open circles) for interactions and temperatures indicated.[]{data-label="fig:Sigma_square"}](Sigma_square.eps){width="0.8\columnwidth"}
These results are compared with QMC performed on computing facilities at Universit[ä]{}t Bonn using the Harsch-Fye algorithm.[@Georges96] Typically, 48 time slices were used, and 2–20$\times 10^6$ MC configurations were recorded. Occasional runs with more time slices or more MC configurations were made to verify error. Between 5 and (for the largest $U$) 40 iterations were needed for convergence of the DMFT loops. The N[é]{}el temperatures were determined from computations of the staggered magnetization. For $U \agt 8t$ and $T$ in or near the magnetic phase boundary, global update techniques (to be described elsewhere) were needed to ensure equilibration of the MC calculation. Error bars are smaller than the symbols shown.
As noted in the previous section, it is expected that the semiclassical approximation becomes accurate in the strong-coupling regime and high temperature. In order to confirm this expectation beyond the harmonic (Gaussian fluctuation) approximation, we first calculate the electronic self-energies in the paramagnetic state and compare them with the QMC results. Recall that for this model the full band-width is $8t$. In Fig. \[fig:Sigma\_square\], shown are the self-energies at $U/t=6$ and $T/t=1/2$ (upper panel), $U/t=8$ and $T/t=1/2$ (middle panel), and $U/t=20$ and $T/t=1/3$ (lower panel) computed by the semiclassical approximation and QMC as functions of imaginary frequency. With increase of $U$, the self-energy at low frequency increases, and for $U > U_c$, diverges indicating the Mott metal-insulator transition. This behavior is well reproduced by the semiclassical approximation, and the agreement with the QMC is remarkable, particularly in the strong coupling regime.
![Density of states of square-lattice half-filled Hubbard model computed using semiclassical method (heavy lines) and QMC (light lines) for paramagnetic \[panels (a) and (c)\] and antiferromagnetic \[panels (b) and (d), $T \sim 0.8 T_N$\] phases at interaction and temperatures indicated. In the antiferromagnetic phase, majority spin density of states is shown by solid line, minority spin by broken line. []{data-label="fig:DOS_square"}](DOS_square.eps){width="1\columnwidth"}
Figure \[fig:DOS\_square\] compares the semiclassical results for a real-frequency quantity, the density of states, to results obtained by maximum entropy analytical continuation of QMC data. This is a rather stringent rest of the method, and agreement is seen to be reasonably good. In the weak coupling, paramagnetic phase \[Fig. \[fig:DOS\_square\] (a)\], the semiclassical approximation underestimates the $\omega=0$ peak (because $\omega=0$ scattering rate is overestimated) and underestimates the weight in the wings (because the high frequency scattering rate is underestimated, see the upper panel of Fig. \[fig:Sigma\_square\]). Below $T_N$, the magnetization is slightly overestimated by the semiclassical approximation (magnetization $\langle m \rangle \sim 0.54$ for the semiclassical approximation and $\langle m \rangle \sim0.4$ for QMC at $U/t=4$ and $T/t=0.2$) as can be seen from the slightly larger gap. DOS is sharper because the self-energy at high-frequency is underestimated as in the paramagnetic state. In the strong coupling regime, agreement between the semiclassical approximation and QMC is quite good as can be expected from the self-energy (see the lower panel of Fig. \[fig:Sigma\_square\]).
![N[é]{}el temperature of square-lattice half-filled Hubbard model as functions of $U$ computed from semiclassical approximation (filled symbols) and QMC (open symbols), HF (light solid line) and large $U$ limit ($T_N = 4t^2/U$) (light broken line).[]{data-label="fig:TN_square"}](TN_square.eps){width="0.8\columnwidth"}
Figure \[fig:TN\_square\] summarizes results for the magnetic phase diagram obtained from the semiclassical approximation, QMC, the HF approximation and a strong-coupling expansion. The semiclassical approximation is seen to give remarkably good results over the whole phase diagram. In the weak-coupling limit, the semiclassical approximation gives correct behavior: $T_N$ asymptotes to the result of HF in the limit of $U \rightarrow 0$. This is natural because the two approximations becomes identical in the $U \rightarrow 0$, $T \rightarrow 0$ limit. At finite but small $U$ region, the reduction of $T_N$ from HF is found to be quite large, because the finite self-energy reduces nesting effect substantially. The present approximation gives correct behavior in the strong-coupling regime; $T_N$ asymptotes to $4t^2/U$, the mean-field result of the strong coupling expansion, and, in particular, is almost identical to the QMC results. It should be noted that at large $U$ and low $T$, the QMC requires a very large amount of Monte-Carlo sampling to reach equilibrium, whereas the semiclassical method is numerically very cheap. Even for $U/t=16$ and $T/t \sim 0.2$ (antiferromagnetic phase), it takes less than 5 mins. on a commercial PC to compute one point in the present method, while it takes $\sim 6$ hrs. by QMC on a similar computer for the same parameters.
Comparison of semiclassical approximation to QMC: FCC lattice
=============================================================
In this section, we apply the semiclassical approximation to the single-band Hubbard model on the FCC lattice in infinite- and three-dimensions, and compare the results to QMC data of Ulmke,[@Ulmke98] and to the “two-site” approximation of Potthoff.[@Potthoff01] We focus on the filling dependence of the magnetism in these models, in which the charge fluctuation plays an important role. At non-integer filling, these models order ferromagnetically because of the large DOS near the bottom of the band, in contrast to the antiferromagnetic ordering found in half-filling square lattice. In addition, near half-filling, the three-dimensional model is found to show an additional competing order: a “layer-type antiferromagnetic” state. We examine whether the semiclassical approximation capture this behavior. In this section, we use the variance of DOS $v$ as an unit of energy.
First, we apply the semiclassical approximation to the infinite-dimensional FCC lattice with DOS given in Eq. (\[eq:Ulmkedos\]), $v=1$. The density of states above the Curie temperature is essentially the same as seen as the solid line in Fig. \[fig:DOS\_xi\], and consists of two structures: one peak just below the Fermi level $\omega=0$ and shoulder at $\omega \sim nU$. These correspond to the lower- and upper-Hubbard bands. (Note that the result shown in Fig. \[fig:DOS\_xi\] is computed at $T<T_C$, but in the paramagnetic phase, by suppressing the magnetic transition.) The separation between the lower- and upper-Hubbard band is somewhat reduced from the bare value of $U$ (by about a factor of mean occupation number $n$), possibly because the neglect of the fluctuation of a charge-field. The two structures continuously evolve into the majority-spin and minority-spin bands below $T_C$ as shown in Fig. \[fig:DOS\_fcc\].
Figure \[fig:DOS\_fcc\] also presents the DOS from QMC. (Temperature and density are chosen so that the magnetization in the two calculations are similar.) In QMC results, we observe a sharp peak below Fermi level for the majority spin band. For the minority spin band, there appears a sharp peak near the Fermi level and a broad hump at $\omega \sim 3$. The former originates from the fact that the system is not fully polarized, and the latter corresponds to the upper-Hubbard band. As noted above, the upper-Hubbard band (broad hump around $\omega=3$ in the minority-spin band) appears slightly low in energy in the semiclassical approximation. Except for this discrepancy, the overall structure of DOS is well reproduced by the semiclassical approximation. In particular, the occupied bands below $\omega=0$ show reasonable agreement.
![Comparison of the density of states of infinite-dimensional FCC Hubbard model with $U=4$ and temperatures chosen so that $\langle m \rangle =0.4$ in each case. Heavy lines: results of the semiclassical approximation at $n=0.6$ and $T=0.07$ ($\sim 13~\%$ below $T_C \sim 0.08$). Light lines: results of QMC at $n=0.58$ and $T=0.04$ ($\sim 20~\%$ below $T_C \sim 0.05$) taken from Ref. . Solid and broken lines are for the majority and minority spin.[]{data-label="fig:DOS_fcc"}](DOS_fcc.eps){width="0.8\columnwidth"}
In Fig. \[fig:TC\_fcc\_inf\], ferromagnetic Curie temperatures $T_C$ for different values of $U$ are shown as functions of electron density. Filled symbols are the results obtained using the semiclassical approximation. For comparison, results of QMC[@Ulmke98] and HF are shown as open symbols and light lines, respectively (Note that the HF $T_C$ shown are reduced by a factor of 10). The semiclassical approximation overestimates $T_C$ than QMC by a factor of $\sim 50~\%$ over a wide range of density. (Of course, the difference near the critical density of QMC is much larger.) It is seen that HF approximation is very poor in all parameter regimes, highly overestimating the ferromagnetic Curie temperature. The critical density $n_c$ is also overestimated, at $n_c \sim 1.2$ at $U=2$ and $n_c \sim 1.5$ at $U=4$. The higher $T_C$ found in the semiclassical approximation may be due to the neglect of the quantum fluctuation of the HS fields. Reduction of the transition temperature due to the quantum fluctuation can be seen in the context of electron-phonon coupling in Ref. . However, good agreement between the semiclassical approximation and QMC indicates that the thermal part of the fluctuation dominates the electron self-energy in the wide region and, thus, the magnetic transition. Critical densities $n_c$ where the ferromagnetism disappears at $T=0$ are found to be slightly higher in the semiclassical approximation than in QMC, $n_c \sim 0.88$ for $U=2$ and $n_c \sim 0.97$ for $U=4$, while QMC gives $n_c \sim 0.7$ for $U=2$ and $n_c \sim 0.88$ for $U=4$. However, in the light of the simplification of the present approximation, the agreement with QMC within 20 % error is remarkable.
We applied the two-site DMFT[@Potthoff01] to the same model to investigate the magnetic phase diagram. The critical densities $n_c$ obtained by two-site DMFT are found to be very similar to those by the semiclassical approximation. However, Curie temperatures are found to be overestimated by a factor of $\sim 6$ compared with QMC, and by a factor of $\sim 3$–$4$ compared with the semiclassical approximation. In the two-site DMFT, $T_C$ at $n=0.5$ is found to be $\sim 0.17$ for $U=2$, and $\sim 0.26$ for $U=4$. The overestimate arises because, in the two-site DMFT, the IAM is composed of only two sites, one is correlated (impurity) and one is non-correlated (bath). The energy difference between these sites is generically large, so the (spin) entropy is underestimated. The two-site DMFT also yields reentrant behavior near the critical concentration, which is not observed in the semiclassical approximation and QMC. Adding more sites in the “bath” part of IAM would remedy these behaviors, but at drastically increased computational expense.
![Ferromagnetic Curie temperature of single-band Hubbard model on an infinite-dimensional FCC lattice as function of electron density for $U=2$ (squares) and 4 (circles). Filled symbols: semiclassical approximation; open symbols: QMC (Ref. ). Curie temperatures ($\times 10^{-1}$) computed by HF approximation to the same model with $U=2(4)$ are shown as a light solid (broken) line. Two-site DMFT results for the transition temperatures are not shown, but representative two-site results for $n=0.5$ are $T_C \sim 0.17 (0.26)$ for $U=2 (4)$. Phase boundary at $T=0$ for $U=2(4)$ computed by the two-site DMFT is shown as a filled (open) triangle. []{data-label="fig:TC_fcc_inf"}](TC_fcc_inf.eps){width="0.8\columnwidth"}
Finally, we investigate the magnetic instability in the more realistic three-dimensional FCC lattice whose free band-dispersion is given in Eq. (\[eq:FCC3d\]). Following Ref. , we take $t'= t/4$ and choose the variance of the DOS $v = \sqrt{12 t^2 + 6 {t'}^2}$ as the energy unit, thus $t \approx 0.284$ and $t' \approx 0.071$. With this parameter set, the bottom of the band is given by $\varepsilon_{min} = -4t+2t' \approx - 0.994$. There is no divergence in the free DOS in contrast to the infinite-dimensional case.
The magnetic phase diagram of the three-dimensional FCC lattice computed by the semiclassical approximation is shown in Fig. \[fig:TC\_fcc\_3d\]. For comparison, QMC[@Ulmke98] and HF results are also plotted. It is seen that the semiclassical-approximation results for $T_C$ are about twice higher than QMC, while HF results are about 20 times higher than QMC. Similar to the infinite-dimensional case, higher critical density $n_c$ is overestimated by $\sim20~\%$. QMC calculations are argued to indicate that there exists a lower critical density $n_c' \sim 0.15$ below which the ferromagnetism disappears.[@Ulmke98] This behavior is ascribed to the absence of the divergence at the bottom of the bare DOS. In the semiclassical approximation, $T_C$ seems to become zero proportional to the electron density similarly to the HF approximation. In the latter, clearly $n_c'=0$. This might be because the semiclassical approximation reduces to the HF at weak coupling (in this case small density) and low temperature limit. However, it is very difficult to judge if $n_c'=0$ or not in the present accuracy for the semiclassical approximation.
![Upper panel: Phase diagram of single-band Hubbard model on a three-dimensional FCC lattice as a function of electron density and temperature for interaction $U=6$ and variance of the density of states $v=1$. Squares and circles are Curie temperature and N[é]{}el temperature for layer-type antiferromagnetic state, respectively. Filled symbols: results of the semiclassical approximation; open symbols: QMC results taken from Ref. . Curie temperature ($\times 10^{-1}$) computed by HF approximation to the same model is shown as a light solid line. Lower panel: Expansion of the phase diagram, showing the region near $n=1$ enclosed by a broken line in the upper panel. The semiclassical results for $T_C$ and $T_N$ indicate the instability of ferromagnetic and antiferromagnetic states, respectively, towards the paramagnetic state. []{data-label="fig:TC_fcc_3d"}](TC_fcc_3d.eps){width="0.8\columnwidth"}
Differently from the infinite-dimensional FCC lattice, QMC calculation found another magnetic state stabilized near $n=1$: a layer-type antiferromagnetic state with a magnetic vector $\vec q = (\pi,0,0)$.[@Ulmke98] The semiclassical approximation is successful in finding the layer-type antiferromagnetic state near $n=1$. Computed N[é]{}el temperature at $n=1$ is $T_N \sim 0.048$ which agrees with the QMC result within the statistical error of QMC. Better agreement in the N[é]{}el temperature than in the Curie temperature may be attributed to the large $U$ used in this calculation and to the suppression of the charge fluctuations near half-filling. According to the two-site DMFT, critical $U_c$ to the Mott transition at half filling is estimated to be $U_c = 6 v$ (correct value is expected to be slightly smaller then this).[@Potthoff01] With the parameter used, $U=6$, the system is in a Mott insulating state at $n=1$, and the charge fluctuation is suppressed. Therefore, the thermal fluctuation of spin field dominates the magnetic transition. In contrast to QMC, ferromagnetic and antiferromagnetic states contact with each other in the semiclassical approximation. (In the semiclassical result, $T_C$ and $T_N$ indicate the instability of the ferromagnetic and antiferromagnetic states, respectively, to the paramagnetic state.) In the light of the overestimation of upper critical density $n_c$ for the ferromagnetism, this failure is also supposed to be from the neglect of the quantum fluctuation of HS field. This point remains to be addressed in the future work.
In the metallic region, one needs to fix the chemical potential according to the density– this must be done at each interaction strength and temperature, which is time consuming. Even in this case, the semiclassical scheme is found to be computationally very cheap. Typical CPU time is less than 5 mins. at $U=2$ and $T=0.05$ using a commercial PC.
Summarizing this section, we investigated the magnetic behavior of the Hubbard model on infinite- and three-dimensional FCC lattices using the semiclassical approximation. The magnetic phase diagram computed by the semiclassical approximation show reasonable agreement with QMC. The ferromagnetic Curie temperature in metallic region is found to be within a factor of $\sim 2$ compared with QMC, and the antiferromagnetic N[é]{}el temperature near $n=1$ is found to agree with QMC within statistical error. Better agreement between the semiclassical approximation and QMC in the N[é]{}el temperature near integer-filling is supposed to be from the fact that the charge fluctuation is suppressed by the correlation, so that thermal spin fluctuations dominate the transition as in the half-filled square-lattice case. Poorer agreement in the Curie temperature is expected to be from the neglect of quantum fluctuation or charge fluctuation.
Application of semiclassical approximation to DCA and fictive-impurity method
=============================================================================
Our semiclassical approximation is easily combined with cluster schemes, such as dynamical cluster approximation[@Hettler98] (DCA) and real-site cluster extension of DMFT.[@Kotliar01; @Biroli02; @Okamoto03] As a simplest example, we use the two-site DCA and fictive-impurity[@Okamoto03] (FI) methods to study the Hubbard model on a square lattice.
We consider half-filling, thus charge fields are absorbed into the chemical potential shift. The Weiss field becomes $2 \times 2$ matrix $\bf a$, and in the paramagnetic phase, this has a form $${\bf a} = a_0 {\bf 1} + a_1 \mbox{\boldmath $\tau$}_x ,$$ with $a_0$ and $a_1$ being the on-site and NN Weiss fields, respectively. $\bf 1$ and $\mbox{\boldmath $\tau$}_x$ are the $2\times2$ matrices acting on orbital (impurity site) space. The HS transformation is performed at each impurity sites $i=1,2$ with spin field $\varphi_i$. The effective potential for HS field becomes $$\begin{aligned}
V(\varphi_1,\varphi_2) \!\! &=& \!\! \frac{1}{4U} \bigl( \varphi_1^2 + \varphi_2^2 \bigr)
\nonumber \\
&& \!\! - T \, {\rm Tr} \, {\rm ln} \biggl[-{\bf a}
- \frac{1}{2}
\biggl(
\begin{array}{cc}
\varphi_1& \\
& \varphi_2
\end{array}
\biggr)
\mbox{\boldmath $\sigma$}_z \biggr].
\label{eq:Veff2}\end{aligned}$$ Here, $\mbox{\boldmath $\sigma$}_z$ is acting on spin space, and Tr is taken over Matsubara-frequency, orbital and spin indices. Then, the partition function is given as $$Z_{approx} = \int d \varphi_1 d \varphi_2 \exp \{ -\beta V (\varphi_1,\varphi_2) \}.$$ Finally, on-site and NN-site Green functions, $G_0$ and $G_1$, respectively, are obtained via $$\begin{aligned}
G_0 = \frac{1}{2} \frac{\delta \ln Z_{approx}}{\delta a_0}, \\
G_1 = \frac{1}{2} \frac{\delta \ln Z_{approx}}{\delta a_1}. \end{aligned}$$ The self-consistency equations for DCA are closed following the scheme presented in Ref. . The self-consistency equations for FI method and the general formalism for the cluster extension of DMFT are given in Ref. .
![Nearest-neighbor spin correlations $- \langle \sigma_{1z} \sigma_{2z} \rangle$ of a square-lattice Hubbard model ($U/t=20$) as functions of $T$ computed using two-site DCA (open circles) and the FI method (open squares), compared to QMC (filled symbols) and high-temperature series expansion(light line). []{data-label="fig:SScomp"}](SScomp.eps){width="0.8\columnwidth"}
Nearest-neighbor spin correlations $-\langle \sigma_{1z} \sigma_{2z} \rangle$ computed by DCA and FI with the semiclassical approximation and QMC as functions of temperature for $U=20t$ are shown in Fig. \[fig:SScomp\]. Similar spin correlation obtained by QMC in both DCA and FI supports the applicability of the semiclassical approximation to the cluster DMFT. For comparison, the same quantity computed using a high-temperature series expansion (HTS) for the $S=1/2$ NN Heisenberg model with $J=t^2/U$ are also shown. It is seen that spin correlation obtained from the DCA is much larger than the HTS result. We believe the origin of the discrepancy between DCA and HTS is that DA is equivalent to imposing a periodic boundary condition in all the directions in a real-space cluster is adopted. Thus, in the two-site DCA, one has a model in which two sites are connected via $z$ bonds \[connectivity $z=4$ in a square lattice\] and the spin correlation is thus overestimated. Interestingly, FI method gives almost identical curves to HTS. (slight deviation can be seen below $T \sim 0.4t$.) A detailed study of multi-site cluster models using semiclassical methods and QMC will be presented elsewhere.[@Fuhrmann05]
Equilibration and partition phase space
=======================================
In this section, we point out an additional advantage of the semiclassical approximation. A key issue in Monte-Carlo simulations is equilibration, which is particularly difficult in systems in which the phase space is partitioned into several nearly equivalent minima, separated by large barriers. Local update techniques require extremely long runs to climb over barriers, while global update techniques are expensive and sometimes inconvenient to implement and bring their own convergence issues. The partitioned phase space phenomenon occurs frequently in strongly-correlated models, and represents a significant obstacle to practical computations. The semiclassical approximation, by contrast, is inexpensive enough that the entire (semiclassical) phase space can be sampled.
For example, Fig. \[fig:P\_DCA\] shows the distribution function of spin HS fields for the two-impurity DCA for the square-lattice half-filled Hubbard model defined by $P(\varphi_1,\varphi_2) = \exp \{-\beta V(\varphi_1,\varphi_2) \}/Z_{approx}$ computed at temperature $T/t=0.5 > T_N$ (see Fig. \[fig:TN\_square\]). The partitioning of phase space is evident. It is seen that the distribution has twofold symmetry, not fourfold, in $\varphi_1-\varphi_2$ plane; $\varphi_1>\varphi_2$ and $\varphi_1<\varphi_2$ are not equivalent. Larger peaks at $(\varphi_1,\varphi_2) \simeq (\pm 20t, \mp 20t)$ than $(\varphi_1,\varphi_2) \simeq (\pm 20t, \pm 20t)$ indicate that the antiferromagnetic correlations prevail far above the N[é]{}el temperature, giving the result shown in Fig. \[fig:SScomp\]. Reliable estimates of $-\langle \sigma_{1z} \sigma_{2z} \rangle$ requires sampling of all four extreme, with correct weights, which is very difficult to achieve in QMC at low $T$ (this is why we do not present data at $T<0.5t$, and statistical errors are evident even at $T=0.6t$). However, the integrals involved in the semiclassical method may be performed with no difficulty.
![Distribution of the spin fields $\varphi_1$ and $\varphi_2$, $P(\varphi_1,\varphi_2)$, computed by two-impurity DCA for the Hubbard model on a square lattice. $P(\varphi_1,\varphi_2)$ is defined by $P(\varphi_1,\varphi_2) = \exp \{-\beta V(\varphi_1,\varphi_2)\}/Z_{approx}$. Parameters are $U/t=20$ and $T/t=0.5$. Larger peaks at $(\varphi_1,\varphi_2) \simeq (\pm 20t, \mp 20t)$ than $(\varphi_1,\varphi_2) \simeq (\pm 20t, \pm 20t)$ indicate antiferromagnetic correlation.[]{data-label="fig:P_DCA"}](P_DCA.eps){width="0.8\columnwidth"}
Summary and discussion
======================
In this paper, we investigated the semiclassical approximation to the continuous Hubbard-Stratonovich transformation as an impurity solver of DMFT method for the correlated-electron models. The Hubbard-Stratonovich transformation introduces two auxiliary fields, coupling to the electron spin and charge density respectively. The semiclassical approximation consists of retaining only the classical (zero-Matsubara-frequency) component in the functional integral over the spin field, and, for each value of the spin field, using a steepest descents approximation to approximate the integral over the charge field by the value which extremizes the action at the given value of the spin field \[see Eq. (\[eq:xi\])\]. This treatment of the charge field was found to be essential for achieving reasonable results: see Fig. \[fig:DOS\_xi\] and the associated discussion. The semiclassical approximation captures the thermal fluctuation of spin field efficiently beyond harmonic (Gaussian fluctuation) approximation and is applicable to both the metallic and insulating region. Estimates obtained using the Gaussian fluctuation approximation indicate that the semiclassical approximation gives reasonable results at larger $U$ (when spin fluctuations are soft) or at high temperature. (Sec. III) We applied the semiclassical approximation to the single-band Hubbard model on a two-dimensional square lattice (half-filling) (Sec. IV) and infinite- and three-dimensional FCC lattices (finite doping) (Sec. V). The semiclassical approximation is found to give reasonable results, with accuracy improving for stronger couplings and in situations where charge fluctuations are suppressed. A particularly attractive finding is that the procedure finds multiple phases when these exist. (Fig. \[fig:TC\_fcc\_3d\]) The key point appears to be that, at stronger correlations, the physically important fluctuations become very soft (justifying a semiclassical approximation) but very anharmonic (requiring an integral over all field configurations). Due to the neglect of quantum effects, the semiclassical approximation fails to reproduce correctly the quasiparticle resonance, which will certainly be important at weak coupling or at low temperature. One possible improvement of this failure would be using a interpolative method[@Savrasov04] to reproduce the low-energy quasiparticle in a paramagnetic metallic region. For the half-filled square lattice, we showed the semiclassical approximation gives reasonable behavior of the self-energy which is comparable to QMC in a wide range of parameters. The density of states and N[é]{}el temperature computed by the semiclassical approximation are found to be in very good agreement with QMC. In the metallic FCC lattice, the DOS has a two-peak structure corresponding to the lower- and upper-Hubbard bands in a paramagnetic phase. These structures evolve into majority- and minority-spin bands as the temperature is decreased through the ferromagnetic transition. Comparison of the DOS computed by the semiclassical approximation and by QMC shows reasonable agreement, especially as concerns the occupied state. Ferromagnetic Curie temperatures computed by the present method are found to be in the range of factor of 2 compared with the QMC. $T=0$ phase boundaries are found to be within the error of 20 %. The semiclassical approximation for the Curie temperature shows better agreement than two-site DMFT, and $T=0$ phase boundaries are found to be in the same range of two-site DMFT. As for the three-dimensional FCC lattice, the semiclassical approximation is able to detect antiferromagnetic state near the integer filling consistent with QMC. N[é]{}el temperature in this case agrees with QMC within the statistic error. In Sec. VI, we apply the semiclassical approximation to two-impurity DCA and real-space cluster DMFT (fictive-impurity) for the square-lattice Hubbard model at half-filling, and investigate how spatial correlation is taken into account in the present approximation. We confirmed that the short-range (nearest-neighbor) spin correlation prevails above the N[é]{}el temperature. Comparison of nearest-neighbor spin correlation between the semiclassical approximation and QMC shows good agreement in ranges where the QMC calculations are well converged. In Sec. VII, equilibration problem in QMC associated with the partition of phase is discussed. This problem does not occur in the semiclassical approximation.
In conclusion, we mention some open scientific questions which can be addressed by the semiclassical method, and mention two areas in which further investigation and improvement would be desirable. A key opportunity involves systems with several sites and/or several orbitals. In such cases, QMC suffers from “sign” problems, and both QMC and ED method scale poorly with system size. The semiclassical method does not suffer from “sign” problem; while there appeared similar problem associated with the imaginary coefficient for the charge-field, this is resolved by applying the saddle-point approximation to the charge-field. When it is applied to systems with $N>1$ orbitals (on one or more sites), computational time for the semiclassical approximation scales of $P_\varphi^N$ with $P_\varphi$ and $N$ being a total number of descretized HS field and total number of sites/orbitals, respectively. By contrast in Monte-Carlo methods, the scaling is $[(\beta U)^2 N_{MC}]^N$ ($N_{MC}$ is a number of Monte-Carlo sampling, typically $N_{MC} \sim 10^5$), and by $4^{N (1+N_{bath})}$ for ED ($N_{bath}$ is a number of “bath” sites per impurity orbital, and $N_{bath}$ should be larger than 2–3 from the overestimation of $T_C$ by two-site DMFT). As can be seen in the Fig. \[fig:P\_DCA\], dominant contribution is known to be from $|\varphi| \sim U$, and if need it would be possible to simplify the $N$-dimensional integral over HS fields. Thus, the semiclassical approximation appears to be an attractive option for studying larger clusters than QMC.
As for the multisite problem, we have only applied the semiclassical approximation to the single-orbital Hubbard model at half-filling with DCA and FI method. In this case, the charge-field is absorbed into the chemical potential. Away from the half-filling, one needs to fix the charge-field at each configuration of the spin-fields according to Eq. (\[eq:xi\]) generalized to the multisite situation. There is no proof that Eq. (\[eq:xi\]) has unique solution in this situation although we have so far not encountered problems. This issue needs further investigation.
As one of the applications of the semiclassical approximation, investigation of the systems with degenerate orbitals like transition-metals oxides with $e_g$ or $t_{2g}$ electrons would be interesting.[@Imada98; @Tokura00] Theoretical studies of the ferromagnetism in these situation have been presented (Refs. ), but these works dealt with spin-fluctuations in the metallic state, fluctuation and ordering associated with the quadrupole moment for $e_g$ and/or $t_{2g}$ orbital was neglected. Hubbard-Stratonivich transformation including spin and quadrupole moment for two-band Hubbard interaction for $e_g$ orbital has been introduced in Ref. . However, pin and orbital transition at finite temperature using the multi-band Hubbard model with full symmetry[@Mizokawa95] remains to be investigated.
Another promising application of the semiclassical approximation would be the spatially inhomogeneous systems, where the computational expense associated with other aspects of the problem renders an inexpensive impurity solver essential. Recently, two of the authors (S.O. and A.J.M.) applied two-site DMFT to the spatially-inhomogeneous heterostructure problem and investigated the evolution of the electronic state (quasiparticle band and upper- and lower-Hubbard bands) as a function of distance from the interface.[@Okamoto04] Applying the semiclassical approximation to such systems to investigate the possible spin (and orbital) orderings[@Nature] is an interesting and urgent task.
The authors acknowledge useful discussions with B. G. Kotliar and M. Potthoff. This research was supported by the JSPS (S.O.) and the DOE under Grant ER 46169 (A.J.M.).
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Electronic address: [email protected]
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Nonrelativistic bound states are studied using an effective field theory. Large logarithms in the effective theory can be summed using the velocity renormalization group. For QED, one can determine the structure of the leading and next-to-leading order series for the energy, and compute corrections up to order $\alpha^8 \ln^3 \alpha$, which are relevant for the present comparison between theory and experiment. For QCD, one can compute the velocity renormalization group improved quark potentials. Using these to compute the renormalization group improved $\bar t t$ production cross-section near threshold gives a result with scale uncertainties of 2%, a factor of 10 smaller than existing fixed order calculations.'
address: |
Department of Physics, University of California at San Diego,\
9500 Gilman Drive, La Jolla, CA 92093-0319
author:
- 'Aneesh V. Manohar and Iain W. Stewart'
title: 'Nonrelativistic Bound States in Quantum Field Theory[^1] '
---
INTRODUCTION
============
Nonrelativistic bound states in QED and QCD provide an interesting and highly nontrivial problem to which effective field theory methods can be applied [@Caswell; @BBL]. The QCD bound states we will consider are heavy $\bar Q Q$ states such as $\bar tt$ bound states or the $\Upsilon$ system. In QED, the classic examples are Hydrogen, muonium ($\mu^+e^-$), and positronium. Each of these systems has three important scales, $m$ the fermion mass, $mv$ the fermion momentum, and $mv^2$, the fermion energy. (For Hydrogen and muonium, $m$ is the electron mass or the reduced mass of the two particles.) The velocity $v$ is of order the coupling constant ($\alpha_s$ or $\alpha$), and we will only consider the case $v\ll1$, $mv^2\gg\lqcd$ so that nonperturbative effects are small.
Multiscale problems with widely separated scales are well suited for study using effective field theories. For example, if the problem has the scales $m_1
\gg m_2 \gg m_3 \ldots$, one first starts with the theory above $m_1$, and matches to an effective theory below $m_1$ in which only modes with masses much smaller than $m_1$ are retained. The effective theory is then scaled using the renormalization group to the next scale $m_2$. At this point, particles with mass $m_2$ are integrated out to construct a new effective theory, and so on. The complicated multiscale computations of the original theory are reduced to a number of simpler single scale computations of matching and running in the effective theory. The effective theory method also allows one to sum logarithms of the ratio of mass scales $\ln m_i/m_{i+1}$ using the renormalization group evolution between $m_i$ and $m_{i+1}$.
The goal is to correctly separate the scale $m$, $mv$ and $mv^2$ for nonrelativistic bound state problems using an effective field theory, and to sum large logarithms using the renormalization group. The large logarithms in this case are $\ln p/m$, $\ln E/m$ and $\ln p/E$ which are proportional to $\ln
v$, and lead to $\ln \alpha$ contributions to bound state energies. Furthermore, for QCD, the effective theory also determines the scale of the strong coupling constant, i.e. whether one should use $\alpha_s(m)$, $\alpha_s(mv)$ or $\alpha_s(mv^2)$. The nonrelativistic effective theory, NRQCD/NRQED, has been studied extensively in the past [@Caswell; @BBL; @Labelle; @LM; @AM; @GR; @LukeSavage; @PS1; @PS2; @PS3; @LMR; @amis; @amis2; @amis3; @amis4; @Brambilla; @Kniehl]. What is new is the precise formulation of the effective theory, and the way in which the renormalization group is scaling is implemented.
The results presented here will be applied to the study of $\bar tt$ production in the threshold region. There is a large ratio of scales, $m_t \sim 175$ GeV, $m_t v \sim 26$ GeV and $m_t v^2 \sim 4$ GeV, where $v \sim 0.15$ is the typical velocity in the nonrelativistic bound state. Clearly $\alpha_s \ln v$ is not small, and summing logarithms is important in this case.
The results are also useful in QED. While $\alpha \ln \alpha$ is small, it is important to compute to high orders because the experiments have high precision. The Hydrogen Lamb shift of 1057.845 MHz is known to an accuracy of 9 KHz [@Lundeen], the Hydrogen hyperfine splitting is measured to be MHz [@Hellwig], and the muonium hyperfine splitting is MHz [@Liu]. The binding energy of Hydrogen, $m_e \alpha^2/2$ is $2 \times 10^{10}$ MHz, so the experimental error in the Lamb shift of 10 ppm is a part in $10^{12}$ of the binding energy. We will be able to compute corrections of order $m_e \alpha^8 \ln^3
\alpha/(4\pi)^2 \sim 5$ KHz to the Lamb shift, which are relevant for the present comparison between theory and experiment. \[The counting of $4\pi$ factors for bound states is a little different than the conventional counting [@GM]. Potential loops give powers of $\alpha$ whereas soft and ultrasoft loops give powers of $\alpha/(4\pi)$.\]
A detailed comparison of theory and experiment for QED can be found in Refs. [@kinoshita; @pachuki; @eides].
NEW RESULTS
===========
There are many interesting new results that have been obtained for QED and QCD [@LMR; @amis; @amis2; @amis3; @amis4; @mss1; @amis5; @hmst1; @hmst2]. For QED, one finds a universal description of $\ln \alpha$ terms. A single renormalization group equation gives the Lamb shift, hyperfine splitting and decay widths for Hydrogen, muonium and positronium. The renormalization group method allows us to compute for the first time the $\alpha^8 \ln^3\alpha$ Lamb shift in positronium and the $\alpha^8 \ln^3\alpha$ Lamb shift in Hydrogen and muonium including recoil corrections. It also resolves a controversy in the literature about the $\alpha^8 \ln^3\alpha$ Hydrogen Lamb shift in the limit $m_p \to
\infty$.
The renormalization group method allows one to understand the structure of the QED perturbation series, and why the $\ln \alpha$ corrections terminate. The leading order series has a single term that contributes at order $\alpha^5 \ln
\alpha$ to the energy, and the next-to-leading order series terminates after three terms, $\alpha^6 \ln \alpha$, $\alpha^7 \ln^2 \alpha$, and $\alpha^8
\ln^3 \alpha$. One also finds some infinite series of terms in QED, but they have the form $(\alpha^3 \ln^2 \alpha)^n$, rather than $(\alpha \ln \alpha)^n$.
In QCD, one is able to sort out the scales for $\alpha_s$, and decide whether the strong coupling is $\alpha_s(m)$, $\alpha_s(mv)$, or $\alpha_s(mv^2)$. We also obtain the renormalization group improved computations of the bound states potentials in QCD. There are numerous applications of these results, and I will show an example of the dramatic improvement one obtains for the $\bar t t$ production cross-section near threshold [@hmst1; @hmst2].
THE PROBLEM
===========
The basic problem can be seen by drawing a few Feynman diagrams. A typical gauge boson exchange in the $t$ channel such as Fig. \[fig:5a\] has momentum transfer of order $p \sim mv$. A wavefunction graph or radiated gauge boson graph such as Figs. \[fig:5b\] have gauge boson momenta of order $E\sim
mv^2$. More interesting diagrams such as those in Figs. \[fig:5c\] involve gauge bosons with momenta of order $p$ and order $E$. In a graph such as Fig. \[fig:5d\], the vacuum polarization insertions make the effective coupling of the two gluons $\alpha_s(mv)$ and $\alpha_s(mv^2)$ respectively.
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One result which should be clear from Fig. \[fig:5d\] is that graphs can involve $\alpha_s(mv)$ and $\alpha_s(mv^2)$ *simultaneously*. We will return to this important point later on.
MOMENTUM REGIONS AND DEGREES OF FREEDOM
=======================================
The Feynman integrals in the full theory can be evaluated using the threshold expansion [@beneke]. The important momentum regions (in Feynman gauge) are referred to in the literature as hard ($E \sim m$, $p \sim m$), potential ($E
\sim mv^2$, $p \sim mv$), ultrasoft ($E \sim mv^2$, $p \sim mv^2$) and soft ($E
\sim mv$, $p \sim mv$). The threshold expansion momentum regions are often used to describe bound state computations; however it is important to note that *the threshold expansion is not an effective field theory*. To construct an effective field theory, one needs to include only modes that can be on-shell. The effective theory therefore has nonrelativistic fermions (which are potential modes), and soft and ultrasoft gauge boson modes. The hard fermion and gauge boson momentum regions, the soft fermion momentum region, and the potential gauge boson momentum region do not require modes in the effective theory.
The desired effective theory is valid for energies and momenta much smaller than the fermion mass $m$. One can try expanding in powers of $E/m$ and $p/m$ as in heavy quark effective theory, so that the expansion parameter is $1/m$. For example, the dispersion relation $E=\sqrt{{\bf p}^2+m^2}$ gives terms in the Lagrangian of the form $$\begin{aligned}
\label{dispersion}
L = \psi^\dagger\left(E-{{\bf p}^2 \over 2m}+{{\bf p}^4 \over 8 m^3}
+\ldots \right)\psi.\end{aligned}$$ The lowest order propagator is $1/(E+i\epsilon)$, which gives $\theta(t)$ in position space. This is the static propagator of HQET: fermions propagate forward in time, but do not move in space. This propagator is acceptable for some calculations involving heavy quarks. For example, one can compute the static potential between fixed sources using this propagator. However, for $\bar t t$ production, the quarks are produced at the same point, and they remain at the same point for all time if the static propagator is used. This is too singular, and the HQET expansion breaks down. In general, it is essential for treating nonrelativistic bound states that the heavy fermions move. For this to occur, the lowest order propagator should be $1/(E-{\bf
p}^2/2m+i\epsilon)$, so that $E$ and ${\bf p}^2/2m$ are of the same order in the effective theory power counting. This implies that the $1/m$ expansion cannot be used; instead one must use an expansion in powers of $v$, where $E$ and ${\bf p}^2/2m$ are both of order $v^2$ [@Caswell; @BBL].
The effective theory expansion parameter is the velocity $v$, and formally, $\alpha$ must also be treated as order $v$. Thus order $\alpha^2$ radiative corrections to the leading term are just as important as order $v^2$ relativistic corrections. The effective theory below the scale $m$ has:
- Nonrelativistic fermions with propagator $${1 \over E - {\bf p}^2/2m + i \epsilon}$$
- Ultrasoft gauge bosons coupled via interactions that are multipole expanded [@GR].
- Potentials $V({\bf p,p^\prime})$ for the scattering of an incoming $Q$ and $\bar Q$ with momenta ${\bf p}$ and $-{\bf p}$ to outgoing $Q$ and $\bar Q$ with momenta ${\bf p^\prime}$ and $-{\bf p^\prime}$.
- Soft gauge bosons. The importance of introducing soft fields in the effective theory was first pointed out by Griesshammer [@griesshammer].
The effective theory has two different gauge boson fields, soft bosons and ultrasoft bosons. This does not lead to any double counting if graphs are evaluated in dimensional regularization.
The static theory is not the $m\to \infty$ limit or the $v \to 0$ limit of the effective theory. For this reason, the static potential and the effective theory potential are not equal.
POWER COUNTING
==============
The power counting parameter of the effective theory is the velocity $v$. If one expands the dispersion relation as in Eq. (\[dispersion\]), then $E$ and ${\bf p}^2/2m$ are both of order $v^2$, and ${\bf p}^4/8m^3$ is of order $v^4$, i.e. of order $v^2$ relative to the leading term.
The potential $V({\bf p},{\bf p^\prime})$ also has an expansion in powers of $v$. The leading term is the Coulomb potential, $V({\bf p},{\bf p^\prime})
\propto \alpha/\vabsq k$, where $k={\bf p^\prime}-{\bf p}$ is the momentum transfer. Since momentum is of order $mv$, the Coulomb potential is naively of order $\alpha/v^2$. However, the potential is a four-fermion operator, whereas the kinetic energy is a two-fermion operator. This leads to an additional factor of $v$ from the power counting factors for the fields, so that the Coulomb potential is of order $\alpha/v$ in the effective theory. One can then determine the power counting for all the other potentials by comparing with the Coulomb potential. The hyperfine interaction $\propto \alpha \mathbf{S}_1 \cdot
\mathbf{S}_2/m^2$ is generated by one-photon exchange, and is of order $v^2$ relative to the Coulomb interaction, so it is of order $\alpha v$ in the power counting, as are the spin-orbit, tensor and contact (Darwin) interactions. At one-loop, there are also potentials that are proportional to odd-powers of $\bf
k$. The first such potential is proportional to $\alpha^2/\vabs k$, and is of order $\alpha^2 v^0$ in the power counting.
A loop graph such as Fig. \[fig:10\] of the time-ordered product of two potentials of order $\alpha^{a_1}v^{b_1}$ and $\alpha^{a_2}v^{b_2}$ is of order $\alpha^{a_1+a_2}v^{b_1+b_2}$.
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One can now see that the static potential differs from the $m\to \infty$ or $v\to 0$ limit of the effective theory potentials. For example, the loop graph of Fig. \[fig:10\] with one $1/(m \vabs k)$ and one Coulomb potential is of order $\alpha^2 v^0 \times \alpha/v = \alpha^3/v$, and is of the same order in $v$ as the Coulomb potential. The two particle intermediate state propagator $1/(E-{\bf p}^2/2m)=2m/(2mE-{\bf p^2})$ produces a factor of $m$ in the numerator, that cancels the $1/m$ at the vertex. In the static theory, the $1/(m \vabs k)$ potential is set to zero before the loop integration, so that the graph of Fig. \[fig:10\] is not present in the static theory. As a result, the NRQCD potential [@amis5] differs from the static potential.
MATCHING CONDITIONS
===================
The method of calculating matching conditions is the same as in any effective theory. One computes the graphs in the full theory at the scale $\mu=m$, and subtracts the corresponding graphs in the effective theory. The graph in Fig. \[fig:13a\] gives the matching condition for the fermion potential. The full theory amplitude, $${
\left[\bar u({\bf p^\prime})\gamma^\mu u({\bf p}) \right]
\left[\bar u(-{\bf p^\prime})\gamma_\mu u(-{\bf p}) \right]
\over \left(\mathbf{p-p^\prime}\right)^2}$$
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is expanded in powers of $\mathbf{p,p^\prime}$, to give the potential in the effective theory. At one-loop, the difference of the full theory and effective theory graphs in Fig. \[fig:13b\] give the one-loop corrections to the
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matching potential. The only difference at this stage between Hydrogen and positronium is that there are annihilation contributions to the positronium potential from graphs such as Fig. \[fig:13c\]. The graphs can have an
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imaginary part, that give the positronium decay width.
RENORMALIZATION GROUP EVOLUTION
===============================
The nonrelativistic bound state system has three important mass scales, $m$, $mv$ and $mv^2$.
Two-stage running
-----------------
The conventional method of implementing the renormalization group is as follows
- Start at $\mu=m$
- Scale $\mu$ from $m$ to $mv$
- Integrate out the soft modes at $mv$
- Scale $\mu$ from $mv$ to $mv^2$
This is referred to as the two-stage method, because there are two-stages of renormalization group evolution. Consider a loop graph involving time-ordered products of potentials, such as Fig. \[fig:14a\].
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This graph contains a logarithm of the form $\ln \sqrt{mE}/\mu$. When $\mu$ is set to $mv$, this logarithm has the from $\ln \sqrt{E/mv^2}$, and is small. Thus the logarithms in the graph are summed by renormalization group evolution of $\mu$ from $m$ to $mv$.
The graph in Fig. \[fig:14b\] involving an ultrasoft photon exchange contains a logarithm of the form $\ln E/\mu$.
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The $\mu$ in this ultrasoft graph is scaled all the way down (in two stages) to $mv^2$, at which point the logarithm is $\ln E/mv^2$, and also small.
However, this two-stage method of implementing the renormalization group turns out to be incorrect for nonrelativistic bound states. The reason is that the scales $mv$ and $mv^2$ are correlated—one cannot be varied independently of the other. Instead one needs to use an alternative one-stage scaling procedure.
One-stage running
-----------------
In one stage running, one introduces two different $\mu$ parameters, $\mu_S$ and $\mu_U$ [@LMR]. In dimensional regularization in $4-2\epsilon$ dimensions, the soft photon coupling is multiplied by $\mu_S^{\epsilon}$, the ultrasoft photon coupling by $\mu_U^{\epsilon}$, and the potentials by $\mu_S^{2\epsilon}$. Note that this is only possible because we have two different photon fields to represent the soft and ultrasoft photons in the effective theory. Then
- Set $\mu_S=m\nu$, $\mu_U=m\nu^2$
- Start at $\nu=1$ and scale to $\nu=v$.
This procedure is referred to as the velocity renormalization group, because one runs in velocity $\nu$ rather than momentum [@LMR]. The logarithms in Figs. \[fig:14a\] and \[fig:14b\] are now $\ln \sqrt{mE}/m \nu$ and $\ln
E/m\nu^2$, which are minimized when $\nu=v$. Thus this method also minimizes logarithms in the diagrams, and sums them by renormalization group evolution.
The difference between the two renormalization group methods can be seen in Fig. \[fig:16\] [@mss1].
(0,0)(10,7.5)(0,0)(10,0) (0,0)(0,6) (9,5)(7,3) (7,3)(5,1) (5,1)(3,1) (3,1)(1,1) (9,5)(5,3) (5,3)(1,1) (7,2.5)[$\gamma_S+\gamma_U$]{} (2.5,3.5)[$\gamma_S+2\gamma_U$]{} (3,0.25)[$\gamma_U$]{} (10.5,0.125)[$\ln \mu_U$]{} (-0.5,6.5)[$\ln \mu_S$]{}
In two-stage running, there is only a single $\mu$, so that $\mu_S=\mu_U=\mu$, and they are lowered together from $m$ to $mv$. At this point, the soft modes are integrated out, and $\mu_U$ for the ultrasoft modes is lowered to $mv^2$. The integration path in Fig. \[fig:16\] is along the lower edges of the triangle. In one-stage running, the integration path is along the diagonal. It is convenient to define two anomalous dimensions, $\gamma_S$ and $\gamma_U$ by taking the derivatives of Green’s functions with respect to $\ln \mu_S$ and $\ln \mu_U$, respectively. One can show by explicit calculation that
- The two paths give different answers. The integration is path dependent because $\nabla \times \gamma \not = 0$.
- One-stage running using the velocity renormalization group agrees with explicit QED calculations at order $\alpha^3\ln^2\alpha$, $\alpha^7 \ln^2
\alpha$ and $\alpha^8 \ln^3 \alpha$.
The moral is that for nonrelativistic bound states, one should run in velocity rather than momentum.
The difference between the two integration methods can be made more precise. In the two-stage method, one first integrates $\gamma_S + \gamma_U$ from $\mu=m$ to $\mu=mv$, and then integrates $\gamma_U$ from $\mu=mv$ to $\mu=mv^2$. In the one-stage method, one integrates $\gamma_S+2\gamma_U$ (since $\ln\mu_U$ runs twice as fast as $\ln \mu_S$) from $\nu=1$ to $\nu=v$. If the anomalous dimensions are constant, the two methods give $$\begin{aligned}
\begin{array}{ll}
\left(\gamma_S + \gamma_U\right) \ln{mv \over m} +
\gamma_U \ln{mv^2 \over mv} & \hbox{\qquad two-stage} \\[10pt]
\left(\gamma_S + 2 \gamma_U\right) \ln{v} & \hbox{\qquad one-stage}
\end{array}\end{aligned}$$ and agree with each other. However, in general anomalous dimensions are not constant, but can depend on coupling constants $V_i$, that themselves run. As a result, one finds that the $\ln v$ terms agree, but the higher order terms differ. For example, consider a $\ln^2 v$ term that depends on the product of $\gamma_S$ and $\gamma_U$. For two-stage running, the contribution is proportional to $\gamma_S \gamma_U + 0 \gamma_U=\gamma_S \gamma_U$ from the two pieces of the path. For one-stage running, the contribution is $\gamma_S \left(2 \gamma_U
\right)$, which differs by a factor of two. Similarly, a $\gamma_S \gamma_U^2
\ln^3 v$ contribution differs by a factor of four, and so on.
RUNNING POTENTIALS
==================
The running potential $V(\mathbf{p,p^\prime})$ has an expansion $$V(\mathbf{p,p^\prime}) = V^{(-1)}+V^{(0)}+V^{(1)}+V^{(2)}+\ldots$$ where $V^{(n)}$ is of order $v^n$ in the velocity power counting. The first three terms in the expansion have the form $$\begin{aligned}
&& V^{(-1)} = {{U}_c \over {\mathbf k}^2}\, ,{\nonumber}\\
&& V^{(0)} ={ {U}_k \over |{\mathbf k}| } \,, \\
&& V^{(1)} = U_2 + U_s\: {\bf S^2} + { U_r ({\mathbf p^2 + \mathbf
p^{\prime 2}}) \over
2 {\mathbf k}^2} {\nonumber}\\
&&\qquad\quad- {i {\mathbf U}_\Lambda \cdot ({\mathbf p^\prime \times
\mathbf p})
\over {\mathbf k}^2 } {\nonumber}\\
&& \qquad\quad + U_t \Big( {\mathbf \bsigma_1 \cdot \bsigma_2}-{3\,{\mathbf k
\cdot \bsigma_1}\, {\mathbf k \cdot \bsigma_2} \over {\mathbf k}^2} \Big )
\,, {\nonumber}\end{aligned}$$ where $V^{(0)} \sim 1/m$, and $V^{(1)} \sim 1/m^2$. In QCD, each of the coefficients can be written as $U \to U^{(1)} 1 \otimes 1 + U^{(T)} T^A
\otimes \bar T^A$, where $1$ and $T^A/\bar T^A$ are color matrices acting on the quark/antiquark lines. The anomalous dimensions for the coefficients $U_c$–$U_t$ have been computed, and the details are given in Refs. [@amis; @amis2; @amis3]. The renormalization group improved static potential was computed in Ref. [@rgstatic]. An important point to note is that graphs can involve both soft and ultrasoft gluons, so that the anomalous dimensions involve *both* $\alpha_s(mv)$ and $\alpha_s(mv^2)$. As an example, the running of $U_2^{(1)}$ is given by $$\begin{aligned}
\label{u2run}
m^2 {U}_2^{(1)}(\nu) &=& \frac{14 C_1}{3}\,
{ \alpha_s(m\nu)}{\alpha_s(m)} \ln\Big({{m\nu}\over {m}}\Big)\nonumber \\
&& -
\frac{32\pi C_1}{3\beta_0}
\, {\alpha_s(m)}
\ln\bigg[ \frac{{\alpha_s(m\nu)}}{{\alpha_s(m\nu^2)}} \bigg]
\,\end{aligned}$$ where $C_1=2/9$ for QCD. Note that Eq. (\[u2run\]) depends on $\alpha_s(m)$, $\alpha_s(m\nu)$, and $\alpha_s(m\nu^2)$. The running coefficients in the singlet channel ($U^{(s)}=U^{(1)}-C_F U^{(T)}$) for $\bar t t$ production are presented in Table \[tab:ttbar\].
Coefficient $\nu=1$ $\nu=v$
--------------------------- -------------- -------------
${{U}_c^{(s)}}$ [$-1.81$]{} [$-2.47$]{}
${{m U}_k^{(s)}}$ [$-0.36 $]{} [$-0.03$]{}
${{m^2 U}_r^{(s)}}$ [$-1.81$]{} [$-1.49$]{}
${{m^2 U}_2^{(s)}}$ [0]{} [0.63]{}
${{m^2 U}_s^{(s)}}$ [0.60]{} [0.53]{}
${{m^2 U}_\Lambda^{(s)}}$ [0.15]{} [0.16]{}
${{m^2 U}_t^{(s)}}$ [2.71]{} [3.11]{}
: Numerical values for the $\bar tt$ singlet potentials. The values at $\nu=1$ are the matching values at $\mu=m_t$. The values at $\nu=v$ are the velocity renormalization group improved values, where $v=0.14$ has been used.[]{data-label="tab:ttbar"}
The renormalization group improved coefficients ${U}_r$ and ${U}_2$ differ significantly from their matching values, because they depend on the ultrasoft scale through $\alpha_s(mv^2)$. The other coefficients only have a soft anomalous dimension, and do not run as much.
The renormalization group improved potentials can be used to calculate the renormalization group improved cross-section for $\bar tt$ production in the threshold region. Fig. \[fig:old\] shows a sample fixed order calculation of $R$, the ratio of $\sigma(e^+e^-\to \bar t t)/\sigma(e^+e^-\to
\mu^+\mu^-)$ [@review]. The scale uncertainty is of order 20%. The renormalization group improved version of the results is shown in Fig. \[fig:new\]. There is a dramatic
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reduction in the scale uncertainty, which is now around 2%, as well as an improvement in convergence for the normalization. The small theoretical uncertainty means that an accurate measurement of the cross-section can be used to study new physics. For example, a standard model Higgs boson of mass around 115 GeV changes the cross-section by $\sim 5$%, and is measurable.
QED
===
The velocity renormalization group method gives very interesting and important results when applied to QED [@amis4]. The basic potentials we will need for QED are summarized in Table \[tab:qed\]. The last column gives the contribution to the bound state energy levels due to the given potential. The fourth column gives the order of a given potential, treating $v$ as order $\alpha$. Since the Coulomb potential is of order unity, one finds the obvious result that the Coulomb potential must be summed to all orders, and cannot be treated as a perturbation. The potentials $V^{(-1)}$, $V^{(1)}$, $V^{(3)}$, are first generated at tree-level, and are of order $\alpha$, whereas the potentials $V^{(0)}$, $V^{(2)}$, $V^{(4)}$, are first generated at one-loop, and are of order $\alpha^2$.
$$\begin{aligned}
\begin{array}{|l|c|c|c|c|}
\hline
& & \hbox{Power Counting} & \hbox{Order} & E \\
\hline
V^{(-1)} & {\alpha \over \mathbf{k}^2} & {\alpha
\over v} & 1 & \alpha^2\\
V^{(0)} & {\alpha \over m\vabs{\mathbf{k}}} &
{\alpha^2 } &
\alpha^2 & \alpha^4 \\
V^{(1)} & {\alpha \over m^2},\ {\alpha \mathbf{S}^2
\over m^2} & {\alpha v}
& \alpha^2 & \alpha^4 \\
V^{(2)} & {\alpha \vabs{\mathbf{k}} \over m^3}
& {\alpha^2 v^2} &
\alpha^4 & \alpha^6 \\
V^{(3)} & {\alpha \mathbf{k}^2 \over m^4} &
{\alpha v^3} & \alpha^4 &
\alpha^6 \\
\vdots & \vdots & \vdots & \vdots & \vdots\\
\hline
\end{array}\end{aligned}$$
The bound state energy levels can be determined to order $\alpha^4$ by computing the matrix elements of $V^{(0)}$ and $V^{(1)}$ between Coulomb wavefunctions. Time-ordered products of two potentials, such as $T\left[
V^{(0)} V^{(0)} \right]$, $T\left[ V^{(0)} V^{(1)} \right]$ and $T\left[
V^{(1)} V^{(1)} \right]$ first contribute at order $\alpha^6$. In principle, to obtain the energy levels to order $\alpha^4$, one also needs the one- and two-loop matching corrections to the Coulomb potential. However, such corrections vanish in QED. As a result, the first correction to the order $\alpha^2$ binding energy is of order $\alpha^4$, and is given by the matrix element of $V^{(0)}+V^{(1)}$. There are no order $\alpha^3$ corrections to the energy levels in QED.
Define the leading and next-to-leading order anomalous dimensions of a potential to be the anomalous dimension from graphs at one and two higher orders in $\alpha$ than the potential itself. For $V^{(-1)}$ and $V^{(1)}$ which are of order $\alpha$, the leading order anomalous dimension is of order $\alpha^2$, and the next-to-leading order anomalous dimension is of order $\alpha^3$. For $V^{(0)}$ which is of order $\alpha^2$, the leading order anomalous dimension is of order $\alpha^3$, and the next-to-leading order anomalous dimension is of order $\alpha^4$. Since different terms in the potential are of different orders in $\alpha$, the terms leading and next-to-leading order are not related to the number of loops.
Integrating the renormalization group equations for $V^{(0)}$ and $V^{(1)}$ using the leading order anomalous dimension gives a series of the form $$\alpha\left (1+ \alpha\ln\alpha + \alpha^2 \ln^2 \alpha + \alpha^3 \ln^3
\alpha + \ldots \right),$$ which contributes $$\label{LOenergy}
\alpha^4
\left(1+ \alpha\ln\alpha + \alpha^2
\ln^2 \alpha + \alpha^3 \ln^3 \alpha + \ldots \right)$$ to the energy. Integrating the next-to-leading order anomalous dimensions gives $$\label{NLOenergy}
\alpha^4 \alpha
\left(1+ \alpha\ln\alpha + \alpha^2
\ln^2 \alpha + \alpha^3 \ln^3 \alpha + \ldots \right)$$ terms in the energy. The next-to-next-to-leading anomalous dimension gives $$\alpha^4 \alpha^2 \left(1+ \alpha\ln\alpha + \alpha^2
\ln^2 \alpha + \alpha^3 \ln^3 \alpha + \ldots \right),$$ terms in the energy, which are the same order as those obtained by using the leading order anomalous dimension for the $V^{(2)}$ and $V^{(3)}$ potentials which first contribute at order $\alpha^6$. Thus one can compute the $$\begin{array}{llll}
\alpha^5 \ln \alpha & \alpha^6 \ln^2 \alpha & \alpha^7 \ln^3 \alpha & \ldots\\
\alpha^6 \ln \alpha & \alpha^7 \ln^2 \alpha & \alpha^8 \ln^3 \alpha & \ldots\\
\end{array}$$ series in the energy using $\gamma_{\rm LO}$, $\gamma_{\rm NLO}$ for $V^{(0,1)}$.
LEADING ORDER
=============
The Coulomb potential and $V^{(0)}$ do not run in QED at leading and next-to-leading order, so one is left with the running of $V^{(1)}$. The anomalous dimensions are evaluated for a particle of mass $m_1$ and charge $-e$ interacting with a second particle of mass $m_2$ and charge $Ze$. Evaluating the graphs in Fig. \[fig:23\] gives
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$$\nu {d U_2 \over d \nu} = {{14 Z^2 \alpha^2 \over 3 m_1 m_2}}+
{{2}\alpha \over 3 \pi}\left({1\over m_1}
+ {Z \over m_2}\right)^2 U_c$$
where the first term is the soft contribution from Fig. \[fig:23\]a and the second is the ultrasoft contribution from Fig. \[fig:23\]b,c. Note that the ultrasoft contribution has been multiplied by two, since the anomalous dimension for the velocity renormalization group is $\gamma_S + 2 \gamma_U$. The other coefficients in $V^{(1)}$ ($U_r$, $U_s$, $U_\Lambda$, $U_t$) have zero anomalous dimension at this order.
Since the Coulomb potential and $\alpha$ do not run in QED, one can combine the two terms, $$\label{20}
\nu {d U_2 \over d \nu} = \gamma_0 U_c$$ which defines $$\gamma_0 = {2 \alpha \over 3 \pi }\left( {1\over m_1^2} + {Z \over 4
m_1 m_2} + {Z^2 \over m_2^2} \right).$$ $\gamma_0$ is a constant in QED since $\alpha$ does not run. Integrating Eq. (\[20\]) gives $$\label{lo}
U_2(\nu) = U_2(1) + \gamma_0 U_c \ln \nu,$$ where $U_2$ is evaluated at $\nu=v=\alpha$. Since $\gamma_0$ is a constant, $U_2(\nu)$ only has a $\ln\nu$ term, and terms of the form $\ln^n \nu$, with $n>1$ vanish. As a result, the leading order energy series Eq. (\[LOenergy\]) terminates after a single term, so one has an $\alpha^5 \ln \alpha$ contribution to the energy, but the $\alpha^6 \ln^2 \alpha$, etc. terms vanish. At low orders, the absence of terms other than $\alpha^5\ln \alpha$ in the leading order series has been noticed before by an explicit examination of Feynman graphs. This is the first general proof that all the terms beyond $\alpha^5\ln \alpha$ in the leading order series vanish
The matrix element of $U_2$ gives the energy shift $$\begin{aligned}
\Delta E &= & \vev{U_2(\nu)} {\nonumber}\\
&=&{\gamma_0} U_c \ln \nu \abs{\psi(0)}^2 \\
&=& -{8 Z^4 \alpha^5 m_R^3 \over 3 \pi n^3}\left( {1\over m_1^2} +
{Z \over 4 m_1 m_2} + {Z^2 \over m_2^2} \right) \ln Z \alpha ,{\nonumber}\end{aligned}$$ where we have used $$\begin{aligned}
\abs{\psi(0)}^2 = {(m_R Z \alpha)^3 \over \pi n^3}\end{aligned}$$ for the $nS$ state, and $m_R$ is the reduced mass. This is the famous $\alpha^5 \ln \alpha$ correction to the Lamb shift first computed by Bethe, including all recoil corrections.
NEXT-TO-LEADING ORDER
=====================
At next-to-leading order, the anomalous dimension for $V^{(1)}$ is $$\begin{aligned}
&& \left. \nu {d U_{2+s} \over d\nu} \right|_{\rm NLO} =
\rho_{ccc}\, U_c^3 + \rho_{cc2}\, U_c^2 \left({ U_{2+s}}
+ U_r
\right) \nonumber \\
&&\quad + {\rho_{c22}}\, U_c\left({U_{2+s}^2}
+2{U_{2+s}} U_r + \frac34 U_r^2
-5 U_t^2 {\bf S^2} \right) \nonumber \\
&& \quad +\rho_{ck}\, U_c U_k +\rho_{k2}\, U_k \left({U_{2+s}} +
U_r/2\right) \nonumber \\
&& \quad + \rho_{c3}\, U_c \left({U_3}+U_{3s} S^2 + {1\over2}U_{rk}\right)
, \label{nlo}\end{aligned}$$ where $U_{2+s}=U_2 + U_s {\mathbf S}^2$, and the coefficients are $$\begin{aligned}
\rho_{ccc}&=& -{m_R^4 \over 64 \pi^2} \left( {1\over m_1^3} + {1\over
m_2^3}\right)^{\!2},{\nonumber}\\
\rho_{c22}&=& -{m_R^2 \over 4 \pi^2},{\nonumber}\\
\rho_{cc2}&=& -{m_R^3 \over 8 \pi^2} \left( {1\over m_1^3} +
{1\over m_2^3} \right),{\nonumber}\\
\rho_{c3} &=& { 2 m_R \over \pi^2},{\nonumber}\\
\rho_{ck} &=& {m_R^2 \over 2 \pi^2}\left({1\over m_1^3} +{1\over m_2^3}
\right), {\nonumber}\\
\rho_{k2} &=& { 2 m_R \over \pi^2}.\end{aligned}$$
The anomalous dimension Eq. (\[nlo\]) can be integrated by substituting the leading order running, Eq. (\[lo\]) for the coefficients on the right-hand side. Since only $U_2$ runs at leading order, the right hand side has at most a $\ln^2 \nu$, so that the integral has at most a $\ln^3 \nu$ term. This implies that the next-to-leading order series Eq. (\[NLOenergy\]) terminates after the first three terms, $\alpha^6\ln \alpha$, $\alpha^7\ln^2 \alpha$, and $\alpha^8\ln^3 \alpha$.
$\ln^3 \alpha$
--------------
The only term that contributes to the $\ln^3\alpha$ correction is the $U_2^2$ term of Eq. (\[nlo\]). Integrating gives a contribution to $U_2(\nu)$ of the form $$\begin{aligned}
\label{a3}
{1\over 3}\, {\gamma_0^2}\,
{\rho_{c22}}\, {U}_c^3(1)\, \ln^3\nu,\end{aligned}$$ which is spin-independent, and has no imaginary part. There is no contribution to the decay width or hyperfine splitting at this order. The Lamb shift at this order is obtained by multiplying Eq. (\[a3\]) by the matrix element of the unit operator, $\abs{\psi(0)}^2$, to give $$\begin{aligned}
\Delta E &=& {64 m_R^5 \alpha^8 Z^6 \over 27 \pi^2 n^3}
\ln^3 \left(Z \alpha \right){\nonumber}\\
&& \times \left(
{1\over m_1^2} + {{Z\over 4 m_1 m_2} + {Z^2\over m_2^2}}
\right)^2\end{aligned}$$ which is approximately 8 KHz for the $2P$–$2S$ Lamb shift in Hydrogen. Substituting $Z=1$ and $m_1=m_2=m_e$ gives the $\alpha^8\ln^3\alpha$ Lamb shift for positronium $$\begin{aligned}
\Delta E &=&
{3 m_e \alpha^8 \ln^3 \alpha \over 8 \pi^2 n^3}.\end{aligned}$$ The positronium Lamb shift is a new result, as are the recoil terms in the Hydrogen Lamb shift. In the limit $m_1/m_2 \to 0$, the Hydrogen Lamb shift has been computed previously by several groups. There is an analytic computation by Karshenboim [@karshenboim] and a numerical computation by Goidenko et al. [@goidenko] that agree with our result. There are also numerical computations by Malampalli and Sapirstein [@malampalli], and by Yerokhin [@yerokhin] which agree with each other, but disagree with the other results. Recently, there has been a computation by Pachucki [@pachuckinew] that agrees with our result. Yerokhin [@yerokhinnew] has emphasized that the complete $\alpha^8 \ln^3
\alpha$ Lamb shift might not be contained in the loop-after-loop calculations of Refs. [@malampalli; @yerokhin].
The other calculations rely on extracting the logarithm from four-loop diagrams such as Fig. \[fig:21\]. The velocity renormalization group factors the graph into the product of a two-loop anomalous dimension $\rho_{c22}$, and the square of a one-loop anomalous dimension $\gamma_0^2$.
=4truecm
$\ln^2 \alpha$
--------------
The $\ln^2 \alpha$ contribution to the hyperfine splitting and decay widths is given by the $U_2(\nu)$ contribution $$\begin{aligned}
{\gamma_0}\, {\rho_{c22}}\, {U}_c^2(1) \left[{U}_2(1)+ {U}_s(1) {\mathbf S}^2
\right] \ln^2 \nu + \ldots.\end{aligned}$$ The spin-dependent term gives the $\alpha^7 \ln^2 \alpha$ hyperfine splitting $$\begin{aligned}
\hbox{HFS} &=& -{64 Z^6 \alpha^7 m_R^5 \mu_1 \mu_2 \over 9m_1m_2 \pi n^3}
\ln^2 (Z \alpha){\nonumber}\\
&&\times \left[{1\over m_1^2} + {Z\over 4 m_1 m_2} + {Z^2\over m_2^2}
\right]\end{aligned}$$ where $\mu_i$ are the magnetic moments normalized to unity for a Dirac fermion. Our result agrees with previous calculations [@karshenboim; @thesis]. Substituting the matching values for $U_s(1)$ for positronium (which differs from Hydrogen because of annihilation contributions), one finds the positronium hyperfine splitting $$\begin{aligned}
\hbox{Ps HFS}= -{7 m_e \over 8 \pi n^3}\ \alpha^7 \ln^2\! \alpha ,\end{aligned}$$ which agrees with a recent computation of Melnikov and Yelkhovsky [@melnikov]. The imaginary parts of the matching coefficients give the decay widths [@karshenboim], $$\begin{aligned}
{\Delta \Gamma\over \Gamma_0} = \gamma_0\, \rho_{c22}\, {U}_c(1)^2 \ln^2 \nu =
-{3 \over 2 \pi}\alpha^3 \ln^2\! \alpha,\end{aligned}$$ for both ortho- and para-positronium.
$\ln \alpha$
------------
The $\ln \alpha$ contributions to the decay width arise from $$\begin{aligned}
&& U_{2+s} \Big[ {\rho_{c22}} U_c \left( {U}_{2+s} +
2 U_r \right) {\nonumber}\\
&&\qquad + \rho_{cc2}
U_c^2 + \rho_{2k} U_k \Big] \ln \nu +\ldots.\end{aligned}$$ which give $$\begin{aligned}
{\Delta \Gamma \over \Gamma_0} &=& \left({m_e^2 \over 2 \pi} \hbox{Re}\, U_{2+s}
- 2 \right)\ln \nu {\nonumber}\\
&=& \left( {7 {\mathbf S}^2 \over 6} - 2 \right)
{\alpha^2} \ln \alpha,\end{aligned}$$ so that $$\begin{aligned}
\left({\Delta \Gamma \over \Gamma_0}\right)_{\rm ortho} &=&
{\alpha^2 \over 3}\ln \alpha \,, {\nonumber}\\
\left({\Delta \Gamma \over \Gamma_0}\right)_{\rm para} &=&
-2{\alpha^2}\ln \alpha \,.\end{aligned}$$ These agree with existing results [@caswell2; @khriplovich].
CONCLUSIONS
===========
The methods presented here give a systematic way of separating scales in nonrelativistic bound state problems. All large logarithms are summed using the velocity renormalization group. The method provides a universal description of QED logarithms. The agreement with known results at order $\alpha^5 \ln
\alpha$, $\alpha^6 \ln \alpha$, $\alpha^7 \ln^2 \alpha$, and $\alpha^8 \ln^3
\alpha$ is a highly non-trivial check of the formalism. In QED, one finds that the leading order series terminates after one term, and the next-to-leading order series terminates after three terms. In addition, the method resolves a controversy about the $\alpha^8 \ln^3 \alpha$ Lamb shift for Hydrogen, and gives the first calculation of the $\alpha^8 \ln^3 \alpha$ energy shift for positronium.
In QCD, one can distinguish $\alpha_s(mv)$ and $\alpha_s(mv^2)$, and both can appear simultaneously in the same anomalous dimension. The renormalization group improved potentials can be used to compute $\bar tt $ production, and reduce the scale uncertainties by a factor of ten.
The velocity renormalization group should also be applicable to other problems with correlated scales. In the bound state problem, one can generate the scale $mv$ in loop graphs from the scale $m$ and $mv^2$, $mv=\sqrt{m \times mv^2}$. Similar effects can occur at finite temperature, where one has the scales $T$, $gT$ and $g^2T$, and some of the ideas described here might be applicable to that problem as well.
This work was supported in part by the Department of Energy under grant DOE-FG03-97ER40546 and by NSERC of Canada.
[99]{}
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A.V. Manohar and I.W. Stewart, Phys. Rev. [**D62**]{} (2000) 074015. A.V. Manohar and I.W. Stewart, hep-ph/0003107. A.V. Manohar and I.W. Stewart, Phys. Rev. Lett. [**85**]{} (2000) 2248. N. Brambilla, A. Pineda, J. Soto and A. Vairo, Nucl. Phys. [**B566**]{} (2000) 275. B. A. Kniehl and A. A. Penin, Nucl. Phys. [**B563**]{} (1999) 200. S.R. Lundeen and F.M. Pipkin, Phys. Rev. Lett. [**46**]{} (1981) 232.
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M.I. Eides, H. Grotch, and V.A. Shelyuto, hep-ph/0002158.
A.V. Manohar, J. Soto, and I.W. Stewart, Phys. Lett. [**B486**]{} (2000) 400. A.V. Manohar and I.W. Stewart, UCSD/PTH 00-24.
A.H. Hoang, A.V. Manohar, I.W. Stewart, and T. Tebuner, UCSD/PTH 00-25.
A.H. Hoang, A.V. Manohar, I.W. Stewart, and T. Tebuner, UCSD/PTH 00-26.
M. Beneke and V.A. Smirnov, Nucl. Phys. [**B522**]{} (1998) 321. H. W. Griesshammer, Nucl. Phys. [**B579**]{} (2000) 313. A. Pineda and J. Soto, hep-ph/0007197. A. H. Hoang [*et al.*]{}, Eur. Phys. J. direct [**C3**]{} (2000) 1. A. H. Hoang, Z. Ligeti and A. V. Manohar, Phys. Rev. Lett. [**82**]{} (1999) 277. A. H. Hoang, Z. Ligeti and A. V. Manohar, Phys. Rev. [**D59**]{} (1999) 074017. A. H. Hoang and T. Teubner, Phys. Rev. [**D60**]{} (1999) 114027. S.G. Karshenboim, Sov. Phys. JETP [**76**]{} (1993) 541.
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[^1]: UCSD/PTH 00-27. Talks presented at Lattice 2000 and SPIN 2000
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We report on measurements of quantized conductance in gate-defined quantum point contacts in bilayer graphene, which show ballistic transport with spin polarized conductance of 6 $e^2/h$ at high in-plane magnetic fields. At the crossings of the Zeeman spin-split subbands of opposite spins, we observe signatures of interaction effects comparable to the 0.7 analog. At zero magnetic field the situation seems to be more complex as the first subband is already splitted with a gap that is close to the expected value for a subband-splitting due to spin-orbit coupling in bilayer graphene, and which can be tuned from 40 to 80 $\mu eV$ by displacement field. Our results suggest that at zero magnetic field there is an interesting interplay between spin-orbit coupling and electron-electron interaction.'
author:
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L. Banszerus$^{1,2}$, B. Frohn$^{1}$, T. Fabian$^3$, S. Somanchi$^{1}$, A. Epping$^{1,2}$, M. Müller$^{1,2}$, D. Neumaier$^{4}$, K. Watanabe$^5$, T. Taniguchi$^5$, F. Libisch$^3$, B. Beschoten$^{1}$, F. Hassler$^{6}$ and C. Stampfer$^{1,2,*}$\
\
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bibliography:
- 'literature.bib'
title: 'Spin-polarized currents in bilayer graphene quantum point contacts'
---
Bilayer graphene (BLG) represents an interesting platform for mesoscopic transport and quantum devices. The possibility of tuning the low-energy electronic bands with a perpendicular electric field is unique to this material [@McCann2013Apr], which allows to open a band gap [@Oos2007Dec; @Zhang2009Jun], to modify band curvatures, and to change the topology of the Fermi surface [@Varlet2014Sep]. As all of this is controlled by external electrostatic gates, it is possible to implement soft-confined one-dimensional channels and quantum dots, where most of the states in BLG are fully depleted. Recent technological advancements – mostly based on the encapsulation of BLG in hexagonal boron nitride (hBN) and on the use of graphite gates – have enabled the observation of spin and valley states in BLG quantum dots [@Eich2018Jul; @Banszerus2018Jun; @Eich2018Jula] and of quantized conductance in gate defined quantum point contacts (QPCs) [@ove18j; @Overweg2018Dec; @Kraft2018Dec; @Lee2019Nov]. BLG is also interesting for spintronics applications [@Datta90; @Gmitra2017Oct; @Island2019Jun] because of its weak hyperfine and spin-orbit interaction [@Yang2011Jul; @Ingla-Aynes2016Aug; @Leutenantsmeyer2018Sep; @Xu2018Sep; @Konschuh12]. Spin-orbit coupling (SOC) is indeed expected to open a gap of only a few tens of $\mu$eV in the low energy spectrum of graphene and BLG [@Kane2005Nov; @Min2006Oct; @Huertas-Hernando2006Oct; @Yao2007Jan; @Konschuh12]. Recent resonance-microwave measurements on graphene on SiO$_2$ give a SOC induced gap around $40\,\mu$eV [@Sichau2019Feb].
![ **(a)** Schematic illustration of the device highlighting the hBN/BLG/hBN heterostructure and the various gates. **(b)** Four-terminal conductance as function of $V_\mathrm{F1}$ for different displacement fields showing steps at multiples of $4\,e^2/h$. **(c)** Finite bias spectroscopy measurements. Different curves correspond to different values of $V_{\rm F1}$, ranging from -9.4 to -10.6 V. A clustering of traces at multiples of $4\,e^2/h$ is visible at low bias voltages, vanishing at high bias. **(d)** Conductance through two QPCs in series separated by 260nm. The conductance is quantized in multiples of $4\,e^2/h$ and depends solely on the QPC with the lowest number of occupied modes.[]{data-label="f1"}](Fig01.pdf){width="\linewidth"}
{width="0.95\linewidth"}
Probing such small energies by transport is challenging but becomes more feasible when having comparable
energy scales in the system, such as small subband spacings in QPCs. In addition, small subband spacing energies enable crossings of spin-split subbands in parallel magnetic fields allowing to observe ballistic spin-polarized transport as well as spin-driven interaction effects such as the 0.7-analog earlier observed in GaAs QPCs [@Graham2003Sep; @Berggren2005Mar; @Weidinger2018Sep].
In this Letter, we report on the observation of highly spin-polarized currents in a quantum point contact (QPC) in BLG. By studying a wide QPC with low subband spacings (0.3-0.5 meV) and comparably large Zeeman energy we are able to observe the crossings of spin-split subbands of opposite spins leading to a regime with six fully spin-down polarized modes at high in-plane magnetic fields. The low subband energies and high energy resolution also allows for the observation (i) of signatures of electron-electron ([*e-e*]{}) interaction at finite magnetic fields (related to the 0.7-analog [@Graham2003Sep; @Berggren2005Mar; @Weidinger2018Sep]) as well as (ii) of indications of SOC, appearing as a feature at $2\,e^2/h$ due to the splitting of the first subband of the QPC at zero magnetic field.
Our device is based on dry-transferred BLG, encapsulated into hBN and placed on a graphite back gate, see Fig. 1(a). We use the combination of two Cr/Au split gates (SG) and the graphite back gate (BG) to apply a perpendicular electric displacement field, $D$, that opens up a band gap and depletes large parts of the BLG, defining a quasi-1D channel with a width of around 250nm, connecting source and drain contacts. In addition, we place 200nm wide finger gates across the channel to locally tune the Fermi energy and thus the number of open modes in the channel [^1]. This forms a QPC below each finger gate. An atomic force microscope image of the device is shown in Fig. 1 of Ref. [@Banszerus2018Jun].
![ **(a)** Zoom of Fig. 2(a) around the first conductance step and low $B_\parallel$, showing the presence of a shoulder at $2e^2/h$ even at $B_\parallel=0$. **(b)** Transconductance as function of $V_\mathrm{F1}$ and $B_\parallel$. The dashed lines mark the evolution of the spin up (white) and spin down bands (black) of the first subband. **(c)** Extracted energy gap $\Delta$ as function of the displacement field.[]{data-label="f3"}](Fig03.pdf){width="0.94\linewidth"}
We perform transport measurements in a He3/He4 dilution refrigerator at a temperature below 30mK, using standard lock-in techniques. The four-terminal conductance as function of finger gate voltage $V_\mathrm{F1}$ [^2] features well-developed plateaus at 4, 8, 12 and $16~e^2/h$ for displacement fields ranging from 0.22 to 0.3V/nm, see Fig. 1(b). The $4\,e^2/h$ step-height indicates four-fold degeneracy (two-fold spin and two-fold valley) and near unity transmission through the QPC. Complete current pinch-off is observed for large $D$-fields, i.e. large band gaps in the BLG. Reducing the displacement field increases the leakage current below the split gates, which leads in turn to an increase in the minimum conductance. Nevertheless, the height of the conductance steps remains nearly unaffected at $4\,e^2/h$. The near unity transmission through the QPC can be demonstrated more explicitly by using the second finger gate (F2), i.e. a second QPC placed 260nm next to the first one, see Fig. 1(a). The conductance of the device as function of both gate voltages $V_\mathrm{F1}$ and $V_\mathrm{F2}$ shows well-developed steps of multiples of $4\,e^2/h$ and depends only on the QPC with the lowest number of open modes, Fig. 1(d). This observation proves that the two QPCs have unity transmission and that the charge carriers travel ballistically through both QPCs, not thermalizing in between them.
To estimate the subband energy spacings of the quasi-1D system we perform finite bias spectroscopy measurements. Fig. 1(c) depicts the four-terminal differential conductance through the QPC in units of $e^2/h$ as function of the DC bias voltage, $V_\mathrm{b}$, applied between the source and the drain contact (see Fig. 1(a)) for different $V_\mathrm{F1}$ at a fixed displacement field of 0.32 V/nm. The conductance traces bunch at multiples of $4~e^2/h$ for low bias voltages. At higher bias the plateaus smear out revealing energy spacings around 0.3meV, 0.4meV and 0.5meV for the first three subbands (see Supplementary Material and below). These energies are a factor 10 smaller than what reported in Ref. [@Kraft2018Dec], because of the large size of our device. Using a hard-wall confinement model for the lowest subband spacing, $\Delta E_{1,2}= \hbar^2 \pi^2 / (2m^*W^2)$, where $m^*=0.033 \,m_\mathrm{e}$ is the effective carrier mass in BLG ($m_\mathrm{e}$ is the electron mass), we estimate the width of the QPC to be $W\approx200\,$nm. This value is in reasonable agreement with the lithographic channel width of 250nm.
We investigate the spin structure of the subbands by studying the evolution of the conductance steps as function of an in-plane magnetic field, $B_\parallel$. In Fig. 2(a), the conductance is shown as a function of $V_\mathrm{F1}$ for fixed $B_\parallel$-fields ranging from $-2$ to $6\,$T. Plateaus at 2, 6, and 10$\,e^2/h$ emerge with increasing magnetic field (see black arrows in Fig. 2(a)), indicating the lifting of the spin degeneracy of the subbands (see Fig. 2(b)). In Fig. 2(c), we plot the transconductance, d$G$/d$V_\mathrm{F1}$, as function of both $V_\mathrm{F1}$ and $B_\mathrm{\parallel}$. The data reveal splittings of all subbands as seen by the negative and positive slopes of spin-up and spin-down bands. Because of the small energy scales of our device, the Zeeman energy matches the subband spacing energy already for magnetic fields between 2 and 4 T, resulting in the crossing of the spin-up bands with the spin-down bands of the next higher subband (crossings are marked by black dots in Figs. 2(c,b)). The feature independent of $B_\parallel$ in Fig. 2(c) corresponds to the spin-down states of the first subband , which are locked to a finger gate voltage slightly above $V_\mathrm{F1} = -10.6\,$V because of quantum capacitance effects. For $|B_\parallel|>5.8\,$T the spin-up states of the first subband cross the spin-down states of the third subband, giving rise to a regime with six fully spin-down polarized modes ($G \sim 6\,e^2/h$; see black arrow in Fig. 2(c)) – an unprecedented high-polarization, making such QPCs interesting for spin polarizers and detectors in ballistic spin transport devices [@Vila2019Oct].
From the data of Fig. 2(c) we can also determine the energy spacing between two neighboring subbands, $\Delta E_{n,n+1}$. At the intersections of the spin-up and spin-down states of adjacent subbands, the Zeeman energy $\Delta E_\mathrm{Z}= g \mu_\mathrm{B}B_\parallel$ is equal to the spacing of the two subbands. Using the fact that in graphene and BLG the Lande factor is $g \sim 2$ [@Tans1997Apr; @Lyon2017Aug] (as confirmed also by direct measurements on our device, see Supplementary Material), we determine the subband spacing $\Delta E_{n,n+1}$ at $B_\parallel=0$. The values determined in this way agree well with those extracted from finite bias measurements (compare gray squares and purple circles in Fig. 2(d)). The energy difference $\Delta E_{1,3}$ extracted from the position of the rotated square in panel (c) coincides with the sum $\Delta E_{1,2}+\Delta E_{2,3}$, further confirming the consistency of the method.
We can investigate the dependence of the subband spacing on displacement field $D$ by performing measurements as those shown in Fig. 2(c) but for different $V_\mathrm{SG}$-$V_\mathrm{BG}$ configurations (see Supplementary Material). The energy spacings between the subbands appear to be independent of the applied displacement field within the margin of the scattering of our data, see Fig. 2(d). This indicates (i) that the electronic width of the transport channel is not strongly affected by the different stray-field contributions at different $V_\mathrm{SG}$ values, and (ii) that the BLG low-energy subband structure does not change appreciably when the band gap increases from $\approx 15\,$ to 35meV [^3].
The data of Fig. 2(c) reveal also the existence of features related to [*e-e*]{} interaction. The discontinuous behaviour of d$G$/d$V_\mathrm{F1}$ at the Zeeman crossings (see labels $\alpha$, $\beta$) – only appearing for the spin-up states – indicate that [*e-e*]{} interactions give rise to a spontaneous spin splitting [@Graham2003Sep; @Berggren2005Mar; @Weidinger2018Sep]. This leads to a spin-driven conductance anomaly, known as 0.7-analog earlier observed in GaAs-based QPCs [@Graham2003Sep]. Although the 0.7-analog is directly connected to the 0.7 anomaly [@Bauer2013Aug; @Weidinger2018Sep], we do not observe any feature at $B_\parallel = 0$ around $4 \times 0.7\,e^2/h$, but rather one at $2\,e^2/h$, which is nearly unaffected by in-plane magnetic field. A close-up of Fig. 2(a) around the first conductance step and at low $B_\parallel$ is shown in Fig. 3(a).
![**(a,b)** Transconductance dG/dV$_\mathrm{F1}$ as function of V$_\mathrm{F1}$ and $B_\perp$ for two different values of the displacement field. **(c,d)** Single-particle model calculations of dG/dV$_\mathrm{F1}$ based on the phenomenologically SOC parameters $\lambda_{\mathrm{lo}}=40\,\mu$eV and $\lambda_{\mathrm{up}}= 80\,\mu$eV (panel c) and $\lambda_{\mathrm{lo}}=\lambda_{\mathrm{up}}=40\,\mu$eV (panel d). The K-valley states are highlighted by dashed blue lines, and the K’-valley states by solid orange lines. They reveal an interesting texture, where the spin-valley coupling enhances the Zeeman splitting in one of the valley (diverging blue lines) but it suppresses in the other one (converging orange lines). The gray-scale background shows an approximation for the differential conductance obtained by gaussian smearing. []{data-label="f4"}](Fig04.pdf){width="\linewidth"}
The presence of the $2\,e^2/h$-feature indicates the energy splitting of the first subband at $B_\parallel=0$. This splitting appears also clearly in transconductance data such as those of Fig. 2(c) and Fig. 3(b). The analysis of the subband spacing performed above allows us to associate an estimated energy scale to the splitting of the lowest subband, $\Delta$, extracted from the transconductance data (see Supplementary Material). Performing this type of analysis for different values of the displacement field, we obtain values of $\Delta$ that range from 40 to 80$\,\mu$eV, with a nearly linear dependence on the $D$-field in the observed parameter range, see Fig. 3(c). This energy scale (together with the $2\,e^2/h$ step) agrees well with what is expected for the SO gap in graphene and BLG [@Kane2005Nov; @Min2006Oct; @Huertas-Hernando2006Oct; @Yao2007Jan; @Konschuh12], and with the experimental value determined by Sichau [*et al.*]{} for graphene on SiO$_2$ [@Sichau2019Feb]. The SOC in BLG is expected to be slightly enhanced by proximity effect when BLG is placed on hBN [@Zollner2019Mar; @PrivCom]. The observed dependence of $\Delta$ on the displacement field can be explained either (i) by an interplay between SOC and [*e-e*]{} interaction or (ii) by a tunable SOC induced gap due to a layer dependent proximity effect that sensitively depends on the microscopic details of the interfaces between BLG and the hBN layers enhancing $\Delta$. We speculate that the missing 0.7-feature at low magnetic field can be explained by a small [*e-e*]{} interaction ($\simeq 1-10\,\mu$eV) such that the SOC dominates over the interaction. At finite in-plane magnetic fields with $B_\parallel\simeq 2-3\,$T, the Zeeman effect quenches the SOC interaction and the [*e-e*]{} interaction becomes relevant at the first subband crossing leading to the 0.7-analog.
Figure 4(a,b) shows the transconductance as function of out-of-plane magnetic field, $B_\perp$, and finger-gate voltage $V_\mathrm{F1}$. In good agreement with earlier work [@Overweg2018Dec; @Kraft2018Dec], we observe the lifting of the valley degeneracy due to nontrivial valley-dependent orbital magnetic moments [@McCann2006Mar] (see dashed lines in Fig. 4(a)), and a characteristic crossing pattern at increasing magnetic fields. The $D$-field dependent splitting of the first subband at $B_\perp=0$ is also clearly visible.
To reproduce most of this pattern – including the splitting of the first subband – we extended the single-particle model developed in Ref. [@Overweg2018Dec] to take into account also the Zeeman term and the effects of SOC but neglecting [*e-e*]{} interaction. In intrinsic graphene and BLG the dominant SOC term is of Kane-Mele (KM) type [@Kane2005Nov] and couples spin and valley degrees of freedom to preserve time reversal symmetry. As for low energies the sublattice and the layer degree of freedom in BLG become equivalent we write the SOC Hamiltonian as $H_{\mathrm{KM}}= \frac 12 [(\lambda_\mathrm{lo}-\lambda_\mathrm{up})\sigma_0 - (\lambda_\mathrm{lo}+\lambda_\mathrm{up})\sigma_z] \tau_\mathrm{z}s_\mathrm{z},$ where $\sigma, \tau$ and $s$ refer to the layer, valley and spin degree of freedom, respectively. Here, we assume that the proximity-enhanced SOC coefficients $\lambda_\mathrm{up,lo}$ are expected to be different in the upper (up) and lower (lo) layer of BLG. As in our device the subband spacing is almost hundred times smaller than the band gap, the electronic modes are all localized in one graphene layer (the upper one), with only a small, $D$-field tunable admixture of the other one. Therefore, we assume that with our model all the $D$-field dependency can be attributed to $\lambda_\mathrm{up}(D)$ whereas $\lambda_\mathrm{lo}$ can be kept constant. Our model does not include Rashba-type SOC terms, as they do not contribute at small energies, i.e. at the K-points [@Konschuh12]. Using an electrostatic simulation of the device to determine the local potential in the BLG (see Supplemental Material), we find overall agreement between theory and experiment (compare Figs. 4(a,b) with 4(c,d)), including the splittings of the first subband both at zero and finite magnetic field (see arrows in Fig. 4(b,d)).
To conclude, we have observed the crossings of spin-split 1D subbands leading (i) to highly spin polarized ballistic currents and (ii) to interaction-driven spontaneous spin splittings at the crossing points. The latter is a halmark of the 0.7-analog structure opening an interesting route for better understanding [*e-e*]{} interaction effects in BLG quantum wires with a unique valley degree of freedom. This becomes even more interesting as at zero magnetic field the [*e-e*]{} interaction seems to be in competition with (proximity tunable) SOC, possibly leading to topologically non-trivial ground states.
**Aknowledgements:** We thank J. Fabian, M. Gmitra, A. Knothe, F. Haupt and B. van Wees for helpful discussions. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 785219 (Graphene Flagship), the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 - 390534769, through DFG (BE 2441/9-1 and STA 1146/11-1), the Austrian WWTF Project No. MA14-002, and by the Helmholtz Nano Facility [@Albrecht2017May]. Growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST(JPMJCR15F3), JST.
[^1]: The finger gates are isolated from the side gates by a 25nm thick Al$_2$O$_3$ layer.
[^2]: Please note that here the finger gate F1 (F2) corresponds to FG2 (FG3) in Ref. [@Banszerus2018Jun]
[^3]: The band gap depends on the displacement field approximately as $D \cdot$ 80meV/(V/nm) [@Zhang2009Jun]
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{
"pile_set_name": "ArXiv"
}
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---
abstract: |
A pore network modeling (PNM) framework for the simulation of transport of charged species, such as ions, in porous media is presented. It includes the Nernst-Planck (NP) equations for each charged species in the electrolytic solution in addition to a charge conservation equation which relates the species concentration to each other. Moreover, momentum and mass conservation equations are adopted and there solution allows for the calculation of the advective contribution to the transport in the NP equations.
The proposed framework is developed by first deriving the numerical model equations (NMEs) corresponding to the partial differential equations (PDEs) based on several different time and space discretization schemes, which are compared to assess solutions accuracy. The derivation also considers various charge conservation scenarios, which also have pros and cons in terms of speed and accuracy. Ion transport problems in arbitrary pore networks were considered and solved using both PNM and finite element method (FEM) solvers. Comparisons showed an average deviation, in terms of ions concentration, between PNM and FEM below $5\%$ with the PNM simulations being over ${10}^{4}$ times faster than the FEM ones for a medium including about ${10}^{4}$ pores. The improved accuracy is achieved by utilizing more accurate discretization schemes for both the advective and migrative terms, adopted from the CFD literature. The NMEs were implemented within the open-source package `OpenPNM` based on the iterative Gummel algorithm with relaxation.
This work presents a comprehensive approach to modeling charged species transport suitable for a wide range of applications from electrochemical devices to nanoparticle movement in the subsurface.\
address:
- 'Department of Chemical Engineering, University of Waterloo, Waterloo, ON, Canada'
- 'Department of Chemical Engineering, McGill University, Montreal, QC, Canada'
- 'Department of Chemical Engineering, University College London, London, United Kingdom'
author:
- Mehrez Agnaou
- Mohammad Amin Sadeghi
- Thomas George Tranter
- Jeff Gostick
bibliography:
- 'references.bib'
title: 'Modeling Transport of Charged Species in Pore Networks: Solution of the Nernst-Planck Equations Coupled with Fluid Flow and Charge Conservation Equations'
---
Porous media ,Nernst-Planck equations ,pore network modeling ,`OpenPNM`
Introduction {#sec:introduction}
============
The Nernst-Planck equations are widely used in the literature to describe the transport of ionic species in electrochemical systems [@meng2014; @metti2016]. With respect to porous media, the equations describe ion transport in a wide variety of applications such as electrochemical cells [@vansoestbergen2010] and certain redox flow batteries [@sadeghi2019b]. They are also used to analyze ion conduction in biological structures of pores [@bolintineanu2009], but probably the most common applications are for the study of ion transport mechanisms in clay soils and concrete. @smith2004 applied the NP equations to the analysis of transport through platy-clay soils and @pivonka2004 analyzed chloride diffusion in concrete for the estimation of structural degradation due to corrosion. Moreover, it has been shown that simulations based on the NP equations accurately predict ionic diffusion coefficients experimentally estimated on concrete [@narsilio2007]. In a more recent work [@azad2016], the transport processes in a system including a concrete plug surrounded by clay stone were modeled using the NP equations.
Another important field where the NP equations are used is modeling transport in capacitive charging and deionization [@biesheuvel2010; @gabitto2015]. Comparisons between simulation results and experimental data [@sharma2015] highlighted the capabilities of the NP based simulations to help in the design of capacitive deionization devices. While the transport of ionic species in the bulk of a solution flowing through a porous medium is generally described using the NP equations, a charge conservation equation is required to close the system. One option, perhaps the most accurate, uses the well-known Poisson equation for the electrostatic potential [@newman2012]. The Poisson equation relates the electric charge density to the Laplacian of the potential and describes the movement of the charged species in solution. This yields the famous Poisson Nernst-Planck system of equations. Charge conservation can also be enforced through a Laplace equation for the potential which allows for further mathematical simplifications under certain assumptions [@newman2012]. In the presence of fluid flow, the solution of the flow problem based on the mass and momentum conservation equations (Stokes or Navier-Stokes) enables the calculation of the advective term in the NP equations.
Solving electrochemical problems in porous media at the pore-scale based on the NP equations is generally carried-out using computational mesh that conforms to the real geometry of the system being analyzed. Different methods have been used to numerically solve the transport equations such as the finite difference [@bolintineanu2009; @meng2014; @sharma2015] and finite element [@samson1999; @narsilio2007; @lu2010; @metti2016; @azad2016]. However, it is well-known that direct numerical simulations (DNS) require significant computational resources. The same logic applies to many other transport problems such as pure diffusion or dispersion in porous media. PNM, as an alternative pore-scale modeling approach, requires substantially lower computing resources (compared to pore-scale DNS) and have been successfully applied to study physics such as diffusion reaction [@gostick2007] and dispersion [@sadeghi2019a] in porous media. However, the use of PNM to study transport of charged species is in its infancy. For instance, in a study of electrokinetic transport through charged porous media [@obliger2014], a steady-state PNM approach was used. This work [@obliger2014] is one of the first modeling electrochemical systems based on PNM. The used pore-scale microscopic transport coefficients were simple analytical relations obtained by solving the NP equations in a cylinder. Recently [@lombardo2019], a pore network model based on the NP equations was used to study porous electrodes in electrochemical devices. However, their approach [@lombardo2019] was based on the upwind scheme, which was recently shown to have high errors when P[é]{}clet number is above unity [@sadeghi2019a].
In this work, a more accurate method was developed and validated to solve the charge conservation NP system in pore networks. This new method will ultimately allow for accurate pore-scale simulation of transport in electrochemical systems with substantially lower computational cost compared to DNS approaches such as FEM. One aim of the present work is to identify the best approach among various options and to establish a numerically accurate and robust algorithm. Future work can then build on this solid foundation.
Although the simplifications related to PNM may induce additional errors into the numerical solution, it has been shown through comparisons between results of advection diffusion simulations, that the PNM approach provides reasonably accurate solutions [@yang2016] compared to those obtained from DNS using lattice Boltzmann and finite volume methods. This work presents a novel PNM framework for the simulation of charged species transport. The framework is based on highly accurate discretization schemes in addition to several charge conservation options. It also supports transient simulations and handles non-linear source terms.
Background {#sec:background}
==========
This work considers single-phase, isothermal, incompressible flow of a dilute electrolytic solution, treated as a Newtonian fluid, in a non-deformable porous medium. Assuming flow in the viscous-dominated regime [@agnaou2016; @agnaou2017], the movement of the electrolytic solution can be described using the following steady-state momentum and mass conservation (Stokes) equations $$\mu \laplacian{\vb*{u}}-\grad{p}=\vb*{0},\label{eq:momentum}$$ and $$\div{\vb*{u}}=0,\label{eq:mass}$$ where $\vb*{u}$ is the velocity of the solution, $p$ its pressure, and $\mu$ its dynamic viscosity and is considered to be constant. Using the NP equation, the flux of ionic species $n$ in the solution is given by [@newman2012; @biesheuvel2010; @sharma2015] $$\vb*{N}^{n}=-D^{n}\grad{c^{n}}+\vb*{u}c^{n}-\frac{D^{n}z^{n}F}{RT}c^{n}\grad{\phi},\label{eq:np_flux}$$ where $c^{n}$ is the ion concentration, $\phi$ is the electrostatic potential, $D^{n}$ is the diffusion coefficient of species $n$ and $z^{n}$ its valence, and $F$ is the Faraday constant. Eq. \[eq:np\_flux\] as written follows several authors [@newman2012; @sharma2015] defining the mobility based on the Nernst-Einstein equation, $u_{mob}^{n}=D^{n}/\qty(RT)$, where $R$ is the universal gas constant and $T$ a constant absolute temperature. The flux as defined by Eq. \[eq:np\_flux\] consists of three terms, representing different transport mechanisms namely, molecular diffusion, bulk advection, and electrostatic migration. Moreover, a mass conservation equation is considered for each of the ionic species $n$ as follows $$\pdv{c^{n}}{t}=-\div{\vb*{N}^{n}}.\label{eq:np_con}$$
Substituting the flux from the Nernst-Planck equation (Eq. \[eq:np\_flux\]) into the conservation equation (Eq. \[eq:np\_con\]), yields an equation for each of the ionic species as follows $$\pdv{c^{n}}{t}=-D^{n}\laplacian{c^{n}}+\vb*{u}\vdot\grad{c^{n}}-\frac{D^{n}z^{n}F}{RT}\div(c^{n}\grad{\phi}).\label{eq:np}$$
The governing equations for fluid flow and concentration of species (Eqs. \[eq:momentum\], \[eq:mass\], \[eq:np\]) are now defined. However, an additional equation is required to close the system of equations since the electrostatic potential is unknown. In this work, three different approaches were considered. Using the Gauss electrostatic theorem [@newman2012], one could relate the distribution of ions in the electrolytic solution to the variation of the electric field through a Poisson equation as follows [@smith2004; @samson1999] $$\div{\qty(\varepsilon \varepsilon_{r} \grad{\phi})}=-F\sum_{n}\qty(z^{n}c^{n}),\label{eq:poisson}$$ such that $\varepsilon$ is the vacuum permittivity and $\varepsilon_{r}$ is the relative permittivity of the electrolytic solution. The quantity on the right-hand side (rhs) of equation \[eq:poisson\] is the electric charge density per unit volume. The solution of the Poisson equation is numerically challenging due to numerical instabilities [@jerome1996; @metti2016] and stabilization techniques are often required [@meng2014]. More stable and simpler alternatives to Eq. \[eq:poisson\] can be used to close the system and enforce charge conservation. However, these alternative equations, discussed in what follows are derived based on specific assumptions and hence, their validity should be limited to specific cases [@macgillivray1968; @macgillivray1969]. In fact, charge conservation can be imposed as follows $$\div{\vb*{i}}=0,\label{eq:charge_conservation_01}$$ where $\vb*{i}$ is the current density and is given by $$\vb*{i}=F\sum_{n}\qty(z^{n}\vb*{N}^{n}).\label{eq:current_density_01}$$ Replacing the flux $\vb*{N}^{n}$ in Eq. \[eq:current\_density\_01\] by its value from Eq. \[eq:np\_flux\] yields $$\vb*{i}=-F\sum_{n}\qty(z^{n}D^{n}\grad{c^{n}})+F\vb*{u}\sum_{n}\qty(z^{n}c^{n})-\frac{F^{2}}{RT}\grad{\phi}\sum_{n}\qty({z^{n}}^{2}D^{n}c^{n}).\label{eq:current_density_02}$$ Then, by virtue of electroneutrality, $\sum_{n}{z^{n}c^{n}}=0$, the second term on the rhs of Eq. \[eq:current\_density\_02\] is zero. Insertion of Eq. \[eq:current\_density\_02\] into Eq. \[eq:charge\_conservation\_01\] gives $$\div{\qty(K \grad{\phi})}=-F\sum_{n}\qty[z^{n}\div{\qty(D^{n}\grad{c^{n}})}],\label{eq:charge_conservation_02}$$ where $K$ is the conductivity of the electrolytic solution and is given by $$K=\frac{{F}^{2}}{RT}\sum_{n}\qty(z^{n\,2}D^{n}{c}^{n}),\label{eq:kappa}$$ Finally, for negligible concentration gradients and assuming a uniform conductivity $K$, Eq. \[eq:charge\_conservation\_02\] reduces to a Laplace equation for the potential as follows $$\laplacian{\phi}=0.\label{eq:laplace}$$
Pore Network Modeling Formulation {#sec:pnm}
=================================
The pore network is a simplified representation of a real porous medium geometry, consisting of pore bodies interconnected by throats. Figure \[fig:conduit\] shows a pore-throat-pore conduit of a pore network. For the sake of simplicity regarding the conservation equations to be considered, idealized shapes are assigned to pores and throats. In this sense, and for a three-dimensional (3D) medium, pores and throats are generally represented by spheres and circular cylinders, respectively. For a two-dimensional (2D) geometry, pores and throats are described by circles and rectangles, respectively.
![\[fig:conduit\] Pore-throat-pore assembly as a single conduit in PNM. Conduit made of throat $ij$ and halves of the neighbor pores $i$ and $j$ of diameters $d_{ij}$, $d_{i}$, and $d_{j}$ and lengths $l_{ij}$, $l_{i}$, and $l_{j}$ and opposing resistances to a transport mechanism $tr$ (from $i$ to $j$ and vice versa) of $1/g^{tr}_{ij}$, $1/g^{tr}_{i}$, and $1/g^{tr}_{j}$, respectively. Conductance of the assembly is given by Eq. \[eq:pnm\_Gtr\].](conduit){width="0.6\linewidth"}
The conductance of the pore-throat-pore assembly or conduit for a given transport mechanism $tr$ (see Fig. \[fig:conduit\]) is given, from the linear resistor theory for resistors in series [@gostick2007], by $$G^{tr}_{ij}=\qty(\frac{1}{g^{tr}_i}+\frac{1}{g^{tr}_{ij}}+\frac{1}{g^{tr}_j})^{-1}.\label{eq:pnm_Gtr}$$ Efficient algorithms for the extraction of pore networks from 2D and 3D images are available in the literature [@dong2009; @rabbani2014; @gostick2017], even for dual networks [@khan2019], within the open-source image analysis package `PoreSpy` [@gostick2019].
This work assumes perfect mixing of the solute within the pore space, unlike more sophisticated approaches to be discussed below. In addition, conservation of physical quantities are enforced in the pores only. Therefore, for a time dependent transport problem, the total void volume of the porous medium is assigned to the pores whereas the throats are considered to have a zero volume. The volume of the throats is distributed among their neighboring pores. This approach offers simplicity and computational efficiency which allows for pore-scale simulations at relatively lower computational costs compared to DNS.
The assumption of perfect mixing is robust for transport problems involving pure diffusion. When additional transport mechanisms such as advection come into play, this assumption remains valid at low P[é]{}clet numbers (P[é]{}clet numbers smaller than unity) where the P[é]{}clet is the ratio of advective to diffusive contributions. The validity of the perfect mixing assumption was extended to pore-scale P[é]{}clet numbers up to $257$ by @mehmani2015 and by @yang2016 in disordered sphere packs and up to $10$ by @sadeghi2019a in cubic networks of random pore sizes. Thus, the mixed-cell method can be used for modeling transport phenomena in disordered porous structures where moderate deviations from pure diffusion exist. The structural disorder refers in the present work to the randomness in the pores and throats sizes and in the coordination number of pores. Deviations from pure diffusion considered in this study (see section \[sec:comparisons\]) result from advective and migrative fluxes. For ordered porous structures, the mixed-cell method should be reserved only for diffusion dominated problems. Furthermore, the mixed-cell method ignores the impact of non-uniform velocity profiles in pores and throats on the transport of chemical species. In the same manner as for the perfect mixing assumption, uniform velocity profiles were found to have a negligible effect on transport in disordered media [@mehmani2015; @yang2016]. Consequently, the PNM method is appropriate to perform pore-scale simulations of advection diffusion problems (and advection diffusion migration problems as will be shown below) in disordered porous media at low computational costs. An alternative to the mixed-cell method when high concentration gradients are expected within the pore space, although computationally more expensive, is the streamline splitting approach [@mehmani2014].
Stokes Flow
-----------
Given steady-state Stokes flow (Eqs. \[eq:momentum\] and \[eq:mass\]) of a Newtonian fluid, corresponding to the electrolytic solution, the mass conservation equation for an arbitrary pore $i$, is $$\begin{aligned}
\sum_{j=1}^{N_i}{G}_{ij}^h\qty(p_i-p_j)=0, & \quad i=1,2,\dots,N_p,\label{eq:pnm_flow}\end{aligned}$$ where the subscripts $i$ and $j$ correspond to the considered pore and the neighboring ones, respectively, and $p_i$ and $p_j$ are the pressure values in pores $i$ and $j$, respectively. In Eq. \[eq:pnm\_flow\], $N_i$ is the number of pores neighboring of pore $i$, $N_p$ is the total number of pores in the network, and $G_{ij}^h$ is the hydraulic conductance of the pore-throat-pore assembly and is given by Eq. \[eq:pnm\_Gtr\] where $tr=h$, $G^{h}_{ij}=\qty({1}/{g^{h}_i}+{1}/{g^{h}_{ij}}+{1}/{g^{h}_j})^{-1}$. The hydraulic conductance of pore $i$, $g_i^h$, can be calculated using the Hagen-Poiseuille model [@sutera1993] as follows $$g_i^h=\frac{\pi}{128\mu}\qty(\frac{d_i^4}{l_i}),\label{eq:pnm_gh_3d}$$ with $d_i$ being the diameter of pore $i$ and $l_i$ its length. It should be noted here that the length of a pore refers to its radius. The hydraulic conductance of throat $ij$ and pore $j$ are computed in the same manner as for pore $i$. Equation \[eq:pnm\_gh\_3d\] is valid for a 3D configuration where the conduit has a cylindrical shape. For a 2D network, where throats are represented by rectangles, the hydraulic conductance is given, from the analytical solution of a plane Poiseuille flow, by $$g_i^h=\frac{1}{12\mu}\qty(\frac{d_i^3}{l_i}),\label{eq:pnm_gh_2d}$$
Nernst-Planck Equations
-----------------------
Special attention was paid to the derivation of the NMEs required to model transport of charged chemical species. In fact, Eq. \[eq:np\] is discretized in both time and space using various schemes with the resulting accuracy assessed in section \[sec:comparisons\]. For the sake of brevity in what follows, only semi-discrete forms are presented. First, with a discretized accumulation term (time discretization) and then, with space discretization. One can easily obtain the NME corresponding to Eq. \[eq:np\] by combining the two semi-discrete forms.
The semi-discrete form of equations \[eq:np\_con\] or \[eq:np\], after time discretization, results in the following species $n$ conservation equation $$\qty[\varphi_b\frac{c^n}{\Delta t}-\varphi_a\qty(-\div{\vb*{N}^{n}})]^{t_1}=\qty[\varphi_b\qty(1-\varphi_a)\qty(-\div{\vb*{N}^{n}})+\varphi_b\frac{c^n}{\Delta t}]^{t_0},\label{eq:np_timediscrete}$$ where $\Delta t$ is the time step, $t_0$ the previous time value, $t_1$ the new time value, and $\varphi_a$ and $\varphi_b$ are constants used to set the time scheme. Values $\varphi_a=1$ and $\varphi_b=1$ result in an implicit, first order accurate, time scheme. Whereas setting $\varphi_a=0.5$ and $\varphi_b=1$ corresponds to the second order accurate Crank-Nicolson scheme. Finally, $\varphi_a=1$ and $\varphi_b=0$ yields the steady-state form of the conservation equation.
Focusing on the space discretization of equation \[eq:np\], the semi-discrete form can be given by, $$\begin{aligned}
\begin{split}
&\sum_{j=1}^{N_i}\qty[{G}_{ij}^{n,d}+\max\qty(q_{ij}-m_{ij}^{n},0)]c_i^n-\\
&\sum_{j=1}^{N_i}\qty[{G}_{ij}^{n,d}+\max\qty(-q_{ij}+m_{ij}^{n},0)]c_j^n={v}_{i}\pdv{c_{i}^{n}}{t},
\end{split}
& \quad i=1,2,\dots,N_p,\label{eq:pnm_np_1}\end{aligned}$$ such that $c_i^n$ and $c_j^n$ are the concentrations of species $n$ at pore $i$ and neighbor pores $j$, respectively, $G_{ij}^{n,d}$ the diffusive conductance (of species $n$) of the pore-throat-pore assembly, $q_{ij}$ is the throat flow rate, $m_{ij}^{n}$ is the migration rate, and $v_{i}$ the volume of pore $i$. Note that the upwind discretization of the advective and migrative terms should be carried-out considering both terms at the same time as done on Eq. \[eq:pnm\_np\_1\]. It was found in this work that considering these terms separately leads to higher errors.
The diffusive conductance $G_{ij}^{n,d}$ of Eq. \[eq:pnm\_np\_1\] can be given, based on Eq. \[eq:pnm\_Gtr\], setting the transport type to $tr=n,d$ to refer to transport of species $n$ via diffusion by $G^{n,d}_{ij}=\qty({1}/{g^{n,d}_i}+{1}/{g^{n,d}_{ij}}+{1}/{g^{n,d}_j})^{-1}$. The pore $i$ diffusive conductance being, for a 3D configuration, $$g_i^{n,d}=\frac{A_{i}D^{n}}{l_{i}},\label{eq:pnm_g_i}$$ such that $A_i$ is the cross-section area of pore $i$, and the diffusion coefficient of species $n$, $D^n$, is considered constant. In the same way as for pore $i$ (Eq. \[eq:pnm\_g\_i\]), the diffusive conductances of pore $j$ and throat $ij$ can be defined. For a 2D configuration, the cross-section area $A_{i}$ in Eq. \[eq:pnm\_g\_i\] should be replaced by the diameter $d_i$. Furthermore, the volumetric flow rate of the electrolytic solution $q_{ij}$, appearing in Eq. \[eq:pnm\_np\_1\], can be calculated as follows, $$q_{ij}=G_{ij}^h\qty(p_i-p_j),\label{eq:pnm_q_ij}$$ and finally, the migration rate of Eq. \[eq:pnm\_np\_1\] can be given under the following form, $$m_{ij}^{n}=G_{ij}^{n,m}\qty(\phi_i-\phi_j),$$ where $G^{n,m}_{ij}=\qty({1}/{g^{n,m}_i}+{1}/{g^{n,m}_{ij}}+{1}/{g^{n,m}_j})^{-1}$ is the migrative conductance and is also defined based on Eq. \[eq:pnm\_Gtr\] where $tr=n,m$ to refer to transport of species $n$ by migration. In these circumstances, the migrative conductance of pore $i$ is $$g_{i}^{n,m}=\frac{z^{n}F}{RT}g_i^{n,d},$$
In equation \[eq:pnm\_np\_1\], while the diffusive flux is discretized based on the central differencing scheme, which is second order accurate in terms of Taylor series expansion, a first order upwind scheme is adopted for both the advective and migration fluxes. However, in a recent work [@sadeghi2019a], a more accurate discretization of the advective and diffusive fluxes was proposed based on the finite difference power-law discretization scheme. Using the power-law discretization for advection and diffusion and the upwind scheme for the migration, the following species conservation equation, that is more accurate than Eq. \[eq:pnm\_np\_1\], can be written $$\begin{aligned}
\begin{split}
&\sum_{j=1}^{N_i}\qty{G_{ij}^{n,d}\max{\qty[\qty(1-\frac{\abs{^{ad}{Pe}^{n}_{ij}}}{10})^{5},0]}+\max{\qty(q_{ij},0)}+\max{\qty(-m_{ij}^{n},0)}}c_i^n-\\
&\sum_{j=1}^{N_i}\qty{G_{ij}^{n,d}\max{\qty[\qty(1-\frac{\abs{^{ad}{Pe}^{n}_{ij}}}{10})^{5},0]}+\max{\qty(-q_{ij},0)}+\max{\qty(m_{ij}^{n},0)}}c_j^n={v}_{i}\pdv{c_{i}^{n}}{t},\\
& \quad i=1,2,\dots,N_p,
\end{split}
\label{eq:pnm_np_2}\end{aligned}$$ where $^{ad}{Pe}^{n}$ is the advective P[é]{}clet number corresponding to species $n$ and is given by the ratio of advective to diffusive contributions as follows $$^{ad}Pe_{ij}^{n}=\frac{q_{ij}}{G_{ij}^{n,d}}.\label{eq:pnm_pe_1}$$
While the discretization given by Eq. \[eq:pnm\_np\_2\] is more accurate than Eq. \[eq:pnm\_np\_1\], the migration term, discretized based on an upwind scheme, is only first order accurate and may be a source of non-negligible errors. Indeed, it was shown by @sadeghi2019a, for advection diffusion problems in pore networks, that the first order upwind discretization of the advective term results in network average relative deviations, in terms of species concentration, of up to $10\%$ compared to FEM simulations. For this reason, an alternative form of the NME was derived where the migration flux was also treated as a power-law. In this form, the advection and migration fluxes in Eq. \[eq:np\_flux\] are grouped together to give rise to a single term that encompasses both advection and migration effects. This leads to an augmented P[é]{}clet number $^{ad,mig}{Pe}^{n}$ which corresponds to the ratio between advective migrative effects and the diffusive ones as follows, $$^{ad,mig}{Pe}_{ij}^{n}=\frac{q_{ij}-m_{ij}^{n}}{G_{ij}^{n,d}}.\label{eq:pnm_pe_2}$$ Accordingly, the species conservation equation takes the following form,
$$\begin{aligned}
\begin{split}
&\sum_{j=1}^{N_i}\qty{G_{ij}^{n,d}\max{\qty[\qty(1-\frac{\abs{^{ad,mig}{Pe}^{n}_{ij}}}{10})^{5},0]}+\max{\qty(q_{ij}-m_{ij}^{n},0)}}c_i^n-\\
&\sum_{j=1}^{N_i}\qty{G_{ij}^{n,d}\max{\qty[\qty(1-\frac{\abs{^{ad,mig}{Pe}^{n}_{ij}}}{10})^{5},0]}+\max{\qty(-q_{ij}+m_{ij}^{n},0)}}c_j^n={v}_{i}\pdv{c_{i}^{n}}{t},\\
& \quad i=1,2,\dots,N_p.
\end{split}
\label{eq:pnm_np_3}\end{aligned}$$
For a 2D problem, the volume ${v}_{i}$, appearing in Eqs. \[eq:pnm\_np\_1\], \[eq:pnm\_np\_2\], and \[eq:pnm\_np\_3\] has to be replaced by the surface area ${s}_{i}$ to ensure units consistency.
Finally, following the same logic, one can define a migrative P[é]{}clet number which corresponds to the ratio of migrative to diffusive effects, $$^{mig}{Pe}_{ij}=\frac{-m_{ij}^{n}}{G_{ij}^{n,d}}.\label{eq:pnm_pe_3}$$ It can be noticed that $^{mig}{Pe}$, unlike $^{ad}{Pe}^{n}$ and $^{ad,mig}{Pe}^{n}$, does not depend on the chemical species $n$. The $^{mig}{Pe}$ will prove useful in section \[sec:comparisons\].
Charge Conservation Laws
------------------------
As stated above, three different approaches for enforcing charge conservation were considered in this work. The PNM form of each approach is described below. These laws describe the relationship between the electrostatic potential of the solution and the spatial distribution of electric charges in the solution.
### Poisson Equation
The discretization of the Poisson equation for the electrostatic potential (Eq. \[eq:poisson\]) is performed based on the second order accurate central differencing scheme. The relative permittivity of the electrolytic solution, $\varepsilon_{r}$, is considered constant and does not depend on the local concentrations. The obtained pore-scale NME, valid for a 3D problem, is given by $$\begin{aligned}
\sum_{j=1}^{N_i}{K^{Poisson}_{ij}}\phi_i-
\sum_{j=1}^{N_i}{K^{Poisson}_{ij}}\phi_j=-{v}_{i}{F}\sum_{n}z^{n}c_{i}^{n},
& \quad i=1,2,\dots,N_p,\label{eq:pnm_poisson}\end{aligned}$$ whereas for a 2D situation, the volume ${v}_{i}$ appearing on Eq. \[eq:pnm\_poisson\] must be replaced by the pore’s surface area ${s}_{i}$. In Eq. \[eq:pnm\_poisson\], $K^{Poisson}_{ij}$ is the ionic conductance of the electrolytic solution for the conduit $ij$. It is given, as on Eq. \[eq:pnm\_Gtr\], by $K^{Poisson}_{ij}=\qty({1}/{k^{Poisson}_i}+{1}/{k^{Poisson}_{ij}}+{1}/{k^{Poisson}_j})^{-1}$ such that the pore $i$ ionic conductance, for a 3D problem, is $${k}^{Poisson}_{i}=\frac{{A}_{i}{\varepsilon\varepsilon_{r}}}{{l}_{i}},\label{eq:pnm_ki_poisson}$$ and, for a 2D configuration, it becomes ${k}^{Poisson}_{i}={{d}_{i}{\varepsilon\varepsilon_{r}}}/{{l}_{i}}$. The ionic conductances of pores $j$ neighboring $i$ and the throat $ij$ is computed in the same way as with Eq. \[eq:pnm\_ki\_poisson\].
### Charge Conservation Equation with Electroneutrality
Charge conservation can also be enforced using Eq. \[eq:charge\_conservation\_02\] assuming electroneutrality. The corresponding NME is given as follows $$\begin{aligned}
\begin{split}
&\sum_{j=1}^{N_i}{K}^{elec}_{ij}\phi_i-
\sum_{j=1}^{N_i}{K}^{elec}_{ij}\phi_j=\\
&-F\sum_{n}{z}^{n}\qty( \sum_{j=1}^{N_i}G_{ij}^{n,d}{c}^{n}_{i}-\sum_{j=1}^{N_i}G_{ij}^{n,d}{c}^{n}_{j}),
\end{split}
& \quad i=1,2,\dots,N_p,\label{eq:pnm_charge_conservation}\end{aligned}$$ where ${K}^{elec}_{ij}$ is the ionic conductance of the electrolytic solution in which electroneutrality is assumed. It is given based on the linear resistor theory for resistors in series (see Eq. \[eq:pnm\_Gtr\]) by $K^{elec}_{ij}=\qty({1}/{k^{elec}_i}+{1}/{k^{elec}_{ij}}+{1}/{k^{elec}_j})^{-1}$ with the ionic conductance for the pore $i$, in a 3D configuration, being, $${k}^{elec}_{i}=\frac{{F}^{2}}{RT}\frac{{A}_{i}}{{l}_{i}}\sum_{n}\qty({z}^{n\,2}{D}^{n}{c}_{i}^{n}),\label{eq:pnm_ki}$$ and for a 2D problem, ${k}^{elec}_{i}=\qty[{{F}^{2}{d}_{i}}/\qty({RT{l}_{i}})]\sum_{n}\qty({z}^{n\,2}{D}^{n}{c}_{i}^{n})$. Conductances of pores $j$ and throats $ij$ are defined in the same manner as in Eq. \[eq:pnm\_ki\]. For the ionic conductance of throat $ij$, $k^{elec}_{ij}$, the concentration of species $n$ at the considered throat, ${c}^{n}_{ij}$, is required. However, since ${c}^{n}_{ij}$ is not solved for, it can be defined based on a volume (or surface for a 2D problem) weighted average using the concentrations at the two neighbor pores. It is given, for a 3D configuration, by $${c}^{n}_{ij}=\frac{{v}_{i}{c}^{n}_{i}+{v}_{j}{c}^{n}_{j}}{{v}_{i}+{v}_{j}},\label{eq:pnm_cij}$$ and, becomes ${c}^{n}_{ij}=\qty({{s}_{i}{c}^{n}_{i}+{s}_{j}{c}^{n}_{j}})/\qty({{s}_{i}+{s}_{j}})$, in a 2D problem.
### Laplace Equation
Finally, another way to enforce charge conservation, is using the Laplace equation for the potential (Eq. \[eq:laplace\]) in situations where the electrolytic solution is electroneutral and the space variations of the concentration are neglected. The corresponding pore-scale NME is given by, $$\begin{aligned}
\sum_{j=1}^{N_i}{K}^{Laplace}_{ij}\phi_i-\sum_{j=1}^{N_i}{K}^{Laplace}_{ij}\phi_j=0, & \quad i=1,2,\dots,N_p.\label{eq:pnm_laplace}\end{aligned}$$ where ${K}^{Laplace}_{ij}$ is the ionic conductance of the electrolytic solution and is given by $K^{Laplace}_{ij}=\qty({1}/{k^{Laplace}_i}+{1}/{k^{Laplace}_{ij}}+{1}/{k^{Laplace}_j})^{-1}$, in the same manner as for other conductances. The ionic conductance for the pore $i$ is $${k}^{Laplace}_{i}=\frac{{A}_{i}}{{l}_{i}},\label{eq:pnm_ki_laplace}$$ for a 3D problem, and becomes ${k}^{Laplace}_{i}={{d}_{i}}/{{l}_{i}}$, for a 2D configuration.
Solution Algorithm {#sec:algorithm}
------------------
![\[fig:algorithm\] Solution algorithm implemented on `OpenPNM` [@gostick2016] to solve time dependent problems of transport of charged chemical species coupled with fluid flow. Fluid flow is described by Eqs. \[eq:momentum\] and \[eq:mass\] and the corresponding NME is Eq. \[eq:pnm\_flow\]. A Nernst-Plank equation, Eq. \[eq:np\] corresponding to NMEs \[eq:np\_timediscrete\] and \[eq:pnm\_np\_1\] or \[eq:pnm\_np\_2\] or \[eq:pnm\_np\_3\], is adopted for every charged species present in the electrolytic solution. Charge conservation is enforced through Eq. \[eq:poisson\] or \[eq:charge\_conservation\_02\] or \[eq:laplace\] and the corresponding NMEs are Eqs. \[eq:pnm\_poisson\] or \[eq:pnm\_charge\_conservation\] or \[eq:pnm\_laplace\], respectively.](algorithm){height="1.25\linewidth"}
The procedure developed in this work to numerically solve the flow problem (Eqs. \[eq:momentum\] and \[eq:mass\]) coupled with the transport of charged species (NP, Eq. \[eq:np\], and charge conservation, Eq. \[eq:poisson\] or \[eq:charge\_conservation\_02\] or \[eq:laplace\] depending on the situation) is described in this section. The solver was implemented within the open-source PNM package `OpenPNM` [@gostick2016]. Although source terms are not considered in sections \[sec:background\] and \[sec:pnm\], the approach followed to handle them is described here. Pore-scale NMEs obtained from the time and space discretization of the PDEs (Eqs. \[eq:momentum\] and \[eq:mass\], Eq. \[eq:np\], and Eq. \[eq:poisson\] or \[eq:charge\_conservation\_02\] or \[eq:laplace\]) are presented in section \[sec:pnm\]. These NMEs yield linear systems of equations solved iteratively based on the algorithm described on Fig. \[fig:algorithm\].
First, the initial and boundary value problem (IBVP), the physical properties of the electrolytic solution, and the solver settings need to be defined. Solver settings include inputs such as the time and space discretization schemes, the different tolerances and maximum number of iterations, type of linear solvers, initial and final time values, the time step, *etc*. Then, the flow problem (Eqs. \[eq:momentum\] and \[eq:mass\] corresponding to NME \[eq:pnm\_flow\]) is solved and a converged steady-state pressure field is obtained (see Fig. \[fig:algorithm\]). Pressure values are used to compute the advective flux in the NP equations.
Subsequently, time marching starts and for each time value, the charge conservation (Eq. \[eq:poisson\] or \[eq:charge\_conservation\_02\] or \[eq:laplace\] corresponding to NMEs \[eq:pnm\_poisson\] or \[eq:pnm\_charge\_conservation\] or \[eq:pnm\_laplace\], respectively) and NP (Eq. \[eq:np\] corresponding to NMEs \[eq:np\_timediscrete\] and \[eq:pnm\_np\_1\] or \[eq:pnm\_np\_2\] or \[eq:pnm\_np\_3\]) system is solved based on the Gummel method [@jerome1996]. Linear systems are decoupled and solved iteratively and may all be subject to Picard iterations [@paniconi1994] in the presence of non-linear source or sink terms. Picard convergence is reached once the value of the solved quantity satisfies the linearized system of equations within a certain tolerance or the maximum number of iterations is reached. The linearization is performed around the value at the previous Picard iteration or the initial value. Gummel iterations are repeated until convergence is obtained or when the maximum number of iterations is reached. A Gummel iteration consists of solving the charge conservation equation, updating the potential values, and solving a NP equation for every species present in the electrolytic solution and finally updating the concentration values. Gummel convergence is achieved when the difference between the values, for both the concentrations and potential, of two successive iterations falls bellow a predefined tolerance. For numerical stability, under-relaxation can be applied to both quantities solved for and/or source or sink terms.
The concentrations and potential fields obtained from the solution of the charge conservation NP system correspond to current time value. The time marching is ended when the predefined final time is reached or if a stationary solution is obtained. Otherwise, a new time iteration will begin after updating all the concentrations and potential values. Stationarity, or transient convergence as shown on Fig. \[fig:algorithm\], is obtained once the variation between both concentrations and potential, at two successive time values falls bellow a given tolerance.
Comparisons with Reference Solutions {#sec:comparisons}
====================================
Ion transport problems over arbitrary disordered porous media were considered here. It is worth recalling that the structural disorder refers to the randomness in pores and throats sizes and in the coordination number of pores. The considered problems were solved numerically based on the PNM approach and, for the sake of comparison, based on the FEM. To assess the accuracy of different NMEs presented in section \[sec:pnm\], PNM simulations were performed using three different NMEs. The NMEs consist of Eqs. \[eq:pnm\_np\_1\], \[eq:pnm\_np\_2\], and \[eq:pnm\_np\_3\] and are referred to as upwind upwind, power-law upwind, and power-law, respectively. Comparisons focused on the concentration fields only and without losing generality, only one charge conservation scenario was considered for brevity.
Initial Boundary Value Problem {#sec:IBVP}
------------------------------
The problem under consideration is that of the transport of saline water over an arbitrary porous medium $\Omega$. The real geometry of the 2D porous medium was modeled using a network of pores as shown on Fig. \[fig:network\]. Despite the fact that the topology of the medium is simplified, analyses based on pore networks were shown to play an important role in diverse applications for the study of flow and transport phenomena in porous media [@xiong2016].
![\[fig:network\] A 2D porous realization $\Omega$ made of $341$ pores in a uniform square lattice and connected by throats. Pores and throats have random sizes and spacing between neighbor pores centers is $1\si{\micro m}$. Four boundary regions are defined; $left$, $right$, $bottom$, and $top$, and one internal region; $internal = \Omega \setminus\qty(left\cup right\cup bottom\cup top)$, with the corresponding initial and boundary conditions.](network){width="0.75\linewidth"}
First, a network was generated with $23 \times 15$ pores, connected by throats, consisting of a square lattice with a spacing of $1\si{\micro m}$. Pores and throats were assigned random sizes based on a uniform distribution. The pores at the corners and the throats connecting the boundary pores one to each other were removed for better agreement with the FEM simulations. Finally, the average coordination number of the network was reduced to an average $3$ by deleting random throats not belonging to the minimum spanning tree found using the Kruskal algorithm with random weights assigned to each throat. This increases the structural randomness to more closely mimic real media while remaining geometrically perfectly known. Four boundary regions were defined, namely, $left$, $right$, $bottom$, and $top$, and an internal region $internal = \Omega \setminus\qty(left\cup right\cup bottom\cup top)$. The electrolytic solution (*i.e.*, saline water) is composed of water (solvent) and salt (electrolyte) dissolved and separated into cations, $Na$, and anions, $Cl$. The physical properties of the solution and its components are reported in Tab. \[tab:properties\].
mixture $Na$ $Cl$
----------------------------------------------- ------------------------ --------------------- ---------------------
Dynamic viscosity ($\mu$) $[\si{Pa\ldotp s}]$ $0.89557\times10^{-3}$ – –
Relative permittivity ($\varepsilon_{r}$) $78.303$ – –
Diffusivity ($D^{n}$) $[\si{m^{2}/s}]$ – $1.33\times10^{-9}$ $2.03\times10^{-9}$
Valence ($z^{n}$) – $+1$ $-1$
: \[tab:properties\] Physical properties of the mixture (saline water) and its components $Na$ and $Cl$ at temperature $T=298.15~\si{K}$ and pressure $p=101325~\si{Pa}$.
The flow of the mixture is described by Eqs. \[eq:momentum\] and \[eq:mass\] whereas the movement of ions is modeled using Eq. \[eq:np\] and the charge conservation is enforced through Eq. \[eq:laplace\]. The initial and boundary conditions associated with this system of equations are included in Fig. \[fig:network\]. Boundary concentrations are $c_{left}=10\si{mol/m^3}$, $c_{right}=20\si{mol/m^3}$, $c^{Na}_{bottom}=c^{Cl}_{top}=5\si{mol/m^3}$, and $c^{Na}_{top}=c^{Cl}_{bottom}=30\si{mol/m^3}$. Although the considered transport problem is arbitrary and is only used for comparisons between different methods, the configuration is comparable to what occurs in a spacer of a desalination unit by capacitive deionization [@hemmatifar2015]. The analysis is performed in terms of the network’s arithmetic mean of the absolute values of the dimensionless numbers $^{ad}{Pe}^{Na}$ and $^{mig}{Pe}$ referred to as $\langle ^{ad}{Pe}^{Na} \rangle$ and $\langle ^{mig}{Pe} \rangle$, respectively. These numbers were varied by considering different values for the pairs $p_{left}$ $p_{right}$ and $\phi_{bottom}$ $\phi_{top}$, respectively. The considered simulation conditions are such that both $\langle ^{ad}{Pe}^{Na} \rangle$ and $\langle ^{mig}{Pe} \rangle$ were varied within a range from $0.1$ to $5$ considering all possible combinations. The network-scale advective forces were always kept acting from $right$ to $left$ by enforcing $p_{right}>p_{left}$. On the other hand, migration influences the transport of ions in a perpendicular direction depending on the ions valence. For $Na$, migration occurs from $bottom$ to $top$ since $\phi_{bottom}>\phi_{top}$.
Numerical Considerations {#sec:numerical}
------------------------
The transport problems were solved numerically based on the PNM approach described on section \[sec:pnm\] using `OpenPNM` [@gostick2016]. The FEM simulations were performed using `COMSOL` [@comsol2018].
![\[fig:mesh\] Computational domain modeling the geometry of Fig. \[fig:network\] with the corresponding grid used for FEM simulations.](mesh){width="0.7\linewidth"}
For FEM simulations, the boundary pores defined on Fig. \[fig:network\] were trimmed at the plan passing through their centers as shown on Fig. \[fig:mesh\]. The boundary conditions are then imposed on the resulting boundary edges. This approach is adopted in order to impose comparable simulation conditions on both the PNM and FEM simulations since boundary conditions are imposed at the pore centers in the PNM simulations. For FEM simulations, the computational domain was meshed, after a mesh sensitivity analysis, using a grid comprised of $97598$ elements for a medium including 341 pores. Triangular and quadrilateral elements were used (see Fig. \[fig:mesh\]).
The FEM simulations were performed using the `Creeping Flow`, `Transport of Diluted Species`, and `Laplace Equation` modules. The system of non-linear equations was solved using Newton’s method and at each of its iterations, the linearized system was solved using the multifrontal massively parallel sparse direct solver MUMPS [@amestoy2006]. For consistency, the same linear solver was used with the PNM simulations.
Simulation Results {#sec:results}
------------------
![\[fig:concentration\] Concentration of $Na$ color map at steady state obtained from PNM simulations based on the power-law NME (Eq. \[eq:pnm\_np\_3\]). Values at the throats are obtained from the interpolation of the neighbor pores concentrations. Simulation conditions: (a) $\langle ^{ad}{Pe}^{Na} \rangle=0.1$, $\langle ^{mig}{Pe} \rangle=0.1$, (b) $\langle ^{ad}{Pe}^{Na} \rangle=0.1$, $\langle ^{mig}{Pe} \rangle=5$, (c) $\langle ^{ad}{Pe}^{Na} \rangle=5$, $\langle ^{mig}{Pe} \rangle=0.1$, and (d) $\langle ^{ad}{Pe}^{Na} \rangle=5$, $\langle ^{mig}{Pe} \rangle=5$.](concentrations){height="0.67\linewidth"}
Figure \[fig:concentration\] shows the $Na$ concentration color map obtained from the solution of the problem defined above (section \[sec:IBVP\]) for some of the considered configurations. These results were obtained based on the PNM approach using the power-law NME (Eq. \[eq:pnm\_np\_3\]). This figure shows that, for the considered problems, when advection and migration forces act with comparable intensities at the network scale, more heterogeneous $Na$ concentration distributions are obtained (Figs. \[fig:concentration\] (a) and (d)). This is due to the fact that boundaries over which these two forces are imposed are at different uniform concentrations. When one of the these two transport mechanisms dominates, a more uniform concentration field is observed (Figs. \[fig:concentration\] (b) and (c)) since uniform concentration values are imposed at the boundaries.
The solutions obtained from the FEM simulations are not shown on Fig. \[fig:concentration\] as they are comparable to the PNM ones with a negligible deviation discussed below. The deviation between PNM and FEM simulations, in terms of concentration of species $n$ at the center of pore $i$ at steady state, is given by $$\begin{aligned}
E^n_i=\frac{\abs{c^n_{i,PNM}-c^n_{i,FEM}}}{c^n_{i,FEM}}, & & \quad n=Na,\,Cl, & & \quad i=1,2,\dots,N_p,\label{eq:error}\end{aligned}$$ where the FEM solution is considered as the reference one. In Eq. \[eq:error\], $c^n_{i,PNM}$ and $c^n_{i,FEM}$ are concentrations of species $n$ at the center of pore $i$ obtained from PNM and FEM simulations, respectively. The analysis of the deviation was carried-out based on the arithmetic mean of $\abs{E^{Na}_i}$ over the entire network and is referred to as $\sigma$.
![\[fig:deviations\] Color map of $\sigma$ versus advective $\langle ^{ad}{Pe}^{Na} \rangle$ and migrative $\langle ^{mig}{Pe} \rangle$ Péclet numbers at steady state. $\sigma$ is the arithmetic mean of the absolute deviation between $Na$ concentrations obtained from PNM and FEM simulations (see Eq. \[eq:error\]). PNM simulations based on the upwind upwind (Eq. \[eq:pnm\_np\_1\]), power-law upwind (Eq. \[eq:pnm\_np\_2\]), and power-law (Eq. \[eq:pnm\_np\_3\]) NMEs. Initial and boundary value problem defined in section \[sec:IBVP\].](deviations){height="1.24\linewidth"}
Values of $\sigma$ obtained using the upwind upwind, power-law upwind, and power-law NMEs are shown on Fig. \[fig:deviations\]. Although the deviation $\sigma$ is always below an acceptable value of $9\%$, local deviations of up to $50\%$ were observed with the two former NMEs for certain configurations. This is consistent with a recent work [@sadeghi2019a] where large deviations between PNM and FEM were observed on dispersion problems in pore networks when the upwind scheme was used in PNM simulations. It was also reported that, for certain advection diffusion problems, the deviation between PNM and FEM simulations increases with the advective Péclet number [@sadeghi2019a]. This is also seen in the results reported on Fig. \[fig:deviations\]. The difference in the dependence of $\sigma$ on advective and migrative Péclet numbers at low values can be attributed to the transport configuration adopted here where the advective, diffusive and migrative driving forces act in different directions in the network.
Analysis of Fig. \[fig:deviations\] also shows that similar behaviors are obtained with the upwind upwind and power-law upwind NMEs although the latter globally presents slightly lower deviations. On the other hand, a significant decrease in $\sigma$ is obtained with the power-law NME. In fact the average deviation is consistently below $5\%$ and marginally exceeds this value when $\langle ^{ad}{Pe}^{Na} \rangle=5$ and $\langle ^{mig}{Pe} \rangle \geq 3.5$. For migration diffusion dominated transport ($\langle ^{ad}{Pe}^{Na} \rangle \leq 0.1$), which is of practical relevance for applications such as battery simulations, a negligible (below $0.4\%$) deviation is observed. The same applies when transport is advection diffusion dominated ($\langle ^{mig}{Pe} \rangle \leq 0.1$), which is of importance for dispersion problems, where $\sigma \leq 1$. It can be concluded from this analysis that the power-law NME should be used when performing PNM simulations to ensure a maximum accuracy.
![\[fig:concentration\_dev\] (a) Concentration of $Na$ color map at steady state obtained from FEM simulations. Color map of the deviation between PNM and FEM simulations $\sigma$ (see Eq. \[eq:error\]) such that PNM simulations are based on the (b) upwind upwind (Eq. \[eq:pnm\_np\_1\]), (c) power-law upwind (Eq. \[eq:pnm\_np\_2\]), and (d) power-law (Eq. \[eq:pnm\_np\_3\]) NMEs. Simulation conditions: $\langle ^{ad}{Pe}^{Na} \rangle=1$, $\langle ^{mig}{Pe} \rangle=1$. Initial and boundary value problem defined in section \[sec:IBVP\].](concentrations_dev){height="0.67\linewidth"}
The source of the deviations between the PNM and FEM simulations resulting from the use of the upwind scheme were discussed in detail in a recent work [@sadeghi2019a]. They were attributed to the fact that in the presence of moderate to important advective effects (*i.e.*, Péclet numbers equal or larger than unity), significant local concentration gradients appear, and the assumption of linear concentration profiles between pores loses accuracy. This behavior also appears in Fig. \[fig:concentration\_dev\]. The considered transport configuration gives rise to a high concentration front on the diagonal of the porous medium from the upper left to the bottom right vertices (see Fig. \[fig:concentration\_dev\] (a)). The high deviation regions coincide with this front for the different NMEs (Figs. \[fig:concentration\_dev\] (b), (c), and (d)).
![\[fig:peclet\] Network scale augmented Péclet number $\langle ^{ad,mig}{Pe}^{Na} \rangle$ versus the advective $\langle ^{ad}{Pe}^{Na} \rangle$ and migrative $\langle ^{mig}{Pe} \rangle$ ones. Péclet numbers obtained from the network’s arithmetic mean of the absolute value of pore-scale Péclet numbers given by Eqs. \[eq:pnm\_pe\_1\], \[eq:pnm\_pe\_2\] and \[eq:pnm\_pe\_3\]. Initial and boundary value problem defined in section \[sec:IBVP\].](peclet){height="0.37\linewidth"}
Finally, the conclusions drawn from the analysis of Fig. \[fig:deviations\], based on $\langle ^{ad}{Pe}^{Na} \rangle$ and $\langle ^{mig}{Pe} \rangle$, can be generalized to be valid when one considers $\langle ^{ad,mig}{Pe}^{Na} \rangle$. In fact, from Fig. \[fig:peclet\], for the considered problems, $\langle ^{ad,mig}{Pe}^{Na} \rangle$ has a quasi-linear dependence upon $\langle ^{ad}{Pe}^{Na} \rangle$ and $\langle ^{mig}{Pe} \rangle$. Hence, the deviation between PNM and FEM increases with $\langle ^{ad,mig}{Pe}^{Na} \rangle$.
Simulation Time {#sec:time}
---------------
![\[fig:times\_and\_performance\] Simulation time using the PNM (`OpenPNM` [@gostick2016]) and FEM (`COMSOL` [@comsol2018]) solvers and their ratio versus the size of the porous medium (*i.e.*, number of pores). Network scale advective and migrative Péclet numbers set to $\langle ^{ad}{Pe}^{Na} \rangle=1$ and $\langle ^{mig}{Pe} \rangle=1$, respectively. Initial and boundary value problem defined in section \[sec:IBVP\]. Simulations run in parallel using two X5650 Intel Xeon CPUs at $2.67\si{GHz}$ with $12$ cores in total.](times_and_performance){height="0.4\linewidth"}
The reduced computational cost of the PNM approach over FEM is staggering. The size of the medium was characterized considering the number of pores included while following the same approach described in section \[sec:IBVP\] to generate the domains. Simulations were run on two X5650 Intel Xeon CPUs at $2.67\si{GHz}$ with $12$ cores in total. The meshing time on the FEM simulations is not included in the comparisons for consistency, although it also requires important computational resources. In fact, meshing the largest domain (includes $10410$ pores), performed in parallel on $12$ cores, took $1219\si{s}$ for a total of $\sim3.04\times{10}^{6}$ grid cells. Whereas generating a cubic network, even with millions of pores is almost instantaneous.
Figure \[fig:times\_and\_performance\] shows the simulation time versus the number of pores, $Np$, for PNM and FEM approaches. For the $Np$ range investigated here, both approaches show a quasi-linear dependence upon $Np$. The simulation time scales as $T_{PNM}(\si{s})=9.08\times{10}^{-5}Np+0.89$ and $T_{FEM}(\si{s})=2.05Np$ with the PNM and FEM solvers considered in the present work, respectively. This means that for the considered range of network sizes, the simulation time increases more than $22.5\times{10}^{3}$ times faster with the FEM solver compared to the PNM one. For the largest computational domain analyzed here, comprising $10410$ pores, solution of the transport problem was performed in just $1.83\si{s}$ using `OpenPNM`. On the other hand, $\sim3.4\si{h}$ were needed for the FEM simulation using `COMSOL`. This result highlights the significant decrease in simulation time which can be achieved adopting the PNM approach described in section \[sec:algorithm\], even for the coupled non-linear multiphysics problem studied here.
The ratio between simulation times using the PNM (`OpenPNM` [@gostick2016]) and FEM (`COMSOL` [@comsol2018]) solvers $T_{FEM}/T_{PNM}$ versus the size of the porous medium is also reported on Fig. \[fig:times\_and\_performance\]. It can be seen that simulation speedup increases with the number of pores reaching a speedup factor of over ${10}^{4}$ for a medium including $\sim{10}^{4}$ pores. The speedup is expected to increase for the same number of pores when considering 3D porous media. In addition to the simulation speedup obtained with the PNM approach compared to the FEM one, the PNM simulations can be run using limited memory resources. In this study, carrying-out the FEM simulation on the largest domain considered (comprising $10410$ pores) required $\sim96\si{GB}$ of memory while only $241.4\si{MB}$ were used on the PNM simulation.
Conclusions {#sec:conclusions}
===========
Ion transport problems in pore networks with random pore sizes and coordination numbers were considered and solved numerically using PNM and FEM solvers. The transport was modeled based on the NP equations for each charged species present in the electrolytic solution in addition to a charge conservation equation which relates the concentration of different species one to each other. In the presence of a fluid flow, the momentum and mass conservation equations, were adopted to describe the fluid flow.
Several time and space discretization schemes were presented to derive the NMEs corresponding to the considered PDEs. The accuracy of each scheme was compared to a reference solution generated by FEM, and best agreement was found when a power-law approach was applied to both the advection diffusion and migration terms. This is consistent with our previous work on advection diffusion [@sadeghi2019a]. These model equations were implemented within the open-source package `OpenPNM` [@gostick2016] based on the Gummel algorithm with relaxation. Comparisons showed a maximum relative deviation, in terms of ions concentration, between PNM and FEM below $\sim5\%$ with the PNM simulations being over ${10}^{4}$ times faster than the FEM ones on a medium including ${10}^{4}$ 2D pores. The speedup is expected to increase for the same number of pores when considering 3D porous media.
The PNM approach allows for simulations with significantly lower computational costs compared to other DNS methods, while retaining reasonable accuracy. This will allow for more effective design and analysis or operation for many electrochemical systems since computation can be performed on large samples while retaining pore-scale resolution. Ultimately, this highly-efficient computational framework could be used for optimization of electrode architectures and cell designs [@fornercuenca2019].
Computer Code Availability {#computer-code-availability .unnumbered}
==========================
The developed solver for transport of charged species in porous media is available on `OpenPNM` [@gostick2016] public repository .
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by CANARIE Canada.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We report measurements on ropes of Single Walled Carbon Nanotubes (SWNT) in low-resistance contact to non-superconducting (normal) metallic pads, at low voltage and at temperatures down to 70 mK. In one sample, we find a two order of magnitude resistance drop below 0.55 K, which is destroyed by a magnetic field of the order of 1T, or by a d.c. current greater than 2.5 $\mu A$. These features strongly suggest the existence of superconductivity in ropes of SWNT.\
address: '$^1$Laboratoire de Physique des Solides, Associé au CNRS, Bât. 510, Université Paris–Sud, 91405, Orsay, France.$^2$ Present address: Starlab, Brussels, Belgium $^3$Institute of Microelectronics Technology and High Purity Materials, Russian Academy of Sciences, Chernogolovka 142432 Moscow Region, Russia.$^4$ Groupe de Dynamique des Phases Condensées, Université Montpellier II 34095 Montpellier France.\'
author:
- |
M. Kociak$^1$, A.Yu. Kasumov$^{1,2,3}$, S. Guéron$^1$, B. Reulet$^1$, I.I. Khodos$^3$,\
Yu.B. Gorbatov$^3$, V.T. Volkov$^3$, L. Vaccarini $^4$ and H. Bouchiat$^1$
title: 'Superconductivity in Ropes of Single-Walled Carbon Nanotubes'
---
[2]{}
Metallic carbon nanotubes are known to be model systems for the study of 1D electronic transport [@Dresselhaus; @Hamada; @Wildoer]. Electronic correlations are expected to lead to a breakdown of the Fermi liquid state. Nanotubes should then be described by Luttinger Liquids (LL) theories [@Egger; @Kane], with collective low energy excitations and no long range order. Proof of the validity of LL description in ropes was given by the measurement of a resistance diverging as a power law with temperature down to 10 K [@Bockrath]. However, this measurement was done on nanotubes separated from measuring leads by tunnel junctions. Because of Coulomb blockade [@Grabert], the low temperature and voltage regime were not explored. In contrast, we have developed a technique in which measuring pads are connected through low contact resistance to suspended nanotubes [@Kasumov2]. We previously showed that when the contact pads are superconducting, a large supercurrent can flow through nanotubes [@Kasumov]. In this letter, we report experimental evidence of intrinsinc superconductivity below 0.55 K in ropes of carbon nanotubes connected to normal contacts.
The samples are ropes of SWNT suspended between normal metal contacts (Pt/Au bilayers). The SWNT are prepared by an electrical arc method with a mixture of nickel and yttrium as a catalyst [@Journet; @Vaccarini]. SWNT with diameters of the order of 1.4 nm are obtained. They are purified by the cross-flow filtration method [@Vaccarini]. The tubes are usually assembled in ropes of a few hundred parallel tubes. Isolation of an individual rope and connection to measuring pads are performed according to the procedure we previously used [@Kasumov2], where ropes are soldered to melted contacts. The contact resistance is low and the tubes can be structurally characterized with a transmission electron microscope (TEM). For the three samples presented here, the contacts were trilayers of sputtered $Al_{2}O_{3}/Pt/Au $ of respective thicknesses 5, 3 and 200 nm. This procedure insures that the tubes do not contain any chemical dopants such as alkalis or halogens. The contacts showed no sign of superconductivity down to 50 mK. The samples were measured in a dilution
![ \[figure1\] Resistance as a function of temperature for the three samples. The length L, number of tubes N and room temperature resistance R of each sample are given in the corresponding panel. a: Sample Pt3. b: Resistance of Pt1 in applied magnetic fields of $\mu_{0}H$= 0, 0.02, 0.04, 0.06, 0.08, 0.1, 0.2, 0.4, 0.6, 0.8 and 1 T from bottom to top. Inset is a zoom of the low temperature region. c: Resistance of Pt2 at $\mu_{0}H$=0, 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1, 1.25, 1.5, 1.75, 2, 2.5 T from bottom to top. Inset: TEM micrograph of sample Pt2, from which we deduce $L_{2}$ and $N_{2}$. $N_{2}$ is estimated from the measured diameter $D_{2}$, through $N_{2} =(D_{2} /(d+e))^2$, where d is the diameter of a single tube (d=1.4 nm), and e is the typical distance between tubes in a rope (e=0.2 nm). The dark spot is a Ni/Y catalyst particle.](fig1.eps){width="8cm"}
refrigerator, at temperatures ranging from 1 K to 0.05 K, through filtered lines [@Reulet2]. Magnetic fields up to 5 T could be applied perpendicularly to the contacts and the tubes. The resistance was measured by applying a small (1 nA to 10 nA, 30 Hz) a.c. current though the sample and measuring the a.c. voltage using lock-in detection.
We select samples with a room temperature (RT) resistance less than 10 k$\Omega$. As is generally observed, we find that the resistance increases as the temperature is lowered between 300 K and 1K . Things change however below 1K, as shown in Fig. 1 for the three samples Pt1, Pt2, and Pt3, measured in magnetic fields ranging from 0 to 2.5 T. At zero field, the zero-bias resistance of Pt3 increases as T is reduced, whereas the resistances of Pt1 and Pt2 decrease drastically below $T_{1}^*= 140$ mK for Pt1 and $T_{2}^*= 550$ mK for Pt2. The resistance of Pt1 is reduced by 30% at 70 mK. That of Pt2 decreases by more than two orders of magnitude, and saturates below 100 mK at a value $R_{r}=74$ $\Omega$. We define a transition temperature $T_{C_{2}}$ by the inflexion point of R(T). $T_{C_{2}}$ is 370 mK at zero field, decreases at higher magnetic fields, and extrapolates to zero at 1.35 T (Fig 4c). At fields above 1.25 T, the resistance increases with decreasing temperature, similarly to Pt3, and becomes independent of magnetic field. The resistance of Pt1 follows qualitatively the same trend, but the full transition did not occur down to 70 mK. Figures 2 and 3 show that in the temperature and field range where the zero-bias resistance drops, the differential resistance is strongly bias-dependent, with lower resistance at low bias. These data suggest that the rope Pt2 (and, to a lesser extent, Pt1) is intrinsically superconducting. Although the experimental data of Pt2 seem similar to those of SWNT connected to superconducting contacts [@Kasumov], there are major differences. In particular the $V(I), dV/dI(I)$ do not show any supercurrent because of the existence of a finite residual resistance.
We now analyse the superconductivity in these systems, taking into account several features: the large normal contacts, the coupling between tubes within the rope, the 1D character of each tube, and their finite length compared to relevant mesoscopic and superconducting scales. The resistance of any superconducting wire measured through normal contacts (an NSN junction) cannot be zero because the number of channels in the wire is much smaller than in the contacts[@Landauer]: a metallic SWNT, with 2 conducting channels, has a contact resistance of half the resistance quantum, $R_{Q}/2$ (where $R_{Q}=h/(2e^2)$=12.9 k$\Omega$), even if it is superconducting. A rope of $N_{m}$ parallel metallic SWNT will have a minimum resistance of $R_{Q}/(2N)$. Therefore we use the residual resistance $R_{r}=74$ $\Omega$ of Pt2 to deduce that Pt2 has at least $N_{m}=R_{Q}/2R_{r}\approx 90$ metallic tubes. This is approximately one quarter of the number of tubes in the rope, measured by TEM (Fig 1c). Similarly, $R_{Q}$ is also the maximum resistance of any phase coherent metallic wire[@Thouless]. As a consequence, the high value (9.2 k$\Omega$) of the resistance at 1K (which corresponds to an average resistance per metallic tube of $9.2$ k$\Omega*N_{m}=830$ k$\Omega=130$ $R_{Q}$) cannot be understood if the nanotubes are independent, unless considering a very short (unphysical) phase coherence length $L_{\varphi}(1K)=L/130=$ 8 nm. On the other hand if the electrons are free to move from tube to tube[@Maarouf], the resistance is simply explained by the presence of disorder. The mean free path is deduced from the RT resistance $R_{2}=4.1$ k$\Omega$ through [@Imry] $l_{e2}\approx \frac{L}{R_{2}}\frac{R_{Q}}{N_{m}}\approx 18$ nm. We conclude that Pt2 is a diffusive conductor with a few hundred conduction channels.
With such a small number of channels, we expect the superconductivity to differ from 3D superconductivity. In particular, we expect to observe a broad resistance drop starting at the 3D transition temperature [@Giordano] $T^* $ and going eventually to $R_{r}$ at zero temperature. This is what is observed in Pt2 (see figure 1.c). We estimate the gap through the BCS relation $\Delta =1.76$ $k_{B}T^*$ : $\Delta \approx 85$ $\mu eV$ for Pt2. We can then deduce the superconducting coherence length along the tube in the diffusive limit $\xi_{2}=\sqrt{\hbar v_{f}l_{e}/\Delta}\approx 0.3$ $\mu$m where $v_{f}$ is the longitudinal Fermi velocity $8\times10^5$ m/s [@Bourbonnais]. Consistent with 1D superconductivity, $\xi_{2}$ is ten times larger than the diameter of the rope.
![ \[figure2\] Differential resistance as a function of current for samples Pt1 and Pt2, in different applied fields. a: Sample Pt1. Fields are 0, 0.02, 0.04, 0.06, 0.08, 0.1, 0.2 and 1 T. b: Sample Pt2. Fields are 0, 0.2, 0.4, 0.6, 0.8, 1, 1.25, 1.5, 1.75, 2, and 2.5 T.](fig2.eps){width="7cm"}
![ \[figure3\] Left panel: Differential resistance of Pt2 vs. current for a larger current amplitude than in Figure 2, at different temperatures. Curves are offset vertically for clarity. Right panel: V(I) and $\frac{dV}{dI}(I)$ curves showing the hysteretic behavior in V(I) at each peak in the $\frac{dV}{dI}(I)$ curve.](fig3.eps){width="8cm"}
Finally, reminiscent of measurements of narrow superconducting metal wires [@Giordano], we find jumps in the differential resistance as the current is increased (Figures 2 and 3). For Pt2 the differential resistance at low currents remains equal to $R_{r}$ up to 50 nA, where it strongly rises but does not recover its normal state value until 2.5 $\mu$A (fig 3a). The jump in resistance at the first step corresponds approximately to the normal state resistance of a length $\xi_{2}$ of Pt2. Each peak corresponds to a hysteretic feature in the V-I curve (fig 3b). Above 1 T the differential resistance is peaked at zero current. This is also the case for Pt3 (data not shown). The variations of the differential resistance of Pt1 are similar to those of Pt2 close to its transition temperature. These jumps are identified as phase slips [@Giordano; @Meyer; @Tinkham], which are the occurrence of normal regions located around defects in the sample. Such phase slips can be thermally activated (TAPS), leading to an exponential decrease of the resistance instead of a sharp transition, in qualitative agreement with our experimental observation (fig 4a). At sufficiently low temperature, TAPS are replaced by quantum phase slips (QPS), which, when tunneling through the sample, contribute an additional resistance to the zero temperature resistance. Moreover, QPS are predicted to supress the transition when the normal state resistance of the sample on the phase coherence scale is larger than $R_{Q}/2$ [@Zaikin](as confirmed by recent experiments [@Bezryadin]). Our data on Pt2 show no evidence of such an effect, even though the normal state resistance, measured above T\*, is 40% larger than $R_{Q}/2 $. The current above which the jumps disappear, 2.5 $\mu A$, is close to the critical current $I_{C}=\Delta /R_{r}e \approx 1$ $ \mu$A of a superconducting wire without disorder and with the same number of conducting channels [@Tinkham]. This large value of critical current would also be the maximum supercurrent in a structure with this same wire placed between superconducting contacts (with gap $\Delta_{S}$), and is much larger than the Ambegaokar-Baratoff
![ \[figure4\] a Resistance of Pt2 plotted on a log scale as a function of the inverse temperature at H=0. We have subtracted the low temperature residual resistance (contact resistance). The slope yields an approximate activation energy of 0.8 K. b Magnetoresistance of Pt2 at 50 mK. We define the critical field as the inflection point of R(H): $\mu_{0}H_{C}$(T= 50 mK)=1.1 T. c Transition line of Pt2 defined in the H,T plane by the inflection point of R(T) or equivalently by the inflection point of R(H). d Field dependence of the critical current of Pt2 defined as the current at which the first resistance jump occurs in the dV/dI curves of Fig. 2. $I_{C}(H)$ extrapolates to a critical field of 1.2 T, in agreement with the linear extrapolation 1.3 T of $T_{C}(H)$. ](fig4.eps){width="7cm"}
prediction $R_{N} I_{C}= \Delta_{S}/e$. This might explain the anomalously large supercurrent measured in a previous experiment [@Kasumov], where nanotubes were connected to superconducting contacts.
We now discuss the effect of the magnetic field. The field at which the resistance saturates to its normal value and at which the critical current vanishes, 1.25 T, coincides with the field obtained by extrapolation of $T_{C}(H)$ to zero temperature (fig. 4b). It is difficult to say what causes the disappearance of superconductivity. The value of Hc(0) should be compared to the depairing field in a confined geometry [@Meservey], and corresponds to a flux quantum $\Phi_{0}$ through a length $\xi$ of an individual SWNT of diameter d, $\mu_{0}H_{C}= \Phi_{0}/(2\sqrt{\pi}d\xi)=1.35$ T. But $H_{C}(0) $is also close to the field $\mu_{0}H_{p}=\Delta /\mu_{B}=1.43$ T at which a paramagnetic state becomes more favorable than the superconducting state [@Clogston; @Chandrasekhar]. Note that this value is of the same order as the critical field that was measured on SWNT connected between superconducting contacts, i.e. much higher than the critical field of the contacts.
We now estimate the superconducting coherence length of the two other samples, to explain the extent or absence of observed transition. Indeed, investigation of the proximity effect at high-transparency NS interfaces has shown that superconductivity resists the presence of normal contacts only if the length of the superconductor is much greater than $\xi$ [@Belzig], i.e. if the wire contains a superconducting reservoir. This condition is nearly fulfilled in Pt2 ($\xi_{2} \approx L_{2}/3$). Using the high temperature resistance values of Pt1 and Pt3, and assuming a gap $\Delta$ and number of metallic tubes equal to those of Pt2 we find $\xi_{1} \approx L_{1}/2$ and $\xi_{3} \approx 2L_{3}$. These values explain qualitatively a reduced transition temperature for Pt1 and the absence of a transition for Pt3. Moreover we can argue that the superconducting transitions we see are not due to a hidden proximity effect : if the $Al_{2}O_{3}/Pt/Au$ contacts were made superconducting by the laser pulse, the shortest nanotube (Pt3) would become superconducting at temperatures higher than the longer tubes (Pt1 and Pt2). The main result, i.e. no visible transition with a short rope, and a visible transition in a long rope, are confirmed by measurements on two other samples which are not presented here.
We now consider the possible mechanism of superconductivity. It has been suggested that coupling with low energy phonons can turn repulsive interactions in a Luttinger liquid into attractive ones and drive the system towards a superconducting phase [@Loss]. Such low energy phonons have been experimentally observed in the form of mechanical bending modes of a suspended SWNT rope [@Reulet]. It was also shown that superconducting fluctuations can dominate at low temperature in ladders such as tubes [@Egger]. In this case the system must be away from half-filling, a condition probably fulfilled in our experimental situation, due to hole doping from the contacts [@Venema; @Odinstov]. Finally, the superconductivity reported here recalls that of graphite intercalated with alkalis (Cs,K), which also occurs between 0.2 and 0.5 K [@Hannay]. Much higher temperatures were observed in alkali doped fullerenes [@Gunnarsson] because of the coupling to higher energy phonons. This suggests the possibility of increasing the transition temperature by chemically doping the nanotubes.
We have shown that ropes of carbon nanotubes are intrinsically superconducting. This is the first observation of superconductivity in a system with such a small number of conduction channels. The understanding of this superconductivity calls for future experimental and theoretical work and motivates in particular a search of superconducting fluctuations in a single SWNT.
A.K. thanks the Russian foundation for basic research and solid state nanostructures for financial support, and thanks CNRS for a visitor’s position. We thank M. Devoret, N. Dupuis, T. Martin, D. Maslov, C. Pasquier for stimulating discussions.
[99]{} M.S. Dresselhaus, G. Dresselhaus, and P.C. Eklund, [*Science of Fullerenes and Carbon nanotubes*]{} (Academic, San Diego, 1996). N. Hamada, S. I. Sawada and A. Oshiyama, Phys. Rev. Lett. [**68**]{}, 1579 (1992). J. W. G. Wildoer [*et al.*]{}, Nature [**391**]{}, 59 (1998). R. Egger, A. Gogolin, Phys. Rev. Lett. [**79**]{}, 5082 (1997). R. Egger, Phys. Rev. Lett. [**83**]{}, 5547 (1999). C. Kane, L. Balents, M. P. Fisher, Phys. Rev. Lett [**79**]{}, 5086 (1997). M. Bockrath [*et al.*]{}, Nature [**397**]{}, 598 (1999). H. Grabert and M. H. Devoret (eds), Single Charge Tunneling (Plenum, New-York, 1992); J. T. Tans [*et al.*]{}, Nature [**386**]{}, 474 (1997). A.Yu. Kasumov, I.I. Khodos, P.M. Ajayan, C. Colliex, Europhys. Lett. [**34**]{}, 429 (1996); A.Yu. Kasumov [*et al.*]{}, Europhys. Lett. [**43**]{}, 89 (1998). A.Yu. Kasumov [*et al.*]{}, Science [**284**]{}, 1508 (1999). C. Journet, [*et al.*]{}, Nature [**388**]{}, 756 (1997). L. Vaccarini, [*et al.*]{}, C.R.Acad.Sci.[**327**]{}, 925 (1999). B. Reulet, H. Bouchiat, and D. Mailly, Europhys. Lett. [**31**]{}, 305 (1995). R. Landauer, IBM Res. Dev. 1, 223 (1957). D. J. Thouless, Phys. Rev. Lett. [**39**]{}, 1967 (1977). Electronic transfer between tubes in a disordered rope is indeed suggested in: A. A. Maarouf, C. L. Kane, and E. J. Mele, Phys. Rev. B[**61**]{}, 11156 (2000); H.R. Shea, R. Martel and Ph. Avouris, Phys. Rev. Lett. [**84**]{}, 4441 (2000). Y. Imry, Europhys. Lett. [**1**]{}, 249 (1986). N. Giordano, Phys. Rev. B [**50**]{}, 160 (1991). Note however that the transverse velocity is certainly much smaller, yielding a transverse coherence length $\xi_{\perp}$ smaller than the longitudinal one, but the 1D character of the transition depicted below indicates that $\xi_{\perp}$ is certainly larger than the diameter of the rope. Such anisotropic transport is observed in organic conductors where a very small coupling between 1D chains restores a 3D behavior of the electrons, and allows superconductivity in a macroscopic sample. (see e.g. C. Bourbonnais, D. Jérôme, in Advance in Synthetic Metals, eds P.Bernier, S. Lefrant and G. Bidan) 206 (1998). J. Meyer, G. V. Minnigerode, Physics Letters, [**38A**]{}, 7, 529 (1972). Tinkham, M., Introduction to superconductivity, McGraw-Hill, 2d Ed. (Singapore, 1996). A.D. Zaikin [*et al.*]{}, Phys. Rev. Lett. [**78**]{}, 1552 (1997). A. Bezryadin, C. N. Lau and M. Tinkham Nature [**404**]{}, 971 (2000).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a code-based public-key cryptosystem, in which we use Reed-Solomon codes over an extension field as secret codes and disguise it by considering its shortened expanded code over the base field. Considering shortened expanded codes provides a safeguard against distinguisher attacks based on the Schur product. Moreover, without using a cyclic or a quasi-cyclic structure we obtain a key size reduction of nearly $45 \%$ compared to the classic [McE]{}liece cryptosystem proposed by Bernstein *et al.*'
address:
- |
Institute of Mathematics\
University of Zurich\
Winterthurerstrasse 190\
8057 Zurich, Switzerland\
- |
Institute of Mathematics\
University of Zurich\
Winterthurerstrasse 190\
8057 Zurich, Switzerland\
- |
Institute of Mathematics\
University of Zurich\
Winterthurerstrasse 190\
8057 Zurich, Switzerland\
author:
- Karan Khathuria
- Joachim Rosenthal
- Violetta Weger
title: 'Encryption Scheme Based on Expanded Reed-Solomon Codes'
---
Introduction {#sec:introduction}
============
In 1978 McEliece [@mc78] presented the first code-based public key cryptosystem. It belongs to the family of very few public-key cryptosystems which are unbroken since decades. The hard problem the McEliece system relies on, is the difficulty of decoding a random (-like) linear code having no visible structure. McEliece proposed to use binary Goppa codes for the encryption scheme. Due to the low error-correcting capacity of Goppa codes, the cryptosystem results in large public key sizes. Several alternative families of codes have been proposed with the aim of reducing the key sizes. Some of the famous families of codes considered are: generalized Reed-Solomon codes [@ba11; @ba15z; @ba19; @be05a; @bo16; @kh18; @ni86a], non-binary Goppa codes [@be10], algebraic geometric codes [@ja96], LDPC and MDPC codes [@ba08p; @mi13], Reed-Muller codes [@si94b] and convolutional codes [@lo12]. Most of them were unsuccessful in hiding the structure of the private code [@co14; @co17; @co17w; @co15a; @la13; @mi07; @ot10; @si92; @wi10].
The motivation to quest for better code-based cryptosystems is mainly due to the advent of quantum computers. In 1994 Peter Shor [@sh94] developed a polynomial time quantum algorithm for factoring integers and solving discrete logarithm problems. This means that most of the currently popular cryptosystems, such as RSA and ECC, will be broken in an era of quantum computers. In the ongoing process of the standardization of quantum-resistant public-key cryptographic algorithms by the National Institute of Standards and Technology (NIST), code-based cryptosystems are one of the most promising candidates. At the time of this writing there are seven code-based cryptosystems included in NIST’s standardization process: BIKE [@ar17] based on quasi-cyclic MDPC codes, classic McEliece [@be08] based on binary Goppa codes, ROLLO [@ca18] based on quasi-cyclic LRPC codes, RQC [@ag18] based on rank metric quasi-cyclic codes, HQC [@ag18] based on Hamming metric quasi-cyclic codes, LEDAcrypt [@ba18b] based on quasi-cyclic LDPC codes and NTS-KEM [@al18] based on binary Goppa codes.
In this paper we present a new variant of the McEliece scheme using expanded Reed-Solomon codes. A linear $[n,k]$ code defined over an extension field ${\mathbb{F}}_{q^m}$ can be expanded, over the base field ${\mathbb{F}}_q$, to a $[mn,mk]$ linear code by expanding each codeword with respect to a fixed ${\mathbb{F}}_q$-linear isomorphism from ${\mathbb{F}}_{q^m}$ to ${\mathbb{F}}_q^m$. In the proposed cryptosystem we hide the structure of an expanded GRS code by puncturing and permuting the columns of its parity check matrix and multiplying by an invertible block diagonal matrix. In order to decode a large number of non-codewords, we use a burst of errors during the encryption step, i.e. we consider error vectors having support in sub-vectors of size $\lambda$. This error pattern comes with a disadvantage: it can be used to speed up the information set decoding (ISD) algorithms. However, for a small degree of extension $m$, the key sizes turn out to be remarkably competitive.
The paper is organized as follows. In Section \[sec:background\], we give the preliminaries regarding the expanded codes. In Section \[sec:cryptosystem\], we describe the proposed cryptosystem which is based on the shortening of an expanded generalized Reed-Solomon code. In Section \[sec:Security\], we provide security arguments for the proposed cryptosystem against the known structural and non-structural attacks. In Section \[sec:KeySize\], we provide parameters of the proposed cryptosystem that achieve a security level of 256-bits against the ISD algorithm.
Background {#sec:background}
==========
Expanded Codes
--------------
Let $q$ be a prime power and let $m$ be an integer. Let $\gamma$ be a primitive element of the field ${\mathbb{F}}_{q^m}$, i.e. ${\mathbb{F}}_{q^m} \cong {\mathbb{F}}_q(\gamma) $. The field ${\mathbb{F}}_{q^m}$ can also be seen as an ${\mathbb{F}}_q$- vector space of dimension $m$ via the following ${\mathbb{F}}_q$-linear isomorphism $$\begin{aligned}
\phi: {\mathbb{F}}_{q^m} & \longrightarrow {\mathbb{F}}_{q}^{m} ,\\
a_0 + a_1 \gamma + \cdots + a_{m-1} \gamma^{m-1} & \longmapsto (a_0,a_1, \ldots, a_{m-1}).\end{aligned}$$ We extend this isomorphism for vectors over ${\mathbb{F}}_{q^m}$ in the following way: $$\begin{aligned}
\phi_n: {\mathbb{F}}_{q^m}^n & \longrightarrow {\mathbb{F}}_{q}^{mn} , \\
(\alpha_0,\alpha_1,\ldots, \alpha_{n-1}) & \longmapsto \left (\phi(\alpha_0), \phi(\alpha_1),\ldots, \phi(\alpha_{n-1}) \right ).\end{aligned}$$ This is clearly an ${\mathbb{F}}_q$-linear isomorphism. Hence this gives us a way to obtain a linear code over ${\mathbb{F}}_q$ from a linear code over ${\mathbb{F}}_{q^m}$.
Let $n,k$ be positive integers with $k \leq n$, let $q$ be a prime power and $m$ be an integer. Let ${\mathcal{C}}$ be a linear code of length $n$ and dimension $k$ over ${\mathbb{F}}_{q^m}$. The expanded code of ${\mathcal{C}}$ with respect to a primitive element $\gamma \in {\mathbb{F}}_{q^m}$ is a linear code over the base field ${\mathbb{F}}_q$ defined as $$\widehat{{\mathcal{C}}} := \lbrace \phi_n(c) : c \in {\mathcal{C}}\rbrace,$$ where $\phi_n$ is the ${\mathbb{F}}_q$-linear isomorphism defined by $\gamma$ as above.
It is easy to see that the expanded code $\widehat{{\mathcal{C}}}$ is a linear code of length $mn$ and dimension $mk$, because $\phi_n$ is an ${\mathbb{F}}_q$-linear isomorphism and\
$|\widehat{{\mathcal{C}}}| = |{\mathcal{C}}| = (q^m)^k = q^{mk}$.
Given a code ${\mathcal{C}}$ with its generator matrix and parity check matrix, the following lemma gives a way to construct a generator matrix and a parity check matrix of the expanded code $\widehat{{\mathcal{C}}}$.
Let ${\mathcal{C}}$ be a linear code in ${\mathbb{F}}_{q^m}^n$.
1. Let ${\mathcal{C}}$ have a generator matrix $G = [g_1, g_2,\ldots,g_k]^\intercal$, where $g_1,g_2,\ldots,g_k$ are vectors in ${\mathbb{F}}_{q^m}^n$. Then the expanded code of ${\mathcal{C}}$ over ${\mathbb{F}}_q$ with respect to a primitive element $\gamma \in {\mathbb{F}}_{q^m}$ has the expanded generator matrix $$\begin{aligned}
\widehat{G} := [\phi_n(g_1),\phi_n(\gamma g_1),\ldots, \phi_n(\gamma^{m-1}g_1),& \phi_n(g_2), \phi_n(\gamma g_2),\ldots, \phi_n(\gamma^{m-1}g_2),\ldots,\\ & \phi_n(g_k),\phi_n(\gamma g_k)\ldots, \phi_n(\gamma^{m-1}g_k)]^\intercal .\end{aligned}$$
2. Let ${\mathcal{C}}$ have a parity check matrix $H = [h_1^\intercal,h_2^\intercal,\ldots,h_n^\intercal]$, where $h_1,h_2,\ldots,h_n$ are vectors in ${\mathbb{F}}_{q^m}^{n-k}$. Then the expanded code of ${\mathcal{C}}$ over ${\mathbb{F}}_q$ with respect to a primitive element $\gamma\in {\mathbb{F}}_{q^m}$ has the expanded parity check matrix $$\begin{aligned}
\widehat{H} := [ & \phi_{n-k}(h_1)^\intercal,\phi_{n-k}(\gamma h_1)^\intercal,\ldots, \phi_{n-k}(\gamma^{m-1}h_1)^\intercal, \phi_{n-k}(h_2)^\intercal,\phi_{n-k}(\gamma h_2)^\intercal,\\ & \ldots, \phi_{n-k}(\gamma^{m-1}h_2)^\intercal, \ldots, \phi_{n-k}(h_n)^\intercal,\phi_{n-k}(\gamma h_n)^\intercal\ldots, \phi_{n-k}(\gamma^{m-1}h_n)^\intercal]. \end{aligned}$$
\[lemma:expand\]
See [@yi11 Theorem 1].
Let ${\mathcal{C}}$ be a linear code in ${\mathbb{F}}_{q^m}^n$ having a generator matrix $G = [g_1, g_2,\ldots,g_k]^\intercal$ and a parity check matrix $H = [h_1^\intercal,h_2^\intercal,\ldots,h_n^\intercal]$. Let $\widehat{G}$ and $\widehat{H}$ be the expanded generator matrix and expanded parity check matrix of $\widehat{{\mathcal{C}}}$, respectively. Then
1. $\phi_n(xG) = \phi_k(x)\widehat{G}$ for all $x \in {\mathbb{F}}_{q^m}^k$,
2. $\phi_{n-k}(Hy^\intercal) = \widehat{H} (\phi_n(y))^\intercal$ for all $y \in {\mathbb{F}}_{q^m}^n$.
\[prop:phipsi\]
Let $x = (x_1,x_2,\ldots,x_k) \in {\mathbb{F}}_{q^m}^k$ and let $x_i = \sum_{j=0}^{m-1}x_{ij}\gamma^j$ for all\
$i \in \{1,2,\ldots,k\}$. Then $$\begin{aligned}
\phi_k(x) \widehat{G} &= \sum_{i=1}^{k} \sum_{j=0}^{m-1} x_{ij} \phi_n(\gamma^jg_i) \\
& = \sum_{i=1}^{k} \phi_n \left( \sum_{j=0}^{m-1} x_{ij} \gamma^j g_i \right) \\
& = \sum_{i=1}^{k} \phi_n(x_i g_i) \\
& = \phi_n \left( \sum_{i=1}^{k} x_i g_i \right) \\
& = \phi_n(xG).\end{aligned}$$ Similarly, $\phi_{n-k}(Hy^\intercal) = \widehat{H} (\phi_n(y))^\intercal$ for all $y \in {\mathbb{F}}_{q^m}^n$.
$\widehat{{\mathcal{C}}}$ can also be determined by the commutativity of the following diagram (as ${\mathbb{F}}_q$-linear maps): $$\begin{tikzcd}
0 \arrow{r}{} & {\mathbb{F}}_{q^m}^k \arrow{r}{G} \arrow[swap]{d}{\phi_k} & {\mathbb{F}}_{q^m}^n \arrow{d}{\phi_n} \arrow{r}{H^\intercal} & \arrow{d}{\phi_{n-k}} {\mathbb{F}}_{q^m}^{n-k} \arrow{r}{} & 0 \\0 \arrow{r}{} & {\mathbb{F}}_q^{mk} \arrow{r}{\widehat{G}}& {\mathbb{F}}_q^{mn} \arrow{r}{\widehat{H}^\intercal} & {\mathbb{F}}_q^{m(n-k)} \arrow{r} & 0
\end{tikzcd}$$
The Cryptosystem {#sec:cryptosystem}
================
In this section we will present the proposed cryptosystem in the Niederreiter version. We consider an expanded GRS code whose parity check matrix can be viewed as $n$ blocks, where each block is of size $m$. In order to destroy the algebraic structure of the code, we choose $2 \leq \lambda \leq m-1$ and shorten it on randomly chosen $m-\lambda$ columns in each block. We then hide the shortened code by multiplying it with an invertible matrix, which preserves the weight of a vector over the extension field ${\mathbb{F}}_{q^m}$.
#### **Key generation:**
Let $q$ be a prime power, $2 \leq \lambda < m$ be positive integers and $k< n \leq q^m$ be positive integers, satisfying $R:= k/n > (1-\lambda/m)$. Consider a GRS code ${\mathcal{C}}=\text{GRS}_{n,k}(\alpha,\beta)$ of dimension $k$ and length $n$ over the finite field ${\mathbb{F}}_{q^m}$ and choose a parity check matrix $H$ of ${\mathcal{C}}$. Let $t$ be the error correction capacity of ${\mathcal{C}}$.
Let $\widehat{H}$ be the expanded parity check matrix of the expanded code $\widehat{{\mathcal{C}}}$ of ${\mathcal{C}}$ with respect to a primitive element $\gamma \in {\mathbb{F}}_{q^m}$. $\widehat{H}$ is an $m(n-k) \times mn$ matrix over ${\mathbb{F}}_{q}$.
#### Shortening $\widehat{{\mathcal{C}}}$
- For each $1\leq i \leq n$, let $S_i$ be a randomly chosen subset of\
$\{ (i-1)m+1,(i-1)m+2,\ldots,im \}$ of size $m-\lambda$ and define $S= \bigcup\limits_{i=1}^n S_i$.
- We puncture $\widehat{H}$ on columns indexed by $S$. Let ${\widehat{H}_S}$ be the resulting\
$m(n-k) \times \lambda n$ parity check matrix and let $\widehat{{\mathcal{C}}}_S$ be the shortened code.
#### Hiding $\widehat{{\mathcal{C}}}_S$
- Choose $n$ random $ \lambda \times \lambda$ invertible matrices $T_1,T_2,\ldots,T_{n}$ over ${\mathbb{F}}_{q}$. Define $T$ to be the block diagonal matrix having $T_1,T_2,\ldots, T_{n}$ as diagonal blocks.
- Now choose a random permutation $\sigma$ of length $n$ and define $P_\sigma$ to be the block permutation matrix of size $ \lambda n \times \lambda n$. It can also be seen as Kronecker product of the $n \times n$ permutation matrix corresponding to $\sigma$ and the identity matrix of size $ \lambda$.
- Define $Q := TP_\sigma$ and compute $H'= {\widehat{H}_S}Q$.
The private key is then $(H,Q,\gamma)$ and the public key is $(H', t, \lambda)$.
#### **Encryption:**
Let $y \in {\mathbb{F}}_{q}^{ \lambda n}$ be a message having support in $t$ sub-vectors each of length $ \lambda$, in particular $$\begin{aligned}
\text{support}(y) \subseteq \left \lbrace \lambda \right.& (i_1-1)+1, \lambda (i_1-1)+2, \ldots, \lambda (i_1), \lambda (i_2-1)+1, \lambda (i_2-1)+2, \\ & \ldots, \lambda (i_2),\ldots,\left. \lambda (i_t-1)+1,\lambda (i_t-1)+2,\ldots, \lambda (i_t) \right \rbrace, \end{aligned}$$ for some distinct $i_1,i_2,\ldots,i_t \in \lbrace1,2,\ldots,n \rbrace$. Then compute the cipher text $$c = H' y^\intercal.$$
#### **Decryption:**
For the decryption we apply $\phi_{n-k}^{-1}$ on $c$, i.e. $$\begin{aligned}
\phi_{n-k}^{-1}(c) & = & \phi_{n-k}^{-1}\left({\widehat{H}_S}Qy^\intercal \right). \end{aligned}$$ Observe that ${\widehat{H}_S}Qy^\intercal = \widehat{H} \bar{y}^\intercal$, where $\bar{y}$ is the embedding of $yQ^\intercal $ to ${\mathbb{F}}_{q^m}$, by introducing zeros on the positions indexed by $S$. From Proposition \[prop:phipsi\] we get $$\begin{aligned}
\phi_{n-k}^{-1}\left(\widehat{H} \bar{y}^\intercal \right) & = & H \left(\phi_n^{-1}(\bar{y})\right)^\intercal.\end{aligned}$$
Due to the block structure of the matrix $Q$, the vector of $Qy^\intercal$ has support in $t$ sub-vectors each of length $\lambda$, thus $ \bar{y}$ has support in $t$ sub-vectors each of length $m$. Henceforth $\text{wt}(\phi_n^{-1}(\bar{y})) \leq t $, and we can decode $\phi_{n-k}^{-1}(c)$ to get $\phi_n^{-1}(\bar{y})$. By applying $\phi_n$ we get $\bar{y}$ and by projecting on positions not indexed by $S$, we get $Qy^\intercal$ and therefore after multiplying by $Q^{-1}$, we recover the message $y$.
Choice of parameters {#choice-of-parameters .unnumbered}
--------------------
For low key sizes it is desirable to use a small degree of extension $m$ and small $\lambda$.
In the case of quadratic extension and in the case of $\lambda=1$, puncturing all but one column from each block results in an alternant code (subfield subcode of a GRS code). Alternant codes are known to be vulnerable to square code attacks [@co17w; @fa13]. Hence, we do not propose to use quadratic extensions or $\lambda=1$.
We therefore propose to use $m=3$ and $m=4$ with $\lambda= 2$.
Security {#sec:Security}
========
In this section we discuss the security of the proposed cryptosystem. We focus on the three main attacks on cryptosystems based on GRS codes. Two of them are structural (or key recovery) attacks, namely the Sidelnikov-Shestakov attack and the distinguisher attack based on the Schur product of the public code. The third one is the best known non-structural attack called information set decoding (ISD).
Sidelnikov and Shestakov attack
--------------------------------
The first code-based cryptosystem using GRS codes as secret codes was proposed by Niederreiter in the same article [@ni86a] as the famous Niederreiter cryptosystem. This proposal was then attacked by Sidelnikov and Shestakov in [@si92a], where they used the fact, that the public matrix is still a generator matrix of a GRS code and they were able to recover the evaluation points and hence the GRS structure of the public matrix.
In the cryptosystem proposed in Section \[sec:cryptosystem\], the secret GRS parity check matrix $H$ over ${\mathbb{F}}_{q^m}$ is hidden in two ways: first by puncturing its expanded parity check matrix $\widehat{H}$ over ${\mathbb{F}}_q$ and then by scrambling the columns of the punctured matrix ${\widehat{H}_S}$. Due to multiplying ${\widehat{H}_S}$ with a block diagonal matrix it is clear that the resulting code is no more equivalent to an evaluation code (or an expanded evaluation code). Hence evaluations (or expanded evaluation column vectors) can not be exploited using the Sidelnikov-Shestakov attack.
Distinguisher attack based on the Schur product
-----------------------------------------------
For the attack based on the Schur product we need to introduce some definitions and notations.
Let $x,y \in {\mathbb{F}}_q^n$. We denote by the Schur product of $x$ and $y$ their component-wise product $$x \star y = (x_1 y_1, \ldots, x_n y_n).$$
The Schur product is symmetric and bilinear. \[remark\_schur\]
Let $\mathcal{A},\mathcal{B}$ be two codes of length $n$. The Schur product of two codes is the vector space spanned by all $a \star b$ with $a \in \mathcal{A}$ and $b \in \mathcal{B}$: $$\langle \mathcal{A} \star \mathcal{B} \rangle = \langle \{ a \star b \bigm| a \in \mathcal{A}, b\in \mathcal{B} \} \rangle.$$ If $\mathcal{A} = \mathcal{B}$, then we call $ \langle \mathcal{A} \star \mathcal{A} \rangle $ the square code of $\mathcal{A}$ and denote it by $ \langle \mathcal{A}^2 \rangle $.
Let $G$ be a $k \times n$ matrix, with rows $(g_i)_{1 \leq i \leq k}$. The Schur matrix of $G$, denoted by $S(G)$, consists of the rows $
g_i \star g_j $ for\
$1 \leq i \leq j \leq k.$
We observe by Remark \[remark\_schur\], that if $G$ is a generator matrix of a code ${\mathcal{C}}$ then its Schur matrix $S(G)$ is a generator matrix of the square code of ${\mathcal{C}}$. Let $s$ be the following map $$\begin{aligned}
s: \mathbb{N} & \to & \mathbb{N} \\
k & \mapsto & \dfrac{1}{2}\left(k^2+k\right).\end{aligned}$$ For a $k \times n$ matrix $A$, we observe that $S(A)$ has the size $s(k) \times n$.\
Various McEliece cryptosystems based on modifications of GRS codes have been proved to be insecure [@co14; @co15a; @ga12]. This is because the dimension of the square code of GRS codes is very low compared to a random linear code of the same dimension. Moreover, other families of codes have also been shown to be vulnerable against the attacks based on Schur products. In [@co17], Couvreur *et al.* presented a general attack against cryptosystems based on algebraic geometric codes and their subcodes. In [@fa13] Faugère *et al.* showed that high rate binary Goppa codes can be distinguished from a random code. In [@co17w], Couvreur *et al.* presented a polynomial time attack against cryptosystems based on non-binary Goppa codes defined over quadratic extensions.
The distinguisher attack is based on the low dimensional square code of the public code (or of the shortened public code). In the following, based on experimental observations, we infer that the public code of the proposed cryptosystem cannot be distinguished using square code techniques.
Let $\widehat{{\mathcal{C}}}_S$ be the public code of the proposed cryptosystem. Note that ${\widehat{{\mathcal{C}}}_S}$ is a shortening of an expanded GRS code $\widehat{{\mathcal{C}}}$.
1. *Squares of expanded GRS codes*: Like in the case of Reed-Solomon codes and their subfield subcodes, the expanded GRS codes also have low square code dimension. To see this, we visualize expanded GRS codes as subfield subcodes of GRS-like codes. Let ${\mathcal{C}}$ be a GRS code of length $n$ and dimension $k$ over ${\mathbb{F}}_{q^m}$ having the following parity check matrix $$H = V_{r}(x,y) := \begin{pmatrix}
y_1 & y_2 & \cdots & y_n \\
y_1 x_1 & y_2 x_2 & \cdots & y_n x_n \\
\vdots & \vdots & \ddots & \vdots \\
y_1 x_1^{r-1} & y_2 x_2^{r-1} & \cdots & y_n x_n^{r-1} \\
\end{pmatrix},$$ where $x = (x_1,\ldots,x_n)$ is a vector of distinct elements in ${\mathbb{F}}_{q^m}$, $y = (y_1,\ldots,y_n)$ is a vector over ${\mathbb{F}}_{q^m}^\ast$ and $r := n-k$. Let $\gamma$ be a primitive element in ${\mathbb{F}}_{q^m}$. We define a new code $\mathcal{B}$ of length $mn$ over ${\mathbb{F}}_{q^m}$ given by the kernel of the following parity check matrix $$H^\prime = \begin{pmatrix}
V_{r}(x,y) & \mid & V_r(x,\gamma y) & \mid & \cdots & \mid & V_r(x,\gamma^{m-1} y)
\end{pmatrix}.$$ Using Lemma \[lemma:expand\], it is easy to observe, that the expanded code $\widehat{{\mathcal{C}}}$ of ${\mathcal{C}}$ with respect to $\gamma$ is permutation equivalent to the ${\mathbb{F}}_q$-kernel of $H^\prime$. In other words $\widehat{{\mathcal{C}}}$ is permutation equivalent to the subfield subcode of $\mathcal{B}$ over ${\mathbb{F}}_q$. Observe that a generator matrix $G^\prime $ of $\mathcal{B}$ is given by $$\begin{pmatrix}
V_k(x,y^\prime) & 0 & \ldots & 0 & 0 \\
0 & V_k(x,\gamma^{-1} y^\prime) & \ldots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \ldots & V_k(x,\gamma^{-(m-2)}y^\prime) & 0 \\
0 & 0 & \ldots & 0 & V_k(x,\gamma^{-(m-1)}y^\prime )\\ \hline
V_r(x,y^{\prime \prime}) & 0 & \ldots & 0 & - V_r(x,\gamma^{-(m-1)} y^{\prime \prime}) \\
0 & V_r(x,\gamma^{-1} y^{\prime \prime}) & \ldots & 0 & - V_r(x, \gamma^{1-(m-1)} y^{\prime \prime}) \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \ldots & V_r(x, \gamma^{-(m-2)} y^{\prime \prime}) & - V_r(x,\gamma^{(m-2)-(m-1)} y^{\prime \prime}) \\
\end{pmatrix},$$ where $y^\prime$ is such that $V_k(x,y^\prime) V_r(x,y)^\intercal = 0$, and $y^{\prime \prime} = (x_1^k,x_2^k,\ldots,x_n^k)\star y^\prime$. One can verify that $G^\prime (H^\prime)^\intercal = 0$. Observe that a generator matrix of $\widehat{{\mathcal{C}}}$ is permutation equivalent to $$\widehat{G} = \begin{pmatrix}
G_1 & 0 & \ldots & 0 \\
0 & G_2 & \ldots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \ldots & G_m \\ \hline
\multicolumn{4}{c}{G_{gv}} \\
\end{pmatrix},$$ where $G_i$ is a generator matrix of the subfield subcode of $V_k(x,\gamma^{-(i-1)}y^\prime)$ over ${\mathbb{F}}_q$, and $G_{gv}$ is a generator matrix of the ${\mathbb{F}}_q$-subfield subcode of the bottom $(m-1)r$ rows of $G^\prime$. The matrix $G_{gv}$ is also known as the glue-vector generator matrix, as in [@va91]. Due to the block structure of $\widehat{G}$ the Schur matrix of $\widehat{G}$ will have many zero rows. As a result the dimension of the square code is not full, given large enough $n$. This may lead to vulnerabilities when using expanded GRS codes directly in the cryptosystem.
2. *Effect of Shortening*: Consider the parity check matrix $\widehat{H}$ of an expanded GRS code as shown in Lemma \[lemma:expand\]. We partition the columns of $\widehat{H}$ into $n$ blocks, each of size $m$. By the definition of $\widehat{H}$, each of these blocks corresponds to a unique column vector of the parity check matrix of the parent GRS code. In order to weaken this correspondence, we puncture (randomly chosen) $m -\lambda$ of the columns from each block of $\widehat{H}$. As a result the correspondence of each block to the parent column vector is inconsistent. In addition we multiply the punctured parity check matrix by an invertible block diagonal matrix $T$. This further destroys the algebraic structure inherited from the parent GRS code. This was evident in our computations of the square code dimension of such shortened codes. Even in the case of $m=3$ we observed that puncturing one column from each block of $\widehat{H}$ results in a full square code dimension.
Information Set Decoding {#sec:ISD}
------------------------
Information set decoding (ISD) algorithms are the best known algorithms for decoding a general linear code. ISD algorithms were introduced by Prange [@pr62] in 1962. Since then several improvements have been proposed for codes over the binary field by Lee-Brickel [@le88], Leon [@le88a], Stern [@st89] and more recently by Bernstein *et al.* [@be11], Becker *et al.* [@be12], May-Ozerov [@ma15]. Several of these algorithms have been generalized to the case of codes over general finite fields, see [@kl17; @hi16; @in18; @ni17; @pe10]. An ISD algorithm in its simplest form first chooses an information set $I$, which is a size $k$ subset of $\lbrace 1,2, \ldots,n \rbrace$ such that the restriction of the parity check matrix on the columns indexed by the complement of $I$ is non-singular. Then Gaussian elimination brings the parity check matrix in a standard form and assuming that the errors are outside of the information set, these row operations on the syndrome will exploit the error vector, if the weight does not exceed the given error correction capacity.
#### ISD for the proposed cryptosystem:
In the proposed cryptosystem we introduce a burst pattern in the error vector, in particular the error vector has support in $t$ sub-vectors each of length $\lambda$. Henceforth, we modify Stern’s ISD algorithm to incorporate such pattern in the error vector.
We first recall the Stern’s algorithm. The algorithm partitions the information set $I$ into two equal-sized subsets $X$ and $Y$, and chooses uniformly at random a subset $Z$ of size $\ell$ outside of $I$. Then it looks for vectors having exactly weight $p$ among the columns indexed by $X$, exactly weight $p$ among the columns indexed by $Y$, and exactly weight 0 in columns indexed by $Z$ and the missing weight $t-2p$ in the remaining indices.
In the proposed cryptosystem we have been given a public code ${\widehat{{\mathcal{C}}}_S}$ of length $\lambda n$ and dimension $k^\prime := mk-(m-\lambda)n$ over ${\mathbb{F}}_q$. We also know that the error vector has support in $t$ sub-vectors of length $\lambda$. Hence we use Stern’s algorithm on the blocks of size $\lambda$. We consider the information set $I$ to have $\left \lfloor k^\prime/\lambda \right \rfloor$ blocks. We partition $I$ into two equal-sized subsets $X$ and $Y$, and choose uniformly at random a subset $Z$ of $\ell$ blocks outside of $I$. Then we look for vectors having support in exactly $p$ blocks in $X$, exactly $p$ blocks in $Y$, and exactly 0 blocks in $Z$.
In Section \[sec:KeySize\] we compute the key sizes of the proposed cryptosystem having 256-bit security against this modified ISD algorithm.
Key size {#sec:KeySize}
========
In this section we compute the key sizes of the proposed cryptosystem having 256-bit security against the ISD algorithm discussed in Section \[sec:ISD\]. Later we compare these key sizes with the key sizes of the McEliece cryptosystem using binary Goppa codes [@be08] and some recently proposed cryptosystems that are using Reed-Solomon codes as secret codes. These are based on the idea of [@ba11; @ba15z] (BBCRS), where the authors proposed to hide the structure of the code using as transformation matrix the sum of a rank $z$ matrix and a weight $w$ matrix. The proposed parameters in [@ba11; @ba15z] with $z=1$ and $w\leq 1+R$ were broken by the square code attack [@co14; @co15a], where $R$ denotes the rate of the code. Two countermeasures were recently proposed in [@ba19; @kh18]. In order to hide the structure of the Reed-Solomon code the authors of [@ba19] use $w>1+R$ and $z=1$ or $w<1+R$ and $z>1$. Whereas in [@kh18] the transformation matrix has weight $w=2$ and rank $z=0$.
In the proposed cryptosystem, the public key is a parity check matrix of a linear code over ${\mathbb{F}}_q$ having length $\lambda n$ and dimension $mk-(m-\lambda) n$. Hence the public key size is $(\lambda n-m(n-k)) \cdot m(n-k) \cdot \log_2(q)$ bits. For a degree of extension $m$, let ${\mathcal{C}}_m$ be the public code.
In Table \[table:m2KS\], we provide the key sizes for different rates of the public code ${\mathcal{C}}_3$ achieving a 256-bit security level against the modified ISD algorithm discussed in Section \[sec:ISD\]. Observe that the smallest key size is achieved at rate $0.82$.
Rate $q$ $n$ $k$ $t$ Key Size (bits)
------ ----- ------ ------ ----- -----------------
0.60 13 1382 829 277 6783627
0.65 13 1270 825 223 5952804
0.70 13 1207 844 182 5339456
0.75 13 1192 894 149 4929077
0.80 13 1230 984 123 4702652
0.82 13 1258 1031 114 4624198
0.85 13 1340 1139 101 4634545
0.87 13 1420 1235 93 4692805
0.90 13 1602 1441 81 4863276
: Comparing key sizes of the proposed cryptosystem with $m=3$ and $\lambda=2$ reaching a $256$-bit security level against the modified ISD algorithm.[]{data-label="table:m2KS"}
In Table \[table:m4KS\], we provide the key sizes for different rates of the public code ${\mathcal{C}}_4$ achieving a 256-bit security level against the modified ISD algorithm discussed in Section \[sec:ISD\]. In this case the smallest key size is achieved at rate $0.89$.
Rate $q$ $n$ $k$ $t$ Key Size (bits)
------ ----- ------ ------ ----- -----------------
0.65 7 2360 1534 413 13134108
0.70 7 1945 1361 292 10191102
0.75 7 1738 1303 218 8480009
0.80 7 1662 1329 167 7448878
0.85 7 1700 1445 128 6815134
0.87 7 1770 1539 116 6785893
0.89 7 1872 1666 103 6754721
0.91 7 2024 1841 92 6814326
: Comparing key sizes of the proposed cryptosystem with $m=4$ and $\lambda=2$ reaching a $256$-bit security level against the modified ISD algorithm.[]{data-label="table:m4KS"}
In conclusion, for a $256$ bit security level we propose to use the cryptosystem with the two sets of parameters $(q=13, m=3, \lambda=2, n=1258 ,k= 1031)$ and $(q=7, m=4, \lambda=2, n=1872, k=1666)$, see Table \[Table:proposed\].
$q$ $m$ $n$ $k$ Key Size (in bits)
-- --------------------- ------ ------ ------ --------- --------------------
Type I 13 3 1258 1031 4624198
Type II 7 4 1872 1666 6754721
2 13 6960 5413 8373911
$w=1.708$ and $z=1$ 1423 1 1422 786 5113520
$w=1.2$ and $z=10$ 1163 1 1162 928 2274160
$w=2$ and $z=0$ 1993 1 1992 1593 6966714
: Comparing the key sizes of the proposed parameters against different cryptosystems.[]{data-label="Table:proposed"}
The proposed parameters for the classic McEliece system using binary Goppa codes by Bernstein *et al.* in [@be08] are $q=2, m=13, n= 6960, k = 5413$, which gives a key size of $8373911$ bits. It achieves a security level of 260-bits with respect to the ball-collision algorithm [@be11].
In comparison to the classic McEliece system, the Type I set of parameters reduces the key size by $44.8 \%$ and the Type II set of parameters reduces the key size by $ 19.3\%$.
Acknowledgement
===============
The authors would like to thank Matthieu Lequesne and Jean-Pierre Tillich for pointing out the square code vulnerability in the case of quadratic extensions. This work has been supported by the Swiss National Science Foundation under grant no. 169510.
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Yingquan Wu. On expanded cyclic and [Reed–S]{}olomon codes. , 57:601 – 620, 03 2011.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Inside the moduli space of curves of genus three with one marked point, we consider the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. These loci have codimension two. We compute the classes of their closures in the moduli space of stable curves of genus three with one marked point. Similarly, we compute the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point. These loci naturally arise in the study of minimal dimensional strata of Abelian differentials.'
address:
- 'Department of Mathematics, Boston College, Chestnut Hill, MA 02467'
- 'Department of Mathematics, University of Utah, Salt Lake City, UT 84112'
author:
- Dawei Chen
- Nicola Tarasca
bibliography:
- 'biblio.bib'
title: Loci of curves with subcanonical points in low genus
---
[^1]
Introduction {#intro}
============
A point $p$ on a smooth curve of genus $g\geq 2$ is called [*subcanonical*]{} if $(2g-2)p$ is a canonical divisor. For example, Weierstrass points of a hyperelliptic curve are subcanonical points. Subcanonical points are the most special Weierstrass points from the perspective of Weierstrass gap sequences ([@MR770932 Exercise 1.E]). The locus of curves admitting a subcanonical point inside the moduli space ${\mathcal M}_g$ of curves of genus $g$ has codimension $g-2$.
Such loci naturally arise from the study of strata of Abelian differentials. Let $\mathscr{H}_g$ be the moduli space of Abelian differentials parameterizing pairs $(C, \omega)$, where $C$ is a smooth curve of genus $g$ and $\omega$ is an Abelian differential on $C$, for $g\geq 2$. Given $\mu = (m_1, \ldots, m_n)$ a partition of $2g-2$, denote by $\mathscr{H}(\mu) \subseteq \mathscr{H}_g$ the stratum of Abelian differentials $(C, \omega)$, where the zeros of $\omega$ are of type $\mu$, namely, $(\omega)_0 = \sum_{i=1}^n m_i p_i$, for distinct points $p_i$. There is a ${\operatorname{GL}}^+_2({\mathbb R})$-action on $\mathscr{H}(\mu)$ that varies the real and imaginary parts of $\omega$ ([@MR2261104]). The study of ${\operatorname{GL}}^+_2({\mathbb R})$-orbits is a major subject in Teichmüller dynamics, with fascinating applications to the geometry of the moduli space of stable curves, such as producing rigid curves, and bounding slopes of effective divisors ([@MR2822219]). Moreover, one can degenerate $(C, \omega)$ to a nodal curve along with a section of the dualizing sheaf, that is, compactify $\mathscr{H}(\mu)$ over the Deligne-Mumford moduli space ${\overline{\mathcal M}}_g$ of stable nodal curves, and study its boundary behavior. This turns out to provide crucial information for invariants in Teichmüller dynamics, such as Siegel-Veech constants, Lyapunov exponents, and classification of Teichmüller curves ([@MR3033521; @MR2910796]).
Among all strata of Abelian differentials, the minimal dimensional stratum is $\mathscr{H}(2g-2)$, parameterizing pairs $(C,\omega)$, where $(\omega)_0 =(2g-2)p$ for a certain point $p$ in $C$. Note that if $(\omega)_0 =(2g-2)p$, then $\mathcal{O}_C((g-1)p)$ is a [*spin structure*]{} on $C$, that is, a square root of the canonical bundle. The parity of a spin structure $\eta$ on a curve $C$ is defined as the parity of $h^0(C,\eta)$.
For $g\geq 4$, the stratum $\mathscr{H}(2g-2)$ has three connected components: the component of hyperelliptic curves $\mathscr{H}(2g-2)^{{\operatorname{hyp}}}$, and the two components $\mathscr{H}(2g-2)^{{\operatorname{even}}}$ and $\mathscr{H}(2g-2)^{{\operatorname{odd}}}$ distinguished by the parity of the corresponding spin structures. For $g=3$, the hyperelliptic and even components coincide, hence $\mathscr{H}(4)$ has two connected components: $\mathscr{H}(4)^{{\operatorname{hyp}}}$ and $\mathscr{H}(4)^{{\operatorname{odd}}}$. For $g=2$, the stratum $\mathscr{H}(2)$ is irreducible. The reader can refer to [@MR2000471] for a complete classification of connected components of $\mathscr{H}(\mu)$, where the minimal dimensional stratum $\mathscr{H}(2g-2)$ plays an important role as a base case for an inductive argument.
One can project $\mathscr{H}(\mu)$ to ${\mathcal M}_g$ via the forgetful map $(C,\omega)\mapsto C$. Alternatively, by marking the zeros of $\omega$, one can lift $\mathscr{H}(\mu)$ (up to rescaling $\omega$) to ${\mathcal M}_{g,n}$. Thus, one obtains a number of interesting subvarieties in ${\mathcal M}_g$ and ${\mathcal M}_{g,n}$. For example, the projection of $\mathscr{H}(2)$ dominates ${\mathcal M}_2$, and the lift of $\mathscr{H}(2)$ is the divisor of Weierstrass points in ${\mathcal M}_{2,1}$.
Motivated by the problem of determining the Kodaira dimension of moduli spaces of stable curves ${\overline{\mathcal M}}_{g,n}$, effective divisor classes in ${\overline{\mathcal M}}_{g,n}$ have been extensively calculated in the last several decades. In contrast, much less is known about higher codimensional cycles in ${\overline{\mathcal M}}_{g,n}$. Recently, there has been growing interest in studying higher codimensional subvarieties of moduli spaces of curves ([@MR2120989; @Hain; @FaberPagani; @MR3109733; @DR]), and in describing cones of higher codimensional effective cycles ([@CC]). Our main result is the explicit computation of classes of closures of loci of curves with subcanonical points in the moduli space of stable curves in low genus.
Let $\mathcal{H}yp_{3,1}$ be the locus of smooth hyperelliptic curves with a marked Weierstrass point in ${\mathcal M}_{3,1}$. Let ${\mathcal F}_{3,1}$ be the locus of smooth non-hyperelliptic curves with a marked hyperflex point in ${\mathcal M}_{3,1}$. The loci $\mathcal{H}yp_{3,1}$ and ${\mathcal F}_{3,1}$ have codimension two in ${\mathcal M}_{3,1}$. They are the lifts of the minimal dimensional stratum components $\mathscr{H}(4)^{{\operatorname{hyp}}}$ and $\mathscr{H}(4)^{{\operatorname{odd}}}$, respectively. In §\[pullbacktoMbar31\] and §\[SF\] we prove the following result in the Chow group $A^2({\overline{\mathcal M}}_{3,1})$ of codimension-two classes on ${\overline{\mathcal M}}_{3,1}$.
\[res31\] The classes of the closures of $\mathcal{H}yp_{3,1}$ and ${\mathcal F}_{3,1}$ in $A^2({\overline{\mathcal M}}_{3,1})$ are $$\begin{aligned}
\overline{\mathcal{H}yp}_{3,1} & \equiv & \psi (18\lambda -2\delta_0 -9\delta_{1,1} -6\delta_{2,1})-\lambda\left(45\lambda -\frac{19}{2}\delta_0 -24\delta_{2,1}\right) -\frac{1}{2}\delta_0^2 -\frac{5}{2}\delta_0\delta_{2,1} -3\delta_{2,1}^2,\\
\overline{\mathcal{F}}_{3,1} & \equiv &{} \psi(77\lambda -3\psi -8\delta_0 -42\delta_{1,1} -19\delta_{2,1}) - \lambda\left(338\lambda -\frac{137}{2}\delta_0 -146 \delta_{2,1}\right) -\frac{7}{2}\delta_0^2\\
&& {} -\frac{31}{2}\delta_0\delta_{2,1} -3\delta_{1,1}^2 -20\delta_{2,1}^2 +3\kappa_2.\end{aligned}$$
As a check, consider the map $p\colon {\overline{\mathcal M}}_{3,1} \rightarrow {\overline{\mathcal M}}_3$ obtained by forgetting the marked point. The push-forward of products of divisor classes in ${\overline{\mathcal M}}_{3,1}$ via $p$ are described in §\[pushfwdtoM3\]. It follows that the push-forward of the class of $\overline{\mathcal{H}yp}_{3,1}$ from Theorem \[res31\] via $p$ is $$p_*\left(\overline{\mathcal{H}yp}_{3,1} \right) \equiv 8\cdot \overline{\mathcal{H}yp}_{3},$$ as expected, where $\overline{\mathcal{H}yp}_{3}$ is the closure of the locus of hyperelliptic curves in ${\overline{\mathcal M}}_3$. Indeed, from [@MR664324] we have $\overline{\mathcal{H}yp}_{3}\equiv 9\lambda-\delta_0-3\delta_1$, and the multiplicity $8$ accounts for the number of Weierstrass points in a hyperelliptic curve of genus $3$. The class of the divisor $p_*(\overline{\mathcal{F}}_{3,1})$ is computed in [@MR1016424], and will be used in the proof of Theorem \[res31\].
Similarly, we can consider the following codimension-two loci in the moduli space of curves of genus $4$. Let ${\mathcal H}_4$ be the locus in ${\mathcal M}_4$ of smooth curves $C$ that admit a canonical divisor of type $6p$. It consists of the three irreducible components $\overline{\mathcal{H}yp}_{4}$, ${\mathcal H}_4^{+}$ and ${\mathcal H}_4^{-}$, defined as the projection images of the minimal dimensional stratum components $\mathscr{H}(6)^{{\operatorname{hyp}}}$, $\mathscr{H}(6)^{{\operatorname{even}}}$, and $\mathscr{H}(6)^{{\operatorname{odd}}}$ to ${\overline{\mathcal M}}_{4}$, respectively.
The class of the closure of $\overline{\mathcal{H}yp}_{4}$ has been first computed in [@MR2120989 Proposition 5] via localization on the moduli space of stable relative maps (see (\[hyp4\])). An alternative proof via test surfaces and admissible covers is given in [@MR3109733]. In §\[h+4\] we obtain the following result in the Chow group $A^2({\overline{\mathcal M}}_4)$.
\[h4+\] The class of the closure of the locus ${\mathcal H}_4^{+}$ in $A^2({\overline{\mathcal M}}_4)$ is $$\begin{aligned}
\overline{\mathcal{H}}^+_4 & \equiv & 2448\lambda^2 -542 \lambda\delta_0 - 1608 \lambda\delta_1 + 276\lambda\delta_2 + 32 \delta_0^2 + 178\delta_0\delta_1 \\
& & {}+ 336\delta_1^2 + 276 \delta_1\delta_2 + 576 \delta_2^2 - 4\delta_{00} - 60 \gamma_1 + 12 \delta_{01a} - 144 \delta_{1|1}. \end{aligned}$$
As an immediate consequence of Theorems \[res31\] and \[h4+\], in §\[ci\] we prove the following geometric result.
\[coro\] The loci $\overline{\mathcal{H}yp}_{3,1}$, $\overline{\mathcal{F}}_{3,1}$, and $\overline{\mathcal{H}}^+_4$ are not complete intersections in their respective spaces.
The class of $\overline{\mathcal{H}yp}_{3,1}$ is known to span an extremal ray of the cone of effective codimension-two classes in ${\overline{\mathcal M}}_{3,1}$ (similarly for the locus $\overline{\mathcal{H}yp}_{4}$ in ${\overline{\mathcal M}}_4$) ([@CC]). It is natural to ask whether the classes of $\overline{\mathcal{F}}_{3,1}$ and $\overline{\mathcal{H}}^+_4$ lie in the interior or in the boundary of the cones of effective codimension-two classes of their respective spaces. In addition, the loci $\overline{\mathcal{H}yp}_{3,1}$, $\overline{\mathcal{F}}_{3,1}$, and $\overline{\mathcal{H}}^+_4$ are images of the strata $\mathscr{H}(4)^{{\operatorname{hyp}}}$, $\mathscr{H}(4)^{{\operatorname{odd}}}$, and $\mathscr{H}(6)^{{\operatorname{even}}}$ in the respective moduli spaces of curves. Thus our results can shed some light on the study of cones of effective higher codimensional cycles, as well as the study of degenerate Abelian differentials appearing in the boundary of these strata.
Let us describe our methods. In order to study the closure of the images of the strata of Abelian differentials inside moduli spaces of curves, one needs a good description of singular hyperelliptic curves, and singular curves with an even or odd spin structure. The closure of the hyperelliptic components can be studied via the theory of admissible covers ([@MR664324]). Moreover, let $\mathcal{S}_g^+$ (respectively, $\mathcal{S}_g^-$) be the moduli space of pairs $[C,\eta]$, where $C$ is a smooth curve of genus $g$, and $\eta$ is an even (respectively, odd) [*theta characteristic*]{} on $C$, that is, a spin structure on $C$. There is a natural map $\mathcal{S}_g^\pm \rightarrow {\mathcal M}_g$ obtained by forgetting the theta characteristic. We use Cornalba’s description of the closure $\overline{\mathcal{S}}_g^\pm \rightarrow {\overline{\mathcal M}}_g$ ([@MR1082361], see also [@MR2779475; @MR2565536; @MR3245010]), and realize the loci $\overline{\mathcal{F}}_{3,1}$ and $\overline{\mathcal{H}}^+_4$ as push-forward of loci in $\overline{\mathcal{S}}_3^-\times_{{\overline{\mathcal M}}_3} {\overline{\mathcal M}}_{3,1}$ and $\overline{\mathcal{S}}_4^+$, respectively.
To perform our computations, we use the basis for $A^2({\overline{\mathcal M}}_{4})$ from [@MR1078265]. It is not clear whether the group of tautological classes $R^2({\overline{\mathcal M}}_{3,1})\subseteq A^2({\overline{\mathcal M}}_{3,1})$ coincides with $A^2({\overline{\mathcal M}}_{3,1})$. Anyhow, it is a consequence of the descriptions in Lemmata \[j3\^\*H\] and \[lemmamnklj\] that the classes of $\overline{\mathcal{H}yp}_{3,1}$ and $\overline{\mathcal{F}}_{3,1}$ are tautological. In §\[basisforMbar31\], we describe a basis for $R^2({\overline{\mathcal M}}_{3,1})$. The classes $\psi, \lambda, \delta_0, \delta_{1,1}, \delta_{2,1}$ form a $\mathbb{Q}$-basis of ${\operatorname{Pic}}({\overline{\mathcal M}}_{3,1})$. Note that the product $\lambda\delta_{1,1}$ is linearly dependent from the other products, see Proposition \[boundaryinA2Mbar31\].
\[basisR2Mbar31\] The following $16$ classes $$\begin{aligned}
\label{bm31gen}
\psi^2, \psi\lambda, \psi\delta_0, \psi\delta_{1,1}, \psi\delta_{2,1}, \lambda^2, \lambda\delta_0, \lambda\delta_{2,1}, \delta_0^2,
\delta_0\delta_{1,1}, \delta_0\delta_{2,1}, \delta_{1,1}^2 , \delta_{1,1}\delta_{2,1}, \delta_{2,1}^2, \delta_{01a}, \kappa_2\end{aligned}$$ form a $\mathbb{Q}$-basis of $R^2({\overline{\mathcal M}}_{3,1})$.
The paper is organized as follows. We collect in §\[ingre\] some results about the enumerative geometry of a general curve. In §\[basisforMbar31\], we prove Proposition \[basisR2Mbar31\] using the following strategy. After [@getzler-looijenga; @yang], it is known that the group $R^2({\overline{\mathcal M}}_{3,1})$ has dimension $16$, equal to the rank of the intersection pairing $R^2({\overline{\mathcal M}}_{3,1})\times R^{5}({\overline{\mathcal M}}_{3,1})$. Hence, one could show that the intersection pairing of classes in (\[bm31gen\]) with classes of complementary dimension has rank $16$. Instead, we reduce the number of computations by using Getzler’s results on $R^2({\overline{\mathcal M}}_{2,2})$ ([@MR1672112]). Let $\vartheta\colon {\overline{\mathcal M}}_{2,2}\rightarrow {\overline{\mathcal M}}_{3,1}$ be the map obtained by attaching a fixed elliptic tail to the second marked point. We first show that the pull-backs of the classes in (\[bm31gen\]) via $\vartheta$ span a $13$-dimensional subspace of the $14$-dimensional space $R^2({\overline{\mathcal M}}_{2,2})$. We compute the $3$-dimensional space of possible relations among the classes in (\[bm31gen\]). Finally, by restricting to three test surfaces, we verify that such relations cannot hold.
In §\[pullbacktoMbar31\] we compute the class of $\overline{\mathcal{H}yp}_{3,1}$ by studying the pull-back of the class $\overline{\mathcal{H}yp}_{4}$ via the map $j_3\colon {\overline{\mathcal M}}_{3,1}\rightarrow {\overline{\mathcal M}}_4$ obtained by attaching a fixed elliptic tail at the marked point of curves in ${\overline{\mathcal M}}_{3,1}$. To compute the classes of $\overline{\mathcal{F}}_{3,1}$ and $\overline{\mathcal{H}}^+_4$, in §\[SF\] and §\[h+4\] we realize each one of these loci as a component of the intersection of two divisors in their respective moduli spaces. We first describe set-theoretically the other components in the intersections. The multiplicity along each component is then computed by restricting to test surfaces, and by studying the push-forward via the map $p\colon {\overline{\mathcal M}}_{3,1}\rightarrow {\overline{\mathcal M}}_3$. The classes of $\overline{\mathcal{F}}_{3,1}$ and $\overline{\mathcal{H}}^+_4$ follow from the computation of the classes of the other components in each intersection. Throughout the paper, we provide various checks on Theorems \[res31\] and \[h4+\] (see §\[checkM31\], (\[kappa2M4\]), Remark \[DR2\], §\[proofF\], (\[checkH4+\])).
We have not succeeded in applying the above ideas to obtain a complete formula for the class of $\overline{\mathcal{H}}^-_4$. One can directly intersect the locus $\overline{\mathcal{H}}^-_4$ with some test surfaces and thus obtain linear relations on the coefficients of the class. While in this way we have produced some relations for such coefficients, at the moment we have not succeeded in computing the whole class. Nevertheless, in §\[det-\] we compute the class of ${\mathcal{H}}^-_4$ in $A^2({\mathcal M}_4)$ (that is, the coefficient of $\lambda^2$) using a determinantal description.
[**Notation.**]{} We use throughout the following notation for divisor classes in ${\rm Pic}({\overline{\mathcal M}}_{g,n})$. For $i=1,\dots,n$, let $\psi_i$ be the cotangent line bundle class at the $i$-th marked point. Let $\lambda$ be the first Chern class of the Hodge bundle, and $\delta_0$ be the class of the locus $\Delta_0$ whose general element is a nodal irreducible curve. For $i=0,\dots,g$ and $S\subseteq \{1,\dots,n\}$, let $\delta_{i,S}$ be the class of the locus $\Delta_{i,S}$ whose general element has a component of genus $i$ containing the points with markings in $S$, and meeting transversally in one point a component of genus $g-i$ containing the remaining marked points. We write $\delta_i:= \delta_{i,\emptyset}$ and $\Delta_i:= \Delta_{i,\emptyset}$. When $n=1$, we write $\psi:= \psi_1$, $\delta_{i,1}:=\delta_{i,\{1\}}$, and $\Delta_{i,1}:=\Delta_{i,\{1\}}$.
We also use the following codimension-two classes. Let $\kappa_2:= (p_{n+1})_*((\psi_{n+1})^3)$, where $p_{n+1} \colon {\overline{\mathcal M}}_{g,n+1} \rightarrow {\overline{\mathcal M}}_{g,n}$ is the natural map obtained by forgetting the last marked point. Let $\delta_{00}$ be the class of the closure of the locus of irreducible curves with two nodes. Let $\gamma_{i,S}$ be the class of the closure of the locus $\Gamma_{i,S}$ of curves with a component of genus $i$ containing the points with markings in $S$ and meeting in two points a component of genus $g-i-1$ containing the remaining marked points. We write $\gamma_i:=\gamma_{i,\emptyset}$ and $\Gamma_i:=\Gamma_{i,\emptyset}$. Let $\delta_{01a}$ be the class of the closure of the locus of curves with a rational nodal tail. Finally, $\delta_{1|1}$ is the class of the closure of the locus of curves with two unmarked elliptic tails.
We work over an algebraically closed field of characteristic $0$. All cycle classes are stack fundamental classes, and all cohomology and Chow groups are taken with rational coefficients.
[**Acknowledgements.**]{} We would like to thank Izzet Coskun, Joe Harris, Scott Mullane, and Anand Patel for helpful conversations on related topics. We are grateful to the referee for many valuable suggestions for improving the exposition of the paper.
Enumerative geometry of general curves {#ingre}
======================================
In this section, we collect some results about the enumerative geometry of general curves for later use.
The difference map
------------------
Let $C$ be a general curve of genus two. Define a generalized Abel-Jacobi map $f_d\colon C\times C\to {\operatorname{Pic}}^1(C)$ as follows: $$(x, y) \mapsto {\mathcal O}_{C}((d+1)x - dy),$$ where $d\in {\mathbb Z}^{+}$. Let $\Delta_{C\times C}\subset C\times C$ be the diagonal and $I\subset C\times C$ be the locus of pairs of points that are conjugate under the hyperelliptic involution. The intersection of $I$ and $\Delta_{C\times C}$ consists of the six Weierstrass points of $C\cong \Delta_{C\times C}$.
\[diff\] Under the above setting, we have:
1. $f_d$ is finite of degree $2 d^2 ( d + 1)^2$;
2. $f_d$ is simply ramified along $\Delta_{C\times C}$ and $I$, away from $I\cap \Delta_{C\times C}$;
3. The ramification order of $f_d$ is $2$ at each point in the intersection $I\cap \Delta_{C\times C}$.
Take a general point $p\in C$. Consider the isomorphism $u\colon {\operatorname{Pic}}^1(C)\to J(C)$ given by $
L\mapsto L\otimes {\mathcal O}_C(-p).
$ Let $h_d = u \circ f_d$. Note that $
h_d(x, y) = {\mathcal O}_C( (d+1)(x-p) - d(y-p)).
$ Let $\theta$ be the fundamental class of the theta divisor in $J(C)$. For $k\in{\mathbb Z}$, the locus of ${\mathcal O}_C(k (x-p))$ for varying $x\in C$ has class $k^2 \theta$ in $J(C)$. Therefore, we conclude that $$\deg f_d = \deg h_d = \deg h_{d*} h_d^{*}([{\mathcal O}_C]) = \deg \left((d+1)^2 \theta \cdot d^2 \theta\right) = 2 d^2 ( d + 1)^2,$$ thus proving the degree part of the proposition.
Next, let $\phi\colon C\to {\mathbb P}^1$ be the hyperelliptic double covering. By [@MR770932 p. 262], the associated (projectivized) tangent space map of $f_d$ at $(x, y)$ can be regarded as ${\mathbb P}^1 \to \overline{\langle\phi(x), \phi(y)}\rangle$, where $\overline{\langle\phi(x), \phi(y)\rangle}$ is the linear span of $\phi(x)$ and $\phi(y)$ in ${\mathbb P}^1$. Therefore, $f_d$ is ramified at $(x,y)$ if and only if $\phi(x) = \phi(y)$, that is, $(x, y)\in \Delta_{C\times C} \cup I$. In particular, $f_d$ is finite away from $\Delta_{C\times C}$ and $I$. Moreover, $f_d(w, w) = {\mathcal O}_C(w)$ for any Weierstrass point $w\in C$, hence $f_d(\Delta_{C\times C})$ and $f_{d}(I)$ cannot be a single point. We thus conclude that $f_d$ is finite.
Finally, via a local computation in a neighborhood of a Weierstrass point of $C$, one verifies the ramification orders along $\Delta_{C\times C}$, $I$, and $\Delta_{C\times C}\cap I$.
Similarly, we consider the following map. Let $p$ be a fixed point on a general curve $C$ of genus two. Fix two positive integers $d_1, d_2$, and let $d =d_1 - d_2 - 1$. Define the map $f_{d_1, d_2}\colon C\times C\rightarrow {\rm Pic}^{d}(C)$ by $$(x, y) \mapsto \mathcal{O}_C(d_1 x - d_2 y - p).$$ The same proof as in Proposition \[diff\] implies the following result.
\[diff2\] The degree of $f_{d_1, d_2}$ is $2 d_1^2 d_2^2$.
The Scorza curve {#Scorza}
----------------
Let $C$ be a curve of genus $g$, and $\eta^+$ an even theta characteristic on $C$. Suppose that $\eta^+$ is not a vanishing theta-null, that is, $h^0(C, \eta^+)=0$. The Scorza curve $T_{\eta^+}$ in $C\times C$ is defined as $$T_{\eta^+} := \{(x,y)\in C\times C \,| \, h^0(\eta^+\otimes \mathcal{O}(x-y))\neq 0 \}$$ (see [@Scorza]). From the assumption, $T_{\eta^+}$ does not meet the diagonal $\Delta_{C\times C}\subset C\times C$. By the Riemann-Roch theorem, $T_{\eta^+}$ is a symmetric correspondence.
We need the following computation: when $T_{\eta^+}$ is reduced, its class in $H^2(C\times C)$ is $$\left[T_{\eta^+}\right] = (g-1) F_1 + (g-1) F_2 + \Delta_{C\times C},$$ where $F_1$ and $F_2$ are the classes of a fiber of the projections $C\times C \rightarrow C$ of the first and second components, respectively (see [@MR1213725 §7.1]). We also use that for a general spin curve $[C,\eta^+]\in \mathcal{S}_g^+$ with $g\geq 2$, the Scorza curve $T_{\eta^+}$ is smooth. In particular, the locus of pairs $(x,y)\in C\times C$ such that $$h^0(C, \eta^+\otimes\mathcal{O}(x-2y))\geq 1 \quad \mbox{and}\quad h^0(C,\eta^+\otimes\mathcal{O}(y-2x))\geq 1$$ is empty (see for instance [@MR3245010 Theorem 4.1]).
Tautological classes on BM31 {#basisforMbar31}
============================
In this section, we show that the classes in (\[bm31gen\]) form a basis of $R^2({\overline{\mathcal M}}_{3,1})$, and thus prove Proposition \[basisR2Mbar31\]. Moreover, in §\[secboundaryinA2Mbar31\] we express certain boundary strata classes in terms of such a basis. We use these results in §\[pullbacktoMbar31\] and §\[SF\].
Proof of Proposition \[basisR2Mbar31\]
--------------------------------------
From [@getzler-looijenga] we know that the dimension of $H^4({\overline{\mathcal M}}_{3,1})$ is $16$ (see also [@MR2433615]). Since the intersection pairing of classes in the tautological group $RH^2({\overline{\mathcal M}}_{3,1})\subseteq H^4({\overline{\mathcal M}}_{3,1})$ with tautological classes of complementary dimension has rank $16$ (see [@yang]), we conclude that $RH^2({\overline{\mathcal M}}_{3,1})=H^4({\overline{\mathcal M}}_{3,1})$. Every relation among codimension-two tautological classes in $RH^2({\overline{\mathcal M}}_{3,1})$ lifts via the cycle map to $R^2({\overline{\mathcal M}}_{3,1})$, hence $R^2({\overline{\mathcal M}}_{3,1})\simeq RH^2({\overline{\mathcal M}}_{3,1})=H^4({\overline{\mathcal M}}_{3,1})$.
In order to prove that the classes in (\[bm31gen\]) form a basis, it is enough to show that the intersection pairing of the classes in (\[bm31gen\]) with classes in complementary dimension has rank $16$. This is the strategy used by Getzler for $H^*({\overline{\mathcal M}}_{2,2})$ in [@MR1672112].
Here we use a different approach, relying on Getzler’s results. Remember that $R^2({\overline{\mathcal M}}_{2,2})$ has dimension $14$, and one has $R^2({\overline{\mathcal M}}_{2,2})\simeq RH^2({\overline{\mathcal M}}_{2,2})=H^4({\overline{\mathcal M}}_{2,2})$. The idea is the following. First we consider the natural map $\vartheta\colon {\overline{\mathcal M}}_{2,2}\rightarrow {\overline{\mathcal M}}_{3,1}$ obtained by attaching a fixed elliptic tail to the second marked point. We show that the pull-backs of the classes in (\[bm31gen\]) via $\vartheta$ span a $13$-dimensional subspace of $R^2({\overline{\mathcal M}}_{2,2})$. Whence we compute the possible three-dimensional space of relations which may still hold among the above classes, depending on three parameters $\alpha, \beta, \gamma$. Finally, using test surfaces we conclude that $\alpha=\beta=\gamma=0$.
A basis for the Picard group of ${\overline{\mathcal M}}_{2,2}$ is given by the following divisor classes $$\begin{aligned}
\psi_1, \quad \psi_2, \quad \delta_{0,\{1,2\}}, \quad \delta_0, \quad \delta_{1,\{1\}}, \quad \delta_{1,\{1,2\}}.\end{aligned}$$ Getzler shows that the following relations hold among products of divisor classes $$\begin{aligned}
\delta_{1,\{1,2\}}\left(12\delta_{1,\{1\}}+12\delta_{1,\{1,2\}}+\delta_0 \right)=\delta_{1,\{1\}}\left(12\delta_{1,\{1\}}+12\delta_{1,\{1,2\}}+\delta_0 \right)=0,\nonumber\\
\delta_{1,\{1\}}\left(\psi_1+\psi_2+\delta_{1,\{1\}} \right)=\psi_1\delta_{0,\{1,2\}}=\psi_2\delta_{0,\{1,2\}}=\delta_{1,\{1\}}\delta_{0,\{1,2\}}=0,\\
\left(\psi_1-\psi_2 \right)\left(10\psi_1+10\psi_2-2\delta_{1,\{1\}}-12\delta_{1,\{1,2\}}-\delta_0 \right)=0,\nonumber\end{aligned}$$ hence a basis for $R^2({\overline{\mathcal M}}_{2,2})$ is given by the following products $$\begin{aligned}
\psi_1\psi_2, \quad \psi_2^2, \quad \psi_1\delta_{1,\{1\}}, \quad \psi_2\delta_{1,\{1\}}, \quad
\psi_1\delta_{1,\{1,2\}}, \quad \psi_2\delta_{1,\{1,2\}} , \quad \psi_1\delta_0,\quad \psi_2\delta_0,\\
\delta_{0,\{1,2\}}^2 , \quad \delta_{1,\{1,2\}}\delta_{0,\{1,2\}} , \quad \delta_0\delta_{0,\{1,2\}}, \quad \delta_0\delta_{1,\{1\}},\quad
\delta_0\delta_{1,\{1,2\}}, \quad \delta_0^2.\end{aligned}$$ The following formulae are well-known $$\begin{aligned}
\vartheta^*(\psi) &= \psi_1, & \vartheta^*(\lambda) &= \lambda=\frac{1}{10}\delta_0+\frac{1}{5}(\delta_{1,\{1\}}+\delta_{1,\{1,2\}}), &
\vartheta^*(\delta_0) &= \delta_0, \\
\vartheta^*(\delta_{1,1}) &= \delta_{1,\{1\}} + \delta_{0,\{1,2\}}, & \vartheta^*(\delta_{2,1}) &= -\psi_2 + \delta_{1,\{1,2\}}. \end{aligned}$$ Moreover, from the computations in [@MR1672112] it follows that $$\begin{aligned}
\label{delta01|12}
\vartheta^*(\delta_{01a}) &=& \delta_{01a}\nonumber\\
&=& 24\psi_2^2 -6\psi_1\delta_{1,\{1\}} -\frac{54}{5}\psi_2\delta_{1,\{1\}}+ 6\psi_1\delta_{1,\{1,2\}} -\frac{114}{5}\psi_2\delta_{1,\{1,2\}} -\frac{12}{5}\psi_2\delta_0\\
&&{}+24\delta_{0,\{1,2\}}^2 +\frac{84}{5}\delta_{1,\{1,2\}}\delta_{0,\{1,2\}} + \frac{12}{5}\delta_0\delta_{0,\{1,2\}} + \frac{47}{50}\delta_0\delta_{1,\{1\}}+ \frac{36}{25}\delta_0\delta_{1,\{1,2\}}+ \frac{3}{25}\delta_0^2.\nonumber\end{aligned}$$ Getzler expresses the basis of $R^2({\overline{\mathcal M}}_{2,2})$ given by products of divisor classes in terms of decorated boundary strata classes. By the inverse change of basis, we obtain the formula (\[delta01|12\]).
It is easy to show that $\vartheta^*(\kappa_2)=\kappa_2$. Note that by Mumford’s relation, one has $$\kappa_2 = \lambda (\lambda+\delta_1) \in A^2({\overline{\mathcal M}}_2).$$ Using the well-known formula $\kappa_i = p_n^*(\kappa_i) +\psi_n^i$, where $p_n\colon {\overline{\mathcal M}}_{g,n}\rightarrow {\overline{\mathcal M}}_{g,n-1}$ is the natural map, we obtain that $$\begin{aligned}
\label{kappa2Mbar22}
\vartheta^*(\kappa_2)=\kappa_2 &=& \lambda(\lambda+\delta_{1,\{1\}}+\delta_{1,\{1,2\}}) +\psi_1^2 + \psi_2^2 +\delta_{0,\{1,2\}}^2 \\
&=& \psi_1^2 + \psi_2^2 +\delta_{0,\{1,2\}}^2 +\frac{3}{25} \delta_0(\delta_{1,\{1\}}+\delta_{1,\{1,2\}}) +\frac{1}{100} \delta_0^2 \in A^2({\overline{\mathcal M}}_{2,2}). \nonumber\end{aligned}$$
Using the above formulae, it is easy to compute that the pull-back via $\vartheta$ of the classes in the statement span a subspace of $R^2({\overline{\mathcal M}}_{2,2})$ of dimension $13$. The only possible relations are given by $$\begin{aligned}
\label{posrel}
\alpha\cdot \left({}-2\psi^2+6\psi\lambda-\frac{1}{2}\psi\delta_0 -2\psi\delta_{1,1} -6\lambda^2 +\frac{1}{2}\lambda\delta_0+6\lambda\delta_{2,1}
-\frac{1}{2}\delta_0\delta_{2,1} -\delta_{1,1}^2 +\kappa_2 \right)\nonumber\\
+\beta \cdot \left({}-\psi^2 +6\psi\lambda -\frac{1}{2}\psi\delta_0 -5\lambda^2 +\frac{1}{2}\lambda\delta_0 -4\lambda\delta_{2,1}
+\frac{1}{2} \delta_0\delta_{2,1} + \delta_{2,1}^2 \right)\\
+\gamma \cdot\left( 120\lambda^2-22\lambda\delta_0 +\delta_0^2 \right)=0 \nonumber\end{aligned}$$ for $\alpha, \beta, \gamma \in \mathbb{Q}$. Restricting this relation to the three test surfaces in §\[TSM31\] yields $\alpha=\beta=\gamma=0$. This completes the proof of the statement. $\square$
Three test surfaces in BM[3,1]{} {#TSM31}
--------------------------------
In the next subsections, we consider the intersections of the classes in Proposition \[basisR2Mbar31\] with three test surfaces. These computations are used in the proof of Proposition \[basisR2Mbar31\]. Note that $\kappa_2$ has zero intersection with the two-dimensional families of curves whose base is a product of two curves.
### {#7}
Let $(E_1, p, q_1)$ be a general two-pointed elliptic curve, and let $(E_2, q_2)$ be a pointed elliptic curve. Consider the surface in ${\overline{\mathcal M}}_{3,1}$ obtained by identifying the points $q_1, q_2$, and by identifying a moving point in $E_1$ with a moving point in $E_2$.
![How the general fiber of the family in §\[7\] moves.](TS7.ps)
The base of the surface is $E_1 \times E_2=:S_1$. Let $\pi_i\colon E_1 \times E_2 \rightarrow E_i$ be the natural projection, for $i=1,2$. The divisor classes restrict as follows $$\begin{aligned}
\psi|_{S_1} &= \pi_1^*[p], & \lambda|_{S_1} &= 0, & \delta_0|_{S_1} &= -2\pi_1^*[q_1] -2\pi_2^*[q_2], &
\delta_{1,1}|_{S_1} &= \pi_1^*[q_1], & \delta_{2,1}|_{S_1} &= \pi_2^*[q_2]. \end{aligned}$$ The classes in Proposition \[basisR2Mbar31\] with nonzero intersection with this test surface are thus the following $$\begin{aligned}
\psi\delta_0|_{S_1} &= -2, & \psi\delta_{2,1}|_{S_1} &= 1, & \delta_0^2|_{S_1} &= 8, &
\delta_0\delta_{1,1}|_{S_1} &= -2, & \delta_0\delta_{2,1}|_{S_1} &= -2, & \delta_{1,1} \delta_{2,1} |_{S_1}&= 1.\end{aligned}$$
### {#10}
Let $(E_1, q_1, r)$ and $(E_2, q_2, p)$ be two general two-pointed elliptic curves. Identify the points $q_1, q_2$; identify the point $r$ with a moving point in $E_1$; finally move the curve $E_2$ in a pencil of degree $12$.
![How the general fiber of the family in §\[10\] moves.](TS10.ps)
The base of this surface is $E_1\times \mathbb{P}^1=:S_2$. Let $x$ be the class of a point in $\mathbb{P}^1$. The divisor classes restrict as follows $$\begin{aligned}
\psi |_{S_2} &= \pi_2^*(x) = \lambda |_{S_2}, & \delta_0 |_{S_2} &= -2\pi_1^*[r] +12\pi_2^*(x), &
\delta_{1,1} |_{S_2} &= -\pi_1^*[q_1] -\pi_2^*(x), & \delta_{2,1} |_{S_2} &= \pi_1^*[r], \end{aligned}$$ and the non-zero restrictions of the generating codimension-two classes are thus $$\begin{aligned}
\psi\delta_0 |_{S_2} &= -2, & \psi\delta_{1,1} |_{S_2} &= -1, & \psi\delta_{2,1} |_{S_2} &= 1, & \lambda \delta_0 |_{S_2} &= -2, \\
\lambda\delta_{2,1} |_{S_2} &= 1, & \delta_0^2 |_{S_2} &= -48, & \delta_0\delta_{1,1} |_{S_2} &= -10, & \delta_0\delta_{2,1}|_{S_2} &= 12, \\
\delta_{1,1}^2 |_{S_2} &= 2, & \delta_{1,1} \delta_{2,1} |_{S_2} &= -1. & &
\end{aligned}$$
### {#11}
Let $(E,p,q,r)$ be a general three-pointed elliptic curve. Consider the surface obtained by identifying the point $r$ with a moving point in $E$, and by attaching at the point $q$ an elliptic tail moving in a pencil of degree $12$.
![How the general fiber of the family in §\[11\] moves.](TS11.ps)
The base of this surface is $E\times \mathbb{P}^1=:S_3$. The divisor classes restrict as follows $$\begin{aligned}
\psi |_{S_3} &= \pi_1^*[p], & \lambda |_{S_3} &= \pi_2^*(x), &
\delta_0 |_{S_3} &= -2\pi_1^*[r] +12\pi_2^*(x), & \delta_{2,1} |_{S_3} &= -\pi_2^*(x). \end{aligned}$$ We deduce that the non-zero restrictions of product of divisor classes are the following $$\begin{aligned}
\psi\lambda |_{S_3} &= 1, & \psi\delta_0 |_{S_3} &= 12, & \psi\delta_{2,1} |_{S_3} &= -1, & \lambda\delta_0 |_{S_3} &= -2, & \delta_0^2 |_{S_3} &= -48, & \delta_0\delta_{2,1} |_{S_3} &=2.\end{aligned}$$ Moreover, we have $\delta_{01a}|_{S_3} = -\pi_1^*[q] \cdot 12\pi_2^*(x)= -12$.
Some equalities in A2BM31 {#secboundaryinA2Mbar31}
-------------------------
In §\[pullbacktoMbar31\] we also use the following result.
\[boundaryinA2Mbar31\] The following equalities hold in $A^2({\overline{\mathcal M}}_{3,1})$ $$\begin{aligned}
\lambda\delta_{1,1} &=& \frac{1}{5}\left({}-\psi+\frac{1}{2}\delta_0 +\delta_{2,1} \right)\delta_{1,1},\\
\delta_{00} &=& -12\psi^2 -24\psi\delta_{1,1}-372\lambda^2 +72\lambda\delta_0 +120\lambda\delta_{2,1} -3\delta_0^2 -12\delta_0\delta_{2,1}\\
&&{}-12\delta_{1,1}^2 -12\delta_{2,1}^2 +12\kappa_2,\\
\delta_{1|1} &=& \frac{1}{2}\psi^2 +\frac{1}{5}\psi\delta_{1,1} +5\lambda^2 -\frac{1}{2}\lambda\delta_0 +4\lambda\delta_{2,1} -\frac{1}{10}\delta_0\delta_{1,1} -\frac{1}{2}\delta_0\delta_{2,1}\\
&&{}-\frac{1}{2} \delta_{1,1}^2 -\frac{6}{5}\delta_{1,1}\delta_{2,1} -\frac{1}{2}\delta_{2,1}^2 +\frac{1}{12}\delta_{01a} -\frac{1}{2}\kappa_2,\\
\gamma_1 &=& \frac{78}{5}\psi\delta_{1,1} +126\lambda^2 -\frac{55}{2}\lambda\delta_0 -78\lambda\delta_{2,1} +\frac{3}{2}\delta_0^2 +\frac{7}{10}\delta_0\delta_{1,1} +\frac{17}{2}\delta_0\delta_{2,1}\\
&&{}+12\delta_{1,1}^2 +\frac{42}{5}\delta_{1,1}\delta_{2,1} +12\delta_{2,1}^2,\\
\gamma_2 &=& \frac{15}{2}\psi^2 -21\psi\lambda +2\psi\delta_0 +\psi\delta_{1,1} +3\psi\delta_{2,1} +\frac{101}{2}\lambda^2 -10\lambda\delta_0 -19\lambda\delta_{2,1}\\
&&{} +\frac{1}{2}\delta_0^2 +2\delta_0\delta_{2,1} +\frac{1}{2}\delta_{1,1}^2 +\frac{5}{2}\delta_{2,1}^2 -\frac{1}{2}\kappa_2.\end{aligned}$$
The idea is to follow the lines of the proof of Proposition \[basisR2Mbar31\]. Let us start with the second equality. Write $\delta_{00}$ as a linear combination of the classes in (\[bm31gen\]) with unknown coefficients. By pulling-back via the map $\vartheta\colon {\overline{\mathcal M}}_{2,2}\rightarrow {\overline{\mathcal M}}_{3,1}$ and using $\vartheta^*(\delta_{00})=\delta_{00}$ and Mumford’s equality $$\delta_{00} = 6\lambda\delta_0 \in R^2({\overline{\mathcal M}}_{2,2}),$$ we are able to determine the coefficients in the statement up to three variables $\alpha, \beta, \gamma$: $$\begin{aligned}
\delta_{00} &=& (-2 \alpha -\beta)\psi^2 +6(\alpha + \beta)\psi\lambda -\frac{\alpha+\beta}{2} \psi\delta_0 -2 \alpha\psi\delta_{1,1}\\
&&{} +(-6 \alpha - 5 \beta + 120 \gamma)\lambda^2 +(6 + \frac{\alpha + \beta}{2} - 22 \gamma)\lambda\delta_0 +(6 \alpha - 4 \beta)\lambda\delta_{2,1}\\
&&{}+ \gamma \delta_0^2 +\frac{\beta-\alpha}{2}\delta_0\delta_{2,1} -\alpha \delta_{1,1}^2 + \beta \delta_{2,1}^2, +\alpha \kappa_2 \in R^2({\overline{\mathcal M}}_{3,1}).\end{aligned}$$ Next, we restrict the above equality to the three test surfaces considered in §\[TSM31\]. It is easy to show that the restriction of $\delta_{00}$ to the first test surface is $0$, while is respectively $$\pi_1^*(-2[r]-[q_1])\cdot\pi_2^*(12x) + \pi_1^*[q_1]\cdot \pi_2^*(12x)=-24$$ and $$\pi_1^*(-2[r]-[p]-[q])\cdot\pi_2^*(12x) + \pi_1^*[p]\cdot\pi_2^*(12x)+ \pi_1^*[q]\cdot\pi_2^*(12x)=-24$$ to the other two test surfaces. It follows that $\alpha=12=-\beta$ and $\gamma= -3$, thus proving the equality for $\delta_{00}$.
The first equality is proven in a similar way. Alternatively, one obtains it as the push-forward of divisor relations via the boundary map ${\overline{\mathcal M}}_{1,2}\times {\overline{\mathcal M}}_{2,1}\rightarrow {\overline{\mathcal M}}_{3,1}$.
For the third equality, note that $$\vartheta^*(\delta_{1|1}) = -\psi_2\delta_{1,\{1,2\}} + \delta_{1|1} \in R^2({\overline{\mathcal M}}_{2,2})$$ and by Getzler’s computations $$\begin{aligned}
\delta_{1|1} &=& \psi_2^2 -\frac{1}{2}\psi_1\delta_{1,\{1\}} -\frac{7}{10}\psi_2\delta_{1,\{1\}}+ \frac{1}{2}\psi_1\delta_{1,\{1,2\}} -\frac{7}{10}\psi_2\delta_{1,\{1,2\}} -\frac{1}{10}\psi_2\delta_0 \\
&&{} + \delta_{0,\{1,2\}}^2 + \frac{1}{5}\delta_{1,\{1,2\}}\delta_{0,\{1,2\}}+ \frac{1}{10}\delta_0\delta_{0,\{1,2\}}+ \frac{3}{50}\delta_0\delta_{1,\{1\}} +\frac{11}{600}\delta_0\delta_{1,\{1,2\}}+ \frac{1}{200}\delta_0^2\end{aligned}$$ in $R^2({\overline{\mathcal M}}_{2,2})$. Finally, the restriction of $\delta_{1|1}\in R^2({\overline{\mathcal M}}_{3,1})$ to the third test surface in §\[TSM31\] is $$\pi_1^*[r] (-\pi_2^*(x))=-1,$$ while the restriction to the other two test surfaces is zero.
For $\gamma_1$ and $\gamma_2$, we use $$\begin{aligned}
\vartheta^*(\gamma_1) &= \gamma_1 + \gamma_{1,\{1\}}, & \vartheta^*(\gamma_2) &= \gamma_{1,\{2\}}.\end{aligned}$$ By Getzler’s computations, we have $$\begin{aligned}
\gamma_1 &= 3\psi_1\psi_2 +3\psi_2^2 -\frac{6}{5}\psi_1\delta_{1,\{1\}} -\frac{9}{5}\psi_2\delta_{1,\{1\}} -\frac{6}{5}\psi_1\delta_{1,\{1,2\}} -\frac{24}{5}\psi_2\delta_{1,\{1,2\}} -\frac{1}{10}\psi_1\delta_0\\
&{} -\frac{2}{5}\psi_2\delta_0 +12\delta_{0,\{1,2\}}^2 +\frac{42}{5}\delta_{1,\{1,2\}}\delta_{0,\{1,2\}} +\frac{7}{10}\delta_0\delta_{0,\{1,2\}} +\frac{3}{25}\delta_0(\delta_{1,\{1\}} +\delta_{1,\{1,2\}}) +\frac{1}{100}\delta_0^2,\\
\gamma_{1,\{1\}} &= 9\psi_2^2-3\psi_1\psi_2 +\frac{6}{5}\psi_1\delta_{1,\{1\}} -\frac{33}{5}\psi_2\delta_{1,\{1\}} +\frac{6}{5}\psi_1\delta_{1,\{1,2\}} -\frac{18}{5}\psi_2\delta_{1,\{1,2\}}
+\frac{1}{10}\psi_1\delta_0 -\frac{3}{10}\psi_2\delta_0, \\
\gamma_{1,\{2\}} &= 9\psi^2_2 -3\psi_1\psi_2 -\frac{24}{5}\psi_1\delta_{1,\{1\}} -\frac{3}{5}\psi_2\delta_{1,\{1\}} +\frac{36}{5}\psi_1\delta_{1,\{1,2\}} -\frac{48}{5}\psi_2\delta_{1,\{1,2\}}
+\frac{3}{5}\psi_1\delta_0 -\frac{4}{5}\psi_2\delta_0\end{aligned}$$ in $R^2({\overline{\mathcal M}}_{2,2})$. The intersection of $\gamma_1 \in R^2({\overline{\mathcal M}}_{3,1})$ with the first test surface in §\[TSM31\] is $$(-\pi_1^*([p]+[q_1])-\pi_2^*[q_2])\cdot(-\pi_1^*[q_1]-\pi_2^*[q_2])+\pi_1^*[p](-\pi_2^*[q_2])=2,$$ while it is zero with the other two test surfaces. Similarly, the intersection of $\gamma_2\in R^2({\overline{\mathcal M}}_{3,1})$ with the first test surface in §\[TSM31\] is $$\pi_1^*[p] (-\pi_2^*[q_2])=-1,$$ while it is zero with the other two test surfaces.
A check {#checkM31}
-------
As a partial check, we compute the expressions for $\delta_{00}$ and $\gamma_1$ in an alternative way. The classes $\delta_{00}$ and $\gamma_1$ in $R^2({\overline{\mathcal M}}_{3,1})$ coincide with the pull-back via $p\colon {\overline{\mathcal M}}_{3,1}\rightarrow {\overline{\mathcal M}}_3$ of the classes $\delta_{00}$ and $\gamma_1$ in $A^2({\overline{\mathcal M}}_3)$. Using the following identities in $A^2({\overline{\mathcal M}}_3)$ $$\begin{aligned}
\delta_{00} &=& {}-372 \lambda^2 +72 \lambda\delta_0 +120\lambda\delta_{1} -3\delta_0^2 -12\delta_0\delta_{1} -12\delta_{1}^2 +12\kappa_2,\\
\gamma_1 &=& 126 \lambda^2 -\frac{55}{2}\lambda\delta_0 -78\lambda\delta_1 +\frac{3}{2}\delta_0^2 +\frac{17}{2}\delta_0\delta_1 +12\delta_1^2\end{aligned}$$ from [@MR1070600 Theorem 2.10], and the following well-known identities $$\begin{aligned}
p^*(\lambda) &= \lambda, & p^*(\delta_0) &= \delta_0, & p^*(\delta_1) &= \delta_{1,1}+\delta_{2,1}, & \kappa_2 &= p^*(\kappa_2) + \psi^2,\end{aligned}$$ one recovers the two formulae in Proposition \[boundaryinA2Mbar31\].
The locus of Weierstrass points on hyperelliptic curves in BM31 {#pullbacktoMbar31}
===============================================================
In this section we compute the class of the closure in ${\overline{\mathcal M}}_{3,1}$ of the locus of Weierstrass points on hyperelliptic curves of genus $3$ $$\mathcal{H}yp_{3,1} := \{[C,p]\in {\mathcal M}_{3,1} \,|\, C \,\,\mbox{is hyperelliptic and}\,\, p \,\,\mbox{is a Weierstrass point in} \,\, C \}.$$ The idea is to consider the pull-back of the hyperelliptic locus in ${\overline{\mathcal M}}_4$ via the clutching map $j_3\colon \overline{\mathcal{M}}_{3,1}\rightarrow {\overline{\mathcal M}}_4$ obtained by attaching a fixed elliptic tail at the marked point of an element $[C,p]\in {\overline{\mathcal M}}_{3,1}$. Using the theory of admissible covers, the following statement is straightforward.
\[j3\^\*H\] We have the following equality in $A^2({\overline{\mathcal M}}_{3,1})$ $$\begin{aligned}
j_{3}^*\left(\overline{\mathcal{H}yp}_4\right) &=& \overline{\mathcal{H}yp}_{3,1}.\end{aligned}$$
Following [@MR1078265], we use the basis of $A^2({\overline{\mathcal M}}_4)$ given by the classes $$\begin{aligned}
\label{basisA2M4}
\lambda^2, \lambda\delta_0, \lambda\delta_1, \lambda\delta_2, \delta_0^2, \delta_0\delta_1, \delta_1^2,
\delta_1\delta_2, \delta_2^2, \delta_{00}, \delta_{01a}, \gamma_1, \delta_{1|1}.\end{aligned}$$ The class $\left[\overline{\mathcal{H}yp}_4 \right]$ in terms of this basis has been computed in [@MR2120989 Proposition 5]: $$\begin{aligned}
\label{hyp4}
\left[ \overline{\mathcal{H}yp}_{4} \right] & = & \frac{51}{4}\lambda^2 -\frac{31}{10}\lambda\delta_0 -\frac{117}{10}\lambda\delta_1 + 3\lambda\delta_2+ \frac{7}{40}\delta_0^2 + \frac{7}{5}\delta_0\delta_1\\
&& {} + \frac{21}{10}\delta_1^2 + 3\delta_1\delta_2 + \frac{9}{2}\delta_2^2+ \frac{1}{40}\delta_{00} -\frac{3}{10}\gamma_1 -\frac{3}{40}\delta_{01a}+ \frac{9}{10}\delta_{1|1}. \nonumber\end{aligned}$$ Note that the above class agrees with the one in [@MR2120989] modulo the relation (\[kappa2M4\]).
Let us compute the pull-back via $j_3\colon {\overline{\mathcal M}}_{3,1}\rightarrow {\overline{\mathcal M}}_4$ of the basis of $A^2({\overline{\mathcal M}}_4)$. We use the notation from §\[basisforMbar31\] for classes in $R^*({\overline{\mathcal M}}_{3,1})$. The pull-back of the basis of ${\operatorname{Pic}}({\overline{\mathcal M}}_4)$ is as follows $$\begin{aligned}
j_3^*(\lambda) &= \lambda, & j_3^*(\delta_0) &= \delta_0, & j_3^*(\delta_1) &= -\psi+\delta_{2,1}, & j_3^*(\delta_2) &= \delta_{1,1}.\end{aligned}$$ Moreover, we have $$\begin{aligned}
j_3^*(\delta_{01a}) &=& \delta_{01a},\\
j_3^*(\delta_{00}) &=& \delta_{00}\\
&=& -12\psi^2 -24\psi\delta_{1,1}-372\lambda^2 +72\lambda\delta_0 +120\lambda\delta_{2,1} -3\delta_0^2 -12\delta_0\delta_{2,1}\\
&&{}-12\delta_{1,1}^2 -12\delta_{2,1}^2 +12\kappa_2,\\
j_3^*(\delta_{1|1}) &=& {}-\psi\delta_{2,1}+\delta_{1|1}\\
&=&{} \frac{1}{2}\psi^2 +\frac{1}{5}\psi\delta_{1,1} -\psi\delta_{2,1}+5\lambda^2 -\frac{1}{2}\lambda\delta_0 +4\lambda\delta_{2,1} -\frac{1}{10}\delta_0\delta_{1,1}\\
&&{} -\frac{1}{2}\delta_0\delta_{2,1} -\frac{1}{2} \delta_{1,1}^2 -\frac{6}{5}\delta_{1,1}\delta_{2,1} -\frac{1}{2}\delta_{2,1}^2 +\frac{1}{12}\delta_{01a} -\frac{1}{2}\kappa_2,\\
j_3^*(\gamma_1) &=& \gamma_1 + \gamma_2\\
&=& \frac{15}{2}\psi^2 -21\psi\lambda +2\psi\delta_0 +\frac{83}{5}\psi\delta_{1,1} +3\psi\delta_{2,1} +\frac{353}{2}\lambda^2 -\frac{75}{2}\lambda\delta_0 -97\lambda\delta_{2,1}\\
&&{} +2\delta_0^2 +\frac{7}{10}\delta_0\delta_{1,1} +\frac{21}{2}\delta_0\delta_{2,1} +\frac{25}{2}\delta_{1,1}^2 +\frac{42}{5}\delta_{1,1}\delta_{2,1}
+\frac{29}{2}\delta_{2,1}^2 -\frac{1}{2}\kappa_2.\end{aligned}$$ For the above equalities we have used Proposition \[boundaryinA2Mbar31\].
As a partial check for the above formulae, remember that the following relation holds in $A^2({\overline{\mathcal M}}_4)$ $$\begin{gathered}
\label{kappa2M4}
60 \kappa_2 - 810 \lambda^2 + 156 \lambda\delta_0 + 252 \lambda \delta_1 -3\delta_0^2 - 24\delta_0\delta_1
{}+24\delta_1^2 -9\delta_{00}+7\delta_{01a} -12\gamma_1-84\delta_{1|1}=0\end{gathered}$$ (see [@MR1078265]). Using the identity $j_3^*(\kappa_2)=\kappa_2$, one can easily verify that the above formulae satisfy the pull-back via $j_3$ of this relation.
As a consequence of Proposition \[j3\^\*H\], by pulling-back via $j_3$ the class of the hyperelliptic locus $\overline{\mathcal{H}yp}_{4}$ in ${\overline{\mathcal M}}_{4}$ (\[hyp4\]) and by means of the above pull-back formulae, we obtain the class of $\overline{\mathcal{H}yp}_{3,1}$.
\[Hhyp31\] The class of $\overline{\mathcal{H}yp}_{3,1}$ in $A^2({\overline{\mathcal M}}_{3,1})$ is $$\overline{\mathcal{H}yp}_{3,1} \equiv \psi (18\lambda -2\delta_0 -9\delta_{1,1} -6\delta_{2,1})-\lambda\left(45\lambda -\frac{19}{2}\delta_0 -24\delta_{2,1}\right) -\frac{1}{2}\delta_0^2 -\frac{5}{2}\delta_0\delta_{2,1} -3\delta_{2,1}^2.$$
\[DR2\] An alternative way to prove Theorem \[Hhyp31\] is to follow the idea used in §\[basisforMbar31\]: the pull-back of $\overline{\mathcal{H}yp}_{3,1}$ via the natural map $\vartheta\colon {\overline{\mathcal M}}_{2,2}\rightarrow {\overline{\mathcal M}}_{3,1}$ is the closure of the locus $\mathcal{DR}_2(2)$ of curves of genus $2$ with two marked Weierstrass points. This locus is a special case of the double ramification locus, and its class in $A^2({\overline{\mathcal M}}_{2,2})$ has been computed in [@DR]: $$\overline{\mathcal{DR}}_2(2) \equiv 6\psi_1\psi_2 -3\psi_2^2 -\frac{12}{5}\psi_1\delta_{1,\{1\}} -\frac{9}{5}\psi_2\delta_{1,\{1\}} -\frac{12}{5}\psi_1\delta_{1,\{1,2\}}+ \frac{6}{5}\psi_2\delta_{1,\{1,2\}} -\frac{1}{5}\psi_1\delta_0+ \frac{1}{10} \psi_2\delta_0.$$ Imposing $\vartheta^*(\overline{\mathcal{H}yp}_{3,1})=\overline{\mathcal{DR}}_2(2)$ determines the class of $\overline{\mathcal{H}yp}_{3,1}$ up to three coefficients. Finally, one can find the values of such coefficients using three test surfaces.
The locus of marked hyperflexes in BM31 {#SF}
=======================================
In this section, we compute the class of the locus of genus-$3$ curves with a marked hyperflex point $$\mathcal{F}_{3,1} := \{[C,p]\in {\mathcal M}_{3,1} \,| \,\,\mbox{$C$ is not hyperelliptic and $p$ is a hyperflex point in $C$} \}.$$ Equivalently, $\mathcal{F}_{3,1}$ is the locus of pointed curves $[C,p]$ in ${\overline{\mathcal M}}_{3,1}$ such that $\mathcal{O}(2p)$ is an odd theta characteristic on $C$.
\[F\] The class of the closure of the locus ${\mathcal{F}}_{3,1}$ in $A^2({\overline{\mathcal M}}_{3,1})$ is $$\begin{aligned}
\overline{\mathcal{F}}_{3,1} &\equiv &{}-3\psi^2+ 77\psi\lambda -8\psi\delta_0 -42\psi\delta_{1,1} -19\psi\delta_{2,1} -338\lambda^2 +\frac{137}{2} \lambda\delta_0\\
&& {} +146 \lambda\delta_{2,1} -\frac{7}{2}\delta_0^2 -\frac{31}{2}\delta_0\delta_{2,1} -3\delta_{1,1}^2 -20\delta_{2,1}^2 +3\kappa_2.\end{aligned}$$
Let $\mathcal{W}_{3,1}$ be the divisor of Weierstrass points in ${\mathcal M}_{3,1}$ $$\mathcal{W}_{3,1} = \{[C,p]\in{\mathcal M}_{3,1} \,\,|\,\, 3p+x\sim K_C, \,\, \mbox{for some}\,\, x\in C \}.$$ The class of its closure has been computed in [@MR1016424] $$\overline{\mathcal{W}}_{3,1} \equiv -\lambda +6 \psi -3\delta_{1,1} -\delta_{2,1} \in {\operatorname{Pic}}({\overline{\mathcal M}}_{3,1}).$$ Let $\Theta_{3,1}$ be the divisor of non-hyperelliptic genus-$3$ curves with a marked point lying on one of the $28$ bitangents to the canonical model of the curve. Equivalently, $$\Theta_{3,1} :=\{[C,p]\in{\mathcal M}_{3,1} \,\,|\,\, p\in {\rm supp}(\eta^-),\,\,\mbox{for some $\eta^-$ an odd theta characteristic} \}.$$ From [@MR2779475 Theorem 0.3] we have $$\overline{\Theta}_{3,1} \equiv 7 \lambda+14\psi -\delta_0 -9\delta_{1,1} -5\delta_{2,1} \in {\operatorname{Pic}}({\overline{\mathcal M}}_{3,1}).$$
In order to prove Theorem \[F\], we show that $\overline{\mathcal{F}}_{3,1}$ is one of the components of the intersection of $\overline{\mathcal{W}}_{3,1}$ and $\overline{\Theta}_{3,1}$. The statement follows after considering all the components with the respective multiplicity in this intersection.
The intersection of [W]{}[3,1]{} and Theta[3,1]{} {#WTh}
-------------------------------------------------
Let us first consider the intersection of ${\mathcal{W}}_{3,1}$ and ${\Theta}_{3,1}$ in the locus of smooth curves ${\mathcal M}_{3,1}$.
The intersection of $\mathcal{W}_{3,1}$ and $\Theta_{3,1}$ consists of two components, corresponding to ${\mathcal{H}yp}_{3,1}$ and ${\mathcal{F}}_{3,1}$.
Let $[C,p]$ be a pointed smooth curve in the intersection of ${\mathcal{W}}_{3,1}$ with ${\Theta}_{3,1}$. Then, there exist $x,y\in C$ such that $3p+x\sim K_C \sim 2p+2y$. If $C$ is hyperelliptic, then we have $x=p$ and $y$ are Weierstrass points. If $C$ is not hyperelliptic, then from $p+x\sim 2y$ we deduce $p=x=y$ and $4p\sim K_C$, hence $p$ is a hyperflex point, and $[C,p]$ is in ${\mathcal{F}}_{3,1}$.
We now consider the components of the intersection of $\overline{\mathcal{W}}_{3,1}$ and the boundary $\Delta={\overline{\mathcal M}}_{3,1}\setminus {\mathcal M}_{3,1}$. Note that the divisor ${\mathcal{W}}_{3,1}$ corresponds to the locus of curves admitting a $\mathfrak{g}^1_3$ with a total ramification at the marked point, and its closure can be studied via admissible covers.
Let $(C,p,x,y)$ be a smooth $3$-pointed curve of genus $2$. If $[C/_{x\sim y}, p]$ admits a triple admissible cover totally ramified at $p$, then there exists $z\in C$ such that $3p\sim x+y+z$. We denote the closure of the locus of such curves by $(\overline{\mathcal{W}}_{3,1})_0$.
Consider a general element inside $\Gamma_1$: a pointed elliptic curve $(E_1,p)$ meeting an elliptic curve $E_2$ at two points. Such a curve admits a triple admissible cover totally ramified at $p$, whose restriction to $E_2$ has degree $2$. It follows that $\Gamma_1$ is in the intersection $\overline{\mathcal{W}}_{3,1}\cap\Delta_0$.
Similarly consider a general element inside $\Gamma_2$: a curve of genus $2$ meeting a rational curve with the marked point in two general points. Such a curve admits a triple admissible cover totally ramified at the marked point, with a simple ramification at one of the two nodes, and no ramification at the other node.
It is easy to see that no other codimension-two boundary locus inside $\Delta_0$ is entirely contained inside $\overline{\mathcal{W}}_{3,1}$, hence we conclude that $$\overline{\mathcal{W}}_{3,1}\cap\Delta_0 = (\overline{\mathcal{W}}_{3,1})_0 \cup \Gamma_1 \cup \Gamma_2.$$
Take an elliptic curve $(E,p,q)$ and attach at $q$ a general genus-$2$ curve $C$. Suppose that such a curve admits a triple admissible cover totally ramified at $p$. There are two cases: if the admissible cover has a simple ramification at $q$, then $q$ is a Weierstrass point in $C$. We denote the closure of the locus of such curves by $(\overline{\mathcal{W}}_{3,1})_{1a}$. If the admissible cover is totally ramified also at $q$, then we have $3p\sim 3q$, that is, $p-q$ is a non-trivial torsion point of order $3$ in ${\rm Pic}^0(E)$. We denote the closure of the locus of such curves by $(\overline{\mathcal{W}}_{3,1})_{1b}$. No other codimension-two boundary locus inside $\Delta_{1,1}$ is contained in $\overline{\mathcal{W}}_{3,1}$, hence we have $$\overline{\mathcal{W}}_{3,1}\cap\Delta_{1,1} = (\overline{\mathcal{W}}_{3,1})_{1a} \cup (\overline{\mathcal{W}}_{3,1})_{1b}.$$
Finally, consider a smooth $2$-pointed genus-$2$ curve $(C,p,q)$ and attach at $q$ an elliptic tail. If such a curve admits a triple admissible cover totally ramified at $p$, then there exists $z\in C\setminus \{p\}$ such that $3p\sim 2q+z$. We denote the closure of this locus by $(\overline{\mathcal{W}}_{3,1})_2$, and we have $$\overline{\mathcal{W}}_{3,1}\cap\Delta_{2,1} = (\overline{\mathcal{W}}_{3,1})_2.$$
Let us now consider the components of the intersection of $\overline{\Theta}_{3,1}$ and the boundary $\Delta$. Let $(C,p,x,y)$ be a smooth $3$-pointed genus-$2$ curve, and suppose that $[C/_{x\sim y},p]$ is inside $\overline{\Theta}_{3,1}$. There are two possibilities, corresponding to the two components of the space $\mathcal{S}^-_{3}$ over $\Delta_0$. One possibility is that $p$ is in the support of a line bundle $\eta$ such that $\eta^{\otimes 2}=K_C\otimes \mathcal{O}(x+y)$. In this case there exists $z\in C$ such that $2p+2z\sim K_C\otimes \mathcal{O}(x+y)$. We denote the closure of the locus of such curves by $(\overline{\Theta}_{3,1})_{0a}$. The other case is when $p$ is in the support of $\eta$ such that $\eta^{\otimes 2}=K_C$. In this case, $p$ is a Weierstrass point in $C$. We denote the closure of the locus of such curves by $(\overline{\Theta}_{3,1})_{0b}$.
Let $E_1, E_2$ be two elliptic curves meeting in two points, with the marked point in $E_1$. Consider the curve $\overline{E}_2$ in ${\overline{\mathcal M}}_{3,1}$ obtained by moving one of the nodes on the curve $E_2$. The intersection of the generating divisor classes is $$\begin{aligned}
\delta_0 |_{\overline{E}_2} &= -2, & \delta_{2,1} |_{\overline{E}_2} &= 1.\end{aligned}$$ We conclude that such a curve has negative intersection with $\overline{\Theta}_{3,1}$. Since a deformation of this curve covers an open subset of the locus $\Gamma_1$, we deduce that $\Gamma_1$ is inside $\overline{\Theta}_{3,1}$.
A similar argument holds for the locus $\Gamma_2$. Alternatively, the canonical model of a general curve in $\Gamma_2$ is an irreducible nodal quartic $C$, and the marked point coincides with the node. It is classically known that when smooth plane quartics specialize to $C$, the $28$ bitangents specialize to $16$ bitangents meeting $C$ away from the node, and $6$ bitangents of multiplicity two passing through the node (see for instance [@MR1949642 §3]). In particular, bitangents can specialize to lines passing through the node, hence blowing up the node we get a general curve in $\Gamma_2$, which implies that $\Gamma_2$ is contained in $\overline{\Theta}_{3,1}$.
It is easy to see that no other codimension-two boundary locus inside $\Delta_0$ is contained in $\overline{\Theta}_{3,1}$, hence we have $$\overline{\Theta}_{3,1}\cap\Delta_0 = (\overline{\Theta}_{3,1})_{0a} \cup (\overline{\Theta}_{3,1})_{0b} \cup \Gamma_1 \cup \Gamma_2.$$
Let $(E,p,q)$ be a $2$-pointed elliptic curve, and attach at $q$ a general curve of genus $2$. Suppose that such a curve is inside $\overline{\Theta}_{3,1}$. Then such a curve admits $((\mathcal{L}_1,\sigma_1),(\mathcal{L}_2,\sigma_2))$ a limit $\mathfrak{g}^0_{2}$, where $\mathcal{L}_1=\eta_1\otimes\mathcal{O}(2q)$ and $\mathcal{L}_2=\eta_2\otimes\mathcal{O}(q)$, $\eta_1,\eta_2$ are theta characteristics on $E,C$ with opposite parity, $\sigma_i\in H^0(\mathcal{L}_i)$, and $p\in {\rm supp}(\sigma_1)$. Note that if $\eta_2=\eta_2^+$ is even, then ${\rm ord}_q (\sigma_2)=0$, hence by compatibility ${\rm ord}_q (\sigma_1)=2$, which contradicts the fact that $\sigma_1$ vanishes also at $p$. Hence we have that $\eta_2=\eta_2^-$ is odd, and $\eta_1=\eta_1^+$ is even. Since $\eta_1^+$ has no sections, we have ${\rm ord}_q(\sigma_2)\geq 1$. There are two cases. If ${\rm ord}_q(\sigma_2)= 2$, then $q$ is a Weierstrass point in $C$. For each $p,q\in E$, we can always find $\eta_1^+$ and $z\in E$ such that $\eta_1^+\otimes \mathcal{O}(2q)=\mathcal{O}(p+z)$. We denote the closure of the locus of such curves by $(\overline{\Theta}_{3,1})_{1a}$. The other case is ${\rm ord}_q(\sigma_2)= 1$. By compatibility we have ${\rm ord}_q(\sigma_1)= 1$, hence $\eta_1^+\otimes \mathcal{O}(2q)=\mathcal{O}(p+q)$. This implies $\eta_1^+=\mathcal{O}(p-q)$. In particular, $2p\sim 2q$. We denote the closure of the locus of such curves by $(\overline{\Theta}_{3,1})_{1b}$. No other codimension-two boundary locus in $\Delta_{1,1}$ is inside $\overline{\Theta}_{3,1}$, hence we have $$\overline{\Theta}_{3,1}\cap\Delta_{1,1} = (\overline{\Theta}_{3,1})_{1a} \cup (\overline{\Theta}_{3,1})_{1b}.$$
Finally, let $(C,p,q)$ be a smooth $2$-pointed curve of genus $2$, and attach at $q$ an elliptic tail $E$. If such a curve is in $\overline{\Theta}_{3,1}$, then it admits $((\mathcal{L}_1,\sigma_1),(\mathcal{L}_2,\sigma_2))$ a limit $\mathfrak{g}^0_{2}$, where $\mathcal{L}_1=\eta_1\otimes\mathcal{O}(2q)$ and $\mathcal{L}_2=\eta_2\otimes\mathcal{O}(q)$, $\eta_1,\eta_2$ are theta characteristics on $E,C$ with opposite parity, $\sigma_i\in H^0(\mathcal{L}_i)$, and $p\in {\rm supp}(\sigma_2)$. There are two cases. If $\eta_2 = \eta_2^-$ is odd and $\eta_1 = \eta_1^+$ is even, then ${\rm ord}_q (\sigma_1)\leq 1$. By compatibility, we have that ${\rm ord}_q (\sigma_2)\geq 1$. Moreover, since $\sigma_2$ vanishes also at $p$, we deduce that ${\rm ord}_q (\sigma_2)= 1$, and $p$ is in the support of $\eta_2^-$, that is, $p$ is a Weierstrass point in $C$. We denote the closure of the locus of such curves by $(\overline{\Theta}_{3,1})_{2a}$. The other case is when $\eta_2 = \eta_2^+$ is even and $\eta_1 = \eta_1^-$ is odd. Since $h^0(\eta_2^+\otimes \mathcal{O}(q-p))\geq 1$, we have that $(p,q)$ belongs to the Scorza curve $T_{\eta_2^+}$ in $C\times C$. We denote the closure of the locus of such curves by $ (\overline{\Theta}_{3,1})_{2b}$. No other codimension-two boundary locus inside $\Delta_{2,1}$ is inside $\overline{\Theta}_{3,1}$, hence we have proven that $$\overline{\Theta}_{3,1}\cap\Delta_{2,1} = (\overline{\Theta}_{3,1})_{2a} \cup (\overline{\Theta}_{3,1})_{2b}.$$
Note that $(\overline{\mathcal{W}}_{3,1})_{1a}=(\overline{\Theta}_{3,1})_{1a}$. We rename this locus as $\overline{\mathcal{W}}_2:=(\overline{\mathcal{W}}_{3,1})_{1a}=(\overline{\Theta}_{3,1})_{1a}$. A general element of each component of $\overline{\Theta}_{3,1}\cap\Delta$ outside $\overline{\mathcal{W}}_2 \cup \Gamma_1 \cup \Gamma_2$ does not admit a triple admissible cover totally ramified at the marked point. Hence, $\overline{\Theta}_{3,1}\cap\Delta$ meets $\overline{\mathcal{W}}_{3,1}$ in codimension higher than two outside $\overline{\mathcal{W}}_2 \cup \Gamma_1 \cup \Gamma_2$. We have thus proven the following result.
\[lemmamnklj\] We have $$\begin{aligned}
\label{mnklj}
\left[ \overline{\mathcal{W}}_{3,1} \right] \cdot \left[\overline{\Theta}_{3,1}\right] = m\cdot \left[\overline{\mathcal{H}yp}_{3,1}\right] + n\cdot \left[\overline{\mathcal{F}}_{3,1} \right]+ k\cdot \left[\overline{\mathcal{W}}_2\right] + l\cdot \gamma_1 +j \cdot \gamma_2 \in A^2({\overline{\mathcal M}}_{3,1})\end{aligned}$$ for some coefficients $m,n,k,l,j$.
Expressing $\lambda\delta_{1,1}$ in terms of the other products of divisor classes (see Proposition \[boundaryinA2Mbar31\]), the left-hand side of (\[mnklj\]) is equal to $$\begin{aligned}
\left[\overline{\mathcal{W}}_{3,1}\right] \cdot \left[\overline{\Theta}_{3,1}\right] &= & 84 \psi^2 +28 \psi\lambda -6 \psi\delta_0 -\frac{468}{5}\psi\delta_{1,1} -44\psi\delta_{2,1} -7\lambda^2 + \lambda\delta_0\\
&& {} -2\lambda\delta_{2,1}+\frac{9}{5} \delta_0\delta_{1,1} +\delta_0\delta_{2,1} +27\delta_{1,1}^2 +\frac{108}{5} \delta_{1,1}\delta_{2,1} +5\delta_{2,1}^2.\end{aligned}$$ The classes $\gamma_1$ and $\gamma_2$ are in Proposition \[boundaryinA2Mbar31\], and the class of $\overline{\mathcal{H}yp}_{3,1}$ comes from Theorem \[Hhyp31\].
The class of $\overline{\mathcal{W}}_2$ in $A^2({\overline{\mathcal M}}_{3,1})$ is $$\left[\overline{\mathcal{W}}_2\right] = {}-\frac{9}{5} \psi\delta_{1,1} -\frac{1}{10} \delta_0\delta_{1,1} -3\delta_{1,1}^2 -\frac{6}{5} \delta_{1,1}\delta_{2,1}.$$
Let $\xi \colon {\overline{\mathcal M}}_{1,2} \times {\overline{\mathcal M}}_{2,1}\rightarrow \Delta_{1,1} \subset {\overline{\mathcal M}}_{3,1}$ be the map obtained by identifying the marked point on a genus-$2$ curve with the second marked point on an elliptic curve. Let $\pi_1 \colon {\overline{\mathcal M}}_{1,2} \times {\overline{\mathcal M}}_{2,1}\rightarrow {\overline{\mathcal M}}_{1,2}$ and $\pi_2 \colon {\overline{\mathcal M}}_{1,2} \times {\overline{\mathcal M}}_{2,1}\rightarrow {\overline{\mathcal M}}_{2,1}$ be the natural projections. Note that $\xi^*(\overline{\mathcal{W}}_2)$ is the pull-back via $\pi_2$ of the Weierstrass divisor in ${\overline{\mathcal M}}_{2,1}$, hence $\xi^*(\overline{\mathcal{W}}_2) \equiv \pi_2^* \left(3 \psi -\frac{1}{10} \delta_0 -\frac{6}{5} \delta_1 \right).$ On the other hand, $A^1(\Delta_{1,1})$ is generated by the classes $\psi\delta_{1,1}, \delta_0\delta_{1,1}, \delta_{1,1}^2, \delta_{1,1}\delta_{2,1}$, and we have the following pull-back formulae $$\begin{aligned}
\xi^*(\psi\delta_{1,1}) &= \pi_1^*(\psi_1), & \xi^*(\delta_0\delta_{1,1}) &= \pi_1^*(\delta_0) +\pi_2^*(\delta_0), \\
\xi^*(\delta_{1,1}^2) &= -\pi_1^*(\psi_2) - \pi_2^*(\psi), & \xi^*(\delta_{1,1}\delta_{2,1}) &= \pi_1^*(\delta_{0,\{1,2\}}) +\pi_2^*(\delta_{1,1}).\end{aligned}$$ Moreover, we have $\psi_i = \frac{1}{12} \delta_0 + \delta_{0,\{1,2\}}$ in $A^1({\overline{\mathcal M}}_{1,2})$, for $i=1,2$. Since the maps $\xi^*\colon A^1(\Delta_{1,1})\rightarrow A^1 ( {\overline{\mathcal M}}_{1,2} \times {\overline{\mathcal M}}_{2,1})$ and $A^1(\Delta_{1,1})\rightarrow A^2({\overline{\mathcal M}}_{3,1})$ are injective, the statement follows.
Test surfaces {#TSMbar31}
-------------
In this section, restricting (\[mnklj\]) to some test surfaces, we deduce linear relations among the coefficients $m,n,k,l,j$.
### {#TS5Mbar31}
Let $(E,q)$ be a pointed elliptic curve. Identify the point $q$ with a moving point $y$ on a general curve $C$ of genus $2$, and consider a moving marked point $x$ in $E$.
\[b\]\[b\][$C$]{} \[b\]\[b\][$E$]{} 
The base of this family is $E\times C=T_1$. Let $\pi_1\colon E\times C \rightarrow E$ and $\pi_2\colon E\times C \rightarrow C$ the natural projections. The non-zero divisors are $$\begin{aligned}
\psi |_{T_1} &= \pi_1^*([q]), & \delta_{1,1} |_{T_1} &= -\pi_1^*([q])-\pi_2^*(K_C), & \delta_{2,1} |_{T_1} &= \pi_1^*([q]).\end{aligned}$$ We deduce $$\begin{aligned}
\psi\delta_{1,1} |_{T_1} &= -2, & \delta_{1,1}^2 |_{T_1} &= 4, & \delta_{1,1} \delta_{2,1} |_{T_1} &= -2,\end{aligned}$$ and $$\begin{aligned}
\left[\overline{\mathcal{W}}_{3,1}\right] \cdot \left[\overline{\Theta}_{3,1}\right] |_{T_1} &= 252, & \left[\overline{\mathcal{H}yp}_{3,1}\right] |_{T_1} &= 18, & \left[\overline{\mathcal{W}}_2 \right] |_{T_1} &= -6, & \gamma_1 |_{T_1} =\gamma_2 |_{T_1} &=0.\end{aligned}$$
Let us consider the restriction of the class of $\overline{\mathcal{F}}_{3,1}$ to this test surface. If a fiber $(E\cup_y C,x)$ of this family is in the intersection with $\overline{\mathcal{F}}_{3,1}$, then it admits $((\mathcal{L}_E, \sigma_E),(\mathcal{L}_C, \sigma_C))$ a limit $\mathfrak{g}^0_{2}$ such that $\mathcal{L}_E=\eta_E\otimes\mathcal{O}(2y)$, $\mathcal{L}_C=\eta_C\otimes\mathcal{O}(y)$, $\eta_E$ and $\eta_C$ are theta characteristics on $E$, $C$ with opposite parity, and $\sigma_E\in H^0(\mathcal{L}_E)$, $\sigma_C\in H^0(\mathcal{L}_C)$. Moreover, we have ${\rm ord}_x \sigma_E =2$. This implies ${\rm ord}_y \sigma_E =0$, hence ${\rm ord}_y \sigma_C =2$. It follows that $\eta_C=\eta_C^-$ is an odd theta characteristic, $y$ is a Weierstrass point on $C$, and $\eta_E=\eta_E^+$ is even. Since $\eta_E^+\otimes \mathcal{O}(2y) = \mathcal{O}(2x)$, we deduce $\eta_E^+=\mathcal{O}(2x-2y)$. The map $E\times E \rightarrow {\rm Pic}^0(E)$ defined as $(x,y)\mapsto \mathcal{O}(2x-2y)$ has degree $4$. We conclude that there are $6\cdot 4 \cdot 3$ fibers of this family in the intersection with $\overline{\mathcal{F}}_{3,1}$. Since this family lies in the locus of compact type, each fiber counts with multiplicity $1$ (see for instance [@MR985853 Lemma 3.4 and the following Remark]). We deduce the following relation $$252 = 18m + 72n -6k.$$
### {#TS9Mbar31}
Let $C$ be a general curve of genus $2$. Let $x,y\in C$ be two moving points in $C$. Consider the surface obtained by attaching an elliptic tail at the point $y$.
\[b\]\[b\][$C$]{} ![How the general fiber of the family in §\[TS9Mbar31\] moves.](TS9.ps "fig:")
The base of this family is $C\times C=:T_2$. Let $\pi_i\colon C\times C \rightarrow C$ be the natural projection, for $i=1,2$. The non-zero restrictions of the divisor classes are $$\begin{aligned}
\psi |_{T_2} &= \pi_1^*(K_C)+\Delta_{C\times C}, & \delta_{1,1} |_{T_2} &= \Delta_{C\times C}, & \delta_{2,1} |_{T_2} &= -\pi_1^*(K_C)-\Delta_{C\times C}. \\\end{aligned}$$ Hence we deduce $$\begin{aligned}
\psi^2 |_{T_2} &= 2, & \delta_{2,1}^2 |_{T_2} &= 2, & \psi\delta_{2,1} |_{T_2} &= -6, & \delta_{1,1}^2 |_{T_2} &= -2.\end{aligned}$$ Note that the restriction of $\kappa_2$ is equal to the restriction of $\kappa_2$ to the surface in ${\overline{\mathcal M}}_{2,2}$ obtained by forgetting the elliptic tail. After (\[kappa2Mbar22\]), the class $\kappa_2$ is equivalent to a product of divisor classes in ${\overline{\mathcal M}}_{2,2}$, hence we easily compute $$\kappa_2 |_{T_2} =2.$$ We deduce $$\begin{aligned}
\left[\overline{\mathcal{W}}_{3,1}\right] \cdot \left[\overline{\Theta}_{3,1}\right] |_{T_2} &= 388, & \left[\overline{\mathcal{H}yp}_{3,1}\right] |_{T_2} &= 30, & \left[\overline{\mathcal{W}}_2 \right] |_{T_2} &= 6, & \gamma_1|_{T_2} =\gamma_2 |_{T_2} &=0.\end{aligned}$$
If a fiber $(E\cup_y C,x)$ of this family is in the intersection with $\overline{\mathcal{F}}_{3,1}$, then it admits a limit linear series $((\mathcal{L}_E, \sigma_E), (\mathcal{L}_C, \sigma_C))$ of type $\mathfrak{g}^0_{2}$ such that $\mathcal{L}_E=\eta_E\otimes\mathcal{O}(2y)$, $\mathcal{L}_C=\eta_C\otimes\mathcal{O}(y)$, $\eta_E$ and $\eta_C$ are theta characteristics on $E$, $C$ with opposite parity, $\sigma_E\in H^0(\mathcal{L}_E)$, $\sigma_C\in H^0(\mathcal{L}_C)$, and ${\rm ord}_x \sigma_C =2$. This implies ${\rm ord}_y \sigma_C =0$ and ${\rm ord}_y \sigma_E =2$, hence $\eta_E=\eta_E^-=\mathcal{O}_E$ is odd and $\eta_C=\eta_C^+$ is even. Note that $\eta_C^+\otimes\mathcal{O}(y)=\mathcal{O}(2x)$. Since the difference map $C\times C\rightarrow {\rm Pic}^0(C)$ defined as $(x,y)\mapsto \mathcal{O}(2x-y)$ has degree $8$ (see Proposition \[diff\]), we deduce that there are $8\cdot 10$ fibers of this family in the intersection with $\overline{\mathcal{F}}_{3,1}$. Each fiber counts with multiplicity one, and we have $$388 = 30m +80n +6k.$$
### {#TS13Mbar31}
Consider a chain of $3$ elliptic curves, with a marked point on an external component. Vary the central elliptic component in a pencil of degree $12$, and vary one of the singular points on the central component.
![How the general fiber of the family in §\[TS13Mbar31\] moves.](TS13.ps)
The base of this surface is the blow-up $T_3$ of $\mathbb{P}^2$ at the $9$ points of intersection of two general cubics. Let $H$ be the pull-back of an hyperplane section in $\mathbb{P}^2$, let $\Sigma$ be the sum of the $9$ exceptional divisors, and $E_0$ one of them. We have $$\begin{aligned}
\lambda |_{T_3} &= 3H-\Sigma, & \delta_0 |_{T_3} &= 36 H-12\Sigma,&
\delta_{1,1} |_{T_3} &= {}-3H +\Sigma -E_0, & \delta_{2,1} |_{T_3} &= {}-3H+\Sigma\end{aligned}$$ (see [@MR1078265 §3 ($\lambda$)]). We deduce $$\begin{aligned}
\delta_0\delta_{1,1} |_{T_3} &= -12, & \delta_{1,1}^2 |_{T_3} &= 1, & \delta_{1,1}\delta_{2,1} |_{T_3} &= 1,\end{aligned}$$ and moreover we have $$\begin{aligned}
\delta_{01a} |_{T_3} &=12, & \kappa_2 |_{T_3} &= 1.\end{aligned}$$ Indeed, there are $12$ fibers of this family contributing to the intersection with $\delta_{01a}$, namely when the central elliptic component degenerates to a rational normal curve and the moving node collides with the other non-disconnecting node. This family has zero intersection with $\delta_{00}$, hence we deduce the restriction of $\kappa_2$ from Proposition \[boundaryinA2Mbar31\]. Finally, we have $$\begin{aligned}
\left[\overline{\mathcal{W}}_{3,1}\right] \cdot \left[\overline{\Theta}_{3,1}\right] |_{T_3} &= 27, & \left[\overline{\mathcal{H}yp}_{3,1}\right] |_{T_3} &= 0, & \left[\overline{\mathcal{W}}_2 \right] |_{T_3} &= -3, & \gamma_1 |_{T_3} &=12, &\gamma_2 |_{T_3} &=0.\end{aligned}$$
Since we can choose the marked point generically in one of the elliptic tails, from an analysis similar to the one in §\[TS5Mbar31\], we deduce that this family is disjoint from $\overline{\mathcal{F}}_{3,1}$. Hence, we have $$27 = -3k +12 l.$$
Push-forward to BM3 {#pushfwdtoM3}
-------------------
In this section, we compute the push-forward of (\[mnklj\]) via the forgetful map $p\colon {\overline{\mathcal M}}_{3,1}\rightarrow {\overline{\mathcal M}}_3$. We use the following formulae $$\begin{aligned}
p_*(\psi^2) &= \kappa_1 = 12 \lambda -\delta_0 - \delta_1, & p_*(\psi\lambda) &= 4\lambda, & p_*(\psi\delta_0) &= 4\delta_0,\\
p_*(\kappa_2) &= p_*(p^*(\kappa_2)+\psi^2) = \kappa_1, & p_*(\psi\delta_{1,1})&= \delta_1, & p_*(\psi\delta_{2,1})&= 3\delta_1,\\
p_*(\delta_{1,1}^2) &= -\delta_1, & p_*(\delta_{1,1}\delta_{2,1})&= \delta_1, & p_*(\delta_{2,1}^2) &= -\delta_1.\end{aligned}$$ All other classes in Proposition \[basisR2Mbar31\] have zero push-forward (see for instance [@MR1953519 Theorem 2.8]). We deduce the following $$p_*(\overline{\mathcal{W}}_{3,1} \cdot \overline{\Theta}_{3,1}) = 1120 \lambda -108 \delta_0 -320\delta_1.$$ Moreover, we have $$\begin{aligned}
p_*(\overline{\mathcal{H}yp}_{3,1}) &= 8(9\lambda-\delta_0-3\delta_1), & p_*(\Gamma_2) &= \delta_0, & p_*(\Gamma_1)=p_*(\overline{\mathcal{W}}_2) &= 0.\end{aligned}$$
The push-forward of the locus $\overline{\mathcal{F}}_{3,1}$ coincides with the push-forward via the forgetful map $\pi\colon \mathcal{S}^-_3\rightarrow {\overline{\mathcal M}}_3$ of the closure of the divisor $\mathcal{Z}_3$ in $\mathcal{S}^-_3$ of curves with an odd spin structure vanishing twice at a certain point. This class has been computed in [@MR1016424 pg. 345]: $$p_*(\overline{\mathcal{F}}_{3,1}) = \pi_*(\overline{\mathcal{Z}}_3) \equiv 308 \lambda -32\delta_0 -76\delta_1 \in {\operatorname{Pic}}({\overline{\mathcal M}}_{3})$$ (see also [@MR3245010 Theorem 0.5]). From equation (\[mnklj\]), we conclude $$\begin{aligned}
m &= 7, & n &= 2, & j &= 12.\end{aligned}$$
Proof of Theorem \[F\] {#proofF}
----------------------
In §\[WTh\], we have realized the locus $\overline{\mathcal{F}}_{3,1}$ as one of the components of the intersection of the divisors $\overline{\mathcal{W}}_{3,1}$ and $\overline{\Theta}_{3,1}$. In order to compute the class of $\overline{\mathcal{F}}_{3,1}$ it remains to find the multiplicities $m,n,k,l,j$ in (\[mnklj\]). From the study of the push-forward to ${\overline{\mathcal M}}_3$ in §\[pushfwdtoM3\], we have found that $m=7$, $n=2$ and $j=12$. In §\[TSMbar31\], using test surfaces we have three linear relations involving $m,n,k,l$. We deduce $k=l=3$, and we have one more relation as a check. The statement follows. $\square$
The locus BH+4 {#h+4}
==============
In this section, we compute the class of the closure in ${\overline{\mathcal M}}_4$ of the locus $$\mathcal{H}^+_4 := \{[C]\in {\mathcal M}_4 \,|\, \mathcal{O}(3x) \,\, \mbox{is an even theta characteristic for some $x\in C$} \}.$$ The strategy is similar to the one used to compute the class of $\overline{\mathcal{F}}_{3,1}$ in §\[SF\], that is, we realize $\overline{\mathcal{H}}^+_4$ as a component of the intersection of two divisors. Namely, we consider the divisor of curves with a vanishing theta-null $$\Theta_{\rm null} :=\{ [C] \in {\mathcal M}_4 \, | \, h^0(\eta^+)>0 \,\, \mbox{for some $\eta^+$ an even theta characteristic} \}$$ and the divisor of curves admitting a $\mathfrak{g}^1_3$ with a total ramification point $$\mathcal{T} := \{ [C]\in {\mathcal M}_4 \, | \, h^0(\mathcal{O}(3x))\geq 2 \,\, \mbox{for some $x\in C$} \}.$$ The classes of the closures of $\Theta_{\rm null}$ and $\mathcal{T}$ in ${\overline{\mathcal M}}_4$ are special cases of divisor classes computed in [@MR937985] and [@MR791679], respectively: $$\begin{aligned}
\overline{\Theta}_{\rm null} & \equiv 34 \lambda - 4 \delta_0 - 14 \delta_1 - 18 \delta_2, &
\overline{\mathcal{T}} & \equiv 264 \lambda - 30 \delta_0 - 96 \delta_1 - 128 \delta_2. \end{aligned}$$
The intersection of Theta[null]{} and T {#intThT}
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Let us first consider the intersection of ${\Theta}_{\rm null}$ and ${\mathcal{T}}$ in ${\mathcal M}_4$.
The intersection of $\Theta_{\rm null}$ and $\mathcal{T}$ in ${\mathcal M}_4$ has two components, corresponding to $\mathcal{H}yp_4$ and $\mathcal{H}_4^+$.
It is clear that $\mathcal{H}yp_4$ and $\mathcal{H}_4^+$ are contained in both $\Theta_{\rm null}$ and $\mathcal{T}$. Conversely, suppose that $C$ is a smooth curve contained in both $\Theta_{\rm null}$ and $\mathcal{T}$. In particular, there exists $x$ in $C$ such that $h^0(\mathcal{O}(3x))\geq 2$. If $C$ is not hyperelliptic, $3x$ admits a unique $\mathfrak{g}^1_3$ equal to its canonical residual, i.e. $6x \sim K_C$, hence $C$ is in $\mathcal{H}_4^+$.
Next, we analyze the intersection of the divisor $\overline{\Theta}_{\rm null}$ and the boundary $\Delta:={\overline{\mathcal M}}_4\setminus {\mathcal M}_4$. Note that the divisor $\overline{\Theta}_{\rm null}$ of genus-$4$ curves with a vanishing theta-null coincides with the Gieseker-Petri divisor of genus-$4$ curves whose canonical model lies on a quadric cone.
Let $(C,x,y)$ be a two-pointed curve of genus $3$, and suppose that $[C/_{x \sim y}]$ is inside $\overline{\Theta}_{\rm null}$. One possibility is that there exists $r$ in $C$ such that $2(p+q+r)\sim K_C\otimes \mathcal{O}(p+q)$. We denote the locus of such curves by $(\overline{\Theta}_{\rm null})_{0a}$. Equivalently, this is the locus of irreducible nodal curves whose canonical model lies on a quadric cone, with the node being away from the vertex of the cone. The other possibility is that $2(p+q)\sim K_C$, that is, $C$ is hyperelliptic. We denote the locus of such curves by $(\overline{\Theta}_{\rm null})_{0b}$. This corresponds to the locus of irreducible nodal curves whose canonical model lies in a quadric cone and the node coincides with the vertex.
Let $\Gamma_1$ be the closure of the locus of curves with an elliptic component meeting a component of genus $2$ in two points. Consider the one-dimensional family $E$ of curves obtained by identifying two general points on a general curve of genus $2$ with a fixed point and a moving point on an elliptic curve. The restriction of the generating divisor classes is $$\begin{aligned}
\delta_0 |_{E} &= -2, & \delta_1 |_{E} &= 1.\end{aligned}$$ It follows that such a family has negative intersection with $\overline{\Theta}_{\rm null}$. Since this family produces a moving curve in $\Gamma_1$, we deduce that $\Gamma_1$ is contained inside $\overline{\Theta}_{\rm null}$.
No other codimension-two boundary component inside $\Delta_0$ is entirely contained in $\overline{\Theta}_{\rm null}$, hence we have $$\overline{\Theta}_{\rm null} \cap \Delta_0 = (\overline{\Theta}_{\rm null})_{0a} \cup (\overline{\Theta}_{\rm null})_{0b}\cup \Gamma_1.$$
Next, let $(C,y)$ be a smooth pointed genus-$3$ curve, and let $(E,y)$ be an elliptic curve. If $[C\cup_y E]$ is in $\overline{\Theta}_{\rm null}$, then it admits a limit linear series $(l_C, l_E) = ((\mathcal{L_C}, V_C), (\mathcal{L}_E, V_E))$ of type $\mathfrak{g}^1_3$, where $\mathcal{L}_C=\eta_C \otimes \mathcal{O}(y)$, $\mathcal{L}_E=\eta_E \otimes \mathcal{O}(3y)$, and $\eta_C, \eta_E$ are theta characteristics on $C,E$, respectively, with same parity. Note that one has $a^{l_C}(y)\geq (0,2)$. If $a_0^{l_C}(y)= 0$ and $a_1^{l_C}(y) \geq 2$, we deduce that $y$ is in the support of $\eta_C$, and $\eta_C=\eta_C^-$ (as well as $\eta_E = \eta_E^-$) is an odd theta characteristic. We denote the locus of such curves by $(\overline{\Theta}_{\rm null})_{1a}$. This corresponds to the locus of cuspidal curves in a quadric cone such that the cusp is not the vertex. If $a^{l_C}(y)= (1,3)$, then $C$ is a hyperelliptic curve, and $y$ is a Weierstrass point in $C$. We denote the locus of such curves by $(\overline{\Theta}_{\rm null})_{1b}$. This corresponds to the locus of cuspidal curves in a quadric cone such that the cusp is the vertex. No other codimension-two boundary component inside $\Delta_1$ is entirely contained in $\overline{\Theta}_{\rm null}$, hence we have $$\overline{\Theta}_{\rm null} \cap \Delta_1 = (\overline{\Theta}_{\rm null})_{1a}\cup (\overline{\Theta}_{\rm null})_{1b}.$$
Suppose $[C_1\cup_y C_2]$ is in $\overline{\Theta}_{\rm null}$, where $C_1$ and $C_2$ are two smooth curves of genus $2$ attached at a point $y$. Then $[C_1\cup_y C_2]$ admits a limit linear series $(l_{C_1}, l_{C_2}) = ((\mathcal{L_{C_1}}, V_{C_1}), (\mathcal{L}_{C_2}, V_{C_2}))$ of type $\mathfrak{g}^1_3$, where $\mathcal{L}_{C_i}=\eta_{C_i} \otimes \mathcal{O}(2y)$ for $i=1,2$, and $\eta_{C_1}, \eta_{C_2}$ are theta characteristics on $C_1, C_2$, respectively, with same parity. If $a^{l_{C_1}}(y)= (0,2)$, then $a^{l_{C_2}}(y)= (1,3)$, and the only other possibility is obtained by switching the two curves. This implies that $y$ is a Weierstrass point on $C_2$. We denote the locus of such curves by $(\overline{\Theta}_{\rm null})_2$. We have $$\overline{\Theta}_{\rm null} \cap \Delta_2 = (\overline{\Theta}_{\rm null})_2.$$
Finally, we consider the intersection of the divisor $\overline{\mathcal{T}}$ and the boundary $\Delta$. The closure of ${\mathcal{T}}$ corresponds to curves admitting a triple admissible cover totally ramified at some nonsingular point $x$.
If $[C/_{x\sim y}]$ is an irreducible nodal curve in $\overline{\mathcal{T}}$, then there exist $x$ and $z$ in $C$ such that $x+y+z\sim 3x$. We denote the locus of such curves by $\overline{\mathcal{T}}_{0}$. Moreover, a general curve inside $\Gamma_1$ has a triple admissible cover totally ramified at a point $x$ in the elliptic component, with a simple ramification at one of the two nodes, and no ramification at the other node. This is the only codimension-two boundary component inside $\Delta_0$ and $\overline{\mathcal{T}}$, hence we have $$\overline{\mathcal{T}} \cap \Delta_0 = \overline{\mathcal{T}}_{0}\cup \Gamma_1.$$
Let $(C,y)$ be a pointed curve of genus $3$, and let $(E,y)$ be an elliptic curve. Suppose that $[C\cup_y E]$ is in $\overline{\mathcal{T}}$, that is, $[C\cup_y E]$ has a triple admissible cover totally ramified at some point $x$. There are two cases. If $x$ is in $C$, then there exists $r$ in $C$ such that $3x\sim 2y+r$. We denote the locus of such curves by $\overline{\mathcal{T}}_{1a}$. If $x$ is in $E$, then $y$ is a Weierstrass point in $C$. Note that $x-y$ is a nontrivial $3$-torsion point in ${\rm Pic}^0(E)$. We denote the locus of such curves by $\overline{\mathcal{T}}_{1b}$. There are no other codimension-two boundary components inside $\Delta_1$ and $\overline{\mathcal{T}}$, hence we have $$\overline{\mathcal{T}} \cap \Delta_1 = \overline{\mathcal{T}}_{1a}\cup \overline{\mathcal{T}}_{1b}.$$
Consider the stable curve $[C_1\cup_y C_2]$ obtained by identifying a point on two smooth curves $C_1$ and $C_2$ of genus $2$. Suppose $[C_1\cup_y C_2]$ admits a triple admissible cover totally ramified at some point $x$ in $C_1$. If the restriction of the cover to $C_2$ has degree $2$, then $y$ is a Weierstrass point in $C_2$. We denote the locus of such curves by $\overline{\mathcal{T}}_{2a}$. Otherwise, if the restriction of the cover to $C_2$ has degree $3$, then $3x\sim 3y$ on $C_1$. This is a codimension-one condition on $(C_1,y)$. We denote the locus of such curves by $\overline{\mathcal{T}}_{2b}$, and we have $$\overline{\mathcal{T}} \cap \Delta_2 = \overline{\mathcal{T}}_{2a}\cup \overline{\mathcal{T}}_{2b}.$$
Note that $\overline{\mathcal{T}}_{2a} = (\overline{\Theta}_{\rm null})_2$. Let us define $\overline{\mathcal{W}}_2:= \overline{\mathcal{T}}_{2a} = (\overline{\Theta}_{\rm null})_2$. A general element of each component of $\overline{\Theta}_{\rm null}\cap\Delta$ outside $\overline{\mathcal{W}}_2 \cup \Gamma_1$ does not admit a triple admissible cover totally ramified at some smooth point. Hence, $\overline{\Theta}_{\rm null}\cap\Delta$ meets $\overline{\mathcal{T}}$ in codimension higher than two outside $\overline{\mathcal{W}}_2 \cup \Gamma_1$. We thus obtain the following result.
We have $$\begin{aligned}
\label{mnkl}
\left[\overline{\Theta}_{\rm null}\right] \cdot \left[\overline{\mathcal{T}}\right] = m \cdot \left[\overline{\mathcal{H}yp}_4\right] + n \cdot \left[\overline{\mathcal{H}}_4^{+} \right] + k \cdot \left[\overline{\mathcal{W}}_2 \right]+ l\cdot \gamma_1 \in A^2 ({\overline{\mathcal M}}_4)\end{aligned}$$ for some coefficients $m,n,k,l$.
Using the relation $(10\lambda - \delta_0 - 2\delta_1 ) \delta_2 = 0$ in $A^2({\overline{\mathcal M}}_4)$, we can write the left-hand side of (\[mnkl\]) as $$\begin{aligned}
\left[\overline{\Theta}_{\rm null}\right] \cdot \left[\overline{\mathcal{T}} \right] & = & 8976\lambda^2 - 2076 \lambda\delta_0 - 6960\lambda\delta_1 + 1416\lambda\delta_2 +
120 \delta_0^2 \\
&&{}+ 804\delta_0\delta_1 + 1344\delta_1^2 + 1416 \delta_1\delta_2 + 2304 \delta_2^2. \end{aligned}$$ The class of $\overline{\mathcal{H}yp}_4$ is in (\[hyp4\]). It remains to compute the class of $\overline{\mathcal{W}}_2$.
The class of $\overline{\mathcal{W}}_2$ in $A^2({\overline{\mathcal M}}_4)$ is $$\left[\overline{\mathcal{W}}_2\right] = {}- \lambda\delta_2 - \delta_1 \delta_2 - 3\delta_2^2 .$$
Let $\xi \colon {\overline{\mathcal M}}_{2,1}\times {\overline{\mathcal M}}_{2,1} \to \Delta_2\subset {\overline{\mathcal M}}_4$ be the gluing morphism, and let $\pi_i\colon {\overline{\mathcal M}}_{2,1}\times {\overline{\mathcal M}}_{2,1} \to {\overline{\mathcal M}}_{2,1}$ be the natural projection on the $i$-th factor, for $i=1,2$. Note that $\xi^{-1}(\overline{\mathcal{W}}_2)$ is the union of the pull-backs of the Weierstrass divisor from both factors. Recall that the Weierstrass divisor in ${\overline{\mathcal M}}_{2,1}$ has class $3\psi - \frac{1}{10}\delta_0 -\frac{6}{5}\delta_1$. This implies that $$\xi^* \left( \left[\overline{\mathcal{W}}_2 \right]\right) = 3 \left( \pi_1^*(\psi) + \pi_2^*(\psi) \right) - \frac{1}{10}\left(\pi_1^*(\delta_0) + \pi_2^*(\delta_{0})\right) - \frac{6}{5} \left(\pi_1^*(\delta_{1}) + \pi_2^*(\delta_{1})\right).$$ Note that $A^1(\Delta_2)$ is generated by $\delta_0\delta_2$, $\delta_1\delta_2$, and $\delta_2^2$. Moreover, $$\begin{aligned}
\xi^{*}(\delta_{0}\delta_2) &= \pi_1^* (\delta_0) + \pi_2^*(\delta_{0}), & \xi^{*} (\delta_{1}\delta_2) &= \pi_1^*(\delta_1) + \pi_2^*(\delta_{1}), & \xi^{*} (\delta_2^2) &= {}- \pi_1^*(\psi) - \pi_2^*(\psi). \end{aligned}$$ We thus conclude that $$\begin{aligned}
\left[\overline{\mathcal{W}}_2 \right] = {}- \frac{1}{10} \delta_0\delta_2 - \frac{6}{5}\delta_1\delta_2 - 3 \delta_2^2 = {}- \lambda\delta_2 - \delta_1 \delta_2 - 3\delta_2^2 \end{aligned}$$ in $A^1(\Delta_2)$. Since $A^1(\Delta_2) \to A^2({\overline{\mathcal M}}_4)$ is injective, the same formula holds in $A^2({\overline{\mathcal M}}_4)$.
Test surfaces {#TS4}
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In this section we restrict (\[mnkl\]) to four test surfaces in order to compute the coefficients $m,n,k,l$.
### {#H4-1}
Let us consider the test surface in $\overline{\mathcal{M}}_4$ obtained by attaching two general curves $C_1$ and $C_2$ of genus $2$ at one point $y$, and moving the point $y$ on both curves. The base of the family of such curves is $C_1\times C_2=: V_1$. We denote by $\pi_i\colon C_1\times C_2 \rightarrow C_i$ the projection to the $i$-th component, for $i=1,2$.
![How the general fiber of the family in §\[H4-1\] moves.](H4-1.ps)
The intersection of this test surface with $\delta_2^2$ is $$\delta_2^2 |_{V_1} = (\pi_1^*K_{C_1}\otimes \pi_2^*K_{C_2})^2 =8$$ while all other generating classes restrict to zero. We deduce that $$\begin{aligned}
\left[\overline{\Theta}_{\rm null}\right] \cdot \left[\overline{\mathcal{T}}\right] |_{V_1} &= 18432, & \left[\overline{\mathcal{H}yp}_4\right] |_{V_1} &= 36, & \left[\overline{\mathcal{W}}_2\right] |_{V_1} &= -24, & \gamma_1 |_{V_1} &=0.\end{aligned}$$
Let us consider the intersection of $\overline{\mathcal{H}}^+_4$ with this test surface. If an element $C_1 \cup_y C_2$ of this family lies in the intersection of $\overline{\mathcal{H}}^+_4$, then it admits $(l_{C_1}, l_{C_2})=((\mathcal{L}_{C_1}, V_{C_1}),(\mathcal{L}_{C_2}, V_{C_2}))$ a limit $\mathfrak{g}^1_{3}$ with $\mathcal{L}_{C_i}=\eta_i\otimes \mathcal{O}(2y)$, for $i=1,2$, where $\eta_1, \eta_2$ are theta characteristics of the same parity, respectively on $C_1, C_2$. Moreover, one of the two linear series $\mathfrak{g}^1_{3}$ has a section vanishing with order $3$ at some point $x$.
Suppose that $x$ specializes to the singular point $y$ of $C_1 \cup_y C_2$. Consider the pointed curve $(C_1 \cup R \cup C_2, x)$ in ${\overline{\mathcal M}}_{4,1}$ lying above $C_1 \cup_y C_2$, where $R$ is a rational component connecting $C_1$ and $C_2$ and containing $x$. If $C_1 \cup_y C_2$ is in $\overline{H}^+_4$, then $(C_1 \cup R \cup C_2, x)$ admits a limit $\mathfrak{g}^3_6$ whose $R$-aspect has a section vanishing with order $6$ at $x$. It is easy to see that this violates the Plücker formula for the total number of ramification points of a linear series on a rational curve. Hence, in the following we assume that $x$ is a smooth point.
Suppose that $x$ is in $C_1$ (the case $x$ in $C_2$ is analogous and will multiply the final answer by a factor $2$). There are two cases.
First, consider the case when $\eta_1, \eta_2$ are even theta characteristics $\eta_1^+, \eta_2^+$. Since $h^0(\eta_1^+\otimes \mathcal{O}(2y-3x))= 1$, we have $a_0^{l_{C_1}}(y)=0$, hence $a_1^{l_{C_2}}(y)=3$. This implies $h^0(\eta_2^+\otimes \mathcal{O}(-y))= 1$, a contradiction, since $\eta_2^+$ is an even theta characteristic on a general curve.
Next, consider the case when $\eta_1, \eta_2$ are odd theta characteristics $\eta_1^-, \eta_2^-$. Similarly as before, we have that $h^0(\eta_1^-\otimes \mathcal{O}(2y-3x))\geq 1$, hence $a_0^{l_{C_1}}(y)=0$, and $a_1^{l_{C_2}}(y)=3$. This implies $h^0(\eta_2^-\otimes \mathcal{O}(-y))= 1$, that is, $y$ is in the support of $\eta_2^-$, that is, $y$ is a Weierstrass point of $C_2$. Since $h^0(\eta_2^-\otimes \mathcal{O}(-y))= 1$, from Riemann-Roch we have $h^0(\eta_2^-\otimes \mathcal{O}(y))=h^0(\eta_2^-\otimes \mathcal{O}(2y))= 2$, hence $a_0^{l_{C_2}}(y)=1$. This implies $a_1^{l_{C_1}}(y)= 2$ (in particular $y$ is not in the support of $\eta_1^-$). Hence there exists a point $z$ in $C_1$ such that $\mathcal{O}(3x)=\mathcal{O}(2y+z)=\eta_1^-\otimes\mathcal{O}(2y)$. This implies $\mathcal{O}(3x-2y)=\eta_1^-=\mathcal{O}(z)$. By Proposition \[diff\], the map $C_1\times C_1 \rightarrow {\rm Pic}^1(C_1)$ given by $(x,y)\mapsto \mathcal{O}(3x-2y)$ is surjective of degree $72$.
Moreover, since $y$ is not in the support of $\eta_1^-$, one has to exclude the pairs $(x,y)$ such that $3x\sim 3y$ and $y$ is in the support of $\eta^-$, that is, $y$ is a Weierstrass point. These conditions imply that $x=y$ is a Weierstrass point. By Proposition \[diff\], the map $C_1\times C_1 \rightarrow {\rm Pic}^1(C_1)$ is generically simply ramified along the diagonal $\Delta\subset C_1\times C_1$ and the locus $I$ of hyperelliptic conjugate pairs, and it admits a triple ramification at the points $\Delta\cap I$. One has to exclude also the cases $x=z$, that is, $2x\sim 2y$ and $x\not=y$, since we do not assume a base point at $x$. We conclude that in this case the number of desired pairs $(x,y)\in C_1\times C_1$ is $(72-3)\cdot 6-6\cdot 5=384$.
Note that, since every element of the test surface $C_1\times C_2$ is a curve of compact type, one can show that the intersection of $\overline{\mathcal{H}}^+_4$ with $C_1\times C_2$ is transverse at every point (see for instance [@MR985853 Lemma 3.4]). It follows that the restriction of $\overline{\mathcal{H}}^+_4$ to this test surface is $$\left[ \overline{\mathcal{H}}^+_4\right] \Big|_{V_1} = 2 \cdot 384\cdot 6 = 4608.$$ Hence, we deduce the following relation $$18432 = 36 \cdot m + 4608 \cdot n -24 \cdot k.$$
### {#H4-2}
Let $C$ be a general curve of genus $2$. Let us consider the surface obtained by attaching two elliptic tails $E_1, E_2$ at two varying points $y_1, y_2$ in $C$. The base of this family is $C\times C=:V_2$. Let $\pi_i\colon C\times C \rightarrow C$ be the natural projection on the $i$-th component, for $i=1,2$.
\[b\]\[b\][$C$]{} \[b\]\[b\][$E_1$]{} \[b\]\[b\][$E_2$]{} ![How the general fiber of the family in §\[H4-2\] moves.](H4-2.ps "fig:")
The non-zero restrictions of the generating classes are $$\begin{aligned}
\delta^2_1 |_{V_2} &=& ({}-\pi_1^* K_C -\pi_2^* K_C -2\Delta_{C\times C} )^2 = 16,\\
\delta^2_2 |_{V_2} &=& \Delta^2_{C\times C}= -2,\\
\delta_{1|1} |_{V_2} &=& ({}-\pi_1^* K_C -\Delta_{C\times C})( {}-\pi_2^* K_C -\Delta_{C\times C} )= 6.\end{aligned}$$ We deduce that $$\begin{aligned}
\left[\overline{\Theta}_{\rm null}\right] \cdot \left[\overline{\mathcal{T}}\right] |_{V_2} &= 16896, & \left[ \overline{\mathcal{H}yp}_4\right] |_{V_2} &= 30, & \left[\overline{\mathcal{W}}_2 \right] |_{V_2} &= 6, & \gamma_1 |_{V_2} &=0.\end{aligned}$$
If an element $E_1 \cup_{y_1} C \cup_{y_2} E_2$ of this family lies in $\overline{\mathcal{H}}^+_4$, then it admits a limit $\mathfrak{g}^1_{3}$ $$(l_{E_1}, l_{C}, l_{E_2})=((\mathcal{L}_{E_1}, V_{E_1}),(\mathcal{L}_{C}, V_{C}),(\mathcal{L}_{E_2}, V_{E_2}))$$ with $\mathcal{L}_{E_i}=\eta_{E_i}\otimes \mathcal{O}(3y_i)$, for $i=1,2$, and $\mathcal{L}_{C}=\eta_{C}\otimes \mathcal{O}(y_1+y_2)$, where $\eta_{E_1}, \eta_C, \eta_{E_2}$ are theta characteristics respectively on $E_1, C, E_2$, either all even, or two odd and one even. Moreover, one of the linear series $\mathfrak{g}^1_{3}$ has a section vanishing with order $3$ at some point $x$. As in §\[H4-1\], $x$ cannot specialize to a singular point.
Suppose that $x$ is in $E_1$ (the case $x$ in $E_2$ is similar, hence the answer will be multiplied by $2$). Since $a_1^{l_{E_1}}(x)=3$, we have $a_0^{l_{E_1}}(y_1)=0$, hence $a_1^{l_C}(y_1)=3$ and $a_0^{l_{C}}(y_2)=0$. It follows that $a_1^{l_{E_2}}(y_2)=3$, hence $h^0(\eta_{E_2})\geq 1$. This implies that $\eta_{E_2}$ is an odd theta characteristic $\eta_{E_2}^-$. There are two cases.
Let us consider the case when $\eta_{E_1}=\eta_{E_1}^+$ is an even theta characteristic and $\eta_{C}=\eta_{C}^-$ is an odd theta characteristic. Since $h^0(\eta_{E_1}^+)=0$, we have $a_1^{l_{E_1}}(y_1)\leq 2$, hence necessarily $a^{l_{E_1}}(y_1)=(0,2)$. This implies $a^{l_{C}}(y_1)=(1,3)$. It follows that $y_1$ is a Weierstrass point in $C$ and $l_C=y_1+|2y_1|$. Then $a^{l_{C}}(y_2)=(0,2)$, hence $y_2$ is also a Weierstrass point in $C$. It follows that $l_{E_2}=y_2+|2y_2|$. Regarding the $E_1$ aspect, there exists $z\not= y_1$ in $E_1$ such that $\mathcal{O}(3x)=\mathcal{O}(2y_1+z)=\eta_{E_1}^+\otimes\mathcal{O}(3y_1)$. Hence, we have $\eta_{E_1}^+=\mathcal{O}(z-y_1)$ and $x$ satisfies $3x\sim 3z$, $x\not=z$. We conclude that this case gives the contribution $2\cdot (6\cdot 5)\cdot 3\cdot 8=1440$ to the intersection of this family with $\overline{\mathcal{H}}^+_4$.
Next, we consider the case when $\eta_{E_1}=\eta_{E_1}^-$ is an odd theta characteristic and $\eta_{C}=\eta_{C}^+$ is an even theta characteristic. Again $a_0^{l_{E_1}}(y_1)=0$, hence $a^{l_{E_1}}(y_1)=(0,3)=a^{l_{C}}(y_1)$ (note that $l_{E_1}$ must have no base points). It follows that $a_0^{l_C}(y_2)=0$. Note that $a_1^{l_C}(y_2)\geq 2$. Since $h^0(\mathcal{L}_C\otimes \mathcal{O}(-3y_1))\geq 1$, from §\[Scorza\] we deduce that $a^{l_C}(y_2)=(0,2)$, hence $l_{E_2}=y_2+|2y_2|$. Moreover, since $h^0(\eta_{C}^+\otimes \mathcal{O}(y_2-2y_1))\geq 1$, we have $\eta_{C}^+=\mathcal{O}(2y_1-y_2)$. From Proposition \[diff\], for each $\eta_C^+$ the number of such pairs is $8$. On $E_1$, since $h^0(\mathcal{O}(3y_1-3x))\geq 1$, we have that $y_1-x$ is a nontrivial $3$-torsion point in ${\rm Pic}^0(E_1)$. Hence the total contribution from this case is $2\cdot 8\cdot 8\cdot 10=1280$.
Let us suppose that $x$ is in $C$. We have necessarily $a_0^{l_C}(y_i)=0$ and $a_1^{l_{E_i}}(y_i)=3$. In particular $h^0(\mathcal{L}_{E_i}\otimes\mathcal{O}(-3y_1))\geq 1$, hence $\eta_{E_i}=\eta_{E_i}^-$ is an odd theta characteristic for $i=1,2$. It follows that $\eta_C=\eta_C^+$ is an even theta characteristic. From $h^0(\mathcal{L}_C\otimes \mathcal{O}(-2y_i))\geq 1$ for $i=1,2$, we deduce that the pair $(y_1,y_2)$ belongs to the Scorza curve $ T_{\eta^+_C} $ (see §\[Scorza\]), hence necessarily $y_1\not= y_2$. Moreover, $x\in C$ satisfies $h^0(\mathcal{L}_C\otimes \mathcal{O}(-3x))\geq 1$, hence $\eta^+_C=\mathcal{O}(3x-y_1-y_2)$. It remains to count the number of triples $(x,y_1,y_2)\in C\times C\times C$ such that $(y_1,y_2)$ belongs to the Scorza curve $T_{\eta^+}$ with $\eta^+=\mathcal{O}(3x-y_1-y_2)$. The class of $T_{\eta^+}$ in $H^2(C\times C)$ is in §\[Scorza\] after [@MR1213725]. Consider $f\colon C \times C \times C \to {\operatorname{Pic}}^1(C)$ sending $(x, y_1, y_2)$ to $\mathcal{O}(3x - y_1 - y_2)$. Let $\pi_{i,j}\colon C \times C \times C \to C\times C$ be the projection to the $i$-th and $j$-th factors, for $i,j\in\{1,2,3\}$. We want to compute $$\deg (\pi_{2,3}^{*} \left[T_{\eta^{+}}\right] \cdot f^{*} [\eta^{+}]) =\deg (f_{*} (\pi_{2,3}^{*} F_1 + \pi_{2,3}^{*} F_2 + \pi_{2,3}^{*} \Delta_{C\times C})\cdot [\eta^{+}]).$$ Note that $f$ restricted to $\pi_{2,3}^{*} F_1$ is the map $C\times C \to {\operatorname{Pic}}^1(C)$ sending $(x, y)$ to $\mathcal{O}(3x - y - q)$ where $q\in C$ is a fixed point. From Proposition \[diff2\] we have that $\deg f_{*} (\pi_{2,3}^{*} F_1) \cdot [\eta^{+}] = 3^2 \cdot 1^2\cdot 2 = 18$. Similarly, we have $\deg f_{*} (\pi_{2,3}^{*} F_2) \cdot [\eta^{+}] = 18$. Finally, $f$ restricted to $\pi_{2,3}^{*} \Delta_{C\times C}$ is the map $C\times C\to {\operatorname{Pic}}^1(C)$ sending $(x, y)$ to $\mathcal{O}(3x - 2y)$, which from Proposition \[diff\] has degree $3^2 \cdot 2^2 \cdot 2 = 72$. We conclude that $$\deg (\pi_{2,3}^{*} \left[T_{\eta^{+}}\right] \cdot f^{*} [\eta^{+}] ) = 18 + 18 + 72 = 108.$$ Moreover, we have to exclude the cases $x=y_1$ or $x=y_2$. The map $f$ is simply ramified at $\pi_{1,2}^*\Delta_{C\times C}$ and $\pi_{1,3}^*\Delta_{C\times C}$, and again by Proposition \[diff\] the degree of the restriction of $f$ to these two loci is $8$. We have to exclude also the cases when $x$ is a base point, that is, $\mathcal{O}(3x)=\mathcal{O}(2y_1+x)=\mathcal{O}(2y_2+x)$ and $\eta_{C}^+=\mathcal{O}(x-y_1+y_2)$. This happens when $x,y_1,y_2$ are different Weierstrass points. We conclude that the contribution given by the case $x\in C$ is $$(108- 2\cdot2\cdot8)\cdot 10-6\cdot 5\cdot 4=640.$$ Note that $a^{l_{E_i}}=(0,3)$ or $(1,3)$, and in each case the aspect $l_{E_i}$ is uniquely determined, for $i=1,2$.
It is easy to see that the fibers of this family over the diagonal $\Delta_{C\times C}$ are disjoint from $\overline{\mathcal{H}}^+_4$. We have thus shown that on this family $$\left[\overline{\mathcal{H}}^+_4 \right] \Big|_{V_2} = 1440 + 1280 + 640 =3360.$$ Hence, we have the following relation $$16896 = 30 \cdot m + 3360 \cdot n + 6 \cdot k.$$
### {#H4-6}
Attach at an elliptic curve $F$ a general curve $C$ of genus $2$ and an elliptic tail. Consider the surface obtained by varying $F$ in a pencil of degree $12$, and by varying one of the singular points on $F$.
![How the general fiber of the family in §\[H4-6\] moves.](H4-6.ps)
The base of this family is the blow-up $V_3$ of $\mathbb{P}^2$ in the nine points of intersection of two general cubics, as in §\[TS13Mbar31\] (see also [@MR1078265 §3 ($\lambda$)]). With the same notation from §\[TS13Mbar31\], we have $$\begin{aligned}
\lambda |_{V_3} &= 3H-\Sigma, & \delta_0 |_{V_3} &= 36 H -12 \Sigma, &
\delta_1 |_{V_3} &= {}-3H+\Sigma, & \delta_2 |_{V_3} &= {}-3H + \Sigma - E_0.\end{aligned}$$ Hence, we deduce $$\begin{aligned}
\delta_1\delta_2 |_{V_3} &= 1, & \lambda\delta_2 |_{V_3} &= -1, & \delta_2^2 |_{V_3} &= 1.\end{aligned}$$ Moreover, we have $$\begin{aligned}
\delta_{01a} |_{V_3} &= 12, & \gamma_1 |_{V_3} &= 12.\end{aligned}$$ Indeed, there are $12$ fibers of this family contributing to the intersection with $\delta_{01a}$, namely when $F$ degenerates to a rational nodal curve and the moving node collides with the other non-disconnecting node. Similarly, there are $12$ fibers contributing to the intersection with $\gamma_1$, namely when $F$ degenerates to a rational nodal curve and the moving node collides with the disconnecting node. Each of these fibers contributes with multiplicity one.
The above restrictions imply $$\begin{aligned}
\left[\overline{\Theta}_{\rm null}\right] \cdot \left[\overline{\mathcal{T}}\right] |_{V_3} &= 2304, & \left[\overline{\mathcal{H}yp}_4 \right] |_{V_3} &= 0, & \left[\overline{\mathcal{W}}_2\right] |_{V_3} &=-3.\end{aligned}$$
Since we assume that the singular point $q$ in $C$ is a general point in $C$, this surface has empty intersection with $\overline{\mathcal{H}}^+_4$. Indeed, suppose an element of this family is in $\overline{\mathcal{H}}^+_4$. Then, such an element admits a limit $\mathfrak{g}^1_{3}$ with one of the aspects vanishing to order $3$ at a certain point $x$. Note that the line bundle of the $C$-aspect is $\mathcal{L}_C=\eta_C\otimes \mathcal{O}(2q)$, for a certain theta characteristic $\eta_C$ of $C$. If $x$ is in $C$, then one has $h^0(\mathcal{L}_C\otimes\mathcal{O}(-3x))\geq 1$, hence $\eta_C=\mathcal{O}(3x-2q)$, a contradiction, since $q$ is general in $C$ (see Proposition \[diff\]). If $x$ is not in $C$, then necessarily $h^0(\mathcal{L}_C\otimes\mathcal{O}(-3q))\geq 1$, hence $q$ is a Weierstrass point in $C$, a contradiction.
We deduce the following relation $$2304 = {}-3\cdot k + 12 \cdot l.$$
### {#H4-8}
Let $(R, q_1, q_2, q_3, q_4, q_5)$ be a $5$-pointed rational curve. Attach at $q_1$ a general curve $C$ of genus $2$, and identify $q_2$ with $q_3$, and $q_4$ with $q_5$. Consider the family of curves obtained by varying the two moduli of the $5$-pointed rational curve.
![The general fiber of the family in §\[H4-8\].](H4-8.ps)
Since the point $q_1$ is general in $C$, this surface is disjoint from $\overline{\mathcal{H}}^+_4$. The argument is similar to the one in §\[H4-6\].
The base of this family is ${\overline{\mathcal M}}_{0,5}=:V_4$. We denote by $D_{i,j}$ the divisor in ${\overline{\mathcal M}}_{0,5}$ of curves with two rational components meeting transversally in a point, with the marked points $i$ and $j$ in a component, and the other three points in the other component, for $i,j\in\{1,2,3,4,5\}$. We denote by $\psi_i$ the cotangent line class in ${\overline{\mathcal M}}_{0,5}$ corresponding to the point marked by $i$, for $i\in \{1,2,3,4,5\}$. Let $j,k$ be two markings different from $i$, and let $l,m$ be the other two markings. The following relation is well-known $$\psi_i = D_{j,k} + D_{i,l} + D_{i,m}.$$
We have $$\begin{aligned}
\lambda |_{V_4} &=& 0,\\
\delta_0 |_{V_4} &\equiv& {}-\psi_2 -\psi_3 -\psi_4 -\psi_5 + D_{1,2} + D_{1,3} + D_{1,4} + D_{1,5} + D_{2,4} + D_{2,5} + D_{3,4} + D_{3,5}\\
&\equiv& {}-2D_{4,5} -D_{2,3} -D_{3,4} -D_{1,2} + D_{1,4},\\
\delta_1 |_{V_4} &\equiv& D_{4,5} + D_{2,3} \\
\delta_2 |_{V_4} &=& -\psi_1.\end{aligned}$$ Using $D_{i,j}^2=-1$, $D_{i,j}\cdot D_{k,l}=1$ if $\{i,j\}\cap \{k,l\}=\emptyset$, and $D_{i,j}\cdot D_{k,l}=0$ otherwise, we have $$\begin{aligned}
\delta_2^2 |_{V_4} &= 1, & \delta_1\delta_2 |_{V_4} &= -2,& \delta_{1|1} |_{V_4} &= 1.\end{aligned}$$ We also have $$\begin{aligned}
\gamma_1 |_{V_4} \equiv (D_{1,2}+ D_{1,3}) ({}-\psi_2 -\psi_3) + (D_{1,4}+ D_{1,5}) ({}-\psi_4 -\psi_5) + D_{2,4}\cdot D_{3,5} + D_{2,5}\cdot D_{3,4} = -2.\end{aligned}$$ Indeed, along the divisors $D_{1,2}, D_{1,3}, D_{1,4}, D_{1,5}$ when the disconnecting node collides with a non-disconnecting node, the fibers of this family all contribute to the intersection with $\gamma_1$. By the excess intersection formula, these contributions equal the restriction of the normal bundle at the two points corresponding to the non-disconnecting node. Moreover, the fibers of this family in the intersections $D_{2,4}\cdot D_{3,5}$ and $D_{2,5}\cdot D_{3,4}$ give additional contributions.
We deduce $$\begin{aligned}
\left[\overline{\Theta}_{\rm null}\right] \cdot \left[\overline{\mathcal{T}}\right] |_{V_4} &= -528, & \left[\overline{\mathcal{H}yp}_4\right] |_{V_4} &= 0, & \left[\overline{\mathcal{W}}_2 \right] |_{V_4} &=-1,\end{aligned}$$ and the following relation follows $$-528 = {}- k -2 \cdot l.$$
Proof of Theorem \[h4+\]
------------------------
In §\[intThT\], we have seen that the locus $\overline{\mathcal{H}}^+_4$ is a component of the intersection of the divisors $\overline{\Theta}_{\rm null}$ and $\overline{\mathcal{T}}$, and we have analyzed the other components of the intersection. To compute the class of $\overline{\mathcal{H}}^+_4$, it remains to compute the coefficients $m,n,k,l$ in (\[mnkl\]). Restricting to the test surfaces in §\[TS4\], we have found the four linear relations $$\begin{aligned}
18432 &=& 36 \cdot m + 4608 \cdot n -24 \cdot k,\\
16896 &=& 30 \cdot m + 3360 \cdot n + 6 \cdot k,\\
2304 &=& {}-3\cdot k + 12 \cdot l,\\
-528 &=& {}- k -2 \cdot l,\end{aligned}$$ whence we deduce that $m = 320$, $n = 2$, $k = 96$, and $l = 216$. The class of $\overline{\mathcal{H}}^+_4$ follows. $\square$
Complete intersections {#ci}
======================
In this section, we prove that the loci $\overline{\mathcal{H}yp}_{3,1}$, $\overline{\mathcal{F}}_{3,1}$, and $\overline{\mathcal{H}}^+_4$ are not complete intersections in their respective spaces.
Modulo the relation $(5 \lambda + \psi - \frac{1}{2}\delta_0 -\delta_{2,1} ) \delta_{1,1} =0$ in $R^2({\overline{\mathcal M}}_{3,1})$ (see Proposition \[boundaryinA2Mbar31\]), the product of two divisor classes in ${\overline{\mathcal M}}_{3,1}$ can be written in terms of the basis of $R^2({\overline{\mathcal M}}_{3,1})$ in Proposition \[basisR2Mbar31\]. The resulting coefficient of $\kappa_2$ is zero. From Theorem \[res31\], it follows that $\overline{\mathcal{F}}_{3,1}$ is not a complete intersection in ${\overline{\mathcal M}}_{3,1}$.
Similarly, modulo the relation $(10\lambda - \delta_0 - 2\delta_1)\delta_2=0$ in $R^2({\overline{\mathcal M}}_4)$, the product of two divisor classes in ${\overline{\mathcal M}}_4$ can be written in terms of the basis in (\[basisA2M4\]). The resulting coefficients of the classes $\delta_{00}$, $\gamma_1$, $\delta_{01a}$, and $\delta_{1|1}$ are zero, hence from Theorem \[h4+\], $\overline{\mathcal{H}}^+_4$ is not a complete intersection in ${\overline{\mathcal M}}_4$.
Finally, suppose that the class of $\overline{\mathcal{H}yp}_{3,1}$ is a product of two [*effective*]{} divisor classes. Imposing the product of two arbitrary divisor classes in ${\overline{\mathcal M}}_{3,1}$ to be equal to the class of $\overline{\mathcal{H}yp}_{3,1}$ in Theorem \[res31\] (modulo $(5 \lambda + \psi - \frac{1}{2}\delta_0 -\delta_{2,1} ) \delta_{1,1} =0$) yields $14$ relations in the $10$ coefficients of the two arbitrary divisor classes. This forces the two divisor classes to be a multiple of $p^*[\overline{\mathcal{H}yp}_{3}]$ and a multiple of $D:=2\psi-5\lambda+\frac{1}{2}\delta_0 + \delta_2$. The class of the Weierstrass divisor $\overline{\mathcal{W}}_{3,1}\equiv 6\psi-\lambda-3\delta_1-\delta_2$ is inside the cone generated by $D$ and the effective classes $\psi$ and $p^*[\overline{\mathcal{H}yp}_{3}]$. This contradicts the extremality of the class of $\overline{\mathcal{W}}_{3,1}$ (see [@MR3034451] or [@MR3071469]). It follows that $D$ is not effective, hence $\overline{\mathcal{H}yp}_{3,1}$ is not a complete intersection in ${\overline{\mathcal M}}_{3,1}$.
Using the class of $\overline{\mathcal{H}yp}_{4}$ computed in [@MR2120989 Proposition 5] (see (\[hyp4\])), the above argument for $\overline{\mathcal{H}}^+_4$ shows that $\overline{\mathcal{H}yp}_{4}$ is also not a complete intersection in ${\overline{\mathcal M}}_4$.
The determinantal description of H4 and H-4
===========================================
As a partial check on Theorem \[h4+\], in this section we compute the coefficient of $\lambda^2$ in the class of $\overline{\mathcal{H}}^+_4$ using a determinant description of ${\mathcal{H}}_4$ in ${\mathcal M}_4$. At the same time, we also compute the coefficient of $\lambda^2$ in the class of $\overline{\mathcal{H}}^-_4$.
The locus H-4 {#det-}
-------------
Let $\mathcal{SH}^-_4$ be the locus in $\mathcal{S}^-_4$ of odd spin genus-$4$ curves $[C,\eta_C^-]$ such that the natural map $$\varphi \colon H^0(C,\eta^-_C) \rightarrow H^0(C, \eta^-_C|_{3x})$$ has rank zero for some point $x$ in $C$. Note that the locus ${{\mathcal H}}^-_4$ in $\mathcal{M}_4$ is the push-forward of $\mathcal{SH}^-_4$ via the natural map $\pi\colon \mathcal{S}^-_4 \rightarrow {\mathcal M}_4$.
Let $p\colon \mathcal{C}\rightarrow \mathcal{S}^-_4$ be the universal curve and $\eta^-\in {\rm Pic}(\mathcal{C})$ be the universal spin bundle of relative degree $g-1$. Note that $p_* \eta^-$ is a line bundle outside a locus of codimension at least $3$ in $\mathcal{S}^-_4$. We can ignore such locus since we will only deal with Chern classes $c_i(p_* \eta^-)$ with $i<3$. In particular $c_2(p_* \eta^-)=0$. The map $\varphi$ globalizes to a map of vector bundles $$\widetilde{\varphi} \colon p^* p_* \eta^- \rightarrow J_2(\eta^-)$$ respectively of rank $1$ and $3$ over $\mathcal{C}$, where $J_2(\eta^-)$ is the second jet bundle of $\eta^-$. We are interested in the locus of curves $C$ where $\widetilde{\varphi}$ has rank zero. By Porteous formula, we have $$[\mathcal{SH}^-_4]= p_* c_3 \left( J_2(\eta^-) - p^* p_* \eta^- \right) \in A^2(\mathcal{S}^-_4).$$
Let us compute the Chern classes of $p^* p_* \eta^-$ and $J_2(\eta^-)$. Since $(\eta^-)^{\otimes 2} \simeq \omega_p$, we deduce $ch(\eta^-)=e^{\frac{1}{2}\psi}$, where $\psi=c_1(\omega_p)$. From Grothendieck-Riemann-Roch, we have $$\begin{aligned}
ch(p_* \eta^-)-ch(R^1 p_* \eta^-) &=& p_* \left( td(\omega_p^\vee) \cdot ch(\eta^-) \right)\\
&=& p_* \left( \frac{\psi}{e^\psi -1} \cdot e^{\frac{1}{2}\psi} \right)\\
&=& p_* \left( (1-\frac{1}{2}\psi +\frac{1}{12}\psi^2 -\frac{1}{720}\psi^4+\cdots) \cdot e^{\frac{1}{2}\psi} \right)\\
&=& p_* \left( 1 -\frac{1}{24}\psi^2 +\frac{7}{5760}\psi^4 +\cdots \right)\\
&=& {}-\frac{1}{24}\kappa_1 +\frac{7}{5760}\kappa_3 +\cdots. \end{aligned}$$ In particular, we have $$2c_1(p_* \eta^-) = {}-\frac{1}{24}\kappa_1 = {}-\frac{1}{2}\lambda.$$ From the standard exact sequence $$0\rightarrow \eta^- \otimes Sym^n \omega_p \rightarrow J_n (\eta^-) \rightarrow J_{n-1} (\eta^-) \rightarrow 0$$ we compute $$\begin{aligned}
ch(J_2(\eta^-)) = ch(\eta^- \otimes (1 \oplus \omega_p \oplus Sym^2 \omega_p))
= e^{\frac{1}{2}\psi} \cdot (1 +e^\psi +e^{2\psi})
= 3 +\frac{9}{2}\psi +\frac{35}{8}\psi^2 +\frac{51}{16}\psi^3 +\!\cdots\end{aligned}$$ whence we deduce $$c (J_2(\eta^-)) = 1 + \frac{9}{2}\psi +\frac{23}{4}\psi^2 + \frac{15}{8}\psi^3.$$ We obtain $$\begin{aligned}
[\mathcal{SH}^-_4] &=& p_* \left(c_3( J_2(\eta^-)) - c_1(p_* \eta^-)c_2( J_2(\eta^-)) +c_1^2(p_* \eta^-)c_1( J_2(\eta^-)) \right)\\
&=& \frac{15}{8} \kappa_2 +\frac{23}{16} \kappa_1\lambda + \frac{27}{16}\lambda^2 = \frac{177}{4}\lambda^2.\end{aligned}$$ In the last equality we have used Mumford’s relation $\kappa_1 = 12\lambda$ and Faber’s relation $\kappa_2=\frac{27}{2}\lambda^2$ on $\mathcal{M}_4$. Since the degree of $\pi$ is $2^{g-1}(2^g-1)$, we deduce $${\mathcal H}^-_4 \equiv 5310 \lambda^2 \in A^2(\mathcal{M}_4).$$
The locus H4
------------
Let us consider the locus $${\mathcal H}_4 :=\{[C]\in \mathcal{M}_4 : h^0(K_C(-6x))\geq 1 \,\, \mbox{for some}\,\, x\in C \}.$$ The locus ${\mathcal H}_4$ consists of curves $C$ of genus $4$ such that the natural map $$\varphi \colon H^0(K_C) \rightarrow H^0(K_C|_{6x})$$ has rank at most $3$, for some point $x$ in $C$. Let $p\colon \mathcal{C}\rightarrow \mathcal{M}_4$ be the universal curve. The map $\varphi$ globalizes to a map of vector bundles $$\widetilde{\varphi}\colon p^* E \rightarrow J_5(\omega_p)$$ where $E:=p_*(\omega_p)$ is the Hodge bundle of rank $4$ and $J_5(\omega_p)$ is the $5^{\rm th}$ jet bundle of $\omega_p$. Using Porteous formula, we have $$[{\mathcal H}_4] = p_* c_3\left(J_5(\omega_p) - p^*E \right) \in A^2(\mathcal{M}_4).$$ Note that $$\begin{aligned}
ch(J_5(\omega_p)) &=& ch(\omega_p\otimes (1\oplus \omega_p \oplus Sym^2 \omega_p \oplus \cdots \oplus Sym^5 \omega_p))
=\sum_{i=1}^6 e^{i\psi}\\
&=& 6 +21\psi +\frac{91}{2}\psi^2 +\frac{441}{6}\psi^3 +\cdots.\end{aligned}$$ Hence $$c(J_5(\omega_p)) = 1 +21\psi +175\psi^2 +735\psi^3 +\cdots$$ and we obtain $$\begin{aligned}
[{\mathcal H}_4] &=& p_* \left( 735\psi^3 -175\psi^2\lambda_1 +21\psi(\lambda_1^2-\lambda_2) \right)
= 735 \kappa_2 -175\kappa_1\lambda_1 +21\cdot (2g-2)(\lambda_1^2-\lambda_2)\\
&=& \frac{15771}{2}\lambda_1^2.\end{aligned}$$ In the last equality we have used the relations $\kappa_1=12\lambda_1$, $\lambda_2=\frac{\lambda_1^2}{2}$, and $\kappa_2=\frac{27}{2}\lambda_1^2$ on $\mathcal{M}_4$.
We have the following equality $$\left[ {\mathcal H}_4 \right] = 10\cdot \left[\mathcal{H}yp_4 \right] + \left[{\mathcal H}^+_4 \right]+ \left[{\mathcal H}^-_4 \right].$$ The multiplicity $10$ is due to the fact that each of the $10$ Weierstrass points gives a contribution. Hence we deduce $$\begin{aligned}
\label{checkH4+}
{\mathcal H}^+_4 \equiv \frac{15771}{2}\lambda_1^2 -10\cdot \frac{51}{4}\lambda^2 - 5310 \lambda^2 = 2448\lambda_1^2 \in A^2(\mathcal{M}_4)\end{aligned}$$ and this checks with Theorem \[h4+\].
[^1]: and NSF CAREER grant DMS-1350396.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Highly polarized QSOs discovered in the Two-Micron All Sky Survey (2MASS) have been observed to determine the source(s) of optical polarization in this near-infrared color-selected sample. Broad emission lines are observed in the polarized flux spectra of most objects, and the polarization of the lines is at about the same level and position angle as the continuum. Generally, the continuum is bluer and the broad-line Balmer decrement is smaller in polarized light than for the spectrum of total flux. Narrow emission lines are much less polarized than the broad lines and continuum for all polarized objects. These properties favor scattering by material close to a partially obscured and reddened active nucleus, but exterior to the regions producing the broad-line emission, as the source of polarized flux in 2MASS QSOs. The largely unpolarized narrow-line features require that the electrons or dust polarizing the light be located at distances from the nucleus not much greater than the extent of the narrow emission-line region. The conclusion that the scattering material is located close to the nucleus is reinforced by the observation in four objects of changes in both the degree and position angle of polarization across the broad H$\alpha\/$ emission-line profile, indicating that the broad emission-line region (BLR) is at least partially resolved at the distance of the scatterers. In addition to known high-polarization objects, four 2MASS QSOs with AGN spectral types of 1.9 and 2 were observed to search for hidden BLRs. Broad lines were detected in polarized light for two of these objects, and the polarizing mechanism appears to be the same for these objects as for the highly polarized QSOs in the sample that readily show broad emission lines in their spectra. The small observed sample of eight Type 1 2MASS QSOs has weak \[\] emission in comparison to optically-selected AGN with similar near-infrared luminosity. The observations also show that starlight from the host galaxy contributes a significant amount of optical flux, especially for the narrow-line objects, and support the suggestion that many 2MASS QSOs are measured to have low polarization simply because of dilution of the polarized AGN light by the host galaxy.'
author:
- 'Paul S. Smith, Gary D. Schmidt, and Dean C. Hines'
- 'Craig B. Foltz'
title: 'Optical Spectropolarimetry of Quasi-Stellar Objects Discovered by the Two-Micron All Sky Survey'
---
Introduction
============
The Two-Micron All Sky Survey [2MASS; @skrutskie97] has revealed previously unknown, primarily low-redshift active galactic nuclei (AGN) whose space density likely exceeds that of AGN selected by their ultraviolet and optical colors [@cutri01]. This large population of radio-quiet AGN was found by @cutri01 using a simple near-IR color criterion ($J - K_s > 2$) with no regard for optical, radio, or X-ray properties. Secondary selection criteria are that the plane of the Galaxy is avoided ($\vert b \vert > 30$[$^{\circ}$]{}) to minimize the contamination of the AGN survey by reddened galactic objects, and that inclusion in the AGN sample requires detection in all three 2MASS near-IR bandpasses. The latter requirement allows for a well-defined color-selected sample, but at the expense of missing even redder AGN given the sensitivity limits of 2MASS. The color criterion ensures that known AGN (as well as stars) are not a major contaminant in the survey since the vast majority of cataloged AGN have bluer $J - K_s\/$ colors.
The simple selection criteria adopted by @cutri01 result in an efficient survey for low-redshift ($z < 0.7$) AGN that are missed in surveys using traditional optical and ultraviolet search methods. Unlike UV-excess AGN samples, the 2MASS objects encompass a large range of AGN optical spectral types. Broad emission-line (Type 1), narrow emission-line (Type 2), and intermediate (Type 1.5–1.9) objects are all well-represented. @cutri01 find $\sim 3\times$ the number of AGN showing broad emission lines in their spectra than Type 2 AGN; over the entire sky this fraction translates to nearly 6000 Type 2 AGN with $K_s \leq 15$. In addition to the red near-IR colors of the 2MASS objects, their optical faintness and red optical colors suggest that the survey is uncovering a large population of dust-obscured AGN.
Follow up X-ray observations and optical broadband polarimetry of 2MASS AGN with QSO-like near-IR luminosities support the contention that the bulk of the near-IR–selected sample is composed of objects at least partially obscured from our direct line of sight. @wilkes02 find large absorbing columns ($N_H \sim 10^{21}$–10$^{23}$ cm$^{-2}$) toward the nuclear X-ray sources. @smith02 find that a large fraction ($>$10%) of luminous 2MASS AGN ($M_{K_s} \lesssim -25$) are highly polarized ($P > 3$%) compared to optically-selected QSOs and broad absorption-line QSOs. In fact, broadband polarizations as high as $\sim 10$% are measured for a few of these near-IR–selected QSOs.
In this paper, optical spectropolarimetry of all highly polarized and some moderately ($P = 1$–3%) polarized QSOs found by @smith02 is presented in an effort to better understand the polarizing mechanism(s) and thus place constraints on the nature of the 2MASS sample and AGN phenomena in general. In particular, it is of interest to test if the polarization properties of near-IR–selected QSOs are consistent with those found in other highly polarized AGN samples where it has been shown that much of the observational data can be explained in the terms of an obscuring dust torus surrounding the active nucleus [e.g., @antonucci93]. For many Seyfert 2 galaxies [@antonucci85; @miller90; @tran92], narrow-line radio galaxies [NLRGs; @tranc95; @ogle97; @cohen99], and hyperluminous infrared galaxies [HIGs; see e.g., @wills92; @young96; @hines01], spectropolarimetry has been able to show that orientation plays a critical role in the classification of AGN. The basic result for all of these narrow emission-line objects is that light from the active nucleus, including the ionizing continuum and the emission from the broad-line region (BLR), is obscured from direct view by dust near the nucleus, but for lines of sight that do not intersect the putative dusty torus, the nuclear radiation energizes the narrow emission-line region (NLR). Dust or electrons located within the NLR, or just outside this region, scatter some of the nuclear flux into our line of sight resulting in the detection of a blue continuum and broad emission lines in the spectrum of polarized light. Spectropolarimetry of these AGN implies that, from the vantage point of the scattering material, the Type 1 analogs to the narrow-line objects would be observed, effectively unifying apparently disparate classes of objects.
With this general picture in mind, several near-IR–selected QSOs classified as Type 2 objects were also observed to search for hidden BLRs and thereby determine if the narrow emission-line QSOs found by 2MASS are higher-luminosity analogs of Seyfert 2 nuclei. @schmidt02 detail the results for one Type 2 2MASS QSO that reveals a hidden BLR in polarized light and shows that for at least some objects, the model that unifies Seyfert nuclei can be extended to near-IR–selected QSOs. The sample of 2MASS QSOs selected for the current study span nearly the full range of AGN spectral type, as well as a large range in broadband optical polarization. After describing the observations (§2) and the data obtained for individual objects (§3), the general trends for this optically diverse sample of QSOs are discussed in §4. We summarize our conclusions in §5. Primary among these are the findings that the polarization properties of 2MASS QSOs are consistent with these objects being obscured by dust to various degrees, and that both the obscuration and the scattering that produces the polarized flux occur near the nucleus.
Observations
============
Optical spectropolarimetry was obtained for 21 AGN discovered by 2MASS (Table 1) using either the 6.5 m MMT located on Mt. Hopkins, AZ, or the 2.3 m Bok Reflector on Kitt Peak, AZ. The data were acquired between 1999 October and 2002 July, providing multi-epoch sampling of several objects. All observations made use of the CCD Spectropolarimeter [@schmidt92b], upgraded with a $1200 \times 800$-pixel, thinned, antireflection-coated, UV-sensitized CCD, and an improved camera lens and half waveplate.
Data were acquired with the MMT both prior to (2001 March/April) and following the deposition of a high-quality aluminum coating on the primary mirror. The original mirror coating suffered from copper contamination that compromised reflectivity ($R \sim 40$% at $\lambda \sim 4000$ Åand $\sim 60$% at $\lambda \sim 8000$ Å) and degraded with time. The coating also resulted in an instrumental polarization of $\sim$1% that required careful calibration and removal from the 2001 March/April data through observations of several interstellar polarization and unpolarized standard stars [@schmidt92a]. The instrumental polarization component was reduced to $<$0.1% and the reflectivity increased to nearly ideal levels with the first fully successful primary mirror aluminization in 2001 November.
All observations utilized a 600 l mm$^{-1}$ grating for high throughput and wide spectral coverage, typically 4400–8800 Å. The full-width at half-maximum (FWHM) resolution with this grating and a slit width of 11–30 is $\sim$17 Å (3 pixels), $\sim$800–1100 [km s$^{-1}$]{}, depending on telescope. A polarimetric measurement sequence involves four separate exposures that sample 16 orientations of the semi-achromatic half waveplate, and totals $\sim$2000–3000 s of integration. Typically, 2–4 such sequences were acquired for each object in a night. Calibration of the degree of polarization ($P\/$) was made with reference to observations of an incandescent light source through a fully polarizing prism. Values quoted for $P\/$ have not been corrected for Ricean statistical bias [@wardle74]. In nearly all cases this correction is inconsequential for the data presented. Polarization position angles were referenced to the equatorial system by means of observations of interstellar polarization standard stars with the identical instrumental setup. The spectral flux distribution is also obtained in the course of the observations and is calibrated relative to spectrophotometric standard stars selected from the IRAF database [@massey88]. Finally, the atmospheric O$_2$ A and B-band absorption features, as well as the H$_2$O feature at $\sim$7200 Å, have been removed from the flux spectra of the AGN by observing early-type stars at nearly the same airmass, or by scaling the results of the stellar observations to the features seen in the AGN spectra.
Table 1 summarizes the observations by listing the observation date, telescope, slit width, total exposure time, and the flux-weighted mean linear polarization within the 5000–8000 Å band (observed frame; unless otherwise noted, wavelengths are given in the observer’s frame). In some cases more than one slit was used for an object during an observing run as dictated by conditions. Nine objects were observed during multiple epochs, and the broadband mean polarization is listed for each observing run. There is no evidence for variability of any of these objects between epochs or between nights of a given run. The data for each object were averaged and these values are also listed in Table 1. The co-added spectropolarimetry is displayed in Figures 1, 2, and 3. The coaddition of the observations was statistically weighted by the data quality at each epoch. Because of the faintness of the targets, $16 \leq B \leq 22$, the polarization spectra can still be noisy, particularly near the limits of the spectral coverage.
The Sample of Objects
---------------------
The 21 objects chosen for spectropolarimetric observation were selected from the sample 89 AGN observed by @smith02. Seventy of these objects meet the criterion of $J - K_s > 2$ to be included in the formal sample of 2MASS red AGN [@cutri01], and so do all but one of the spectropolarimetric targets. Because all of the 2MASS red AGN have $M_{K_s}\/$ that fall comfortably within the $K_s\/$-band luminosity range of optically-selected QSOs, @smith02 classified these objects as QSOs in their own right even though they are generally underluminous in the optical. All 10 of the QSOs found to have optical broadband polarizations $>$3% by Smith [et al. ]{}were selected for follow up spectropolarimetry. Of these objects, spectropolarimetry of 2MASSI J151653.2+190048 (2M151653; hereafter in the text, the identification of objects will take the form: 2M[*hhmmss*]{}, where the decimal seconds of the J2000 Right Ascension have been truncated and the Declination has been omitted) and of 2M165939 using the Bok Reflector have been reported by @smith00. We include these objects for completeness, and in the case of 2M165939, add new data obtained with the MMT.
In addition to the highly polarized QSOs in the 2MASS sample, spectropolarimetry of 2M135852 is presented. This object does not quite meet the 2MASS red AGN selection criteria with $J - K_s = 1.8$, but was found by @smith02 to have an $R\/$-band polarization of nearly 5%.
Four 2MASS QSOs having spectra dominated by narrow emission lines (2M100121, 2M105144, 2M130005, and 2M222554) were observed ostensibly to search for broad emission lines in their polarized spectra. All of these objects have measured broadband polarizations of $<$2%. The positive result for 2M130005 is reported and discussed by @schmidt02, and these data are included here for completeness and for comparison to other objects in the sample. Supplementing the observations of highly polarized and low-polarization narrow-line targets are six other 2MASS QSOs with moderate broadband polarization ($P \sim 1$–3%). These objects were observed as time and conditions allowed.
Except for 2M004118 (§3.2.1), Galactic interstellar polarization (ISP) in the sight lines to the targets has been ignored. The high Galactic latitude of the sample and the high polarization of many of the objects virtually ensure that Galactic ISP is not a significant contributor to the observed polarization. @smith02 directly show this to be true for six spectropolarimetry targets since only upper limits can be be set for the polarization of objects near these sight lines with measurement uncertainties of $\lesssim$0.5%. The high quality flux spectra that are a byproduct of the spectropolarimetry allow a re-examination of the optical spectral classification of the 2MASS AGN that was based on the original, confirming spectroscopy [@cutri01] listed in @smith02. We use the AGN classification guidelines of @ho97 that were also used by Cutri [et al. ]{}in generating the 2MASS AGN sample. This scheme is very close to the classification criteria of @veilleux87. Some of the original classifications were ambiguous because H$\beta\/$ was not available for various reasons. This emission line is clearly identified in all of the 21 spectropolarimetry targets and allows for a more definitive classification of four objects. 2M135852 and 2M150113 are now classified as Type 1 AGN and 2M163700 (see Figure 2) is Type 1.5 owing to the prominent narrow-line component of H$\beta\/$. @schmidt02 reclassified 2M130005 as a Type 2 AGN.
Four other objects were reclassified based on the new spectra. 2M105144 is Type 1.9 as opposed to the original LINER classification because it possesses a broad H$\alpha\/$ component (Figure 3). 2M222221 (Figure 2) is reclassified as Type 1.5 because of the strong, narrow emission-line component seen in the Balmer-line profiles and the relatively high ratio of \[\]$\lambda$5007 to H$\beta\/$ flux. In contrast, the smaller \[\]$\lambda$5007/H$\beta\/$ flux ratio, absence of a distinguishable narrow emission-line component for H$\beta\/$, and very strong optical features favor classifying 2M230307 as Type 1. The original Starburst classification is somewhat problematic for 2M100121. The emission-line ratios are consistent with either Starburst or Seyfert 2 designation, but given its high QSO-like luminosity in the near infrared, it is difficult to imagine that hot stars power the nuclear emission. Therefore, we reclassify 2M100121 as a Type 2 AGN. Table 1 summarizes the adopted AGN spectral classifications for the spectropolarimetry sample.
Results
=======
Spectropolarimetric Measurements
--------------------------------
The spectropolarimetric results are presented in Figures 1–3. For all objects, the sequence of four panels depicts, from top to bottom: the polarization position angle $\theta\/$, the rotated Stokes parameter $q'\/$ for a coordinate system aligned with the mean polarization of the source in the 5000–8000 Åbandpass ($q' = q \cos 2\theta + u \sin 2\theta\/$), the polarized (or “Stokes”) flux $q' \times F_\lambda\/$, and finally, the total spectral flux $F_\lambda\/$. Two artifacts should be noted. First, even though the data have been corrected for terrestrial absorption features, the O$_2$ and H$_2$O bands are sufficiently deep that increased noise can be noted in the polarized flux spectra around the wavelengths of these features. Second, fringing sharply modulates the CCD quantum efficiency for $\lambda \gtrsim 8000$ Å. When combined with a small amount of flexure in the instrument, the result is that measured quantities show rapid oscillations as a function of wavelength around their mean values for a few objects. 2M010607 and 2M135852 (Figure 1) show examples of this effect.
Objects have been divided into three groups: Type 1, Type 1.5, and Types 1.8–2 to compare and contrast characteristics over a wide range of emission-line properties. Summarized in Table 2 are the equivalent width (EW in Å) and line flux (in units of 10$^{-14}$ erg cm$^{-2}$ s$^{-1}$) for H$\beta\/$, H$\alpha\/$ (for objects with $z < 0.34$), and \[\]$\lambda$5007. The line widths of H$\alpha\/$ and H$\beta\/$ (FWHM in [km s$^{-1}$]{}) are also given. Because the emission lines from the NLR are generally unresolved, we do not list the \[\]$\lambda$5007 line width, but instead tabulate the \[\]$\lambda$5007 luminosity ($H_0 = 75$ [km s$^{-1}$]{} Mpc$^{-1}$, $q_0 = 0$, and $\Lambda = 0$ are assumed throughout). Measurements for all of the available quantities are given for both the total flux spectrum ($F_\lambda\/$) and the polarized flux spectrum ($q' \times F_\lambda\/$) in the observed frame. A colon after an entry signifies that the measurement is difficult and the reason for the resulting uncertainty is identified in the last column of Table 2.
Measurements of H$\alpha\/$ are contaminated to various degrees by \[\]$\lambda\lambda$6548,6583. The contribution to the H$\alpha\/$ flux by \[\] is not large except for the Type 1.8–2 QSOs, where the \[\] lines may emit $\gtrsim$1/2 of the flux and broaden the feature. Flux from \[\]$\lambda\lambda$6717,6731 was excluded from measurements of H$\alpha\/$ + \[\]. Given the spectral resolution of the observations, the values listed in Table 2 for H$\alpha\/$ include the contributions from both the narrow and broad emission-line components as well as \[\]. Likewise, the measurements of H$\beta\/$ include both broad- and narrow-line components of the emission feature.
Optical emission is detected in all of the Type 1 and 1.5 objects except 2M165939. Since measurements of H$\beta\/$ and \[\]$\lambda$5007 are affected by strong multiplets in this wavelength region, the emission was removed from $F_\lambda\/$ and $q' \times F_\lambda\/$ (if detected in polarized light) before these lines were measured. An optical template based on the narrow-line QSO I Zw 1 [@boroson92] was broadened to the approximate H$\beta\/$ FWHM, scaled to the estimated flux of the feature observed at $\sim$4450–4750 Å (rest frame), and subtracted from the 2MASS QSO spectra.
The continuum properties of the 2MASS QSOs are characterized in Table 3. For more than half of the sample, a significant contribution to the observed flux is made by starlight originating in the host galaxy of the QSO. We have estimated the amount of starlight falling within the observing aperture for all objects showing stellar absorption features in their spectra. For three other objects that do not show obvious stellar features, an estimate of the host galaxy flux is provided by high resolution imaging obtained by @marble03 using the Wide Field/Planetary Camera 2 (WFPC2) and F814W filter aboard the [*Hubble Space Telescope*]{} ([*HST*]{}). The elliptical galaxy spectrum of NGC 3379 [@kennicutt92] was used as a template to extrapolate the measured stellar contribution within the F814W filter to shorter wavelengths. This same template also reasonably accounts for the observed absorption features of the 11 objects where the host galaxy is directly detected in the total flux spectra. The ratio of starlight from the host galaxy to total flux at 5500 Å in the rest frame is listed in Table 3, along with the polarization of the remaining light in the observed 6000–7000 Å band after the subtraction of the assumed unpolarized stellar component.
Power-law fits to the continua for both the polarized flux spectrum and the host galaxy-subtracted total flux spectrum were made to characterize the continuum properties of the sample. The power-law index, $\beta\/$, where $F_\lambda \propto \lambda^\beta$, is given in Table 3. The power-law fits avoid regions of exceptionally high noise, major emission lines, and obvious features if present.
Following $\beta\/$ in Table 3 are the strengths of \[\]$\lambda$5007, H$\alpha\/$, and relative to the H$\beta\/$ flux. If possible, entries for these line ratios are listed for both the total and polarized flux spectra. As in Table 2, the measurement of $F_{{\rm H}\alpha\/}$ includes flux from \[\]$\lambda\lambda$6548,6583. The strength of the optical emission is characterized by the its flux within the rest frame 4450–4750 Å band.
Finally, the last three columns in Table 3 list the line and continuum polarizations for H$\beta\/$ and H$\alpha\/$ and an estimate of the polarization of the NLR based on the measurements of \[\]$\lambda$5007. The polarizations of the permitted lines are calculated using the measured total and polarized line fluxes. The continuum polarization at H$\alpha\/$ and H$\beta\/$ is derived from the power-law fits to the continua described above. Throughout this paper, we assume that the generally low observed polarization of the NLR is well-represented by the strength of narrow-line features in the polarized flux spectrum.
Type 1 Objects
--------------
About 3/4 of the 2MASS red AGN sample show broad emission lines in their total flux spectra [@cutri01]. In the spectropolarimetric sample, Type 1 objects correspond to those with $F_{\rm [O~III]}$/$F_{{\rm H}\beta} < 0.5$. Spectropolarimetry of seven Type 1 2MASS QSOs and one highly polarized, broad emission-line AGN found by 2MASS (2M135852) is shown in Figure 1.
### 2MASSI J004118.7+281640
Optical unfiltered polarimetry of 2M004118 yields $P = 2.2\% \pm 0.3$% at $\theta$ = 104[$^{\circ}$]{}$\pm$4[$^{\circ}$]{} [@smith02]. Our spectropolarimetry confirms the level of polarization, and we measure a flux-weighted polarization position angle of 97[$^{\circ}$]{} in the 5000–8000 Å bandpass. A portion of this originates as Galactic ISP, since @smith02 found $P = 0.74$%, $\theta = 117$[$^{\circ}$]{} for a nearby field star. The data displayed for 2M004118 in Figure 1 and Tables 2 and 3 have been corrected for this ISP, assuming that it follows the standard Serkowski law [@serkowski75] with a maximum polarization of 0.74% at 5500 Å.
Although the polarization of 2M004118 is diminished by the ISP correction, it is clear that the object is intrinsically polarized. The spectrum of $q'$ shows that the H$\alpha$, H$\beta$, and \[\] emission lines are not polarized to the same degree as the surrounding continuum, as would be expected if ISP were the sole polarizing mechanism. The ISP correction yields a mean polarization in the 5000–8000 Å bandpass for 2M004118 of $P \sim 1.7$% at $\theta \sim 89$[$^{\circ}$]{}.
2M004118 is one of the bluest objects in the sample, with $B - K_s = 3.4$ and $\beta_{F_\lambda} = -2.2$, and has strong optical emission features. The polarized flux spectrum shows a continuum nearly as blue as the total flux spectrum. Hydrogen $\alpha$ exhibits changes in $q'\/$ and $\theta\/$ across its profile. The core of the line marks a transition between the red wing of the line that is polarized to the degree and position angle of the continuum, and the less polarized blue wing. The position angle swings through $\sim$90[$^{\circ}$]{} near the line core. A similar signature in $q'\/$ is identified for H$\beta$ and the strong feature blended with \[\]$\lambda$5007 at 5980 Å. The complex structure of the H$\alpha$ polarized flux profile is reminiscent of some highly polarized Seyfert 1 nuclei [e.g., @goodrich94; @smith95; @smith97; @martel97; @martel98; @smithj02].
### 2MASSI J010607.7+260334
In contrast to 2M004118, 2M010607 is a much redder object optically ($\beta_{F_\lambda} = 1.3$; $B - K_s > 6$), more than 3 mag fainter, and more highly polarized. 2M010607 shows emission features nearly as strong relative to H$\beta$ as in 2M004118. Unfortunately, the redshift of 2M010607 places H$\alpha$ outside the sensitivity range of the spectropolarimeter. The primary feature of the polarized flux spectrum is the presence of broad ($\sim$1900 [km s$^{-1}$]{} FWHM) H$\beta$ with roughly the same equivalent width as seen in the total flux spectrum. There is no evidence of \[\]$\lambda$5007 in the polarized flux spectrum and the polarized continuum is quite red ($\beta_{q' \times F_\lambda} = 0.5$).
### 2MASSI J125807.4+232921
Spectropolarimetry of 2M125807 confirms the low $R\/$-band polarization measurement reported in @smith02. The object is $\sim$1% polarized and the appropriate quantities listed in Table 3 have been measured after subtracting an elliptical host galaxy spectrum that accounts for 13% of the total light received at 5500 Å in the rest frame. Because stellar spectral features are not apparent in the spectrum, the contribution of the host galaxy was estimated from an [*HST*]{} WFPC2 F814W image [@marble03]. In this case, the host galaxy has little impact on the measured polarization and other quantities.
The spectral slopes of total flux and polarized flux continua are similar to those of 2M004118, as is the Balmer decrement in the total flux spectrum. Subtraction of the template from the total flux spectrum accounts for all of emission peaks between H$\beta\/$ and $\sim$7000 Å. There is no evidence of \[\] emission from 2M125807, making this the only object in spectropolarimetric sample not exhibiting emission lines from this species. No definitive features are seen in the polarized flux spectrum.
### 2MASSI J132917.5+121340
Although 2M132917 is polarized only $\sim$1% around H$\alpha$, its polarization rises rapidly to the blue. Indeed, the object exhibits one of the bluest polarized flux spectra of this sample ($\beta_{q' \times F_\lambda} = -2.0$), much bluer than the total flux continuum ($\beta_{F_\lambda} = -$0.5). The feature seen in the polarized flux spectrum near \[\]$\lambda$5007 is much narrower ($\sim$1 pixel) than the instrumental resolution and could simply be a noise spike due to a cosmic ray. The flux in this narrow feature is used as an upper limit to the polarized flux from \[\]$\lambda$5007. Comparison of the H$\beta$/\[\]$\lambda$5007 flux ratio for both $F_\lambda\/$ and $q' \times F_\lambda\/$ suggests that the polarization of the BLR is at least four times higher than that of the NLR.
The Balmer lines reveal a pronounced red asymmetry in their profiles in the total flux spectrum. Much like 2M004118, the H$\alpha$ polarized flux is emitted in the red wing of the line. Near the line core and in the blue wing $q'\/$ falls well below 0. Since there is no corresponding feature seen in the rotated $u\/$ Stokes parameter, the polarization of the blue wing of H$\alpha$ is orthogonal to that of the red wing. The two polarized components would then tend to cancel near the line core, as seen in the $q'\/$ and $q' \times F_\lambda\/$ panels for 2M132917 in Figure 1. The line center of H$\beta$ in polarized flux is shifted $\sim$1000 [km s$^{-1}$]{} to the red of the line peak in total flux. This could be caused by the same polarization structure seen in H$\alpha$, although the spectropolarimetry does not have the signal-to-noise ratio (S/N) to clearly show these features.
### 2MASSI J135852.5+295413
This object is not a member of the 2MASS red QSO sample because $J - K_s = 1.8$ in revised 2MASS photometry. Nevertheless, 2M135852 has a $K_s\/$ luminosity well within the range of the 2MASS QSOs and $B - K_s = 4.15$, redder than the median $B - K_s\/$ color index of the Type 1 objects observed by @smith02. In addition, $R$-band imaging polarimetry shows $P \sim 4.8$%, resulting in the object being added to the list of 2MASS AGN for follow up observation.
Spectropolarimetry confirms the highly polarized nature of 2M135852. The polarization reaches $\sim$8% at 4600 Å and has a position angle that is constant with wavelength. The spectrum reveals stellar absorption features that can be removed with a starlight fraction of $\sim$60% in the observing aperture at 5500 Å (rest frame). This implies an intrinsic polarization in the 6000–7000 Å band double of that observed, and well over 10% for $\lambda < 5000$ Å. The host galaxy also leads to higher observed polarization in the Balmer lines than in the continuum since the relative dilution in the lines is reduced. Balmer line widths in $F_\lambda$ and $q' \times F_\lambda$ are about the same. Even with the correction to $F_\lambda$ for a red stellar component, the polarized flux continuum is bluer than the nuclear light in the total flux spectrum.
The $q'\/$ panel in Figure 1 for 2M135852 indicates that the polarization at \[\]$\lambda$5007 is much diminished from the surrounding continuum. Estimating the \[\]$\lambda$5007 flux in the polarized spectrum yields a polarization for the NLR of $\lesssim$2.5%. Unlike several other Type 1 objects, 2M135852 shows no structure in $\theta\/$ across H$\alpha$ in the polarized flux spectrum.
### 2MASSI J150113.1+232908
2M150113 is optically the reddest of the eight Type 1 objects with $\beta_{F_\lambda} = 2.3$, and the H and K break is seen in its spectrum. The stellar features imply that the host galaxy contributes $\sim$60% of the light at 5500 Å (rest frame) within the $1\farcs 5 \times 3\farcs 8$ observation aperture. This, in turn, suggests that the polarization of the nuclear light in the $R\/$-band is $\sim$8%, in contrast to the $P = 3$–4% measured by the spectropolarimetry and by @smith02. The increased polarization seen at H$\alpha\/$ and H$\beta\/$ is explained by unpolarized starlight included in the aperture, as is also the case for 2M135852.
The noise level of the polarized flux spectrum makes it is difficult to determine if \[\]$\lambda$5007 is present in polarized light. An upper limit to the \[\]$\lambda$5007 polarized flux suggests that the emission from the NLR cannot be polarized more than $\sim$1/3 of the polarization measured for the continuum and BLR.
The degree of polarization of 2M150113 rises to the blue as indicated by the spectral index of the polarized continuum relative to the total nuclear continuum (after the host galaxy spectrum has been subtracted). In addition, the Balmer decrement is large in both the total flux spectrum ($F_{{\rm H}\alpha}/F_{{\rm H}\beta} \sim 12$) and the the polarized flux spectrum ($F_{{\rm H}\alpha}/F_{{\rm H}\beta} \sim 7$). The large decrement in $q' \times F_\lambda\/$ indicates that that even the light scattered into our view is substantially reddened.
### 2MASSI J151653.2+190048
@smith00 describe the polarization properties of this object. We include it here to compare with the rest of the sample and to tabulate the measurements listed in Tables 2 and 3. The object is the most luminous at $K_s\/$ of the Type 1 QSOs in the spectropolarimetry sample.
The strong optical emission features are observed in the spectrum of polarized flux at about the same strength relative to H$\beta$ as in the total flux spectrum. The only other QSO in this sample to unambiguously show polarized features is 2M091848 (see §3.3.1). The broad emission lines are polarized at about the same level as the continuum, which exhibits a strong rise in polarization from 8600 Å ($P \sim 7$%) to 4600 Å ($P \sim 14$%). The bluer polarized flux spectrum relative to the total flux spectrum is accompanied by a Balmer decrement in polarized light nearly a factor of 2 smaller than for the total flux spectrum.
The NLR is unpolarized as indicated by the absence of \[\]$\lambda\lambda$4959,5007 in the polarized flux spectrum. Structure is observed in $q'\/$ and $\theta$ across H$\alpha\/$, and to a lesser degree, H$\beta\/$ (Figure 1). The red wings of the Balmer lines are more highly polarized than the blue wings. This same signature is seen for H$\alpha\/$ in the Type 1 objects 2M004118 and 2M132917.
### 2MASSI J230307.2+254503
This object is the strongest emitter relative to H$\beta\/$ of the QSOs in this sample. Unfortunately, H$\alpha\/$ falls very close to the red end of the spectrum, making measurements of this emission line somewhat problematic. The S/N of the spectropolarimetry is too low around the H$\alpha\/$ line for any meaningful measurements. Despite the faintness of this QSO and its relatively low optical polarization, H$\beta\/$ can be seen in emission in polarized flux. Another feature apparent in Figure 1 is the decline of polarization to $q' \sim 0$ at \[\]$\lambda$5007.
Intermediate Objects
--------------------
In this section we discuss QSOs classified as Type 1.5. The narrow line-dominated Type 1.8 and 1.9 objects, also considered “intermediate” AGN spectral types, have been lumped together with the Type 2 2MASS QSOs (§3.4). This division also separates objects in the sample by their \[\]$\lambda$5007/H$\beta\/$ flux ratios, with Type 1.5 QSOs in this sample having $0.5 \leq F_{\rm [O~III]}/F_{{\rm H}\beta} \leq 3$. Figure 2 displays the results for these objects.
### 2MASSI J091848.6+211717
The polarization of 2M091848 increases strongly to the blue, reaching over 10% for $\lambda < 5000$ Å, and has a position angle that is constant with wavelength. The Balmer lines, including H$\gamma\/$, are seen in polarized flux, as is \[\]$\lambda$5007.
2M091848 along with 2M151653 are the only two objects to clearly show polarized optical emission features. The emission on either side of H$\beta\/$ in 2M091848 is more prominent in polarized flux than in the total flux spectrum largely because of the significant amount of host galaxy starlight included in the total flux. Stellar absorption features in $F_\lambda\/$ suggest a stellar–to–total light ratio at 5500 Å (rest frame) of $\sim$0.4. This contribution by the host galaxy implies that the intrinsic $R\/$-band polarization of the QSO is $\sim 10$%, as opposed to $6.3 \pm 0.1$% measured by @smith02 without correction for starlight.
The polarized flux spectrum of 2M091848 shows \[\]$\lambda$5007, though its strength relative to H$\beta\/$ is much reduced and the weaker \[\]$\lambda$4959 line does not rise above the S/N level in polarized light. The \[\]$\lambda$5007 measurements yield a polarization for this line of $\sim$2%, which is a factor of 4–6 lower than the polarization of H$\beta\/$ and the continuum at this wavelength.
### 2MASSI J134915.2+220032
2M134915 is the least luminous QSO in the sample in the near infrared ($M_{K_s} = -24.9$) and has the smallest near-IR-to-optical flux ratio inferred from $B - K_s\/$. Although the observed polarization for 2M134915 is much lower than in 2M091848, many of the polarization properties of these two QSOs are similar. Both exhibit redder continua in total flux than in polarized flux. Both objects clearly show a polarized \[\]$\lambda$5007 line. In 2M134915, \[\]$\lambda$4959 is also seen in $q' \times F_\lambda\/$. Based on the \[\] measurements, the NLR is polarized by $<$1%; $<$1/4 the polarization of the continuum at this wavelength.
Unlike most of the Type 1 objects, which typically show Balmer lines in polarized light that have about same FWHM as in total flux, polarized H$\beta\/$ in 2M134915 has a width $\sim$2700 [km s$^{-1}$]{} larger than measured in the total flux spectrum. To a large extent, this reflects the lessened contribution from the NLR in the polarized H$\beta\/$ line profile compared to that for the line in total flux. The increased polarization seen in the Balmer lines is caused by the substantial contribution from the host galaxy within the observing aperture. From the stellar features observed in the spectrum, nearly half of the light within the aperture at 5500 Å (rest frame) is starlight from the host galaxy.
### 2MASSI J163700.2+222114
The H and K break is readily apparent in the spectrum 2M163700. The stellar features imply a flux contribution by the host galaxy of $\sim$0.5 at 5500 Å (rest frame) that in turn suggests an intrinsic level of polarization of the AGN of over 4%. Although the continuum slope of the polarized flux spectrum is bluer than that of the total flux, both are red ($\beta_{F_\lambda} \sim 2.0$; $\beta_{q' \times F_\lambda} \sim 0.9$). Large Balmer decrements in both polarized and total flux spectra further confirm the highly reddened nature of this QSO. In fact, these decrements are by far the largest for the Type 1.5 QSOs, and only 2M150113 challenges 2M163700 in this parameter among the Type 1 objects. There is no evidence for \[\] (or \[\]$\lambda$3727) in the polarized flux spectrum.
### 2MASSI J165939.7+183436
Spectropolarimetry of this object obtained at the Bok Reflector is reported and discussed by @smith00. Subsequently, an MMT observation of similar quality was obtained of 2M165939 on 2001 March 31. No significant differences are seen between these observations and the co-added results from the two telescopes are displayed in Figure 2. Like 2M134915 and 2M163700, the host galaxy of 2M165939 contributes to the observed flux. Various slit widths (1–3) were employed for the observations of 2M165939 and a starlight-to-total flux ratio of 0.26 in the rest frame $V\/$-band is adopted for the composite spectrum. This ratio is based on the identification of a weak absorption feature at $\sim$6060 Å with b.
Narrow emission lines seen in the total flux spectrum are not seen in polarized light. In contrast, H$\alpha\/$ and H$\beta\/$ are prominent in the polarized spectrum and are much broader than in either $F_\lambda\/$ or $q' \times F_\lambda\/$ for any of the Type 1 objects. The asymmetric profile of the broad component of H$\alpha\/$ in the total flux spectrum is somewhat reproduced in polarized light with the blue wing of the line being sharper than the red wing. The broad hump identified with H$\beta\/$ in the polarized spectrum has an impressive FWHM of nearly 16,000 [km s$^{-1}$]{}.
### 2MASSI J170003.0+211823
2M170003 has the highest redshift, $z = 0.596$, of the 2MASS QSOs and is one of the reddest objects in the sample. It also shows the highest observed optical broadband polarization [$P \sim 11$%; @smith02]. The spectropolarimetry confirms the high level of polarization and reveals that the polarized spectrum is about as red as the extremely red continuum observed in total light. Again, the major difference between the spectra of total and polarized light is that the emission from the NLR region is absent in the polarized flux spectrum. Broad H$\beta\/$ and H$\gamma\/$ are detected in polarized spectrum and are polarized to about the same degree as the continuum. The position angle is constant across the entire spectrum.
### 2MASSI J222202.2+195231
The highly polarized QSO 2M222202 is the most luminous ($M_{K_s} = -28.6$) of the sample of 70 2MASS red QSOs observed by @smith02 and it displays a very rich emission-line spectrum that includes emission lines of H, , , , \[\], \[\], \[\], \[\], and \[\] (Figures 2 and 4). Weak optical features are also present. Remarkably, with the possible exception of [H$\beta\/$]{}, none of these features are seen in the polarized flux spectrum. The rotated Stokes parameter shows complex structure across the spectrum with dramatic decreases in polarization at each emission line. Indeed, significant decreases in $q'\/$ are even seen at the locations of lines with small equivalent widths, such as $\lambda$4686 and \[\]$\lambda$3727. The continuum polarization increases from $\sim$10% at the red end of the spectrum to nearly 18% at $\sim$5370 Å, in the gap between H$\epsilon\/$ and \[\]$\lambda$3869 + $\lambda$3889 + H$\zeta\/$. The continuum polarization then decreases from this point until the blue end of the observed spectrum at 4200 Å where $q'\/$ is again $\sim$10%.
The position angle spectrum is not as dramatic as $q'\/$, but is of sufficient S/N to show interesting structure as well (Figure 4). Generally from $\sim$5600 Å to 8600 Å, $\theta\/$ is nearly constant at around 120[$^{\circ}$]{}. However, a 10[$^{\circ}$]{} rotation can be seen at the location of \[\]$\lambda$5007 and more tentatively in the core of [H$\beta\/$]{} and at [H$\gamma\/$]{} + \[\]$\lambda$4363. A smaller rotation in the same sense can be discerned at \[\]$\lambda$4959. Although $q'\/$ is based on a position angle of 120[$^{\circ}$]{}, these $\lesssim$10[$^{\circ}$]{} rotations in $\theta\/$ are too small to significantly affect the spectrum of $q'\/$ presented in Figure 4.
In addition to the discrete rotations in $\theta\/$ in some of the emission lines, a broad feature is observed centered near 5000 Å. The rotation has about the same amplitude as seen for \[\]$\lambda$5007, and although the S/N is much diminished at the blue end of the spectrum, $\theta\/$ appears to recover back to $\sim$120[$^{\circ}$]{} by 4400 Å. This feature coincides with the decrease in the continuum polarization in the blue and the emergence of the “3000 Å bump” in the flux spectrum [see e.g., @wills85]. At these wavelengths, the 3000 Å bump is primarily emission from high-level Balmer lines and the Balmer continuum. In 2M222202, this feature, \[\]$\lambda\lambda$4959,5007, and other emission lines must be polarized since $\theta\/$ cannot be affected by unpolarized light. The continuum is polarized to a much higher degree than the emission from the NLR since the rotation in $\theta\/$ at strong NLR features is only $\sim$10[$^{\circ}$]{}. Measurement of the \[\]$\lambda$5007 line implies a polarization for the NLR light of $P \sim 0.8$% at $\theta \sim 85$[$^{\circ}$]{}. Assigning the same polarization to the 3000 Å bump does not readily account for the polarization observed near the Balmer limit, although the decomposition of the spectrum into two polarized components is highly dependent on uncertain choices of the continuum strength and polarization in this spectral region. To be consistent with the results at \[\]$\lambda$5007, either the 3000 Å bump must be a larger contributor to the total flux around 5000 Å than implied by a simple extrapolation from longer wavelengths, or the continuum polarization at $\sim$5000 Åis not as high as indicated by the rapid rise in $P\/$ observed at longer wavelengths.
The deviations in $\theta\/$ across the spectrum make it more difficult to interpret the spectrum of polarized light. The emission lines seen in $F_\lambda\/$ are absent in $q' \times F_\lambda\/$. There is an abrupt break in the polarized flux spectrum around H$\beta\/$/\[\]$\lambda\lambda$4959,5007 where the continuum flattens from $\beta_{q' \times F_\lambda\/} \sim -2.3$ for $\lambda \gtrsim 6800$ Å to $\beta_{q' \times F_\lambda\/} \sim -0.4$ for $\lambda \lesssim 6800$ Å. A very low contrast emission feature is tentatively identified at the location of H$\beta\/$ and \[\]$\lambda\lambda$4959,5007. If this feature is actually polarized flux from H$\beta\/$, the line is polarized at only $\sim$1/2 the level of the continuum. An identification with H$\beta\/$ also implies that the scattered line profile is even broader ($\sim 17,000$ [km s$^{-1}$]{} FWHM) than the polarized H$\beta\/$ profile observed in 2M165939 (§3.3.4). The “feature” could also simply be a manifestation of the position angle rotation seen in the emission lines. We have included measurements of the feature in Tables 2 and 3 under the assumption that it is H$\beta\/$.
The ambiguity in the identification of polarized H$\beta\/$, and the lack of other lines in the polarized flux spectrum of 2M222202, results in greater uncertainty in identifying the mechanism responsible for the high continuum polarization. The polarization of the narrow emission-lines must either be caused by scattering from material far enough away from the NLR to result in a net polarization of the light, or dichroic absorption in the sight line between us and the NLR of 2M222202. Dichroic absorption is essentially ruled out as the cause of the continuum polarization because an enormous amount of extinction would be required to account for polarization approaching 20%. Scattering by dust or electrons requires that the particles be located very close to the nuclear continuum source. To account for the very weak (or nonexistent) broad-line features in the polarized flux, the scatterers need to be intermixed with the gas in the BLR, or located just exterior to the line-emitting region. Polarized flux produced in such close proximity to the BLR probably favors electrons as the scatterers since the environment is likely to be too harsh for the survival of dust grains (however, see e.g., @goodrich95 for evidence that dust can exist in the BLR).
If indeed the polarized flux spectrum of 2M222202 is featureless, synchrotron radiation is a possible source of the highly polarized continuum. In this case, the optical flux and polarization would be expected to strongly vary as observed for OVV quasars and BL Lacertae objects. In fact, there is no evidence for variability in 2M222202 from the three epochs of spectropolarimetry obtained between 1999 September and 2002 July. 2M222202 is detected as a 5 mJy radio source at 1.4 GHz in the NRAO VLA Sky Survey [@condon98], but the object is $\sim 100-1000\times$ less luminous in the radio than objects that show strong and variable optical synchrotron continua.
### 2MASSI J222221.1+195947
The polarization of 2M222221 is only $\sim$1% even after subtracting the host galaxy contribution to the observed spectrum estimated from [*HST*]{} imaging [@marble03 $F_{\rm gal}/F_{\rm Total} \sim 0.3$ in the rest frame $V\/$-band]. Despite the low level of polarization, broad H$\alpha\/$ and H$\beta\/$ are apparent in $q' \times F_\lambda$. The lines in polarized flux are blue-shifted by $\sim$1000 [km s$^{-1}$]{} relative to the narrow Balmer-line components in the total flux spectrum. Also detected in the polarized flux is \[\]$\lambda$5007, although at a reduced strength relative to H$\beta\/$. This yields an estimate of the NLR polarization of only $\sim$0.3%. One other feature of note for 2M222221 is a large rotation in $\theta\/$ across H$\alpha\/$. The position angle is $\sim$160[$^{\circ}$]{} in the blue wing of the line, $\sim$180[$^{\circ}$]{} around the line core, and then rotates to 130[$^{\circ}$]{}–110[$^{\circ}$]{} in the red wing.
Type 1.8, 1.9, and 2 Objects
----------------------------
The objects described in this section would traditionally not be classified as QSOs on the basis of their optical spectra since their narrow emission lines far outshine any weak broad permitted lines. Indeed, their H$\beta\/$ widths are $\sim$1000 [km s$^{-1}$]{} (FWHM) or less. All but one show stellar absorption features with equivalent widths suggesting that well over half of the optical continuum flux is starlight from the galaxy hosts. The 2MASS results tell a much different story for these red QSOs: all have $M_{K_s} < -25.5$ and have large near-IR/optical flux ratios ($B - K_s > 5$). Spectropolarimetry of the Type 1.8–2 QSOs is displayed in Figure 3.
Combining the Type 1.8–2 objects separates QSOs with $F_{\rm [O~III]}/F_{{\rm H}\beta} > 3$ from the rest of the sample. Their total flux spectra generally have larger Balmer decrements than those measured for the Type 1s and 1.5s. This is due in part to the inclusion of \[\]$\lambda\lambda$6548,6563 in the spectral measurements of H$\alpha\/$, since the \[\] flux in several objects is comparable to H$\alpha\/$. The presence of the \[\] lines also yields widths in total flux that are roughly double those measured for H$\beta\/$. Optical emission is not detected in these six objects.
### 2MASSI J010835.1+214818
2M010835 is the only object besides 2M171559 (§3.4.5) among the 15 Type 1.8–2 AGN in the 2MASS red QSO sample that shows broadband optical polarization $>$3% [@smith02]. Spectropolarimetry confirms that the polarization of the continuum is $\sim$5% and reveals stellar features in the total flux spectrum. The strength of the features suggests that about 80% of the continuum at $\sim$7000 Å is starlight, requiring that the true continuum polarization of the AGN be over 20% redward of $\lambda$5007 and $<$10% at the blue end of the spectrum. However, given the faintness of the continuum, this estimate is very uncertain. A fainter host galaxy would, of course, result in lower overall polarization as well as a smaller relative decrease in the polarization in the blue since the spectrum of the AGN would be redder.
Permitted and forbidden lines are observed in the polarized flux spectrum. Although H$\alpha$ is near the red end of the observed spectrum where the S/N is low, the line (plus possibly \[\]) can be seen in polarized flux. Both \[\] lines are apparent in $q' \times F_\lambda\/$, but at much smaller equivalent widths than in total flux. The ratio of the polarized-to-total \[\] flux implies a polarization of $\sim$1.5%, substantially less than for the continuum. Relative to \[\], the Balmer lines are stronger in polarized light than in $F_\lambda\/$, indicated a higher polarization for the permitted lines.
### 2MASSI J100121.1+215011 and 2MASSI J222554.2+195837
The Type 2 QSOs 2M100121 and 2M222554 are polarized at a very low level ($<$1%) and no features can be discerned in the spectrum of polarized light. Stellar absorption features in the total flux spectra, including the break, indicate that the host galaxy contributes $\sim$80% of the light from these objects at 5500 Å (rest frame) within the 11$\times\sim$4 apertures employed for the spectropolarimetry. Correcting the data for even this large amount of unpolarized flux only elevates the polarization of 2M222554 to $\sim$1%.
### 2MASSI J105144.2+353930
2M105144, like 2M100121 and 2M222554, is not highly polarized and exhibits prominent stellar features in its spectrum. The ratio of starlight to AGN light in the 15$\times$46 aperture used is estimated to be $\sim$2.3 in the rest-frame $V\/$-band. Subtraction of the assumed elliptical galaxy spectral template gives an $R\/$-band polarization of 2–3% for the remaining light.
Measurements of H$\alpha\/$ reported in Tables 2 and 3 are uncertain because the redshift places the line center at the position of the O$_2$ A-band absorption. Despite this unfortunate coincidence, 2M105144 earns its classification of Type 1.9 because H$\alpha\/$ possesses an $\sim$18,000 [km s$^{-1}$]{} (full width at zero intensity) component. This broad feature is also detected in the polarized flux spectrum, indicating that some of the flux from the inner BLR is scattered into our line of sight. No other features are identified in the polarized spectrum.
### 2MASSI J130005.3+163214
@schmidt02 present and discuss the MMT observations of the Type 2 QSO 2M130005 that are redisplayed in Figure 3. The dominant feature in the polarized spectrum of 2M130005, as for 2M105144, is very broad ($\sim$18,000 [km s$^{-1}$]{} FWHM) H$\alpha$. The S/N of the data is much higher than for 2M105144, and @schmidt02 are able to deduce the polarization of the continuum, broad H$\alpha$, and the narrow emission lines. In comparison to other QSOs in the sample, 2M130005 has by far the reddest total flux (after subtracting the substantial contribution of the host galaxy) and polarized flux spectra. Not all of the D feature is stellar in origin, since it can be seen in $q' \times F_\lambda\/$.
In addition to the red polarized continuum, the Balmer decrement is possibly very large since H$\beta\/$ is not definitively detected in polarized flux. However, assuming that H$\beta\/$ is as broad as H$\alpha\/$ in polarized flux, the extreme width of the line hinders its identification. A rough limit on the strength of a possible broad H$\beta\/$ feature was estimated by fitting a power-law to the polarized continuum redward of 5900 Å and avoiding H$\alpha\/$. Subtraction of this fit from $q' \times F_\lambda\/$ reveals emission from $\sim$5700 Å to the blue edge of the spectrum at 4200 Å that could be the broad, blended lines of H$\beta\/$, H$\gamma\/$, and higher-order Balmer lines. The possible emission excess may indicate that the H$\beta\/$ flux is as much as $0.14 \times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$, implying a polarized Balmer decrement of only $\sim$3.
### 2MASSI J171559.7+280717
This object is the most distant of the Type 1.8–2s and one of the most luminous QSOs in the sample ($M_{K_s} = -28.1$). Its redshift of $z = 0.524$ has shifted H$\alpha\/$ out of the observed spectral range. Weak, broad H$\beta\/$ is detected and we therefore classify 2M171559 as Type 1.8.
The object is extremely faint ($V \sim 21.4$), but the spectropolarimetry confirms that 2M171559 does indeed join 2M010835 in showing a broadband polarization $>$3%. @marble03 find significant extended flux around the AGN in the WFPC2 F814W image. Using this observation to estimate the contribution of starlight in the spectropolarimetry suggests that about half of the light at a rest frame wavelength of 5500 Å is from the host galaxy. This choice of host galaxy is quite uncertain as there are no discernible stellar features superimposed on the faint continuum. A host galaxy contribution this large implies an intrinsic polarization within the $R\/$-band of over 15% for the AGN.
There is no evidence of the strong, narrow emission lines in the polarized flux spectrum. Close inspection of the spectrum of $q'\/$ reveals decreases in the polarization at \[\]$\lambda\lambda$4959,5007, \[\]$\lambda$3727, and the narrow component of H$\beta\/$. Some of the increase in polarized flux between the location of narrow H$\beta\/$ and the \[\] lines could be due to polarized broad H$\beta\/$. The S/N is insufficient to unambiguously identify broad H$\beta\/$ in polarized light and we have not attempted to measure the width and strength of this possible feature. Although the measurements are uncertain, the polarized flux continuum (ignoring the region around H$\beta\/$) is much redder than that of the host galaxy-subtracted total flux.
Discussion
==========
Spectral Properties
-------------------
Although nearly the full range of AGN spectral types is represented by the 21 objects described in the previous section, they may not be representative of the 2MASS AGN sample as a whole. The objects chosen for observation are primarily highly polarized. This bias translates into a sample that is generally more luminous in the near-IR and has higher near-IR–to–optical flux ratios than a “typical” 2MASS QSO [@smith02]. Caution should therefore be exercised in extrapolating the spectroscopic results for the highly polarized objects to the entire sample. The 2MASS sample is generally compared to the low-redshift ($z < 0.6$) members of the Palomar-Green (PG) QSO sample [@mschmidt83] in the following discussion. The choice of the PG sample to represent optically-selected QSOs is primarily dictated by the large amount of data available in the literature for these objects.
### \[\] Luminosity
The 2MASS AGN discussed in the previous section have been classified as QSOs based on the fact that they are as luminous at 2.2 $\mu$m as QSOs in other samples, not because of their observed optical luminosity or color. Indeed, nearly half of the spectropolarimetry sample shows evidence that starlight from the host galaxy contributes more than 50% of the continuum light observed in the 1–3 apertures employed. The red colors and high fraction of optical host galaxy starlight are consistent with the nuclear regions being obscured and reddened by dust in our line of sight, resulting in a QSO sample that is apparently optically underluminous and redder than previous cataloged QSOs. The high polarizations observed in the 2MASS sample are also consistent with a sample of objects partially hidden from direct view by dust [@smith02]. Since the dust extinction at $K\/$ is much less than that experienced at optical wavelengths, it stands to reason that a near-IR search would uncover QSOs missed by surveys that rely on optical color and luminosity selection criteria, and that the intrinsic luminosity of these AGN would be more accurately estimated from a near-IR brightness than by optical magnitude.
Data at longer wavelengths offer some support for the assumption that the absolute $K_s\/$ magnitude, $M_{K_s}\/$, can be used to distinguish QSOs from AGN of lower luminosity [@smith02]. Of the objects detected at 60 $\mu$m and 1.4 GHz, there is no large systematic difference between 2MASS QSOs and optically-selected QSOs for a given 2.2 $\mu$m luminosity. The validity of using $M_{K_s}\/$ as a measure of intrinsic luminosity can also be verified optically by inspecting the \[\]$\lambda$5007 luminosities. In the context of current ideas on the role that viewing orientation plays in determining the observed properties of AGN [see e.g., @antonucci93], emission from the kpc-sized NLR is generally thought to be more isotropic than the emission from the BLR and ionizing continuum, although there is evidence for anisotropy in at least the higher ionization narrow emission lines in some AGN [see e.g., @jackson90; @hes93; @baker97]. Despite these examples, the \[\]$\lambda$5007 flux should, to first order, scale with the luminosity of the ionizing nuclear continuum.
Figure 5 compares the \[\]$\lambda$5007 luminosity of the 2MASS spectropolarimetry sample with low-redshift PG QSOs. Following @boroson92, the \[\]$\lambda$5007 luminosity is given by $$M_{\rm [O~III]} = M_V - 2.5 \log ({\rm EW}_{\rm [O~III]}),$$ where $M_V\/$ is calculated from the rest frame 5500 Åflux density. For both samples of objects, a clear trend of increasing \[\]$\lambda$5007 luminosity with increasing $M_{K_s}\/$ is seen. In addition, the strength of the \[\]$\lambda$5007 emission from the 2MASS QSOs is within the large range exhibited by the optically-selected QSOs. For the predominantly highly polarized 2MASS sample, the Type 1.5, 1.8 (2M171559), and 1.9 (2M010835) objects cannot be distinguished from the low-polarization, Type 1 PG QSOs. The three Type 2 QSOs and 2M105144 are clustered around $M_{K_s} \sim -26$, $M_{\rm [O~III]} \sim -24.5$; consistent with PG QSOs in having fainter \[\] emission for this lower near-IR brightness.
It is true that, in comparison with the PG sample, the Type 1 2MASS QSOs typically have a lower \[\] luminosity for a given brightness at $K_s\/$. Only seven of the eight Type 1 QSOs are included in Figure 5 since \[\]$\lambda$5007 is not detected in 2M125807. Such weak \[\] is also seen in some PG QSOs, as four of 74 objects in the @boroson92 study do not show emission from the NLR, and there are some PG QSOs with \[\] strengths as low as the 2MASS objects. However, it is striking that the majority of PG QSOs show stronger \[\] emission than the highly polarized Type 1 2MASS objects for a given near-IR luminosity.
A proper comparison of the \[\] emission between the near-IR and optically-selected samples awaits measurements for a larger sample of Type 1 2MASS objects, and the small number of objects presented here is strongly biased toward those showing high optical polarization. With these qualifications in mind, it appears that 2MASS is adept at finding red QSOs that, although exhibiting broad permitted emission lines in their total flux spectra, are underluminous in \[\]. A possible explanation for a link between the adopted red near-IR color selection and weak \[\] strength may be that 2MASS finds AGN with a larger dust covering factor than is typical for optically-selected QSOs. @borosonm92 suggest a similar situation for the low-ionization broad absorption-line QSOs (BALQSOs) within the sample of infrared-selected AGN discovered by the [*Infrared Astronomical Satellite*]{} [[*IRAS*]{}; @low88].
At the same time, there is no evidence from the objects of intermediate spectral type in this small sample that suggests a [*fundamental*]{} difference with UV-excess QSOs in terms of near-IR and \[\] luminosity. For these AGN, it would appear that we have a relatively unobstructed view of the NLR that is powered by an continuum source indistinguishable from an optically-selected QSO if viewed from the line-emitting region.
### Optical and Near-IR Continua
Although dramatic differences are not seen between the 2MASS and PG QSO samples in terms of \[\] and near-IR luminosity, clear trends are observed in the optical spectral index ($\beta_{\rm OPT}\/$). Figure 6 plots both the near-IR spectral index ($\beta_{\rm IR}\/$; determined from the 2MASS $JHK_s\/$ photometry) and the $B - K_s\/$ color index against $\beta_{\rm OPT}\/$. The optical continuum is much redder for the 2MASS objects than for the comparison sample of PG QSOs with $z < 0.6$ measured by @neugebauer87. The displacement of the 2MASS QSOs away from the PG QSOs in Figure 6 is generally consistent with that expected from reddening by dust, though for consistency with the PG data, $\beta_{\rm OPT}\/$ has not been corrected for the flux contribution made by the host galaxy. It is also the case that, for most of the 2MASS objects discussed here, a significant amount of the observed AGN flux is scattered into our line of sight. The scattered light does not appear to experience the same amount of reddening as the direct, unscattered light from the nucleus (see §4.2), and dust scattering typically results in a much bluer scattered light spectrum. Therefore, it is very difficult to ascribe a single reddening value to account for the spectral energy distribution throughout the entire optical/near-IR spectral region.
### Balmer Decrement
Another reddening indicator is given by the H$\alpha$/H$\beta\/$ flux ratio. This quantity can be measured in 17 of the 21 objects observed. Generally, H$\alpha$/H$\beta$ is much larger than typically found for optically-selected QSOs, and in several objects it is measured to be $>$10. It can be seen in Figure 7 that, at least for Type 1 and 1.5 QSOs, there is a trend between the Balmer decrement and the slope of optical continuum of the AGN. As expected from dust extinction, larger Balmer decrements tend to be observed for objects with redder optical continua.
For Figure 7, the estimated contributions from host galaxy starlight have been subtracted from the spectrum. The fluxes of the Balmer lines include both broad and narrow-line components, and \[\]$\lambda\lambda$6548,6583 has not been deblended from H$\alpha\/$. The \[\] lines are only significant relative to H$\alpha\/$ for the Types 1.9 and 2 objects, and even reducing the measured H$\alpha\/$ flux by a factor of 2 to account for the blended forbidden lines still implies a large Balmer decrement for these QSOs.
Figure 7 also includes a rough reference point for comparison of the 2MASS QSOs with optically-selected QSOs. The star in the figure represents the median high frequency power-law fit to PG QSOs with $z < 0.6$ [@neugebauer87] and H$\alpha$/H$\beta$ (=3.7) measured from the QSO template spectrum constructed from the [*Sloan Digital Sky Survey*]{} [SDSS; @vandenberk01]. Roughly half of the Type 1 2MASS QSOs and all of the Type 1.5-2 objects have much larger Balmer decrements than the adopted optically-selected composite QSO.
For each of the Type 1 and 1.5 objects that have their Balmer decrement measured from the $q' \times F_\lambda\/$ spectrum, H$\alpha\/$/H$\beta\/$ is smaller in polarized light than observed in the total flux spectrum (Table 3). However, a proper comparison between the polarized and total flux Balmer decrements requires that only the broad Balmer-line components be used to determine the decrement since the polarized flux spectra of the 2MASS QSOs are generally devoid of NLR features. The narrow-line contribution to H$\alpha\/$ and H$\beta\/$ is not significant for the Type 1 objects, but by definition, the Type 1.5 QSOs exhibit Balmer lines with distinct broad-line and narrow-line components. After deblending the components and subtracting an estimate of the \[\]$\lambda\lambda$6548,6583 flux from $F_{{\rm H}\alpha}\/$, a broad-line Balmer decrement is derived. The resolution of the observations does not permit direct measurement of \[\] and we have assumed that the forbidden-line flux equals that of narrow H$\alpha\/$, close to the average line ratio for AGN [see e.g., @veilleux87]. It is clear from this exercise that broad H$\alpha\/$ dominates the line flux in the five Type 1.5 QSOs that have measured Balmer decrements. The broad-line Balmer decrements are found to be roughly the same as those given in Table 3 for the total flux spectra, and therefore, the fact that the decrement is smaller in polarized light is not caused by the inclusion of narrow-line flux.
The smaller polarized Balmer decrements suggest that the scattered light may undergo less reddening than the total AGN light. In fact, since $F_{{\rm H}\alpha}/F_{{\rm H}\beta} < 4$ for several objects, either there is little reddening from the nucleus to the scatterers and from the scatterers to us, or the scattering efficiency is higher at shorter wavelengths. In the latter case, electron scattering would be ruled out.
Polarization Properties
-----------------------
Because all of the highly polarized 2MASS QSOs are included in this study, these objects are largely responsible for the correlations between the degree of polarization, near-IR luminosity, and $B - K_s\/$ found by @smith02. That is, AGN with high near-IR luminosity and near-IR–to–optical flux ratios tend to be highly polarized. These trends are consistent with a model that consists of 1) unpolarized, reddened AGN light that is observed directly, 2) unpolarized starlight of the host galaxy, and 3) AGN light polarized by scattering into our line of sight. Smith [et al. ]{}argue that the majority of Type 1 objects are not highly polarized because the unscattered nuclear light dominates any source of polarized light. From our analysis of selected objects, it appears clear that the Type 2 QSOs show little broadband polarization because of host galaxy dilution, despite the fact that their extremely red colors imply heavy extinction of nuclear light along our line of sight. It may also be that the obscuration is so pervasive in the Type 2 QSOs that the light from the scattering region is also obscured. The intermediate QSOs show the highest mean polarization. For these, @smith02 suggest that our direct view of the nucleus is more heavily obscured than for a typical Type 1 2MASS QSO, so that AGN light scattering into our view is not swamped by direct nuclear light. The fact that the QSOs of higher intrinsic luminosity tend to be the highly polarized objects in the sample is a consequence of the ability of the luminous AGN to better illuminate scattering material.
### Continuum Polarization
Starlight from the host galaxies of many 2MASS QSOs results in continuum polarization being substantially less than if the AGN light could be observed in isolation. The 6000–7000 Å polarizations for 14 QSOs after the subtraction of the best estimates for the diluting stellar continua are listed in Table 3. The observed polarizations of the remaining objects are also listed since these objects show no evidence of significant “contamination” by the host galaxy. Not all objects exhibit high polarization even after the stellar continuum is taken into account, but these results support the assertion by @smith02 that the distribution of polarization of the 2MASS sample is even more distinct from optically-selected samples than suggested from raw broadband polarimetry. A striking example of the effects of dilution is given by 2M130005 [Figure 3 and @schmidt02]. Both the broadband measurement and the subsequent spectropolarimetry yield $P < 3$% in the $R\/$-band, but correction for the obvious late-type stellar spectrum implies an intrinsic polarization of the nuclear light of around 10%.
A wide range of continuum slopes are observed for the polarized flux and the host galaxy-subtracted total flux spectra. The power-law spectral indices for both $F_\lambda\/$ and $q' \times F_\lambda\/$ are listed in Table 3. It can be seen in Figure 8 that for most 2MASS QSOs the polarized flux spectra are much bluer than observed for the optical AGN total flux, but $\beta_{q' \times F_\lambda}\/$ is still generally redder than the median optical continuum for the PG QSOs ($\beta_{\rm OPT} \sim -1.6$; Figure 7). Two factors are most likely responsible for bluer polarized flux continua: the scattered light from the nuclear region experiences less reddening than the light directly seen from the AGN, and/or the scattering is more efficient in the blue than at longer wavelengths. The second effect, if applicable, rules out electron scattering and strongly hints that the scattering particles are dust grains. Although $\beta_{q' \times F_\lambda}\/$ tends to be bluer than $\beta_{F_\lambda}\/$, the polarized flux can be very red. Again, 2M130005 provides an extreme case of a red polarized continuum [@schmidt02], and 2M170003 also shows $\beta_{q' \times F_\lambda} > 2$. The similarity of polarized and total flux spectral indices for each of these QSOs suggests that the light from the scattering regions is reddened by an amount comparable to the total flux spectrum.
Five objects have measured spectral indices for the polarized continuum that are redder than those for the total flux spectrum of the AGN. Each of these measurements, however, is much more uncertain than indicated by the formal error bars displayed in Figure 8. The error bars do not fold in the uncertainty in the fraction of host galaxy starlight. Of the five QSOs, polarizations $\lesssim$1.0% are observed for 2M100121, 2M125807, and 2M222221, making measurement of the polarized continuum difficult. Inspection of Figure 2 for 2M222221 does show that the polarized continuum between H$\beta\/$ and H$\alpha$ is redder than $F_\lambda\/$ over the same wavelength range. Low polarization also hampers measurement of $\beta_{q' \times F_\lambda}\/$ for 2M105144, 2M132917, and 2M222554; objects showing blue polarized continua relative to their total flux spectra. 2M010835 and 2M171559 are the two remaining QSOs measured to have redder polarized continua, but the value of $\beta_{F_\lambda}\/$ is highly dependent on the choice of AGN-to-host galaxy flux ratio. This is true for all objects with large fractions of starlight in their spectra, mostly the Type 1.8–2 QSOs. The fraction of host galaxy flux is highly uncertain for both 2M010835 and 2M171559 simply because of the weak continua observed for these QSOs.
### Broad-Line Polarization and Hidden BLRs
The vast majority of 2MASS QSOs included in this study exhibit broad Balmer lines in their polarized flux spectra. Some objects show polarization structure across the line profiles (see below), but in general the line polarizations are similar to that of the continuum in degree and position angle. This fact excludes synchrotron radiation in general as the source of polarization. There are possibly only five QSOs in the sample that do not have polarized broad permitted lines. Three objects, 2M100121, 2M125807, and 2M222554 are very weakly polarized, making the identification of spectral features difficult. The only Type 1 object that shows no evidence for polarized emission lines is 2M125807, and it has the lowest polarization of the eight objects observed. The case of 2M222202 is detailed in §3.3.6 and the available data do not allow an unambiguous identification of weak polarized H$\beta\/$ despite the high S/N polarized flux spectrum (Figures 2 and 4). For this reason, a highly polarized synchrotron continuum cannot be ruled out for 2M222202, but observations at three epochs spanning 2.5 yr have not shown the object to be variable. Emission from H$\beta\/$ also cannot be identified in the much lower S/N polarized flux spectrum of the optically faint QSO 2M171559.
Given the resolution and S/N of the spectropolarimetry, the polarized and total flux line widths of the Balmer lines are roughly equivalent for the Type 1–1.5 QSOs. This implies that the scattering material is illuminated by an emission-line spectrum that is not much different from the BLR spectrum observed in our line of sight. The Balmer line widths tabulated in Table 2 identify four Type 1.5 QSOs with seemingly broader Balmer lines in polarized flux than in total flux. The decreased FWHM of the Balmer lines measured in the total flux spectrum of 2M134915, 2M165939, and 2M222221 is caused by the prominent narrow-line component largely absent from the polarized line profile. A Balmer emission-line component as broad as the polarized emission features can be seen in the total flux spectrum for all three of these objects. In the case of 2M222202, H$\beta\/$ is tentatively measured to be very broad (FWHM $\sim$ 17,000 [km s$^{-1}$]{}), although the emission feature is very weak and may even be misidentified.
The Type 1.8–2 objects with measured Balmer-line widths for both total and polarized flux spectra are represented by 2M010835, 2M105144, and 2M130005. The latter two objects were originally added to the sample to search for hidden BLRs, and both, in fact, exhibit very broad H$\alpha$ lines in polarized flux. This broad-line component can also be seen in the high S/N total flux spectrum of 2M105144 (Figure 3) and has been noted in 2M130005 by @schmidt02.
2M130005 presents a dramatic example of a situation where emission from high-velocity clouds in the BLR is scattered into our line of sight. Though the broad H$\alpha\/$ line is extremely difficult to discern in the total flux spectrum, it appears as a very prominent, broad feature in polarized flux (Schmidt [et al. ]{}2002). Of course, hidden BLRs have been found in many Seyfert 2s [@antonucci85; @miller90; @tran95], NLRGs [@ogle97; @cohen99], and HIGs [@hinesw93; @hines95; @young96; @goodrich96], and these discoveries have greatly promoted the idea that orientation of the nuclear region to the line of sight largely determines the “type” of AGN that we perceive. The results from 2M105144 and 2M130005 suggest that near-IR–selected AGN will also provide a number of similar objects; many with the intrinsic luminosity of a QSO.
Large polarized Balmer decrements and/or red polarized continua are observed for 2M010607, 2M150113, 2M130005, 2M163700, and 2M170003, and this may imply that the scattering region, as well as the nucleus, is highly reddened along the line of sight, or that light from the nucleus is reddened before reaching the scatterers. It may be common for the scattering regions in 2MASS QSOs to be reddened either by the dust torus that obscures the direct view to the nucleus, or by dust in the body of the host galaxy. A practical consequence of significant extinction of the extended scattering regions for Type 2 objects is that hidden BLRs will be more difficult to uncover polarimetrically.
Several 2MASS QSOs show higher polarization at H$\alpha\/$ and/or H$\beta\/$ compared to the local continuum. Good examples are 2M130005 and 2M134915, and this property can be attributed to dilution of the polarized light by an unpolarized continuum. Because the flux of the emission lines is not as heavily diluted as the polarized continuum, the polarization rises with the line profile. The same effect is often seen in highly polarized Seyfert 2s and radio galaxies, but for many of these objects, careful subtraction of the stellar light still leaves the broad emission lines more highly polarized than the continuum [see e.g., @tran95; @cohen99]. Another unpolarized, featureless continuum (FC2) of unknown origin, is generally invoked to bring the intrinsic continuum polarization to parity with that of the BLR, avoiding the physically untenable situation of having nuclear light, originating from a very compact region, be less polarized than the light from a more extended region surrounding the continuum source. Within the S/N of the spectropolarimetry of this optically faint sample, division of $q' \times F_\lambda\/$ by the total flux spectra after subtraction of the estimated stellar continua yields polarization levels at H$\alpha\/$ and H$\beta\/$ consistent with the local continuum polarization. Therefore, the existence of FC2 in the 2MASS sample is not supported by the observations.
Beyond the effect of the host galaxy continuum on the polarization of the broad Balmer lines relative to the continuum polarization, four 2MASS QSOs exhibit variations in polarization across the Balmer-line profiles. 2M004118 and 2M151653 most clearly show changes in both $P\/$ and $\theta\/$ at the positions of the BLR features [Figure 1 and @smith00]. In both cases, the blue wings of the lines are less polarized than the red wings, but inspection of Figure 1 reveals that the behavior of $\theta\/$ across H$\alpha\/$ is different for the two QSOs. For 2M132917 (Figure 1), deviations of $P\/$ and $\theta\/$ from the level of the local continuum appear to be restricted to the line core of H$\alpha\/$. 2M222221 (Figure 2) also shows complex rotations of $\theta\/$ across H$\alpha\/$ and, in contrast to 2M004118 and 2M151653, diminished polarization in the line’s red wing.
The wavelength dependence of the polarization across the BLR emission features implies a degree of complexity for the BLR and/or scattering regions. It also implies that the scattering material is located quite close to the BLR for these 2MASS QSOs, since polarization structure is very difficult to explain if the BLR is unresolved from the vantage point of the scattering clouds. Spectropolarimetry of several other Type 1 AGN have shown similar features. @goodrich94 and @smith97 have identified cases among Seyfert 1 nuclei [see also @martel97; @smithj02], and @cohen99 have done the same from a small sample of broad-line radio galaxies (BLRGs; namely, 3C 227 and 3C 445). Apparently, the phenomenon is independent of the radio power of the AGN. @cohen99 suggest a scenario to explain the complex polarization position angle rotations observed in broad H$\alpha\/$ for 3C 445 using the assumption that the gas motion within the BLR is not chaotic. For instance, the BLR emission could be scattered off of the inner wall of a dusty torus that is coplanar with the orbiting BLR clouds. If the emission line is broadened by the orbital motion of the BLR gas, and not the thermal motion of the gas nor the motion of the scatterers, the position angle difference between the red- and blue-shifted line emission can be explained by the fact that the two line components illuminate the scattering material from different directions. @cohen99 point out that such a scenario works over a restricted range of inclination angles. The BLR also needs to be viewed directly since broad lines are prominent in the total flux spectrum, but if the inclination of the dust torus is too high, the scattering region that gives rise to the wavelength dependence of $\theta\/$ across the line will be obscured by the torus. A similar scenario with narrower spectral features than 3C 445 could be applicable to 2M004118 and 2M222221, but the variety of wavelength dependences in $P\/$ and $\theta\/$ precludes a single geometry in all Type 1 objects. For example, the Seyfert 1 Mrk 486 [@smith97] also requires at least two polarized emission-line components to describe the polarization at H$\alpha\/$, but instead of being shifted in wavelength, the components have different widths.
### The Polarization of the NLR and the Location of the Scattering Material
Emission from the NLR is observed to have low polarization in a wide variety of polarized AGN samples where scattering of nuclear light into our line of sight is thought to be the polarizing mechanism. This is interpreted as the result of the scattering material being located within, or just exterior to, the NLR. In this respect, the highly polarized 2MASS QSOs resemble for example, polarized Seyferts, BLRGs, and HIGs since they all show much reduced polarization in their prominent narrow emission lines. Close proximity of the scatterers to the nuclear region is also required for the QSOs that show polarization variations with wavelength across their broad emission-line profiles (§4.2.2). Besides giving an indication of the location of the scattering regions, the differentiation of the NLR polarization from the continuum and BLR validates the assumption made in §2 that Galactic ISP is not the source of polarized flux for the sample.
Polarized \[\]$\lambda$5007 is convincingly detected in six of the 2MASS QSOs: 2M010835, 2M091848, 2M130005, 2M134915, 2M222202, and 2M222221, plus 2M135852. The level of polarization for the NLR is $<$3% for all of these QSOs (last column of Table 3). All spectral classes are represented in the small number of objects showing polarized \[\]$\lambda$5007. In general, the NLR shares a common polarization position angle with the continuum and BLR, although the polarization of the NLR in 2M222202 is deduced from the rotation in $\theta\/$ seen in the prominent narrow emission-lines. The level of polarization of the NLR is low enough that transmission through aligned dust grains within the body of the host galaxy cannot be ruled out as the polarizing mechanism, but the common polarization position angle of the NLR and the continuum observed in most objects suggests that the material scattering the continuum and BLR light also scatters and polarizes the inner NLR. 2M130005 provides evidence suggestive of the scattering material being located throughout the NLR [@schmidt02]. A polarization of $\sim$1.5% is found for \[\]$\lambda$5007, whereas the lower ionization \[\]$\lambda\lambda$6716,6731 lines are unpolarized. In addition, the relative strengths of narrow H$\alpha\/$ and \[\]$\lambda\lambda$6548,6583 are noticeably different in polarized light [*vs.*]{} total flux, with H$\alpha\/$ being stronger the polarized light. These observations are consistent with the regions producing the \[\] and narrow H$\alpha\/$ lines being located closer to the nucleus, and with the scattering material being possibly intermixed with the unpolarized NLR gas. Similar evidence for such a stratified NLR is also seen in the HIG [*IRAS*]{} P09104+4109 [@hines99; @tran00].
If the continuum is polarized by scattering in 2M222202 (§3.3.6), then the scatterers must be located closer to the nucleus than in the other 2MASS QSOs, since the polarized spectrum is nearly featureless. The scattering medium in this case may be electrons, given that the scatterers are likely to be intermixed with the BLR gas. Scattering by dust further out in the NLR may be responsible for the observed polarization in the rest of the sample. Dust would naturally explain the diminished Balmer decrements and blue continua of the polarized flux spectra, although electron scattering coupled with reduced reddening along the line of sight to the scattering region is also consistent with the observations. Ultraviolet spectropolarimetry may hold the key to identifying the scattering material in 2MASS QSOs since the shape of the spectrum of UV polarized flux has been used to identify dust as the scatterers in other reddened AGN [e.g., @hines01].
Host Galaxies
-------------
Direct imaging with [*HST*]{} of 29 2MASS QSOs by @marble03 shows that the sample exhibits a large range of AGN–to–host galaxy flux ratios. In addition, @marble03 find that a wide variety of host galaxy morphologies and luminosities are represented, and that no clear differences are seen between the 2MASS QSO hosts and the galaxies hosting UV-excess QSOs. Although there is little overlap between the samples, the spectropolarimetry also finds objects that show little or no evidence for a significant contribution to their spectra from starlight, as well as objects where the host galaxy dominates the observed continuum. Type 1.8–2 QSOs generally show the largest host galaxy-to-total flux ratios in this sample, while stellar absorption features cannot be definitively identified for most of the eight Type 1 objects. These results are consistent with the suggestion by @smith02 that, unlike optically-selected QSO samples, dilution of the polarized flux by starlight has a major effect on the observed distribution of broadband polarization in the 2MASS QSO sample (see §4.2.1).
Conclusions and Summary
=======================
Spectropolarimetry of 21 highly polarized and narrow-line QSOs discovered by 2MASS reveals several important aspects of this near-IR color-selected sample:
1\. The observations are consistent with scattering of nuclear continuum and emission from the BLR by dust or electrons located exterior to the BLR as the polarizing mechanism. In general, the low polarization of the NLR implies that the scatterers cannot be situated much further from the ionizing continuum source than the extent of the NLR. The close proximity of the scattering material to the BLR in some objects is also indicated by the observation of wavelength-dependent polarization across their broad emission-line profiles. For one object, 2M222202, the scatterers must be located very close to the nucleus because the broad emission lines are essentially unpolarized.
2\. Nearly all of the polarized 2MASS QSOs have broad Balmer emission lines detected in their polarized flux spectra.
3\. In addition to displaying a wide array of AGN spectral types, the 2MASS QSOs exhibit a huge range of optical continuum slopes and Balmer decrements. These properties are consistent with various amounts of reddening and extinction of AGN light for these objects. In general, the polarized flux continua are bluer than the total flux spectra (i.e., $P\/$ increases to the blue), and the broad-line Balmer decrement is smaller in the polarized light. This trend can be explained if the scattering efficiency increases with decreasing wavelength (small dust grain scattering), or if the particles are electrons and the scattered light experiences less reddening than the direct light from the nucleus. Although the polarized flux spectrum tends to be bluer than the total flux spectrum for these QSOs, the polarized continuum can be extremely red and large Balmer decrements can be measured for the polarized flux spectrum. For several objects, these properties indicate that there is a significant amount of reddening of the scattering regions situated around the nucleus.
4\. The 2MASS sample includes a few objects with hidden BLRs. Spectropolarimetry of the Type 2 QSO 2M130005 reveals an extremely broad H$\alpha\/$ emission line in polarized flux [@schmidt02]. A similarly broad feature is also detected in both the polarized and total flux spectra of 2M105144.
5\. At optical wavelengths, $\gtrsim$50% of the observed continuum for many objects is from stars in the host galaxy. The largest host galaxy contributions to the optical flux are generally found for the Type 1.8–2 QSOs. This result is consistent with the finding of @marble03 from [*HST*]{} imaging, that the fraction of host galaxy-to-AGN light is generally larger for Type 1.8–2 objects than for Type 1 and 1.5 QSOs, and implies that the nuclear regions of Type 1.8–2 2MASS QSOs are typically more highly obscured in our line of sight. The spectropolarimetry also supports the inference made by @smith02 that the low level of broadband polarization observed for Type 1.8–2 QSOs is primarily due to a large amount of unpolarized light included in the observation aperture. In particular, correction for the host galaxy starlight yields intrinsic optical polarizations of the nuclear continuum for some Type 1.8–2 QSOs of $\sim$10–20%.
6\. The sample of eight Type 1 2MASS QSOs observed have \[\]$\lambda$5007 luminosities systematically lower than PG QSOs of similar $M_{K_s}\/$. Since the sample is biased toward highly polarized objects, this may not be a property of the 2MASS QSOs as a whole. However, at least for the selected QSOs, this finding suggests that dust obscuration extends over a larger solid angle than is typically inferred to account for the relative numbers of optical spectral types in simple unification schemes [e.g., @antonucci93]. No difference in \[\] luminosity is seen between presumably more heavily obscured Type 1.5–1.9 2MASS QSOs and unobscured optically-selected QSOs.
It appears that the same basic model of an AGN surrounded by a dusty torus, with scattering material located above and below the toroidal plane, can be applied to Seyfert 1 and 2 nuclei, narrow- and broad-line radio galaxies, HIGs, and now a large sample of red AGN discovered by 2MASS. Upon close examination, the 2MASS objects do not appear to differ systematically from previous optically-selected AGN in a variety of fundamental parameters. The primary difference seems to be the amount of extinction along our line of sight to the nucleus. Optical spectropolarimetry and \[\] luminosities suggest that the obscuring material does not cover all of the sky as seen by the central engine, and that many 2MASS QSOs would appear as typical UV/optical QSOs if viewed from a different perspective.
In retrospect, it may not be surprising that many AGN are obscured to some degree from our direct view, since the nuclei are often hosted by galaxies that contain large amounts of dust. This situation was foreseen in Seyfert galaxies by @rowan77, but the magnitude of the nuclear obscuration was not appreciated until the first IR surveys for AGN. An exciting and challenging aspect of the new near-IR sample is that it represents a large, possibly dominant, low-redshift AGN population [@cutri01]. It is clear from even the small number of objects observed in this study that the new search techniques are exploring larger ranges of inclination angle and obscuration of the active nucleus and its immediate environment than have been probed by traditional surveys.
While the 2MASS QSOs so far identified add to our understanding of AGN in the local universe, it is likely that more highly obscured objects are excluded by the selection criteria. At sufficiently high column depths, the $J\/$-band flux from the nucleus will fall below the 2MASS sensitivity limit or below that of the host galaxy, making color selection inefficient. Surveys at longer wavelengths would alleviate this problem. The [*Space Infrared Telescope Facility*]{} ([*SIRTF*]{}) has the potential to uncover many more obscured AGN, but its coverage of the sky will be limited.
We thank the National Aeronautics and Space Administration (NASA) and the Jet Propulsion Laboratory (JPL) for support through [*SIRTF*]{}/MIPS and Science Working Group contracts 960785 and 959969 to The University of Arizona. Polarimetric instrumentation at Steward Observatory is maintained, in part, through support by National Science Foundation (NSF) grants AST 97–30792 and AST 98–03072. We also thank an anonymous referee for suggestions that improved the manuscript. This publication makes use of data products from the Two-Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by NASA and the NSF.
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[lrcrrrlccrrr]{}
004118.7+281640 & 0.194 & 1 & $-$27.29 & 12.50 & 3.40 & 1999 Oct 14 & Bok & 3.0 & 7200 & 2.18$\pm$0.02 & 97.9$\pm$0.3\
& & & & & & 2002 Jul 6 & MMT & 1.1 & 1920 & 2.28$\pm$0.02 & 95.2$\pm$0.3\
& & & & & & ave. & & & 9120 & 2.23$\pm$0.01 & 96.9$\pm$0.2\
010607.7+260334 & 0.411 & 1 & $-$27.58 & 14.61 & $>$6.4 & 2000 Jan 9, 10 & Bok & 2.0,3.0 & 19200 & 7.84$\pm$0.14 & 118.4$\pm$0.5\
010835.1+214818 & 0.285 & 1.9 & $-$27.64 & 13.46 & 6.54 & 1999 Oct 13, 15 & Bok & 2.0 & 22400& 5.07$\pm$0.11 & 118.9$\pm$0.6\
091848.6+211717 & 0.149 & 1.5 & $-$26.65 & 12.55 & 5.95 & 2000 Jan 9, 10 & Bok & 3.0 & 24800 & 6.49$\pm$0.02 & 153.4$\pm$0.1\
100121.1+215011 & 0.248 & 2 & $-$25.79 & 14.68 & 5.52 & 2002 Feb 15 & MMT & 1.1 & 3840 & 0.73$\pm$0.16 & 150.5$\pm$6.0\
105144.2+353930 & 0.158 & 1.9 & $-$25.77 & 13.54 & 5.06 & 2002 Feb 15 & MMT & 1.5 & 3840 & 1.18$\pm$0.11 & 30.8$\pm$2.7\
125807.4+232921 & 0.259 & 1 & $-$27.09 & 13.45 & 3.85 & 2000 Jan 10 & Bok & 3.0 & 4800 & 1.00$\pm$0.03 & 107.7$\pm$1.0\
130005.3+163214 & 0.080 & 2 & $-$25.84 & 11.86 & 5.24 & 2001 Apr 1 & MMT & 1.1 & 3840 & 2.58$\pm$0.03 & 50.8$\pm$0.3\
& & & & & & 2002 Jan 5 & Bok & 3.0 & 2400 & 1.81$\pm$0.77 & 46.5$\pm$1.2\
& & & & & & 2002 Jul 4, 5 & MMT & 1.1 & 8640 & 2.51$\pm$0.01 & 43.7$\pm$0.2\
& & & & & & ave. & & & 14880 & 2.76$\pm$0.01 & 45.3$\pm$0.1\
132917.5+121340 & 0.203 & 1 & $-$25.78 & 14.12 & 4.58 & 2001 Mar 31 & MMT & 1.1 & 3840 & 1.37$\pm$0.05 & 13.8$\pm$1.0\
134915.2+220032 & 0.062 & 1.5 & $-$24.87 & 12.24 & 3.27 & 2000 May 7 & Bok & 3.0 & 9600 & 1.84$\pm$0.03 & 108.2$\pm$0.5\
& & & & & & 2001 Mar 30 & MMT & 1.5 & 8640 & 1.64$\pm$0.03 & 104.0$\pm$0.5\
& & & & & & ave. & & & 18240 & 1.78$\pm$0.02 & 106.9$\pm$0.3\
135852.5+295413 & 0.113 & 1 & $-$25.58 & 12.85 & 4.15 & 2000 May 7 & Bok & 3.0 & 12800 & 4.59$\pm$0.02 & 23.0$\pm$0.1\
150113.1+232908 & 0.258 & 1 & $-$27.24 & 13.46 & 5.84 & 2001 Mar 30, Apr 1 & MMT & 1.5,1.1 & 12480 & 3.40$\pm$0.04 & 156.0$\pm$0.3\
151653.2+190048 & 0.190 & 1 & $-$28.35 & 11.41 & 4.39 & 2000 May 9 & Bok & 3.0 & 6720 & 9.27$\pm$0.01 & 106.5$\pm$0.1\
163700.2+222114 & 0.211 & 1.5 & $-$26.44 & 13.59 & 5.41 & 2002 Jul 5 & MMT & 1.1 & 7200 & 2.49$\pm$0.04 & 116.4$\pm$0.4\
165939.7+183436 & 0.170 & 1.5 & $-$26.59 & 12.91 & 5.29 & 1999 Oct 13, 14, 15 & Bok & 3.0,2.0 & 9600 & 5.34$\pm$0.06 & 158.4$\pm$0.3\
& & & & & & 2001 Mar 31 & MMT & 1.1 & 960 & 5.44$\pm$0.09 & 160.0$\pm$0.5\
& & & & & & ave. & & & 10560 & 5.33$\pm$0.03 & 158.8$\pm$0.2\
170003.0+211823 & 0.596 & 1.5 & $-$28.30 & 14.88 & 7.21 & 2001 Mar 31 & MMT & 1.1 & 5760 & 10.96$\pm$0.15 & 108.4$\pm$0.4\
& & & & & & 2002 Jul 4 & MMT & 1.1 & 7200 & 11.10$\pm$0.08 & 108.4$\pm$0.2\
& & & & & & ave. & & & 12960 & 11.06$\pm$0.04 & 108.7$\pm$0.1\
171559.7+280717 & 0.524 & 1.8 & $-$28.14 & 14.63 & $>$6.4 & 2002 Feb 19 & MMT & 1.1 & 1920 & 3.62$\pm$1.11 & 0.3$\pm$8.8\
& & & & & & 2002 Jul 6 & MMT & 1.1 & 7200 & 5.03$\pm$0.18 & 1.4$\pm$1.0\
& & & & & & ave. & & & 9120 & 5.16$\pm$0.08 & 1.5$\pm$0.4\
222202.2+195231 & 0.366 & 1.5 & $-$28.60 & 13.30 & 6.20 & 1999 Oct 13, 15 & Bok & 2.0 & 19200 & 11.41$\pm$0.11 & 118.1$\pm$0.3\
& & & & & & 2002 Jan 6 & Bok & 2.0,3.0 & 5760 & 12.51$\pm$0.25 & 116.9$\pm$0.6\
& & & & & & 2002 Jul 4 & MMT & 1.1 & 4800 & 11.59$\pm$0.03 & 118.8$\pm$0.1\
& & & & & & ave. & & & 29760 & 11.04$\pm$0.04 & 118.8$\pm$0.1\
222221.1+195947 & 0.211 & 1.5 & $-$27.10 & 12.92 & 4.58 & 1999 Oct 15 & Bok & 3.0 & 6400 & 1.15$\pm$0.04 & 156.4$\pm$0.9\
& & & & & & 2002 Jul 8 & MMT & 1.5 & 1920 & 0.96$\pm$0.03 & 165.2$\pm$0.8\
& & & & & & ave. & & & 8320 & 1.02$\pm$0.02 & 161.7$\pm$0.6\
222554.2+195837 & 0.147 & 2 & $-$25.65 & 13.49 & 5.31 & 2002 Jul 5 & MMT & 1.1 & 7200 & 0.28$\pm$0.03 & 0.7$\pm$3.4\
230307.2+254503 & 0.331 & 1 & $-$26.70 & 14.50 & 6.21 & 1999 Oct 14, 16 & Bok & 2.0 & 22400 & 3.31$\pm$0.09 & 137.3$\pm$0.8\
& & & & & & 2002 Jul 8 & MMT & 1.1 & 2400 & 4.66$\pm$0.19 & 138.3$\pm$1.2\
& & & & & & ave. & & & 24800 & 3.47$\pm$0.04 & 138.2$\pm$0.3\
[lcccccccccc]{} 004118.7+281640 & 81/& 500/130: & 8/& 2280/& 1960/3200: & 8.07/& 29.5/0.16: & 0.72/& 41.95/& 1,2\
010607.7+260334 & 71/87 & N/A & 20/& 1740/1880 & N/A & 0.20/0.02 & N/A & 0.06/& 41.71/&\
125807.4+232921 & 25/& 145/& /& 2350/& 2010/& 1.09/& 4.30/& /& /&\
132917.5+121340 & 75/180 & 300/& 29/$<$15 & 3660/4610 & 3610/& 1.18/0.05 & 4.75/& 0.45/$<$0.01 & 41.79/$<$39.7 & 1\
135852.5+295413 & 36/68 & 275/550 & 14/6 & 6720/6590 & 5440/5430 & 2.21/0.18 & 15.00/1.14 & 0.75/0.02 & 41.41/39.77 &\
150113.1+232908 & 32/85 & 225/890 & 8/$<$7 & 2530/3170 & 2540/2580 & 0.23/0.03 & 2.71/0.18 & 0.06/$<$0.01 & 41.19/$<$39.8 &\
151653.2+190048 & 87/92 & 645/440 & 16/& 4040/4840 & 3680/4100 & 18.47/1.95 & 110.00/6.35 & 3.36/& 42.60/& 1\
230307.2+254503 & 49/40 & 80:/& 10/& 2300/1550 & 1800:/& 0.22/0.01 & 0.6:/& 0.04/& 41.33/& 3\
091848.6+211717 & 44/74 & 240/230 & 62/18 & 1980/2270 & 1790/1800 & 0.85/0.11 & 6.54/0.35 & 1.19/0.03 & 41.89/40.21 & 4\
134915.2+220032 & 63/260 & 310/480 & 185/73 & 1900/4580 & 1800/2620 & 2.69/0.17 & 15.90/0.49 & 7.44/0.05 & 41.82/39.66 & 5\
163700.2+222114 & 25/49 & 220/320 & 42/& 1270/1930 & 2910/3700 & 0.12/0.01 & 2.47/0.09 & 0.32/& 41.69/&\
165939.7+183436 & 45/117 & 240/500 & 112/& 4110/15700 & 4270/7430 & 1.02/0.14 & 6.27/0.49 & 2.48/& 42.34/& 5,6\
170003.0+211823 & 123/90 & N/A & 64/& 2630/2980 & N/A & 0.35/0.03 & N/A & 0.19/& 42.74/& 6\
222202.2+195231 & 206/70: & N/A & 173/& 2040/17000: & N/A & 1.22/0.06: & N/A & 1.03/& 42.84/& 7\
222221.1+195947 & 118/240 & 880/1110 & 65/25 & 4260/7910 & 3390/6860 & 8.35/0.17 & 46.80/0.57 & 4.49/0.02 & 42.84/40.37 & 1\
010835.1+214818 & 56/43 & 430/480: & 360/120 & 870/1320 & 1730/3100: & 0.27/0.01 & 2.21/0.16: & 2.43/0.04 & 42.91/41.10 & 3,5\
100121.1+215011 & 7/& 190/& 24/& 1090/& 1830/& 0.03/& 0.80/& 0.09/& 41.30/& 3,5\
105144.2+353930 & 5/& 270:/& 97/& 930/& 2000:/9000: & 0.04/& 2.09:/0.10: & 0.55/& 41.61/& 5,8\
130005.3+163214 & 5/& 57/415 & 57/56 & 1210/& 2060/18400 & 0.18/$<$0.14 & 3.67/0.45 & 1.61/0.03 & 41.40/39.61 & 5\
171559.7+280717 & 130/& N/A & 450:/& 1090/& N/A & 0.19/& N/A & 0.6:/& 43:/& 8\
222554.2+195837 & 4/& 72/& 43/& 960/& 2190/& 0.08/& 1.06/& 0.58/& 41.56/& 4,5\
[lcrrccccccc]{} 004118.7+281640 & & 2.35$\pm$0.02 & 96.3$\pm$0.2 & $-$2.2/$-$1.9 & 0.1/& 3.7/& 1.4/& /2.3 & 0.5:/2.1 &\
010607.7+260334 & & 7.73$\pm$0.20 & 117.3$\pm$0.8 & +1.3/+0.5 & 0.3/& N/A & 0.8/& 9.1/8.6 & N/A &\
125807.4+232921 & 0.1: & 1.05$\pm$0.06 & 111.4$\pm$1.7 & $-$2.2/$-$1.0: & /& 3.9/& 2.1/& /2: & /3: &\
132917.5+121340 & & 1.32$\pm$0.07 & 11.4$\pm$1.6 & $-$0.5/$-$2.0: & 0.4/$<$0.1 & 4.0/& 0.7/& 4.1/2: & /3: & $<$1\
135852.5+295413 & 0.57 & 9.10$\pm$0.07 & 23.4$\pm$0.2 & +0.7/$-$0.9 & 0.3/0.1 & 6.8/6.1 & 0.2/& 8.5/12.8 & 7.6/7.9 & 2.4\
150113.1+232908 & 0.59 & 8.25$\pm$0.14 & 157.0$\pm$0.5 & +2.3/+0.0 & 0.3/$<$0.1 & 11.6/7.1 & 0.7/& 10.7/12.2 & 6.5/6.1 & $<$3\
151653.2+190048 & & 9.24$\pm$0.02 & 107.0$\pm$0.1 & $-$0.9/$-$2.1 & 0.2/& 6.0/3.3 & 1.1/1.0 & 10.6/10.7 & 5.8/7.5 &\
230307.2+254503 & & 2.93$\pm$0.05 & 138.8$\pm$0.5 & +0.9/$-$0.3 & 0.2/& 3:/& 3.3/& 3.7/7.0 & /5.0 &\
091848.6+211717 & 0.37 & 10.01$\pm$0.07 & 153.3$\pm$0.2 & +1.0/$-$0.6 & 1.4/0.2 & 7.7/3.3 & 1.1/1.4 & 12.4/12.7 & 5.4/8.0 & 2.1\
134915.2+220032 & 0.47 & 2.62$\pm$0.04 & 108.0$\pm$0.4 & +0.8/+0.0 & 2.8/0.3 & 5.9/2.9 & 0.8/& 6.2/4.7 & 3.1/3.7 & $<$1\
163700.2+222114 & 0.51 & 4.46$\pm$0.10 & 115.6$\pm$0.6 & +2.0/+0.9 & 1.6/& 12.6/9.2 & 1.1/& 8.3/7.4 & 3.6/5.4 &\
165939.7+183436 & 0.26 & 7.14$\pm$0.12 & 158.8$\pm$0.5 & $-$0.3/$-$0.7 & 2.4/& 6.2/3.5 & /& 14.0/8.6 & 7.8/7.6 &\
170003.0+211823 & & 11.40$\pm$0.05 & 107.9$\pm$0.1 & +2.1/+2.0 & 0.5/& N/A & 0.4/& 8.0/12.4 & N/A &\
222202.2+195231 & & 10.40$\pm$0.05 & 118.9$\pm$0.1 & +0.6/$-$1.0 & 0.8/& N/A & 0.3/& 8:/14.1 & N/A & 0.8\
222221.1+195947 & 0.3: & 1.15$\pm$0.04 & 163.7$\pm$1.0 & $-$1.8/$-$0.2: & 0.5/0.1 & 5.6/3.4 & 0.2/& 2.0/2: & 1.2/3: & $<$1\
010835.1+214818 & 0.8: & 22:$\pm$1: & 120.0$\pm$1.2 & $-$2.3:/$-$0.6 & 9.1/3.5 & 8.2/15: & /& 4.1/24: & 7:/40: & 1.6\
100121.1+215011 & 0.84 & 2.58$\pm$1.11 & 152.7$\pm$12.4 & $-$0.7/+0.5: & 3.3/& 31/& /& /20: & /30: &\
105144.2+353930 & 0.71 & 2.72$\pm$0.55 & 38.3$\pm$5.8 & +2.4/+0.6: & 15/& 57:/& /& /20: & 5:/10: &\
130005.3+163214 & 0.81 & 9.50$\pm$0.07 & 44.7$\pm$0.2 & +3.6/+3.0 & 8.9/& 20/$>$3 & /& /11.9 & 12.2/9.8 & 1.6\
171559.7+280717 & 0.5: & 18.5:$\pm$0.4: & 2.4$\pm$0.7 & $-$0.0:/+0.8: & 3:/& N/A & /& /17: & N/A &\
222554.2+195837 & 0.80 & 1.00$\pm$0.24 & 7.2$\pm$6.8 & +1.2/+0.4: & 7.8/& 14/& /& /7: & /5: &\
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Yukiko <span style="font-variant:small-caps;">Omori</span>, Masahisa <span style="font-variant:small-caps;">Tsuchiizu</span>, and Yoshikazu <span style="font-variant:small-caps;">Suzumura</span>'
title: |
Possible Metastable State Triggered by Competition\
of Peierls State and Charge Ordered State
---
Introduction
============
Electron correlation in quasi-one-dimensional molecular conductors has been studied extensively [@Jerome; @Gruner_rev] where the spin density wave (SDW) state with the momentum 2${k_{\mathrm{F}}}$ (${k_{\mathrm{F}}}$ denotes the Fermi momentum) originates from the combined effect of repulsive interaction and the nesting of the Fermi surface. The former effect is relatively large in molecular conductors and the latter effect becomes perfect for the one-dimensional band. In particular, the quarter-filled band systems, which have been the main subject in molecular conductors, supply a rich variety of electronic states, [@Emery1979] e.g., the charge ordering (CO) [@Seo1997; @Yoshioka2000] with a periodic array of charge disproportionation (+1, 0, +1, 0, $\cdots$). In addition to the pure 2${k_{\mathrm{F}}}$ SDW, the coexistent state of the 2${k_{\mathrm{F}}}$ SDW and 2${k_{\mathrm{F}}}$ charge-density wave (CDW) has been observed in the X-ray experiment on (TMTSF)$_2$PF$_6$,[@Pouget; @Kagoshima] where the coexistent state has a purely electronic origin due to the absence of lattice distortion. It has been clarified theoretically that the next-nearest repulsive interaction gives rise to such a coexistence,[@Kobayashi1998; @Tomio2000JPSJ] and that the state undergoes a first-order phase transition into the normal state with increasing temperature $T$.[@Tomio2001JPCS]
The molecular compound (EDO-TTF)$_2$PF$_6$, which is another salt showing a typical quasi-one-dimensional system, is a recent topic of the photoinduced phase transition. [@Chollet2005; @Ota2002; @Drozdova2004; @Onda2005] It has been reported that the first-order phase transition from metal to the Peierls insulator occurs at $T \simeq$ 278 K and is followed by the fourfold periodicity of the charge-rich sites and the charge-poor sites with the spatial variation of (+1, 0, 0, +1, $\cdots$). The large hysteresis is accompanied by the lattice distortion indicating the strong coupling to the lattice through the electron-phonon (e-p) interaction. This Peierls state shows a distinctive feature of the $2{k_{\mathrm{F}}}$ CDW where the amplitude of the lattice distortion takes a maximum between the hole-rich sites of the EDO-TTF molecules (i.e., the bond ordering). Although the spatial variation of the charge density is similar to the conventional CDW, [@Pouget; @Kagoshima; @Kobayashi1998; @Tomio2000JPSJ; @Tomio2001JPCS] the spatial pattern of the lattice distortion found in (EDO-TTF)$_2$PF$_6$ has not been fully understood in the context of the previous theory. [@Kuwabara2003] Furthermore, the conductivity of (EDO-TTF)$_2$PF$_6$ exhibits a gigantic photoresponse, and the ultrafast insulator-to-metal transition is induced by the weak laser photoexcitation. The mechanism of this behavior has been discussed by assuming the existence of a metastable state without lattice distortion. However, it remains unclear if such a metastable state can be understood using the Peierls model in the presence of the electronic correlation.
In the present paper, we examine the possible origin of the metastable state, which arises from the interplay of the Peierls state and the CO state. In §2, the model with both the repulsive interactions and the e-p interaction is described and the formulation is given within the mean-field theory. In §3 and §4, the ground state and the metastable state are calculated to obtain the phase diagram. The possible parameter region for the metastable state is estimated. Section 5 is devoted to summary and discussions.
Formulation
===========
We consider the 1/4-filled Peierls-Hubbard Hamiltonian given by [@Su1979; @Fukuyama1985] $$\begin{aligned}
\label{eq:hamiltonian}
H = &
- t
\sum_{j=1}^{N}
\sum_{\sigma= \uparrow, \downarrow}
(c_{j,\sigma}^\dagger c_{j+1,\sigma} + \mathrm{H.c.})
\nonumber \\
& + U\sum_j n_{j,\uparrow} n_{j,\downarrow}
+ V\sum_j n_j n_{j+1}
\nonumber \\
& + t
\sum_{j\sigma}u_j(c_{j,\sigma}^\dagger c_{j+1,\sigma} + \mathrm{H.c.} )
\nonumber \\
&
+ \frac{K}{2}\sum_j u_j^2
+ \frac{\delta K}{2}\sum_{\mathrm{odd} \, j} u_j^2 , \end{aligned}$$ where $c_{j,\sigma}$ (spin $\sigma = \uparrow, \downarrow$) is the annihilation operator of the electron at the site $j$ and $ n_j = \sum_{\sigma} c_{j,\sigma}^\dagger c_{j,\sigma}$. We impose the periodic boundary condition, $c_{j+N, \sigma} = c_{j,\sigma}$. Quantities $U$ and $V$ denote the on-site and nearest-neighbor-site repulsive interactions, respectively. The term proportional to $u_j$ (corresponding to the lattice distortion) denotes the e-p interaction and the last two terms represent the elastic energy with the elastic constant $K$ and $K+\delta K$ for even and odd $j$, respectively. The alternating elastic constant given by $\delta K $ plays a crucial role in the present paper. As for the lattice distortion $u_j$, we take into account only the modulation with the momentum $2{k_{\mathrm{F}}}(=\pi/2)$ (corresponding to the lattice tetramerization), which is relevant to the Peierls state in the (EDO-TTF)$_2$PF$_6$, and $u_j$ is rewritten as $$\begin{aligned}
u_j = {u_{\mathrm{t}}}\cos \left( \frac{\pi}{2}j+\xi \right),
\label{eq:distortion}\end{aligned}$$ where ${u_{\mathrm{t}}}$ is the amplitude of the lattice tetramerization. The elastic constant $K$ is scaled so as to include the e-p coupling constant. In the present paper, we do not consider the possibility of the lattice dimerization with the momentum $4k_F(=\pi)$. [@Kuwabara2003; @Clay] Its detail is discussed in §5.
After applying the Fourier transform, we define order parameters ($ m=0,1,2,3$) as [@Tomio2001JPCS] $$\begin{aligned}
S_{mQ_0} =& \frac{1}{N} \sum_{\sigma=\uparrow,\downarrow}
\sum_{-\pi < k \leq \pi } \mathrm{sgn} (\sigma)
\left<c_{k,\sigma}^\dagger c_{k+mQ_0,\sigma}
\right>_{\mathrm{MF}} ,
\label{OPS}
\\
D_{mQ_0} =& \frac{1}{N} \sum_{\sigma=\uparrow,\downarrow}
\sum_{-\pi < k \leq \pi }
\left<c_{k,\sigma}^\dagger c_{k+mQ_0,\sigma}
\right>_{\mathrm{MF}}
, \label{OPD} \end{aligned}$$ where $Q_0 = 2{k_{\mathrm{F}}}= \pi/2$, $S_0=0$, $D_0=1/2$, and $$\begin{aligned}
& S_{Q_0}=S^*_{3Q_0}\equiv S_1 \mathrm{e}^{\mathrm{i} \theta} ,
\quad D_{Q_0}=D^*_{3Q_0} \equiv D_1 \mathrm{e}^{\mathrm{i} \theta'} ,
\nonumber \\
& S_{2Q_0}=S^*_{2Q_0} \equiv S_2 ,
\quad D_{2Q_0}=D^*_{2Q_0} \equiv D_2 .
\nonumber
\end{aligned}$$ We note that $S_{1}$, $S_{2}$, $D_{1}$, and $D_{2}$ (which are real numbers) correspond to the amplitudes for the 2${k_{\mathrm{F}}}$ SDW, 4${k_{\mathrm{F}}}$ SDW, 2${k_{\mathrm{F}}}$ CDW, and 4${k_{\mathrm{F}}}$ CDW, respectively, and $\theta$ and $\theta'$ are the phases for the 2${k_{\mathrm{F}}}$ SDW and 2${k_{\mathrm{F}}}$ CDW. It has been found that the relation $\theta' = \pi/2 + \theta$ holds for ${u_{\mathrm{t}}}= 0$.[@Tomio2000JPSJ] In terms of these order parameters, the mean-field Hamiltonian is written as
$$\begin{aligned}
\label{eq:MF_Hamiltonian}
H_{\mathrm{MF}} =&
\sum_{k,\sigma}
\left\{
\left( \epsilon_k+\frac{U}{4}+V \right)
c_{k,\sigma}^\dagger c_{k,\sigma}
+
\left[
\left( \Delta_{Q_0,\sigma}
+ \frac{1}{2} {u_{\mathrm{t}}}{\mathrm e}^{\mathrm{i} \xi}
(1-\mathrm{i})(\cos k - \sin k) \right)
c_{k+Q_0,\sigma}^\dagger c_{k,\sigma} + \mathrm{H.c.} \right]
+
\Delta_{2Q_0,\sigma}
c_{k,\sigma}^\dagger c_{k+2Q_0,\sigma}
\right\}
\nonumber \\
&
- \frac{N}{4} \Biggl[ U \left(
\frac{1}{4} + 2D_1^2- 2S_1^2 + D_2^2 - S_2^2 \right)
+ V ( 1-4D_2^2 ) {\Biggr]}
+ \frac{N}{4}\left( K + \delta K \sin^2 \xi \right) u_{\mathrm t}^2 ,\end{aligned}$$
where $\epsilon_k = -2t\cos k$ and $$\begin{aligned}
\Delta_{ Q_0,\sigma}
&= \frac{U}{2}D_{Q_0}-{\mathrm{sgn}}(\sigma)\frac{U}{2}S_{Q_0},
\label{eq:S}\\
\label{eq:D}
\Delta_{2Q_0,\sigma}
&=
\left( \frac{U}{2}-2V \right) D_{2Q_0}-{\mathrm{sgn}}(\sigma)\frac{U}{2}S_{2Q_0} .\end{aligned}$$ The total energy is given by $ E({u_{\mathrm{t}}}, \xi) = \langle H_{\mathrm{MF}} \rangle_{\mathrm{MF}}/N$, where $\langle \cdots \rangle_{\mathrm{MF}}$ is the expectation value over the mean-field Hamiltonian. The ground-state energy is obtained by minimizing the total energy with respect to ${u_{\mathrm{t}}}$ and $\xi$, where the corresponding equations are obtained as $$\begin{aligned}
& ( K + \delta K \sin^2 \xi ) {u_{\mathrm{t}}}\nonumber \\
& = {\mathrm e}^{\mathrm{i} \xi} \frac{-1+\mathrm{i}}{N} \sum_{k,\sigma}
(\cos k - \sin k)
\left<c_{k\sigma}^\dagger c_{k+Q_0,\sigma}
\right>_{\mathrm{MF}}
+ \mathrm{c.c.} ,
\label{eq:ut}
\\
& - \frac{\delta K}{2} {u_{\mathrm{t}}}\sin (2\xi)
\nonumber \\
& = {\mathrm e}^{\mathrm{i} \xi} \frac{1+ \mathrm{i}}{N} \sum_{k,\sigma}
(\cos k - \sin k)
\left<c_{k\sigma}^\dagger c_{k+Q_0,\sigma}
\right>_{\mathrm{MF}} + \mathrm{c.c.} .
\label{eq:xi}\end{aligned}$$ The parameters $D_1, D_2, S_1, S_2$, ${u_{\mathrm{t}}}$, $\theta$, $\theta'$, and $\xi$ are evaluated self consistently using eqs. (\[OPS\]), (\[OPD\]), (\[eq:ut\]), and (\[eq:xi\]). We also examine $E({u_{\mathrm{t}}},\xi)$ to investigate the possible metastable state at ${u_{\mathrm{t}}}=0$. The present calculation has been performed for $U=3$, where we take the hopping energy $t$ as unity.
{width="8cm"}
In Fig. 1, we present the states obtained in our Hamiltonian. Two kinds of phases of $\xi$ and $\theta$ correspond to those of order parameters for the lattice tetramerization and the $2{k_{\mathrm{F}}}$ SDW, respectively: $$\begin{aligned}
{u_{\mathrm{t}}}\cos \left(\frac{\pi}{2} j + \xi\right), \quad
S_1 {\mathrm e}^{\mathrm{i} \theta} .
\nonumber
\end{aligned}$$ The symbol $*$ in Fig. 1 implies the state with ${u_{\mathrm{t}}}=0$. The SDW + CO ($D_2 + S_1$) is the charge-ordered state with the alternation of the charge-rich/poor sites. The pure SDW ($S_1$) has a spin amplitude, the maximum of which is located on the bonds in order to gain the transfer energy. Both the SDW+CO state and the SDW state do not have any lattice distortion, namely, ${u_{\mathrm{t}}}= 0$. The bond-centered CDW and site-centered CDW states are the Peierls states, i.e., the nonmagnetic states with finite lattice distortion (tetramerization) and with $2{k_{\mathrm{F}}}$ charge modulations. The lattice tetramerization takes the maximum amplitude on the bonds in the former state, while it does so on the sites in the latter state. We note that all these states exhibit the insulating behavior due to the presence of the $2{k_{\mathrm{F}}}$ density wave ($S_1$ or $D_1$), which yields the gap in the dispersion at the Fermi energy.
We note the relevance of the present analysis to the state of the quasi-one-dimensional (EDO-TTF)$_2$PF$_6$ compound. The Peierls state observed at low temperatures is precisely of the bond-centered CDW type. [@Chollet2005] Therefore, in the present paper, such a type of the CDW is mainly examined to find the possible metastable state, which could be the origin of the photoinduced phase realized after releasing the lattice distortion.
Effect of Alternating Elastic Constant on Ground State
======================================================
In order to clarify the starting point of the present work, we begin to examine the case for $\delta K = 0$, in which some results can be compared with the previous works.
First, we briefly recall the electronic state in the absence of the e-p interaction.[@Seo1997] There is a critical value $V_{\mathrm{c}} \simeq 0.28$ where the state with $S_1 \not=0$ and $D_2 =0$ is obtained for $V < V_{\mathrm{c}}$, and that with $S_1 \not=0$ and $D_2 \not= 0$ is obtained for $V > V_{\mathrm{c}}$. The CO state for $V > V_{\mathrm{c}}$ has a charge disproportionation with two-fold periodicity (Fig. 1). The phase transition at $V=V_{\mathrm{c}}$ is of the first order where the both order parameters $D_2$ and $S_1$ exhibit discontinuous change. The difference in SDW ($S_1$) between the case of $V<V_{\mathrm{c}}$ and that of $V>V_{\mathrm{c}}$ is that the maximum of the spin amplitude is on the bonds in the former case while it is on the sites in the latter case. Such a difference can be understood from the commensurability energy for the $2{k_{\mathrm{F}}}$ SDW ($S_{Q_0} \equiv S_1 {\mathrm e}^{\mathrm{i}\theta}$) at quarter filling, which decreases continuously and vanishes only at $V=V_{\mathrm{c}}$. Actually, the $\theta$ dependence of eq. (\[eq:MF\_Hamiltonian\]) takes the form [@Suzumura1997] $$\begin{aligned}
\label{eq:Eg}
E_g(\theta) = \mathrm{const.} + C(V) \, \cos 4 \theta ,
\end{aligned}$$ where the $V$ dependence of $C(V)$ is given by $C(V)= C_{00} - C_{01} V - C_{03} V^3 + \cdots $ with $C_{0j}$ being positive numbers. Thus, one finds $C(V) > 0$, i.e., $\theta = \pi/4$ (bond-centered density wave), for $V < V_{\mathrm{c}}$, while one finds $C(V) <0 $, i.e., $\theta = 0$ (site-centered density wave), for $V > V_{\mathrm{c}}$. We note that, due to such a sign change of $C(V)$, the collective mode for the phase fluctuation exhibits the vanishing of the commensurability gap at $V=V_{\mathrm{c}}$ leading to the metallic property. In fact, a noticeable behavior emerges in the optical conductivity $\sigma (\omega)$ ($\omega$ being the frequency), where the static conductivity $\sigma (0)$ becomes finite at $V=V_{\mathrm{c}}$ even for the commensurate SDW state. [@Tomio2002JPSJ]
Next, we consider the case with the e-p coupling but with $\delta K =0$. In Fig. 2, we show the phase diagram of the ground states obtained from eqs. (\[OPS\]), (\[OPD\]), (\[eq:ut\]), and (\[eq:xi\]). The bold line denotes the boundary for the first-order phase transition, which is estimated by comparing the minimum energy of $E(u_{\rm t},\xi)$. For large $K$ (i.e., for small e-p coupling), we obtain ${u_{\mathrm{t}}}= 0$ and there is a critical value $V_{\mathrm{c}} \simeq 0.28$, where the SDW state is obtained for $V < V_{\mathrm{c}}$ and the SDW+CO state is obtained for $V > V_{\mathrm{c}}$. With decreasing $K$, the state with finite lattice tetramerization (${u_{\mathrm{t}}}\neq 0$) appears. There is the $D_1 + S_1$ state in between the SDW state and the bond-centered CDW state, where both boundaries do not depend on $V$. Such a $V$-independent result is ascribed to the mean-field treatment in which $V$ does not contribute to the mean field for $S_1$ and $D_1$ \[see eqs. (\[eq:S\]) and (\[eq:D\])\]. For the intermediate $K$ and small $V$, there exists the region of the bond-centered CDW. This state resembles the BCDW state suggested by Clay *et al*. (Fig. 3(b) and Fig. 4 in ref. ) and the DM+SP state suggested by Kuwabara *et al*. (Fig. 2 in ref. ), in the sense that the amplitude of the lattice tetramerization takes a maximum on the bonds; i.e., bond-centered state.
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Here, we note the effects of the quantum fluctuation and the lattice dimerization, which are not taken into account in the present mean-field treatment. It is possible that the magnetic state of the SDW+CO turns into a nonmagnetic state in the presence of the quantum fluctuation and the e-p coupling. Compared with the case in ref. , the mean field stabilizes the CO and the charge disproportionation in which the strong correlation and the quantum fluctuations take important roles. Thus, in the mean-field treatment, the boundary between the bond-centered CDW state and the site-centered CDW state is shifted to the weak coupling region, and the SDW+CO state is stabilized for relatively small $V$. It is also expected that, due the quantum effect, the boundary between the SDW+CO and site-centered CDW states becomes a crossover and the nonmagnetic state would be realized in both regions. Furthermore the lattice dimerization can be expected to coexist with the bond-centered CDW state. [@Clay; @Kuwabara2003] In the sense of the variational principle, the region for the bond-centered CDW is extended by introducing the lattice dimerization.
The bond-centered CDW state originates from the facts that the on-site repulsive interaction $U$ separates two electrons being on the same site, and that the bond-centered ordering gives rise to the large gain of the transfer energy. For large $V$, the site-centered CDW state is realized in order to avoid the increase in the energy of $V$. As seen from Fig. 1, the site-centered CDW state takes a maximum of the charge on the sites, where the gain of the transfer energy is made favorable owing to the electron hopping into both nearest sites. This state would correspond to the 4${k_{\mathrm{F}}}$ CDW-SP state suggested by Clay *et al.* (Fig. 3(d) in ref. ) and the CO+SP state suggested by Kuwabara *et al.*,[@Kuwabara2003] since the lattice tetramerization takes the maximum amplitude on the sites. We note that, in the present case, the maximum of charge is located on the same position as that of lattice tetramerization (i.e., the amplitude of $D_1$ is larger than that of $D_2$). However, for the state in ref. , the maximum is obtained at the same position as that of the $4{k_{\mathrm{F}}}$ CDW (i.e., the amplitude of $D_2$ is larger than $D_1$). This may originate from our choice of relatively small $U$ and then our site-centered CDW state is expected to move to the CO+SP state for large $U$.
{width="8.5cm"}
Now we show how the site-centered CDW ($D_1+D_2$) diminishes when $\delta K$ increases from zero. In Fig. 3, the variations of the $D_1+D_2$ state and the bond-centered CDW ($D_1$) are shown as a function of $\delta K$. The SDW+CO ($D_2+S_1$) remains unchanged since ${u_{\mathrm{t}}}=0$. The addition of the $\delta K$ term increases the energy of the $D_1+D_2$ state with $\xi \not= 0$, but does not affect on the $D_1$ state with $\xi = 0$. Thus, the change from the $D_1+D_2$ state into the $D_1$ state occurs with increasing $\delta K$. In the phase diagram of Fig. 4, the boundaries between the $D_1+D_2$ state and the $D_1$ state are shown for the choices of $\delta K = 0.05$ and $0.1$. The region of bond-centered CDW increases rapidly with increasing $\delta K$, i.e., a small amount of $\delta K$ is enough to obtain the bond-centered CDW state as the ground state. It is noticed that the boundary of the direct transition from the bond-centered CDW state to the SDW+CO state emerges even for $\delta K =0.05$. Such a boundary is in contrast to that in Fig. 2.
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Here, we compare the experimental findings in the (EDO-TTF)$_2$PF$_6$ compound with the results of the present calculation. The Peierls state observed in (EDO-TTF)$_2$PF$_6$ is of the type of the bond-centered CDW, [@Drozdova2004] and the photoinduced phase transition from the Peierls insulator to a metal takes place by the weak laser photoexcitation. This behavior has been discussed by assuming the metastable state without lattice distortion. [@Chollet2005] In our calculation, the bond-centered CDW state is actually reproduced in Figs. 2 and 4. One can expect the metastable state in the region near the boundary between the the bond-centered CDW state and the undistorted state, since the the phase transitions are of the first order. Thus, we focus on the properties of the first-order phase transitions from the bond-centered CDW state to the SDW+CO state. From the inset of Fig. 2, it can be seen that the system does not show a direct transition between the bond-centered CDW state and the SDW+CO state, but there are intermediate states between these two states. Actually, the following three types of transitions occur when the bond-centered CDW state moves to the SDW+CO state with increasing $K$. (i) For $0.67 \lesssim V \lesssim 1.06$, the bond-centered CDW state shows a first-order phase transition to the $D_1+D_2+S_1$ state and undergoes a second-order transition to the SDW+CO state. (ii) For $0.49 \lesssim V \lesssim 0.67$, the bond-centered CDW state shows a second-order transition to the $D_1+S_1$ state followed by the successive transitions of a first-order one into the $D_1+D_2+S_1$ state and a second-order one into the SDW+CO state. (iii) For $0.28 \lesssim V \lesssim 0.49$, the bond-centered CDW state shows a second-order transition to the $D_1+S_1$ state and then undergoes a first-order transition to the SDW+CO state. The first-order phase transition in case (i) can be ascribed to the difference between the symmetry in the bond-centered CDW ($\xi=0$) and that in the $D_1+D_2+S_1$ state ($\xi=-\pi/4$), where both states have finite lattice distortion. The origin of the first-order transition in case (ii) is also the same as that in case (i). In these cases, the second-order transition between the $D_1 + D_2 + S_2$ state with $\xi=-\pi/4$ and the SDW + CO state implies that the state around ${u_{\mathrm{t}}}=0$ is unstable and turns out to be irrelevant to (EDO-TTF)$_2$PF$_6$. For case (iii), the first-order transition is due to the difference in the locking of the phase $\theta$, i.e., $\theta=\pi/4$ in the $D_1+S_1$ state and $\theta=0$ in the SDW+CO state. This case is also irrelevant to the state of the (EDO-TTF)$_2$PF$_6$, since the spin ordering is absent in the Peierls state. Thus, both the metastable state at ${u_{\mathrm{t}}}=0$ and the ground state of the bond-centered CDW (${u_{\mathrm{t}}}\not= 0$) cannot be explained within the model of $\delta K=0$, suggesting a new mechanism for the photoinduced phase observed in the (EDO-TTF)$_2$PF$_6$ compound. Our idea is the introduction of the alternation of the elastic constant, which enables us to reproduce the experimental findings. By increasing $\delta K$, the region of the bond-centered CDW state is enhanced and moreover, there is a direct first-order phase transition to the lattice-undistorted SDW+CO state, as seen from Fig. 4. Such a behavior comes from the fact that the $\delta K$ has a role of fixing the phase $\xi=0$, which can be obtained from eq. (\[eq:MF\_Hamiltonian\]). On the basis of such a consideration, we examine the bond-centered CDW state and the mechanism of the emergence of the metastable state at ${u_{\mathrm{t}}}=0$ by setting $\xi=0$ in the next section. We discuss the relevance to the (EDO-TTF)$_2$PF$_6$ compound in §5.
Phase Diagram and Metastable State for Large $\delta K$
=======================================================
In the following calculation, we examine the case of $\xi = 0$ corresponding to the bond-centered lattice distortion, which is realized by choosing a moderate magnitude of $\delta K \geq 0.2$ as shown in the preceding section. The neglect of the lattice dimerization can be justified by considering the large $\delta K$, as will be discussed in §5.
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In order to understand the role of the e-p interaction in the correlated system, we first examine eq. (\[eq:MF\_Hamiltonian\]) by treating ${u_{\mathrm{t}}}$ as an external field, i.e., by discarding the condition of eqs. (\[eq:ut\]) and (\[eq:xi\]). We calculate the order parameters $S_j$ and $D_j$ ($j=1$, $2$) as a function of ${u_{\mathrm{t}}}$. The CDW, whose order parameter $D_1$ is always finite due to the presence of the external field ${u_{\mathrm{t}}}$, is called the extrinsic 2${k_{\mathrm{F}}}$ charge-density wave (ECDW). In Fig. 5, we explain how the state varies with increasing ${u_{\mathrm{t}}}$. When $V=0$, the order parameter $S_1$ decreases monotonically and becomes zero at ${u_{\mathrm{t}}}\approx 0.5$. For $V=1$, both the order parameters $D_2$ and $S_1$ take finite values for ${u_{\mathrm{t}}}=0$, but as ${u_{\mathrm{t}}}$ increases, $D_2$ decreases rapidly and becomes zero at ${u_{\mathrm{t}}}\approx 0.26$ and $S_1$ becomes zero at ${u_{\mathrm{t}}}\approx 0.5$. The $D_1+S_1$ state is understood by noting that $S_1$ is rather robust compared with $D_2$ because $U > V$. With increasing ${u_{\mathrm{t}}}$ ($0.28 < V$), the phase of the SDW, $\theta$, increases from zero and reaches $\pi/4$ at which $D_2$ vanishes. Such a variation of $\theta$ comes from the competition of $D_2$ and $D_1$. We obtain $\theta' = \theta + \pi/2$ for the $D_1+S_1$ state and the $D_1$ state. For larger values of $V(\gtrsim 1.5)$, the strong competition of $D_2$ and ${u_{\mathrm{t}}}$ results in the first-order transition, which is followed by the discontinuous changes of $S_j$ and $D_j$ ($j=1$, $2$). Furthermore, for $V (\gtrsim 1.7)$, the vanishing of $D_2$ and $S_1$ takes place simultaneously due to the strong effect of ${u_{\mathrm{t}}}$.
The phase diagram is summarized in Fig. 5. The second-order and first-order transitions take place at the boundaries of the thin and bold lines, respectively, where the point with ${u_{\mathrm{t}}}=0$ and $V=V_{\mathrm{c}}(\simeq 0.28)$ is the singular one showing the first-order transition of $D_2$. For the coexistent state of $D_1+D_2+S_1+S_2$, both the amplitudes $S_1$ and $D_2$ are larger than that of $D_1$. In the $D_1+S_1$ state, the amplitude of $S_1$ is larger than that of $D_1$. For large ${u_{\mathrm{t}}}$, the pure ECDW ($D_1$) state appears, e.g., for ${u_{\mathrm{t}}}\gtrsim 0.5$ and $V \lesssim 1.7$. The boundary between the $D_1+S_1$ state and the $D_1+D_2+S_1+S_2$ state in the region of $0.5 \lesssim V \lesssim 1.5$ is given by $ V \simeq 4 {u_{\mathrm{t}}}$, indicating a direct competition between ${u_{\mathrm{t}}}$ and $V$. On the boundary corresponding to the vanishing of $D_2$, one sees a point where the second-order transition changes into the first-order transition with increasing $V$. This variation resembles that found in the phase diagram in the $V$ versus temperature (instead of ${u_{\mathrm{t}}}$) plane. [@Tomio2001JPCS] Such a fact may be explained by the existence of the term proportional to $D_2 S_1^2$ in the energy expansion with respect to the order parameter. [@Tomio2001JPCS] The solid line of the phase boundary between the $D_1$ state and the $D_1+D_2+S_1+S_2$ state is given by ${u_{\mathrm{t}}}\simeq \frac{2}{3}V -0.8$ for $V\gtrsim 1.7$. We found that $D_1$ is always finite due to the external field ${u_{\mathrm{t}}}$, but is strongly suppressed in the presence of $D_2$ (not shown), i.e., the strong competition between $D_2$ and $D_1$, which is a characteristic of 1/4-filled systems. Also note that $\theta = \pi/4$ if $D_2 = 0$; i.e., the maximum density is located on the bonds in the absence of CO.
Next we examine the ${u_{\mathrm{t}}}$ dependence of the mean-field energy $E({u_{\mathrm{t}}})$, which is defined by $E({u_{\mathrm{t}}},\xi=0)$. The minimum of this energy gives the true ground-state energy. In Fig. 6, the energy difference, $\delta E({u_{\mathrm{t}}}) \equiv E({u_{\mathrm{t}}}) - E(0)$, is shown as a function of ${u_{\mathrm{t}}}$ for $V=0.1$ with some choices of $K$. This behavior denotes the conventional second-order phase transition except for its having several kinds of order parameters, $S_1$ and $D_1$. For $K =1.06$, the minimum is obtained at ${u_{\mathrm{t}}}=0$ corresponding to the pure SDW state. The closed circle at ${u_{\mathrm{t}}}\simeq 0.5$ denotes the point where the $S_1$ state vanishes. Thus, we obtain the SDW ($S_1$) state for $K \gtrsim 1.05$, the $D_1+S_1$ state for $1.03 \lesssim K \lesssim 1.05 $, and the pure bond-centered CDW ($D_1$) state for $K \lesssim 1.03$. Figure 7 shows the case for $V=1$. The novel feature in $\delta E({u_{\mathrm{t}}})$ compared with that of Fig. 6 is the emergence of a local minimum at ${u_{\mathrm{t}}}=0$. For $0.79 \lesssim K \lesssim 1.01$, we find a metastable state where the energy of the local minimum $E({u_{\mathrm{t}}}=0)$ is larger than the energy of the true minimum at finite ${u_{\mathrm{t}}}$. This fact implies that, with decreasing $K$, the system exhibits the first-order phase transition from the SDW+CO ($D_2+S_1$) state into the bond-centered CDW state at $K \approx 1.01$. We note that, for $ 0.28 \lesssim V \lesssim 0.70$, there is a very narrow region of $K$ around $K \approx 1.03$ for the $D_1+S_1$ state (not shown), as found in Fig. 4. For a larger choice of $V = 1.6$, the local minimum at ${u_{\mathrm{t}}}=0$ is found for $0.6 \lesssim K \lesssim 0.89$.
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![ $V$ dependence of $\alpha$ and $\beta$ in eq. (\[eq:energy\_expansion\]), where $\theta = \pi/4$ ($0$) for $V < V_{\mathrm{c}}$ ($V>V_{\mathrm{c}}$). The inset denotes $ - 1/\beta$, which is proportional to $(V - V_{\mathrm{c}})$ for $V > V_{\mathrm{c}}$. ](fig8.eps){width="7cm"}
Here, we examine the origin of the metastable state found in Fig. 7. The mean-field energy of eq. (\[eq:MF\_Hamiltonian\]) can be expanded in terms of ${u_{\mathrm{t}}}$ as $$\begin{aligned}
\label{eq:energy_expansion}
E({u_{\mathrm{t}}}) = E(0) + \left(\alpha + \frac{1}{4} K \right) {u_{\mathrm{t}}}^2
+ \beta {u_{\mathrm{t}}}^4 + \cdots .\end{aligned}$$ The coefficients, $\alpha$ and $\beta$, can be estimated numerically and are shown in Fig. 8 as a function of $V$. With increasing $V$, there is a jump in both $\alpha$ and $\beta$ at $V=V_{\mathrm{c}}(\simeq 0.28)$, at which the pure SDW state moves to the SDW+CO state; i.e., the $D_2$ state emerges. It can be found that $\alpha < 0$ for arbitrary $V$, and that $\beta < 0$ ($\beta > 0$) for $V>V_{\mathrm{c}}$ ($V<V_{\mathrm{c}}$). The result of $\alpha < 0$ is quite reasonable when we note that ${u_{\mathrm{t}}}$ acts as the external field. In order to understand why $\beta$ is negative and diverges as $V \rightarrow V_{\mathrm{c}} + 0$, we examine eq. (\[eq:MF\_Hamiltonian\]) in terms of the phase degrees of freedom $\theta$. [@Tomio2001JPCS] Actually, from eq. (\[eq:MF\_Hamiltonian\]), the energy with fixed ${u_{\mathrm{t}}}$ and $\theta$ is expressed as $$\begin{aligned}
\label{eq:energy_expansion2}
E({u_{\mathrm{t}}},\theta) = &
E_g(\theta)
+ \left( \alpha' - \gamma \sin 2 \theta + \frac{1}{4}K \right){u_{\mathrm{t}}}^2
\nonumber \\
& {} + \beta' {u_{\mathrm{t}}}^4 + \cdots ,\end{aligned}$$ where $E_g(\theta)$ is given by eq. (\[eq:Eg\]). Quantities $ \alpha'(<0)$, $\beta'(>0)$, and $\gamma (>0)$ are estimated numerically from eq. (\[eq:MF\_Hamiltonian\]). The $\gamma$-term, which is proportional to $S_1^2$, denotes the interaction between the bond-centered CDW ($D_1$) state and the SDW ($S_1$) state. The coefficients are estimated as $ \alpha' \simeq -0.197$, $\beta'=0.0068$, and $\gamma \simeq 0.056$ for $V=1$, where $\beta' \simeq 0.0145$ for $V < V_{\mathrm{c}}$. There is a small jump of $\gamma$ at $V = V_{\mathrm{c}}$, where the $V$ dependence of $\gamma$ is much smaller than that of $\alpha.$ The locking position of $\theta$ can be determined in order to minimize eq. (\[eq:energy\_expansion2\]). The commensurability potential $E_g(\theta)$ for $V>V_{\mathrm{c}}$ favors the locking position $\theta = 0$, while the $\gamma$-term favors $\theta=\pi/4$. For $V>V_{\mathrm{c}}$, by minimizing eq. (\[eq:energy\_expansion2\]) with respect to $\theta$, we obtain eq. (\[eq:energy\_expansion\]) with the coefficients $$\begin{aligned}
\alpha = \alpha' ,
\qquad
\beta = \beta' - \frac{\gamma^2}{8 |C(V)|} .
\label{eq:energy_expansion3}\end{aligned}$$ We can immediately reproduce the anomalous behavior $-1/\beta \propto (V-V_{\mathrm{c}})$ at $V=V_{\mathrm{c}}+0$, by noting that the coefficient $C(V)$ follows $C(V)\propto (V-V_{\mathrm{c}})$. This is the reason why the coefficient of the fourth-order term $\beta$ can become negative for $V>V_{\mathrm{c}}$. It is shown that the result similar to eq. (\[eq:energy\_expansion3\]) can be obtained from the bosonization scheme, in which the spin degree of freedom is also taken into account. [@Tsuchiizu2007_preprint] For $V< V_{\mathrm{c}}$, the $\gamma$-term does not contribute to the ${u_{\mathrm{t}}}^4$ term since both the commensurability potential and the $\gamma$ term favors the locking position $\theta = \pi/4$ (one simply obtains $\alpha=\alpha'-\gamma$). Thus, the conventional second-order transition is reproduced. From these arguments, the conditions for the metastable state are summarized as follows: [ *(i) the existence of the CO for $V>V_{\mathrm{c}}$ (the commensurability energy leading to $\theta=0$), (ii) the coupling between the SDW state ($S_1$) and the bond-centered CDW state (${u_{\mathrm{t}}}$), which is given by the $\gamma$ term in eq. (\[eq:energy\_expansion2\]), and (iii) the mechanism to fix $\xi=0$ as discussed in §3.* ]{}
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Finally we show the phase diagram of the ground state on the plane of $V$ and $K$ in Fig. 9. For $K \gtrsim 1.05$ corresponding to the weak e-p coupling, we obtain either the pure SDW ($V<V_{\mathrm{c}}$) with $\theta = \pi/4$ or the SDW+CO ($V>V_{\mathrm{c}}$) with $\theta = 0$. With decreasing $K$ (i.e., increasing the e-p coupling), the bond-centered CDW state with $\theta= \pi/4$ appears for $K <K_{\mathrm{c}}$ where $K_{\mathrm{c}}$ is a critical value of the first-order phase transition for the Peierls state. The state with ${u_{\mathrm{t}}}=0$ becomes metastable in the large interval region of $K < K_{\mathrm{c}}$ as shown by the shaded area. The lower boundary of the shaded region is given by $$\begin{aligned}
K = - 4 \alpha ,
\end{aligned}$$ which is verified from $\alpha$ in Fig. 8. There is also another metastable state with the local minimum at ${u_{\mathrm{t}}}\not=0$, which is located in a certain region of $K > K_{\mathrm{c}}$ (not shown in Fig. 9).
Summary and Discussion
======================
We examined the Peierls-Hubbard model at quarter-filling with the intersite interaction ($V$), and obtained the ground-state phase diagram within the mean-field theory. A noticeable finding is the undistorted state (${u_{\mathrm{t}}}=0$), which becomes metastable in the bond-centered CDW state close to the boundary between the bond-centered CDW state and the SDW+CO state. The obtained bond-centered CDW ground state and its metastable state may share the following common features with the Peierls state of the (EDO-TTF)$_2$PF$_6$ compound. The spatial variation of the Peierls state is the same, and the metastable state for ${u_{\mathrm{t}}}=0$ could be related to the photoinduced phase found in the experiment on the EDO-TTF compound.
In the present analysis of §4, we assumed $\xi = 0$ in order to stabilize the bond-centered CDW state and examined the metastable state at ${u_{\mathrm{t}}}=0$, which may be relevant to the property of the EDO-TTF compound. Actually, in the normal state of this compound, there is the alternation of the bending of the molecule for every two sites along the the one-dimensional chain.[@Ota2002] In addition to the ground state (§3), we discuss this assumption for the metastable state on the basis of the energy, which is expanded in terms of ${u_{\mathrm{t}}}$. The energy with fixed ${u_{\mathrm{t}}}$, $\theta$, and $\xi$ can be explicitly written as $$\begin{aligned}
\label{eq:xi2}
E({u_{\mathrm{t}}}, \theta,\xi)
=& \,
C(V) \cos 4 \theta
+ \left( \alpha' - \gamma \sin (2 \theta - 2 \xi) \right) {u_{\mathrm{t}}}^2
\nonumber \\
&+ \left( \frac{K}{2} + \frac{\delta K}{4} \sin^2 \xi \right) {u_{\mathrm{t}}}^2
+ \cdots , \end{aligned}$$ where the $\delta K$ term denotes an increase in elastic energy for $\xi \not= 0$. Here, we note that, even for $V>V_{\mathrm{c}}$, there is no competition between $\theta$ and $\xi$ for $\delta K=0$, since the minimum of eq. (\[eq:xi2\]) is obtained by choosing $\xi$ so as to satisfy $\sin (2 \theta - 2 \xi) = 1$. In this case, the $\gamma^2/C(V)$ term in $\beta$ \[see eq. (\[eq:energy\_expansion3\])\] is absent and the energy $E({u_{\mathrm{t}}}, \theta,\xi)$ \[eq. (\[eq:xi2\])\] takes a true minimum at ${u_{\mathrm{t}}}=0$ (i.e., $\beta > 0$). When $\delta K \neq 0$, the energy $E({u_{\mathrm{t}}}, \theta,\xi)$ as a function of $\xi$ increases and the metastable state around ${u_{\mathrm{t}}}=0$ can be expected (i.e., $\beta < 0$). Actually, for $\delta K>0.46$, the energy $E({u_{\mathrm{t}}}, \theta,\xi)$ takes a local minimum at ${u_{\mathrm{t}}}=0$ for all values of $\xi$, in the case of $K=0.9$ and $V=1$. Thus, the assumption $\xi=0$ can be justified when the degree of alternation of the elastic constant $\delta K$ becomes larger than a critical value.
In the present analysis, we did not consider the possibility of lattice dimerization ($u_j=(-1)^j u_{\mathrm{d}}$, where $u_{\mathrm{d}}$ is the amplitude of the lattice dimerization). Since the elastic energy of the lattice dimerization is given by $(N/2)(K+\delta K/2) u_{\mathrm{d}}^2 $, $\delta K$ also has an effect of suppressing the lattice dimerization. From these arguments, we expect that the effect of $\delta K$ will arise from the bending freedom of the molecules in the (EDO-TTF)$_2$PF$_6$ compound. Such a bending plays important roles in the photoinduced cooperative transition, although the origin of the molecular property still remains an open question from a microscopic view point.
Here, we note states at finite temperatures. There are some theoretical works on finite-temperature properties. Quite recently, it has been pointed out that the Peierls-Hubbard model with $\delta K = 0$ exhibits the first-order transition from the dimer-Mott state into the spin-Peierls state,[@Seo2007JPSJ] where both states show the insulating behavior. On the other hand, the purely electronic model shows the transition from the metallic state into the insulating state of the 2${k_{\mathrm{F}}}$ SDW + 4${k_{\mathrm{F}}}$ SDW + 2${k_{\mathrm{F}}}$ CDW state. [@Tomio2001JPCS] Compared with the state of the (EDO-TTF)$_2$PF$_6$ compound, the normal state at high temperatures is different from the dimer-Mott state obtained from the former model and the lattice deformation cannot be reproduced in the latter model. It is expected that the parameters in the shaded region in Fig. 9 will lead the first-order phase transition into the normal state with increasing temperature, and that such a transition will be strongly enhanced by the effect of the large bending of molecules in the (EDO-TTF)$_2$PF$_6$.
The authors thank Professors H. Yamochi, G. Saito, S. Koshihara, K. Yonemitsu, J.-F. Halet, and L. Ouahab, and Drs. Y. Nakano, K. Onda, A. Ota, and H. Seo, for useful discussions and comments. The present work has been financially supported by a Grant-in-Aid for Scientific Research on Priority Areas of Molecular Conductors (No. 15073213) from the Ministry of Education, Culture, Sports, Science and Technology, Japan, and by the JSPS Core-to-Core Program Project \`\`Multifunctional Molecular Materials and Device Applications".
[99]{} D. Jérome and H. J. Schulz: Adv. Phys. **31** (1982) 299. G. Grüner: Rev. Mod. Phys. **66** (1994) 1. V. J. Emery: [*Highly Conducting One-Dimensional Solids*]{}, eds. J. T. Devreese, R. P. Evrard, and V. E. van Doren (Plenum, New York, 1979). H. Yoshioka, M. Tsuchiizu, and Y. Suzumura: J. Phys. Soc. Jpn. **69** (2000) 651. H. Seo and H. Fukuyama: J. Phys. Soc. Jpn. **66** (1997) 1249. J. P. Pouget and S. Ravy: J. Phys. I France **6** (1996) 1501, Synth. Metals **85** (1997) 1523. S. Kagoshima, Y. Saso, M. Maesato, R. Kondo, and T. Hasegawa: Solid State Commun. **110** (1999) 479. N. Kobayashi, M. Ogata, and K. Yonemitsu: J. Phys. Soc. Jpn. **67** (1998) 1098. Y. Tomio and Y. Suzumura: J. Phys. Soc. Jpn. **69** (2000) 796. Y. Tomio and Y. Suzumura: J. Phys. Chem. Solids **62** (2001) 431. M. Chollet, L. Guerin, N. Uchida, S. Fukaya, H. Shimoda, T. Ishikawa, K. Matsuda, T. Hasegawa, A. Ota, H. Yamochi, G. Saito, R. Tazaki, S. Adachi, and S. Koshihara: Science **307** (2005) 86. A. Ota, H. Yamochi, and G. Saito: J. Mater. Chem. **12** (2002) 2600. O. Drozdova, K. Yakushi, K. Yamamoto, A. Ota, H. Yamochi, G. Saito, H. Tashiro, and D.B. Tanner: Phys. Rev. B **70** (2004) 075107. K. Onda, T. Ishikawa, M. Chollet X. Shao, H. Yamochi, G. Saito, and S. Koshihara: J. Phys. Conf. Series **21** (2005) 216. M. Kuwabara, H. Seo, and M. Ogata: J. Phys. Soc. Jpn. **72** (2003) 225. W. P. Su, J. R. Schrieffer, and A. J. Heeger: Phys. Rev. Lett. **42** (1979) 1698. H. Fukuyama and H. Takayama: in [*Dynamical Properties of Quasi-One-Dimensional Conductors - Phase Hamiltonian Approach in Electronic Properties of Inorganic Quasi-One-Dimensional Compounds -*]{}, ed. P. Monceau (D. Reidel Publishing 1985), p. 41. R.T. Clay, S. Mazumdar, and D. K. Campbell: Phys. Rev. B **67** (2003) 115121. Y. Suzumura: J. Phys. Soc. Jpn. **66** (1997) 3244. Y. Tomio and Y. Suzumura: J. Phys. Soc. Jpn. **71** (2002) 2742. M. Tsuchiizu and Y. Suzumura: arXiv:0801.1891. H. Seo, Y. Motome, and T. Kato: J. Phys. Soc. Jpn. **76** (2007) 013707.
|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'Business intelligence (BI) tools for database analytics have come a long way and nowadays also provide ready insights or visual query explorations, e.g. QuickInsights by Microsoft Power BI, SpotIQ by ThoughtSpot, Zenvisage, etc. In this demo, we focus on providing insights by examining periodic spreadsheets of different reports (aka views), without prior knowledge of the schema of the database or reports, or data information. Such a solution is targeted at users without the familiarity with the database schema or resources to conduct analytics in the contemporary way.'
author:
- 'Medha Atre, Anand Deshpande, Reshma Godse, Pooja Deokar, Sandip Moharir'
- |
Dhruva Ray, Akshay Chitlangia, Trupti Phadnis, Yugansh Goyal\
\
\
title: 'Needles in the ‘Sheet’stack: Augmented Analytics to get Insights from Spreadsheets'
---
at (current page.north west) [{height="2cm"}]{};
Introduction {#sec:introduction}
============
Business Intelligence (BI) tools built on top of database systems provide excellent data analytics capabilities. E.g., trends of revenues of different companies over the past six months, or different departmental store products’ sales comparison over the past three months etc. However, user needs a fair bit of knowledge of the underlying database schema and acquaintance with the BI dashboard. This can be particularly challenging for users and organizations that are not tech savvy or do not have time and resources to do so. A recent BI tools survey[^1] highlights that despite the market presence of a large variety of BI tools, about 38% of the users continue to use spreadsheets as the main reporting and analytics tool. The use of BI tools remains low at an average of 10-15% of the users. While spreadsheet tools like Microsoft Excel provide advanced analytics abilities, e.g., pivot tables, visual charts, ready-to-use functions etc, they still have the bottleneck that the user requires learning about these techniques and knowledge of the database and spreadsheet schema. Also it is our observation that spreadsheet reports are often generated targeting a wide variety of audiences within the organization making them bulky with tens of columns and thousands of rows.
*Augmented analytics* proposes to take this *learning* effort off the user’s shoulder, with the help of machine learning techniques for automated data cleaning, preparation, and insight discovery[^2]. For example, if a user wants to find the sharpest sales trend (rising or falling) among thousands of departmental store products, an augmented analytics tool can provide such *insights* automatically. Existing BI tools have indeed started moving in this direction, such as QuickInsights by Microsoft Power BI [@quickinsights17; @quickinsights19], SpotIQ by ThoughtSpot etc. These tools, in their present form, are tightly coupled with an existing BI dashboard interface and a backend database. However, as we noted above, a sizable percent of users still use spreadsheets as the main reporting and analytics tool.
Our main contributions through this demo proposal are:
1. Finding *insights* from a series of spreadsheets of a report, without requiring pre-training or prior knowledge of the spreadsheet schema.
2. Allowing users to interact with the systems using a chatbot and semi-structured English commands to set individual preferences, and give *personalized insights* from the same spreadsheets there onward for different users.
For instance, CEO and marketing manager of a company can both get different personalized insights from the same series of spreadsheets. Our goal is to build a system similar to the modern social media, where different newsfeeds are generated for different users from the same underlying data. Our system’s focus is on the personalized newsfeed of insights generated from the *same* spreadsheets of organization reports, consumed over emails, chatbots, RSS etc.
Architecture {#sec:architecture}
============
![Architecture[]{data-label="fig:arch"}](./images/archactins)
Figure \[fig:arch\] shows the high level architecture of our system. *Insights Generator* runs continuously accepting spreadsheets of different reports. In Figure \[fig:arch\], $R1, R2, R3$ are three different types of reports, e.g., $R1$ can be product sales report, $R2$ new joinee and attrition report, and $R3$ bug report. $R1_1, R1_2...$ represent periodic spreadsheets of report type $R1$ and so on. The engine does not impose any limit on the types of reports, the number of spreadsheets of each report, or their periodicity. It treats each report type and its spreadsheets independently.
The insight generator has three main components – (1) backend analysis unit, (2) frontend for insights delivery and user interaction through chatbot, (3) a middle layer for communicating backend insights in JSON format to the frontend. Frontend in turn consists of two subcomponents – (a) an insight delivery mechanism (currently through email), and (b) user interaction and personalization through chatbot using semi-structured English language commands. Note that through this personalization we enable different users to get different insights in the successive spreadsheets of the same report type. We have deployed components of our engine using Microsoft Azure framework, but they can be deployed on any other suitable cloud or independent architecture.
Data Cleaning and Preparation {#sec:dataprep}
-----------------------------
Our insight generator consumes reports in the spreadsheet format, which may not be strictly structured, and may have free text in the top rows or leftmost columns. We extract the *most significant table* from such sheets as follows – from the top of the sheet, designate the first row with maximum non-empty columns as the header, discard any leftmost empty columns from this header, and extract the table under.
Without losing generality, let the spreadsheets of a report type $R1$ have columns $c_1, c_2, c_3...$. We categorize these into four mutually exclusive sets – (1) primary key attributes (K), (2) categorical attributes (C), (3) numeric attributes (N), and (4) all others. Presently we disregard \#4 all other attributes. Due to space constraints, we briefly describe our method of identifying these four attribute types using heuristic measures.
For primary key attributes (K), we assume that typically they appear in the first five columns of a spreadsheet, and identify them by considering the total number of unique values in these columns and the total number of rows in the spreadsheet. In case if no such attribute or combination of them exist, we keep primary key attributes empty. We identify categorical attributes (C) using heuristic measures of the *data-type* of the column and the ratio of total non-empty rows in that column to the unique values in it ($\#nonemptyrows/ \#uniq val$). If the spreadsheet has more than five categorical attributes we sort all the categorical attributes in the descending order of ($\#nonemptyrows/ \#uniq val$), and uniformly pick five attributes from this sorted order. From these five categorical attributes, we create ${5 \choose 2} + {5 \choose 1} = 15$ attribute combinations for analysis. Note that our engine allows end user to change this choice or combination for future analysis as elaborated later. We consider all the numeric attributes (N) without discarding any.
**Timeseries:** Let us consider $R1_1, R1_2...R1_s$ spreadsheets of report $R1$, where $R1_1$ has the oldest timestamp and $R1_s$ the latest. We first choose $R1_1$, and consider each primary key (composite) value $k_i$ of $K$ attributes. This is a composite key if $K$ has more than one attribute. We consider each numeric attribute $n_j \in N$, and create a timeseries key $(k_i,
n_j)$ (key-attribute KA pair). E.g., if $R1$ has *Product-ID* as the primary key, with *Sales* as a numeric attribute, the timeseries key for product with ID $P1234$ is *(P1234, Sales)*. Let the value of *Sales* for $P1234$ in $R1_1$ be $y_1$, in $R1_2$ be $y_2$ and so on. Thus from $R1_1...R1_s$ spreadsheets, we form a time series for key *(P1234, Sales)* $\rightarrow [(t_1, y_1), (t_2, y_2)...(t_s, y_s)]$, where $t_1...t_s$ are the timestamps of the spreadsheets. A spreadsheet, say $R1_4$, may not have the specific key $P1234$, in which case we do not enter $(t_4, y_4)$ value in the timeseries, thus accommodating for disappearing and reappearing entities. This is done for each unique primary key value and unique numeric attribute in all the spreadsheets. We call these timeseries numeric attribute timeseries or **NTS**.
Next, for each spreadsheet $R1_i$, we consider each numeric column $n_j \in N$, and order $n_j$’s values corresponding to each primary key value and add a numeric column of these ranks, we call this $\mathit{n_j\mhyphen rank}$. E.g., within the *Sales* column in $R1_1$, let $P1234$ be *\$1000*, $P2345$ be *\$500*, and $P3456$ be *\$1200*, then the relative ascending order of $P1234, P2345, P3456$ for *Sales* column is $2, 1, 3$ and we add a column *Sales-rank*. Using the same procedure described above for NTS timeseries, we create rank timeseries **RTS** for each hybrid timeseries key $(k_i, \mathit{n_j\mhyphen rank}) \rightarrow [(t_1,
r_1), (t_2, r_2)...(t_s, r_s)]$, where $r_1...r_s$ are the values corresponding to $k_i$ in $\mathit{n_j\mhyphen rank}$ column of each spreadsheet. In all we get $(2 * |\bigcup_{R1_1...R1_s} uniq(K) \times N|)$ NTS and RTS timeseries over all the spreadsheets[^3].
Recall that we pick five categorical attributes, if more than five are present, and prepare maximum fifteen combinations of them. For each spreadsheet, we consider each $c_k$ categorical attribute combination of these maximum $15$ combinations. We compute the total number of rows $u$ for a unique value $v_l
\in c_k$. This is a composite value if $c_k$ has two categorical attributes. We do this for each spreadsheet, and form a timeseries for $(v_l, c_k) \rightarrow
[(t_1, u_1), (t_2, u_2)...(t_s, u_s)]$ for each $(v_l, c_k)$ value-attribute (VA) pair across all the spreadsheets. We call these categorical attribute timeseries or **CTS**. Thus given timestamp ordered spreadsheets of a particular report, we form three types of timeseries – NTS, RTS, CTS.
Analytics {#sec:analytics}
---------
For the analytical processing, we consider only timeseries that have more than five points in them. Shorter timeseries are discussed after this under “*LT5 Mean, Variance*”.
**Linear Regression:** Recall that our engine neither assumes any information about the spreadsheet schema and data domain, nor is it pre-trained on any existing corpus of schemas such as [schema.org](schema.org). We use unsupervised learning methods based on – (1) trend analysis (linear regression), (2) mean squared error (MSE) of the fitted trend line, and (3) outlier scores of data points within the fitted trend, i.e., Cook’s Distance[^4]. As given in Section \[sec:dataprep\], we form NTS, RTS, CTS consisting of several timeseries based on the unique values of primary keys and categorical attributes in the spreadsheets. For each timeseries in RTS, NTS, CTS, we do linear regression and fit a trend line. We compute the mean squared error (MSE) of this fit, and compute Cook’s Distance of each data point $(t_i, y_i)$, $(t_i, r_i)$, and $(t_i, u_i)$ in each time series. For each time series, we pick the data point with highest Cook’s Distance. Thus at the end of this exercise, each timeseries in NTS, RTS, and CTS has a (1) line equation with slope $m$ and intercept $b$, (2) MSE $mse$, and (3) a point with maximum Cook’s Distance $mcd$[^5]. We use these three features for deciding relative ranking of timeseries within each of NTS, RTS, CTS as follows. The procedure is applied same way for each of NTS, RTS, CTS group timeseries and hence we do not mention group names explicitly. We sort timeseries in the – (1) descending order of $m^2$ (sharpness of the slope irrespective of whether the slope is rising or falling), (2) descending order of $mse$, and (3) descending order of $mcd$. In all, we get *nine* orderings of timeseries, three each for NTS, RTS, CTS..
**LT5 Mean, Variance:** When the length of a timeseries is less than or equal to five, we only compute the mean and variance of the timeseries points, for each timeseries in NTS, RTS, CTS. Thus for each shorter timeseries in NTS, RTS, CTS we compute – (1) mean $\mu$, and (2) variance $\alpha^2$.
**Diff of the latest two:** We consider only the latest two spreadsheets of a report type, and compute the difference between them. This is achieved by using timeseries of NTS, RTS, CTS, and considering the last two points within them, if their timestamps correspond to the latest two reports. E.g., if a timeseries within NTS group is $(k_i, n_j) \rightarrow (t_1,
y_1)...(t_9, y_9), (t_{10}, y_{10})$,then we compute $(y_{10} - y_9)^2$ for the given $(k_i, n_j)$ value, if $t_{10}$ is of the latest spreadsheet. We repeat the same process for RTS and CTS timeseries.
**New Entities and Attributes:** Considering only the latest spreadsheet of a report type, say $R1$, we compute if there are any new primary key values or attributes added to it. In our observation, the report schema can undergo slight changes over time. Our procedure of timeseries composition outlined above can accommodate such changes, because we form timeseries for each unique key-attribute (KA) or value-attribute (VA) pair. From these various metrics computed on timeseries and spreadsheets, we compute *insights* (the most significant observations) as described in Section \[sec:insights\].
Insights {#sec:insights}
--------
We generate insights in four categories – (1) Overall highlight, (2) most significant sharp and flat trends, (3) most significant outlier, (4) most significant difference of the latest two spreadsheets (Delta).
**Highlight**: Recall from Section \[sec:analytics\] that after linear regression we sort timeseries by $m^2$, $mse$, and $mcd$ to get three sorted orders of timeseries in NTS, RTS, CTS each. Within the NTS timeseries, we compute a *composite sorted order* of each timeseries by multiplying sorted indices of it in each of the three sort orders, i.e., for the timeseries of $(k_i, n_j)$, its composite rank is $o_{m^2} * o_{mse} * o_{mcd}$, where $o_{m^2},o_{mse}, o_{mcd}$ are indices of $(k_i, n_j)$ in the sorted orders of $m^2, mse, mcd$ respectively. From the composite rank we pick the top one. In case of a tie, we pick one randomly. We repeat this process for RTS, CTS group of timeseries too, and pick the timeseries with the top composite rank. Intuitively, the highlighted insight from NTS, RTS, CTS are timeseries that have sharp trend (rising or falling), have high fluctuations, and an outlier with relatively higher residual error.
**Trend:** Disregarding the timeseries picked in the Highlight (to avoid redundancy), next we pick the top timeseries from the $m^2$ sorted order for each of the NTS, RTS, CTS group. Additionally we also pick the last timeseries from $m^2$ order. Thus the Trend insights have sharp rising or falling and the flattest trends.
**Outlier:** Next, disregarding the timeseries picked in Highlight and Trend, we pick the top timeseries from the sorted order of $mcd$ (Cook’s Distance) for NTS, RTS, CTS each.
**Delta:** Considering the ‘*Diff of the latest two*’ as given in Section \[sec:analytics\], we pick the timeseries that shows the maximum change (Delta) in the latest two reports.
Note that we compute relative order of *all* the timeseries formed over *all* the numeric and selected categorical attributes (ref. Section \[sec:analytics\]), and pick the top ones for insights. However, different users might be interested in focusing on different attributes. For this we provide a user interaction interface for *personalization* as given in Section \[sec:userinterface\].
{height="2.5in" width="3.35in"}
{height="2.5in" width="3.35in"}
User Interaction {#sec:userinterface}
----------------
In spreadsheets with tens of columns and thousands of rows, the number of timeseries can run in several thousands, e.g., when we tested with Stock Exchange archive spreadsheets [@nse], each spreadsheet has several numeric attributes and a couple of thousand rows, making around 20,000 timeseries. Since our engine does not assume any prior knowledge of schema or data domain, we treat all of them at par and choose the top insights. But *user-1* may be interested in insights from specific attributes, which may get overshadowed in the ranking from other insights. Hence we let user interact with the engine and set personalized configuration and preferences using chatbot. Through this, for the next round of insights, each user will get insights according to their previously set preferences.
Moving Window
-------------
Recall from Figure \[fig:arch\] that our engine can continuously churn periodically arriving spreadsheets, and generate a fresh set of insights after every new spreadsheet. We surmise that a typical user is most interested in finding out insights from the latest few spreadsheets. Thus we provide a configurable option to have a *moving window* over spreadsheets sorted by their timestamp. E.g., for report $R1$, a user wants to consider only the latest 10 spreadsheets for insight generation. Thus when a new spreadsheet of $R1$ arrives, we move the window by discarding the $10^{th}$ *oldest* spreadsheet and adding the latest in the window.
Setup {#sec:setup}
=====
Our engine’s three main components (1) analysis or spreadsheet processing, (2) frontend for insight delivery and user feedback, (3) middle layer for communicating insights from analysis to the frontend using JSON, are implemented using Python 3.0. Currently our setup is deployed using Microsoft Azure infrastructure and works as follows.
We assume that a spreadsheet of a report is sent to several people in the organization over email. In addition to these people, the spreadsheet is also sent to an email robot listening service for our engine, e.g., *[email protected]*. Treating the subject of the email as the report type, the service creates a separate storage for each report, runs the procedure given in sections \[sec:dataprep\], \[sec:analytics\], and \[sec:insights\], and sends the first set of insights to all the recipients of the report over an email, without any inputs or configuration asked from the user. In Figure \[fig:adcard1\] and \[fig:adcard2\], we have shown example of such insights generated on COVID-19 data acquired from [@covidva]. Figure \[fig:adcard1\] shows top insights generated on the spreadsheets between February 1–12 and Figure \[fig:adcard2\] shows insights for March 1–15. Comparing the two, we can note that for February, all the top insights came from the Chinese provinces, whereas by March when COVID-19 spread in other countries, and its spread in China started dampening (due to the lock-down), the top insights started coming from Italy and Spain[^6]. ***Note:** these insights were generated by our engine prior knowledge of the data, schema, or user inputs.*
Clicking the ‘*Explore*’ button opens a Microsoft Teams chatbot. Users can interact with the engine using simple semi-structured English commands, personalize the configuration by choosing only some attributes according to their preference, and instantaneously get updated insights. The user can save this personal configuration. When the new spreadsheet arrives, the next insights are personalized using individual user’s configuration.
Related Work
============
Microsoft Power BI’s QuickInsights [@quickinsights17; @quickinsights19], ThoughtSpot’s SpotIQ [@spotiq], and other augmented analytics tools offered by the BI tools come closest to our system. However, all those systems are integrated with an existing BI tool, and thus require access to the customer database. In comparison, our tool is *lightweight* which can work with spreadsheets without access to the full database or its schema. SeeDB [@seedb], Zenvisage [@zenvisage16], and their successor systems [@shapesearch] mainly focus on insightful visualizations of the underlying data. While their focus is on insightful visualizations, our focus is on insights presented in the English language. These systems expect knowledge of the underlying database schema to be able to identify the *dimension* and *measure* attributes. Since our system is targeted at users without the knowledge or resources to provide this information, we use heuristics to identify these attributes in the spreadsheets.
Future Work
===========
In the ongoing enhancements, we are working on providing insights into identifying attribute correlations, to be able to provide insights into *correlated outliers*, e.g., a sudden fall or rise in the sales of particular products due to the change in the store staff attendance (e.g., COVID-19 pandemic or weather patterns). Attributes which are dependent on other attributes will have similar timeseries, and generate redundant insights. Similar timeseries can be identified using a variety of techniques such as Earth Mover’s Distance, Euclidean Distance, Dynamic Time Warp etc., and they can be clustered so as to efficiently provide *unique* insights. We plan to analyse *semantics* of attributes using NLP techniques, so as to make better decisions in deciding how to process them. Finally, while we keep the focus of the insights on the latest moving window, we intend to give a historical perspective for a particular timeseries.
[^1]:
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|
{
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|
---
abstract: 'Harish-Chandra’s volume formula shows that the volume of a flag manifold $G/T$, where the measure is induced by an invariant inner product on the Lie algebra of $G$, is determined up to a scalar by the algebraic properties of $G$. This article explains how to deduce Harish-Chandra’s formula from Weyl’s law by utilizing the Euler-Maclaurin formula. This approach leads to a mystery that lies under the Atiyah-Singer index theorem.'
address: 'Mathematisches Institut, Busenstra[ß]{}e 3–5, D-37073 Göttingen, Germany'
author:
- Seunghun Hong
title: |
Harish-Chandra’s volume formula via\
Weyl’s Law and Euler-Maclaurin formula
---
Introduction
============
Harish-Chandra’s volume formula [@harishchandra1]\*[Lem. 4, p. 203]{} calculates the volume of a flag manifold $G/T$ by algebraic means. To wit, suppose the Lie algebra $\mathfrak{g}$ of $G$ is equipped with an $\operatorname{Ad}(G$)-invariant inner product, where $\operatorname{Ad}$ denotes the adjoint representation of $G$ on $\mathfrak{g}$. The inner product determines a unique invariant metric on $G$. Endow the quotient measure[^1] on $G/T$. Then $${\mathrm{vol}}(G/T) =\prod_{\alpha\in\Phi^+}\frac{2\pi}{\langle\alpha,\rho\rangle}.$$ Here $\Phi^+$ is the set of (selected) positive roots of $G$; $\rho$ is half the sum of the positive roots; and $\langle \cdot , \cdot \rangle$ is the inner product on the dual space ${\mathfrak{t}}^*$ of the Lie algebra of $T$ induced by the inner product on ${\mathfrak{g}}$. Since an invariant inner product on $\mathfrak{g}$ is unique up to a scalar factor, the formula shows that the volume is essentially determined by the algebraic properties of $G$. Apart from this, a significance of the volume of $G/T$ can be found in its appearance in the integration formula [@weyl1]\*[$\S$ 6]{}.
Proofs for the volume formula other than Harish-Chandra’s own can be found in , and [@bgv]\*[Cor. 7.27, p. 230]{}, and [@duistermaatkolk]\*[Eq. 3.14.13, p. 192]{}, [@fegan]\*[Thm. 1.4, p. 591]{}, Flensted-Jensen [@flensted-jensen]\*[Eq. 3.9, p.116]{}, and [@macdonald]\*[p. 95]{}. The ones that are close to our approach would be that of and , and , as they explicitly depend on the property of the heat trace. But and exploits the integration formula, while relies on the summation formula. Our approach differs from them in that we utilize the Euler-Maclaurin formula; as the Euler-Maclaurin formula relates sums with integrals, the integration formula also plays a natural role.
Here is the outline of our proof. The starting point is what is known as *Weyl’s law* for closed manifolds, namely that, if $\Delta$ is the Laplacian of a closed Riemannian manifold $M$ of dimension $n$, then its *heat trace* $$Z(t):= \operatorname{tr}(e^{t\Delta}),$$ for $t>0$, satisfies the asymptotic equality: $$(4\pi t)^{n/2}Z(t) = {\mathrm{vol}}(M) + O(t),\label{eq:weylslaw}$$ for $t\to 0+$ ( and Pleijel [@minak]). Now, as a result of Peter-Weyl theorem and Schur’s lemma, the spectrum of the Laplacian is parametrized by the unitary dual $\hat G$ of $G$, and we have: $$Z(t) =\sum_{u\in\hat G} \dim(u)^2 e^{tC_u},$$ where $C_u$ is a constant that depends on $u$. According to the representation theory of compact Lie groups, there is a one-to-one correspondence between $\hat G$ and $\Lambda\cap K$, where $\Lambda$ and $K$, respectively, are a certain lattice and cone in $\mathfrak{t}^*$. Hence, $$Z(t) =\sum_{\lambda\in\Lambda\cap K} d(\lambda)^2 e^{t\Omega_\lambda},$$ where $d(\lambda)$ is some function on $\mathfrak{t}^*$, and $\Omega_\lambda$ is a constant that depends on $\lambda$. Applying the Euler-Maclaurin formula, we obtain the asymptotic equality: $$t^{n/2}Z(t)= t^{n/2}I(t) +O(t) \label{eq:httreumc}$$ for $t\to0+$, where $$I(t)= \frac{{\mathrm{vol}}(T)}{(2\pi
)^{\dim(T)}}\int_{K} d(\lambda)^2e^{-t\|\lambda\|^2}\,d\lambda.$$ Owing to the invariance of $d(\lambda)^2$ and $\|\lambda\|$ relative to the Weyl group action, the domain of the above integral can be extended to whole $\mathfrak{t}^*$. Then, with the aid of Weyl integration formula, we arrive at: $$I(t) = \frac{{\mathrm{vol}}(T)^2}{(2\pi)^{\dim(T)}{\mathrm{vol}}(G)}\Bigl(\prod_{\alpha\in\Phi^+}\frac{1}{\langle\alpha,\rho\rangle^2}\Bigr)
\int_{{\mathfrak{g}}} e^{-t\|X\|^2} \,dX.$$ The last Gaussian integral is easily evaluated. Comparing the two asymptotic equations and then yields Harish-Chandra’s formula.
The first significance of this line of reasoning would be that once we are equipped with few key results in Lie theory and differential geometry, namely, the representation theory of compact Lie groups and Weyl’s law, we can attain Harish-Chandra’s formula by an elementary means of the Euler-Maclaurin formula. A greater significance might still lie ahead in the context of index theory. We briefly indicate this in the concluding remarks.
Acknowledgement
===============
This work was done as a partial fulfillment of the requirements for the author’s doctoral degree. He wishes to express his heartfelt gratitude to his advisor Nigel Higson.
Preliminary Remarks {#par:cptlgpstrth1}
===================
Throughout this article, $G$ is a compact connected Lie group, and $T$ is a maximal torus of $G$. The measure on $G/T$ under consideration is the quotient measure coming from an invariant measure on $G$. An invariant measure on $G$ is unique up to a scalar factor; moreover, it is induced by a bi-invariant metric on $G$, which is in turn induced by an $\operatorname{Ad}(G)$-invariant inner product on $\mathfrak{g}$. Henceforth we fix such an inner product and denote it by $\langle \cdot , \cdot \rangle$, and endow $G$ and $T$ the invariant metrics induced by it. Then we have $${\mathrm{vol}}(G/T)=\frac{{\mathrm{vol}}(G)}{{\mathrm{vol}}(T)}.$$
We point out that, as far as ${\mathrm{vol}}(G/T)$ is concerned, we may further assume, without loss of generality, that the compact connected Lie group $G$ is semisimple and simply connected, for the following reasons. By the general theory of compact Lie groups, every compact connected Lie group $G$ satisfies an isomorphism $$G\cong (R\times S)/F, \label{eq:strcthmcclieg}$$ where $R$ is a torus, $S$ is a compact, connected, simply connected, semisimple Lie group, and $F$ is a finite abelian subgroup of $R\times S$ (see Knapp [@knapp]\*[Thm. 4.29, p. 250]{}). Thus $F$ is contained in a maximal torus $\widetilde T$ of $R\times S$. Then $\widetilde T/F$ is a maximal torus of $(R\times S)/F$; let $T$ be the corresponding maximal torus of $G$ under the isomorphism . The Lie groups $G$, $R\times S$, and $(R\times S)/F$ all have isomorphic Lie algebras. Hence the $\operatorname{Ad}(G)$-invariant inner product on ${\mathfrak{g}}$ induces a bi-invariant metric on the Lie groups $G$, $R\times S$, $(R\times S)/F$, and their respective maximal tori. Their volumes satisfy: $$\frac{{\mathrm{vol}}(R\times S)}{{\mathrm{vol}}(\widetilde T)}=\frac{{\mathrm{vol}}((R\times S)/F)}{{\mathrm{vol}}(\widetilde T/F)}=\frac{{\mathrm{vol}}(G)}{{\mathrm{vol}}(T)}.$$ Therefore, for our purpose, we may assume that $F$ is trivial, so that $G\cong R\times S$. Now the maximal torus $\widetilde T$ of $R\times S$ is of the form $R\times T_S$, where $T_S$ is a maximal torus of $S$. Hence, $$\frac{{\mathrm{vol}}(R\times S)}{{\mathrm{vol}}(\widetilde T)}= \frac{{\mathrm{vol}}(R\times S) }{{\mathrm{vol}}(R\times T_S)}=\frac{{\mathrm{vol}}(S)}{{\mathrm{vol}}(T_S)}.$$ This shows that we may as well assume that $G=S$, that is, $G$ is semisimple and simply connected.\[par:cptlgpstrth2\]
Euler-Maclaurin Formula
=======================
Let $f$ be a smooth function on the real line. The Euler-Maclaurin formula relates a sum $\sum_{x=0}^nf(x)$ to the integral $\int^n_0f(x)\,dx$ (Euler [@eulerm1; @eulerm2], Maclaurin [@emaclaurin]); precisely stated, for $N\in{\mathbb{N}}$, $$\sum_{x=0}^nf(x)=\int^n_0 f (x)\,dx
+\sum^N_{q=1}(-1)^q\frac{B_{q}}{q!} \bigl(f^{(q-1)}(n)-f^{(q-1)}(0)\bigr) +R_N,\label{eq:eulermac}$$ where the coefficients $B_{q}$ are the Bernoulli numbers defined by the power series $$\operatorname{Td}(x):= \frac{x}{1-e^{-x}}=\sum^\infty_{q=0}(-1)^q\frac{B_{q}}{q!}x^q,$$ and $R_N$ is the remainder term, which can be estimated by $$|R_N|\le \frac{2\zeta(N)}{(2\pi)^{N}} \int^n_0\bigl|f^{(N)}(x)\bigr|\,dx.\label{eq:eulermaclaruinrem}$$ Here $\zeta$ is the Riemann zeta function. Suppose $\sum^{\infty}_{x=0}f(x)$ exists and $\lim_{x\rightarrow\infty}f^{(q)}(x)=0$ for all $q\in{\mathbb{N}}$. Then we may set $n=\infty$ in Equation , which yields: $$\sum_{x=0}^\infty f(x)=\int^\infty_0 f (x)\,dx
+\sum^N_{q=1}(-1)^q\frac{B_{q}}{q!} f^{(q-1)}(0) +R_N.\label{eq:eulermacinfty}$$ If, furthermore, $R_N\rightarrow0$ as $N\rightarrow\infty$ (for instance, when $f$ is a polynomial), we have: $$\sum_{x=0}^\infty f(x) = \Bigl.\operatorname{Td}\Bigl(\frac{\partial}{\partial h} \Bigl)\Bigr|_{h=0}\int^\infty_{-h}f(x)\,dx.$$ This expression for the Euler-Maclaurin formula first appeared in and [@tdeulmac].
The following lemma can be proved using the Euler-Maclaurin formula. We shall make use of it later on.
\[lem:eulmacprlm\] Let $A$ and $B$ be real numbers with $A>0$, and let $m$ be any nonnegative integer. Let $$f_t(x) := x^{2m}e^{-t(Ax^2+Bx)}.$$ Consider the sum $$S(t) :=\sum_{x=0}^\infty f_t(x)$$ for $t>0$. Then, $$t^{m+1/2}S(t)= t^{m+1/2}\int^\infty_0 f_t(x)\,dx + O(t^{1/2})$$ for $t\rightarrow0+$.
By Equation with $N=1$, $$S(t)=\int^\infty_0 f_t (x)\,dx
- B_{1} f_t(0) +R_1.$$ By the estimate , the absolute value of $R_1$ is bounded by, up to a constant factor, $$\int^\infty_0 \Bigl|
2mx^{2m-1}e^{-t(Ax^2+Bx)} -t (2Ax^{2m+1}+Bx^{2m})e^{-t(Ax^2+Bx)}\Bigr|\,dx.$$ By a change of variable of the type $x\mapsto \alpha x+\beta$, $\alpha>0$, the above integral can be recast in the following form: $$\int^\infty_\alpha \Bigl|P_1(x)e^{-tx^2} +tP_2(x)e^{-tx^2}\Bigr|\, dx,$$ where $P_1(x)$ and $P_2(x)$ are polynomials of degree $2m-1$ and $2m+1$, respectively. This integral is bounded by, up to a constant factor, $$\int^\infty_{-\infty} |P_1(x)|e^{-tx^2}dx +t\int^\infty_{-\infty}|P_2(x)|e^{-tx^2}\, dx.
\label{eq:eulmaclem}$$ It is known that, for a nonnegative integer $n$, $$\int^\infty_{-\infty} |x^n|e^{-tx^2}dx = \frac{1}{t^{(n+1)/2}}\Gamma\Bigl(\frac{n+1}{2}\Bigr),$$ where $\Gamma$ denotes the Gamma function. Hence, we see that the integral is bounded by a quantity that is of $O(t^{-m})$ for $t\rightarrow0+$. Meanwhile, the integral $\int_{0}^\infty f_t(x)\,dx$ is of $O(t^{-m-1/2})$. Hence, $$t^{m+1/2}S(t) = t^{m+1/2}\int^{\infty}_0f_t(x)\,dx + O(t^{1/2}).\qedhere$$
Proof of Harish-Chandra’s Formula {#par:introgencase}
=================================
We use the following notation for the dimensions of $G$, $T$, and $G/T$: $$n:=\dim G,\quad r:=\dim T,\quad 2m:=\dim (G/T)=n-r.$$ We denote by $\Phi^+$ the selected set of positive roots of $\mathfrak{g}$, and set $\rho=\frac{1}{2}\sum_{\alpha\in\Phi^+}\alpha$. The selection of $\Phi^+$ amounts to choosing a Weyl chamber in $\mathfrak{t}^*$, which we designate by $K$ and refer to as the fundamental Weyl chamber. The fundamental weights are denoted by $\{\lambda_i\}_{i=1}^r$. The lattice of (analytically) integral weights is represented by $\Lambda$. For each $\lambda\in\Lambda\cap K$, we denote by $V_\lambda$ the irreducible $G$-vector space with highest weight $\lambda$. (For reference on the terminologies, see and [@duistermaatkolk]\*[Chs. 3–4]{}.)
Let $\mathfrak{p}$ be the orthogonal complement of $\mathfrak{t}$ in $\mathfrak{g}$. The complexification $\mathfrak{p}_{\mathbb{C}}$ of $\mathfrak{p}$ is the direct sum of root spaces of $\mathfrak{g}$. Since half of the roots are positive, we have $$m = |\Phi^+|.
\label{eq:dimgmtposrt}$$
\[prop:cptliehttrpr\] For the heat trace $Z(t)$ of the Laplacian on $G$, we have: $$Z(t) =\sum_{\lambda\in \Lambda\cap K}\dim(V_\lambda)^2 e^{tC_\lambda},$$ where $$C_\lambda=-\|\lambda+\rho\|^2+\|\rho\|^2.$$ (Here $\|\cdot \|$ is the norm on $\mathfrak{t}^*$ induced by the inner product on $\mathfrak{g}$.)
Owing to the representation theory of compact Lie groups, there is a one-to-one correspondence between the unitary dual $\hat G$ and $\Lambda\cap K$. Now the Peter-Weyl theorem states that we have a $(G\times G)$-equivariant Hilbert space isomorphism ( and [@peterweyl]): $$L^2(G,{\mathbb{C}}) \cong \bigoplus_{\lambda\in\Lambda\cap K} {V_\lambda^*\otimes V_\lambda}.
\label{eq:peterweyl}$$ Here $L^2(G,{\mathbb{C}})$ denotes the space of square-integrable complex-valued functions on $G$. As a consequence of the invariance of the metric on $G$, the action induced on the right-hand side of by the Laplacian is that of $\mathbf{1}\otimes \Omega$, where $\Omega$ denotes the Casimir element in the universal enveloping algebra of $\mathfrak{g}$. By Schur’s lemma, the action of $\Omega$ on $V_\lambda$ is by a constant, say $C_\lambda$, whose value is well-known to be as asserted in the proposition (see, for instance, Knapp [@knapp]\*[p. 295]{}).
\[prop:modihtr2\] We have: $$Z(t) = e^{t\|\rho\|^2}\sum_{\lambda\in \Lambda\cap K} d(\lambda)^2 e^{-t\|\lambda\|^2},$$ where $d$ is a function on $\mathfrak{t}^*$ defined by $$d(\lambda) := \frac{\prod_{\alpha\in\Phi^+}\langle \alpha,\lambda\rangle}{\prod_{\alpha\in\Phi^+}\langle \alpha,\rho\rangle}.$$ (Here we are using, with a slight abuse of notation, $\langle \cdot , \cdot \rangle$ to denote the inner product on $\mathfrak{t}^*$ induced by that on $\mathfrak{g}$.)
The argument presented below is from Fegan [@fegan]\*[p. 594]{}:
In terms of the function $d$, we may express the Weyl dimension formula as: $$\dim(V_\lambda) = d(\lambda+\rho).$$ Hence, by Proposition \[prop:cptliehttrpr\], we have $$Z(t) = e^{t\|\rho\|^2}\sum_{\lambda\in \Lambda\cap K} d(\lambda+\rho)^2 e^{-t\|\lambda+\rho\|^2}.$$ Note that $$\sum _{\lambda\in \Lambda\cap K} d(\lambda+\rho)^2 e^{-t\|\lambda+\rho\|^2}
=\sum _{\lambda\in (\Lambda\cap K)+\rho} d(\lambda)^2 e^{-t\|\lambda\|^2}.$$ The shifted index set $(\Lambda\cap K)+\rho$ is the set of weights that lie in the interior $K^\circ$ of the fundamental Weyl chamber. Hence, $$Z(t) = e^{t\|\rho\|^2}\sum_{\lambda\in \Lambda\cap K^\circ} d(\lambda)^2 e^{-t\|\lambda\|^2}.
\label{eq:cptlghtrmd}$$ But since the boundary $\partial K$ of the fundamental Weyl chamber is contained in the hyperplanes orthogonal to the roots, the restriction of $d$ to $\partial K$ is zero. Hence we may replace the index set $\Lambda\cap K^\circ$ of the sum to $\Lambda\cap K$. This proves the proposition.
\[lem:htrldt\] Let $\mu_{{\mathfrak{t}}^*}$ denote the Lebesgue measure on ${\mathfrak{t}}^*$ induced by the inner product $\langle \cdot , \cdot \rangle$. Let $P$ be the fundamental parallelepiped in $\mathfrak{t}^*$ formed by the fundamental weights of $G$.
1. \[item:htrldt\] For $t\rightarrow0+$, we have $$t^{n/2} Z(t) = t^{n/2} I(t) +O(t), \label{eq:httremaprx}$$ where $$I(t)= \int_{K} d(\lambda)^2e^{-t\|\lambda\|^d}\,\frac{\mu_{{\mathfrak{t}}^*}(\lambda)}{{\mathrm{vol}}(P)}. \label{eq:httrapprxint}$$
2. \[item:torusvol\] The volume of the maximal torus is related to ${\mathrm{vol}}(P)$ by $${\mathrm{vol}}(T)= \frac{(2\pi)^r}{{\mathrm{vol}}(P)}. \label{eq:torusvol}$$
(\[item:htrldt\]) By Proposition \[prop:modihtr2\] we have: $$t^{n/2}Z(t) =t^{n/2}\bar Z(t) + O(t),$$ where $$\bar Z(t) := \sum_{\lambda\in \Lambda\cap K} d(\lambda)^2 e^{-t\|\lambda\|^2}.$$ Let $(x_1,\dotsc,x_r)$ denote the component variables on ${\mathfrak{t}}^*$ relative to the fundamental weights $\{\lambda_i\}_{i=1}^r$, so that an arbitrary element $\lambda\in\mathfrak{t}^*$ is expressed as $\lambda=\sum_{i=1}^rx_i\lambda_i$. Then $\bar Z(t)$ is of the form: $$\bar Z(t) =\sum_{x_1=0}^\infty\dotsb\sum_{x_r=0}^\infty d(x_1,\dotsc,x_r)^2 e^{-tq(x_1,\dotsc,x_r)},$$ where $d$ and $q$ are homogeneous polynomials of degree $m$ and $2$, respectively. Applying Lemma \[lem:eulmacprlm\] iteratively to $\bar Z(t)$, we have: $$t^{n/2}\bar Z(t) = t^{n/2}I(t) +O(t),$$ where $$I(t) = \int^\infty_{0}\dotsi\int^\infty_{0} d(x_1,\dotsc,x_r)^2 e^{-tq(x_1,\dotsc,x_r)}\,dx_1\dotsm dx_r.$$ This is the integral .
(\[item:torusvol\]) Recall our assumption that $G$ is compact, connected, simply connected, and semisimple. In this case the fundamental weights $\{\lambda_i\}_{i=1}^r$ form a basis for $\Lambda$. Let $\{ H_i \}_{i=1}^r$ be the simple coroots, that is, the vectors in $\mathfrak{t}$ that are dual to the fundamental weights relative to the inner product. Let $\hat \Lambda$ be the ${\mathbb{Z}}$-lattice spanned by the simple coroots. Then $$2\pi\hat\Lambda:= \exp^{-1}\{e\}\cap\mathfrak{t},$$ where $\exp:\mathfrak{g}\to G$ is the exponential map, and $e$ is the identity in $G$.
Let $Q$ be the fundamental parallelepiped in ${\mathfrak{t}}$ formed by the simple coroots. Let $\mu_{\mathfrak{t}}$ be the Lebesgue measure on ${\mathfrak{t}}$ induced by the inner product. Since $\exp(2\pi H_i)=e$ for all simple coroots $H_i$, we have $${\mathrm{vol}}(T) = (2\pi)^r{\mathrm{vol}}(Q).$$ Meanwhile, because the lattice $\hat\Lambda$ and $\Lambda$ are dual to each other, we have $${\mathrm{vol}}(P) = \frac{1}{{\mathrm{vol}}(Q)}.$$ Thus we have Equation .
Let $G$ be a compact connected Lie group and $T$ its maximal torus. Let ${\mathfrak{g}}$ and ${\mathfrak{t}}$ be their Lie algebras, respectively. Let $\langle \cdot , \cdot \rangle$ denote an $\operatorname{Ad}(G)$-invariant inner product on ${\mathfrak{g}}$ and also the induced inner product on the dual space ${\mathfrak{g}}^*$. Endow $G$ and $T$ with the measures induced by the bi-invariant metric that is generated by $\langle \cdot , \cdot \rangle$. Then, with the quotient measure on $G/T$, we have: $${\mathrm{vol}}(G/T) =\prod_{\alpha\in\Phi^+}\frac{2\pi}{\langle \alpha,\rho\rangle},$$ where $\Phi^+$ is the set of the selected positive roots of $G$ and $\rho=\frac{1}{2}\sum\limits_{\alpha\in\Phi^+}\alpha$.
As explained in Section \[par:cptlgpstrth1\], we may assume that $G$ is simply connected and semisimple. Let $W$ denote the Weyl group of $G$. The action of $W$ on $\mathfrak{t}^*$ preserves the set of roots and the inner product. This implies that $d(\lambda)^2$ and $\|\lambda\|$, which appear in the integral , are preserved under the $W$-action; so it is possible to extend the domain of integration to all of ${\mathfrak{t}}^*$ as follows: $$I(t) = \frac{1}{|W|} \int_{{\mathfrak{t}}^*} d(\lambda)^2 e^{-t\|\lambda\|^2}\frac{\mu_{{\mathfrak{t}}^*}(\lambda)}{{\mathrm{vol}}(P)}.$$ Then, by Equation , $$I(t) = \frac{1}{|W|} \frac{{\mathrm{vol}}(T)}{(2\pi)^r} \int_{{\mathfrak{t}}^*} d(\lambda)^2 e^{-t\|\lambda\|^2}\mu_{{\mathfrak{t}}^*}(\lambda).$$
We now switch the domain of integration from ${\mathfrak{t}}^*$ to ${\mathfrak{t}}$ via the linear isomorphism $$\begin{array}{ccc}
{\mathfrak{t}}^*&\rightarrow&{\mathfrak{t}},\\
\lambda&\mapsto&X_\lambda,
\end{array}$$ where $X_\lambda$ is the vector that is dual to $\lambda$ relative to the inner product. This isomorphism is precisely through which the inner product on ${\mathfrak{t}}$ was transferred to ${\mathfrak{t}}^*$; in particular, the Jacobian determinant of this isomorphism is $1$. Moroever, we have: $$d(\lambda)^2= \frac{\prod_{\alpha\in\Phi^+}\langle \alpha,\lambda\rangle^2}{\prod_{\alpha\in\Phi^+}\langle \alpha,\rho\rangle^2} = \frac{\prod_{\alpha\in\Phi^+}\alpha(X_\lambda)^2}{\prod_{\alpha\in\Phi^+}\langle \alpha,\rho\rangle^2},$$ and $$e^{-t\|\lambda\|^2}=e^{-t\|X_\lambda\|^2}.$$ Hence, $$I(t) =\frac{1}{|W|} \frac{{\mathrm{vol}}(T)}{(2\pi)^r}\Bigl(\prod_{\alpha\in\Phi^+}\frac{1}{\langle \alpha,\rho\rangle^2}\Bigr) \int_{{\mathfrak{t}}} \Bigl(\prod_{\alpha\in\Phi^+}\alpha(X)^2\Bigr) e^{-t\|X\|^2}\,\mu_{{\mathfrak{t}}}(X), \label{eq:volfmlder1}$$ where $\mu_{\mathfrak{t}}$ is the Lebesgue measure on ${\mathfrak{t}}$ induced by the inner product.
Recall the Weyl integration formula, which states that $$\int_{{\mathfrak{g}}} f(X)\, \mu_{\mathfrak{g}}(X) = \frac{1}{|W|}\frac{{\mathrm{vol}}(G)}{{\mathrm{vol}}(T)}\int_{\mathfrak{t}}f(X)\, \Bigl(\prod_{ \alpha\in\Phi^+}\alpha(X)^2\Bigr)\,\mu_{\mathfrak{t}}(X) \label{eq:wifexplct}$$ for any function $f$ that is invariant relative to the adjoint action of ${\mathfrak{g}}$ on itself (see and [@duistermaatkolk]\*[Thm. 3.14.1, p. 185]{}). Here the measure $\mu_{\mathfrak{g}}$ on ${\mathfrak{g}}$ is again the Lebesgue measure induced by the inner product. Substitution of $f(X)=e^{-t\|X\|^2}$ in Equation and some rearranging of terms leads us to: $$\int_{\mathfrak{t}}e^{-t\|X\|^2} \Bigl(\prod_{ \alpha\in\Phi^+}\alpha(X)^2\Bigr)\,\mu_{\mathfrak{t}}(X) = |W|\frac{{\mathrm{vol}}(T)}{{\mathrm{vol}}(G)} \int_{{\mathfrak{g}}} f(X)\, \mu_{\mathfrak{g}}(X).$$ Implementing this on Equation , we get $$I(t) = \frac{{\mathrm{vol}}(T)^2}{(2\pi)^r{\mathrm{vol}}(G)}\Bigl(\prod_{\alpha\in\Phi^+}\frac{1}{\langle \alpha,\rho\rangle^2}\Bigr)
\int_{{\mathfrak{g}}} e^{-t\|X\|^2} \,\mu_{\mathfrak{g}}(X). \label{eq:sslvolform}$$ The last integral is just a Gaussian integral, which has the value: $$\int_{{\mathfrak{g}}} e^{-t\|X\|^2} \,\mu_{\mathfrak{g}}(X) = \Bigl( \frac{\pi}{t} \Bigr)^{n/2}.$$ Therefore, $$I(t)= \frac{{\mathrm{vol}}(T)^2}{(2\pi)^r{\mathrm{vol}}(G)}\Bigl(\prod_{\alpha\in\Phi^+}\frac{1}{\langle \alpha,\rho\rangle^2}\Bigr)\Bigl( \frac{\pi}{t} \Bigr)^{n/2}.\label{eq:heattrint}$$ Inserting this expression into Equation and invoking Weyl’s law (Equation ), we get $$\frac{{\mathrm{vol}}(G)^2}{{\mathrm{vol}}(T)^2}=(2\pi)^{2m}\prod_{\alpha\in\Phi^+}\frac{1}{\langle \alpha,\rho\rangle^2}.$$ The inner product between two positive roots is nonnegative. Hence, $$\frac{{\mathrm{vol}}(G)}{{\mathrm{vol}}(T)}=(2\pi)^{m}\prod_{\alpha\in\Phi^+}\frac{1}{\langle \alpha,\rho\rangle}.$$ By Equation , we have $$\frac{{\mathrm{vol}}(G)}{{\mathrm{vol}}(T)}=\prod_{\alpha\in\Phi^+}\frac{2\pi}{\langle \alpha,\rho\rangle}.$$ This proves the formula.
Concluding Remarks
==================
As we have mentioned in the introduction, one significance of the present argument is that, having equipped with some key results of Lie theory and differential geometry, namely, the representation theory of compact Lie groups and Weyl’s law, one can easily access Harish-Chandra’s formula by an elementary means of the Euler-Maclaurin formula.
What seems to be of greater significance, however, lies in the question that it stirs up when one brings this approach to the Atiyah-Singer index theorem. Suppose $D$ is a Dirac operator on a closed spin manifold $M$. Let $\operatorname{Ind}(D)$ be the (graded) index of $D$. In its simplest setting, the Atiyah-Singer index theorem states that $$\operatorname{Ind}(D) =\int_{M}\hat A, \label{eq:atiyahsinger}$$ where $\hat A$ is the Hirzebruch $\hat A$-class of $M$ (Atiyah and Singer [@atiyahsinger]). Meanwhile, it is also true that $$\operatorname{Ind}(D) = \operatorname{Str}(e^{tD^2}),$$ where $\operatorname{Str}$ denotes the super trace (trace over the even domain minus the trace over the odd domain); this is known as the McKean-Singer formula (McKean and Singer [@mckeansinger]\*[p. 61]{}). Therefore, $$\operatorname{Str}(e^{tD^2})=\int_M\hat A.\label{eq:atsimc}$$ At this point we may apply on the left-hand side the Euler-Maclaurin formula, which involves the power series $$\operatorname{Td}(x) = \frac{x}{1-e^{-x}}.$$ But this power series is what produces the $\hat A$-class, which is on the right-hand side of Equation , under the Chern-Weil homomorphism for the tangent bundle of $M$. It is a mystery (at least for me) whether this is just a fluke; or can the Euler-Maclaurin formula explain the appearance of the $\hat A$-class in the index formula? The author welcomes suggestions towards answering this question.
[^1]: Let $p\colon G\to G/T$ be the canonical projection. If $\mu_G$ is a measure on $G$, then the quotient measure $\mu_{G/T}$ on $G/T$ is defined by $\mu_{G/T}(U) = \mu_G(p^{-1}(U))$ for every open subset $U$ of $G/T$.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- '\'
bibliography:
- 'IEEEabrv.bib'
- 'Mybib.bib'
title: On the Matrix Inversion Approximation Based on Neumann Series in Massive MIMO Systems
---
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We explore the use of the optimal statistical interpolation (OSI) data assimilation method for the statistical tracking of emerging epidemics and to study the spatial dynamics of a disease. The epidemic models that we used for this study are spatial variants of the common susceptible-infectious-removed (S-I-R) compartmental model of epidemiology. The spatial S-I-R epidemic model is illustrated by application to simulated spatial dynamic epidemic data from the historic “Black Death” plague of 14th century Europe. Bayesian statistical tracking of emerging epidemic diseases using the OSI as it unfolds is illustrated for a simulated epidemic wave originating in Santa Fe, New Mexico.
Bayesian statistical tracking, emerging epidemics, spatial S-I-R epidemic model, data assimilation, ensemble Kalman filter, optimal statistical interpolation
author:
- 'Ashok Krishnamurthy[^1]'
- 'Loren Cobb${}^{*}$'
- 'Jan Mandel${}^{*}$'
- 'Jonathan Beezley${}^{*}$'
bibliography:
- 'epienkf.bib'
title: 'Bayesian Tracking of Emerging Epidemics Using Ensemble Optimal Statistical Interpolation (EnOSI)'
---
=1
Introduction\[intro\]
=====================
Mathematical models have been used since 1927 to describe the rise and fall of infectious disease epidemics [@Diekmann-2000-MEI; @Castillo-2002-MAE1; @Castillo-2002-MAE2; @Ma-2009-MUI]. A majority of the models are based on the three-compartment nonlinear Susceptible-Infected-Removed (S-I-R) model developed by [@Kermack-1927-ACM]. A person occupies the *susceptible* or *infectious* compartments if he or she can contract or transmit the disease, respectively. The *removed* compartment includes those who have died, have been quarantined, or have recovered from the disease and become immune. The state variables are the number of susceptible ($S$), the infectious ($I$), and the removed ($R$) in a closed population. S-I-R models often perform surprisingly well in modeling the temporal evolution of real-world epidemics, and their spatial variants can produce a traveling-wave spatial dynamics similar to that displayed by some epidemics. Traveling waves are solutions to the spatial models such that the distribution of infected at time ($t+1$) is approximately a translation of the distribution at time $t$.
Tracking and forecasting the full spatio-temporal evolution of a new epidemic is notoriously difficult. Often the model itself is incorrect in unknown ways, observational data may be affected by many sources of error, and new data arrives on an irregular schedule. This is a well-known problem in a variety of empirical areas of high importance, such as storm and wildfire forecasting. The general category of tracking techniques that incorporate error-prone data as they arrive by sequential statistical estimation is known as *data assimilation* [@Kalnay-2003-AMD]. Use of the statistical methods of data assimilation can increase the accuracy and reliability of epidemic tracking by incorporating data as it arrives, with weighting factors that reflect the observed reliability of the observations. A few applications of data assimilation in epidemiology already exist [@Kalivianakis-1994-RSV; @Cazelles-1997-UKF; @Costa-2005-MPA; @Bettencourt-2007-TRT; @Bettencourt-2008-RTB; @Jegat-2008-EDA; @Rhodes-2009-VDA; @Dukic-2009-TFE; @Mandel-2010-DDC].
The aim of this paper is to study the use of a spatial variant of the S-I-R model to track newly emerging epidemics using a data assimilation technique called optimal statistical interpolation (OSI). This is already a popular data assimilation method in the literature of meteorology and oceanography. When coupled with a spatial dynamic model, the OSI method can be used to forecast the spatio-temporal evolution of an epidemic, and to adjust those forecasts appropriately as sparse and error-prone data arrives.
This paper is organized as follows. In Section \[epi\], we present a stochastic spatial epidemic model and use it to reproduce the spatio-temporal disease spread map of the 14th century Black Death. In Section \[track\], we illustrate the Bayesian tracking of emerging epidemics using OSI with a simulated epidemic wave originating in Santa Fe, New Mexico. In Section \[results\], we provide computational results for the stochastic spatial epidemic model with epidemic tracking method presented in Section \[track\]. Finally, in Section \[conclusion\], we provide some concluding remarks and future directions.
A Stochastic Spatial Epidemic Model\[epi\]
==========================================
Epidemic Dynamics
-----------------
For this study we use a discretized stochastic version of the [@Hoppensteadt-1975-PDE] spatial S-I-R epidemic model. As with almost all spatial epidemic models since [@Bailey-1957-MTE; @Bailey-1967-SSE; @Kendall-1965-MMS], we assume that individuals are continuously distributed on a spatial domain. This model uses three variables to define the state of the epidemic at each $(x,y)$ coordinate:
$\qquad S(x,y,t)=$ density (per unit area) of the susceptible population,
$\qquad I(x,y,t)=$ density of the infected population, and
$\qquad R(x,y,t)=$ density of the removed population.
Thus, each of these variables is a scalar field that evolves with time.
In continuous time the epidemic dynamics are defined by a system of three partial differential equations for the state variables. There are no vital dynamics in this model, meaning that there are no new births or non-disease related deaths in any of the three compartments. Following [@Hoppensteadt-1975-PDE], we assume that the rate of new infections at location $(x,y)$ depends on the density of infection at that point, and in nearby locations that have been weighted with a kernel function that drops off exponentially with Euclidean distance. The effective (i.e. weighted) density of infection at $(x,y)$ is given by $$\begin{aligned}
J(x,y,t) & =\int\int I(x-\phi,y-\theta)K(\phi,\theta)d\phi d\theta,\\
K(\phi,\theta) & \propto\exp\left( -\alpha\sqrt{\phi^{2}+\theta^{2}}\right) ,\end{aligned}$$ where the integral is taken over the entire surface area under study, and the proportionality constant is given by the condition that $\int\int
K(\phi,\theta)d\phi d\theta=1$. Then, the three partial differential equations for the *deterministic* evolution of the spatial epidemic are:$$\begin{aligned}
\partial_{t}S & =-\beta SJ,\\
\partial_{t}I & =\beta SJ-\gamma I,\\
\partial_{t}R & =\gamma I.\end{aligned}$$
In this model, $\beta$ is the rate of infection from infected to susceptibles, given homogeneous mixing, $\alpha$ is an intensity measure of infectiousness of the disease, given by the product of mixing rate and the infection rate and $\gamma$ is the rate of removal of infected persons through death, recovery with immunity, and quarantine. To make this model stochastic, we assume that the quantities $\beta SJ$ and $\gamma I$ are the intensities of two independent Poisson processes. For simulations in discrete time and space, we use the following approximation: The number of newly infected and newly removed persons over the time interval $(t,t+\Delta t)$, within a box centered at position $(x,y)$ with $\Delta x\Delta y$ units of area, are given by
Number newly infected $\sim Poisson(\beta S(x,y,t)J(x,y,t)\Delta
x\Delta y\Delta t)$,
Number newly removed $\sim Poisson(\gamma I(x,y,t)\Delta x\Delta
y\Delta t).$
Thus a susceptible individual, at a particular location, may become infected when he/she comes in contact with an infected individual from within a neighboring area, with a monotonically decreasing weighting function that declines exponentially with distance. If this contact causes sufficient secondary infections then a new epidemic focus will develop at that new location. The simulation evolves on a two-dimensional discretized spatial domain with a total of $n\times m$ grid cells.
Example: The Black Death in Europe
----------------------------------
The “Black Death” bubonic plague epidemic that hit Europe in 1347 killed somewhere between 30% and 60% of the population of Europe over the course of about four years. The virulence of the disease back then was severe, which explains the extremely high number of deaths. Plague recurred in various regions of Europe for another 300 years before gradually withdrawing from Europe. There have been several attempts to reconstruct the movement of the wavefront of the epidemic [@Langer-1964-TBD; @Noble-1974-GTD; @Christakos-2005-NST; @Christakos-2007-RRS; @Gaudart-2010-DAS], as it swept across the continent; one such attempt using our spatial S-I-R model is shown in Figure 1b; others are similar. The disease arrived from Asia into Europe by trading ships, appearing first in Constantinople (modern Istanbul, Turkey) in 1347. From there it was carried by ship to Italy, France, Spain, and Croatia. Once ashore it moved inland (mainly in pneumonic form) at a speed that has been estimated at between 100 and 400 miles per year.
If it is to have credibility, a spatial epidemic model should be able to reproduce the principal historical features of the Black Death: its movement into the interior of Europe from the coastal cities, especially Marseilles, and movement up the island of Britain after its arrival in Bristol and London. Figure 1b shows the population density of infected people, using modern population densities in place of the unknown medieval population pattern, at roughly the beginning of 1350.
\[ptb\]
The R statistical computing language [@R-2010-LES], freely available from [www.cran.r-project.org](www.cran.r-project.org), was used to carry out the simulations for the spatial spread of the epidemic. Modern day population density data were downloaded as GPW (Gridded Population of the World) data files from the Center for International Earth Science Information Network at the Columbia University and converted to the array-oriented Network Common Data Form (NetCDF) format. These datasets were then loaded into R using the built-in package *ncdf*.
Comparison to Historical Data
-----------------------------
In historical reality, the zone from Romania to Poland to Russia suffered only lightly from the first wave of the Black Death. It is quite plain that the S-I-R simulation does not conform to the historical reconstruction in that zone. The explanation may lie with the very low population density and geographic mobility in Eastern Europe in the 14th century. A smaller discrepancy occurs in the city of Milan, Italy, which brutally stopped the spread of the plague by boarding up infected people in their homes. Despite differences in population distribution (France was then much more populous than Germany, for example), we believe that the simulation performs reasonably well in all areas except Eastern Europe.
Statistical Tracking Using Data Assimilation Techniques\[track\]
================================================================
We use data assimilation for the statistical tracking of emerging epidemics as they are unfolding. This involves two basic components: a dynamic model to forecast the state of the epidemic between arrivals of new data, and observations that are used to update an ensemble of state estimates. Data assimilation requires estimating the uncertainty both for model and observations forecasts. Our goal in this paper is to incorporate sparse and noisy observational epidemic data over space and time into a dynamic statistical model so as to produce an optimal Bayesian estimate of the current state of the infected population, and to forecast the progress of the real epidemic.
The Kalman Filter (KF)\[kf\]
----------------------------
The Kalman Filter (KF) was first presented by [@Kalman-1960-NAL] and [@Kalman-1961-NRL] as a method for tracking the state of a linear dynamic system perturbed by Gaussian white noise. In mathematical terms, this means that the errors are drawn from a zero-mean distribution with diagonal covariance matrix.
The full state of a discretized spatial epidemic model is a grid of $n\times
m$ cells, each of which contains a characterization of the population currently within the limits of the cell. To apply the Kalman filtering method, we represent the $n\times m$ values of the *Infected* variable on this grid as a single long vector $x$ with $p=n\times m$ elements, which for the purposes of data assimilation is the dimensionality of the state space. If we could observe this state vector without error, our observations would be another vector $y$ that satisfies $y=Hx$. Now consider the situation in which we have a forecast of the current state, $x^{f}$, and a newly arrived vector of noisy observations $y=Hx+\varepsilon$, where $\varepsilon\sim N\left(
0,R\right) $. We need to update the forecast by optimally assimilating these new observations. The result will be called the *analysis* estimate of state, $x^{a}$. The superscripts $f$ and $a$ are used to denote the forecast (prior) and analysis (posterior) estimate of the current state, respectively.
In the classical Kalman filter, the underlying dynamics are assumed to be linear, e.g.$$\begin{aligned}
x_{t} & =F_{t-1}x_{t-1}+u_{t-1},\\
u_{t} & \sim N\left( 0,\Sigma\right) ,\end{aligned}$$ and the analysis estimate of the state vector is calculated from$$\begin{aligned}
x_{t}^{f} & =F_{t-1}x_{t-1}^{f}\\
x_{t}^{a} & =x_{t}^{f}+K_{t}(y_{t}-Hx_{t}^{f}),\\
Q_{t}^{a} & =(I-K_{t}H)Q_{t}^{f},\end{aligned}$$ where$$\begin{aligned}
K_{t} & =Q_{t}^{f}H^{T}(HQ_{t}^{f}H^{T}+R)^{-1},\text{ and}\\
Q_{t}^{f} & =F_{t-1}Q_{t-1}^{f}F_{t-1}^{\prime}+\Sigma.\end{aligned}$$ Here $K_{t}$ is the Kalman gain matrix at time $t$, and $Q_{t}^{f}$ is the covariance matrix for the forecast state vector.
The extended Kalman filter (EKF) [@Julier-1995-ANA] was an early attempt to adapt the basic KF equations for nonlinear problems, through linearization. However, the EKF has it’s own disadvantages: if the model is strongly nonlinear at the time step of interest, linearization errors can turn out to be non-negligible, which leads to filter divergence [@Evensen-1992-UEK]. The EKF is not suitable for high-dimensional 2D and 3D data assimilation problems. Other commonly used Bayesian tracking techniques for nonlinear problems include the unscented Kalman filter (UKF) and the particle filter (PF) [@Gordon-1993-NAN]. Particle filtering is a versatile Monte Carlo technique that can handle nonlinearities in the model and represents the Bayesian posterior probability density function by a set of samples drawn at random with associated weights.
Ensemble Kalman Filter (EnKF)\[enkf\]
-------------------------------------
The KF algorithm described above is based on assimilating only one initial state ignoring the uncertainties in the model. In the following, we account for this state-dependent uncertainty by taking an ensemble approach to data assimilation. To see the problem, consider a 2D simulation of a scalar field that has been discretized on a $10^{3}\times10^{3}$ grid. The state vector of this system has $10^{6}$ elements, and its covariance matrix has $10^{6}\times10^{6}=10^{12}$ elements, requiring eight terabytes just to store. The EnKF solves this storage-and-retrieval problem by (in effect) calculating the covariances from the members of the ensemble as they are needed. The result is in an elegant Bayesian update algorithm with dramatically improved efficiency and storage requirements [@Mandel-2010-DDC].
The ensemble Kalman filter (EnKF) was introduced by [@Evensen-1994-SDA], modified to provide correct covariance by [@Burgers-1998-ASE], and improved by [@Houtekamer-1998-DAE]. The EnKF is a popular sequential Bayesian data assimilation technique that uses a collection of almost-independent simulations (known as an ensemble) to solve the covariance problem of Kalman filtering for systems with very high-dimensional state vectors. It does this using a two-step process: estimate of the covariance matrix, followed by an ensemble update. The covariance of a single state estimate in the KF is replaced by the sample covariance computed from the ensemble members. This sample covariance of ensemble forecasts is then used to calculate the Kalman gain matrix. There are two basic approaches to the EnKF update: stochastically perturbed observations (Monte Carlo), and square-root filters (deterministic). Both approaches adopt the same covariance estimate step, but differ in the ensemble update step. Regardless of the specific approach employed, the goal is to obtain a Bayesian estimate of the state as efficiently as possible. In many real-world examples these two approaches perform quite similarly. A more detailed description EnKF may be found in the book by [@Evensen-2009-DAE].
The EnKF analysis update equations are analogous to the classical KF equations, except that they use the covariance of the forecast ensemble to substitute for the matrix $Q$, which in a high-dimensional system is too large to store. Let $X$ be a random ensemble matrix of dimension $p\times N$ whose columns are realizations sampled from the prior distribution of the system state of dimension $p$ with ensemble size $N$. Then the EnKF update formula is:$$X^{a}=X^{f}+K_{e}(Y-HX^{f}),$$ where $Y$ is the observed ensemble data matrix whose columns are the true state perturbed by random Gaussian error. $H$ is, as before, the linear operator that maps the state vector onto the observational space. The deviation $Y-HX^{f}$ is commonly referred to as the observed-minus-forecast residual or simply as the innovation. In the above equation $K_{e}$ is the ensemble Kalman gain matrix given by$$K_{e}=Q^{f}H^{T}(HQ^{f}H^{T}+R)^{-1},$$ where $Q^{f}$ is the forecast-error covariance matrix of dimension $p\times
p$, and $R$ is the symmetric and positive-definite observational (measurement) error covariance matrix. The EnKF technique contains two sources of randomness: the random model input, and the measurement errors. Assuming that these two sources of randomness are uncorrelated, the analysis-error covariance matrix of dimension $p\times N$ can be computed from the equation$$Q^{a}=(I-K_{e}H)Q^{f}.$$
Optimal Statistical Interpolation (OSI)\[osi\]
----------------------------------------------
In the EnKF, the model error covariance matrix is evolved fully at each data assimilation step using an MCMC method. In contrast, Optimal Statistical Interpolation (OSI) is a data assimilation technique based on statistical estimation theory in which the model error covariance matrix is pre-determined empirically and is assumed to be time-invariant. The model error covariance matrix is dependent only on the distance between spatial grid cells. The correlation length is ad hoc and adjusted empirically.
OSI was derived by [@Eliassen-1954-PRC]. This method has been referred to as Statistical Interpolation, Optimal Interpolation, or Objective Analysis. OSI is called univariate if the observations are of a single scalar field, and multivariate if the observations of one or more scalar fields are used for estimating another scalar field [@Talagrand-2003-BEO]. Multivariate OSI was developed independently by [@Gandin-1965-OAM] for the analysis of meteorological fields in the former Soviet Union. It requires the specification of the cross-covariance matrix between the observed scalar fields and the scalar field to be estimated. The ensemble OSI (EnOSI), used here, requires much less computational effort than the EnKF, because the model error covariance matrix is fixed. The EnOSI approach may provide a practical and cost-effective alternative to the EnKF for tracking epidemics. The stationary model error covariance matrix in our epidemic simulation used a version with the correlation function having an exponential decay along the off-diagonal entries. The ensemble Kalman gain matrix was then calculated using this time-invariant covariance matrix with a fixed structure. The accuracy of the EnOSI process will be affected if the approximate covariance matrix differs substantially from the true covariance matrix. Therefore, one disadvantage of the OSI is the need for a fixed spatial covariance structure that can reasonably represent the epidemic dynamics throughout the whole domain at all times.
The EnOSI analysis update equation, using the stationary covariance, is given by$$X^{a}=X^{f}+K_{OSI}(Y-HX^{f}) \label{eq:OSI}$$ where$$K_{OSI}=Q_{OSI}^{f}H^{T}(HQ_{OSI}^{f}H^{T}+R)^{-1}.$$
Example: An Epidemic Wave Originating In New Mexico
---------------------------------------------------
To test the performance of EnOSI and other tracking algorithms designed for high-dimensional state vectors, we constructed a spatial simulation of an epidemic that originates in Santa Fe, New Mexico, and spreads outwards towards Albuquerque and Denver, Colorado. In this simulation the epidemic moves smoothly towards Albuquerque, but jumps suddenly to to Denver as if carried by a traveler in an automobile or airplane. Properly detecting and assimilating a new feature far from an existing focus is a serious challenge for EnKF algorithms [@Beezley-2008-MEK; @Mandel-2010-DDC].
To improve the realism of the test for the case in which an entirely new disease emerges for the first time, we initialized all members of the tracking ensemble so that they contain no disease whatsoever. New data in the form of an empirical scalar field arrives at time steps 10, 20, 30, 40, and 50. These data are complete in the sense that in this case $H$ is just the identity matrix, but they are substantially modified by independent Poisson-distributed random errors. The tracking algorithm forecasts the state up until the time when data is received, and then it assimilates this data into the forecast.
Results\[results\]
==================
We have applied the EnOSI for the New Mexico example mentioned above to the epidemic model described in Section 2 with an ensemble of size 25. For this example the *Infected* state of the model is the output of the observation function. Synthetic data were simulated from a model and initialized in exactly the same way as the ensemble.
In our epidemics application, the perturbed observations $Y$ in (\[eq:OSI\]) were obtained by sampling from the Poisson distribution with the intensity equal to the data value and rounding to integer, consistently with the stochastic character of the model, instead of Gaussian perturbations as in the EnKF. This is the key feature behind the successful use of our method and it also guarantees that $Y$ has nonnegative entries and thus the columns of $Y$ are meaningful as the *Infected* variable. In general, one may have to guarantee that the entries of the members of the analysis ensemble $X^{a}$ are also nonnegative, e.g. by censoring. This, however, was not needed in the results reported here.
The result for each member of the ensemble advanced in time by 10 model time units is a Bayesian update of the forecast scalar field, which is referred to as the analysis (i.e. the posterior estimate). We assume that data arrive only once every 10 time steps, with errors. A total of 5 assimilation cycles were performed in this manner. The mean and standard deviation (not reported here) of the ensemble analysis values in each cell of the scalar field gives the EnOSI estimate of the state of the epidemic, with its uncertainty quantified. The following figures present a spatial “image” of the number of infected persons over the planar domain considered in the New Mexico example.
\[ptb\]
\[ptb\]
\[ptb\]
\[ptb\]
Figure 2a shows the epidemic position in Santa Fe, New Mexico, at time 10. The initial forecast (2b) is empty, as it should be for the first appearance of any newly emergent disease. The EnOSI analysis (2c) shows that the arriving data have been partially assimilated, with a resulting picture that is indistinct and less than fully accurate. Figure 3a (time 20) shows the growth of the epidemic towards Albuquerque, and a very small new focus of infection in Denver. The forecast handles the movement towards Albuquerque quite well, but is devoid of any infection in Denver. After assimilation of the data, the analysis now also reflects a small focus in Denver. Figure 4 (time 30) shows the epidemic gaining size, and beginning a major expansion within both Albuquerque and Denver. Figure 5 (time 50) shows the epidemic gaining definition within the most heavily populated urban regions. The analysis steps, after data assimilation, are now tracking the epidemic quite well.
Conclusion\[conclusion\]
========================
The spread of newly emerging infectious diseases pose a serious challenge to public health in every country of the world. Tracking the spread of an epidemic in real-time can help public health officials to plan their emergency response and health care. The purpose of this paper has been to present the some preliminary results on the statistical tracking of emerging epidemics of infectious diseases using a Bayesian data assimilation technique called ensemble optimal statistical interpolation (EnOSI). Our simulation results confirms that EnOSI can be used to track the spatio-temporal patterns of emerging epidemics. We found that EnOSI can efficiently adjust its estimated spatial distribution of the number of infected, if and when the epidemic jumps from city to city, and with data that are sparse and error-ridden. The tracking accuracy in our simulations provides evidence of the good performance of the EnOSI approach, therefore the assumption that the model error covariance is time-invariant is reasonable. However, the effect of this assumption needs more rigorous theoretical justification.
The EnOSI as presented here requires the manipulation of the state covariance matrix, which gets very demanding if stored as a full matrix - computational grid of 200 by 200 points results in 40,000 variables and thus 40,000 by 40,000 covariance matrix, which requires a supercomputing cluster. Sparsification of the covariance matrix can decrease the computational cost somewhat, but it is still significant and the implementation grows complicated. For this reason, we plan to investigate a version of OSI by [@Mandel-2010-O] with the covariance implemented by the Fast Fourier Transform (FFT), similarly as in the FFT EnKF [@Mandel-2010-DDC; @Mandel-2010-DAM; @Mandel-2010-FFT].
Our research has set the groundwork for further efforts to incorporate the ideas of data assimilation to track diseases in real-time. Our future work includes extending the R framework that we have developed for employing an ensemble of spatial simulations to track diseases to test and compare a other variants of the EnKF [@Anderson-2001-EAK; @Tippett-2003-ESR; @Beezley-2008-MEK; @Mandel-2009-DAW; @Ott-2004-LEK; @Hunt-2007-EDA]. These variants aim to enhance the performance of the ensemble filters by representing the underlying model error statistics in an efficient manner. However, since the ensemble size required can be large (easily hundreds) [@Evensen-2009-DAE] for the approximation to converge [@Mandel-2009-CEK], the amount of computations in the ensemble-based methods can be significant, and so special localization techniques, such as tapering, need to be employed to suppress spurious long-range correlations in the ensemble covariance matrix [@Furrer-2007-EHP]. Thus the EnKF (and its variants) may not work well for problems with sharp coherent features, such as the traveling waves found in some epidemics. Choosing a small ensemble size, so small that it is not statistically representative of the state of a system, leads to underestimation of the analysis error covariances. Choosing a really large ensemble size may not be computationally feasible and cost efficient. Finally, methods for incorporating long-distance human movements to track the rapid geographical spread of infectious diseases have been proposed in the literature [@Brockmann-2009-HMS; @Belik-2009-IHM; @Merler-2010-RPH; @Balcan-2010-MSS; @Belik-2010-HMS]. In the future, we plan to explore such spatially extended epidemic models to track emerging epidemics.
[^1]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Ping Cao, Jingui Xie,\
School of Management, University of Science and Technology of China,\
[email protected], [email protected]
title: A New Condition for the Existence of Optimal Stationary Policies in Denumerable State Average Cost Continuous Time Markov Decision Processes with Unbounded Cost and Transition Rates
---
[**Abstract**]{} This paper presents a new condition for the existence of optimal stationary policies in average-cost continuous-time Markov decision processes with unbounded cost and transition rates, arising from controlled queueing systems. This condition is closely related to the stability of queueing systems. It suggests that the proof of the stability can be exploited to verify the existence of an optimal stationary policy. This new condition is easier to verify than existing conditions. Moreover, several conditions are provided which suffice for the average-cost optimality equality to hold.\
*Keywords*: Markov decision processes; Average-cost criterion; Unbounded transition rates; Optimal stationary policy
Introduction
============
Queueing systems have wide applications in computer communication networks, manufacturing processes and customer service platforms [@cmp]. There exists a lot of literature studying issues such as system stability, cost and performance analysis under a given service principle, e.g., [@awz; @xhz08]. In order to cut down the operational cost and better serve customers, the queueing models should be controlled in a way such that the operational cost is minimized, e.g., [@h84]. A lot of controlled queueing models can be analyzed as continuous-time Markov decision processes (CTMDP) [@ws87]. The buffer of the queue model is often unlimited, and transition rates might be dependent on the system state. Therefore, the corresponding CTMDP often has denumerable states and the transition rates are unbounded. Moreover, the state-dependent cost rates are also unbounded. A question naturally arises that whether an optimal stationary policy for such a CTMDP exists or not. For the discounted cost CTMDP, it is often relatively easy to verify whether an optimal stationary policy exists [@guoher; @guoz02a]. However, for the average-cost CTMDP, more conditions should be imposed to ensure the existence of an optimal stationary policy [@guoher; @guoz02b]. This paper provides a new condition under which an average-cost optimal stationary policy exists, which is different from existing conditions.
In 2002, [@guoz02b] presents a set of conditions under which an average-cost optimal stationary policy exists. Their conditions requires constructing a series of functions which satisfy their proposed assumptions. It is not straightforward to construct these functions for most problems we encounter. Later in 2009, [@guoher] gives a sufficient condition for the existence of optimal stationary policies, which also requires finding a function satisfying several conditions. However, this function is often problem specific. Without adequate research of the specific CTMDP, it is not easy to find an appropriate function. It will be valuable if we can find a way to bypass seeking for such a function. Focused on discrete-time Markov decision processes (DTMDP), [@sen99] gives several conditions under which an average cost optimal stationary policy exists. [@sen99] has also mentioned that CTMDP can be transformed into DTMDP if the transition rate is uniformly bounded by employing uniformization method. However, the uniformization method cannot be applied if the transition rate is unbounded. Therefore, it is quite necessary to analyze the CTMDP with unbounded transition rates separately.
In this paper, a new condition is provided to ensure the existence of an average-cost optimal stationary policy for denumerable state CTMDP with unbounded cost and transition rates. This condition concerns that whether the expected time and expected cost of a first passage from any state to a given state is finite or not under a given controlled policy. The former is related to the stability of the queueing system, while the latter may be seen as a generalized stability if we notice that the expected cost is equal to the expected time if the cost rate is 1. A lot of literature has focused on discussing the stability of the queueing system, e.g., [@t75; @xhz08]. Their results can help us verify whether an average-cost optimal stationary policy exists or not.
This paper is organized as follows. In Section 2, we introduce CTMDP and present our main result. Section 3 gives the proof of the main result. In Section 4, we give conditions under which the average-cost optimality inequality (ACOI) becomes equality.
Model and Main Result
=====================
Consider a continuous-time Markov decision process $\{x(t): t\geq 0\}$ consisting of four-element tuple $\{S, (A(i), i\in S), q(j|i,a), c(i,a)\}$:
1. The state space $S$ is denumerable;
2. Each action space $A(i)$ is a subset of the finite action space $A$;
3. The transition rate $q(j|i,a)$ satisfies $q(j|i,a)\geq 0$, $\forall\ i\neq j$, $i, j\in S$, $a\in A(i)$ and $\sum_{j\in S}q(j|i,a)=0$, $\forall\ i\in S, a\in A(i)$.
4. The cost rate function $c(i,a)\geq 0$, $\forall\ i\in S, a\in A(i)$.
Let $\Pi$ be the set of all randomized Markov policies and $F$ the set of all stationary policies [@guoher]. Given $\pi=(\pi_t)\in\Pi$ and the discount factor $\alpha>0$, we define the expected discounted cost function (with initial state $ i $) $$J_{\alpha}(i,\pi)=\int_{0}^{\infty}e^{-\alpha t}E_{i}^{\pi}[c(x(t),\pi_t)]dt, \forall\ i\in S, \pi\in\Pi, \label{eq_dis}$$ and the corresponding optimal discounted cost function $J_{\alpha}^{*}(i)=\inf_{\pi\in\Pi}J_{\alpha}(i,\pi), \forall\ i\in S,$ where $c(i,\pi_t)$ is the expected cost rate at state $i$ using policy $\pi_t$ at time $t$, which is defined as $c(i,\pi_t)=\int_{A(i)}c(i,a)\pi_t(da|i)$.
It is straightforward to show that for any stationary policy $f\in F$, we have $$\alpha J_{\alpha}(i,f)=c(i,f(i))+\sum_{j\in S}J_{\alpha}(j)q(j|i,f(i)). \label{eq_dis_dp_f}$$
Since $A(i)$ is finite, [@guoher] states that $J_{\alpha}^{*}(i)$ is well defined and satisfies the discounted-cost optimality equation $$\alpha J_{\alpha}^{*}(i)=\min_{a\in A(i)}\left\{c(i,a)+\sum_{j\in S}J_{\alpha}^{*}(j)q(j|i,a)\right\}. \label{eq_dis_dp}$$
Moreover, we define the long run expected average cost function $$J_c(i,\pi)=\lim\sup_{T\rightarrow\infty}\frac{1}{T}\int_0^TE_{i}^{\pi}[c(x(t),\pi_t)]dt, \forall\ i\in S, \pi\in\Pi, \label{eq_ave}$$ and the corresponding optimal average cost function $J_{c}^{*}(i)=\inf_{\pi\in\Pi}J_{c}(i,\pi), \forall\ i\in S.$
One of the most important questions is whether an average-cost optimal stationary policy exists for the CTMDP. Before we state our main results, we propose the following definition [@sen99].
[Let $d$ be a (randomized) stationary policy. Then $d$ is a $i_0$-standard policy if the Markov process induced by $d$, $\{x^d(t):t\geq 0\}$ satisfies that for any $i\in S$, the expected time $m_{i,i_0}(d)$ of a first passage from $i$ to $i_0$ (during which at least one transition occurs) is finite and the expected cost $c_{i,i_0}(d)$ of a first passage from $i$ to $i_0$ (during which at least one transition occurs) is finite.]{}
[**Remark 1:**]{} Note that $x(t)=x(t+)$, a.e.. Thus, if we define the first passage time $\tau_{i,i_0}$ as $\tau_{i,i_0}=\inf\{t>0: x(t)=i_0|x(0)=i\}$, then $\tau(i,i_0)=0$ a.e. if $i=i_0$. Hence we impose additional constraint that at least one transition occurs on the definition of the first passage time.
[**Remark 2:**]{} If the cost rate function is bounded, then $m_{i,i_0}(d)<\infty$ can implies $c_{i,i_0}(d)<\infty$. In this case, $d$ is a $i_0$-standard policy if the Markov process induced by $d$ is ergodic (i.e., irreducible and positive recurrent).
The following lemma is extensively used for analysing the stability of a queueing systems, of which the proof is omitted for brevity.
[\[lem\_1\] Assume that $m_{i,i_0}<\infty$, $\forall\ i\in S$. Assume that there exists a (finite) nonnegative function $r$ on $S$ and a finite subset $H^*$ containing $i_0$ such that $$\sum_{j}q(j|i)r(j)<\infty, i\in H^*, \label{eq_lem_1}$$ and $$c(i)+\sum_{j}q(j|i)r(j)\leq 0, i\notin H^*. \label{eq_lem_2}$$ Then there exists a (finite) nonnegative constant $F$ such that $c_{i,i_0}\leq r(i)-r(i_0)+Fm_{i,i_0}$, $\forall\ i\neq i_0$. Especially, if $H^*=\{i_0\}$, then $c_{i,i_0}\leq r(i)$, $\forall\ i\neq i_0$. ]{}
Let $ S = \{0,1,2,\dots\} $. Now we propose our main result.
\[the\_2\] Assume that $J_{\alpha}^*(i)$ is increasing in $i$ for $\alpha>0$. If there exists a $0$-standard policy $d$, then there exists a constant $g^*\geq 0$, a stationary policy $f^*$, and a real-valued function $h^*$ (which is increasing in $i$) such that:
\(i) There exists a sequence $\{\alpha_n, n\geq 1\}$ tending to zero (as $n\rightarrow\infty$) such that $\forall\ i\in S$, $$f^*(i)=\lim_{k\rightarrow\infty}f_{\alpha_k}^*(i), g^*=\lim_{k\rightarrow\infty}\alpha_k J_{\alpha_k}^*(0),\label{eq_limit_fg}$$ and $$h^*(i)=\lim_{k\rightarrow\infty}h_{\alpha_k}(i), \label{eq_limit_h}$$ where $h_{\alpha}(i):=J_{\alpha}^*(i)-J_{\alpha}^*(0)$.
\(ii) $(g^*, f^*, h^*)$ satisfy the following average-cost optimality inequality (ACOI): $$\begin{aligned}
g^* &\geq& c(i,f^*)+\sum_{j\in S}h^*(j)q(j|i,f^*) \label{eq_acoe}
\\ &= & \min_{a\in A(i)}\left\{c(i,a)+\sum_{j\in S}h^*(j)q(j|i,a)\right\}, \forall i\in S, \nonumber\end{aligned}$$ and $f^*$ is an average-cost optimal stationary policy.
[**Remark 1:**]{} The above result still holds when the state is a vector rather than a scalar.
[**Remark 2:**]{} The monotonicity of the discounted value function $J_{\alpha}^*(i)$ is often satisfied, e.g., in queueing systems more customers staying in the queue implies more waiting.
[**Remark 3:**]{} The $0$-standard policy $d$ is not required to be optimal. It can be any policy which is easy to be constructed and analyzed.
[**Remark 4:**]{} This theorem closely relates the existence of an average-cost optimal stationary policy to the stability of the queueing system under a given service policy. The queueing system is called to be stable under a given service policy if the induced Markov process is ergodic (irreducible and positive recurrent). Positive recurrence implies that for any $i\in S$, the expected time $m_{i,0}$ of a first passage from $i$ to $0$ is finite. To prove (positive recurrence) the finiteness of the expected time $m_{i,0}$, a Lyapunov function $r(\cdot)$ might be constructed in order to apply Lemma \[lem\_1\] with $c(i)=1$ and $H^*=\{0\}$ (noticing that the expected cost is expected time if the cost rate is 1). This method can also be found in Theorem 1.18 in [@c91]. In many situations, with slightly modification of the Lyapunov function $r(\cdot)$ constructed for proving $m_{i,0}<\infty$, another Lyapunov function can be constructed to satisfy (\[eq\_lem\_2\]) and thus the finiteness of the expected cost $c_{i,0}$ can be proved. That is to say, the discussion of stability of the queueing system can help prove the existence of an average-cost optimal stationary policy.
Proof of Theorem \[the\_2\]
===========================
[@guoher] proposes the following assumptions to ensure the existence of an average-cost optimal stationary policy, which can be seen as a continuous-time counterpart of (SEN) assumptions proposed in [@sen99].
[**Assumptions A:**]{} For some decreasing sequence $\{\alpha_n, n\geq 1\}$ tending to zero (as $n\rightarrow\infty$) and some state $i_0\in S$,
(A1) $\alpha_n J_{\alpha_n}^*(i_0)$ is bounded in $n$.
(A2) There exists a nonnegative (finite) function $H$ such that $h_{\alpha_n}(i)\leq H(i)$, $\forall\ i\in S, n\geq 1$, where $h_{\alpha}(i)=J_{\alpha}^*(i)-J_{\alpha}^*(i_0)$.
(A3) There exists a nonnegative constant $L$ such that $-L\leq h_{\alpha_n}(i)$, $\forall\ i\in S, n\geq 1$.
Before proving Theorem \[the\_2\], we give some results of the Markov process $\{x(t): t\geq 0\}$. Let $J_{i}(t)=\frac{1}{t}E\left[\int_0^t c(x(s))ds|x(0)=i\right], \forall\ i\in S.$ We have the following result.
\[pro\_5\] Let $R$ be a positive recurrent class.
\(i) For $i\in R$, $\lim_{t\rightarrow\infty}J_i(t)$ exists and equals the (finite or infinite) constant $J_R=: \sum_{j\in R}\pi_j c(j)$, where $\pi_j$ is the steady sate probability of being in state $j$.
\(ii) For $i\in R$, we have $J_R=c_{i,i}/m_{i,i}$.
\(iii) $J_R=\sum_{j\in R}\pi_j E[c(x(t))|x(0)=j]$, $\forall\ t\geq 0$.
Let $e_{i,j}$ be the expected time of visits to $j$ during a first passage from $i$ to $i$. Then $\pi_j=e_{i,j}/m_{i,i}$. Therefore, $J_R=\sum_{j\in R}\pi_j c(j)=\sum_{j\in R}c(j)e_{i,j}/m_{i,i}=c_{i,i}/m_{i,i}$ and thus (ii) holds.
Note that $J_{i}(t)=\sum_{j}c(j)E[\int_0^ t 1(x(s)=j)ds|x(0)=i]t^{-1}$ and $\lim_{t\rightarrow\infty}E[\int_0^ t 1(x(s)=j)ds|x(0)=i]t^{-1}=\pi_j$. By Fatou lemma, it follows that $\lim\inf_{t\rightarrow\infty}J_i(t)\geq J_R$. Thus, if $J_R=\infty$, the limit exists and equals $\infty$, $\forall\ i\in R$. If $J_R<\infty$, then (i) follows from the renewal reward theorem (See [@ross96]).
Next we prove (iii). Note that $E[c(x(t))|x(0)=j]=\sum_{k\in S}p(j,k,t)c(k)$, where $p(j,k,t)=P[x(t)=k|x(0)=j]$. Since $\sum_{j\in R}\pi_j p(j,k,t)=\sum_{j\in S}\pi_j p(j,k,t)=\pi_k$ (noting that $\pi_i=0$ for $i\in S-R$), we have $$\begin{aligned}
&&\sum_{j\in R}\pi_j E[c(x(t))|x(0)=j]=\sum_{j\in R}\pi_j\sum_{k\in S}p(j,k,t)c(k)
\\&=& \sum_{k\in S}c(k)\sum_{j\in R}\pi_j p(j,k,t)= \sum_{k\in S}c(k)\pi_k=\sum_{k\in R}c(k)\pi_k=J_R,\end{aligned}$$ where the interchange of the order of summation is valid as all terms are nonnegative.
[Suppose that $d$ is a $i_0$-standard policy with positive recurrent class $R$. Let $J_R(d)$ and $\pi_i(d)$ be defined as in Proposition \[pro\_5\], then $$J_R(d)=\alpha\sum_{i\in R}\pi_i(d)J_{\alpha}(i,d), \forall\ \alpha>0. \label{eq_standard}$$ ]{}
It follows from (\[eq\_dis\]) and Proposition \[pro\_5\](iii) that $$\begin{aligned}
&&\alpha\sum_{i\in R}\pi_i(d)J_{\alpha}(i,d)=\alpha\sum_{i\in R}\pi_i(d)\int_{0}^{\infty}e^{-\alpha t}E^{d}[c(x(t),d)|x(0)=i]dt
\\&=&\alpha\int_{0}^{\infty}e^{-\alpha t}\left[\sum_{i\in R}\pi_i(d)E^{d}[c(x(t),d)|x(0)=i]\right]dt=J_R(d),\end{aligned}$$ where the interchange of the summation and integration is valid as all terms are nonnegative.
[\[pro\_3\] Assume that $J_{\alpha}^*(i_0)<\infty$, for some $\alpha>0$. Given $i\neq i_0$, assume that there exists a policy $\theta_i$ such that both the expected time and expected cost of a first passage from $i$ to $i_0$ are finite. Then $h_{\alpha}(i)\leq c_{i,i_0}(\theta_i)$, and hence (A2) holds for $i_0$ with $H(i)=c_{i,i_0}(\theta_i)$.]{}
If the process begins in state $i\neq i_0$ and follows policy $\theta_i$, it will reach state $i_0$ at some time in the future, which is denoted by $T$. Let the policy $\psi$ follow $\theta_i$ until $i_0$ is reached, then follow an $\alpha$ discounted optimal policy $f_{\alpha}$.
Then we have $$\begin{aligned}
J_{\alpha}^*(i)&\leq & J_{\alpha}(i,\psi) \nonumber
\\&=&E^{\psi}\left[\int_0^Te^{-\alpha t}c(x(t), a(t))dt|x(0)=i\right]+E^{\psi}\left[e^{-\alpha T}|x(0)=i\right]J_{\alpha}^*(i_0) \nonumber
\\&\leq & E^{\psi}\left[\int_0^Tc(x(t), a(t))dt|x(0)=i\right]+J_{\alpha}^*(i_0) \nonumber
\\&\leq & c_{i,i^0}(\theta_i)+J_{\alpha}^*(i_0). \label{eq_com}\end{aligned}$$ The result follows by subtracting $J_{\alpha}^*(i_0)$ from both sides.
[**Remark:**]{} Proposition \[pro\_3\] gives a way to construct a function $H(i)$. From the remark below Proposition \[pro\_2\], it is known that $c_{i,i_0}(\theta_i)$ is a quite good choice for $H(i)$.
[**Proof of Theorem \[the\_2\]:** ]{} We only need to prove that (A1-3) hold under conditions in Theorem \[the\_2\]. Let $i_0=0$. It follow from (\[eq\_standard\]) that $J_R(d)\geq \alpha\pi_{0}(d)J_{\alpha}(0,d)\geq \alpha\pi_{0}(d)J_{\alpha}^*(0)$. Hence $\alpha J_{\alpha}^*(0)\leq J_R(d)/\pi_{0}(d)=c_{0,0}(d)$. Therefore, (A1) holds. From Proposition \[pro\_3\] we know that (A2) holds with $H(i)=c_{i,0}(d)$ for $i\neq 0$ and $H(0)=0$. Since $J_{\alpha}^*(i)$ is increasing in $i$, it follows that $h_{\alpha}(i)\geq 0$, and hence (A3) holds with $L=0$. It follows from (\[eq\_limit\_h\]) and the fact that $h_{\alpha}(i)$ is increasing in $i$ that $h^*(i)$ is increasing in $i$. $\hfill\square$
Sufficient Conditions for ACOE to Hold
======================================
Proposition 5.11, [@guoher] has given an example to demonstrate that (SEN-C) is not sufficient to claim that the average-cost optimality equality (ACOE) holds, i.e., ACOI might be strict. [@guoher] gives one condition under which the ACOE holds, i.e., the inequality in (\[eq\_acoe\]) is in fact equality. However, in many situations it is hard to verify this condition and even in some cases it fails to hold due to improper choice of the function $H(i)$.
In this section, we give conditions under which the ACOE holds. We first develop some notations. Let $\mathfrak{R}(i,G)$ be the class of policies $\theta$ satisfying $$P_{\theta}(x(t)\in G \mbox{ for some } t>0, \mbox{at least one transition occurs}|x(0)=i)=1,$$ and the expected time $m_{i,G}(\theta)$ of a first passage from $i$ to $G$ (during which at least one transition occurs) is finite. Let $\mathfrak{R}^*(i,G)$ be the class of policies $\theta\in\mathfrak{R}(i,G)$ such that the expected cost $c_{i,G}(\theta)$ of a first passage from $i$ to $G$ (during which at least one transition occurs) is finite. If $G=\{x\}$, then $\mathfrak{R}(i,G)$ is denoted by $\mathfrak{R}(i,x)$ (respectively, $\mathfrak{R}^*(i,G)$ by $\mathfrak{R}^*(i,x)$).
[\[pro\_2\] Assume that the Assumptions (A1-3) hold, and for some state $i$ and nonempty set $G$, there exists a policy $\theta\in\mathfrak{R}(i,G)$ such that $\sum_{j\in G}H(j)P_{\theta}(x(T)=j)<\infty$, where $T$ is the first passage time from $i$ to $G$ and $H$ is the function from (A2). Then for any limit function $h^*$, we have $$h^*(i)\leq c_{i,G}(\theta)-g^*m_{i,G}(\theta)+E_{\theta}[h^*(x(T))|x(0)=i]. \label{eq_recur}$$ ]{}
In a derivation very similar to that in (\[eq\_com\]), we have $$J_{\alpha}^*(i)\leq c_{i,G}(\theta)+E_{\theta}[e^{-\alpha T}J_{\alpha}^*(x(T))|x(0)=i],$$ which can be written as $$h_{\alpha}(i)\leq c_{i,G}(\theta)-\alpha J_{\alpha}^*(i_0)\left(\frac{1-E_{\theta}[e^{-\alpha T}|x(0)=i]}{\alpha}\right)+E_{\theta}[e^{-\alpha T}h_{\alpha}(x(T))|x(0)=i]. \label{eq_com_2}$$ Note that $$\frac{1-E_{\theta}[e^{-\alpha T}|x(0)=i]}{\alpha}=E_{\theta}\left[\int_0^Te^{-\alpha s}ds|x(0)=i\right]. \label{eq_com_1}$$
The term $\int_0^Te^{-\alpha s}ds$ is decreasing in $\alpha$. It follows from monotone convergence theorem that the limit of the left side of (\[eq\_com\_1\]) exists and equals to $E_{\theta}[T|x(0)=i]=m_{i,G}(\theta)$.
Choose a discount factor sequence $\{\alpha_n, n\geq 1\}$ tending to zero such that (\[eq\_limit\_fg\]) and (\[eq\_limit\_h\]) hold. Taking the limit of both sides of (\[eq\_com\_2\]) as $\alpha_n\rightarrow 0^+$ yields $$h^*(i)\leq c_{i,G}(\theta)-g^*m_{i,G}(\theta)+\lim_{n\rightarrow\infty}E_{\theta}[e^{-\alpha_n T}h_{\alpha_n}(x(T))|x(0)=i].$$
Note that $e^{-\alpha_n T}h_{\alpha_n}(x(T))$ converges to $h^*(x(T))$ as $n\rightarrow\infty$. Since $e^{-\alpha_n T}h_{\alpha_n}(x(T))$ is bounded by $\max(L, H(x(T)))$ from (A2) and (A3), and $E_{\theta}[\max(L, H(x(T)))]\leq L+ E_{\theta}H(x(T))=L+\sum_{j\in G}H(j)P_{\theta}(x(T)=j)<\infty$, by dominated convergence theorem it is known that $$\lim_{n\rightarrow\infty}E_{\theta}[e^{-\alpha_n T}h_{\alpha_n}(x(T))|x(0)=i]=E_{\theta}[h^*(x(T))|x(0)=i].$$
Therefore, (\[eq\_recur\]) holds.
Now we give sufficient conditions under which the ACOE holds.
\[the\_1\] Assume that the Assumptions (A1-3) hold, and let $e$ be a stationary policy realizing the minimum in the ACOI. Define the nonnegative discrepancy function $\Phi$ to satisfy $$g^*=c(i,e)+\Phi(i)+\sum_{j\in S}q(j|i,e)h^*(j), i\in S. \label{eq_ave_op_dis}$$
Then $\Phi(i)=0$, and hence the ACOE holds at the particular state $i$ under any of the following conditions:
\(i) There exists a nonempty set $G$ such that $e$ satisfies $e\in \mathfrak{R}(i,G)$ and $\sum_{j\in G}H(j)P_{\theta}(x(T)=j)<\infty$, where $T$ is the first passage time from $i$ to $G$.
\(ii) $e\in\mathfrak{R}(i,i_0)$.
\(iii) The Markov process induced by $e$ is positive recurrent at $i$.
\(iv) $\sum_{j\in S}|q(j|i,a)|H(j)<\infty$ for $a\in A(i)$. This conditions typically hold when the jump size at each state $i$ is bounded and thus there are finite number of $j$ such that $q(j|i,a)>0$ for each $i\in S$.
To prove equality under (i), let the process operate under $e$, and suppress the initial state $i$. Since the first passage time from $i$ to $G$, $T$, is a stopping time such that $E_e[T]=m_{i,G}(e)<\infty$ as $e\in \mathfrak{R}(i,G)$, it follows from Dynkin’s formula (see [@os05]) that $$E^e[h^*(x(T))]=h^*(i)+E^e\left[\int_0^T \sum_{j\in S}q(j|x(s),e)h^*(j)ds\right].$$
From (\[eq\_ave\_op\_dis\]) it is known that $$E_e[h^*(x(T))]=h^*(i)+E^e\left[\int_0^T (g^*-c(x(s),e)-\Phi(x(s))ds\right],$$ and thus $$c_{i,G}(e)-g^*m_{i,G}(e)+E^e\left[\int_0^T\Phi(x(s))ds\right]+E^e[h^*(x(T))]=h^*(i), \label{eq_ave_op_dis_2}$$ which implies that $c_{i,G}(e)<\infty$, and hence $e\in \mathfrak{R}^*(i,G)$. Therefore, we can apply Proposition \[pro\_2\], which yields $$c_{i,G}(e)-g^*m_{i,G}(e)+E^e[h^*(x(T))]\geq h^*(i).$$ Comparing the above equation with (\[eq\_ave\_op\_dis\_2\]) and keeping in mind that $\Phi$ is nonnegative, we know that $\Phi=0$ during the first passage from $i$ to $G$. Specially, we have $\Phi(i)=0$ and thus the ACOE holds at state $i$.
\(ii) follows from (i) by choosing $G=\{i_0\}$ and the fact $h^*(i_0)=0$.
\(iii) follows from (i) by noting that if the Markov process induced by $e$ is positive recurrent at $i$, then $e\in \mathfrak{R}(i,i)$.
\(iv) follows from the same argument in Theorem 5.9 in [@guoher].
[**Remark:**]{} If starting from an arbitrary initial state $i$, in a finite expected amount of time the Markov process induced by $e$ reaches a finite set $G$, then the ACOE holds.
A Queueing Example
==================
[**Example 1.**]{} A single-server, 2-buffer queueing model. Consider a server serving two types of customers: type 1 and type 2 customers. Type 1 and 2 customers form queue 1 and queue 2, respectively. Type 1 and 2 customers arrive according to two independent Poisson processes with parameter $\lambda_1$ and $\lambda_2$, respectively. Buffers of both queues are assumed to be infinitely large. The service times of type 1 and 2 customers are exponentially distributed with parameters $\mu_1$ and $\mu_2$, respectively. While waiting in queue, a type 1 customer may change to a type 2 customer after a random time T, which is exponentially distributed with parameter $\lambda_T$. The holding cost of a customer in queue 1 and 2 per unit time is $h_1$ and $h_2$, respectively. When a type 1 customer upgrades, the cost of transferring from queue 1 to queue 2 is $c$ per unit. The server should decide which buffer to serve to minimize the average cost.
The state can be denoted by $\mathbf{q}=(q_1,q_2)$, where $q_i$ is the length of queue $i$, $i=1,2$. For each state $\mathbf{q}$, we have the corresponding action set $$A(\mathbf{q})=\left\{\begin{array}{ll}\{0\}, & \mbox{ if } \mathbf{q}=(0,0),
\\ \{1\}, & \mbox{ if } \mathbf{q}=(q_1,0), q_1>0,
\\ \{2\}, & \mbox{ if } \mathbf{q}=(0,q_2), q_2>0,
\\ \{1,2\}, & \mbox{ otherwise. }
\end{array}
\right.$$
And the corresponding transition rate is $$q(\mathbf{q}'|\mathbf{q}, 1)=\left\{\begin{array}{ll}\mu_1, & \mbox{ if } \mathbf{q}'=(q_1-1,q_2),
\\ \lambda_1, & \mbox{ if } \mathbf{q}'=(q_1+1,q_2),
\\ \lambda_2, & \mbox{ if } \mathbf{q}'=(q_1,q_2+1),
\\ q_1\lambda_T, & \mbox{ if } \mathbf{q}'=(q_1-1,q_2+1),
\\ -(\mu_1+\lambda_1+\lambda_2+q_1\lambda_T), & \mbox{ if } \mathbf{q}'=(q_1,q_2),
\\ 0, & \mbox{ otherwise;}
\end{array} \mbox{ for } q_1\geq 1,
\right.$$ and $$q(\mathbf{q}'|\mathbf{q}, 2)=\left\{\begin{array}{ll}\mu_2, & \mbox{ if } \mathbf{q}'=(q_1,q_2-1),
\\ \lambda_1, & \mbox{ if } \mathbf{q}'=(q_1+1,q_2),
\\ \lambda_2, & \mbox{ if } \mathbf{q}'=(q_1,q_2+1),
\\ q_1\lambda_T, & \mbox{ if } \mathbf{q}'=(q_1-1,q_2+1),
\\ -(\mu_2+\lambda_1+\lambda_2+q_1\lambda_T), & \mbox{ if } \mathbf{q}'=(q_1,q_2),
\\ 0, & \mbox{ otherwise;}
\end{array} \mbox{ for } q_1\geq 1, q_2\geq 1,
\right.$$
$$q(\mathbf{q}'|\mathbf{q}, 2)=\left\{\begin{array}{ll}\mu_2, & \mbox{ if } \mathbf{q}'=(q_1,q_2-1),
\\ \lambda_1, & \mbox{ if } \mathbf{q}'=(q_1+1,q_2),
\\ \lambda_2, & \mbox{ if } \mathbf{q}'=(q_1,q_2+1),
\\ -(\mu_2+\lambda_1+\lambda_2), & \mbox{ if } \mathbf{q}'=(q_1,q_2),
\\ 0, & \mbox{ otherwise;}
\end{array} \mbox{ for } q_1=0, q_2\geq 1,
\right.$$
Moreover, $q(\mathbf{q}'|\mathbf{q}, 0)=0$.
Let $x(t)=(x_1(t),x_2(t))$ be state at time $t$, and $T_{r}^\pi(t)$ be the total number of transferred customers till time $t$ under policy $\pi$, where $x_i(t)$ is the length of queue $i$ at time $t$, $i=1,2$. The expected discounted cost function under policy $\pi$ in this example can be formulated as $$\begin{aligned}
J_{\alpha}(\mathbf{q},\pi)&=&E_{\mathbf{q}}^{\pi}\left[\int_{0}^{\infty}e^{-\alpha t}(dT_{r}^\pi(t)+(h_1x_1(t)+h_2x_2(t))dt)\right]
\\&=& \int_{0}^{\infty}e^{-\alpha t}E_{\mathbf{q}}^{\pi}(h_1x_1(t)+h_2x_2(t)+c\lambda_T x_1(t))dt.\end{aligned}$$ Hence, the cost rate function is $c(\mathbf{q},1)=c(\mathbf{q},2)=c(\mathbf{q})=h_1q_1+h_2q_2+cq_1\lambda_T$.
We have the following result for Example 1.
[Suppose that $\lambda_1+ \lambda_2< \min(\mu_1,\mu_2)$. There exists an average-cost optimal stationary policy for Example 1 and the ACOE holds. ]{}
We apply Theorem \[the\_2\] by proving that
\(i) $J_{\alpha}^{*}(\mathbf{q})$ is increasing in $\mathbf{q}$;
\(ii) The priority service (PS) policy is a $\mathbf{0}=(0,0)$-standard policy. The PS policy specifies that the server will always choose a customer in (nonempty) queue 2 to serve at each decision epoch. If queue 2 is empty, the serve will serve customers in queue 1, if there is any. If the server is serving a type 1 customer when a type 2 customer arrives, the type 1 customer is pushed back to queue 1 and the server begins to serve the type 2 customers. The interrupted type 1 customer will resume or repeat its service if the server is available to serve type 1 customers. If the system is empty, the server will be idle.
To prove (i), denote the optimal stationary policy by $\pi^*$. At state $(q_1,q_2)$, we add a virtual customer of type 1 at queue 1. He has the same transfer rate as the ordinary customer of type 1. However, he has no holding cost and transferring cost. For this queueing system $G(q_1,q_2;1,0)$, policy $\pi^*$ can still be used and by comparing each realized trajectory we know that the resulting expected discounted cost $C(G(q_1,q_2;1,0))$ is less than $J_{\alpha}^*(q_1+1,q_2)$. On the other hand, the queueing system $G(q_1,q_2;1,0)$ is in fact a queueing system with state $(q_1,q_2)$ and since policy $\pi^*$ for system $G(q_1,q_2;1,0)$ might not be an optimal policy for queueing system with state $(q_1,q_2)$ we have that $C(G(q_1,q_2;1,0))\geq J^*(q_1,q_2)$. Therefore, $J_{\alpha}^*(q_1+1,q_2)\geq J_{\alpha}^*(q_1,q_2)$ and thus $J_{\alpha}^*(q_1,q_2)$ is increasing in $q_1$. Similarly, $J_{\alpha}^*(q_1,q_2)$ is increasing in $q_2$. Thus, $J_{\alpha}^{*}(\mathbf{q})$ is increasing in $\mathbf{q}$.
Let $\epsilon=\mu_2-\lambda_1-\lambda_2>0$ and $d$ be the PS policy. From [@xhz08] it is known that the Markov process induced by $d$ is ergodic, and thus $m_{\mathbf{q},\mathbf{0}}(d)<\infty$, $\forall\ \mathbf{q}\in S$. Next we prove that $c_{\mathbf{q},\mathbf{0}}(d)<\infty$, $\forall\ \mathbf{q}\in S$.
Inspired by [@xhz08], we choose the Lyapunov function $r(\mathbf{q})=Kr_1^{q_1}r_2^{q_2}$ and then apply Lemma \[lem\_1\] with $H^*=\{\mathbf{0}\}$. Here the constants $K$, $r_1$, $r_2$ are left to be specified later. (\[eq\_lem\_2\]) requires that $$\begin{aligned}
&& c(\mathbf{q})+Kr_1^{q_1}r_2^{q_2}\left[\mu_2\left(\frac{1}{r_2}-1\right)+\lambda_1(r_1-1)+\lambda_2 (r_2-1)+q_1\lambda_T\left(\frac{r_2}{r_1}-1\right)\right]\leq 0, \nonumber \\ && \hspace{9cm}q_1\geq 0,q_2\geq 1,\label{eq_con_1}\end{aligned}$$ and $$\begin{aligned}
&& c(q_1,0)+Kr_1^{q_1}\left[\mu_1\left(\frac{1}{r_1}-1\right)+\lambda_1(r_1-1)+\lambda_2 (r_2-1)+q_1\lambda_T\left(\frac{r_2}{r_1}-1\right)\right]\leq 0, \nonumber
\\ &&\hspace{9cm} q_1\geq 1.\label{eq_con_2}\end{aligned}$$
Choose $r_2=r_1$ and $r_1>1$ such that $r_1>\frac{\min(\mu_1,\mu_2)}{\lambda_1+\lambda_2}$. Denote $\delta=\left(\frac{\min(\mu_1,\mu_2)}{r_1}-\lambda_1-\lambda_2\right)\cdot(r_1-1)$. By the choice of $r_1$ it is known that $\delta>0$. Choose $K$ such that $K(r_1-1)>\max(h_1+c\lambda_T, h_2)$. Therefore, we have $$\begin{aligned}
&&c(\mathbf{q})+Kr_1^{q_1}r_2^{q_2}\left[\mu_2\left(\frac{1}{r_2}-1\right)+\lambda_1(r_1-1)+\lambda_2 (r_2-1)+q_1\lambda_T(\frac{r_2}{r_1}-1)\right]
\\&\leq& c(\mathbf{q})-Kr_1^{q_1+q_2}\delta
\\&\leq&h_1q_1+h_2q_2+cq_1\lambda_T -K(r_1-1)\delta(q_1+q_2)
\\&\leq& 0,\end{aligned}$$ and thus (\[eq\_con\_1\]) holds. Similarly, (\[eq\_con\_2\]) also holds. Therefore, it follows from Lemma \[lem\_1\] that $c_{\mathbf{q},\mathbf{0}}(d)<0$ for $\mathbf{q}\neq \mathbf{0}$. Besides, it is easily seen that $$c_{\mathbf{0},\mathbf{0}}=\frac{\lambda_1}{\lambda_1+\lambda_2}c_{(1,0),\mathbf{0}}(d)+\frac{\lambda_2}{\lambda_1+\lambda_2}c_{(0,1),\mathbf{0}}(d)<\infty.$$ Therefore, the PS policy $d$ is a $\mathbf{0}$-standard policy, and thus (ii) is proved.
It follows from Theorem \[the\_2\] that an average-cost optimal stationary policy exists and the ACOI is satisfied. Besides, condition (iv) in Theorem \[the\_1\](iv) is satisfied as there are only finite $j$ such that $q(j|i,a)\neq 0$ for any $i\in S, a\in A(i)$. Therefore, the ACOE holds for any $ i\in S$.
[**Remark:**]{} From the proof we know that the result still holds if the cost rate function is increasing and polynomial in $\mathbf{q}$.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors gratefully acknowledge that this work was supported by NSFC under grant 71201154, NSFC major program (Grant No. 71090401/71090400) and CPSF under grants 2012M521260 and 2013T60627.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we study the effects of different prior and likelihood choices for Bayesian matrix factorisation, focusing on small datasets. These choices can greatly influence the predictive performance of the methods. We identify four groups of approaches: Gaussian-likelihood with real-valued priors, nonnegative priors, semi-nonnegative models, and finally Poisson-likelihood approaches. For each group we review several models from the literature, considering sixteen in total, and discuss the relations between different priors and matrix norms. We extensively compare these methods on eight real-world datasets across three application areas, giving both inter- and intra-group comparisons. We measure convergence runtime speed, cross-validation performance, sparse and noisy prediction performance, and model selection robustness. We offer several insights into the trade-offs between prior and likelihood choices for Bayesian matrix factorisation on small datasets—such as that Poisson models give poor predictions, and that nonnegative models are more constrained than real-valued ones.'
author:
- Thomas Brouwer
- |
Pietro Lió\
Computer Laboratory, University of Cambridge, United Kingdom\
bibliography:
- 'bibliography.bib'
title: Prior and Likelihood Choices for Bayesian Matrix Factorisation on Small Datasets
---
Introduction
============
Matrix factorisation methods have become very popular in recent years, and used for many applications such as collaborative filtering [@Mnih2008; @Chen2009] and bioinformatics [@Gonen2012; @Brouwer2017a]. Given a matrix relating two entity types, such as movies and users, matrix factorisation decomposes that matrix into two smaller so-called factor matrices, such that their product approximates the original one. Matrix factorisation is often used for predicting missing values in the datasets, and analysing the resulting factor values to identify biclusters or features.
Most models can be categorised as being either non-probabilistic, such as the popular models by [@Lee2000], or Bayesian. The former seek to minimise an error function (such as the squared error) between the original matrix and the approximation. In contrast, Bayesian variants treat the two smaller matrices as random variables, place prior distributions over them, and find the posterior distribution over their values after observing the data. A likelihood function, usually Gaussian, is used to capture noise in the dataset. Previous work [@Brouwer2017a] has demonstrated that Bayesian variants are much better for predictive tasks than non-probabilistic versions, which tend to overfit to noise and sparsity.
Matrix factorisation techniques can also be grouped by their constraints on the values in the factor matrices. Firstly, many approaches place no constraints, using real-valued factor matrices (commonly done in the Bayesian literature [@Salakhutdinov2008; @Gonen2012]). Instead, we could constrain them to be nonnegative (as is popular in the non-probabilistic literature [@Lee2000; @Tan2013]), limiting its applicability to nonnegative datasets, but making it easier to interpret the factors and potentially also making the method more robust to overfitting. Thirdly, semi-nonnegative variants constrain one factor matrix to be nonnegative, leaving the other real-valued [@FeiWangTaoLi2008; @Ding2010]. Finally, some versions work only on count data.
In the Bayesian setting, the first three groups of methods all generally use a Gaussian likelihood for noise, and place either real-valued or nonnegative priors over the matrices. For the former, Gaussian is a common choice [@Salakhutdinov2008; @Gonen2012; @Virtanen2011; @Virtanen2012], and for the latter options include the exponential distributions [@Schmidt2009]. The fourth group uses a Poisson likelihood to capture count data [@Gopalan2014; @Gopalan2015; @Hu2015]. These models are often extended by using complicated hierarchical prior structures over the factor matrices, giving additional behaviour (such as automatic model selection). This paper offers the first systematic comparison between different Bayesian variants of matrix factorisation. Similar comparisons have been provided in other fields, such as for the regression parameter in Bayesian model averaging [@Ley2009; @Eicher2011], which demonstrated that the choice of prior can greatly influence the predictive performance of these models. However, a similar study for Bayesian matrix factorisation is still missing. More strikingly, many papers that introduce new matrix factorisation models do not provide a thorough comparison with competing approaches, or popular non-probabilistic ones such as [@Lee2000]—for example, the seminal paper by [@Salakhutdinov2008] compares their approach with only one other matrix factorisation method; although [@Gopalan2015] compares with three others.
We give an overview of the different approaches that can be found in the literature, including hierarchical priors, and then study the effects of these different Bayesian prior and likelihood choices. We aim to make general statements about the behaviour of the four different groups of methods on small real-world datasets (up to a million observed entries), by considering eight datasets across three different applications—four drug effectiveness datasets, two collaborative filtering datasets, and two methylation expression datasets. Our experiments consider convergence speed, cross-validation performance, sparse and noisy prediction performance, and model selection effectiveness. This study offers novel insights into the differences between the four approaches, and the effects of popular hierarchical priors.
We note that there is a rich literature of Bayesian nonparametric matrix factorisation models, which learn the size of the factor matrices automatically. However, these models often require complex inference approaches to find good solutions, and hence their predictive performance is more determined by the inference method than the precise model choices (such as likelihood and prior). In this paper we therefore focus on parametric matrix factorisation models, to isolate the effects of likelihood and prior choices.
Finally, we acknowledge that the models we study were generally introduced for a specific application domain, and that this makes it hard to make general statements about the behaviour of these methods on different datasets. However, we believe that it is essential to provide a cross-application comparison of the different approaches, as this teaches us valuable lessons for the applications studied, and they are likely to apply to different areas as well. The lack of other studies exploring the trade-offs between likelihood and prior choices for Bayesian matrix factorisation make this a novel and essential study.
Bayesian Matrix Factorisation
=============================
In this section we introduce the different matrix factorisation models that we study. Formally, the problem of matrix factorisation can be defined as follows. Given an observed matrix $ {\boldsymbol R}\in \mathbb{R}^{I \times J} $, we want to find two smaller matrices $ {\boldsymbol U}\in \mathbb{R}^{I \times K} $ and $ {\boldsymbol V}\in \mathbb{R}^{J \times K} $, each with $K$ so-called factors (columns), to solve $ {\boldsymbol R}= {\boldsymbol U}{\boldsymbol V}^T + {\boldsymbol E}$, where noise is captured by matrix $ {\boldsymbol E}\in \mathbb{R}^{I \times J} $. Some entries in ${\boldsymbol R}$ may be unobserved, as given by the set $ \Omega = \left\{ (i,j) \text{ $ \vert $ $ R_{ij} $ is observed} \right\} $. These entries can then be predicted by ${\boldsymbol U}{\boldsymbol V}^T$.
In the Bayesian treatment of matrix factorisation, we express a likelihood function for the observed data that captures noise (such as Gaussian or Poisson). We treat the latent matrices as random variables, placing prior distributions over them. A Bayesian solution for matrix factorisation can then be found by inferring the posterior distribution $p({\boldsymbol \theta}|D)$ over the latent variables ${\boldsymbol \theta}$ (${\boldsymbol U}$, ${\boldsymbol V}$, and any additional random variables in our model), given the observed data $ D = \lbrace R_{ij} \rbrace_{i,j \in \Omega} $. This posterior distribution is often intractable to compute exactly, but several methods exist to approximate it (see Section \[Inference\]).
In Section \[Models\] we introduce a wide range of models from the literature, and categorise them into four groups. The model names are highlighted in bold in the text.
Probability Distributions
-------------------------
We introduce all notation and probability distributions in the paper below.
$\text{diag}(\boldsymbol \lambda^{-1})$ is a diagonal matrix with entries $\lambda_1^{-1}, .., \lambda_K^{-1}$ on the diagonal.
$ \mathcal{N} (x|\mu,\tau^{-1}) = \tau^{\frac{1}{2}} (2\pi)^{-\frac{1}{2}} \exp \left\{ -\frac{\tau}{2} (x - \mu)^2 \right\} $ is a Gaussian distribution with precision $ \tau $.
$ \mathcal{N} (\boldsymbol x|\boldsymbol \mu,\boldsymbol \Sigma) = \vert \boldsymbol \Sigma \vert^{-\frac{1}{2}} (2\pi)^{-\frac{K}{2}} \exp \left\{ -\frac{1}{2} (\boldsymbol x - \boldsymbol \mu)^T \boldsymbol \Sigma^{-1} (\boldsymbol x - \boldsymbol \mu) \right\} $ is a $K$-dimensional multivariate Gaussian distribution.
$ \mathcal{G} (\tau | \alpha_{\tau}, \beta_{\tau} ) = \frac{{\beta_{\tau}}^{\alpha_{\tau}}}{\Gamma(\alpha_{\tau})} x^{\alpha_{\tau} -1} e^{- \beta_{\tau} x} $ is a Gamma distribution, where $ \Gamma(x) = \int_{0}^{\infty} x^{t-1} e^{-x} dt $ is the gamma function.
$ \mathcal{NIW} ( \boldsymbol{\mu}, \boldsymbol{\Sigma} | \boldsymbol{\mu_0}, \beta_0, \nu_0, \boldsymbol{W_0} ) = \mathcal{N}(\boldsymbol \mu | \boldsymbol{\mu_0}, \frac{1}{\beta_0} \text{\textbf I} ) \mathcal{W}^{-1} ( \boldsymbol \Sigma | \nu_0, \boldsymbol{W_0} ) $ is a normal-inverse Wishart distribution, where $\mathcal{W}^{-1} ( \boldsymbol \Sigma | \nu_0, \boldsymbol{W_0} ) $ is an inverse Wishart distribution, and $\textbf I$ the identity matrix.
$\mathcal{L} ( x | \mu, \rho ) = \frac{1}{2 \rho} \exp \left\{ -\frac{ |x - \mu| }{\rho} \right\} $ is a Laplace distribution.
$ \mathcal{IG}(x|\mu, \lambda) = \frac{ \lambda }{2 \pi x^3} \exp \left\{ - \frac{\lambda (x - \mu)^2}{2 \mu^2 x} \right\} $ is an inverse Gaussian.
$ \mathcal{E} ( x | \lambda ) = \lambda \exp \left\{ - \lambda x \right\} u(x) $ is an exponential distribution, where $u(x)$ is the unit step function. $$\mathcal{TN} ( x | \mu, \tau ) = \left\{
\begin{array}{ll}
\displaystyle \frac{ \sqrt{ \frac{\tau}{2\pi} } \exp \left\{ -\frac{\tau}{2} (x - \mu)^2 \right\} }{ 1 - \Phi ( - \mu \sqrt{\tau} )} & \mbox{if } x \geq 0 \\
0 & \mbox{if } x < 0
\end{array}
\right.$$ is a truncated normal: a normal distribution with zero density below $ x = 0 $ and renormalised to integrate to one. $ \Phi(\cdot) $ is the cumulative distribution function of $ \mathcal{N}(0,1) $.
Category Name Likelihood Prior ${\boldsymbol U}$ Prior ${\boldsymbol V}$ Hierarchical prior
------------- ------------------ --------------------------------------------------------------------------- -------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------
Real-valued GGG $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $\mathcal{N} ( {\boldsymbol{U_i}}| \boldsymbol 0, \lambda^{-1} \text{\textbf I} )$ $\mathcal{N} ( {\boldsymbol{V_j}}| \boldsymbol 0, \lambda^{-1} \text{\textbf I} )$ -
GGGU $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $\mathcal{N} ( {\boldsymbol{U_i}}| \boldsymbol 0, \lambda^{-1} \text{\textbf I} )$ $\mathcal{N} ( {\boldsymbol{V_j}}| \boldsymbol 0, \lambda^{-1} \text{\textbf I} )$ -
GGGA $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $\mathcal{N} ( {\boldsymbol{U_i}}| \boldsymbol 0, \text{diag}(\boldsymbol \lambda^{-1}) )$ $\mathcal{N} ( {\boldsymbol{V_j}}| \boldsymbol 0, \text{diag}(\boldsymbol \lambda^{-1}) )$ $\lambda_k \sim \mathcal{G} ( \alpha_0, \beta_0 )$
GGGW $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $\mathcal{N} ( {\boldsymbol{U_i}}| \boldsymbol{\mu_U}, \boldsymbol{\Sigma_U})$ $\mathcal{N} ( {\boldsymbol{V_j}}| \boldsymbol{\mu_V}, \boldsymbol{\Sigma_V})$ $(\boldsymbol{\mu_U}, \boldsymbol{\Sigma_U}) \text{ and } (\boldsymbol{\mu_V}, \boldsymbol{\Sigma_V}) $
$\sim \mathcal{NIW} ( \boldsymbol{\mu_0}, \beta_0, \nu_0, \boldsymbol{W_0} )$
GLL $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $\mathcal{L} ( U_{ik} | 0, \eta ) $ $\mathcal{L} ( V_{jk} | 0, \eta ) $ -
GLLI $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $\mathcal{L} ( U_{ik} | 0, \eta^U_{ik} ) $ $\mathcal{L} ( V_{jk} | 0, \eta^V_{jk} ) $ $\eta^U_{ik} \text{ and } \eta^V_{jk} \sim \mathcal{IG}(\mu, \lambda) $
GVG $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $p({\boldsymbol U}) \propto $ $\mathcal{N} ( {\boldsymbol{V_j}}| \boldsymbol 0, \lambda^{-1} \text{\textbf I} )$ -
$\exp \lbrace - \gamma \det ({\boldsymbol U}^T {\boldsymbol U}) \rbrace$
Nonnegative GEE $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $\mathcal{E} ( U_{ik} | \lambda )$ $\mathcal{E} ( V_{jk} | \lambda )$ -
GEEA $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $\mathcal{E} ( U_{ik} | \lambda_k )$ $\mathcal{E} ( V_{jk} | \lambda_k )$ $\lambda_k \sim \mathcal{G} ( \alpha_0, \beta_0 )$
GTT $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $\mathcal{TN} ( U_{ik} | \mu_U, \tau_U )$ $\mathcal{TN} ( V_{jk} | \mu_V, \tau_V )$ -
GTTN $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $\mathcal{TN} ( U_{ik} | \mu_U, \tau_U )$ $\mathcal{TN} ( V_{jk} | \mu_V, \tau_V )$ $p(\mu^U_{ik}, \tau^U_{ik} | \mu_{\mu}, \tau_{\mu}, a, b) \propto $
$\frac{1}{\sqrt{\tau^U_{ik}}} \left( 1 - \Phi ( - \mu^U_{ik} \sqrt{\tau^U_{ik}} ) \right)$
$\mathcal{N} (\mu^U_{ik} | \mu_{\mu}, \tau_{\mu}^{-1} ) \mathcal{G} (\tau^U_{ik} | a, b)$
G$\text{L}^2_1 $ $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $p({\boldsymbol U}) \propto \exp $ $p({\boldsymbol V}) \propto \exp $ -
$ \lbrace -\frac{\lambda}{2} \sum_i ( \sum_k U_{ik} )^2 \rbrace $ $ \lbrace -\frac{\lambda}{2} \sum_j ( \sum_k V_{jk} )^2 \rbrace $
with $U_{ik} \geq 0$ with $V_{jk} \geq 0$
Semi- GEG $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ $\mathcal{E} ( U_{ik} | \lambda )$ $\mathcal{N} ( {\boldsymbol{V_j}}| \boldsymbol 0, \lambda^{-1} \text{\textbf I} )$ -
nonnegative GVnG $\mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$ GVG with $U_{ik} \geq 0$ $\mathcal{N} ( {\boldsymbol{V_j}}| \boldsymbol 0, \lambda^{-1} \text{\textbf I} )$ -
Poisson PGG $\mathcal{P} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}})$ $\mathcal{G} ( U_{ik} | a, b )$ $\mathcal{G} ( V_{jk} | a, b )$ -
PGGG $\mathcal{P} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}})$ $\mathcal{G} ( U_{ik} | a, h^U_i )$ $\mathcal{G} ( V_{jk} | a, h^V_j )$ $h^U_i \text{ and } h^V_j \sim \mathcal{G} (a', \frac{a'}{b'})$
Models {#Models}
------
There are three types of choices we make that determine the type of matrix factorisation model we use: the likelihood function, the priors we place over the factor matrices ${\boldsymbol U}$ and ${\boldsymbol V}$, and whether we use any further hierarchical priors. We have identified four different groups of Bayesian matrix factorisation approaches based on these choices: Gaussian-likelihood with real-valued priors, nonnegative priors (constraining the matrices ${\boldsymbol U}, {\boldsymbol V}$ to be nonnegative), semi-nonnegative models (constraining one of the two factor matrices to be nonnegative), and finally Poisson-likelihood approaches. Models within each group use different priors and hierarchical priors, and many choices can be found in the literature. In this paper we consider a total of sixteen models, as summarised in Table \[overview\_bmf\_models\]. We have focused on fully conjugate models (meaning the prior and likelihood are in the same family of distributions) to ensure inference for each model is guaranteed to work well, so that all performance differences in Section \[Experiments\] come entirely from the choice of likelihood and priors.
The first three groups all use a Gaussian likelihood for noise, by assuming each value in ${\boldsymbol R}$ comes from the product of ${\boldsymbol U}$ and ${\boldsymbol V}$, $R_{ij} \sim \mathcal{N} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1} )$, with Gaussian noise added of precision $\tau$, for which we use a Gamma prior $\mathcal{G} (\tau | \alpha_{\tau}, \beta_{\tau} )$. The last group instead opt for a Poisson likelihood, $ R_{ij} \sim \mathcal{P} (R_{ij} | {\boldsymbol{U_i}}{\boldsymbol{V_j}})$. This only works for nonnegative count data, with $ {\boldsymbol R}\in \mathbb{N}^{I \times J} $, but has been studied extensively in the literature due to the popularity and prevalence of datasets like the Netflix Challenge.
#### Real-valued matrix factorisation
The most common approach is to use independent zero-mean Gaussian priors for ${\boldsymbol U}, {\boldsymbol V}$ [@Salakhutdinov2008; @Gonen2012; @Virtanen2011; @Virtanen2012], which gives rise to the **GGG** model. The **GGGU** model is identical but uses a univariate posterior for inference (see supplementary materials).
The first hierarchical model (**GGGA**) uses the Bayesian automatic relevance determination (ARD) prior, which helps with model selection. The main idea is to replace the $\lambda$ hyperparameter by a factor-specific variable $\lambda_k$, which has a further Gamma prior. This causes all entries in columns of ${\boldsymbol U}$ and ${\boldsymbol V}$ to go further to zero if only a few values in that column are high, effectively making the factor inactive. This prior has been used for real-valued [@Virtanen2011; @Virtanen2012] and nonnegative matrix factorisation [@Tan2013].
Another hierarchical model (**GGGW**) was introduced in the seminal paper of [@Salakhutdinov2008]. Instead of assuming independence of each entry in ${\boldsymbol U}, {\boldsymbol V}$, we assume each row of ${\boldsymbol U}$ comes from a multivariate Gaussian with row mean $\boldsymbol{\mu_U}$ and covariance $\boldsymbol{\Sigma_U}$, and similarly for ${\boldsymbol V}$. We then place a further Normal-Inverse Wishart prior over these parameters.
An alternative to the Gaussian prior is to use the Laplace distribution [@Jing2015], which has a much more pointy distribution than Gaussian around $x=0$. This leads to more sparse solutions, as more factors are set to low values. The basic model (**GLL**) can be extended with a hierarchical Inverse Gaussian prior over the $\eta$ parameter (**GLLI**), which they claim helps with variable selection.
The final model (**GVG**) was introduced by [@Arngren2011]. They used a volume prior for the ${\boldsymbol U}$ matrix, with density $p({\boldsymbol U}) \propto \exp \lbrace - \gamma \det ({\boldsymbol U}^T {\boldsymbol U}) \rbrace $. The $\gamma$ hyperparameter determines the strength of the volume penalty (higher means stronger prior).
#### Nonnegative matrix factorisation
These models all place nonnegative prior distributions over entries in ${\boldsymbol U}$ and ${\boldsymbol V}$, and as a result can only deal with nonnegative datasets.
[@Schmidt2009] introduced a model using exponential priors over the factor matrices (**GEE**). This model can also be extended with ARD [@Brouwer2017b] (**GEEA**). Another option is to use the truncated normal distribution (**GTT**), which can also be extended by placing a hierarchical prior over the mean and precision $\mu_U, \tau_U, \mu_V, \tau_V$ (**GTTN**), as done by [@MikkelN.Schmidt2009]. This nontrivial prior cannot be sampled from directly, but will be useful for inference.
Finally, we can use a prior inspired by the $\text{L}^2_1$ norm for both ${\boldsymbol U}$ and ${\boldsymbol V}$ (**G$\boldsymbol{\text{L}^2_1}$**), as we will discuss in Section \[PriorsNorms\].
#### Semi-nonnegative matrix factorisation
Instead of forcing nonnegativity on both factor matrices, we could place this constraint on only one, as was done in [@FeiWangTaoLi2008; @Ding2010]. In the Bayesian setting we place a real-valued prior over one matrix, and a nonnegative prior over the other. The major advantage is that we can handle real-valued datasets, while still enforcing some nonnegativity. However, we will see in Section \[Experiments\] that its performance is identical to the real-valued approaches.
Firstly we can use an exponential prior for entries in ${\boldsymbol U}$, and a Gaussian for ${\boldsymbol V}$, effectively combining the GGG and GEE models into one (**GEG**). Another semi-nonnegative model (**GVnG**) comes from constraining the volume prior in the GVG model to also be nonnegative: $p({\boldsymbol U}) = 0$ if any $U_{ik} < 0$.
#### Poisson likelihood
The standard Poisson matrix factorisation model (**PGG**) uses independent Gamma priors over the entries in ${\boldsymbol U}$ and ${\boldsymbol V}$, with hyperparameters $a, b$ [@Gopalan2014; @Gopalan2015; @Hu2015]. This model can also be extended with a hierarchical prior (**PGGG**), by replacing $b$ with $h^U_i, h^V_j$ and placing a further Gamma prior over these parameters [@Gopalan2015].
Priors and Norms {#PriorsNorms}
================
The prior distributions in Bayesian models act as a regulariser that prevents us from overfitting to the data, preventing poor predictive performance. We can write out the expression of the log posterior of the parameters, which for a Gaussian likelihood and no hierarchical priors becomes $$\begin{aligned}
{1}
& \log p({\boldsymbol \theta}|D) = \log p(D|{\boldsymbol \theta}) + \log p({\boldsymbol \theta}) + C_1 \\
&\hspace{25pt} = {\sum_{(i,j) \in \Omega}}\log p(R_{ij}|{\boldsymbol{U_i}}{\boldsymbol{V_j}}, \tau^{-1}) + \log p({\boldsymbol U}, {\boldsymbol V}) + C_2 \\
&\hspace{25pt} = - \frac{\tau}{2} {\sum_{(i,j) \in \Omega}}( R_{ij} - {\boldsymbol{U_i}}{\boldsymbol{V_j}})^2 + \log p({\boldsymbol U}, {\boldsymbol V}) + C_3
\end{aligned}$$ for some constants $C_i$. Note that this last expression is simply the negative Frobenius norm (squared error) of the training fit, plus a regularisation term over the matrices ${\boldsymbol U}, {\boldsymbol V}$. This training error is frequently used in the nonprobabilistic matrix factorisation literature [@Lee2000; @Pauca2004; @Pauca2006], where different regularisation terms are used. These are often based on row-wise **matrix norms**, such as $$\begin{aligned}
{2}
& \text{L}_1 = \sum_{i=1}^I \sum_{k=1}^K U_{ik}
\quad\quad && \text{L}_2 = \sum_{i=1}^I \sqrt{\sum_{k=1}^K U_{ik}} \\
& \text{L}^2_1 = \sum_{i=1}^I (\sum_{k=1}^K U_{ik})^2
\quad\quad && \text{L}^2_2 = \sum_{i=1}^I \sum_{k=1}^K U_{ik}^2
\end{aligned}$$ This offers some interesting insights: the $\text{L}^2_2$ norm is equivalent to an independent Gaussian prior (GGG), due to the square in the exponential of the Gaussian prior; the $\text{L}_1$ norm is equivalent to a Laplace prior distribution (GLL); if we constrain ${\boldsymbol U}, {\boldsymbol V}$ to be nonnegative then the $\text{L}_1$ norm is equivalent to an exponential prior distribution (GEE); and finally, the $\text{L}^2_1$ norm can be formulated as a nonnegative prior distribution, which we use for the $\text{GL}^2_1$ model (see Table \[overview\_bmf\_models\]).
In other words, the type of priors chosen for Bayesian matrix factorisation determine the type of regularisation that we add to the model. Additionally, we can use hierarchical priors to model further desired behaviour (such as ARD).
Model Discussion
================
Inference {#Inference}
---------
In this paper we use Gibbs sampling (see Section \[Inference\]), because it tends to be very accurate at finding the true posterior [@Brouwer2017b], but other methods like variational Bayesian inference are also possible. The Gibbs sampling algorithms, together with their time complexities, are given in the supplementary materials.
Hyperparameters {#Hyperparameters}
---------------
![Plots of the prior distributions with hyperparameters from Section \[Hyperparameters\].[]{data-label="priors_plot"}](priors.png){width="\columnwidth"}
The hyperparameter values we choose for each model can influence their performance, especially when the data is sparse. The hierarchical models try to automatically choose the correct values, by placing a prior over the original hyperparameters. This introduces new hyperparameters, but the models are generally less sensitive to these.
However, in our experience even the models without hierarchical priors are not very sensitive to this choice, as long as we use fairly weak priors. In particular, we used $\lambda = 0.1$ (GGG, GGGU, GEE, GTT, $\text{GL}^2_1$, GEG), $\eta = \sqrt{10}$ (GLL), and $a = 1, b = 1$ (PGG). The distributions with these hyperparameter values are plotted in Figure \[priors\_plot\].
{width="0.255\columnwidth"} {width="0.255\columnwidth"} {width="0.255\columnwidth"} {width="0.255\columnwidth"} {width="0.255\columnwidth"} {width="0.255\columnwidth"} {width="0.255\columnwidth"} {width="0.255\columnwidth"}
For the other models we used: $\alpha_{\tau} = \beta_{\tau} = 1 $ (Gaussian likelihood); $\alpha_0 = \beta_0 = 1 $ (GGGA, GEEA); $\boldsymbol{\mu_0} = \boldsymbol{0}, \beta_0 = 1, \nu_0 = K, \boldsymbol{W_0} = \text{\textbf{I}} $ (GGGW); $\mu = \lambda = K $ (GLLI), $ \mu_{\mu} = 0, \tau_{\mu} = 0.1, a = b = 1 $ (GTTN), $a = a' = b' = 1$ (PGGG).
We did find that the volume prior models (GVG, GVnG) were very sensitive to the hyperparameter choice $\gamma$. The following values were chosen by trying a range on each dataset and choosing the best one: $ \gamma = 10^{\lbrace-30,-20,-10,-10,0,0,0,0\rbrace}$ for {GDSC,CTRP,CCLE $IC_{50}$,$EC_{50}$,MovieLens 100K,1M,GM,PM}.
Software
--------
Implementations of all models, datasets, and experiments, are available at <https://github.com/Anonymous/>.
Datasets
========
We conduct our experiments on a total of eight real-world datasets across three different applications, allowing us to see whether our observations on one dataset or application also hold more generally. We will focus on one or two datasets at a time for more specific experiments. Also note that we make sure all datasets contain only positive integers, so that we can compare all four groups of Bayesian matrix factorisation approaches.
Dataset Rows Columns Fraction obs.
----------------- ------ --------- ---------------
GDSC $IC_{50}$ 707 139 0.806
CTRP $EC_{50}$ 887 545 0.801
CCLE $IC_{50}$ 504 24 0.965
CCLE $EC_{50}$ 502 24 0.632
MovieLens 100K 943 1473 0.072
MovieLens 1M 6040 3503 0.047
Gene body meth. 160 254 1.000
Promoter meth. 160 254 1.000
: Overview of the four drug sensitivity, two MovieLens, and two methylation datasets, giving the number of rows (cell lines, users, genes), columns (drugs, movies, patients), and the fraction of entries that are observed.[]{data-label="summary_datasets"}
The first comes from bioinformatics, in particular predicting missing values in drug sensitivity datasets, each detailing the effectiveness ($IC_{50}$ or $EC_{50}$ values) of a range of drugs on different cancer and tissue types (cell lines). We consider the Genomics of Drug Sensitivity in Cancer (GDSC v5.0 [@Yang2013], $IC_{50}$), Cancer Therapeutics Response Portal (CTRP v2 [@Seashore-Ludlow2015], $EC_{50}$), and Cancer Cell Line Encyclopedia (CCLE [@Barretina2012], both $IC_{50}$ and $EC_{50}$) datasets. We preprocessed these datasets by: undoing the natural log transform of the GDSC dataset; capping high values to 100 for GDSC and CTRP; and then casting them as integers. We also filtered out rows and columns with only one or two observed datapoints.
The second application is collaborative filtering, where we are given movie ratings for different users (one to five stars) and we wish to predict the number of stars a user will give to an unseen movie. We use the MovieLens 100K and 1M datasets [@Harper2015], with 100,000 and 1,000,000 ratings respectively.
Finally, another bioinformatics application, this time looking at methylation expression profiles [@Koboldt2012]. These datasets give the amount of methylation measured in either the body region of 160 breast cancer driver genes (gene body methylation) or the promoter region (promoter methylation) for 254 different patients. We multiplied all values by twenty and cast them as integers.
The datasets are summarised in Table \[summary\_datasets\], and the distribution of values for each dataset is visualised in Figure \[dataset\_plots\]. This shows us that the drug sensitivity datasets tend to be bimodal, whereas the MovieLens and methylation datasets are more normally distributed. We can also see that the MovieLens datasets tend to be large and sparse, whereas the others are well-observed and relatively small.
Experiments {#Experiments}
===========
We conducted experiments to compare the four different groups of approaches. In particular, we measured their convergence speed, cross-validation performance, sparse prediction performance, and model selection effectiveness. We sometimes focus on a selection of the methods for clarity. To make the comparison complete, we also added a popular non-probabilistic nonnegative matrix factorisation model (NMF) [@Lee2000] as a baseline. The results are discussed in Section \[Discussion\].
Convergence
-----------
Firstly we compared the convergence speed of the models on the GDSC and MovieLens 100K datasets. We ran each model with $K=20$, and measured the average mean squared error on the training data across ten runs. We plotted the results in Figure \[convergences\], where each group is plotted as the same colour: red for real-valued, blue for nonnegative, green for semi-nonnegative, yellow for Poisson, and grey for the non-probabilistic baseline. Runtime speeds are given in the supplementary materials.
{width="\columnwidth"}
{width="1\columnwidth"}
{width="0.495\columnwidth"} {width="0.495\columnwidth"}
{width="\columnwidth"}
Cross-validation {#Cross-validation}
----------------
Our first predictive experiment was to measure the 5-fold cross-validation performance on each of the eight datasets. We used the hyperparameter values from Section \[Hyperparameters\], and used 5-fold nested cross-validation to choose the dimensionality $K$. The average mean squared error of predictions are given in Figure \[crossvalidation\] for all eight datasets. The average dimensionality found in nested cross-validation can be found in the supplementary materials.
Noise test
----------
We then measured the predictive performance when the datasets are very noisy. We added different levels of Gaussian noise to the data, with the noise-to-signal ratio being given by the ratio of the variance of the Gaussian noise we add, to the standard deviation of the generated data. For each noise level we split the datapoints randomly into ten folds, and measured the predictive performance of the models on one held-out set at a time. We used $K=5$ for all methods. The results for the GDSC drug sensitivity dataset are given in Figure \[noise\_gdsc\], where we plot the ratio of the variance of the data to the mean squared error of the predictions—higher values are better, and using the row average gives a performance of one.
{width="70.00000%"} {width="90.00000%"}
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
Sparse predictions
------------------
Next we measured the predictive performances when the sparsity of the data increases. For different fractions of unobserved data, we randomly split the data based on that fraction, trained the model on the observed data, and measured the performance on the held-out test data. We used $K=5$ for all models. The average mean squared error of ten repeats is given in Figure \[sparsity\_gm\_gdsc\], showing the performances on both the methylation GM and GDSC drug sensitivity datasets.
Model selection
---------------
We also measured the robustness of the models to overfitting if the dimensionality $K$ is high. As a result, most models will fit very well to the training data, but give poor predictions on the test data. Here, we vary the dimensionality $K$ for each of the models on the GDSC drug sensitivity dataset, randomly taking out $10\%$ as test data, and repeating ten times. The results are given in Figure \[model\_selection\_gdsc\]—in the supplementary materials we look at two more datasets.
Discussion {#Discussion}
==========
From the results shown in the previous section, we were able to draw the following conclusions.
Observation 1: Poisson likelihood methods perform poorly compared to the Gaussian likelihood—they overfit quickly (Figures \[model\_selection\_gdsc\]a), give worse predictive performances in cross-validation (Figure \[crossvalidation\]) and under noisy conditions (Figure \[noise\_gdsc\]), presumably because they cannot converge as deep as the other methods (Figure \[convergences\]). At high sparsity levels they can start to perform better (Figure \[sparsity\_gm\_gdsc\]d). Some papers [@Gopalan2015] claim that Poisson models offer better predictions, but for small and well-observed datasets we found the opposite to be true.
Observation 2: Nonnegative models are more constrained than the real-valued ones, causing them to converge less deep (Figure \[convergences\]), and to be less likely to overfit to high sparsity levels (\[sparsity\_gm\_gdsc\]c, \[sparsity\_gm\_gdsc\]h) than the standard GGG model. However, the right hierarchical prior for a real-valued model (such as Wishart) can bridge this gap.
Observation 3: There is no difference in performance between real-valued and semi-nonnegative matrix factorisation, as shown in the model selection and sparsity experiments (Figures \[sparsity\_gm\_gdsc\]e, \[sparsity\_gm\_gdsc\]j, and \[model\_selection\_gdsc\]e): the performance for GGG and GEG, as well as GVG and GVnG, are nearly identical.
Observation 4: There is no difference in predictive performance between univariate and multivariate posteriors (GGG, GGGU), as shown in Figures \[sparsity\_gm\_gdsc\]b and \[sparsity\_gm\_gdsc\]g.
Observation 5: The automatic relevance determination and Wishart hierarchical priors are effective ways of preventing overfitting, as shown in Figures \[model\_selection\_gdsc\]b and \[model\_selection\_gdsc\]c: the GGGA, GGGW, and GEEA models keep the line down as $K$ increase, whereas the GGG and GEE models start overfitting more. This has been shown before for nonnegative models [@Brouwer2017b] but the effect is even stronger for the real-valued ones.
Overvation 6: Similarly, the Laplace priors are good at reducing overfitting as the dimensionality grows (Figure \[model\_selection\_gdsc\]b), without requiring additional hierarchical priors.
Observation 7: Some other hierarchical priors do not make a difference, such as with GLLI, GTTN, PGGG—Figures \[sparsity\_gm\_gdsc\]d, \[sparsity\_gm\_gdsc\]i, and \[model\_selection\_gdsc\]d show little difference in performance. They can help us automatically choose the hyperparameters, but in our experience the models are not very sensitive to this choice anyways.
Although these observations are specific to the applications and dataset sizes studied, we believe that general insights can be drawn from them about the behaviour of the four different groups of Bayesian matrix factorisation models. The behaviour of Poisson models is especially interesting, because they are often claimed to be better than Gaussian models for large datasets, but for smaller ones this does not hold. We hope that these insights will assist future researchers in their model design.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report results from an extensive set of measurements of the $\beta$-decay response in liquid xenon. These measurements are derived from high-statistics calibration data from injected sources of both $^{3}$H and $^{14}$C in the LUX detector. The mean light-to-charge ratio is reported for 13 electric field values ranging from 43 to 491 V/cm, and for energies ranging from 1.5 to 145 keV.'
author:
- 'D.S. Akerib'
- 'S. Alsum'
- 'H.M. Araújo'
- 'X. Bai'
- 'J. Balajthy'
- 'A. Baxter'
- 'P. Beltrame'
- 'E.P. Bernard'
- 'A. Bernstein'
- 'T.P. Biesiadzinski'
- 'E.M. Boulton'
- 'B. Boxer'
- 'P. Brás'
- 'S. Burdin'
- 'D. Byram'
- 'M.C. Carmona-Benitez'
- 'C. Chan'
- 'J.E. Cutter'
- 'L. deViveiros'
- 'E. Druszkiewicz'
- 'S.R. Fallon'
- 'A. Fan'
- 'S. Fiorucci'
- 'R.J. Gaitskell'
- 'J. Genovesi'
- 'C. Ghag'
- 'M.G.D. Gilchriese'
- 'C. Gwilliam'
- 'C.R. Hall'
- 'S.J. Haselschwardt'
- 'S.A. Hertel'
- 'D.P. Hogan'
- 'M. Horn'
- 'D.Q. Huang'
- 'C.M. Ignarra'
- 'R.G. Jacobsen'
- 'O. Jahangir'
- 'W. Ji'
- 'K. Kamdin'
- 'K. Kazkaz'
- 'D. Khaitan'
- 'E.V. Korolkova'
- 'S. Kravitz'
- 'V.A. Kudryavtsev'
- 'E. Leason'
- 'B.G. Lenardo'
- 'K.T. Lesko'
- 'J. Liao'
- 'J. Lin'
- 'A. Lindote'
- 'M.I. Lopes'
- 'A. Manalaysay'
- 'R.L. Mannino'
- 'N. Marangou'
- 'M.F. Marzioni'
- 'D.N. McKinsey'
- 'D.-M. Mei'
- 'M. Moongweluwan'
- 'J.A. Morad'
- 'A.St.J. Murphy'
- 'A. Naylor'
- 'C. Nehrkorn'
- 'H.N. Nelson'
- 'F. Neves'
- 'A. Nilima'
- 'K.C. Oliver-Mallory'
- 'K.J. Palladino'
- 'E.K. Pease'
- 'Q. Riffard'
- 'G.R.C. Rischbieter'
- 'C. Rhyne'
- 'P. Rossiter'
- 'S. Shaw'
- 'T.A. Shutt'
- 'C. Silva'
- 'M. Solmaz'
- 'V.N. Solovov'
- 'P. Sorensen'
- 'T.J. Sumner'
- 'M. Szydagis'
- 'D.J. Taylor'
- 'R. Taylor'
- 'W.C. Taylor'
- 'B.P. Tennyson'
- 'P.A. Terman'
- 'D.R. Tiedt'
- 'W.H. To'
- 'M. Tripathi'
- 'L. Tvrznikova'
- 'U. Utku'
- 'S. Uvarov'
- 'A. Vacheret'
- 'V. Velan'
- 'R.C. Webb'
- 'J.T. White'
- 'T.J. Whitis'
- 'M.S. Witherell'
- 'F.L.H. Wolfs'
- 'D. Woodward'
- 'J. Xu'
- 'C. Zhang'
title: 'Improved Measurements of the $\beta$-Decay Response of Liquid Xenon with the LUX Detector'
---
Introduction {#section:Intro}
============
The Large Underground Xenon experiment (LUX) was a liquid-xenon (LXe) time-projection chamber (TPC). Before it was decommissioned in 2016, LUX was located in the Davis cavern of the Sanford Underground Research Facility (SURF) in Lead, South Dakota, on the 4,850’ level [@lux_2012]. In total, it contained about 370 kg of xenon, 250 kg of which was active. Energy deposits in the sensitive volume were detected using two arrays of 61 photomultiplier tubes (PMTs) at the top and bottom of the detector.
LUX was initially designed as a WIMP dark-matter detector. The LUX full exposure of 3.35 $\times \ 10^4$ kg days combines the first data-taking run (WS2013), which took place from April to August 2013 [@lux_2014; @lux_2016], with the second data-taking run (WS2014-2016), which ran from September 2014 until May 2016 [@lux_2017]. A 50 GeV/c$^2$ WIMP with a cross section greater than 1.1 $\times \ 10^{-46}$ cm$^2$ is excluded with 90% confidence. Recently, stronger limits have been placed by the XENON-1T and PandaX experiments [@xenon_1t; @pandax].
As a two-phase TPC, LUX was sensitive to light and charge signals via primary ($S1$) and secondary ($S2$) scintillation, respectively. The light and charge yields ($Ly$ and $Qy$) are defined as the average number of quanta (photons and electrons) per keV of energy deposited in the LXe. These depend upon the energy of the event, the magnitude of the electric field applied at the event’s location, and whether the interaction leads to a nuclear recoil (NR) or an electron recoil (ER). The yields of an electron recoil may also depend upon further specifics of the interaction. For instance, interactions of $\beta$-particles in LXe may produce different yields from those of gammas, because the latter has some energy-dependent probability of photoabsorption, while the former does not [@nest2]. Such variations will be seen in section \[sec:discussion\] when comparing values of $Qy$ from $\beta$ interactions with those from $^{83m}$Kr and $^{131m}$Xe decays.
The most prevalent and problematic backgrounds in current and future LXe dark matter experiments are $\beta$-decays of Rn daughters [@lz_tdr; @lz_sensitivity; @xenon_1t; @pandax]. It is therefore important to understand the $\beta$-decay-induced light and charge yields in LXe as a function of electric field and energy. Previous measurements of these yields using LUX WS2013 data, including a set of measurements using a $^{3}$H injection source at both 105 V/cm and 180 V/cm were reported in Ref. [@lux_tritium; @DQyields; @Evanyields]. In this article we use the data from a novel $^{14}$C injection and a high statistics $^{3}$H calibration, which were conducted after WS2014-2016, to extend the previous results over a much wider range of energies and electric fields.
The electric field in the WS2014-2016 LUX detector was highly non-uniform, ranging from less than 50 V/cm to over 500 V/cm. In this article we divide the detector into thirteen electric field bins, with central values spanning from 43 to 491 V/cm, and obtain measurements of the light and charge yields for each associated field value [@thesis]. The use of a $^{14}$C injection source in addition to $^{3}$H increases the energy range of our measurements by nearly an order of magnitude. The radioactive isotope $^{14}$C $\beta$-decays to the ground state of $^{14}$N with a Q-value of 156 keV, which is 8.6 times greater than the $^{3}$H Q-value of 18.1 keV [@C14_Kuzminov; @C14_Wietfeldt; @lux_tritium].
The $^{14}$C calibration was performed at the end of the LUX operational lifetime and just before decommissioning in September, 2016. We will refer to this period as “post-WS”. The activities used in post-WS calibrations were allowed to be significantly greater than previous calibrations because maintaining low detector backgrounds was no longer a requirement. This resulted in a data set of roughly 2 million $^{14}$C events. After fiducial cuts, each of the thirteen electric field bins has between 60,000 and 120,000 events. A separate post-WS $^{3}$H dataset is also analyzed, which has about one third of the number of events as the $^{14}$C set.
Data Selection
==============
An interaction in the sensitive LXe produces primary scintillation photons and ionization electrons. The primary scintillation is collected by the PMTs and constitutes the $S1$ signal. The electrons are transported through an electric drift field to the top of the detector, where the electrons are extracted from the liquid surface into a region of gaseous xenon and produce the $S2$ signal through electroluminescence. The $S2$ signal is proportional to the number of electrons extracted.
The low-energy electronic depositions studied in this work have very short track lengths ($\sim$0.3 mm) [@lxe_detectors], and are treated as occurring at points in space. The signals caused by these depositions will therefore have a single $S1$ followed by a single $S2$. The $S2$ light generated by an extracted electron is highly localized in the x-y plane at the top of the detector, so the x-y position of an event can be reconstructed by analyzing the relative size of the pulses in the top PMT array. The drifting electrons take many microseconds longer to reach the liquid surface than the $S1$ photons take to be detected. The resulting difference in time between the $S1$ and $S2$ is referred to as the “drift time”. In a detector with a uniform electric drift field, the electrons drift vertically at a constant velocity, so drift time gives a direct measurement of the z-position of an event [@lux_posrec].
The charge and light collection efficiencies have some dependence upon position due to the attachment of drifting electrons to impurities and detector geometry. These effects are measured using the response to $^{83m}$Kr and $^{3}$H [@lux_2017]. A new set of efficiency-corrected data has been produced in which these effects are corrected for. In this article, $S1$ and $S2$ refer to these corrected values unless otherwise specified.
To avoid edge effects in our analysis, we reject events near the boundaries of the sensitive volume. Events within about 3 cm from the walls of the detector were rejected using a radial cut which is described in section 4.2.2 of Ref. [@thesis]. For the same reason, events with drift times greater than 330 $\mu$s or less than 10 $\mu$s were also rejected.
The simplest selection cut used to isolate single site events is to reject any event that does not contain exactly one $S1$ and one $S2$, with the $S1$ occurring before the $S2$. The efficiency of this cut is found to decrease at higher energies due to correlated pile-up in both $S1$ and $S2$. In this work we use a modified version of this cut which is described in detail in section 4.2.1 of Ref. [@thesis]. We require a selected event have at least one $S2$, and at least one $S1$ before the first $S2$. The first $S1$ and $S2$ pulses are required to contain at least 93% of the total $S1$ and $S2$ area, respectively. This modified selection cut increases the acceptance of $^{131m}$Xe events from 61% to 92%, and improves the acceptance of $^{14}$C events to more than 90% across the entire energy spectrum [@thesis]. We use $^{131m}$Xe as a test of our selection cut because it is a mono-energetic peak at 163.9 keV, just above the $^{14}$C Q-value. The background rate remains very small in comparison to the injected sources, so the additional leakage of noise events due to the relaxed cut is negligible.
During and after WS2014-2016, the electric fields in LUX were highly non-uniform. A comprehensive study of the drift-field was performed and is detailed in Ref. [@lux_efield]. This study produced high-resolution maps of the electric field. Using a 3-dimensional linear interpolation of these maps, we assign a specific field value to every event in our data sets. This enables us to define 13 bins in electric field whose centers range from 43 to 491 V/cm, and where each bin extends 10% above and below its central value. We also limit the range of drift times that are drawn from so that a bin does not extend past the central drift time values of its adjacent bins. These bins are described in greater detail in section 4.2.3 of Ref. [@thesis].
Calibration Source Injections
=============================
After WS2014-2016 was completed in June of 2016, the detector was exercised with a variety of ER and NR calibration sources. The usual ER calibrations of $^{83m}$Kr [@lux_kr1; @lux_kr2; @lux_kr3] and tritiated methane (CH$_3$T) [@lux_tritium] were performed, along with NR calibrations using the deuterium-deuterium (DD) neutron generator [@lux_dd1; @lux_dd2]. In addition to these standard calibrations, novel techniques and sources were implemented. The timeline and activities of these injections can be seen in Figure \[fig:DAQrate\].
{width="\linewidth"}
\[fig:DAQrate\]
$^{131m}$Xe and $^{37}$Ar
-------------------------
The isotope $^{131m}$Xe de-excites through a gamma transition to its ground state with an energy of 163.93 keV and a half life of 11.84 days. It is generated using a commercially available $^{131}$I pill and is introduced into the primary xenon circulation path using the $^{3}$H injection system described in Ref. [@lux_tritium].
An injection of $^{37}$Ar was also deployed. This isotope decays via electron capture to $^{37}$Cl with a half-life of 35 days. In 90% of these decays, a K-shell electron is captured, followed by the emission of x-rays and Auger electrons which total to 2.82 keV. There are also non-zero branching ratios for the capture of L- and M-shell electrons. $^{37}$Ar can be produced through stimulated $\alpha$ emission of a $^{40}$Ca target using a neutron beam. The $^{37}$Ar sample used in LUX was produced by irradiating an aqueous solution of CaCl$_2$ with neutrons from an AmBe source. The gas above this solution was then collected and purified to obtain the gaseous sample of $^{37}$Ar [@pixey_ar37]. This sample was injected into the LUX xenon circulation using the same system as the $^{83m}$Kr calibrations [@lux_kr2].
$^{14}$C and $^{3}$H
--------------------
The $^{3}$H injection system and procedure was described in detail in Ref. [@lux_tritium]. In the post-WS injection campaign, it was used to deploy a high statistics $^{3}$H injection, as well as a novel $^{14}$C injection into the LUX detector. The $^{14}$C was in the form of radio-labeled methane which is chemically identical to the tritiated-methane used in Ref. [@lux_tritium] and was also synthesized by Moravek Biochemical [@moravek]. The methane, and therefore the $^{14}$C activity, is removed from the LUX xenon in the same manner as $^{3}$H via circulation through a heated zirconium getter.
Model of the LUX Post-WS $\beta$-Decay Data
===========================================
Smearing of Continuous $\beta$ Spectra {#sec:desmear}
--------------------------------------
Measurements of energy-dependent parameters from continuous $\beta$ spectra are affected in nontrivial ways by finite detector resolution. We obtain measurements of yields and recombination by dividing the $^{14}$C and $^3$H into reconstructed energy bins. Finite detector resolution impacts our measurements by smearing some events into a reconstructed energy bin, whose true energies lie outside of the bin. If we know both the spectral shape and the energy-dependent detector resolution, this effect can be accounted for by integrating the contribution to the $i^{th}$ bin from each point in the spectrum. This type of analysis was done analytically in the previous $^{3}$H results [@lux_tritium; @attila]. However, the $S2$ tails described in section \[sec:s2tails\] make the analytic calculations unwieldy, so in this article we perform the integration numerically using Monte Carlo (MC) data.
In order to estimate these smearing corrections, the charge and light yield is initially taken from the NEST model [@nest1; @nest2; @nest3], and an initial set of MC $S1$ and $S2$ data is generated. This data set is used to make preliminary measurements of $L_y$ and $Q_y$, after which the MC data are regenerated using these newly measured yields. This new set of MC data is used to re-measure the smearing corrections, giving us the final measurements of $L_y$ and $Q_y$.
Combined Energy Model {#sec:combE}
---------------------
We adapt the combined energy model for ER events [@nest1; @aprile_doke_LXe], which relies on a simplified Platzman equation [@platzman]: $$\label{eq:combe1}
E_{ER}=W(n_*+n_i),$$ where $n_*$ is the initial number of excitons generated by an event, and $n_i$ is the initial number of ions prior to recombination. For ER events, the work function, $W$, has been measured to be $13.7 \ \pm0.2$ eV/quantum [@dahl]. Recombination converts some ions into excitons so that the observable number of photons ($n_{\gamma}$) and electrons ($n_{e}$) is given by: $$\label{eq:combe2}
\begin{split}
n_{\gamma}&=n_*+rn_i\\
&=(\alpha+r)n_i\\
n_e&=(1-r)n_i,
\end{split}$$ where the exciton-ion ratio $\alpha \equiv n_{*}/n_{i}$, has been measured to be about 0.06-0.20 [@doke2002; @attila] for ER events and is typically assumed to be constant with energy. For ER events, we assume a constant value of $\alpha=0.18$, which is taken from Ref. [@attila]. For each event, the number of ions that recombine, $R$, is randomly distributed about an expected value that is equal to the mean recombination probability, $r$, times the number of ions: $$\langle R \rangle = r N_i .$$ The mean recombination probability depends on both the energy deposited and the applied electric field.
We model this process using a modified version of the NEST simulation software [@nest1; @nest2; @nest3]. The total number of quanta in a simulated event ($N_q$) is equal to the event energy, divided by the work function. The apportionment of these quanta into exitons and ions ($N_*$ and $N_i$) is treated as a binomial process, where the probability that a quantum is an ion is equal to $\frac{1}{1+\alpha}$. Recombination is modeled by drawing the number of electrons from a normal distribution with mean equal to $(1-r) \cdot N_i$ and standard deviation equal to $\sigma_R$, where $r$ and $\sigma_R$ depend on the energy and field of the simulated event. The number of photons ($N_{\gamma}$) is then taken to be the number of quanta, minus the number of electrons.
Measuring Average Charge and Light Collection Efficiencies
----------------------------------------------------------
Reconstructed energy is an observable quantity that fluctuates around the true energy deposited in the LXe during an event. In terms of the observable $S1$ and $S2$ signals, the reconstructed energy of an event is given by: $$\label{eq:combe3}
E_{rec}=W(\frac{S1}{g_1}+\frac{S2}{g_2}),$$ where $g_1$ and $g_2$ are average gain factors. These values are used to convert from $S1$ to $n_{\gamma}$ and $S2$ to $n_{e}$, accounting for the total efficiency of the detector. Equation \[eq:combe3\] also provides a useful tool in measuring the efficiency factors, $g_1$ and $g_2$, through a method introduced by Doke *et al.* in Ref. [@doke2002].
For a set of ER events with a constant energy $E$, Equation \[eq:combe3\] can be used to write: $$\left(\frac{W\overline{S1}}{E}\right)=-\frac{g_1}{g_2}\left(\frac{W\overline{S2}}{E}\right)+g_1,$$ where $\overline{S1}$ and $\overline{S2}$ are the average $S1$ and $S2$ signal observed. This linear expression may then be plotted in the usual way such that $y=\left(\frac{W\overline{S1}}{E}\right)$ and $x=\left(\frac{W\overline{S2}}{E}\right)$ are both in terms of either measurable quantities or known constants. The efficiency factors, $g_1$ and $g_2$, can then be obtained by fitting a line through a set of (x,y) values measured at different energies and fields.
For this analysis, the LUX post-WS values of $g_1$ and $g_2$ are measured by dividing the $^{83m}$Kr and $^{131m}$Xe data into separate drift-time regions and plotting the average $S1$ and $S2$ values in each region. The results of this analysis are shown in Figure \[fig:Doke\_plot\]. To test for remaining position dependence in $g_1$ and $g_2$, we also calculate a two-point Doke plot using the $^{83m}$Kr and $^{131m}$Xe values from each drift-time region. The systematic uncertainty is taken to be the standard deviation of these two-point values.
![Doke-style plot of post-WS $^{83m}$Kr and $^{131m}$Xe data. The uncertainties on the $g_1$ and $g_2$ values include systematic variation in drift time, and were calculated as described in the text. The $g_1$ and $g_2$ values are highly correlated, with a Pearson correlation coefficient of $-$0.90.[]{data-label="fig:Doke_plot"}](Doke_plot){width="\linewidth"}
The values of $g_1$ and $g_2$ we measure are $0.0931~(\pm0.0012)$, and $18.6~(\pm0.9)$, respectively. The uncertainties of these values are dominated by the systematic deviation described above. These are broadly consistent with the values found in Ref. [@lux_2017], although our measured $g_1$ is about 3-$\sigma$ below the lowest value found there. This discrepancy is likely due to a continued decrease in light collection efficiency over the three months between the end of WS2014-2016 and the beginning of the injection campaign.
Empirical Model of $S2$ Tail Pathology {#sec:s2tails}
--------------------------------------
Figures \[fig:Ar\_spec\] and \[fig:Xe\_spec\] show the spectra of the $^{37}$Ar and $^{131m}$Xe decays measured during the post-WS calibration campaign. These combined energy spectra have clear non-Gaussian tails toward high energy. The tails in the energy spectra stem from underlying tails in the individual $S2$ spectra, which result from a pathological effect in the $S2$ signals. These $S2$ tails are much more pronounced in the WS2014-2016 and the post-WS data than in WS2013 data. The exact pathology is unknown, but there is some evidence that the tails are caused by either photoionization of impurities or “electron trains”. An electron train occurs when electrons from a previous large event fail to be immediately extracted from the liquid surface and instead are emitted into the gas over a millisecond time scale.
![Comparison of measured $^{37}$Ar energy spectrum (black) versus two simulated spectra; one with the $S2$ tails modeled (red) and one without (blue). The data shown are taken from the full WIMP-search fiducial volume corresponding to electron drift times of 40 to 300 microseconds.[]{data-label="fig:Ar_spec"}](Ar_spec){width="\linewidth"}
To correctly account for smearing effects on the $^{14}$C and $^3$H spectra, we use an empirical model of the $S2$ tails. We begin with simulated $S1$ and $S2$ areas from MC events generated without tails, assuming Gaussian detector resolution. The effect of the $S2$ tails is modeled by assigning additional $S2$ area to a fraction of the simulated events. The additional tail area for a chosen event is drawn from an exponential distribution whose mean is proportional to uncorrected $S2$ size. This model of the $S2$ tails is generated using three steps and three fitting parameters, which are assumed to be independent of position, energy, and field. First, a “true” $g_2$ value ($g_{2,true}$) is used to generate an initial set of tail-less MC events. The value of $g_{2,true}$ is less than the one measured above, since the observed value of $g_2$ includes both the “true” $S2$ area, as well as the tail area. Next, a fraction of events, $R$, is selected to be assigned additional tail area. Finally, for each of the selected events, a random number of tail electrons is drawn from an exponential distribution with a mean of $b \cdot n_{e,LS}$, where $b$ is a fitting parameter, and $n_{e,LS}$ is the number of simulated electrons that reach the liquid surface. The parameters are tuned in order to reproduce the energy spectrum of the $^{37}$Ar and $^{131m}$Xe data. The best fit values of $b$, $R$, and $g_{2,true}$ are found to be $0.112~(\pm 0.003)$, $0.73~(\pm 0.02)$, and $17.60~(\pm 0.05)$, respectively.
![Comparison of measured $^{131m}$Xe energy spectrum (black) versus two simulated spectra; one with the $S2$ tails modeled (red) and one without (blue). The data shown are taken from the full WIMP-search fiducial volume corresponding to electron drift times of 40 to 300 microseconds. []{data-label="fig:Xe_spec"}](Xe_spec){width="\linewidth"}
Figures \[fig:Ar\_spec\] and \[fig:Xe\_spec\] show the best fit spectra for $^{37}$Ar and $^{131m}$Xe. Figures \[fig:H3\_spec\] and \[fig:C14\_spec\] show the best fit model applied to $^{3}$H and $^{14}$C, respectively. The agreement of all of the simulated spectra with data are significantly improved after the addition of the tail model. The $^{37}$Ar MC spectrum over-predicts the amplitude of the data at low energy ($<$2 keV); however, the same discrepancy is not observed in the $^{3}$H spectrum.
![Comparison of measured $^{3}$H energy spectrum (black) versus two simulated spectra; one with the $S2$ tails modeled (red) and one without (blue). The data shown are taken from the full WIMP-search fiducial volume corresponding to electron drift times of 40 to 300 microseconds.[]{data-label="fig:H3_spec"}](H3_spec){width="\linewidth"}
![Comparison of measured $^{14}$C energy spectrum (black) versus two simulated spectra; one with the $S2$ tails modeled (red) and one without (blue). The data shown are taken from the full WIMP-search fiducial volume corresponding to electron drift times of 40 to 300 microseconds.[]{data-label="fig:C14_spec"}](C14_spec){width="\linewidth"}
Recombination Fluctuations {#sec:sigr_mod}
--------------------------
The fluctuations in the recombination fraction, $\sigma_R$, are known to deviate significantly from those of a binomial process [@dahl; @attila; @lux_tritium]. In Ref. [@lux_tritium] we reported that the fluctuations are approximately linear in $n_i$ with a slope of about 0.067 quanta per ion. We do not attempt to obtain an absolute measurement of $\sigma_R$ using the post-WS data because the $S2$ tails are correlated with the recombination fluctuations in nontrivial ways. The correlation makes it impractical to separate detector resolution and recombination fluctuations as was done for the WS2013 $^{3}$H data. We instead apply an adjustment to the linear model and compare the resulting MC spectrum to data. We find the data are best described by a Gaussian adjustment to the linear model: $$\label{eq:sigrmod}
\begin{split}
\sigma_{R}(r)^2=r&(1-r)\cdot n_i +\\
&\left(F_0\exp \left(\frac{-(r-F_1)^2}{2F_2^2}\right)\right)^2n_i^2,
\end{split}$$ where $F_0$, $F_1$ and $F_2$ are the constant fitting parameters: $$\label{eq:sigrbestfit}
\begin{split}
F_0=0.075 \pm 0.005\\
F_1=0.413 \pm 0.024\\
F_2=0.243 \pm 0.024 .
\end{split}$$ These parameters were optimized using post-WS $^3$H and $^{14}$C data using using a grid search method described in section 5.5.3 of Ref. [@thesis]. A set of measurements of $\sigma_R$ from Ref. [@dahl] were also used to help constrain the low-recombination side of the model. When $N_i$ is large and $r$ is close to $F_1$, Equation \[eq:sigrmod\] reduces to a linear model with a slope of $0.075 (\pm 0.005)$ quanta per ion. The first term on the right side of Equation \[eq:sigrmod\] mimics a binomial variance and prevents $\sigma_R$ from going to zero at extreme values or r. In our best fit model, the binomial term is negligible across the range of energies and fields tested.
The width of the MC spectra are compared to data in Figure \[fig:C14\_sigR\_sim\]. We find that by adding the $S2$ tail model described in section \[sec:s2tails\] and a model of recombination fluctuations that follows Equation \[eq:sigrmod\] to our simulation, we are able to reproduce the widths of the $S1$ and $S2$ spectra for $^{14}$C $\beta$-decay events across all of the electric fields tested. Figure \[fig:sigR\_run03\_comp\] shows a comparison to the $\sigma_R$ measurements from WS2013. We find the new model matches data better than the linear model. The upward kink in the WS2013 measurements at 16 keV is due to an error that will be described in section \[sec:discussion\].
There is still tension remaining between simulated and measured widths with this new model included. It may be that this is due to an underlying field dependence in the recombination fluctuations that has not been unaccounted for. This would be a third-order effect in our measurements of the yields, and we are able to reproduce the measured $^3$H and $^{14}$C spectra without accounting for this possible field dependence. We therefore elect to proceed using the model as described above.
![This figure shows the best-fit Gaussian width of the $^{14}$C $S1$ (right) and $S2$ (left) bands for a selection of electric field bins. In these plots, the red dashed lines indicate simulated widths using the model described in sections \[sec:s2tails\] and \[sec:sigr\_mod\], while the black lines indicate the widths observed in data.[]{data-label="fig:C14_sigR_sim"}](C14_sigR_sim "fig:"){width="\linewidth"}\[h\]
![Comparison of the recombination model developed in section \[sec:sigr\_mod\] (blue line) to the model from [@lux_tritium] (black dashed line). The black markers indicate the $\sigma_R$ data presented in [@attila].[]{data-label="fig:sigR_run03_comp"}](sigR_run03_comp){width="\linewidth"}
Results and Discussion
======================
Photon-Electron Fraction
------------------------
Assuming that for ER the total number of generated quanta is fixed, as described in section \[sec:combE\], we can reduce $L_y$, $Q_y$, and $r$ to a single quantity: $$\rho \equiv \frac{n_{\gamma}}{n_e}.$$ Using the relations: $$L_y\equiv \frac{n_{\gamma}}{E}, \ Q_y\equiv \frac{n_{e}}{E},\text{ and }
L_y+Q_y=\frac{1}{W},$$ we can reconstruct the individual yields from $\rho$: $$\label{eq:translate_rho1}
L_y=\frac{1}{W}\frac{\rho}{1+\rho} \text{ and } Q_y=\frac{1}{W}\frac{1}{1+\rho}.$$ Further, using the relations: $$n_Q=(1+\alpha)n_i=(1+\rho)n_e \text{ and }
r=\frac{n_i-n_e}{n_i},$$ where $n_Q$ is the total number of quanta, we can reconstruct the average recombination probability: $$\label{eq:translate_rho2}
r=\frac{\rho-\alpha}{1+\rho}.$$
Here we report the measured results of $\rho\equiv n_{\gamma}/n_e$. Figure \[fig:C14\_rho\_final\_z\] shows the results for post-WS $^{14}$C, and figure \[fig:H3\_rho\_final\_z\] shows the results for post-WS $^{3}$H. The measurements of $n_{\gamma}$ and $n_e$, along with the reconstructed energy have been numerically de-smeared following the procedure laid out in section \[sec:desmear\], using the model described in sections \[sec:s2tails\] and \[sec:sigr\_mod\]. The sizes of these smearing corrections are taken as systematic uncertainty on their respective measurements. The uncertainties in $g_1$ and $g_2$ are also included in the systematic error. The systematic uncertainties are combined in quadrature and are shown as the light gray error bars in figures \[fig:C14\_rho\_final\_z\] and \[fig:H3\_rho\_final\_z\]. The statistical fitting uncertainty and the uncertainty due to bin width are shown as the black error bars.
Discussion {#sec:discussion}
----------
Our results are in good agreement with previous measurements from WS2013, as is shown in figure \[fig:QY\_run03\_comp\]. In this figure we compare measurements of $Q_y$ from WS2013 $^{3}$H [@lux_tritium] and from the $^{127}$Xe electron capture [@DQyields; @Evanyields] with those from this work at similar electric fields. We find that the measurements agree within systematic error. When comparing our measurements of $Q_y$ from interactions of $\beta$’s in LXe to those from the $^{83m}$Kr and $^{131m}$Xe decays we find a disagreement of about 2-$\sigma$ [@Evanyields]. This is likely due to a difference in yields between $\beta$-decay interactions and those involving composite decays (such as $^{83m}$Kr) or photo-absorption [@nest2].
![Comparison of LUX post-WS $Q_y$ measurements using $^{3}$H (red) and $^{14}$C (orange) to WS2013 measurements, which were taken at 105 and 180 V/cm. The green diamonds show the WS2013 $^{3}$H measurements [@lux_tritium], and the blue X’s and open magenta squares indicate WS2013 measurements of $Q_y$ from $^{127}$Xe electron capture [@DQyields; @Evanyields]. The open circles and diamonds indicate WS2013 measurements of $Q_y$ from the $^{83m}$Kr and $^{131m}$Xe decays, respectively [@Evanyields]. The black line shows the final model used to generate the smearing corrections [@thesis].[]{data-label="fig:QY_run03_comp"}](QY_run03_comp){width="\linewidth"}
{width="\linewidth"}
{width="\textwidth"}
It should be noted that the results we present here are in disagreement with our previous $^3$H yields and recombination measurements from Ref. [@lux_tritium] above 16 keV. In the previous work, an error in the implementation of the energy smearing correction resulted in some of the data being over-corrected. The error has only a minimal effect for most of the results reported in Ref. [@lux_tritium], but it is manifest at the endpoint of the tritium spectrum as a kink in the yields. A detailed discussion of this error can be found in section 5.3.2 of Ref. [@thesis].
Summary
=======
We have presented improved measurements of the response of liquid xenon to $\beta$-decays in the LUX detector, which were taken after WS2014-2016 was completed. We describe the various sources used, along with the time-line and the respective activities of the calibrations. We use the $^{83m}$Kr and $^{131m}$Xe lines to measure the average $g_1$ and $g_2$ efficiency factors and to characterize the positional variation thereof.
The $^{37}$Ar and $^{131m}$Xe calibration data are used to alter the existing model of detector resolution to account for the $S2$ tail pathology in the $S2$ spectra. We used this updated model to numerically calculate the effect of smearing on the $^{3}$H and $^{14}$C $\beta$-decay spectra. We also found it necessary to update the empirical model of recombination fluctuations presented in [@lux_tritium] to better match the data above 20 keV.
We present measurements of the photon-to-electron ratio of $\beta$ events in liquid xenon from $^{3}$H and $^{14}$C. These measurements can be used to calculate the charge yield, light yield, and recombination probability over a wide range of electric fields and energies. This is the most extensive dataset of the quantities for $\beta$-decay in liquid xenon, and are directly relevant for understanding the dominant background in future dark matter experiments.
This work was partially supported by the U.S. Department of Energy (DOE) under award numbers DE-FG02-08ER41549, DE-FG02-91ER40688, DE-FG02-95ER40917, DE-FG02-91ER40674, DE-NA0000979, DE-FG02-11ER41738, DE-SC0006605, DE-AC02-05CH11231, DE-AC52-07NA27344, DE-SC0019066, and DE-FG01-91ER40618; the U.S. National Science Foundation under award numbers PHYS-0750671, PHY-0801536, PHY-1004661, PHY-1102470, PHY-1003660, PHY-1312561, PHY-1347449; the Research Corporation grant RA0350; the Center for Ultra-low Background Experiments in the Dakotas (CUBED); and the South Dakota School of Mines and Technology (SDSMT). LIP-Coimbra acknowledges funding from Fundação para a Ciência e a Tecnologia (FCT) through the project-grant PTDC/FIS-PAR/28567/2017. Imperial College and Brown University thank the UK Royal Society for travel funds under the International Exchange Scheme (IE120804). The UK groups acknowledge institutional support from Imperial College London, University College London and Edinburgh University, and from the Science & Technology Facilities Council for PhD studentship ST/K502042/1 (AB). The University of Edinburgh is a charitable body, registered in Scotland, with registration number SC005336. This research was conducted using computational resources and services at the Center for Computation and Visualization, Brown University.
We gratefully acknowledge the logistical and technical support and the access to laboratory infrastructure provided to us by the Sanford Underground Research Facility (SURF) and its personnel at Lead, South Dakota. SURF was developed by the South Dakota Science and Technology Authority, with an important philanthropic donation from T. Denny Sanford, and is operated by Lawrence Berkeley National Laboratory for the Department of Energy, Office of High Energy Physics.
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10.1109/TNS.2015.2481322), [****, ()](\doibase 10.1103/RevModPhys.82.2053) @noop [****, ()]{} **, @noop [Ph.D. thesis]{}, () [****, ()](http://stacks.iop.org/1347-4065/41/i=3R/a=1538)
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[**[A note on monomial ideals]{}**]{}[^1]
.5truecm
[Margherita Barile\
Dipartimento di Matematica, Università di Bari, Via E. Orabona 4,\
70125 Bari, Italy\
e-mail:[email protected]]{}
1truecm [**Abstract**]{} We show that the number of elements generating a squarefree monomial ideal up to radical can always be bounded above in terms of the number of its minimal monomial generators and the maximal height of its minimal primes. 0.5 truecm
Introduction {#introduction .unnumbered}
============
Let $R$ be a commutative Noetherian ring with non-zero identity. If $I$ is an ideal of $R$, we say that the elements $a_1,\dots, a_s\in R$ [*generate*]{} $I$ [*up to radical*]{} if ${\rm Rad}\,(a_1,\dots, a_s)={\rm Rad}\,I$. The minimum number $s$ with this property is called the [*arithmetical rank*]{} of $I$. If $\mu(I)$ is the minimum number of generators for $I$, then obviously $$\label{ara}{\rm ara}\,I\leq\mu(I).$$ This inequality yields a trivial upper bound for the arithmetical rank of $I$. Moreover, it is well-known that $$\label{height}h(I)\leq {\rm ara}\,I,$$ where $h(I)$ is the height of $I$. If equality holds in (\[ara\]) and in (\[height\]), $I$ is called a [*complete intersection*]{}. The following result by Schmitt and Vogel [@SV], p. 249, shows that (\[ara\]) can be improved if special combinatorial conditions on the generating set are fulfilled.
\[Schmitt\] Let $P$ be a finite subset of elements of $R$. Let $P_0,\dots, P_r$ be subsets of $P$ such that
$\bigcup_{i=0}^rP_i=P$;
$P_0$ has exactly one element;
if $p$ and $p'$ are different elements of $P_i$ $(0<i\leq r)$ there is an integer $i'$ with $0\leq i'<i$ and an element in $P_{i'}$ which divides $pp'$.
We set $q_i=\sum_{p\in P_i}p^{e(p)}$, where $e(p)\geq1$ are arbitrary integers. We will write $(P)$ for the ideal of $R$ generated by the elements of $P$. Then we get $${\rm Rad}\,(P)={\rm Rad}\,(q_0,\dots,q_r).$$
We will apply Lemma \[Schmitt\] in the polynomial ring $R=K[x_1,\dots, x_n]$, where $K$ is a field, in order to determine upper bounds for the arithmetical rank of ideals generated by monomials, i.e., by products of indeterminates. Since ideals with the same radical have the same arithmetical rank, without loss of generality we can restrict our investigation to radical monomial ideals. These are the so-called [*squarefree*]{} monomial ideals, i.e., the ideals that are generated by products of pairwise distinct indeterminates. The aim of this paper is to replace inequality (1) by a more precise one in the case where $I$ is a squarefree monomial ideal. We will derive a closed expression, depending only on $\mu(I)$ and the heights of the minimal primes of $I$, which can be put on the right-hand of inequality (\[ara\]) in order to obtain a better general upper bound. This new upper bound is still sharp when $I$ is a complete intersection. Moreover it improves the general upper bound conjectured by Lyubeznik in [@L2] when $I$ has a small number of generators. Recall that every monomial ideal $I$ has a unique monomial generating set of cardinality $\mu(I)$: it is the one formed by the monomials in the ideal that are minimal with respect to the divisibility relation.
A general upper bound
=====================
Let $I$ be a squarefree monomial ideal of $R$, and let $M$ be the set of $\mu(I)$ monomial generators for $I$. For all $i=1,\dots, n$ let $$\label{mi}M_i=\{f\in M\mid x_i\mbox{ divides }f\}.$$ Let $I_1, \dots, I_r$ be the minimal primes of $I$, so that $I=\cap_{j=1}^rI_j$. For all $j=1,\dots, r$ set $$\label{nuj}\nu_j=\max\{\vert M_i\vert\mid x_i\in I_j\},$$ and let $x_{i_j}\in I_j$ be such that $\nu_j=\vert M_{i_j}\vert$. Here the bars indicate the cardinality. Finally, let $$\label{nu}\nu(I)=\min\{\nu_j\mid j=1,\dots, r\}.$$ From definitions (\[mi\]), (\[nuj\]) and (\[nu\]) it follows that $\nu(I)\leq\vert M\vert=\mu(I)$.
\[mylemma\] ara$I\leq\mu(I)-\nu(I)+1$.
For the sake of simplicity we put $\mu=\mu(I)$ and $\nu=\nu(I)$. We are going to explicitly construct a set of at most $\mu-\nu+1$ polynomials which generate $I$ up to radical. For all $k=1,\dots, \mu$ let $$L_k=\{\prod_{i=1}^kf_i\mid f_1, \dots, f_k\mbox{ are pairwise distinct elements of $M$}\}.$$ Further, let $$p_0=\prod_{j=1}^rx_{i_j}\in\displaystyle \cap_{j=1}^rI_j=I.$$ Then put $$\label{p0} P_0=\{p_0\},$$ and, for all $i=1,\dots, \mu-\nu$, $$P_i=L_{\mu-\nu+1-i}.$$ We show that, for all $i=1,\dots, \mu-\nu$, $$\label{condition}\mbox{ the product of any two distinct elements of $P_i$ is divisible by some element of $P_{i-1}$}.$$ We first show it for $i>1$. Set $k=\mu-\nu+1-i$ and let $p,p'$ be distinct elements of $P_i=L_k$. Then there are two different $k$-subsets $\{f_1,\dots, f_k\}$ and $\{f_1',\dots, f_k'\}$ of $M$ such that $$p=\prod_{i=1}^kf_i\qquad\qquad\mbox{ and }\qquad\qquad p'=\prod_{i=1}^kf_i'.$$ Up to a change of indices we may assume that $f_1'\not\in\{f_1,\dots, f_k\}$. Then $pp'$ is divisible by $$\prod_{i=1}^kf_i\cdot f_1'\in L_{k+1}=P_{i-1}.$$ Now consider two distinct elements $p,p'$ of $P_1=L_{\mu-\nu}$. By the above argument, $pp'$ is divisible by a product $\pi$ of at least $\mu-\nu+1$ distinct elements of $M$. By definition, for all $j=1,\dots, r$, exactly $\nu_j$ elements of $M$ are divisible by $x_{i_j}$, in particular, by definition (\[nu\]), at least $\nu$ elements of $M$ are divisible by $x_{i_j}$, or, equivalently, at most $\mu-\nu$ are not divisible by $x_{i_j}$. It follows that the product $\pi$ necessarily involves, as a factor, an element of $M$ which is divisible by $x_{i_j}$. Hence $pp'$ is divisible by $l_0$.
This proves condition (\[condition\]) and implies that $P_0, \dots, P_{\mu-\nu}$ fulfill the assumption (iii) of Lemma \[Schmitt\] with $i'=i-1$. Moreover, we have that $P_{\mu-\nu}=L_1=M$, and $P_i\subset (M)$ for all $i=0,\dots,\mu-\nu$, so that $\left(\cup_{i=0}^{\mu-\nu}P_i\right)=(M)=I$. Thus assumption (i) of Lemma \[Schmitt\] is satisfied by $P=M$. Finally, assumption (ii) is trivially true by (\[p0\]). Therefore, if we set $$q_i=\sum_{p\in P_i}p\qquad\qquad\mbox{ for }i=0,\dots, \mu-\nu,$$ by Lemma \[Schmitt\] we have that $$I=(M)={\rm Rad}\,(q_0,\dots, q_{\mu-\nu}),$$ which implies the claim.
\[remark1\] [The construction of the sets $P_i$ given in the proof of Lemma \[mylemma\] does not give rise to an efficient algorithm, since it requires the computation of $2^{\mu(I)}-1$ polynomial products. The output can be simplified if one reduces each set $P_i$ (with $i\geq1$) to the set of its elements which are minimal with respect to the divisibility relation. This will not affect condition (\[condition\]), and will possibly produce a smaller number of polynomials $q_i$. Sets $P_i$ fulfilling (7) and assumptions (i) and (ii) of Lemma \[Schmitt\] can also be constructed by the following method, which is taken from [@B], and is more convenient from a computational point of view. We first put $\Gamma_1=M$, and for all indices $i\geq2$, we recursively define $\Gamma_i$ as the set of all elements of $$\{\lcm(f,g)\mid f, g\in \Gamma_{i-1}, f\ne g\}$$ which are are minimal with respect to the divisibility relation. The procedure stops at step $N$ as soon as there is some $p_0\in I$ which divides all elements of $\Gamma_N$. At this point we set $P_0=\{p_0\}$ and $P_i=\Gamma_{N-i}$ for all $i=1,\dots, N-1$. Then the above condition (\[condition\]) is trivially true.The different approaches discussed here are not equivalent: one can find examples where they yield distinct numbers of polynomials generating $I$ up to radical.]{}
Next we show that number $\mu(I)$ can be compared with $\nu(I)$. Let $$\tau(I)=\max\{h(I_j)\mid j=1,\dots, r\}.$$ Then we have
\[mylemma2\] $\mu(I)\leq\nu(I)\tau(I)$.
We adopt the simplified notation used in the preceding proof and we also set $\tau=\tau(I)$. In view of (\[nu\]) we have to prove that $\mu\leq\nu_j\tau$ for all $j=1,\dots, r$. We show it for $j=1$. Up to renaming the indeterminates we may assume that $I_1=(x_1,\dots, x_s)$ for some $s\leq\tau$. Since $M\subset I_1$, by (\[mi\]) we have $M=\cup_{i=1}^s M_i$, whence $$\mu=\vert M\vert \leq \sum_{i=1}^s\vert M_i\vert\leq s\nu_1\leq\tau\nu_1,$$ where the second inequality follows from (\[nuj\]) for $j=1$. This completes the proof.
The next result is an immediate consequence of Propositions \[mylemma\] and \[mylemma2\].
\[mycorollary\] Let $J$ be an ideal of $R$ such that $I=$Rad$(J)$ is a (squarefree) monomial ideal. Then $$\label{ara2}{\rm ara}\,J\leq \mu(I)-\displaystyle\frac{\mu(I)}{\tau(I)}+1.$$
\[ci\][The new upper bound given in Corollary \[mycorollary\] is, like the one in (\[ara\]), sharp if $I=J$ and $I$ is a complete intersection: in that case $\mu(I)=h(I)$ and it is well-known that $I$ is pure-dimensional, i.e., all its minimal primes have the same height (see, e.g., [@BH], Corollary 5.1.5), so that $h(I)=\tau(I)$. ]{}
\[jaballah\][For a squarefree monomial ideal $I$, the number $\mu(I)$ can be bounded above in terms of the number $n$ of indeterminates of $R$ and/or of the heights of the minimal primes of $I$. Some of these formulas can be found in [@J] or in [@J0]. If we replace them in (\[ara2\]) we obtain similar upper bounds for ara$I$. For example, Lemma (4) (a) in [@J] yields $${\rm ara}\,I\leq \left(1-\frac1{\tau(I)}\right){n\choose{\lbrack\frac{n}2\rbrack}}+1,$$ where the square brackets denote the integer part, and, according to Satz (5) (a) in [@J], we have $${\rm ara}\,I\leq \left(1-\frac1{\tau(I)}\right){n\choose{h(I)-1}}+1,$$ whenever $h(I)\geq\frac{n+1}2$. ]{}
On a conjecture by Lyubeznik
============================
It was conjectured by Lyubeznik in [@L2] that for every pure-dimensional ideal $J$ of $R$ of height $t$, $$\label{Lyubeznik}{\rm ara}\,J\leq n-\left[\frac{n-1}{t}\right],$$ so that, in particular, for every pure-dimensional monomial ideal $I$, $$\label{Lyubeznik2}{\rm ara}\,I\leq n-\left[\frac{n-1}{\tau(I)}\right].$$ Inequality (\[Lyubeznik\]) has been proven by the same author in [@L3] for all ideals $J$ in the localized ring $R_{(x_1,\dots, x_n)}$ (if $K$ is infinite), whereas (\[Lyubeznik2\]) was established in [@L1], Theorem 6, for monomial ideals $I$ with $\tau(I)=2,3$ (see also [@B], Section 2, for a different proof in the case $\tau(I)=2$). Other special cases were examined in [@SeV] and [@SV]. In [@B] it is conjectured that (\[Lyubeznik2\]) holds for every squarefree monomial ideal $I$. The results in the previous section allow us to show, and even improve, inequality (\[Lyubeznik2\]) for a new class of monomial ideals. Corollary \[mycorollary\] implies the following:
\[mycorollary2\] Let $J$ be an ideal of $R$ such that $I=$Rad$(J)$ is a monomial ideal generated by at most $n-1$ (squarefree) monomials. Then $${\rm ara}\,J\leq n-\frac{n-1}{\tau(I)}.$$
[In the class of squarefree monomial ideals $I$ such that $\mu(I)\leq n-1$ the upper bounds given in Proposition \[mylemma\] and in Corollary \[mycorollary\] are in general strictly better than the one in (\[Lyubeznik2\]), as the next example shows. ]{}
\[example1\][Suppose that char$K=0$ and in $R=K[x_1,\dots, x_9]$ consider the ideal $$I=(x_1x_2,\ x_1x_3,\ x_2x_4,\ x_4x_5,\ x_4x_6,\ x_2x_7,\ x_6x_8,\ x_6x_9),$$ with $\mu(I)=8$. Its minimal primes can be quickly computed by CoCoA [@CoC]: $$I_1=(x_1, x_2, x_4, x_6),\qquad I_2=(x_2, x_3, x_4, x_6),\qquad I_3=(x_1, x_2, x_5, x_6),$$ $$I_4=(x_2, x_3, x_5, x_6),\qquad I_5=(x_1, x_4, x_6, x_7),\qquad I_6=(x_1, x_2, x_4, x_8, x_9),$$ $$I_7=(x_2, x_3, x_4, x_8, x_9),\qquad I_8=(x_1, x_4, x_7, x_8, x_9).$$ Thus we have $\tau(I)=5$, so that $n-\lbrack\frac{n-1}{\tau(I)}\rbrack=9-\lbrack\frac85\rbrack=8$. Hence (\[Lyubeznik2\]) yields the trivial inequality (\[ara\]). From Corollary \[mycorollary\] we derive that ara$I\leq 8-\frac85+1=7.4$, i.e., ara$I\leq 7$. Moreover, $$\vert M_3\vert=\vert M_5\vert=\vert M_7\vert=\vert M_8\vert=\vert M_9\vert=1,\qquad\vert M_1\vert =2,$$ $$\vert M_2\vert=\vert M_4\vert=\vert M_6\vert=3,$$ so that, by definition (\[nuj\]), $\nu_j=3$ for all $j=1,\dots, 8$. Therefore, by definition (\[nu\]), $\nu(I)=3$. Hence Proposition \[mylemma\] gives us ara$I\leq 8-3+1=6$. The method from [@B] that we mentioned in Remark \[remark1\] also yields ara$I\leq6$. In fact ara$I=5$: on the one hand, by virtue of Lemma \[Schmitt\], $I$ is generated up to radical by the 5 polynomials $$x_1x_2,\ x_1x_3+x_2x_4,\ x_2x_7+x_4x_6,\ x_4x_5+x_6x_8,\ x_6x_9;$$ on the other hand, according to [@L4], Theorem 1, and [@H], Example 2, p. 414, we have pd$(R/I)\leq$ara$I$, where pd denotes the projective dimension, and CoCoA tells us that here pd$(R/I)=5$. ]{}
In the previous example, the best upper bound for the arithmetical rank is the one derived from Proposition \[mylemma\]. Sometimes this proposition yields the exact value of the arithmetical rank. An interesting example of this kind can be obtained by a slight modification of the ideal in Example \[example1\].
[Suppose that char$K=0$ and in $R=K[x_1,\dots, x_9]$ consider the ideal $$I=(x_1x_2,\ x_1x_3,\ x_2x_4,\ x_4x_5,\ x_4x_6,\ x_6x_7,\ x_6x_8,\ x_6x_9),$$ with $\mu(I)=8$. The minimal primes of $I$ are $$I_1=(x_1, x_4, x_6),\qquad I_2=(x_2, x_3, x_4, x_6),\qquad I_3=(x_1, x_2, x_5, x_6),$$ $$I_4=(x_2, x_3, x_5, x_6),\qquad I_5=(x_1, x_4, x_7, x_8, x_9),\qquad I_6=(x_2, x_3, x_4, x_7, x_8, x_9).$$ Now $\tau(I)=6$, so that $n-\lbrack\frac{n-1}{\tau(I)}\rbrack=9-\lbrack\frac86\rbrack=8$. As in the previous example, (\[Lyubeznik2\]) yields a trivial upper bound. From Corollary \[mycorollary\] we deduce that ara$I\leq 8-\frac86+1=7.\bar6$, i.e., ara$I\leq 7$. Since $\nu(I)=3$, by Proposition \[mylemma\] we again have ara$I\leq6$. The same result is obtained if we use the method from [@B], and also if we apply Lemma \[Schmitt\] directly to the minimal monomial generators of $I$: $I$ is generated up to radical by the 6 polynomials $$x_2x_4,\ x_1x_2+x_4x_6,\ x_4x_5+x_6x_7,\ x_1x_3,\ x_6x_8,\ x_6x_9.$$ This time pd$(R/I)=6$, so that ara$I=6$. ]{}
[SV]{} Barile, M. (2005). On ideals whose radical is a monomial ideal. [*Commun. Algebra*]{} 33:4479–4490. Bruns, W., Herzog, J. (1993). [*Cohen-Macaulay rings*]{}; Cambridge University Press, Cambridge. CoCoATeam, CoCoA, a system for doing Computations in Commutative Algebra. Available at . Hartshorne, R. (1968). Cohomological dimension of algebraic varieties. [*Ann. Math.*]{} 88:403–450. Jaballah, A. (1988). Minimale Erzeugendensysteme und minimale Primteiler von monomialen Radikalidealen. [*Math. Ann.*]{} 280:683–686. Jaballah, A. (1990). Generating radical monomial ideals. [*Arch. Math.*]{} 55:533–536. Lyubeznik, G. (1984). Set-theoretic intersections and monomial ideals, Ph.D. Thesis, Columbia University. Lyubeznik, G. (1984). On the local cohomology modules $H^i_{\cal A}(R)$ for ideals ${\cal A}$ generated by monomials in an $R$-sequence. In: [*Complete Intersections*]{}, Lectures given at the 1$^{\rm st}$ 1983 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), Acireale, Italy, June 13–21, 1983; Greco, S., Strano, R., Eds.; Springer: Berlin Heidelberg, 1984. Lyubeznik, G. (1985). Some algebraic sets of high local cohomological dimension in $P^n$. [*Proc. Amer. Math. Soc.*]{} 95:9–10. Lyubeznik, G. (1988). On the arithmetical rank of monomial ideals. [*J. Algebra*]{} 112:86–89. Schenzel, P., Vogel, W. (1977) On set-theoretic intersections. [*J. Algebra*]{} 48:401–404. Schmitt, Th., Vogel, W. (1979). Note on set-theoretic intersections of subvarieties of projective space. [*Math. Ann.*]{} 245:247–253.
[^1]: MSC 2000: 13A10, 13A15, 13F55
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{
"pile_set_name": "ArXiv"
}
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[**Why [*one-size-fits-all*]{} vaso-modulatory interventions fail to control glioma invasion: [*in silico*]{} insights**]{}
J. C. L. Alfonso$^{1,*}$, A. Köhn-Luque$^{1,2,*}$, T. Stylianopoulos$^{3}$, F. Feuerhake$^{4,5}$, A. Deutsch$^{1}$ and H. Hatzikirou$^{1,6,\dag}$\
$^{(1)}$ Department for Innovative Methods of Computing, Center for Information Services and High Performance Computing, Technische Universität Dresden, 01062 Dresden, Germany.\
$^{(2)}$ Department of Biostatistics, Institute of Basic Medical Sciences, Faculty of Medicine, University of Oslo, 0317 Oslo, Norway.\
$^{(3)}$ Cancer Biophysics Laboratory, Department of Mechanical and Manufacturing Engineering, University of Cyprus, 1678 Nicosia, Cyprus.\
$^{(4)}$ Institute of Pathology, Medical School Hannover, 30625 Hannover, Germany.\
$^{(5)}$ Institute of Neuropathology, University Clinic Freiburg, 79117 Freiburg, Germany.\
$^{(6)}$ Department of Systems Immunology and Braunschweig Integrated Centre of Systems Biology, Helmholtz Center for Infectious Research, 38124 Braunschweig, Germany.\
$*$ These authors contributed equally to this work.\
$\dag$ Corresponding author: [email protected]\
The authors have declared that no competing interest exists.
Abstract {#abstract .unnumbered}
========
There is an ongoing debate on the therapeutic potential of vaso-modulatory interventions against glioma invasion. Prominent vasculature-targeting therapies involve functional tumour-associated blood vessel deterioration and normalisation. The former aims at tumour infarction and nutrient deprivation mediated by vascular targeting agents that induce occlusion/collapse of tumour blood vessels. In contrast, the therapeutic intention of normalising the abnormal structure and function of tumour vascular networks, e.g. via alleviating stress-induced vaso-occlusion, is to improve chemo-, immuno- and radiation therapy efficacy. Although both strategies have shown therapeutic potential, it remains unclear why they often fail to control glioma invasion into the surrounding healthy brain tissue. To shed light on this issue, we propose a mathematical model of glioma invasion focusing on the interplay between the migration/proliferation dichotomy (Go-or-Grow) of glioma cells and modulations of the functional tumour vasculature. Vaso-modulatory interventions are modelled by varying the degree of vaso-occlusion. We discovered the existence of a critical cell proliferation/diffusion ratio that separates glioma invasion responses to vaso-modulatory interventions into two distinct regimes. While for tumours, belonging to one regime, vascular modulations reduce the tumour front speed and increase the infiltration width, for those in the other regime the invasion speed increases and infiltration width decreases. We show how these [*in silico*]{} findings can be used to guide individualised approaches of vaso-modulatory treatment strategies and thereby improve success rates.
[**Keywords:**]{} glioma invasion; go-or-grow mechanism; vaso-modulatory interventions; vascular occlusion and normalization; invasion speed and infiltration width; mathematical modelling.
Introduction {#introduction .unnumbered}
============
Malignant gliomas are aggressive brain tumours typically associated with a poor prognosis, sharp deterioration in the patients’ quality of life and markedly low survival rates, making this disease a challenge to treat. According to the World Health Organization (WHO) [@Louis2007], gliomas are classified into different categories varying from low-grade (slow-growing) to high-grade (rapidly-growing) tumours depending on their proliferative capacity and invasiveness, glioblastoma multiforme (GBM) being the most malignant form. Despite advances in surgical and medical neuro-oncology [@Stupp2005; @Weller2010], complete tumour resection is unlikely and subsequent recurrence is almost inevitable. A major obstacle to cure this devastating type of brain tumours is attributed to its highly invasive nature. Glioma cells have a remarkable capacity to infiltrate the surrounding normal brain tissue and migrate long distances from the tumour bed, which enables them to escape surgical resection, radiation exposure and chemotherapy [@Giese2003; @Westphal2011; @Cuddapah2014]. The persistently poor prognosis and high treatment failure rates demand more effective therapeutic strategies that should be based on a deeper mechanistic understanding of the key events triggering tumour invasion.
The influence of the microenvironment on the behaviour of glioma cells plays a crucial role in the resulting diffusive tumour growth and infiltration into the adjacent brain tissue. Hypoxia, the presence of abnormal and sustained low oxygen levels in the tumour tissue, strongly correlates with glioma malignancy [@Evans2004]. At higher glioma cell densities, tumours contain hypoxic regions with an inadequate oxygen supply due to tumour-induced vascular abnormalities. Under such oxygen-limiting conditions, glioma cells develop a wide variety of rescue mechanisms to survive and sustain proliferation. These include recruitment of new blood vessels driven by secretion of pro-angiogenic factors, modulations of cell oxygen consumption and activation of cellular migratory mechanisms to escape from poorly oxygenated regions [@AllalunisTurner1999; @Turcotte2002; @Hatzikirou2012; @Hardee2012]. In particular, the ability of glioma cells to switch phenotype in response to metabolic stress may have important implications for tumour progression and resistance to therapies. For instance, the mutually exclusive switching between proliferative and migratory phenotypes experimentally observed, and known as the migration/proliferation dichotomy (or Go-or-Grow mechanism), is considered to significantly increase invasiveness in response to low oxygen levels [@Giese1996; @Giese2003; @Hatzikirou2012; @Bottger2012; @Bottger2015]. However, how the dynamical interplay between glioma cells and their microenvironment leads to development of hypoxic regions, as well as their global impact on glioma invasion are still not fully understood.
A particularly important component of the tumour microenvironment is the vasculature. There exist various positive and negative feedback mechanisms between glioma cells and the vasculature. Gliomas are reported as highly vascularised neoplasias [@Jain2007; @Swanson2011], where excessive vascularisation is induced by a wide range of pro-angiogenic factors [@Carmeliet2011; @Weis2011]. However, over-expression of pro-angiogenic factors produced by hypoxic glioma cells is commonly observed and results in local vascular hyperplasia with defective blood vessels. Such morphological abnormalities in the vasculature are a common feature of gliomas, where blood vessels have significantly larger diameters and thicker basement membranes than those in normal brain tissue [@Jain2007], see Figure \[fig1\](A,B). Moreover, vaso-occlusive events have been reported to initiate a hypoxia/necrosis cycle influencing the dynamical balance between migration and proliferation of glioma cells. In fact, different pathological and experimental observations suggest that vaso-occlusion could readily explain the rapid peripheral expansion and diffusely infiltrative growth behaviour of malignant gliomas [@Brat2004b; @Rong2009]. Blood vessel occlusion can mainly occur due to increased mechanical pressure exerted on them by tumour cells or induced by intravascular pro-thrombotic mechanisms [@Brat2004a; @Stamper2010], see Figure \[fig1\](C,D). Occluded or collapsed blood vessels induce perivascular tumour hypoxia and favour glioma cell migration towards better oxygenated regions. This fact has been linked to waves of hypoxic glioma cells actively migrating away from oxygen-deficient regions leading to pseudopalisade formation [@Brat2004a; @Brat2004b; @Rong2006; @Rong2009]. Since hypoxia-induced migration is recognised to support further neoplastic dissemination, investigating the overall effect of vaso-modulatory interventions on the tumour front speed and infiltration width turns crucial.
![**Histological images of functional and occluded blood vessels in malignant gliomas.** (A) From right to left brain tissue infiltrated by glioma cells with meningeal blood vessels of normal size and anatomy. (B) Atypical and not occluded intratumoural blood vessels with activated endothelium and thicker/plumper muscular layers than the normal brain vessels. (C) A longitudinal section of a large intratumoural blood vessel with a not obliterated part filled with blood (left) and an occluded part (right). (D) Thrombotic occlusion in small intratumoural blood vessels. The arrowheads point to blood vessels which are magnified in the corresponding subfigures.[]{data-label="fig1"}](Figure1.pdf){width="95.00000%"}
The high degree of angiogenesis and vascular pathologies observed in malignant gliomas have been the target of several vaso-modulatory strategies [@Jain2014a; @Wick2015]. Current clinical and preclinical findings suggest that angiogenesis inhibitors alone, with the potential to starve glioma cells, have limited efficacy in terms of tumour shrinkage, functional vasculature destruction and patient survival [@Ebos2011; @Jayson2012; @Jain2013]. Furthermore, anti-angiogenic factors as inhibitors of neovascularisation are also restricted by transient effects and development of therapy resistance [@Duda2007]. Instead, improved tumour vascularisation, either via normalisation or due to a stress alleviation strategy based on reopening compressed blood vessels, is an emerging concept expected to reduce tumour hypoxia, improve perfusion and enhance the delivery of cytotoxic drugs and radiotherapy efficacy [@Jain2001; @Jain2005; @Stylianopoulos2013; @Jain2014a]. Recent evidences indicate that judicious application of an anti-angiogenic therapy may normalise the structure and function of tumour vasculature [@Jain2001; @Jain2005; @Jain2013], where potential benefits are schedule- and patient-dependent [@Sorensen2012; @Batchelor2013]. Although vasculature-targeting interventions could provide therapeutic benefits, further mechanistic insights into glioma invasion responses are still needed to improve treatment outcomes and patient survival [@Stylianopoulos2013; @Jain2014a].
In this work, we propose a mathematical model of reaction-diffusion type that is based on well-supported biological assumptions for the growth of vascularised gliomas. In particular, we focus on the interplay between the migration/proliferation dichotomy of glioma cells and modulations of functional tumour vasculature. Mathematical modelling has the potential to improve our understanding of the complex biology of tumours and their interactions with the microenvironment, as well as may help to design more effective and personalised therapeutic strategies [@Anderson2008; @Byrne2010; @Chauviere2012; @Martinez2012; @Baldock2013; @Alfonso2014a; @Alfonso2014; @Hatzikirou2015; @Reppas2015]. Several mathematical models have been developed to identify mechanisms that facilitate proliferation and migration of glioma cells [@Tracqui1995; @Woodward1996; @Burgess1997; @Swanson2000; @Swanson2002a; @Swanson2002b; @Swanson2003; @Frieboes2007; @Swanson2008; @Swanson2011; @Martinez2012; @Gerlee2012], see also [@Hatzikirou2005; @Harpold2007] for reviews. Most of these models have been formulated to study glioma invasion based exclusively on cell diffusion and proliferation rates [@Tracqui1995; @Woodward1996; @Burgess1997; @Swanson2000; @Swanson2002b]. Among modelling results, interpretation of glioma growth patterns compared to clinical data [@Swanson2000; @Swanson2002b], as well as plausible predictions of the success or failure of different treatment techniques have been reported [@Tracqui1995; @Woodward1996; @Swanson2002a; @Swanson2003; @Swanson2008; @Rockne2010]. Recently, different models including the influence of tumour microenvironmental conditions such as hypoxia, necrosis and angiogenesis have been developed [@Swanson2011; @Martinez2012; @Gerlee2012]. However, the role of vaso-occlusion in glioma invasion, considering the Go-or-Grow mechanism, has not been addressed so far. Accordingly, we intend to generate insights into the effects of vaso-modulatory interventions on tumour front speed and infiltration width. The main aim is to use the better understanding to investigate the potential of personalised therapeutic protocols. To that end, we begin by defining the biological assumptions taken into account when developing our glioma-vasculature interplay model. We then investigate the effect of modulations of cell oxygen consumption and vaso-occlusion rates in glioma invasion. We show that one-size-fits-all vaso-modulatory interventions should be expected to fail to control glioma growth and lead to a trade-off between tumour front speed and infiltration width. The model results provide a better understanding of glioma-microenvironment interactions, and it is therefore suited for analysing the potential success or failure of vaso-modulatory treatment strategies. We conclude with a discussion of the main implications of our model results in designing novel personalised therapeutic protocols.
Materials and Methods {#materials-and-methods .unnumbered}
=====================
A glioma-vasculature interplay model {#a-glioma-vasculature-interplay-model .unnumbered}
------------------------------------
The mathematical model we develop describes the growth of vascularised gliomas focusing on the interplay between the migration/proliferation dichotomy and vaso-occlusion at the margin of viable tumour tissue. The system variables are density of glioma cells $\rho(x,t)$ and functional tumour vasculature $v(x,t)$, as well as concentrations of oxygen $\sigma(x,t)$ and pro-angiogenic factors $a(x,t)$ in the tumour microenvironment, where $(x,t)\in \mathbb{R}^d\times \mathbb{R}$ and $d$ is the dimension of the system. Figure \[fig2\](A) shows a diagram of the system interactions/assumptions considered, which are summarised as follows:\
[**\[A1\]**]{} Glioma cells switch phenotypes between proliferative (normoxic) and migratory (hypoxic) depending on the oxygen concentration in the tumour microenvironment [@Giese1996; @Giese2003; @Hatzikirou2012; @Bottger2012; @Bottger2015].\
[**\[A2\]**]{} Under hypoxia conditions glioma cells secrete large amounts of pro-angiogenic factors [@Carmeliet2011; @Weis2011; @Jain2007; @Jain2014a].\
[**\[A3\]**]{} Pro-angiogenic factors drive new blood vessel formation and vasculature remodelling [@Weis2011; @Jain2013].\
[**\[A4\]**]{} Endothelial cells uptake pro-angiogenic factors [@Weis2011; @Nakayama2013].\
[**\[A5\]**]{} Functional tumour-associated vasculature releases oxygen [@Jain2007; @Carmeliet2011; @Weis2011].\
[**\[A6\]**]{} Oxygen availability is essential for glioma growth and progression [@Carmeliet2011; @Weis2011; @Jain2014b].\
[**\[A7\]**]{} Glioma cells consume oxygen provided by the existing functional vascular network [@Carmeliet2011; @Rockne2015].\
[**\[A8\]**]{} Prothrombotic factors and high mechanical pressure induce vaso-occlusion in gliomas [@Brat2004b; @Padera2004; @Rong2006; @Jain2014b].
![**Modelling logic and hierarchy**. (A) Diagram of the interactions between glioma cells, oxygen, functional tumour-associated vasculature and pro-angiogenic factors. (B) From left to right model complexity increases with respect to the interactions between system variables: density of glioma cells $\rho$, density of functional tumour vasculature $v$ and oxygen concentration $\sigma$. $\sigma_0$ and $v_0$ represent a constant oxygen concentration and functional tumour vascularisation. The model parameters $g_2$ and $h_2$ are the rates of vaso-occlusion and glioma cell oxygen consumption, respectively (see equations (\[eq15\])-(\[eq16\])).[]{data-label="fig2"}](Figure2.pdf){width="80.00000%"}
### Density of glioma cells {#density-of-glioma-cells .unnumbered}
Based on the migration/proliferation dichotomy [@Giese1996; @Giese2003; @Hatzikirou2012; @Bottger2012; @Bottger2015], we assume that glioma cells $\rho(x,t)$ switch between two different cell phenotypes, migratory $\rho_1(x,t)$ (hypoxic) and proliferative $\rho_2(x,t)$ (normoxic), depending on the concentration of oxygen in the tumour microenvironment described by $\sigma(x,t)$. More precisely, we consider two linear switching functions, $f_{21}(\sigma) = \lambda_{1} - \sigma$ and $f_{12}(\sigma) = \lambda_{2} \sigma$, that represent the rate at which glioma cells change from migratory to proliferative and vice versa, respectively. The parameters $\lambda_{1}$ and $\lambda_{2}$ are positive constants, see the Supplementary Material for further details. Cell motility is modelled as a diffusive process mimicking the net invasion of glioma cells into the surrounding brain tissue, while a logistic growth term is considered for tumour cell proliferation. Accordingly, the system of equations governing the dynamics of migratory and proliferative glioma cells is given by
$$\begin{aligned}
\label{eq1} \frac{\partial \rho_{1}}{\partial t} &=& D_{\rho} \nabla^{2}\rho_{1} - f_{12}(\sigma)\rho_{1} + f_{21}(\sigma)\rho_{2}, \\
\label{eq2} \frac{\partial \rho_{2}}{\partial t} &=& b_{\rho} \hspace{0.5mm} \rho_{2}\left( 1- (\rho_{1}+\rho_{2})/N \right) + f_{12}(\sigma)\rho_{1} - f_{21}(\sigma)\rho_{2},\end{aligned}$$
where the temporal $t$ and spatial $x$ coordinates in the arguments of variables have been omitted for notational simplicity. $D_{\rho}$ and $b_{\rho}$ are the diffusion and proliferation rates of migratory and proliferative glioma cells, respectively. $N$ represents the brain tissue carrying capacity, i.e. the maximum number of cells that can be located within a domain element. The model parameters $D_{\rho}$, $b_{\rho}$ and $N$ are positive constants.
The system (\[eq1\])-(\[eq2\]) is reduced to a single equation for the total density of glioma cells $\rho = \rho_{1} + \rho_{2}$ by assuming that $f_{12}(\sigma)\rho_{1} = f_{21}(\sigma)\rho_{2}$. This assumption implies that each phenotypic switching event is faster compared to migration and proliferation cell processes, which allows to express $\rho_{1}$ and $\rho_2$ as a function of $\rho$ in the following form
$$\label{eq3} \rho = \left( 1 + \frac{f_{12}(\sigma)}{f_{21}(\sigma)} \right) \rho_{1} = \left( 1 + \frac{f_{21}(\sigma)}{f_{12}(\sigma)} \right) \rho_{2},$$
where we have that
$$\label{eq4} \rho_{1} = \left( \frac{1}{1 + f_{12}(\sigma) / f_{21}(\sigma)} \right) \rho$$
and
$$\label{eq5} \rho_{2} = \left( \frac{1}{1 + f_{21}(\sigma) / f_{12}(\sigma)} \right) \rho.$$
Summing equations (\[eq1\]) and (\[eq2\]), and substituting the expressions above for $\rho_{1}$ and $\rho_2$, we obtain the governing equation for the total (migratory and proliferative) density of glioma cells as follows
$$\label{eq6} \frac{\partial \rho}{\partial t} = D_{\rho} \nabla^{2}(\alpha(\sigma)\rho) + b_{\rho} \hspace{0.5mm} \beta(\sigma)\rho\left(1-(\alpha(\sigma) + \beta(\sigma)) \rho / N \right),$$
where the oxygen-dependent functions $\alpha(\sigma)$ and $\beta(\sigma)$ are given by
$$\label{eq7} \alpha(\sigma) = \frac{1}{1+ f_{12}(\sigma) / f_{21}(\sigma)} = \frac{\lambda_{1} - \sigma}{(\lambda_{2} - 1) \sigma + \lambda_{1}},$$
and
$$\label{eq8} \beta(\sigma) = \frac{1}{1+ f_{21}(\sigma) / f_{12}(\sigma)} = \frac{\lambda_{2} \sigma}{(\lambda_{2} - 1) \sigma + \lambda_{1}}.$$
Then, taking into account that $\alpha(\sigma) + \beta(\sigma)=1$, we can rewrite equation (\[eq6\]) as
$$\label{eq9} \frac{\partial \rho}{\partial t} = D_{\rho} \nabla^{2}(\alpha(\sigma)\rho) + b_{\rho} \hspace{0.5mm} \beta(\sigma) \rho \left( 1- \rho / N \right).$$
We notice that equation (\[eq9\]) is a generalisation of the widely studied Fisher-Kolmogorov model to describe glioma invasion [@Murray2002; @Harpold2007]. The nonlinear terms $\alpha(\sigma)$ and $\beta(\sigma)$ in equation (\[eq9\]) modify the rates of cell diffusion and proliferation according to oxygen availability. Under hypoxic conditions cell diffusion increases, while proliferation decreases, i.e. glioma cells become more migratory and less proliferative. On the contrary, for normal oxygen levels glioma cells become more proliferative and less invasive. Let $\sigma{_0} > 0$ be the physiological concentration of oxygen in the host brain tissue. Then, by normalising $D_{\rho} = D/\alpha(\sigma_{0})$ and $b_{\rho} = b/\beta(\sigma_{0})$ the classical Fisher-Kolmogorov equation is recovered under the assumption of a constant oxygen concentration
$$\label{eq10} \frac{\partial \rho}{\partial t} = D \nabla^{2}\rho + b \hspace{0.5mm} \rho \left( 1- \rho / N \right),$$
where $D$ and $b$ are positive constants denoting respectively the intrinsic rates of diffusion and proliferation of glioma cells. Equation (\[eq10\]) has been extensively used to predict untreated glioma invasion kinetics, as well as to estimate patient-specific parameters based on standard medical imaging [@Swanson2002b; @Harpold2007; @Swanson2011; @Hawkins2013]. Furthermore, this model allowed for suitable estimations of glioma recurrence after surgical resection [@Swanson2003] and simulations of tumour responses to conventional therapeutic modalities as chemo- [@Swanson2002a] and radiation therapy [@Rockne2010].
### Pro-angiogenic factor concentration {#pro-angiogenic-factor-concentration .unnumbered}
Neovascularisation in tumours takes place when pro-angiogenic factors overcome anti-angiogenic stimuli. However, in gliomas there is a wide range of pro- and anti-angiogenic factors involved, each of them acting through different vascularisation mechanisms [@Jain2007; @Jain2013; @Jain2014a]. While not explicitly considering the vascular endothelial growth factor (VEGF) or any other pro-angiogenic molecule, we assume a generic effective pro-angiogenic factor concentration $a$ at quasi-steady state. In fact, we suppose that an over-expression of pro-angiogenic factors instantaneously promotes formation of functional tumour vasculature. We further assume that pro-angiogenic factors are only produced by glioma cells under hypoxic conditions at a rate proportional to tumour cell density, and therefore neglect hypoxia-independent pathways. Moreover, pro-angiogenic factors are consumed by endothelial cells and undergo natural decay. The equation for the effective pro-angiogenic factor concentration $a(x,t)$ is given by
$$\label{eq11} 0 = k_{\text{1}} \hspace{0.5mm} \rho \hspace{0.5mm} \tilde{\text{H}}_{\theta}(\sigma-\sigma_a^{*}) - k_{2}av - k_{3}a,$$
where
$$\label{eq12} a = \frac{k_{1} \hspace{0.5mm} \rho \hspace{0.5mm} \tilde{\text{H}}_{\theta}(\sigma - \sigma_a^{*})}{k_{2}v+k_{3}}.$$
The positive constants $k_{1}$, $k_{2}$ and $k_{3}$ represent the production, consumption and natural decay rates, respectively, where $0 < \sigma_a^{*} < \sigma_0$ is the hypoxic oxygen threshold for production of pro-angiogenic factors by glioma cells. The function $\tilde{\text{H}}_{\theta}(\sigma - \sigma_a^{*})$ is a continuous approximation of the Heaviside decreasing step function $H(\xi)$, which is defined as $H(\xi) = 1$ if $\xi \leq 0$ and $H(\xi) = 0$ if $\xi > 0$, and given by
$$\label{eq13} \tilde{\text{H}}_{\theta}(\sigma - \sigma_a^{*})=1-\frac{1}{1+e^{-2\theta (\sigma - \sigma_a^{*})}},$$
where $\theta$ is a positive constant that controls the steepness of $\tilde{\text{H}}_{\theta}$ at $(\sigma - \sigma_a^{*})$.
### Density of functional tumour vasculature {#density-of-functional-tumour-vasculature .unnumbered}
Several experimental findings support that vascular structure and function become markedly abnormal in brain tumours [@Carmeliet2011; @Weis2011; @Jain2014b]. Malignant gliomas, and particularly glioblastomas, have blood vessels of increased diameter, high permeability, thickened basement membranes and highly proliferative endothelial cells [@Jain2007], see also Figure \[fig1\](B). Due to such abnormalities, a significant fraction of the tumour-associated vascular network does not constitute functional blood vessels [@Jain2007]. Based on these observations, we exclusively account for functional vascularisation instead of modelling the total density of tumour blood vessels. Accordingly, we assume that the density of functional tumour vasculature $v(x,t)$ is a dimensionless and normalised quantity with values in the interval $[0, 1]$. The normal density of functional vascularisation in the host brain tissue is taken equal to $v = 1/2$. The limit case $v = 0$ represents an avascular tissue, while on the contrary $v = 1$ describes a hypothetical scenario characterised by excessive vascularisation.
Blood vessels in gliomas are not stable, being continuously formed, occluded and destroyed. Neovascularisation takes place by different angiogenic and vasculogenic processes induced by complex signalling mechanisms that are not well understood [@Hardee2012; @Kohn2013; @Sugihara2015]. For simplicity, we assume that tumour blood vessels are created when pro-angiogenic factors prevail anti-angiogenic stimuli, i.e. for $a > 0$, leading to development of new functional vasculature according to a logistic growth term. The rate at which such vasculature is generated follows the Michaelis-Menten kinetics depending on the pro-angiogenic factor concentration, where diffusive vascular dispersal at a constant rate is assumed. On the other hand, mechanical or chemical cues in regions of high glioma cell density induce blood vessel occlusion or collapse [@Brat2004b; @Padera2004; @Rong2006]. Vaso-occlusion is then modelled by an exponential term depending on the density of glioma cells. The equation for the density of functional tumour vasculature $v(x,t)$ is given by
$$\label{eq14} \frac{\partial v}{\partial t} = D_{v}\nabla^{2}v + g_{1}\frac{a}{\mu+a}v\left(1-v\right) - g_{2} v \rho^{n},$$
where the temporal $t$ and spatial $x$ coordinates in the arguments of variables have been omitted for notational simplicity. $D_{v}$ is the diffusion coefficient representing the net dispersal of functional tumour vasculature, $g_{1}$ is the maximum formation rate of functional blood vessels, $\mu$ is the pro-angiogenic factor concentration at which $g_{1}$ is half-maximal, $g_{2}$ is the vaso-occlusion rate and $n$ is a parameter that controls the degree of vaso-occlusion depending on the density of glioma cells. The model parameters $D_{v}$, $g_{1}$, $\mu$, $g_{2}$ and $n$ are positive constants.
Plugging equation (\[eq12\]) for effective pro-angiogenic factor concentration into equation (\[eq14\]), and assuming that the decay rate of $a$ is much smaller than the consumption rate by endothelial cells, i.e. $k_3 \ll k_2$ [@KohnLuque2013], we obtain that
$$\label{eq15} \frac{\partial v}{\partial t} = D_{v}\nabla^{2}v + g_{1}\dfrac{ \dfrac{\rho}{v}\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a})} {K + \dfrac{\rho}{v}\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a})}v\left(1-v\right) - g_{2} v \rho^{n},$$
where $K = \mu k_{2} / k_{1}$ is a positive constant denoting the concentration of pro-angiogenic factors at which the functional tumour vasculature formation rate is half-maximal, see the Supplementary Material for more details.
### Oxygen concentration {#oxygen-concentration .unnumbered}
Oxygen is delivered to the host brain tissue via functional blood vessels, spreads into the tumour mass and is consumed by glioma cells. Transport of oxygen within tissues occurs by diffusion and convection [@Jain1999]. For simplicity, we neglect the convective contributions and only consider that after transvascular exchange oxygen molecules move exclusively by diffusion. Oxygen supply is modelled by assuming that the supply rate is proportional to the functional vascularisation and the difference between the physiological oxygen concentration in the host brain tissue and that in the tumour interstitium. These assumptions result in the following equation for the oxygen concentration $\sigma(x,t)$
$$\label{eq16} \frac{\partial \sigma}{\partial t} = D_{\sigma}\nabla^{2}\sigma + h_{1} v \left(\sigma_{0} - \sigma \right) - h_{2}\rho \sigma,$$
where the temporal $t$ and spatial $x$ coordinates in the arguments of variables have been omitted for notational simplicity. $D_{\sigma}$ is the oxygen diffusion coefficient, $h_{1}$ is the permeability coefficient of functional blood vessels, $\sigma_{0}$ is the physiological oxygen concentration in the host brain tissue and $h_{2}$ is the oxygen consumption rate by glioma cells. The model parameters $D_{\sigma}$, $h_{1}$, $\sigma_{0}$ and $h_{2}$ are positive constants. Similar assumptions have been previously considered to model oxygen-related mechanisms in tumour growth [@Stamper2010].
### Model formulation, boundary and initial conditions {#model-formulation-boundary-and-initial-conditions .unnumbered}
The proposed model of glioma-vasculature interplay comprises the following system of coupled partial differential equations
$$\begin{aligned}
\label{eq17} \frac{\partial \rho}{\partial t} &=& D_{\rho} \nabla^{2}(\alpha(\sigma)\rho) + b_{\rho} \hspace{0.5mm} \beta(\sigma)\rho\left(1- \rho/N \right), \\
\label{eq18} \frac{\partial v}{\partial t} &=& D_{v}\nabla^{2}v + g_{1}\dfrac{ \dfrac{\rho}{v}\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a})} {K + \dfrac{\rho}{v}\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a})}v\left(1-v\right) - g_{2} v \rho^{n}, \\
\label{eq19} \frac{\partial \sigma}{\partial t} &=& D_{\sigma}\nabla^{2}\sigma + h_{1}v \left(\sigma_{0} - \sigma \right) - h_{2} \rho \sigma,\end{aligned}$$
where the oxygen-dependent functions $\alpha(\sigma)$ and $\beta(\sigma)$ are given by equations (\[eq7\])-(\[eq8\]), respectively. The system of equations above is closed by imposing the following initial conditions
$$\begin{aligned}
\rho(x,0) &=& \rho_{0} \tilde{H}_{\gamma}(x-\epsilon) = \rho_{0} \left( 1 - \frac{1}{1+e^{-2 \gamma (x-\epsilon)}} \right), \hspace{5mm} 0 \leq x \leq L, \\
v(x,0) &=& v_{0}, \hspace{62mm} 0 \leq x \leq L, \\
\sigma(x,0) &=& \sigma_{0}, \hspace{62mm} 0 \leq x \leq L,\end{aligned}$$
where the positive constants $\rho_{0}$, $\sigma_{0}$ and $v_{0}$ are the initial density of glioma cells located in a small segment of length $\epsilon$, density of functional tumour vasculature and oxygen concentration, respectively. The length of the one-dimensional simulation domain is represented by $L > 0$, and $\gamma$ is a positive constant that controls the steepness of $\tilde{H}_{\gamma}$ at $(x-\epsilon)$ with $\epsilon > 0$. Moreover, we consider an isolated host tissue in which all behaviours arise due to the interaction terms. This assumption results in no-flux boundary conditions of the form
$$\begin{aligned}
\begin{split}
\rho_{x}(0,t) &=& v_{x}(0,t) &=& \sigma_{x}(0,t) &=& 0, \hspace{5mm} 0 \leq t \leq T_{f}, \\
\rho_{x}(L,t) &=& v_{x}(L,t) &=&\sigma_{x}(L,t) &=& 0, \hspace{5mm} 0 \leq t \leq T_{f},
\end{split}\end{aligned}$$
where $T_{f} > 0$ is an arbitrary time. These boundary conditions also imply that no cell or molecule leaves the system through the tissue/domain boundaries.
Modelling hierarchy {#modelling-hierarchy .unnumbered}
-------------------
The glioma-vasculature interplay model given by equations (\[eq17\])-(\[eq19\]), and referred to as [*model III*]{}, is a generalisation of two simpler models which are also of interest for the study of glioma invasion. As shown in Figure \[fig2\](B), such simpler models are obtained under the assumptions of constant density of functional tumour vasculature $v(x,t) = v_0$ ([*model II*]{}), and also constant oxygen concentration $\sigma(x,t) = \sigma_{0}$ ([*model I*]{}). Specifically, [*model II*]{} is obtained from [*model III*]{} by setting $g_{1} = g_{2} = 0$ in equation (\[eq18\]), i.e. assuming neither formation nor occlusion/collapse of tumour blood vessels. In turn, [*model I*]{} is obtained from [*model II*]{} by setting $h_{2} = 0$ in equation (\[eq19\]), i.e. assuming a constant concentration of oxygen in the tumour microenvironment.
[*Model I*]{} is similar to the classical Fisher-Kolmogorov equation (\[eq10\]), for which a large number of theoretical and simulation results are known [@Murray2002; @Harpold2007]. [*Model II*]{} given by equations (\[eq17\]) and (\[eq19\]) contains an extended version of the Fisher-Kolmogorov equation with nonlinear glioma cell diffusion and proliferation terms. Both nonlinearities depend on the oxygen concentration in the tumour microenvironment, which is governed by a reaction-diffusion equation with linear diffusion and nonlinear reaction terms. In addition, the dynamics of the glioma cell population are modelled by considering the migration/proliferation dichotomy (Go-or-Grow). As in [*model II*]{} the supply of oxygen is assumed constant, the blood perfusion is stable and we neglect tumour-induced vascular pathologies. The latter is a a reasonable assumption, especially for low grade gliomas, where abnormal vasculature is not prominent [@Swanson2011]. A natural extension of [*model II*]{} is to consider tumour-associated vascularisation dynamics. Accordingly, [*model III*]{} is formulated to investigate the effects of vaso-modulatory interventions on glioma invasion. Taking into account the huge amount of results reported from [*model I*]{}, we analyse [*model II*]{} as an intermediate step towards the study of [*model III*]{}, see Figure \[fig2\](B). In particular, we focus on the impact of glioma cell oxygen consumption and vaso-occlusion modulations on tumour front speed and infiltration width. In the Supplementary Material we provide details about model simulations and the numerical implementation.
Model observables {#model-observables .unnumbered}
-----------------
We characterise glioma invasion by the tumour front speed and infiltration width, see Figure S1 in the Supplementary Material. The tumour front speed is estimated by the rate of change given by the point of maximum slope in $\rho(x,t)$ at the end of numerical simulations $T_f$. In turn, the infiltration width is defined by the difference between the points where glioma cell density is $80\%$ and $2\%$ of the maximum cell density at time $T_f$. These tumour invasion properties have been reported crucial to determine glioma malignancy and therapeutic failure rates [@Swanson2003; @Harpold2007; @Swanson2011].
Unlike the mathematical model given by the classical Fisher-Kolmogorov equation (\[eq10\]), in our glioma invasion model given by equations (\[eq17\])-(\[eq19\]) cell processes are regulated by oxygen availability. Thus, we distinguish intrinsic cell diffusion $D$ and proliferation $b$ rates from effective rates which take into account the oxygen concentration in the tumour microenvironment. Accordingly, the effective diffusion $D_{\mbox{eff}}$ and proliferation $b_{\mbox{eff}}$ rates are defined as
$$\label{eq20} D_{\mbox{eff}} = D_{\rho} \hspace{1mm} L^{-1} \int\limits_{L} \alpha(\sigma(x,t)) \hspace{1mm} dx$$
and
$$\label{eq21} b_{\mbox{eff}} = b_{\rho} \hspace{1mm} L^{-1} \int\limits_{L} \beta(\sigma(x,t)) \hspace{1mm} dx,$$
where $L$ represents the length of the one-dimensional domain of simulation, $D_{\rho} = D/\alpha(\sigma_{0})$ and $b_{\rho} = b/\beta(\sigma_{0})$, where $D$ and $b$ are the intrinsic rates of glioma cell diffusion and proliferation, respectively. The parameter $\sigma{_0}$ is the physiological concentration of oxygen in the host brain tissue. In the following, we investigate the dependence of $D_{\mbox{eff}}$ and $b_{\mbox{eff}}$ at time $T_f$, as well as the tumour front speed and infiltration width, for different ranges of model parameters $h_2$ (cell oxygen consumption) and $g_2$ (vaso-occlusion).
Model parameterisation {#model-parameterisation .unnumbered}
----------------------
Model parameter values are taken from published data wherever possible or estimated to approximate physiologic conditions based on appropriate physical arguments, see Table \[tab1\] and the Supplementary Material for further details. For parameters of special interest, a wide range of values is considered to explore their effects on the resulting glioma invasion.
[**Parameter**]{} [**Description**]{} [**Value**]{} [**Source**]{}
------------------- -------------------------------------------------- ---------------------------------------------------------------------------------- ------------------------------------------------ --
Glioma Cells
$D$ Intrinsic diffusion rate of glioma cells \[$2.73 \times 10^{-3}$, $2.73 \times 10^{-1}$\] mm$^2$ day$^{-1}$ [@Harpold2007; @Swanson2011; @Badoual2014]
$b$ Intrinsic proliferation rate of glioma cells \[$2.73 \times 10^{-4}$, $2.73 \times 10^{-2}$\] day$^{-1}$ [@Harpold2007; @Swanson2011; @Badoual2014]
$N$ Brain tissue carrying capacity $10^{2}$ cells mm$^{-1}$ [@Eikenberry2009; @McDaniel2013]
$\sigma_0$ Physiological oxygen concentration 1.0 nmol mm$^{-1}$ [@Hoffman1996; @Carreau2011]
$\lambda_1$ Phenotypic switching parameter $^{(\dagger)}$ 2.0 nmol mm$^{-1}$ [*Model specific*]{}
$\lambda_2$ Phenotypic switching parameter $^{(\ddagger)}$ $\{0.5,~1.0,~2.0\}$ [*Model specific*]{}
Oxygen
$D_{\sigma}$ Diffusion rate of oxygen $1.51 \times 10^{2}$ mm$^2$ day$^{-1}$ [@Matzavinos2009; @Stamper2010; @Powathil2012]
$h_{1}$ Oxygen supply rate $3.37 \times 10^{-1}$ day$^{-1}$ [@Eggleton1998; @Goldman2000; @Kelly2006]
$h_{2}$ Oxygen consumption rate $[5.73 \times 10^{-3},~1.14 \times 10^{-1}]$ mm cell$^{-1}$ day$^{-1}$ [@Vaupel1989; @Grimes2014]
Vasculature
$D_{v}$ Vasculature dispersal rate $5.0 \times 10^{-4}$ mm$^2$ day$^{-1}$ [@Anderson1998; @Stamper2010; @Swanson2011]
$g_{1}$ Vasculature formation rate $10^{-1}$ day$^{-1}$ [@Shaifer2010; @Stamper2010; @Scianna2013]
$\sigma^{*}_{a}$ Oxygen concentration threshold for hypoxia $2.5 \times 10^{-1}$ nmol mm$^{-1}$ [@Cardenas2004; @Vaupel2007; @Powathil2012]
$K$ Half-maximal pro-angiogenic factor concentration 1.0 nmol mm$^{-1}$ [*Estimated*]{}
$g_2$ Vaso-occlusion rate \[$5.0 \times 10^{-13}$, $1.5 \times 10^{-11}$\] cell$^{-n}$ mm$^{n}$ day$^{-1}$ [*Estimated*]{}
$n$ Dimensionless vaso-occlusion degree 6 [*Estimated*]{}
: Model parameter values (see the Supplementary Material).[]{data-label="tab1"}
Results {#results .unnumbered}
=======
Increasing cell oxygen consumption and vaso-occlusion result in more diffusive and less proliferative gliomas {#increasing-cell-oxygen-consumption-and-vaso-occlusion-result-in-more-diffusive-and-less-proliferative-gliomas .unnumbered}
-------------------------------------------------------------------------------------------------------------
The glioma-vasculature interplay model given by equations (\[eq17\])-(\[eq19\]) is first used to investigate the effects of cell oxygen consumption and vaso-occlusion modulations on the effective behaviour of gliomas. Figures \[fig3\](A,B) and \[fig4\](A,B) provide simulation maps of effective diffusion $D_{\mbox{eff}}$ and proliferation $b_{\mbox{eff}}$ rates, as defined in equations (\[eq20\]) and (\[eq21\]), for gliomas characterised by different combinations of intrinsic cell coefficients $D$ and $b$. Model simulations in Figure \[fig3\](A,B) are obtained under the assumption of constant functional vasculature density, i.e. neither formation nor occlusion/collapse of tumour blood vessels, for increasing oxygen consumption rates by glioma cells. In turn, Figure \[fig4\](A,B) provides simulation maps for a fixed oxygen consumption rate considering tumour vascularisation dynamics and increasing vaso-occlusion rates.
Comparative simulation maps in Figures \[fig3\](A,B) and \[fig4\](A,B) illustrate that increasing the rate of oxygen consumption by glioma cells $h_2$ and vaso-occlusion $g_2$ result in more diffusive and less proliferative tumours. Modulations of the oxygen consumption rate have major impact on highly infiltrative and rapidly growing gliomas. At high values of both parameters, $h_2$ and $g_2$, the oxygen concentration in the tumour microenvironment significantly decreases. The lack of oxygen limits the proliferative capacity of glioma cells, and in turn enhances the hypoxia-induced cell migration towards better oxygenated brain tissue regions. The precise way in which such changes in glioma cell dynamics affect invasion responses are predicted to depend on the intrinsic tumour features.
![**Oxygen consumption effects on glioma invasion for constant functional tumour vasculature.** Simulation maps with respect to the intrinsic proliferation $b \in [2.73 \times 10^{-4},~2.73 \times 10^{-2}]$ days$^{-1}$ and diffusion $D \in [2.73 \times 10^{-3},~2.73 \times 10^{-1}]$ mm$^2$ days$^{-1}$ rates of glioma cells. (A) Effective diffusion, (B) effective proliferation, (C) tumour front speed and (D) infiltration width for different oxygen consumption rates $h_2 = \{ 5.73 \times 10^{-4},~5.73 \times 10^{-3},~5.73 \times 10^{-2}\}$ mm cell$^{-1}$ day$^{-1}$ in simulation maps I-III respectively. (A-D) Differences between simulation maps are provided. The other model parameters are as in Table \[tab1\].[]{data-label="fig3"}](Figure3.pdf){width="84.00000%"}
![**Vaso-occlusion effects on glioma invasion.** Simulation maps with respect to the intrinsic proliferation $b \in [2.73 \times 10^{-4},~2.73 \times 10^{-2}]$ days$^{-1}$ and diffusion $D \in [2.73 \times 10^{-3},~2.73 \times 10^{-1}]$ mm$^2$ days$^{-1}$ rates of glioma cells. (A) Effective diffusion, (B) effective proliferation, (C) tumour front speed and (D) infiltration width for a fixed oxygen consumption $h_2 = 5.73 \times 10^{-3}$ mm cell$^{-1}$ day$^{-1}$ and different vaso-occlusion $g_2 = \{5.0 \times 10^{-13},~5.0 \times 10^{-12},~1.5 \times 10^{-11}\}$ cells$^{-n}$ mm$^{n}$ day$^{-1}$ rates in simulation maps I-III respectively. (A-D) Differences between simulation maps are provided. The other model parameters are as in Table \[tab1\].[]{data-label="fig4"}](Figure4.pdf){width="84.00000%"}
Modulations of cell oxygen consumption and vaso-occlusion rate result in opposing effects on glioma invasion {#modulations-of-cell-oxygen-consumption-and-vaso-occlusion-rate-result-in-opposing-effects-on-glioma-invasion .unnumbered}
------------------------------------------------------------------------------------------------------------
Figures \[fig3\](C,D) and \[fig4\](C,D) show simulation maps of tumour front speed and infiltration width with respect to different combinations of intrinsic cell coefficients $D$ and $b$. In particular, these glioma invasion properties are determined by a non-linear relationship between effective diffusion $D_{\mbox{eff}}$ and proliferation $b_{\mbox{eff}}$ of glioma cells. For instance, in the simplest case of [*model I*]{} similar to the classical Fisher-Kolmogorov equation (\[eq10\]), the tumour front speed is proportional to $\sqrt{D_{\mbox{eff}}\hspace{1.0mm}b_{\mbox{eff}}}$ and infiltration width $\sqrt{D_{\mbox{eff}} / b_{\mbox{eff}}}$. Model simulations predict that, depending on the intrinsic tumour features, modulations of cell oxygen consumption and vaso-occlusion rates produce opposing effects on the resulting front speed and infiltration width. These findings are counter-intuitive and might have important implications for possible modulatory interventions targeting cell oxygen consumption and vaso-occlusion in gliomas, as discussed below.
Cell oxygen consumption variations reveal a critical proliferation rate for glioma invasion {#cell-oxygen-consumption-variations-reveal-a-critical-proliferation-rate-for-glioma-invasion .unnumbered}
-------------------------------------------------------------------------------------------
Model analysis, under the assumption of constant density of functional tumour vasculature, reveals that modulations of the rate $h_2$ at which glioma cells consume oxygen produce opposing effects on the tumour front speed. More precisely, Figure \[fig3\](C) reveals that there exists a critical proliferation rate $b^{*}$ for which the front speed of gliomas characterised by $b > b^{*}$ decreases at higher values of $h_2$, while on the contrary tumours with $b < b^{*}$ invade faster. Assuming that tumour front speed is proportional to the product of effective diffusion and proliferation rates, we can easily understand the afore-mentioned results for variations of $h_2$. In particular, above the critical proliferation rate $b^{*}$ effective diffusion and proliferation negate each other and leave the resulting front speed almost invariant. For $b < b^{*}$, the effective tumour proliferation remains intact, but the effective diffusion capacity increases for raising $h_2$ values inducing higher front speeds.
The flatness/steepness of tumour fronts is determined by a relation dependent on the ratio of effective diffusion and proliferation rates. When oxygen is not limited, highly diffusive tumours evolve with large and flat fronts, whereas increased cell proliferation results in short and steep fronts. However, under oxygen-limiting conditions this relation is markedly influenced by the specific rate at which glioma cells consume oxygen. Figure \[fig3\](D) shows that variations in the cell oxygen consumption rate have always the same overall impact on the tumour infiltration width. Comparative simulation maps reveal that whatever the intrinsic tumour features, an arbitrary increase (decrease) in the cell oxygen consumption rate produces larger (smaller) infiltrative responses. Indeed, the effective glioma proliferation capacity is reduced for increasing oxygen consumption rates and in turn hypoxia-induced effective migration is enhanced, yielding more infiltrative tumour growth patterns.
Modulation of tumour vaso-occlusion reveals a critical cell proliferation/diffusion ratio for glioma invasion {#modulation-of-tumour-vaso-occlusion-reveals-a-critical-cell-proliferationdiffusion-ratio-for-glioma-invasion .unnumbered}
-------------------------------------------------------------------------------------------------------------
Model simulations show that for rising vaso-occlusion rates $g_2$, the front speed is affected differently depending on the intrinsic diffusion and proliferation rates of glioma cells. In this case, glioma invasion is additionally influenced by vascularisation mechanisms. Comparative simulation maps in Figure \[fig4\](C) suggest that tumours with features inside a region delimited by a critical proliferation rate $b^{+}$ and an approximate ratio between cell diffusion and proliferation rates $\Lambda^{+} = b/D$ invade faster as $g_2$ increases. The tumour front speed out of such region decreases or remains invariant. Gliomas characterised by $b < b^{+}$ evolve at low cell density and thus vaso-occlusive events hardly occur. On the other hand, increasing vaso-occlusion rates for $b > b^{+}$ enhances effective migration towards better vascularised brain tissue areas. Although vaso-occlusion limits the proliferative activity of glioma cells, faster front speeds are obtained as long as the induced migratory responses dominate.
The infiltration width of gliomas with $b < b^{+}$ is almost unaffected for increasing vaso-occlusion rates as shown in Figure \[fig4\](D). However, gliomas characterised by $b > b^{+}$ are also separated by an approximated linear relationship between cell coefficients $D$ and $b$ with respect to variations in the infiltration width. In particular, increasing vaso-occlusive events results in larger flat fronts for gliomas with cell proliferation/diffusion ratios above the critical one, while the infiltration width decreases in the remaining cases.
Discussion {#discussion .unnumbered}
==========
In this work, we developed a deterministic mathematical model of glioma invasion which is formulated as a system of reaction-diffusion equations. The model accounts for the dynamics of normoxic and hypoxic glioma cells based on the Go-or-Grow mechanism and influenced by the functional tumour-associated vasculature, as well as concentrations of pro-angiogenic factors and oxygen in the tumour microenvironment. Specifically, we focused on the effects of cell oxygen consumption and vascular modulations on relevant properties of glioma invasion, i.e. tumour front speed and infiltration width. The main simulation results of the model are summarised in Figure \[fig5\].
![**Overview of model simulation results.** (A) Modulations of cell oxygen consumption under the assumption of constant functional vasculature density reveal a critical proliferation rate $b^{*}$ in glioma invasion responses ([*model II*]{}). (B) Modulations of functional tumour-associated vasculature reveal a critical proliferation/diffusion ratio $\Lambda^{+} = b/D$ for proliferation rates higher than $b^{+}$ in glioma invasion responses ([*model III*]{}). Colour gradients from low to high represent the increase of cell oxygen consumption and vaso-occlusion rates. The purple and black wedges/bars represent the resulting effects on tumour front speed and infiltration width, for increasing/decreasing cell oxygen consumption and vaso-occlusion rates.[]{data-label="fig5"}](Figure5.pdf){width="75.00000%"}
The model analysis reveals that increasing cell oxygen consumption and vaso-occlusion rates result in more diffusive and less proliferative gliomas. In both scenarios, the average oxygen concentration in the tumour microenvironment decreases which limits cell proliferation and enhances hypoxia-induced migration. However, the extent to which such oxygen-mediated cell responses to vasculature-targeting treatment interventions influence glioma invasion depends on the specific intrinsic tumour features. Modulations of the functional tumour-associated vasculature reveals the existence of a critical cell proliferation/diffusion ratio for glioma invasion responses, see Figure \[fig5\](B). This fact is observed for gliomas evolving with sufficiently high cell proliferation rates for variations in the oxygen concentration, due to vaso-occlusion or normalisation, significantly influences tumour cell dynamics. In such cases, tumour vascular modulations are predicted to produce opposing effects on front speed and infiltration width. Moreover, we found that depending on the intrinsic tumour features two distinct regimes can be identified where invasive behaviours in responses to vaso-modulatory interventions are different. A pro-thrombotic treatment is predicted to increase front speeds, but in turn reduces infiltration capacity of gliomas characterised by a cell proliferation/diffusion ratio below the critical threshold. On the contrary, gliomas in the other regime under the same treatment strategy become increasingly infiltrative and slowly growing. Analogously, vascular normalisation therapies produce opposing results for the corresponding parameter regimes.
Recently, it has been shown that the migration/proliferation dichotomy introduces a critical glioma cell density threshold separating tumour growth and extinction dynamics, a phenomenon called Allee effect [@Bottger2015]. Here, we also identify critical parameter values that distinguish different glioma invasive behaviours with respect to variations of cell oxygen consumption or vaso-occlusion. Interestingly, this is an emergent consequence of the Go-or-Grow plasticity, since in its absence (see *model I*) no critical behaviour is observed. Assuming or not tumour vasculature dynamics, the Go-or-Grow induced criticality is expressed either in the form of a critical intrinsic proliferation/diffusion ratio $\Lambda^+$ or an intrinsic proliferation rate $b^*$, respectively, see Figure \[fig5\]. This result highlights the importance of further investigating the clinical effects of the Go-or-Grow phenomenon on glioma invasion.
The above *in silico* findings demonstrate that [*one-size-fits-all*]{} vaso-modulatory interventions should be expected to fail to control glioma invasion due to the complexity of the involved mechanisms and the heterogeneity of patient- and tumour-related factors. This study proves the value of personalised treatment strategies based on a precise tumour profiling and provides a modelling framework with the potential to parametrise model predictions based on biopsy measurements. In particular, individual estimation of intrinsic proliferation and diffusion rates, for instance via biopsy tumour sample analysis, would be crucial components of such future tailored approaches to personalised glioma therapy. Moreover, this work substantially expands the current theoretical concepts in glioma invasion, showing that any vasculature-targeting therapeutic intervention will inevitably lead to a trade-off between tumour front speed and infiltration width. This finding suggests that vaso-modulatory therapies should be embedded in personalised combination therapy regimens, in which anti-angiogenesis might be integrated with individually adjusted other modules targeting proliferation, metabolism or tumour immunology. For instance, in the case of gliomas characterised by a high intrinsic proliferation/diffusion ratio, a pro-thrombotic or an anti-vasogenic treatment technique may reduce tumour invasion speed, but at the same time leads to highly infiltrative responses that makes this therapeutic strategy rather inappropriate. However, selecting a blood vessel normalisation strategy results in faster growing gliomas as a bulk with less-infiltrating morphologies. Thus, surgical resection could be considered to remove such compact tumours. In turn, the benefits of conventional treatments such as chemotherapy, radiotherapy and immunotherapy might increase in better-vascularised tumours [@Jain2001; @Jain2005; @Stylianopoulos2013; @Jain2014a]. Therefore, an accurate glioma patient stratification during clinical decision-making is predicted relevant for the efficacy of vasculature-targeting therapies, based on either tumour-associated blood vessel deterioration or normalisation.
This work provides a mathematical framework for exploring novel approaches to rational combination therapies or regimens composed of subsequent periods of vaso-modulatory interventions and potentially other therapeutic modules. In our model the vaso-occlusion term is rather phenomenological and more accurate modelling is required. Furthermore, the migration/proliferation dichotomy has been modelled in the simplest possible way and more informed models could be integrated. In turn, intra-tumour genetic diversity is not directly considered, but we take into account phenotypic diversity depending on the oxygen availability, that is crucial for therapeutic outcomes. The latter is supported by evidences that genetic diversity is tumour-subtype specific and not significantly affected during treatment, while phenotypic heterogeneity is different before and after therapy [@Almendro2014]. Despite the fact that the model involves a large number of model parameters, their values were defined independently from each other based on published experimental data. For those parameters estimated, a parametric analysis was performed and we concluded that variations of their values do not affect the general conclusions of this study. At this stage, we restrict the modelling strategy to the effects of vasculature-targeting therapies, however, we are aware of the fact that further aspects of tumour biology may play a crucial role. In fact, we aim to investigate the interactions between the immune system and angiogenesis as an additional level of complexity given the potential success of immunomodulatory therapies. In particular, macrophages are likely to be involved in relevant mechanisms and will be included in future developments of the current approach. This is particularly relevant in the light of recent advanced in molecular classification of malignant gliomas [@Ceccarelli2016]. Mathematical modelling provides an integrative approach for conventional radiological, biopsy and molecular tumour characterization, allowing for the prediction of glioma treatment responses and translation into clinical decision-making.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by the Free State of Saxony and European Social Fund of the European Union (ESF, grant GlioMath-Dresden). J. C. L. Alfonso, F. Feuerhake and H. Hatzikirou gratefully acknowledge the funding support of the German Federal Ministry of Education and Research (BMBF) for the eMED project SYSIMIT (01ZX1308D). A. Deutsch acknowledges the support by Deutsche Krebshilfe. Authors also thank the Center for Information Services and High Performance Computing (ZIH) at TU Dresden for generous allocations of computational resources.
Supplementary Material {#supplementary-material .unnumbered}
======================
1.1 Numerical implementation {#numerical-implementation .unnumbered}
----------------------------
Numerical solutions of the proposed glioma-vasculature interplay model are obtained by implementing the finite element method and backward Euler scheme for spatial and temporal discretisation, respectively [@Larsson2008; @Johnson2012]. The system of coupled partial differential equations (17)-(19) is first transformed into a weak formulation, which results in a system of ordinary differential equations with respect to time. The one-dimensional domain of simulation over which such equations are numerically solved is divided into a finite number of distinct and non-overlapping linear elements. The integrals involved in the weak form of the system are calculated on each domain element by means of a Gaussian quadrature formula, which exactly integrates the resulting polynomials [@Khursheed2012]. The backward Euler scheme is then used to obtain a temporal discretisation that results in a nonlinear system of equations solved at each instant of time by the Newton-Raphson method [@Larsson2008]. Model simulations were carried out using MATLAB software ([*www.mathworks.com*]{}) in a SuSE Linux Enterprise Server 11 with 5888 core AMD Opteron 6274 2.2GHz, 92 nodes each with 64 cores and 64 to 512 GB of memory.
1.2 Simulation domain {#simulation-domain .unnumbered}
---------------------
The system of equations (17)-(19) is solved in a one-dimensional domain $\Omega$ of length $L = 200$ mm for a total simulation time of 3 years, i.e. $T_{f} = 1095$ days. The independent system variables are time $t$ and space $x$ with $0 \leq x \leq L$ and $0 \leq t \leq T_f$. The $x$-axis can be thought of as a two-dimensional domain which is spatially averaged in one direction. The simulation domain, either inside the region occupied by glioma cells or outside representing the host brain tissue, is discretised into an irregular grid varying from a minimum segment length of $2.5 \times 10^{-3}$ mm to a maximum one of $2.5 \times 10^{-2}$ mm. The time step is taken equal to $0.25$ day, i.e. 6 hours. Both, segment length and time step are properly selected to ensure numerical stability.
1.3 Model observables {#model-observables-1 .unnumbered}
---------------------
We characterise glioma invasion by the tumour front speed and infiltration width. The front speed is estimated by the rate of change given by the point of maximum slope in $\rho(x,t)$ at the end of numerical simulations $T_f$, see Figure S1. In turn, the infiltration width is defined by the difference between the points where glioma cell density is $80\%$ and $2\%$ of the maximum cell density $\overline{\rho}$ at time $T_f$.
\[figure: diagrams\] ![Model observables. The tumour front is defined by the point of maximum slope in $\rho(x,t)$ (green) and infiltration width (red).[]{data-label="figS1"}](FigureS1.pdf "fig:"){width="40.00000%"}
2 Model parameterisation {#model-parameterisation-1 .unnumbered}
------------------------
### 2.1 Initial conditions {#initial-conditions .unnumbered}
Density of functional tumour vasculature and oxygen concentration are initialised in the domain $\Omega$ as $v_0 = 1/2$ and $\sigma_0 = 1.0$ nmol mm$^{-1}$, respectively. In turn, the initial number of glioma cells, $p_0 = 40$ cells mm$^{-1}$, is modulated by the continuous approximation of the Heaviside decreasing step function $\tilde{H}_{\gamma}(x-\epsilon) = 1 - \left(1/ \left( 1 + e^{-2 \gamma (x - \epsilon)} \right) \right)$ for $x \in \Omega$, with $\gamma = 1.0 \times 10^{1}$ and $\epsilon = 0.5$. The latter choice provides continuity on the model initial conditions and guarantees numerical stability. At both extremes of the simulation domain $\Omega$, no-flux boundary conditions are imposed.
### 2.2 Density of glioma cells, $\rho(x, t)$ {#density-of-glioma-cells-rhox-t .unnumbered}
**- Intrinsic diffusion rate of glioma cells $D$** (in mm$^2$ day$^{-1}$). Several studies using a data-driven Fisher-Kolmogorov model support that the diffusion rate of glioma cells is a patient-specific parameter [@Harpold2007; @Swanson2008; @Wang2009; @Rockne2010; @Swanson2011]. Estimates of $D$ vary from $2.73 \times 10^{-3}$ to $2.73 \times 10^{-1}$ mm$^2$ day$^{-1}$, which is supposed to cover low to high grade gliomas [@Harpold2007; @Swanson2008; @Swanson2011; @Badoual2014]. Notice that $D_{\rho} = D/\alpha(\sigma_0)$, see equation (4) where $\alpha(\sigma)$ is defined.
**- Intrinsic proliferation rate of glioma cells $b$** (in day$^{-1}$). Similar as reported for the diffusion rate of glioma cells, $b$ is also suggested to be patient-specific [@Harpold2007; @Swanson2008; @Wang2009; @Rockne2010; @Swanson2011]. Estimates of $b$ vary from $2.73 \times 10^{-4}$ to $2.73 \times 10^{-2}$ day$^{-1}$, which is supposed to cover low to high grade gliomas [@Harpold2007; @Swanson2008; @Swanson2011; @Badoual2014]. Notice that $b_{\rho} = b/\beta(\sigma_0)$, see equation (5) where $\beta(\sigma)$ is defined.
**- Brain tissue carrying capacity $N$** (in cells mm$^{-1}$). This model parameter describes the limiting concentration of glioma cells that a volume of host brain tissue can hold. Considering an average glioma cell diameter of about 10 $\mu$m [@Swanson2011], the one-dimensional carrying capacity is about $10^{2}$ cells mm$^{-1}$. This estimate is in line with previous values considered for modelling of glioma growth [@Eikenberry2009; @McDaniel2013].
**- Physiological oxygen concentration in the host brain tissue $\sigma_0$** (in nmol mm$^{-1}$). Although [*in vivo*]{} estimates of oxygen pressure in the brain tissue may vary with respect to measurement methods and other factors, a suitable experimental value for $\sigma_0$ is 40 mmHg [@Maas1993; @Meixensberger1993; @Hoffman1996; @Carreau2011]. Henry’s law [@Henry1803] is used to obtain the concentration of oxygen in the brain tissue as follows
$$\sigma_0 = 40/k_H \approx 2.068~\mbox{nmol}~\mbox{mm}^{-3},$$
where $k_H = 1.93420922505 \times 10^{10}$ mm$^3$ mmHg mol$^{-1}$ is the Henry’s law constant for oxygen at normal body temperature.
To convert the three-dimensional oxygen concentration in the host brain tissue into its equivalent one-dimensional concentration, we multiply by the area of a transversal section of the tumour. We assume that such transversal section is equivalent to the surface area of a sphere of radius $r$, where $A = 4 \pi r^2$. Moreover, we consider that $r = 200~\mu\mbox{m} = 2.0 \times 10^{-1}$ mm is the characteristic nutrient diffusion length, which is consistent with the observed thickness of viable rims of tumour cells in spheroids [@Frieboes2006; @Cristini2008; @Hatzikirou2015]. Then, $A = 4 \pi (2 \times10^{-1})^2 = 16 \pi \times 10^{-2}$ mm$^2$ and we obtain that
$$\sigma_0 = 2.068~\mbox{nmol}~\mbox{mm}^{-3} \cdot (16 \pi \cdot 10^{-2}~\mbox{mm}^2) \approx 1.0~\mbox{nmol}~\mbox{mm}^{-1}.$$
**- Phenotypic switching parameter (proliferative to migratory) $\lambda_1$** (in nmol mm$^{-1}$). We take $\lambda_1 = \sigma_{M}$ in the phenotypic switching function $f_{21} = \lambda_1 - \sigma$ of glioma cells, where $\sigma_{M}$ is the maximum oxygen concentration in the host brain tissue. In normal brain tissues, oxygen tension has been estimated to range from 10 to 80 mmHg [@Luoto2013; @Crawford2013]. Accordingly, we consider that $\sigma_{M} = 2.0$ nmol mm$^{-1}$, i.e. for an oxygen pressure equal to 80 mmHg, which is two times higher than the assumed physiological oxygen concentration $\sigma_0$.
**- Phenotypic switching parameter (migratory to proliferative) $\lambda_2$** (dimensionless). The effect of $\lambda_2$ on glioma invasion is investigated by considering the following overall proliferation rate of glioma cells
$$B = b \hspace{0.5mm} \frac{\beta(\sigma)}{\beta(\sigma_0)} = b \hspace{0.5mm} \frac{(\lambda_2 - 1)\sigma_0 + \lambda_1}{(\lambda_2 - 1)\sigma + \lambda_1} \frac{\sigma}{\sigma_0},$$
where taking into account that $\lambda_1 = \sigma_{M}$, we can distinguish the following three representative cases:
\(i) If $0 < \lambda_2 < 1$, then $B = b \hspace{0.5mm} \dfrac{\sigma_{M} - |\lambda_2 - 1| \sigma_0}{\sigma_{M} - |\lambda_2 - 1| \sigma} \hspace{0.5mm} \dfrac{\sigma}{\sigma_0} \propto \dfrac{\sigma}{\dfrac{\sigma_{M}}{|\lambda_2 - 1|} - \sigma}$.
\(ii) If $\lambda_2 = 1$, then $B = b \hspace{0.5mm} \dfrac{\sigma}{\sigma_0} \propto \sigma$.
\(iii) If $\lambda_2 > 1$, then $B = b \hspace{0.5mm} \dfrac{|\lambda_2 - 1| \sigma_0 + \sigma_{M}}{|\lambda_2 - 1| \sigma + \sigma_{M}} \hspace{0.5mm} \dfrac{\sigma}{\sigma_0} \propto \dfrac{\sigma}{\dfrac{\sigma_{M}}{|\lambda_2 - 1|} + \sigma}$.
According to (i)-(iii), we reduce model simulations to the following three parameter values $\lambda_2 = \{0.5,~1.0,~2.0\}$, see Figure S2. Notice that in the limiting case of $\lambda_2 = 0$ glioma cells do not proliferate, and therefore we neglect this scenario. Although numerical simulations are obtained for the phenotypic switching parameter $\lambda_2 = 1.0$, we report in Section 3 the effect of $\lambda_2$ variations on glioma invasion, see Figures S4 and S5.
\[figure: diagrams\] ![Oxygen-dependant phenotypic switching functions based on the migration/proliferation dichotomy of glioma cells.[]{data-label="figS2"}](FigureS2.pdf "fig:"){width="40.00000%"}
### 2.3 Oxygen concentration, $\sigma(x, t)$ {#oxygen-concentration-sigmax-t .unnumbered}
**- Diffusion rate of oxygen $D_{\sigma}$** (in mm$^2$ day$^{-1}$). Based on experimental data, the oxygen diffusion rate in tumour tissues at 37 $^{\circ}$C has been reported equal to $1.75 \times 10^{-5}$ cm$^2$ s$^{-1}$ [@Grote1977]. Thus, we consider that $D_{\sigma} = 1.51 \times 10^{2}$ mm$^2$ day$^{-1}$, which is in agreement with previous estimates of the oxygen diffusion rate [@Matzavinos2009; @Stamper2010; @Powathil2012].
**- Oxygen supply rate $h_1$** (in day$^{-1}$). Experimental estimates of transvascular permeability to oxygen $Pm_{O2}$ have been reported in the range $3 \times 10^{-5}$ to $3 \times 10^{-4}$ m s$^{-1}$ [@Kelly2006]. In turn, the ratio of capillary surface area to volume $\frac{S}{V}$ has been observed to vary between 0.13 and 0.33 m$^{-1}$ [@Kelly2006]. Then, $Pm_{O2} \cdot \frac{S}{V}$ lies in the range $4.0 \times 10^{-6}$ to $1.0 \times 10^{-4}$ s$^{-1}$, which is equivalent to model parameter $h_1$ in the oxygen supply term of equation (15) [@Kelly2006]. These estimates are also in line with other oxygen supply rates reported [@Eggleton1998; @Goldman2000], i.e. $h_1 = 3.5 \times 10^{-6}$ and $4.0 \times 10^{-6}$ s$^{-1}$. Accordingly, we consider that $h_1 = 3.37 \times 10^{-1}$ day$^{-1}$, which is in the range of $h_1$ values above.
**- Oxygen consumption rate $h_2$** (in mm cell$^{-1}$ day$^{-1}$). Oxygen consumption rates by tumour cells have been reported to vary from 2 to 40 $\mu$l g$^{-1}$ min$^{-1}$ [@Vaupel1989; @Grimes2014]. Considering the average mass of a cancer cell equal to $10^{-9}$ kg [@Powathil2012] and taking into account that $1~\mu\mbox{l} = 1$ mm$^3$, we have that $h_2$ is in the range $2.88 \times 10^{-3}$ to $5.76 \times 10^{-2}$ mm$^{3}$ cells$^{-1}$ day$^{-1}$. As explained above for the estimation of the physiological oxygen concentration in the host brain tissue $\sigma_0$, we convert the three-dimensional oxygen consumption rate into its equivalent one-dimensional rate dividing by the area of a transversal section of the tumour equal to $A = 16 \pi \cdot 10^{-2}$ mm$^2$. Thus, we have that $h_2$ is in the range $5.73 \times 10^{-3}$ to $1.14 \times 10^{-1}$ mm cell$^{-1}$ day$^{-1}$.
### 2.4 Density of functional tumour vasculature, $v(x, t)$ {#density-of-functional-tumour-vasculature-vx-t .unnumbered}
**- Vasculature dispersal rate $D_{v}$** (in mm$^2$ day$^{-1}$). Experimental estimates of endothelial cell motility rate in different conditions have been reported between $10^{-3}$ and $10^{-4}$ mm$^2$ day$^{-1}$ [@Stokes1991; @Kouvroukoglou2000]. We consider that $D_{v} = 5.0 \times 10^{-4}$ mm$^2$ day$^{-1}$, which is in line with previous models of vascularised tumour growth [@Anderson1998; @Stamper2010; @Swanson2011].
**- Vasculature formation rate $g_1$** (in day$^{-1}$). We consider that $g_1 = 1.0 \times 10^{-1}$ day$^{-1}$ by assuming that blood vessels are formed in a timescale of hours [@Shaifer2010; @Stamper2010; @Scianna2013]. We remark that variations in the value of $g_1$ change the results only quantitatively, while qualitative phenomena are conserved.
**- Oxygen concentration threshold for hypoxia $\sigma^{*}_{a}$** (in nmol mm$^{-1}$). Although no consensus has been achieved for hypoxic thresholds, tumour tissues with $\mbox{P}_{\mbox{O2}}$ levels below 10 mmHg are usually considered under hypoxia [@Cardenas2004; @Vaupel2007; @Powathil2012]. Indeed, tissues with oxygen tension between 5.0 and 7.5 mmHg are considered under moderate hypoxia, and less than or equal to 2.5 mmHg under severe hypoxia [@Cardenas2004]. Accordingly, we assume that $\sigma^{*}_{a} = 2.5 \times 10^{-1}$ nmol mm$^{-1}$, see also the derivation of the physiological oxygen concentration in the host brain tissue $\sigma_0$ for further details.
**- Half-maximal pro-angiogenic factor concentration $K$** (in nmol mm$^{-1}$). We assume that the natural decay rate of pro-angiogenic factors is much smaller than the consumption rate by endothelial cells, i.e. $k_3 \ll k_2$ [@KohnLuque2013]. Then, taking into account the equation of effective pro-angiogenic factor concentration
$$a = \frac{k_{1} \hspace{0.5mm} \rho \hspace{0.5mm} \tilde{\text{H}}_{\theta}(\sigma-\sigma_a^{*})}{k_{2}v+k_{3}},$$
the Michaelis-Menten kinetics on the density of functional tumour vasculature in equation (10) is as follows
$$\frac{a}{\mu + a} = \dfrac{\dfrac{k_{1}\rho\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a})}{k_{2}v+k_{3}}}{\mu+\dfrac{k_{1}\rho\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a})}{k_{2}v+k_{3}}} = \dfrac{ \dfrac{k_{1}}{k_{2}} \dfrac{\rho}{v}\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a})} {\mu+\dfrac{k_{1}}{k_{2}} \dfrac{\rho}{v}\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a})} = \dfrac{ \dfrac{\rho}{v}\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a})} {K + \dfrac{\rho}{v}\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a})},$$
where $K=\mu k_{2} / k_{1}$ is a positive constant denoting the concentration of pro-angiogenic factors at which the functional tumour vasculature formation rate is half-maximal. $\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a})$ is a continuous approximation of the Heaviside decreasing step function $H(\xi)$, defined as $H(\xi) = 1$ if $\xi \leq 0$ and $H(\xi) = 0$ if $\xi > 0$, and given by
$$\tilde{\text{H}}_{\theta}(\sigma-\sigma^{*}_{a}) = 1 - \frac{1}{1+e^{-2\theta (\sigma-\sigma^{*}_{a})}},$$
where $\theta = 1.0 \times 10^{1}$ and $K = 1.0 \times 10^{1}$ nmol mm$^{-1}$. We remark that variations in the value of $K$ slightly change the results quantitatively, while qualitative phenomena are conserved.
**- Vaso-occlusion term $G(v, \rho) = g_2 v \rho^n$**. Figure S3(A) shows a schematic representation of vaso-occlusion, see also equation (14). We assume that occlusion of tumour blood vessels only occurs for glioma cell densities greater than $N/2$, where $N$ is the brain tissue carrying capacity [@Padera2004]. Accordingly, we can distinguish the following two representative cases:
\(i) If $\rho \leq N/2$, then $G(v, \rho) = g_2 v \rho^n = g_2 \dfrac{N^n}{2^{n+1}} \approx 0$.\
(ii) If $\rho > N/2$, then $G(v, \rho) = g_2 v \rho^n > g_2 \dfrac{N^n}{2^{n+1}} > 0$.\
Considering the functional tumour vasculature at normal density, i.e. $v = 1/2$, we have that to satisfy the above assumption on vaso-occlusion induced by glioma cell density low and high values of $g_2$ and $n$ are required, respectively. Therefore, we take $n = 6$ and consider the following values of $g_2 = \{5.0 \times 10^{-13},~5.0 \times 10^{-12},~1.5 \times 10^{-11}\}$ cell$^{-n}$ mm$^{n}$ day$^{-1}$. We remark that lower values of $n$ do not reproduce the experimental observation that vaso-occlusion starts to occur at tumour cell densities greater than $N/2$. Figure S3(B) shows the dependence of the vaso-occlusion term $G(v, \rho)$ in equation (14) on the density of glioma cells $\rho$ for $n = 6$, $v = 1/2$ and values of $g_2$ considered. In turn, Figure S3(C) provides simulation maps of the vaso-occlusion percentage depending on the diffusion and proliferation rates of glioma cells at the end of numerical simulations $T_{f} = 3$ years. This percentage is obtained as the ratio between the integral of $v(x,T_f)$ from $x = 0$ to the point $x_v$ where $v = 1/2$ and the area of the rectangle given by $x_v/2$. We observe that vaso-occlusion increases as the proliferation rate of glioma cells becomes higher, see Figure S3(C).
The term $G(v, \rho) = g_2 v \rho^n$ in equation (14) is selected to model vaso-occlusion because from our experience extensive tumour blood vessel collapse is taking place when solid stress exceeds a critical value [@Stylianopoulos2013a; @Stylianopoulos2013b]. Prior to this critical stress threshold, blood vessel collapse is moderate [@Stylianopoulos2013b]. We remark that the use of a different expression for $G(v, \rho)$ would change the results only quantitatively and it is not expected to affect the general conclusions of this study.
![(A) Schematic representation of vaso-occlusion. (B) Dependence of the vaso-occlusion term $G(v, \rho) = g_2 v \rho^n$ on density of glioma cells $\rho$ for $v = 1/2$, $n = 6$ and different values of $g_2$. (C) Vaso-occlusion percentage, at the end of numerical simulations $T_{f} = 3$ years, with respect to intrinsic diffusion and proliferation rates of glioma cells for a fixed oxygen consumption rate $h_2 = 5.73 \times 10^{-3}$ mm cell$^{-1}$ day$^{-1}$ and $g_2 = \{5.0 \times 10^{-14},~5.0 \times 10^{-13},~5.0 \times 10^{-12},~1.5 \times 10^{-11}\}$ cells$^{-n}$ mm$^{n}$ day$^{-1}$ in simulation maps I-IV, respectively. Other model parameters are as in Table 1.[]{data-label="figS3"}](FigureS3.pdf){width="70.00000%"}
3 Effect of phenotypic switching parameter $\lambda_2$ on model observables {#effect-of-phenotypic-switching-parameter-lambda_2-on-model-observables .unnumbered}
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Numerical simulations are obtained for the phenotypic switching parameter $\lambda_2 = 1.0$. Thus, in order to complete the model analysis we investigate the effect of different values of $\lambda_2 = \{0.5,~1.0,~2.0\}$ on glioma invasion. Indeed, this set of $\lambda_2$ values covers the three representative cases discussed above, see also Figure S2. As shown in Figures S4 and S5, for increasing values of $\lambda_2$ the tumour front speed increases, while the infiltration width decreases. We further note that such changes in glioma invasion are similar with respect to the intrinsic tumour features. Based on these results, we can state that glioma invasion in response to variations of $\lambda_2$ is only quantitatively influenced, while qualitative phenomena are conserved.
![**Model observables with respect to parameter $\lambda_2$ for constant functional tumour vasculature.** Simulation maps with respect to the intrinsic proliferation $b \in [2.73 \times 10^{-4},~2.73 \times 10^{-2}]$ days$^{-1}$ and diffusion $D \in [2.73 \times 10^{-3},~2.73 \times 10^{-1}]$ mm$^2$ days$^{-1}$ rates of glioma cells. (A) tumour front speed and (B) infiltration width for a fixed oxygen consumption $h_2 = 5.73 \times 10^{-3}$ mm cell$^{-1}$ day$^{-1}$ rate, and values of $\lambda_2 = \{0.5,~1.0,~2.0\}$ in simulation maps I-III, respectively. (A-B) Differences between simulation maps are provided. The other parameters are as in Table 1.[]{data-label="figS4"}](FigureS4.pdf){width="90.00000%"}
![**Model observables with respect to parameter $\lambda_2$.** Simulation maps with respect to the intrinsic proliferation $b \in [2.73 \times 10^{-4},~2.73 \times 10^{-2}]$ days$^{-1}$ and diffusion $D \in [2.73 \times 10^{-3},~2.73 \times 10^{-1}]$ mm$^2$ days$^{-1}$ rates of glioma cells. (A) tumour front speed and (B) infiltration width for fixed oxygen consumption $h_2 = 5.73 \times 10^{-3}$ mm cell$^{-1}$ day$^{-1}$ and vaso-occlusion $g_2 = 5.0 \times 10^{-12}$ cells$^{-n}$ mm$^{n}$ day$^{-1}$ rates, and values of model parameter $\lambda_2 = \{0.5,~1.0,~2.0\}$ in simulation maps I-III, respectively. (A-B) Differences between simulation maps are provided. The other parameters are as in Table 1.[]{data-label="figS5"}](FigureS5.pdf){width="90.00000%"}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Let $F$ be a field of prime characteristic $p$ containing ${{\mathbb F}}_{p^n}$ as a subfield. We refer to $q(X)=X^{p^n}-X-a\in F[X]$ as a generalized Artin-Schreier polynomial. Suppose that $q(X)$ is irreducible and let $C_{q(X)}$ be the companion matrix of $q(X)$. Then $ad\, C_{q(X)}$ has such highly unusual properties that any $A\in{{\mathfrak {gl}}}(m)$ such that $ad\, A$ has like properties is shown to be similar to the companion matrix of an irreducible generalized Artin-Schreier polynomial.
We discuss close connections with the decomposition problem of the tensor product of indecomposable modules for a 1-dimensional Lie algebra over a field of characteristic $p$, the problem of finding an explicit primitive element for every intermediate field of the Galois extension associated to an irreducible generalized Artin-Schreier polynomial, and the problem of finding necessary and sufficient conditions for the irreducibility of a family of polynomials.
address:
- 'Av. Corrientes 3985 6A, (1194) Buenos Aires, Argentina'
- 'Department of Mathematics and Statistics, University of Regina, Canada'
author:
- 'N. H. Guersenzvaig'
- Fernando Szechtman
title: 'Generalized Artin-Schreier polynomials'
---
[^1]
Introduction
============
If ${{\mathfrak g}}$ is a finite dimensional semisimple Lie algebra over an algebraically closed field of characteristic 0 then the finite dimensional indecomposable (i.e., irreducible) ${{\mathfrak g}}$-modules are well understood (see [@H]). However, the problem of classifying the finite dimensional indecomposable modules of an arbitrary finite dimensional Lie algebra over any given field is in general unattainable. This is explained in [@GP] when ${{\mathfrak g}}$ is abelian with $\dim({{\mathfrak g}})>1$ and, more generally, in [@M] for virtually any Lie algebra over an algebraically closed field of characteristic 0 that is not semisimple or 1-dimensional. Nevertheless, significant progress can be made by restricting attention to certain types of indecomposable modules. In a sense, the simplest type of indecomposable module other than irreducible is a uniserial module, i.e., one admitting a unique composition series. A systematic study of uniserial modules for perfect Lie algebras of the form ${{\mathfrak s}}\ltimes V$, where ${{\mathfrak s}}$ is complex semisimple and $V$ is a non-trivial irreducible ${{\mathfrak s}}$-module, is carried out in [@CS], which culminates with a complete classification of uniserial modules for ${{\mathfrak {sl}}}(2)\ltimes V(m)$, with $m>0$.
The problem of classifying all uniserial modules of an abelian Lie algebra over an arbitrary field $F$ is studied in [@CS2]. The classification is achieved under certain restrictions on $F$, as this subtle problem is intimately related to a delicate and sharpened version of the classical theorem of the primitive element. Such versions have been studied by Nagell [@N1] and [@N2], Kaplansky [@K], Isaacs [@Is] and many other authors (see [@CS2] and references therein). Suppose $F$ contains a copy of ${{\mathbb F}}_{p^n}$ as subfield, where $p$ is a prime and $n\geq 1$. Refer to $q(X)=X^{p^n}-X-a\in F[X]$ as a generalized Artin-Schreier polynomial. In [@CS2], Example 2.3, irreducible generalized Artin-Schreier polynomials are used to exhibit the limitations of some of the modified versions of the theorem of the primitive element.
In [@CS3] attention is focused on the classification of uniserial and far more general indecomposable modules over a family of 2-step solvable Lie algebras ${{\mathfrak g}}$. A complete classification is achieved when the underlying field $F$ has characteristic 0. The properties of the operator $ad\,A:{{\mathfrak {gl}}}(m)\to{{\mathfrak {gl}}}(m)$, given by $B\mapsto [A,B]$, where $A\in{{\mathfrak {gl}}}(m)$ is the companion matrix of a polynomial $q\in F[X]$, play a significant role in this classification. If $F$ has prime characteristic $p$ and $q\in F[X]$ is an irreducible generalized Artin-Schreier polynomial with companion matrix $C_q$, then $ad\; C_q$ has exceptional properties which are used in [@CS3], Note 3.18, to furnish examples of uniserial ${{\mathfrak g}}$-modules that fall outside of the above classification. Some of the easily verified properties enjoyed by the companion matrix $C_q\in{{\mathfrak {gl}}}(p)$ of an irreducible classical Artin-Schreier polynomial, are as follows:
$\bullet$ All eigenvalues of $ad\; C_q$ are in $F$;
$\bullet$ The eigenvalues of $ad\; C_q$ form a subfield of $F$;
$\bullet$ The centralizer of $C_q$ is a subfield of $M_p(F)$;
$\bullet$ All eigenvectors of $ad\; C_q$ are invertible in $M_p(F)$;
$\bullet$ All eigenspaces of $ad\; C_q$ have the same dimension;
$\bullet$ $ad\; C_q$ is diagonalizable with minimal polynomial $X^p-X$.
In this paper we find all matrices $A\in{{\mathfrak {gl}}}(m)$ such that $ad\,
A:{{\mathfrak {gl}}}(m)\to{{\mathfrak {gl}}}(m)$ has like properties.
\[uno\] Let $F$ be a field and let $A\in {{\mathfrak {gl}}}(m,F)$. Then
(C1) All eigenvalues of $ad\, A$ are in $F$,
(C2) The eigenvalues of $ad\, A$ form a subfield of $F$,
(C3) The centralizer of $A$ is a subfield of $M_m(F)$,
if and only if
(C4) $F$ has prime characteristic $p$,
(C5) $A$ is similar to the companion matrix of a monic irreducible polynomial $h\in F[X]$ of degree $m$,
(C6) If $q\in F[X]$ is the separable part of $h$, i.e., $h(X)=q(X^{p^e})$, $e\geq 0$, and $q$ is separable, then $q=X^{p^n}-X-a$, where $a\in F$, $n\geq 1$, and ${{\mathbb F}}_{p^n}$ is a subfield of $F$.
Moreover, if (C4)-(C6) hold then: $q$ is irreducible; the subfield of $F$ formed by the eigenvalues of $ad\, A$ is precisely ${{\mathbb F}}_{p^n}$; all eigenvectors of $ad\, A$ are invertible in $M_m(F)$; all eigenspaces of $ad\, A$ have dimension $p^{n+e}$; $ad\, A$ is diagonalizable if and only if $h$ itself is separable; the invariant factors of $ad\, A$ are $$X^{p^{n+e}}-X^{p^e},\dots,X^{p^{n+e}}-X^{p^e},\text{ with
multiplicity }p^{n+e},$$ so, in particular, the minimal polynomial of $ad\, A$ is $X^{p^{n+e}}-X^{p^e}$.
The most challenging part of the proof of Theorem \[uno\] is to find the invariant factors of $ad\, A$, as this is depends on the solution to the following problem.
Let $L=\langle x\rangle$ be a 1-dimensional Lie algebra over a field $F$ and let $V$ and $W$ be indecomposable $L$-modules of respective dimensions $n$ and $m$ upon which $x$ acts with at least one eigenvalue from $F$.
[Question.]{} How does the $L$-module $V\otimes W$ decompose as the direct sum of indecomposable $L$-modules?
When $F$ has characteristic 0 we may derive an answer from the Clebsch-Gordan formula by imbedding $L$ into ${{\mathfrak {sl}}}(2)$. A direct computation in the complex case already appeared in [@Ro] in 1934. The results in characteristic 0 fail, in general, in prime characteristic $p$, which is the case we require. The analogue problem for a cyclic $p$-group when $F$ has prime characteristic $p$ was solved by B. Srinivasan [@S]. Her solution is of an algorithmic nature. Since then several algorithms have appeared. We mention [@Ra], [@Re] and, most recently, [@I], although the literature is quite vast on this subject. For information on the decomposition of the exterior and symmetric squares of an indecomposable module of a cyclic $p$-group in prime characteristic $p$ see [@GL]. What we need to be able to compute the invariant factors of $ad\, A$ in Theorem \[uno\] is a closed formula for the decomposition of the $L$-module $V\otimes
W$ when $n\leq m=p^e$, $e\geq 0$. This is achieved in §\[ma4\].
The proof of Theorem \[uno\] is given in §\[ma3\], making use of preliminary results from §\[sec:pre\] and §\[ma4\].
In §\[alfaH\] we give an application to Galois theory of the polynomials appearing in Theorem \[uno\]. Let $F$ be a field of prime characteristic $p$ containing ${{\mathbb F}}_{p^n}$ as a subfield and suppose that $q(X)=X^{p^n}-X-a\in F[X]$ is irreducible. Let $K/F$ be the corresponding Galois extension. Here $K=F[\alpha]$, where $\alpha\in K$ is a root of $q(X)$. Given an arbitrary intermediate field $E$ of $K/F$ we find a primitive element $\alpha_E$ such that $E=F[\alpha_E]$. We actually give a recursive formula to write $\alpha_E$ as a polynomial in $\alpha$ with coefficients in ${{\mathbb F}}_{p^n}$. This is achieved by means of the so-called Dickson invariants, discovered by L. E. Dickson [@D] in 1911.
Finally, in §\[hector\] we discuss the actual existence of irreducible polynomials $q(X)=X^{p^n}-X-a\in F[X]$, with $F$ as in the previous paragraph. As explained in §\[alfaH\], if $n>1$ and $q(X)$ is irreducible then $a$ must be transcendental over ${{\mathbb F}}_p$. This fact, together with the more general polynomials $q(X^{p^e})$ considered in Theorem \[uno\], lead us to study the irreducibility of polynomials of the form $$h(X)=X^{p^{n+e}}-X^{p^e}-g(Z^r)\in F[X],$$ where $X$ and $Z$ are algebraically independent elements over an arbitrary field $K$ of prime characteristic $p$, $n>0$, $r>0$, $e\geq 0$, $F=K(Z)$, and $g(Z)\in K[Z]$ is a non-zero polynomial of degree relatively prime to $p$. Using results from [@MS] and [@G], we obtain in §\[hector\] necessary and sufficient conditions for the irreducibility of $h(X)$. In particular, $X^{p^n}-X-g(Z^r)\in F[X]$ is irreducible for any $n>0$, $r>0$ and non-zero $g\in K[Z]$ whose degree relatively prime to $p$. This limitation on $\deg(g)$ is needed, as the example $X^{p^n}-X-(Z^{p^n}-Z)$ shows.
Eigenvalues {#sec:pre}
===========
Let $F$ be a field. For $A\in M_m(F)$ let $\chi_A$ and $\mu_A$ denote the characteristic and minimal polynomials of $A$. If $b\in
F$ is an eigenvalue of $A$ we write $E_b(A)$ for the corresponding eigenspace. If $B\in M_m(F)$ we write $A\sim B$ whenever $A$ and $B$ are similar. The companion matrix to a monic polynomial $g\in
F[X]$ of degree $m$ will be denoted by $C_g$.
\[l1\] Let $A\in M_m(F) $ and let $C$ be the centralizer of $A$ in $M_m(F)$. Then $C$ is a subfield of $M_m(F)$ if and only if $A$ is similar to the companion matrix of a monic irreducible polynomial in $F[X]$ of degree $m$.
If $A$ is similar to the companion matrix of a monic polynomial of degree $m$ -necessarily $\mu_A$- it is well-known [@J], §3.11, that $C=F[A]$. If, in addition, $\mu_A$ is irreducible, then $C=F[A]\cong F[X]/(\mu_A)$ is a field.
Assume that $C$ is a field. Then $K=F[A]$ is a field, so $\mu_A$ is irreducible and the column space $V=F^m$ is a vector space over $K$. As such, $C=End_K(V)$. If $\dim_K(V)>1$ then $End_K(V)$ is not a field. Thus $\dim_K(V)=1$, so $\dim_F(V)=[K:F]=\deg(\mu_A)$, whence $A$ is similar to the companion matrix of $\mu_A$.
\[l2\] Let $A\in{{\mathfrak {gl}}}(m)$ and let $K$ be a splitting field of $\mu_A$ over $F$. Then $\mu_{ad\, A}$ splits over $K$. Moreover, if $S_A$ and $S_{ad\,
A}$ denote the sets of eigenvalues of $A$ and $ad\, A$ in $K$, respectively, then $S_{ad\, A}=\{\alpha-\beta\,|\, \alpha,\beta\in
S_A\}$.
According to [@H], §4.2, we have $A=D+N$, where $D,N\in {{\mathfrak {gl}}}(m,K)$, $D$ is diagonalizable, $N$ is nilpotent, and $[D,N]=0$. In particular, $\chi_D=\chi_A$. Moreover, $ad\, A= ad\; D+ad\; N$, where $ad\; D$ is diagonalizable, $ad\; N$ is nilpotent, $[ad\; D,ad\; N]=0$. As above, $\chi_{ad\; D}=\chi_{ad\, A}$. It thus suffices to prove the statement for $D$ instead $A$, a well-known result also found in [@H], §4.2.
\[gam\] Let $b\in F$ and let $f\in F[X]$ be a monic polynomial of degree $m\geq 1$. Then $C_{f(X)}\sim C_{f(X+b)}+bI$.
Since the minimal polynomial of $C_{f(X+b)}$ is $f(X+b)$, that of $C_{f(X+b)}+ b I$ is $f(X)$. As $C_{f(X)}$ and $C_{f(X+b)}+b I$ have the same minimal polynomial, whose degree is the size of these matrices, they must be similar.
A more general result than Lemma \[gam\], found in [@GS], reads as follows.
Let $f,g\in F[X]$, where $f$ is monic of degree $m\geq 1$, and $g$ has degree $d\geq 1$ and leading coefficient $a$. Then $$\label{ind2}
g(C_{a^{-m}}f(g(X)))\sim C_f\oplus\cdots\oplus
C_f,\quad d\text{ times}.$$
It is not difficult to verify (see [@CS3]) that if $S\in{{\mathrm {GL}}}_m(F)$ is the Pascal matrix $$S=\left(
\begin{array}{cccccc}
1 & b & b^2 & b^3 & \dots & b^{m-1} \\
0 & 1 & 2b & 3 b^2 & \dots & {{m-1}\choose{1}}b^{m-2}\\
0 & 0 & 1 & 3 b & \dots & {{m-1}\choose{2}}b^{m-3}\\
0 & 0 & 0 & 1 & \dots & {{m-1}\choose{3}}b^{m-4}\\
\vdots & \vdots & \vdots & & \ddots & \vdots\\
0 & 0 & 0 & \dots & \dots & 1 \\
\end{array}
\right)$$ then $$\label{ewq} S^{-1}(C_{f(X+b)}+b I)S=C_{f(X)}.$$ We leave it to the reader to determine if, in analogy with (\[ewq\]), it is possible to choose a similarity transformation in (\[ind2\]) that depends on $g$ but not on $f$.
\[l3\] Let $A\in {{\mathfrak {gl}}}(m)$ and suppose that the centralizer $C$ of $A$ is a subfield of $M_m(F)$. Suppose further that $ad\, A$ has at least one non-zero eigenvalue $b\in F$. Then $F$ has prime characteristic $p$, every $b$-eigenvector of $ad\, A$ is invertible in $M_m(F)$, and $E_b(ad\, A)$ has the same dimension as $C=E_0(ad\, A)$.
Let $K$ be a splitting field for $\mu_A$ over $F$. By Lemma \[l2\] and assumption, there are eigenvalues $\alpha,\beta\in K$ of $A$ such that $\alpha=\beta+b$ for some $b\in F$. By Lemma \[l1\], $\mu_A$ is irreducible. By [@J], Theorem 4.4, there is an automorphism of $K/F$ such that $\beta\mapsto\beta+b$, where $|\mathrm{Aut}(K/F)|\leq [K:F]$ is finite. Since $b\neq 0$, $F$ must have prime characteristic $p$. Alternatively, $\mu_A(X)$ and $\mu_A(X+b)$ have a common root $\beta$. Since they are irreducible in $F[X]$, they must be equal. It follows that $\mu_A(X)=\mu_A(X+ib)$, and hence $\alpha+ib$ is a root of $\mu_A(X)$, for every $i$ in the prime field of $F$. This forces the prime field of $F$ to be finite.
Now $X\in E_b(ad\, A)$ if and only if $$AX-XA=bX,$$ i.e., $$\label{es} AX=X(A+bI).$$ By Lemma \[l1\], $A\sim C_{\mu_A(X)}$. Hence $\mu_A(X)=\mu_A(X+b)$ and Lemma \[gam\] imply $A\sim A+b I$. Thus, there is $S\in{{\mathrm {GL}}}_m(F)$ such that $A+bI=SAS^{-1}$. Replacing this in (\[es\]), yields $$AX=XSAS^{-1},$$ i.e., $$AXS=XSA.$$ Since $A$ is cyclic, this equivalent, by [@J], §3.11, to $XS\in F[A]$, or $X=f(A)S^{-1}$, for some $f\in F[X]$. But $F[A]$ is a field, so the result follows.
Decomposition numbers {#ma4}
=====================
Let $F$ be a field and let $L=\langle x\rangle$ be a 1-dimensional Lie algebra over $F$. Let $V$ be an $L$-module of dimension $n$ and let $x_V$ be the linear operator that $x$ induces on $V$. Suppose that $x_V$ has at least one eigenvalue in $F$ and that $V$ is an indecomposable $L$-module. This means that there is a basis $B$ of $V$ relative to which the matrix $M_B(x_V)$ of $x_V$ is the upper triangular Jordan block $J_n(\alpha)$, where $\alpha\in F$ is the only eigenvalue of $x_V$. Suppose next that $W$ is an indecomposable $L$-module of dimension $m$ and that $x_W$ has eigenvalue $\beta\in F$. As above, there is a basis $C$ of $W$ relative to which $M_C(x_V)=J_m(\beta)$.
As usual, we may view $V\otimes W$ as an $L$-module via $$x(v\otimes w)=xv\otimes w+v\otimes xw,\quad v,w\in V.$$ Let $x_{V\otimes W}$ be the linear operator that $x$ induces on $V\otimes W$. It is easy to see that the minimal polynomial of $x_{V\otimes W}$ splits in $F$ and a single eigenvalue, namely $\alpha+\beta$. This follows from the well-known formula: $$\label{car} (x-(\alpha+\beta)\cdot 1)^k(v\otimes
w)=\underset{0\leq i\leq k}\sum{{k}\choose{i}}(x-\alpha\cdot
1)^{k-i}(v)\otimes (x-\beta\cdot 1)^i(w).$$
\[q3\] How does $V\otimes W$ decompose as a direct sum of indecomposable $L$-modules?
That is, what is the length $\ell$ of $V\otimes W$ and what are the decomposition numbers $d_1\geq \cdots\geq d_\ell$ such that $$x_{V\otimes W}\sim J_{d_1}(\alpha+\beta)\oplus\cdots\oplus
J_{d_\ell}(\alpha+\beta)?$$ Replacing $x_V$ by $x_V-\alpha\cdot 1$, $x_W$ by $x_W-\beta\cdot
1$, and $x_{V\otimes W}$ by $x_{V\otimes W}-(\alpha+\beta)\cdot
1$, we see that $\ell$ and the decomposition numbers $d_1\geq
\cdots\geq d_\ell$ are independent of $\alpha$ and $\beta$, and can be computed when $\alpha=0=\beta$.
When $F$ has characteristic 0 and $n\leq m$ then $\ell=n$, with decomposition numbers $$m+n-1,m+n-3,\dots,m-n+3,m-n+1.$$ This can obtained by imbedding $L$ into ${{\mathfrak {sl}}}(2)$ and using the Clebsch-Gordan formula [@H], §22.5. The analogue of Question \[q3\] for a cyclic $p$-group and $F$ of prime characteristic $p$ was solved by B. Srinivasan [@S]. The answer is given recursively, rather than as a closed formula. Alternative algorithms can be found in [@Ra] and [@Re]. Presumably, Srinivasan’s results translate to our present set-up mutatis mutandis.
This section furnishes a closed formula in answer to Question \[q3\], albeit only in the special case $m=p^e$, where $F$ has prime characteristic $p$ and $e\geq 0$, as required in the proof of Theorem \[uno\].
\[fin0\] Let $p$ be a prime and let $e\geq 0$. Then $p|{{p^e}\choose{i}}$ for any $0<i<p^e$.
Since $i{{p^e}\choose{i}}=p^e{{p^e-1}\choose{i-1}}$ and the largest power of $p$ dividing $i$ is at most $p^{e-1}$, it follows that $p$ divides ${{p^e}\choose{i}}$.
\[fin\] Let $F$ be a field of prime characteristic $p$ and let $e\geq 0$. Let $L=\langle x\rangle$ be a 1-dimensional Lie algebra over $F$. Let $V$ and $W$ be indecomposable $L$-modules of dimensions $n$ and $p^e$, respectively, where $n\leq p^e$. Suppose that $x$ has eigenvalues $\alpha,\beta\in F$ when acting on $V$ and $W$, respectively. Then the $L$-module $V\otimes W$ decomposes as the direct sum of $n$ isomorphic indecomposable $L$-modules, each of which has dimension $p^e$ and is acted upon by $x$ with a single eigenvalue $\alpha+\beta$. In symbols, $$\ell=n\text{ and }d_1=\dots=d_{p^e}=p^e.$$
As mentioned above, we may assume without loss of generality that $\alpha=0$ and $\beta=0$, and we will do so, mainly for simplicity of notation. For the same reason, we let $m=p^e$.
Let $B=\{v_1,\dots,v_n\}$ and $C=\{w_1,\dots,w_m\}$ be bases of $V$ and $W$ relative to which $M_B(x_V)=J_n(0)$ and $M_C(x_W)=J_m(0)$.
Since $n\leq m$, we have $$x_V^m=0\text{ and }x_W^m=0.$$ Therefore, Lemma \[fin0\] and (\[car\]) imply $$x_{V\otimes W}^m=0.$$ We next view $M=V\otimes W$ as a module for the polynomial algebra $F[X]$ via $x_{V\otimes W}$. We wish to show that $M$ has elementary divisors $X^m,\dots,X^m$, with multiplicity $n$.
It follows from (\[car\]) that $$x^{m-1}(v_1\otimes w_m)=v_1\otimes w_1\neq 0.$$ Let $N_1$ be the $F[X]$-submodule of $M$ generated by $v_1\otimes
w_m$. Then $N_1$ has a single elementary divisor, namely $X^m$.
Suppose that $1\leq i<n$ and the $F[X]$-submodule of $M$, say $N_i$, generated by $v_1\otimes w_m,\dots,v_i\otimes w_m$ has elementary divisors $X^m,\dots,X_m$, with multiplicity $i$. Using (\[car\]) we see that $v_{i+1}\otimes w_1$ appears in $x^{m-1}(v_{i+1}\otimes w_m)$ with coefficient 1. Since $v_{i+1}\otimes w_1\notin N_i$, the minimal polynomial of the vector $v_{i+1}\otimes w_m+N_i\in M/N_i$ is $X^m$. The theory of finitely generated modules over a principal ideal domain implies that the $F[X]$-submodule of $M$ generated by $v_1\otimes
w_m,\dots,v_i\otimes w_m,v_{i+1}\otimes x_m$ has elementary divisors $X^m,\dots,X_m$, with multiplicity $i+1$. The result now follows.
Unlike what happens in characteristic 0, the decomposition of $V\otimes W$ for $L$ is not, in general, the same as for ${{\mathfrak {sl}}}(2)$. Indeed, suppose $F$ has characteristic 2 and let $V=W$ be the natural module for ${{\mathfrak {sl}}}(2)$. Then $V\otimes W$ is an indecomposable ${{\mathfrak {sl}}}(2)$-module, but decomposes as the direct sum of two indecomposable $L$-modules for $L=\langle x\rangle$, where $x,h,y$ is the standard basis of ${{\mathfrak {sl}}}(2)$.
Resuming our prior discussion, let $F$ be a field and let $L=\langle x\rangle$ be a 1-dimensional Lie algebra over $F$. Let $V_1,\dots,V_s$ be $L$-modules with bases $B_1,\dots,B_s$ relative to which $M_{B_i}(x_{V_i})=J_{m_i}(\alpha_i)$, where $1\leq i\leq
s$ and $\alpha_i\in F$. Consider the $L$-module $V=V_1\oplus\cdots\oplus V_s$. We may view ${{\mathfrak {gl}}}(V)$ as an $L$-module via: $$x\cdot f=x_V f-fx_V.$$ Thus $x$ acts on ${{\mathfrak {gl}}}(V)$ via $ad\, x_V$. By Lemma \[l2\], the eigenvalues of $ad\, x_V$ are $\alpha_i-\alpha_j$, where $1\leq
i,j\leq s$. We can view ${{\mathfrak {gl}}}(V)$ as the direct sum of the $L$-submodules $$\mathrm{Hom}(V_j,V_i)\cong V_j^*\otimes V_i,\quad 1\leq i,j\leq s.$$ Here $V_j^*$ is an indecomposable $L$-module upon which $x$ acts with eigenvalue $-\alpha_j$. It is then clear that the generalized eigenspace of $ad\, x_V$ for a given eigenvalue $\gamma$ is the sum of all $\mathrm{Hom}(V_j,V_i)$ such that $\alpha_i-\alpha_j=\gamma$.
\[co1\] Keep the above notation and suppose, further, that $F$ has prime characteristic $p$ and all $L$-modules $V_i$ have the same dimension $p^e$, for some $e\geq 0$. Let $S=\{\alpha_i-\alpha_j\,|\, 1\leq i,j\leq s\}$, the set of distinct eigenvalues of $ad\, x_V$. Then the minimal polynomial of $ad\, x_V$ is $\underset{\gamma\in
S}\Pi(X-\gamma)^{p^e}$. Moreover, the elementary divisors of $ad\,
x_V$ are $(X-\gamma)^{p^e}$, $\gamma\in S$, with multiplicity $p^em(\gamma)$, where $$m(\gamma)=|\{(i,j)\,|\, 1\leq i,j\leq s,\;
\alpha_i-\alpha_j=\gamma\}|.$$
Proof of Theorem \[uno\] {#ma3}
========================
Suppose conditions (C4)-(C6) hold. Then $q$ is irreducible, since so is $h$. Let $K$ be a splitting field for $h$ over $F$. Since $q$ is separable, the number of distinct roots of $h$ in $K$ is exactly $p^n$.
Let $\beta\in K$ be a root of $h$. We readily verify that $\beta+b$ is a root of $h$ for every $b\in {{\mathbb F}}_{p^n}$. It follows that $\beta+b$, $b\in {{\mathbb F}}_{p^n}$, are the distinct roots of $h$ in $K$ and each has multiplicity $p^e$. By Lemma \[l2\], $\mu_{ad\,
A}$ splits in $K$ and the set of eigenvalues of $ad\, A$ is precisely ${{\mathbb F}}_{p^n}$. Moreover, by Lemma \[l1\], the centralizer of $A$ is a subfield of $M_m(F)$. In particular, conditions (C1)-(C3) hold.
Furthermore, by Lemma \[l3\], all eigenvectors of $ad\, A$ are invertible in $M_m(F)$ and all eigenspaces of $ad\, A$ have dimension $p^{n+e}$. Thus, the sum of the dimensions of all eigenspaces of $ad\, A$ is $p^{2n+e}$. This equals the dimension of ${{\mathfrak {gl}}}(m)$, namely $m^2=p^{2(n+e)}$, if and only if $e=0$. Therefore, $ad\, A$ is diagonalizable if and only if $h$ is separable.
Regardless of whether $e=0$ or not, we claim that the invariant factors of $ad\, A$ are $X^{p^{n+e}}-X^{p^e},\dots,X^{p^{n+e}}-X^{p^e}$, with multiplicity $p^{n+e}$. For this purpose, we may assume without loss of generality that $F=K$. Hence $\mu_{ad\, A}$ splits in $F$, by Lemma \[l2\]. Thus $A$ is similar to the direct sum of the companion matrices to $X^{p^e}-\beta^{p^e}=(X-\beta)^{p^e}$, as $\beta$ runs through the $p^n$ distinct roots of $h$ in $F$. Hence, we may assume without loss of generality that $A$ is the direct sum of the $p^n$ Jordan blocks $J_{p^e}(\beta)$, with $\beta$ as above. Now, the eigenvalues of $ad\, A$ form the subfield ${{\mathbb F}}_{p^n}$ of $F$. Moreover, for each $b\in {{\mathbb F}}_{p^n}$, we have $$|\{(\beta_1,\beta_2)\,|\, h(\beta_1)=0=h(\beta_2),\,
\beta_1-\beta_2=b\}|=p^n.$$ It follows from Corollary \[co1\] that the elementary divisors of $ad\, A$ are $(X-b)^{p^e}$, $b\in {{\mathbb F}}_{p^n}$, each with multiplicity $p^{n+e}$. Since $\underset{b\in
{{\mathbb F}}_{p^n}}\Pi(X-b)=X^{p^n}-X$, the claim follows.
Suppose conversely that (C1)-(C3) hold. Since the eigenvalues of $ad\, A$ form a subfield of $F$, we see that $ad\, A$ has a non-zero eigenvalue in $F$. It follows from Lemma \[l3\] that (C4) holds. Since the centralizer of $A$ is a subfield of $M_m(F)$, Lemma \[l1\] shows that (C5) holds.
Let $K$ be a splitting field for $\mu_A$ over $F$. Let $q$ be the separable part of $\mu_A$, so that $\mu_A(X)=q(X^{p^e})$ for some $e\geq 0$.
Let ${{\mathbb F}}_{p^n}$ be the subfield of $F$ formed by the roots of $ad\, A$ ensured by (C3). Let $\beta\in K$ be a root of $\mu_A$. It follows from Lemma \[l2\] that the distinct roots of $\mu_A$ in $K$ are $\beta+b$, for all $b\in {{\mathbb F}}_{p^n}$. Since $\mu_A$ and its separable part must have the same number of distinct roots, we deduce that $q$ has degree $p^n$. Let $\alpha=\beta^{p^e}\in K$. Then $\alpha$ is a root of $q$. Moreover, if $c\in {{\mathbb F}}_{p^n}$ then $\alpha+c$ is also a root of $q$. Indeed, $b\mapsto b^{p^e}$ is an automorphism of ${{\mathbb F}}_{p^n}$, so $c=b^{p^e}$ for some $b\in
{{\mathbb F}}_{p^n}$. Therefore, $$q(\alpha+c)=\mu_A(\beta+b)=0.$$ Thus $F[\alpha]$ is a splitting field for $q$ over $F$. Since $q$ is separable, we deduce that $F[\alpha]/F$ is a finite Galois extension, whose Galois group we denote by $G$. We claim that $\alpha^{p^n}-\alpha\in F$. To see this, it suffices to show that $\alpha^{p^n}-\alpha\in F$ is fixed by every $\sigma\in G$. Let $\sigma\in G$. Then $\sigma(\alpha)$ must be a root of $q$, so $\sigma(\alpha)=\alpha+b$ for some $b\in {{\mathbb F}}_{p^n}$. Therefore, $$\sigma(\alpha^{p^n}-\alpha)=(\alpha+b)^{p^n}-(\alpha+b)=\alpha^{p^n}-\alpha,$$ as required. Thus $\alpha^{p^n}-\alpha=a\in F$, so $X^{p^n}-X-a\in
F[X]$ has $\alpha$ as root. Hence $q|(X^{p^n}-X-a)$. Since these polynomials have the same degree and are monic, they must be equal. This completes the proof of the theorem.
Primitive elements of intermediate fields in a Galois extension {#alfaH}
===============================================================
Let $F$ be a field of prime characteristic $p$ having ${{\mathbb F}}_{p^n}$ as a subfield and consider the generalized Artin-Schreier polynomial $q=X^{p^n}-X-a\in F[X]$. Let $\alpha$ be a root of $q$ in a field extension of $F$. Then $\alpha+b$, $b\in {{\mathbb F}}_{p^n}$, are all roots of $q$, so $F[\alpha]$ is a splitting field for $q$ over $F$. Moreover, all roots of $q$ have the same degree over $F$, since $F[\alpha]=F[\alpha+b]$ for any $b\in {{\mathbb F}}_{p^n}$. Thus, all irreducible factors of $q$ in $F[X]$ have the same degree. Let $G$ be the Galois group of $F[\alpha]/F$. We claim that $G$ is elementary abelian $p$-group. Indeed, let $\sigma,\tau\in G$. Then $\sigma(\alpha)=\alpha+b$ and $\tau(\alpha)=\alpha+c$ for some $b,c\in {{\mathbb F}}_{p^n}$. Therefore $\sigma^p=1$ and $\sigma\tau=\tau\sigma$, as claimed. Since $|G|=[F[\alpha]:F]$, which is the degree the minimal polynomial of $\alpha$ over $F$, we deduce that all irreducible factors of $q$ in $F[X]$ have degree $p^m$ for a unique $0\leq m\leq n$
If $a=0$ it is obvious that $m=0$. If $a\neq 0$ and $F={{\mathbb F}}_{p^n}$ then $m=1$. More generally, if $F$ algebraic over ${{\mathbb F}}_{p^n}$ and there is no $b\in F$ such that $q(b)=0$ then $m=1$. Indeed, let $f$ be the minimal polynomial of $\alpha$ over $F$ and let $p^m$ be its degree. By assumption $m>0$. Let $E$ be the subfield of $F$ obtained by adjoining $a$ and the coefficients of $f$ to ${{\mathbb F}}_{p^n}$. Then $f$ is irreducible and a factor of $q$ in $E[X]$. By above, $\mathrm{Gal}(E[\alpha]/E)$ is an elementary abelian group of order $p^m$. Since $E$ is a finite field, $\mathrm{Gal}(E[\alpha]/E)$ is cyclic, so $m=1$.
Assume henceforth that $q$ is actually irreducible. Then $K=F[\alpha]$ is a splitting field for $q$ over $F$ and $G=\mathrm{Gal}(K/F)$ is an elementary abelian $p$-group of order $p^n$. More explicitly, for $b\in {{\mathbb F}}_{p^n}$, let $\sigma_b\in G$ be defined by $\sigma_b(\alpha)=\alpha+b$. Then $b\mapsto
\sigma_b$ defines a group isomorphism ${{\mathbb F}}_{p^n}^+\to G$. In particular, $G$ has normal subgroups of all possible orders.
Suppose that $m$ satisfies $0\leq m\leq n$. Let $H$ be a subgroup of $G$ of order $p^{m}$. Then the fixed field $E=K^H$ of $H$ satisfies $[K:E]=p^m$. Since $F[\alpha]=E[\alpha]$, the minimal polynomial $\mu_{\alpha,E}$ of $\alpha$ over $E$ must have degree $p^m$. In fact, $$\mu_{\alpha,E}(X)=\underset{\sigma\in H}\Pi(X-\alpha^\sigma).$$ Since $\mu_{\alpha,E}(X)$ divides $q=X^{p^n}-X-a\in E[X]$, it follows that all irreducible factors of $q$ in $E[X]$ have degree $p^m$. In fact, $$X^{p^n}-X-a=\underset{\sigma\in\mathrm{Gal}(E/F)}\Pi\mu_{\alpha,E}(X)^{\sigma}.$$ Let $$\label{al3} \alpha_H=\underset{\sigma\in H}\Pi\alpha^\sigma.$$ Since every $\sigma\in G$ is of the form $\sigma_b$ for $b\in
{{\mathbb F}}_{p^n}$, it follows that $\alpha_H$ is a monic polynomial in $\alpha$ of degree $p^m$ with coefficients in ${{\mathbb F}}_{p^n}$. Since $\alpha$ has degree $p^n$ over $F$, the degree of $\alpha_H$ over $F$ is at least $p^{n-m}$. But clearly $\alpha_H\in E$, where $[E:F]=p^{n-m}$. It follows that $E=F[\alpha_H]$.
As just noted, $\alpha_H$ as an ${{\mathbb F}}_{p^n}$-linear combination of powers of $\alpha$. In fact, we may use the so-called Dickson invariants, found by L. E. Dickson [@D] in 1911, to obtain a sharper result. These invariants have been revisited numerous times (see, for instance, [@H2] and [@St]).
Consider the polynomial $\Phi_m$ in the polynomial algebra $F[A,B_1,\dots,B_m]$, defined as follows: $$\label{al4} {{\Phi}}_m(A,B_1,\dots,B_m)=\underset{s_1, \dots,s_m\in
{{\mathbb F}}_p}\Pi(A+s_1B_1+\cdots+s_mB_m).$$ Clearly $\Phi_m$ is ${{\mathrm {GL}}}_m({{\mathbb F}}_p)$-invariant. Dickson showed that $$\Phi_m=A^{p^m}+f_{m-1}(B_1,\dots,B_m)A^{p^{m-1}}+\cdots+f_{1}(B_1,\dots,B_m)A^{p}+f_{0}(B_1,\dots,B_m)A,$$ where $f_0,\dots,f_{m-1}\in {{\mathbb F}}_p[B_1,\dots,B_m]$ are algebraically independent and generate ${{\mathbb F}}_p[B_1,\dots,B_m]^{{{\mathrm {GL}}}_m({{\mathbb F}}_p)}$. Moreover, $\Phi_m$, and hence $f_{m-1},\dots,f_0$, can be recursively computed from $$\label{rec0} \Phi_0=A,$$ $$\label{rec1}
{{\Phi}}_i={{\Phi}}_{i-1}(A,B_1,\dots,B_{i-1})^p-{{\Phi}}_{i-1}(B_i,B_1,\dots,B_{i-1})^{p-1}{{\Phi}}_{i-1}(A,B_1,\dots,B_{i-1}).$$
Let $\sigma_{b_1},\dots,\sigma_{b_m}$ be generators of $H$. This means that $\sigma_{b_1},\dots,\sigma_{b_m}$ are in $H$ and that $b_1,\dots,b_m$ are linearly independent over ${{\mathbb F}}_p$. It follows from (\[al3\]) that $$\alpha_H=\underset{s_1,\dots,s_m\in
{{\mathbb F}}_p}\Pi(\alpha+s_1b_1+\cdots+s_m
b_m)=\Phi_m(\alpha,b_1,\dots,b_m).$$ This, together with (\[rec0\]) and (\[rec1\]) allows us to recursively find $c_0,\dots,c_{m-1}\in {{\mathbb F}}_{p^n}$ such that $$\alpha_H=\alpha^{p^m}+c_{m-1}\alpha^{p^{m-1}}+\cdots+c_{1}\alpha^{p}+c_0\alpha.$$
In certain special cases we actually have $c_0,\dots,c_{m-1}\in
{{\mathbb F}}_p$, in which case we will say that $H$ has property $P$.
Let $R$ be the subgroup of ${{\mathbb F}}_{p^n}^+$ that corresponds to $H$ under ${{\mathbb F}}_{p^n}^+\to G$. Thus, $R$ is an ${{\mathbb F}}_p$-subspace of ${{\mathbb F}}_{p^n}$, namely the ${{\mathbb F}}_p$-span of $b_1,\dots,b_m$.
$H$ has property $P$ if and only $R$ is invariant under the Frobenius automorphism of ${{\mathbb F}}_{p^n}$, in which case $$\alpha_H=f_R(\alpha),$$ where $$f_R(Y)=\underset{b\in R}\Pi(Y+b).$$
We start by showing that $H$ has property $P$ if and only if $f_R\in {{\mathbb F}}_{p^n}[Y]$ has coefficients in ${{\mathbb F}}_p$.
Suppose first that there exist $c_{m-1},\dots,c_0\in {{\mathbb F}}_p$ such that $$\alpha_H=\alpha^{p^m}+c_{m-1}\alpha^{p^{m-1}}+\cdots+c_1\alpha^p+c_0\alpha.$$ Set $$\label{efe} f(Y)=Y^{p^m}+c_{m-1}Y^{p^{m-1}}+\cdots+c_1 Y^p+c_0Y\in
{{\mathbb F}}_p[Y].$$ Let $b\in R$. Since $\alpha_H^{\sigma_b}=\alpha_H$, it follows that $$f(b)=0.$$ Therefore $f_R=f\in {{\mathbb F}}_p[Y]$. Conversely, if $f_R\in {{\mathbb F}}_p[X]$ then $$\alpha_H=\underset{b\in R}\Pi(\alpha+b)=f_R(\alpha),$$ which is an ${{\mathbb F}}_p$-linear combination of $\alpha^{p^m},\dots,\alpha^p,\alpha$ with first coefficient 1.
Let $\tau$ be the Frobenious automorphism of ${{\mathbb F}}_{p^n}$. Then $f_R\in {{\mathbb F}}_p[Y]$ if and only if $$f_R(Y)=f_R(Y)^\tau=\underset{b\in R}\Pi(Y+b^\tau )=\underset{b\in
R^\tau}\Pi (Y+b)=f_{R^\tau}(Y),$$ which is equivalent to $R=R^\tau$.
Suppose that $R$ is actually a subfield of ${{\mathbb F}}_{p^n}$. Then $R$ is certainly invariant under $b\mapsto b^p$. Moreover, $f_R(Y)=Y^{p^m}-Y$. Therefore, in this case, $$\alpha_H=\alpha^{p^m}-\alpha.$$ In particular, $\alpha_G=a$.
\[indep2\] If $b_1,\dots,b_m\in {{\mathbb F}}_{p^m}$ are linearly independent over ${{\mathbb F}}_p$ then $$f_0(b_1,\dots,b_m)=-1,\text{ whereas }f_j(b_1,\dots,b_m)=0,\text{
if }1\leq j\leq m-1.$$
Corollary \[indep2\] is not true, in general, if $b_1,\dots,b_m\in R$ are linearly dependent over ${{\mathbb F}}_p$, as the case $m=2$ will confirm by taking $b_1=1=b_2$ for $j=0,1$.
Here we furnish examples of subspaces $R$ of ${{\mathbb F}}_{p^n}$ that are invariant under $b\mapsto b^p$ but are not subfields of ${{\mathbb F}}_{p^n}$, even when $m|n$.
Suppose first $m=1$, where $(p-1)|n$ and $p$ is odd. Take $c\in
{{\mathbb F}}_p$, $c\neq 0,1$, and set $$f(Y)=Y^p-cY\in {{\mathbb F}}_p[Y].$$ Since $Y^{p^{p-1}}\equiv Y\mod f$, it follows that $f$ splits in ${{\mathbb F}}_{p^{p-1}}$ and hence in ${{\mathbb F}}_{p^n}$. The roots of $f$ in ${{\mathbb F}}_{p^n}$ form a $1$-dimensional, Frobenius-invariant, subspace $R$ of ${{\mathbb F}}_{p^n}$, which is not a field, since ${{\mathbb F}}_p$ is the only $1$-dimensional subfield of ${{\mathbb F}}_{p^n}$ and 0 is the only element of ${{\mathbb F}}_p$ that is a root of $f$.
Suppose next that $m=2$, where $2p|n$ and $p$ is odd. Let $$\label{terra} f(Y)=\underset{j\in {{\mathbb F}}_p}\Pi(Y^p-Y-j).$$ Thus, $f$ is the product of all Artin-Schrier polynomials in ${{\mathbb F}}_p[Y]$. This readily implies that the roots of $f$ form a 2-dimensional, Frobenius-invariant, subspace $R$ of ${{\mathbb F}}_{p^n}$ containing ${{\mathbb F}}_p$. In particular, $f(cY)=f(Y)$ for all $0\neq
c\in {{\mathbb F}}_p$. Since $Y|f(Y)$ and $f(Y)$ has degree $p^2$, it follows that $$f(Y)=Y^{p^2}+a Y^{p}+b Y,$$ where $a,b\in {{\mathbb F}}_p$. Using (\[terra\]) and $p>1$ to compute $a,b$ reveals that $$f(Y)=Y^{p^2}-Y^{p}-Y.$$ However, the subfield of ${{\mathbb F}}_{p^n}$ obtained by adjoining $R$ to ${{\mathbb F}}_p$ is ${{\mathbb F}}_{p^p}$. Since $p>2$, it follows that $R$ is not a subfield of ${{\mathbb F}}_{p^n}$.
Existence of irreducible generalized Artin-Schreier polynomials {#hector}
===============================================================
We begin this section by giving an elementary example of an irreducible Artin-Schreier polynomial. We then furnish substantially more general examples, which require the use of preliminary results from [@G] and [@MS]. In fact, we give necessary and sufficient conditions for a polynomial $$q(X)=X^{p^{n+e}}-X^{p^e}-g(Z^r)\in F[X]$$ to be irreducible, where $X$ and $Z$ are algebraically independent elements over an arbitrary field $K$ of prime characteristic $p$, $n>0$, $r>0$, $e\geq 0$, $F=K(Z)$, and $g(Z)\in K[Z]$ is non-zero of degree relatively prime to $p$.
Recall that an element $\pi$ of and an integral domain $D$ is irreducible if $d$ is neither 0 nor a unit and whenever $\pi=ab$ with $a,b\in D$ then $a$ or $b$ is a unit.
It is easy to see that $X^{p^n}-X-Z$ is irreducible in $K[X][Z]$, hence in $K[X,Z]$ and therefore in $K[Z][X]$. It follows from Gauss’ Lemma (see [@J], §2.16) that $X^{p^n}-X-Z$ is irreducible in $F[X]$.
\[nhg1\] [(see [@MS], Proposition 1.8.9)]{}. Let $D$ be an integral domain. Let $X, Z$ be algebraically independent elements over $D$. Let $f\in D[X]$ and $g\in D[Z]$. If $\gcd(deg(f),
\deg(g))=1$, then $h(X, Z)= f(X)-g(Z)$ is irreducible in $D[X,
Z]$.
\[nhg2\] [(see [@G], Theorem 1.1)]{}. Let $D$ be a unique factorization domain. Let $t$ be any positive integer and let $f$ be an irreducible polynomial in $D[Z]$ of positive degree $m$, leading coefficient $a$ and nonzero constant term $b$. Suppose that for each prime $p$ dividing $t$ and any unit $u$ of $D$ at least one of the two following statements is true:
[(A)]{} $ua\notin D^p$;
[(B)]{} [(i)]{} $(-1)^mub\notin D^p$ and [(ii)]{} $ub\notin D^2$, if $4|t$.
[*Then $$\text{$f(Z^t)$ is irreducible in
$D[Z]$.}$$* ]{}
\[nhg3\] [(see [@G], Corollary 4.6 (b))]{}. Let $D$ be a unique factorization domain of prime characteristic $p$. Let $f(Z)\in D[Z]$ be an arbitrary polynomial of positive degree that is irreducible in $D[Z]$, and let $s$ be any positive integer. Then $$\text{$f\bigl(Z^{p^s}\bigr)$ \!is reducible \!in
$D[Z]\!\!\!\iff$\!\!\! there exists a unit $u$ of $D$ such that
$uf(Z)\!\in \!D^p[Z]$.}$$
We can now prove the following result.
Let $K$ be a field of prime characteristic $p$. Let $X,Z$ be algebraically independent elements over $K$ and set $F=K(Z)$. Let $n, e, r$ be integers such that $n>0,
r>0$ and $e\geq 0$. Let $g(Z)=c_d Z^d+\cdots+c_1 Z+c_0\in K[Z]$ be any polynomial whose degree $d$ is coprime with $p$. Then $h(X,Z)=X^{p^{n+e}}-X^{p^e}-g(Z^r)$ is irreducible in $F[X]$ if and only if at least one of the following conditions is satisfied:
[(i)]{} $p\nmid r$, [(ii)]{} $e =0$, [(iii)]{} $g(Z)\notin K^p[Z]$.
By unique factorization in $\mathbb Z$ there exist a positive integer $r_0$ coprime with $p$ and a nonnegative integer $s$ such that $r=r_0p^s$, so $p|r$ if and only if $s\ge 1$.
Suppose first none of the conditions (i)-(iii) is fulfilled. Thus $e\ge 1$, $s\ge 1$ and $g(Z)\in K^p[Z]$. The last two conditions imply $g(Z^r)=Q^p(Z)$ for some $Q(Z)$ in $K[Z]$, and now the first condition implies that $h(X,Z)$ is a $p$th power in $F[X]$.
Suppose next that at least one of the conditions (i)-(iii) holds. We wish to show that $h(X,Z)$ is irreducible in $F[X]$.
Case (i). Suppose $p\nmid r$. Therefore $p\nmid dr$. Since $g(Z^{r})$ has degree $dr$, from the case $D=K$, $f(X)=
X^{p^{n+e}}-X^{p^e}$ of Theorem \[nhg1\], with $g(Z^{r})$ instead of $g(Z)$, we see that $h(X, Z)$ is irreducible in $K[X,
Z]$, hence in $K[Z][X]$, and therefore in $F[X]$ by Gauss’ Lemma.
Case (ii). Suppose $e=0$. The previous case guarantees that $X^{p^n}-X-g(Z^{r_0})$ is irreducible in $F[X]$, so we can suppose $s\ge 1$. Let $D=K[X]$. Therefore $f(Z)=X^{p^n}-X-g(Z^{r_0})$ is an irreducible polynomial in $D[Z]$ of degree $m=dr_0$ and constant term $b=X^{p^n}-X-c_0$. Since $X^{p^n}-X-c_0$ has no repeated roots, for any $u\in K^*$ (i.e., for any unit $u$ of $D$) we have $(-1)^{m}ub\notin D^p$ as well as $ub\notin D^2$ if $4|p^s$ (i.e., if $p=2$ and $s\ge 2$). Thus part (B) of Theorem \[nhg2\] is satisfied with $D=K[X]$ and $t=p^s$. We conclude that $f(Z^{p^s})$ is irreducible in $D[Z]$, and therefore in $K[Z][X]$. Hence $h(X,Z)=f(Z^{p^s})$ is irreducible in $F[X]$ by Gauss’ Lemma.
Case (iii). Assume $g(Z)\notin K^p[Z]$. From the cases (i) and (ii) we can assume $s\ge 1$ and $e\ge 1$. Suppose, if possible, that $h(X, Z)$ is reducible in $F[X]$. Letting $D=K[X]$ we get, via Gauss’ Lemma, that $h(X, Z)$ is reducible in $K[Z][X]$, and therefore in $D[Z]$. Letting $f(Z)=X^{p^n}-X-g(Z^{r_0})$ we get that $f(Z^{p^s})= h(X, Z)$ is reducible in $D[Z]$. But, as seen above, $f(Z)=X^{p^n}-X-g(Z^{r_0})$ is irreducible in $D[Z]$ of positive degree $m=dr_0$. Hence, by Theorem \[nhg3\], there exists a unit $u$ of $D$ and $Q\in D^p[Z]$ such that $uf(Z)=Q(Z)$. In other words, there exist $u\in K^*$ and $Q_0, Q_1, \dots ,
Q_m\in K[X]$ such that
$$u\biggl(X^{p^{n+e}} - X^{p^e} - \sum_{0\le k\le
d}c_kZ^{kr_0}\biggr) = \sum_{0\le k\le m} Q_k^p Z^k.$$ Equating coefficients of like monomials we obtain $$\text{$u(X^{p^{n+e}} - X^{p^e}-c_0) = Q_0^p$ and $uc_k= Q_{kr_0}^p\in K^p$ for $k=1, \dots , d$.}$$ Since $Q_0$ has degree $m_0= p^{n+e-1}$, there must exist $d_0,
d_1, \dots , d_{m_0}$ in $K$ such that $Q_0= \sum_{0\le k\le m_0}
d_kX^k$, whence $$\label{nhg4}
u(X^{p^{n+e}} - X^{p^e}) -uc_0 = \sum_{0\le k\le m_0}d_k^pX^{kp}.$$ Equating leading coefficients yields $u = d_{m_0}^p\in K^p$, so $c_k\in K^p$ for $k=1, \dots , d$. But $$u(X^{p^{n+e}} - X^{p^e})= \biggl(d_{m_0}X^{p^{e-1}}(X-1)^{p^{n-1}}\biggr)^{\!\!p},$$ so (\[nhg4\]) gives $$uc_0= u(X^{p^{n+e}} - X^{p^e})- Q_0^p= \biggl(d_{m_0}X^{p^{e-1}}(X-1)^{p^{n-1}} -Q_0\biggr)^{\!\!p}\in K[X]^p\cap K=K^p,$$ and a fortiori $c_0\in K^p$. Hence all $c_i\in K^p$, against the fact that $g(Z)\notin K^p[Z]$.
[**Acknowledgments.**]{} We are grateful to D. Stanley for fruitful conversations, R. Guralnick and A. Zalesski for useful references, and the referee for valuable comments.
[FH]{}
L. Cagliero and F. Szechtman, *The classification of uniserial ${{\mathfrak {sl}}}(2)\ltimes V(m)$-modules and a new interpretation of the Racah-Wigner 6j-symbol*, J. Algebra 386 (2013) 142-–175.
L. Cagliero and F. Szechtman, [*On the theorem of the primitive element with applications to the representation theory of associative and Lie algebras* ]{}, Canad. Math. Bull., to appear.
L. Cagliero and F. Szechtman, [*Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0*]{}, submitted.
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[^1]: The second author was supported in part by an NSERC discovery grant
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'A Bragg medium in the nonlinear Kerr regime, submitted to incident cw-radiation at a frequency in a band gap, switches from total reflection to transmission when the incident energy overcomes some threshold. We demonstrate that this is a result of [*nonlinear supratransmission*]{}, which allows to prove that : i) the threshold incident amplitude is simply expressed in terms of the deviation from the Bragg resonance, ii) the process is not the result of a shift of the gap in the nonlinear dispersion relation, iii) the transmission does occur by means of gap soliton trains, as experimentally observed \[D. Taverner [*et al.*]{}, Opt Lett 23 (1998) 328\], iv) the required energy tends to zero close to the band edge.'
author:
- 'J. Leon'
- 'A. Spire'
title: Gap soliton formation by nonlinear supratransmission in Bragg media
---
#### Introduction. {#introduction. .unnumbered}
Light propagation in dielectric media with periodically varying index acquires intriguing properties when Kerr nonlinearities come into play. From a linear theory [@Brill] we know that such a medium is a [*photonic band gap*]{} structure, or Bragg medium, which is totally reflective when the frequency of the incoming wave falls in one of the gaps.
However, for sufficiently energetic irradiation, the Kerr effect becomes sensible and the Bragg medium may become transparent in the gap, switching from total reflection to [*high transmissivity*]{}, which has been predicted in 1979 [@winful] by using the stationary coupled mode approach of [@kog], and which has been experimentally realized for the first time in 1992 [@sankey].
In 1987 the existence of effective wave propagation in a gap has been associated to the existence of [*gap solitons*]{} derived from slowly varying envelope limits of the Maxwell equation in a periodic structure [@chen; @mills]. For unidirectionnal propagation, fast Kerr nonlinearity and slowly varying evelope approximation, one of these limit models is the [*coupled mode system*]{} [@desterke] which describes the evolution of the envelopes of the contrapropagating electric fields. A rigorous derivation of from the anharmonic Maxwell-Lorentz equations by the method of multiple scales is given in [@goodman]. The explicit soliton-like solution of the coupled mode equation has been discovered in 1989 [@christo], and at the same time generalized to a two-parameter family in [@aceves]. This two-parameter gap soliton solution, that we use here, has motivated many experimental searches of the [*Bragg soliton*]{}.
After the pioneering experiments of [@sankey] in steady-state regime, the first experimental observations of nonlinear pulse [*propagation*]{} in a fiber Bragg grating have been performed in 1996 [@eggleton] under laser pulse irradiation at a frequency near (but not inside) the photonic band gap. In that case the nonlinearity acts as a source of [*modulationnal instability*]{} generating trains of pulses [@eggleton2]. Particularly interesting is the observation of pulse propagation at a velocity less than the light velocity. Later in 1998 [@taverner] a convincing experiment with a fiber Bragg grating demonstrated the repeated formation of gap soliton under quasi-constant wave irradiation [*inside*]{} a band gap, this is the problem we are interested in.
There has been a great deal of discussions concerning the exact role of the gap soliton in the switching to high transmissivity, see e.g. [@souk]. A rather natural intuition is that the nonlinearity shifts the gap and hence allows the medium to become transparent and to create a [*gap soliton*]{}. We shall discover that this intuition is wrong as the process at the origin of high transmissivity in Bragg media is the [*nonlinear supratransmission*]{} recently discovered in the sine-Gordon chain [@jg-alex; @nst-prl] and experimentally realized (together with application to Josephson junctions arrays) in [@nst-jpc]. As a consequence, the switching to high transmissivity is indeed accomplished by means of gap soliton generation as a result of a fundamental instability [@instab] which are effectively the objects experimentally detected in the output in [@taverner].
This result allows us to predict analytically by formula the threshold of incident energy of a cw-radiation above which high transmissivity is reached, as an explicit simple function of the departure (denoted by the dimensionless angular frequency $\Omega$) from the Bragg frequency (gap center). In particular we obain that the process requires less energy for frequencies close to the band edge $\Omega=1$, a result which was previously attributed to pulse shape properties. Interestingly enough, in the limit $\Omega\to 1$ the required energy vanishes, opening the way to experimental realizations.
#### The model and its solution. {#the-model-and-its-solution. .unnumbered}
Our starting point is the basic coupled mode system [@desterke; @goodman] governing the forward $E_f$ and backward $E_b$ slowly varying envelopes of the electric field $$E(Z,T)=[E_fe^{ik_0Z}+E_be^{-ik_0Z}]e^{-i\omega_0 T}\ .$$ $E(Z,T)$ is the transverse polarized component propagating in the direction $Z$ at frequency close to the Bragg frequency $\omega_0$. Following [@goodman] we write the coupled mode system in reduced units as $$\begin{aligned}
\label{coup-sys}
i[\frac{\partial e}{\partial t}+\frac{\partial e}{\partial z}]+ f
+(\frac12|e|^2+|f|^2)e=0\ ,\nonumber\\
i[\frac{\partial f}{\partial t}-\frac{\partial f}{\partial z}]+ e
+(\frac12|f|^2+|e|^2)f=0\ .\end{aligned}$$ The reduced units are $z=\kappa Z$ and $t=\kappa c T$ where $c$ is the light velocity in the medium and $\kappa$ the coupling constant. The reduced field variables are $e=E_f\sqrt{2\Gamma/\kappa}$ and $f=E_b\sqrt{2\Gamma/\kappa}$ where $\Gamma$ is the nonlinear factor. The (linear) dispersion relation of , namely $\omega^2=1+k^2$, possess the gap $[-1,+1]$.
The solitary wave solution of given in [@aceves] has a simple stationnary expression $$\begin{aligned}
e=\sqrt{2/3}
\lambda \,e^{-i(\Omega t -\phi)}\cosh^{-1}[\lambda(z-z_0)-\frac i2q]
\ ,\label{e-sol}\\
f=-\sqrt{2/3}
\lambda \,e^{-i(\Omega t -\phi)}\cosh^{-1}[\lambda(z-z_0) +\frac i2q]
\ .\label{f-sol}\end{aligned}$$ The real valued parameters determining this [*‘gap soliton’*]{} are $\Omega$ (frequency), $\phi$ (initial phase) and $z_0$ (center), and we have $$\Omega^2=1-\lambda^2\ ,\quad \Omega=\cos q\ .$$ Then $\lambda$ plays the role of the [*‘wave number’*]{} of the evanescent wave.
#### Nonlinear supratransmission threshold. {#nonlinear-supratransmission-threshold. .unnumbered}
For a Bragg medium, extending in the region $z\in[0,L]$, and initially [*in the dark*]{}, we set the initial data $$\label{init}
e(z,0)=0\ ,\quad f(z,0)=0\ .$$ The boundary value problem that mimics the scattering of an incident radiation at frequency $\omega_0+\kappa c\Omega$ on the medium in $z=0$ is $$\label{bound}
e(0,t)=A\,e^{-i\Omega t}\ ,\quad f(L,t)=0\ ,$$ where the second requirement means no backward wave incident from the right in $z=L$. The constant $A$ (in general complex valued) is the dimensionless amplitude of the incoming radiation. Equations and constitute a well posed initial-boundary value problem for the PDE .
In the linear case such a boundary forcing would simply generate the evanescent wave $e(z,t)=A\,e^{-i\Omega t-\lambda z}$. In the nonlinear case however this boundary forcing generates the solution for the value of $\phi$ and $z_0<0$ such that the amplitude in $z=0$ be precisely $A$. Then, for each fixed forcing frequency $\Omega$, there exists a maximum value $A_s$ of $A$ beyond which there is no solution $\{\phi,z_0\}$. This threshold $A_s$ is given by $|e(0,t)|$ from evaluated at $z_0=0$, namely by $$\label{threshold}
A_s=2\sqrt{2/3}\,\sin(\frac12\arccos \Omega)\ .$$ This is the threshold amplitude of the incident envelope above which the system develops an instability and generates a propagating nonlinear mode.
#### Nonlinear dispersion relation prediction. {#nonlinear-dispersion-relation-prediction. .unnumbered}
By seeking a solution of $e=a\exp[i(kz-\omega t)]$, $f=b\exp[i(kz-\omega t)]$ one gets a nonlinear algebraic system for the unknowns $a$ (incident amplitude) and $\omega$ (incident frequency) expressed in terms of the wave number $k$ once the amplitude $b$ has been eliminated. This gives the [*nonlinear dispersion relation*]{} $\omega(a,k)$ as a solution of a third order algebraic equation. One of the 3 solutions must be discarded as it is singular in the linear limit $a\to0$, the other two giving the deformations of the linear branches $\omega(k)=\pm\sqrt{1+k^2}$. It can be shown that they correspond to the relations $b=-a$ (upper branch) and $b=a$ (lower branch).
Being interested in the shift of the gap, it is sufficient to study the solutions at $k=0$ where the system simplifies, and to stick with the upper branch (as we look at right-going incident waves) for which $b=-a$. Then readily provides the value $\omega(0,a)=1-\frac32\,a^2$ of the gap opening.
For our purpose it is more convenient to express the value $A_m$ of the incident amplitude $a$ for which the incident frequency $\Omega$ touches the nonlinear gap edge $\omega(0,a)$, namely $$\label{nln-predic}
A_m=\sqrt{\frac23(1-\Omega)}\ .$$ This formula provides a prediction of [*transparency*]{} by nonlinear shift of the gap. It is now compared to our prediction by means of numerical simulations.
#### Numerical simulations. {#numerical-simulations. .unnumbered}
The system is solved by means of an semi-implicit third order finite difference scheme and boundary values are taken into account at first order. Explicitely we first rewrite for two functions $u(z,t)$ and $v(z,t)$ and replace the operators $\partial_t +\partial_z$ by the set of differences for $u_n(t)=u(nh,t)$ $$\begin{aligned}
& \dot u_1 + \frac1h(u_1-u_0),\quad
\dot u_2+\frac1{2h}(u_2-u_0),\quad\cdots \\
& \dot u_n+\frac1{h}\left[\frac23(u_{n+1}-u_{n-1})-
\frac1{12}(u_{n+2}-u_{n-2})\right],\quad\cdots\\
& \dot u_{N-1}+\frac1{2h}(u_{N+1}-u_{N-1}),\quad
\dot u_{N}+\frac1{h}(u_{N}-u_{N-1}),\end{aligned}$$ so as for $v_n(t)=v(nh,t)$. The length is $L=hN$ and overdot means time differentiation. Equation results as a system of $2 N$ coupled ODE then solved through the subroutine [dsolve]{} of the [MAPLE8]{} software package that uses a Fehlberg fourth/fifth order Runge-Kutta method. Finally the solution, e.g. $e(z,t)$, is obtained from $u_n(t)$ by $e(z,t)=[u_{n+1}(t)+u_n(t)]/2$. Although such code sends some numerical noise in the solution, it is uneffective for sufficiently small $h$ depending on the required time of integration.
Next, in order to avoid initial shock, the boundary condition $e(0,t)$ is smoothly turned on and smoothly turned off by assuming instead of $$\label{bound-smooth}
e(0,t)=\frac A2\left[\tanh(p(t-t_0))-\tanh(p(t-t_1))\right]\ .$$ A practical interest of such an incident wave is that it reproduces the [*quasi-constant wave*]{} irradiation of the experiments of [@taverner].
Most of the results presented here are obtained with $N=120$ spatial mesh points, $h=0.05$ grid spacing over (hence a length $L=6$ normalize units) a time of integration $t_m=200$. The parameters of the boundary field are a real-valued amplitude $A$, a slope $p=0.2$, an ignition time $t_0=20$ and an extiction time $t_1=180$.
We display in figure \[fig:breather\] an instance of two different numerical solutions (the modulus of the right-going envelope) for an incident frequency $\Omega=0.8$ and amplitudes $A=0.51$ (no transmission) and $A=0.52$ (gap soliton generation) when the theoretical threshold predicted by is $A_s=0.5164$.
Using this simple diagnostic, the bifurcation predicted by expression is numerically checked for a series of frequency values (in the range $[0.1,0.995]$) and we obtain the figure \[fig:bif\] where the dots represent the smallest value (with absolute precision of $10^{-2}$) of the amplitude $A$ for which nonlinear supratransmission is seen to occur. The expression is also plotted (dashed line) which shows that the nonlinear shift provides a wrong answer.
#### Bifurcation of transmitted energy. {#bifurcation-of-transmitted-energy. .unnumbered}
The system possess the conservation law $$\label{cons}
\partial_t(|e|^2+|f|^2)+\partial_z(|e|^2-|f|^2)=0.$$ For a given boundary condition (i.e. for fixed $A$ and $\Omega$), we define the incident ($I$), reflected ($R$) and transmitted ($T$) energies at given arbitrary time $t_m$ by $$\begin{aligned}
I(A,\Omega)=\int_0^{t_m}
dt\,|e(0,t)|^2,\\ R(A,\Omega)=\int_0^{t_m} dt\,|f(0,t)|^2,\\
T(A,\Omega)=\int_0^{t_m} dt\,|e(L,t)|^2.\end{aligned}$$ Then, upon integration of on the length $[0,L]$ and time $[0,t_m]$ we obtain $$R+T-I+\int_0^Ldz\,\left(|e(z,t_m)|^2+|f(z,t_m)|^2\right)=0\ .$$ If $t_m$ is larger than the irriadiation duration, the energy injected eventually radiates out completely, namely $e(z,t_m)$ and $f(z,t_m)$ vanish, and we are left with the conservation relation $R+T=I$ which can be written $$\rho(A,\Omega)+\tau(A,\Omega) =1,\quad \rho=R/I,\quad
\tau=T/I.$$
The figure \[fig:energy\] shows a typical numerical simulation where, for $\Omega=0.95$, we have computed the reflection and transmission factors, together with their sum, for 50 different values of the amplitude $A$ in the range $[0.20,\, 0.45]$.
These simulations showt the sudden energy flow through the medium, as a result of nonlinear supratransmission. Note from the sum $\rho+\tau$ before the bifurcation on figure \[fig:energy\] that the numerical code conserves the energy with a precision of $2\%$.
The transmissivity is due to the generation and propagation of light pulses shown figure \[fig:outflux\] representing the energy density $|e(L,t)|^2$ measured at the output as a function of time. We have also plotted the input energy density given in with parameters $t_m=100$, $t_0=20$, $t_1=90$ and $p=2$.
We show now that these are [*gap solitons*]{} travelling at a fraction of the light velocity (slow light pulses).
#### Travelling gap solitons. {#travelling-gap-solitons. .unnumbered}
Although the system is not Lorentz invariant, the propagating solution of [@aceves] can still be written in terms of the boosted variables [@barash]. As we are intersted here only in the energy flux, we write the soliton solution $|e(z,t)|^2$ moving at velocity $v<1$ with frequency $\cos q$ as $$\begin{aligned}
\label{e-mov}
|e(z,t)|^2=\frac{2\sin^2 q\,\left(\frac{1-v^2}{3-v^2}\right)
\left(\frac{1+v}{1-v}\right)^{1/2}}
{\left|\cosh\left(\frac{\sin q}{\sqrt{1-v^2}}(z-z_0-vt)-\frac i2q\right)
\right|^{2}}.\end{aligned}$$ Such an expression allows to fit a given simulation by seeking the two parameters $v$ and $q$, and the initial position $z_0$, that reproduce with the explicit solution the results of the numerical simulation.
An instance of such a fit is displayed in figure \[fig:fit\] that shows the analytic soliton $|e(z,t)|^2$ compared with the numerical simulation for two fixed values of time. This furnishes the following velocities $$\label{tab}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$\Omega$& 0.95 & 0.9 & 0.8 & 0.7 & 0.6 & 0.5 \\
\hline
$ A $ & 0.28 & 0.38 & 0.53 & 0.65 & 0.75 & 0.83 \\
\hline
\hline
$ v $ & 0.25 & 0.34 & 0.42 & 0.50 & 0.58 & 0.66 \\
\hline
\end{tabular}$$ These are [*slow light pulses*]{} and naturally, small velocities are obtained close to the gap edge where the required energy is small.
#### Conclusion. {#conclusion. .unnumbered}
The property of a Bragg medium in the nonlinear Kerr regime to sustain nonlinear supratransmission has allowed us to obtain the analytic expression that fixes the threshold amplitude of an incident cw-beam in terms of its departure from the Bragg resonance.
This constitutes a practical tool to investigate switching properties of a Bragg medium and allows at the same time to understand the process at the origin of sudden transmissivity of the Bragg mirror. It is the nonlinear supratansmission that results from a nonlinear instability intrinsic to boundary value problems [@instab].
We expect our result to be usefull for experiments by the ability of the Bragg medium to become partly transparent by gap soliton generation for an incident beam of [*low energy*]{} if its frequency is chosen close to the gap edge.
#### Acknowledgments. {#acknowledgments. .unnumbered}
This study was initiated during a stay of one of us (J.L.) at the department of applied mathematics, Boulder University. It is a pleasure to aknowledge invitation and enlighting discussions with M.J. Ablowitz.
This work received support from [*Fundação para à Ciência e à Tecnologia*]{}, grant BPD/5569/2001, and from [*Açoes Integradas Luso-Francesas*]{}, grant F-4/03.
[aaa]{}
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|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Mixed $^3$He-$^4$He droplets created by hydrodynamic instability of a cryogenic fluid-jet may acquire angular momentum during their passage through the nozzle of the experimental apparatus. These free-standing droplets cool down to very low temperatures undergoing isotopic segregation, developing a nearly pure $^3$He crust surrounding a very $^4$He-rich superfluid core. Here, the stability and appearance of rotating mixed helium nanodroplets are investigated using Density Functional Theory for an isotopic composition that highlights, with some marked exceptions related to the existence of the superfluid inner core, the analogies with viscous rotating droplets.'
author:
- Martí Pi
- Francesco Ancilotto
- José María Escartín
- Ricardo Mayol
- Manuel Barranco
title: 'Rotating mixed $^3$He-$^4$He nanodroplets'
---
Ordinary liquids are known to form droplets held together by surface tension. When they are set into rotation, their spherical shape experiences large deformations, evolving from oblate to prolate and 2-lobed, eventually fissioning if the rotational velocity is large enough.[@Bro80] Experiments carried out by Plateau on olive oil droplets immersed in a mixture of water and alcohol with nearly the same density, disclosed the sequence of droplet shapes as the angular velocity of the rotating shaft to which they were attached increased.[@Pla63] The appearance and stability of, e.g., rotating celestial bodies,[@Cha65] atomic nuclei[@Coh74] and tektites,[@Bal15] to cite some quite different objects, has been found to bear similarities with rotating classical droplets, adding an extrinsic interest to their study.
Helium, in its two isotopes $^4$He and $^3$He, is the only element in nature that may remain liquid and form droplets at temperatures $(T)$ close to absolute zero.[@Toe04] Both isotopes may be superfluid, with normal-to-superfluid transition temperatures of 2.17 K ($^4$He) and 2.7 mK ($^3$He). At the temperatures of helium droplets experiments, 0.37 K for $^4$He[@Har95] and 0.15 K for $^3$He,[@Sar12] $^3$He is a normal fluid whereas $^4$He is superfluid. They constitute an ideal testground to study how superfluidity affects rotation, as they are isolated quantum systems formed by atoms subject to the same bare interaction.
Rotating superfluid $^4$He droplets made of $N_4= 10^8-10^{10}$ atoms, produced by hydrodynamic instability of a cryogenic fluid-jet, have been studied by coherent x-rays scattering from a free-electron laser,[@Gom14] revealing the presence of vortex lattices through the observation of Bragg patterns produced by Xe clusters captured by the vortex lines. Coherent diffractive imaging experiments using extreme ultraviolet pulses have also been carried out, aimed at providing informations about the droplet shapes.[@Lan18] Surprisingly, these studies have shown that superfluid $^4$He droplets follow the same shape sequence of rotating viscous droplets made of normal fluid. It has been shown that this is due to the presence of quantized vortices and capillary waves, whose interplay confers to the superfluid droplet the appearance of a classical rotating object.[@Anc18; @Oco19] Until now, a deeper knowledge of how superfluid droplets rotate has been hampered by the experimental difficulty of determining their angular momentum, which is usually unknown.[@Oco19] This prevents a detailed comparison with theoretical models and the disclosure of the precise quantum nature of such rotation. Similar studies have been conducted very recently for rotating pure $^3$He droplets.[@QFC2019] These droplets are non-superfluid and, as shown by Density Functional Theory (DFT) calculations,[@Pi19] behave very much as classical rotating droplets.
![ Dimensionless angular velocity $\Omega$ vs. dimensionless angular momentum $\Lambda$. The numbers close to the symbols indicate the ${\cal L}$ value. Referring to the appearance of the outer surface of the $^3$He shell: open circles, oblate configurations \[$\Omega(\Lambda)$ rising branch\]; triangles, prolate configurations \[$\Omega(\Lambda)$ falling branch\]. Open squares, 3-lobed $^4$He core configurations. The cross indicates the oblate-to-prolate bifurcation point. The solid black line –drawn as a guide to the eye– connects the stable configurations for give ${\cal L}$, and the dotted red line shows the region of metastable fissioned $^4$He core configurations. The dashed blue line is the DFT result for pure $^3$He droplets.[@Pi19] []{data-label="fig1"}](fig1){width="1.0\linewidth"}
In liquid helium mixtures characterized by the $^3$He fraction $x_3= N_3/N$, with $N=N_3+N_4$, the normal-to-superfluid transition temperature decreases with increasing $x_3$.[@Edw92] At low $T$, the mixture undergoes a two-phase separation where a pure $^3$He phase coexists with a very $^4$He-rich mixture.[@Edw92] These properties are transferred to the mixed droplets, which at the experimental $T$ –sensibly that of pure $^3$He droplets[@Gre98]– experience a two-phase separation yielding a core-shell structure, with a crust made of $^3$He atoms in the normal state and a superfluid core mostly made of $^4$He atoms.[@Bar06] This segregation has been instrumental in finding the minimum number of $^4$He atoms needed to display superfluidity.[@Gre98]
The ability of forming self-bound isolated droplets made of a superfluid core enveloped by a normal fluid shell is a unique characteristic of liquid helium mixtures at very low temperatures. Indeed, Bose-Einstein condensates (BEC) immersed in a Fermi sea have been observed,[@Sch01] but these systems are not self-bound, and only exist confined by an external trap to which the droplets adapt their shape; self-bound droplets made of mixtures of bosonic cold gases have been also observed,[@Cab18; @Sem18] and self-bound Bose-Fermi droplets have been studied theoretically.[@Rak19] However, these cold-gas mixtures do not exist as phase-separated Bose-Fermi droplets, as they must remain in a mixed configuration to be self-bound.[@Pet15]
The mixed normal fluid-superfluid structure of $^3$He-$^4$He droplets is expected to affect their structural properties as they are set into rotation. A recent study addresses theoretically the classical rotation of droplets made of two immiscible viscous fluids.[@But20] When one component is superfluid, as in the case studied here, then a quantum description is in order. We provide here such description.
![ Referring to the appearance of the outer surface of the $^3$He shell, $b_y^3/V$ ratio vs. $a_x/c_z$, where $V$ is the volume of the spherical droplet. Symbols and lines have the same meaning as in Fig. \[fig1\]. The numbers close to the symbols indicate the ${\cal L}$ value. The pictograms represent calculated $^3$He (top) and $^4$He (bottom) 2D densities for selected configurations. []{data-label="fig2"}](fig2-pic){width="1.0\linewidth"}
Results {#results .unnumbered}
=======
We describe the stability and shape transitions of mixed He droplets within the DFT approach[@Bar06] taking as case of study a $^4$He$_{1500}$-$^3$He$_{6000}$ nanodroplet ($x_3=80\%$). The size and composition of this nanodroplet, which can be seen in Supplementary Fig. 1, have been chosen (within the limitations imposed by the unavoidable computational cost of the calculations) for physical reasons: a thick $^3$He crust is needed to model a deformable container inside which the superfluid $^4$He core may undergo different shape transitions; a thin crust would just adapt to the deforming core and one should not expect a phenomenology much different from that of pure $^4$He droplets.[@Anc18] At the same time, the superfluid $^4$He inner droplet must be large enough to host a number of quantized vortices.[@Anc18] In superfluid liquid helium mixtures the vortex cores are filled with $^3$He and, depending on $x_3$, their radius can be up to five times larger than for pure $^4$He,[@Jez97] as shown in Supplementary Fig. 2. For the chosen $N_4$ value, the number of vortices ($n_v$) is expected to be small.
The calculations have been carried out as a function of the angular momentum per atom around the rotation $z$-axis, ${\cal L} = (L_3+L_4)/N = {\cal L}_3+{\cal L}_4$, expressed in $\hbar$ units throughout this work. For a given ${\cal L}$, the stable configuration is that with the lowest energy including the rotational energy (Routhian).[@Bro80]
We characterize the appearance of the $^4$He core and $^3$He crust by the distance of their sharp surfaces (defined, for each isotope, by the locus at which the density equals that of the bulk liquid divided by two[@Anc18; @Pi19]) to the center of mass of the droplet, denoting the distances along the $x, y$ and $z$ axes as $a_x$, $b_y$ and $c_z$, respectively. Dimensionless angular momentum $\Lambda$ and angular velocity $\Omega$ variables have been defined in Supplementary Section 2. As in pure droplets,[@Bro80; @Bal15; @Oco19; @But11] these variables are very useful to scale the results to droplets of different size for a given composition.[@But20]
The detailed energetics and morphologic characteristics of rotating mixed helium nanodroplets are collected in Figs. \[fig1\]-\[fig4\] and Supplementary Table 1. Figure \[fig1\] shows the stability diagram and constitutes the main result of this work. Figure \[fig2\] provides information on the shapes of the droplets through relationships between the geometrical parameters $a_x$, $b_y$, and $c_z$ that characterize the shape of the outer $^3$He surface.[@Bal15; @Lan18] Lastly, Fig. \[fig3\] connects the shapes of the droplets with the dimensionless angular momentum $\Lambda$.
Our study has unveiled a rich variety of stable and metastable configurations. As $\Lambda $ increases from zero, the $^3$He crust becomes oblate as it happens in normal fluids.[@Bro80] At variance, the superfluid $^4$He core remains spherical, becoming axisymmetric only when the angular momentum of the droplet is large enough. Since the superfluid $^4$He core cannot be set into rotation around the symmetry axis because it is quantum-mechanically forbidden, all the angular momentum is stored in the $^3$He crust. This is in stark contrast with the case of pure $^4$He droplets, where oblate configurations can exist because they host quantized vortices.[@Anc15; @Anc18; @Oco19] In rotating mixed droplets, the $^4$He core may remain axisymmetric because the angular momentum is mainly stored in the $^3$He shell, which acts as a rotating deformable container. We have found that this happens up to the oblate-to-prolate bifurcation point at $(\Lambda,\Omega)=(0.99, 0.52)$, as shown in Fig. \[fig1\]. Thus, for this composition, oblate stable configurations are vortex-free for droplets of the size studied here. We recall that for pure $^3$He droplets the bifurcation point is at $(\Lambda, \Omega)=(1.28, 0.57)$.[@Pi19]
![ $a_x/c_z$ ratio of the outer surface of the $^3$He shell vs $\Lambda$. Symbols and lines have the same meaning as in Fig. \[fig1\]. The numbers close to the symbols indicate the ${\cal L}$ value. The pictograms represent calculated $^3$He (top) and $^4$He (bottom) 2D DFT densities for selected configurations. []{data-label="fig3"}](fig3-pic){width="1.0\linewidth"}
In the prolate branch, the outer surface of the $^3$He crust is triaxial ellipsoid-like up to $\Lambda \sim 1.64$, where it becomes 2-lobed. At variance, the superfluid $^4$He core becomes 2-lobed at $\Lambda \sim1.05$, i.e. immediately after bifurcation. This is due to the small surface tension of the $^3$He-$^4$He interface, 0.016 KÅ$^{-2}$, as compared to that of the $^3$He free surface, 0.113 KÅ$^{-2}$. Pure $^3$He droplets become 2-lobed at $\Lambda=1.85$.[@Pi19]
Prolate stable configurations with simply connected $^4$He cores have been found only in a narrow angular momentum range $3.0 \leq {\cal L} \leq 3.5$, where the core shape evolves from spheroidal to triaxial to 2-lobed. Due again to the small surface tension of the $^3$He-$^4$He interface, the $^4$He core undergoes fission when the $^3$He crust is still triaxial ellipsoid-like. The resulting stable prolate configuration consist of a fissioned $^4$He core inside a rotating triaxial $^3$He crust. The transition from simply connected to fissioned $^4$He core configurations appears as a jump in the $\Omega(\Lambda)$ curve on Fig. \[fig1\] at $\Lambda = 1.31$. Notice that the jump to fissioned inner $^4$He core (and the associated hysteresis loop as well) could not be predicted out of simple models based on energy minimization restricted to simply connected shapes.
We have looked for prolate configurations with a fissioned $^4$He core hosting a vortex in each moiety. After phase-imprinting them,[@Anc18] vortices are eventually expelled in the course of the numerical relaxation; we conclude that these configurations are not stable for up to the largest $\Lambda$ addressed in this study, $\Lambda = 1.97$, a fairly large value for current experiments.
In spite of the fact that the lowest-energy configurations of the inner $^4$He nanodroplet are mostly vortex-free (most of the angular momentum being efficiently stored in the $^3$He shell), we have found a number of vortex-hosting *metastable* configurations in our calculations. Remarkably, along the oblate branch (in the metastable region beyond the bifurcation point) we have found that configurations with $n_v=1$ have lower energy than vortex-free configurations. In this region, configurations hosting up to 4 vortices appear at ${\cal L>}$ 3.5. All these configurations are metastable, as they may decay to prolate configurations lying at much lower energy (see Supplementary Table 1 for details).
{width="1.0\linewidth"}
In the oblate branch, we have also addressed multiply-charged quantum vortices with charge (quantum circulation) $m=2-4$, and their relative stability with respect to configurations with $n_v=2-4$ and $m=1$. Multiply charged vortices in BEC have been found that are stabilized temporarily by the external confining potential.[@Shi04; @Oka07] In mixed helium droplets, the self-bound thick $^3$He shell yields a confining potential that plays the same role. We have found (see Supplementary Table 1) that, depending on ${\cal L}$, multiply charged single vortex configurations with charge $m$ are more stable than $n_v=m$ singly-charged vortices, and hence cannot decay into them as it would happen in pure $^4$He. This is likely due to the presence of $^3$He in the expanded vortex cores and the $^3$He crust, which together define a region similar to the rotating annulus used to study quantized superfluid states and vortices in the first experiments on quantized circulation in superfluid liquid $^4$He.[@Vin61; @Don91]
Along the prolate branch, metastable $^4$He configurations with $n_v=1-2$ have been found (see Supplementary Table 1) where the angular momentum of the $^4$He core is shared between vortices and capillary waves as shown in Supplementary Fig. 3. This is similar to what happens in spinning $^4$He droplets.[@Anc18; @Oco19] When the $^4$He core becomes 2-lobed, the neck connecting them gets thinner as ${\cal L}$ increases and eventually the most stable configuration is the fissioned one.
Three-lobed configurations were predicted to appear in classical droplets rotating at high velocities.[@Bro80] These configurations are metastable with respect to prolate 2-lobed configurations and were not expected to be accessible experimentaly. However, 3-lobed configurations were obtained,[@Ohs00] stabilized by forcing the droplet into large amplitude periodic oscillations –thus not in gyrostatic equilibrium. Charged $^4$He drops magnetically levitated have been studied displaying $\ell$ =2-4 oscillation modes induced by a rotating deformation of the droplet.[@Whi98] More recently, triangular-shaped magnetically levitated water droplets have been found[@Hil08] where the amplitude of the surface oscillation is small and the equilibrium shape could be observed clearly for about 100 revolutions.
In the water droplet experiments, some surfactant was added to water to decrease the surface tension, helping the 3-lobed configurations to be formed. In the case of mixed helium droplets, the surface tension of the $^3$He-$^4$He interface is already small, hence it is possible to have metastable 3-lobed $^4$He core configurations while the outer surface of the $^3$He crust is still oblate. Indeed, we have found such configurations in the $3.64 \leq {\cal L} \leq 5$ range, one of which is shown in Fig. \[fig4\] (see also the central pictogram in Fig. \[fig2\]). The 3-lobed bifurcation sets in at $(\Lambda, \Omega)= (1.20, 0.58)$. We have looked for metastable 4-lobed configurations but have not found any, the droplet always decaying into fissioned $^4$He prolate configurations.
Discussion {#discussion .unnumbered}
==========
The presence of a superfluid $^4$He core inside a normal fluid $^3$He droplet produces remarkable changes in its rotational properties. Oblate configurations are affected as the superfluid core cannot participate in the rotation because of its symmetry around the rotational axis. At variance with rotating pure $^4$He nanodroplets, stable vortex-hosting oblate configurations are absent as it is energetically more favorable for the droplet to store angular momentum in the deformable $^3$He crust than in the superfluid $^4$He core. Vortex-hosting configurations are absent as equilibrium configurations in the prolate branch as well. While this could be due to the nanoscopic size of the droplets studied here, we want to stress that, as Supplementary Table 1 shows, the energy difference between vortex-hosting and vortex-free configurations can be very small and likely separated by small energy barriers, so one should not discard that both types of configurations are found in actual experiments.
As for pure $^4$He droplets, the presence of vortices can be experimentally tested by doping the droplets with heliophilic impurities.[@Gom14; @Oco19] These impurities are captured by the droplet and sink into the $^4$He core.[@Bar06] Coherent x-ray scattering would reveal the space distribution of the impurities, which might arrange along the vortex cores if vortex arrays are present,[@Gom14; @Oco19] producing otherwise interference patterns very different from those of vortex-hosting droplets.
In the prolate branch, due to the small surface tension of the $^3$He-$^4$He interface, the $^4$He core fissions already at moderate angular velocities and superfluid, multiple cores configurations are the more stable ones. Diffractive imaging of multiple connected $^4$He cores for droplets of the size addressed in this work is challenging at present because they are small and the contrast is expected to be small, but likely not for the $N=10^8-10^{11}$ droplets in ongoing experiments.
It is less obvious whether metastable 3-lobed $^4$He core configurations may be experimentally detected; in the case of classical droplets, they were identified[@Ohs00; @Hil08] twenty years after being predicted.[@Bro80] These configurations are highly unstable; for instance, Supplementary Table 1 shows that a clear 3-lobed configuration as that at ${\cal L}=4$ can either decay to the metastable oblate $n_v=1$ vortex configuration, $\sim 8$K below it, or to the stable prolate, fissioned core configuration, about $37$K below it.
An investigation on viscous immiscible two-fluid droplets has been conducted recently[@But20] that complements the present study on droplets made of two quantum fluids of limited solubility. Last but not least, our results can be used as benchmark for the applicability of these classical calculations to the quantum domain.
Methods {#methods .unnumbered}
=======
We have considered in the present work mixed helium droplets at “zero temperature”, this meaning a temperature so low (a few mK) that thermal effects on the energetics and morphology of the droplet are negligible, $^3$He is in the normal phase, and $^4$He is superfluid.
The droplet is described within the DFT approach[@Anc17] using the density functional of Ref. . Due to the large number of $^3$He atoms in this study, $^3$He can be treated semiclassically in the Thomas-Fermi approximation.[@Pi19] The DFT equations have been solved adapting the 4He-DFT BCN-TLS computing package[@Pi17] to the case of helium mixtures. Further details are given in Supplementary Section 1.
Code availability {#code-availability .unnumbered}
=================
The results have been obtained adapting the 4He-DFT BCN-TLS computing package which is freely available.[@Pi17]
We are most indebted to Andrey Vilesov for informations on their ongoing experiments with mixed helium droplets that have motivated and clarified some aspects of this work, and to Sam Butler for useful exchanges. This work has been performed under Grant No FIS2017-87801-P (AEI/FEDER, UE). J.M.E. acknowledges support from Ministerio de Ciencia e Innovación of Spain through the Unidades de Excelencia “María de Maeztu” grant MDM-2017-0767. M.B. thanks the Université Fédérale Toulouse Midi-Pyrénées for financial support throughout the “Chaires d’Attractivité 2014” Programme IMDYNHE.
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[**Supplementary Information for “Rotating mixed $^3$He-$^4$He nanodroplets”**]{}
S1.- The DFT approach for mixed $^3$He-$^4$He nanodroplets {#s1.--the-dft-approach-for-mixed-3he-4he-nanodroplets .unnumbered}
==========================================================
The DFT equations obtained by functional variation of the energy density are formulated in a rotating frame of reference with constant angular velocity $\omega$ around the $z$ axis.[@Anc18] In terms of the DFT Hamiltonians ${\cal H}_3[\rho_3,\rho_4]$ and ${\cal H}_4[\rho_3,\rho_4]$,[@Bar97] $$\begin{aligned}
&\left\{{\cal H}_3[\rho_3,\rho_4] \,- \frac{m_3}{2}\,\omega^2 (x^2+y^2)\right\} \,\Psi_3(\mathbf{r}) = \,\mu_3 \, \Psi_3(\mathbf{r})
\\
&\left\{{\cal H}_4[\rho_3,\rho_4] \,-\omega \hat{L}_4\right\} \,\Psi_4(\mathbf{r}) = \,\mu_4 \, \Psi_4(\mathbf{r})
\label{eq1}
\end{aligned}$$ where $\mu_3$($\mu_4$) is the $^3$He($^4$He) chemical potential, $\hat{L}_4$ is the $^4$He angular momentum operator, and $\Psi_3(\mathbf{r})$ and $\Psi_4(\mathbf{r})$ are the real $^3$He and complex $^4$He effective wavefunctions related to the atom densities as $\Psi_3^2(\mathbf{r})=\rho_3(\mathbf{r})$ and $|\Psi_4(\mathbf{r})|^2=\rho_4(\mathbf{r})$. These equations are solved imposing a given value of the total angular momentum per atom, which requires finding iteratively the value of $\omega$. Classically, this corresponds to torque-free droplets with an initially prescribed rotation, as they are isolated.
Vortices are nucleated in the $^4$He core using the imprinting procedure[@Anc18] by which $n_v$ vortex lines parallel to the $z$ axis are initially created, *i.e.*, one starts the iterative solution of equation (\[eq1\]) from the wave function $$\Psi_4(\mathbf{r})=\sqrt{\tilde\rho_4(\mathbf{r})}\,
\prod _{j=1}^{n_v} {(x-x_j)+\imath (y-y_j) \over \sqrt{(x-x_j)^2+(y-y_j)^2}}
\label{eq2}$$ where $(x_j, y_j)$ is the initial position of the $j$-vortex linear core with respect to the $z$-axis of the droplet, and $\tilde\rho_4(\mathbf{r})$ is the vortex-free $^4$He density. The initial vortex positions are guessed and during the functional minimization of the total energy, both the vortex positions and droplet density are allowed to change to provide at convergence the lowest energy configuration for the chosen value of $L_z$.
S2.- Scaled angular momentum and angular velocity {#s2.--scaled-angular-momentum-and-angular-velocity .unnumbered}
=================================================
Classical rotating droplets subject to surface tension and centrifugal forces are characterized by two dimensionless variables, angular momentum $\Lambda$ and velocity $\Omega$ that allow to describe the sequence of droplet shapes in a universal phase diagram, independently of the droplet size.[@Bro80; @But11] These variables have also been used to characterize pure $^4$He and $^3$He droplets within the DFT approach.[@Anc18; @Pi19]
The definitions of $\Lambda$ and $\Omega$ have been generalized to the case of droplets made of two immiscible fluids as follows.[@But20] An effective particle density $\rho_{\rm eff}$ and mass $m_{\rm eff}$ are introduced such that the moment of inertia of a spherical droplet with such effective mass and density coincides with that of a spherical droplet of inner radius $R_i$ and atom density $\rho_4$ surrounded by a spherical shell of outer radius $R_o$ and atom density $\rho_3$: $$m_{\rm eff} \rho_{\rm eff} = \frac{ m_4 \rho_4 R_i ^5 + m_3 \rho_3 (R_o^5 - R_i^5) }{R_o^5}
\label{eq10}$$ together with an effective surface tension $$\gamma_{\rm eff} = \frac{\gamma_{34} R_i + \gamma_3 R_o}{R_o}
\label{eq11}$$ where $\gamma_{34}$ is the surface tension of the $^3$He-$^4$He interface and $\gamma_3$ that of the $^3$He free surface. The scaled variables $\Lambda$ and $\Omega$ are then written as for pure droplets [@Bro80; @But11] but replacing $m$ and $\gamma$ with $m_{\rm eff}$ and $\gamma_{\rm eff}$, respectively: $$\Lambda = \frac{ N \hbar}{\sqrt{8 \,\gamma_{\rm eff}\, R_o^7 \, m_{\rm eff} \, \rho_{\rm eff} }}\; {\cal L}
\label{eq12}$$ $$\Omega= \sqrt{\frac{m_{\rm eff}\, \rho_{\rm eff} \, R_o^3}{8\, \gamma_{\rm eff}}} \; \omega \;.
\label{eq13}$$ Note that $\Lambda$ is written in terms of the total angular momentum per atom in $\hbar$ units, ${\cal L} = (L_3+L_4)/(N_3+N_4) = L/N$.
The values of the magnitudes entering the above equations are: $\gamma_3= 0.113$KÅ$^{-2}$, $\gamma_{34}= 0.016$KÅ$^{-2}$, $\hbar^2/(2m_3) = 8.0418$KÅ$^2$, $\hbar^2/(2m_4) = 6.0597$KÅ$^2$, $\rho_3 = 0.016347$Å$^{-3}$, and $\rho_4 = 0.021836$Å$^{-3}$. For the $N_3=6000$, $N_4=1500$ droplet, one gets $R_i= 25.40$Å and $R_o= 47.02$Å, and hence $\Lambda = {\cal L}/3.042$ and $\Omega = 10.61 \; \hbar \omega$, if $\hbar \omega$ is given in K.
---------- ----------- ---------- ------------- --------------- --------------------- --------------------- ------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
$\Lambda$ $\Omega$ ${\cal L} $ ${\cal L}_3 $ ${\cal L}_4 $ $ \hbar \omega$ ${\cal R}$
($\hbar$) ($\hbar$) ($\hbar$) ($\times 10^{-2}$K) (K) $a_x$ (Å) $b_y$ (Å) $c_z$ (Å) $a_x/c_z$ $b_y^3/V$ $a_x$ (Å) $b_y$ (Å) $c_z$ (Å)
[****]{} 0.164 0.105 0.5 0.50 0.00 0.99 -22488.5 47.36 47.36 46.95 1.009 0.243 25.29 25.26 25.35
[****]{} 0.329 0.206 1 1.00 0.00 1.94 -22433.3 47.86 47.86 45.96 1.041 0.251 25.24 25.22 25.43
O1v 0.329 0.169 1 0.80 0.20 1.59 -22385.4 47.87 47.87 45.93 1.042 0.251 29.15 29.15 –
[****]{} 0.657 0.381 2 2.00 0.00 3.59 -22223.4 49.64 49.63 42.65 1.164 0.280 25.26 25.19 25.31
O1v 0.657 0.352 2 1.80 0.20 3.32 -22198.8 49.49 49.49 42.81 1.156 0.278 28.90 28.90 –
[****]{} 0.986 0.517 3 3.00 0.00 4.87 -21903.87 52.02 52.02 38.57 1.349 0.322 25.57 25.57 24.31
O1v 0.986 0.493 3 2.80 0.20 4.65 -21897.4 51.83 51.83 38.57 1.344 0.319 28.67 28.67 –
O1v2m 0.986 0.474 3 2.60 0.40 4.47 -21875.2 51.67 51.67 38.77 1.333 0.316 32.59 32.59 –
O1v3m 0.986 0.451 3 2.40 0.60 4.25 -21851.2 51.61 51.61 38.90 1.327 0.315 35.77 35.77 –
O 1.151 0.571 3.5 3.50 0.00 5.38 -21711.4 53.33 53.33 36.50 1.461 0.347 25.89 25.89 23.47
O1v 1.151 0.550 3.5 3.50 0.20 5.18 -21713.0 53.12 53.12 36.33 1.462 0.343 28.67 28.67 –
O1v2m 1.151 0.533 3.5 3.10 0.40 5.02 -21697.0 52.93 52.93 36.51 1.449 0.339 32.48 32.48 –
O 1.315 0.618 4 4.00 0.00 5.82 -21501.1 54.66 54.66 34.48 1.585 0.374 26.36 26.36 22.47
O1v 1.315 0.597 4 3.80 0.20 5.63 -21510.2 54.46 54.46 34.17 1.594 0.370 28.80 28.80 –
O1v2m 1.315 0.582 4 3.60 0.40 5.49 -21499.8 54.27 54.27 34.26 1.584 0.366 32.42 32.42 –
O1v3m 1.315 0.568 4 3.40 0.60 5.35 -21488.6 54.10 54.10 34.43 1.571 0.363 35.73 35.73 –
O1v4m 1.315 0.550 4 3.20 0.80 5.18 -21474.8 54.01 54.01 34.59 1.561 0.361 38.38 38.38 –
O2v 1.315 0.589 4 3.70 0.30 5.56 -21496.8 54.18 54.26 34.62 1.565 0.366 30.87 29.23 18.72
O1v 1.644 0.673 5 4.80 0.20 6.34 -21059.9 57.17 57.17 30.10 1.899 0.428 29.42 29.42 –
O1v2m 1.644 0.660 5 4.60 0.40 6.22 -21059.4 56.99 56.99 29.87 1.908 0.424 32.56 32.56 –
O1v3m 1.644 0.648 5 4.40 0.60 6.11 -21057.0 56.81 56.81 29.93 1.898 0.420 35.76 35.76 –
O1v4m 1.644 0.637 5 4.20 0.80 6.00 -21053.5 56.65 56.65 30.08 1.883 0.416 38.55 38.55 –
O2v 1.644 0.660 5 4.63 0.37 6.22 -21057.7 57.27 56.76 29.94 1.913 0.419 33.73 30.13 –
O3v 1.644 0.650 5 4.50 0.50 6.13 -21050.2 56.88 56.89 30.01 1.895 0.422 34.36 34.30 3.05
O4v 1.644 0.644 5 4.39 0.61 6.07 -21039.6 56.77 56.77 30.14 1.884 0.419 36.47 36.47 11.80
O4v 1.972 0.696 6 5.39 0.71 6.56 -20565.5 59.44 59.44 25.90 2.295 0.481 38.13 38.13 7.34
3L 1.197 0.584 3.64 3.628 0.008 5.50 -21655.7 53.74 53.62 35.94 1.495 0.353 – – –
3L 1.315 0.607 4 3.880 0.120 5.72 -21502.4 54.44 54.68 34.37 1.584 0.374 – – –
3L 1.479 0.638 4.5 4.278 0.222 6.01 -21282.9 54.79 56.85 32.20 1.702 0.421 – – –
3L 1.644 0.655 5 4.541 0.459 6.17 -21056.0 55.63 58.28 30.12 1.847 0.453 – – –
P 0.986 0.516 3 2.994 6.3$\times 10^{-3}$ 4.86 -21903.86 52.51 51.54 38.57 1.361 0.313 27.83 23.36 24.22
[****]{} 1.019 0.523 3.1 3.060 0.040 4.93 -21867.1 53.78 50.72 38.13 1.410 0.299 31.38 20.20 23.62
[****]{} 1.052 0.528 3.2 3.116 0.084 4.98 -21829.9 55.34 49.64 37.69 1.468 0.280 34.17 18.05 22.85
[****]{} 1.068 0.529 3.25 3.141 0.109 4.99 -21811.2 56.27 48.96 37.47 1.502 0.269 35.51 17.18 22.39
[****]{} 1.151 0.523 3.5 3.264 0.236 4.93 -21718.0 61.37 45.20 36.31 1.690 0.211 41.44 14.46 19.94
P1v 1.151 0.549 3.5 3.299 0.201 5.17 -21712.9 54.52 51.73 36.34 1.500 0.317 29.39 27.97 –
P2v 1.151 0.544 3.5 3.239 0.261 5.12 -21697.7 53.78 52.31 36.47 1.475 0.328 30.95 29.93 –
Pf 1.151 0.483 3.5 2.897 0.603 4.55 -21705.5 63.24 43.04 36.14 1.750 0.183 – – –
[****]{} 1.233 0.506 3.75 3.385 0.365 4.77 -21627.0 66.12 41.90 35.18 1.879 0.168 46.69 12.94 17.47
P1v 1.233 0.528 3.75 3.488 0.262 4.98 -21617.4 63.96 44.00 35.20 1.817 0.195 37.43 22.87 –
P2v 1.233 0.528 3.75 3.329 0.421 4.98 -21603.1 61.76 45.70 35.40 1.745 0.219 41.05 23.98 14.27
Pf 1.233 0.470 3.75 3.003 0.747 4.43 -21621.6 67.08 40.53 34.98 1.918 0.152 52.74 – –
P 1.315 0.488 4 3.499 0.501 4.60 -21539.1 70.04 39.35 34.08 2.055 0.139 51.45 11.65 15.04
P1v 1.315 0.492 4 3.422 0.578 4.64 -21527.8 69.08 40.00 34.13 2.024 0.091 51.06 13.76 –
P2v 1.315 0.498 4 3.389 0.611 4.69 -21512.3 67.75 41.10 34.31 1.975 0.159 49.41 19.45 11.31
[****]{} 1.315 0.455 4 3.115 0.885 4.29 -21539.8 70.77 38.20 33.80 2.094 0.128 – – –
[****]{} 1.644 0.386 5 3.640 1.360 3.64 -21242.8 83.55 30.56 28.86 2.895 0.065 – – –
[****]{} 1.972 0.385 6 4.317 1.683 2.69 -20999.7 97.61 20.19 19.90 4.905 0.019 – – –
---------- ----------- ---------- ------------- --------------- --------------------- --------------------- ------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
{width="0.8\linewidth"}
{width="0.8\linewidth"}
{width="0.8\linewidth"}
[99]{}
F. Ancilotto, M. Pi & M. Barranco, Spinning superfluid $^4$He nanodroplets. [*Phys. Rev. B*]{} **97,** 184515 (2018).
M. Barranco, M. Pi, S.M. Gatica, E.S. Hernández & J. Navarro, Structure and energetics of mixed $^4$He-$^3$He drops. [*Phys. Rev. B*]{} **56,** 8997 (1997).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Two important enhanced sampling algorithms, simulated (ST) and parallel (PT) tempering, are commonly used when ergodic simulations may be hard to achieve, e.g, due to a phase space separated by large free-energy barriers. This is so for systems around first-order phase transitions, a case still not fully explored with such approaches in the literature. In this contribution we make a comparative study between the PT and ST for the Ising (a lattice-gas in the fluid language) and the BEG (a lattice-gas with vacancies) models at phase transition regimes. We show that although the two methods are equivalent in the limit of sufficiently long simulations, the PT is more advantageous than the ST with respect to all the analysis performed: convergence towards the stationarity; frequency of tunneling between phases at the coexistence; and decay of time-displaced correlation functions of thermodynamic quantities. Qualitative arguments for why one may expect better results from the PT than the ST near phase transitions conditions are also presented.'
author:
- 'Carlos E. Fiore'
- 'M. G. E. da Luz'
title: Comparing parallel and simulated tempering enhanced sampling algorithms at phase transition regimes
---
Introduction
============
A keystone procedure to obtain macroscopic thermodynamics quantities (e.g., energy, specific heat, magnetization, phase transition points, etc) of statistical systems is to perform appropriate averages over their microscopic configurations. In practice, however, such systems usually have a prohibitive number of states for a full covering. Therefore, approaches relying on proper representative samplings must be considered and so Monte Carlo tools become fundamental for calculations. By a proper sampling we mean that for a given instance a method should satisfactorily: (i) represent the way the system actually evolves throughout the different microstates (among the whole set $\mathcal{S}$ of microstates in the system); and (ii) generate a set $\Omega$ of visited microstates that indeed gives a good picture of all the relevant microstates which describe the problem at that particular situation.
Within this framework, an important issue is to know under what conditions the above criteria are fullfiled. For example, biased values for physical quantities may arise when the system displays local free-energy minima and the dynamics used to generate the microscopic configurations either is not able to cross such barriers or it does so, but only after too long times. Consequently, we have broken ergodicity for finite (even large) simulations [@palmer; @neirotti], leading to metastability and thus to poor estimates for the system properties due to a non-representative $\Omega$. Metastability and broken ergodicity appear in several problems like; spin-glasses; protein folding, biomolecules; and random search, to name just a few [@examples-metastability]. Moreover, they are not restricted only to complex systems, also being present in simpler contexts like in lattice-gas models displaying first-order phase transitions [@cluster1; @cluster2; @fiore8]. As noted, in such case the sampling dynamics may present difficulties to cross the energetic barriers. Then, the system can develop hysteresis by passing back and forth the phase frontiers as we change the parameter control [@fiore8].
Different alternative ideas have been considered to overcome [@wang] or even circumvent [@cluster1; @cluster2] entropic barriers, thus restoring the ergodic behavior. In particular, enhanced sampling algorithms, such as parallel tempering (PT) [@nemoto; @geyer; @hansmann1] – also known as multiple replica exchange – and simulated tempering (ST) [@marinari; @pande], have recently attracted a lot of attention, specially due to their simplicity and generality compared to other Monte Carlo algorithms [@cluster1; @fiore8]. Briefly, in the PT method, microscopic configurations in higher temperatures are used to assure an ergodic free walk in lower temperatures: one simulates replicas of the same system at distinct $T$’s, allowing the exchange of temperature between the replicas. For the ST, on the other hand, an unique replica is considered, however, the system occasionally undergoes temperature changes along its evolution.
Given the different tempering implementation in the two approaches, a natural question is how they compare to each other [@park2; @ma; @pande2]. For example, the rate of temperatures switching is higher for the ST [@park2; @ma; @pande2]. So, usually one could expect a larger number of distinct phase space regions visited when using the ST, thus a possible advantage over the PT. But as we discuss in Section II.C, near phase transition conditions this is not always the case. Therefore, it still an open query if indeed one method is systematically superior in all situations. With the above in mind, here we compare the PT and ST efficiencies when applied to phase transitions, specially to the first order case.
A short comment regarding the comparison between the PT and ST for first order phase transitions is in order. In principle, for a true first order transition, i.e., for systems in the thermodynamic limit, the energy descontinuous gap would lead to a small probability of accepting exchanges between the PT replicas [@nemoto]. But in concrete calculations, one is always dealing with finite sizes $L$, where the actual thermodynamics properties are described by continuous functions. Also, these functions are smooth and tend to the correct asymptotic behavior (for $L \rightarrow \infty$) only if the state space is properly sampled [@cluster2; @wang], what has been shown to be the case for the PT [@fiore8]. Thus, in practice the above mentioned difficulty for the PT is not an issue and the method is indeed an appropriate tool to study first order transitions, as discussed and exemplified in different works [@fiore8; @pt-first-order; @hansmann2]. Hence, the PT and ST (this latter rarely considered in such regime, few exceptions being Refs. [@st-first-order]) can be analyzed at the same footing. So, possible convergence differences can be associated just to the way the algorithms generate the sets $\Omega$, and not to the approaches eventual instrinsic distinctions (recall that conceptually they are similar [@berg]).
In this contribution we first revisit the simplest Ising spin model displaying a well understood second order phase transition. This is an instructive example because in a recent work [@ma], it has been shown that through an improved version of the ST, the frequency of successful exchanges (measured in terms of transition decay rates) is higher for the ST than for the PT method. However, the comparison was not carried near the critical temperature. By analyzing time correlation functions, defined as $$C_{w}(\tau) =
\langle w(t) - {\bar w} \rangle \langle w(t + \tau) - {\bar w} \rangle,
\label{acf}$$ for $w$ relevant thermodynamic quantities (like energy and magnetization) of mean $\bar w$ and $\langle \ \rangle$ denoting time averages, one no longer gets a better performance of the ST around $T_c$. In fact, we find that the PT leads to faster decaying $C$’s.
Then, we move to the main focus of this contribution: the harder situation of strong first-order phase transitions, where the use of one-flip algorithms like Metropolis often gives rise to poor numerical simulations. As the specific case study, we consider the lattice gas model with vacancies (a spin-1 model in the magnetic systems jargon) [@BEGMODEL]. This class of problems has been extensively studied under different alternative methods [@cluster1; @pla; @cluster2; @fiore8; @fiore4]. Hence, the many available results can help to benchmark those obtained from the PT and ST. We show that although both, PT and ST, lead to equivalent good results in the limit of long simulations, the PT displays a faster convergence towards stationarity. Moreover, for the PT, the tunneling between different phases at the coexistence is more frequent and the generated microscopic configurations uncorrelate faster.
The work is organized as the following. In Sec. II we review the PT and ST methods, discussing distinct implementations. We also give reasons why the PT may outperform ST near phase transition conditions. In Sec. III we consider a spin system displaying a second-order phase transition. The lattice-gas model and its comparative study with the PT and ST methods – addressing a first order phase transition – are presented in Sec. IV. Finally, in Sec. V we draw our last remarks and the conclusion.
The PT and ST sampling algorithms
=================================
The central idea behind a tempering enhanced sampling algorithm is try to guarantee ergodicity by means of appropriate temperature changes during the simulations, thus allowing efficient and uniform visits to a fragmented multiple regions phase space [@berg]. Suppose we shall study a system at a given $T_0$. We assume $T_1 = T_0$ and define a set of $N$ distinct temperatures $T_1 < T_2 < \ldots < T_N$, with $\Delta T = T_N - T_1$. There are different ways to implement tempering [@tempering-implementation], two important ones being the PT and ST, which we describe next.
Parallel Tempering
------------------
The PT approach combines a standard algorithm (e.g., Metropolis) with the simultaneous evolution of $N$ copies of the system (each at a different $T_n$), occasionally allowing the replicas to exchange their temperatures. Fixing relevant parameters, the method is implemented by first running $M_{eq}$ times (to assure equilibration of all the $N$ copies) a two parts procedure, (a) and (b), discussed below. After that, for each (a)-(b) composite MC step (repeated $M_{a,b}$ times) we calculate the thermodynamics quantities at the temperature of interest $T = T_1$. The average over the $M_{a,b}$ partial values give the final results. In fact, we further improve the calculations and estimate the statistical deviations by performing this procedure (after relaxation) $M_{rep}$ times, so that in total the number of (a)-(b) MC steps is $M_{tot} = M_{eq} + M_{a,b} \times M_{rep}$.
In (a), for each replica (at a distinct $T_n$), a site lattice $l$ is chosen randomly. Then, its occupation variable $\sigma_l$ may change to a new value $\sigma_{l}'$ according to the Metropolis prescription $P = \rm min\{1, \, \exp[-\beta \Delta \cal H]\}$ [@metr], where $\Delta{\cal H} = {\cal H}(\sigma') - {\cal H}(\sigma)$ is the energy variation due to the occupation change. This is done until a full lattice covering and the process is repeated all over again $M$ times. (b) In the second part, arbitrary pairs of replicas (say, at $T_{n'}$ and $T_{n''}$ and with microscopy configurations $\sigma'$ and $\sigma''$) can undergo temperatures switchings, with probability ($\beta_n = (k_{B} T_{n})^{-1}$) $$p_{n' \leftrightarrow n''} =
\min \{1, \, \exp[(\beta_{n'} - \beta_{n''})
({\cal H}(\sigma') - {\cal H}(\sigma''))] \}.
\label{p-pt}$$ The PT algorithm is schematic represented in Fig. 1 (a).
Although the above prescription is rather simple, few technical aspects should be observed. First, it is necessary to find a good compromise between the $p$’s values (which increase with $\Delta T/N$ decreasing) and the replicas number $N$. This is so to guarantee relatively frequent exchanges, while keeping the computational efforts low. Hence, extra procedures have been proposed [@hansmann2; @helmut; @predescu; @doll; @bittner]. Here we use only the ones explained above. However we mention that for our present systems, one of us has tested some of these extra implementations [@fiore8] (always assuming arbitrary $n'$’s and $n''$’s for the step (b) above), not finding any significant difference. Second, the system size ($L$) also imposes restrictions on the $N$’s. For small systems, a few number of replicas is enough to assure rapid convergence. On the other hand, by increasing $L$ the exchange probabilities (Eq. (\[p-pt\])) decreases, so the inclusion of extra copies becomes necessary. Such care has been explicit taken in our simulations. Finally, we observe that most works that use the PT method implement the switching attempts only between adjacent replicas (i.e., at $T_{n'}$ and $T_{n'' = n'+1}$), in principle because the probability of exchanges decreases for increasing $T_{n''} - T_{n'}$. Nevertheless, it has been shown [@fiore8] that non-adjacent exchanges are essential to speed up the crossing of high free-energy barriers (what we discuss in more details in Section II.C). Therefore, here we will allow exchanges between first ($\delta = 1$), second ($\delta = 2$), etc, neighbor replicas, meaning those between $T_n$ and $T_{n + \delta}$.
Simulated Tempering
-------------------
For the ST, a single realization of the model is considered, however, during the dynamics its temperature can assume the different values $T_n$’s. The implementation is similar to that for the PT in Section II.A, but applied only to one copy of the system. Therefore, the previous step (b) now reads: A change $T_{n'} \rightarrow T_{n''}$ may take place for the system according to the probability (with $\sigma$ its configuration) $$p_{n' \rightarrow n''} =
\min \{ 1, \, \exp[(\beta_{n'} - \beta_{n''}){\cal H}(\sigma)
+ (g_{n''} - g_{n'})] \}.$$ The ST algorithm is illustrated in Fig. 1 (b).
Note that $p_{n' \rightarrow n''}$ depends on the weights $g$’s. Moreover, for a better sampling, the evolution should uniformly visit all the established temperatures. This is just the case when $g_{n} = \beta_{n} \, f_{n}$, with $f_n$ the system free energy at $T_n$ [@pande; @park2; @pande2]. To obtain $f$ is not an easy task. For instance, in Ref. [@ma] its exact (numerical) values follows from $f_n = - \ln[Z_n] / (V \beta_n)$, with the partition function $Z_n$ computed by an involving recursive procedure. Here, $V$ is the system volume, which in a regular square lattice reads $V = L^2$. In our examples we will consider this same protocol, but using a simpler numerical implimentation for $Z_n$. Indeed, in the thermodynamic limit $$Z_n = - (\lambda^{(0)}_n)^L,
\label{e4}$$ where $\lambda^{(0)}_n$ is the largest eigenvalue of the transfer matrix ${\mathcal T}$ at $T_n$ (for details see, e.g., Ref. [@sauerwein]). By its turn, $\lambda^{(0)} = \langle {{\mathcal T}(S_{k}, S_{k})} \rangle /
\langle \delta_{S_{k}, S_{k+1}} \rangle$ can be calculated from straightforward Monte Carlo simulations [@sauerwein], where $S_{k}$ is the lattice $k$-layer configuration $\sigma_{1,k}, \sigma_{2,k}, \ldots, \sigma_{L,k}$ and $\delta_{S_{k}, S_{k+1}} = 1$ ($= 0$) if the $k$ and $k+1$ layers are equal (different). A central point is that in principle Eq. (\[e4\]) would hold true only for infinite size systems. However, if $L$ is not too small, the above relation is extremely accurate and for any practical purpose gives the correct $Z_n$, as we show in the next Section. Such way to determine $p_{n' \rightarrow n''}$ will be named the ST (exact) free-energy method, ST-FEM.
Finally, we observe that approximations for $g$ are equally possible. One implementation being [@pande] $$g_{n+1} - g_{n} \approx
(\beta_{n+1} - \beta_{n}) (U_{n+1}+U_{n})/2,$$ with $U_n = \langle {\cal H}_n \rangle$ ($n=1, 2, \ldots, N$) the average energy at $T_n$. The $U$’s can be evaluated from direct auxiliary simulations. For completeness we will also consider this ST approximated method, which we call ST-AM.
The PT and ST methods near phase transition regimes
---------------------------------------------------
The sampling of a statistical system when the phase space has a complicated landscape full of free-energy valleys and hills [@mauro1] is particularly delicate: one needs to uniformly visit different regions of $\mathcal{S}$ [@debenedetti] (those more important for the given parameters), but which are separated by many entropic barriers [@helmut]. In this case, the particular way in which a method evolves throughout the microstates space to generate $\Omega$ – even with the use of enhanced procedures – may crucially determine the final outcome of sampling. For instance, non-ergodic “probing” of the multiple domains [@williams] can prevent the proper relaxation to equilibrium.
The previous comments fit perfectly well first-order phase transitions, where the minima of the free-energy are separated by large barriers. Nevertheless, we observe that for second-order phase transitions, the divergence of time and spatial length correlations creates strongly correlated configurations [@landau-binder]. It leads to a certain clusterization of relevant parts of $\mathcal{S}$ at the critical point, with independent and unbiased $\Omega$ difficult to obtain. So, although associated to different mechanisms, near both first and second order transitions we can expect a “fragmented” phase space. Hence, even if the PT and ST are not crucially distinct in usual situations (in fact, the ST being slight better than the PT in few instances [@ma]), here we argue qualitatively that in such cases the PT can outperform the ST.
Thus, for the above contexts of multiple basins [@mauro2], the Fig. 2 schematically represents “stretches” of typical dynamical paths generated by the ST and PT algorithms. The successively visited $\omega$’s until leaving the domain – delimited by high local free-energy barriers (or cluster walls) – can form a very sinuous trajectory on that particular region of $\mathcal{S}$ due to a complex topography.
Thus, consider first the ST, Fig. 2 (a). The initial microstate $\omega_0$ evolves (at $T = T_1$) in a very tortuous path, but in average towards the border of the domain, reaching $\omega_a$ after $M$ steps. Then, it undergoes a temperature change $T_1 \rightarrow T_3$ and again evolves $M$ steps getting to $\omega_a'$, this time in a more straight trajectory because the higher $T$ (note if there was no temperature change, the path would follow the dashed line displayed in the plot). Finally, there is a second successful attempt to change $T$, $T_3 \rightarrow T_j > T_3$, and after $M$ steps the system ends up very close to the barrier separating the basins.
In Fig. 2 (b) we observe the PT dynamics, where just one successful temperature exchange takes place (between the only two replicas depicted). The microstate $\omega_b$ ($\omega_d$) is obtained from $\omega_0$ after $2 M$ steps at $T = T_1$ ($T = T_j$). Obviously, $\omega_a'$ in the ST must be in average closer to (farther from) the domain border than $\omega_b$ ($\omega_d$) in the PT implementation. Then, there is an exchange of temperatures and the evolution of $\omega_d$ at $T_1$, after $m < M$ steps, already makes the replica to cross the basin barrier to the microstate $\omega_d'$. Furthermore, after $\Delta t = M$ the state $\omega_b$ at $T_j$ leads to a $\omega_b'$ close to the border.
The above illustration – although certainly not extinguishing all the possibilities – is already representative of why the PT can be more efficient in sampling a space full of energetic valleys and hills (e.g., at phase transition regimes). It is so for the following reasons: (i) In the PT, the existence of replicas at all the interval $\Delta T$ of temperatures generate some paths which more quickly will approach the domain borders, as seen in Fig. 2 (b) for $\omega_0 \rightarrow \omega_d$ at $T_j$. Moreover, the microstates along such trajectories at higher $T$’s of course are usually more energetic. (ii) So, when finally there is an exchange of temperature, a microstate of high energy, even if now at lower $T$’s, will demand a smaller number of steps to cross a barrier (like $\omega_d \rightarrow \omega_d'$ in Fig. 2 (b)), and thus to start visiting other basins. On the other hand, trajectories of microstates of low energy, that during a certain $\Delta t$ have evolved under small values of $T$’s, e.g. $\omega_0 \rightarrow \omega_b$ in Fig. 2 (b), when shifting to higher temperatures will speed up their ways towards the barrier ($\omega_b \rightarrow \omega_b'$). Note, nevertheless, that this is possible [*only*]{} if non-adjacent exchanges are allowed, the case we are assuming here. (iii) The above collective dynamics makes possible many of the replicas successfully leave a domain after fairly similar number of steps. Hence, once in another basin region, this “parallel” process can proceed in the same fashion. (iv) By its turn, we can face the ST as a “serial” process, then a faster drift towards the domain walls takes place only when $T$ increases. As a consequence, the eventual more frequent temperature exchange for the ST [@park2; @ma; @pande2] not necessarily constitutes an advantage in complex $\mathcal{S}$ landscapes (as illustrated in Fig. 2). (v) Lastly, a not critical issue but which also may give some small advantage for the PT over the ST is that in the former, often the replicas (even at smaller $T$’s) cross the domain high barriers more or less at the same time. Thus, once leaving a certain basin we already have a sample of microstates at $T_1$ to make averages for the PT. As displayed in the Fig. 2 (a), for the ST it may happen that when the system reaches a microstate configuration able to cross the barrier, it is not at $T_1$. Hence, an extra time is necessary for the system (naturally from the algorithm dynamics) to come back to $T_1$ and so the averages to be performed.
We finally observe that when the relevant space is more homogeneous in energy (e.g., far away from phase transitions), one should not expect so high increase of the trajectories sinuosity as we diminish $T$. Then, it is not difficult to realize that the listed differences between the PT and ST methods might not be important.
The previous discussion is based on qualitative arguments. Of course, they should be corroborated by concrete quantitative studies. Next we analyze two systems near phase transition conditions. We will explicit show through detailed numerical simulations that indeed the PT algorithm is more efficient, specially in the case of first order phase transitions.
The Ising model
===============
The model is defined by the following Hamiltonian $${\cal H} = - J \sum_{<i,j>} \sigma_{i} \, \sigma_{j}
- H \sum_{i=1}^{V} \sigma_i,$$ where $<i,j>$ denotes nearest-neighbors pairs $i$ and $j$ of a $d$-dimensional lattice of $V = L^{d}$ sites. At each site $i$, the spin variable assumes the values $\sigma_i = \pm 1$. $J$ is the interaction energy and $H$ is the magnetic field. The Ising model displays a second-order phase transition (ferromagnetic–paramagnetic) at $T_c \approx 2.269$ and $H=0$. For a square lattice ($d=2$), the transfer matrix diagonal elements are $${\mathcal T}(S_{k}, S_{k}) = \exp\left[ \beta \, \big(
\sum_{l=1}^{L} J \,
(1 + \sigma_{l,k} \, \sigma_{l+1,k}) + H \, \sigma_{l,k} \big) \right].
\label{e17}$$
![For the Ising model with $H=0$, $L=32$, and units of $J/k_B$, comparison between the partition function versus $T$ calculated exactly [@ferdinand] and from Eq. (\[e4\]).](fig-extra-new.eps)
Our interest are in the energy $u = \langle {\cal H}\rangle/V$ and modulus of the magnetization (which is the order parameter) $m = \langle |\sum_{i=1}^{V} \sigma_{i}| \rangle / V$ per volume. For their auto-correlation functions, we just set $w = u$ and $w = m$ in Eq. (\[acf\]). Regarding the parameters, we choose $H = 0$ and a square lattice of $L = 32$. All the results are given in units of $J/k_B$. To test the accuracy of the transfer matrix largest eigenvalue method in obtaing $Z$, in Fig. 3 we compare the exact partition function (obtained from the solution in Ref. [@ferdinand]) with that calculated from Eq. (\[e4\]) for the Ising model and the above parameters. The agreement is indeed remarkable, indicating that even for $L = 32$, $Z$ and consequently $f$ is already very close to the thermodynamic limit value.
Figure 4 displays $C_m$ and $C_u$ for $T_1 = T_c$. In the simulations we use only two replicas (with $T_2 = 2.4$) and $M=1$. From the plots we see that the auto-correlations decay faster when calculated by the PT than by both the ST-AM and ST-FEM methods. In Fig. 5 we compare the time evolution of the thermodynamic quantities starting from a “hard” initial condition, i.e., a configuration very different from the ones representative of the steady state. Thus, we consider a fully ordered configuration, which obviously is not typical at $T = T_c$. This is a way of testing how efficient is a certain approach to drive the system to the stationary state. The Ising model at the transition temperature evolves to the equilibrium basically in the same fashion either when simulated by the PT or by both the ST’s.
So, we have that for a continuous phase transition (at least for the Ising model) the performances of the two tempering methods are essentially equivalent. Although at $T_c$ the PT shows faster auto-correlation decays (in contrast with the results of Ref. [@ma] for the same model, however calculated far away from the critical temperature), the stationary state is characterized by equivalent values of $m$ and $u$ for all methods.
The lattice-gas model with vacancies (BEG)
==========================================
Model
-----
The lattice-gas model (of size $V = L^d$) with vacancies is characterized by the Hamiltonian $${\cal H} = - \sum_{<i,j>} \sum_{r,s} \epsilon_{r,s} \, N_{r,i} \, N_{s,j}
- \sum_{r} \sum_i \mu_r \, N_{r,i} .
\label{e1}$$ Here, $r$ and $s$ run over the species labels $A$ and $B$, the $\epsilon_{r s}$’s are the coupling energies ($\epsilon_{AA}$, $\epsilon_{BB}$, $\epsilon_{AB}$ and $\epsilon_{BA}$), $N_{r, i} = 0, 1$ is the occupation numbers at site $i$ for species $r$, and $\mu_r$ is the species $r$ chemical potential. The above model is equivalent to the Blume-Emery-Griffiths (BEG) spin-1 ${\cal H}$ [@BEGMODEL]. Indeed, defining (with $\sigma_i = 0, \pm 1$ the possible values for the spin variable) $$N_{A,i} = (\sigma_i^2 + \sigma_i)/2, \qquad
N_{B,i} = (\sigma_i^2 - \sigma_i)/2,
\label{e2}$$ associating $\sigma_i = 1$ (-1) with the species $A$ ($B$) and $\sigma_i = 0$ with a vacancy, and setting $\epsilon_{AA} = \epsilon_{BB}$ and $\epsilon_{AB} = \epsilon_{BA}$, we get the BEG Hamiltonian $${\cal H} =
-\sum_{<i,j>} (J \, \sigma_{i} \, \sigma_{j} + K \,
\sigma_{i}^{2} \, \sigma_{j}^{2})
- \sum_{i} (H \, \sigma_i - D \, \sigma_i^2),
\label{e3}$$ for $$\begin{aligned}
& & {\it H} = (\mu_{A} - \mu_{B})/2, \qquad D = - (\mu_{A} + \mu_{B})/2,
\nonumber \\
& & J = (\epsilon_{AA} - \epsilon_{AB})/2, \ \ \ \
K = (\epsilon_{AA} + \epsilon_{AB})/2.
\label{e7} \end{aligned}$$
We will consider a square lattice with periodic boundary conditions. In this case, the transfer matrix diagonal elements read $$\begin{aligned}
{\mathcal T}( S_{k},S_{k}) &=&
\exp\Big[ \beta \sum_{l=1}^{L}
\Big( (H + J \, \sigma_{l+1,k}) \, \sigma_{l,k} \nonumber \\
& & + (J - D + K (1 + \sigma_{l+1,k}^{2})) \sigma_{l,k}^{2}
\Big) \Big].
\label{e18}\end{aligned}$$ The model has two order parameters, $q$ and $m$, defined by $q = \langle \sum_{i=1}^{V}(N_{A,i}+N_{B,i} ) \rangle/V$ and $m = \langle \sum_{i=1}^{V} (N_{A,i}-N_{B,i}) \rangle/V$. Also important is the quantity energy per volume, given by $u = \langle {\cal H} \rangle/V$. The auto-correlation are then obtained from $w = q$, $w = m$ and $w = u$ in Eq. (\[acf\]).
Results
-------
For fixed $K/J$, $H$ and $T$, the characteristic of the phase space is determined by $D$. In the regime we are interested, there are two phases if $D$ is small, one rich in species A and the other in species B. For high values of $D$, the model displays a single gas phase, rich in vacancies. A strong first-order phase transition between these two situations takes place at $D = D^{*}$, which obviously depends on $K/J$, $H$ and $T$. For definiteness, in the following we study the BEG Hamiltonian assuming $K/J = 3$, $H = 0$ and $T = T_1 = 1.4$ (for other parameter values, see Sec. V). In such case, $D^{*} = 8.000$ in the thermodynamic limit [@fiore8]. All the results will be presented in units of $J/k_B$.
It is well known that for different lattice-gas systems, approaches based on cluster algorithms [@cluster1] are very appropriate to deal with metastability arising in first-order phase transitions. So, next we will compare results obtained from both tempering methods with those available from cluster calculations [@cluster1]. Regarding the parameters values, unless otherwise explicit mentioned, in the simulations we consider $L = 20$, $D = 8.000$ and the replicas in the temperature interval $\Delta T = 0.6$. Also, whenever necessary we perform in total up to $M_{tot} = 8 \times 10^{7}$ simulation steps (see Sec. II.A) to evaluate the sought quantities. Furthermore, we always use $M=1$.
As the first comparative analysis, in Fig. 5 we plot the order parameter $q$ probability distribution histogram for a long simulation run of $10^{7}$ MC steps. As the chemical potential we set $D = 8.004$, instead of $D = 8.000$, since it leads to a same high for the two peaks of the bimodal order parameter probability distribution (we mention, nevertheless, that $D = 8.000$ gives the same qualitative results). The agreement of the two tempering with the cluster method [@cluster2] is similar (in fact, a little better for the PT case). Such calculations show that for a long enough time, both the PT and ST are able to circumvent the metastable states, allowing the system to cross the free-energy barriers separating the different phases at the coexistence.
Despite the previous agreement, the PT and ST do present differences when other aspects are analyzed. For instance, we show in Fig. 6 the time evolution of $q$ towards the steady state, starting from a fully random initial configuration. We also consider distinct number of replicas $N$ and temperature intervals $\Delta T $. We find that under the same simulation conditions, generally the PT converges faster, being closer to the cluster results than the ST (ST-FEM and ST-AM). However, for the lower value of $\Delta T = 0.25$, in all cases the system (up to $10^4$ MC steps) cannot even escape the region near the initial random configuration. On the other hand, by increasing $\Delta T = 0.6$ – although the probability for temperature exchanging decreases – the system starts to move towards the stationary regime. Furthermore, the larger the number of replicas $N$, the faster the convergence. Finally we mention that the steady value of $q = 2/3$ at $D = D^{*} = 8.000$ can be understood recalling that at the phase coexistence, two liquid phases ($q \approx 1$) coexist with one gas phase ($q \approx 0$). Since their weights are equal (1/3), we have $q \approx 2/3$ for any system size.
Another interesting test is to perform the numerical simulations when the system is already at the steady state. In Fig. 7 we show the time evolution of the “magnetization” $m$ for both tempering methods at the phase coexistence. In the plots the time is shifted so to discard the $M_{eq}$ initial MC steps necessary for equilibration. We see that the tunneling between the three different phases is substantially more frequent for the PT than for the ST. It being true along the whole evolution, as we have checked for an interval of $10^{7}$ MC steps (in the Fig. 7 we show only two distinct simulation stretches). Actually, the PT tunneling pattern presents the same behavior than that observed in the notorious accurate cluster algorithm [@cluster1], Fig. 8. Such results concrete exemplify some of the qualitative arguments given in Sec. II.C to explain why the PT should be more efficient than the ST around first-order phase transitions.
A different efficiency for the methods is observed not just at the phase coexistence, but also for other values of the chemical potential $D$ around $D^{*}$. Figure 9 plots the order parameter $q$ versus $D$ for the PT and ST implementations, evaluating the averages at each $M_{a,b} = 10^{4}$ MC steps. Note that overall the PT is already quite close to the values obtained from the cluster algorithm, whereas both ST still show some discrepancy, specially for $D > D^{*}$. If now the averages are calculate each $M_{a,b} = 5 \times 10^{4}$ MC steps, the ST also becomes closer to the cluster’s (inset of Fig. 9). Once more such results can be understood in terms of the tunneling between the phases. For $D \sim D^{*}$, we still can expect high free energy barriers. With the ST, the system does not cross such barriers a sufficient number of times if $M_{a,b} = 10^{4}$. By increasing the number of MC steps for the averages, we generate a more representative $\Omega$ and thus a better estimation for $m$.
As a last efficiency measure, we consider the two relevant auto-correlation functions, $C_{q}(\tau)$ and $C_{u}(\tau)$, shown in Fig. 10. We should note that although time displaced correlation functions are more commonly studied in the context of continuous phase transitions, in the present case they are an interesting auxiliary tool to compare the PT and ST performances. As it should be, the ST-FEM uncorrelates faster than the ST-AM. Nevertheless, we see that the $C$’s decay even faster for the PT method (in fact, with a very drastic difference in the case of $C_{u}(\tau)$).
Usually, the frequency (measured in terms of a probability $p^{*}$) in which a given tempering method changes the system temperature is taken as a good indication of its efficiency. For the PT and ST algorithms, such quantity respectively reads [@juan] $p^{*} = \langle \min \{1,\exp[(\beta_i-\beta_j)({\cal H}
(\sigma_{i})-{\cal H}(\sigma_{j})] \} \rangle$ and $p^{*} = \langle \min \{1,\exp[(\beta_{i}-\beta_{j}){\cal H}(\sigma)
+ g_{j}-g_{i}] \}\rangle$. The averages are over $T_1, \ldots, T_N$, such that $p^{*}$ of order $\delta$ is the mean from all the exchanges among $T_{n}$ and $T_{n+\delta}$ (see Sec. II.A).
In Fig. 11 we display $p^{*}$ as function of $T = T_1$ for the PT and ST-FEM (the ST-AM being similar to the latter), with $N = 12$ and $\Delta T = 0.55$. As it can be seen, for any $\delta$ the ST always presents a higher probability of acceptance than the PT, in agreement with previous studies [@ma; @pande2]. Such findings are in contrast with our results here. Indeed, larger $p^{*}$’s do not translate into a better performance of the ST, at least in the case of phase transitions as argued in Sec. II-C. Therefore, exchange probabilities alone should be faced with care when trying to characterize the best tempering method for a certain context.
Finally, we show in Figs. 12 and 13 finite size analysis for the total density $q$ and the isothermal susceptibility $\chi_{T} = \beta L^{2}(\langle q^{2} \rangle-\langle q \rangle^{2})$ from the PT and ST-FEM. Continuous lines correspond to fitting curves by a method proposed in Ref. [@cluster2]. At the phase coexistence, thermodynamic quantities scale with the system volume [@rBoKo; @challa]. A discontinuous phase transition is characterized by a jump in the order parameter or even a delta function-like singularity for the susceptibility or specific heat. But this is so only at the thermodynamic limit. For finite systems not only the order parameter, but also other quantities are described by continuous functions [@fiore8; @wang; @cluster2]. We should emphasizes that smooth curves are obtained only when one uses a simulation dynamics which correctly yields an appropriate sampling. For instance, from simple Metropolis algorithms, neither the crossing among isotherms nor accurate finite size analysis for smooth curves are possible. It is due to the presence of hysteresis effects [@fiore8; @cluster1; @cluster2], which hence demand tempering enhanced algorithm. From the plots we see that both the PT and ST give fairly good results. However, the cluster continuous curve [@cluster2] is smoother and better fitted in the PT case, specially for the larger $L = 30$ value.
![$q$ versus $D$ for $L$ equal to 10 (circle), 20 (square) and 30 (triangle), calculated from the (a) PT and (b) ST-FEM. Continuous lines are fitting results [@fiore8; @cluster2]. The curves collapse if plotted as $q \times (D-D^{*}) L^2$ (insets).](figure12.eps)
Remarks and Conclusion
======================
In this paper we have presented a comparative study between two important enhanced sampling methods, namely, simulated (ST) and parallel (PT) tempering, considering spin-lattice models at phase transition conditions. Special attention has been payed to first-order phase transitions at low temperatures (for the BEG model). In such regimes, more standard algorithms often give poor results because their difficulties to overcome the large free-energy barriers in the phase space, leading, e.g., to ergodicity breaking and artificial algorithm-induced hysteresis. We also have investigated the less critical case of second order-phase transition – for which no free-energy barriers exist but there is the formation of strongly correlated clusters (basin regions) [@landau-binder] – for the well understood Ising model.
As for the tempering implementations, we have followed the usual PT procedure, but allowing temperature exchanges between non-adjacent replicas. For the temperature change probability weights $g$ in the ST, we have assumed a recent proposed approximation [@pande] (ST-AM) and a new alternative exact approach (ST-FEM), based on the eigenvalues of the transfer matrix [@sauerwein]. The ST-FEM here is formally similar to that in Ref. [@ma], but avoids the necessity to implement more complicated recursive procedures to estimate the partition function.
Different comparative analysis, both at the transient regime and already at the steady state, have been carried out. Despite the facts that: (i) after long times (thus demanding large computational effort) the final results from the PT and ST are similar; and (ii) the PT displays a smaller exchange probability than the ST; we have found that for discontinuous phase transitions the PT is always more efficient in any verified aspect. The main reason for this is basically that the PT enables the system to cross free-energy barriers more frequently than the ST: either at or near phase coexistence conditions (as explicit illustrated, e.g., in Figs. (7) and (8)). Furthermore, besides the quantitative numerical results, we also have presented heuristic arguments for why it should be expected.
Results for the instructive Ising model at the critical temperature (second-order phase transition) have also agreed with our qualitative predictions. Indeed, far away from $T_c$ it has been reported a faster convergence for the ST [@ma]. We have shown that for $T \sim T_c$ just the opposite takes place, with the auto-correlations decaying faster for the PT.
For completeness, we also have analyzed other values of $K/J$ for the BEG model (not shown), in particular for $K/J=0$, the so called Blume-Capel model. The calculations at the first-order transition ($T_1 = 0.4$ and $D=1.9968$) have corroborated the higher efficiency of the PT over the ST. More specifically, until $M_{tot} = 3 \times 10^{7}$, the system when simulated with the ST-AM has not reached the steady state, whose values for the thermodynamic quantities were different from those obtained by the ST-FEM, PT and cluster algorithms. Furthermore, the ST-FEM have agreed with the PT and cluster only for long $M_{tot}$’s. Time-displaced correlation functions decays and actual thermodynamic quantities convergence were always faster for the PT.
A second contribution of this work has been an (numerically simpler) alternative way to calculate the exact $g$ in the ST method. When comparing the ST-AM with the ST-FEM, we have found that the ST-FEM allows the system to converge to steady regime quicker than the ST-AM (see above). In addition, at the steady state, configurations generated by ST-FEM uncorrelate faster than those by the ST-AM. On the other hand, with respect to the frequency in which the system tunnels between different phases at the coexistence and the final sough thermodynamic quantities, both implementations are similar, but the latter only for long $M_{tot}$’s.
Summarizing, at phase transition regimes the PT and ST provide the same results for long (sometimes even costly) simulations. However, we find that for all the tested measures, the parallel converges faster than the simulated tempering. Also, even in such situation of a better performance from the PT, still the rate of temperature switching is higher for the ST. Thus, another message from our work is that alone, the switching rates are not sufficient to characterize the efficiency of a tempering enhanced sampling algorithm.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge researcher grants by CNPq. Financial support is also provided by CNPq-Edital Universal, Fundação Araucária and Finep/CT-Infra.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We prove the existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds provided there are barriers.'
author:
- Christian Enz
title: The scalar curvature flow in Lorentzian manifolds
---
Introduction
============
We want to find a closed spacelike hypersurface of prescribed scalar curvature in a globally hyperbolic Lorentzian manifold $N$ having a compact Cauchy hypersurface $\mc S_0$. Looking at the Gau[ß]{} equation for a spacelike hypersurface $M$, we deduce that its scalar curvature $R$ satisfies $$\lae{0.1}
R=-[H^2-|A|^2]+\bar{R}+2\bar{R}_{\alpha\beta}\nu^\alpha \nu^\beta.$$ Denoting the curvature operator defined by $H_2$ by $F$, then this equation is equivalent to $$\lae{0.2}
R=-2F(h_{ij})+\bar R+2\bar R_{\al\bet}\n^{\al}\n^{\bet}.$$ Thus, we have to allow that the right-hand side $f$ of the equation $$\lae{0.3}
F_{|_M}=f$$ is defined in $T(N)$, or more precisely, after choosing a local trivialization of $T(N)$, that $f$ depends on $x\in N$ and timelike vectors $\n\in T_x(N)$, and we look for a closed spacelike hypersurface $M$ satisfying $$\lae{0.4}
F_{|_M}=f(x,\n)\qq\A x\in M.$$
In Gerhardt solved this problem by using the method of elliptic regularization. We give a new existence proof, based on the curvature estimates in , by showing that the scalar curvature flow exists for all time, and that the leaves $M(t)$ of the flow converge to a solution of .
To give a precise statement of the existence result we need a few definitions and assumptions. First, we assume that $\Om$ is a precompact, connected, open subset of $N$, that is bounded by two compact, connected, spacelike hypersurfaces $M_1$ and $M_2$ of class $C^{6,\al}$, where $M_1$ is supposed to lie in the past of $M_2$.
Let $F=H_2$ be the scalar curvature operator defined on the open cone $\C_2\su \R[n]$, and $f=f(x,\n)$ be of class $C^{4,\al}$ in its arguments such that $$\begin{aligned}
0<c_1&\le f(x,\n)\qq\tup{if}\q\spd\n\n=-1,\lae{0.5}\\[\cma]
\nnorm{f_\bet(x,\n)}&\le c_2 (1+\nnorm\n^2),\lae{0.6}\\
\intertext{and}
\nnorm{f_{\n^\bet}(x,\n)}&\le c_3 (1+\nnorm\n),\lae{0.7}\end{aligned}$$ for all $x\in\bar\Om$ and all past directed timelike vectors $\n\in T_x(\Om)$, where $\nnorm{\cdot}$ is a Riemannian reference metric that will be detailed in .
The condition is reasonable as is evident from the Einstein equation $$\lae{0.8}
\bar R_{\al\bet}-\tfrac12 \bar R \msp \bar g_{\al\bet}=T_{\al\bet},$$ where the energy-momentum tensor $T_{\al\bet}$ is supposed to be positive semi-definite for timelike vectors (weak energy condition, cf. ); but it would be convenient, if the estimate in would be valid for all timelike vectors. In fact, we may assume this without loss of generality: Let $\vt$ be a smooth real function such that $$\lae{0.9}
\frac{c_1}2\le \vt\q\tup{and}\q \vt(t)=t\q \A\,t\ge c_1,$$ then, we can replace $f$ by $\vt\circ f$ and the new function satisfies our requirements for all timelike vectors. We therefore assume in the following that the relation holds for all timelike vectors $\n\in T_x(N)$ and all $x\in \bar\Om$.
We suppose that the boundary components $M_i$ act as barriers for $(F,f)$.
$M_2$ is an upper barrier for $(F,f)$, if $M_2$ is admissible, i.e. its principal curvatures $(\ka_i)$ with respect to the past directed normal belong to $\C_2$, and if $$\lae{0.10}
\fv F{M_2}\ge f(x,\n)\qq\A\,x\in M_2.$$ $M_1$ is a lower barrier for $(F,f)$, if at the points $\Si\su M_1$, where $M_1$ is admissible, there holds $$\lae{0.11}
\fv F\Si \le f(x,\n)\qq\A\,x\in \Si.$$ $\Si$ may be empty.
Now, we can state the main theorem.
Let $M_1$ be a lower and $M_2$ an upper barrier for $(F,f)$, where $F=H_2$. Then, the problem $$\fmo M= f(x,\n)$$ has an admissible solution $M\su \bar\Om$ of class $C^{6,\al}$ that can be written as a graph over $\mc S_0$ provided there exists a strictly convex function $\chi\in C^2(\bar\Om)$.
As proved in the existence of a strictly convex function $\chi$ is guaranteed by the assumption that the level hypersurfaces $\{x^0=\tup{const}\}$ are strictly convex in $\bar\Om$, where $(x^\al)$ is a Gaussian coordinate system associated with $\so$.
Looking at Robertson-Walker space-times it seems that the assumption of the existence of a strictly convex function in the neighbourhood of a given compact set is not too restrictive: in Minkowski space e.g. $\chi=-\abs{x^0}^2 +\abs x^2$ is a globally defined strictly convex function.
Notations and preliminary results
=================================
The main objective of this section is to state the equations of Gau[ß]{}, Codazzi, and Weingarten for hypersurfaces $M$ in a Lorentzian space $N$. Geometric quantities in $N$ will be denoted by $(\bar g_{\al\bet}),(\riema \al\bet\ga\de)$, etc., and those in $M$ by $(g_{ij}), (\riem
ijkl)$, etc. Greek indices range from $0$ to $n$ and Latin from $1$ to $n$; the summation convention is always used. Generic coordinate systems in $N$ resp. $M$ will be denoted by $(x^\al)$ resp. $(\x^i)$. Covariant differentiation will simply be indicated by indices, only in case of possible ambiguity they will be preceded by a semicolon, i.e. for a function $u$ in $N$, $(u_\al)$ will be the gradient and $(u_{\al\bet})$ the Hessian, but e.g., the covariant derivative of the curvature tensor will be abbreviated by $\riema \al\bet\ga{\de;\e}$. We also point out that $$\lae{1.1}
\riema \al\bet\ga{\de;i}=\riema \al\bet\ga{\de;\e}x_i^\e$$ with obvious generalizations to other quantities.
Let $M$ be a spacelike hypersurface, i.e. the induced metric is Riemannian, with a differentiable normal $\n$ that is timelike. In local coordinates, $(x^\al)$ and $(\x^i)$, the geometric quantities of the spacelike hypersurface $M$ are connected through the following equations $$\lae{1.2}
x_{ij}^\al= h_{ij}\n^\al$$ the so-called Gau[ß]{} formula. Here, and also in the sequel, a covariant derivative is always a full tensor, i.e. $$\lae{1.3}
x_{ij}^\al=x_{,ij}^\al-\ch ijk x_k^\al+\cha \bet\ga\al x_i^\bet x_j^\ga.$$ The comma indicates ordinary partial derivatives. In this implicit definition the second fundamental form $(h_{ij})$ is taken with respect to $\n$. The second equation is the Weingarten equation $$\lae{1.4}
\n_i^\al=h_i^k x_k^\al,$$ where we remember that $\n_i^\al$ is a full tensor. Finally, we have the Codazzi equation $$\lae{1.5}
h_{ij;k}-h_{ik;j}=\riema\al\bet\ga\de\n^\al x_i^\bet x_j^\ga x_k^\de$$ and the Gau[ß]{} equation $$\lae{1.6}
\riem ijkl=- \{h_{ik}h_{jl}-h_{il}h_{jk}\} + \riema \al\bet\ga\de x_i^\al x_j^\bet x_k^\ga
x_l^\de.$$
Now, let us assume that $N$ is a globally hyperbolic Lorentzian manifold with a compact Cauchy surface. $N$ is then a topological product $\R[]\times \mc
S_0$, where $\mc S_0$ is a compact Riemannian manifold, and there exists a Gaussian coordinate system $(x^\al)$, such that the metric in $N$ has the form $$\lae{1.7}
d\bar s_N^2=e^{2\psi}\{-{dx^0}^2+\s_{ij}(x^0,x)dx^idx^j\},$$ where $\s_{ij}$ is a Riemannian metric, $\psi$ a function on $N$, and $x$ an abbreviation for the spacelike components $(x^i)$, see , and . We also assume that the coordinate system is future oriented, i.e. the time coordinate $x^0$ increases on future directed curves. Hence, the contravariant timelike vector $(\x^\al)=(1,0,\dotsc,0)$ is future directed as is its covariant version $(\x_\al)=e^{2\psi}(-1,0,\dotsc,0)$.
Let $M=\graph \fv u\so$ be a spacelike hypersurface $$\lae{1.8}
M=\set{(x^0,x)}{x^0=u(x),\,x\in\mc S_0},$$ then the induced metric has the form $$\lae{1.9}
g_{ij}=e^{2\psi}\{-u_iu_j+\s_{ij}\},$$ where $\s_{ij}$ is evaluated at $(u,x)$, and its inverse $(g^{ij})=(g_{ij})^{-1}$ can be expressed as $$\lae{1.10}
g^{ij}=e^{-2\psi}\{\s^{ij}+\frac{u^i}{v}\frac{u^j}{v}\},$$ where $(\s^{ij})=(\s_{ij})^{-1}$ and $$\lae{1.11}
\begin{aligned}
u^i&=\s^{ij}u_j\\[\cma]
v^2&=1-\s^{ij}u_iu_j\equiv 1-\abs{Du}^2.
\end{aligned}$$ Hence, $\graph u$ is spacelike if and only if $\abs{Du}<1$. The covariant form of a normal vector of a graph looks like $$\lae{1.12}
(\n_\al)=\pm v^{-1}e^{\psi}(1, -u_i).$$ and the contravariant version is $$\lae{1.13}
(\n^\al)=\mp v^{-1}e^{-\psi}(1, u^i).$$ Thus, we have Let $M$ be spacelike graph in a future oriented coordinate system. Then, the contravariant future directed normal vector has the form $$\lae{1.14}
(\n^\al)=v^{-1}e^{-\psi}(1, u^i)$$ and the past directed $$\lae{1.15}
(\n^\al)=-v^{-1}e^{-\psi}(1, u^i).$$ In the Gau[ß]{} formula we are free to choose the future or past directed normal, but we stipulate that we always use the past directed normal. Look at the component $\al=0$ in and obtain in view of $$\lae{1.16}
e^{-\psi}v^{-1}h_{ij}=-u_{ij}-\cha 000\mspace{1mu}u_iu_j-\cha 0j0
\mspace{1mu}u_i-\cha 0i0\mspace{1mu}u_j-\cha ij0.$$ Here, the covariant derivatives are taken with respect to the induced metric of $M$, and $$\lae{1.17}
-\cha ij0=e^{-\psi}\bar h_{ij},$$ where $(\bar h_{ij})$ is the second fundamental form of the hypersurfaces $\{x^0=\const\}$. An easy calculation shows $$\lae{1.18}
e^{-\psi}\bar h_{ij}=-\tfrac{1}{2}\dot\s_{ij} -\dot\psi\s_{ij},$$ where the dot indicates differentiation with respect to $x^0$.
Next, let us state under which condition a spacelike hypersurface $M$ can be written as a graph over the Cauchy hypersurface $\mc S_0$. In Kröner proved
Let $N$ be globally hyperbolic, $\mc S_0\nobreak\
\su\nobreak\ N$ a compact, connected Cauchy hypersurface, and $M\su N$ a compact, connected spacelike hypersurface of class $C^m, m\ge 1$. Then, $M=\graph \fu
u{\mc S}0$ with $u\in C^m(\mc S_0)$.
Sometimes, we need a Riemannian reference metric, e.g. if we want to estimate tensors. Since the Lorentzian metric can be expressed as $$\lae{1.19}
\bar g_{\al\bet}dx^\al dx^\bet=e^{2\psi}\{-{dx^0}^2+\s_{ij}dx^i dx^j\},$$ we define a Riemannian reference metric $(\tilde g_{\al\bet})$ by $$\lae{1.20}
\tilde g_{\al\bet}dx^\al dx^\bet=e^{2\psi}\{{dx^0}^2+\s_{ij}dx^i dx^j\}$$ and we abbreviate the corresponding norm of a vectorfield $\h$ by $$\lae{1.21}
\nnorm \h=(\tilde g_{\al\bet}\h^\al\h^\bet)^{1/2},$$ with similar notations for higher order tensors.
For a spacelike hypersurface $M=\graph u$ the induced metrics with respect to $(\bar g_{\al\bet})$ resp. $( \tilde g_{\al\bet})$ are related as follows $$\lae{1.22}
\begin{aligned}
\tilde g_{ij}&= \tilde g_{\al\bet}x^\al_i x^\bet_j=e^{ 2\psi}[u_i u_j +\s_{ij}]\\[\cma]
&= g_{ij}+2 e^{2\psi} u_i u_j.
\end{aligned}$$ Thus, if $(\x^i)\in T_p(M)$ is a unit vector for $(g_{ij})$, then $$\lae{1.23}
\tilde g_{ij}\x^i \x^j= 1+2 e^{2\psi}\abs {u_i\x^i}^2,$$ and we conclude for future reference
Let $M=\graph u$ be a spacelike hypersurface in $N$, $p\in M$, and $\x\in
T_p(M)$ a unit vector, then $$\lae{1.24}
\nnorm{x^\bet_i\x^i}\le c (1+\abs{u_i\x^i})\le c \tilde v,$$ where $\tilde v=v^{-1}$.
Curvature functions
===================
Let $\C\su\R[n]$ be an open cone containing the positive cone $\C_+$, and $F\in C^{2,\al}(\C)\ii C^0(\bar\C)$ a positive symmetric function satisfying the condition $$\lae{2.1}
F_i=\pd F\ka i>0\; ,$$ then, $F$ can also be viewed as a function defined on the space of symmetric matrices $\msc S_\Gamma$, the eigenvalues of which belong to $\C $, namely, let $(h_{ij})\in
\msc S_\Gamma$ with eigenvalues $\ka_i,\,1\le i\le n$, then define $F$ on $\msc S_\Gamma$ by $$\lae{2.2}
F(h_{ij})=F(\ka_i).$$ If we define $$\begin{aligned}
F^{ij}&=\pde F{h_{ij}}\\
\intertext{and}
F^{ij,kl}&=\pddc Fh{{ij}}{{kl}}\end{aligned}$$ then, $$\lae{2.5}
F^{ij}\x_i\x_j=\pdc F\ka i \abs{\x^i}^2\q\A\, \x\in\R[n],$$ in an appropriate coordinate system, $$\lae{2.6}
F^{ij} \,\text{is diagonal if $h_{ij}$ is diagonal,}$$ and $$\lae{2.7}
F^{ij,kl}\h_{ij}\h_{kl}=\pddc F{\ka}ij\h_{ii}\h_{jj}+\sum_{i\ne
j}\frac{F_i-F_j}{\ka_i-\ka_j}(\h_{ij})^2,$$ for any $(\h_{ij})\in \msc S$, where $\msc S$ is the space of all symmetric matrices. The second term on the right-hand side of is non-positive if $F$ is concave, and non-negative if $F$ is convex, and has to be interpreted as a limit if $\ka_i=\ka_j$.
The preceding considerations are also applicable if the $\ka_i$ are the principal curvatures of a spacelike hypersurface $M$ with metric $(g_{ij})$. $F$ can then be looked at as being defined on the space of all symmetric tensors $(h_{ij})$ the eigenvalues of which belong to $\C$. Such tensors will be called admissible; when the second fundamental form of $M$ is admissible, then, we also call $M$ admissible.
For an admissible tensor $(h_{ij})$ $$\lae{2.8}
F^{ij}=\pdc Fh{{ij}}$$ is a contravariant tensor of second order. Sometimes it will be convenient to circumvent the dependence on the metric by considering $F$ to depend on the mixed tensor $$\lae{2.9}
h_j^i=g^{ik}h_{kj}.$$ Then, $$\lae{2.10}
F_i^j=\pdm Fhji$$ is also a mixed tensor with contravariant index $j$ and covariant index $i$. Such functions $F$ are called curvature functions.
Important examples are the elementary symmetric polynomials of order $k$, $H_k$, $1\le k\le n$, $$\lae{2.11}
H_k(\ka_i)=\sum_{i_1<\cdots <i_k} \ka_{i_1}\cdots \ka_{i_k}.$$ They are defined on an open set $\C_k$ that can be characterized as the connected component of $\{H_k>0\}$ that contains the positive cone $\C_+$. The $\C_k$ are cones, $\C_n=\C_+$, and in it is proved that $$\lae{2.12}
\C_k\su\C_{k-1}.$$ Huisken and Sinestrari in gave an equivalent characterisation of $\C_k$ by showing that $$\lae{2.13}
\C_k=\{(\ka_i)\in \R[n]: H_1(\ka_i)>0, H_2(\ka_i)>0, \dots , H_k(\ka_i)>0\}.$$ They also proved that $\C_k$ is convex. The $H_k$ are strictly monotone in $\C_k$, cf. , and the the k-th roots $$\lae{2.14}
\s_k=H_k^\frac1k$$ are also concave, cf. .
Since we have in mind that the $\ka_i$ are the principal curvatures of a hypersurface, we use the standard symbols $H$ and $\abs A$ for $$\begin{aligned}
H&=\sum_i\ka_i,\\
\intertext{and}
\abs A^2&=\sum_i \ka^2_i.\end{aligned}$$ We note that $$\lae{2.17}
\tfrac1nH^2\le\abs A^2.$$ The scalar curvature function $F=H_2$ can be expressed as $$\lae{2.18}
F=\tfrac12{(H^2-\abs A^2)},$$ and we deduce that for $(\ka_i)\in \C_2$ $$\begin{aligned}
\abs A^2&\le H^2,\lae{2.19}\\[\cma]
F&\le \tfrac12 H^2,\lae{2.20}\\[\cma]
F_i&=H-\ka_i,\lae{2.21}\\
\intertext{and hence,}
H&>\ka_i,\lae{2.22}\\[\cma]
H F_i&\ge F,\lae{2.23}\end{aligned}$$ for is equivalent to $$\lae{2.24}
H\ka_i\le \tfrac12 H^2 + \tfrac12 \abs A^2,$$ which is obviously valid.
The evolution problem
=====================
To prove the existence of hypersurfaces of prescribed curvature $F$ for $F=\s_2$ we look at the evolution problem $$\begin{aligned}\lae{3.1}
\dot x&=(F-f)\n,\\[\cma]
x(0)&=x_0,
\end{aligned}$$ where $\n$ is the past-directed normal of the flow hypersurfaces $M(t)$, $F=\s_2$ the curvature evaluated at $M(t)$, $x=x(t)$ an embedding and $x_0$ an embedding of an initial hypersurface $M_0$, which we choose to be the upper barrier $M_2$.
Since $F$ is an elliptic operator, short-time existence, and hence, existence in a maximal time interval $[0,T^*)$ is guaranteed, cf. . If we are able to prove uniform a priori estimates in $C^{2,\al}$, long-time existence and convergence to a stationary solution will follow immediately.
But before we prove the a priori estimates, we want to show how the metric, the second fundamental form, and the normal vector of the hypersurfaces $M(t)$ evolve. All time derivatives are total derivatives. We shall omit the proofs, which can be found in .
The metric, the normal vector and the second fundamental form of $M(t)$ satisfy the evolution equations $$\begin{aligned}
\dot g_{ij}&=2( F- f)h_{ij},\lae{3.2}\\[\cma]
\dot \n&=\nabla_M( F- f)=g^{ij}( F- f)_i x_j,\lae{3.3}\\[\cma]
\dot h_i^j&=( F- f)_i^j- ( F- f) h_i^k h_k^j-( F- f) \riema
\al\bet\ga\de\n^\al x_i^\bet \n^\ga x_k^\de g^{kj},\lae{3.4}\\[\cma]
\dot h_{ij}&=(F-f)_{ij}+( F- f)h_i^k h_{kj}-(F-f)\riema
\al\bet\ga\de\n^\al x_i^\bet \n^\ga x_j^\de.\lae{3.5}\end{aligned}$$ The term $( F- f)$ evolves according to the equation $$\begin{gathered}
\lae{3.6}
{( F- f)}^\prime- F^{ij}( F- f)_{ij}=\msp[3]-
F^{ij}h_{ik}h_j^k ( F- f)
- f_\al\n^\al ( F- f)\\-f_{\n^\al}x^\al_i(F-f)_jg^{ij}
- F^{ij}\riema \al\bet\ga\de\n^\al x_i^\bet \n^\ga x_j^\de ( F- f).\end{gathered}$$ From we deduce with the help of the Ricci identities a parabolic equation for the second fundamental form, cf. . The mixed tensor $h_i^j$ satisfies the parabolic equation $$\lae{3.7}
\begin{aligned}%\raisetag{12pt}
\dot h_i^j- &F^{kl}h_{i;kl}^j\\
&=- F^{kl}h_{rk}h_l^rh_i^j
+f h_i^kh_k^j\\
&\hp{+}- f_{\al\bet} x_i^\al x_k^\bet g^{kj}- f_\al\n^\al
h_i^j-f_{\al\n^\bet}(x^\al_i x^\bet_kh^{kj}+x^\al_l
x^\bet_k h^{k}_i\, g^{lj})\\ &\hp{=}
-f_{\n^\al\n^\bet}x^\al_lx^\bet_kh^k_ih^{lj}-f_{\n^\bet} x^\bet_k h^k_{i;l}\,g^{lj}
-f_{\n^\al}\n^\al h^k_i h^j_k\\ &\hp{=}+
F^{kl,rs}h_{kl;i}h_{rs;}^{\hphantom{rs;}j}+2 F^{kl}\riema
\al\bet\ga\de x_m^\al x_i ^\bet x_k^\ga x_r^\de h_l^m g^{rj}\\
&\hp{=}- F^{kl}\riema \al\bet\ga\de x_m^\al x_k ^\bet x_r^\ga x_l^\de
h_i^m g^{rj}- F^{kl}\riema \al\bet\ga\de x_m^\al x_k ^\bet x_i^\ga x_l^\de h^{mj} \\
&\hp{=}- F^{kl}\riema \al\bet\ga\de\n^\al x_k^\bet\n^\ga x_l^\de h_i^j+ f
\riema \al\bet\ga\de\n^\al x_i^\bet\n^\ga x_m^\de g^{mj}\\
&\hp{=}+ F^{kl}\bar R_{\al\bet\ga\de;\e}\{\n^\al x_k^\bet x_l^\ga x_i^\de
x_m^\e g^{mj}+\n^\al x_i^\bet x_k^\ga x_m^\de x_l^\e g^{mj}\}.
\end{aligned}$$ In view of the maximum principle, we immediately deduce from that the term $( F- f)$ has a sign during the evolution if it has one at the beginning, i.e., if the starting hypersurface $M_0$ is the upper barrier $M_2$, then $( F- f)$ is non-negative $$\lae{3.8}
F\ge f.$$
Lower order estimates
=====================
Since the two boundary components $M_1, M_2$ of $\pa\Om$ are compact, connected spacelike hypersurfaces, they can be written as graphs over the Cauchy hypersurface $\so$, $M_i=\graph u_i$, $i=1,2$, and we have $$\lae{4.1}
u_1\le u_2,$$ for $M_1$ should lie in the past of $M_2$.
Let us look at the evolution equation with initial hypersurface $M_0$ equal to $M_2$ defined on a maximal time interval $I=[0,T^*)$, $T^*\le
\un$. Since the initial hypersurface is a graph over $\so$, we can write $$\lae{4.2}
M(t)=\graph\fu{u(t)}S0\q \A\,t\in I,$$ where $u$ is defined in the cylinder $Q_{T^*}=I\times \so$.
We then deduce from , looking at the component $\al=0$, that $u$ satisfies a parabolic equation of the form $$\lae{4.3}
\dot u=-e^{-\psi}v^{-1}(F- f),$$ where we use the notations in , and where we emphasize that the time derivative is a total derivative, i.e. $$\lae{4.4}
\dot u=\pde ut+u_i\dot x^i.$$ Since the past directed normal can be expressed as $$\lae{4.5}
(\n^\al)=-e^{-\psi}v^{-1}(1,u^i),$$ we conclude from , and $$\lae{4.6}
\pde ut=-e^{-\psi}v(F- f).$$ Thus, $\pde ut$ is non-positive in view of .
Next, let us state our first a priori estimate, . Suppose that the boundary components act as barriers for $(F,f)$, then the flow hypersurfaces stay in $\bar
\Om$ during the evolution. For the $C^1$-estimate the term $\tilde v=v^{-1}$ is of great importance. It satisfies the following evolution equation. Consider the flow in the distinguished coordinate system associated with $\so$. Then, $\tilde v$ satisfies the evolution equation $$\lae{4.7}
\begin{aligned}
\dot{\tilde v}- F^{ij}\tilde v_{ij}=&- F^{ij}h_{ik}h_j^k\tilde v
-f\h_{\al\bet}\n^\al\n^\bet\\
&-2 F^{ij}h_j^k x_i^\al x_k^\bet \h_{\al\bet}- F^{ij}\h_{\al\bet\ga}x_i^\bet
x_j^\ga\n^\al\\
&- F^{ij}\riema \al\bet\ga\de\n^\al x_i^\bet x_k^\ga x_j^\de\h_\e x_l^\e g^{kl}\\
&- f_\bet x_i^\bet x_k^\al \h_\al g^{ik} -f_{\n^\bet}x^\bet_k h^{ik}x^\al_i\h_\al,
\end{aligned}$$ where $\h$ is the covariant vector field $(\h_\al)=e^{\psi}(-1,0,\dotsc,0)$.
The proof uses the relation $$\lae{4.8}
\tilde v=\h_\al \n^\al$$ and is identical to that of having in mind that presently $f$ also depends on $\n$. Let $M(t)=\graph u(t)$ be the flow hypersurfaces, then, we have $$\lae{4.9}
\begin{aligned}
\dot u-F^{ij}u_{ij}=e^{-\psi}\tilde v f+\cha 000\, F^{ij}u_i u_j
+2\msp
F^{ij}\cha 0i0\,u_j +F^{ij}\cha ij0,
\end{aligned}$$ where all covariant derivatives are taken with respect to the induced metric of the flow hypersurfaces, and the time derivative $\dot u$ is the total time derivative, i.e. it is given by . We use the relation together with . As an immediate consequence we obtain The composite function $$\lae{4.10}
\f=e^{\m e^{\lam u}},$$ where $\m,\lam$ are constants, satisfies the equation $$\lae{4.11}
\begin{aligned}
\dot\f -F^{ij}\f_{ij}=&fe^{-\psi} \tilde v\msp \m \lam \msp[2] e^{\lam u} \,\f + F^{ij}
u_i u_j \,\cha000\,\m \lam\,e^{\lam u}\f\\[\cma]
&+2\msp F^{ij} u_i \cha 0j0\,\m \lam\msp[2] e^{\lam u}\,\f+ F^{ij}\cha
ij0\,\m\lam\msp[2] e^{\lam u}\,\f\\[\cma]
&-[1+\m\msp[2] e^{\lam u}] F^{ij} u_i u_j\,\m \lam^2\msp[2] e^{\lam u}\,\f.
\end{aligned}$$ Before we can prove the $C^1$- estimates we need two more lemmata. There is a constant $c=c(\Om)$ such that for any positive function $0<\e=\e(x)$ on $\so$ and any hypersurface $M(t)$ of the flow we have $$\begin{aligned}
\nnorm{\n}&\le c \msp \tilde v,\lae{4.12}\\[\cma]
g^{ij}&\le c\msp \tilde v^2\msp \s^{ij},\lae{4.13}\\[\cma]
F^{ij}&\le F^{kl} g_{kl}\msp g^{ij},\lae{4.14}\end{aligned}$$
$$\lae{4.15}
\begin{aligned}
\abs{F^{ij}h^k_j x^\al_i x^\bet_k \msp\h_{\al\bet}}\le \frac\e 2F^{ij}h^k_i
h_{kj}\msp
\tilde v + \frac c{2\e} F^{ij}g_{ij}\msp \tilde v^3,
\end{aligned}$$
$$\lae{4.16}
\abs{F^{ij}\h_{\al\bet\ga} x^\bet_i x^\ga_j \n^\al}\le c\msp \tilde v^3 F^{ij}g_{ij},$$
$$\lae{4.17}
\abs{F^{ij}\riema \al\bet\ga\de \n^\al x^\bet_i x^\ga_k x^\de_j\h_\e x^\e_l g^{kl}}\le
c\msp
\tilde v^3 F^{ij}g_{ij}.$$
Let $M\su \bar \Om$ be a graph over $\so$, $M=\graph u$, and $\e=\e(x)$ a function defined on $\so$, $0<\e <\frac12$. Let $\f$ be defined through $$\lae{4.18}
\f=e^{\m e^{\lam u}},$$ where $0<\m$ and $\lam<0$. Then, there exists $c=c(\Om)$ such that $$\lae{4.19}
\begin{aligned}
2\abs{F^{ij} \tilde v_i \f_j}&\le c\msp F^{ij}g_{ij} \tilde v^3 \abs{\lam} \m e^{\lam u}
\f +(1-2\e) F^{ij} h^k_i h_{kj} \tilde v \f\\[\cma]
&\hp{\le} +\frac1{1-2\e} F^{ij} u_i u_j \m^2 \lam^2 e^{2\lam u} \tilde v \f.
\end{aligned}$$
A proof of and can be found in .
Applying to the evolution equation for $ \tilde v$ we conclude There exists a constant $c=c(\Om)$ such that for any function $\e$, $0<\e=\e(x)<1$, defined on $\so$ the term $ \tilde v$ satisfies an evolution inequality of the form $$\lae{4.20}
\begin{aligned}
\dot {\tilde v} -F^{ij} \tilde v_{ij}&\le -(1-\e) F^{ij} h^k_i h_{kj} \tilde v -f
\h_{\al\bet} \n^\al \n^\bet\\[\cma]
&\hp{\le} +\frac{c}\e F^{ij} g_{ij} \tilde v^3 +c\msp \nnorm{f_\bet} \tilde v^2+
f_{\n^\bet} x^\bet_l h^{kl} u_k e^\psi.
\end{aligned}$$ We are now ready to prove the uniform boundedness of $ \tilde v$. Assume that there are positive constants $c_i$, $1\le i\le 3$, such that for any $x\in \Om$ and any past directed timelike vector $\n$ there holds $$\begin{aligned}
-c_1&\le f(x,\n),\lae{4.21}\\[\cma]
\nnorm{f_\bet(x,\n)}&\le c_2 (1+\nnorm{\n}),\lae{4.22}\\
\intertext{and}
\nnorm{f_{\n^\bet}(x,\n)}&\le c_3.\lae{4.23}\end{aligned}$$ Then, the term $ \tilde v$ remains uniformly bounded during the evolution $$\lae{4.24}
\tilde v\le c=c(\Om, c_1, c_2, c_3).$$ We show that the function $$\lae{4.25}
w= \tilde v\f,$$ $\f$ as in , is uniformly bounded, if we choose $$\lae{4.26}
0<\m<1\q \tup{and}\q \lam<<-1,$$ appropriately, and assume furthermore, without loss of generality, that $u\le
-1$, for otherwise replace $u$ by $(u-c)$, $c$ large, in the definition of $\f$. With the help of , and we derive from the relation $$\lae{4.27}
\dot w - F^{ij} w_{ij}=[\dot{ \tilde v}- F^{ij} \tilde v_{ij}] \f+
[\dot\f-F^{ij}\f_{ij}] \tilde v-2 F^{ij} \tilde v_i \f_j$$ the parabolic inequality $$\lae{4.28}
\begin{aligned}
\dot w -F^{ij} w_{ij}&\le -\e\msp[2] F^{ij} h^k_i h_{kj} \tilde v \f +
c[\e^{-1}+\abs\lam \m e^{\lam u}]F^{ij} g_{ij} \tilde v^3 \f\\[\cma]
&\hp{\le} +[\frac1{1-2\e}-1] F^{ij} u_i u_j \m^2\lam^2 e^{2\lam u} \tilde v
\f \\[\cma]
&\hp{\le}-F^{ij} u_i u_j \m\lam^2 e^{\lam u} \tilde v \f \\[\cma]
&\hp{\le} +f
[-\h_{\al\bet}\n^\al \n^\bet+e^{-\psi}
\m
\lam e^{\lam u} \tilde v^2] \f\\[\cma]
&\hp{\le} + c\msp[2] \nnorm{f_\bet} \tilde v^2 \f +f_{\n^\bet} x^\bet_l h^{kl} u_k
e^\psi
\f ,
\end{aligned}$$ where we have chosen the same function $\e=\e(x)$ in resp. . We claim that $w$ is uniformly bounded provided $\m$ and $\lam$ are chosen appropriately. We shall use the maximum principle, therefore let $0<T<T^*$ and $x_0=x(t_0,\x_0)$ be such that $$\lae{4.29}
\sup_{[0,T]}\sup_{M(t)}w=w(t_0,\x_0).$$ To exploit the good term $$\lae{4.30}
-\e \msp[2] F^{ij} h^k_i h_{kj} \tilde v \f,$$ we use the fact that $Dw(x_0)=0$, or, equivalently $$\lae{4.31}
\begin{aligned}
- \tilde v_i&= \m \lam e^{\lam u} \tilde v u_i\\[\cma]
&= e^\psi h^k_iu_k -\h_{\al\bet}\n^\al x^\bet_i,
\end{aligned}$$ where the second equation follows from and the definition of the covariant vectorfield $\h=e^\psi (-1,0,\dotsc,0)$. Next, we choose a coordinate system $(\x^i)$ such that in the critical point $$\lae{4.32}
g_{ij}=\de_{ij}\qq\tup{and}\qq h^k_i=\ka_i \de^k_i,$$ and the labelling of the principal curvatures corresponds to $$\lae{4.33}
\ka_1\le \ka_2\le \dotsb \le \ka_n.$$ Then, we deduce from $$\lae{4.34}
e^\psi \ka_i u_i=\m\lam e^{\lam u} \tilde v u_i + \h_{\al\bet}\n^\al x^\bet_i.$$ Assume that $ \tilde v(x_0)\ge 2$, and let $i=i_0$ be an index such that $$\lae{4.35}
\abs{u_{i_0}}^2\ge \frac1n\norm{Du}^2.$$ Setting $(e^i)=\pde{}{\x^{i_0}}$ and assuming without loss of generality that $0< u_ie^i$ in $x_0$ we infer from $$\lae{4.36}
\begin{aligned}
e^\psi \ka_{i_0} u_ie^i&=\m\lam e^{\lam u} \tilde v u_i e^i+\h_{\al\bet}\n^\al
x^\bet_ie^i \\[\cma]
&\le \m\lam e^{\lam u} \tilde v u_ie^i +c \tilde v^2,
\end{aligned}$$ and we deduce further in view of , and that $$\lae{4.37}
\begin{aligned}
\ka_{i_0}\le [\m\lam e^{\lam u}+ c] \tilde v e^{-\psi}
\le \frac12\m\lam e^{\lam u} \tilde v e^{-\psi},
\end{aligned}$$ if $\abs\lam$ is sufficiently large, i.e. $\ka_{i_0}$ is negative and of the same order as $ \tilde v$. The Weingarten equation and yield $$\lae{4.38}
\nnorm{\n^\bet_iu^i}=\nnorm{h^k_iu^i x^\bet_k}\le c \tilde v [h^k_iu^i
h_{kl}u^l]^{\frac12},$$ and therefore, we infer from $$\lae{4.39}
\nnorm{\n^\bet_i u^i}\le c \m\abs\lam e^{\lam u} \tilde v^3$$ in critical points of $w$, and hence, that in those points, the term involving $f_{\tilde \n^\bet}$ on the right-hand side of inequality can be estimated from above by $$\lae{4.40}
\abs{f_{ \n^\bet}\n^\bet_i u^i e^\psi
\f}\le c c_3 \m\abs\lam e^{\lam u} \tilde v^3\f.$$ Next, let us estimate the crucial term in . Using the particular coordinate system , as well as the inequalities , together with the fact that $\ka_{i_0}$ is negative, we conclude $$\lae{4.41}
-F^{ij}h^k_i h_{kj}\le -\sum_{i=1}^{i_0} F^i_i
\ka^2_i\le -\sum_{i=1}^{i_0} F^i_i \ka^2_{i_0}.$$ $ F$ is concave, and therefore, we have in view of $$\lae{4.42}
F^1_1\ge F^2_2\ge\dotsb\ge F^n_n,$$ cf. . Hence, we conclude $$\lae{4.43}
-\sum_{i=1}^{i_0}F^i_i\le -F^1_1\le -\frac1n\sum_{i=1}^nF^i_i.\\$$ Using , , , and , we deduce further $$\lae{4.44}
\begin{aligned}
-F^{ij} h^k_i h_{kj}&\le-cF^{ij}g_{ij} \m^2\lam^2 e^{2\lam u} \tilde v^2\\[\cma]
&\le -c \m^2\lam^2 e^{2\lam u} \tilde v^2
\end{aligned}$$ Inserting this estimate, and the estimate in in , with $\e=e^{-\lam u}$, we obtain $$\lae{4.45}
\begin{aligned}
\dot w-F^{ij} w_{ij}&\le-c F^{ij}g_{ij}\m^2\lam^2 e^{\lam u} \tilde v^3\f+cF^{ij}g_{ij}\m\abs\lam e^{\lam u} \tilde v^3\f
\\[\cma]
&\hp{\le}
+\frac2{1-2\e}\msp[2]F^{ij} u_i u_j \m^2\lam^2 e^{\lam u} \tilde v
\f -F^{ij} u_i u_j \m\lam^2 e^{\lam u} \tilde v \f \\[\cma]
&\hp{\le}+ c \msp c_1 \m \abs\lam e^{\lam u} \tilde v^2 \f + c\msp c_2 \tilde v^3 \f
+c\msp c_3 \m \abs\lam e^{\lam u} \tilde v^3 \f,
\end{aligned}$$ where $\abs\lam$ is chosen so large that $$\lae{4.46}
e^{-\lam u}\le \frac14.$$ Choosing $\m=\tfrac14$ and $\abs\lam$ sufficient large, we see that in view of the right-hand side of the preceding inequality is negative, contradicting the maximum principle, i.e. the maximum of $w$ cannot occur at a point where $ \tilde v\ge 2$. Thus, the desired uniform estimate for $w$ and hence $ \tilde v$ is proved. Notice that the preceding $C^1$-estimate is valid for any curvature function $F$ that is monotone, concave and homogeneous of degree 1. Let us close this section with an interesting observation that is an immediate consequence of the preceding proof, we have especially and in mind.
Suppose $F=\s_2$ is evaluated at a point $(\ka_i)$, and assume that $\ka_{i_0}$ is a component that is either negative or the smallest component of that particular $n$- tupel, then $$\lae{4.47}
\sum_{i=1}^n F_i\ka^2_i\ge \frac1n \sum_{i=1}^n F_i \msp[2] \ka^2_{i_0}.$$
Curvature estimates
===================
We want to prove that the principal curvatures of the flow hypersurfaces are uniformly bounded. Let us first prove an a priori estimate for $F$. Let $M(t)$, $0\le t<T^*$, be solutions of the evolution problem with $M(0)=M_2$ and $F=\s_2$. Then, $F$ is bounded from above during the evolution provided the $M(t)$ are uniformly spacelike, i.e. uniform $C^1$- estimates are valid. Let $0<T<T^*$ and $x_0=x(t_0,\x_0)$ be such that $$\lae{5.1}
\sup_{[0,T]}\sup_{M(t)}(F-f)=(F-f)(x_0)> 0.$$ Applying the maximum principle we deduce from $$\lae{5.2}
0\le -F^{ij}h_{ik}h^{kj}+c(1+F^{ij}g_{ij}),$$ where we have estimated bounded terms by a constant $c$.
Then, we infer from , and $$\lae{5.3}
0\le -\tfrac{1}{2n}FH+c(1+F^{-1}H),$$ which is equivalent to $$\lae{5.4}
0\le -\tfrac{1}{2n}F^2+c(FH^{-1}+1).$$ Thus, in view of , we obtain an a priori estimate for $F$. Let $\chi$ be the strictly convex function. Its evolution equation is $$\lae{5.5}
\begin{aligned}
\dot\chi- F^{ij}\chi_{ij}&=f\chi_\al\nu^\al -F^{ij}\chi_{\al\bet}x^\al_ix^\bet_j\\
&\le f\chi_\al\nu^\al -c_0 F^{ij}g_{ij},
\end{aligned}$$ where $c_0>0$ is independent of $t$. Under the assumptions of Lemma 5.1 the principal curvatures $\ka_i, 1\le i\le n$, of the flow hypersurfaces are uniformly bounded during the evolution provided there exists a strictly convex function $\chi\in C^2(\bar\Om)$. Let $\zeta$ and $w$ be respectively defined by $$\begin{aligned}
\zeta&=\sup\set{{h_{ij}\h^i\h^j}}{{\norm\h=1}},\lae{5.6}\\[\cma]
w&=\log\zeta+\lam \chi,\lae{5.7}\end{aligned}$$ where $\lam>0$ is supposed to be large. We claim that $w$ is bounded, if $\lam$ is chosen sufficiently large.
Let $0<T<T^*$, and $x_0=x_0(t_0)$, with $ 0<t_0\le T$, be a point in $M(t_0)$ such that $$\lae{5.8}
\sup_{M_0}w<\sup\set {\sup_{M(t)} w}{0<t\le T}=w(x_0).$$ We then introduce a Riemannian normal coordinate system $(\x^i)$ at $x_0\in
M(t_0)$ such that at $x_0=x(t_0,\x_0)$ we have $$\lae{5.9}
g_{ij}=\delta_{ij}\q \tup{and}\q \zeta=h_n^n.$$ Let $\tilde \h=(\tilde \h^i)$ be the contravariant vector field defined by $$\lae{5.10}
\tilde \h=(0,\dotsc,0,1),$$ and set $$\lae{5.11}
\tilde \zeta=\frac{h_{ij}\tilde \h^i\tilde \h^j}{g_{ij}\tilde \h^i\tilde \h^j}\raise 2pt
\hbox{.}$$ $\tilde\zeta$ is well defined in neighbourhood of $(t_0,\x_0)$.
Now, define $\tilde w$ by replacing $\zeta$ by $\tilde \zeta$ in ; then, $\tilde w$ assumes its maximum at $(t_0,\x_0)$. Moreover, at $(t_0,\x_0)$ we have $$\lae{5.12}
\dot{\tilde \zeta}=\dot h_n^n,$$ and the spatial derivatives do also coincide; in short, at $(t_0,\x_0)$ $\tilde \zeta$ satisfies the same differential equation as $h_n^n$. For the sake of greater clarity, let us therefore treat $h_n^n$ like a scalar and pretend that $w$ is defined by $$\lae{5.13}
w=\log h_n^n+\lam \chi.$$ We assume that the section curvatures are labelled according to .
At $(t_0,\xi_0)$ we have $\dot w \ge0$, and, in view of the maximum principle, we deduce from , , and $$\lae{5.14}
\begin{aligned}
0&\le-\tfrac12F^{ij}h_{ki}h^k_j+c h^n_n+cF^{ij}g_{ij}+\lam c-\lam c_0F^{ij}g_{ij}\\
&\hp{\le}\;+F^{ij}(\log h^n_n)_i(\log h^n_n)_j+\frac2{\ka_n-\ka_1} \sum_{i=1}^n(F_n-F_i)(h_{ni;}^{\hp{ni;}n})^2 (h^n_n)^{-1},\\
\end{aligned}$$ where we have estimated bounded terms by a constant $c$, and assumed that $h^n_n$ and $\lam$ are larger than $1$. We distinguish two cases
$1$.Suppose that $$\lae{5.15}
\abs{\ka_1}\ge \e_1 \ka_n,$$ where $\e_1>0$ is small. Then, we infer from $$\lae{5.16}
F^{ij}h_{ki}h^k_j\ge \tfrac1n F^{ij}g_{ij}\e_1^2\ka_n^2,$$ and $$\lae{5.17}
F^{ij}g_{ij}\ge F(1,\dots,1),$$ for a proof see .
Since $Dw=0$, $$\lae{5.18}
D\log h^n_n=-\lam D\chi,$$ hence $$\lae{5.19}
F^{ij}(\log h^n_n)_i(\log h^n_n)_j\le \lam^2F^{ij}\chi_i\chi_j.$$ Hence, we conclude that $\ka_n$ is a priori bounded in this case.
$2$.Suppose that $$\lae{5.20}
\ka_1\ge -\e_1\ka_n,$$ then, the last term in inequality is estimated from above by $$\lae{5.21}
\begin{aligned}
&\frac2{1+\e_1} \sum_{i=1}^n(F_n-F_i)(h_{ni;}^{\hp{ni;}n})^2 (h^n_n)^{-2}&\\
&\le\frac2{1+2\e_1} \sum_{i=1}^n(F_n-F_i)(h_{nn;}^{\hp{nn;}i})^2 (h^n_n)^{-2}\\
&\qq+c(\e_1)\sum_{i=1}^{n}(F_i-F_n)\ka_n^{-2},
\end{aligned}$$ where we used the Codazzi equation. The last sum can be easily balanced.
The terms in containing the derivative of $h^n_n$ can therefore be estimated from above by $$\lae{5.22}
\begin{aligned}
&-\frac{1-2\e_1}{1+2\e_1} \sum_{i=1}^nF_i(h_{nn;}^{\hp{nn;}i})^2 (h^n_n)^{-2}\\
&+\frac2{1+2\e_1}F_n\sum_{i=1}^n(h_{nn;}^{\hp{nn;}i})^2 (h^n_n)^{-2}\\
&\le 2 F_n\sum_{i=1}^n(h_{nn;}^{\hp{nn;}i})^2 (h^n_n)^{-2}\\
&= 2\lam^2 F_n \norm{D\chi}^2.\\
\end{aligned}$$ Hence, we infer $$\lae{5.23}
\begin{aligned}
0\le - \tfrac12 F_n\ka_n^2&+\lam^2c F_n+c\ka_n+cF^{ij}g_{ij} \\
&+\lam c-\lam c_0F^{ij}g_{ij}.
\end{aligned}$$ From , and we deduce $$\lae{5.24}
F^{ij}g_{ij}\ge c\ka_n,$$ thus, taking into account, we obtain an a priori estimate $$\lae{5.25}
\ka_n\le \const,$$ if $\lam$ is chosen large enough. Notice that $\e_1$ is only subject to the requirement $0<\e_1<\frac12$. In view of and , we conclude that the principal curvatures of the flow hypersurfaces stay in a compact subset of $\C$.
Existence of a solution
=======================
We shall show that the solution of the evolution problem exists for all time, and that it converges to a stationary solution. The solutions $M(t)=\graph u(t)$ of the evolution problem with $F=\s_2$ and $M(0)=M_2$ exist for all time and converge to a stationary solution provided $f\in C^{4,\al}$ satisfies the conditions , and . Let us look at the scalar version of the flow as in $$\lae{6.1}
\pde ut=-e^{-\psi}v(F- f).$$ This is a scalar parabolic differential equation defined on the cylinder $$\lae{6.2}
Q_{T^*}=[0,T^*)\times \so$$ with initial value $u(0)=u_2\in C^{4,\al}(\so)$.
In view of the a priori estimates, which we have established in the preceding sections, we know that $$\lae{6.3}
{\abs u}_\low{2,0,\so}\le c$$ and $$\lae{6.4}
F\,\tup{is uniformly elliptic in}\,u$$ independently of $t$, in view of . Thus, we can apply the known regularity results, see e.g. , where even more general operators are considered, to conclude that uniform $C^{2,\al}$-estimates are valid. Therefore, the maximal time interval is unbounded, i.e. $T^*=\un$.
Now, integrating with respect to $t$, and observing that the right-hand side is non-positive, yields $$\lae{6.5}
u(0,x)-u(t,x)=\int_0^te^{-\psi}v(F- f)\ge c\int_0^t(F- f),$$ i.e., $$\lae{6.6}
\int_0^\un \abs{F- f}<\un\qq\A\msp x\in \so.$$ Hence, for any $x\in\so$ there is a sequence $t_k\rightarrow \un$ such that $(F- f)\rightarrow 0$.
On the other hand, $u(\cdot,x)$ is monotone decreasing and therefore $$\lae{6.7}
\lim_{t\rightarrow \un}u(t,x)=\tilde u(x)$$ exists and is of class $C^{2,\al}(\so)$.We conclude that $\tilde u$ is a stationary solution, and that $$\lae{6.8}
\lim_{t\rightarrow \un}(F- f)=0.$$ Now, we can deduce that uniform $C^{6,\al}$-estimates are valid, cf . Hence, we conclude that the functions $u(t,\cdot)$ converge in $C^6(\so)$ to $\tilde u \in C^{6,\al}(\so)$. We want to solve the equation $$\lae{6.9}
\s_{2|_M}=f^{\frac12}(x,\nu),$$ where $f$ satisfies the conditions of , and . Thus we would like to apply the preceding existence result. But, unfortunately, the derivatives $f_\beta$ resp. $f_{\nu^\beta}$ grow quadratically resp. linear in $\nnorm{\nu}$ contrary to the assumptions in . Therefore, we define a smooth cut-off function $\tht\in C^\un(\R[]_+)$, $0<\tht\le2k$, where $k\ge k_0>1$ is to be determined later, by $$\lae{6.10}
\tht(t)=
\begin{cases}
t, &0\le t\le k,\\
2k , &2k\le t,
\end{cases}$$ such that $$\lae{6.11}
0\le \dot \tht \le 4$$ and consider the problem $$\lae{6.12}
\s_{2|_M}=f^{\frac12}(x,\tilde\nu),$$ where for a spacelike hypersurface $M=\graph u$ with past directed normal vector $\nu$ we set $$\lae{6.13}
\tilde\nu=\tht(\tilde v)\tilde v^{-1}\nu.$$ Then $$\lae{6.14}
\nnorm{\tilde \nu}\le ck,$$ so that the assumptions in are certainly satisfied. The constant $k_0$ should be so large that $\tilde \nu=\nu$ in case of the barriers $M_i, i=1,2.$ is therefore applicable leading to a solution $M_k=\graph u_k$ of .
From we then deduce that there exists a constant $m$ such that $$\lae{6.15}
\tilde v=(1-\abs{Du_k}^2)^{-\frac12}\le m \qquad \A k.$$ Hence, $M_k=\graph u_k$ is a solution of , if we choose $k\ge \max(2m, k_0)$.
Acknowledgement {#acknowledgement .unnumbered}
===============
I would like to thank my advisor Prof. Dr. Claus Gerhardt for offering me the opportunity to work on this beautiful subject, for great support, and for many interesting discussions.
[99]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the peeling of Dirac and Maxwell fields on a Schwarzschild background following the approach developed by the authors in [@MaNi2009] for the wave equation. The method combines a conformal compactification with vector field techniques in order to work out the optimal space of initial data for a given transverse regularity of the rescaled field across null infinity. The results show that analogous decay and regularity assumptions in Minkowski and in Schwarzschild produce the same regularity across null infinity. The results are valid also for the classes of asymptotically simple spacetimes constructed by Corvino-Schoen / Chrusciel-Delay.'
---
[ ]{}
Introduction
============
Zero rest-mass fields on asymptotically flat spacetimes admit a peeling-off, or peeling, property, discovered by Sachs in 1961 [@Sa61] in the flat case. It can be described in terms of principle null directions : for a zero rest-mass field $\phi_{AB...F} = \phi_{(AB...F)}$ with $n$ indices, the part of the field falling-off like $r^{-k-1}$, $0\leq k \leq n$, along outgoing null geodesics, has $n-k$ of its principle null directions aligned along the generator of the geodesics. On Minkowski spacetime, this can be stated simply in terms of components in a well-chosen spin-frame. Consider the Newman-Penrose tetrad $$l = \frac{1}{\sqrt{2}} (\partial_t + \partial_r) \, ,~ n = \frac{1}{\sqrt{2}} (\partial_t - \partial_r) \, ,~ m = \frac{1}{r\sqrt{2}} (\partial_\theta + \frac{i}{\sin \theta} \partial_\varphi ) \, ,~\bar{m}$$ and the associated spin-frame $\{ o \, ,~ \iota \}$ (unique modulo overall sign), then the component $\phi_{n-k}$, which is the contraction of $\phi$ with $n-k$ $\iota$’s and $k$ $o$’s, falls off like $r^{-k-1}$ along the integral curves of $l$.
It is essential to note that the property as we have brutally stated it is wrong. One can consider initial data on a spacelike slice that are exponentially increasing at infinity and the associated solution will not fall-off at all along outgoing null geodesics. The peeling is true in flat space-time for smooth compactly supported initial data and for certain classes of data that satisfy adequate regularity and fall-off assumptions. The situation is expected to be similar on generic asymptotically flat spacetimes but it has been speculated that the conditions on the initial data may be more stringent due to the more complicated asymptotic structure.
In 1965, Penrose [@Pe65] presented a new derivation of the peeling based on conformal compactifications. Using the conformal embedding of Minkowski spacetime into the Einstein cylinder and the conformal invariance of zero rest-mass field equations, he showed that the peeling property is equivalent to the continuity of the rescaled field at null infinity (${{\mathscr I}}$). Then he argued that the peeling should be a generic behaviour of zero rest-mass fields on asymptotically flat spacetimes. He went on to define a class of spacetimes, referred to as asymptotically simple spacetimes, providing a generic model of asymptotic flatness as well as a framework in which the peeling should occur under reasonable conditions. These spacetimes are classifed according to the regularity of their conformal metric at null infinity, which encodes the information of the peeling of the Weyl tensor.
His results raised criticisms regarding the genericity of asymptotically simple spacetimes and the peeling property. The jist of the arguments put forward was that the asymptotic structure of the Schwarzschild metric would only allow peeling for a considerably smaller family of initial data than in Minkowski spacetime. Indeed, asymptotically simple spacetimes are meant to be physically reasonable and as such contain mass-energy, which means that the physical metric differs from the flat one at first approximation by a Schwarzschild-type behaviour in $m/r$. The asymptotic structure of Schwarzschild’s spacetime is substantially different from Minkowski’s in particular with a singular conformal structure at spatial infinity. It was unclear as to whether this would impose more stringent hypotheses on initial data to ensure peeling; the singularity at spatial infinity could interact with the tail of the falloff of the initial data at spatial infinity and prevent the peeling that might otherwise take place if the data were compactly suppported. The genericity of the peeling for zero rest-mass fields being questioned, the peeling for linearized gravity on Schwarzschild’s spacetime was also doubted and this made the asymptotically simple spacetime model appear as anything but generic. The question of regularity of null infinity and asymptotic simplicity has now been resolved in various ways, Christodoulou-Klainerman [@ChriKla], Corvino [@Co2000], Chrusciel and Delay [@ChruDe2002; @ChruDe2003], Corvino-Schoen [@CoScho2003], Friedrich (see [@HFri2004] for a survey of his contributions) and Klainerman-Nicolò [@KlaNi; @KlaNi2002; @KlaNi2003]. However, even in the simple case of the Schwarzschild metric, it was not at all clear, until the authors provided a first element of answer in [@MaNi2009], whether zero rest-mass fields admit peeling properties for reasonably large classes of initial data.
Penrose’s constructions in [@Pe65] provide powerful techniques to analyze the constraints on the initial data implicit in the peeling property. A $t=$ [*constant*]{} slice of Minkowski spacetime corresponds to a $3$-sphere with a point removed on the Einstein cylinder. Physical initial data that once rescaled extend as smooth functions on the whole $3$-sphere give rise, as ensured by Leray’s theorem, to a rescaled solution that is smooth on the whole Einstein cylinder and consequently the physical solution satisfies the peeling property. This is a little crude since the peeling really means continuity of the rescaled solution at null infinity and the class of data considered above provides solutions that are ${\cal C}^\infty$ across ${{\mathscr I}}$. A more detailed understanding requires to consider intermediate regularities, but ${\cal C}^k$ spaces are not adapted to the Cauchy problem for hyperbolic equations ; a typical example is the wave equation on Minkowski spacetime : for initial data $f\vert_{t=0} \in {\cal C}^k ({\mathbb{R}}^3 )$ and $\partial_t f \vert_{t=0} \in {\cal C}^{k-1} ({\mathbb{R}}^3 )$, the solution $f$ is generally not in ${\cal C}^k ({\mathbb{R}}\times {\mathbb{R}}^3 )$. It is better to use Sobolev spaces instead since they are naturally controlled by energy estimates. In a previous paper [@MaNi2009], the authors used a combination of Penrose’s conformal techniques and geometric energy estimates (or vector field methods) to characterize completely the spaces of initial data ensuring peeling for the scalar wave equation on the Schwarzschild metric. A new definition of peeling with specifyable order of regularity was given in terms of (weighted) Sobolev spaces. The classes of data were shown to have analogous fall-off properties as those obtained in the flat case using the conformal embedding in the Einstein cylinder. This established that at least for the wave equation, the different asymptotic structure of the Schwarzschild spacetime does not change the classes of data ensuring peeling. To gain a more complete understanding of the question, it is necessary to study other types of fields on a Schwarzschild background, such as higher spin zero rest-mass fields, or solutions of non linear equations, and then to extend the results to more general asymptotically flat spacetimes.
The present paper uses similar techniques to investigate the peeling for Dirac and Maxwell fields on the Schwarzschild spacetime. It is organized as follows. Section \[GeoSet\] describes the geometric ingredients of the method : the conformal (partial) compactification of Schwarzschild’s spacetime with choices of Newman-Penrose tetrads on the physical and rescaled spacetimes, the corresponding rescaling of spin-coefficients, the neighbourhood of spacelike infinity in which we establish our estimates, a choice of foliation and the Morawetz vector field. This vector field is as crucial for the Maxwell case as it was for the wave equation, but it does not play any part in the construction for Dirac fields thanks to the existence of a conserved current independent of a choice of observer. In section \[DirMaxFields\], we focus on the field equations, their conformal invariance, the proof of the equivalence between the peeling and the continuity of the rescaled field at null infinity as Penrose gave it in [@Pe65], the conserved quantities (current for Dirac and stress-energy tensor for Maxwell), the associated energies on the hypersurfaces we work with and the generic method for obtaining energy estimates for a perturbed equation (explained simply in the Dirac case). Section \[Peeling\] contains the peeling results and their proof for Dirac and Maxwell fields. We establish energy estimates for fields supported away from spacelike infinity and then use these to construct our spaces of initial data by completion in the norms obtained on the initial hypersurface. We get a full set of function spaces with all degrees of regularity ; for data in these spaces, we have estimates both ways between the norm on the initial hypersurface and a corresponding norm on ${{\mathscr I}}^+$ involving transverse derivatives as well as angular ones. The main difficulty is of course the estimates on transverse derivatives. The control of angular derivatives is straightforward thanks to the spherical symmetry ; in the case of Dirac fields, it can be obtained elegantly by commuting into the equation the Dirac operator on the sphere ; for the Maxwell system, this cannot be done because we have four equations and three unknowns (this is also manifest in the spin and boost weights of the components of the field and their transformation under the Geroch-Held-Penrose angular operators), so we use a set of three Killing vectors on the sphere instead. The main results, given in theorems \[DiracPeeling\] and \[PeelingMaxwell\], are the energy estimates and the function spaces on the initial data surface that one can infer from them. The interpretation of the results, namely the comparison between our classes of initial data and the classes obtained in Minkowski spacetime using the full conformal embedding in the Einstein cylinder, is given in section \[Interpretation\], together with a remark on the constraints. All the calculations in this paper are done using partial derivatives of the field components. One may prefer using covariant derivatives of the fields and then taking components. This requires to know the different curvature spinors of the rescaled Schwarzschild spacetime. The calculation of these quantities and the derivation of an energy estimate for a transverse derivative using this approach is given for Dirac fields in the appendix. Although the calculations are more involved, the error terms in the conservation law for the transverse derivative are much simpler than if we use partial derivatives (see remark \[PurelyTransverse\] at the end of the paper).
Since the main nontrivial results of this paper concern the behaviour of fields in a neighbourhood of space-like infinity and its intersection with null infiinity, the results are valid for the classes of spacetimes of Corvino-Schoen / Chrusciel-Delay.
[**Notations.**]{} Throughout the paper, we use the formalisms of abstract indices, $2$-component spinors, Newman-Penrose and Geroch-Held-Penrose.
Geometric setting {#GeoSet}
=================
Rescaled Schwarzschild spacetime
--------------------------------
We work on the Schwarzschild metric $$\begin{gathered}
g = F(r) {\mathrm{d}}t^2 - F(r)^{-1} {\mathrm{d}}r^2- r^2 {\mathrm{d}}\omega^2 \, , ~m>0 \, ,\\
F(r) = 1-2m/r \, ,~{\mathrm{d}}\omega^2 = {\mathrm{d}}\theta^2 + \sin^2 \theta {\mathrm{d}}\varphi^2 \, ,\end{gathered}$$ in the region outside the black-hole ${\mathbb{R}}_t \times ]2m ,+\infty [_r \times S^2_\omega$. Introducing the variables $u=t-r_*$, where $r_*= r +2m \log (r-2m)$ is the Regge-Wheeler coordinate, and $R = 1/r$, we obtain the following expression for the metric $g$ conformally rescaled using the conformal factor $R$ : $$\label{RescMet}
\hat{g}= R^2 g= R^2F{\mathrm{d}}u^2-2{\mathrm{d}}u{\mathrm{d}}R-{\mathrm{d}}\omega^2 \, ,~ \mbox{still denoting } F = F(r) = 1-2mR \, .$$ These choices of conformal rescaling and variables allow to define naturally future null infinity (${{\mathscr I}}^+$) as ${\mathbb{R}}_u \times \{ R=0 \} \times S^2_\omega$. The Levi-Civita symbols must be rescaled accordingly : $$\label{RescEpsilon}
\hat{\varepsilon}_{AB} = R \, \varepsilon_{AB} \, .$$ We make on the exterior of the black hole the following choice of unitary (for $\hat{g}$) Newman-Penrose tetrad : $$\hat{l}^a \partial_a = -\sqrt\frac{F}{2}\, \partial_R \, ,~ \hat{n}^a \partial_a = \sqrt\frac{2}{F} \left( \partial_u + \frac{R^2 F}{2} \partial_R \right) \, , ~ \hat{m}^a \partial_a = \frac{1}{\sqrt{2}} \left( \partial_\theta + \frac{i}{\sin \theta} \partial_\varphi \right) \, , \label{NPTetrad}$$ with corresponding dual tetrad $$\hat{l}_a {\mathrm{d}}x^a = \sqrt\frac{F}{2}\, {\mathrm{d}}u \, ,~ \hat{n}_a {\mathrm{d}}x^a = \sqrt\frac{2}{F} \left( \frac{R^2 F}{2} {\mathrm{d}}u - {\mathrm{d}}R \right) \, , ~
\hat{m}_a {\mathrm{d}}x^a = -\frac{1}{\sqrt{2}} \left( {\mathrm{d}}\theta + i\sin \theta {\mathrm{d}}\varphi \right) \, . \label{NPDualTetrad}$$ This is in fact a simple rescaling of a classic Newman-Penrose tetrad for the metric $g$ : $$\label{OriginalTetrad}
l^a \partial_a = \frac{1}{\sqrt{2F}} \left( \partial_t + \partial_{r_*} \right) \, ,~ n^a \partial_a = \frac{1}{\sqrt{2F}} \left( \partial_t - \partial_{r_*} \right) \, ,~ m^a \partial_a = \frac{1}{r\sqrt{2}} \left( \partial_\theta + \frac{i}{\sin \theta} \partial_\varphi \right) \, ,$$ since we have $$\hat{l}^a = r^2 l^a \, ,~ \hat{n}^a = n^a \, ,~ \hat{m}^a = r m^a \, .$$ The indices for the rescaled frame vectors are lowered with the rescaled metric, so we have the following link with the unrescaled frame co-vectors : $$\hat{l}_a = l_ a \, ,~ \hat{n}_a = R^2 n_a \, ,~ \hat{m}_a = R m_a \, .$$ In terms of associated spin-frames, this corresponds to the rescaling $$\label{RescDyad}
\hat{o}^A = r o^A \, ,~ \hat\iota^A = \iota^A \, ,~ \hat{o}_A = o_A \, ,~ \hat{\iota}_A = R \iota_A \, .$$ We shall use the standard Newman-Penrose notations $D$, $D'$, $\delta$ and $\delta'$ for the directional derivatives $l^a\nabla_a$, $n^a \nabla_a$, $m^a\nabla_a$ and $\bar{m}^a \nabla_a$ ; similarly, we denote by $\hat{D}$, $\hat{D}'$, $\hat{\delta}$ and $\hat{\delta}'$ the directional derivatives along $\hat{l}$, $\hat{n}$, $\hat{m}$ and $\bar{\hat{m}}$.
The $4$-volume measure associated with the metric $\hat{g}$ is given by $$\label{4Vol}
\mathrm{dVol}^4 = i \hat{l} \wedge \hat{n} \wedge \hat{m} \wedge \bar{\hat{m}} = - {\mathrm{d}}u \wedge {\mathrm{d}}R \wedge {\mathrm{d}}^2 \omega \, .$$ where ${\mathrm{d}}^2 \omega = i \hat{m}\wedge \bar{\hat{m}}$ is the euclidian measure on the $2$-sphere.
[**Note.**]{} We have denoted by $\hat{l}$, $\hat{n}$, $\hat{m}$ and $\bar{\hat{m}}$ the $1$-forms $\hat{l}_a {\mathrm{d}}x^a$, $\hat{n}_a {\mathrm{d}}x^a$, $\hat{m}_a {\mathrm{d}}x^a$ and $\bar{\hat{m}}_a {\mathrm{d}}x^a$. We shall use this convention again.
Rescaling of spin coefficients
------------------------------
The rescaling of spin-coefficients under a general conformal rescaling $$\hat{g} = \Omega^2 g \, ,~ \hat{o}^A = \Omega^{-1} o^A \, ,~ \hat{\iota}^A = \iota^A \, ,~ \hat{o}_A = o_A \, ,~ \hat{\iota}_A = \Omega \iota_A \, ,$$ is described in [@PeRi84] vol. 1 p. 359. The rescaled coefficients are obtained from the original ones by multiplication by a power of $\Omega$ with, for some coefficients, some additional terms that involve the derivatives of $\omega = \log \Omega$ along the original frame vectors. In the special case of the conformal rescaling (\[RescMet\]), (\[RescDyad\]), $\Omega =R$ and these terms take the form $$\begin{aligned}
D \omega &=& l^a \nabla_a \omega = \frac{1}{\sqrt{2F}} \left( \frac{\partial}{\partial t} + F \frac{\partial}{\partial r} \right) \left( - \log r \right) = -\sqrt{\frac{F}{2}} R \, , \\
\delta ' \omega &=& \bar{m}^a \nabla_a \omega = \frac{1}{\sqrt{2r}} \left( \frac{\partial}{\partial \theta} - \frac{i}{\sin \theta} \frac{\partial}{\partial \varphi} \right) \left( - \log r \right) = 0 \, , \\
\delta \omega &=& m^a \nabla_a \omega = \frac{1}{\sqrt{2r}} \left( \frac{\partial}{\partial \theta} + \frac{i}{\sin \theta} \frac{\partial}{\partial \varphi} \right) \left( - \log r \right) = 0 \, , \\
D' \omega &=& n^a \nabla_a \omega = \frac{1}{\sqrt{2F}} \left( \frac{\partial}{\partial t} - F \frac{\partial}{\partial r} \right) \left( - \log r \right) = \sqrt{\frac{F}{2}} R \, ,\end{aligned}$$ and we have the following relations between the original and rescaled spin-coefficients : $$\begin{array}{|c|c|c|} \hline {\hat{\kappa} = r^3 \kappa} & {\hat{\varepsilon} = r^2 \varepsilon} & {\hat{\pi} = r \pi} \\ \hline {\hat{\rho} = r^2 \rho + \sqrt{\frac{F}{2}} r} & {\hat{\alpha} = r \alpha} & {\hat{\lambda} = \lambda} \\ \hline {\hat{\sigma} = r^2 \sigma} & {\hat{\beta} = r \beta} & {\hat{\mu} = \mu + \sqrt{\frac{F}{2}} R} \\ \hline {\hat{\tau} = r \tau} & {\hat{\gamma} = \gamma - \sqrt{\frac{F}{2}} R} & {\hat{\nu} = R \nu } \\ \hline \end{array}$$ The spin coefficients in the original tetrad (\[OriginalTetrad\]) have been calculated in [@Ni1997]. Using the array above, we obtain : $$\begin{gathered}
\hat{\kappa} = \hat{\sigma} = \hat{\lambda} = \hat{\tau} = \hat{\nu} = \hat{\pi} = \hat{\rho} = \hat{\mu} = 0 \, , \nonumber \\
\hat{\varepsilon} = \frac{m}{2 \sqrt{2F}} \, ,~ \hat{\gamma} = \frac{5m R^2 -2R}{2\sqrt{2F}} \, ,~ \hat{\beta} = -\hat{\alpha} = \frac{\cot \theta}
{2 \sqrt{2}} \, . \label{RescSpinCoeff}\end{gathered}$$ Note that the coefficients $\rho$ and $\mu$ were not zero for the original tetrad[^1].
Neighbourhood of spacelike infinity
-----------------------------------
We work in the following domain for a given $u_0 << -1$ $$\Omega_{u_0}^+ := \left\{ (u,R,\omega) \, ;~ u \leq u_0 \, ,~ 0\leq t \leq +\infty \, , ~ \omega \in S^2 \right\} \, .$$ We foliate this neighbourhood of $i^0$ by the hypersurfaces (which are spacelike except for ${\cal H}_0$ which is null) $${\cal H}_{s} = \{ u = -s r_* \, ;~ u \leq u_0 \} \, ,~0 \leq s \leq 1 \, .$$ For $s=1$, the hypersurface ${\cal H}_{1}$ is the part of the $\{ t=0 \}$ surface inside $\Omega_{u_0}^+$ and for $s=0$, ${\cal H}_{0}$ also denoted ${{\mathscr I}}^+_{u_0}$ is the part of ${{\mathscr I}}^+$ inside $\Omega_{u_0}^+$. The level hypersurfaces of $u$ within $\Omega^+_{u_0}$ will be denoted by ${\cal S}_u$, they are null. Given $0\leq s_1 < s_2 \leq 1$, we will denote by ${\cal S}_u^{s_1,s_2}$ the portion of ${\cal S}_u$ between ${\cal H}_{s_1}$ and ${\cal H}_{s_2}$.
We need an identifying vector field between the hypersurfaces ${\cal H}_{s}$ when decomposing $4$-volume integrals over $\Omega_{u_0}^+$ using the foliation. We use $$\label{VectVHs}
\nu=r_*^2R^2(1-2mR)|u|^{-1}{\partial}_R \, .$$ It is tangent to the $u =$ constant surfaces and is naturally associated to the parameter $s$ in that $\nu (s) = 1$. The splitting of the $4$-volume measure $\mathrm{dVol}^4$ corresponding to the foliation $\left\{ { \cal H}_{s} \right\}_{0\leq s \leq 1}$ with identifying vector field $\nu$ is the product of ${\mathrm{d}}s$ (being the measure along the integral lines of $\nu^a$) and $\nu {\lrcorner}\mathrm{dVol}^4 |_{{\cal H}_s}= r_*^2 R^2 (1-2mR) |u|^{-1} {\mathrm{d}}u {\mathrm{d}}^2\omega |_{{\cal H}_s}$ (which is the resulting $3$-volume measure on each ${\cal H}_s$).
We recall from [@MaNi2009] the controls we can infer on $u$, $R$ and $r_*$ in the domain $ \Omega_{u_0}^+$ for $|u_0|$ large enough.
\[ApproxCloseI0\] Let $\varepsilon >0$, then for $u_0<0$, $|u_0|$ large enough, in the domain $\Omega_{u_0}^+$, we have $$r< r_* <r(1+\varepsilon ) \, ,~ 1 < Rr_* < 1+\varepsilon \, ,~ 0 < R|u| < 1+\varepsilon \, ,~ 1-\varepsilon < 1-2mR <1 \, ,$$ and of course $$0\leq s=\frac{|u|}{r_*} \leq 1 \, .$$ The factor $r_*^2R^2(1-2mR)|u|^{-1}$ appearing in the expression of the vector field $\nu$ satisfies $$\frac{1-\varepsilon}{|u|} < r_*^2R^2(1-2mR)|u|^{-1} < \frac{(1+\varepsilon )^2}{|u|} \, .$$
The Morawetz vector field
-------------------------
The name “Morawetz vector field” is slightly inadequate. It refers to a vector field that is timelike in the neighbourhood of spacelike infinity and is transverse to ${{\mathscr I}}^+$. It is constructed from the actual Morawetz vector field in flat spacetime (see [@Mo1962]) by expressing it in a coordinate system resembling our $u,R,\omega$ coordinates and brutally keeping the expression on the rescaled Schwarzschild spacetime. More precisely, the Morawetz vector field on Minkowski spacetime is defined by $$K = (r^2+t^2) \partial_t + 2tr\partial_r$$ and finds its simplest expression in the coordinates $u=t-r$, $v=t+r$ : $$K = u^2 \partial_u + v^2 \partial_v \, .$$ This is a conformal Killing vector field of Minkowski spacetime and is precisely Killing for the Minkowski metric $\eta$ rescaled using the conformal factor $\Omega = 1/r$ (i.e. $\hat\eta =(1/r^2) \eta$). If we use the coordinates $u=t-r$ and $R=1/r$, the vector $K$ takes the form $$K = u^2 \partial_u -2 (1+uR) \partial_R \, .$$ We define the “Morawetz vector field” on the rescaled Schwarzschild spacetime in the coordinates $u=t-r_*$, $R=1/r$, as $$\label{Morawetz}
T^a \partial_a := u^2 \partial_u -2 (1+uR) \partial_R \, .$$ It has the following decomposition on the tetrad $\hat{l}$, $\hat{n}$, $\hat{m}$, $\bar{\hat{m}}$ : $$\label{MorawetzTetrad}
T^a = \sqrt{\frac{2}{F}} \left( 2(1+uR) + \frac{1}{2} (uR)^2 F \right) \hat{l}^a + u^2\sqrt{\frac{F}{2}} \, \hat{n}^a \, .$$ Since $K$ is Killing for the rescaled Minkowski metric $R^2 \eta$ and since the Schwarzchild metric is asymptotically flat, the vector field $T$ should provide an approximate Killing vector field, near $i^0$ and ${{\mathscr I}}$, for the rescaled metric $\hat{g}$, which is precisely the reason why it was introduced in [@MaNi2009] to study the peeling of scalar fields on the Schwarzschild metric. A calculation of its Killing form shows that it is indeed a good approximation of a Killing vector in the vicinity of $i^0$ and ${{\mathscr I}}$ : $$\nabla_{(a} T_{b)} {\mathrm{d}}x^a {\mathrm{d}}x^b = 4mR^2(3+uR){\mathrm{d}}u^2 = 8mR^2F^{-1} (3+uR ) \hat{l}_a \hat{l}_b {\mathrm{d}}x^a {\mathrm{d}}x^b \, .$$ Note that a similar construction, based on the null coordinates $(u =t-r_* \, , ~v=t+r_*)$ instead of $(u, R)$, and also referred to as the Morawetz vector field, was used by Dafermos and Rodnianski in [@DaRo].
Dirac and Maxwell fields {#DirMaxFields}
========================
A Dirac spinor field is the direct sum of a neutrino part $\chi^{A'}$ and an anti-neutrino part $\psi_A$ ; in the massless case, the two parts decouple and Dirac’s equation reduces to the Weyl anti-neutrino equation $$\label{WeylEq}
\nabla^{AA'} \psi_A = 0 \, .$$ Similarly, in the source-free case, the anti-self-dual part $\phi_{AB} = \phi_{(AB)}$ and the self-dual part $\bar{\phi}_{A'B'}$ of the electromagnetic field decouple and Maxwell’s equations are equivalent to the equations for $\phi_{AB}$ : $$\label{MaxwellEq}
\nabla^{AA'} \phi_{AB} = 0 \, .$$ Both equations are conformally invariant : spinor-valued distributions $\psi_A$ and $\phi_{AB} = \phi_{(AB)}$ satisfy respectively equations and on the exterior of the black hole if and only if the rescaled quantities $\hat{\psi}_A= \Omega^{-1} \psi_A = r \psi_A$ and $\hat{\phi}_{AB} = \Omega^{-1} \phi_{AB} = r\phi_{AB}$ satisfy on the same domain the rescaled equations $$\begin{aligned}
\label{RescWeylEq}
\hat{\nabla}^{AA'} \hat\psi_A = 0 \, , \\
\label{RescMaxwellEq}
\hat{\nabla}^{AA'} \hat\phi_{AB} = 0 \, ,\end{aligned}$$ where $\hat{\nabla}$ is the Levi-Civita connection for the rescaled metric $\hat{g}$.
Rescaling of the field components
---------------------------------
We decompose the physical Dirac field $\psi_A$ and Maxwell field $\phi_{AB}$ onto the spin-frame $\{ o^A , \iota^A \}$ and the rescaled fields $\hat{\psi}_A = r\psi_A$ and $\hat{\phi}_{AB}= r\phi_{AB}$ onto the rescaled spin-frame $\{ \hat{o}^A , \hat{\iota}^A \}$. The decomposition is as follows : $$\begin{aligned}
\psi_A &=& \psi_1 o_A -\psi_0 \iota_A \, , \\
\hat{\psi}_A &=& r \psi_A = r \psi_1 o_A - r \psi_0 \iota_A \\
&=& \hat{\psi}_1 \hat{o}_A -\hat{\psi}_0 \hat{\iota}_A = \hat{\psi}_1 o_A -\hat{\psi}_0 R \iota_A\, ,\end{aligned}$$ Hence, $$\label{DiracCompRescaled}
\hat{\psi}_0 = r^2 \psi_0 \, ,~\hat{\psi}_1 = r \psi_1 \, .$$ A simpler version is the following : $$\hat{\psi}_0 = \hat{\psi}_A \hat{o}^A = r \psi_A r o^A = r^2 \psi_0 \, ,~ \hat{\psi}_1 = \hat{\psi}_A \hat{\iota}^A = r \psi_A \iota^A = r \psi_1 \, .$$ As for the Maxwell field : $$\label{MaxwellCompRescaled}
\hat{\phi}_0 = r^3 \phi_0 \, ,~ \hat{\phi}_1 = r^2 \phi_1 \, ,~ \hat{\phi}_2 = r \phi_2 \, .$$
The rescaled equations
----------------------
The rescaled Weyl equation can be expressed using the tetrad (\[NPTetrad\]) and the associated spin-coefficients as follows (see S. Chandrasekhar [@Cha]) : $$\left\{ \begin{array}{l}
{ \hat{D}' \hat{\psi}_0 - \hat{\delta} \hat{\psi}_1 + (\hat{\mu} - \hat{\gamma} ) \hat{\psi}_0 + (\hat{\tau} - \hat{\beta} )
\hat{\psi}_1 = 0 \, , } \\ \\
{ \hat{D} \hat{\psi}_1 - \hat{\delta}' \hat{\psi}_0 + (\hat{\alpha} - \hat{\pi} )\hat{\psi}_0 + (\hat{\varepsilon} -
\hat{\rho} ) \hat{\psi}_1 = 0 \, ,} \end{array} \right.$$ the link being $$\begin{aligned}
0 = \hat{\nabla}^{AA'} \hat{\psi}_A &=& \left( \hat{D}' \hat{\psi}_0 - \hat{\delta} \hat{\psi}_1 + (\hat{\mu} - \hat{\gamma} ) \hat{\psi}_0 + (\hat{\tau} - \hat{\beta} )
\hat{\psi}_1 \right) \bar{\hat{o}}^{A'} \nonumber \\
&& + \left( \hat{D} \hat{\psi}_1 - \hat{\delta}' \hat{\psi}_0 + (\hat{\alpha} - \hat{\pi} )\hat{\psi}_0 + (\hat{\varepsilon} -
\hat{\rho} ) \hat{\psi}_1 \right) \bar{\hat\iota}^{A'} \, . \label{RescWeylEqComponents}\end{aligned}$$ This gives us the system ( with $F = 1-2mR)$: $$\left\{ \begin{array}{l}
{ \sqrt{\frac{2}{F}} \left( \partial_u + \frac{1}{2} R^2 F \partial_R \right) \, \hat{\psi}_0 - \frac{1}{\sqrt{2}} \left( \partial_\theta + \frac{1}{2} \cot \theta + \frac{i}{\sin \theta} \partial_\varphi \right) \, \hat{\psi}_1 - \frac{5m R^2 -2R}{2\sqrt{2F}} \hat{\psi}_0 = 0 \, , } \\ \\
{ -\sqrt{\frac{F}{2}} \partial_R \hat{\psi}_1 - \frac{1}{\sqrt{2}} \left( \partial_\theta + \frac{1}{2} \cot \theta - \frac{i}{\sin \theta} \partial_\varphi \right) \, \hat{\psi}_0 + \frac{m}{2 \sqrt{2F}} \hat{\psi}_1 = 0 \, .} \end{array} \right.$$ This can be simplified as follows : $$\label{WeylEqGHP}
\left\{ \begin{array}{l}
{ \hat{\mbox{\th}}' \hat{\psi}_0 - \hat{\eth} \hat{\psi}_1 = 0 \, , } \\ \\
{ \hat{\mbox{\th}}\hat{\psi}_1 - \hat{\eth}' \hat{\psi}_0 = 0 \, ,} \end{array} \right.$$ where $\hat{\mbox{\th}}$, $\hat{\mbox{\th}}'$, $\hat{\eth}$ and $\hat{\eth}'$ are the weighted differential operators of the GHP formalism (Geroch-Held-Penrose [@GHP], also referred to as compacted spin-coefficient formalism in Penrose and Rindler [@PeRi84] Vol.1 section 4.12), which, applied to $\hat{\psi}_0$ and $\hat{\psi}_1$ take the form $$\begin{aligned}
\hat{\mbox{\th}}' \hat{\psi}_0 = \sqrt{\frac{2}{F}} \left( \partial_u + \frac{R^2 F}{2} \partial_R + \frac{2R-5m R^2}{4} \right) \, \hat{\psi}_0 &,& \hat{\mbox{\th}}\hat{\psi}_1 = -\sqrt{\frac{F}{2}} \left( \partial_R - \frac{m}{2F} \right) \hat{\psi}_1 \, ,\\
\hat{\eth} \hat{\psi}_1 = \frac{1}{\sqrt{2}} \left( \partial_\theta + \frac{1}{2} \cot \theta + \frac{i}{\sin \theta} \partial_\varphi \right) \, \hat{\psi}_1 &,& \hat{\eth}' \hat{\psi}_0 = \frac{1}{\sqrt{2}} \left( \partial_\theta + \frac{1}{2} \cot \theta - \frac{i}{\sin \theta} \partial_\varphi \right) \, \hat{\psi}_0 \, .\end{aligned}$$ For the rescaled anti-self-dual Maxwell system, we have $$\begin{aligned}
0=\nabla^{AA'} \hat\phi_{AB} &=& \left( \hat{D}' \hat{\phi}_0 - \hat{\delta} \hat{\phi}_1 + (\hat{\mu} - 2\hat{\gamma} ) \hat{\phi}_0 + 2 \hat{\tau}
\hat{\phi}_1 - \hat\sigma \hat\phi_2 \right) \bar{\hat{o}}^{A'} \hat{o}_B \\
&&- \left( \hat{D} \hat{\phi}_1 -\hat{\delta}' \hat{\phi}_0 + (2 \hat{\alpha} - \hat{\pi} )\hat{\phi}_0 - 2 \hat{\rho} \hat{\phi}_1 + \hat\kappa \hat\phi_2 \right) \bar{\hat{o}}^{A'} \hat{\iota}_B \\
&& + \left( \hat{D}' \hat{\phi}_1 - \hat{\delta} \hat{\phi}_2 - \hat{\nu} \hat{\phi}_0 + 2\hat{\mu} \hat{\phi}_1 + (\hat\tau - 2\hat\beta ) \hat\phi_2 \right) \bar{\hat{\iota}}^{A'} \hat{o}_B \\
&& - \left( \hat{D} \hat{\phi}_2 - \hat{\delta}' \hat{\phi}_1 + \hat{\lambda} \hat{\phi}_0 - 2 \hat\pi \hat\phi_1 + (2 \hat{\varepsilon} - \hat{\rho} ) \hat{\phi}_2 \right) \bar{\hat{\iota}}^{A'} \hat{\iota}_B \, ,\end{aligned}$$ so equation is equivalent to the system $$\left\{ \begin{array}{l}
{ \hat{D}' \hat{\phi}_0 - \hat{\delta} \hat{\phi}_1 + (\hat{\mu} - 2\hat{\gamma} ) \hat{\phi}_0 + 2 \hat{\tau}
\hat{\phi}_1 - \hat\sigma \hat\phi_2 = 0 \, , } \\ \\
{ \hat{D} \hat{\phi}_1 - \hat{\delta}' \hat{\phi}_0 + (2 \hat{\alpha} - \hat{\pi} )\hat{\phi}_0 - 2 \hat{\rho} \hat{\phi}_1 + \hat\kappa \hat\phi_2 = 0 \, ,} \\ \\
{ \hat{D}' \hat{\phi}_1 - \hat{\delta} \hat{\phi}_2 - \hat{\nu} \hat{\phi}_0 + 2\hat{\mu} \hat{\phi}_1 + (\hat\tau - 2\hat\beta ) \hat\phi_2 = 0 \, , } \\ \\
{ \hat{D} \hat{\phi}_2 - \hat{\delta}' \hat{\phi}_1 + \hat{\lambda} \hat{\phi}_0 - 2 \hat\pi \hat\phi_1 + (2 \hat{\varepsilon} - \hat{\rho} ) \hat{\phi}_2 = 0 \, . } \end{array} \right.$$ In the Geroch-Held-Penrose formalism, this takes on the simpler expression $$\label{GHPMaxwell}
\left\{ \begin{array}{l}
{ \hat{\mbox{\th}}' \hat{\phi}_0 - \hat{\eth} \hat{\phi}_1 = 0 \, , } \\ \\
{ \hat{\mbox{\th}}\hat{\phi}_1 - \hat{\eth}' \hat{\phi}_0 = 0 \, ,} \\ \\
{ \hat{\mbox{\th}}' \hat{\phi}_1 - \hat{\eth} \hat{\phi}_2 = 0 \, , } \\ \\
{ \hat{\mbox{\th}}\hat{\phi}_2 - \hat{\eth}' \hat{\phi}_1 = 0 \, ,} \end{array} \right.$$ with $$\begin{aligned}
\hat{\mbox{\th}}' \hat\phi_0 = \sqrt{\frac{2}{F}} ( \partial_u + \frac{1}{2} R^2F \partial_R ) \hat\phi_0 -\frac{5mR^2 -2R}{\sqrt{2F}} \hat{\phi}_0 &,& \hat{\mbox{\th}}\hat\phi_1 = -\sqrt{\frac{F}{2}} \partial_R \hat{\phi}_1 \, ,\\
\hat{\mbox{\th}}' \hat\phi_1 = \sqrt{\frac{2}{F}} ( \partial_u + \frac{1}{2} R^2F \partial_R ) \hat{\phi}_1 &,& \hat{\mbox{\th}}\hat\phi_2 = -\sqrt{\frac{F}{2}} \partial_R \hat{\phi}_2 + \frac{m}{\sqrt{2F}} \hat{\phi}_2 \, , \\
\hat{\eth} \hat\phi_1 = \frac{1}{\sqrt{2}} ( \partial_\theta + \frac{i}{\sin \theta} \partial_\varphi ) \hat{\phi}_1 &,& \hat{\eth}' \hat\phi_0 = \frac{1}{\sqrt{2}} ( \partial_\theta + \cot \theta - \frac{i}{\sin \theta} \partial_\varphi ) \hat{\phi}_0 \, , \\
\hat{\eth} \hat\phi_2 = \frac{1}{\sqrt{2}} ( \partial_\theta + \cot \theta + \frac{i}{\sin \theta} \partial_\varphi ) \hat{\phi}_2 &,& \hat{\eth}' \hat\phi_1 = \frac{1}{\sqrt{2}} ( \partial_\theta - \frac{i}{\sin \theta} \partial_\varphi ) \hat{\phi}_1 \, .\end{aligned}$$
Conserved quantity and estimates for perturbed equations
--------------------------------------------------------
### The Weyl equation
The conserved current for the Weyl equation is $J^a = \hat{\psi}^A \bar{\hat{\psi}}^{A'}$, which gives the following closed $3$-form by contraction with the $4$-volume measure : $$\begin{aligned}
\omega &:=& * J_a {\mathrm{d}}x^a = J {\lrcorner}{\mathrm{dVol}}^4\nonumber \\
& = & \left( \left| \hat{\psi}_1 \right|^2 \hat{l}^a \partial_a + \left| \hat{\psi}_0 \right|^2 \hat{n}^a \partial_a - \hat{\psi}_1 \overline{\hat{\psi}_0} \hat{m}^a \partial_a - \overline{\hat{\psi}_1} \hat{\psi}_0 \bar{\hat{m}}^a \partial_a \right) {\lrcorner}{\mathrm{dVol}}^4 \nonumber \\
&=& -\vert \hat\psi_1 \vert^2 \hat{l} \wedge {\mathrm{d}}^2 \omega + \vert \hat\psi_0 \vert^2 \hat{n} \wedge {\mathrm{d}}^2 \omega - i \hat{\psi}_1 \overline{\hat{\psi}_0} \, \hat{l} \wedge \hat{n} \wedge \hat{m} + i \hat{\psi}_0 \overline{\hat{\psi}_1} \, \hat{l} \wedge \hat{n} \wedge \bar{\hat{m}} \nonumber \\
&=& -\sqrt{\frac{2}{F}} \left| \hat{\psi}_0 \right|^2 {\mathrm{d}}R \wedge {\mathrm{d}}^2 \omega - \sqrt{\frac{F}{2}} \left( \left| \hat{\psi}_1 \right|^2 - R^2 \left| \hat{\psi}_0 \right|^2 \right) {\mathrm{d}}u \wedge {\mathrm{d}}^2 \omega \nonumber \\
&& + \sqrt{2} \Re \left( \hat{\psi}_0 \overline{\hat{\psi}_1}\right) {\mathrm{d}}u \wedge {\mathrm{d}}R \wedge \sin \theta {\mathrm{d}}\varphi + \sqrt{2} \Re \left( i \hat{\psi}_0 \overline{\hat{\psi}_1}\right) {\mathrm{d}}u \wedge {\mathrm{d}}R \wedge {\mathrm{d}}\theta \, . \label{DiracClosed3Form}\end{aligned}$$ On a given hypersurface ${\cal H}_{s}$, we have $${\mathrm{d}}R = \frac{FR^2}{s} {\mathrm{d}}u$$ and the conserved quantity takes the form $$\begin{aligned}
{\cal E}_{{\cal H}_s} (\hat{\psi} ) := \int_{{\cal H}_s} \omega &=& \int_{{\cal H}_s} \left( \left( \frac{2}{s} - 1 \right) R^2 \left| \hat{\psi}_0 \right|^2 + \left| \hat{\psi}_1 \right|^2 \right) \sqrt{\frac{F}{2}} {\mathrm{d}}u \, {\mathrm{d}}^2 \omega \nonumber \\
&=& \int_{{\cal H}_s} \left( \left( \frac{2r_*}{|u|} - 1 \right) R^2 \left| \hat{\psi}_0 \right|^2 + \left| \hat{\psi}_1 \right|^2 \right) \sqrt{\frac{F}{2}} {\mathrm{d}}u \, {\mathrm{d}}^2 \omega \, .\end{aligned}$$ On ${\cal S}_u$, $$\begin{aligned}
{\cal E}_{{\cal S}_{u}} (\hat{\psi} ) := \int_{{\cal S}_{u}} \omega &=& \int_{{\cal S}_{u}} \sqrt{\frac{2}{F}} \left| \hat{\psi}_0 \right|^2 {\mathrm{d}}R \, {\mathrm{d}}^2 \omega \, .\end{aligned}$$
The energies on ${\cal S}_u$, $u\leq u_0$ and ${\cal H}_s$, $0 \leq s \leq 1$ have the following simpler equivalents (meaning that there are constants independent of $u\leq u_0$, $0\leq s \leq 1$ and the smooth spinor field $\hat{\psi}_A$ such that the energies on ${\cal S}_u$ and ${\cal H}_s$ are controlled above and below by these constants times the simpler expressions) : $$\begin{aligned}
{\cal E}_{{\cal S}_{u}} (\hat{\psi} ) &\simeq & \int_{{\cal S}_{u}} \left| \hat{\psi}_0 \right|^2 {\mathrm{d}}R \, {\mathrm{d}}^2 \omega \, , \\
{\cal E}_{{\cal H}_s} (\hat{\psi} ) &\simeq & \int_{]-\infty , u_0 [_u \times S^2_\omega} \left( \frac{R}{\vert u \vert} \left| \hat{\psi}_0 \right|^2 + \left| \hat{\psi}_1 \right|^2 \right) {\mathrm{d}}u \, {\mathrm{d}}^2 \omega \, .\end{aligned}$$
[**Proof.**]{} This is a direct consequence of lemma \[ApproxCloseI0\] and of the fact that $$\frac{1}{s} \leq \frac{2}{s} -1 \leq \frac{2}{s} \, . \qed$$
The closedness of the $3$-form $\omega$ gives for any smooth solution $\hat{\psi}_A$ of with compactly supported initial data : $$\label{L2EnEq}
{\cal E}_{{\cal H}_{s_1}} (\hat{\psi} ) + {\cal E}_{{\cal S}_{u_0}^{s_1,s_2}} (\hat{\psi} ) = {\cal E}_{{\cal H}_{s_2}} (\hat{\psi} ) \mbox{ for any } 0\leq s_1<s_2\leq 1\, .$$ Now, consider a Dirac equation with error terms of two types, a potential $P$ and a source $Q$ : $$\label{DiracErrorTerm}
\nabla^{AA'} \hat{\psi}_A = P^{AA'} \hat{\psi}_A + Q^{A'} \, .$$ Then, differentiating the current $1$-form $J_a$, we get $$\nabla^{AA'} \left( \hat{\psi}_A \bar{\hat{\psi}}_{A'} \right) = \left( P^{AA'} + \bar{P}^{AA'} \right) \hat{\psi}_A \bar{\hat{\psi}}_{A'} + 2 \Re \left( Q^{A'} \bar{\hat{\psi}}_{A'} \right)$$ which is an approximate conservation law. When integrating this over the $4$-volume $\Omega_{u_0}^{s_1 ,s_2}$ bounded by hypersurfaces ${\cal H}_{s_1}$, ${\cal H}_{s_2}$, for $0\leq s_1<s_2\leq 1$, and the part ${\cal S}^{s_1,s_2}_{u_0}$ of ${\cal S}_{u_0}$ between ${\cal H}_{s_1}$ and ${\cal H}_{s_2}$, we obtain for any smooth solution of with compactly supported initial data : $$\begin{gathered}
\left\vert {\cal E}_{{\cal H}_{s_1}} (\hat{\psi} ) - {\cal E}_{{\cal H}_{s_2}} (\hat{\psi} ) + {\cal E}_{{\cal S}_{u_0}^{s_1,s_2}} (\hat{\psi} ) \right\vert = \left\vert \int_{\Omega_{u_0}^{s_1 ,s_2}} \left( \left( P^{AA'} + \bar{P}^{AA'} \right) \hat{\psi}_A \bar{\hat{\psi}}_{A'} + 2 \Re \left( Q^{A'} \bar{\hat{\psi}}_{A'} \right) \right) {\mathrm{dVol}}^4 \right\vert \\
\leq (1+\varepsilon )^2 \int_{s_1}^{s_2} \int_{{\cal H}_s} \left\vert \left( P^{AA'} + \bar{P}^{AA'} \right) \hat{\psi}_A \bar{\hat{\psi}}_{A'} + 2 \Re \left( Q^{A'} \bar{\hat{\psi}}_{A'} \right) \right\vert \frac{1}{\vert u \vert} {\mathrm{d}}u {\mathrm{d}}^2 \omega {\mathrm{d}}s \, .\end{gathered}$$ Energy estimates will be established using the Gronwall inequality provided the integrand on the right-hand side can be estimated by the energy density on ${\cal H}_s$, in a sufficiently uniform way so as not to prevent integrability in $s$.
### The Maxwell system
An anti-self-dual Maxwell field $\hat{\phi}_{AB}$ has a stress-energy tensor given by the following expression $$T_{ab} = \hat{\phi}_{AB} \bar{\hat{\phi}}_{A'B'} \, .$$ In order to define an energy current, we need to choose a timelike vector field to contract the stress-energy tensor with. A natural timelike vector field would be the Killing vector $\partial_u$ which is equal to $\partial_t$ in the Schwarzschild coordinate system, but $\partial_u$ becomes null on ${{\mathscr I}}^+$ and we require more control there in order to establish peeling results. So we use the Morawetz vector field $$T^a \partial_a = u^2 \partial_u - 2(1+uR ) \partial_R \, .$$ The associated energy current is the vector field $V^a = T^{ab} T_b$ whose decomposition on the Newman-Penrose tetrad $\hat{l}, \hat{n}, \hat{m}, \bar{\hat{m}}$ is given by $$\begin{aligned}
V^a &=& \left( \sqrt{\frac{2}{F}} (2 + 2uR + \frac{(uR)^2}{2} F ) \vert \hat\phi_1 \vert^2 + \sqrt{\frac{F}{2}} u^2 \vert \hat\phi_2 \vert^2 \right) \hat{l}^a \\
&& + \left( \sqrt{\frac{2}{F}} (2 + 2uR + \frac{(uR)^2}{2} F ) \vert \hat\phi_0 \vert^2 + \sqrt{\frac{F}{2}} u^2 \vert \hat\phi_1 \vert^2 \right) \hat{n}^a \\
&& - \left( \sqrt{\frac{2}{F}} (2 + 2uR + \frac{(uR)^2}{2} F ) \hat\phi_1 \overline{\hat{\phi}_0} + \sqrt{\frac{F}{2}} u^2 \hat\phi_2 \overline{\hat{\phi}_1} \right) \hat{m}^a \\
&& - \left( \sqrt{\frac{2}{F}} (2 + 2uR + \frac{(uR)^2}{2} F ) \hat\phi_0 \overline{\hat{\phi}_1} + \sqrt{\frac{F}{2}} u^2 \hat\phi_1 \overline{\hat{\phi}_2} \right) \hat{\bar{m}}^a \, .\end{aligned}$$ Since $T^a$ is not an exact Killing vector, $V^a$ is not divergence free and it satisfies merely an approximate conservation law $$\label{MaxwellErrorTerm}
\nabla_a V^a = \nabla_{(a} T_{b)} T^{ab} = 8mR^2 F^{-1} (3+uR) T_{ab} l^a l^b = 8mR^2 F^{-1} (3+uR) \vert \hat{\phi}_0 \vert^2 \, .$$ The energy $3$-form is the Hodge dual of the energy current $$\begin{aligned}
\omega &:=& * (V_a {\mathrm{d}}x^a ) = V \lrcorner {\mathrm{dVol}}^4 \\
&= & \left( \sqrt{\frac{2}{F}} (2 + 2uR + \frac{(uR)^2}{2} F ) \vert \hat\phi_1 \vert^2 + \sqrt{\frac{F}{2}} u^2 \vert \hat\phi_2 \vert^2 \right) (-\hat{l} \wedge {\mathrm{d}}^2 \omega ) \\
&& + \left( \sqrt{\frac{2}{F}} (2 + 2uR + \frac{(uR)^2}{2} F ) \vert \hat\phi_0 \vert^2 + \sqrt{\frac{F}{2}} u^2 \vert \hat\phi_1 \vert^2 \right) (\hat{n} \wedge {\mathrm{d}}^2 \omega ) \\
&& + \left( \sqrt{\frac{2}{F}} (2 + 2uR + \frac{(uR)^2}{2} F ) \hat\phi_1 \overline{\hat{\phi}_0} + \sqrt{\frac{F}{2}} u^2 \hat\phi_2 \overline{\hat{\phi}_1} \right) (-i\hat{l} \wedge \hat{n} \wedge \hat{m}) \\
&& + \left( \sqrt{\frac{2}{F}} (2 + 2uR + \frac{(uR)^2}{2} F ) \hat\phi_0 \overline{\hat{\phi}_1} + \sqrt{\frac{F}{2}} u^2 \hat\phi_1 \overline{\hat{\phi}_2} \right) (i\hat{l} \wedge \hat{n} \wedge \bar{\hat{m}}) \, ,\end{aligned}$$ (recall that ${\mathrm{d}}^2 \omega = i \hat{m} \wedge \bar{\hat{m}}$ is the euclidian measure on $S^2$).
On a $u=$constant hypersurface ${\cal S}_u$, the energy is given by $$\label{MEnergySu}
{\cal E}_{{\cal S}_u} (\hat{\phi} ) := \int_{{\cal S}_u} \omega = \int_{{\cal S}_u} \left( \sqrt{\frac{2}{F}} (2 + 2uR + \frac{(uR)^2}{2} F ) \vert \hat\phi_0 \vert^2 + \sqrt{\frac{F}{2}} u^2 \vert \hat\phi_1 \vert^2 \right) {\mathrm{d}}R \, {\mathrm{d}}^2 \omega \, .$$ On a $u=-sr_*$ hypersurface ${\cal H}_s$, recall that $${\mathrm{d}}R = \frac{FR^2}{s}{\mathrm{d}}u$$ and therefore (for $0<s\leq 1$) $$\begin{aligned}
{\cal E}_{{\cal H}_s} (\hat{\phi} ) := \int_{{\cal H}_s} \omega &=& \int_{{\cal H}_s} \left( \left( \frac{2}{s}-1 \right) R^2 \left( 2+2uR + \frac{(uR)^2}{2} F \right) \vert \hat{\phi}_0 \vert^2 \right. \nonumber \\
&& \hspace{0.5in} \left. + \left( 2 + 2uR + \frac{(uR)^2F}{s} \right) \vert \hat{\phi}_1 \vert^2 + \frac{F}{2} u^2 \vert \hat{\phi}_2 \vert^2 \right) {\mathrm{d}}u \, {\mathrm{d}}^2 \omega \, .
\label{MEnergyHs}\end{aligned}$$ As $s\rightarrow 0$, this expression simplifies to give the energy on ${{\mathscr I}}^+$ $$\label{MEnergyScriPlus}
{\cal E}_{{{\mathscr I}}^+_{u_0}} (\hat{\phi} ) := \int_{{{\mathscr I}}^+_{u_0}} \omega = \int_{{{\mathscr I}}^+_{u_0}} \left( 2 \vert \hat{\phi}_1 \vert^2 + \frac{u^2}{2} \vert \hat{\phi}_2 \vert^2 \right) {\mathrm{d}}u \, {\mathrm{d}}^2 \omega \, ,$$ using the fact that $$\frac{(uR)^2}{s} = (-uR) Rr_* \rightarrow 0 \mbox{ as } r \rightarrow +\infty \mbox{ with } u \mbox{ bounded }$$ and similarly $R^2 /s \rightarrow 0$ as $r\rightarrow +\infty$.
We have the following equivalent simpler expressions fo the energies on ${\cal S}_u$, $u\leq u_0$ and ${\cal H}_s$, $0\leq s \leq 1$ $$\begin{aligned}
{\cal E}_{{\cal S}_u} (\hat{\phi} ) & \simeq & \int_{{\cal S}_u} \left( \vert \hat\phi_0 \vert^2 + u^2 \vert \hat\phi_1 \vert^2 \right) {\mathrm{d}}R \, {\mathrm{d}}^2 \omega \, , \\
{\cal E}_{{\cal H}_s} (\hat{\phi} ) &\simeq & \int_{{\cal H}_s} \left( \frac{R}{\vert u \vert} \vert \hat{\phi}_0 \vert^2 + \vert \hat{\phi}_1 \vert^2 + u^2 \vert \hat{\phi}_2 \vert^2 \right) {\mathrm{d}}u \, {\mathrm{d}}^2 \omega \, .\end{aligned}$$
[**Proof.**]{} We notice that $2+2uR + \frac{(uR)^2}{2} F $ vanishes for $$uR = -\frac{2}{F} ( 1 \pm \sqrt{2mR} ) \, .$$ We know from lemma \[ApproxCloseI0\] that in $\Omega^+_{u_0}$, $-1-\varepsilon < uR \leq 0$ with $0\leq \varepsilon <<1$ and also $$2(1-\varepsilon ) < \frac{2}{F} ( 1 \pm \sqrt{2mR} ) < 2 (1+\varepsilon ) \, .$$ It follows that $2+2uR + \frac{(uR)^2}{2} F$ vanishes nowhere for $0< s \leq 1$ and tends to $2$ as $uR \rightarrow 0$, so this quantity is also bounded below away from zero and above, uniformly on $\Omega^+_{u_0}$. The lemma then follows from lemma \[ApproxCloseI0\].
Peeling {#Peeling}
=======
Peeling for Dirac
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We have already established energy estimates for $\hat\psi_A$ between ${{\mathscr I}}^+_{u_0}$, ${\cal S}_{u_0}$ and ${\cal H}_1$. Now we establish estimates for successive derivatives of $\hat\psi_A$. We do not need to commute all directional derivatives into the equation ; as our goal is to control transverse regularity on ${{\mathscr I}}^+$ we focus on derivatives in the direction of $\partial_R$, i.e. of $\hat{l}^a$. We denote by $D_R \hat{\psi}_A$ the spinor $$D_R \hat{\psi}_A := \partial_R \hat{\psi}_1 o_A - \partial_R \hat{\psi}_0 \iota_A$$ and we work out the equation satisfied by $D_R \hat{\psi}$. In order to obtain a more useable expression for this equation we multiply the first line of by $\sqrt{2F}$, keep the second as it is and commute $\partial_R$ into the resulting system. We get : $$\left\{ \begin{array}{l}
{ \left( 2 \partial_u + R^2 F \partial_R \right) \, \partial_R \hat{\psi}_0 - \sqrt{2F} \, \hat{\eth} \partial_R \hat{\psi}_1 + R \left( 1- \frac{5mR}{2} \right) \partial_R \hat{\psi}_0 } \\
{ \hspace{1in} = -2R \left( 1-3mR \right) \partial_R \hat{\psi}_0 - m \sqrt{\frac{2}{F}} \hat{\eth} \hat{\psi}_1 - \left( 1- 5mR \right)\hat{\psi}_0 \, ,} \\ \\
{ -\sqrt{\frac{F}{2}} \left( \partial_R - \frac{m}{2F} \right) \partial_R \hat{\psi}_1 - \hat{\eth}' \partial_R \hat{\psi}_0 = -\frac{m}{\sqrt{2F}} \partial_R \hat{\psi}_1 - \frac{m^2}{(2F)^{3/2}} \hat{\psi}_1 \, .} \end{array} \right.$$ This can be re-written as $$\left\{ \begin{array}{l}
{ \sqrt{\frac{2}{F}} \left( \partial_u + \frac{R^2 F}{2} \partial_R + \frac{2R-5mR^2}{4} \right) \, \partial_R \hat{\psi}_0 - \hat{\eth} \partial_R \hat{\psi}_1 } \\
{ \hspace{1in} = -\sqrt{\frac{2}{F}} R (1-3mR) \partial_R \hat{\psi}_0 - \frac{m}{F} \hat{\eth} \hat{\psi}_1 - \frac{1- 5mR}{\sqrt{2F}} \hat{\psi}_0 \, ,} \\ \\
{ -\sqrt{\frac{F}{2}} \left( \partial_R - \frac{m}{2F} \right) \partial_R \hat{\psi}_1 - \hat{\eth}' \partial_R \hat{\psi}_0 = -\frac{m}{\sqrt{2F}} \partial_R \hat{\psi}_1 - \frac{m^2}{(2F)^{3/2}} \hat{\psi}_1} \end{array} \right.$$ and as a spinorial equation takes the form $$\begin{aligned}
\hat{\nabla}^{AA'} \left( D_R \hat{\psi}_A \right) &=& - \left( \sqrt{\frac{2}{F}} R (1-3mR) \partial_R \hat{\psi}_0 + \frac{m}{F} \hat{\eth} \hat{\psi}_1 + \frac{1- 5mR}{\sqrt{2F}} \hat{\psi}_0 \right) o^{A'} \nonumber \\
&& - \left( \frac{m}{\sqrt{2F}} \partial_R \hat{\psi}_1 + \frac{m^2}{(2F)^{3/2}} \hat{\psi}_1 \right) \iota^{A'} \, .\label{WeylEqDrPsi}\end{aligned}$$ The conservation law associated with equation is the following $$\begin{aligned}
\hat{\nabla}^{AA'} \left[ \left( D_R \hat{\psi}_A \right) \left( D_R \bar{\hat{\psi}}_{A'} \right) \right] &=& 2 \Re \left[ \left( \hat{\nabla}^{AA'} D_R \hat{\psi}_A \right) D_R \bar{\hat{\psi}}_{A'} \right] \\
&=& - 2 \Re \left[ \left( \sqrt{\frac{2}{F}} R (1-3mR) \partial_R \hat{\psi}_0 + \frac{m}{F} \hat{\eth} \hat{\psi}_1 + \frac{1- 5mR}{\sqrt{2F}} \hat{\psi}_0 \right) \overline{\partial_R \hat{\psi}_0} \right. \\
&& \left. + \left( \frac{m}{\sqrt{2F}} \partial_R \hat{\psi}_1 + \frac{m^2}{(2F)^{3/2}} \hat{\psi}_1 \right) \overline{\partial_R \hat{\psi}_1} \right]\end{aligned}$$ In order to obtain energy estimates for $D_R \hat{\psi}$ using Gronwall’s inequality, we need to estimate the right-hand side by the energy densities for $\hat{\psi}$ or $D_R \hat{\psi}$. Two types of terms present a difficulty : those involving angular derivatives and the term involving $ \hat{\psi}_0 $ and $\partial_R \hat{\psi}_0 $ without a factor of $R$. In order to control the angular terms, we must commute angular derivatives into the equation ; this will turn out to give us the additional control on $\hat{\psi}_0$ that we need.
The commutation of angular derivatives into the Weyl equation is best described, and performed, using the GHP formalism, i.e. we use the form of the Weyl equation. In our framework, using the values of the spin coefficients, we have the following identities $$\label{GHPCommutators}
\left[ \hat{{\mbox{\th}}} \, ,~\hat{\eth} \right] \eta =0 \mbox{ for } \eta \mbox{ of weight } \{ -1, 0 \} \mbox{ and } \left[ \hat{{\mbox{\th}}}' \, ,~\hat{\eth}' \right] \eta =0 \mbox{ for } \eta \mbox{ of weight } \{ 1, 0 \} \, .$$ So when we apply $\eth'$ to the first equation and $\eth$ to the second, we obtain $$\label{WeylEqEthPsi}
\left\{ \begin{array}{l}
{ \hat{{\mbox{\th}}} ' \hat{\eth}' \hat{\psi}_0 - \hat{\eth}' \hat{\eth} \hat{\psi}_1 =0 } \\
{ \hat{{\mbox{\th}}} \hat{\eth} \hat{\psi}_1- \hat{\eth} \hat{\eth}' \hat{\psi}_0 =0 \, .} \end{array} \right.$$ Note that $\hat{\eth}' \hat{\psi}_0$ has weight $\{ 0,1 \}$ and $\hat{\eth} \hat{\psi}_1$ weight $\{ 0, -1 \}$, so putting $$(D_\omega \hat{\psi} )_{A'} = \hat{\eth} \hat{\psi}_1 \hat{o}_{A'} - \hat{\eth}' \hat{\psi}_0 \hat{\iota}_{A'} \, ,$$ equation can be written as a conjugate Weyl equation $$\nabla^{AA'} (D_\omega \hat{\psi} )_{A'} =0 \, ,$$ i.e. $D_\omega$ is a symmetry operator for the Dirac equation on the rescaled Schwarzchild metric, sending anti-neutrino fields to neutrino fields.
Now consider some smooth solution $\hat{\psi}_A$ of with compactly supported initial data. Both $\hat{\psi}$ and $D_\omega \hat{\psi}$ satisfy the energy equality , which entails for any $s \in [0,1]$ : $$\begin{aligned}
\int_{{\cal H}_1} \left( \frac{R}{\vert u \vert} \vert \hat{\psi}_0 \vert^2 + \vert \hat{\psi}_1 \vert^2 \right) {\mathrm{d}}u {\mathrm{d}}^2 \omega & \simeq & \int_{{\cal H}_s} \left( \frac{R}{\vert u \vert} \vert \hat{\psi}_0 \vert^2 + \vert \hat{\psi}_1 \vert^2 \right) {\mathrm{d}}u {\mathrm{d}}^2 \omega \\
&&+ \int_{{\cal S}_{u}^{s,1}} \left| \hat{\psi}_0 \right|^2 {\mathrm{d}}R {\mathrm{d}}^2 \omega\, , \\
\int_{{\cal H}_1} \left( \frac{R}{\vert u \vert} \vert \hat{\eth}' \hat{\psi}_{0} \vert^2 + \vert \hat{\eth} \hat{\psi}_{1} \vert^2 \right) {\mathrm{d}}u {\mathrm{d}}^2 \omega & \simeq & \int_{{\cal H}_s} \left( \frac{R}{\vert u \vert} \vert \hat{\eth}' \hat{\psi}_{0} \vert^2 + \vert \hat{\eth} \hat{\psi}_{1} \vert^2 \right) {\mathrm{d}}u {\mathrm{d}}^2 \omega \\
&&+ \int_{{\cal S}_{u}^{s,1}} \left| \hat{\eth}' \hat{\psi}_{0} \right|^2 {\mathrm{d}}R {\mathrm{d}}^2 \omega\, . \\\end{aligned}$$ Note that this immediately gives us a control of the $4$-volume $L^2$ norm of $\hat{\psi}_0$ and $\hat{\eth}' \hat{\psi}_0$ in terms of the energy on ${\cal H}_1$ using the foliation by ${\cal S}_u$ with the identifying vector field $\partial_u$. However, this works only once we have obtained the estimates, it does not allow to control a perturbed equation with an error term $( \hat{\psi}_0)^2$ in the conservation law.
Integrating on $\Omega^{s_1,s_2}_{u_0}$ the conservation law associated with equation , we obtain $$\begin{gathered}
\left\vert {\cal E}_{{\cal H}_{s_1}} (D_R \hat{\psi}) + {\cal E}_{{\cal S}^{s_1,s_2}_{u_0}} (D_R \hat{\psi}) - {\cal E}_{{\cal H}_{s_2}} (D_R \hat{\psi}) \right\vert \\
\leq 2\int_{s_1}^{s_2} \int_{{\cal H}_{s}} \left( \left\vert -\sqrt{\frac{2}{F}} R (1-3mR) \partial_R \hat{\psi}_0 - \frac{m}{F} \hat{\eth} \hat{\psi}_1 - \frac{1- 5mR}{\sqrt{2F}} \hat{\psi}_0 \right\vert \vert \partial_R \hat{\psi}_0\vert \right. \\
\left. + \left\vert -\frac{m}{\sqrt{2F}} \partial_R \hat{\psi}_1 - \frac{m^2}{2F\sqrt{2F}} \hat{\psi}_1 \right\vert \vert \partial_R \hat{\psi}_1\vert \right) \frac{1}{\vert u \vert} {\mathrm{d}}u {\mathrm{d}}^2 \omega {\mathrm{d}}s\end{gathered}$$ The last two error terms are trivially controlled by the energies of $\hat{\psi}_A$ and $D_R \hat{\psi}_A$ ; the first error term, thanks to the $1/\vert u \vert$ coming from the Leray measure, is exactly controlled by the energy of $D_R \hat{\psi}_A$. The difficulties are with the second and third error terms. For the second term, we use the control we have obtained over angular derivatives as follows $$\begin{aligned}
\int_{s_1}^{s_2} \int_{{\cal H}_{s}} \left\vert \frac{m}{F} \hat{\eth} \hat{\psi}_1 \right\vert \vert \partial_R \hat{\psi}_0 \vert \frac{1}{\vert u \vert} {\mathrm{d}}u {\mathrm{d}}^2 \omega {\mathrm{d}}s & \lesssim & \int_{s_1}^{s_2} \frac{1}{\sqrt{s}}\int_{{\cal H}_{s}} \left\vert \frac{m}{F} \hat{\eth} \hat{\psi}_1 \right\vert \vert \partial_R \hat{\psi}_0 \vert \sqrt{\frac{R}{\vert u \vert}} {\mathrm{d}}u {\mathrm{d}}^2 \omega {\mathrm{d}}s \\
& \lesssim & \int_{s_1}^{s_2} \frac{1}{\sqrt{s}} \int_{{\cal H}_{s}} \left( \vert \hat{\eth} \hat{\psi}_1 \vert^2 + \frac{R}{\vert u \vert} \vert \partial_R \hat{\psi}_0 \vert^2 \right) {\mathrm{d}}u {\mathrm{d}}^2 \omega {\mathrm{d}}s \\
& \lesssim & \int_{s_1}^{s_2} \frac{1}{\sqrt{s}} \left( {\cal E}_{{\cal H}_{s}} (D_\omega \hat{\psi}) + {\cal E}_{{\cal H}_{s}} (D_R \hat{\psi}) \right) {\mathrm{d}}s \, ,\end{aligned}$$ which allows to apply a Gronwall inequality since $1/\sqrt{s}$ is integrable on $[0,1]$. The third term is the trickiest. We use the fact that the lowest eigenvalue of $\eth'$ on weighted scalar fields of weight $\{ 1 , 0 \}$ is positive (see [@PeRi84] section 4.15) ; this implies $$\int_{S^2} \vert \hat\psi_0 \vert^2 {\mathrm{d}}^2 \omega \lesssim \int_{S^2} \vert \hat\eth' \hat\psi_0 \vert^2 {\mathrm{d}}^2 \omega$$ uniformly on $\Omega^+_{u_0}$, whence using the same method as for the previous term, $$\int_{s_1}^{s_2} \int_{{\cal H}_{s}} \left\vert \frac{1- 5mR}{\sqrt{2F}} \hat{\psi}_0 \right\vert \vert \partial_R \hat{\psi}_0 \vert \frac{1}{\vert u \vert} {\mathrm{d}}u {\mathrm{d}}^2 \omega {\mathrm{d}}s \lesssim \int_{s_1}^{s_2} \frac{1}{\sqrt{s}} \int_{{\cal H}_{s}} ( \left\vert \hat{\eth}' \hat{\psi}_0 \right\vert^2 + \frac{R}{\vert u \vert} \vert \partial_R \hat{\psi}_0 \vert^2 ) {\mathrm{d}}u {\mathrm{d}}^2 \omega {\mathrm{d}}s$$ and since $\hat\eth' \hat{\psi}_0 = \hat{\mbox{\th}} \hat{\psi}_1$ which is controlled uniformly in $\Omega^+_{u_0}$ by $\vert \partial_R \hat{\psi}_1 \vert + \vert \hat{\psi}_1 \vert$, we can once again apply Gronwall’s inequality. Successive applications of $\partial_R$ will produce error terms which can be controlled by similar techniques using angular derivatives and lower order norms. We get the following result :
\[DiracPeeling\] There exist positive constants $C_n$, $n\in {\mathbb{N}}$ such that for any smooth compactly supported data on ${\cal H}_1$, the associated solution $\hat{\psi}_A$ of Dirac’s equation satisfies $$\begin{gathered}
\sum_{p=0}^n \sum_{k=0}^p {\cal E}_{{{\mathscr I}}^+_{u_0}} (D^k_\omega D^{p-k}_R \hat\psi_A ) \leq C_n \sum_{p=0}^n \sum_{k=0}^p {\cal E}_{{\cal H}_1} (D^k_\omega D^{p-k}_R \hat\psi_A ) \, , \\
\sum_{p=0}^n \sum_{k=0}^p {\cal E}_{{\cal H}_1} (D^k_\omega D^{p-k}_R \hat\psi_A ) \leq C_n \sum_{p=0}^n \sum_{k=0}^p \left( {\cal E}_{{{\mathscr I}}^+_{u_0}} (D^k_\omega D^{p-k}_R \hat\psi_A ) + {\cal E}_{{\cal S}_{u_0}} (D^k_\omega D^{p-k}_R \hat\psi_A ) \right) \, .\end{gathered}$$ This extends to the spaces of initial data $\mathfrak{h}^n ({\cal H}_1 )$ obtained by completion of ${\cal C}^\infty_0 ({\cal H}_1 )$ in the norms $$\Vert \hat\psi_A \Vert_{\mathfrak{h}^n ({\cal H}_1 )} = \left( \sum_{p=0}^n \sum_{k=0}^p {\cal E}_{{\cal H}_1} (D^k_\omega D^{p-k}_R \hat\psi_A ) \right)^{1/2} \, .$$
Peeling for Maxwell
-------------------
Contrary to the case of the Dirac equation, we already have a problem with the basic energy estimate since the energy current $V$ does not satisfy an exact conservation law. However, the error term in the approximate conservation law satisfied by $V$ is easily controlled by the energy density on the hypersurfaces ${\cal H}_{s}$. This gives us the following result :
\[BasicEstimateMax\] There exists a positive positive constant $C$, such that for any smooth data on ${\cal H}_1$ satisfying the constraints and supported away from $i^0$, the associated solution $\hat{\phi}_{AB}$ of Maxwell’s equations satisfies for any $0\leq s<1$ $$\begin{gathered}
{\cal E}_{{\cal H}_s} (\hat\phi_{AB} ) \leq C {\cal E}_{{\cal H}^1} (\hat\phi_{AB} ) \, , \\
{\cal E}_{{\cal H}_1} ( \hat\phi_{AB} ) \leq C \left( {\cal E}_{{\cal H}_s} (\hat\phi_{AB} ) + {\cal E}_{{\cal S}^{s,1}_{u_0}} (\hat\phi_{AB} ) \right) \, .\end{gathered}$$
We now obtain similar estimates for successive derivatives of the Maxwell field, starting with the derivative with respect to $R$. We multiply by $\sqrt{2F}$ the first and third equations of and commute $\partial_R$ into the system. We obtain $$\left\{ \begin{array}{l}
{ \left( 2\partial_u + R^2 F \partial_R - (5mR^2 -2R ) \right) \partial_R \hat{\phi}_0 - \sqrt{2F} \, \hat\eth \partial_R \hat{\phi}_1 } \\
{ \hspace{1in} = -2R(1-3mR) \partial_R \hat\phi_0 -2 (1-5mR) \hat\phi_0 - \sqrt{\frac{2}{F}} m \hat\eth \hat\phi_1 \, , } \\ \\
{ -\sqrt{\frac{F}{2}} \partial_R^2 \hat{\phi}_1 - \hat{\eth}' \partial_R \hat{\phi}_0 = -\frac{m}{\sqrt{2F}} \partial_R \hat\phi_1 \, ,} \\ \\
{ (2 \partial_u + R^2F\partial_R ) \partial_R \hat{\phi}_1 - \sqrt{2F} \hat{\eth} \partial_R \hat{\phi}_2 = -2R(1-3mR) \partial_R \hat\phi_1 - \sqrt{\frac{2}{F}} m \eth \hat\phi_2 \, , } \\ \\
{ -\sqrt{\frac{F}{2}} \partial_R^2 \hat{\phi}_2 + \frac{m}{\sqrt{2F}} \partial_R \hat\phi_2 - \hat{\eth}' \partial_R \hat{\phi}_1 = -\frac{m}{\sqrt{2F}} \partial_R \hat\phi_2 - \frac{m^2}{(2F)^{3/2}} \hat\phi_2 \, , } \end{array} \right.$$ which we rewrite as $$\left\{ \begin{array}{l}
{ \hat{\mbox{\th}}' \partial_R \hat{\phi}_0 - \hat\eth \partial_R \hat{\phi}_1 = -\sqrt{\frac{2}{F}} R(1-3mR) \partial_R \hat\phi_0 -\sqrt{\frac{2}{F}} (1-5mR) \hat\phi_0 - \frac{m}{F} \hat\eth \hat\phi_1 \, , } \\ \\
{ \hat{\mbox{\th}}\partial_R \hat{\phi}_1 - \hat{\eth}' \partial_R \hat{\phi}_0 = -\frac{m}{\sqrt{2F}} \partial_R \hat\phi_1 \, ,} \\ \\
{ \hat{\mbox{\th}}' \partial_R \hat{\phi}_1 - \hat{\eth} \partial_R \hat{\phi}_2 = -\sqrt{2}{F} R(1-3mR) \partial_R \hat\phi_1 - \frac{m}{F} \eth \hat\phi_2 \, , } \\ \\
{ \hat{\mbox{\th}}\partial_R \hat{\phi}_2 - \hat{\eth}' \partial_R \hat{\phi}_1 = -\frac{m}{\sqrt{2F}} \partial_R \hat\phi_2 - \frac{m^2}{(2F)^{3/2}} \hat\phi_2 \, , } \end{array} \right.$$ Putting $$D_R \hat\phi_{AB} := \partial_R \hat\phi_0 \hat\iota_A \hat\iota_B - \partial_R \hat\phi_1 (\hat{o}_A \hat{\iota}_B + \hat\iota_A \hat{o}_B ) + \partial_R \hat\phi_2 \hat{o}_A \hat{o}_B \, ,$$ the above system is the perturbed Maxwell equation for $D_R \hat\phi_{AB}$ : $$\begin{aligned}
\nabla^{AA'} D_R \hat\phi_{AB} &=& - \left( \sqrt{2} FR (1-3mR) \partial_R \hat\phi_1 + \frac{m}{F} \hat\eth \hat\phi_2 \right) \bar{\hat{o}}^{A'} \hat{o}_B \\
&& + \left( \sqrt{\frac{2}{F}} R (1-3mR) \partial_R \hat\phi_0 + \sqrt{\frac{2}{F}} (1-5mR) \hat\phi_0 + \frac{m}{F} \hat\eth \hat\phi_1 \right) \bar{\hat{o}}^{A'} \hat\iota_B \\
&& - \left( \frac{m}{\sqrt{2F}} \partial_R \hat\phi_2 + \frac{m^2}{(2F)^{3/2}} \hat{\phi}_2 \right) \hat\iota^{A'} \hat{o}_B + \frac{m}{\sqrt{2F}} \partial_R \hat{\phi}_1 \hat\iota^{A'} \hat\iota_B \, .\end{aligned}$$ The associated approximate conservation law is therefore $$\begin{gathered}
\nabla^{AA'} \left( T^{BB'} (D_R \hat\phi_{AB}) (D_R \bar{\hat\phi}_{A'B'}) \right) = \nabla^{(a} T^{b)} (D_R \hat\phi_{AB}) (D_R \bar{\hat\phi}_{A'B'}) \\
+ 2 \Re \left\{ T^{BB'} \left[ - \left( \sqrt{2} FR (1-3mR) \partial_R \hat\phi_1 + \frac{m}{F} \hat\eth \hat\phi_2 \right) \bar{\hat{o}}^{A'} \hat{o}_B \right. \right.\\
+ \left( \sqrt{\frac{2}{F}} R (1-3mR) \partial_R \hat\phi_0 + \sqrt{\frac{2}{F}} (1-5mR) \hat\phi_0 + \frac{m}{F} \hat\eth \hat\phi_1 \right) \bar{\hat{o}}^{A'} \hat\iota_B \\
\left. \left. - \left( \frac{m}{\sqrt{2F}} \partial_R \hat\phi_2 + \frac{m^2}{(2F)^{3/2}} \hat{\phi}_2 \right) \hat\iota^{A'} \hat{o}_B + \frac{m}{\sqrt{2F}} \partial_R \hat{\phi}_1 \hat\iota^{A'} \hat\iota_B \right] D_R \bar{\hat{\phi}}_{A'B'} \right\} \, ,\end{gathered}$$ which in full details, using the decomposition of $T^a$ on the rescaled tetrad, reads $$\begin{aligned}
&&\nabla^{AA'} \left( T^{BB'} (D_R \hat\phi_{AB}) (D_R \bar{\hat\phi}_{A'B'}) \right) = 8mR^2 F^{-1} (3+uR) \vert \partial_R \hat\phi_0 \vert^2 \nonumber \\
&& \hspace{1in} - \frac{4}{F} \left( 2(1+uR) + \frac{(uR)^2F}{2} \right) (1-3mR) R \vert \partial_R \hat\phi_0 \vert^2 \nonumber \\
&& \hspace{1in} - \left( 2 F^{3/2} (1-3mR) \vert Ru\vert \vert u \vert + \frac{m}{F} \left( 2(1+uR) + \frac{(uR)^2F}{2} \right) \right) \vert \partial_R \hat\phi_1 \vert^2 \nonumber \\
&& \hspace{1in} - mu^2 \vert \partial_R \phi_2 \vert^2 \nonumber \\
&& \hspace{1in} - 2 \Re \left[ \frac{m}{\sqrt{2F}} u^2 \hat\eth \hat\phi_2 \partial_R \bar{\hat{\phi}}_1 + \frac{m^2}{4F} u^2 \hat\phi_2 \partial_R \bar{\hat\phi}_2 \right. \nonumber \\
&& \hspace{1.5in} + \frac{2}{F} (1-5mR) \left( 2(1+uR) + \frac{(uR)^2F}{2} \right) \hat\phi_0 \partial_R \bar{\hat\phi}_0 \nonumber \\
&& \hspace{1.5in} \left. + \frac{\sqrt{2} m}{F^{3/2}} \left( 2(1+uR) + \frac{(uR)^2F}{2} \right) \hat\eth \hat\phi_1 \partial_R \bar{\hat{\phi}}_0 \right] \, . \label{MaxDRConsLaw}\end{aligned}$$ As was the case for the Weyl equation, it is necessary to control the angular derivatives first before gaining a control on the derivative with respect to $R$. However, the weights of $\hat\phi_0$, $\hat\phi_1$ and $\hat\phi_2$ are $\{ 2,0 \}$, $\{ 0,0 \}$ and $\{ -2 , 0 \}$ and for an anti-self-dual field $\hat\phi_{A'B'}$ we would get the weights $\{ 0,2 \}$, $\{ 0,0 \}$ and $\{ 0,-2 \}$ ; since the weights of $\hat\eth$ and $\hat\eth '$ are $\{ 1, -1 \}$ and $\{ -1 , 1 \}$, by commuting these operators into the Maxwell system we do not recover any of the adequate weights of the components. So we cannot hope to commute a well-chosen arrangement of $\hat\eth$ and $\hat\eth '$ into the (anti-self-dual) Maxwell system and obtain another (self-dual) Maxwell system. But the Schwarzschild metric is spherically symmetric, we have a $3$-dimensional space of Killing vector fields tangent to the sphere which generate all rotations. We choose a basis of this space, denoted $X$, $Y$, $Z$, such that $-X^2 - Y^2 - Z^2$ is controlled below and above by the positive Laplacian on $S^2$. Since $X$, $Y$ and $Z$ will commute with Maxwell’s equations, this gives us a control analogous to proposition \[BasicEstimateMax\] over angular derivatives of any order.
Integrating the error term of over $\Omega^+_{u_0} \cap \{ \tau \leq s \leq1 \}$, for $0<\tau \leq 1$, and splitting the result as an integral in $s$ of integrals over the hypersurfaces ${\cal H}_s$, we gain a factor $1/u$ in the $3$-volume measure. With this, the integrals of all terms on ${\cal H}_s$ are controlled for each $s$ by the sum $${\cal E}_{{\cal H}_s} (\hat\phi_{AB} ) + {\cal E}_{{\cal H}_s} (D_R \hat\phi_{AB} ) + {\cal E}_{{\cal H}_s} (D_X \hat\phi_{AB} ) + {\cal E}_{{\cal H}_s} (D_Y \hat\phi_{AB} ) + {\cal E}_{{\cal H}_s} (D_Z \hat\phi_{AB} ) \, ,$$ except for the last two terms. The treatment for both terms is similar to the Dirac case. For the first, we use the fact that the $L^2$ norm of $\hat\phi_0$ on the $2$-sphere is (uniformly in $\Omega_{u_0}^+$) controlled[^2] by that of $\hat\eth' \hat\phi_0$ and then the second equation of the Maxwell system giving us the equality of $\hat\eth' \hat\phi_0$ and $\hat{\mbox{\th}}\hat\phi_1$. So we obtain for $0<s \leq 1$ $$\begin{gathered}
2\int_{{\cal H}_s} \left\vert \frac{2}{F} (1-5mR) \left( 2(1+uR) + \frac{(uR)^2F}{2} \right) \hat\phi_0 \partial_R \bar{\hat\phi}_0 \right\vert \frac{1}{\vert u \vert } {\mathrm{d}}u {\mathrm{d}}^2 \omega \\
\hspace{0.5in} \lesssim \frac{1}{\sqrt{s}} \int_{{\cal H}_s} \vert \hat\phi_0 \partial_R \bar{\hat\phi}_0 \vert \sqrt\frac{R}{\vert u \vert } {\mathrm{d}}u {\mathrm{d}}^2 \omega \\
\hspace{0.5in} \lesssim \frac{1}{\sqrt{s}} \left( \int_{{\cal H}_s} \vert \hat\eth '\hat\phi_0 \vert^2 {\mathrm{d}}u {\mathrm{d}}^2 \omega + \int_{{\cal H}_s} \vert \partial_R \bar{\hat\phi}_0 \vert^2 \frac{R}{\vert u \vert } {\mathrm{d}}u {\mathrm{d}}^2 \omega \right) \\
\hspace{0.5in} \lesssim \frac{1}{\sqrt{s}} \left( \int_{{\cal H}_s} \left( \vert \hat\phi_1 \vert^2 + \vert \partial_R \hat\phi_1 \vert^2 \right) {\mathrm{d}}u {\mathrm{d}}^2 \omega + \int_{{\cal H}_s} \vert \partial_R \bar{\hat\phi}_0 \vert^2 \frac{R}{\vert u \vert } {\mathrm{d}}u {\mathrm{d}}^2 \omega \right) \\
\hspace{0.5in} \lesssim \frac{1}{\sqrt{s}} \left( {\cal E}_{{\cal H}_s} (\hat\phi_{AB} ) + {\cal E}_{{\cal H}_s} (D_R \hat\phi_{AB} ) \right)\, .\end{gathered}$$ The second is simpler to deal with : $$\begin{gathered}
2\int_{{\cal H}^s} \left\vert \frac{\sqrt{2} m}{F^{3/2}} \left( 2(1+uR) + \frac{(uR)^2F}{2} \right) \hat\eth \hat\phi_1 \partial_R \bar{\hat{\phi}}_0 \right\vert \frac{1}{\vert u \vert } {\mathrm{d}}u {\mathrm{d}}^2 \omega \\
\hspace{0.5in} \lesssim \frac{1}{\sqrt{s}} \int_{{\cal H}^s} \vert \hat\eth \hat\phi_1 \partial_R \bar{\hat\phi}_0 \vert \sqrt\frac{R}{\vert u \vert } {\mathrm{d}}u {\mathrm{d}}^2 \omega \\
\hspace{0.5in} \lesssim \frac{1}{\sqrt{s}} \left( {\cal E}_{{\cal H}^s} (D_X \hat\phi_{AB} ) + {\cal E}_{{\cal H}^s} (D_Y \hat\phi_{AB} ) + {\cal E}_{{\cal H}^s} (D_Z \hat\phi_{AB} ) + {\cal E}_{{\cal H}^s} (D_R \hat\phi_{AB} ) \right)\, .\end{gathered}$$
Note that since $\hat\phi_1$ is the component of $\hat\phi_{AB}$ whose weight is $\{ 0,0 \}$, $\hat\eth \hat\phi_1$ is merely $\hat\delta \hat\phi_1 = \nabla_{\hat{m}} \hat\phi_1$, it involves no spin-coefficient. And since $m$ is a bounded vector field on $S^2$, the $L^2$ norm of $\hat\eth \hat\phi_1$ on $S^2$ is controlled (uniformly on $\Omega^+_{u_0}$) by the sum of the $L^2$ norms of $D_X \hat\phi_1$, $D_Y \hat\phi_1$ and $D_Z \hat\phi_1$.
The successive derivatives with respect to $R$ will be controlled in a similar way, controlling first the angular derivatives. This gives the theorem :
\[PeelingMaxwell\] There exist positive constants $C_n$, $n\in {\mathbb{N}}$ such that for any smooth data on ${\cal H}_1$ compactly supported, the associated solution $\hat{\phi}_{AB}$ of Maxwell’s equations satisfies $$\begin{gathered}
\sum_{k_1+k_2+k_3+k_4\leq n} {\cal E}_{{{\mathscr I}}^+_{u_0}} (D^{k_1}_X D^{k_2}_Y D^{k_3}_Z D^{k_4}_R \hat\phi_{AB} ) \leq C_n \sum_{k_1+k_2+k_3+k_4\leq n} {\cal E}_{{\cal H}_1} (D^{k_1}_X D^{k_2}_Y D^{k_3}_Z D^{k_4}_R \hat\phi_{AB} ) \, , \\
\sum_{k_1+k_2+k_3+k_4\leq n} {\cal E}_{{\cal H}_1} (D^{k_1}_X D^{k_2}_Y D^{k_3}_Z D^{k_4}_R \hat\phi_{AB} ) \leq C_n \sum_{k_1+k_2+k_3+k_4\leq n} \left( {\cal E}_{{{\mathscr I}}^+_{u_0}} (D^{k_1}_X D^{k_2}_Y D^{k_3}_Z D^{k_4}_R \hat\phi_{AB} ) \right. \\
\hspace{4in} \left.+ {\cal E}_{{\cal S}_{u_0}} (D^{k_1}_X D^{k_2}_Y D^{k_3}_Z D^{k_4}_R \hat\phi_{AB} ) \right) \, .\end{gathered}$$ This extends to the spaces of initial data $\mathfrak{h}^n ({\cal H}_1 )$ obtained by completion of ${\cal C}^\infty_0 ({\cal H}_1 )$ in the norms $$\Vert \hat\phi_{AB} \Vert_{\mathfrak{h}^n ({\cal H}_1 )} = \left( \sum_{k_1+k_2+k_3+k_4\leq n} {\cal E}_{{{\mathscr I}}^+_{u_0}} (D^{k_1}_X D^{k_2}_Y D^{k_3}_Z D^{k_4}_R \hat\phi_{AB} ) \right)^{1/2} \, .$$
Interpretation {#Interpretation}
==============
In this section, we check that for a given order of transverse regularity at ${{\mathscr I}}^+$, our classes of data ensuring that the rescaled solution has at least this regularity are not smaller than they are in the flat case when the full embedding in the Einstein cylinder is used. Of course, since we have only worked in a neighbourhood of $i^0$ in Schwarzschild, what we really mean by this is a comparison of the asymptotic constraints on the fall-off of initial data. In the case of Dirac and Maxwell fields, this comparison is made easier than for the wave equation (see [@MaNi2009] for the analogous interpretation for the wave equation) because the energy on a spacelike slice is conformally invariant.
Let us consider a $4$-dimensional globally hyperbolic (and therefore admitting a spin-structure) spacetime $({\cal M} , g)$, $\Sigma$ a spacelike hypersurface, $\nu^a$ its future-oriented unit normal vector field, $\psi_A$ a solution to and $\Phi_{AB}$ a solution to . Let $\Omega$ be a smooth positive function on $\cal M$, and put $$\hat{g} := \Omega^2 g \, ,~ \hat{\psi}_A := \Omega^{-1} \psi_A \, ,~ \hat{\phi}_{AB} := \Omega^{-1} \phi_{AB} \, .$$ The unit normal to $\Sigma$ for $\hat{g}$ is now $$\hat{\nu}^a = \Omega^{-1} \nu^a$$ and if we denote by $\mu$ (resp. $\hat{\mu}$) the measure induced on $\Sigma$ by $g$ (resp. $\hat{g}$), then $$\hat{\mu} = \Omega^3 \mu \, .$$ The energy of the rescaled Weyl field on $\Sigma$ is given by (in this section, we shall denote by $\hat{\cal E}$ the energies for the rescaled metric and $\cal E$ the energies for the unrescaled metric) $$\hat{\cal E}_\Sigma (\hat\psi ) = \int_\Sigma \hat{\nu}^a \hat{\psi}_A \bar{\hat{\psi}}_{A'} {\mathrm{d}}\hat{\mu} = \int_\Sigma {\nu}^a {\psi}_A \bar{\psi}_{A'} {\mathrm{d}}{\mu} = {\cal E}_\Sigma (\psi ) \, .$$ For the Maxwell field, we need a choice of observer (or merely of timelike vector field) $t^a$ in the neighbourhood of $\Sigma$ to define the energy and then $$\hat{\cal E}_\Sigma (\hat\phi ) = \int_\Sigma \hat{\nu}^a t^b \hat{\phi}_{AB} \bar{\hat{\phi}}_{A'B'} {\mathrm{d}}\hat{\mu} = \int_\Sigma {\nu}^a t^b {\phi}_{AB} \bar{\phi}_{A'B'} {\mathrm{d}}{\mu} = {\cal E}_\Sigma (\phi ) \, .$$ Note that the observer is not rescaled.
Dirac fields
------------
We now compare the classes of data for Dirac fields in the flat case and in the Schwarzschild spacetime. In the flat case ($m=0$), the conformal embedding of Minkowski spacetime into the Einstein cylinder is realized using the conformal factor $$\Omega = \frac{2}{\sqrt{1+(t+r)^2} \sqrt{1+(t-r)^2}}$$ and the vector field used for increasing the regularity in the energy estimates is the time translation along the Einstein cylinder $$\frac{\partial}{\partial \tau} = \frac{1}{2} \left( (1+t^2+r^2) \frac{\partial}{\partial t} + 2tr \frac{\partial}{\partial r} \right) \, .$$ This is a Killing vector field on the Einstein cylinder, i.e. a conformal Killing vector field of Minkowski spacetime. Let $\psi_A$ be a Weyl field on Minkowski spacetime, denoting by $\Sigma$ the $t=0$ hypersurface and choosing the spin-frame given by the choice of Newman-Penrose tetrad $$\label{FlatNP}
l = \frac{1}{\sqrt{2}} (\partial_t + \partial_r ) \, ,~ n = \frac{1}{\sqrt{2}} (\partial_t + \partial_r ) \, ,~ m=\frac{1}{\sqrt{2}} (\partial_\theta +\frac{i}{\sin \theta} \partial_\varphi ) \, ,$$ the future-oriented unit (for the unrescaled metric) normal to $\Sigma$ is $\partial_t = \frac{1}{\sqrt{2}} (l+n )$ and we have $$\hat{\cal E}_\Sigma (\hat\psi ) = {\cal E}_\Sigma (\psi ) = \int_\Sigma \nu^a \psi_A \bar\psi_{A'} {\mathrm{d}}^3 x = \frac{1}{\sqrt{2}} \int_\Sigma (\vert \psi_0 \vert^2 + \vert \psi_1 \vert^2 ) {\mathrm{d}}^3 x \, .$$ On the Einstein cylinder, we have the following energy equality $$\hat{\cal E}_\Sigma (\hat\psi ) = \hat{\cal E}_{{{\mathscr I}}^+} (\hat\psi )$$ and commuting $\partial_\tau^k$ into the equation we get $$\hat{\cal E}_\Sigma (\partial_\tau^k \hat\psi ) = \hat{\cal E}_{{{\mathscr I}}^+} (\partial_\tau^k \hat\psi ) \, .$$ The right-hand side is a measure of transverse regularity at ${{\mathscr I}}^+$.
We work out the constraint on initial data corresponding to the first level of regularity. Using the expression of $\partial_\tau$ in terms of $(t,r)$ variables at $t=0$, the left-hand side for $k=1$ can be rewritten as $$\hat{\cal E}_\Sigma (\frac{1+r^2}{2} \partial_t \hat\psi ) ={\cal E}_\Sigma (\frac{1+r^2}{2} \partial_t \psi ) \, .$$ Using the ellipticity of the spacelike part of the Dirac equation (i.e. using the Bochner-Lichnerowicz-Weitzenböck formula on ${\mathbb{R}}^3$), this corresponds, modulo lower order terms, to an $L^2$ control over $$(1+r^2) \partial_r \psi \mbox{ and } \frac{1+r^2}{r} \nabla_{S^2} \psi$$ independently. This is what defines the scale of weighted Sobolev spaces obtained on $\Sigma$ by requiring the finiteness of the energies of $\partial_\tau^k \hat\psi$.
Note that this is not the original conformal peeling construction by Penrose, but it is the closest equivalent in terms of Sobolev spaces (see [@MaNi2009] for a more detailed presentation for the wave equation).
In the Schwarzschild case, the energy equality is now $$\hat{\cal E}_{{\cal H}^1} (\hat\psi ) = \hat{\cal E}_{{\cal S}_{u_0}} (\hat\psi ) + \hat{\cal E}_{{{\mathscr I}}^+_{u_0}} (\hat\psi )$$ and to raise the regularity up to order $k$, we use all combinations up to order $k$ of $\partial_R$ and $D_{\omega}$. At first order, the quantity defining the space of data on ${\cal H}^1$ is $$\label{FirstOrderNorm}
\hat{\cal E}_{{\cal H}^1} (\hat\psi ) + \hat{\cal E}_{{\cal H}^1} (\partial_R \hat\psi ) + \hat{\cal E}_{{\cal H}^1} (D_\omega \hat\psi ) \, ,$$ where $D_\omega$ can be replaced by $\nabla_{S^2}$, the former acting on the components, the latter on the full spinor. First of all, using the conformal invariance of the energy, for the unrescaled spin-frame associated with the tetrad , $$\hat{\cal E}_{{\cal H}^1} (\hat\psi ) = \int_{{\cal H}^1} (\vert \psi_0 \vert^2 + \vert \psi_1 \vert^2) F^{1/2} r^2 \sin \theta {\mathrm{d}}r {\mathrm{d}}\theta {\mathrm{d}}\varphi \, ,$$ which gives a control equivalent to the flat $L^2$ norm. Now up to lower order terms, using the conformal invariance of the energy, is equivalent to the same expression with unrescaled energies for the unrescaled field $\psi$. The last term in bounds the $L^2$ norm of $\nabla_{S^2} \psi$ with no weight, which is weaker than in the flat case, but the second term will impose a constraint similar to the flat case, though in a more mixed manner. Indeed, using the equation satisfied by $\psi$ in components (see for example [@Ni1997]) $$\partial_t \psi + F\left( \begin{array}{cc} {-1} & 0 \\ 0 & 1 \end{array} \right) (\partial_r + \frac{1}{r} + \frac{F'}{4F} ) \psi + \frac{F^{1/2}}{r} D_\omega \psi =0 \, ,$$ we find that $$\begin{aligned}
\partial_R \psi &=& \frac{r^2}{F} (\partial_t + \partial_{r_*} ) \psi = \frac{2r^2}{F} \left( \begin{array}{c} {\partial_{r_*} \psi_0 } \\ 0 \end{array} \right) - r F^{-1/2} D_\omega \psi + \mbox{ lower order terms,} \\
&=& \left( \begin{array}{c} {2r^2 \partial_{r} \psi_0 } \\ 0 \end{array} \right) - r F^{-1/2} D_\omega \psi + \mbox{ lower order terms.}\end{aligned}$$ We see that in terms of weights, this is similar to the flat case. The gain is that we only control $r^2 \partial_r$ applied to $\psi_0$ and not to $\psi_1$. This is not surprising since $\psi_0$ is the component of $\psi$ that propagates dominantly to the left (towards the black hole) and which, were its fall-off at infinity not strong enough, might therefore propagate singularities along ${{\mathscr I}}^+$ from $i^0$. Also the weight in front of the angular derivatives in the expression of $\partial_R \psi$ is the same as in the flat case, but the term is not controlled on its own, only in combination with $r^2 \partial_{r} \psi_0$. This is what defines the scale of weighted Sobolev spaces obtained from our energy estimates in the Schwarzschild case.
The conclusion is that the constraints for peeling in the Schwarzschild case are weaker than the ones obtained in Minkowski using the full conformal embedding in the Einstein cylinder. In other words, peeling in Schwarzschild at any order is valid for a class of data slightly larger than the usual class in the flat case.
This does not mean that the asymptotic structure of Schwarzschild allows peeling for more general data than in Minkowski. This only shows that our definition of peeling is more general than the one usually considered. The uniformity of our norms in the mass in any compact interval $[0,M]$ shows that for our definition, the classes in Minkowski and Schwarzschild are the same.
Maxwell fields
--------------
In the Maxwell case, the essential ingredients are the same as in the Dirac case : conformal invariance of the norm, same vector fields used to raise the regularity and controlling better the part of the data propagating to the left, plus one more ingredient, the Morawetz vector field. What is not clear for Maxwell fields is that the basic energies should give equivalent controls. The reason why this is true is that in flat spacetime, the Morawetz vector field and the time translation along the Einstein cylinder differ by a constant multiple of $\partial_t$ (see [@MaNi2009]) which gives a weaker norm than the two others, plus the local equivalence of the norms in the mass $m$. Let us describe this more explicitely.
On the Einstein cylinder, we use the vector field $\partial_\tau$ both for defining the conserved current for the Maxwell field and for raising the regularity in the energy equalities. Using the conformal invariance of the energy and the expression of $\partial_\tau$, decomposing the unrescaled field in the spin-frame associated with , we get $$\hat{\cal E}_\Sigma (\hat\phi ) = {\cal E}_\Sigma (\phi ) = \int_\Sigma ( \vert \phi_0 \vert^2 + 2 \vert \phi_1 \vert^2 + \vert \phi_2 \vert^2 )\frac{1+r^2}{4} {\mathrm{d}}^3 x \, .$$ In the Schwarzschild case, we define the energy using the Morawetz vector field given in terms of the vectors of the rescaled tetrad by . The initial energy therefore is given by (using the fact that the future unit normal to the hypersurface $\{ t=0 \}$ for the metric $g$ is $\frac{1}{\sqrt{2}} (l+n)$ and also $u=-r_*$ on ${\cal H}^1$) $$\begin{aligned}
\hat{\cal E}_{{\cal H}^1} (\hat\phi ) = {\cal E}_{{\cal H}^1} (\phi ) &=& \int_{{\cal H}^1} \left( \frac{1}{\sqrt{F}} ( 2(1-r_*R) + \frac{(r_*R)^2F}{2} ) r^2 \vert \phi_0 \vert^2 \right. \\
&& + ( \frac{1}{\sqrt{F}} (2(1-r_*R) + \frac{(r_*R)^2F}{2} ) r^2 + \frac{r_*^2 \sqrt{F}}{2} )\vert \phi_1 \vert^2 \\
&& \left. + \frac{\sqrt{F} r_*^2}{2} \vert \phi_2 \vert^2 \right) \sqrt{F} r^2 {\mathrm{d}}r_* {\mathrm{d}}\omega \, ,\end{aligned}$$ which in $\Omega_{u_0}^+$ is equivalent to $$\int_{{\cal H}^1} \left( \vert \phi_0 \vert^2 + \vert \phi_1 \vert^2 + \vert \phi_2 \vert^2 \right) r^4 {\mathrm{d}}r {\mathrm{d}}\omega$$ and therefore gives the same control as the flat norm involving $\partial_\tau$ instead of the Morawetz vector field. When raising the regularity in the energy estimates, the principles are the same as for Dirac. On Minkowski spacetime, a $\partial_t$ applied to $\phi$ gives a control on the full gradient on ${\mathbb{R}}^3$ because Maxwell equations entail that the d’Alembertian of the field vanishes and this ensures the ellipticity of the spacelike part. So the flat spacetime energy at the first level of regularity satisfies $${\cal E}_\Sigma ( \phi ) + {\cal E}_\Sigma (\frac{1+r^2}{2} \partial_t \phi ) \simeq \int_\Sigma \left( \vert \phi \vert^2 + (1+r^2) \vert \nabla_{{\mathbb{R}}^3} \phi \vert^2 \right) (1+r^2) {\mathrm{d}}^3 x \, .$$ In the Schwarzschild case, the first level of regularity is controlled by the energy $$\hat{\cal E}_{{\cal H}^1} (\hat\phi ) + \hat{\cal E}_{{\cal H}^1} (\partial_R \hat\phi ) + \hat{\cal E}_{{\cal H}^1} (\nabla_{S^2} \hat\phi ) \, ,$$ which is equivalent to $${\cal E}_{{\cal H}^1} (\phi ) + {\cal E}_{{\cal H}^1} (\partial_R \phi ) + {\cal E}_{{\cal H}^1} ( \nabla_{S^2} \phi ) \, .$$ Similarly to the Dirac case, we can evaluate $\partial_R \phi$ using $$\partial_R = \frac{r^2}{F} (\partial_t + \partial_{r_*} )$$ and Maxwell’s equations. We have : $$\begin{aligned}
\partial_R \phi_0 &=& \frac{r^2}{F} ( \sqrt{2F} m^a \partial_a \phi_1 + 2 \partial_{r_*} \phi_0 )+\mbox{ lower order terms,} \\
\partial_R \phi_1 &=& \sqrt{\frac{2}{F}} r^2 \bar{m}^a \partial_a \phi_0 +\mbox{ lower order terms,} \\
\partial_R \phi_2 &=& \sqrt{\frac{2}{F}} r^2 \bar{m}^a \partial_a \phi_1 +\mbox{ lower order terms.}\end{aligned}$$ So we observe exactly the same phenomenon as for Dirac, a similar control as in the flat case, except that the control on radial derivatives is weaker : only the component propagating to the left is explicitely controlled on the initial hypersurface.
A remark on the constraints
---------------------------
This paper is concerned about the asymptotic assumptions on initial data ensuring peeling at a certain order and on comparing these asumptions in Schwarzschild and in flat spacetime. There is one aspect, for Maxwell fields, which may affect the classes of data leading to a peeling at a certain order in different ways in Schwarzschild and Minkowski : the constraints. We do not address this question in detail here but we simply make a remark on the integrability of the constraints from infinity in a reasonably large class of functions and the compatibility between the constraints and our energy norms. Let us consider the Maxwell system in the unrescaled Schwarzschild spacetime using the tetrad . The energy at $t=0$ associated with the Morawetz vector field in the $r_* > -u_0$ region is equivalent to $$\int_{r_* >-u_0} (\vert \phi_0 \vert^2 + \vert \phi_1 \vert^2 + \vert \phi_2 \vert^2 ) r^4 {\mathrm{d}}r_* {\mathrm{d}}^2 \omega \, .$$ Taking the difference of the second and third equations of the system, we get $$F \left( \partial_r + \frac{2}{r} \right) \phi_1 = \frac{1}{r} \left( \eth \phi_2 - \eth' \phi_0 \right) \, ;$$ rescaling the field components by $r^2$, $$\tilde{\phi}_i := r^2 \phi_i \, ,~i=0,1,2,$$ this reads $$\label{RescConstraints}
F \partial_r \tilde{\phi}_1 = \frac{1}{r} \left( \eth \tilde\phi_2 - \eth' \tilde\phi_0 \right) \, .$$ Decomposing the field into spin-weighted spherical harmonics, the operators $\eth$ and $\eth'$ will turn into multiplication operators by constant factors. If we consider a simplified situation where the behaviour of the field at infinity is in powers of $r^{-1}$, the finiteness of the Morawetz energy implies that the rescaled field components fall off at least like $r^{-1}$. Assuming such a behaviour for $\tilde\phi_0$ and $\tilde\phi_2$ and integrating from infinity starting from the value zero, we recover the same fall-off for $\tilde\phi_1$.
So heuristically, the constraints are compatible with the Morawetz energy in both the Schwarzschild and Minkowski spacetimes. This heuristic argument does not address the question of polyhomogeneous solutions of the constraints. Our sole purpose here is to show that in Schwarzschild and Minkowski, there will be large classes of solutions of the constraints compatible with our function spaces and that the constraints will not introduce in the Schwarzschild case any additional restriction compared to the flat case.
Covariant derivative approach {#App1}
=============================
Curvature spinors
-----------------
We calculate the curvature spinors for the rescaled metric. Recall that given a spacetime $({\cal M},g)$ with a spin structure and equipped with the Levi-Civitta connection, the Riemann tensor $R_{abcd}$ can be decomposed as follows (see [@PeRi84]) : $$R_{abcd} = X_{ABCD} \, \varepsilon_{A'B'} \varepsilon_{C'D'} + \Phi_{ABC'D'} \, \varepsilon_{A'B'} \varepsilon_{CD} + \bar{\Phi}_{A'B'CD} \, \varepsilon_{AB} \varepsilon_{C'D'} + \bar{X}_{A'B'C'D'} \, \varepsilon_{AB} \varepsilon_{CD} \, ,$$ where $X_{ABCD}$ is a complete contraction of the Riemann tensor in its primed spinor indices $$X_{ABCD} = \frac{1}{4} R_{abcd} {\varepsilon}^{A'B'} {\varepsilon}^{C'D'}$$ and ${\Phi}_{ab} = {\Phi}_{(ab)}$ is the trace-free part of the Ricci tensor multiplied by $-1/2$ : $$2{\Phi}_{ab} = 6 {\Lambda} {g}_{ab} - {R}_{ab} \, ,~ {\Lambda} = \frac{1}{24} \mathrm{Scal}_{g} \, .$$ It is convenient when dealing with conformal rescalings to use instead of $\Phi_{ab}$ the curvature $2$-form $$P_{ab} = \Phi_{ab} - \Lambda g_{ab}$$ because of its simpler transformation law. Also, it is usual to isolate the totally symmetric part $\Psi_{ABCD} $ of $X_{ABCD}$, referred to as the Weyl spinor, which describes the conformally invariant part of the curvature : $$X_{ABCD} = \Psi_{ABCD} + \Lambda \left( \varepsilon_{AC} \varepsilon_{BD} + \varepsilon_{AD} \varepsilon_{BC} \right) \, ,~ \Psi_{ABCD} = X_{(ABCD)} \, .$$ The scalar $\Lambda$ and the curvature spinors $P$ and $\Psi$ have simple rules of transformation under a conformal rescaling $\hat{g} = \Omega^{2} g$, given by (see [@PeRi84] p. 120-123) : $$\begin{gathered}
\hat{\Psi}_{ABCD} = \Psi_{ABCD} \, , \\
\hat{\Lambda} = \Omega^{-2} \Lambda + \frac{1}{4} \Omega^{-3} \square \Omega \, , ~\square = \nabla^a \nabla_a \, , \\
\hat{P}_{ab} = P_{ab} - \nabla_b \Upsilon_a + \Upsilon_{AB'} \Upsilon_{BA'} \, ,~ \mbox{with } \Upsilon_a = \Omega^{-1} \nabla_a \Omega = \nabla_a \log \Omega \, .\end{gathered}$$
For the rescaled Schwarzschild metric (\[RescMet\]), the values of $\Lambda$ and $\hat{\Phi}_{ab}$ are : $$\begin{aligned}
\hat{\Lambda} &=& mR/2 \, ,\\
\hat{\Phi}_{ab} {\mathrm{d}}x^a {\mathrm{d}}x^b &=& \frac{1-3mR}{2} \left( R^2 F {\mathrm{d}}u^2 -2 {\mathrm{d}}u {\mathrm{d}}R + {\mathrm{d}}\omega^2 \right) \\
&=& \left( 1-3mR \right) \left( \frac{1}{2} \hat{g} + {\mathrm{d}}\omega^2 \right) \, .\end{aligned}$$
[**Proof.**]{} The value of $\hat{\Lambda}$ was calculated in [@MaNi2009]. In order to evaluate $\hat{\Phi}_{ab}$, we determine $\hat{P}_{ab}$. First note that the Schwarzschild metric is Ricci flat, whence $\Phi_{ab}=P_{ab}=R_{ab}=0$. Hence, $$\hat{P}_{ab} = - \nabla_b \Upsilon_a + \Upsilon_{AB'} \Upsilon_{BA'} \, .$$ Since $\Omega = R = 1/r$, $$\Upsilon_a {\mathrm{d}}x^a = -\frac{{\mathrm{d}}r}{r} \, .$$ We need to determine its spinor components. We do so in the dyad $\{ o^A , \iota^A \}$, denoting $x^0 = t$, $x^1 =r$, $x^2 = \theta$, $x^3 = \varphi$ : $$\Upsilon_{AA'} = \frac{-1}{r} g^1_{AA'} = \frac{-1}{r} g^{11} \varepsilon_{AB} \varepsilon_{A'B'} g_1^{BB'} = \frac{F}{r} \varepsilon_{AB} \varepsilon_{A'B'} g_1^{BB'}$$ and $$g_1^\mathbf{BB'} = \left( \begin{array}{cc} {n_1} & {-\bar{m}_1} \\ {-m_1} & {l_1} \end{array} \right) = \frac{1}{\sqrt{2F}} \left( \begin{array}{cc} {1} & {0} \\ {0} & {-1} \end{array} \right) \, .$$ It follows that $$\Upsilon_\mathbf{AA'} = -\frac{1}{r} \sqrt{\frac{F}{2}} \left( \begin{array}{cc} {1} & {0} \\ {0} & {-1} \end{array} \right) \, .$$ The non zero components of $\alpha_{ab} := \Upsilon_{AB'} \Upsilon_{BA'}$ are therefore $$\begin{gathered}
\alpha_{00'00'} = \Upsilon_{00'} \Upsilon_{00'} = \alpha_{11'11'} = \Upsilon_{11'} \Upsilon_{11'} = \frac{F}{2r^2} \, , \\
\alpha_{01'10'} = \Upsilon_{00'} \Upsilon_{11'} = \alpha_{10'01'} = \Upsilon_{11'} \Upsilon_{00'} = -\frac{F}{2r^2}\, .\end{gathered}$$ and the $2$-form $\alpha_{ab}$ reads : $$\begin{aligned}
\Upsilon_{AB'} \Upsilon_{BA'}{\mathrm{d}}x^a {\mathrm{d}}x^b &=& \frac{F}{2r^2} \left( l_a l_b + n_a n_b - m_a \bar{m}_b - \bar{m}_a m_b \right) {\mathrm{d}}x^a {\mathrm{d}}x^b \\
&=& \frac{F}{2} \left( R^2 \hat{l}_a \hat{l}_b + R^{-2} \hat{n}_a \hat{n}_b - \hat{m}_a \bar{\hat{m}}_b - \bar{\hat{m}}_a \hat{m}_b \right) {\mathrm{d}}x^a {\mathrm{d}}x^b \\
&=& \frac{R^2F^2}{2} d u^2 -F {\mathrm{d}}u {\mathrm{d}}R + \frac{{\mathrm{d}}R^2}{R^2} - \frac{F}{2} {\mathrm{d}}\omega^2 \, .\end{aligned}$$ We now calculate $\nabla_b \Upsilon_a$ : $$\nabla_b \Upsilon_a {\mathrm{d}}x^a {\mathrm{d}}x^b = \nabla_b \left( -\frac{{\mathrm{d}}r}{r} \right) {\mathrm{d}}x^b = \frac{{\mathrm{d}}r^2}{r^2} + \frac{1}{r} \Gamma^1_\mathbf{ab} {\mathrm{d}}x^\mathbf{a} {\mathrm{d}}x^\mathbf{b} \, ,$$ and among the Christoffel symbols $$\Gamma^1_\mathbf{ab} = \frac{1}{2} g^{1\mathbf{c}} \left( \frac{\partial g_\mathbf{ac}}{\partial x^\mathbf{b}} + \frac{\partial g_\mathbf{bc}}{\partial x^\mathbf{a}} - \frac{\partial g_\mathbf{ab}}{\partial x^\mathbf{c}} \right) = -\frac{F}{2} \left( \frac{\partial g_{\mathbf{a}1}}{\partial x^\mathbf{b}} + \frac{\partial g_{\mathbf{b}1}}{\partial x^\mathbf{a}} - \frac{\partial g_\mathbf{ab}}{\partial r} \right) \, ,$$ the non-zero ones are $$\Gamma^1_{00} = \frac{FF'}{2} \, ,~ \Gamma^1_{11} = -\frac{F'}{2F} \, ,~ \Gamma^1_{22} = -Fr \, ,~ \Gamma^1_{33} = -Fr \sin^2 \theta \, .$$ Hence $$\begin{aligned}
\nabla_b \Upsilon_a {\mathrm{d}}x^a {\mathrm{d}}x^b &=& \frac{{\mathrm{d}}r^2}{r^2} + \frac{FF'}{2r} {\mathrm{d}}t^2 -\frac{F'}{2rF} {\mathrm{d}}r^2 -F {\mathrm{d}}\omega^2 \\
&=& \frac{{\mathrm{d}}R^2}{R^2} +\frac{m}{r^3}\left( F{\mathrm{d}}t^2 - F^{-1} {\mathrm{d}}r^2 \right) - F {\mathrm{d}}\omega^2 \\
&=& FmR^3 {\mathrm{d}}u^2 -2mR {\mathrm{d}}u {\mathrm{d}}R + \frac{{\mathrm{d}}R^2}{R^2} -F {\mathrm{d}}\omega^2 \, .\end{aligned}$$ From this we can infer the value of $\hat{\Phi}_{ab}$ : $$\begin{aligned}
\hat{\Phi}_{ab} {\mathrm{d}}x^a {\mathrm{d}}x^b &=& \hat{P}_{ab} {\mathrm{d}}x^a {\mathrm{d}}x^b + \hat{\Lambda} \hat{g} \\
&=& \Upsilon_{AB'} \Upsilon_{BA'} - \nabla_b \Upsilon_a + \frac{mR}{2} \hat{g} \\
&=& \frac{1-3mR}{2} \left( R^2 F {\mathrm{d}}u^2 -2 {\mathrm{d}}u {\mathrm{d}}R + {\mathrm{d}}\omega^2 \right) \, .\end{aligned}$$ This concludes the proof.
Energy estimate for the transverse derivative of a Dirac field
--------------------------------------------------------------
First we set up a general formula for the commutation of a directional covariant derivative into the Dirac equation, then we apply it to the derivative along $\partial_R$.
Consider any vector field $V^a$ and commute ${\hat{\nabla}}_V$ into the Weyl equation : $$\begin{aligned}
0= V^a {\hat{\nabla}}_a {\hat{\nabla}}^{BB'} \hat{\psi}_B &=& -V^a {\hat{\nabla}}_a \hat\varepsilon^{B'C'} {\hat{\nabla}}_{CC'} \hat{\psi}^C \\
&=& - \hat\varepsilon^{B'C'} \left( V^a \Delta_{ac} \hat{\psi}^C + {\hat{\nabla}}_{CC'} \left( V^a {\hat{\nabla}}_a \hat{\psi}^C \right) - \left( {\hat{\nabla}}_c V^a \right) {\hat{\nabla}}_a \hat{\psi}^C \right) \\
&=& - \hat\varepsilon^{B'C'} V^a \Delta_{ac} \hat{\psi}^C + {\hat{\nabla}}^{BB'} \left( {\hat{\nabla}}_V \hat{\psi}_B \right) - \left( {\hat{\nabla}}^b V^a \right) {\hat{\nabla}}_a \hat{\psi}_B\end{aligned}$$ where $\Delta_{ab} = {\hat{\nabla}}_a {\hat{\nabla}}_b - {\hat{\nabla}}_b {\hat{\nabla}}_a$. The first and third terms in the right-hand side can be calculated more explicitely : $$V^a \hat{\Delta}_{ac} \hat{\psi}^C = V^a \left[ \hat{\varepsilon}_{A'C'} {\hat X}_{ACE}^{{\hspace{0.09in}}{\hspace{0.09in}}{\hspace{0.09in}}C} + \hat{\varepsilon}_{CA} \hat{\Phi}_{A'C'E}^{{\hspace{0.135in}}{\hspace{0.135in}}{\hspace{0.09in}}C} \right] \hat{\psi}^E \, .$$ The symmetries of the Riemann tensor imply that $${\hat X}_{ACE}^{{\hspace{0.09in}}{\hspace{0.09in}}{\hspace{0.09in}}C} = 3 \hat{\Lambda} \hat{\varepsilon}_{AE} = \frac{3mR}{2} \hat{\varepsilon}_{AE} \, ;$$ whence $$- \varepsilon^{B'C'} V^a \hat{\varepsilon}_{A'C'} \hat{X}_{ACE}^{{\hspace{0.09in}}{\hspace{0.09in}}{\hspace{0.09in}}C} \hat{\psi}^E = \varepsilon^{B'C'} \frac{3mR}{2} V^A_{C'} \hat{\psi}_A = \frac{3mR}{2} V^{AB'} \hat{\psi}_A\, .$$ The term involving $\hat\Phi_{ab}$ can be written $$- \varepsilon^{B'C'} \hat{\varepsilon}_{AC} \hat{\Phi}_{A'C'E}^{{\hspace{0.135in}}{\hspace{0.135in}}{\hspace{0.09in}}C} \hat{\psi}^E = -V_{CA'} \hat{\Phi}^{EB'CA'} \hat{\psi}_E = -V_a \hat{\Phi}^{BB'a} \hat{\psi}_B \, .$$ It follows that the equation satisfied by ${\hat{\nabla}}_V \hat{\psi}^B$ is $$\label{DirEqNablaVPsi}
{\hat{\nabla}}^{BB'} \left( {\hat{\nabla}}_V \hat{\psi}_B \right) = \left( {\hat{\nabla}}^b V^a \right) {\hat{\nabla}}_a \hat{\psi}_B + V_a \hat{\Phi}^{ab} \hat{\psi}_B - \frac{3 mR}{2} V^b \hat{\Psi}_B \, .$$ If the vector field is $V^a \partial_a = \partial_R$, then $${\hat{\nabla}}^\mathbf{a} V^\mathbf{b} = \hat{g}^{{\mathbf{a}}{\mathbf{d}}} \partial_\mathbf{d} V^\mathbf{b} + \hat{g}^{{\mathbf{a}}{\mathbf{d}}} \hat{\Gamma}^{{\mathbf{b}}}_{{\mathbf{d}}{\mathbf{c}}} V^{\mathbf{c}}= \hat{g}^{{\mathbf{a}}{\mathbf{d}}} \hat{\Gamma}^{{\mathbf{b}}}_{{\mathbf{d}}1}$$ and the only non-zero coefficient is $$\hat{\Gamma}^1_{01} = R (1-3mR) \, .$$ Hence, $${\hat{\nabla}}^a V^b \partial_a \partial_b = -R (1-3mR) \partial_R \otimes \partial_R$$ and $$\left( {\hat{\nabla}}^b V^a \right) {\hat{\nabla}}_a \hat{\psi}_B = -R (1-3mR) V^b {\hat{\nabla}}_R \hat{\psi}_B \, .$$ The spinor ${\hat{\nabla}}_R \hpsi_A$ thus satifies $$\begin{gathered}
\label{DirEqNablaRPsi}
{\hat{\nabla}}^{AA'} \left( {\hat{\nabla}}_R \hat{\psi}_A \right) = \left[ {\hat{\nabla}}^{AA'} , {\hat{\nabla}}_R \right] \hat{\psi}_A = -R V^a \left( (1-3mR) {\hat{\nabla}}_R \hat{\psi}_A + \frac{3 m}{2} \hat{\Psi}_A \right) - \hat{\Phi}^{0a} \hat{\psi}_A \, , \\ \mbox{and } \hat{\Phi}^{0a} \partial_a = -\frac{1-3mR}2 \partial_R \, . \nonumber\end{gathered}$$ The equation satisfied by higher order radial derivatives is obtained by means of the commutator expansion : $$\begin{aligned}
{\hat{\nabla}}^{AA'} \left( {\hat{\nabla}}_R^k \hat{\psi}_A \right) &=& \left[ {\hat{\nabla}}^{AA'} , {\hat{\nabla}}_R^k \right] \hat{\psi}_A \nonumber \\
&=& \sum_{p=0}^{k-1} {\hat{\nabla}}_R^{k-p-1} \left[ {\hat{\nabla}}^{AA'} , {\hat{\nabla}}_R \right] {\hat{\nabla}}_R^p \hat{\psi}_A \nonumber \\
&=& -\sum_{p=0}^{k-1} {\hat{\nabla}}_R^{k-p-1} \left( R V^a (1-3mR) {\hat{\nabla}}_R^{p+1} \hat{\psi}_A \right) \nonumber \\
&& + \sum_{p=0}^{k-1} {\hat{\nabla}}_R^{k-p-1} \left( \frac{1-6 mR}{2} V^a {\hat{\nabla}}_R^p\hat{\psi}_A \right) \, . \label{DirEqHigherRDerivative}\end{aligned}$$ It is useful to obtain the explicit expression of the action of $\hat{\Phi}^{0a}$ and $\partial_R$ on $\hat{\psi}_A$ by contraction. We simply need the Infeld-Van der Waerden symbol : $$\hat{g}_1^\mathbf{AA'} = \left( \begin{array}{cc} {\hat{n}_1} & {-\bar{\hat{m}}_1} \\ {-\hat{m}_1} & {\hat{l}_1} \end{array} \right) = -\sqrt{\frac{2}{F}} \left( \begin{array}{cc} 1 & 0\\ 0 & 0 \end{array} \right) \, .$$ It follows $$\begin{aligned}
\left( \partial_R \right)^a \hat{\psi}_A &=& \hat{g}_1^{AA'} \hat{\psi}_A = - \sqrt{\frac2F} \hat{\Psi}_0 o^{A'} \, , \\
\hat{\Phi}^{0a} \hat{\psi}_A &=& -\frac{1-3mR}2 \left( \partial_R \right)^a \hat{\psi}_A = \frac{1-3mR}{\sqrt{2F} }\hat{\Psi}_0 o^{A'} \, , \\\end{aligned}$$
The conservation law for $\hat{\nabla}_R \hat{\psi}_A$ is the following $$\begin{aligned}
{\hat{\nabla}}^{AA'} \left( ( {\hat{\nabla}}_R \hat{\psi}_A ) ({\hat{\nabla}}_R \bar{\hat{\psi}}_{A'} ) \right) &=& 2 \Re \left( -R (1-3mR) \left( \partial_R \right)^a ({\hat{\nabla}}_R \hat{\psi}_A ) {\hat{\nabla}}_R \bar{\hat{\psi}}_{A'} ) \right. \\
&& \left. + \frac{1-6mR}{2} \left( \partial_R \right)^a \hat{\psi}_A {\hat{\nabla}}_R \bar{\hat{\psi}}_{A'} ) \right) \\
&=& \sqrt{\frac{2}{F}} \Re \left( 2R (1-3mR) \left\vert ({\hat{\nabla}}_R \hat\Psi )_0 \right\vert^2 - (1-6mR) \hat{\psi}_0 \overline{({\hat{\nabla}}_R \hat{\Psi})_{0} } \right) \, .\end{aligned}$$ The $L^2$ norm of $\hat\psi_0$ on a $2$-sphere is controlled uniformally on $\Omega^+_{u_0}$ by that of $\eth' \hat\psi_0$. This in turn is controlled by the $L^2$ norms of $\hat\psi_1$ and $\partial_R \hat\psi_1$ using the second part of Dirac’s equation. And since $$\begin{aligned}
({\hat{\nabla}}_R \hat\Psi )_1 &=& ({\hat{\nabla}}_R \hat\Psi_A ) \hat\iota^A \\
&=& - \sqrt{\frac{2}{F}} \left( \hat{D} \hat{\psi}_A \right) \hat\iota^A \\
&=& - \sqrt{\frac{2}{F}} \left( \hat{D} \left( \hat{\psi}_1 \hat{o}_A - \hat{\psi}_0 \hat\iota_A \right) \right) \hat\iota^A \\
&=& - \sqrt{\frac{2}{F}} \left( \hat{\psi}_1 \hat\iota^A \hat{D} \hat{o}_A + \hat{D} \hat\psi_1 - \hat\psi_0 \hat\iota^A \hat{D} \hat\iota_A \right) \\
&=& - \sqrt{\frac{2}{F}} \left( \hat{\varepsilon} \hat{\psi}_1 + \hat{D} \hat\psi_1 - \hat{\pi} \hat\psi_0 \right) = \partial_R \hat\psi_1 - \frac{m}{2F} \hat\psi_1 \, ,\end{aligned}$$ we can estimate the $L^2$ norm of $\hat\psi_0$ on $S^2$ uniformly on $\Omega^+_{u_0}$ by the sum of the $L^2$ norms of $\hat\psi_1$ and $({\hat{\nabla}}_R \hat\psi)_1$. Then we deal with the error term as we did in section \[Peeling\].
\[PurelyTransverse\] Note that the error term is much simpler than it was when we used partial derivatives. This is to be expected but the practical upshot here is that we do not need to control angular derivatives in order to get estimates on the derivative transverse to ${{\mathscr I}}^+$. This remains true for higher orders. It can be useful to bear this in mind if one is interested in controlling transverse derivatives with low angular regularity. The same is very probably true for Maxwell.
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S. Klainerman & F. Nicol[ò]{}, [*On local and global aspects of the Cauchy problem in general relativity*]{}, Class. Quantum Grav. [**16**]{} (1999), p. R73-R157.
S. Klainerman & F. Nicol[ò]{}, The Evolution Problem in General Relativity, Progress in Mathematical Physics Vol. 25 (2002), Birkhaüser.
S. Klainerman & F. Nicol[ò]{}, [*Peeling properties of asymptotically flat solutions to the Einstein vacuum equations*]{}, Class. Quantum Grav. [**20**]{} (2003), p. 3215-3257.
L.J. Mason, & J.-P. Nicolas, [*Conformal scattering and the Goursat problem*]{}, Journal of Hyperbolic Differential Equations, [**1**]{} (2) (2004), p. 197–233.
L.J. Mason, & J.-P. Nicolas, [*Regularity at spacelike and null infinity*]{}, J. Inst. Math. Jussieu [**8**]{} (2009), 1, 179–208.
C.S. Morawetz, [*The decay of solutions of the exterior initial-boundary value problem for the wave equation*]{}, Comm. Pure Appl. Math. [**14**]{} (1961), p. 561–568.
E.T. Newman, R. Penrose [*An approach to gravitational radiation by a method of spin coefficients*]{}, J. Mathematical Phys. [**3**]{} (1962), 566–578.
J.-P. Nicolas, [*Global exterior Cauchy problem for spin $3/2$ zero rest-mass fields in the Schwarzschild space-time*]{}, Commun. in PDE 22 (1997), 3&4, p. 465-502.
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[^1]: The property that $\hat\rho =0$ was clear without calculation since $\hat\rho$ represents the geodesic expansion along the flow of $\hat{l}$ (which is a geodesic flow). This is clearly zero since for $\hat{g}$ the surface of the $2$-spheres orthogonal to $\hat{l}$ and $\hat{n}$ is constant. As for $\hat\mu$, it is equal to $-\hat\rho'$, i.e. corresponds to the geodesic contraction along the flow of $\hat{n}$. It is therefore also obviously zero for similar reasons.
[^2]: The first eigenvalue of $\hat\eth'$ on weighted scalars of weight $\{ 2 , 0 \}$ is again positive, see [@PeRi84] section 4.15.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Ofer Dekel, Ran Gilad-Bachrach, Ohad Shamir and Lin Xiao\
Microsoft Research\
`{oferd,rang,ohadsh,lin.xiao}@microsoft.com`
bibliography:
- 'mybib.bib'
title: Robust Distributed Online Prediction
---
Robust Learning with a Decentralized Architecture {#sec:async}
=================================================
In the previous section, we discussed asynchronous algorithms based on a master-workers paradigm. Using off-the-shelf fault tolerance methods, one can design simple and robust variants, capable of coping with dynamic and heterogeneous networks.
That being said, this kind of approach also has some limitations. First of all, access to a shared database may not be feasible, particularly in massively distributed environments. Second, utilizing leader election algorithms is potentially wasteful, since by the time a new master is elected, some workers or local worker groups might have already accumulated more than enough gradients to perform a gradient update. Moreover, what we really need is in fact more complex than just electing a random node as a master: electing a computationally weak or communication-constrained node will have severe repercussions. Also, unless the communication network is fully connected, we will need to form an entire DAG (directed acyclic graph) to relay gradients from the workers to the elected master. While both issues have been studied in the literature, it complicates the algorithms and increases the time required for the election process, again leading to potential waste. In terms of performance guarantees, it is hard to come up with explicit time guarantees for these algorithms, and hence the effect on the regret incurred by the system is unclear.
In this section, we describe a robust, fully decentralized and asynchronous version of DMB, which is not based on a master-worker paradigm. We call this algorithm *asynchronous* DMB, or ADMB for brevity. We provide a formal analysis, including an explicit regret guarantee, and show that ADMB shares the advantages of DMB in terms of dependence on network size and communication latency.
Description of the ADMB Algorithm
---------------------------------
We assume that communication between nodes takes place along some bounded-degree acyclic graph. In addition, each node has a unique numerical index. We will generally use $i$ to denote a given node’s index, and let $j$ denote the index of some neighboring node.
Informally, the algorithm works as follows: each node $i$ receives examples, accumulates gradients with respect to its current predictor (which we shall denote as $w_i$), and uses batches of $b$ such gradients to update the predictor. Note that unlike the MaWo-DMB algorithm, here there is no centralized master node responsible for performing the update. Also, for technical reasons, the prediction themselves are not made with the current predictor $w_i$, but rather with a running average $\bar{w}_i$ of predictors computed so far.
Each node occasionally sends its current predictor and accumulated gradients to its neighboring nodes. Given a message from a node $j$, the receiving node $i$ compares its state to the state of node $j$. If $w_i=w_j$, then both nodes have been accumulating gradients with respect to the same predictor. Thus, node $i$ can use these gradients to update its own predictor $w_i$, so it stores these gradients. Later on, these gradients are sent in turn to node $i$’s neighbors, and so on. Each node keeps track of which gradients came from which neighboring nodes, and ensures that no gradient is ever sent back to the node from which it came. This allows for the gradients to propagate throughout the network.
An additional twist is that in the ADMB algorithm, we no longer insist on all nodes sharing the exact same predictor at any given time point. Of course, this can lead to each node using a different predictor, so no node will be able to use the gradients of any other node, and the system will behave as if the nodes all run in isolation. To prevent this, we add a mechanism, which ensures that if a node $i$ receives from a neighbor node $j$ a “better” predictor than its current one, it will switch to using node $j$’s predictor. By “better”, we mean one of two things: either $w_j$ was obtained based on more predictor updates, or $j<i$. In the former case, $w_j,\bar{w}_j$ should indeed be better, since they are based on more updates. In the latter case, there is no real reason to prefer one or the other, but we use an order of precedence between the nodes to determine who should synchronize with whom. With this mechanism, the predictor with the most gradient updates is propagated quickly throughout the system, so either everyone starts working with this predictor and share gradients, or an even better predictor is obtained somewhere in the system, and is then quickly propagated in turn - a win-win situation.
We now turn to describe the algorithm formally. The algorithm has two global parameters:
- $b$: As in the DMB algorithm, $b$ is the number of gradients whose average is used to update the predictor.
- $t$: This parameter regulates the communication rate between the nodes. Each node $i$ will send message to its neighbor every $t$ time–units.
Each node $i$ maintains the following data structures:
- A *node state* $S_i=(w_i,\bar{w}_i,v_i)$, where
- $w_i$ is the current predictor.
- $\bar{w}_i$ is the running average of predictors actually used for prediction.
- $v_i$ counts how many predictors are averaged in $\bar{w}_i$. This is also the number of updates performed according to the online update rule, in order to obtain $w_i$.
- A vector $g_i$ and associated counter $c_i$, which hold the sum of gradients computed from inputs serviced by node $i$.
- For each neighboring node $j$, a vector $g_i^j$ and associated counter $c_i^j$, which hold the sum of gradients received from node $j$.
When a node $i$ is initialized, all the variables discussed above are set to zero, The node then begins the execution of the algorithm. The protocol is composed of executing three event-driven functions: the first function (Algorithm \[alg:asyncfunc\] below) is executed when a new request for prediction arrives, and handles the processing of that example. The second function (Algorithm \[alg:asyncsend\]) is executed every $t$ time–units, and sends messages to the node’s neighbors. The third function (Algorithm \[alg:asyncreceive\]) is executed when a message arrives from a neighboring node. Also, the functions use a subroutine `update_predictor` (Algorithm \[alg:updatepredictor\]) to update the node’s predictor if needed. For simplicity, we will assume that each of those three functions is executed atomically (namely, only one of the function runs at any given time). While this assumption can be easily relaxed, it allows us to avoid a tedious discussion of shared resource synchronization between the functions.
Predict using $\bar{w}_i$ Receive input $z$, suffer loss and compute gradient $\nabla_{w} f(w_i,z)$ $g_i:=g_i+\nabla_{w} f(w_i,z)$ , $c_i:=c_i+1$
For each neighboring node $j'$, send message $\left(i,S_i,g_i+\sum_{j\neq j'}g_i^j,c_i+\sum_{j\neq j'}c_i^j\right)$
Let $(j,S_j,g,c)$ be the received message
use averaged gradient $\frac{g_i+\sum_j g_i^j}{c_i+\sum_j c_i^j}$ to compute updated predictor $w_{i}$ $\bar{w}_i ~:=~ \frac{v_i}{v_i+1} \bar{w}_i+\frac{1}{v_i+1}w_i$ $v_i:= v_i+1$ , $g_i:= 0$ , $c_i:= 0$ $\forall j$ $g_i^j:= 0$ , $c_i^j:= 0$
It is not hard to verify that due to the acyclic structure of the network, no single gradient is ever propagated to the same node twice. Thus, the algorithm indeed works correctly, in the sense that the updates are always performed based on independent gradients. Moreover, the algorithm is well-behaved in terms of traffic volume over the network, since any communication link from node $i$ to node $j$ passes at most $1$ message every $t$ time–units, where $t$ is a tunable parameter.
As with the MaWo-DMB algorithm, the ADMB algorithm has some desirable robustness properties, such as heterogeneous nodes and adding/removing new nodes, and communication latencies. Moreover, it is robust to network failures: even if the the network is split into two (or more) partitions, it only means we end up with two (or more) networks which implement the algorithm in isolation. The system can continue to run and its output will remain valid, although the predictor update rate will become somewhat slower, until the failed node is replaced. Note that unlike the MaWo-DMB algorithm, there is no need to wait until a master node is elected.
Analysis {#subsec:analysis}
--------
We now turn to discuss the regret performance of the algorithm. Before we begin, it is important to understand what kind of guarantees are possible in such a setting. In particular, it is not possible to provide a total regret bound over all the examples fed to the system, since we have not specified what happens to the examples which were sent to malfunctioning nodes - whether they were dropped, rerouted to a different node and so on. Moreover, even if nodes behave properly in terms of processing incoming examples, the performance of components such as interaction with neighboring nodes might vary over time in complex ways, which are hard to model precisely.
Instead, we will isolate a set of “well-behaved” nodes, and focus on the regret incurred on the examples sent to these nodes. The underlying assumption is that the system is mostly functional for most of the time, so the large majority of examples are processed by such well-behaved nodes. The analysis will focus on obtaining regret bounds over these examples.
To that end, let us focus on a particular set of $k'$ nodes, which form a connected component of the communication framework, with diameter $d'$. We will define the nodes as *good*, if all those nodes implement the ADMB algorithm at a reasonably fast rate. More precisely, we will require the following from each of the $k'$ nodes:
- Executing each of the three functions defining the ADMB algorithm takes at most one time–unit.
- The communication latency between two adjacent nodes is at most one time–unit.
- The $k'$ nodes receive at most $M$ examples every time–unit.
As to other nodes, we only assume that the messages they send to the good nodes reflect a correct node state, as specified earlier. In particular, they may be arbitrarily slow or even completely unresponsive.
First, we show that when the nodes are good, up-to-date predictors from any single node will be rapidly propagated to all the other nodes. This shows that the system has good recovery properties (e.g. after most nodes fail).
\[lem:predprop\] Assume that at some time point, the $k'$ nodes are good, and at least one of them has a predictor based on at least $v$ updates. If the nodes remain good for at least $(t+2)d'$ time–units, then all nodes will have a predictor based on at least $v$ updates.
Let $i$ be the node with the predictor having at least $v$ updates. Counting from the time point defined in the lemma, at most $t+2$ time–units will elapse until all of node $i$’s neighbors will receive a message from node $i$ with its predictor, and either switch to this predictor (and then will have a predictor with $v$ updates), or remain with the same predictor (and this can only happen if its predictor was already based on $v$ updates). In any case, during the next $t+2$ time–units, each of those neighboring nodes will send a message to its own neighbors, and so on. Since the distance between any two nodes is at most $d'$, the result follows.
The next result shows that when all nodes are good and have a predictor based on at least $v$ updates, not too much time will pass until they will all update their predictor.
\[thm:fastupdates\] Assume that at some time point, the $k'$ nodes are good, and every one of them has a predictor with $\geq v$ updates (not necessarily the same one). Then after the nodes process at most $$b+2(t+2)d'M$$ additional examples, all $k'$ nodes will have a predictor based on at least $v+1$ updates.
Consider the time point mentioned in the theorem, where every one of the $k'$ nodes, and in particular the node $i_0$ with smallest index among them, has a predictor with $\geq v$ updates. We now claim that after processing at most $$\label{eq:timespan2}
(t+2)d' M$$ examples, either some node in our set had a predictor with $\geq v+1$ updates, or every node has the same predictor based on $v$ updates. The argument is similar to , since everyone will switch to the predictor propagated from node $i_0$, assuming no predictor obtained a predictor with more updates. Therefore, at most $(t+2)d'$ time–units will pass, during which at most $(t+2)d' M$ examples are processed.
So suppose we are now at the time point, where either some node had a predictor with $\geq v+1$ updates, or every node had the same predictor based on $v$ updates. We now claim that after processing at most $$\label{eq:timespan3}
b+(t+2)d'M$$ examples, any node in our set obtained a predictor with $\geq v+1$ updates. To justify , let us consider first the case where every node had the same predictor based on $v$ updates. As shown above, the number of time–units it takes any single gradient to propagate to all $k'$ nodes is at most $(t+2)d'$. Therefore, after $T$ time–units elapsed, each node will accumulate and act upon all the gradients computed by all nodes up to time $T-(t+2)d'$. Since at most $M$ examples are processed each time–unit, it follows that after processing at most $b+(t+2)d'M$ examples, all nodes will update their predictors, as stated in .
We still need to consider the second case, namely that some good node had a predictor with $\geq v+1$ updates, and we want to bound the number of examples processed till all nodes have a predictor with $\geq v+1$ updates. But this was already calculated to be at most $(t+2)d'M$, which is smaller than . Thus, the time bound in covers this case as well.
Adding and , the theorem follows.
With these results in hand, we can now prove a regret bound for our algorithm. To do so, define a *good time period* to be a time during which:
- All $k'$ nodes are good, and were also good for $(t+2)d'$ time–units prior to that time period.
- The $k'$ nodes handled $b+2(t+2)d'M$ examples overall.
As to other time periods, we will only assume that at least *one* of the $k'$ nodes remained operational and implemented the ADMB algorithm (at an arbitrarily slow rate).
\[thm:asyncregret\] Suppose the gradient-based update rule has the serial regret bound ${\psi}({\sigma}^2, m)$, and that for any ${\sigma}^2$, $\frac{1}{m}{\psi}({\sigma}^2,m)$ decreases monotonically in $m$.
Let $m$ be the number of examples handled during a sequence of non-overlapping good time periods. Then the expected regret with respect to these examples is at most $$\sum_{j=1}^{\ceil{m/\mu}}\frac{\mu}{j}{\psi}\left(\frac{{\sigma}^2}{b},j\right),$$ where $\mu=b+2(t+2)d'M$. Specifically, if ${\psi}({\sigma}^2, m)=2D^2{L}+2D{\sigma}\sqrt{m}$, then the expected regret bound is $$2D^2L(b+2(t+2)d'M)(1+\log(m))+
4D\sigma\sqrt{\left(1+\frac{2(t+2)d'M}{b}\right)m}$$
When the batch size $b$ scales as $m^{\rho}$ for any $\rho\in (0,1/2)$, we get an asymptotic regret bound of the form $4D\sigma\sqrt{m}+o(\sqrt{m})$. The leading term is virtually the same as the leading term in the serial regret bound. The only difference is an additional factor of $2$, essentially due to the fact that we need to average the predictors obtained so far to make the analysis go through, rather than just using the last predictor.
Let us number the good time periods as $j=1,2,\ldots$, and let $\bar{w}_j$ be a predictor used by one of the nodes at the beginning of the $j$-th good time period. From and , we know that the predictors used by the nodes were updated at least once during each period. Thus, $\bar{w}_j$ is the average of $j'\geq j$ predictors $w_1,w_2,\ldots,w_{j'}$, where each $w_{p+1}$ was obtained from the previous $w_{p}$ using $b_p\geq b$ gradients each, on some examples which we shall denote as $z_{p,1},z_{p,2},\ldots,z_{p,b_p}$. Since $w_p$ is independent of these examples, we get $$\E\left[\frac{1}{b_p}\sum_{q=1}^{b_p}f(w_p,z_{p,q})-f(w^\star,z_{p,q})~\big| w_p\right] = \E[f(w_p,z)-f(w^\star,z)\big|w_p].$$ Based on this observation and Jensen’s inequality, we have $$\begin{aligned}
&\E\left[f(\bar{w}_j,z)-f(w^\star,z)\right]\notag\\
&\leq \frac{1}{j'}\E\left[\sum_{p=1}^{j'}f(w_p,z)-f(w^\star,z)\right]\notag\\
&= \frac{1}{j'}\E\left[\sum_{p=1}^{j'}\frac{1}{b_p}
\sum_{q=1}^{b_p}f(w_p,z_{p,q})-f(w^\star,z_{p,q})\right].\label{eq:jen}\end{aligned}$$ The online update rule was performed on the averaged gradients obtained from $z_{p,1},\ldots,z_{p,b_p}$. This average gradient is equal to the gradient of the function $\frac{1}{b_p}\sum_{q=1}^{b_p}f(w_p,z_{p,q})$. Moreover, the variance of this gradient is at most ${\sigma}^2/b_p\leq {\sigma}^2/b$. Using the regret guarantee, we can upper bound by $$\frac{1}{j'}{\psi}\left(\frac{{\sigma}^2}{b},j'\right).$$ Since $j'\geq j$, and since we assumed in the theorem statement that the expression above is monotonically decreasing in $j'$, we can upper bound it by $$\frac{1}{j}{\psi}\left(\frac{{\sigma}^2}{b},j\right).$$ From this sequence of inequalities, we get that for *any* example processed by one of the $k'$ nodes during the good time period $j$, it holds that $$\label{eq:epochregret}
\E\left[f(\bar{w}_j,z)-f(w^\star,z)\right]\leq \frac{1}{j}{\psi}\left(\frac{{\sigma}^2}{b},j\right).$$ Let $\mu=b+2(t+2)d'M$ be the number of examples processed during each good time period. Since $m$ examples are processed overall, the total regret over all these examples is at most $$\label{eq:regretfinal}
\sum_{j=1}^{\ceil{m/\mu}}\frac{\mu}{j}{\psi}\left(\frac{{\sigma}^2}{b},j\right).$$
To get the specific regret form when ${\psi}({\sigma}^2,m)=2D^2L+2D\sigma\sqrt{m}$, we substitute into , and substitute $\mu=b+2(t+2)d'M$ to get $$\begin{aligned}
&\sum_{j=1}^{\ceil{m/\mu}}\left(2D^2L\frac{\mu}{j}+\frac{2D\sigma\mu}{\sqrt{b}}
\frac{1}{\sqrt{j}}\right)\\
&\leq 2D^2L\mu(1+\log(m))+
\frac{4D\sigma\mu}{\sqrt{b}}\sqrt{\frac{m}{\mu}}\\
&= 2D^2L(b+2(t+2)d'M)(1+\log(m))+
4D\sigma\sqrt{\left(1+\frac{2(t+2)d'M}{b}\right)m}.\end{aligned}$$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the role of deformation on the fusion probability around the barrier using the Time-Dependent Hartree-Fock theory with a full Skyrme force. We obtain a distribution of fusion probabilities around the nominal barrier due to the different contributions of the various orientations of the deformed nucleus at the touching point. It is also shown that the long range Coulomb reorientation reduces the fusion probability around the barrier.'
author:
- 'C. Simenel'
- 'M. Bender'
- 'Ph. Chomaz'
- 'T. Duguet'
- 'G. de France'
title: 'Quantum calculation of Coulomb reorientation and near-barrier fusion'
---
[ address=[DSM/DAPNIA, CEA SACLAY, F-91191 Gif-sur-Yvette, France]{} ,altaddress=[NSCL, MSU, East Lansing, Michigan 48824, USA]{} ]{}
[ address=[DSM/DAPNIA, CEA SACLAY, F-91191 Gif-sur-Yvette, France]{} ,altaddress=[NSCL, MSU, East Lansing, Michigan 48824, USA]{} ]{}
[ address=[GANIL, B.P. 55027, F-14076 CAEN Cedex 5, France]{} ]{}
[ address=[NSCL, MSU, East Lansing, Michigan 48824, USA]{} ,altaddress=[Physics and Astronomy Department, MSU, East Lansing, Michigan 48824, USA]{} ]{}
[ address=[GANIL, B.P. 55027, F-14076 CAEN Cedex 5, France]{} ]{}
Introduction
============
Fusion of massive nuclei has recently drawn a lot of interest, especially at energies around the fusion barrier generated by the competition between the Coulomb and nuclear interactions. In this energy domain, the reaction mechanisms may depend strongly on the structure of the collision partners. The proper description of near-barrier fusion is thus a challenging N-body quantum dynamical problem involving the competition between various reaction channels. For example, the coupling between the internal degrees of freedom and the relative motion may generate a fusion barrier distribution [@das83]. Such couplings are needed to reproduce the sub-barrier fusion [@das98].
One of the internal degrees of freedom which can strongly affect the fusion is the static deformation [@rie70; @jen70]. First, the fusion probability depends on the orientation of the deformed nucleus at the touching point. Second, a reorientation can occur under the torque produced by the long-range Coulomb force [@hol69; @wil67; @sim04; @uma06b]. Such a reorientation is a consequence of the excitation of rotational states. It induces an anisotropy in the orientation distribution, thus modifying the near-barrier fusion [@bab00].
In this work we study the fusion of a spherical and a prolate deformed nucleus within the Time-Dependent Hartree-Fock (TDHF) theory. We first show the effect of the orientation at the touching point on the fusion probability. Then we include the long range Coulomb excitation of rotational states and study its effect on fusion. The results give a useful interpretation of full coupling channels calculations.
Time-Dependent Hartree-Fock theory
==================================
Let us first recall briefly some aspects of TDHF theory and of its numerical applications to nuclear collisions. TDHF is a mean field quantum dynamical theory [@har28; @foc30; @dir30] . It describes the evolution of occupied single particle wave functions in the mean field generated by all the particles. The total wave function of the system is constrained to be a Slater determinant at any time which assures an exact treatment of the Pauli principle during the dynamics. All standard applications of TDHF neglect pairing correlations so far. Like all mean-field methods, TDHF is best suited to desctibe average values of one-body operators. Such quantities are determined from the one-body density matrix $
{\hat{\rho}}=\sum_{n=1}^{N}{\left|}{\varphi}_{n}{\rangle}{\langle}{\varphi}_{n}{\right|}$ where ${|}{\varphi}_n {\rangle}$ denotes an occupied single particle state. In TDHF, its evolution is determined by a Liouville-von Neumann equation, $
i{\hbar}\partial_t {\hat{\rho}}=[{\hat{h}}({\rho}),{\hat{\rho}}]
$ where ${\hat{h}}({\rho})$ is the mean-field Hamiltonian.
The great advantage of TDHF is that it treats the static properties [*and*]{} the dynamics of nuclei within the same formalism, i.e. using the same effective interaction (usually of the Skyrme type [@sky56]). The initial state is obtained through static Hartree-Fock (HF) calculations which are known to reproduce rather well nuclear binding energies and deformations. TDHF can be used in two ways to describe nuclear reactions:
- A single nucleus is evolved in an external field [@vau72], simulating for instance the Coulomb field of the collision partner [@sim04].
- The evolution of two nuclei, initially with a zero overlap, is treated in the same box with a single Slater determinant [@bon76; @neg82].
The first case is well suited for the description of inelastic scattering, like Coulomb excitation of vibrational and rotational states. The second case is used for more violent collisions like deep-inelastic and fusion reactions. In such cases, the lack of a collision term in TDHF might be a drawback. At low energy, however, the fusion is mainly driven by the one-body dissipation because the Pauli blocking prevents nucleon-nucleon collisions. The system fuses mainly by transfering relative motion into internal excitation via one-body mechanisms well treated by TDHF.
Another important advantage of TDHF concerning its application to near-barrier reaction studies is that it contains implicitely all types of couplings between the relative motion and internal degrees of freedom whereas in coupling channels calculations one has to include them explicitely according to physical intuition which is not always straightforward for complex mechanisms. The only condition in TDHF is that the symmetries corresponding to the internal degrees of freedom of interest are relaxed. This is now the case with the latest TDHF codes in 3 dimensions (3D) which use a full Skyrme force [@kim97; @uma06a]. However, TDHF gives only classical trajectories for the time-evolution and expectation values of one-body observables. In particular, TDHF does not include tunneling of the global wave function.
We use the TDHF code built by P. Bonche and coworkers [@kim97] using a Skyrme functional [@sky56]. This code computes the evolution of each occupied single-particle wave function in a 3D box assuming one symmetry plane. The step size of the network is 0.8 fm and the step time 0.45 fm/c. We use the SLy4$d$ parametrization [@kim97] of the Skyrme force which is a variant of the SLy4 one specifically designed for TDHF calculations.
Fusion with a deformed nucleus
==============================
Effect of the static deformation
--------------------------------
Many nuclei exhibit static deformation, that is well described by mean-field calculations. Static deformation breaks the rotational invariance of the Slater determinant, which introduces an intrinsic frame of the nucleus. TDHF calculations of nuclear collisions, however, are performed in the laboratory frame, and one is left with an ambiguity concertning the relative orientation of the deformed nuclei. This is a critical point, because different orientations might ultimately lead to different reaction paths.
To illustrate this point we consider central collisions of a prolate deformed $^{24}$Mg ($\beta_2=0.4$) with a spherical $^{208}$Pb. For symmetry reasons, the reaction mechanism will depend only on the energy and the angle between the deformation axis and the collision axis noted ${\varphi}$. Fig. \[fig:Tall\] shows the time evolution of the density for two different initial orientations. We see that with an initial orientation ${\varphi}= 0 {{^\circ}}$ the nuclei fuse whereas with ${\varphi}= 37.5 {{^\circ}}$ the two fragments separate after a deep-inelastic collision.
![Density plots of head-on $^{208}$Pb+$^{24}$Mg collisions at $E_{CM}=95$ MeV with an initial orientation at 20 fm of 0$^\circ$ (left) and 37.5$^\circ$ (right). The time step between each figure is 135 fm/c.[]{data-label="fig:Tall"}](./Tall.eps){height=".62\textheight"}
The technic used to overcome the ambiguity of the initial orientation is based on two prescriptions [@sim04; @uma06b]:
1. It is necessary to assume an initial distribution of orientations.
2. Interferences between different orientations are neglected. Then each Slater determinant evolves in its own mean field.
Let us first assume an isotropic distribution of the orientations at the initial time, corresponding to a distance $D=20$ fm between the two centers of mass. This means that the $^{24}$Mg is supposed to be initially in its $0^+$ ground state and that all kind of long range Coulomb excitations are neglected up to this distance. Then, using the above prescriptions we get the fusion probability $$P_{fus}(E)=\frac{1}{2}\int_0^\pi {\!\!\!}{{\mbox d}}{\varphi}{\,\,\,}\sin {\varphi}{\,\,\,}P_{fus}(E,{\varphi})$$ where $P_{fus}(E,{\varphi})=0$ or 1. The solid line in Fig. \[fig:pfus\_e\]-a shows the resulting fusion probability as function of the center of mass energy. Below 93 MeV no orientation leads to fusion and above 99 MeV, all of them fuse. Between these two values, the higher the energy, the more orientations lead to fusion. As shown in Fig. \[fig:Tall\], configurations with small ${\varphi}$ are the first to fuse, even below the nominal barrier which would correspond to a spherical $^{24}$Mg case (dotted line). To conclude, sub-barrier fusion is described in TDHF through couplings between static deformation and relative motion.
![Fusion probability for a head-on collision or “penetrability” of $^{208}$Pb+$^{24}$Mg as a function of the c.m. energy. $a)$ TDHF results. Isotropic distribution of the initial orientations is supposed at 20 fm (solid line) and at 220 fm (dashed line). Step function expected in case of a spherical $^{24}$Mg (dotted line). $b)$ CCFULL results without (solid line) and with (dashed line) Coulomb excitation.[]{data-label="fig:pfus_e"}](./pfus_e){height=".2\textheight"}
Long range Coulomb reorientation
--------------------------------
As a consequence of numerical limitations, actual TDHF calculations for collisions are performed in relatively small boxes and are started with internuclear distances of a few Fermi. However, the Coulomb interaction starts playing a role much earlier in the reaction process [@alder]. It is weaker than the nuclear interaction, but integrated over a long time it may induce a polarization, and then modify the reaction mechanism [@sim04; @row06]. Long range Coulomb reorientation has been studied in Ref. [@sim04] with TDHF. The results have been interpreted using the classical formalism [@alder; @broglia] where the motion of a deformed rigid projectile is considered in the Coulomb field of the target. An important conclusion of this work is that the reorientation, although being the result of a Coulomb effect, depends neither on the charges nor on the energy. Let us illustrate this phenomenon with a simple example. Consider a system at time $t$ formed by a deformed projectile at the distance $D(t)$ from the target. Increasing artificially the charge of one of the nuclei at this time has two effects. First, the Coulomb interaction increases and then the torque applied on the deformed nucleus should [*increase*]{} too. On the other hand, the distance $D(t)$ between the projectile and the target is larger because of the stronger Coulomb repulsion between the centers of mass. The latter effect leads to a [*decrease*]{} of the effective torque at time $t$ and both effects overall cancel exactly. One is left with a charge independent reorientation. The same argument applies for the energy.
To study the effect of reorientation on fusion we calculate the reorientation in the approach phase between $D=220$ fm and 20 fm with TDHF using the technic described in Ref. [@sim04]. Assuming an isotropic distribution of orientations at 220 fm we get a new distribution at 20 fm which includes the reorientation coming from long range Coulomb excitation. The new fusion probability distribution (dashed line in Fig. \[fig:pfus\_e\]-a) is obtained with two additional assumptions:
- The rotational speed of the $^{24}$Mg is neglected at the initial time of the TDHF calculation (corresponding to $D=20$ fm), i.e. only a static reorientation is considered.
- The effect of the excitation energy on the relative motion is neglected, i.e. we assume a Rutherford trajectory before $D=20$ fm.
We observe in Fig. \[fig:pfus\_e\]-a a fusion hindrance up to 20$\%$ which is due to higher weights on orientations leading to compact configurations (${\varphi}\sim 90{{^\circ}}$ at the touching point) because of the reorientation [@sim04; @row06].
The previous study is helpful to interpret coupling channels results. Calculations on the same system have been performed with the code CCFULL [@hag99] including coupling to the five first excited states of $^{24}$Mg rotational band. The fusion probability, or “penetrability” of the fusion barrier is given by the relation $ P_{fus}=\frac{{{\mbox d}}{\left(}{\sigma}E{\right)}}{{{\mbox d}}E}\pi
R_B^2 $ where $R_B=11.49$ fm is the barrier position. Fig. \[fig:pfus\_e\]-b shows the fusion probability obtained from CCFULL including nuclear (solid line) and nuclear+Coulomb (dashed line) couplings. As with TDHF, an hindrance of the fusion due to Coulomb couplings is observed. However the shape of TDHF and CCFULL distributions are quite different. This is due to the fact that quantum mechanical effects are missing in TDHF. This point out the importance of improving the theory. It is also striking to see that TDHF “misses” the nominal barrier by about 15$\%$. TDHF is known to overestimate the fusion cross sections. One possible issue might be the time odd terms in the Skyrme energy functional. Their importance on fusion have been stressed recently [@uma06a; @mar06].
conclusion
==========
To summarize, we performed a TDHF study of near-barrier fusion between a spherical and a deformed nucleus. The calculations show that, around the barrier, different orientations lead to different reaction path. Considering all possible orientations leads to a distribution of fusion probabilities interpreted as an effect of the coupling between the static deformation and the relative motion. We then included the long range Coulomb coupling which induces a [*charge*]{} and [*energy*]{} independent reorientation of the deformed nucleus. The effect of the reorientation is to hinder the near-barrier fusion. Finally the TDHF study have been used to interpret coupling channels calculations with the code CCFULL which show also an hindrance of near-barrier fusion due to Coulomb couplings. We also note some drawbacks of TDHF which, in one hand, underestimates the fusion barrier, and, in the other hand, miss important quantum effects.
We warmly thank Paul Bonche for providing his TDHF code. This work has been partially supported by NSCL, Michigan State University and the National Science Foundation under the grant PHY-0456903.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The seesaw mechanism to derive the light masses of left-handed neutrinos using heavy masses of right-handed neutrinos gives rise to a connection between low-energy measurables and GUT-scale mechanism. We expresses the neutrino mixing angles in terms of a single variable $\sin\theta_{13}$, whose size was measured recently. The lepton asymmetry from heavy neutrinos via Yukawa coupling is described by CP phases in both Dirac and Majorana type. It is shown that the seesaw scale relevant to the lepton asymmetry can be constrained by CP phase in this minimal model.
PACS numbers
: 11.30.Fs, 14.60.Pq, 14.60.St
Keywords
: leptogenesis, neutrino mass, CP asymmetry, seesaw mechanism
author:
- Kim Siyeon
date: 'November 14, 2016'
nocite: '[@*]'
title: Seesaw Scale and CP Phases in a Minimal Model of Leptogenesis
---
\[sec:level1\]Introduction
==========================
The transformation between three active neutrinos of Standard Model(SM) and massive neutrinos is almost understood from the measurements of three mixing angles except a CP phase [@atm][@SNO][@An:2012eh][@Ahn:2012nd]. The current global analysis presents the following best fits: $\sin^2\theta_{12}=3.08\times 10^{-1}, ~\sin^2\theta_{23}=4.37\times 10^{-1},$ and $\sin^2\theta_{13}=2.34\times 10^{-2}$ for normal hierarchy mass ordering(NH), i.e., assuming $m_1 < m_2 < m_3$ [@Beringer:1900zz][@Capozzi:2013csa][@Tortola:2012te]. The current knowledge on neutrino masses is still nothing but the mass-squared differences, $m_3^2-m_1^2=2.43\times 10^{-3}\mathrm{eV}^2$ and $m_2^2-m_1^2=7.54\times 10^{-5}\mathrm{eV}^2$ for atmospheric neutrinos and solar neutrinos, respectively. Recent measurements of the third mixing angle $\theta_{13}$ have been naturally followed by search of the CP phase in Pontecorvo-Maki-Nakagawa-Sakata(PMNS) . The CP conservation at $3\sigma$ confidence level(CL) has been excluded by the result of T2K [@Escudero:2016odp]. A next-generation oscillation experiment is rushing to narrow down the range of $\delta$, the CP phase of PMNS matrix [@Acciarri:2015uup].
Leptogenesis is a theory regarded as an explanation of the Baryon-antibaryon Asymmetry in Universe(BAU) [@lep][@Harvey:1990qw][@Kolb:qa], in which an indirect test is possible. If the decays of heavy Majorana neutrinos via Yukawa couplings are the sources of leptonic CP violation and the heavy neutrinos are the elements in seesaw mechanism [@Barger:2003gt][@Luty:un], some parameters can be tested phenomenologically [@Endoh:2002wm]. It is worthwhile to examine the implication of the sizes of $\theta_{13}$ and $\delta$ in a canonical leptogenesis model. Seesaw mechanism is a top-down approach in which the light neutrino masses are obtained through Grand Unified Theory(GUT)-scale mechanism [@Gell-Mann:vs]. In this work, bottom-up approach is taken to probe the lepton asymmetry of heavy-neutrino decays in high-energy scale by using the masses and the mixing angles of light neutrinos in low energy.
In order to see the effect from the definite size of $\theta_{13}$ on the high-energy asymmetry, all the elements of PMNS matrix are expressed in terms of a single variable $\sin\theta_{13}$ except CP phase. One of the minimal choices is made for seesaw mechanism, which is that Yukawa matrix is constructed only with two right-handed neutrinos. We found the way to express $3 \times 2$ Dirac mass matrix by matching the light neutrino mass matrix obtained from the seesaw mechanism to the low energy neutrino mass matrix in weak interaction basis. Our previous work on the seesaw mechanism in bottom-up approach was also based on $3 \times 2$ structure for Dirac mass, although a texture zero in matrix was forced [@Lee:2005cda]. This work is outlined as follows: The neutrino mass matrix is constructed with masses and unitary transformation in Section II. We derive a mass matrix using the seesaw mechanism and CP asymmetry from the decay of right-handed neutrino via Yukawa coupling in Section III. In Section IV, We examine the dependency of the lepton asymmetry on low-energy parameters. It is worthwhile to discuss the correlation of the lepton asymmetry with CP phases in PMNS matrix. In conclusion, we summarize the relation of the lepton asymmetry with low-energy measurables and the possibility to narrow down the models of seesaw mechanism.
Low-energy constraint
=====================
The PMNS mixing matrix for 3 generations of Majorana neutrinos is given by $$\begin{aligned}
\tilde{U} &=& R\left(\theta_{23}\right)
R\left(\theta_{13},\delta\right)
R\left(\theta_{12}\right)P(\varphi_2, \varphi_3)
\label{fulltrans}
\end{aligned}$$ where each $R$ is a rotation matrix with a mixing angle $\theta_{ij}$ between $i$-th and $j$-th generations. According to the standard parametrization, the Dirac phase is combined with $\theta_{13}$ as $R\left(\theta_{13},\delta\right)$ in the PMNS matrix. A diagonal phase transformation $P(\varphi_2, \varphi_3)$ is given by Diag$\left(1,e^{i\varphi_2/2},e^{i\varphi_3/2}\right)$. The Majorana phases $\varphi_2$ and $\varphi_3$ can be a part of the mass matrix of light neutrinos in the following way: $$M_{\nu} = U Diag(m_1,\check{m}_2,\check{m}_3) U^T,
\label{umu}$$ where $U \equiv \tilde{U}P^{-1}$, $\check{m}_2 \equiv
m_2e^{i\varphi_2}$ and $\check{m}_3 \equiv m_3e^{i\varphi_3}$. The standard parametrization for Cabibbo-Kobayashi-Maskawa(CKM) matrix can be taken for transformation matrix $U$ in the PMNS such as
$$\begin{aligned}
U=
\left(\begin{array}{ccc}
c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta}\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta} &
c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta} &
s_{23}c_{13} \\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta} &
-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta} &
c_{23}c_{13}
\end{array}\right),\label{ckm}
\end{aligned}$$
where $s_{ij}$ and $c_{ij}$ denotes $\sin{\theta_{ij}}$ and $\cos{\theta_{ij}}$.
The measurement of $\theta_{13}$ from recent reactor anti-neutrino detection allows the following conditions: $$\begin{aligned}
s_{12}=\frac{1}{\sqrt{2}}-s_{13}, \label{s12}\\
s_{23}=\frac{1}{\sqrt{2}}-s_{13}^2. \label{s23}
\end{aligned}$$ Such deviation from bi-maximal mixing scheme was introduced in a number of models. By substituting Eq.(\[s12\]) and Eq.(\[s23\]) into Eq.(\[ckm\]) the neutrino mixing matrix in the leading order can be written as; $$\begin{aligned}
U &\approx&
\left(\begin{array}{ccc}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\
-\frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}} \\
\frac{1}{2} & -\frac{1}{2} & \frac{1}{\sqrt{2}}
\end{array}\right) \label{xorder} \\
&+&
s_{13} \left( \begin{array}{ccc}
1 & -1 & e^{-i\delta} \\
\frac{1}{\sqrt{2}}-\frac{1}{2}e^{+i\delta} & \frac{1}{\sqrt{2}}-\frac{1}{2}e^{+i\delta} & 0 \\
-\frac{1}{\sqrt{2}}-\frac{1}{2}e^{+i\delta} & -\frac{1}{\sqrt{2}}-\frac{1}{2}e^{+i\delta} & 0
\end{array} \right)
+ \mathcal{O}(s_{13}^2). \nonumber\end{aligned}$$ Suppose $m_1=0$ in an extreme case of normal hierarchical mass ordering. Since only the difference of each Majorana phase relative to overall phase becomes physical, one of the Majorana phases in Eq.(\[fulltrans\]) can be removed due to vanishing $m_1$. The symmetric neutrino mass matrix in Eq.(\[umu\]) appears as
$$\begin{aligned}
M_\nu = && ~ \tilde{m}_3 \left(
\begin{array}{ccc}
s_{13}^2 e^{-2 \imath\delta} & \frac{1}{\sqrt{2}} s_{13} e^{-\imath\delta} & \frac{1}{\sqrt{2}} s_{13} e^{-\imath\delta} \\
\checkmark & \frac{1}{2} & \frac{1}{2} \\
\checkmark & \checkmark & \frac{1}{2} \\
\end{array}\right) ~+ \label{lowmass3} \\
&& m_2 \left(\begin{array}{ccc}
\frac{1}{2}- \sqrt{2} s_{13} ~ & \frac{1}{2\sqrt{2}}\left(1 - \sqrt{2}s_{13} \right) \left(1 - \sqrt{2}s_{13} + s_{13} e^{\imath\delta}\right) & -\frac{1}{2\sqrt{2}}\left(1 - \sqrt{2}s_{13} \right) \left(1- \sqrt{2}s_{13} + s_{13} e^{\imath\delta}\right) \\
\checkmark & \frac{1}{4} \left(1 - \sqrt{2}s_{13} + s_{13} e^{\imath\delta}\right)^2 & -\frac{1}{4}\left(1 - \sqrt{2}s_{13} \right) \left(1+ \sqrt{2}s_{13} + s_{13} e^{\imath\delta}\right) \\
\checkmark & \checkmark & \frac{1}{4}\left(1+ \sqrt{2}s_{13} + s_{13} e^{\imath\delta}\right)^2
\end{array} \right), \nonumber
\end{aligned}$$
where $\tilde{m}_3 \equiv m_3e^{i\varphi}$ with $\varphi=\varphi_3-\varphi_2$.
The quantities related to the imaginary part of PMNS are Jarlskog invariant $J_{CP}$ and effective electron neutrino mass $<m_{ee}>$. Jarlskog invariant evaluates the magnitude of CP asymmetry from Dirac phase, which can be estimated from oscillation probability in appearance experiments. Thus, from $U$ in Eq.(\[xorder\]), the size of $J_{CP}$ is expressed in a simple combination of $s_{13}$ and Dirac phase $\delta$ as follows, $$\begin{aligned}
J_{CP}&=&\mathrm{Im}[U_{e1}U_{\tau 3}U_{\tau 1}^*U_{e3}^*] \\
&=& \frac{1}{4}s_{13}\sin\delta - \frac{1}{2}s_{13}^3\sin\delta + \mathcal{O}(s_{13}^5),
\end{aligned}$$ The effective electron neutrino mass $<m_{ee}>$ is the only neutrino-dependent factor to determine the amplitude of neutrinoless double beta decay. $$\begin{aligned}
<m_{ee}> &\equiv& |\sum^3_{i=1}U_{ei}^2m_ie^{i\varphi_i}| \\
&=& m_2\left( \frac{1}{2}-\sqrt{2}s_{13} + s_{13}^2 \right) \\
&+& m_3\left( \cos\left(2\delta+\varphi\right)s_{13}^2 +\frac{m_3}{m_2}\sin^2\left(2\delta+\varphi\right)s_{13}^4 \right), \nonumber
\end{aligned}$$ Since the $<m_{ee}>$ of normal hierarchical masses is far below the sensitivity pursued by on-going neutrinoless double-beta decay experiments, we do not discuss it any longer in this work.
Yukawa Interaction and Seesaw Mechanism
=======================================
Canonical model of seesaw mechanism to suppress neutrino masses below 1 eV using heavy neutrino mass scale requires $SU(2)$ Higgs doublet, whose existence was definitely confirmed. The lagrangian density for seesaw mechanism consists of Yukawa couplings of leptons and lepton-number-violating Majorana mass terms of right-handed heavy neutrinos. $$\begin{aligned}
- \mathcal{L} = H \mathcal{Y}_\ell L_e \bar{e}_R
+ H \mathcal{Y}_\nu L_e \bar{N}_R + \frac{1}{2} M_R N_R N_R,
\label{lagrangian}
\end{aligned}$$ The minimal model that assumes CP violation from heavy neutrino decay through Yukawa coupling requires at least two right-handed neutrinos. We consider the following particle contents: $N_R=(N_1, N_2)$ in the basis mass matrix $M_R$ is diagonal, and $\nu_l = (\nu_e, \nu_\mu, \nu_\tau)$ in the basis Yukawa coupling of charged leptons $\mathcal{Y}_\ell$ is diagonal. The $\mathcal{L}$ consists of a $3 \times 3$ matrix $\mathcal{Y}_\ell$, a $3 \times
2$ matrix $\mathcal{Y}_\nu$ and a $2 \times 2$ matrix $M_R$, which naturally result in zero mass for one of light neutrinos through the seesaw mechanism[@Gell-Mann:vs], $M_\nu = - v^2 \mathcal{Y}_\nu M_R^{-1} \mathcal{Y}_\nu^T$ in top-down approach.
In bottom-up approach, it is possible to trace the matrix $\mathcal{Y}_\nu$ in terms of light neutrino masses and mixing angles using the seesaw mechanism in opposite direction. When the $3 \times 2$ Yukawa matrix is given by $$\mathcal{Y}_\nu \equiv
\left(
\begin{array}{cc}
y_{11} & y_{12} \\
y_{21} & y_{22} \\
y_{31} & y_{32}
\end{array} \right),
\label{dirac}$$ the Yukawa couplings are imbedded in symmetric neutrino mass matrix in the following way: $$\begin{aligned}
M_\nu &=& \frac{v^2}{M_1}
\left(\begin{array}{ccc}
y_{11}^2 & y_{11}y_{21} & y_{11}y_{31} \\
\checkmark & y_{21}^2 & y_{21} y_{31} \\
\checkmark & \checkmark & y_{31}^2
\end{array}\right) \nonumber \\
&+& \frac{v^2}{M_2}
\left(\begin{array}{ccc}
y_{12}^2 & y_{12} y_{22} & y_{12} y_{32} \\
\checkmark & y_{22}^2 & y_{22} y_{32} \\
\checkmark & \checkmark & y_{32}^2
\end{array}\right)\label{m2seesaw},
\end{aligned}$$ where $M_1$ and $M_2$ are the masses of $N_1$ and $N_2$, respectively, and $v$ is the vacuum expectation value of $H$.
In comparison of the $M_\nu$ in terms of low-energy physical parameters Eq.(\[lowmass3\]) with the $M_\nu$ in terms of Yukawa couplings Eq.(\[m2seesaw\]), we obtained the following relations; $$\begin{aligned}
&& \left( \begin{array}{c} y_{11} \\ y_{21} \\ y_{31} \end{array} \right)=
\sqrt{\frac{m_3}{m_2}}e^{\imath\varphi/2} \left( \begin{array}{l} s_{13} e^{-\imath\delta} \\
\frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} \end{array} \right) \label{yuk11} \\
&& \left( \begin{array}{c} y_{12} \\ y_{22} \\ y_{32} \end{array} \right)=
\sqrt{\frac{M_2}{M_1}} \left( \begin{array}{l}
\frac{1}{\sqrt{2}} - s_{13} \\
\frac{1}{\sqrt{2}} \left( \frac{1}{\sqrt{2}}-s_{13} + \frac{1}{\sqrt{2}}s_{13} e^{\imath\delta}\right) \\
-\frac{1}{\sqrt{2}} \left( \frac{1}{\sqrt{2}}+s_{13} + \frac{1}{\sqrt{2}}s_{13} e^{\imath\delta}\right)\end{array} \right). \label{yuk12}
\end{aligned}$$ All Yakawa couplings are determined in terms of 5 physical values: $s_3, \delta, \varphi,\sqrt{m_3/m_2}$, and $\sqrt{M_2/M_1}$, implying there can be a way to test the model and constrain the scale of seesaw mechanism from experimental measurements.
Once we have the neutrino Yukawa matrix with the couplings in Eq.(\[yuk11\]) and Eq.(\[yuk12\]), we can calculate the magnitude of CP asymmetry $\epsilon_i$ in decays of heavy Majorana neutrinos [@Kolb:qa; @Luty:un], $$\begin{aligned}
\epsilon_i
&=&\frac{\Gamma (N_i \to \ell H)
- \Gamma (N_i \to \bar{\ell} H^*)}
{\Gamma (N_i \to \ell H)
+ \Gamma (N_i \to \bar{\ell} H^*)},
\label{aacp}\end{aligned}$$ where $i$ denotes a generation. If the scale of $M_1$ is far below $M_2$, the CP asymmetry is solely obtained from the decay of $M_1$ [@Kolb:qa; @Luty:un]. Hence, the asymmetry is replaced by just $\epsilon_1$ whose magnitude is given by $$\begin{aligned}
\epsilon_1 &=& \frac{1}{8\pi}
\frac{{\rm Im}\left[(\mathcal{Y}_\nu^\dagger \mathcal{Y}_\nu)_{12}^2\right]
} {(\mathcal{Y}_\nu^\dagger \mathcal{Y}_\nu)_{11}}
f\left(\frac{M_2}{M_1}\right)\;, \label{cp1}\end{aligned}$$ where $f\left(M_2/M_1\right)$ represents loop contribution to the decay width from vertex and self energy. In the Standard Model, it is given by $$f(x) = x\left[1-(1+x^2)\ln \frac{1+x^2}{x^2} +
\frac{1}{1-x^2}\right],$$ which can be approximated to $3/2x$ for sufficiently large $x$. Then the asymmetry $\epsilon_1$ in Eq.(\[cp1\]) can be written as $$\begin{aligned}
&& \epsilon_1 = \frac{3}{16\pi}\Delta_1 \frac{M_1}{M_2}, \label{finalCP}
\end{aligned}$$ for $M_1 \ll M_2$. The $\Delta_1$ factor that depends on Yukawa couplings is $$\begin{aligned}
&& \Delta_1 \equiv \frac{{\rm Im}\left[(\mathcal{Y}_\nu^\dagger \mathcal{Y}_\nu)_{12}^2\right]}
{(\mathcal{Y}_\nu^\dagger \mathcal{Y}_\nu)_{11}} \label{delta1} \\
&& = \frac{Im \left[( y_{11}^* y_{12} + y_{21}^* y_{22} + y_{31}^* y_{32})^2\right]}
{|y_{11}|^2 + |y_{21}|^2 +|y_{31}|^2 },
\nonumber \\
&& = \frac{\mu^2 s_{13}^2}{1+s_{13}^2}
\left( \sqrt{2}\sin\left(\varphi-\delta\right)-\frac{1}{2}\sin\left(\varphi-2\delta\right)-\sin\varphi \right), \nonumber \end{aligned}$$ with $$\begin{aligned}
\mu^2=\frac{M_2}{M_1}.
\end{aligned}$$ Then, the ratio of heavy Majorana neutrino masses is eliminated in $\epsilon_1$, so that one can have $\epsilon_1=\epsilon_1(\delta, \varphi, s_{13})$ regardless of the relative mass scale of heavy neutrinos.
Lepton Asymmetry
================
The baryon density of our universe $\Omega_B h^2 = 0.02240$ from nine-year Wilkinson Microwave Anisotropy Probe(WMAP) data indicates the observed baryon asymmetry in the Universe[@WMAP1], $$\eta_B^{CMB}=
\frac{n_B-n_{\bar{B}}}{n_\gamma}= 6.5 \times 10^{-10},
\label{baryon}$$ where $n_B, n_{\bar{B}}$ and $n_\gamma$ are number density of baryon, anti-baryon and photon, respectively. The baryon asymmetry Eq.(\[baryon\]) can be rephrased by $$Y_B = \frac{n_B-n_{\bar{B}}}{s} \simeq
9.8\times 10^{-11}. \label{cosmo}$$ with the entropy density $s$ in order to take into account the number density with respect to a co-moving volume element. The baryon asymmetry produced through sphaleron process is related to the lepton asymmetry [@Harvey:1990qw] by $Y_B = \frac{a}{a-1} Y_L$ with $ a \equiv (8 N_F + 4 N_H) / (22 N_F + 13 N_H)$. For the Standard Model(SM), $a=28/79$ with three generations of fermions and a single Higgs doublet. The lower bound of $Y_B=9.8\times 10^{-11}$ in Eq.(\[cosmo\]) can be replaced by $Y_L=1.8 \times 10^{-10}$. The generation of a lepton asymmetry requires the CP-asymmetry and out-of-equilibrium condition. The $Y_L$ is explicitly parameterized by two factors, $\epsilon$, the size of CP asymmetry, and $\kappa$, the dilution factor from washout process. $$\begin{aligned}
Y_L = \frac{(n_L - n_{\overline{L}})}{s} = \kappa
\frac{\epsilon_i}{ g^*} \label{aalepto}\end{aligned}$$ where $g^*\simeq 110$ is the number of relativistic degree of freedom.
The $\kappa$ in Eq.(\[aalepto\]) is determined by solving the full Boltzmann equations. The $\kappa$ can be simply parameterized in terms of $K$, which is defined as the ratio of $\Gamma_1$ the tree-level decay width of $N_1$ to $H$ the Hubble parameter at temperature $M_1$, $\Gamma_1 / H$. The condition $K<1$ describes processes out of thermal equilibrium and $\kappa<1$ describes washout effect[@Harvey:1990qw], $$\begin{aligned}
\kappa = \frac{0.3}{K \left(\ln K \right)^{0.6}}
~&~\rm{for}~&~ 10 \lesssim K \lesssim 10^6, \label{largek} \\
\kappa = \frac{1}{2 \sqrt{K^2+9}}
~&~\rm{for}~&~ 0 \lesssim K \lesssim 10. \label{smallk}\end{aligned}$$ The decay width of $N_1$ by the Yukawa interaction at tree level and Hubble parameter in terms of temperature $T$ and the Planck scale $M_{pl}$ are $\Gamma_1 = (\mathcal{Y}_\nu^\dagger \mathcal{Y}_\nu)_{11}
M_1 / (8 \pi) $ and $H = 1.66 g^{1/2}_* T^2 / M_{pl}$, respectively. At temperature $T = M_1$, the ratio $K$ is $$\begin{aligned}
K = \frac{M_{pl}}{1.66 \sqrt{g^*}(8 \pi)}
\frac{(\mathcal{Y}_\nu^\dagger \mathcal{Y}_\nu)_{11}}{M_1}, \label{yukawak}
\end{aligned}$$ which reduces to $$\begin{aligned}
K = \frac{M_{pl}/M_1}{1.66 \sqrt{g^*}(8 \pi)} \frac{m_3}{m_2},
\label{kay}
\end{aligned}$$ since $(\mathcal{Y}_\nu^\dagger
\mathcal{Y}_\nu)_{11}=\sum|y_{i1}|^2 = m_3/m_2 $ up to the leading order of $s_{13}$ from Eq.(\[yuk11\]), Then the dilution factor $\kappa$ depends on the relative scale of $M_{pl}/M_1$ and the ratio of light masses $m_3/m_2$, so that $$\begin{aligned}
Y_L =
\frac{1}{g^*}~\kappa~(\frac{M_{pl}}{M_1},\frac{m_3}{m_2})
~\epsilon_1(\delta,\varphi,s_{13}), \label{YL_inputs}
\end{aligned}$$ which implies that the thermal effect is a matter of mass scales while the CP asymmetry is a matter of complex mixing of particles.
In this framework of bottom-up approach, some of low energy quantities can be fixed by best fit in global analysis, $$\begin{aligned}
\frac{m_2}{m_3} &=& \sqrt{\frac{\Delta m_{21}^2}{\Delta m_{31}^2}} ~\approx~ 0.176, \\
s_{13} &=& \sin\theta_{13} ~\approx~ 0.157, \label{bestfit}
\end{aligned}$$ where their $3\sigma$-ranges are given by $(0.164~-~0.192)$ and by $(0.133~-~0.171)$, respectively. The curves in Fig. \[fig2:fig\_theta13\_contours\] present the contours of lepton asymmetry $Y_L=\pm 1.8\times 10^{-10}$ for $m_2/m_3$ given in Eq.(\[bestfit\]) and a selected scale of $M_1/M_{pl}$. The region below the size of $Y_L$ derived from the cosmological bound $Y_B$ is shaded, excluding the corresponding combination of $\varphi$ and $\delta$ . Both signs of asymmetry are taken into account. It is clear that the amount of lepton asymmetry cannot be reached to the cosmological bound without Dirac phase, although they are not proportional to each other. In this model, $\delta=0$ is excluded regardless other variables in Eq.(\[YL\_inputs\]) as long as their values are considered within phenomenologically allowed ranges. It is also shown that the variation of $\sin\theta_{13}$ within $3\sigma$ range does not draw a remarkable change in lepton asymmetry $Y_L$, and neither does the variation of $m_2/m_3$ in $3\sigma$. The changes in $Y_L$ affected by $m_2/m_3$ is barely visible. The experiments of long-baseline oscillation began obtaining results on Dirac CP phase. The red shades in Fig.\[fig2:fig\_theta13\_contours\] and Fig.\[fig3:fig\_seesaw\_scale\] present the ranges of $\delta$ at 68% and 90% CL obtained by T2K with best fit $-0.5\pi$. The CP conservation is ruled out at 90% CL.
The amount of lepton asymmetry in Eq.(\[aalepto\]) is now given as a function of $\delta$ and $\varphi$ as well as $M_1/M_{pl}$ such as $$\begin{aligned}
Y_L =
\frac{1}{g^*}\kappa\left(\frac{M_1}{M_{pl}}\right)
\epsilon_1\left(\delta,\varphi\right), \label{finalYL}
\end{aligned}$$ because the variation in $m_2/m_3$ and that in $s_{13}$ do barely affect $Y_L$. It turns out that the washout effect of asymmetry is mainly affected by the lightest mass of heavy Majorana neutrino $M_1$, in other word, the scale of seesaw mechanism. It appears as the ratio to the planck scale $M_1/M_{pl}$. According to Eq.(\[smallk\]), there is an upper bound 17% to the dilution factor $\kappa$ no matter how strong the out-of-equilibrium condition is. The relation of $K$ with $M_1$ in Eq.(\[yukawak\]) implied that the lower $M_1$ scale becomes, the more asymmetry is washed out, as shown in Fig.\[fig3:fig\_seesaw\_scale\]. The areas enclosed by different contours indicate the cosmological bound derived from different scales of $M_1$. As the seesaw scale $M_1$ decreases, the region for sufficient asymmetry becomes narrower. For example, the scale $M_1$ below $10^{-6}M_{pl}$ is not compatible with the Dirac CP phase within T2K 68% CL.
Conclusion
==========
In a minimal seesaw model with two right-handed neutrinos, the lepton asymmetry $Y_L$ for Baryogenesis can be probed by low-energy phenomenology. For sufficient $Y_L$, the model requires non-zero $\delta$. Although the size of $Y_L$ depends on the values of $m_2/m_3$ and $s_{13}$, its variations within current uncertainties of those parameters are almost invisible. Once mixing angles are fixed, the asymmetry is described by seesaw-scale factor and by experimentally measurable factor, as shown in Eq.(\[finalYL\]). One can expect that the precise measurement of $\delta$ in future experiments can constrain the right-handed neutrino mass for seesaw mechanism.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the order statistics of one dimensional branching Brownian motion in which particles either diffuse (with diffusion constant $D$), die (with rate $d$) or split into two particles (with rate $b$). At the critical point $b=d$ which we focus on, we show that, at large time $t$, the particles are collectively bunched together. We find indeed that there are two length scales in the system: (i) the diffusive length scale $\sim \sqrt{Dt}$ which controls the collective fluctuations of the whole bunch and (ii) the length scale of the gap between the bunched particles $\sim \sqrt{D/b}$. We compute the probability distribution function $P(g_k,t|n)$ of the $k$th gap $g_k = x_k - x_{k+1}$ between the $k$th and $(k+1)$th particles given that the system contains exactly $n>k$ particles at time $t$. We show that at large $t$, it converges to a stationary distribution $P(g_k,t\to \infty|n) = p(g_k|n)$ with an algebraic tail $p(g_k|n) \sim 8(D/b) g_k^{-3}$, for $g_k \gg 1$, independent of $k$ and $n$. We verify our predictions with Monte Carlo simulations.'
author:
- Kabir Ramola
- 'Satya N. Majumdar'
- Grégory Schehr
title: Universal Order and Gap Statistics of Critical Branching Brownian Motion
---
The statistics of the global maximum of a set of random variables finds applications in several fields including physics, engineering, finance and geology [@gumbel] and the study of such extreme value statistics (EVS) has been growing in prominence in recent years [@katz; @embrecht; @bouchaud_mezard; @dean_majumdar; @monthus; @gutenburg]. In many real world examples where EVS is important, the maximum is not independent of the rest of the set and there are strong correlations between near-extreme values. Examples can be found in meteorology where extreme temperatures are usually part of a heat or cold wave [@robinson] and in earthquakes and financial crashes where extreme fluctuations are accompanied by foreshocks and aftershocks [@omori; @utsu; @lillo; @peterson]. Near-extreme statistics also play a vital role in the physics of disordered systems where energy levels near the ground state become important at low but finite temperature [@bouchaud_mezard]. In this context, the distribution of the $k$th maximum $x_k$ of an ordered set $\{x_1 > x_2 > x_3 ...\}$ (order statistics [@order_book]) and the gap between successive maxima $g_k = x_k - x_{k+1}$ provides valuable information about the statistics near the extreme value. Such near-extreme distributions have recently been of interest in statistics [@pakes] and physics [@sabhapandit_majumdar; @schehr_majumdar; @mounaix; @perret]. Although the order and gap statistics of independent identically distributed (i.i.d.) variables are fully understood [@order_book], very few exact analytical results exist for strongly correlated random variables. In this context, random walks and Brownian motion offer a fertile arena where near-extreme distributions for correlated variables can be computed analytically [@racz; @schehr_majumdar; @mounaix; @perret].
Another interesting system where order statistics plays an important role is the branching Brownian motion (BBM). In BBM, a single particle starts initially at the origin. Subsequently, in a small time interval $dt$, the particle splits into two independent offsprings with probability $b\, dt$, dies with probability $d\, dt$ and with the remaining probability $(1- (b+d)\,dt)$ it diffuses with diffusion constant $D$. A typical realization of this process is shown in Fig. \[Fig1\]. BBM is a prototypical model of evolution, but has also been extensively used as a simple model for reaction-diffusion systems, disordered systems, nuclear reactions, cosmic ray showers, epidemic spreads amongst others [@brunet_derrida_epl; @brunet_derrida_jstatphys; @mezard; @derrida_spohn; @demassi; @takayasu; @harris; @golding; @fisher; @sawyer; @bailey; @mckean; @bramson; @majumdar_pnas; @derrida_brunet_simon]. In one dimension, the position of the existing particles at time $t$ constitute a set of strongly correlated variables that are naturally ordered according to their positions on the line with $x_1(t) > x_2(t) > x_3(t)
\hdots$. The particles are labelled sequentially from right to left as shown in Fig. \[Fig1\]. One dimensional BBM then provides a natural setting to study the order and the gap statistics for strongly correlated variables.
![A realization of the dynamics of branching Brownian motion with death (left) in the supercritical regime ($b > d$) and (right) the critical regime ($b = d$). The particles are numbered sequentially from right to left as shown in the inset.[]{data-label="Fig1"}](walk_together.eps){width="\linewidth"}
The number of particles $n(t)$ present at time $t$ in this process is a random variable with different behavior depending on the relative magnitude of the rates of birth $b$ and death $d$. When $b < d$ ([*[subcritical]{}*]{} phase), the process dies eventually and on an average there are no particles at large times. In contrast, for $b > d$ ([*[supercritical]{}*]{} phase), the process is explosive and the average number of particles grows exponentially with time. In the borderline $b=d$ ([*[critical]{}*]{}) case, the probability $P(n,t)$ of having $n$ particles at time $t$, starting with a single particle initially, has a well known expression [@feller] (a simple derivation is provided in [@supplementary]) $$\begin{aligned}
P(0,t)=\frac{bt}{1+bt} \;, \; P(n\ge1,t)=\frac{(bt)^{n-1}}{(1+bt)^{n+1}} \;.
\label{particle_probabilities}\end{aligned}$$ The probability that there are no particles tends to $1$ as $1-1/(bt)$ while the probability that there are $n \ge 1$ particles tends to $0$ as $1/(bt)^2$. The average number of particles is independent of time with $\langle n(t) \rangle =\sum_{n=1}^{\infty}n P(n,t)=1$. There are thus strong fluctuations at the critical point which causes most of the realizations of this process to have no particles at large times.
In the supercritical phase, in particular for $d=0$, the statistics of the $k$-th maximum $x_k(t)$ has been studied extensively in mathematics and physics literature with direct relevance to polymer [@derrida_spohn] and spin-glass physics [@mezard]. For example, the first maximum $x_1(t)\sim v t$ typically increases linearly with $t$ and its cumulative distribution satisfies a nonlinear Fisher-Kolmogorov-Petrovky-Piscounov equation [@fisher; @kpp] with a traveling front solution with velocity $v$ [@mckean; @bramson]. The statistics of this first maximum, in the supercritical phase, also appears in numerous other applications in mathematics [@lalley_sellke; @arguin] and physics [@brunet_derrida_epl; @brunet_derrida_jstatphys; @majumdar_pnas]. More recently, the statistics of the gaps between successive maxima have also been studied in the supercritical phase [@brunet_derrida_epl; @brunet_derrida_jstatphys] and the average gap between the $k$-th and $(k+1)$-th maximum was shown to tend to a $k$-dependent constant, independent of time $t$, at large $t$. The stationary probability distribution function (PDF) of the first gap was also computed numerically and an analytical argument was given to explain its exponential tail [@brunet_derrida_epl; @brunet_derrida_jstatphys]. However, an exact analytical computation of the stationary PDFs of these gaps in the supercritical phase still remains an open problem. Much less is known about the order statistics at the critical point ($b = d$) which is relevant to several systems including population dynamics, epidemics spread, nuclear reactions etc. [@majumdar_pnas; @lalley; @sagitov; @aldous]. In this Letter, we show that, in contrast to the supercritical case, the order and the gap statistics can be computed exactly for the critical case $b=d$. In the critical case where $\langle n(t)\rangle=1$ at all times, to make sense of the gaps between particles, it is necessary to work in the fixed particle number sector, i.e., condition the process to have exactly $n(t)=n$ particles at time $t$, with their ordered positions denoted by $x_1(t) > \hdots x_k(t) \hdots > x_n(t)$. We show that a typical trajectory of the critical process is characterized by two length scales at late times: (i) each particle $\langle |x_k(t)| \rangle \sim \sqrt{4Dt/\pi}$ for all $1 \le k \le n$, implying an effective bunching of the particles into a single cluster that diffuses as a whole and (ii) within this bunch, the gap $g_k(t)= x_k(t) - x_{k+1}(t)$ between successive particles tends to a time-independent random variable of $\sim O(1)$. We compute analytically the PDF of this gap (conditioned to be in the fixed $n$-particle sector) and show that it becomes stationary at late times $P(g_k=z,t\to \infty|n)\to p(z|n)$ independent of $k$. Moreover, quite [*remarkably*]{}, $p(z|n)$ has an [*universal*]{} algebraic tail, $p(z|n)\sim 8(D/b)/{z}^3$, independent of $k$ and $n$.
[*Statistics of the Maximum:*]{} We first analyze the behavior of the rightmost particle at time $t$. A convenient quantity is the joint probability that there are $n \ge 1$ particles at time $t$, with all of them lying to the left of $x$: $Q(n;x,t)={\rm Prob.}[n(t)=n,
x_n(t)<x_{n-1}(t)<\ldots< x_1(t)<x]$. It evolves via a backward Fokker-Planck (BFP) equation which can be derived by splitting the time interval $[0,t+\Delta t]$ into $[0,\Delta t]$ and $[\Delta t, t + \Delta t]$ and considering all events that take place in the first small interval $[0,\Delta t]$. In this small interval, the single particle at the origin can: i) with a probability $b \Delta t$ split into two independent particles which give rise to $r$ and $n-r$ particles at the final time respectively; ii) die with the probability $d \Delta t$ and therefore not contribute to the probability at subsequent times; or iii) diffuse by a small amount $\Delta x$ with probability $1-(b +d)\Delta t$, effectively shifting the entire process by $\Delta x$. Summing these contributions, taking the $\Delta t\to 0$ limit and setting $b=d$, we get [@supplementary] $$\begin{aligned}
\nonumber
\frac{\partial Q(n;x,t)}{\partial t} =
D\frac{\partial^{2}Q(n;x,t)}{\partial x^{2}} - 2 b Q(n;x,t)\\
+ 2 b\, P(0;t)Q(n;x,t)+ b\, \sum_{r = 1}^{n-1}Q(r;x,t)Q(n-r;x,t),
\label{BFP_Qn}\end{aligned}$$ starting from the initial condition $Q(n;x,0)= \delta_{n,1}$ for all $x>0$ and satisfying the boundary conditions: $Q(n;-\infty,t)=0$ and $Q(n;\infty,t)=P(n,t)$. Next, we consider the conditional probability $Q(x,t|n)=Q(n;x,t)/P(n,t)$, i.e., the cumulative probability of the maximum given $n$ particles at time $t$. Using (\[BFP\_Qn\]) and the explicit expression of $P(n,t)$ in (\[particle\_probabilities\]), we find that $Q(x,t|n)$ evolves via $$\begin{aligned}
\nonumber
\frac{\partial Q(x,t|n)}{\partial t} +\frac{n-1}{t(1+bt)} Q(x,t|n) =
D\frac{\partial^{2}Q(x,t|n)}{\partial x^{2}}\\ +
\frac{1}{t(1+bt)} \sum_{r = 1}^{n-1}Q(x,t|r)Q(x,t|n-r).
\label{cond.n}\end{aligned}$$ This is a linear equation for $Q(x,t|n)$ for a given $n$ that involves, as source terms, the solutions $Q(x,t|k)$ with $k<n$. Hence it can be solved recursively for any $n$, starting with $n=1$. For $n=1$, one obtains an explicit solution [@supplementary]: $Q(x,t|1)=\frac{1}{2}\,
\mathrm{erfc}\left(\frac{-x}{\sqrt{4Dt}}\right)$, where $\mathrm{erfc}(x)=\frac{2}{\sqrt{\pi}}\,\int_x^{\infty} e^{-u^2}\, du$ is the complementary error function. Consequently, the PDF of the maximum $x_1(t)$ in the single particle sector, $P(x_1,t|1)= \partial_{x_1}Q(x_1,t|1)= \frac{1}{\sqrt{4 \pi D t}} \exp
\left(-\frac{x_1^2}{4 D t} \right)$, is a simple Gaussian. The particle thus exhibits free diffusion, implying that the effect of branching exactly cancels the effect of death. For later purpose, we note that $P(1;x,t)=\partial_x
Q(1;x,t)=P(1,t) \partial_x Q(x,t|1)$, i.e. the probability density of having one particle at position $x$ at time $t$, reads $$P(1;x,t)= \frac{1}{(1+bt)^2}\, \frac{1}{\sqrt{4\pi Dt}}\, e^{-x^2/{4Dt}}.
\label{p1}$$ Finally, feeding the one particle solution $Q(x,t|1)$ into (\[cond.n\]) for $n=2$, one can also obtain $Q(x,t|2)$ (see [@supplementary]) and recursively $Q(x,t|n)$ for higher $n$.
For general $n>1$, one can estimate easily the late time asymptotic solution. Since $Q(x,t|n)$ is bounded as $0 < Q(x,t|n) < 1$, Eq. (\[cond.n\]) reduces, for large $t$, to a simple diffusion equation which does not contain $n$ explicitly, implying $Q(x,t|n)\sim Q(x,t|1)$. Hence, the PDF of the maximum for any $n\ge 1$ particle sector behaves as $P(x_1,t|n) \approx \frac{1}{\sqrt{4 \pi D t}}
\exp \left(-\frac{x_1^2}{4 D t} \right)$ for large $t$. By symmetry, the minimum $x_n$ is also governed by the same distribution. This illustrates an important feature of BBM at criticality: [*the maximum and minimum of $n$ particles both behave as a free diffusing particle at large $t$*]{}. The rest of the particles are confined between these two extreme values ($x_1(t) > \hdots x_k(t) \hdots > x_n(t)$) and hence also behave diffusively, $\langle |x_k| \rangle \sim \sqrt{4Dt/\pi} $, independent of $k$ and $n$ for large $t$, leading to the bunching of the particles. The gap between the particles $g_k(t)= x_k(t)-x_{k+1}(t)$ thus probes the sub-leading large $t$ behavior of the particle positions $x_k(t)$, which we consider next.
[*Gap Statistics:*]{} We start with the first gap $g_1(t)=x_1(t)-x_2(t)$ between the rightmost and the preceding particle in the particle number $n\ge 2$ sector. To probe this gap, it is convenient to study the joint PDF $P(n;x_1, x_2,t)$ that there are $n$ particles at time $t$ with the first particle at position $x_1$ and the second at position $x_2<x_1$. We first analyze the simplest case $n=2$ and argue later that the behavior of $g_1$ in this $n=2$ sector is actually quite generic and holds for higher $n$ as well. Using a similar BFP approach outlined before, we find the following evolution equation (for detailed derivation see [@supplementary]) $$\begin{aligned}
\nonumber
\frac{\partial P(2;x_{1},x_{2},t)}{\partial t} = D\left( \frac{\partial}{\partial x_1}
+ \frac{\partial}{\partial x_2}\right)^2
P(2;x_{1},x_{2},t)\\
-\frac{2b}{1+bt}P(2;x_{1},x_{2},t)+ 2 b P(1;x_1,t) P(1;x_2,t)
\label{2part_FP}\end{aligned}$$ where $P(1;x,t)$ is given in (\[p1\]). This linear equation for $P(2;x_1,x_2,t)$ can be solved explicitly [@supplementary]. Consequently, the conditional probability $P(x_1,x_2,t|2)= P(2;x_1,x_2,t)/P(2,t)$ (with $P(2,t)= bt/(1+bt)^3$ given in (\[particle\_probabilities\])), denoting the joint PDF of $x_1$ and $x_2$ given $n=2$ particles, can also be obtained explicitly. The solution is best expressed in terms of the variables, $s=(x_1+x_2)/2$ (center of mass) and $g_1=x_1-x_2$ (gap): $P(x_1,x_2,t|2)\to P(s,g_1,t|2)$ and reads [@supplementary] $${P}(s,g_1,t|2) = \left(\frac{1 + b t}{2 \pi D t}\right)
\int_{0}^{t} \frac{dt'}{(1+ b t')^2} \frac{e^{- \frac{g_1^2}{8 D t'}
- \frac{s^2}{2D(2 t- t')}}}{\sqrt{t'(2t-t')}}.
\label{2part_distribution}$$ The marginal PDF of the centre of mass $P(s,t|2)=\int_{0}^{\infty} P(s,g_1,t|2)
dg_1$ is easily obtained by integrating over the gap $g_1$ and for large $t$, $P(s,t|2) \sim \frac{1}{\sqrt{4 \pi D t}} \exp \left(-\frac{s^2}{4 D t} \right)$, as expected from the free diffusive behavior of the clustered particles. Similarly, by integrating over $s$ we obtain the marginal PDF of the gap at any $t$ $$P(g_1,t|2) = \left(\frac{1 + b t}{b t}\right) \int_{0}^{t} \frac{b dt'}{(1+ b t')^2}
\frac{\exp(- \frac{g_1^2}{8 D t'})}{\sqrt{2 \pi D t'}}.
\label{gap_distribution}$$ At large times $P(g_1,t|2)$ converges to a stationary distribution $P(g_1,t \to \infty|2) = p(g_1|2)$ (Fig. \[2part\_approach\]), which can be computed explicitly. It can be expressed as $p(g_1|2) = (4\sqrt{D/b})^{-1} f[g_1/(4\sqrt{D/b})]$ with $$\begin{aligned}
\label{scaling_f}
f(x) = -4x + \sqrt{2\pi} \, e^{2x^2}(1+4x^2)\,{\rm erfc}(\sqrt{2} \,x) \;.\end{aligned}$$ This distribution (\[scaling\_f\]) has a very interesting relation to the PDF of the (scaled) $k$-th gap between extreme points visited by a single random walker found in Ref. [@schehr_majumdar] \[the scaling function found there (see Eq. (1) of [@schehr_majumdar]) is exactly $-f'(x)/\sqrt{2 \pi}$\]. It behaves asymptotically as $$\begin{aligned}
p(g_1|2) \sim
\begin{cases}
\sqrt \frac{\pi b}{8 D} \;, \; g_1 \to 0,\\
\left(\frac{8 D}{b} \right) {g_1^{-3}} \;, \; g_1 \to \infty \;.
\end{cases}
\label{large_g_behaviour}\end{aligned}$$ This function $p(g_1|2)$ describes the typical fluctuations of the gap $g_1$, which are of order $\sqrt{D/b}$. However, because of the algebraic tail, only the first moment of the gap is dominated by the typical fluctuations, $\langle g_1 \rangle = \sqrt{{2 \pi D}/{b}}$. The higher moments instead get contributions from the time dependent far tail of the PDF in (\[gap\_distribution\]): $\langle g_1^2 \rangle \sim \ln(t)$ and $\langle g_1^m \rangle \sim t^{\frac{m}{2}-1}$ for $m >2$.
![Exact gap PDF in the two particle sector (Eq. \[gap\_distribution\]) at different times, showing the approach to the stationary behavior at large times. The solid line indicates the expected power law decay for $t \to \infty$. Here $D = 1$ and $b ={1}/{2}$.[]{data-label="2part_approach"}](Theory_2part.eps){width="\linewidth"}
In Fig. \[2part\_approach\], we plot $P(g_1,t|2)$ at different times showing the approach to the stationary distribution with a power law tail at large times.
The computation for the first gap $g_1$ for $n=2$ outlined above can be generalized to the $n>2$ sector. Once again using the BFP approach, we find that the joint PDF $P(n;x_{1},x_{2},t)$ obeys $$\begin{aligned}
\nonumber
&&\frac{\partial P(n;x_{1},x_{2},t)}{\partial t} = D\left( \frac{\partial}{\partial x_1} +
\frac{\partial}{\partial x_2}\right)^2 P(n;x_{1},x_{2},t)\\
&&~~~~~~~~~~~-\frac{2b}{1+bt}P(n;x_{1},x_{2},t)+ b \mathcal{S}(n;x_1,x_2,t).
\label{n_particle_diffusion}\end{aligned}$$ Here $\mathcal{S}(n;x_1,x_2,t)$ is a source term that arises from the branching at the first time step. It can be computed explicitly in terms of spatial integrals involving $P(k; x_1, x_2, t)$ with $k < n$ – the resulting expression being however a bit cumbersome [@supplementary]. However Eq. (\[n\_particle\_diffusion\]) can still be solved recursively to obtain the exact distribution of the first gap $g_1 = x_1 - x_2$ in the $n$ particle sector. We have solved these equations exactly up to $n=4$ [@supplementary]. These computations are quite instructive as they allow us to analyze Eq. (\[n\_particle\_diffusion\]) in the large $t$ and large gap $g_1$ limit for generic $n$ as follows. The solution of (\[n\_particle\_diffusion\]) is a linear combination of solutions arising from individual terms present in the source function ${\cal S}$. From this one can show that the PDF of the first gap in the $n$-particle sector converges to a stationary distribution $P(g_1,t\to \infty|n)=p(g_1|n)$. While the full PDF $p(g_1|n)$ depends on $n$ (see also Fig. \[Npartfit\]), its tail is universal. This follows from the fact that the leading contribution to ${\cal S}$ in (\[n\_particle\_diffusion\]) when the gap $g_1 = x_1-x_2 \gg 1$ is large tends to $2 b P(1;x_1,t) P(1;x_2,t)$ at large $t$ [@supplementary]. This is precisely the source term for the two-particle case analyzed in Eq. (\[2part\_FP\]). One can show that all other terms in ${\cal S}$ involve a larger gap between particles generated by the same offspring walk and are thus suppressed by a factor $\int_{g_1}^{\infty} p(g'|k) dg'$, $k<n$ [@supplementary]. Therefore, when $g_1 \to \infty$ the tail of the PDF of the first gap in the $n$ particle sector converges to that of the two-particle case, $p(g_1|n) \sim \left(\frac{8 D}{b} \right) {g_1^{-3}}$, for all $n$.
![Time-integrated PDF for the first gap $g_1 = x_1 - x_2$ in different particle sectors computed from Monte Carlo simulations. [**Inset**]{}: Time-integrated PDF for the $k$-th gap $g_k = x_k - x_{k+1}$ in the $10$-particle sector, showing the approach to the same asymptotic value. The lines have a slope of $-3$. Here $D= 1, b = {1}/{2}$, and $t =10^4$.[]{data-label="Npartfit"}](N_final_fit.eps){width="\linewidth"}
A similar analysis yields the asymptotic behavior of the $k$-th gap $g_k(t) = x_{k}(t) - x_{k+1}(t)$. In this case, we study $P(n;x_k,x_{k+1},t)$, the joint PDF that there are $n$ particles at time $t$ with the $k$-th particle at position $x_k$ and the $(k+1)$-th particle at position $x_{k+1}$. This PDF once again satisfies a diffusion equation with a source term similar to (\[n\_particle\_diffusion\]), from which we can show that the PDF of the $k$th gap reaches a stationary distribution $P(g_k,t\to \infty|n)=p(g_k|n)$. In the large gap limit, the dominant term in the source function is the one in which the first $k$ particles belong to one of the offsprings generated at the first time step, and the subsequent $n-k$ particles belong to the other. This term tends to $2 b P(1;x_k,t) P(1;x_{k+1},t)$ at large $t$, as it involves the minimum of the first process being at $x_k$ and the maximum of the other process being at $x_{k+1}$. As noticed before for $g_1$, all other terms involve a large gap between particles generated by the same offspring process and are hence suppressed. This in turn leads to the large gap stationary behavior $p(g_k|n) \sim \left(\frac{8 D}{b} \right) g_k^{-3}$ for all $k$ and $n$. [*Monte Carlo Simulations:*]{} We have directly simulated the critical BBM process and we have computed the PDFs of the gap. To obtain better statistics we compute the time-integrated PDF $S(g_k,t|n) = \frac{1}{t}\int_{0}^{t} P(g_k,t'|n) dt'$, which has the same stationary behavior as $P(g_k,t|n)$, $S(g_k,t\to \infty|n) = p(g_k|n)$. In Fig. \[Npartfit\] we plot $S(g_1,t|n)$, corresponding to the first gap, for different values of $n = 1, \cdots, 8$ and $t=10^4$. The different curves show an approach to the same asymptotic, large $g_1$, behavior (note that the approach to the stationary state gets slower as $n$ increases). In the inset of Fig. \[Npartfit\] we show a plot of $S(g_k,t|n)$ for $n=10$ and $t = 10^4$ for different values of $k=1, \cdots, 5$. This also shows a convergence to the same large $g_k$ behavior $\sim \left(\frac{8 D}{b} \right) g_k^{-3}$. Numerical results for short times (up to $n=4$), not shown here [@supplementary], show a perfect agreement with the solution of Eq. (\[n\_particle\_diffusion\]).
[*Conclusion*]{}: We have obtained exact results for the order statistics of critical BBM. We showed that the statistics of the near extreme points displays a quite rich behavior characterized by a stationary gap distribution with a universal algebraic tail. This presents a physically relevant instance of strongly correlated random variables for which order statistics can be solved exactly. It will be interesting to extend the BFP method developed here to compute exactly the gap statistics in the supercritical case.
KR acknowledges helpful discussions with Shamik Gupta. SNM and GS acknowledge support by ANR grant 2011-BS04-013-01 WALKMAT and in part by the Indo-French Centre for the Promotion of Advanced Research under Project 4604-3. GS acknowledges support from Labex-PALM (Project Randmat).
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****[^1]\
[ Shixin Zhu, Binbin Pang, Zhonghua Sun]{}
*Department of Mathematics, Hefei University of Technology, Hefei 230009, Anhui, P.R.China*
**Abstract:** In this paper, by investigating the factor of the $x^n+1$, we deduce that the structure of the reversible negacyclic code over the finite field $\mathbb{F}_{q}$, where $q$ is an odd prime power. Though studying $q-$cyclotomic cosets modulo $2n$, we obtain the parameters of negacyclic BCH code of length $n=\frac{q^\ell+1}{2}$ , $n=\frac{q^m-1}{2(q-1)}$ and $n=\frac{q^{t\cdot2^\tau}-1}{2(q^t+1)}$. Some optimal linear codes from negacyclic codes are given. Finally, we discuss a class of MDS LCD negacyclic codes.\
*Keywords*: Negacyclic codes, Reversible code, Negacyclic BCH codes, MDS code. The negacyclic codes have been well studied in literatures. And the definition of negacyclic BCH code was given in \[4\]. Dinh established the structure of negacyclic codes of length $n$, where gcd$(n,q)=1$ \[11\]. Aydin, Sliap and Ray-chaudhuri gave the BCH bound for the constacyclic codes \[3\]. LCD codes were initiated by Massey \[6\], he also showed the existence of the asymptotically good LCD codes. The condition of the LCD codes was given by Yang and Massey \[7\]. Hou and Oggier acquired the construction and properties of a lattice from LCD codes \[9\]. Lina and Nocon constructed some special LCD codes and confirmed that permutation equivalence of codes preserves the LCD-ness of codes. Ding and Li constructed several classes of reversible cyclic codes over finite fields and analyzed their parameters \[1\], and they also showed the parameters of some reversible BCH codes \[5\]. Güneri and Özkaya studied the quasi-cyclic complementary dual code by using their concatenated structure, and they also constructed quasi-cyclic complementary dual code from codes over larger alphabets \[13\]. The existence of the MDS Hermitian self-orthogonal and self-dual was obtained by Yang and Cai \[12\].
We will study the reversible negacyclic codes over finite fields. In this paper, using the method of investigate LCD cyclic codes, we deduce the condition of reversible negacycylic codes. The structure of LCD negacyclic codes is determined, and in the special case, the quantity of reversible negacyclic codes is gained. We discuss the parameters of negacyclic BCH codes when the length $n=\frac{q^\ell+1}{2}$, $n=\frac{q^m-1}{2(q-1)}$ and $n=\frac{q^{t\cdot2^\tau}-1}{2(q^t+1)}$, and a class of MDS LCD negacyclic codes.
Throughout this paper, let $\mathbb{F}_q$ be a finite field of size $q$, where $q$ is an odd prime power. A linear $[n,k,d]$ code $C$ over $\mathbb{F}_q$ is called negacyclic if $(c_0,c_1,\ldots,c_{n-1})\in C$ implies its negacyclic shift $(-c_{n-1},c_0,\ldots,c_{n-2})\in C$. Let $C$ be an $[n,k]$ linear code over $\mathbb{F}_q$, its dual code $C^\perp$ is defined by $C^{\perp}=\{{\textbf{u}\in \mathbb{F}_q^n\mid \textbf{u}\cdot \textbf{c}=\textbf{0},\forall\ \textbf{c}\in C}\}$. By identifying any vector $(c_0,c_1,\ldots,c_{n-1})\in \mathbb{F}_q$ corresponds to a polynomial $c_0+c_1x+,\cdots,+c_{n-1}x^{n-1}\in \mathbb{F}_q[x]/\langle x^n+1\rangle$, then a linear negacyclic code over $\mathbb{F}_q$ is an ideal of ring $\mathbb{F}_q[x]/\langle x^n+1\rangle$. In fact every ideal in $\mathbb{F}_q[x]/\langle x^n+1\rangle$ is a principal ideal, so every negacyclic code $C$ has generator polynomial $g(x)$. Let $C=\langle g(x)\rangle$, where $g(x)$ is a unique monic and has minimal degree polynomial in $C$. And $h(x)=(x^n+1)/g(x)$ is referred to as the check polynomial of $C$. The dual code of $C$ is also negacyclic code and has generator polynomial $g^\perp(x)=x^{\textrm{deg}h}h(x^{-1})$.
In this paper, we always assume that gcd$(n,q)=1$, and note that $x^n+1$ has no repeated root over $\mathbb{F}_q$ if and only if gcd$(n,q)=1$. Let $\gamma$ be a primitive $2n-$th root of unity in $\mathbb{F}_{q^m}$, where $m$ is the multiplicative of $q$ modulo $2n$, i.e. $m=\textrm{ord}_{2n}(q)$. Then the roots of $x^n+1$ are $\gamma^{1+2i}$, $0\leq i\leq n-1$. Let $\mathbb{Z}_{2n}=\{0,1,\cdots,2n-1\}$. For any $s\in \mathbb{Z}_{2n}$, the $q-$cyclotomic coset$$C_s=\{s,sq,sq^2,\cdots,sq^{d_s-1}\}\ \textrm{mod}2n\subseteq \mathbb{Z}_{2n}$$ where $d_s$ is the smallest positive integer such that $sq^{d_s}\equiv s\ \textrm{mod}2n$, and the size of the $C_s$. Let $T=\{1+2i\mid0\leq i\leq n-1\}$, containing all the odd integers of $\mathbb{Z}_{2n}$, obviously, $T\subseteq \mathbb{Z}_{2n}$ and $|T|=\frac{1}{2}|\mathbb{Z}_{2n}|$.\
**Lemma 2.1.** For any $s\in \mathbb{Z}_{2n}$, $T\cap C_s=C_s\ \textrm{or}\ \ \emptyset$, where $C_s$ denotes the $q-$cyclotomic coset modulo $2n$.\
Let $T_s=T\cap C_s$, if $T\cap C_s=C_s$. Let $X_{(n,q)}$ be a set of all the coset leaders of $T_s$, we have any $s,t\in X_{(n,q)}, s\neq t$, $T_s\cap T_t=\emptyset$, and $$\large{\bigcup_{s\in X_{(n,q)}}T_s=T}\tag{2.1}$$ **Lemma 2.2.** The cardinality $d_s$ of $T_s$ is a divisor of $m=\textrm{ord}_{2n}(q)$ which is equivalent to $d_1=|T_1|$.\
**Theorem 2.3.** Let $n$ be an odd integer such that gcd$(n,q)=1$. For any $s\in X_{(n,q)}$, then $|T_s|=|C_{2s}|$.\
**Proof**. From the definition of $T_s$, $T_s=C_s=\{s,sq,sq^2,\cdots,sq^{d_s-1}\}\ \textrm{mod}2n$, where $d_s$ is the smallest positive integer such that $sq^{d_s}\equiv s\ \textrm{mod}2n$. We easily obtain $2sq^{d_s}\equiv 2s\ \textrm{mod}2n$. where $d_s$ must also be the smallest positive integer. Otherwise, there exist an integer $d'_s$, and $d'_s<d_s$, such that $2sq^{d'_s}\equiv 2s\ \textrm{mod2}n$, we get that there exist an integer $k$ such that $2sq^{d'_s}-2s=2nk$. Since $s,q,n$ are all odd integers, we deduce $4|(2sq^{d'_s}-2s)$, then $4|2nk$. Thus $sq^{d'_s}\equiv s\ \textrm{mod}2n$, which is contrary to the definition of $d_s$, the conclusion is obtained.\
The following is immediate from \[1\] Lemma 2.\
**Lemma 2.4.** Let $q^{\lfloor m/2\rfloor}/2<n\leq (q^m-1)/2$ be a positive integer such that gcd$(n,q)=1$, where $m=\textrm{ord}_{2n}(q)$. Then the cardinality of $T_s=T\cap C_s=\{sq^i\ \textrm{mod}2n|0\leq i\leq m-1\}$ is equal to $m$ for $\forall s\in X_{(n,q)}$ in the range $1\leq s\leq 2nq^{\lfloor m/2\rfloor}/(q^m-1)$. In this case every $s$ with $s\not\equiv 0\ \textrm{mod}q$ is a coset leader of $T_s$.\
Let $\alpha$ be a generator of $\mathbb{F}_{q^m}^\ast$, where $m=\textrm{ord}_{2n}(q)$. Put $\beta=\alpha^{(q^m-1)/2n}$, then $\beta$ is a primitive $2n-$th root of unity in $\mathbb{F}_{q^m}$, the minimal polynomial $m_i(x)$ of $\beta^i, i\in X_{(n,q)}$ over $\mathbb{F}_q$ is given by $m_i(x)=\Pi_{j\in T_i}(x-\beta^j)$. Summarizing the equality (2.1) gets $$x^n+1=\prod_{i\in X_{(n,q)}}m_i(x)\tag{2.2}$$ which is the canonical factorization of $x^n+1$ over $\mathbb{F}_q$. This is vital for studying of negacyclic codes. A linear code has complementary dual (or LCD code for short) if Hull$(C)=C\cap C^\perp=\{\textbf{0}\}$, which is equivalent to $C+C^\perp=\mathbb{F}_q^n$.\
Let $h(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0\in \mathbb{F}_q[x]$, with $a_n\neq 0$ and $ a_0\neq 0$, the reciprocal polynomial $h^\ast(x)$ of $h(x)$ is defined by $$h^\ast(x)=a_0^{-1}x^nh(x^{-1}).$$ **Lemma 3.1.** Let $h(x), f(x)\in \mathbb{F}_q[x]$. Then
1\) If $\textrm{deg}h\geq \textrm{deg}f$, then $(h(x)+f(x))^\ast=h^\ast(x)+x^{\textrm{deg}h-\textrm{deg}f}f^\ast(x)$,
2\) $(h(x)f(x))^\ast=h^\ast(x)f^\ast(x)$.\
A code $C$ is called reversible if $(c_0,c_1,\ldots,c_{n-1})\in C$ implies that $(c_{n-1},c_{n-2},\ldots,c_0)\in C$. Note that a negacyclic code $C$ is reversible if and only if the generator polynomial of $C$ is self-reciprocal.\
**Theorem 3.2.** Let $C$ be a negacyclic code over $\mathbb{F}_q $ with generator polynomial $g(x)$, then the following conclusions are equivalent.
1\) $C$ is a LCD code,
2\) $g(x)$ is self-reciprocal $(g(x)=g^\ast(x))$,
3\) An element $\beta$ in the splitting field of $g(x)$, if $g(\beta)=0$, then $g(\beta^{-1})=0$.\
**Proof**. 1) is equivalent to $C+C^\perp=\mathbb{F}_q^n$, if and only if $C=\langle g(x)\rangle$ and $C^\perp=\langle h(x)\rangle$, where $h(x)=(x^n+1)/g(x)$. We get that $C$ and $C^\perp$ are both reversible. It is equivalent to 2) and 3).\
**Theorem 3.3.** The negacyclic code over $\mathbb{F}_q$ of length $n$ is reversible, if $-1$ is a power of $q$ $2n$.\
**Proof**. By the definition of $T_s$, we get that there exist $\ell$ such that $sq^\ell\equiv -s\ \textrm{mod}2n$, thus $-s\in T_s$. Hence every irreducible factor of $x^n+1$ is self-reciprocal, the corresponding negacyclic code is reversible.\
**Theorem 3.4.** The irreducible polynomial $m_s(x)$ is self-reciprocal if only and if $2n-s\in T_s$.\
Let $Y_{(n,q)}$ be a set such that $\{T_s\cup T_{2n-s}|s\in Y_{(n,q)}\}$ is a partition of $T$. Notice that there are different choices for $Y_{(n,q)}$.\
**Theorem 3.5.** There are $2^{|Y_{(n,q)}|}-1$ reversible negacyclic codes over $\mathbb{F}_q$ of length $n$, and those generator polynomials are as follows $$g(x)=\prod_{s\in S}\textrm{lcm}(m_s(x),m_{2n-s}(x))$$ which $S\subseteq Y_{(n,q)}$ and $S\neq \emptyset$.\
**Example 3.6.** Let $n=7,q=3$, from the definition of $C_s$ we can deduce $$C_0=\{0\}\ \ \ C_1=\{1,3,5,9,11,13\}$$ $$C_7=\{7\}\ \ \ C_2=\{2,4,6,8,10,12\}$$\
Notice that $T=\{1,3,5,7,9,11,13\}$, $T_1=T\cap C_1=C_1$, $T_7=T\cap C_7=C_7$, from the equality $(2.2)$, we get that $x^7+1=m_1(x)m_7(x)$, where $m_1(x)=x^6+2x^5+x^4+2x^3+x^2+2x+1$, $m_7(x)=x+1$. In this case, $m_1(x)$ and $m_7(x)$ are both self-reciprocal. And we obtain $X_{(n,q)}=Y_{(n,q)}=\{1,7\}$. Thus, the number of the reversible ternary negacyclic codes of length 7 is 3.\
**Lemma 3.7.** Let $m\geq1$ and $b>1, b\in \mathbb{Z}$ . Then $$\begin{aligned}
\textrm{gcd}(b^n+1,b^m-1)=
\left\{ {{\begin{array}{ll}
{1}, & {\textrm{if}\ \frac{m}{\textrm{gcd}(n,m)}}\ \textrm{is odd and}\ b \textrm{ is even},\\
{2}, & {\textrm{if}\ \frac{m}{\textrm{gcd}(n,m)}}\ \textrm{is odd and}\ b \textrm{ is odd}, \\
{b^{\textrm{gcd}(n,m)}+1}, & {\textrm{if}\ \frac{m}{\textrm{gcd}(n,m)}}\ \textrm{is even}. \\
\end{array} }} \right .\end{aligned}$$ **Theorem 3.8.** Let $q=p^\ell$, where $p$ is an odd prime integer. Put $n=\frac{(q^m-1)}{2}$ be an odd integer. Then
1\) If $m$ is odd integer, $x+1$ is the only self-reciprocal irreducible factor of $x^n+1$ over $\mathbb{F}_q$,
2\) If $m$ is odd prime, there are $2^{\frac{q^m+(m-1)q+m}{4m}}-1$ reversible negacyclic codes of length $n$ over $\mathbb{F}_q$.\
**Proof**. 1). From Lemma 3.7, we get any $0\leq i\leq m-1$, gcd$(q^i+1,q^m-1)=2$, thus $s(1+q^i)\equiv 0\ \textrm{mod}2n$ if and only if $s=0$ or $s=n$, however, $n\in T$ and $0\not\in T$. Hence, $x+1$ is the only self-reciprocal irreducible factor of $x^n+1$ over $\mathbb{F}_q$.\
2). Since $m$ is odd prime, the cardinality of $T_s$ is either 1 or $m$. Since gcd$(q-1,q^m-1)=q-1$, there are exactly $(q-1)$ cycloyomic cosets modulo $2n$ have size 1. By Theorem 2.3, any $s\in X_{(n,q)}$, $|T_s|=|C_{2s}|$. We then deduce that the number of $|T_s|=1$ is $\frac{q-1}{2}$, and the number of $|T_s|=m$ is half of the number of $|C_a|=m$, where $a\in \mathbb{Z}_{2n}$. Summarizing the 1) we get $$|Y_{(n,q)}|=\frac{\frac{q^m-1}{2}-\frac{q-1}{2}}{2m}+\frac{\frac{q-1}{2}-1}{2}+1=\frac{q^m+(m-1)q+m}{4m}$$ Thus, there are $2^{\frac{q^m+(m-1)q+m}{4m}}-1$ reversible negacyclic codes of length $n=\frac{q^m-1}{2}$ over $\mathbb{F}_q$\
Let $n$ be a positive integer such that gcd$(n,q)=1$, and we know $x^n+1=\Pi_{s\in X_{(n,q)}}m_s(x)$. Let $\alpha$ be a generator of $\mathbb{F}_{q^m}^\ast$, where $m=\textrm{ord}_{2n}(q)$. Put $\beta=\alpha^{(q^m-1)/2n}$, $m_s(x)$ is the minimal polynomial of $\beta^s, s\in X_{(n.q)}$, the reversible negacyclic code over $\mathbb{F}_q$ of length $n$ has generator polynomial $g(x)=\prod_{s\in S}\textrm{lcm}(m_s(x),m_{2n-s}(x))$, where $S\subseteq Y_{(n,q)}$.
From the BCH bound for constacyclic codes$^{[3]}$, if $C$ is a negacyclic code and let $g(x)$ be a generator polynomial of $C$ and $g(x)$ has roots $\{\beta^{1+2i}, 0\leq i\leq d-2\}$, where $\beta$ is a primitive $2n-$th root of unity, we deduce that the minimum distance of the code is at least $d$.
From the definition of negacyclic BCH codes$^{[4]}$, let $C$ be a negacyclic code with generator $g_{(q,n,\delta,b)}(x)$, then there exist an odd integer $b\geq1$ and $\delta\geq2$ such that $g(\beta^b)=g(\beta^{b+2})=\cdots=g(\beta^{b+2(\delta-2)})=0$. Then the minimum distance of the code is at least $\delta$. Denoted $C_{(q,n,\delta,b)}$ by the code with generator polynomial $g_{(q,n,\delta,b)}(x)$.\
In this section, we always assume that $n=\frac{q^\ell+1}{2}$. Every negacyclic code of length $n$ is reversible from the Theorem 3.3, we will study the parameters of these codes.\
**Lemma 4.1.** $\textrm{ord}_{2n}(q)=2\ell=m$.\
**Proof**. It is easy to prove by Lemma 3.7.\
**Lemma 4.2.$^{[1]}$** Let $\ell\geq2$, then any odd integer $s\in T$, $s\leq q^{\lfloor (\ell-1)/2\rfloor}+1$ and $s\not\equiv0\ \textrm{mod}q$ is a coset leader and $|T_s|=2\ell$.\
**Theorem 4.3.** Let $n=\frac{q^\ell+1}{2}$ and $m=2\ell$. Then the minimum distance of code $C_{(q,n,\delta,1)}, \ d\geq2\delta-1$.\
**Proof**. Let $\alpha$ be a generator of $\mathbb{F}_{q^m}^\ast$, and put $\beta=\alpha^{q^\ell-1}$ is a primitive $2n-$th root of unity. The generator polynomial $g_{(q,n,\delta,1)}(x)$ of code $C_{(q,n,\delta,1)}$ has roots $\beta^i, i\in \{1,3,\cdots,1+2(\delta-2)\}$ . The code $C_{(q,n,\delta,1)}$ is reversible from the Theorem 3.3. Hence we deduce that the $g_{(q,n,\delta,1)}(x)$ has roots $\beta^i, i\in \{2n-1-2(\delta-2),\cdots,2n-1,1,3,\cdots,1+2(\delta-2)\}$ . By the negacyclic BCH bound, we get $d\geq2\delta-1$.\
According to this theorem we can obtain the parameters of the code in this case.\
**Theorem 4.4.** Let $n=\frac{q^\ell+1}{2}$. For any integer $2\leq\delta\leq \frac{q^{\lfloor(\ell-1)/2\rfloor}}{2}+2$, then the negacyclic BCH code $C_{(q,n,\delta,1)}$ has parameters $$\begin{aligned}
\left\{ {{\begin{array}{ll}
{[\frac{q^\ell+1}{2},\frac{q^\ell+1}{2}-2\ell(\delta-1-\lfloor\frac{2\delta-3}{2q}\rfloor),d\geq2\delta-1]}, & {\textrm{if} \ 2\delta-3=\varepsilon+2qi},\\
{[\frac{q^\ell+1}{2},\frac{q^\ell+1}{2}-2\ell(\delta-1-\lceil\frac{2\delta-3}{2q}\rceil),d\geq2\delta-1]}, & { \ \textrm{otherwise}}. \\
\end{array} }} \right .\end{aligned}$$ which $\varepsilon=2k+1<q,k\in \mathbb{Z},k\geq0$ and $i\in \mathbb{Z},i\geq0$, with generator polynomial $$\prod_{0\leq a\leq \delta-2, 1+2a\not\equiv0\ \textrm{mod}q}m_{1+2a}(x)$$ **Proof**. Since $1\leq 2\delta-3\leq q^{\lfloor(\ell-1)/2\rfloor}+1$, for any integer $a, 0\leq a\leq\delta-2$ and $1+2a\not\equiv0\ \textrm{mod}q$ is the coset leader and others are not from the Lemma 4.2. Hence we deduce that the number of $0\leq a\leq\delta-2$ and $1+2a\equiv0\ \textrm{mod}q$ is equal to $\lfloor\frac{2\delta-3}{2q}\rfloor$ if $2\delta-3=\varepsilon+2qi$, where $\varepsilon=2k+1<q,k\in \mathbb{Z},k\geq0$, and $i\in \mathbb{Z},i\geq0$, otherwise $\lceil\frac{2\delta-3}{2q}\rceil$. By the definition of negacyclic BCH code, we get that the generator polynomial of the code is $g(x)=\Pi_{0\leq a\leq \delta-2, 1+2a\not\equiv0\ \textrm{mod}q}m_{1+2a}(x)$. According to the negacyclic code theory, the dimension of the code is equal to $n-\textrm{deg}g(x)$. Again by Lemma 4.2 and 4.3, we obtain the conclusion.\
**Corollary 4.5.** From Theorem 4.4, let $q=3$ then we have the following table\
code parameters generator polynomial
------------- ----------------- --------------------------------------------------------- ----------------------
$\ell\geq3$ $C_{(3,n,3,1)}$ $[\frac{3^\ell+1}{2},\frac{3^\ell+1}{2}-2\ell,d\geq5]$ $m_1(x)$
$\ell\geq3$ $C_{(3,n,4,1)}$ $[\frac{3^\ell+1}{2},\frac{3^\ell+1}{2}-4\ell,d\geq7]$ $m_1(x)m_5(x)$
$\ell\geq4$ $C_{(3,n,6,1)}$ $[\frac{3^\ell+1}{2},\frac{3^\ell+1}{2}-6\ell,d\geq11]$ $m_1(x)m_5(x)m_7(x)$
\
**Example 4.6.** From Corollary 4.5, we deduce\
code parameters
------------------ ----------------- --------------------------------------------
$\ell=\{3,4,5\}$ $C_{(3,n,3,1)}$ $[14,8,5],[41,33,d\geq5],[122,112,d\geq5]$
$\ell=\{3,4\}$ $C_{(3,n,4,1)}$ $[14,2,d\geq7],[41,25,d\geq7]$
$\ell=\{4,5\}$ $C_{(3,n,6,1)}$ $[41,17,d\geq11],[122,92,d\geq11]$
\
The codes in the second row of the table are optimal linear codes from Database.\
Let $n=(q^m-1)/2(q-1)$ be an integer. For studying we give the following definition. Put $\delta\geq2$ be an integer $$K_{(q,n,\delta)}=\bigcup_{0\leq i\leq \delta-1}T_{1+2i}$$ and $$-K_{(q,n,\delta)}=\{2n-a|a\in K_{(q,n,\delta)}\}$$ where $T_{1+2i}=T\cap C_{1+2i}$, and $C_{1+2i}$ is the cyclotomic coset modulo $2n$.\
**Lemma 5.1.$^{[1]}$** Let $\delta=\lceil\frac{q^e}{2}\rceil$, where $e=\lfloor \frac{m-1}{2}\rfloor$, then $K_{(q,n,\delta)}\cap(-K_{(q,n,\delta)})=\emptyset$.\
**Theorem 5.2.** Let $1\leq \delta\leq \frac{q^{\lfloor(m-1)/2\rfloor}+1}{2}$ be an integer. Then the dimension of the negacyclic BCH code $C_{(q,n,2\delta+1,1-2\delta)}$ of length $n=(q^m-1)/2(q-1)$ is equal to $$k=n-2m\large\large\lceil(2\delta-1)(q-1)/2q\rceil,$$\
and the minimum distance of code $d\geq2\delta+1$.\
**Proof**. Let the negacyclic BCH code $C_{(q,n,\delta+1,1)}$ has generator polynomial $g_{(q,n,\delta+1,1)}(x)$, by Lemma 2.4, we get $$\textrm{deg}g_{(q,n,\delta+1,1)}(x)=m\large\large\lceil(2\delta-1)(q-1)/2q\rceil.$$ The generator polynomial $g_{(q,n,2\delta+1,1-2\delta)}(x)$ of the code $C_{(q,n,2\delta+1,1-2\delta)}$ is given by $$g_{(q,n,2\delta+1,1-2\delta)}(x)=\textrm{lcm}(g_{(q,n,\delta+1,1)}(x),g_{(q,n,\delta+1,1)}^\ast(x)),$$ which $g_{(q,n,\delta+1,1)}^\ast(x))$ is the reciprocal polynomial of $g_{(q,n,\delta+1,1)}(x)$.\
Since $1\leq \delta\leq \frac{q^{\lfloor(m-1)/2\rfloor}+1}{2}$, from Lemma 5.1, then $$g_{(q,n,2\delta+1,1-2\delta)}(x)=g_{(q,n,\delta+1,1)}(x)g_{(q,n,\delta+1,1)}^\ast(x).$$ Hence the code $C_{(q,n,2\delta+1,1-2\delta)}$ has dimension $$k=n-2m\large\large\lceil(2\delta-1)(q-1)/2q\rceil.$$ And by the negacyclic BCH bound, we have $d\geq2\delta+1$.\
**Example 5.3.** From the Theorem 5.2, we obtain the following table\
code parameters
------------------------ --------------------------------- ------------------
$(q,m,\delta)=(3,4,2)$ $C_{(q,n,2\delta+1,1-2\delta)}$ $[20,12,d\geq5]$
$(q,m,\delta)=(5,4,3)$ $C_{(q,n,2\delta+1,1-2\delta)}$ $[78,62,d\geq7]$
$(q,m,\delta)=(5,4,4)$ $C_{(q,n,2\delta+1,1-2\delta)}$ $[78,54,d\geq9]$
\
\
**Lemma 5.4.** Let $m\geq0$ be an even integer, $\delta=\frac{q^{m/2}+1}{2}$, put $\ell=(q^{m/2}-1)/(q-1)$ when $1\leq s\leq q-1$ and $s\ell$ is an odd integer, then $s\ell$ is the coset leader of $T_{s\ell}$, $T_{s\ell}=-T_{s\ell}$ and $|T_{s\ell}|=m$, what is more, $$K_{(q,n,\delta)}\cap(-K_{(q,n,\delta)})=\bigcup_{1\leq s\leq q-1}T_{s\ell},$$ where $s\ell$ is an odd integer.\
**Proof**. It follows from the definition of $T_i=T\cap C_i$ and \[1\] Lemma 27.\
**Lemma 5.5.** Let $m\geq4$ be an even integer and $n=(q^{m/2}-1)/2(q-1)$. Put $a$ is an odd integer, $q^{(m-2)/2}\leq a\leq q^{m/2}$ and $a\not\equiv0\ \textrm{mod}q$. Then\
(1) When $q=3$, $a$ must be a coset leader.\
(2) When $q\geq3$ is an odd integer, $a$ is a coset leader except that $a=1+i+i\frac{q^{m/2}-q}{q-1}, i\in I$.
1\) When $m=4s, s\in \mathbb{Z}^\ast, I=\large\{\frac{q+1}{2},\frac{q+1}{2}+1,\cdots,\frac{q+1}{2}+\frac{q-5}{2}\}$.
2\) When $m=4s+2, s\in \mathbb{Z}^\ast$, i). $q=4t+1,t\in \mathbb{Z}^\ast, I=\large\{\frac{q+1}{2}+1,\frac{q+1}{2}+3,\cdots,\frac{q+1}{2}+\frac{q-7}{2}\}$
ii). $q=4t-1,t\in \mathbb{Z}^\ast, I=\large\{\frac{q+1}{2},\frac{q+1}{2}+2,\cdots,\frac{q+1}{2}+\frac{q-7}{2}\}$\
what is more, $a$ is a coset leader, then $$\begin{aligned}
|T_a|=
\left\{ {{\begin{array}{ll}
{\frac{m}{2}}, & {\textrm{if} \ a=\frac{q^m+1}{2}},\\
{m}, & {\textrm{otherwise.}} \\
\end{array} }} \right .\end{aligned}$$\
**Proof**. By the definition of $T_i=T\cap C_i$, we get the case which $a$ is an odd integer in the Lemma $28^{[1]}$. By binomial theorem, we deduce if $q=4t+1, t\in \mathbb{Z}^\ast$,$$\frac{q^{m/2}-q}{q-1}=\frac{m}{2}-1+\binom{\frac{m}{2}}{2}4t+\cdots+4t^{(\frac{m}{2}-1)}$$ and if $q=4t-1, t\in \mathbb{Z}^\ast$, $$\frac{q^{m/2}-q}{q-1}=\frac{m}{2}-1+\binom{\frac{m}{2}}{2}(4t-2)+\cdots+(4t-2)^{(\frac{m}{2}-1)}$$ Again classify to $m$, we will obtain the desired conclusion of lemma.\
**Theorem 5.6.** Let $1\leq \delta\leq \frac{q^{m/2}+1}{2}$ be an integer and $m\geq4$ be an even integer. Define $\omega=\lfloor\frac{2(\delta-1)(q-1)}{q^{m/2}-1}\rfloor$. Then the negacyclic BCH code $C_{(q,n,\delta+1,1)}$ of length $n=(q^m-1)/2(q-1)$ has minimum distance $d\geq\delta$, and dimension\
(1) if $\omega<\lfloor\frac{q-1}{2}\rfloor$, $k=n-m\large\lceil(2\delta-1)(q-1)/2q\rceil$\
(2) if $\omega\geq\lfloor\frac{q-1}{2}\rfloor$, $s,t\in \mathbb{Z}^\ast$, $$\begin{aligned}
k=
\left\{ {{\begin{array}{ll}
{n-m\large\large\lceil(2\delta-1)(q-1)/2q\rceil+(2\omega-q+2)\frac{m}{2}}, & {\textrm{if}\ m=4s}, \\
{n-m\large\large\lceil(2\delta-1)(q-1)/2q\rceil+(\omega-(q-1)/2)\frac{m}{2}}, & {\textrm{if}\ m=4s+2,q=4t+1,\omega\ \textrm{is odd}},\\
{n-m\large\large\lceil(2\delta-1)(q-1)/2q\rceil+(\omega-(q-3)/2)\frac{m}{2}}, & {\textrm{if}\ m=4s+2,q=4t+1,\omega\ \textrm{is even}},\\
& {\textrm{or}\ m=4s+2,q=4t-1,\omega\ \textrm{is odd}},\\
{n-m\large\large\lceil(2\delta-1)(q-1)/2q\rceil+(\omega-(q-5)/2)\frac{m}{2}}, & {\textrm{if}\ m=4s+2,q=4t-1,\omega\ \textrm{is even}}.\\
\end{array} }} \right .\end{aligned}$$\
**Proof**. (1) If $\omega<\frac{q-1}{2}$, it means $2(\delta-1)\leq(q^{m/2}-1)/2$. By the Lemma 2.4 and 5.5, we obtain that every odd integer $a$ with $1\leq a\leq2\delta-1$, $a\not\equiv0\ \textrm{mod}q$ is a coset leader with $|T_a|=m$, hence $k=n-m\large\large\lceil(2\delta-1)(q-1)/2q\rceil$\
(2) If $\omega\geq\frac{q-1}{2}$, again by Lemma 2.4 and 5.5. Let $s,t\in \mathbb{Z}^\ast$, we know, $$\mid\{\ b\mid b\ \textrm{is odd integer and noncoset leader with}\ (q^{m/2}+1)\leq b\leq2\delta-1\ \}\mid=\mid I\mid$$.
1\) $m=4s$, $|I|=\omega-\frac{q+1}{2}+1=\omega-\frac{q-1}{2}$
2\) $m=4s+2$,
i). $q=4t+1$, and $\omega$ is even, then $|I|=\frac{\omega-\frac{q+1}{2}-1}{2}+1=\frac{2\omega-q+1}{4}$
ii). $q=4t+1$, and $\omega$ is odd, then $|I|=\frac{\omega-\frac{q+1}{2}-1-1}{2}+1=\frac{2\omega-q-1}{4}$
iii). $q=4t-1$, and $\omega$ is even, then $|I|=\frac{\omega-\frac{q+1}{2}}{2}+1=\frac{2\omega-q+3}{4}$
iv). $q=4t-1$, and $\omega$ is odd, then $|I|=\frac{\omega-\frac{q+1}{2}-1}{2}+1=\frac{2\omega-q+1}{4}$\
Again by Lemma 2.4 and 5.5, when $a$ is odd with $1\leq a\leq2\delta-1$, the number of coset leader is equal to $\lceil(2\delta-1)(q-1)/2q\rceil-|I|$. And put $\widetilde{a}=(q^{m/2}+1)/2\in T$, $|T_{\widetilde{a}}|=\frac{m}{2}$, then the proof of the theorem is completed.\
**Theorem 5.7.** Let $\delta=\frac{q^{m/2}+1}{2}$ be an integer and $m\geq4$ be an even integer, then the negacyclic BCH code $C_{(q,n,\delta,1)}$ with length $n=(q^m-1)/2(q-1)$ has minimum distance $d\geq\delta$ and dimension\
$$\begin{aligned}
k=
\left\{ {{\begin{array}{ll}
{n-\large\frac{m(q-1)}{2}q^{(m-2)/2}+\frac{qm}{2}}, & {\textrm{if}\ m=4s},\\
{n-\large\frac{m(q-1)}{2}q^{(m-2)/2}+\frac{(q+1)m}{4}}, & {\textrm{if}\ m=4s+2,q=4t+1},\\
{n-\large\frac{m(q-1)}{2}q^{(m-2)/2}+\frac{(q+3)m}{4}}, & {\textrm{if}\ m=4s+2,q=4t-1}.\\
\end{array} }} \right .\end{aligned}$$\
where $s,t\in \mathbb{Z}^\ast$\
**Proof**. In this case, $\lceil(2\delta-1)(q-1)/2q\rceil=\frac{(q-1)}{2}q^{(m-2)/2}$, $\omega=q-1>\frac{q-1}{2}$ is a even integer, the proof of dimension follows from Theorem 5.6, and by the negacyclic BCH bound , we have minimum distance $d\geq\delta$.\
**Theorem 5.8.** Let $1\leq \delta\leq \frac{q^{m/2}+1}{2}$ be an integer and $m\geq4$ be an even integer. Define $\omega=\lfloor\frac{2(\delta-1)(q-1)}{q^{m/2}-1}\rfloor, \varpi=\lfloor\frac{(2\delta-1)(q-1)}{q^{m/2}-1}\rfloor$. Then the reversible negacyclic BCH code $C_{(q,n,2\delta+1,1-2\delta)}$ of length $n=(q^m-1)/2(q-1)$ has minimum distance $d\geq2\delta+1$ and dimension\
(1) if $\omega<\lfloor\frac{q-1}{2}\rfloor$, $$\begin{aligned}
k=
\left\{ {{\begin{array}{ll}
{n-m\large\lceil(2\delta-1)(q-1)/2q\rceil}, & {\textrm{if}\ m=4s},\\
{n-m\large\lceil(2\delta-1)(q-1)/2q\rceil+\lceil \varpi/2\rceil m}, & {\textrm{if}\ m=4s+2}. \\
\end{array} }} \right .\end{aligned}$$\
(2) if $\omega\geq\lfloor\frac{q-1}{2}\rfloor$, $s,t\in \mathbb{Z}^\ast$, $$\begin{aligned}
k=
\left\{ {{\begin{array}{ll}
{n-2m\large\large\lceil(2\delta-1)(q-1)/2q\rceil+(2\omega-q+2)m}, & {\textrm{if}\ m=4s}, \\
{n-2m\large\large\lceil(2\delta-1)(q-1)/2q\rceil+(\omega-(q-1)/2+\lceil \varpi/2\rceil)m}, & {\textrm{if}\ m=4s+2,q=4t+1,\omega\ \textrm{is odd}},\\
{n-2m\large\large\lceil(2\delta-1)(q-1)/2q\rceil+(\omega-(q-3)/2+\lceil \varpi/2\rceil)m}, & {\textrm{if}\ m=4s+2,q=4t+1,\omega\ \textrm{is even}},\\
& {\textrm{or}\ m=4s+2,q=4t-1,\omega\ \textrm{is odd}},\\
{n-2m\large\large\lceil(2\delta-1)(q-1)/2q\rceil+(\omega-(q-5)/2+\lceil \varpi/2\rceil)m}, & {\textrm{if}\ m=4s+2,q=4t-1,\omega\ \textrm{is even}}.\\
\end{array} }} \right .\end{aligned}$$\
**Proof**. Let the negacyclic BCH code $C_{(q,n,\delta+1,1)}$ has generator polynomial $g_{(q,n,\delta+1,1)}(x)$, then we have\
1) if $\omega<\frac{q-1}{2}$, deg$g_{(q,n,\delta+1,1)}(x)=m\large\lceil(2\delta-1)(q-1)/2q\rceil$\
2) if $\omega\geq\frac{q-1}{2}$ and $s,t\in \mathbb{Z}^\ast$, $$\begin{aligned}
\textrm{deg}g_{(q,n,\delta+1,1)}(x)=
\left\{ {{\begin{array}{ll}
{m\lceil(2\delta-1)(q-1)/2q\rceil+(2\omega-q+2)\frac{m}{2}}, & {\textrm{if}\ m=4s}, \\
{m\lceil(2\delta-1)(q-1)/2q\rceil+(\omega-(q-1)/2)\frac{m}{2}}, & {\textrm{if}\ m=4s+2,q=4t+1,\omega\ \textrm{is odd}},\\
{m\lceil(2\delta-1)(q-1)/2q\rceil+(\omega-(q-3)/2)\frac{m}{2}}, & {\textrm{if}\ m=4s+2,q=4t+1,\omega\ \textrm{is even}},\\
& {\textrm{or}\ m=4s+2,q=4t-1,\omega\ \textrm{is odd}},\\
{m\lceil(2\delta-1)(q-1)/2q\rceil+(\omega-(q-5)/2)\frac{m}{2}}, & {\textrm{if}\ m=4s+2,q=4t-1,\omega\ \textrm{is even}}.\\
\end{array} }} \right .\end{aligned}$$\
From the definition of $C_{(q,n,2\delta+1,1-2\delta)}$, which has generator polynomial $g_{(q,n,2\delta+1,1-2\delta)}(x)$,$$g_{(q,n,2\delta+1,1-2\delta)}(x)=\textrm{lcm}(g_{(q,n,\delta+1,1)}(x),g_{(q,n,\delta+1,1)}^\ast(x))$$ where $g_{(q,n,\delta+1,1)}^\ast(x)$ is the reciprocal polynomial of $g_{(q,n,\delta+1,1)}(x)$.\
It follows from Lemma 5.4, we deduce, $$\begin{aligned}
\textrm{deg}(\textrm{gcd}(g_{(q,n,\delta+1,1)}(x),g_{(q,n,\delta+1,1)}^\ast(x)))=
\left\{ {{\begin{array}{ll}
{0}, & {\textrm{if}\ m=4s},\\
{\lceil\varpi/2\rceil}, & {\textrm{if}\ m=4s+2}. \\
\end{array} }} \right .\end{aligned}$$\
Hence, $$\textrm{deg}g_{(q,n,2\delta+1,1-2\delta)}(x)=2\textrm{deg}g_{(q,n,\delta+1,1)}(x)-\textrm{deg}(\textrm{gcd}(g_{(q,n,\delta+1,1)}(x),g_{(q,n,\delta+1,1)}^\ast(x))).$$ Then the dimension can be obtained, and by the negacyclic BCH bound, we have minimum distance $d\geq2\delta+1$.\
**Theorem 5.9.** Let $\delta=\frac{q^{m/2}+1}{2}$ be an integer and $m\geq4$ be an even integer, then the negacyclic BCH code $C_{(q,n,2\delta+1,1-2\delta)}$ with length $n=(q^m-1)/2(q-1)$ has minimum distance $d\geq2\delta+1$ and dimension\
$$\begin{aligned}
k=
\left\{ {{\begin{array}{ll}
{n-(q-1)q^{(m-2)/2}m+(q+1)m}, & {\textrm{if}\ m=4s+2, q=4t-1},\\
{n-(q-1)q^{(m-2)/2}m+qm}, & {\textrm{otherwise}}.\\
\end{array} }} \right .\end{aligned}$$\
Where $s, t\in \mathbb{Z}^\ast$\
**Proof**. In this case, $\lceil(2\delta-1)(q-1)/2q\rceil=\frac{(q-1)}{2}q^{(m-2)/2}$, $\omega=q-1>\frac{q-1}{2}$ is an even integer, and $\varpi=q-1$. The dimension follows from Theorem 5.8, and by the negacyclic BCH bound, we have minimum distance $d\geq2\delta+1$.\
**Example 5.10.** From the Theorem 5.9, we have following table\
code parameters
------------------------- --------------------------------- --------------------
$(q,m,\delta)=(3,6,14)$ $C_{(q,n,2\delta+1,1-2\delta)}$ $[182,98,d\geq29]$
$(q,m,\delta)=(5,4,13)$ $C_{(q,n,2\delta+1,1-2\delta)}$ $[78,18,d\geq27]$
\
\
Let $n=\frac{q^{t\cdot2^\tau}-1}{2(q^t+1)}$ be an integer, where $t,\tau\in \mathbb{Z}^\ast, \tau\geq2, t\geq2$. We study those negacyclic BCH codes of length $n$.\
**Lemma 5.11.** Let $m=\textrm{ord}_{2n}(q)$, then $m=t\cdot2^\tau$.\
**Proof**. Since $2n\mid (q^{t\cdot2^\tau}-1)$, then $m\mid t\cdot2^\tau$. We get $m\in\{2^i,\ t,\ t\cdot2^i \mid 0\leq i\leq\tau\}$. Write $2n$ to the base $q$ as $2n=\sum_{j=0}^{s_{2n}}jq^j$, where $0\leq j\leq q-1$. As $2n=\frac{q^{t\cdot2^\tau}-1}{q^t+1}\geq\frac{q^{t\cdot2^\tau}-1}{2q^t}$, then $s_{2n} \geq (t\cdot2^\tau-t-1)$. And since $2n\mid (q^m-1)$, hence $m\geq(t\cdot2^\tau-t-1)> t\cdot2^{\tau-1}$. Thus $m=t\cdot2^\tau$.\
**Lemma 5.12.** For any odd integer $s\in T$, $s\leq q^{\lfloor (t\cdot2^{\tau-1}-1)/2\rfloor}+1$ and $s\not\equiv0\ \textrm{mod}q$ is a coset leader and $|T_s|=t\cdot2^\tau$.\
**Proof**. It follows from the Lemma 4.2.\
**Theorem 5.13.** Let $n=\frac{q^{t\cdot2^\tau}-1}{2(q^t+1)}$, where $t,\tau\in \mathbb{Z}^\ast, \tau\geq2, t\geq2$. For any integer $\delta$, $1\leq\delta\leq \frac{q^{\lfloor (t\cdot2^{\tau-1}-1)/2\rfloor+1}}{2}$, then the dimension of negacyclic BCH code $C_{(q,n,2\delta+1, 1-\delta)}$, $$k=n-t\cdot2^{\tau+1}\large\large\lceil(2\delta-1)(q-1)/2q\rceil,$$ and $d\geq2\delta+1$.\
**Proof**. From Theorem 5.2, Lemma 5.11 and 5.12, we can obtain the conclusion immediately.\
**Example 5.14.** We get the following table\
code parameters
------------------------------- --------------------------------- --------------------
$(q,t,\tau,\delta)=(3,2,2,2)$ $C_{(q,n,2\delta+1,1-2\delta)}$ $[328,312,d\geq5]$
$(q,t,\tau,\delta)=(3,2,2,3)$ $C_{(q,n,2\delta+1,1-2\delta)}$ $[328,296,d\geq7]$
\
In this section, we study the MDS LCD negacyclic codes over $\mathbb{F}_q$. A $[n,k,d]$ code is called maximum distance separable (abbreviated MDS) if $d=n-k+1$. Let $\beta$ be a primitive $2n-$th root of unity in $\mathbb{F}_{q^m}$, where $m=\textrm{ord}_{2n}(q)$. Let $C$ be a negacyclic code over $F_q$ of length $n$. Let $S$ be the defining set of $C$. Then $S$ is the union of some $T_i$, where $T_i$ is defined above. Then we have following theorem.\
**Theorem 6.1.** Let $n$ be an even integer and $n\mid(q-1)$. Let $C$ be the negacyclic code over $F_q$ of length $n$ with defining set $S\subset T$. Put $0\leq\rho<\frac{n}{2}-1$. Then the $C$ is a MDS LCD negacyclic code if $$S=\{1+2i\mid i=\frac{n}{2}+j\ \textrm{and}\ \frac{n}{2}-k, 0\leq j\leq\rho,\ 1\leq k\leq \rho+1\}\ (\textrm{mod}2n)$$ moreover $C$ has parameters $[n,n-2(\rho+1),2\rho+3]$\
**Proof**. We can easily get that $S$ has $2(\rho+1)$ elements, and then the dimension of $C$ is $n-2(\rho+1)$. Put $A=\{i\mid i=\frac{n}{2}+j\ \textrm{and}\ \frac{n}{2}-k, 0\leq j\leq\rho,\ 1\leq k\leq \rho+1\}$. Then $A$ has $2(\rho+1)$ consecutive integers. From the negacyclic BCH bound and the singleton bound, we can get that the minimum distance of $C$ is $2\rho+3$, and the code $C$ is a MDS code.
Next we illustrate that the code $C$ is a LCD code. $\forall\ a\in S$, there exist two integer $j_0$,$k_0$. $0\leq j_0\leq\rho$ or $1\leq k_0\leq\rho+1$ such that $a=n+2j_0+1$ or $a=n-2k_0+1$. If $a=n+2j_0+1$, then $2n-a=1+2(\frac{n}{2}-(j_0+1))$, and $1\leq j_0+1\leq(\rho+1)$. Hence, $2n-a\in S$. If $a=n-2k_0+1$, then $2n-a=1+2(\frac{n}{2}+(k_0-1))$, and $0\leq k_0-1\leq\rho$. Hence, $2n-a\in S$. Summarizing the two cases get $\forall\ a\in S$, then $2n-a\in S$. By Lemma 3.4, we deduce that $C$ is a LCD code.\
**Example 6.2.** We list some MDS LCD negacyclic codes over some finite fields in the following table.\
$q$ $n$ $\rho$ parameters $q$ $n$ $\rho$ parameters $q$ $n$ $\rho$ parameters
----- ----- -------- ------------ ----- ----- -------- ------------- ----- ----- -------- -------------
5 4 0 $[4,2,3]$ 11 10 3 $[10,2,9]$ 17 8 1 $[8,4,5]$
7 6 0 $[6,4,3]$ 13 6 0 $[6,4,3]$ 17 8 2 $[8,2,7]$
7 6 1 $[6,2,5]$ 13 6 1 $[6,2,5]$ 17 16 0 $[16,14,3]$
9 4 0 $[4,2,3]$ 13 12 0 $[12,10,3]$ 17 16 1 $[16,12,5]$
9 8 0 $[8,6,3]$ 13 12 1 $[12,8,5]$ 17 16 2 $[16,10,7]$
9 8 1 $[8,4,5]$ 13 12 2 $[12,6,7]$ 17 16 3 $[16,8,9]$
9 8 2 $[8,2,7]$ 13 12 3 $[12,4,9]$ 17 16 4 $[16,6,11]$
11 10 0 $[10,8,3]$ 13 12 4 $[12,2,11]$ 17 16 5 $[16,4,13]$
11 10 1 $[10,6,5]$ 17 4 0 $[4,2,3]$ 17 16 6 $[16,2,15]$
11 10 2 $[10,4,7]$ 17 8 0 $[8,6,3]$
\
We have studied the reversible negacyclic codes over finite fields. In section 2, we deduce the condition of reversible negacycylic codes. In section 3, the structure of LCD negacyclic codes is determined, and in the special case, the quantity of reversible negacyclic codes is gained. In section 4 and 5, we discuss the parameters of negacyclic BCH codes when the length $n=\frac{q^\ell+1}{2}$ , $n=\frac{q^m-1}{2(q-1)}$ and $n=\frac{q^{t\cdot2^\tau}-1}{2(q^t+1)}$. In section 6, we study a class of MDS LCD negacyclic codes.\
\
\
\
[99]{} C. S. Ding, C. J. Li and S. X. Li, LCD cyclic Codes over Finite Fields, arXiv:1608.02170v1\[cs.IT\].
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G. G. La Guardia, On negacyclic MDS-convolutional codes, Linear Algebra and its Applications, 448 (2014), 85-96.
C. J. Li, C. S. Ding and H. Liu, Parameters of two classes of LCD BCH codes, arXiv:1608.02670v1\[cs.IT\].
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X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual, Discrete Math, vol. 126(1994), pp.391-393.
N. Sendrier, Linear codes with complementary duals meet the Gilbert-Varshamov bound, Discrete Math, vol. 285(2004), pp.345-347. X. Hou, and F. Oggier, On LCD codes and lattices, IEEE International Symposium on Information Theory, 2016, pp.1501-1505.
E. R. Lina Jr. and E. G. Nocon, On the construction of some LCD codes over finite fields, the DLSU Research Congress, vol.4(2016).
H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans, Inf. Theory, vol.50, No.8(2004), pp.1728-1744. Y. S. Yang and W. C. Cai, On self-dual constacyclic codes over finite fields, Des. Codes Cryptogr, vol.74(2015), pp.335-364. C. Güneri, B. Özkaya and P. Solé, Quasi-cyclic complementary dual codes, Finite Fields and Their Applications, vol.42 (2016), pp.67-80.
[^1]: E-mail addresses: [email protected](S.Zhu), [email protected](B.Pang), [email protected](Z. Sun). This research is supported by the National Natural Science Foundation of China (No.61370089; No.61572168).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Subsurface projection is indispensable to studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called *relative twisting*. We give two alternate versions of relative twisting for the outer automorphism group of a free group. We use this to describe sufficient conditions for when a folding path enters the [*thin*]{} part of Culler-Vogtmann’s Outer space. As an application of our condition, we produce a sequence of fully irreducible outer automorphisms whose axes in Outer space travel through graphs with arbitrarily short cycles; we also describe the asymptotic behavior of their translation lengths.'
address:
- |
Dept. of Mathematics\
Allegheny College\
Meadville, PA 16335
- |
Dept. of Mathematics\
University of British Columbia\
Vancouver, BC V6T 1Z2
author:
- Matt Clay
- Alexandra Pettet
bibliography:
- 'bibliography.bib'
title: Relative twisting in Outer space
---
[^1]
Introduction {#sc:intro}
============
Culler and Vogtmann gave the first account of [*Outer space $CV_k$*]{} in their 1986 paper [@Culler-Vogtmann]: elements are finite marked projectivized metric graphs with fundamental group $F_k$, the rank $k$ non-abelian free group, and two graphs are close when the lengths of some finite collection of elements of $F_k$ are close. By considering the universal covers of the marked graphs, $CV_k$ is also described as the space of free simplicial minimal isometric actions of $F_k$ on ${\mathbb{R}}$–trees. Topologically, $CV_k$ has the structure of a contractible simplicial complex (missing some faces) on which $\operatorname{Out}F_k$ acts properly and simplicially by changing markings. Metrically, however, Outer space remains largely a mystery. Much of the conjectural picture for Outer space geometry comes from Teichmüller theory, where the Teichmüller metric, the Weil-Petersson metric, and the Thurston metric have been defined and extensively studied. Unfortunately Outer space lacks much of the structure that paves the way for these metrics; perhaps most notably, $CV_k$ is [*not*]{} a manifold.
Of the three metrics on Teichmüller space mentioned above, only the third, the Thurston metric, has been interpreted in the Outer space setting; there it is more commonly referred to as the [*Lipschitz metric*]{}. Features of this metric were recorded by Francaviglia-Martino in [@un:FM1; @un:FM2].
Algom-Kfir (see also Hamenstädt [@un:Hamenstadt]) proved that axes of fully irreducible elements of $\operatorname{Out}F_k$ are [*strongly contracting*]{}, so that $CV_k$ exhibits a characteristic of negative curvature in these directions. Her result was anticipated by a theorem of Minsky [@ar:Minsky], which showed that Teichmüller geodesics contained in the $\epsilon$-[*thick*]{} part of Teichmüller space are strongly contracting, uniformly depending on $\epsilon$. Algom-Kfir’s contraction constants depend on the outer automorphisms to which they belong. The question of whether these constants only depend on the geometry of the graphs along the axes has not been addressed.
For $\epsilon > 0$, we define $CV_k^\epsilon$ as the subset of $CV_k$ consisting of graphs that contain a cycle of length less than $\epsilon$. We should perhaps resist calling $CV_k^\epsilon$ the “thin part” of Outer space as it is not clear that Algom-Kfir’s theorem extends uniformly to geodesics in the complement of $CV_k^\epsilon$. Nevertheless, this set does hold some nice properties analogous to those of the thin part of Teichmüller space; for instance, the cusps of the quotient $CV_k/ \operatorname{Out}F_k$ are contained in $CV_k^\epsilon$, and the quotient $(CV_k -CV_k^\epsilon)/ \operatorname{Out}F_k$ of the complement is quasi-isometric to $\operatorname{Out}F_k$.
The main results of this paper provide conditions, akin to those of Rafi [@ar:Rafi05] in the setting of Teichmüller space, that guarantee that a geodesic or an axis of a fully irreducible element travel through $CV_k^\epsilon$. Our criteria are based on a notion of [*relative twisting*]{} in Outer space. We come at this from two different points of view, each motivated by the quest to find satisfactory analogues of subsurface projection and relative twisting from the theory of mapping class groups [@ar:MM00; @ar:FLM01]. [**Geometric:**]{} Our first approach to relative twisting directly adapts the original geometric definition to free groups. We give a pairing $\tau_a(G,G')$ between two graphs $G,G' \in
CV_k$ relative to some nontrivial $a \in F_k$, which we define by means of the *Guirardel core* [@ar:Gu05]. This is a certain 2–complex associated to the graphs that provides a means of selecting a geometry for $F_k$ that “sees” both $G$ and $G'$. We obtain a condition on the graphs that, when satisfied, enables us to construct a connecting geodesic between them, traveling through $CV_k^\epsilon$.
Suppose $G,G' \in CV_k$ with $d = d_L(G,G')$ such that $\tau_a(G,G')
\geq n+2$ for some $a \in F_k$. Then there is a geodesic $\alpha{\colon\thinspace}[0,d] \to CV_k$ such that $\alpha(0) = G$ and $\alpha(d) = G'$ and for some $t \in [0,d]$, we have $\ell_{\alpha(t)}(a) \leq 1/n$. In other words, $\alpha([0,d]) \cap CV_k^{1/n} \neq \emptyset$.
As a corollary, we get the following lower bound the distance between two marked graphs in $CV_k$.
Suppose $G,G' \in CV_k$ and $G'$ does not have a cycle of length less than $\epsilon$. Then: $$d_L(G,G') \geq \log \sup_{1 \neq a \in F_k} \epsilon\tau_a(G,G')$$
Let $a \in F_k$ be nontrivial. If $\tau_a(G,G') \geq n$, then by Theorem \[th:geo-twist\], there is a geodesic $\alpha{\colon\thinspace}[0,d] \to
CV_k$ such that $\alpha(0) = G$ and $\alpha(d) = G'$, and for some $t
\in [0,d]$ that $\ell_{\alpha(t)}(a) \leq 1/n$. As $G'$ does not have a cycle of length less than $\epsilon$, it will follow from Proposition \[prop:loops\] that $d_L(\alpha(t),G') \geq
\frac{\epsilon}{1/n} = \epsilon n$. As $\alpha(t)$ is on a geodesic from $G$ to $G'$, the corollary holds.
The similar lower bound for Teichmüller space is a special case of a theorem of Rafi [@ar:Rafi07].
[**Algebraic:**]{} The second point of view to relative twisting gives a pairing $\tau_a(T,\Lambda)$ between a tree $T \in \overline{CV}_k$ and an algebraic lamination $\Lambda$ of $F_k$ relative to some nontrivial $a \in F_k$. This pairing measures how the axes of $a$ in $T$ overlap with the leaves of the lamination. It is similar to the notion of “twisting” used by Alibegović [@ar:Al02]. We obtain a criterion that implies that the axis of a fully irreducible element travels through $CV_k^\epsilon$ in terms of its unstable tree and lamination.
Suppose $\phi \in \operatorname{Out}F_k$ is fully irreducible, with unstable tree $T_-$ and lamination $\Lambda_-$ such that $\tau_a(T_-,\Lambda_-) \geq n+4$ for some $a \in F_k$. Then given any train-track $G$, there is an axis ${\mathcal{L}}_\phi$ for $\phi$ that contains $G$ and a graph $G_0$ with $\ell_{G_0}(a) \leq 1/n$. In other words, ${\mathcal{L}}_\phi
\cap CV_k^{1/n} \neq \emptyset$.
As an application of Theorem \[th:alg-twist\], we examine outer automorphisms of $F_k$ that are products of powers of two Dehn twists $\delta_1$ and $\delta_2$ which “fill” in an appropriate sense. We show (Section \[sc:example\]) that axes for $\delta_1^n\delta_2^{-n}$ travel through graphs with a cycle of length $\sim 1/n$. Moreover, we can estimate their translation lengths on $CV_k$; we compute that they grow logarithmically in $n$ (Theorem \[th:twist-translation\]).
The proofs of Theorems \[th:geo-twist\] and \[th:alg-twist\] are similar. In both cases we show that large relative twist implies the existence of a certain path that contains a large power of $a$ (Propositions \[prop:ntwist-vp\] and \[prop:ncover-vp\]). These paths, called vanishing paths, are folded, either in the map $G \to
G'$ or in a train-track map representing $\phi$, and are homotopically trivial in the image. The most efficient way to fold over a loop representing $a$ several times is to first make $a$ a short loop (Proposition \[prop:vpimpliessmallcycle\]).
It appears likely that our definition of algebraic twist (at least as used in Theorem \[th:alg-twist\]) is a special case of our definition of geometric twist. We anticipate investigating this relationship in a further paper.
The paper is organized as follows. In Section \[sc:prelim\] we review some of the basic theory of Outer space and the Lipschitz metric, irreducible outer automorphisms, train-track maps, and laminations; only Section \[ssc:nvp\] contains some new material. As this section is already lengthy, some background, such as a summary of currents for free groups, is suppressed until it is needed in Section \[sc:example\]. In Section \[sc:gtwist\], following an outline of some properties of Guirardel’s core and a brief review of relative twisting for the mapping class group, the first, “geometric,” analogue of relative twisting for $\operatorname{Out}F_k$ is given. Section \[sc:atwist\] is concerned with the second, “algebraic,” notion of relative twisting. Each of Sections \[sc:gtwist\] and \[sc:atwist\] conclude with a proposition essential to the proofs of the main theorems, found in Section \[sc:smallcycle\]. In Section \[sc:example\], we bring together results from Section \[sc:smallcycle\] and previous papers of the authors [@un:CP2; @ar:CP] to describe a method for constructing geodesic axes of fully irreducible elements which enter the thin part of Outer space.
[**Acknowledgments**]{} The authors would like to thank Kasra Rafi and Juan Souto for helpful conversations concerning this project. Additionally, the authors thank the referee for a careful reading of this work with helpful suggestions.
Preliminaries {#sc:prelim}
=============
Outer space {#ssc:cv}
-----------
We begin by fixing a generating set of the free group $F_k = \langle
x_1, \ldots, x_k \rangle$. Let $G$ be a simplicial graph, i.e., a one-dimensional cell complex, with $\pi_1(G)$ isomorphic to $F_k$. Let $R$ be a wedge of $k$ (oriented) circles, with each circle identified to one of the generators of $F_k$. Then by a [*marking*]{} of $G$ we mean a homotopy equivalence $\rho{\colon\thinspace}R \to G$. From the map $\rho_\ast{\colon\thinspace}\pi_1(R) \to \pi_1(G)$, we then have an identification of $F_k$ with $\pi_1(G)$. Given two marked metric graphs $\rho_1{\colon\thinspace}R
\to G_1$ and $\rho_2{\colon\thinspace}R \to G_2$, a map $f: G_1 \to G_2$ is a [ *change of marking*]{} if it is linear on edges, and if $f \circ
\rho_1{\colon\thinspace}R \to G_2$ is homotopic to $\rho_2{\colon\thinspace}R \to G_2$. A [ *topological representative*]{} of $\phi \in \operatorname{Out}F_k$ is a marked graph $\rho {\colon\thinspace}R \to G$, together with a self homotopy equivalence $g{\colon\thinspace}G
\to G$, so that the homotopy equivalence $\rho{^{-1}}\circ g \circ \rho
{\colon\thinspace}R \to R$ induces $\phi$ on $\pi_1(R) = F_k$.
We denote by $cv_k$ the [*unprojectivized (Culler–Vogtmann) Outer space*]{} consisting of marked metric graphs $G$, where $\pi_1(G) =
F_k$ and the degree of every vertex of $G$ is at least 3. Two points $\rho_1{\colon\thinspace}R \to G_1$ and $\rho_2{\colon\thinspace}R \to G_2$ in $cv_k$ are equivalent if there is an isometry $\iota{\colon\thinspace}G_1 \to G_2$ so that $\iota \circ \rho_1$ is homotopic to $\rho_2$. An alternate description of $cv_k$ is as the space of free minimal isometric $F_k$–actions on simplicial trees, and we will alternate freely between treating Outer space as a space of trees and as a space of graphs. There is a right action of $\operatorname{Out}F_k$ given by precomposing the marking (or $F_k$–action) by a representative of the outer automorphism. Outer space is defined as the projectivization of $cv_k$: $CV_k = cv_k/{\mathbb{R}}_{>0}$; it can be identified with the subspace of $cv_k$ consisting of marked graphs whose edge lengths sum to 1.
To simplify the notation for elements in Outer space, we denote a marked metric graph $\rho{\colon\thinspace}R \to G$ simply by $G$. A [*path*]{} in $G$ is a continuous map $\alpha{\colon\thinspace}I \to G$, where $I$ is an interval of ${\mathbb{R}}$. For convenience, and when it is clear from context, $\alpha$ may denote either the map or its image in $G$; while $[\alpha]$ will denote the image of $\alpha$ after “pulling it tight,” i.e., the image of any immersed homotopy (relative endpoints) representative of $\alpha$. We then use $L_G(\alpha)$ to denote the length of $[\alpha]$ in $G$. For an element $a \in F_k$, we write $\ell_G(a)$ to denote the minimal length of a loop in $G$ representing the conjugacy class of $a$.
For points $x,y \in T$, we use $[x,y]$ to denote the image of the unique tight path connecting $x$ and $y$ in $T$. For an $F_k$–tree $T$ and an element $a \in F_k$, we write $\ell_T(a)$ to denote the minimal translation length of $a$ in $T$. If $\ell_T(a) \neq 0$, then $a$ has an invariant axis $T^{{\langle}a {\rangle}}$ and $\operatorname{vol}(T^{{\langle}a {\rangle}}/{\langle}a {\rangle}) =
\ell_T(a)$. If $G$ is a graph with fundamental group $F_k$, then $\widetilde{G}$ denotes the universal cover of $G$, with a chosen base point so that there is an $F_k$–action on $\widetilde{G}$. Clearly $\ell_G = \ell_{\widetilde G}$.
Using the description as a space of tree actions, $cv_k$ is topologized via the the *axes topology*. That is, a tree $T \in
cv_k$ is identified with a point in ${\mathbb{R}}^{F_k}$ by the coordinates $(\ell_T(g))_{g \in F_k}$ [@ar:CM87]. Cohen and Lustig proved that the space of very small actions on ${\mathbb{R}}$–trees contains the closure $\overline{cv}_k$ of $cv_k$ [@ar:CL95]. The converse, that every very small minimal action on an ${\mathbb{R}}$–tree is the limit of free minimal simplicial actions, was shown by Bestvina and Feighn [@un:BF]. Recall that an action of $F_k$ on an ${\mathbb{R}}$–tree is [ *very small*]{} if arc stabilizers are trivial or maximal cyclic, and the stabilizer of any tripod is trivial.
Given two points $G_1$ and $G_2$ in the projectivized Outer space $CV_k$, let $f{\colon\thinspace}G_1 \to G_2$ be a change of marking, and denote by $\sigma(f)$ the maximal slope of $f$ (recall that $f$ is linear on edges). We have the following proposition, due to White (see [@un:Algom-Kfir; @un:Bestvina]):
\[prop:loops\] Let $G_1,G_2$ be two graphs in $CV_k$. Then: $$\inf\{\sigma(f) \ | \ f{\colon\thinspace}G_1 \to G_2 \ \mbox{\rm change of marking} \} =
\sup_{1 \neq a \in F_k} \frac{\ell_{\widetilde{G}_2}(a)}{\ell_{\widetilde{G}_1}(a)}$$ Moreover both inf and sup are realized.
For $G_1$ and $G_2$ in $\operatorname{Out}F_k$, let $\sigma(G_1,G_2)$ be the value in Proposition \[prop:loops\]. We define a function $d_L: CV_k
\times CV_k \to {\mathbb{R}}_{\geq 0}$ by $$d_L(G_1,G_2) = \log \sigma(G_1,G_2).$$ Its only failure to be a distance is that it is not symmetric; it is not hard to construct examples of $G_1, G_2 \in CV_k$ with $d_L(G_1,G_2) \neq d_L(G_2,G_1)$ (see [@un:A-KB]). In spite of this anomaly, we will refer to $d_L$ as the [*Lipschitz metric*]{} on $CV_k$. We remark that it is known that the minimal Lipschitz constant, taken over all continuous maps $f{\colon\thinspace}G_1 \to G_2$ such that $f
\circ \rho_1$ is homotopic to $\rho_2$, is achieved by a map that is linear on edges ([@un:Algom-Kfir; @un:FM2]).
\[ex:twist\] We present an example of computing distances in $CV_k$ that will be relevant to those examples constructed in Section \[sc:example\]. Fix a basis ${\mathcal{T}}= {\mathcal{A}}\cup \{ t \}$ of $F_k$ and an element $c \in {\langle}{\mathcal{A}}{\rangle}$ that is cyclically reduced with respect to ${\mathcal{T}}$. Consider the Cayley tree $T$ and the marked graph $G = T/F_k$, metrized so that all edge lengths are equal to $1/k$.
Let $\delta$ be the automorphism that sends $t$ to $ct$ and acts as the identity on ${\langle}{\mathcal{A}}{\rangle}$. Then there is a change of marking map $f {\colon\thinspace}G \to G\delta^n$ defined by subdividing the edge corresponding to $t$ into $n+1$ edges and sending each of the first $n$ edges over the edge path for $c$ and the last edge over the edge corresponding to $t$. Therefore, the image of the edge $t$ has length: $$n\ell_{G}(c) + \frac{1}{k} = \frac{nk\ell_G(c)
+ 1}{k}$$ and hence the edge $t$ has been stretched by $nk\ell_G(c) + 1$. Since the edge corresponding to $t$ is the only edge stretched and since it is mapped to a tight loop we have that: $$d_L(G,G\delta^n) = \log (nk\ell_G(c) + 1).$$ In the terminology from the proof of Theorem 2.1 in [@un:Algom-Kfir], the loop $t$ is the subgraph $G_f \subset G$ and it is a legal loop.
The automorphism $\delta$ is an example of a [*Dehn twist automorphism*]{} (see Section \[sc:example\]); in this case corresponding to the Bass–Serre tree arising from the HNN-extension ${\langle}{\mathcal{A}},c_0, t \, | \, t{^{-1}}ct = c_0
{\rangle}$. We refer to such a tree as a *cyclic tree*. For the case of a cyclic tree dual to an amalgamated free product ${\langle}{\mathcal{A}}{\rangle}*_{{\langle}c {\rangle}} {\langle}c,{\mathcal{B}}{\rangle}$ and its associated Dehn twist ($a
\mapsto a$, $b \mapsto cbc{^{-1}}$), one can also show, using similar methods, that the distance from $G$ to $G\delta^n$ is approximately $\log n$. In this case though, the obvious map sending edges corresponding to elements $b \in {\mathcal{B}}$ to $c^nbc^{-n}$ is not the optimal map. Instead one sends $b$ to $c^{n/2}bc^{-n/2}$ and edges corresponding to elements $a \in {\mathcal{A}}$ to $c^{-n/2}ac^{n/2}$.
We remark for use in Section \[sc:example\], that $d_L(G\delta^n,G) = d_L(G,G\delta^n)$. Indeed, $d_L(G\delta^n,G) =
d_L(G,G\delta^{-n})$ and the same argument as above shows that $d_L(G,G\delta^{-n}) = \log(nk\ell_G(c^{-1}) + 1)$. But of course $\ell_G(c) = \ell_G(c^{-1})$.
Bounded backtracking {#ssc:bbt}
--------------------
Suppose that $f {\colon\thinspace}T \to T'$ is a continuous map, where $T$ and $T'$ are trees. We say that $f$ has *bounded backtracking* if there is a constant $C$ such that for any path $[x,y] \subset T$ from $x$ to $y$ in $T$, and any $z \in [x,y]$, necessarily $d_{T'}([f(x),f(y)],f(z)) \leq C$. We denote by $BBT(f)$ the minimal such constant $C$. We note that for any given $T \in cv_k$ and $T' \in
\overline{cv}_k$, that any $F_k$–equivariant map $f {\colon\thinspace}T \to T'$ has bounded backtracking. Moreover $BBT(f) \leq \operatorname{Lip}(f) \operatorname{vol}_T(T/F_k)$, where $\operatorname{Lip}(f)$ is the Lipschitz constant of the map $f$ [@ar:BFH97]. In particular, if $T$ and $T' $ are contained in the projectivized space $CV_k$, then $BBT(f) \leq \operatorname{Lip}(f)$.
For a path $\alpha \subset T$, denote by $\alpha\dagger_L$ the path obtained by deleting the extremal paths of length $L$. The following is an easy consequence of bounded backtracking.
\[lm:surviving-subsegment\] Suppose that $T,T' \in \overline{cv}_k$, that $f{\colon\thinspace}T \to T'$ is an $F_k$–equivariant map that has bounded backtracking, and that $\ell{\colon\thinspace}{\mathbb{R}}\to T$ is a parametrized geodesic. If $L > BBT(f)$, and if for some interval $I \subset {\mathbb{R}}$, a tight path $\alpha
\subset T'$ is contained in $[f(\ell(I))]\dagger_L$, then necessarily $\alpha \subset [f(\ell(I'))]$ for any interval $I'
\supset I$.
Let $I = [x,y] \subset {\mathbb{R}}$ and assume that the hypotheses of the lemma hold, so that $d_{T'}(f(\ell(x)),\alpha) \geq L$ and $d_{T'}(f(\ell(y)),\alpha) \geq L$. Next let $x' \in I$ be such that $f(\ell(x'))$ is the endpoint of $\alpha$ closest to $f(\ell(x))$, and let $y' \in I$ be such that $f(\ell(y'))$ is the endpoint of $\alpha$ closest to $f(\ell(y))$.
Now suppose that $\alpha \not\subset [f(\ell(I'))]$ for some interval $I'$ that contains $I$. As $T'$ is a tree, either there exists an $x'' \in I' - I$ such that $f(\ell(x'')) =
f(\ell(x'))$ or there exists an $y'' \in I' - I$ such that $f(\ell(y'')) = f(\ell(y'))$; without loss of generality, we assume the former. Then the path $[\ell(x'),\ell(x'')]
\subset T$ and the point $\ell(x) \in [\ell(x'),\ell(x'')]$ violate bounded backtracking, as $$d_{T'}([f(\ell(x'),f(\ell(x'')],f(\ell(x)))
\geq L > BBT(f).$$
Irreducible elements and train-tracks {#ssc:iwip-tt}
-------------------------------------
Let $G$ be a graph. A [*turn*]{} is an unordered pair of oriented edges that share a common initial vertex. Letting $\bar{e}$ denote the edge $e$ with opposite orientation, we say that an edge path $\alpha$ [*crosses*]{} a turn $\{ e_1, e_2 \}$ if it contains an occurrence of either $\bar{e}_1 {e}_2$ or $\bar{e}_2 e_1$.
Let $g{\colon\thinspace}G \to G$ be a homotopy equivalence that is linear on edges, mapping edges to edge paths. Then $g$ induces a map on the set of turns of $G$, as follows. Let $v$ be the common initial vertex of the edges of a turn $\{ e_1,e_2 \}$. Some initial segment of $e_1$ is mapped onto an edge $e_1'$ based at $g(v)$, while some initial segment of $e_2$ onto an edge $e_2'$, also based at $g(v)$. We assign $g(\{ e_1, e_2 \}) = \{ e_1',e_2' \}$.
The homotopy equivalence $g{\colon\thinspace}G \to G$ is a [*train-track map*]{} if there is a collection ${\mathcal{L}}{\mathcal{T}}$ of turns such that:
1. ${\mathcal{L}}{\mathcal{T}}$ is closed under iteration of $g$ and
2. for an edge $e \subset G$, any turn crossed by $g(e)$ is in ${\mathcal{L}}{\mathcal{T}}$.
The unordered pairs of ${\mathcal{L}}{\mathcal{T}}$ are called *legal turns*, while an unordered pair of turns not in ${\mathcal{L}}{\mathcal{T}}$ is called an *illegal turn*. A path is *legal* if it only crosses legal turns. We will regularly refer to the underlying graph $G$ as a “train-track.”
An element of $\phi \in \operatorname{Out}F_k$ is [*reducible*]{} if some conjugacy class of a proper free factor of $F_k$ is $\phi$–periodic; otherwise $\phi$ is [*irreducible*]{}. Bestvina and Handel proved that every irreducible element of $\operatorname{Out}F_k$ has a topological representative that is a train-track map [@ar:BH92]. If $g{\colon\thinspace}G \to G$ is a train-track map representing an irreducible element of $\operatorname{Out}F_k$, there is a metric on $G$ such that $g$ linearly expands each edge of $G$ by the same factor $\lambda$, called the *expansion factor*. This factor is the Perron–Frobenius eigenvalue of the transition matrix for $g$; a positive eigenvector for this eigenvalue specifies the metric on $G$. Bounded cancellation implies that there is a bound on the amount of cancellation when tightening $g(\alpha \cdot \beta)$ where $\alpha$ and $\beta$ are legal paths. We denote the optimal constant by $BCC(g)$. As such, when $\alpha,\beta$ and $\gamma$ are legal paths, if $length(\beta) > \frac{2BCC(g)}{\lambda - 1}$ then the length of $[g^k(\alpha \cdot \beta \cdot \gamma)]$ goes to infinity as $k \to \infty$. The number $\frac{2BCC(g)}{\lambda -1}$ is called the *critical constant* for the map $g$.
All proper powers of a [*fully irreducible*]{} element $\phi$ of $\operatorname{Out}F_k$ are irreducible. A fully irreducible element $\phi$ has the property that its minimal displacement in the Lipschitz metric is related to its expansion factor $\lambda_\phi$ by $$\min_{G \in CV_k}d_L(G,G\phi) = \log(\lambda_\phi).$$ Moreover, this minimum is realized by a train-track map for $\phi$. This relationship is one reason we choose not to symmetrize the Lipschitz metric, for typically the expansion factor of a fully irreducible element is not equal to that of its inverse. See for example [@ar:HM07].
For a fully irreducible element $\phi \in \operatorname{Out}F_k$ and any tree $T \in CV_k$, the sequence $T\phi^n$ has a well-defined limit $T_+(\phi)$ in $\overline{CV}_k$, called the *stable tree* of $\phi$ [@ar:BFH97; @ar:LL03]. The [*unstable tree*]{} for $\phi$, denoted by $T_-(\phi)$, is the stable tree for $\phi{^{-1}}$, i.e., $T_-(\phi) = T_+(\phi{^{-1}})$. For an explicit description see [@un:HM]. We further note that if $T \in
\overline{CV}_k - \{ T_-(\phi)\}$, then $T\phi^n$ converges to $T_+(\phi)$ [@ar:LL03].
A element $\phi \in \operatorname{Out}F_k$ is [*hyperbolic*]{} if no conjugacy class of $F_k$ if $\phi$-periodic. Note that the set of hyperbolic elements of $\operatorname{Out}F_k$ is distinct from the set of fully irreducible elements. Also note that no power of a hyperbolic element of $\operatorname{Out}F_k$ is [*geometric*]{}; that is, no power is induced by a surface homeomorphism.
There is a notion of a *rotationless train-track map* and a *rotationless* element of $\operatorname{Out}F_k$ defined by Feighn and Handel [@un:FH], given as an analogue in $\operatorname{Out}F_k$ of a pure mapping class. In what follows, the precise definition of rotationless is not required, we need only the following property:
\[prop:rotationless\] For all $k \geq 1$, there exists an $N_k > 1$ such that $\phi^{N_k}$ is rotationless for all $\phi \in \operatorname{Out}F_k$.
Geodesics in Outer space {#ssc:axes}
------------------------
Next, for an interval $I \subset {\mathbb{R}}$, we describe paths $I \to CV_k$ known as [*folding lines*]{}. We will be concerned with two types of such paths: those which connect two points $G_1$ and $G_2$ in the interior $CV_k$, and those which are axes of fully irreducible elements. The latter were studied by Algom-Kfir in [@un:Algom-Kfir].
For $G_1, G_2 \in CV_k$, let $f{\colon\thinspace}G_1 \to G_2$ be a change of marking map whose Lipschitz constant realizes $\sigma(G_1,G_2)$. Find a path $\tilde{\alpha}_1$ based at $G_1$ contained in an open simplex of (unprojectivized) $cv_k$, along which edges of $G_1$ shrink just until the map induced by $f$ stretches every edge of the resulting graph by $\sigma(G_1,G_2)$; note that the lengths of those edges of $G_1$ that are stretched by exactly $\sigma(G_1,G_2)$ do not change along $\tilde{\alpha}_1$. Let the endpoint of the corresponding path $\alpha_1$ in (projectivized) $CV_k$ be $H_1$, with the change of marking map $h{\colon\thinspace}H_1 \to G_2$ induced by $f$. We choose a parameterization $\alpha_1{\colon\thinspace}[0,d_L(G_1,H_1)] \to CV_k$ by arclength.
Now we construct a path $\alpha_2: [0,d_L(H_1,G_2)] \to CV_k$ with $\alpha_2(0) = H_1$ and $\alpha_2(d_L(H_1,G_2)) = G_2$. First subdivide the edges of $H_1$ to obtain a graph $H_1'$ so that the preimage of vertices in $G_2$ consists of vertices in $H_1'$, while the induced map $h {\colon\thinspace}H_1' \to G_2$ remains cellular. Select a vertex $v$ of $H_1'$ at which two edges $e_1$ and $e_2$ identified by $h$ are based. Let $H(t)$ be the graph obtained from $H_1'$ by folding the initial segments of length $(1-e^{-t})$ of $e_1$ and $e_2$; let $\alpha_2(t)$ be the graph in $CV_k$ obtained from $H(t)$ by “forgetting” valence 2 vertices. Note that $d_L(H_1,\alpha_2(t)) =
t$. Define $\alpha_2(t)$ in this way until $e_1$ and $e_2$ are completely identified; then repeat the above with the resulting graph. Continue this process until the map induced by $f$ is an immersion in $G_2$; this is a finite process as $H_1'$ has a finite number of vertices. Note that the immersion is necessarily an isometry as every edge is stretched by the same factor; thus the fold line just constructed connects $H_1$ to $G_2$ in $CV_k$. Finally, let $\alpha$ be the concatenation of $\alpha_1$ and $\alpha_2$; a path based at $G_1$ and terminating at $G_2$. Francaviglia and Martino [@un:FM2 Theorem 5.5] proved that $\alpha {\colon\thinspace}[0,d_L(G_1,G_2)] \to CV_k$ is a geodesic.
We can now describe a geodesic axis for a fully irreducible element $\phi$ of $\operatorname{Out}F_k$. Let $\lambda_\phi$ be the expansion factor of $\phi$, and let $G$ be a train-track. To obtain a parametrized geodesic axis for $\phi$, first find a folding path $\alpha: [0,\log(\lambda_\phi)]
\to CV_k$ as above, connecting $G$ to $G\phi$. Then define the graph $\alpha(t) = \alpha(t-n(t) \log(\lambda_\phi)) \phi^{n(t)}$, where $n(t)$ is the integer $\lfloor \frac{t}{\log(\lambda_\phi)} \rfloor$, and let $\mathcal{L}_\phi$ denote the image of $\alpha$. Algom-Kfir [@un:Algom-Kfir Proposition 3.5] showed that $\alpha: {\mathbb{R}}\to CV_k$ is a geodesic parametrized by arclength.
Nielsen and vanishing paths {#ssc:nvp}
---------------------------
Suppose $G$ is a graph with a homotopy equivalence $g {\colon\thinspace}G \to G$. We make note of two special types of paths in $G$ and collect some relevant results that will be useful for us in the sequel.
First, a path $\alpha \subset G$ is a *Nielsen path* if $[g(\alpha)] =
[\alpha]$; it is *indivisible* if it is not a concatenation of nontrivial Nielsen paths, so that any Nielsen path is a concatenation of indivisible Nielsen paths.
\[th:Np-or-vp\] Suppose that $\phi \in \operatorname{Out}F_k$ is fully irreducible with stable tree $T_+ = T_+(\phi)$, and that $g {\colon\thinspace}G \to G$ is a train-track representative. Then there is a surjective $F_k$–equivariant map $f_g {\colon\thinspace}\widetilde{G} \to T_+$ such that if $[x,y] \subset \widetilde{G}$ is a lift of a path $\alpha \subset G$ and $f_g(x) = f_g(y)$, then for some $m \geq 0$, the path $[g^m(\alpha)]$ is either a Nielsen path or trivial.
\[def:ncover\] Let $a \in F_k$ and $T \in \overline{cv}_k$ be such that $\ell_T(a)
\neq 0$. We say that a path $\alpha \subset T$ (possibly infinite) *n–covers* $a$ if $L_T(\alpha \cap T^{{\langle}a {\rangle}}) \geq
n\ell_T(a)$. In other words, $\alpha$ overlaps with a segment of the axis of $a$ for length at least $n\ell_T(a)$; there is a point $x \in \alpha$ such that $a^nx \in \alpha$ as well.
Similarly, given $G \in CV_k$, we say a path $\alpha \subset G$ *n–covers* $a$ if a lift of $\alpha$ to $\widetilde{G}$ $n$–covers $a$. In other words, $\alpha$ decomposes into $\alpha =
\beta \cdot \alpha_0 \cdot \beta'$, where $\alpha_0$ is the loop representing the conjugacy class of $a^n$.
\[lem:onlyiNP\] Let $\phi$ and $g{\colon\thinspace}G \to G$ be as in Theorem \[th:Np-or-vp\]. Suppose that a Nielsen path $\alpha$ in $G$ $n$–covers $a$, for some $a \in F_k$ with $\ell_{T_+}(a) > 0$ and $n \geq 2$. Then there is a subpath $\alpha_0$ of $\alpha$ which $n$–covers $a$ and which is contained in $\alpha' \dagger_\epsilon$ for some indivisible Nielsen path $\alpha'$ and $\epsilon > 0$.
Suppose that $\alpha_0$ is a shortest subpath of $\alpha$ that $n$–covers $a$, but is not contained in the interior of an indivisible Nielsen path.
We can express $\alpha$ as a concatenation $\alpha_1 \alpha_2 \ldots
\alpha_r$ of indivisible Nielsen paths $\alpha_i$, $i=1,\ldots,r$. The $g$-fixed points of $\alpha$ are precisely the endpoints of the $\alpha_i$’s. Then since $\alpha_0$ is not contained in an indivisible Nielsen path, it must contain one of one of these fixed points $p$. It therefore contains at least $n$ copies of the fixed point $p$. Therefore some sequence $\alpha_i \ldots
\alpha_j$ forms a Nielsen path and corresponds to the conjugacy class of $a$. A closed Nielsen path corresponds to a periodic loop. It follows that $\ell_{T_+}(a) = 0$, contradicting the hypothesis on $a$.
A path $\alpha \subset G$ is a *vanishing path* of $g$ if $[g^m(\alpha)]$ is trivial (i.e., is a point) for some $m \geq 1$. We record the following observation from [@ar:BBC10]:
\[lem:vp-in-Np\] Let $\phi$ and $g{\colon\thinspace}G \to G$ be as in Theorem \[th:Np-or-vp\], and suppose that $\alpha$ is an indivisible Nielsen path. Then any subpath $\beta \subseteq \alpha \dagger_\epsilon$ for some $\epsilon >0$ is contained in a vanishing path.
An indivisible Nielsen path can be decomposed into a sequence of legal paths as $\alpha = \alpha_0
\cdot \beta_0 \cdot \overline{\beta}_1 \cdot \overline{\alpha}_1$, where $g(\alpha_i) = \alpha_i \cdot \beta_i$ and $g(\beta_0) =
g(\beta_1)$ [@ar:BH92]. Hence for $\epsilon >0$, with large enough $n$, the path $[g^n(\alpha\dagger_\epsilon)]$ is contained in $\beta_0 \cdot
\overline{\beta}_1$, and hence $[g^{n+1}(\alpha\dagger_\epsilon)]$ is trivial.
Putting together Theorem \[th:Np-or-vp\] and Lemmas \[lem:onlyiNP\] and \[lem:vp-in-Np\], we have the following:
\[prop:find-vp\] Suppose that $\phi \in \operatorname{Out}F_k$ is fully irreducible with stable tree $T_+=T_+(\phi)$, that $g
{\colon\thinspace}G \to G$ is a train-track representative of $\phi$, and that $a\in F_k$ is such that $\ell_{T_+}(a) > 0$. Then there is a surjective $F_k$–equivariant map $f_g {\colon\thinspace}\widetilde{G} \to T_+$ such that if $[x,y] \subset \widetilde{G}$ is a lift of a reduced path $\alpha
\subset G$ that $n$–covers $a$, for some $n \geq 2$, and if $f_g(x) =
f_g(y)$, then $\alpha$ contains a vanishing path that $n$–covers $a$.
Laminations and the map Q {#ssc:lamination-Q}
-------------------------
There are several notions of a “lamination” on a free group. For a full discussion on three different approaches and the relations between them, see [@ar:CHL08-I; @ar:CHL08-II; @ar:CHL08-III]. We will only briefly describe the elements of the theory we need here.
The group $F_k$ is hyperbolic and hence has a boundary ${\partial}F_k$. We denote: $${\partial}^2 F_k = \{ (x_1,x_2) \in {\partial}F_k \times {\partial}F_k \; | \; x_1
\neq x_2 \}$$ This set is naturally identified with the space of oriented *bi-infinite geodesics* in a tree $T \in cv_k$ as we explain now.
An oriented bi-infinite geodesic is an isometric embedding $\ell {\colon\thinspace}{\mathbb{R}}\to T$ considered up to reparametrization preserving the orientation. Any geodesic has two distinct endpoints in ${\partial}T$, denoted $\ell(\infty)$ and $\ell(-\infty)$. We can thus identify the geodesic $\ell$ with endpoints $(\ell(\infty),\ell(-\infty)) \in {\partial}^2
T = \{ (x_1,x_2) \in {\partial}T \times {\partial}T \; | \; x_1 \neq x_2 \}$, which, via the action of $F_k$, is naturally identified with ${\partial}^2
F_k$. Conversely, a point $(x_1,x_2) \in {\partial}^2 F_k$ determines two distinct points $x_1',x_2' \in {\partial}T$. Between these two points, there is a unique (up to orientation preserving reparametrization) oriented geodesic $\ell {\colon\thinspace}{\mathbb{R}}\to T$ such that $\ell(\infty) = x_1'$ and $\ell(-\infty) = x_2'$. We will use this identification without further remark.
There is fixed point free involution on ${\partial}^2 F_k$ defined by $\sigma{\colon\thinspace}(x_1,x_2) \to (x_2,x_1)$, corresponding to reversing a geodesic’s orientation in $T$.
A *lamination* is a closed $F_k$–invariant and $\sigma$–invariant subset $\Lambda \subseteq {\partial}^2 F_k$. The set of algebraic laminations inherits a Hausdorff topology from ${\partial}^2 F_k$, which is described in [@ar:CHL08-I]. A nontrivial element $a \in
F_k$ determines a [*minimal rational*]{} lamination: $$\Lambda(a) =
\{ (ga^{-\infty}, ga^{+\infty}) \cup (ga^{+\infty}, ga^{-\infty}) \ |
\ g \in F_k \}$$ Note that the set $\Lambda(a)$ depends only on the conjugacy class of $a$. Although we will not need them here, we mention that the set of [*rational*]{} laminations consists of finite unions of minimal rational laminations. The most important example of a lamination in what follows is the *stable lamination* $\Lambda_+(\phi)$ associated to a fully irreducible element $\phi \in \operatorname{Out}F_k$, as defined in [@ar:BFH97].[^2] The *unstable lamination* $\Lambda_-(\phi)$ associated to $\phi$ is the stable lamination of $\phi^{-1}$, so that $\Lambda_-(\phi) =
\Lambda_+(\phi^{-1})$.
In what follows, the most important example of a lamination is the *dual lamination*, $L^2(T)$, of a tree $T \in \overline{cv_k}$. The case of most interest is when $T$ is the stable tree of a fully irreducible element $\phi \in \operatorname{Out}F_k$. Therefore, the definition we give below is not the standard definition of $L^2(T)$, but equivalent in the setting we consider. See [@ar:CHL08-II] for other approaches and their equivalences.
Let $g {\colon\thinspace}G \to G$ be a train-track representative of $\phi$ with expansion factor $\lambda$. After passing to a power of $g$ if necessary, we can assume that $g$ has a fixed point $x$ contained in the interior of an edge. For some small $\epsilon$–neighborhood $U$ of $x$, we have that $g(U) \supset U$. Fix an isometry $\ell {\colon\thinspace}(-\epsilon,\epsilon) \to U$ and extend this to the unique isometric immersion $\ell {\colon\thinspace}{\mathbb{R}}\to G$ such that $\ell(\lambda^n t) = g^n(
t)$. This immersion lifts to a collection of geodesics $\tilde{\ell}
{\colon\thinspace}{\mathbb{R}}\to \widetilde{G}$. Using the identification mentioned above between ${\partial}^2 F_k$ and the space of geodesics in $\widetilde{G}$, the collection of all geodesics (called *leaves*) constructed as above determines a closed $F_k$–invariant subset of ${\partial}^2 F_k$ called the *stable lamination*. It is proved in [@ar:BFH97] that this set is well-defined independent of $g$. The leaves of $\Lambda_+(\phi)$ are *quasi-periodic* [@ar:BFH97], so that for every $L > 0$ there is an $L' > L$ such that for every interval $I$ of length $L$ and every interval $I'$ of length $L'$ there is an element $x \in F_k$ such that $x\ell(I) \subseteq \ell(I')$.
Given a basis ${\mathcal{A}}$ of $F_k$ and a tree $T \in \overline{cv}_k$, define the set $L^1_{\mathcal{A}}(T)$ as the set of right infinite reduced words $x = x_1x_2x_3\cdots$ in the basis ${\mathcal{A}}$ such that for some $p \in T$, the sequence of points $ (x_1x_2 \cdots x_i)p$ is bounded. The identification of right infinite reduced words in ${\mathcal{A}}$ with ${\partial}F_k$ identifies $L^1_{\mathcal{A}}(T)$ with a subset $L^1(T) \subseteq {\partial}F_k$ that is well-defined independent of the choice of basis. Bounded backtracking ensures the existence of a well-defined injective map ${\mathcal{Q}}{\colon\thinspace}{\partial}F_k - L^1(T) \to {\partial}T$. Using the injectivity of ${\mathcal{Q}}$ on ${\partial}F_k - L^1(T)$, we associate to any oriented bi-infinite geodesic $\alpha = (x_1,x_2) \in
{\partial}^2 F_k - (L^1(T))^2$ an oriented bi-infinite geodesic $\alpha_T \subset T$, namely $\alpha_T = ({\mathcal{Q}}(x_1),{\mathcal{Q}}(x_2))
\in {\partial}^2 T$, if neither endpoint of $\alpha$ is in $L^1(T)$; otherwise we define $\alpha_T$ to be the empty set. In the latter case, following [@ar:LL03], we say that the geodesic $\alpha$ is *$T$–bounded*.
In certain cases, the map ${\mathcal{Q}}{\colon\thinspace}{\partial}F_k - L^1(T) \to {\partial}T$ extends to a map on ${\partial}F_k$.
\[prop:Q\] Suppose $T \in \overline{cv}_k$ has dense orbits and trivial arc stabilizers (e.g., the stable tree for a fully irreducible outer automorphism). There exists a map ${\mathcal{Q}}{\colon\thinspace}L^1(T) \to \overline{T}$ to the metric closure $\overline{T}$ of $T$ such that, for any $f
{\colon\thinspace}T_0 \to T$, where $T_0 \in cv_k$, and any ray $\rho$ in $T_0$ representing $x \in L^1(T)$, the point ${\mathcal{Q}}(x)$ belongs to the closure of $f(\rho)$ in $\overline{T}$.
Combining this with the previous discussion, we have a map ${\mathcal{Q}}{\colon\thinspace}{\partial}F_k \to \overline{T} \cup {\partial}T$ whenever $T$ has dense orbits and trivial arc stabilizers.
The relation between stable trees and laminations is illustrated by the following.
\[prop:stableQ\] Suppose that $\phi \in \operatorname{Out}F_k$ is fully irreducible with stable tree $T_+$ and unstable lamination $\Lambda_-$. Let ${\mathcal{Q}}{\colon\thinspace}{\partial}F_k \to
\overline{T} \cup {\partial}T$ be the map defined following Proposition \[prop:Q\]. Then for any leaf $\ell \in \Lambda_-$, we have ${\mathcal{Q}}(\ell(\infty)) =
{\mathcal{Q}}(\ell(-\infty))$.
In light of the above propositions, we can define for any tree $T \in
\overline{cv}_k$ with dense orbits and trivial arc stabilizers [@ar:CHL08-II]: $$L^2_{\mathcal{Q}}(T) = \{ (x_1,x_2) \in {\partial}^2 F_k \; | \; {\mathcal{Q}}(x_1) = {\mathcal{Q}}(x_2) \}$$ where ${\mathcal{Q}}{\colon\thinspace}{\partial}F_k \to \overline{T} \cup {\partial}T$ is the map from Proposition \[prop:Q\]. With this definition, Proposition \[prop:stableQ\] states that $\Lambda_-(\phi) \subseteq
L^2_{{\mathcal{Q}}}(T_+^\phi)$ where $\phi \in \operatorname{Out}F_k$ is fully irreducible. If $\phi$ is hyperbolic (i.e., $\phi$ does not have nontrivial periodic conjugacy class) then $L^2_{{\mathcal{Q}}}(T_+^\phi)$ is the “diagonal closure” of $\Lambda_-(\phi)$ [@un:KL].
Missing from the above is a discussion of *measured geodesic currents* and *Dehn twist automorphisms* needed for Section \[sc:example\]. We defer their discussion until needed.
Geometric relative twisting {#sc:gtwist}
===========================
Our first definition of relative twisting for $\operatorname{Out}F_k$ follows closely the original geometric notion for the mapping class group, upon replacing a surface with a suitable 2–complex. This complex, the [*Guirardel Core*]{} for two $F_k$–trees $T,T'$, is a certain $F_k$–invariant subspace ${\mathcal{C}}\subset T \times T'$ (with the diagonal action). We will not need the precise definition of the complex for our purposes; rather, we record in Proposition \[prop:core\] just those properties of ${\mathcal{C}}$ we do require, together with references.
In the following, if $p$ is a point in $T$, then ${\mathcal{C}}_{p} = \{ x' \in
T' \, | \, (p,x') \in {\mathcal{C}}\}$; similarly, for $p' \in T'$, we have ${\mathcal{C}}_{p'} = \{ x \in T \, |
\,(x,p') \in {\mathcal{C}}\}$. These sets are called the *slices of the core*.
\[prop:core\] Suppose $T,T' \in CV_k$.
1. The core ${\mathcal{C}}\subset T \times T'$ is nonempty, connected, closed, [CAT(0)]{}, $F_k$–invariant and has convex fibers, i.e., the slices ${\mathcal{C}}_p$ and ${\mathcal{C}}_{p'}$ are each convex for all $p \in T$ and $p'
\in T'$. Moreover, ${\mathcal{C}}$ is the minimal (with respect to inclusion) subset of $T \times T'$ with these properties [@ar:Gu05 Main Theorem].
2. The quotient ${\mathcal{C}}/F_k$ has finite volume [@ar:Gu05 Theorem 8.1]. This volume is called the *intersection number*, denoted $i(T,T')$.
3. For any $p' \in T'$ that is not a vertex, any arc $\gamma
\subset {\mathcal{C}}_{p'}$ is contained in a vanishing path of any change of marking map $f{\colon\thinspace}T \to T'$ [@ar:BBC10 Lemma 3.7 & Remark 5.3].
For the complete definition of the core, along with examples, see [@ar:BBC10; @ar:Gu05].
Before going further we briefly recall relative twisting for curves on a surface. Let $S$ be a surface of genus at least two, equipped with a hyperbolic metric. We can consider $\pi_1(S)$ as a discrete group of isometries of $\mathbb{H}^2$, so that $S = \mathbb{H}^2/\pi_1(S)$. Fix three simple closed curves, $\alpha,\beta,\gamma$, so that $\beta$ and $\gamma$ both intersect $\alpha$. Each of these curves corresponds to a conjugacy class of an element in $\pi_1(S)$, and we can assume that all three are geodesics on $S$. Let $S_\alpha$ be an annular cover of $S$ corresponding to $\alpha$; that is, the quotient of $\mathbb{H}^2$ by the cyclic group generated by a representative of $\alpha$ in $\pi_1(S)$. We let $\alpha_c$ denote the unique lift of $\alpha$ to $S_\alpha$ that is closed. The *twist of $\beta$ and $\gamma$ relative to $\alpha$* is defined as the maximum geometric intersection number between $\beta'$ and $\gamma'$ that intersect $\alpha_c \subset S_\alpha$, where $\beta'$ and $\gamma'$ range over lifts of $\beta$ and $\gamma$ to $S_\alpha$.
We can reformulate this in terms of the universal cover $\widetilde{S}$, defining the relative twist as follows. Fixing a lift $\tilde{\alpha}$ of $\alpha$ to $\tilde{S}$, the twist of $\beta$ and $\gamma$ relative to $\alpha$ is the maximum number of $\alpha$–translates of $\tilde{\gamma}$ that intersect $\tilde{\beta}$, over all choices of lifts $\tilde{\beta},
\tilde{\gamma}$ of $\beta,\gamma$ that intersect $\tilde{\alpha}$. See Figure \[fig:rtwist\]. This interpretation can be extended to trees in Outer space, using the Guirardel core of $F_k$–trees $T$ and $T'$ in place of $\tilde{S}$.
$\tilde{\alpha}$ \[l\] at 132 220 $\tilde{\beta}$ \[l\] at 159 180 $\tilde{\gamma}$ \[l\] at 41 154 $\alpha\tilde{\gamma}$ \[l\] at 52 208 $\alpha^2\tilde{\gamma}$ \[l\] at 81 244 $\alpha^3\tilde{\gamma}$ \[l\] at 110 265 $\alpha^{-1}\tilde{\gamma}$ \[l\] at 56 92 $\alpha^{-2}\tilde{\gamma}$ \[l\] at 192 47 $\alpha^{-3}\tilde{\gamma}$ \[l\] at 169 21 ![The relative twist of $\beta$ and $\gamma$ with respect to $\alpha$ is 5.[]{data-label="fig:rtwist"}](rtwist "fig:")
The role of the simple closed curves $\beta,\gamma$ is filled by tracks on ${\mathcal{C}}$, and of the simple closed curve $\alpha$ by the axis of an element of $F_k$ in ${\mathcal{C}}$. A [*track for $T$*]{} in ${\mathcal{C}}$ is the set $\{ p\} \times {\mathcal{C}}_{p}$ where $p$ is the midpoint of some edge of $T$; a track for $T'$ is defined similarly. We will also use [*track*]{} to refer to the image of a track in the quotient ${\mathcal{C}}/F_k$. We record some elementary properties of tracks.
1. Every track separates ${\mathcal{C}}$.
2. Every track is a finite subtree.
3. Every track is a convex subset of ${\mathcal{C}}$.
As ${\mathcal{C}}$ is CAT(0), and every nontrivial element acts hyperbolically, the minset of a nontrivial element $g \in F_k$ is isometric to a product $Y \times {\mathbb{R}}$, where $Y$ is a convex subset of ${\mathcal{C}}$ [@bk:BH99]. An [*axis*]{} of a nontrivial element $a \in F_k$ is a subset of the minset of $a$ of the form $\{ y_0 \} \times {\mathbb{R}}$. The element $a$ acts by translation on any of its axes.
Lemmas \[lem:gtwist-wd\] and \[lem:wd\] describe the extent to which the intersection number between tracks and axes is well-defined.
\[lem:gtwist-wd\] Let $a \in F_k$ be a nontrivial element, $T,T' \in CV_k$ and consider the core ${\mathcal{C}}\subset T \times T'$. If a track $\tau$ in ${\mathcal{C}}$ intersects an axis of $a$ in ${\mathcal{C}}$, then it intersects every axis of $a$ in ${\mathcal{C}}$.
Let $\tau = \{ p \} \times {\mathcal{C}}_p$ be a track for $T$ that intersects an axis for $a$. Let $Y \times {\mathbb{R}}\subset {\mathcal{C}}$ be the minset for $a$. As tracks and axes are convex, their intersection is convex as well, and hence connected. Moreover, as tracks are finite, there are $s,t \in {\mathbb{R}}$ such that $\left(Y \times {\mathbb{R}}\right) \cap
\tau \subset Y \times [s,t]$.
Let $\{ y_0 \} \times {\mathbb{R}}\subset Y \times {\mathbb{R}}$ be an axis of $a$ that intersects $\tau$. As $\left(\{ y_0 \} \times {\mathbb{R}}\right) \cap \tau$ is connected, we have that $\{ y_0 \} \times (-\infty,s)$ and $\{y_0 \} \times (t
,\infty)$ project to different components of $T - \{ p \}$. Hence $\tau$ separates $\{ y_0 \} \times \{ s-1 \}$ from $\{ y_0 \} \times
\{ t+1 \}$ in ${\mathcal{C}}$. If there were an axis, say $\{ y_1 \} \times
{\mathbb{R}}$, that did not intersect $\tau$, then the concatenation of the fiber-wise paths $$\{y_0 \} \times \{ s-1 \} \to \{y_1 \} \times \{ s-1 \} \to \{y_1
\} \times \{ t+1 \} \to \{y_0 \} \times \{ t+1 \}$$ would be a path that connected $\{ y_0 \} \times \{ s-1 \}$ to $\{y_0 \} \times \{
t+1 \}$ avoiding $\tau$, which is a contradiction as $\tau$ separates these points. Thus every axis for $a$ intersects $\tau$.
\[lem:wd\] Let $a \in F_k$ be a nontrivial element, $T,T' \in CV_k$ and consider the core ${\mathcal{C}}\subset T \times T'$. Fix a track $\tau'$ for $T'$ that intersects an axis of $a$ in ${\mathcal{C}}$. Let $\tau_0$ and $\tau_1$ be two tracks for $T$ that intersect an axis of $a$. Then: $$\Bigl| |\tau' \cap {\langle}a {\rangle}\tau_0 | - |\tau' \cap {\langle}a {\rangle}\tau_1 | \Bigr| \leq 1.$$
Suppose $|\tau' \cap {\langle}a
{\rangle}\tau_0| = n$. We will show that $|\tau' \cap {\langle}a {\rangle}\tau_1|
\geq n-1$. The statement of the lemma follows after interchanging $\tau_0$ and $\tau_1$ and applying the same argument.
Since a track for $T$ and a track for $T'$ can intersect at most once in ${\mathcal{C}}$, the track $\tau'$ intersects exactly $n$ ${\langle}a
{\rangle}$–translates of $\tau_0$. We claim that there is an $i$ such that $\tau'$ intersects $a^{i+j} \tau_0$ for $j = 0,\ldots, n-1$. Indeed, this follows as $\tau'$ is connected, and as $a^m \tau_0$ separates $a^{m-r}\tau_0$ from $a^{m+s}\tau_0$ for all $m$ and positive $r$ and $s$. Replacing $\tau_0$ by $a^i\tau_0$, we can assume that $\tau'$ intersects $\tau_0,\ldots a^{n-1}\tau_0$.
If $\tau_1 = a^i\tau_0$ for some $i$, then the statement is obvious. Otherwise, as $|\tau' \cap {\langle}a {\rangle}\tau_1 |$ only depends on the orbit of $\tau_1$ under $a$, we can replace $\tau_1$ by $a^i \tau_1$ for some $i$ to assume that $\tau_1$ separates $\tau_0$ from $a\tau_0$.
We claim that $\tau'$ intersects $\tau_1,\ldots, a^{n-2}\tau_1$. Indeed, since $\tau'$ is connected and since $a^i \tau_1$ separates $a^i\tau_0$ from $a^{i+1}\tau_0$, both of which intersect $\tau'$ for $i = 0,\ldots n-2$, the track $\tau'$ must intersect $a^i \tau_1$.
As $|\tau' \cap {\langle}a {\rangle}\tau| = |{\langle}a {\rangle}\tau' \cap \tau|$, Lemma \[lem:wd\] shows that if $\tau'_0$ and $\tau'_1$ are tracks for $T'$ that intersect an axis of $a$, and likewise $\tau_0$ and $\tau_1$ are tracks for $T$ that intersect an axis of $a$, then: $$\Bigl| |\tau_0' \cap {\langle}a {\rangle}\tau_0 | - |\tau_1' \cap {\langle}a {\rangle}\tau_1 | \Bigr| \leq 2.$$ With this bound we can define the relative twist number. By $Tr_a(T)$ we denote the set of tracks for $T$ in ${\mathcal{C}}$ that intersect an (and hence every) axis of $a$. We define the set $Tr_a(T')$ similarly.
\[def:gtwist\] Given $T,T' \in CV_k$ and a nontrivial element $a \in F_k$, define the *twist of $T$ and $T'$ relative to $a$* as: $$\tau_a(T,T') = \max_{\tau' \in Tr_a(T'),\tau \in Tr_a(T)}
|\tau' \cap {\langle}a {\rangle}\tau|.$$
We remark that this number is always finite. Indeed, as tracks are finite, the quantities we are maximizing over are finite. Then by the above discussion, there are only finitely many distinct possibilities for these numbers.
The significance of the relative twist number to the geodesic in $CV_k$ connecting two marked graphs is the following, to be used in Section \[sc:smallcycle\] to prove Theorem \[th:geo-twist\]:
\[prop:ntwist-vp\] Suppose $G,G' \in CV_k$ with $d = d_L(G,G')$ such that $\tau_a(G,G')
\geq n$ for some nontrivial $a \in F_k$. Then for every change of marking map $g {\colon\thinspace}G \to G'$, there is a vanishing path $\gamma
\subset G$ that $n$–covers $a$.
Let $\widetilde{G}$ and $\widetilde{G}'$ be the universal covers of $G$ and $G'$ respectively, and consider the core ${\mathcal{C}}\subset
\widetilde{G} \times \widetilde{G}'$. Fix an axis of $a$, and tracks $\tau \in Tr_a(\widetilde{G}), \, \tau' \in
Tr_a(\widetilde{G}')$ such that $\tau_a(G,G') = |\tau' \cap {\langle}a {\rangle}\tau|$.
Let $m_0$ and $m_1$ be the least and greatest integer, respectively, such that $\tau' \cap a^{m_0}\tau \neq \emptyset$ and $\tau' \cap a^{m_1}\tau \neq
\emptyset$. Thus $\tau_a(G,G') = m_1 - m_0 \geq n$. Denote $x_0 =
\tau' \cap a^{m_0}\tau$ and $x_1 = \tau' \cap a^{m_1}\tau$ and let $\rho$ be the path in $\tau'$ connecting $x_0$ to $x_1$. As $\tau'
= {\mathcal{C}}_{p'} \times \{ p' \}$ for some point $p' \in \widetilde{G}'$, we can consider $\rho$ as a path in ${\mathcal{C}}_{p'} \subset
\widetilde{G}$. Notice that the endpoints of $\rho$, also denoted $x_0$ and $x_1$, are on the axis for $a$, and that $a^{m_1-m_0}x_0 = x_1$. Thus $a$ is $n$–covered by $\rho$. By Proposition \[prop:core\](3), the path $\rho$ is contained in a vanishing path $\gamma$ for any change of marking map $G \to G'$. As $\gamma$ contains $\rho$, the vanishing path $\gamma$ $n$–covers $a$ as well.
Algebraic relative twisting {#sc:atwist}
===========================
In this section we give our algebraic interpretation of relative twisting and develop some consequences that will be useful for applications in later sections. The key result is Proposition \[prop:ncover-vp\], which is used to prove Theorem \[th:alg-twist\].
\[def:woMeasure\] Given $T \in \overline{cv}_k$, a lamination $\Lambda
\subset \partial^2 F_k$, and an element $a \in F_k$, if $\ell_T(a) \neq 0$, then we define the *twist of $T$ and $\Lambda$ relative to a* to be: $$\tau_a(T,\Lambda) = \sup \left\{
\frac{L_T(\alpha_T \cap T^{{\langle}a {\rangle}})}{\ell_T(a)} \, \Big| \,
\alpha \in \Lambda \right\}.$$ If $\ell_T(a) = 0$, then define $\tau_a(T,\Lambda) = 0$.
Recall that given $\alpha = (x_1,x_2) \in {\partial}^2 F_k$, we have that $\alpha_T$ is equal to $({\mathcal{Q}}(x_1),{\mathcal{Q}}(x_2)) \in {\partial}^2 T$ if $\alpha$ is not $T$–bounded, and is equal to the empty set otherwise. We insist that $L_T(\emptyset) = 0$.
We allow for the possibility that $\tau_a(T,\Lambda) = \infty$. This occurs in particular for the rational lamination $\Lambda(a)$, when $\ell_T(a) \neq 0$.
As the above definition only uses the support of $\mu$ and does not take into account the actually measure, it might seem like relative twisting should be defined between a tree and a lamination. Currents are easier to work with computationally, as is evident in Section \[sc:example\].
Central to our analysis is the following proposition:
\[prop:lowersemicontinuous\] Suppose $a \in F_k$, $T \in \overline{cv}_k$, and that $\Lambda$ is a lamination containing no $T$–bounded geodesic. If $\{ T_i \}$ is a sequence of trees in $cv_k$ converging to $T$, and $\{ \Lambda_i
\}$ is a sequence of laminations converging to $\Lambda$, then: $$\lim_{i \to \infty} \tau_a(T_i,\Lambda_i) \geq \tau_a(T,\Lambda).$$
The proposition is obviously true when $\ell_T(a) = 0$, and so we assume that $\ell_T(a) > 0$.
We proceed with the following:
If $T \in \overline{cv}_k$, and if $\alpha, \beta \in {\partial}^2 F_k$ are not $T$–bounded and $\alpha(\infty),\alpha(-\infty),\beta(\infty),\beta(-\infty)$ are four distinct points, then for sufficiently close $T' \in cv_k$, we have $L_{T'}(\alpha_{T'} \cap \beta_{T'})$ close to $L_T(\alpha_T \cap \beta_T)$.
As ${\mathcal{Q}}$ is injective on ${\partial}F_k - L^1(T)$, we have that $\alpha_T \cap \beta_T$ is compact set.
Fix a tree $T_0 \in cv_k$, a map $f{\colon\thinspace}T_0 \to T$ that is linear on edges, and elements $a_i^\pm \in T_0$ so that $[a_i^-,a_i^+]
\to \alpha_{T_0}$. Then $\alpha_i = [f(a_i^-),f(a_i^+)] \to
\alpha_T$; in particular, the overlap of $\alpha_i$ and $\alpha_T$ can be made arbitrarily large. Similarly define $b_i^\pm \in T_0$ and $\beta_i = [f(b_i^-),f(b_i^+)]$ so that, as before, we have $\beta_i \to \beta_T$.
Fix a $T' \in cv_k$ and an equivariant map $f' {\colon\thinspace}T_0 \to T'$, linear on edges. As before, we have $\alpha'_i =
[f'(a_i^-),f'(a_i^+)] \to \alpha_{T'}$ and $\beta'_i =
[f'(b_i^-),f'(b_i^+)] \to \beta_{T'}$.
Now choose $n$ large enough so that each of $\alpha_n \cap
\alpha_T$ and $\beta_n \cap \beta_T$ contains $\alpha_T \cap
\beta_T$. Then increase $n$ if necessary so that $\alpha_n' \cap
\alpha_{T'}$ and $\beta_n' \cap \beta_{T'}$ each contain $\alpha_{T'} \cap \beta_{T'}$. For sufficiently close trees $T,
T',$ the lengths $L_T(\alpha_n \cap \beta_n)$ and $L_{T'}(\alpha'_n \cap \beta'_n)$ of the overlaps are close [@ar:Clay05; @ar:GL07]. By choice of $n$, we have $L_T(\alpha_n
\cap \beta_n) = L_T(\alpha_T \cap \beta_T)$ and $L_{T'}(\alpha'_n
\cap \beta'_n) = L_{T'}(\alpha_{T'} \cap \beta_{T'})$. The claim follows.
We are now prepared to complete the proof of the proposition. First assume that $\tau_a(T,\Lambda) \neq \infty$. This implies that no geodesic in $\Lambda$ has $a^{+\infty}$ or $a^{-\infty}$ as an endpoint and so we can use the above claim. Let $\epsilon$ be small and choose a geodesic $\alpha \in \Lambda$ so that: $$L_T(\alpha_T \cap T^{{\langle}a {\rangle}})/\ell_T(a) > \tau_a(T,\Lambda) - \epsilon.$$ Then by the above: $$\frac{L_{T'}(\alpha_{T'} \cap T'^{{\langle}a {\rangle}})}{\ell_{T'}(a)} >
\frac{L_T(\alpha_T \cap T^{{\langle}a {\rangle}})}{\ell_T(a)} - \epsilon >
\tau_a(T,\Lambda) - 2\epsilon$$ for $T'$ sufficiently close to $T$. For $\Lambda'$ sufficiently close to $\Lambda$, there exists $\alpha' \in \Lambda'$ so that $\alpha_{T'} \cap T'^{\langle a
\rangle} \subset \alpha'_{T'}$. Thus: $$\tau_a(T',\Lambda') \geq \frac{L_{T'}(\alpha_{T'} \cap T'^{{\langle}a
{\rangle}})}{\ell_{T'}(a)} > \tau_a(T,\Lambda) - 2\epsilon$$ for $T'$ sufficiently close to $T$ and $\Lambda'$ sufficiently close to $\Lambda$. Since we obtain such an inequality for every $\epsilon$, the proposition holds.
Suppose $\tau_a(T,\Lambda) = \infty$. If $a^{+\infty}$ or $a^{-\infty}$ is an endpoint of a geodesic in $\Lambda$, then $\tau_a(T',\Lambda) = \infty$ for all trees $T' \in cv_k$. Else, we have that for, for every $M >0$ there is a geodesic $\alpha \in
\Lambda$ so that: $$\infty > L_T(\alpha_T \cap T^{{\langle}a {\rangle}})/\ell_T(a) > M.$$ Then arguing in a similar fashion as above, we have for $T'$ sufficiently close to $T$ and $\Lambda'$ sufficiently close to $\Lambda$: $$\tau_a(T',\Lambda') \geq \frac{L_{T'}(\alpha_{T'} \cap T'^{{\langle}a
{\rangle}})}{\ell_{T'}(a)} > \frac{M}{2}.$$ Since we obtain such an inequality for every $M$, the proposition holds.
\[rm:T-bounded\] Examples of tree, lamination pairs satisfying the hypotheses of Proposition \[prop:lowersemicontinuous\] are:
1. $T_+(\phi),\Lambda_+(\phi)$ where $\phi \in \operatorname{Out}F_k$ is fully irreducible,
2. $T,\Lambda$ where $T \in cv_k$ and $\Lambda$ is any lamination, and
3. $T,\Lambda(a)$ whenever $\ell_T(a) \neq 0$.
\[rm:lowersemicontinuous\] The inequality in Proposition \[prop:lowersemicontinuous\] can be strict; we present an example here. Consider the element of $\phi
\in \operatorname{Out}F_2$ represented by the automorphism $a \mapsto ab$, $b
\mapsto bab$. This element is fully irreducible and, as it is a positive automorphism, has a train-track representative on the graph with a single vertex and two edges marked $a$ and $b$ respectively. We will refer to these edges as $a$ and $b$. By $T \in CV_2$ denote the universal cover of this graph. The only illegal turn is the turn between the terminal segment of $a$ and the terminal segment of $b$. Let $g = aba^{-1}b^{-1}$ and $h = ga$. Notice that $\phi^i(h)
= g\phi^i(a)$. Thus $\tau_g(T\phi^i,\Lambda(\phi^i(h))) \geq 1$ as the leaves of $\Lambda(\phi^i(h))$ in $T\phi^i$ are the axes of conjugates of $\phi^{2i}(h)$, a subsegement of which is a fundamental domain for an axis for $g$. We have $\Lambda(\phi^i(h))
\to \Lambda_+(\phi)$ [@ar:BFH97]. However, as $\ell_{T_+(\phi)}(g) = 0$, we have $\tau_g(T_+(\phi),\Lambda_+(\phi)) = 0$. Notice that there is a sequence of laminations $\Lambda_i \to \Lambda_+(\phi)$ such that $\lim_{i \to \infty} \tau_g(T\phi^i,\Lambda_i) = 0 =
\tau_g(T_+(\phi),\Lambda_+(\phi))$, namely $\Lambda_i =
\Lambda(\phi^i(a))$ as $\tau_g(T\phi^i,\Lambda(\phi^i(a)))=0$.
For a fully irreducible element $\phi$ with large twist $\tau_a(T_-(\phi),\Lambda_-(\phi))$ for some nontrivial $a \in F_k$, our goal is to locate a train-track $G_0$ of $\phi$ with a vanishing path that $n$–covers $a$, similar to Proposition \[prop:ntwist-vp\]. Our tool to produce such a path is Proposition \[prop:find-vp\]. First we see how to use the lamination to get the required setup.
\[lm:extend-segment\] Suppose $\phi \in \operatorname{Out}F_k$ is fully irreducible, $g{\colon\thinspace}G \to G$ is a train-track representative for $\phi$, $T_+ \in \overline{cv}_k$ is the stable tree for $\phi$, $f_g{\colon\thinspace}\widetilde{G} \to T_+$ is the induced map from Theorem \[th:Np-or-vp\], and $\ell{\colon\thinspace}{\mathbb{R}}\to
\widetilde{G}$ is a leaf of the unstable lamination $\Lambda_-$. Then for all $I \subset {\mathbb{R}}$, there exists $I' = [x,y] \subset {\mathbb{R}}$ such that $I \subseteq I'$ and $f_g(\ell(x)) = f_g(\ell(y))$.
We claim that for any $L \geq 0$, there is an interval $[a,b] \subseteq {\mathbb{R}}$ such that $|b - a| \geq L$ and $f_g(\ell(a)) =
f_g(\ell(b))$. The lemma follows: by the quasi-periodicity of $\ell$, there is then an interval $I_0
= [a,b] \subset {\mathbb{R}}$ such that $f_g(\ell(a)) = f_g(\ell(b))$ and $x\ell(I) \subseteq \ell(I_0)$. Setting $I' =
\ell^{-1}(x^{-1}\ell(I_0))$ completes the proof. We must then just establish the claim.
Since $\ell$ is a leaf of the unstable lamination, we have ${\mathcal{Q}}(\ell(-\infty)) = {\mathcal{Q}}(\ell(\infty))$. There are sequences $a_i
\to -\infty$ and $b_i \to \infty$ such that $f_g(\ell(a_i)) \to
{\mathcal{Q}}(\ell(-\infty)) = {\mathcal{Q}}(\ell(\infty))$ and $f_g(\ell(b_i)) \to
{\mathcal{Q}}(\ell(\infty))$ [@ar:LL03 Lemma 3.4]. Now we have two cases, either the sequences $\{f_g(\ell(a_i)) \}$ and $\{
f_g(\ell(b_i)) \}$ are in the same component of $\overline{T}_+ - \{
{\mathcal{Q}}(\infty)\}$ or they are not.
If the sequences are in the same component, choose $n$ with $|b_n -
a_n| > L$. The arc $\alpha$ connecting $f_g(a_n)$ to $f_g(b_n)$ is then disjoint from ${\mathcal{Q}}(\ell(\infty))$, and there is a unique point $p \in \alpha$ which is closest to ${\mathcal{Q}}(\ell(\infty))$. As $\overline{T}_+$ is an ${\mathbb{R}}$–tree, $p$ is on the geodesic $[f_g(a_n),{\mathcal{Q}}(\ell(-\infty))]$. Then by continuity of $f\ell$, there is an $a' \leq a_n$ such that $f_g(a') = p$. Likewise, there is a $b' \geq b_n$ such that $f_g(b') = p$. Thus the inteveral $[a',b']$ satisfies the claim.
Now suppose the sequences are not in the same component. If $f_g(\ell({\mathbb{R}}))$ crosses the point ${\mathcal{Q}}(\ell(\infty))$ infinitely many times, then we can find a sequence of points $a_i,b_i \in {\mathbb{R}}$ such that $f_g(\ell(a_i)) = f_g(\ell(b_i)) = {\mathcal{Q}}(\ell(\infty))$ such that $|b_i - a_i| \to \infty$. Indeed, there is a lower bound on the distance between two pre-images of ${\mathcal{Q}}(\infty)$ in ${\mathbb{R}}$ as every edge of $\widetilde{G}$ is isometrically embedded by $f_g$. For large enough $i$, the interval $[a_i,b_i]$ satisfies the claim.
Finally, suppose $f_g(\ell({\mathbb{R}}))$ crosses ${\mathcal{Q}}(\ell({\mathbb{R}}))$ only finitely many times. Let $a$ be the smallest number such that $f_g(\ell(a)) = {\mathcal{Q}}(\ell(\infty))$. Then arguing as in the first case using sequences $a_i \to -\infty$ and $b_i \to a$ ($b_i < a$) we can find the desired inteveral.
In the next proposition, we find a candidate vanishing path in a train-track $G$ that folds over $a$ several times. The technicalities in its proof arise from the fact that, as the hypothesis concerns the [ *unstable*]{} lamination, we must first find a large power of $a$ covered by a leaf of the lamination in a train-track map for the [ *inverse*]{} $\phi^{-1}$ of $\phi$. Care is then required in mapping this leaf over to a train-track for $\phi$, as there might be excessive cancellation. We resolve this difficulty by applying powers of $\phi^{-1}$, so that the length of $a$ dominates any such cancellation.
\[prop:unstable\] Suppose that $\phi$ is fully irreducible with unstable tree $T_-$ and lamination $\Lambda_-$, with $\tau_a(T_-,\Lambda_-) \geq n + 2$ for some $a \in F_k$. Then there exists a train-track graph $G \in CV_k$ for $\phi$ and a leaf of the unstable lamination $\ell {\colon\thinspace}{\mathbb{R}}\to
\widetilde{G}$ such that for all $L > 0$, there is a finite interval $I \subset {\mathbb{R}}$ such that $[\ell(I)]\dagger_L$ $n$–covers $a$.
Let $H \in CV_k$ be a train-track graph for $\phi{^{-1}}$ with train track representative $h {\colon\thinspace}H \to H$ and $G \in CV_k$ a train track graph for $\phi$ with train-track map $g{\colon\thinspace}G \to G$. Fix Lipschitz homotopy equivalences $\kappa {\colon\thinspace}H \to G$ and $\kappa' {\colon\thinspace}G \to H$ representing the change in markings. Thus the following diagram is commutative up to homotopy: $$\xymatrix{G \ar[r]^g & G \ar[d]^{\kappa'} \\
H \ar[u]^\kappa & H \ar[l]^h}$$ Notice that $\kappa$ lifts to $\widetilde{\kappa}{\colon\thinspace}\widetilde{H}
\to \widetilde{G}$, with bounded backtracking. In particular, we can pick a constant $C$ such that if a path $\gamma \subset H$ has length at least $C$, then $\kappa(\gamma)$ is not homotopically trivial relative to its endpoints.
As $\tau_a(T_-,\Lambda_-) \geq n + 2$, Proposition \[prop:lowersemicontinuous\] implies that $\tau_a(\widetilde{H}_0,\Lambda_-) \geq n + 2$, where $H_0 =
H\phi^{-M}$ for some large $M$. Define $G_0 = G\phi^{-M}$. Now there is a leaf $\ell {\colon\thinspace}{\mathbb{R}}\to \widetilde{H}_0$ of the unstable lamination and an interval $I_0 \subset {\mathbb{R}}$ such that $\ell(I_0)
\subset \widetilde{H}_0^{{\langle}a {\rangle}}$ and $L_{\widetilde{H}_0}(\ell(I_0)) \geq (n +
2)\ell_{\widetilde{H}_0}(a)$. Notice that this implies that the loop representing the conjugacy class of $a$ in $H_0$ is legal with respect to $h$. Then if we let $\lambda$ be the expansion factor for $\phi{^{-1}}$, and let $d = \ell_{\widetilde{H}_0}(a)$, we have $\ell_{\widetilde{H}_0\phi^{-m}}(a) = \lambda^md$. Let $N$ be such that $\lambda^Nd \geq C$. Define $G_1 = G_0\phi^{-N}$ and $H_1 =
H_0\phi^{-N}$.
Fix $L>0$ and let $L'$ be such that if we have paths $\gamma \subset
\gamma'$ in $H_0$ and $\gamma$ has length at least $L'$ then the path $\kappa(\gamma)$ intersects $[\kappa(\gamma')]$ in a path of length at least $L$ (necessarily a subpath of $[\kappa(\gamma')]$, but not necessarily a subpath of $[\kappa(\gamma)]$). Indeed, such an $L'$ exists as $\kappa$ is a quasi-isometry and the graph $G_0$ has valence bounded by $2k$. Finally, extend $I_0$ to $I = I_1 \cup I_0 \cup
I_2 \subset {\mathbb{R}}$ by including intervals $I_1$ and $I_2$ of length at least $(L' + C)/\lambda^N$.
Notice that the lengths of $h^N(\ell(I_1))$ and $h^N(\ell(I_2))$ in $H_1$ are at least $L' + C$. Moreover, the initial subsegment $\iota \subset h^N(\ell(I_1))$ of length $L'$ maps by $\kappa$ to a segment in $G_1$ that intersects $[\kappa h^N\ell(I)]$ in a path of length at least $L$, so that $\kappa(\iota)$ does not cancel with $\kappa h^N\ell(I_0)$, as $\iota$ and $h^N\ell(I_0)$ are separated in $h^N\ell(I)$ by a subsegment of length at least $C$. Similarly for the terminal subsegment of $h^N(I_2)$.
Recall $\ell(I_0)$ $(n+2)$–covers $a$ in $H_0$. This clearly implies that $h^N\ell(I_0)$ $(n+2)$–covers $a$ in $H_1$ as all of the paths are legal with respect to $h$. Thus $h^N\ell(I_0)$ contains a subpath whose image in $H_1$ represents the conjugacy class of $a^{n+2}$. As $\ell_{\widetilde{H}_1}(a) \geq C$, a subsegment of length $n
\ell_{\widetilde{G}_1}(a)$ in $[\kappa h^N\ell(I_0)]$ survives after tightening $[\kappa h^N\ell(I_1)] \cdot [\kappa h^N\ell(I_0)] \cdot
[\kappa h^N\ell(I_2)]$ to get $[\kappa h^N\ell(I)]$. Now tighten $\kappa\ell$ to get a leaf of the unstable lamination $\ell_1{\colon\thinspace}{\mathbb{R}}\to \widetilde{G}_1$. Thus $L_{\widetilde{G}_1}([\ell_1(I)]\dagger_L \cap \widetilde{G}_1^{{\langle}a {\rangle}}) \geq n\ell_{\widetilde{G}_1}(a)$ and hence $[\ell_1(I)]\dagger_L$ $n$–covers $a$.
Proposition \[prop:ncover-vp\] will now follow relatively quickly once we observe the following consequence of Proposition \[prop:unstable\].
\[co:extend-point\] Suppose that $\phi$ is fully irreducible with unstable tree $T_-$ and lamination $\Lambda_-$, such that $\tau_a(T_-,\Lambda_-) \geq n
+ 2$ for some $a \in F_k$. Then there exists a train-track $G \in
CV_k$ for $\phi$, with train-track map $g{\colon\thinspace}G \to G$, and a path $\gamma = [x,y] \subset \widetilde{G}$ such that:
1. $\gamma$ $n$–covers $a$; and
2. $f_g(x) = f_g(y)$ where $f_g {\colon\thinspace}\widetilde{G} \to T_+$ is the induced map (see Theorem \[th:Np-or-vp\]) from $\widetilde{G}$ to $T_+$, the stable tree for $\phi$.
Let $G$ be the train-track graph given by Proposition \[prop:unstable\], and let $f_g {\colon\thinspace}\widetilde{G} \to T_+$ be the induced map. Let $\ell {\colon\thinspace}{\mathbb{R}}\to \widetilde{G}$ be the leaf of the unstable lamination and $I \subset {\mathbb{R}}$ the interval produced by Proposition \[prop:unstable\], for $L = BBT(f_g) + 1$. For this $I$, let $I'$ be the interval given by Lemma \[lm:extend-segment\].
We claim that $\gamma = [\ell(I')]$ satisfies the conclusion of the corollary. Indeed by Lemma \[lm:surviving-subsegment\], as $[\ell(I)]\dagger_L$ contains a path in the axis of $a$ of length $n\ell_{\widetilde{G}}(a)$, so does $\gamma = [\ell(I')]$. By construction, $f_g$ identifies the endpoints of $\gamma$.
Proposition \[prop:find-vp\] applied to the train-track $G$ and the path $\gamma$ of Corollary \[co:extend-point\] give the following:
\[prop:ncover-vp\] Suppose $\phi$ is a fully irreducible element with unstable tree $T_-$ and lamination $\Lambda_-$ such that $\tau_a(T_-,\Lambda_-)
\geq n + 2$ for some $a \in F_k$. Then there exists a train-track $G
\in CV_k$ for $\phi$, with train-track map $g{\colon\thinspace}G \to G$, and a vanishing path $\gamma \subset G$ that $n$–covers $a$.
Finding small cycles {#sc:smallcycle}
====================
With our definition of relative twist, we can prove the analogue of a special case of Rafi’s characterization of short curves along geodesics in Teichmüller space [@ar:Rafi05].
\[prop:vpimpliessmallcycle\] Suppose $G,G' \in CV_k$, $f{\colon\thinspace}G \to G'$ is a change of marking map with minimal slope, $d = d_L(G,G')$ and $a \in F_k$. If there is a vanishing path $\gamma \subset G$ that $(n+2)$–covers $a$, then there is a geodesic $\alpha{\colon\thinspace}[0,d] \to CV_k$ such that $\alpha(0)
= G$, $\alpha(d) = G'$ and for some $t \in [0,d]$, we have $\ell_{\alpha(t)}(a) \leq 1/n$.
Shrinking the edges of $G$ such that each edge is stretched by exactly $e^d$ results in a marked graph $G_0$ and provides a geodesic $\alpha{\colon\thinspace}[0,d_0] \to CV_k$ such that $\alpha(0) = G$, $\alpha(d_0) = G_0$ and $d = d_L(G_0,G') + d_0$. Denote the induced map $G_0 \to G'$ by $f$.
Consider the graph $H_a = \widetilde{G}_0^{{\langle}a {\rangle}}/{\langle}a {\rangle}$ and the map $h_{G'}{\colon\thinspace}H_a \to G'$ which is the composition of the immersion $H_a \to G_0$ with the map $f{\colon\thinspace}G_0 \to G'$. By appropriately subdividing and folding the graph $H_a$, after pruning we obtain a graph immersion $H'_a \to G'$. Now choose a folding path based at $G_0$ whose folds correspond to the folds performed on $H_a$. The end of the folding path is a graph $G_1$ in which $H'_a$ is immersed.
Let $d_1 = d_L(G_0,G_1)$. As $f$ has minimal slope, the above path extends $\alpha$ to a geodesic $\alpha{\colon\thinspace}[0,d_1] \to CV_k$ such that $\alpha(0) = G$ and $\alpha(d_1) = G_1$ [@un:FM2 Theorem 5.5]. Further $d_L(G_1,G') = d - d_1$. Denote the induced map $G_1 \to G'$ by $f_1$.
The geodesic segment $\alpha$ can further be extended to a geodesic by folding $G_1$. The image of $\gamma$ in $G_1$ (which we denote by $\gamma_1$) is a vanishing path for the map $f_1$.
The path $\gamma_1 \subset G_1$ $n$–covers $a$.
Consider the graph $\widetilde{G}_0/{\langle}a{\rangle}$. This graph consists of a collection of trees attached to $H_a$. We consider the graph $H_a$ as oriented counterclockwise and decompose $H_a$ into subsegments $\delta_1\epsilon_1\cdots\delta_\ell\epsilon_\ell$ where the $\delta_i$ are the maximal subsegments that remain upon folding $H_a \to H'_a$ and pruning. The images of the $\epsilon_i$ are what get pruned. There is a lift of $\gamma$ to $\widetilde{G}_0/{\langle}a {\rangle}$ that decomposes into subpaths $\gamma
= \beta_0 \alpha \beta_1$ where $\beta_0$ and $\beta_1$ are embedded and $\alpha$ is the immersed path that covers $H_a$ $n+2$ times.
When folding the segments $\epsilon_i$, the initial part of $\beta_1$ may (by equivariance) become identified with some portion of $H_a$. We are interested in bounding how much is identified with the terminal portion of $\alpha$ as this could reduce the amount of $\gamma_1$ that covers $a$. We will show that the portion of $\alpha$ identified is a segment of $H_a$. In other words, we can reduce this amount by at most 1.
Without loss of generality we assume that $\beta_1$ only intersects $H_a$ in a single vertex. Suppose this vertex is in $\delta_i$ and consider performing the folds in $\epsilon_i$. After folding $\epsilon_i$, the subsegment of the terminal part of $\alpha$ identified with the initial part of $\beta_1$ either heads counterclockwise from $v_0$ (which we are not concerned with as this adds to the amount by which $\gamma_1$ covers $a$) or it is contained in the union of $\delta_i$ and $\epsilon_i$. Indeed, we can just check locally that when folding two edges $e_1$ and $e_2$ in $\epsilon_i$ together in $H_a$ that $\beta_1$ cannot fold past (in the clockwise direction) the image of their terminal vertices. This involves a few cases depending on the relative positions of $\beta_1$, $e_1$ and $e_2$; all of which are easily verified.
Similarly, if $\beta_1$ only intersects $H_a$ in a vertex of $\epsilon_i$, then we find the the subsegment of the terminal portion of $\alpha$ that is identified with $\beta_1$ either heads counterclockwise from $v_0$ or it is contained in $\epsilon_i$.
Thus when performing the folds in $H_a \to H'_a$, the initial portion of $\beta_1$ is identified at most one copy of $H_a$. Likewise, the same holds for the terminal portion of $\beta_0$. Therefore the image path $\gamma_1$ $n$–covers $a$.
As a consequence of the claim, we have $\ell_{G'}(a^n) \leq BBT(f_1)$. Since $H'_a$ is immersed in every graph along the folding path between $G_1$ and $G'$, we have $\ell_{G'}(a^n) = \operatorname{Lip}(f_1)\ell_{G_1}(a^n)$, so that $$\ell_{G_1}(a^n) = \ell_{G'}(a^n)/\operatorname{Lip}(f_1) \leq
BBT(f_1)/\operatorname{Lip}(f_1) \leq 1$$ and so $\ell_{G_1}(a) \leq \frac{1}{n}$.
Combining Proposition \[prop:vpimpliessmallcycle\] with Proposition \[prop:ntwist-vp\] we get the first of the main results of this paper.
\[th:geo-twist\] Suppose $G,G' \in CV_k$ with $d = d_L(G,G')$ such that $\tau_a(G,G')
\geq n+2$ for some $a$. Then there is a geodesic $\alpha{\colon\thinspace}[0,d] \to
CV_k$ such that $\alpha(0) = G$ and $\alpha(d) = G'$ and for some $t
\in [0,d]$, we have $\ell_{\alpha(t)}(a) \leq 1/n$. In other words, $\alpha([0,d]) \cap CV_k^{1/n}(a) \neq \emptyset$.
Additionally, combining Proposition \[prop:vpimpliessmallcycle\] with Proposition \[prop:ncover-vp\] we get the second of the main results of this paper.
\[th:alg-twist\] Suppose $\phi \in \operatorname{Out}F_k$ is fully irreducible, with unstable tree $T_-$ and lamination $\Lambda_-$ such that $\tau_a(T_-,\Lambda_-) \geq n+4$ for some $a \in F_k$. Then given any train-track $G$, there is an axis ${\mathcal{L}}_\phi$ for $\phi$ that contains $G$ and a graph $G_0$ such that $\ell_{\widetilde{G}_0}(a) \leq 1/n$. In other words, ${\mathcal{L}}_\phi
\cap CV_k^{1/n}(a) \neq \emptyset$.
Example {#sc:example}
=======
Here we present an application of Theorem \[th:alg-twist\] in which we describe the asymptotic behavior of the translation length in $CV_k$ of certain elements of $\operatorname{Out}F_k$, given as products $\phi_n =
\delta_1^n \delta_2^{-n}$ of powers of Dehn twists $\delta_1,
\delta_2$. These types of outer automorphisms were considered in [@ar:CP] and used in [@un:CP2] to show that there is no homological obstruction to full irreducibility. We briefly recall the setup here; for more details consult either of the references [@un:CP2; @ar:CP].
Dehn twists {#ssc:Dehn-Twist}
-----------
A *cyclic* tree is a Bass–Serre tree associated to a splitting of $F_k$ over ${\mathbb{Z}}$, either as an amalgamated free product or as an HNN-extension. To such a tree is an associated (outer) automorphism called a *Dehn twist*. Given $F_k = A *_{{\langle}c {\rangle}} B$ we define the Dehn twist automorphism $\delta_c$ of $F_k$ by: $$\begin{aligned}
\forall a \in A \qquad & \delta_c(a) = a \\
\forall b \in B \qquad & \delta_c(b) = cbc^{-1}.\end{aligned}$$ Likewise, given $F_k = A *_{\mathbb{Z}}= {\langle}A, t \ | \ t^{-1}ct = c' {\rangle}$ for $c,c' \in A$, we define the Dehn twist $\delta_c$ of $F_k$ by: $$\begin{aligned}
\forall a \in A \qquad & \delta_c(a) = a \\
& \delta_c(t) = ct.\end{aligned}$$
Two cyclic trees $T_1$ and $T_2$ *fill* if their associated Dehn twists $\delta_1$, $\delta_2$ do not have any common invariant conjugacy classes of proper free factors or cyclic subgroups. As mentioned above, if $T_1$ and $T_2$ fill, then for large enough $n$, the element $\delta_1^n\delta_2^{-n}$ is fully irreducible (and hyperbolic) [@ar:CP Theorem 5.3].
Currents {#ssc:currents}
--------
A *(measured geodesic) current* on $F_k$ is an $F_k$–invariant and $\sigma$–invariant positive Radon measure on ${\partial}^2 F_k$ (refer to Section \[ssc:lamination-Q\]). Such measures where originally considered by Bonahon [@col:Bonahon91], see also [@col:Kapovich06]. Given a tree $T \in CV_k$, there is an identification between ${\partial}F_k$ and ${\partial}T$ used to interpret a current as a measure on the set of (bi-infinite) geodesics in $T$. Given a tight path $\alpha \subset T$, the *two-sided cylinder* $Cyl_T(\alpha)$ is the collection of geodesics that contain $\alpha$; such sets determine a basis for ${\partial}^2T$, and so in turn for ${\partial}^2 F_k$. When $\alpha$ is a fundamental domain for the action of $a \in F_k$ on $T^{{\langle}a {\rangle}}$, we will denote $Cyl_T(\alpha)$ by $Cyl_T(a)$. For a current $\nu \in Curr(F_k)$, define ${\langle}a, \nu {\rangle}_T = \nu(Cyl_T(a))$. As $\nu$ is $F_k$–invariant, this is well-defined. The current is uniquely defined by the values ${\langle}a , \nu {\rangle}_T$. If $c \in F_k$ is not a proper power, then we define the [*counting current*]{} $\eta_c$ of $c$ by: $${\langle}a, \eta_c {\rangle}_T = \#\mbox{ of axes of conjugates of $c$ in }
Cyl_T(a)$$ If $b = c^m$ where $c$ is not a proper power, then $\eta_b
= m\eta_c$.
Recall that an element $\phi \in \operatorname{Out}F_k$ is [*hyperbolic*]{} if it has no nontrivial periodic conjugacy classes in $F_k$; all such elements are necessarily non-geometric, in the sense that they are not induced by a surface homeomorphism. A hyperbolic fully irreducible element of $\operatorname{Out}F_k$ acts on the projectivized space of currents $\mathbb{P}Curr(F_k)$ with North–South dynamics [@thesis:Martin95]. In particular, there are both *stable* $[\mu_+(\phi)]$ and *unstable* $[\mu_-(\phi)]$ fixed projectivized currents associated to such an element. A similar statement holds for non-hyperbolic fully irreducible elements as well, after restricting to the subspace of $\mathbb{P}Curr(F_k)$ consisting just of those currents in the closure of the set of counting currents of primitive elements in $F_k$ [@KL_Boundary].
The *support* $Supp(\nu)$ of a current $\nu$ is the closure of the union of all open sets $U$ such that $\nu(U) > 0$. The support of a current is a lamination. The relationship between the stable currents and stable laminations of a fully irreducible element of $\operatorname{Out}F_k$ is given by the proposition below. The result is probably well-known, but to our knowledge, its proof does not appear in the literature. See also [@un:KL] for closely related results.
\[prop:lamination-current\] Suppose $\phi \in \operatorname{Out}F_k$ is fully irreducible with stable and unstable laminations $\Lambda_+, \Lambda_-$ and stable and unstable currents $\mu_+, \mu_-$. We have $Supp(\mu_\pm) = \Lambda_\pm$.
Let $g: G \to G$ be a train-track representative of $\phi$. Let $a \in F_k$ be a primitive element and $\alpha \subset G$ the reduced loop representing its conjugacy class. Then $\alpha$ is the union of $N$ legal paths in $G$ for some $N$, so that for all $m \geq 0$, the closed path $g^m(\alpha)$ consists of $N$ segments of leaves of the stable lamination $\Lambda_+$.
The set of cylinders $Cyl_{\tilde{G}}(\gamma)$, $\gamma$ a reduced path in $\tilde{G}$, not containing any leaf of $\Lambda_+$ give a cover of the complement of $\Lambda_+$. Choose one such cylinder $Cyl_{\tilde{G}}(\gamma)$, so that $\gamma$ is not a subsegment of any leaf of $\Lambda_+$. Then for any $m \geq 0$, the reduced loop $[g^m(\alpha)]$ contains at most $N$ copies of the image of $\gamma$ in $G$, and hence $\eta_{\phi^m(a)}(Cyl_{\tilde{G}}(\gamma)) \leq N$. Recall that, because $a$ was chosen to be primitive, we have the convergence of $[\eta_{\phi^m(a)}] \to [\mu_+]$. Now for a sequence $\lambda_m$ to give the convergence of $\frac{1}{\lambda_m} \eta_{\phi^m(a)} \to \mu_+$, it is necessary that $\lambda_m \to \infty$ [@ar:KL10 Theorem 1.2]. Thus we have $\mu_+(Cyl_{\tilde{G}}(\gamma)) = 0$.
We have shown that $Supp(\mu_+) \subseteq \Lambda_+$, a nonempty sublamination of a minimal lamination [@ar:BFH97; @ar:KL10]. The claim of the proposition is verified.
Axes of products of Dehn twists {#ssc:example}
-------------------------------
Let $k \geq 3$ and fix two filling cyclic trees $T_1$, $T_2$ with Dehn twists $\delta_1$ and $\delta_2$. Let $c_1$, $c_2$ denote the respective edge stabilizers. We assume that the set $\{c_1,c_2\}$ is not *separable*, i.e., no conjugates of $c_1$ and $c_2$ are contained in a proper free factor of $F_k$, nor in complementary free factors. Further, we assume that $c_1$ and $c_2$ are not *simultaneously elliptic* in $\overline{CV}_k$, i.e., $\ell_T(c_1) + \ell_T(c_2) \neq 0$ for all $T \in \overline{cv}_k$. These conditions can be guaranteed, for instance, by requiring $c_1$ and $c_2$ to be primitive elements sufficiently far apart in the free factor complex [@un:CR].
For the remainder of this section, elements $\phi_n \in \operatorname{Out}F_k$ denote the outer automorphisms induced by the automorphisms $\delta_1^n\delta_2^{-n}$. For large enough $n$, the elements $\phi_n$ are fully irreducible and hyperbolic [@ar:CP Theorem 5.3]. From [@un:CP2 Theorem 5.2], we understand the limiting behavior of the stable and unstable currents: $[\mu_+(\phi_n)] \to [\eta_{c_1}]$ and $[\mu_-(\phi_n)] \to [\eta_{c_2}]$. Using this, together with the parabolic behavior of Dehn twists on $\overline{CV}_k$ [@ar:CL95], we show that the sequence of stable and unstable trees likewise converge to the expected trees (cf., [@un:CP2 Remark 5.3]).
\[th:treeconvergence\] The trees $T_+(\phi_n) \in \overline{CV}_k$ converge to $T_2$. Similarly, the trees $T_-(\phi_n)$ converge to $T_1$.
Denote $\psi_n = \delta_2^{-n} \delta_1^n$ so that $\phi_n =
\delta_2^n \psi_n \delta_2^{-n}$. Then as the outer automorphisms are conjugate by $\delta_2^n$, we have $T_+(\phi_n) =
T_+(\psi_n)\delta_2^{-n}$.
Recall that in [@un:CP2 Theorem 5.2], we determined that $\lim_{n \to \infty}[\mu_-(\psi_n)] = [\eta_{c_1}]$. The continuity of the Kapovich–Lustig intersection form (see [@ar:KL10] for its definition and properties) implies that $c_1$ has a fixed point in an accumulation point of the sequence $\{T_+(\psi_n)\}$ (see [@un:CP2 Remark 5.3]). Therefore as $c_1$ and $c_2$ are not simultaneously elliptic, $c_2$ has positive translation length in any such accumulation point.
As $\overline{CV}_k$ is compact, some subsequence of $\{ T_+(\phi_n) \}$ converges. Consider such a convergent subsequence $\{ T_+(\phi_{n_m})
\} \subseteq \{T_+(\phi_n)\}$. By passing to a further subsequence, we can assume that both $\{ T_+(\phi_{n_{m_\ell}})\}$ and $\{ T_+(\psi_{n_{m_\ell}})
\}$ converge. Let $T_\infty$ denote the limit of the latter sequence. By the above remark, $c_2$ has positive translation length on the tree $T_\infty$.
Let $U \subset \overline{CV}_k$ be a neighborhood of $T_2$. As the set $\{ T_+(\psi_{n_{m_\ell}}) \} \cup \{ T_\infty \}$ is compact, and as $c_2$ has positive translation length on every tree therein, by [@ar:CL95 Theorem 13.2], there is an $N$ such that for $\ell
\geq N$ we have $T_+(\phi_{n_{m_\ell}}) =
T_+(\psi_{n_{m_\ell}})\delta_2^{-n_{m_\ell}} \in U$. Therefore the subsequence $\{ T_+(\phi_{n_m}) \}$ converges to $T_2$. As this is true for every convergent subsequence of $\{
T_+(\phi_n) \}$, and as $\overline{CV}_k$ is compact, we have the convergence of $T_+(\phi_n) \to T_2$. Applying the same argument to $\phi_n^{-1} =
\delta_2^n\delta_1^{-n}$ we see that $T_-(\phi_n) \to T_1$ as well.
Fix bases $\mathcal{T}_1$ and $\mathcal{T}_2$ for $F_k$, obtained from the vertex group(s) (and possibly a choice of stable letter in the case of an HNN-extension) of the Bass-Serre trees $T_1$ and $T_2$, respectively; to see how this is done, we refer to Section 3.1 of [@ar:CP]. Let $T_{\mathcal{T}_1}$ and $T_{\mathcal{T}_2}$ be the Cayley trees for the basis $\mathcal{T}_1$ and $\mathcal{T}_2$, respectively. See [@un:CP2; @ar:CP] for the details underlying these constructions.
\[prop:ntwist-T2\] For sufficiently large $n$, we have: $$\tau_{c_2}(T_{{\mathcal{T}}_2},\Lambda_-(\phi_n)) \geq \frac{n}{2}.$$
The proposition follows from a slight modification of the arguments from Theorem 5.2 in [@un:CP2]. In its proof (equation (5.9)), we showed that for for every $\epsilon >0 $ and integer $r > 0$, there is an $N > 0$ such that for $n \geq N$: $$\label{eqn:5.9}
\frac{ {\langle}c_2^r, \mu_-(\phi_n) {\rangle}_{T_{{\mathcal{T}}_2}} }
{\omega_{T_{{\mathcal{T}}_2}}(\mu_-(\phi_n))} > 1 - \epsilon.$$ The $\omega_{T_{{\mathcal{T}}_2}}(\cdot)$ in the demoninator is a normalization factor whose only relevant value to the present discussion is $\omega_{T_{{\mathcal{T}}_2}}(\mu_-(\phi_n))$; it may thus be treated as a positive constant.[^3] Equation (\[eqn:5.9\]) shows that there is a leaf of $\Lambda_-(\phi_n)=Supp(\mu_-(\phi_n))$ contained in the cylinder $Cyl_T(c_2^{r})$, and hence $\tau_{c_2}(T_{{\mathcal{T}}_2},\Lambda_-(\phi_n)) \geq r$.
Following the same analysis as in [@un:CP2 Theorem 5.2], fixing $r=n/2$, one can show: $$\label{eqn:5.9ii}
\left| 1 - \frac{{\langle}c_2^{n/2}, \mu_-(\phi_n) {\rangle}_{T_{{\mathcal{T}}_2}}}
{\omega_{T_{{\mathcal{T}}_2}}(\mu_-(\phi_n))} \right| <
\left|\frac{\frac{1}{2}An^2 + A_1n + A_2}{An^2 - B_1n - B_2} \right|
+ \frac{\epsilon}{2}$$ for some fixed positive constants $A,A_1,A_2,B_1$ and $B_2$.[^4] Thus for large enough $n$, we have that ${\langle}c^{n/2},\mu_-(\phi_n) {\rangle}_T > 0$, and hence there is a leaf of $\Lambda_-(\phi_n) = Supp(\mu_-(\phi_n))$ contained in the cylinder $Cyl_T(c_2^{n/2})$. This implies that $\tau_{c_2}(T_{{\mathcal{T}}_2},\Lambda_-(\phi_n)) \geq n/2$, as claimed.
We can use the fact that the sequence of trees $\{ T_-(\phi_n) \}$ converges to $T_1$ to show that the twist of $T_-(\phi_n)$ with $\Lambda_-(\phi_n)$ relative to $c_2$ is also approximately at least $n$.
\[prop:ntwist\] There exists a constant $D \geq 1$ such that for sufficiently large $n$: $$\tau_{c_2}(T_-(\phi_n),\Lambda_-(\phi_n)) \geq \frac{n}{D}.$$
As before, let $T_{{\mathcal{T}}_2}$ be the Cayley tree corresponding to the basis ${\mathcal{T}}_2$. By Proposition \[prop:ntwist-T2\], for each $n$ there is a leaf $\ell_n {\colon\thinspace}{\mathbb{R}}\to T_{{\mathcal{T}}_2}
$ of $\Lambda_-(\phi_n)$ that intersects the axis of $c_2$ in a segment of length at least $n\ell_T(c_2)/2$. We must verify that this overlap is not significantly reduced when mapping to $T_-(\phi_n)$.
Fix an $F_k$–equivariant map $f{\colon\thinspace}T_{{\mathcal{T}}_2} \to T_1$ and scale the metric on $T_1$ so that $\operatorname{Lip}(f) \leq 1$, and thus $BBT(f) \leq 1$ (we assume that the volume of $T_{{\mathcal{T}}_2}/F_k$ is 1). By scaling the metrics on $T_-(\phi_n)$ we have the convergence of $T_-(\phi_n) \to T_1$ from Theorem \[th:treeconvergence\]. Thus for large enough $n$, we can choose equivariant maps $f_n {\colon\thinspace}T_{{\mathcal{T}}_2} \to T_-(\phi_n)$ so that $\operatorname{Lip}(f_n) \leq 2$ and so $BBT(f_n) \leq 2$. As convergence is in the space of length functions, and as $\ell_{T_1}(c_2) > 0$, there is $\delta > 0$ such that $0 < \delta < \ell_{T_-^n}(c_2) < 1/\delta$ for all $n$.
Now let $x_n \in \ell_n({\mathbb{R}}) \cap T_{{\mathcal{T}}_2}^{{\langle}c_2 {\rangle}}$ be such that $y_n =
c_2^{n/2}x_n \in \ell_n({\mathbb{R}}) \cap T_{{\mathcal{T}}_2}^{{\langle}c_2 {\rangle}}$. Thus the path $[f_n(x_n), f_n(y_n)]$ contains an arc of the axis of $c_2$ in $T_-(\phi_n)$ of length at least $\frac{n}{2}\ell_{T_-(\phi_n)}(c_2)$. Further notice that the distance from either $f_n(x_n)$ or $f_n(y_n)$ to this arc is at most 2 (an upper bound for the bounded back tracking constant).
As $\ell_n {\colon\thinspace}{\mathbb{R}}\to T_{{\mathcal{T}}_2}$ is a leaf of the unstable lamination, after tightening its image in $T_-(\phi_n)$ we obtain a geodesic $[f_n(\ell_n({\mathbb{R}}))]$, and the same statement in the previous paragraph for the segment $[f_n(x_n),f_n(y_n)]$ and the axis of $c_2$ holds in turn for $[f_n(x_n),f_n(y_n)]$ and the geodesic $[f_n(\ell_n({\mathbb{R}}))]$. Hence the leaf of $\Lambda_-(\phi_n)$ whose image in $T_-(\phi_n)$ is $[f_n(\ell_n({\mathbb{R}})]$ intersects the axis of $c_2$ along a segment of length at least: $$\begin{aligned}
\frac{n\ell_{T_-(\phi_n)}(c_2)}{2} - 4 &> \frac{n\ell_{T_-(\phi_n)}(c_2)}{2} -
\frac{4\ell_{T_-(\phi_n)}(c_2)}{\delta} \\
&= \ell_{T_-(\phi_n)}(c_2)\left(\frac{n\delta - 8}{4\delta} \right) \\
& = \ell_{T_-(\phi_n)}(c_2)\frac{n}{D}
\end{aligned}$$ for some constant $D > 0$, provided $n > 8/\delta$. Thus $$\tau_{c_2}(T_-(\phi_n),\Lambda_-(\phi_n)) \geq \frac{n}{D}.$$
It follows from the proposition, together with Theorem \[th:alg-twist\], that for each element $\phi_n$, there is a train-track map $g_n{\colon\thinspace}G_n \to G_n$ such that $\ell_{G_n}(c_2) \leq D'/n$ for some constant $D'$. Note that we are using the fact that every graph on the axis of $\phi$ represents a train-track of $\phi$.
Recall that we assumed that $\{ c_1, c_2 \}$ is not separable in $F_k$.
\[lm:separable\] If $\{c_1, c_2 \}$ is not separable, then for large enough $n$, neither is $\{c_2,\delta^n_1(c_2)\}$.
This is easy to see using Whitehead graphs. Since the set $\{
c_1,c_2 \}$ is not separable, the union of their Whitehead graphs is connected and does not have a cut vertex (in an appropriate basis) [@col:Stallings99]. As cancellation is bounded, for large enough $n$, the subword representing $c_1$ will appear as a subword of $\delta_1^n(c_2)$. Hence the union of the Whitehead graphs of $c_2$ and $\delta_1^n(c_2)$ will cover the union of the Whitehead graphs of $c_1$ and $c_2$. In particular, their union will be connected and will not have a cut vertex. This implies the set $\{ c_2, \delta_1^n(c_2) \}$ is not separable.
Note that $\phi_n(c_2) = \delta_1^n(c_2)$ so that, as a consequence of the lemma, every edge in the track-track graph $G_n$ must be crossed by either $c_2$ or $\phi_n(c_2)$. Therefore, the length of $\phi_n(c_2)$ is at least $1 - D'/n = (n-D')/n$, and thus the Lipschitz constant for $\phi_n$ is at least $$\frac{(n-D')/n}{D'/n} = n/D' -1.$$ In particular, we have now shown that for some constant $K_1>0$: $$\frac{1}{K_1} \log n \leq tr_{CV_k}(\phi_n)$$ where $tr_{CV_k}(\phi) = \min \{ d_L(G,G\phi) \, | \, G \in CV_k \}$ is the minimal translation length of the element $\phi$.
We obtain the corresponding upper bound on $tr_{CV_k}(\phi_n)$ by explicitly constructing a path by piecing together geodesic segments such as those constructed in Example \[ex:twist\].
As before, let $T_{\mathcal{T}_1}$ and $T_{\mathcal{T}_2}$ be the Cayley trees for the basis $\mathcal{T}_1$ and $\mathcal{T}_2$, respectively. We consider these trees as points in $CV_k$, with every edge of each tree having length $1/k$. We first connect $T_{\mathcal{T}_2}\delta_2^{n}$ to $T_{\mathcal{T}_2}$ by a geodesic of length $\sim \log n$. Then we follow an optimal path $P$ from $T_{\mathcal{T}_2}$ to $T_{\mathcal{T}_1}$, and then connect $T_{\mathcal{T}_1}$ to $T_{\mathcal{T}_1}\delta_1^{n}$ with a geodesic which has length $\sim
\log n$. Finally, using the $\delta_1^{n}$–translate $P
\delta_1^{n}$ of $P$, we connect $T_{\mathcal{T}_1}\delta_1^{n}$ to $T_{\mathcal{T}_2}\delta_1^{n}$ (see Figure \[fig:path\]). As the length of $P$ is independent of $n$, translating the entire path by $\delta_2^{-n}$, we have for all $n$: $$d_L(T_{\mathcal{T}_2},T_{\mathcal{T}_2}\phi_n ) \leq K_2\log n$$ for some $K_2>0$.
at 0 42 at 210 42 at 410 42 at 135 -10 at 335 -10 at 95 35 at 230 -3 ![A path from $T_{\mathcal{T}_2}\delta_2^n$ to $T_{\mathcal{T}_2}\delta_1^n$.[]{data-label="fig:path"}](path "fig:")
Combining this upper bound with the previous lower bound, we have established the following:
\[th:twist-translation\] Let $T_1,T_2$ be two cyclic trees that fill with associated Dehn twist automorphisms $\delta_1$ and $\delta_2$ and let $c_1$, $c_2$ denote the respective edge stabilizers. Suppose that $\{ c_1,c_2 \}$ is not separable and that $c_1$ and $c_2$ are not simultaneouly elliptic in $\overline{CV}_k$. For $n \geq 1$, let $\phi_n$ be the outer automorphism induced by $\delta_1^n\delta_2^{-n}$. Then there is a constant $K = K(T_1,T_2)$ such that for large enough $n$:
1. there is a train-track representative $g_n{\colon\thinspace}G_n \to G_n$ such that $\ell_{G_n}(c_2) \leq K/n$, and
2. $\frac{1}{K} \log n \leq tr_{CV_k}(\phi_n) \leq K \log n$.
THIS IS MOVED FROM ABOVE, SINCE WE ARE TRYING LAMINATIONS INSTEAD OF CURRENTS...
It seems like we can define geometric relative twist for $T'
\in \partial cv_k$ if we know that $Tr_{T'}$ was a finite set... even so it could be that the twist is infinite. This happens when $a$ fixes a point in $T'$... I think. I guess the claim is that $\tau_a(T,T') = \tau_a(T,L^2(T'))$ where the first is geometric relative twisting and the second is analytic relative twisting?
Generalized relative twisting {#sc:grt}
=============================
\[def:twist\] Given an indivisible $a \in F_k$, a tree $T$ as above and current $\mu \in Curr(F_k)$ define $$\tau_a(T, \, \mu) = \max\{ \{ n \in {\mathbb{N}}\, | \, \mu(Cyl_T(a^n)) > 0
\} \cup \{ 0 \} \}.$$
I think this is the natural setting for the converse:
\[th:gthin-converse\] Suppose $\phi$ is fully irreducible, $T_-$ and $\mu_-$ are the unstable tree and current for $\phi$ and $H$ is an indivisible subgroup. Then if there is a train-track graph $G \in CV_k$ for $\phi$ such that $\operatorname{vol}_{\widetilde{G}}(H) \leq 1/n$ (i.e., ${\mathcal{L}}_\phi
\cap CV_k^{1/n}(H) \neq\emptyset$) and $H$ is “maximal” with respect to this property (need to be clear on what this means), then $\tau_H(T_, \, \mu_-) \geq n$.
Briefly why is this true? Some part of the unstable lamination crosses though $H$ and if $H$ is small, then the unstable lamination must cover $H$ several times. Now push this over to a train-track graph for $\phi{^{-1}}$ and map it to $T_-$. This should translate into a large intersection of a subtree of $T_-^H$ with some leaf of the unstable lamination.
Ann Arbor
=========
We would like to prove the following, after making it precise:
Suppose that $s$ is a segment of an edge of $G$ and $e$ is an edge in $G$. Suppose that $g: G \to G$ is a train-track map for $\phi$ with expansion factor $\lambda$, and that $g(s) = e$. Then (after possibly passing to a power) there exists a vanishing path that covers $s$ approximately $\lambda/
(3k-3)$ times.
We also identified the other steps to completing the proof. These are:
1. \# of $a$’s in a vanishing path $\geq n \implies$ we can find a small graph with a large portion of the vanishing path.
2. large portion of vanishing path in a subgraph $\implies$ big relative twist
3. other direction: big twist $\implies$ small graph
Perron-Frobenious
=================
Let $A$ be the matrix for which the $ij$th entry $a_{ij}$ is the number of times $f(e_i)$ passes over $e_j$. The (right) eigenvector $\vec{w}$ of $A$ with entries adding to $1$ corresponding to the eigenvalue $\lambda=\rho(A)$ gives the length of each of the edges of the graph $G$ of the train-track map $g: G \to G$. Now let $\vec{v}$ be the (right) eigenvector corresponding to the eigenvalue $\lambda$ of $A^T$ such that $\vec{v}^T \vec{w} =1$, and let $
\vec{u}$ be the vector all of whose entries are $1$. Then the $i$th entry of $A^T\vec{u}$ determines how many times edges get mapped over $e_i$ by $g$; the $i$th entry of $(A^T)^n \vec{u}$ the number of times edges get mapped over $e_i$ by $g^n$. Then Perron-Frobenious theory tells us that $\lim_{n \to
\infty} (A^T/\lambda)^n$ is equal to $M = \vec{v}\vec{w}^T$. Therefore we have that for any $\epsilon$ there is $N$ such that if $n \geq N
$, then the length of $$(A^T)^n\vec{u} - \lambda^n M\vec{u}$$ is less than $\epsilon$, so that $(A^T)^n\vec{u}$ is approximately equal to $
\lambda^n \vec{v}$. After many iterations, the number of edges that get mapped over $e_i$ is approximately $\lambda^n v_i$.
[^1]: The first author is partially supported by NSF grant DMS-1006898. The second author was partially supported by NSF grant DMS-0856143, NSF RTG DMS-0602191, EPSRC grant EP/D073626/2, and an NSERC Discovery Grant.
[^2]: Note that in [@un:HM], the stable lamination is called the “expanding lamination” and denoted by $\Lambda_-$ as it is more naturally associated to $T_-(\phi)$. See Proposition \[prop:stableQ\].
[^3]: On the other hand, to recognize (\[eqn:5.9\]) from equation (5.9) in the proof of [@un:CP2 Theorem 5.2], it should be observed that $\omega_{T_{{\mathcal{T}}_2}}(\eta_{c_2})=\langle c_2^r, \eta_{c_2} \rangle_{T_{{\mathcal{T}}_2}}$.
[^4]: Compare this to equation (5.9) from [@un:CP2 Theorem 5.2] where the numerator of the righthand side is linear in $n$, and note that the constants $\beta_1, \beta_2$ there depend on $r$. Here in (\[eqn:5.9ii\]), the numerator is quadratic because of the choice of $r = n/2$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Given a finitely generated free monoid $X$ and a morphism $\phi : X\to X$, we show that one can construct an algebra, which we call an iterative algebra, in a natural way. We show that many ring theoretic properties of iterative algebras can be easily characterized in terms of linear algebra and combinatorial data from the morphism and that, moreover, it is decidable whether or not an iterative algebra has these properties. Finally, we use our construction to answer several questions of Greenfeld, Leroy, Smoktunowicz, and Ziembowski by constructing a primitive graded nilpotent algebra with Gelfand-Kirillov dimension two that is finitely generated as a Lie algebra.'
address: |
University of Waterloo\
Department of Pure Mathematics\
Waterloo, Ontario\
Canada N2L 3G1\
author:
- 'Jason P. Bell and Blake W. Madill'
title: Iterative algebras
---
Introduction
============
A rich class of examples in ring theory is provided by monomial algebras, which have the advantage of having the property that many ring theoretic properties can be understood purely combinatorially in terms of forbidden subwords and other notions from combinatorics on words (cf. [@Belov]). In addition, the study of monomial algebras via combinatorial methods has played a key role in many results from ring theory dealing with growth and other properties due to the intimate connection with noncommutative Gröbner bases [@Drensky; @Giambruno]. In this paper, we study a certain subclass of monomial algebras which we call *iterative algebras* (see §2 for the definition). One of the problems with the class of monomial algebras is that it is in general a very large and unwieldy class. One can, in some settings, restrict one’s attention to finitely presented monomial algebras, and in this case one has graph theoretic and other combinatorial tools at one’s disposal for their study (cf. [@Belov Chapter 5]). On the other hand, this class of algebras can often be too well-behaved to give one a sense of the types of pathologies that can occur in noncommutative algebra. Iterative algebras are, in general, non-finitely presented monomial algebras produced via iterated morphisms of monoids. They retain many of the nice combinatorial properties of finitely presented algebras, but at the same time can sometimes possess bizarre properties due to the iterative nature of their definition.
Informally, an iterative algebra is obtained by taking a field $k$ and then forming the quotient of $k\{x_1,\ldots ,x_d\}/I$, where $I$ is the ideal generated by all words over the alphabet $\{x_1,\ldots ,x_d\}$ that do not occur as a subword of a right-infinite word $w$ over the alphabet $\{x_1,\ldots ,x_d\}$ that is a fixed point of an endomorphism $\phi$ of the free monoid on $\{x_1,\ldots ,x_d\}$. In particular, the right-infinite word $w$, being a fixed point of an endomorphism, has many self-similarity properties that make studying the structure theory of the iterative algebras we construct fairly straightforward. In fact, we are able to show that ring theoretic properties such as begin prime, semiprime, satisfying a polynomial identity and being noetherian are all decidable in terms of the combinatorics of the endomorphism $\phi$ (see Theorem \[thm: char\] and §5). We are also able to show that the algebras we construct have Gelfand-Kirillov dimension in the set $\{1,2,3\}$ (see Theorem \[thm: GK\]). The impetus for the construction of these algebras, however, came from our attempt to provide answers to some questions of Greenfeld, Leroy, Smoktunowicz, and Ziembowski [@GLSZ]. These questions have to do with graded nilpotent algebras. These are graded algebras $A$ with the property that if $S$ is a subset of $A$ consisting of homogeneous elements of the same degree, then there is some natural number $N=N(S)$ such that $S^N=(0)$. Using our iterative algebra construction, we are able to present an example or a ring that simultaneously provides answers to three questions of Greenfeld *et al.* [@GLSZ].
\[thm: main\] Let $k$ be a field. Then there exists a finitely generated $k$-algebra $R$ that is a prime, graded nilpotent algebra of Gelfand-Kirillov dimension two that has trivial Jacobson radical and is finitely generated as a Lie algebra.
This paper is organized as follows. In §2, we describe morphic words and give the construction of an iterative algebra. In §3, we show that the Gelfand-Kirillov dimension of these algebras is either one, two, or three. In §3, we give characterizations of various ring theoretic properties in terms of combinatorial data of the underlying morphism and associated infinite word; then in §4, we discuss decidability of these properties. Finally, in §5 we prove Theorem \[thm: main\] (see Theorem \[thm: main2\]). We conclude with §6 by giving some open problems and making some remarks.
Pure morphic words and iterative algebras
=========================================
In this section, we give our construction of iterative algebras and give some of the necessary background on morphic words.
Let $\Sigma$ be a finite alphabet. We let $\Sigma^*$ denote the free monoid on $\Sigma$ (we let $\varepsilon$ denote the empty word, which serves as the identity of this monoid). Then a morphism $\phi :\Sigma^*\to \Sigma^*$ is determined by the images of the elements of $\Sigma$. We say that $a\in\Sigma$ is *mortal* if there is some $j$ such that $\phi^j(a)=\varepsilon$, and we let $X$ denote the submonoid of $\Sigma^*$ generated by the mortal letters.
We then say that the morphism $\phi$ is *prolongable* on the letter $b\in \Sigma$ if $\phi(b)=bx$ with $x\in \Sigma^*\setminus X$. In this case, we can create an infinite sequence $$b,\phi(b)=bx,\phi^2(b)=bx\phi(x), \phi^3(b)=bx\phi(x)\phi^2(x),\ldots$$ by repeatedly iterating $\phi$ and starting with the letter $b$. Notice that the limit of this sequence is an infinite word $$\phi^{\omega}(b):=bx\phi(x)\phi^2(x)\cdots.$$ We then say that a right-infinite word $w$ over the alphabet $\Sigma$ is a *pure morphic word* if there exists a morphism $\phi:\Sigma^*\to \Sigma^*$ that is prolongable on some letter $b\in \Sigma$ such that $w=\phi^{\omega}(b)$. In general, a *morphic word* is obtained from a pure morphic word by applying a *coding*; that is a letter-by-letter substitution of $w$ by a map $\tau: \Sigma\to \Delta$ where $\Delta$ is another finite alphabet and $\tau$ is not necessarily one-to-one. For our purposes, we only work with pure morphic words.
We now introduce some terminology, which will give important classes of morphic words.
*Let $w$ be a right-infinite morphic word that is associated to a morphism $\phi:\Sigma^*\to \Sigma^*$ that is prolongable on $b\in \Sigma$ for some finite alphabet $\Sigma$.*
1. We say that $w$ is a $d$-*uniform* morphic word if $\phi(a)$ is a word in $\Sigma^*$ of length $d$ for every $a\in \Sigma$.
2. We say that $w$ is *primitive* if for every $a,a'\in \Sigma$ we have that there is some natural number $m$, depending on $a$ and $a'$, such that $a'$ appears as a letter in $\phi^m(a)$.
We point out that primitivity of a pure morphic word is decidable (see §5 for more details). First, observe that by removing letters from $\Sigma$ if necessary, we may always assume without any loss of generality that every $a\in \Sigma$ occurs in $w$. Then if $\phi$ is prolongable on $b\in \Sigma$ and $w=\phi^{\omega}(b)$ then if $a'\in \Sigma$ then $a'$ appears in $\phi^n(b)$ for some $n$. Hence if $b$ occurs as a letter in $\phi^m(a)$ for some $m$ then $a'$ will occur as a letter in $\phi^{n+m}(a)$. Thus to check primitivity, it suffices to check that for every $a\in \Sigma$, there is some $n$, depending upon $a$, such that $b$ occurs in $\phi^n(a)$. We are now able to define an iterative algebra. (We do not use the term morphic algebra, as that term has been used to name an unrelated class of rings)
Given a pure morphic word on an alphabet $\Sigma=\{x_1,\ldots ,x_m\}$, we can associate an $m\times m$ matrix, $M(w)$, called the *incidence matrix*. The $(i,j)$-entry of $M(w)$ is the number of occurrences of $x_i$ in $\phi(x_j)$. Given a word $u\in \Sigma^*$, we can then associate an $m\times 1$ integer vector $\theta(u)$ whose $j$-th coordinate is the number of occurrences of $x_j$ in $u$. Then [@AS Proposition 8.2.2] shows that we have the relationship $$\theta(\phi^n(u)) = M(w)^n \theta(u).\label{eq: theta}$$
[ *Let $k$ be a field, let $\Sigma=\{x_1,\ldots ,x_m\}$ be a finite alphabet, and let $w$ be a right-infinite pure morphic word over $\Sigma$. We define the *iterative algebra* $A_w$ associated to $w$ to be the quotient $k\{x_1,\ldots ,x_m\}/I$, where $I$ is the ideal generated by all finite words over $\Sigma$ that do not appear as a subword of $w$.*]{}
These algebras form a subclass of the monomial algebras studied in [@Belov Chapter 3].
Gelfand-Kirillov dimension
==========================
In this section, we discuss the growth of iterative algebras, showing that their Gelfand-Kirillov dimension is either $1$, $2$, or $3$. We recall that given two maps $f,g :\mathbb{N}_0\to \mathbb{R}_+$, we say that $f(n)=\Theta(g(n))$ if there exist positive constants $C_1$ and $C_2$ such that $$C_1g(n) \le f(n) \le C_2 g(n)$$ for all $n$ sufficiently large.
Given a field $k$ and a finitely generated $k$-algebra $A$, we recall that the *Gelfand-Kirillov* dimension of $A$, denoted ${\rm GKdim}(A)$, is given by $${\rm GKdim}(A):=\limsup_{n\to \infty} \frac{\log\,{\rm dim}(V^n)}{\log\, n},$$ where $V$ is a finite-dimensional subspace of $A$ that contains $1$ and generates $A$ as a $k$-algebra. This quantity does not depend upon the choice of subspace $V$ having these properties. For more information on Gelfand-Kirillov dimension, we refer the reader to the book of Krause and Lenagan [@KL]. We note that GK dimension one is equivalent to ${\rm dim}(V^n)=\Theta(n)$ (i.e., linear growth) by Bergman’s gap theorem [@KL Theorem 2.5] and that an important subclass of algebras of GK dimension two are given by those of *quadratic growth*; these are algebras for which ${\rm dim}(V^n)=\Theta(n^2)$.
In the case of iterative algebras, there is an intimate connection between the Gelfand-Kirillov dimension of the algebra and the *subword complexity* (also called factor complexity in the literature) of the associated pure morphic word $w$. We recall that given a right-infinite word $w$ over a finite alphabet $\Sigma$, the *subword complexity function* is defined by taking $p_w(n)$ to be the number of distinct subwords of $w$ of length $n$.
For a pure morphic word $w$, a result of Pansiot [@BR Theorem 4.7.1] shows that $p_w(n)$ is either $O(1), \Theta(n),\Theta(n \log\log n), \Theta(n\log n)$, or $ \Theta(n^2)$; moreover, each of these possibilities can be realized by some purely morphic word and $p_w(n)=O(1)$ if and only if $w$ is eventually periodic. If $A_w$ is the iterative algebra corresponding to the purely morphic word $w$ then if $V$ is the image of the space $k+\sum_{a\in \Sigma} ka$ in $A_w$ then $V^n$ has a basis consisting of subwords of $w$ of length at most $n$. In particular, ${\rm dim}(V^n)=\sum_{j=0}^n p_w(j)$. If $p_w(n)=\Theta(n), \Theta(n\log n),$ or $\Theta(n \log\log n)$, then we respectively have $\sum_{j=0}^n p_w(j)=\Theta(n^2), \Theta(n^2\log n),$ or $\Theta(n^2 \log\log n)$, and so $$\frac{\log\,{\rm dim}(V^n)}{\log\, n}\to 2$$ as $n\to \infty$. On the other hand, if $p_w(n)=\Theta(n^2)$ then $$\frac{\log\,{\rm dim}(V^n)}{\log\, n}\to 3$$ as $n\to \infty$. Finally, if $p_w(n)={\rm O}(1)$ then ${\rm dim}(V^n)$ grows at most linearly with $n$ and since $A_w$ is infinite-dimensional, we see that the GK dimension of $A_w$ is one in this case. Putting these observations together we obtain the following result.
\[thm: GK\] Let $k$ be a field and let $A_w$ be an iterative $k$-algebra. Then ${\rm GKdim}(A_w)\in \{1,2,3\}$ and the Gelfand-Kirillov dimension is equal to one if and only if $w$ is eventually periodic.
We point out that in the case that $w$ is either primitive or $d$-uniform for some $d$, we can say more.
Let $k$ be a field and let $w$ be a pure morphic word that is either primitive or $d$-uniform for some $d\ge 2$. Then $A_w$ has either linear or quadratic growth and if $w$ is not eventually periodic then the growth is quadratic. \[prop: quadratic\]
By [@AS Theorem 10.4.12 and Corollary 10.3.2] we see that if $w$ is primitive or $d$-uniform[^1] and $w$ is not eventually periodic then $p_w(n)=\Theta(n)$ and so the result now follows from the remarks made earlier in this section. If $w$ is eventually periodic, then $p_w(n)=O(1)$ and so $A_w$ has linear growth.
We note that the prime spectra of monomial algebras of quadratic growth are particularly well-behaved [@BS].
Ring theoretic properties in terms of combinatorics of words
============================================================
In this section, we give characterizations of some basic ring theoretic properties of iterative algebras in terms of the associated morphic words.
Let $k$ be a field, let $w$ be a right-infinite morphic word over an alphabet $\Sigma=\{x_1,\ldots ,x_m\}$, and let $A_w$ be the iterative $k$-algebra associated to $w$. We assume that every letter in $\Sigma$ appears as a letter in $w$. Then the following characterizations hold:
1. $A_w$ is prime if and only if the first letter of $w$ occurs at least twice;
2. $A_w$ is semiprime if and only if it is prime;
3. $A_w$ is just infinite if and only if $w$ is uniformly recurrent (that is, given a subword $v$ of $w$ there exists some $N=N(v)$ such that every block of $N$ consecutive letters in $w$ contains $v$ as a subword);
4. $A_w$ satisfies a polynomial identity if and only if $w$ is eventually periodic;
5. $A_w$ is noetherian if and only if $w$ is eventually periodic;
\[thm: char\]
Since $A_w$ is a monomial algebra, we know that if $A_w$ fails to be prime then there must exist subwords $v,v'$ of $w$ such that $vuv'$ is not a subword of $w$ for any $u\in \Sigma^*$. Thus we can never have $v'$ occurring after $v$ in $w$ and since both $v'$ and $v$ occur as subwords of $w$ we see that there is some subword $u$ of $w$ that has $v'$ as a prefix and $v$ as a suffix. By assumption $w=\phi^{\omega}(b)$ for some $b\in \Sigma$ such that $\phi$ is prolongable on $b$, and so there is some $n$ such that $\phi^n(b)$ contains the subword $u$. If $b$ occurs at least twice in $w$ then there is some word $u'$ such that $bu'b$ occurs as a prefix of $w$ and so $\phi^n(b)\phi^n(u')\phi^n(b)$ is also a prefix of $w$. But this now means that $uu''u$ occurs as a subword of $w$ and this contradicts the fact that no element from $v\Sigma^* v'$ occurs as a subword of $w$. Thus if $A_w$ is not prime then the first letter of $w$ occurs only once.
On the other hand if $b$ is the first letter of $w$ and it only occurs once in $w$ then $bub$ cannot be a subword of $w$ for any word $u$ and so the image of $b$ in $A_w$ generates a nilpotent ideal. This proves (1) and (2).
Next, by [@Belov Theorem 3.2], $A_w$ is just infinite if and only if $w$ is a uniformly recurrent word, and so this gives (3). To see (4), note that if $w$ is eventually periodic, then $A_w$ has Gelfand-Kirillov dimension one by Theorem \[thm: GK\] and hence is PI [@SSW]. Conversely, suppose that $A_w$ satisfies a polynomial identity. If we take the ideal $I$ of $A_w$ generated by the images of all subwords of $w$ that only appear finitely many times in $w$ then by construction the algebra $B=A_w/I$ is a prime monomial algebra and satisfies a polynomial identity. It is straightforward to see that there is some recurrent word $u$ such that the images of the subwords of $u$ in $B$ form a basis for $B$. By [@Belov Remark, page 3523], we then have that $u$ is periodic since $B$ satisfies a polynomial identity. We let $q(n)$ denote the collection of subwords of $w$ of length $n$ that appear infinitely often in $w$. Then by construction $q(n)$ is precisely the number of distinct subwords of $u$ of length $n$ and so $q(n)={\rm O}(1)$. In particular, there is some $d$ such that $q(d)\le d$. Let $v_1,\ldots ,v_m$ denote the distinct subwords of $w$ of length $d$. By assumption, at most $d$ of these words occur infinitely often and so we may write $w=vw'$ where $v$ is finite and $w'$ has at most $d$ subwords of length $d$. We then have that $w'$ is eventually periodic [@AS Theorem 10.2.6] and so $w$ is eventually periodic. This gives (4).
Finally, by [@Belov Corollary 5.40], $A_w$ is noetherian if and only if $A_w$ has GK dimension one. But this occurs if and only if $w$ is eventually periodic by Theorem \[thm: GK\].
Decidability
============
One of the advantages of working with iterative algebras is that many of the ring theoretic properties described in the preceding section are decidable for these algebras; that is, one can give an algorithm which inputs the data associated with the morphism and tells whether or not the corresponding algebra has one of the ring theoretic properties described earlier. As Theorem \[thm: char\] shows, many of the relevant properties for iterative algebras can be described in terms of a handful of properties of the corresponding pure morphic word. In particular, it suffices to understand whether a pure morphic word is eventually periodic, uniformly recurrent, or whether certain letters occur twice. In fact, these are well-understood for pure morphic words and we give a summary of some of what is known.
Let $w$ be a pure morphic word over a finite alphabet $\Sigma$. Then it is decidable whether or not $w$ has the following properties:
1. $w$ is eventually periodic;
2. $w$ is primitive;
3. $w$ is uniformly recurrent;
4. $a\in \Sigma$ occurs at least twice in $w$;
Decidability of the eventual periodicity question was answered by Harju and Linna [@HL] and independently by Pansiot [@Pa]. Durand [@Durand] showed that uniform recurrence was decidable. Primitivity and questions about occurrence of letters are much more straightforward. Let $b$ be the first letter of $w$, let $M(w)$ be the incidence matrix of $w$ and let $\phi$ be the underlying morphism. Then there is some fixed vector ${\bf u}$ such that the number of occurrences of $a$ in $\phi^n(b)$ is given by $f(n)={\bf u}^T M(w)^n\theta(b)$. Thus if $a$ occurs at most once, then $f(n)=0,1$ for all $n$. Now $f(n)$ satisfies a linear recurrence that can be computed explicitly using the Cayley-Hamilton theorem and by computing the first $d$ terms of $f(n)$, where $d=\#\Sigma$. Solving the recurrence, it is straightforward to decide whether or not this is the case. Primitivity can be decided analogously.
Construction
============
In this section, we give an example of an iterative algebra, which gives answer to several questions of Greenfeld, Leroy, Smoktunowicz, and Ziembowski [@GLSZ]. (In particular, Question 31 and 36 have the answer ‘no’ and Question 32 has the answer ‘yes’.) The questions we consider deal with *graded nilpotent* algebras. These are graded algebras $A$ with the property that if $S$ is a subset of $A$ consisting of homogeneous elements of the same degree, then there is some natural number $N=N(S)$ such that $S^N=(0)$. We will show how one can use iterative algebras to construct such rings. Iterative algebras are unital and thus will not be graded nilpotent without some alteration; we will show, however, that by taking the positive part of an iterative algebra then in certain cases one can find a grading that gives a graded nilpotent algebra. Questions 31 and 32 of [@GLSZ] ask respectively whether a graded nilpotent algebra must be Jacobson radical and whether it can have GK dimension two; Question 36 asks whether a graded nilpotent algebra that is finitely generated as a Lie algebra must be nilpotent. We give answers to these questions with a single example. Specifically, we construct a graded nilpotent algebra of quadratic growth that is finitely generated as a Lie algebra and whose Jacobson radical is trivial.
Let $\Sigma=\{x_1,\ldots, x_6,y_1,\ldots ,y_6\}$ and let $\phi : \Sigma^*\to \Sigma^*$ be the morphism given by $$\left. \begin{matrix} x_1\mapsto x_1x_2y_1y_2& x_2\mapsto x_1x_3 y_1y_3 & x_3 \mapsto x_1x_4y_1y_4
\\
x_4\mapsto x_1x_5y_1y_5 & x_5\mapsto x_1x_6y_1y_6 & x_6\mapsto x_2x_3y_2y_3 \\
y_1\mapsto x_2x_4y_2y_5 & y_2\mapsto x_2x_5y_3y_4 & y_3\mapsto x_2x_6y_2y_6 \\
y_4\mapsto x_3x_4y_3y_5 & y_5 \mapsto x_3x_5y_3y_6 & y_6\mapsto x_3x_6y_4y_5. \end{matrix}\right.$$ Let $w$ be the unique right infinite word whose first letter is $x_1$ and that is a fixed point of $\phi$. We note that $w$ is $4$-uniform.
Then a straightforward computer computation shows that the incidence matrix $M(w)$ has characteristic polynomial $$P_w(x):=x^{12} - x^{11} - 8 x^{10} - 16 x^9 - 2 x^8 + 5 x^7 + 5 x^6 + 21 x^5 + 31 x^4 - 10 x^3 - 8 x^2.
\label{eq: charpoly}$$ We now put a grading on $A_w$ by declaring that $x_1$ has degree one and that all other letters in $\Sigma$ have degree two. We let $W_n$ denote the degree of $\phi^n(x_1)$. Then Equation (\[eq: theta\]) gives $$W_n={\bf u} M(w)^n \theta(x_1),$$ where ${\bf u}=[1,2,\ldots ,2]$. A straightforward computer calculation shows that for $n=0,1,\ldots ,12$, we get the sequence of values $$\label{eq: list}
1,9,40, 162,655,2627,10487,41987,167922, 671648, 2686840, 10746875, 42987905$$ for $W_n$.
Let $d$ be a positive integer. Then with the grading described above, the algebra $A_w$ has the property that the homogeneous component of degree $d$ is nilpotent. \[prop: grnilpotent\]
We let ${\mathcal}{S}=\{s_0,s_1,s_2,\ldots\}$ denote the subset of $\mathbb{N}_0$ constructed as follows. We define $s_0=0$ and for $i\ge 1$, we define $s_i=s_{i-1}+{\rm deg}(a_i)$, where $a_i\in\Sigma$ is the $i$-th letter of $w$. Then since $w=x_1x_2y_1y_2x_1x_3y_1y_3\cdots $, we see that ${\mathcal}{S}=\{0,1,3,5,7,8,10,12,14,\ldots\}$. We now let $(A_w)_d$ denote the homogeneous component of $A_w$ of degree $d$. Since $(A_w)_d$ is spanned by subwords of $w$ of degree $d$, we see that if $(A_w)_d$ is not nilpotent, then for every $r\ge 1$ there must exist some subword of $w$ of the form $u_1u_2\cdots u_r$ where $u_1,\ldots ,u_r$ are words of degree $d$. In particular, ${\mathcal}{S}$ must contain an arithmetic progression of the form $a,a+d,a+2d,\ldots ,a+(r-1)d$ for every $r\ge 1$.
We now show that this cannot occur. We suppose, towards a contradiction, that there is some $d\ge 1$ such that ${\mathcal}{S}$ contains arbitrarily long arithmetic progressions of the form $$\lbrace a, a+d, a+2d, \ldots, a+bd\rbrace,$$ where $a,b\in {\mathbb{N}}$. Then for every $n\ge 2$ there exists a subword $u=u_1u_2\cdots u_{r}$ of $w$ such that each $u_i$ is a subword of $w$ of degree $d$ and $r>4^{n+1}$. Now $u$ has length at least $4^{n+1}$ and since $w$ is a fixed point of $\phi$, we then see that $u$ must have a subword of the form $\phi^n(z)$ for some $z\in \Sigma$. It follows that there exist natural numbers $i$ and $j$ with $i>1$ and $i+j<r$ such that $\phi^n(z)=v u_i\cdots u_{i+j} v'$, where $v$ is a proper (possibly empty) suffix of $u_{i-1}$ and $v'$ is a proper (possibly empty) prefix of $u_{i+j+1}$. Every $z\in \Sigma$ has the property that $\phi^2(z)$ begins with $x_1$ and so $\phi^n(z)$ begins with $\phi^{n-2}(x_1)$. In particular, since $\phi^{n-2}(x_1)$ is a prefix of $w$ of length $4^{n-2}$ and it begins $vu_iu_{i+1}\cdots$, we see that ${\mathcal}{S}$ contains a progression of the form $t,t+d,t+2d,\ldots ,t+sd$ with $t<d$ and $i+(s+1)d$ strictly greater than the length of $\phi^{n-2}(x_1)$. By assumption, ${\mathcal}{S}$ contains arbitrarily long arithmetic progressions of length $d$ and so there are infinitely many natural numbers $n$ for which $\phi^n(x_1)$ contains a progression of the form $t,t+d,t+2d,\cdots t+sd$ for some $t<d$ and $t+(s+1)d$ strictly greater than the length of $\phi^n(x_1)$. In particular, there is some fixed $t<d$ for which there are infinitely many natural numbers $n$ with this property. Thus we see ${\mathcal}{S}$ contains arbitrarily long arithmetic progressions of the form $t,t+d,t+2d,\ldots ,t+rd$ for some fixed $t<d$. It follows that ${\mathcal}{S}$ contains an infinite arithmetic progression $t,t+d,t+2d,\ldots $.
Now $$\phi^{n+2}(x_1)=\phi^{n+1}(x_1x_2y_1y_2)=\phi^{n+1}(x_1)\phi^n(x_1x_3y_1y_3)\phi^{n+1}(y_1y_2).$$ Thus $\phi^{n+1}(x_1)\phi^n(x_1)$ is a prefix of $w$ for every $n\ge 1$.
Without loss of generality $d\in {\mathbb{N}}$ is minimal with respect to having the property that ${\mathcal}{S}$ has an infinite arithmetic progression of the form $a+d{\mathbb{N}}$, where $a< d$. Define $$T:=\lbrace a: 0\leq a< d, \{a+dn\colon n\ge 0\}\subseteq {\mathcal}{S}\rbrace.$$ We write $T=\lbrace i_1, i_2, \dots, i_{p}\rbrace,$ where $i_1<i_2<\dots<i_{p}$. Notice that there exists a positive integer $N$ such that if $j\in \{0,\ldots ,d-1\}\setminus T$ then there is some $m$, depending upon $j$, such that $j+md\le N$ and $j+md\not\in {\mathcal}{S}$.
Now let $a_j:=i_{j+1}-i_{j}$ for $j=1,\ldots ,p$, where we take $i_{p+1}=i_1+d$. Consider $${\bf a}:=(a_1,a_2,\dots, a_p)\in {\mathbb{N}}^p.$$ Let $\sigma=(1,2,3,\dots,p)\in S_p$, the symmetric group on $p$ letters. For $\pi\in S_p$, we let $\pi({\bf a})$ denote $(a_{\pi(1)}, a_{\pi(2)},\dots, a_{\pi(p)})$.
We claim that no non-trivial cyclic permutation of ${\bf a}$ can be equal to ${\bf a}$. Assume, towards a contradiction, that $\sigma^m(a)=a$ for some $m\in \{1,\ldots ,p-1\}$. Let $\pi=\sigma^m$ so that we then have that $a_i=a_{\pi(i)}$. We see that, by definition, ${\mathcal}{S}$ contains the set $$\{i_1,i_2,\dots, i_p, i_1+d,i_2+d,\dots, i_p+d, i_1+2d,\dots\}$$ Moreover, the differences between successive terms of this sequence are given by the sequence $$(a_1,a_2,\dots, a_p, a_1,a_2,\dots, a_p,a_1,\dots).$$ By repeatedly applying the identity $\sigma^m(a)=a$, we have that $$(a_1,a_2,\dots, a_p, a_1,a_2,\dots, a_p,a_1,\dots)=(a_1,a_2,\dots, a_{m},a_1,a_2,\dots, a_{m},a_1,\dots).$$ Therefore ${\mathcal}{S}$ contains an infinite arithmetic progression of the form $j+(a_1+\dots +a_{m}){\mathbb{N}}$, and $a_1+\cdots +a_m<d$. Moreover, by the argument we used earlier, where we observed that $\phi^n(x_1)$ occurs infinitely often in $w$ for every $n$, we see that we can take $j<a_1+\cdots +a_m$. But this contradicts the minimality of $d$. Therefore $\sigma^m(a)\neq a$ for any $m\in \{1,\ldots ,p-1\}$, as claimed.
Now for every $n\ge 2$, we have $\phi^{n+1}(x_1)\phi^n(x_1)$ is a prefix of $w$. Moreover, by assumption the infinite arithmetic progressions in ${\mathcal}{S}$ with difference $d$ all appear in the subset $$X:=\{i_1,i_2,\ldots ,i_p,i_1+d,i_2+d,\ldots \}.$$ We recall that we let $W_m$ denote the weight of $\phi^m(x_1)$ for each $m\ge 0$. Then given a positive integer $n> N$, there exists a unique $s\in \{1,\ldots ,p\}$ and a unique $r\ge 0$ such that $i_s+dr$ is the largest positive integer in the set $\{i_q+d \ell \colon 1\le q\le p, \ell\ge 0\}$ that is less than or equal to $W_n$. Since $\phi^n(x_1)\phi^{n-1}(x_1)$ is a prefix of $w$, we see that the part of the set $$\{i_1,i_2,\ldots ,i_p, i_1+d,i_2+d\ldots ,i_s+dr,W_n+i_1, W_n+i_2,\ldots , W_n+i_p,W_n+i_1+d,\ldots \}$$ in $[0,W_{n}+W_{n-1}]$ is entirely contained in ${\mathcal}{S}$.
For $j=1,\ldots ,p$, define $i_{s+j}$ to be $i_{s+j-p}+d$ if $s+j>p$. Then by definition of $T$, we have that $\{i_{s+1}+dr, i_{s+2}+dr,\ldots , i_{s+p}+dr,i_{s+1}+d(r+1),\ldots \}\cap [0,W_{n}+W_{n-1}]\subseteq {\mathcal}{S}$ and so subtracting $W_n$, the weight of $\phi^n(x_1)$, and using the fact that the prefix $\phi^n(x_1)$ in $w$ is then followed by $\phi^{n-1}(x_1)$ in $w$, we see that $$\{i_{s+1}+dr-W_n,i_{s+2}+dr-W_n,\ldots\}\cap [0,W_{n-1}]\subseteq {\mathcal}{S}.$$ Since $n-1\ge N$ and $i_{s+j}+dr-W_n\in \{0,\ldots ,d-1\}$ for $j=0,\ldots ,p$, we see from tour choice of $N$ that we must have $i_{s+j}+dr-W_n = i_j$ for $j=1,\ldots ,p$. Notice also that $(i_{s+j+1}+dr-W_n)-(i_{s+j}+dr-W_n)=a_{s+j}$ for $j=1,\ldots ,p$, where we take $a_{s+j}=a_{s+j-p}$ if $s+j>p$. Since $i_{s+j}+dr-W_n = i_j$, we see that $a_j=i_{j+1}-i_{j}=a_{s+j}$ for $j=1,\ldots ,p$. Since no non-trivial cyclic permutation of ${\bf a}$ is equal to ${\bf a}$, we have that $s=p$ and so taking $j=1$ in the equation $i_{s+j}+dr-W_n = i_j$ gives $i_1+dr+d-W_n = i_1$. In particular, $W_n\equiv 0~(\bmod~d)$ for all $n> N$.
We now show that there is no $d>1$ such that $d|W_n$ for all sufficiently large $n$, and we will then obtain the desired result since ${\mathcal}{S}$ cannot contain an infinite arithmetic progression with difference one.
Using Equation (\[eq: charpoly\]) and the Cayley-Hamilton theorem, we have that $W_n$ satisfies the recurrence $$\begin{aligned}
&~& W_n - W_{n-1} - 8W_{n-2} - 16 W_{n-3} - 2W_{n-4} + 5 W_{n-5} + 5W_{n-6} +21 W_{n-7}\nonumber \\
&~& +31W_{n-8} -10W_{n-9} - 8 W_{n-10}=0 \label{eq: recurrence}\end{aligned}$$ for $n\ge 12$. In particular, $W_n \equiv W_{n-1}+ W_{n-5} + W_{n-6} +W_{n-7} +W_{n-8} ~(\bmod\, 2)$ for all $n\ge 12$. We can now show that there are infinitely many $n$ for which $W_n$ is not divisible by $2$. To see this, suppose that this were not the case. Then there would exist some largest natural number $\ell$ such that $W_{\ell}$ is odd. From Item (\[eq: list\]), we see that $\ell\ge 4$ and so $\ell+8\ge 12$. But now Equation (\[eq: recurrence\]) gives that $W_{\ell+8} + W_{\ell+7}+ W_{\ell+3} + W_{\ell+2} +W_{\ell+1} \equiv W_{\ell} ~(\bmod\, 2)$, which is a contradiction since the left-hand side is even and the right-hand side is odd. It follows that if $d|W_n$ for all sufficiently large $n$ then $d$ must be odd. Now suppose towards a contradiction that $d$ is odd and that $d|W_n$ for all sufficiently large $n$ and that $d>1$. Then there is some odd prime $p$ divides $d$. Then we have $p|W_n$ for all $n$ sufficiently large. Let $\ell$ be the largest natural number for which $p$ does not divide $W_{\ell}$. Then since $\gcd(W_4,W_5)=1$, we see that $\ell\ge 4$. But now, since $\ell+10\ge 12$, Equation (\[eq: recurrence\]) gives that $8W_{\ell}$ is a $\mathbb{Z}$-linear combination of elements of the form $W_{\ell+i}$ with $i=1,\ldots ,10$ and so we see that if $p|W_n$ for $n>\ell$ then $p|8W_{\ell}$. Since $p$ is odd and $p$ does not divide $W_{\ell}$ we get a contradiction. It follows that $d=1$ and so ${\mathcal}{S}$ must contain the progression $\{0,1,2,3,\ldots\}$, but this is clearly false. The result follows.
We next show that the algebra $A_w$ is finitely generated as a Lie algebra.
Let $v$ be a subword of $w$ of length at least two. Then some cyclic permutation of $v$ is not a subword of $w$. \[lem: cyclic\]
Let $d$ denote the length of $v$. We prove this by induction on $d$. Our base cases are when $d=2,3,4$. We consider each of these cases separately. In each case, we suppose towards a contradiction that there exists a word $v$ of length $d$ all of whose cyclic permutations are subwords of $w$ and we derive a contradiction. *Case I: d=2*. For the case when $d=2$, observe that a subword of $w$ of length two is either of the form $x_ix_j$ with $i<j$ or $y_iy_j$ with $i<j$ or of the form $x_k y_{\ell}$ or $y_{\ell}x_k$. It is immediate that if our subword of length two is of the form $x_i x_j$ or $y_i y_j$ then $i<j$ and so $x_j x_i$ and $y_j y_i$ cannot be subwords of $w$ in this case. Thus for the case when $d=2$, it only remains to show that if $x_i y_j$ is a subword of $w$ then $y_j x_i$ cannot be. Observe that any subword of $w$ of length two is either a subword of $\phi(a)$ for some $a\in \Sigma$ or it is a subword of $\phi(ab)$, with $a,b\in \Sigma$, consisting of the last letter of $\phi(a)$ followed by the first letter of $\phi(b)$. In the case that we have a word of the form $x_i y_j$, then we see that it must be the second and third letters of $\phi(a)$ for some $a\in \Sigma$. We are also assuming that $y_jx_i$ is a subword of $w$. In this case, we have that there are letters $b,c\in \Sigma$ such that $bc$ is a subword of $w$ and $y_j$ is the last letter of $\phi(b)$ and $x_i$ is the first letter of $\phi(c)$.
Since the second letter of any $\phi(a)$ must be in $\{x_2,x_3,x_4,x_5,x_6\}$, we see that $i\neq 1$. Similarly, the first letter of any $\phi(c)$ must be in $\{x_1,x_2,x_3\}$ and so $i\neq 4,5,6$. Thus $i\in \{2,3\}$. Notice that the last letter of $\phi(b)$ can never be $y_1$ and so $j\in \{2,3,4,5,6\}$. By assumption, there exists some $a\in \Sigma$ such that $x_iy_j$ is a subword of $\phi(a)$ with $i\in \{2,3\}$ and $j>1$. Looking at the map $\phi$, we see that the only possibility is $i=3$, $j=2$ coming from $a=x_6$. But then $y_jx_i = y_2 x_3$. By assumption, there exist $b$ and $c$ in $\Sigma$ such that $bc$ is a subword of $w$ and such that $y_2$ is the last letter of $\phi(b)$ and $x_3$ is the first letter of $\phi(c)$. But $x_1$ is the only letter in $\Sigma$ with the property that applying $\phi$ to it gives a four-letter word ending in $y_2$. Thus $b=x_1$. Similarly, since $\phi(c)$ begins with $x_3$ we see $c\in \{y_4,y_5,y_6\}$. But this then means that $x_1y_k$ is a subword of $w$ for some $k\ge 4$, which is impossible since $x_1$ is always followed by an element from $\{x_2,\ldots ,x_6\}$ in $w$. *Case II: d=3*. Since we can never have three consecutive letters from $\{x_1,\ldots ,x_6\}$ occurring in $w$, we see that there are at most two letters from $x_1,\ldots ,x_6$ in $v$. Similarly, there are at most two letters from $y_1,\ldots ,y_6$ occurring in $v$. Thus either $v$ has exactly two letters from $\{x_1,\ldots ,x_6\}$ and one letter from $\{y_1,\ldots ,y_6\}$ or it has exactly two letters from $\{y_1,\ldots ,y_6\}$ and one letter from $\{x_1,\ldots ,x_6\}$. We consider the first case, as the other case is identical. In the first case, $v$ has some cyclic permutation of the form $x_i y_j x_k$, which cannot be a subword of $w$ since we must always have a block of two consecutive $y_j$’s between blocks of $x_i$’s. *Case III: d=4.* In this case, we can argue as in Case II to show that some cyclic permutation of $v$ is of the form $x_i x_j y_k y_{\ell}$. Any subword of $w$ of the form $x_i x_j y_k y_{\ell}$ must be $\phi(a)$ for some $a\in \Sigma$. In particular, $i<j$ and $k<\ell$. But by assumption $y_k y_{\ell}x_i x_j$ is also a subword of $w$ and so there exist $b,c\in \Sigma$ such that $bc$ is a subword of $w$, $y_k y_{\ell}$ are the last two letters of $\phi(b)$ and $x_ix_j$ are the last two letters of $\phi(c)$. But the first two letters of $\phi(d)$ for $d\in \Sigma$ completely determine $d$, and similarly for the last two letters. In particular, $b=c=a$ and so $bc=a^2$ is a subword of $w$. But we now see this is impossible from Case I.
We now complete the induction argument. Suppose that $d\ge 5$ is such that if $m\in \{2,\ldots ,d-1\}$ then any subword $u$ of $w$ of length $m$ has the property that some cyclic permutation of $u$ is not a subword of $w$. Let $v$ be a word of length $d$. Then arguing as in Case II, we have that some cyclic permutation $v'$ of $v$ is of the form $x_{i_1}x_{i_2}y_{j_1}y_{j_2}x_{i_3}\cdots $. Notice that if $d$ is $1$ or $2~(\bmod 4)$ then $v'$ ends with some $x_k$ and so if we move this $x_k$ to the beginning of $v'$ then we get a cyclic permutation of $v$ with three consecutive letters $x_k x_{i_1}x_{i_2}$, which cannot be a subword of $w$. If $d$ is $3~(\bmod 4)$, then the last two letters of $v'$ are of the form $x_i y_j$ and so if we shift these two letters to the beginning of $v'$ we see that $v$ has a cyclic permutation with three consecutive letters $x_iy_jx_{i_1}$, which cannot be a subword of $w$. Thus we have $d=4m$ with $m\ge 2$. Also $v'=\phi(a_1)\cdots \phi(a_m)$ for some $a_1,\ldots ,a_m\in \Sigma$. But then if $u=a_1\ldots a_m$ and $u'$ is a cyclic permutation of $u$ then $\phi(u')$ is a cyclic permutation of $\phi(u)=v'$. Moreover, since $\phi(u')$ begins with two consecutive letters from $x_i$ and $x_j$, we see that $\phi(u')$ is a subword of $w$ if and only if $u'$ is a subword of $w$. In particular, if all cyclic permutations of $v$ are subwords of $w$ then all cyclic permutations of $u$ are subwords of $w$. But $u$ has length $m\in \{2,\ldots ,d-1\}$ and so we see that this cannot occur by the induction hypothesis. The result now follows.
The algebra $A_w$ is finitely generated as a Lie algebra.\[prop: fg\]
Let $B$ denote the Lie subalgebra of $A_w$ that is generated by the elements $$\{1,x_1,\ldots ,x_6,y_1,\ldots ,y_6\}.$$ Since $A_w$ is spanned by the images of all subwords of $w$, it suffices to show that the image of every subword of $w$ is in $B$. Suppose that $u$ is a subword of $w$ whose image is not in $B$. We may assume that we pick $u$ of minimal length $d$ with respect to having this property. Since all subwords of $w$ of length $\le 1$ have images in $B$, $d$ must be at least two. By Lemma \[lem: cyclic\], $u$ has some cyclic permutation that is not a subword of $w$. In particular, we may decompose $u=ab$ so that $a$ and $b$ are subwords of $w$ but such that $ba$ is not a subword of $w$; i.e., $ba$ has zero image in $A_w$. Then $u=[a,b]\in A_w$. But now $a$ and $b$ have length less than $d$ and so by minimality of $d$, we have that the images of $a$ and $b$ are in $B$ and so the image of $u=[a,b]$ is in also in $B$, a contradiction. The result follows.
We need one last result to obtain our example.
\[lem: evper\] The word $w$ is uniformly recurrent.
It is straightforward to show that $\phi^2(u)$ begins with $x_1$ for each $u\in \Sigma$. Since $w$ can be written as $\phi^2(u_1)\phi^2(u_2)\cdots$ with $u_1,u_2,\ldots \in \Sigma$, and since each $\phi^2(u_i)$ has length $16$, we see that each subword of $w$ of length at least $16$ contains $x_1$. Let $v$ be a subword of $w$ and let $n$ be a natural number. Write $w=a_1a_2\cdots$ with $a_i\in \Sigma$. Then we have shown that whenever $a_i=x_1$ there is some $N\le 16$ such that $a_{i+N}=x_1$. Since $w=\phi^n(a_1)\phi^n(a_2)\cdots$ and each $\phi^n(a_j)$ has length $4^n$, we see that if $\phi^n(x_1)$ occurs in $w$ then there is another occurrence of $\phi^n(x_1)$ beginning at most $16\cdot 4^n$ places later. In particular, since each subword of $w$ is a subword of $\phi^n(x_1)$ for some $n$, we see that $w$ is uniformly recurrent.
Putting these results together, we obtain the following result, which answers questions 31, 32, and 36 of [@GLSZ].
\[thm: main2\] Let $R$ denote the positive part of the algebra $A_w$ with the grading described above. Then $R$ is just infinite, graded nilpotent, has quadratic growth, has trivial Jacobson radical, and is finitely generated as a Lie algebra.
The fact that $R$ is graded nilpotent follows from Proposition \[prop: grnilpotent\]. By Lemma \[lem: evper\] and Theorem \[thm: char\] (4), we see that $A_w$ is just infinite and hence $R$ is just infinite. To prove the remaining claims, we first show that $A_w$ does not satisfy a polynomial identity. To see this, by Theorem \[thm: char\] it suffices to show that $w$ is not eventually periodic. Towards a contradiction, suppose that $w$ is eventually periodic. Then by Theorem \[thm: char\], $A_w$ is an infinite-dimensional graded noetherian PI algebra. It is also prime since $A_w$ is just infinite. But now the graded version of Goldie’s theorem [@GS] gives that $A_w$ must have a regular homogeneous element of positive degree, a contradiction. Thus $A_w$ is not PI.
Since $w$ is not eventually periodic, we now see that $R$ has quadratic growth by Proposition \[prop: quadratic\], since $w$ is $4$-uniform. To show that $R$ has trivial Jacobson radical, it suffices to show that $A_w$ is semiprimitive. Since $w$ is uniformly recurrent and $w$ is not eventually periodic, Corollary 3.11 and Proposition 3.8 of [@Belov] then gives that the Jacobson radical of $A_w$ is zero. Finally, $R$ is finitely generated as a Lie algebra by Proposition \[prop: fg\]. This completes the proof.
In fact, since $A_w$ is just infinite, it is prime and so by a result of Okni[ń]{}ski [@Ok], the algebra $A_w$ must be primitive, since it is not PI and has zero Jacobson radical.
Concluding questions
====================
There are lots of natural questions one can ask about iterative algebras. We restrict ourselves to posing just a small number of questions, which we have not investigated in this article, but which we feel could be of independent interest or of later use. The first, and most compelling question, has to do with understanding ${\sf QGr}(A_w)$, the category of $\mathbb{Z}$-graded right $A_w$-modules modulo the full subcategory consisting of modules that are the sum of finite-dimensional submodules. Holdaway and Smith [@HS] investigated this category for finitely presented algebras, and it is natural to investigate it in this setting.
Can one give a concrete description of the category ${\sf QGr}(A_w)$ for an iterative algebra $A_w$?
A second question deals with when, precisely, can an iterative algebra be given a grading that makes the algebra graded nilpotent.
Can one determine when an iterative algebra associated to a pure morphic word $w$ over an alphabet $\Sigma$ has a grading, induced by making the letters of $\Sigma$ homogeneous of some positive degrees, such that the positive part of $A_w$ is graded nilpotent?
Can one determine for which pure morphic words $w$ the Jacobson radical of $A_w$ is trivial?
By a result of Okni[ń]{}ski [@Ok], a prime monomial algebra is either primitive, satisfies a polynomial identity, or has a nonzero locally nilpotent Jacobson radical. Thus if one could decide whether the Jacobson radical is nonzero then one can decide whether or not the algebra is primitive by Theorem \[thm: char\].
We are also intrigued by the fact that there are five different types of growth of iterative algebras, as shown by Theorem \[thm: GK\]. We pose the following question.
Can one give purely ring theoretic characterizations that determine which of the five possible growth types an iterative algebra has?
This question is admittedly vague, but we observe that $A_w$ has linear growth if and only if $A_w$ satisfies a polynomial identity. We wonder if, in a similar vein, one can find ring theoretic characterizations for the other growth types.
For the last question, we note that decidability of whether a morphic word is eventually periodic is a longstanding open problem (cf. [@HHKR p. 2]). In terms of algebras this corresponds to looking at graded subalgebras (generated in degree one) of $A_w$ and asking whether or not one can decide if the subalgebra has Gelfand-Kirillov dimension one. Motivated by these problems from theoretical computer science, we pose the following questions.
Given an iterative algebra $A_w$ can one determine the possible values for the GK dimension of a subalgebra $B$ that is generated by elements of degree one? Is it decidable whether or not $B$ has GK dimension one?
Finally, there are interesting questions about finite presentation and Hilbert series of iterative algebras.
Given an iterative algebra $A_w$ can one characterize the words for which $A_w$ has rational Hilbert series with the standard grading? Can one characterize the words for which $A_w$ is finitely presented?
[99]{}
[^1]: Corollary 10.3.2 of [@AS] refers to the subword complexity of automatic words. A result of Cobham (see [@AS Theorem 6.3.2]) gives that a $d$-uniform pure morphic word is automatic.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'S. Chaty'
- 'G. Dubus'
- 'A. Raichoor'
date: 'Received August 15, 2010; accepted February 15, 2011'
title: 'Near-infrared jet emission in the microquasar $\xtejqcq$[^1] '
---
[Microquasars are accreting Galactic sources that are also observed to launch relativistic jets. A key signature of the ejection is non-thermal radio emission. The level of this jet component at high frequencies is still poorly constrained.]{} [The X-ray binary and microquasar black hole candidate $\xtejqcq$ exhibited a faint X-ray outburst in April 2003 during which it stayed in the X-ray low/hard state. We took optical and near-infrared (NIR) observations with the ESO/NTT telescope during this outburst to distinguish the various contributions to the spectral energy distribution (SED) and investigate the presence of a jet component.]{} [Photometric and spectroscopic observations allowed us to construct an SED and also to produce a high time-resolution lightcurve.]{} [The SED shows an abrupt change of slope from the NIR domain to the optical. The NIR emission is attributed to non-thermal synchrotron emission from the compact, self-absorbed jet that is known to be present in the low/hard state. This is corroborated by the fast variability, colours, lack of prominent spectral features and evidence for intrinsic polarisation. The SED suggests the jet break from the optically thick to the thin regime occurs in the NIR.]{} [The simultaneous optical-NIR data allow an independent confirmation of jet emission in the NIR. The transition to optically thin synchrotron occurs at NIR frequencies or below, which leads to an estimated characteristic size $\ga 2\times 10^8$ cm and magnetic field $\la 5$T for the jet base, assuming a homogeneous one-zone synchrotron model.]{}
Introduction
============
X-ray binary systems are composed of a companion star and a compact object – a black hole or a neutron star. In low-mass X-ray binaries (LMXBs), the companion star is a late-type star filling its Roche lobe. Matter transiting through the Lagrange point forms an accretion disk around the compact object. The LMXBs spend most of their time in a quiescent state with a low X-ray luminosity. Outbursts occasionally occur, owing to an instability in the accretion disk, during which the X-ray luminosity increases by several orders of magnitude. Those LMXBs that additionally show non-thermal radio emission that is sometimes spatially resolved into jets are called microquasars (see e.g. @chaty:2006a [-@chaty:2006a]; @chaty:2006b [-@chaty:2006b]; @fender:2006 [-@fender:2006]; @mirabel:1998b [-@mirabel:1998b]).
Several canonical states for LMXBs have been defined according to their X-ray emission properties (see e.g. @belloni:2010 [-@belloni:2010]; @remillard:2006 [-@remillard:2006]), the main ones being:
- [*the high/soft state*]{}, characterized in the X-rays by a high luminosity, dominated by the thermal emission of the accretion disk, with a peak temperature $\sim 1-1.5 \keV$, emitting as a multicolour blackbody from optical to X-rays,
- [*the low/hard state*]{}, with the X-rays dominated by a power-law component, the accretion disk being weak in the X-ray band, at a temperature $\sim 0.01-0.5 \keV$; the hard state is invariably associated with strong, flat spectrum radio emission that is attributed to a compact, self-absorbed jet (with an extension of $\sim 10^{-6}$$\asec$, see e.g. @fender:2006 [-@fender:2006]).
- [*the quiescent state*]{}, when the X-ray luminosity is low and the optical/infrared emission is dominated by the emission of the companion star.
We also point out the existence of [*the intermediate state*]{}. During the transition between the low/hard and the high/soft state, discrete ejections are most of the time observed in radio, with an extension $\sim 0.1-10 \asec$ (see e.g. @fender:2006 [-@fender:2006]).\
$\xtejqcq$ was discovered as a transient X-ray binary in September 1998 by the All Sky Monitor (ASM) onboard the [*Rossi-XTE*]{} satellite [@remillard:1998]. Optical [@orosz:1998; @jain:1999] and radio [@campbell-wilson:1998] counterparts were promptly identified, classifying $\xtejqcq$ as a microquasar. The companion star is in a 1.541 day orbit [@jain:2001a]. It has been shown spectroscopically to be a G8IV–K4III at a distance of 5.3kpc; the compact object, with a mass of $10.5 \pm 1.0 \Msol$, is a black hole candidate [@orosz:2002]. Five main outbursts have been observed in $\xtejqcq$ (see Figure \[ASM\_overview\] left part). The first one, which started in September 1998 and lasted about 200 days, was the most powerful: at its maximum, the 2-10 keV flux reached almost $500$ASM counts/s (=6.8Crabs). During this outburst, the source had a complex behaviour and transited through all canonical X-ray spectral states [@homan:2001]. In 2000, the source exhibited another outburst, the 2-10 keV flux reaching around 1 Crab, and the source transiting again through different spectral states. In 2001, 2002, and 2003, the source showed three less powerful outbursts, during which it remained in the low/hard state [@arefev:2004; @sturner:2005]. Such “mini” outbursts following a major eruption have been seen in other X-ray binaries (e.g. @simon:2010) and in some dwarf novae [@kuulkers:1996]. The mechanism triggering these mini-outbursts is not understood within the framework of the standard LMXB disk instability model [@dubus:2001b]. They could be related to the accretion disk becoming eccentric during outburst in binary systems with small mass ratios [@hellier:2001].
Overlapping contributions from the companion star, the accretion disk, and from the relativistic jet make the optical to near-infrared (NIR) wavelength range particularly important to study. For instance, a compilation of optical and NIR observations of the black hole candidate X-ray binary GX339-4 showed that its NIR emission during low/hard states was non-thermal, likely synchrotron radiation emanating from the compact jet of this microquasar [@corbel:2002a]. Its spectral energy distribution (SED) showed the typical signature of a compact jet, namely a clear change in slope in the optical-NIR domain with an inverted power-law at lower frequencies. Similar signatures have been seen in other LMXBs [@kalemci:2005; @migliari:2006].
Here, we report on the results of optical and NIR observations of the microquasar $\xtejqcq$ during the 2003 outburst. $\xtejqcq$ remained in the low-hard state during the outburst. Our goal was to assess whether emission from the compact jet extended to the NIR. The weakness of the X-ray ouburst made it potentially more favourable to detect a jet contribution, because the accretion luminosity is expected to decrease faster than the jet luminosity [@heinz:2003]. The optical and NIR observations and data reduction are described in Sect. \[observations\]. The results are presented in Sect. \[results\] and discussed in Sect. \[conclusions\].
Observations and data reduction {#observations}
===============================
Photometry and spectroscopy
---------------------------
Our observations were performed during the night between 2003 April $21^{st}$ and $22^{nd}$ (Figure \[ASM\_overview\] right shows when our observations took place during the X-ray outburst), using the NTT (New Technology Telescope) telescope on the La Silla observatory of ESO (European Southern Observatory), as part of a Target of Opportunity (ToO) programme (PI S. Chaty). They took place on the declining phase of the 2003 outburst of $\xtejqcq$ and consist of optical and NIR photometry (both deep and rapid), NIR spectroscopy and polarimetry. The results from the NIR polarimetry were reported in [@dubus:2006b].
The NIR data were obtained in the J, H, and K$_s$ filters with the spectro-imager SoFI (Son oF Isaac), using the large field imaging (field of view of $4\aminp92 \times 4\aminp92$ and image scale of $0
\asecp 288$/pixel). For each filter of the NIR deep photometric observations, we observed the source at nine different positions with 60 s exposure time each, to estimate and substract the thermal sky emission, with a standard shift-and-combine jitter procedure. We also performed rapid photometry in the K$_{\rm s}$ filter, observing the source at different positions for three hours, with an integration time of 2 s for each exposure. The readout mode was double correlated read, leading to an overhead time of $\sim$50%; therefore we reached a time resolution of nearly 3 s for this rapid photometry.
To calibrate the photometric observations, we observed two photometric standard stars of the Persson catalogue [@persson:1998]: sj9136 and sj9146. Concerning the NIR spectroscopy, we took 24 spectra of 60 s each, half with the blue grism ($0.95-1.64 \microns$) and half with the red grism ($1.53-2.52
\microns$); we also took spectra of the telluric standard Hip63689 to correct the spectra of $\xtejqcq$ from the atmospheric absorption.
The optical data were obtained in the B, V, R, I and Z filters with the spectro-imager EMMI (Extraordinaire Multi-Mode Imager), using the large field imaging (field of view of $9\aminp9 \times 9\aminp1$, binning $2 \times 2$ for a better sensitivity and image scale of $0
\asecp 332$/pixel). We observed the source for 300 s in the B-band, for 3 s and 60 s in the V-band and for 60 s in the R-, I- and Z-band. We also observed the photometric standard stars PG1633 and PG1657. We performed rapid photometry in the V filter (a series of 10 s exposures during two hours). Concerning the optical spectroscopy, we took three spectra of 300 s each and a spectrum of a spectro-photometric standard star (LTT7379).
We used the Image Reduction and Analysis Facility (IRAF) suite to perform data reduction, carrying out standard procedures of optical and NIR image reduction, including flat-fielding and NIR sky subtraction. For the standard photometry, we used a median filter before carrying out aperture photometry with the [*noao.daophot*]{} package. We obtained the apparent magnitudes $m_{app}$ from the instrumental magnitudes $m_{inst}$ through the following formula, where $Z_p$ is the zero-point, $ext$ the extinction coefficient, and $airmass$ the airmass at the time of the observations: $m_{app} = m_{inst} - Z_p - ext \times airmass$
We used the characteristic extinction coefficients at La Silla and obtained the zero points given in Table \[coeff-ext-Zp\] by averaging the values obtained with the different standard stars. For the Z filter, the apparent magnitudes of the standard stars were not available, so we did not calibrate m$_{app}$(Z).
Concerning the spectra, we used the IRAF [*noao.twodspec*]{} package to extract the spectra and perform wavelength calibration. Since we did not observe any spectro-photometric standard stars, we did not perform flux calibration. We divided the NIR spectra of $\xtejqcq$ by the spectra of the telluric standard and multiplied it by the spectra of a 4600 K blackbody (we took the average of the effective temperature range of $\xtejqcq$ companion star given by @orosz:2002 [-@orosz:2002]: between 4100K and 5100K). We point out that the signal-to-noise ratio of the optical spectra of $\xtejqcq$ was too faint to securely identify any feature. Fortunately the NIR spectra were more exploitable, but not in the whole waveband coverage though, due to absorption.
We give in Figure \[champ\_xte\] a finding chart in NIR wavelengths for $\xtejqcq$.
Polarimetry {#section:polarimetry}
-----------
We also performed polarimetric observations on August 1 and 2, 2007. We collected a series of 10 second K$_s$ band exposures of the field around $\xtejqcq$ at the ESO NTT using SOFI in polarimetric mode. A Wollaston prism splits the incoming light into two images with perpendicular polarisation. $\xtejqcq$ was jittered along the mask for sky subtraction. Images taken at four different angles were used to compensate for the instrumental polarisation and this was checked against observations of unpolarised standards (see @dubus:2008 for details).
In addition, we also used polarimetric observations to derive the K$_{\rm s}$ magnitude in quiescence. Since we did not observe any photometric standard star during this run, we performed relative photometric calibration, thanks to isolated and bright stars of the 2MASS catalogue [@cutri:2003], close to $\xtejqcq$. We obtained the following apparent magnitude for $\xtejqcq$: K$_s = 16.25 \pm 0.05 \mags$.
Filter ext. Z$_{p}$
--------- ------- --------------------
B 0.214 0.069 $\pm$ 0.002
V 0.125 -0.584 $\pm$ 0.002
R 0.091 -0.741 $\pm$ 0.003
I 0.051 -0.231 $\pm$ 0.003
J 0.08 2.062 $\pm$ 0.010
H 0.03 2.232 $\pm$ 0.006
K$_{s}$ 0.05 2.799 $\pm$ 0.008
: Characteristic extinction coefficients at La Silla and derived zero-points for the different optical and NIR bands.
\[coeff-ext-Zp\]
Results
=======
Photometry and extinction {#photometry}
-------------------------
We have estimated the interstellar absorption using the column density on the line of sight derived from [*Chandra*]{} observations: $\nh = 0.88 \pm 0.1 \times 10^{22}\cmmoinsdeux$ [@corbel:2006]. This value is somewhat lower than, but still consistent with, the HI column density integrated through the whole Galaxy, given by both the Leiden/Argentine/Bonn ($\nh = 1.01 \times 10^{22} \cmmoinsdeux$) and Dickey & Lockman ($\nh = 0.897 \times 10^{22}\cmmoinsdeux$) surveys. The interstellar absorption in the V-band A$_V$ is then deduced from the relation $\Av = 5.59 \times 10^{-22} \nh$ [@predehl:1995] and the different A$_\lambda$ using the relations established by @cardelli:1989. Table \[magapp\] lists the apparent magnitudes we derived from our observations, together with the values obtained for A$_\lambda$ (taking into account the 1.6 $\sigma$ uncertainty on $\nh$) and, finally, the dereddened apparent magnitudes (taking into account the uncertainty on A$_\lambda$). The observed fluxes before and after corrections are plotted in the left part of Fig. \[sed\_2003\].
There is clearly a change of slope in between the NIR and visible wavelengths. The reddened NIR and optical spectral slopes are consistent with powerlaws of spectral index 1.7 and 3.9 respectively, and the dereddened NIR and optical with spectral index of 0.3 and -1.3 respectively[^2]. The best-fitting powerlaws are indicated in Figure \[sed\_2003\] (left) by dotted lines.
In order for both the NIR and optical slopes to be roughly compatible, one would have to decrease the value of the column density to A$_V$=3.5 (corresponding to $\nh \sim 0.6 \times 10^{22} \cmmoinsdeux$), which is clearly well below the value derived from X-ray observations, and also from the HI surveys, including uncertainty. And even by doing this, the optical I magnitude point is never aligned with the NIR and optical slopes, confirming that a change of slope is clearly present between the optical and NIR power laws, with two different spectral indices. Because the NIR spectrum is optically thin with a positive spectral index, the jet break must be located either in the NIR (around the H-band as suggested by right panel of Fig. \[sed\_2003\]), or towards longer wavelengths, in the MIR domain. Only contemporaneous observations from optical to MIR domain would allow us to constrain the exact location of the jet break.
Filter m$_{app}$ A$_\lambda$ m$_{app}$-A$_\lambda$
--------- ------------------ ----------------- -----------------------
B 20.00 $\pm$ 0.04 6.57 $\pm$ 0.75 13.43 $\pm$ 0.79
V 18.48 $\pm$ 0.03 4.92 $\pm$ 0.56 13.56 $\pm$ 0.59
R 17.35 $\pm$ 0.01 3.69 $\pm$ 0.42 13.56 $\pm$ 0.43
I 16.33 $\pm$ 0.01 2.36 $\pm$ 0.27 13.97 $\pm$ 0.28
J 14.50 $\pm$ 0.01 1.39 $\pm$ 0.16 13.11 $\pm$ 0.17
H 13.46 $\pm$ 0.02 0.94 $\pm$ 0.11 12.52 $\pm$ 0.13
K$_{s}$ 12.40 $\pm$ 0.01 0.56 $\pm$ 0.06 11.84 $\pm$ 0.07
: Apparent magnitudes, interstellar absorption, and dereddened apparent magnitudes for various wavelengths.
\[magapp\]
SED {#SED}
---
The broad-band SED of $\xtejqcq$ from radio to X-rays is shown in the right part of Fig. \[sed\_2003\], including our optical/NIR observations taken during the mini-outburst and also during quiescence. The optical/NIR data are dereddened from interstellar absorption. The line in the lower right indicates the flux and spectral index of the simultaneous X-ray data of $\xtejqcq$ during the 2003 mini-outburst, as observed by ASM/[*Rossi-XTE*]{} [@arefev:2004]. We also include radio data obtained during the 2002 mini-outburst with a similar X-ray flux, reported with the line in the left. Although no contemporary radio observations could be found, this is indicative of the radio emission that could have been expected from the compact jet during the low/hard X-ray mini-outburst of 2003 (see Sect. \[introduction\]).
The optical data are consistent with the Rayleigh-Jeans tail of a multicolour blackbody, which is characteristic of the emission coming from the outer part of the accretion disk, whereas the NIR data suggest a non thermal, inverted spectra, which is characteristic of synchrotron emission and is usually associated with a compact radio jet (see e.g. @corbel:2002a [-@corbel:2002a] for $\gx$ and @chaty:2003b [-@chaty:2003b] for $\xtejodh$). The level of NIR emission is consistent with the extrapolation to high frequencies of the (non-contemporary) flat/inverted radio spectrum and the extrapolation to low frequencies of the X-ray spectrum (with photon index $\approx 1.6$, @arefev:2004 [@sturner:2005]).
{width="6.5cm"} {width="6.5cm"}
Colour-magnitude diagrams {#CMD}
-------------------------
The absolute magnitudes of $\xtejqcq$ during the mini-outburst are reported in the colour-magnitude diagrams (CMD) in Figure \[jk\_k\]. The absolute magnitude was computed via $$M_\lambda = m_\lambda + 5 - 5 \times \log(d(pc)) - A_\lambda.$$
In both CMDs, the big asterisk indicates the position of $\xtejqcq$ optical/NIR counterpart, and the small asterisks surrounding it represent the parameter space of the source, taking into account the uncertainty on its distance ($d = 5.3
\pm 2.3 \kpc$; @orosz:2002 [-@orosz:2002]) and on the column density ($\nh = 0.88 \pm 0.1 \times 10^{22} \cmmoinsdeux$; @corbel:2006 [-@corbel:2006], as reported in Table \[magapp\]). According to @orosz:2002, the companion star has a type from G8IV to K4III, hence the quiescent state lies in the lower left part of the red giant branch. We indicate in Figure \[jk\_k\] (right) the position of the $\xtejqcq$ quiescent magnitudes.
The NIR CMD (Fig. \[jk\_k\] right part) shows that the source is redder in outburst than in quiescence. The companion star contributes more flux to the $J$ band than to the K$_{\rm s}$ band. Hence, the significantly redder colour implies an additional contribution with a flat or inverted spectrum.
The optical CMD (Fig. \[jk\_k\] left part) shows that the source was bluer in these bands in outburst compared to quiescence, which is consistent with the dominant contribution of the accretion disk in the optical. Indeed, when a multicolour blackbody spectrum is evolving with a higher flux and temperature, the flux increases more in the B-band than in the V-band, which implies a decrease in the (B-V) colour.
Both CMDs confirm that the optical and NIR fluxes evolve differently between the mini-outburst and quiescence.
{width="6.5cm"} {width="6.5cm"}
Rapid photometry {#rapidphot}
----------------
We performed rapid photometry in the V and K$_{s}$ filters on $\xtejqcq$ and on stars present in the field of view that were of comparable brightness to $\xtejqcq$. We then averaged the fluxes of these stars to get a mean flux, and we finally divided the flux of $\xtejqcq$ by this mean flux. These corrected and normalized fluxes are shown in Figure \[phot\_rpd\_Ks\] in the optical and NIR respectively, where we can see the intrinsic variations of the X-ray source and the surrounding stars. While rapid photometry in the NIR is rarely performed on LMXBs, it allows us to constrain rapid phenomena occuring on short timescale, either in the accretion disk, or related to the jet.
The rapid photometry shows that the source presents variations of amplitude in the NIR greater than those of the surrounding stars, whereas in the optical, the source behaviour is comparable with the surrounding stars. The standard deviations of this optical and NIR rapid photometry for $\xtejqcq$ and the surrounding stars are presented in Table \[tab\_phot\_rpd\]. Again, this suggests that the origin of the NIR and optical emission are different. However, note that variations in the optical of a similar amplitude to those in the NIR would probably not have been detected with these observations.
------------ ------- ------------ -------
$\xtejqcq$ 0.099 $\xtejqcq$ 0.074
Star \#1 0.020 Star \#1 0.051
Star \#2 0.024 Star \#2 0.065
Star \#3 0.021 Star \#3 0.098
------------ ------- ------------ -------
: Standard deviations taken from the rapid photometry lightcurves of $\xtejqcq$ and of various stars from the field of view. The stars have a brightness similar to the one of $\xtejqcq$.
\[tab\_phot\_rpd\]
Spectroscopy
------------
We took two IR spectra in the blue and red grisms, both are very absorbed, with a low S/N ratio. As shown in Figure \[spectro\_xte\], we only detect a faint emission line corresponding to Br$\gamma$ transition at 2.166$\microns$, very likely produced by the accretion disk. Apart from this, both NIR spectra are featureless, consistent with non-thermal emission emanating from the compact jet.
![Corrected NIR spectra of $\xtejqcq$ (the y-axis is in arbitrary units). Apart from the faint Br$\gamma$ emission line visible at 2.166$\microns$, both blue and red grisms are featureless.[]{data-label="spectro_xte"}](15589fg6.eps){width="9cm"}
Polarimetry {#polarimetry}
-----------
The NIR polarimetry of $\xtejqcq$, taken on the same night as part of the same observing programme, was reported earlier [@dubus:2006b]. The polarimetry showed an excess polarisation in $\xtejqcq$ compared with other stars in the field-of-view. Here, we report on polarimetric observations collected well after the mini-outburst described in this paper. The goal was to obtain a better measure of the polarisation signal from field stars in order to find the absolute value of the NIR polarisation from $\xtejqcq$ during its 2003 mini-outburst.
As described in Section \[section:polarimetry\], $\xtejqcq$ was detected in quiescence at a magnitude K$_s = 16.25 \pm 0.05 \mags$, but the low S/N prevented a meaningful polarisation measurement. The mean polarisation of the bright stars within $1\amin$ of $\xtejqcq$ (see right panel of Fig. 1 in @dubus:2006b [-@dubus:2006b]) was found to be $\approx 1.4\%$. The corresponding reduced Stokes parameter values are $q=-0.8\pm0.3$% and $u=1.1\pm0.5$% with the errors derived from the scatter of the stars $q$ and $u$ values. Assuming the polarisation of the field stars has not changed between 2003 and 2007, we correct for this mean polarisation and find that the absolute K$_s$ band polarisation during the 2003 mini-outburst of $\xtejqcq$ was about 2.4%. This confirms that the polarisation fraction was in excess of the interstellar polarisation that could be expected with E(B-V)=0.7 ($\approx 0.7$%, see @dubus:2006b). We were unable to determine the angle correction. The few polarised standards available to this effect are very bright. Several were observed by defocusing the telescope, but the resulting photometry proved too unreliable to be of use. Hence, we cannot give the orientation of the polarisation angle with respect to the jet axis.
Discussion {#conclusions}
==========
The spectral break
------------------
We detected a break between the NIR and optical wavelengths during a mini-outburst of $\xtejqcq$, with a positive spectral index in NIR, suggesting optically thin emission consistent with a jet spectrum. The break is clearly visible in the spectral energy distribution and colour-magnitude diagrams. Inaccuracies in the column density used to deabsorb the fluxes cannot account for this break. The NIR spectrum is featureless, apart from a faint Br$\gamma$ emission line. The K$_{\rm s}$ lightcurve shows $\approx 10$% variability on short timescales, which is suggestive of a non-thermal component. Evidence for an intrinsic IR polarisation during the outburst also points towards synchrotron emission [@dubus:2006b]. The overall SED is reminiscent of the low/hard state SED of GX 339-4 and XTE J1118-480 where the infrared emission was attributed to the compact jet (see §1). Our data lead us to conclude that synchrotron jet emission dominated in the NIR during the 2003 mini-outburst of $\xtejqcq$.
A dominant jet contribution in NIR was also put forward by [@russell:2010] to explain the correlations between the $H$ band and X-ray fluxes during the 2000 outburst of $\xtejqcq$. The 3-10 keV flux during our observations was about $6\times 10^{-10}$ erg cm$^{-1}$ s$^{-1}$. Comparing our results with Fig. 1 of @russell:2010, we find that with $H=13.46$ the source was in NIR about 50% brighter during our observations than during the decline of the 2000 outburst, at the time when the source reached the same X-ray flux. Our observations are consistent with the picture of an increasing jet contribution in the NIR as the source becomes harder and fainter in X-rays [@russell:2006].
The $V$ flux in the 2003 mini-outburst is brighter than that seen in 2000 for similar X-ray luminosities [@jain:2001b; @russell:2010]. The optical spectrum we measure is compatible with a Rayleigh-Jeans tail. This requires the temperature of the outer disk radius to be $\ga 10^4$ K. Otherwise, the flat part of the disk blackbody spectrum ($F_\nu\propto \nu^{1/3}$) should be visible[^3]. Here, we assume the temperature distribution as a function of disk radius $R$ is $T_{\rm disk}\propto R^{-3/4}$, which is adequate for an accretion disk in outburst [@dubus:2001b]. The temperature is high enough to ionize hydrogen in the outer disk, as expected in outburst. The data during the 2000 decline did not show a break in the spectra between $H$, $I$ and $V$ bands. The weaker $V$ flux in 2000 may have been due to a hotter disk temperature: this would have placed the Rayleigh-Jeans tail at higher frequencies.
Physical conditions at the jet base
-----------------------------------
The infrared fluxes decrease progressively from 12.0 mJy in $K$ to 6.6 mJy in $I$ before increasing again. The spectral steepening suggests the transition in the compact jet to optically thin synchrotron emission occurs at IR frequencies. The optically thin emission is dominated by the emission from the innermost region in self-absorbed jet models [@blandford:1979; @hjellming:1988; @kaiser:2006]. This is usually the case because the optically thin flux along the jet decreases rapidly with distance. The turnover frequency decreases along the jet and the summed contribution produces the flat spectrum at lower frequencies.
Assuming the innermost region with cross-section radius $R_0$ and length $H_0$ is seen sideways, the transition from thick to thin synchrotron emission occurs at $\tau_\nu=\alpha_\nu R_0\approx 1$. The synchrotron emission and absorption coefficients have analytical expressions for a power-law distribution of electrons with an index $p$. Further assuming that the energy density in non thermal electrons is a fraction $\xi$ of the magnetic energy density $B_0^2/8\pi$ in the region, this gives a relationship between the peak frequency, $\xi$, $R_0$ and $B_0$. The approximate flux at the peak frequency can be derived and depends on $\xi$, $R_0$, $H_0$ and $B_0$. We find the following relationships for $B_0$ and $R_0$ (see Appendix A) $$\begin{aligned}
B_0&\approx& 5\times 10^4\ \nu_{14} S_{10}^{-1/9} \xi^{-2/9} {h}^{1/9} d_{5}^{-2/9} {\rm \ G} \label{eq2}\\
R_0&\approx& 2.5\times 10^8\ \nu_{14}^{-1} S_{10}^{17/36} \xi^{-1/18} {h}^{-17/36} d_{5}^{17/18} {\rm \ cm}
\label{br}\end{aligned}$$ where we have taken a minimum Lorentz factor for the electrons $\gamma_{\rm min}=1$, $p=2.5$ (optically thin spectral index of 0.75), $\nu_{\rm peak}=10^{14} \nu_{14}$ Hz, $S_{\rm peak}=10\ S_{10}$ mJy, $d=5\ d_5$ kpc and $H_0=h R_0$. Similar results are obtained for $p=2$ or $p=3$. These equations apply equally to black hole or neutron star LMXBs. The magnetic field depends most sensitively on the turnover frequency.
Self-absorbed models usually make the assumption that adiabatic cooling is dominant over the radiative timescales. The adiabatic timescale is $t_{\rm ad}\ga R_0/c\approx 8$ ms. The synchrotron timescale is $t_{\rm sync}\propto \gamma^{-1} B^{-2}_0 \approx 120 \gamma^{-1}$ ms with the magnetic field derived above. Since NIR emission requires electron Lorentz factors $\gamma\approx 20$, it means that this is marginally verified at the jet base. We find variability on a few second timescales. $K$ band variability on $\approx 200$ ms timescales was also reported in GX 339-4 during a low/hard state [@casella:2010] but the above suggests that there could be NIR variability down to 10 ms timescales. Beyond the jet base, $t_{\rm sync}/t_{\rm ad}$ increases rapidly because $B\sim z^{-1}$ along the jet axis $z$ whereas $R\sim z^\beta$ with $\beta\approx 0.5$ to reproduce flat spectra (e.g. @hjellming:1988 [@kaiser:2006]). Self-Compton cooling can be ignored because a luminosity ratio of Compton to synchrotron emission $L_{\rm ic}/L_{\rm sync}\approx 0.2$ is inferred using the $R_0$ and $B_0$ given above. It can be shown that $L_{\rm ic}/L_{\rm sync}\propto \nu_{\rm peak} S_{\rm peak}^{-5/18}$. Synchrotron self-Compton emission from the jet base will be negligible unless $\nu_{\rm peak}$ moves into the visible.
@russell:2010 speculate that the X-ray emission in the hard state becomes fully jet-dominated when the 3-10 keV flux is below a few 10$^{-10}$ erg cm$^{-2}$ s$^{-1}$. Our observations do show that the NIR emission lies close to the extrapolated X-ray spectrum. If the cutoff at $\approx$ 100 keV in INTEGRAL is caused by synchrotron emission (but see @zdziarski:2004), then the maximum electron Lorentz factor is $\gamma_{\rm max}\approx 6000$. The X-ray emitting electrons should be radiatively cooled and the X-ray spectrum below the cutoff should have a photon index $\ga 2$ when an index $\approx 1.7$ is observed.
Finally, a magnetic field of a few teslas at the jet base is inevitable regardless of the detailed model if the NIR break is due to self-absorption. For comparison, the equipartition magnetic field with thermal pressure in the accretion disk at a radius close to the compact object is $$B_{\rm eq}\approx 5\times 10^7\ \eta^{-1/2} M_1^{-1/2} \dot{m}^{1/2} r^{-5/4} {\rm \ G,}$$ where $r$ is the radius in units of the last stable orbit, $M_1$ is the mass of the compact object in solar masses, $\dot{m}$ is the accretion rate in units of Eddington and $\eta=H/R\rightarrow 1$ for Bondi-Hoyle or radiatively inefficient accretion. The X-ray luminosity during our observations is $\approx 10^{-3} L_{\rm Edd}$, which implies $\dot{m}\ga 10^{-3}$. Therefore, the magnetic field at the jet base in $\xtejqcq$ represents at most 1% of the equipartition magnetic field at the innermost radius.
Curiously, compact jets in AGNs typically have turnovers in the 1-100 GHz range [@kellermann:1981], exactly as expected if the ratio $B/B_{\rm eq}$ were constant from microquasars to quasars. If this ratio is also constant in an object, then the turnover frequency will move to longer wavelengths as the mass accretion rate decreases. This can be tested observationally in microquasars when they decline from outbursts.
Conclusion
==========
We have obtained simultaneous NIR to optical coverage of the microquasar $\xtejqcq$ during a mini-outburst. Our dataset shows a break in the SED from the NIR to the optical. The optical emission is compatible with the Rayleigh-Jeans tail of the accretion disk. The lack of prominent spectral feature in the NIR, the fast variability and the evidence for intrinsic polarisation lead us to attribute the NIR emission to synchrotron radiation from the compact jet. Based on correlations between IR and X-ray fluxes during its 2000 outburst, @russell:2010 also interpreted the NIR emission from $\xtejqcq$ as jet emission. Evidence for NIR or optical jet emission from $\xtejqcq$ was also suggested by @jain:2001b [@corbel:2001; @russell:2007b]. The NIR luminosity represents about 1.7% of the X-ray luminosity. The jet contribution appears to be more important, in terms of the NIR to X-ray ratio, during the faint 2003 mini-outburst than during the 2000 outburst.
The SED shows a steepening from $K$ to $I$, suggesting the transition from optically thick to thin synchrotron emission occurs around 10$^{14}$ Hz. If this interpretation is correct, then the magnetic field at the jet base is at most a few teslas, or about 1% of the equipartition magnetic field in the accretion disk close to the black hole as in AGN compact jets. The NIR emission region must be small and sub-second variability can be expected.
Our data provide only a snapshot of the SED during an outburst. The evolution of the jet break during an outburst can provide important diagnostics of the jet physics [@heinz:2003; @markoff:2003]. Good sampling of the optical to NIR SED both in time and frequency, ideally in combination with polarisation measurements, is required to identify this break independently of the radio or X-ray observations and to test models that suggest jet emission can dominate the X-ray emission.
SC thanks the ESO staff for performing service observations, and SC and GD are grateful to an anonymous referee who helped to improve the paper. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the National Science Foundation. [*Rossi-XTE*]{} Results were provided by the ASM/[*Rossi-XTE*]{} teams at MIT and at the [*Rossi-XTE*]{} SOF and GOF at NASA’s GSFC. This research has made use of NASA’s Astrophysics Data System Bibliographic Services. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This work was supported by the Centre National d’Etudes Spatiales (CNES), based on observations obtained with MINE –the Multi-wavelength INTEGRAL NEtwork–; and by the European Community via contract ERC-StG-200911.
Magnetic field and size of the jet base
=======================================
We assume that the synchrotron-emitting region at the jet base is a homogeneous cylinder of radius $R_0$ and height $H_0=h R_0$. The electrons follow a power-law distribution $dN=K_0\gamma^{-p}d\gamma$; the magnetic field is $B_0$. The standard formula for the synchrotron absorption coefficient $\alpha_\nu$ is [@rybicki:1979] $$\alpha_\nu=\frac{\sqrt{3}e^3}{8\pi m^2_e c^2}\left(\frac{3 e}{2\pi m_e c}\right)^{\frac{p}{2}} \Gamma\left(\frac{3p+2}{12}\right) \Gamma\left(\frac{3p+22}{12}\right)K_0 B_0^{\frac{p+2}{2}} \nu^{-\frac{p+4}{2}}.$$ Similarly, the optically thin emissivity $j_\nu$ is $$j_\nu=\frac{\sqrt{3} e^3}{2\pi m_e c^2}\left(\frac{m_e c}{3e}\right)^{-\frac{p-1}{2}}\Gamma\left(\frac{3p+19}{12}\right)\Gamma\left(\frac{3p-1}{12}\right) \frac{K_0 B_0^{\frac{p+1}{2}} }{p+1}\nu^{-\frac{p-1}{2}},$$ so that the flux from the region can be written as $S_\nu=(1/2) (R_0/d)^2 H_0 j_\nu$, with $d$ the distance to the source. The synchrotron self-absorbed emission peaks at the frequency $\nu_{\rm peak}$ where $\tau_\nu=\alpha_\nu R_0\approx 1$. The jet emission transits from the flat optically thick part to the optically thin part at $\nu_{\rm peak}$, regardless of the detailed emission further down the jet (which only affects emission at frequencies below $\nu_{\rm peak}$). Inverting $S_\nu=S_{\rm peak}$ and $\tau_{\rm peak}=1$ gives two equations on $R_0$ and $B_0$, as functions of $S_{\rm peak}$, $\nu_{\rm peak}$, $h$, $p$ and $K_0$. We assume the energy in non-thermal electrons $\epsilon_e$ is a fraction $\xi$ of the magnetic field energy density $$\epsilon_e\equiv \int_{\gamma_{\rm min}}^{\gamma_{\rm max}} K_0 \gamma^{1-p}m_e c^2 d\gamma= \xi \frac{B_0^2}{8\pi}.$$ $K_0$ can be expressed as a function of the equipartition fraction $\xi$, $B_0$, $p$, $\gamma_{\rm min}$ and $\gamma_{\rm max}$. We have assumed $p=2.5$ and $\gamma_{\rm max}\gg \gamma_{\rm min}=1$ in deriving Eq. \[br\].
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[^1]: Based on observations collected at the European Southern Observatory, Chile, through programs 071.D-0071 and 079.D-0623.
[^2]: The spectral index $\alpha$ is defined as $F_{\nu} \propto \nu^{-\alpha}$
[^3]: Steeper accretion disk spectra can be expected when the disk is irradiated [@hynes:2005] but this is unlikely to be the case here given the weak X-ray flux in outburst.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We discuss methods of quantum state tomography for solid-state systems with a large nuclear spin $I=3/2$ in nanometer-scale semiconductors devices based on a quantum well. Due to quadrupolar interactions, the Zeeman levels of these nuclear-spin devices become nonequidistant, forming a controllable four-level quantum system (known as quartit or ququart). The occupation of these levels can be selectively and coherently manipulated by multiphoton transitions using the techniques of nuclear magnetic resonance (NMR) \[Yusa *et al.*, Nature (London) [**434**]{}, 101 (2005)\]. These methods are based on an unconventional approach to NMR, where the longitudinal magnetization $M_z$ is directly measured. This is in contrast to the standard NMR experiments and tomographic methods, where the transverse magnetization $M_{xy}$ is detected. The robustness against errors in the measured data is analyzed by using the condition number based on the spectral norm. We propose several methods with optimized sets of rotations yielding the highest robustness against errors, as described by the condition number equal to 1, assuming an ideal experimental detection. This robustness is only slightly deteriorated, as given by the condition number equal to 1.05, for a more realistic “noisy” $M_z$ detection based on the standard cyclically-ordered phase sequence (CYCLOPS) method.'
author:
- Adam Miranowicz
- 'Şahin K. Özdemir'
- Jiří Bajer
- Go Yusa
- Nobuyuki Imoto
- Yoshiro Hirayama
- Franco Nori
title: 'Quantum state tomography of large nuclear spins in a semiconductor quantum well: Optimal robustness against errors as quantified by condition numbers'
---
Introduction
============
Quantum state engineering has been attracting increasing attention in fundamental physics research as well as in applications in quantum cryptography, quantum communication and, potentially, quantum information processing (QIP) [@SchleichBook]. Quantum state engineering provides methods for synthesis of quantum states, and their coherent control and characterization. The latter task can be realized by quantum state and process tomographic methods.
*Quantum state tomography* (QST) is a method for reconstruction of a quantum state in a series of measurements performed on an ensemble of identical quantum states. *Quantum process tomography* (QPT) is a method, closely related to QST, which enables a complete characterization of the dynamics of a quantum system. Both QST and QPT have been applied widely to QIP in finite- and infinite-dimensional optical systems (for reviews see Refs. [@ParisBook; @DAriano03] and references therein). In particular, much work has been on QST of polarization states of photons (see, e.g., Refs. [@James01; @Altepeter05; @Burgh08; @Adamson10; @Miran14]), homodyne QST [@Vogel89], and homodyne QPT [@Lobino08; @Wang13] probed with coherent states. Other examples include QST in superconducting circuits [@You05; @Liu04]. QST has also been applied in nuclear-spin systems using nuclear magnetic resonance (NMR) spectroscopy, which was motivated by quantum information interest [@Jones11; @Vandersypen04]. The NMR QST and NMR QPT were first developed for liquid-state nuclear spin-1/2 systems [@Vandersypen04], and only later applied both to liquid- and solid-state systems of quadrupolar nuclei of spin-3/2 [@Bonk04; @Kampermann05; @Auccaise08; @Teles12] and spin-7/2 [@Teles07].
The use of multi-level systems (so-called *qudits*) instead of two-level systems (qubits) is an alternative paradigm [@Nori; @Lanyon] in QIP, which has attracted attention in recent years. Standard examples of higher-order nuclear spins $I$ include: $I=3/2$ for the isotopes $^{69}$Ga, $^{71}$Ga, and $^{75}$As (in, e.g., GaAs), $I=5/2$ for $^{27}$Al (in, e.g., AlN) or $^{121}$Sb (in, e.g., FeSb$_2$), $I=7/2$ for $^{123}$Sb (also in FeSb$_2$), as well as $I=9/2$ for $^{113}$In and $^{115}$In (in, e.g., InAs, InSb, and InP), and $^{73}$Ge. Note that large nuclear spins occur also in molecular magnets, i.e., clusters of spins, which can be applied for QIP [@Leuenberger01; @Ardavan07]. As another example, the superconducting circuits in Ref. [@Nori] have up to five levels and can model rotations of spin-1 and spin-3/2. Here, we will focus solely on QIP using quadrupolar nuclei with spin-3/2, which are equivalent to a four-level system.
Another motivation for the application of qudits for QIP is related to an important question concerning the scalability of two qubits to many qubits. If one simply plans to increase the number $N$ of qubits, then the required numbers of levels scales up exponentially. This becomes very hard to implement when $N$ is large. However, some ordinary classical computers are not assembled with simple AND, NAND, OR, NOT gates, but instead they are constructed using higher-level logic gates. Similarly, quantum computers might be constructed with slightly more complex logic gates, rather than, e.g., only single-qubit gates and CNOT gates. In this direction, multi-level systems are helpful.
A word of caution: Replacing qubits by qudits causes a faster exponential divergence in the number of levels, thus one loses the advantage of “using a single multi-level system” over “using many two-level systems.” Thus, this approach should be applied carefully. Indeed, using qudits could reduce the complexity of quantum computers (see Ref. [@Nori] and the justifications given there). Moreover, for qudits, there are optimal recipes for gate operations (see Sec. III) and quantum tomography methods to be discussed in the following sections.
Various two-qubit quantum state engineering methods and quantum algorithms have been realized in NMR experiments with spin-3/2 systems. Examples include: the demonstration of classical [@Khitrin00; @Sinha01; @Kumar02] and quantum [@Sarthour03; @Kampermann02; @Bonk04; @Kampermann05] gates, generation of Bell states [@Sarthour03; @Kampermann02], the quantum Fourier transform [@Kampermann05], and implementations of simple quantum algorithms (i.e., the two-qubit Grover search algorithm [@Ermakov02; @Kampermann05] and the Deutsch-Jozsa algorithm [@Das03a; @Kampermann05]). The existence of quantum correlations (as revealed by quantum discord) was also experimentally demonstrated in spin-3/2 systems (see, e.g., [@Soares10]). It is worth noting that a prerequisite for the realization of all these gates and algorithms is the preparation of pseudo-pure states (see also Refs. [@Hirayama06; @Jones11; @Tan12] and references therein).
Quadrupolar nuclei with spin-7/2 have also attracted increasing interest, as it is highly desirable to scale QIP beyond two (real or virtual) qubits. A few NMR experiments were performed with spin-7/2 systems, e.g., the preparation of effective pure states [@Khitrin01a], a quantum simulation [@Khitrin01b], a half-adder and subtractor operations [@Murali02; @Kumar02], a test of phase coherence in electromagnetically-induced transparency [@Murali04], and three-qubit Deutsch-Jozsa algorithms [@Das06; @Teles07; @Gopinath08].
A complete verification of the generated states and/or performed algorithms in the aforementioned experiments requires the application of QST.
In this article, we describe QST methods for an unconventional approach to NMR (sometimes referred to as “exotic NMR”) in semiconductor nanostructures [@Machida03; @Yusa05; @Hirayama06; @Ota07; @decoherence], which is based on the measurement of the longitudinal magnetization $M_{z}$.
In contrast to this approach, the vast majority of the NMR tomographic methods are based on conventional (standard) NMR experiments, where the transverse magnetization $M_{xy}$ is detected. Indeed, a very tiny magnetic field produced by the nuclear spin rotation in the $xy$-plane with a resonant frequency is picked up by a surrounding coil. In this method, the $M_{xy}$ component is measured by using induction detection ($M_{xy}$ detection). However, this widely used conventional NMR suffers from low sensitivity arising from induction detection, so one should prepare large volume samples occasionally reaching a qubic centimeter (at least a qubic millimeter). In the application to semiconductor (solid-state) systems (see Ref. [@Hirayama09] and references therein), multiple-layer quantum wells with 10-100 layers should be prepared to detect clear signals with a sufficient noise-to-signal ratio. A main advantage of semiconductor (solid-state) qubits is its precise controllability by using gate operations. Such gate operation is based on a single quantum well and nanostructure so conventional NMR is obviously not appropriate for these systems. Since the mid-2000s, highly sensitive NMR methods suitable for semiconductor hetero- and nanosystems have been developed by using electrical [@Machida03; @Yusa05] and optical [@Kondo08] means. However, they all relied on a direct measurement of the nuclear spin magnetization, i.e., $M_{z}$ detection. Therefore, it is important for semiconductor (solid-state) nuclear-spin qubits to develop QST appropriate for the direct detection of $M_{z}$.
Here we study NMR tomography of solid-state four-level quantum systems, also known as quartits or ququarts. The main result of this paper is the proposal of various QST methods based on $M_z$ detection, which are the most robust against errors as quantified by a condition number equal (or almost equal) to 1. Note that the proposed QST methods can be applied not solely to solid-state systems but also to liquid-state quartits. Moreover, these methods can be generalized for QST of qudits. There has also been interest in the generation and state tomography of other systems, especially optical qudits, including qutrits (i.e., three-level quantum systems) (see, e.g., Ref. [@Thew02]).
The paper is organized as follows: In Sec. II, we specify the quadrupolar interaction model. In Sec. III, we describe sequences of NMR pulses for implementing qubit gates in qudits. In Sec. IV, we present the key aspects of the $M_z$-based QST of a spin-3/2 system. We also briefly discuss a nanometer-scale all-electrical resistively-detected NMR device [@Machida03; @Yusa05], where the $M_z$-magnetization can be measured. In Sec. V, we discuss the linear reconstruction of density matrices in relation to condition numbers describing how these methods are robust against errors. In Sec. VI, we specify the $M_z$ detection approaches to be applied in the next sections. These include three approaches: a theoretical approach, as well as both ideal and non-ideal (noisy) experimental approaches. The main results of this paper are presented in Secs. VII–IX. Specifically, we propose various sets of rotations, which enable optimal reconstructions of all the diagonal (in Sec. VII) and off-diagonal (in Sec. VIII) elements of a spin-3/2 density matrix. These two reconstructions are combined in Sec. IX. In Sec. X, we show how to construct sets of operationally-optimized rotations by finding single-photon replacements for multiphoton rotations. We conclude in Sec. XI. In the Appendices, for completeness and clarity, we define selective rotations, and briefly compare the $M_{xy}$ and $M_{z}$ detections.
Interaction model
=================
First, we describe a model for large nuclear spins, in a semiconductor quantum well, which are interacting with radio-frequency (RF) pulses. A general description of such an interaction can be found in standard textbooks on NMR (see, e.g., Refs. [@AbragamBook; @ErnstBook]). In particular, the model described in detail by Leuenberger [@Leuenberger02], which was directly applied in the experiment of Yusa [@Yusa05], can also be adapted here.
Specifically, we analyze an ensemble of quadrupolar nuclei (with spin $I=3/2$) in a semiconductor quantum well interacting with $N$ RF pulses of the carrier frequency $\omega_{_{\rm RF}}^{(k)}$, phase $\phi_{_{\rm RF}}^{(k)}$, and magnetic-field amplitude ${B}_{k}$ ($k=1,2,...,N$) in the presence of a strong magnetic field ${B}_{0}$. The effective total Hamiltonian in the laboratory frame reads [@AbragamBook; @ErnstBook; @Leuenberger02; @Yusa05] $$\begin{aligned}
{\cal H}&=&{\cal H}_{0}+{\cal H}_{\rm int},\label{H}\end{aligned}$$ being a sum of the free term $$\begin{aligned}
{\cal H}_{0} &=& {\cal H}_{\rm Z}+{\cal H}_{Q}
= \hbar\omega_{0}{I}_{z}+\frac{\hbar\omega_{Q}}{3}
[3{I}_{z}^{2}-I(I+1)], \label{H0}\end{aligned}$$ and the term describing the interaction of the nuclei with $N$ pulses: $$\begin{aligned}
{\cal H}_{\rm int} = \sum_{k=1}^N
\frac{\hbar\omega_{k}}{2}\left[{I}_{+}{\rm e}^{-i(\omega_{_{\rm RF}}^{(k)}t+\phi_{_{\rm
RF}}^{(k)})} + {I}_{-} {\rm e}^{i(\omega_{_{\rm RF}}^{(k)}t+\phi_{_{\rm
RF}}^{(k)})} \right]\; \\
= \sum_{k=1}^N
\hbar\omega_{k}\!\left[{I}_{x}\cos(\omega_{_{\rm RF}}^{(k)}t+\phi_{_{\rm
RF}}^{(k)}) + {I}_{y} \sin(\omega_{_{\rm RF}}^{(k)}t+\phi_{_{\rm
RF}}^{(k)}) \right]. \label{Hint}
\nonumber\end{aligned}$$ Here, ${\cal H}_{\rm Z}=\hbar\omega_{0}{I}_{z}$ and ${\cal H}_{Q}$ describe, respectively, the Zeeman and quadrupole splittings (see Fig. 1). The operator ${I}_{\alpha}$ (for $\alpha=x,y,z$) is the $\alpha$-component of the spin angular momentum operator, and $
I_{\pm}= I_x\pm i I_y$. Moreover, $\omega_{0}=-\gamma {B}_{0}$ is the nuclear Larmor frequency, and $\omega_{k}=-\gamma {B}_{k}$ is the amplitude (strength) of the $k$th pulse, where $\gamma$ is the gyromagnetic ratio. For the example of the nuclei ${}^{69}{\rm
Ga}$ and ${}^{71}{\rm As}$ of spin $I=3/2$ in semiconductor GaAs, we can choose the gyromagnetic ratios to be $\gamma({}^{69}{\rm
Ga})=1.17 \times 10^7 \,{\rm s}^{-1} {\rm T}^{-1}$ and $\gamma({}^{71}{\rm As})=7.32 \times 10^6\, {\rm s}^{-1} {\rm
T}^{-1}$, which are estimated from the spectra measured in Ref. [@Yusa05]. The Hamiltonian ${\cal H}_{Q}={\cal
H}_{Q}^{(1)}+{\cal H}_{Q}^{(2)}+...$ describes the quadrupolar interaction as a sum of the first- and second-order quadrupolar terms (as shown in Fig. 1), but also higher-order terms. The first-order quadrupolar splitting parameter (quadrupolar frequency) $2\omega_{Q}$ is given for solids by [@LevittBook]: $$\omega_{Q}\equiv \omega_{Q}^{(1)}=\frac{3\pi C_{Q}}{4I(2I-1)}
(3\cos^{2}\theta_{Q}-1),$$ where $C_{Q}$ is the quadrupolar coupling constant, and $\theta_{Q}$ is the angle between the direction of the field ${B}_{0}$ and the principle axis of the electric-field gradient tensor. We assume a uniaxial electric-field gradient tensor, i.e., the biaxiality parameter is zero ($\eta_{Q}=0$). Under the secular approximation, which is valid for relatively small $\omega_{Q}$, the effective interaction is described solely by the first-order quadrupolar Hamiltonian, as we have assumed in Eq. (\[H0\]).
The quadrupolar frequencies are typically of the order of 10–100 kHz. For example, the values for the isotopes in semiconductor GaAs can be found in Refs. [@Salis01; @Leuenberger02; @Yusa05]. In our numerical simulations, we set the following values of the quadrupolar frequencies $\omega_{Q}({}^{69}{\rm Ga})=15.2$ kHz and $\omega_{Q}({}^{71}{\rm As})=26.9$ kHz. These values were estimated from the experimental spectra reported in Ref. [@Yusa05]. Moreover, we also choose in our simulations the same values of parameters as those measured or estimated in the experiment with nanometer-scale device in Ref. [@Yusa05]. Namely, ${B}_0=6.3$ T and ${B}_k=\;$0.2–1.4 mT, and decoherence time is $T_2\approx 1$ ms.
Let us denote the eigenvalues and eigenvectors of ${\cal H}_0$ by $\epsilon_{m}$ and $|m\rangle$, respectively, i.e., ${\cal
H}_0|m\rangle=\epsilon_{m}|m\rangle.$ If the condition $|\omega_k|
\ll |\omega_Q| \ll |\omega_{0}|$ is satisfied, one can apply a selective RF pulse resonant with a transition $|m\rangle\leftrightarrow|n\rangle$, i.e., $\hbar\omega_{_{\rm
RF}}^{(k)} =\epsilon_{m}-\epsilon_{n}$, where $m,n=0,1,...$. One can also analyze $N$-photon resonant transitions, which correspond to the condition $N\hbar\omega_{_{\rm
RF}}^{(k)}=\epsilon_{m}-\epsilon_{n}$, where $k=1,2,...$ (see Fig. 2). Note that Eq. (\[Hint\]) can still be used, even if the $\omega_{_{\rm RF}}^{(k)}$ are slightly detuned from the resonant frequencies by $\delta\omega_k$.
In the more general case when $N$ RF pulses of different frequencies $\omega_{_{\rm RF}}^{(k)}$ are applied simultaneously, then clearly the [*standard*]{} rotating frame is not useful to transform the time-dependent Hamiltonian, given by Eq. (\[Hint\]), into a time-independent form. However, if the quadrupolar splitting $2\hbar\omega_Q$ is much larger than the detuning energies $\hbar\delta\omega_k$, then one can still transform Eq. (\[H\]) into a completely time-independent Hamiltonian in a [*generalized*]{} rotating frame, as described in, e.g., Ref. [@Leuenberger02].
The Hamiltonian ${\cal H}$ can be transformed to the rotating frame as follows $$\begin{aligned}
{\cal H}_{\rm rot} &=& U {\cal H} U^{\dagger}
-i\hbar U\frac{\partial U^{\dagger}}{\partial t}.
\label{HrotU}\end{aligned}$$ Let us assume that only a single pulse ($k=1$) is applied of strength $\omega_1$, frequency $\omega_{_{\rm RF}}\equiv
\omega_{_{\rm RF}}^{(1)}$, and phase $\phi\equiv \phi_{_{\rm
RF}}^{(1)}$. Then the time-dependent Hamiltonian, given by Eq. (\[H\]), in the frame rotating with angular frequency $\omega_{_{\rm RF}}$, becomes the well-known time-independent Hamiltonian: $$\begin{aligned}
{\cal H}_{\rm rot}&=&
\hbar\Delta\omega{I}_{z} +{\cal
H}_{Q} +\hbar\omega_{1} {I}_{\phi}\,,\label{Hrot}
\\
{I}_{\phi}&=&{I}_x\cos\phi+{I}_y\sin\phi
=\frac12 ({I}_+{\rm e}^{-i\phi}+{I}_-{\rm e}^{i\phi}),
\nonumber\end{aligned}$$ where $\Delta\omega=\omega_{0}-\omega_{_{\rm RF}}$ is the frequency offset. Equation (\[Hrot\]) is obtained from Eqs. (\[H\]) and (\[HrotU\]) for $ U =\exp(-i\omega_{_{\rm
RF}} I_z t)$.
The initial state (before applying pulses) of the spin system at a high temperature $T$ can be described by $$\begin{aligned}
\rho = Z^{-1} \exp(-\beta {\cal H}_0) \approx Z^{-1}(1-\beta {\cal
H}_0),
\label{rho0}\end{aligned}$$ where $Z$ is the partition function and $\beta=1/(k_B T)$. The term $\beta {\cal H}_0/Z$ corresponds to a deviation density matrix. Thus, the initial state $ \rho$ of a spin-$3/2$ system can be approximated by $$\begin{aligned}
\rho \approx \frac14(1 - \hbar\omega_0\beta I_z),
\label{rho0b}\end{aligned}$$ if $|\omega_Q|\ll |\omega_0|$.
The evolution of a state, given by $ \rho(t_0)$, during the application of a single pulse of strength $\omega_1$ and duration $t_p$ is described in the rotating frame by: $$\rho(t+t_p) = {\cal U}(\omega_1,t_p) \rho(t)
{\cal U}^\dagger(\omega_1,t_p),
\label{U1}$$ where the evolution operator is $${\cal U}(\omega_1,t) = \exp[-(i/\hbar) {\cal H}_{\rm rot}
t].
\label{U}$$ The evolution of $ \rho(t)$ in the absence of pulses from the time $t$ to $t+\Delta t$ is given by: $$\rho(t+\Delta t) = {\cal U}(0,\Delta t) \rho(t) {\cal U}^\dagger(0,\Delta t). \label{U2}$$ Analytical expressions for the evolution operator ${\cal
U}(\omega_1,t)$ and the corresponding density matrices can be obtained by finding eigenvalues and eigenstates of ${\cal H} _{\rm
rot}$. For example, by assuming an RF pulse to be resonant with the central line (i.e., $\omega_{_{\rm
RF}}=\omega_{12}=\omega_0$), and by setting $\phi=0$, we find the following eigenvalues of ${\cal H} _{\rm rot}$: $${\rm eig}({\cal H}_{\rm rot})=\left[
\frac{\omega_1}2 + \Omega_{-},\frac{\omega_1}2 - \Omega_{-},
-\frac{\omega_1}2 - \Omega_{+},-\frac{\omega_1}2 +
\Omega_{+}\right]
\label{eigval}$$ where $\Omega_{\pm}=\sqrt{\omega_1^2\pm\omega_1
\omega_Q+\omega^2_Q}$. The corresponding eigenvectors of ${\cal
H}_{\rm rot}$ for $m=1,2$ and $n=3,4$ are equal to: $$\begin{aligned}
|V_{m}\rangle &=& {\cal N}_{m}[
\sqrt{3}\omega_1(|3\rangle+|0\rangle)+
y_{m}(|1\rangle+|2\rangle)],
\nonumber\\
|V_{n}\rangle &=& {\cal N}_n [
\sqrt{3}\omega_1(|3\rangle-|0\rangle)+
z_{n}(|1\rangle-|2\rangle)],
\label{eigvec}\end{aligned}$$ where ${\cal N}_{m}$ and ${\cal N}_{n}$ are normalization constants, and $$\begin{aligned}
y_m &=& \omega_1 + 2(-1)^m\Omega_{-}-2\omega_Q,
\nonumber\\
z_n &=& \omega_1 - 2(-1)^n \Omega_{+}+2\omega_Q.
\label{eigvec2}\end{aligned}$$ The general solution for ${\cal U}(\omega_1,t)$ is quite lengthy. However by assuming that $|\omega_0|\gg |\omega_Q| \gg|\omega_1|$, it can be effectively reduced to a form corresponding to all ideal selective rotations as defined in Appendix A. This can be shown by expanding the elements of the matrix ${\cal U}(\omega_1,t)$ in a power series of the parameter $\epsilon=|\omega_1|/|\omega_Q|$, and, finally, keeping only the first term of this expansion.
For example, if the pulse is resonant with the central transition, then the evolution operator ${\cal U}(\omega_1,t)$ can be approximated by $${\cal U}_{12}(\omega_1,t_p)= {{\left[\begin{array}{cccc} \delta^* & 0 & 0 & 0 \\ 0 & \delta \cos(\omega_1 t_p) & -i\delta \sin(\omega_1 t_p) & 0 \\ 0 & -i\delta \sin(\omega_1 t_p) & \delta \cos(\omega_1 t_p) & 0 \\ 0 & 0 & 0 & \delta^* \end{array} \right]}},
\label{Uapprox}$$ where $\delta=\exp(i\omega_Q t_p)$. Note that ${\cal
U}_{12}(\omega_1,t)$ reduces to the perfect selective rotation ${\cal X}_{12}(\theta)= R^{(X)}_{12}(\theta)$, with $\theta=2\omega_1 t_p$, if the pulse duration is chosen such that $\omega_Q t_p$ is a multiple of $2\pi$. Analogously, other rotations $R^{(i)}_{mn}(\theta)$, given by Eq. (\[A4\]), can be implemented for $i=X,Y,Z$ and $m,n=0,...,3$ with $m\neq n$.
Implementing gates in spin-3/2 system
=====================================
Here, we discuss how to implement single- and two-qubit gates in systems with spin-3/2. This can enable formally simple implementations of arbitrary multi-qubit quantum algorithms by applying sequences of NMR pulses in multi-level spin systems. We focus on various NMR QST methods for a system with spin-3/2 nuclei but our analysis can be easily generalized for larger spins.
Due to the Zeeman and quadrupolar interactions (shown in Fig. 1), a spin-3/2 system is described in an external magnetic field by a non-equidistant four-level energy spectrum. Thus, this system can be referred to as a *quartit* (also called ququart or four-level qudit). The basic set of eigenfunctions of the system can be described with the states $|mn\>\equiv|m\>_A |n\>_B$ of two logical (or virtual) qubits $A$ and $B$ corresponding to an ensemble of identical spin-1/2 pairs: $$\begin{aligned}
|{{\textstyle{\frac{3}{2}}}}, {{\textstyle{\frac{3}{2}}}}\> &\equiv& |0\> \equiv |00\> , \quad
|{{\textstyle{\frac{3}{2}}}},-{{\textstyle{\frac{1}{2}}}}\> \equiv |2\> \equiv |10\> ,
\nonumber \\
|{{\textstyle{\frac{3}{2}}}}, {{\textstyle{\frac{1}{2}}}}\> &\equiv& |1\> \equiv |01\> , \quad
|{{\textstyle{\frac{3}{2}}}},-{{\textstyle{\frac{3}{2}}}}\> \equiv |3\> \equiv |11\> .\label{N01}\end{aligned}$$ A pure state of a quartit can be written in this basis states as $$\begin{aligned}
{\mbox{$|\psi\rangle$}}&=& c_0 {\mbox{$|0\rangle$}}+ c_1 {\mbox{$|1\rangle$}}+c_2 {\mbox{$|2\rangle$}}+c_3 {\mbox{$|3\rangle$}}
\label{N02}\end{aligned}$$ in terms of the normalized complex amplitudes $c_i$, so an arbitrary mixed state of a quartit is described by a density matrix $\rho=[\rho_{nm}]_{4\times 4}$.
Our discussion in this section is based on a fundamental theorem in quantum information according to which any quantum gate can be constructed from single-qubit rotations and any nontrivial two-qubit gate, e.g., the CNOT gate [@SchleichBook]. In a quartit, rotations of a virtual qubit $A$ and $B$, denoted, respectively, by $R^A(\theta)$ and $R^B(\theta)$, can be implemented by the application of two pulses: $$\begin{aligned}
R^A(\theta
) &=& R_{02}(\theta) R_{13}(\theta) ,
\nonumber \\
R^B(\theta) &=& R_{01}(\theta) R_{23}(\theta) ,
\label{Q1}\end{aligned}$$ where $R_{mn}(\theta)$ (with $R=X,Y,Z$) is a selective rotation resonant with a transition between levels $|m\>$ and $|n\>$ as defined in Appendix A (see also Fig. 2).
Note that realizations of *single* virtual qubit gates in a qudit are more complicated than those for real qubits. In contrast to those, usually *two* virtual qubit gates can be realized much simply, e.g., a CNOT-like gate can be implemented by applying a *single* $\pi$-pulse, e.g., $$\begin{aligned}
S_{23}\equiv U'_{\rm CNOT} ={{\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{array} \right]}}
= {\cal Y}_{23}(\pi).\quad
\label{Q2}\end{aligned}$$ Similarly, a SWAP-like gate can also be implemented easily by a *single* $\pi$-pulse: $$\begin{aligned}
S_{12}\equiv U'_{\rm SWAP } &=& {{\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 &-1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]}}
= {\cal Y}_{12}(\pi).
\label{Q3}\end{aligned}$$ The above CNOT-like and SWAP-like gates can be related to the standard CNOT and SWAP gates as follows: $$\begin{aligned}
U_{\rm CNOT} &=& DU'_{\rm CNOT},
\nonumber \\
U_{\rm SWAP} &=& U'_{\rm SWAP}D,
\label{Q4}\end{aligned}$$ where $D={\rm diag}([1,1,-1,1])$. One can define a SWAP-like gate $S_{nm}$ between any levels ${\mbox{$|n\rangle$}}$ and ${\mbox{$|m\rangle$}}$ in a quartit, simply as $$S_{nm} = {\cal Y}_{nm}(\pi),
\label{Q5}$$ which in special cases reduce to Eqs. (\[Q2\]) and (\[Q3\]).
It is worth noting that any unitary operator that can create entanglement between a pair of qubits (or virtual qubits) is universal. Thus, the standard SWAP gate $U_{\rm SWAP}$ is not universal, as its entangling power is zero. In contrast to this gate, the SWAP-like gate *$U'_{\rm SWAP}$ is universal*, as it can entangle qubits.
Principles of $M_z$-based QST
=============================
NMR quantum state tomography is a method for the complete reconstruction of a given density matrix $\rho$ in a series of NMR measurements. In general, to completely reconstruct a density matrix $\rho$ for a quartit or two qubits, we need to determine 16 real parameters. Note that if the efficiency of a given detection system is known then the 16th element can typically be found from the normalization condition. Single NMR readout can only give some elements of $\rho$: either diagonal (in case of $M_z$ detection) or off-diagonal elements (for $M_{xy}$ detection), as discussed in Appendix B. The remaining matrix elements of the original density matrix $\rho$ can be obtained by rotating it through properly chosen rotational operations $R^{(k)}$, which change $\rho$ as follows: $$\begin{aligned}
\rho^{(k)} &\equiv& R^{(k)} \rho (R^{(k)})^\dagger.
\label{N07}\end{aligned}$$ These operations are performed before NMR readout measurements. Thus, the reconstruction of a given density matrix is possible by transforming $\rho$ through various rotations $R^{(k)}$ in such a way that all the elements of $\rho$ go over into measurable ones in a given detection method.
In the standard NMR $M_{xy}$ detection, one can directly determine some of the off-diagonal elements of the density matrix. In contrast to the $M_{xy}$ detection, one directly determines only diagonal elements in the $M_z$ detection. It is worth noting that the spectrum of a spin-3/2 system obtained via the $M_z$ detection contains less information than the spectra obtained by the $M_{xy}$ detection as discussed in Appendix B: the $M_{xy}$ detection of a spin-3/2 system yields six real values, which correspond to three peaks of real and those of imaginary parts of the spectrum. Note that the $M_{xy}$ detection of a coupled two spin-1/2 system can yield even more values if one could detect signals from ensembles of two different spins simultaneously.
Tomography based on the measurements of the $M_{z}$ and $M_{xy}$ magnetizations of spin-3/2 systems has been performed in experiments reported in Refs. [@Bonk04] and [@Kampermann05], respectively.
An implementation of $M_z$ detection in a nanometer-scale device
----------------------------------------------------------------
Here, we briefly describe an implementation of the NMR detection of the longitudinal magnetization of a small ensemble of quadrupolar spins-3/2, which is beyond the detection limits of conventional NMR techniques [@Machida03; @Yusa05; @Hirayama06; @Ota07; @decoherence; @Kondo08].
This NMR detection was developed and applied in Refs. [@Yusa05] to an on-chip semiconductor device based on a quantum-well structure shown in Fig. 3. This nanometer-scale device is composed of a monolithic GaAs quantum well integrated with a point contact channel and an antenna gate, where an RF field can be locally applied. The GaAs layer effectively forms a two-dimensional electron gas. The point-contact channel is composed of isotopes $^{69}$Ga, $^{71}$Ga, and $^{75}$As having total ground-state spin $I$=3/2. The nuclear spins in the channel can be selectively polarized by flowing current, while the spins in the other regions are kept in thermal equilibrium. These interactions between electron and nuclear spins are enhanced when an external static magnetic field ${B}_0$ is applied to set the system at the spin phase transition of the Landau level filling factor 2/3 [@Hashimoto02]. The polarization is followed by RF pulses applied through the antenna gate, which enable manipulation of the nuclear spins. This coherent manipulation results in oscillations of the resistance of the point-contact channel, which are directly related to the oscillations in the longitudinal magnetization $M_z$. Reference [@Yusa05] observed clear oscillations reflecting all possible transitions between the four nuclear-spin states (see Fig. 2) of each nuclide ($^{69}$Ga, $^{71}$Ga, and $^{75}$As). This novel device, exhibiting extremely low decoherence [@Yusa05; @decoherence], opens new perspectives to study characteristics of nuclear spins in nanoscale semiconductors, but also to precisely control nuclear-spin states. The arbitrary control of the superposition of the four spin-3/2 states enables the implementation of two-qubit coherent operations [@Hirayama06; @Ota07]. Thus, the device offers new possibilities to perform single- and two-qubit quantum gates, or even to test simple quantum-information processing algorithms. A fabrication of analogous device based on InAs and InSb [@Liu10], instead of GaAs, where the isotopes $^{113}$In and $^{115}$In have spin $I=9/2$ (a ten-level qudit) and the isotope $^{123}$Sb has spin $I=7/2$ (an eight-level qudit), would enable the implementation of three-qubit quantum gates and algorithms. But it must be admitted that the devices are not easily scalable for much higher number of virtual and/or real qubits.
The initialization of the described device is relatively easy. We can realize the effective pure state ${\mbox{$|3\rangle$}}$ by using current-induced nuclear spin polarization with randomizing pulses of $\omega_{01}$ and $\omega_{12}$. Once the state ${\mbox{$|3\rangle$}}$ is realized, it is transferred to ${\mbox{$|0\rangle$}}$, ${\mbox{$|1\rangle$}},$ and ${\mbox{$|2\rangle$}}$ by applying a respective $\pi$ pulse as described by us in Refs. [@Hirayama06; @Ota07].
The estimated polarization of the nuclear spins is quite high. Therefore, obtaining initial states with high purity should be practically quite simple — at least to start with the pseudo-pure states $|0\>$ or $|4\>$.
There are many different possibilities to measure spectra. The following is the simplest example applied in experiments described in Ref. [@Hirayama06]. After the preparation of a desired state, we apply a pulse with a duration corresponding to a $\pi$ pulse and measure how a resistance changes as a function of frequency. In case of the constant pulse-current amplitude, the length of the $\pi$-pulse changes $1/\sqrt{3}:1/\sqrt{4}:1/\sqrt{3}$ for $\omega_{01}$, $\omega_{12}$ and $\omega_{23}$, respectively. (These differences can be ignored in a simple experiment.) From this spectrum, we can estimate a population difference between neighboring states, i.e., $\rho_{11}-\rho_{00}$, $\rho_{22}-\rho_{11}$, and $\rho_{33}-\rho_{22}$ (see Sec. VI.B). Another experimental observation approach will be described in Sec. VI.C.
Linear reconstruction and the error robustness
==============================================
Various numerical procedures for reconstructing an unknown density matrix $\rho$ from experimental data have been developed (see, e.g., Refs. [@ParisBook; @DAriano03] and references therein).
The simplest and most intuitive QST is based on the inversion of a linear system, $$\begin{aligned}
Ax = b,
\label{Axb}\end{aligned}$$ where the real vector $x={\rm vec}(\rho)$ corresponds to the state $\rho$ to be reconstructed. This vector can be defined in various ways. Here, for a quartit state, we define $x$ as $$x={\rm vec}(\rho) = [\rho_{00},{\rm Re} \rho_{01},{\rm Im}
\rho_{01},{\rm Re} \rho_{02},{\rm Im} \rho_{02}, ...,\rho_{33}]^T,
\label{Na1}$$ where $\rho_{ij}$, for $i\le j$, are only included. Thus, a density matrix $\rho$ can be expressed via the elements of the vector $x$ as follows $$\begin{aligned}
\rho = \left[
\begin{array}{cccc}
x_{1} & x_{2}+i x_{3} & x_{4}+i x_{5} & x_{6}+i x_{7} \\
x_{2}-i x_{3} & x_{8} & x_{9}+i x_{10} & x_{11}+i x_{12} \\
x_{4}-i x_{5} & x_{9}-i x_{10} & x_{13} & x_{14}+i x_{15} \\
x_{6}-i x_{7} & x_{11}-i x_{12} & x_{14}-i x_{15} & x_{16} \\
\end{array}
\right]\!\!. \label{rho}\end{aligned}$$ Moreover, in Eq. (\[Axb\]), ${b}$ is the *observation vector*, which contains the measured data; and $A$ is the *coefficient matrix*, which is also referred to as the rotation matrix, or the data matrix in a more mathematical context. Thus, the element $A_{j i}$ is the coefficient of $x_i$ in the $j$th equation ($j=1,...,N_{\rm eqs}$) for a chosen measurement rotation. In our context, the observation vector $b_{j}$ corresponds to the integrated area of the NMR spectra. The number $N_{\rm eqs}$ of equations is given by $N_{\rm r}\times
N_{\rm vals}$, assuming $N_{\rm r}$ readouts (for a given measurement), where each of them yields $N_{\rm vals}$ values corresponding, e.g., to the number of peaks of an NMR spectrum (including both real and imaginary parts). Usually, an extra equation is added for the normalization condition, $\tr\rho=1$. Thus, for a quartit, the observation vector has $N_{\rm eqs}$ elements and the coefficient matrix $A$ is of dimensions $N_{\rm
eqs}\times 16$.
Usually, there are more equations than unknowns. Such *overdetermined* problems can be solved as $$\begin{aligned}
{C} x=\tilde b,
\label{N09}\end{aligned}$$ where ${C}\equiv[ {C}_{ij}]_{16 \times 16}={A}^\dagger {A}$ and $\tilde b \equiv[\tilde b_{j}]_{16 \times 1}={A}^\dagger {b}$. Equation (\[N09\]) results from the standard least-squares-fitting analysis based on the minimalization of $\chi^{2}=||Ax-b||^2$. Thus, one can easily calculate the solution $x = {C}^{-1}\tilde b$ and, finally, reconstruct the sought density matrix as $$\rho={\rm vec}^{-1}(x)={\rm vec}^{-1}\left({C}^{-1}\tilde b\right),
\label{invX}$$ as the inverse of Eq. (\[Na1\]).
Dozens of different linear-inversion-based QST protocols have been proposed and applied (see, e.g., [@ParisBook; @DAriano03] and references citing those). Then the question arises: Which of them are preferable for certain goals and tasks?
As an indicator of the quality of a linear-inversion-based QST method, or more precisely its error robustness (or error sensitivity), one can apply the so-called *condition number* defined as [@AtkinsonBook; @HighamBook; @GolubBook]: $${\rm cond}_{\alpha,\beta}(C) = \Vert C \Vert_{\alpha,\beta}\; \Vert C^{-1}
\Vert_{\beta,\alpha}\ge 1,
\label{N11}$$ where $C$ is a nonsingular square matrix and the convention is used that ${\rm cond}_{\alpha,\beta}(C) = +\infty$ for a singular matrix $C$. Moreover, $\Vert \cdot \Vert_{\alpha,\beta}$ denotes the subordinate matrix norm, which can be defined via the vector norms: $ \Vert C \Vert_{\alpha,\beta} = \max_{x\neq 0} \Vert Cx
\Vert_{\beta}/\Vert x \Vert_{\alpha}.$ Clearly, the condition numbers depend on the applied norm. Here, we apply the spectral norm only.
The spectral norm (also refereed to as the 2-norm) is given by the largest singular value of $C$, i.e, $\Vert C \Vert_{2}\equiv
\Vert C \Vert_{2,2}=\max[{\rm svd}(C)]\equiv \sigma_{\max}(C)$, where the function ${\rm svd}(C)$ gives the singular values of $C$. Then this condition number is simply given by $$\begin{aligned}
\kappa(C)\equiv {\rm cond}_{22}(C) = \frac{\sigma_{\max}(C)}{\sigma_{\min}(C)},
\label{kappa2}\end{aligned}$$ where we have used $\Vert C^{-1} \Vert_{2}= \max[{\rm
svd}(C^{-1})]= \{\min[{\rm svd}(C)]\}^{-1}\equiv
\sigma^{-1}_{\min}(C).$ There are various geometrical, algebraic, and physical interpretations of condition numbers (see Ref. [@Miran14] and references therein, in addition to Refs. [@AtkinsonBook; @HighamBook; @GolubBook; @MeyerBook]). In particular, according to the Gastinel-Kahan theorem, the inverse of a condition number corresponds to the relative distance of a nonsingular square matrix $C$ to the set of singular matrices. Another, more physical interpretation can be given as follows [@AtkinsonBook]: Let us assume errors $\delta\tilde b$ in the observation vector $\tilde b$, which cause errors $\delta
x$ in the reconstructed vector $x$: $$\begin{aligned}
C(x+\delta x) &=& \tilde b+\delta \tilde b,
\label{Atkinson1}\end{aligned}$$ then the following inequalities hold: $$\frac{1}{{\rm cond_{\alpha,\beta}}(C)}
\frac{||\delta \tilde b||}{||\tilde b||} \le \frac{||\delta x||}{||x||} \le {\rm cond_{\alpha,\beta}}(C)
\frac{||\delta \tilde b||}{||\tilde b||}.
\label{Atkinson3}$$ It is clear that when a condition number ${\rm
cond_{\alpha,\beta}}(C)\approx 1$, then small relative changes in the observation vector $\tilde b$ cause equally small relative changes in the reconstructed state $x$. This interpretation can be generalized to include also errors $\delta C$ in the coefficient matrix $C$.
Thus, by applying this general theorem, given in Eq. (\[Atkinson3\]), to QST, we can conclude that if ${\rm
cond}_{\alpha,\beta}(C)$ is small (large), then the coefficient matrix $C$ and the corresponding QST method are called *well-conditioned* (*ill-conditioned*), which means that the method is robust (sensitive) to errors in the observation vector $\tilde b$. For ill-conditioned QST, even a minor error in $\tilde b$ can cause a large error in $x$. Some instructive numerical examples of ill-conditioned problems are given in Refs. [@AtkinsonBook; @Miran14].
Condition numbers were applied to estimate the quality of optical tomographic reconstructions in, e.g., Refs. [@Bogdanov10; @Miran14]. A condition number was also calculated for the NMR tomography of two qubits (two spins-1/2) [@Roy10]. However, to our knowledge, these parameters have not been applied yet to analyze the quality of QST of any *qudit* systems. More importantly, none of the previous NMR tomographic methods exhibits the optimum robustness against errors as described by a condition number equal or almost equal to 1. Below we propose a few NMR QST protocols and compare their error robustness based on the condition numbers to show that some of our methods are optimal.
Note that the smallest singular value (or, equivalently, eigenvalue) $\sigma_{\min}({C}) = \min[{\rm svd}({C})] =
||{C}^{-1}||_2$ of ${C}$ is also sometimes used as an error-robustness parameter. This approach was applied in the analysis of an NMR QST method in, e.g., Ref. [@Long01]. In comparison to $\sigma_{\min}({C})$, the condition numbers are much better parameters of the error robustness as discussed in, e.g., Ref. [@Miran14].
Observation approaches
======================
Here we specify three observation approaches based on the $M_z$ detection to be studied in detail in the next sections.
Theoretical approach
--------------------
In an ideal $M_z$ detection, one can directly access all the diagonal elements $$\begin{aligned}
b_n^{(k)} &=& \rho^{(k)}_{nn}
\label{OA1}\end{aligned}$$ of any rotated density matrix $\rho^{(k)}\equiv R^{(k)}\rho\,
(R^{(k)})^{\dagger}$ for $k=1,...,N_{\rm r}$, where $N_{\rm r}$ is the number of readouts (operations or sets of rotations). We refer to this purely theoretical method as the *theoretical approach*.
Ideal experimental approach
---------------------------
In a more realistic observation approach, the information is gathered from the $M_z$-spectra, where one can roughly estimate the population differences ($\rho_{11}-\rho_{00}$, $\rho_{22}-\rho_{11}$, and $\rho_{33}-\rho_{22}$) from the amplitude of the signals by integrating the area of the peaks centered at $\omega_{01}$, $\omega_{12}$, and $\omega_{23}$, respectively. Thus, on including the normalization condition, we have the following set of equations: $$\begin{aligned}
b_n^{(k)} &=& \rho^{(k)}_{n+1,n+1}-\rho^{(k)}_{nn}
= \tr(I_z^{(n+1,n)}\rho^{(k)}),
\nonumber \\ 1 &=& \tr \rho^{(k)}
\label{OA2}\end{aligned}$$ for each rotated density matrix $\rho^{(k)}$, where $I_z^{(n+1,n)}={\mbox{$|n+1\rangle$}}{\mbox{$\langlen+1|$}}-{\mbox{$|n\rangle$}}{\mbox{$\langlen|$}}$ is the fictitious spin-1/2 operator for general spin. Note that $b_n^{(k)}$ can be rescaled as $\bar b_n^{(k)}{\cal N}$, where the constant ${\cal N}$ is usually chosen so the thermal equilibrium magnetization vector is equal to a unit vector along the $z$ axis [@LevittBook]. By referring to the ideal experimental approach, we mean that based on Eq. (\[OA2\]).
Alternatively, the measured resistance in experiments performed in, e.g., Refs. [@Yusa05; @Hirayama06; @Ota07], can be proportional to the longitudinal magnetization $M_z \propto
\tr[\rho I_z]$ defined in terms of the total angular momentum operator $I_z={\rm diag} ([\frac32,\frac12,-\frac12,-\frac32])$ for spin $I=3/2:$ $$M_z^{(k)}\propto\tr[\rho^{(k)} I_z]=\frac{1}{2}
(3\rho^{(k)}_{00}+\rho^{(k)}_{11}-\rho^{(k)}_{22}-3\rho^{(k)}_{33}).
\label{OA2extra}$$ However, instead of studying this approach based on Eq. (\[OA2extra\]), we apply a more practical observation method based on the standard cyclically-ordered phase sequence (CYCLOPS) technique.
Non-ideal experimental approach using CYCLOPS
---------------------------------------------
Here, we study a practical measurement method by applying the CYCLOPS to a $\pi/20$ reading pulse and receiver [@FreemanBook]. This method was used in, e.g., the experiment on QST for quadrupolar nuclei of a liquid crystal by Bonk *et al.* [@Bonk04]. In this observation approach, the NMR spectra were obtained from free induction decay (FID) averaged over each phase ($x,-y,-x,y$). This enables the suppression of receiver imperfections and, thus, the cancellation of artifacts from the NMR spectra. The intensities $b_{n}^{(k)}$ of the three ($n=1,2,3$) peaks of the averaged NMR spectrum for the rotated deviation matrices $$\Delta\rho^{(k)}\equiv \rho^{(k)}-\tfrac14 {\cal I}, \quad \text{for}\;k=1,...,N_{\rm
r},
\label{deviation}$$ together with the normalization conditions are described by the following set of equations for, e.g., the quartit: $$\begin{aligned}
[b_{1}^{(k)},b_{2}^{(k)},b_{3}^{(k)}]^T&=&V {\rm
diag}(\Delta\rho^{(k)}), \nonumber \\ 0 &=& \tr
(\Delta\rho^{(k)}), \label{OA3}\end{aligned}$$ where [@Bonk04]: $$V=\left[\begin{array}{cccc}
\sqrt{3}e_{11}e_{12} & - \sqrt{3}e_{12}e_{22} & - \sqrt{3}e_{23}e_{13} & - \sqrt{3}e_{13}e_{14}\\
2e_{13}e_{12}& 2e_{22}e_{23}& - 2e_{23}e_{22}& - 2e_{13}e_{12}\\
\sqrt{3}e_{13}e_{14} & \sqrt{3} e_{13}e_{23} & \sqrt{3}
e_{12}e_{22} & - \sqrt{3} e_{11}e_{12}
\end{array}\right].
\label{OA3a}$$ The $n$th NMR peak corresponds to the transition between levels $|n-1\rangle$ and $|n\rangle$. Above, ${\rm
diag}(\Delta\rho^{(k)})$ denotes a column vector of the diagonal elements of $\Delta\rho^{(k)}$. The coefficients $e_{ij}$ are the absolute values of the $\pi/20$ hard-reading pulse given by: $$[e_{ij}]=\frac 14 \left[
\begin{array}{cccc}
c_{31} & s_{zz} & c_{z,-z} & s_{3,-1} \\
s_{zz} & c_{13} & s_{-1,3} & c_{z,-z} \\
c_{z,-z} & s_{-1,3} & c_{13} & s_{zz} \\
s_{3,-1} & c_{z,-z} & s_{zz} & c_{31}
\end{array}
\right], \label{eij}$$ where $c_{xy}=x \cos \left(\frac{\pi }{40}\right)+y \cos
\left(\frac{3 \pi }{40}\right)$, $s_{xy}=x \sin \left(\frac{\pi
}{40}\right)+y \sin \left(\frac{3 \pi }{40}\right)$, and $z=\sqrt{3}$. It seems that this method results in the coefficient matrices, which are completely different from those obtained in the ideal experimental approach. However, we will show that they are practically very similar.
Optimal reconstruction of the diagonal elements of $\rho$
=========================================================
Here we analyze the error robustness based on the condition number $\kappa$ for the reconstruction of only diagonal terms $\rho_{nn}$ of a quartit density matrix $\rho$ for the three observation approaches using various sets of rotations.
Theoretical approach
--------------------
In the theoretical observation approach, we assume a direct access to all the diagonal terms of $\rho$: $$\begin{aligned}
b^{(1)}_1 &= \rho_{00} \equiv x_1, \quad \;
b^{(1)}_2 = \rho_{11} \equiv x_8,
\nonumber \\
b^{(1)}_3 &= \rho_{22} \equiv x_{13}, \quad
b^{(1)}_4 = \rho_{33} \equiv x_{16},
\label{N35a}\end{aligned}$$ where $\rho=\rho^{(1)}$. This implies that this partial tomography is perfectly robust against errors, as described by the condition number $\kappa=1$. Obviously, this robustness does not guarantee that complete tomographic methods can also be perfectly robust against errors.
Ideal experimental approach
---------------------------
The set of equations (\[OA2\]) directly leads to the coefficient matrices, which are in general different from those obtained by a direct measurement of all the diagonal elements of $\rho$. Nevertheless, from Eq. (\[OA2\]), one can easily determine all the diagonal elements $\rho^{(k)}$, e.g., as follows: $$\begin{aligned}
\rho^{(k)}_{00} &=& {{\textstyle{\frac{1}{4}}}}- {{\textstyle{\frac{1}{4}}}} (3 b^{(k)}_1 +2 b^{(k)}_2 +b^{(k)}_3),
\nonumber \\
\rho^{(k)}_{11} &=& {{\textstyle{\frac{1}{4}}}}+ {{\textstyle{\frac{1}{4}}}} (b^{(k)}_1 -2 b^{(k)}_2 -b^{(k)}_3),
\nonumber \\
\rho^{(k)}_{22} &=& {{\textstyle{\frac{1}{4}}}}+ {{\textstyle{\frac{1}{4}}}} (b^{(k)}_1 +2 b^{(k)}_2 -b^{(k)}_3),
\nonumber \\
\rho^{(k)}_{33} &=& {{\textstyle{\frac{1}{4}}}}+ {{\textstyle{\frac{1}{4}}}} (b^{(k)}_1 +2 b^{(k)}_2 +3 b^{(k)}_3).
\label{X8}\end{aligned}$$ We can rewrite this problem in a matrix form, given by Eq. (\[Axb\]), with its solution $x=(A^{\rm temp1}_{\rm
diag})^{-1}b$, where $$\begin{aligned}
A^{\rm temp1}_{\rm diag}&=&\left[
\begin{array}{llll}
-1 & 1 & 0 & 0 \\
0 &-1 & 1 & 0 \\
0 & 0 &-1 & 1 \\
1 & 1 & 1 & 1
\end{array}
\right],
\label{X9} \\
x &=& [\rho_{00},\rho_{11},\rho_{22},\rho_{33}]^T,
\label{X9a} \\
b &=& [b_1^{(1)},b_2^{(1)},b_3^{(1)},1]^T,
\label{X9b}\end{aligned}$$ where, as usual, $b_n^{(1)}$ corresponds to the $n$th peak resulting from the transition between the levels ${\mbox{$|n-1\rangle$}}$ and ${\mbox{$|n\rangle$}}$ in the original matrix $\rho=\rho^{(1)}$. The condition number reads $\kappa(A^TA)=6.83$, for $A\equiv A^{\rm temp1}_{\rm
diag}$. Thus, this direct application of the ideal experimental observation method to reconstruct *only* the diagonal terms of $\rho$ for a quartit can magnify the relative error in the observation vector $b$ by almost one order of magnitude.
Nevertheless, in the following, we show how to achieve $\kappa(A^TA)=1$, even if the diagonal matrix elements are not directly measured. It is worth noting that the condition number $\kappa(A^TA)$ can also be equal to 1 for analogous $M_z$-based QST of the diagonal elements of a density matrix for two spatially-separated qubits. This is because the diagonal matrix elements can be directly measured, so no reconstruction of these elements is required.
Note that this coefficient matrix $A^{\rm temp1}_{\rm diag}$ is unbalanced, which implies that some elements of $\rho$ are measured more often than others. Specifically, there are only two nonzero elements in the first and last columns of $A^{\rm
temp1}_{\rm diag}$ (corresponding to $\rho_{00}$ and $\rho_{33}$) and three nonzero elements in the other columns of $A$ (corresponding to $\rho_{11}$ and $\rho_{22}$). To overcome this problem, let us apply the pulse $S_{13}\equiv {\cal Y}_{13}(\pi)$, which corresponds to the SWAP-like gate. Then, we measure only the first peak (corresponding to the transition ${\mbox{$|0\rangle$}}\leftrightarrow {\mbox{$|1\rangle$}}$) of the rotated density matrix $S_{13}\rho S_{13}^\dagger$. Thus, by adding this equation to $A^{\rm temp1}_{\rm diag}$, one obtains $$A^{\rm temp2}_{\rm diag}=[A^{\rm temp1}_{\rm diag};(-1,0,0,1)].
\label{X10}$$ Then the condition number becomes $\kappa(A^TA)=2$ for $A=A^{\rm
temp2}_{\rm diag}$, which is much smaller than that for $A=A^{\rm
temp1}_{\rm diag}$. One can then obtain a more balanced coefficient matrix $A^{\rm opt}_{\rm diag}$ by adding two equations to $A^{\rm temp2}_{\rm diag}$, which correspond to the first peak of the rotated density matrices $S_{12}\rho
S_{12}^\dagger$ and $S_{03}\rho S_{03}^\dagger$. Thus, we have $$\begin{aligned}
A_{\rm diag}^{\rm opt1} &=& \begin{pmatrix}
-1 & 1 & 0 & 0 \\
0 &1 & -1 & 0 \\
0 & 0 &-1 & 1 \\
-1 & 0 & 0 & 1 \\
-1 & 0 & 1 & 0 \\
0 & 1 & 0 &-1 \\
s & s & s & s \
\end{pmatrix}.
\label{Adiag_opt1}\end{aligned}$$ Note that we have multiplied the second row in Eq. (\[X9\]) by the factor (-1) to obtain Eq. (\[Adiag\_opt1\]), which enables us to slightly simplify the following Eq. (\[Nfig4\]). This operation does not affect the corresponding condition numbers. As usual, the last row (equation) in Eq. (\[Adiag\_opt1\]) corresponds to the normalization condition, where $s$ is the scaling factor, which is set here as $s=1$. Then we find that $C_{\rm diag}^{\rm opt}=(A_{\rm diag}^{\rm opt1})^\dagger A_{\rm
diag}^{\rm opt1}=4I_4,$ where $I_4$ is the four-dimensional identity operator. In general, this factor $s$ determines the contribution of the last equation to the whole set $Ax=b$ of equations and can be chosen such that $\kappa$ is minimized.
Thus, we have shown that the condition number $\kappa(C_{\rm
diag}^{\rm opt})=1$ indicates the optimality of the coefficient matrix $A_{\rm diag}^{\rm opt}$ for the ideal experimental observation approach. This matrix $A_{\rm diag}^{\rm opt}$ can be obtained from the following set of rotations: $$R^{\rm opt1}_{\rm diag}=[I,S_{02},S_{13}S_{02},S_{13},S_{12},S_{03}],
\label{Nfig4}$$ as shown in Fig. 4, using the SWAP-like gates $S_{nm}\equiv {\cal
Y}_{nm}(\pi)$. Here we assume that only the first peak is measured, while the other two peaks are ignored, in the $M_z$-spectra of the rotated density matrices $\rho^{(k)}=R^{(k)}\rho\,(R^{(k)})^\dagger$ for all the rotations $R^{(k)}$ in Eq. (\[Nfig4\]).
Alternatively, we can swap the elements $\rho_{nn}$ in such a way that only the central peak, corresponding to ${\mbox{$|1\rangle$}}\leftrightarrow {\mbox{$|2\rangle$}}$, is measured. Namely, one can use the following set of rotations $$\begin{aligned}
R^{\rm opt2}_{\rm diag}&=&[I,S_{02},S_{13},S_{01}S_{23},S_{01},S_{23}]
\label{Nfig5a}\end{aligned}$$ as shown in Fig. 5, which leads to the following coefficient matrix $$\begin{aligned}
A_{\rm diag}^{\rm opt2} &=& \left(
\begin{array}{cccc}
0 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 \\
0 & 0 & 1 & -1 \\
-1 & 0 & 0 & 1 \\
-1 & 0 & 1 & 0 \\
0 & -1 & 0 & 1 \\
s & s & s & s \\
\end{array}
\right), \label{Adiag_opt2}\end{aligned}$$ where the last row corresponds, as usual, to the normalization condition. By setting $s=1$, one finds that $C_{\rm diag}^{\rm
opt}=(A_{\rm diag}^{\rm opt2})^\dagger A_{\rm diag}^{\rm
opt2}=4I_4,$ the same as for $(A_{\rm diag}^{\rm opt1})^\dagger
A_{\rm diag}^{\rm opt1}$. This property holds since $A_{\rm
diag}^{\rm opt1}$ and $A_{\rm diag}^{\rm opt2}$ differ only in the order of their rows and in the opposite sign of all the elements of some of their rows. Therefore, these coefficient matrices can be considered equivalent, $$A_{\rm diag}^{\rm opt1}\cong A_{\rm diag}^{\rm opt2}.
\label{equivalent1}$$ Moreover, the two-photon rotations, given in Eq. (\[Nfig5a\]), can be replaced by single-photon transitions, e.g., $S_{02}$ can be replaced by $S_{01}S_{12}$, and $S_{13}$ by $S_{12}S_{23}S_{12}$, as will be described in general terms in Sec. X. This might be an advantage from the experimental point of view.
It is worth noting that the state vector $x$ is defined as in Eq. (\[X9a\]) for both methods, based on the rotations $R^{\rm
opt1}_{\rm diag}$ and $R^{\rm opt2}_{\rm diag}$. However, the observation vectors $b$ are defined as follows: (i) For the rotations $R^{\rm opt1}_{\rm diag}$, one measures $$\begin{aligned}
b &=& [b_1^{(1)},b_1^{(2)},...,b_1^{(6)},s]^T,
\label{b1}\end{aligned}$$ where $b_1^{(k)}$ corresponds to the first peak obtained for the rotated density matrix $\rho^{(k)}=R^{\rm opt1}_{{\rm
diag},k}\rho\,(R^{\rm opt1}_{{\rm diag},k})^\dagger$. (ii) For the rotations $R^{\rm opt2}_{\rm diag}$, the observation vector reads $$\begin{aligned}
b &=& [b_2^{(1)},b_2^{(2)},...,b_2^{(6)},s]^T,
\label{b2}\end{aligned}$$ where $b_2^{(k)}$ corresponds to the second peak obtained for $\rho^{(k)}=R^{\rm opt2}_{{\rm diag},k}\rho\,(R^{\rm opt2}_{{\rm
diag},k})^\dagger$.
Non-ideal experimental approach using CYCLOPS
---------------------------------------------
The set of Eqs. (\[OA3\]) can be rewritten in the matrix form $Ax=b$, where $$\begin{aligned}
A^{\rm temp3}_{\rm diag} &=& [V;(s,s,s,s)],
\label{X15} \\
x &=& [\rho_{00}-\tfrac{1}{4},\rho_{11}-\tfrac14,\rho_{22}-\tfrac14,\rho_{33}-\tfrac14]^T,\quad\quad
\label{X15b}\\
b &=& [b_1^{(1)},b_2^{(1)},b_3^{(1)},0]^T.
\label{X15c}\end{aligned}$$ Then, we find the condition number to be $\kappa(A^{T}A)=98.46$, if $A=A^{\rm temp3}_{\rm diag}$ and $s=1$. It means that the determination of all the diagonal terms of $\rho$ for a quartit using the standard CYCLOPS method can be relatively sensitive to errors. Indeed, the relative errors in the observation vector $b$ can be magnified in the reconstructed vector $x$ by almost two orders of magnitude. Thus, one could conclude that the reconstruction of all (not only diagonal) elements of $\rho$ can be worse by at least two orders of magnitude, in comparison to the corresponding QST methods but assuming the ideal experimental observation approach. In contrast to these tentative conclusions, we will show below that, in fact, the error robustness can be described by the condition number $\kappa\approx 1$ for both partial and complete tomographic methods.
Analogously to the theoretical approach, we can also optimize the set of rotations of $\rho$ and measure only some peaks of the $M_z$-spectra if some non-ideal experimental observation approach is applied. Here we analyze the optimization of rotations for the CYCLOPS method.
First, we optimize the value of the scaling factor $s$ in $A\equiv
A^{\rm temp3}_{\rm diag}$. By choosing $s\in(0.1,0.25)$, we find that $\kappa(A^{T}A)=6.1375$, which is almost one order smaller than $\kappa(A^{T}A)$ for $s=1$. Now, we apply the sets of rotations $R^{{\rm opt}1}_{\rm diag}$ and $R^{{\rm opt}2}_{\rm
diag}$ to obtain the coefficient matrices $\bar A^{{\rm
opt}1}_{\rm diag}$ and $\bar A^{{\rm opt}2}_{\rm diag}$ assuming the scaling factors $s=0.2318$ and $s=0.3043$, respectively. Specifically, the optimal value of $s$ for a given coefficient matrix $A$ is chosen here as $\max_{i,j}A_{ij}'$, where $A'$ is the matrix $A$ but without the last row (i.e., with the nonzero elements equal to $s$). Note that this last equation is added to include the normalization condition for measuring the $n$th peak ($n=1,2$) using the CYCLOPS method. Although our precise expressions for the coefficient matrices $\bar A^{{\rm opt}1}_{\rm
diag}$ and $\bar A^{{\rm opt}2}_{\rm diag}$ are quite lengthy, and thus not shown here, we find that $$\bar A_{\rm diag}^{\rm opt1} \approx A_{\rm diag}^{\rm opt1}\cong A_{\rm diag}^{\rm opt2} \approx \bar A_{\rm diag}^{\rm opt2},
\label{equivalent2}$$ where Eq. (\[equivalent1\]) was used. Our precise calculations result in the following condition numbers $$\begin{aligned}
\kappa(A^TA) &=& 1.0371 \quad {\rm for}\; A=\bar A^{\rm opt1}_{\rm diag},
\nonumber \\
\kappa(A^TA) &=& 1.0384 \quad {\rm for}\; A=\bar A^{\rm opt2}_{\rm
diag},
\label{kkk1}\end{aligned}$$ which are very close to one.
Reconstruction of the off-diagonal elements of $\rho$
=====================================================
Now we propose several sets of rotations for the reconstruction of all the off-diagonal terms $\rho_{nm}$ (with $n\neq m$) for the three observation approaches and study the robustness of these methods against errors.
Theoretical approach
--------------------
Our first temporary proposal for QST of a spin-3/2 system is based on a natural choice of 12 rotations (see also Fig. 6): $$\begin{aligned}
R^{\rm temp}_{\rm offdiag} = [Y_{01},X_{01},Y_{12},X_{12},Y_{23},X_{23},\nonumber \\
Y_{02},X_{02},Y_{13},X_{13},Y_{03},X_{03}],
\label{Nfig6}\end{aligned}$$ where hereafter $X_{mn}\equiv {\cal X}_{mn}({{\textstyle{\frac{\pi}{2}}}})$ and $Y_{mn}\equiv {\cal Y}_{mn}({{\textstyle{\frac{\pi}{2}}}})$ as special cases of the selective rotations ${\cal X}_{mn}(\theta)$ and ${\cal
Y}_{mn}(\theta)$ defined in Appendix A. Thus, the method is based on 6 single-photon, 4 two-photon, and 2 three-photon transitions.
In the $M_z$ detection approach, we can determine the diagonal elements $[\rho_{00}, \rho_{11} , \rho_{22}, \rho_{33}]$ of a density matrix $\rho$. By denoting the diagonal elements of $\rho^{(k)}$ as ${\rm diag}(\rho^{(k)}) \equiv
(\rho_{nn}^{(k)})_n$, the following elements: $$\begin{aligned}
{\rm
diag}(\rho^{(1)})&=&[f_{01}^{(22)},f_{01}^{(00)},\rho_{22},\rho_{33}],
\nonumber \\
{\rm
diag}(\rho^{(2)})&=&[f_{01}^{(13)},f_{01}^{(31)},\rho_{22},\rho_{33}],
\nonumber \\
{\rm
diag}(\rho^{(3)})&=&[\rho_{00},f_{12}^{(22)},f_{12}^{(00)},\rho_{33}],
\nonumber \\
{\rm
diag}(\rho^{(4)})&=&[\rho_{00},f_{12}^{(13)},f_{12}^{(31)},\rho_{33}],
\nonumber \\
{\rm
diag}(\rho^{(5)})&=&[\rho_{00},\rho_{11},f_{23}^{(22)},f_{23}^{(00)}],
\nonumber \\
{\rm
diag}(\rho^{(6)})&=&[\rho_{00},\rho_{11},f_{23}^{(13)},f_{23}^{(31)}],
\label{Nfig6rho} \\
{\rm
diag}(\rho^{(7)})&=&[f_{02}^{(22)},\rho_{11},f_{02}^{(00)},\rho_{33}],
\nonumber \\
{\rm
diag}(\rho^{(8)})&=&[f_{02}^{(13)},\rho_{11},f_{02}^{(31)},\rho_{33}],
\nonumber \\
{\rm
diag}(\rho^{(9)})&=&[\rho_{00},f_{13}^{(22)},\rho_{22},f_{13}^{(00)}],
\nonumber \\
{\rm
diag}(\rho^{(10)})&=&[\rho_{00},f_{13}^{(13)},\rho_{22},f_{13}^{(31)}],
\nonumber \\
{\rm diag}(\rho^{(11)})
&=&[f_{03}^{(22)},\rho_{11},\rho_{22},f_{03}^{(00)}],
\nonumber \\
{\rm diag}(\rho^{(12)})
&=&[f_{03}^{(13)},\rho_{11},\rho_{22},f_{03}^{(31)}], \nonumber\end{aligned}$$ are found for the set of rotations given by Eq. (\[Nfig6\]), where the auxiliary function $f_{mn}^{(kl)}$ is defined by $$\begin{aligned}
f_{mn}^{(kl)} &=& \frac 12 (\rho_{mm}+i^k \rho_{mn}+i^l \rho_{nm}+
\rho_{nn}).
\label{N22}\end{aligned}$$ In the theoretical approach, all these equations can determine the coefficient matrices ${A}=A^{\rm temp}_{\rm offdiag}$, which are based on the set of $N_{\rm eqs}=48$ equations given by Eq. (\[OA1\]). The singular values of ${C}=A^TA$ are found to be ${\rm svd}(C)=\{12,8^{\otimes 3},2^{\otimes 12}\}$, where our compact notation $\sigma_i^{\otimes n}$ denotes that $\sigma_i$ occurs $n$ times. Thus, we can determine the condition number describing the error robustness of the QST method as $\kappa(C)=6$, which is clearly not optimal. However, by analyzing Eq. (\[Nfig6rho\]), one can find that all the off-diagonal elements of $\rho$ can directly be determined as follows: $$\begin{aligned}
&2x_2=f_{01}^{(00)}-f_{01}^{(22)},
&2x_3=f_{01}^{(31)}-f_{01}^{(13)},
\nonumber \\ \nonumber
&2x_9=f_{12}^{(00)}-f_{12}^{(22)},
&2x_{10}=f_{12}^{(31)}-f_{12}^{(13)},
\\ \nonumber
&2x_{14}=f_{23}^{(00)}-f_{23}^{(22)},
&2x_{15}=f_{23}^{(31)}-f_{23}^{(13)},
\label{x_offdiag}\\
&2x_{4}=f_{02}^{(00)}-f_{02}^{(22)},
&2x_{5}=f_{02}^{(31)}-f_{02}^{(13)},
\\ \nonumber
&2x_{11}=f_{13}^{(00)}-f_{13}^{(22)},
&2x_{12}=f_{13}^{(31)}-f_{13}^{(13)},
\\ \nonumber
&2x_{6}=f_{03}^{(00)}-f_{03}^{(22)},
&2x_{7}=f_{03}^{(31)}-f_{03}^{(13)}.\end{aligned}$$ Thus, the corresponding condition number can be decreased to 1.
Ideal experimental approach
---------------------------
The $M_z$-based tomography for the rotations, given by Eq. (\[Nfig6\]), in the ideal experimental observation approach can be understood as follows: When we apply the $Y_{01}$ pulse after a certain photon operation, we obtain the diagonal components including $\rho_{01}$ and $\rho_{10}$. In our $M_z$ detection method, one of the three signals, which corresponds to the differences of the populations between four spin states, is proportional to $\RE(\rho_{01})+\RE(\rho_{10})$. Because $\rho_{mn}=\rho_{nm}^*$, we can obtain $\RE(\rho_{01})=\RE(\rho_{10})$. Similarly, $Y_{12}$, $Y_{23}$, $Y_{02}$, $Y_{13}$, and $Y_{03}$ give us other elements $\RE(\rho_{mn})=\RE(\rho_{nm})$. Imaginary parts are also estimated by applying the $X_{mn}$ pulse by noting that $\IM(\rho_{mn})=-\IM(\rho_{nm})$. Although QST needs a few multiphoton operations, this method looks simple and easy to interpret.
In the ideal experimental approach, corresponding to Eq. (\[OA2\]), we obtain ${C}=A^TA$, where $A=A^{\rm temp}_{\rm
offdiag}$, having the following singular values: $$\begin{aligned}
{\rm svd}(C)=\{48., 24.25, 16.17, 9.97, 6., 5.45, 5^{\otimes
2},\nonumber \\ 4.91, 4.37, 3^{\otimes 2}, 2.92, 2.26, 2, 1.71\},\end{aligned}$$ which yield the condition number $\kappa(C)=28.14$, which is far from being optimal.
However, by analyzing the equations in Eq. (\[Nfig6rho\]), one can conclude that (at least) some of the off-diagonal terms of $\rho$ can be measured directly, i.e., $$\begin{aligned}
&b_1^{(1)}= 2x_2, \quad\; b_1^{(2)}=2 x_3,\, \quad\;\; b_2^{(3)}= 2x_9,
\nonumber \\
&b_2^{(4)}=2 x_{10},\quad b_3^{(5)}=2 x_{14}, \quad\; b_3^{(6)}= 2x_{15},
\label{M7}\end{aligned}$$ where $b_n^{(k)}$ corresponds to the $n$th peak of the $M_z$ spectrum obtained in the CYCLOPS method for the rotated density matrix $\rho^{(k)}=R_k\rho R_k^\dagger$, where the rotation $R_k$ is given by the $k$th element in Eq. (\[Nfig6\]).
In order to directly measure other off-diagonal elements of $\rho$, one can swap some quartit levels, say ${\mbox{$|k\rangle$}}$ and ${\mbox{$|l\rangle$}}$, by applying the $\pi$-pulse $S_{kl}\equiv{\cal
Y}_{kl}(\pi)$. For example, one can use the following set of rotations: $$\begin{aligned}
R^{\rm opt0}_{\rm offdiag} &=&
[Y_{01},X_{01},Y_{12},X_{12},Y_{23},X_{23},Y_{01}S_{12},X_{01}S_{12}, \nonumber\\
&&Y_{12}S_{23},X_{12}S_{23},Y_{01}S_{13},X_{01}S_{13}],\quad\quad
\label{Nfig7}\end{aligned}$$ as shown in Fig. 7. Then, all the off-diagonal terms can be directly measured including $$\begin{aligned}
&b_1^{(7)}=- 2x_4,\quad\quad b_1^{(8)}=-2 x_5, \quad b_2^{(9)}= -2x_{11}, \nonumber \\
& b_2^{(10)}=-2x_{12},\quad b_1^{(11)}= -2x_{6}, \quad b_1^{(12)}=-2x_{7},
\label{M9}\end{aligned}$$ in addition to those given in Eq. (\[M7\]). Thus, the corresponding coefficient matrix becomes $$\begin{aligned}
A^{\rm opt0}_{\rm offdiag} =
2\left(
\begin{array}{cccccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & \bar1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \bar1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \bar1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \bar1 & 0 & 0 \\
0 & 0 & 0 & 0 & \bar1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \bar1 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right), \label{Aoffdiag0}\end{aligned}$$ where $\bar1=-1$. The reconstructed vector for all the off-diagonal elements reads $$x=[x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{9}, x_{10},
x_{11}, x_{12}, x_{14}, x_{15}]^T,$$ while the observation vector is $$\begin{aligned}
b&=&[
b_1^{(1)},
b_1^{(2)},
b_2^{(3)},
b_2^{(4)},
b_3^{(5)},
b_3^{(6)},
\nonumber \\ &&
b_1^{(7)},
b_1^{(8)},
b_2^{(9)},
b_2^{(10)},
b_1^{(11)},
b_1^{(12)}]^T,\end{aligned}$$ as implied by Eqs. (\[M7\]) and (\[M9\]). This observation vector is obtained by measuring only a properly-chosen single peak in a given $M_z$ spectrum, while the other two peaks are ignored. Specifically, to determine a chosen term $b_n^{(k)}$, from those in Eqs. (\[M7\]) and (\[M9\]), one should only measure the $n$th peak of the $M_z$-spectra corresponding to the transition ${\mbox{$|n-1\rangle$}}\leftrightarrow {\mbox{$|n\rangle$}}$ of the rotated density matrix $\rho^{(k)}=R_k\rho R_k^\dagger$, where $R_k=R^{\rm opt0}_{{\rm
offdiag},k}$.
We can operationally simplify the problem by requiring that always the same $n$th peak is measured in the all $M_z$-spectra. For example, to measure always the first peak, one can perform the SWAP-like operations. Thus, we propose the following optimal (in terms of $\kappa=1$) set of rotations $$\begin{aligned}
R^{\rm opt1}_{\rm offdiag} &=&
[Y_{01},X_{01},S_{02}Y_{12},S_{02}X_{12},Y_{01}S_{13}S_{02},\nonumber \\
&&X_{01}S_{13}S_{02},Y_{01}S_{12},X_{01}S_{12},S_{02}Y_{12}S_{23},\nonumber\\
&&S_{02}X_{12}S_{23},Y_{01}S_{13},X_{01}S_{13}],
\label{Nfig8}\end{aligned}$$ as shown in Fig. 8, which corresponds to the following coefficient matrix $$\begin{aligned}
A^{\rm opt1}_{\rm offdiag} = 2\left(
\begin{array}{cccccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \bar1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \bar1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & \bar1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \bar1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & \bar1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \bar1 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right). \label{Aoffdiag1}\end{aligned}$$ In this approach only the first peak is measured. Alternatively, we can swap the quartit levels in such a way that only the central peak is measured. Then, the optimal tomography can be achieved for the following set of rotations $$\begin{aligned}
R^{\rm opt2}_{\rm offdiag} &=&
[Y_{12}S_{02},X_{12}S_{02},Y_{12},X_{12},Y_{12}S_{13},\nonumber \\
&&X_{12}S_{13},Y_{12}S_{01},X_{12}S_{01},Y_{12}S_{23},\nonumber \\
&&X_{12}S_{23},Y_{12}S_{01}S_{23},X_{12}S_{01}S_{23}],
\label{Nfig9}\end{aligned}$$ as shown in Fig. 9, which corresponds to the following coefficient matrix $$\begin{aligned}
A^{\rm opt2}_{\rm offdiag} = 2\left(
\begin{array}{cccccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \bar1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \bar1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \bar1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \bar1 & 0 & 0 \\
0 & 0 & 0 & 0 & \bar1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \bar1 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right). \label{Aoffdiag2}\end{aligned}$$ The advantage of the rotations $R^{\rm opt2}_{\rm offdiag}$, in comparison to $R^{\rm opt1}_{\rm offdiag}$, resides in the lower number of two-photon rotations. In addition, the remaining two-photon SWAP-like rotations $S_{02}$ and $S_{13}$, listed in Eq. (\[Nfig9\]), can be replaced by a sequence of single-photon rotations as described in Sec. X.
It is seen that $$A^{\rm opt0}_{\rm offdiag} \cong A^{\rm opt1}_{\rm offdiag} \cong A^{\rm opt2}_{\rm
offdiag},
\label{equivalent3}$$ are equivalent up to an irrelevant multiplication of some of their rows by the factor (-1).
In conclusion, we find that the proposed sets of rotations for the reconstruction of all the off-diagonal density-matrix elements in this ideal observation approach are optimal, as leading to the lowest value of the condition number $$\kappa \left[(A^{{\rm opt},l}_{\rm offdiag})^T A^{{\rm opt},l}_{\rm offdiag}\right]
= 1,
\label{kappa_offdiag}$$ for $l=0,1,2$.
Non-ideal experimental approach via CYCLOPS
-------------------------------------------
The experimental observation approach based on the CYCLOPS method can be considered an imperfect (or noisy) version of the above ideal experimental approach based on Eq. (\[OA2\]). For example, the first peak $b_1^{(1)}$ in the $M_z$ spectrum after the rotation $Y_{01}$ corresponds to: $$b_1^{(1)} = c_1x_1-c_2x_2+c_8x_8-c_{13}x_{13}-c_{16}x_{13}
\approx -c_2 x_2,
\label{1row}$$ where $c_{1}\approx c_{8}\approx 0.0014$, $c_{2}\approx 0.4608$, $c_{13}\approx 0.0029$, and $c_{16}\approx 9\times 10^{-16}$. Note that the element $x_2$ is dominant, at least, by three orders of magnitude in comparison to the other elements, as $c_2\approx
161\times \max_{i\neq 2}c_i$. Then, we find that the condition number $\kappa(A^{\rm opt}_{\rm offdiag})\approx 1.000$ for the rotations $R^{{\rm opt},l}_{\rm offdiag}$ (for $l=0,1,2$) assuming the CYCLOPS measurement. This approximate calculation is performed by ignoring the contributions of the diagonal terms $x_1$, $x_8$, $x_{13}$, and $x_{16}$.
Optimal reconstruction of all the elements of $\rho$
====================================================
Theoretical approach
--------------------
By analyzing Eqs. (\[N35a\]) and (\[x\_offdiag\]), all of the 16 elements of $x$ (and, thus, $\rho$) can be accessed directly in this theoretical observation approach. Therefore, the corresponding coefficient matrix is diagonal, and the reconstruction of $x$ is trivial. Thus, this QST is perfectly robust against errors, as the condition number is equal to 1.
Ideal experimental approach
---------------------------
In order to reconstruct all the elements of $\rho$ we can combine the optimal reconstructions for the off-diagonal elements (based on the optimal coefficient matrix $A_{\rm offdiag}^{{\rm opt},k}$ corresponding to the rotations $R_{\rm offdiag}^{{\rm opt},k}$ for $k=0,1,2$) and diagonal elements (described by $A_{\rm diag}^{{\rm
opt},l}$ corresponding to $R_{\rm diag}^{{\rm opt},l}$ for $l=1,2)$. For example, the combined coefficient matrices of the dimensions $19\times 16$ can read $$\begin{aligned}
A_{\rm opt1}&=&[A_{\rm offdiag}^{\rm opt1};A_{\rm diag}^{\rm
opt1}],
\nonumber \\
A_{\rm opt2}&=&[A_{\rm offdiag}^{\rm opt2};A_{\rm diag}^{\rm
opt2}], \label{Afull2}\end{aligned}$$ corresponding to the sets of rotations $$\begin{aligned}
R_{\rm opt1}&=&[R_{\rm offdiag}^{\rm opt1};R_{\rm diag}^{\rm
opt1}],
\nonumber \\
R_{\rm opt2}&=&[R_{\rm offdiag}^{\rm opt2};R_{\rm diag}^{\rm
opt2}], \label{Rfull2}\end{aligned}$$ respectively. The last row in the matrices in Eq. (\[Afull2\]) corresponds to the normalization condition with the scaling factor $s=1$. Thus, we obtain the total optimal coefficient matrix $$\begin{aligned}
A_{\rm opt1}=
\begin{pmatrix}
0 \, 2 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \\
0 \, 0 \, 2 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \\
0 \, 0 \, 0 \, \bar2 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \\
0 \, 0 \, 0 \, 0 \, \bar2 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \\
0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 2 \, 0 \, 0 \\
0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 2 \, 0 \\
0 \, 0 \, 0 \, \bar2 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \\
0 \, 0 \, 0 \, 0 \, \bar2 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \\
0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 2 \, 0 \, 0 \, 0 \, 0 \, 0 \\
0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 2 \, 0 \, 0 \, 0 \, 0 \\
0 \, 0 \, 0 \, 0 \, 0 \, \bar2 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \\
0 \, 0 \, 0 \, 0 \, 0 \, 0 \, \bar2 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \\
\bar1 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 1 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \\
0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 1 \, 0 \, 0 \, 0 \, 0 \, \bar1 \, 0 \, 0 \, 0 \\
0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, \bar1 \, 0 \, 0 \, 1 \\
\bar1 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 1 \\
\bar1 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 1 \, 0 \, 0 \, 0 \\
0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 1 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, \bar1 \\
s \, 0 \, 0 \, 0 \, 0 \, 0 \, 0 \, s \, 0 \, 0 \, 0 \, 0 \, s \, 0 \, 0 \, s \\
\end{pmatrix}
, \label{Acomplete}\end{aligned}$$ where $\bar{1}=-1$ and $\bar{2}=-2$. For brevity, the analogous coefficient matrix $A_{\rm opt2}$ is not presented here. A simple calculation shows that $C_{{\rm opt},l}=(A_{{\rm opt},l})^\dagger
A_{{\rm opt},l}$ (for $l=1,2$) is proportional to the identity operator, so the condition number $\kappa(C_{{\rm opt},l})=1$. Thus, the proposed tomographic methods are optimal concerning their robustness against errors assuming the ideal experimental observations.
Non-ideal experimental approach using CYCLOPS
---------------------------------------------
In the non-ideal experimental approach, we can follow the analysis for the ideal experimental approach. In particular, we can apply Eq. (\[Acomplete\]), but for the properly chosen scaling factors. It should be stressed that Eq. (\[Acomplete\]) is only an approximation of $\bar A_{\rm opt1}$ for the non-ideal case. For example, the first row for exact $\bar A_{\rm opt1}$ is given by Eq. (\[1row\]). However, we observe that $$\bar A_{\rm opt1} \approx A_{\rm opt1} \cong A_{\rm opt2} \approx \bar A_{\rm opt2},
\label{AAAfull}$$ where the combined coefficient matrices $\bar A_{{\rm opt},l}$ (for $l=1,2)$ are obtained for the sets of rotations $R_{{\rm
opt},l}$, given in Eq. (\[Rfull2\]), analogously to $A_{{\rm
opt},l}$, given in Eq. (\[Afull2\]). Note that the last row in $\bar A_{{\rm opt}1}$ and $\bar A_{{\rm opt}2}$ corresponds to the normalization condition with the scaling factors $s=0.2304$ and $s=0.3043$, respectively. Moreover, in the solution $x=A^{-1}b$, the observation vector $b$ is equal to $[b_l^{(1)},b_l^{(2)},...,b_l^{(18)},0]^T$ for $l=1,2$, respectively, and the reconstructed state vector $x=[x_1,x_2,...,x_{16}]^T-\frac14$ is related to $\rho$ by Eq. (\[Na1\]). By performing precise numerical calculations, we conclude that the condition numbers are very close 1, i.e., $$\begin{aligned}
\kappa(A^TA) &=& 1.0592\;\approx\; 1 \quad {\rm for}\; A=\bar A_{\rm opt1},
\nonumber \\
\kappa(A^TA) &=& 1.0528\;\approx\; 1 \quad {\rm for}\; A=\bar A_{\rm opt2}.
\label{kkk2}\end{aligned}$$ Thus, even the non-ideal QST, as based on the CYCLOPS method, can be almost perfectly robust to errors, as described by their condition numbers $\kappa$.
Single-photon replacements for multiphoton rotations
====================================================
The error-robustness analysis is based on the properties of the coefficient matrices $A$ and, thus, enables to find experimental setups for the reliable QST even without specific experimental data. The optimization in our approach resides in replacing degenerate multiphoton (multi-quantum) rotations by single-photon ones.
For example, some of the discussed sets of rotations for QST include single-photon $X$ rotations ($X_{01}, X_{12}, X_{23}$), two-photon $X$ rotations ($X_{02}, X_{13}$), and a three-photon $X$ rotation ($X_{03}$) together with analogous $Y$ rotations. Especially the three-photon transitions are not the simplest to be realized experimentally due to the degeneracy between $\omega_{03}/3$ and $\omega_{12}$ if the second order quadrupolar shifts are neglected (see Fig. 1). Namely, we want to perform the three-photon rotations $Y_{03}$ and $X_{03}$ between levels ${\mbox{$|0\rangle$}}$ and ${\mbox{$|3\rangle$}}$ (for brevity, we say the 0-3 rotation) solely without changing populations between levels ${\mbox{$|1\rangle$}}$ and ${\mbox{$|2\rangle$}}$. We can effectively rotate 1-2 without rotating 0-3, but we are not able to rotate 0-3 without rotating 1-2. So a feasible tomographic method should be described without direct rotations 0-3. Under this requirement, it is easy to show analytically that one needs combinations of at least two rotations for some of the operations for complete reconstruction. Then, unfortunately, the above interpretation of the tomographic operations, given by Eq. (\[Nfig6rho\]), loses its clarity.
First, we calculate replacements for multiphoton $X$ rotations. By inspection, we find that $$\begin{aligned}
{\cal X}_{0n}(\theta ) &=&S_{1n}{\cal X}_{01}(\theta ){S}^\dag
_{1n} \nonumber \\
&=&{S}_{01}\,{\cal X}_{1n}(-\theta )\,{S}^\dag_{01} \nonumber \\
&=&{S}_{02}\,{\cal X}_{2n}(-\theta )\,{S}^\dag_{02} \label{N23} \\
&=&{S}_{2n}\,{\cal X}_{02}(\theta )\,{S}^\dag_{2n} \nonumber \\
&=&{S}_{01}\,{S}_{2n}\,{\cal X}_{12}(-\theta
)\, {S}^\dag _{2n}\,{S}^\dag_{01}, \nonumber\end{aligned}$$ given in terms of SWAP-like operations $S_{kl}\equiv{\cal
Y}_{kl}(\pi).$ Note that ${\cal Y}_{k,n}^{T}(\theta ) ={\cal
Y}_{k,n}^{\dag }(\theta )={\cal Y} _{k,n}(-\theta )$, so $S^{\dag}_{kl}={\cal Y}_{kl}(-\pi).$ Analogously, other replacements can be found. By repeatedly applying the first relation in Eq. (\[N23\]) we obtain a general formula for any two levels $k<n-1$: $$\begin{aligned}
{\cal X}_{kn}(\theta )=S_{k+1,n}\,{\cal X}_{k,k+1}(\theta)\,{S}^\dag
_{k+1,n}. \label{N24}\end{aligned}$$ Alternatively, one can apply the second relation in Eq. (\[N23\]) to obtain: $$\begin{aligned}
{\cal X}_{kn}(\theta ) ={S}^{(k,n)}\,{\cal X}_{n-1,n}[(-1)^{n-k+1}\theta
]\,(S^{(k,n)})^\dag, \label{N25}\end{aligned}$$ where $$\begin{aligned}
{S}^{(k,n)}\equiv {S}_{k,k+1}\,{S}_{k+1,k+2} \cdots{S}_{n-2,n-1}.
\label{N26}\end{aligned}$$ In the same way, we find replacements for multiphoton $Y$ rotations for any $\theta$ and $k<n-1$: $$\begin{aligned}
{\cal Y}_{kn}(\theta ) &=& S_{k+1,n}\,{\cal Y}_{k,k+1}(\theta
)\,S^\dag_{k+1,n}
\nonumber \\
&=& {S}^{(k,n)}\,{\cal Y}_{n-1,n}[(-1)^{n-k+1}\theta]\,(S^{(k,n)})^\dag,
\label{N27}\end{aligned}$$ in terms of the pulse sequences given by Eq. (\[N26\]).
In a special case, for a given QST method of the quartit system, the $X$ rotations based on three-photon (0-3) and two-photon (0-2 and 1-3) transitions can be replaced by various sequences of rotations requiring only single-photon transitions, e.g., $$\begin{aligned}
{\cal X}_{03}(\theta) &=& {S}_{01}\, {S }_{23}\, \,{\cal
X}_{12}(-\theta) \,{S}_{23}^\dag\, {S}_{01}^\dag,
\nonumber \\
{\cal X}_{02}(\theta) &=&{S}_{01} \,{\cal X}_{12}(-\theta)\,
{S}_{01}^\dag, \label{N28}
\\
{\cal X}_{13}(\theta) &=& {S}_{12}\, {\cal X}_{23}(-\theta) \,
{S}_{12}^\dag, \nonumber\end{aligned}$$ and analogously for the multi-quantum $Y$ rotations.
Finally, we point out some practical aspects in the described realization of a nanometer-scale device in a relation to the problem of degeneracy between $\omega_{03}/3$ and $\omega_{12}$. The rotation frequency is proportional to the first Bessel function of the oscillation field strength for the coherent rotation between levels ${\mbox{$|1\rangle$}}$ and ${\mbox{$|2\rangle$}}$, but proportional to the third Bessel function for that between ${\mbox{$|0\rangle$}}$ and ${\mbox{$|3\rangle$}}$. Therefore, the 0-3 rotation becomes negligible if the applied field is weak. Moreover, it is possible to select the oscillating field strength, which satisfies some angle rotation for 0-3, which differs from a multiple of 2$\pi$ rotation for 2-3. Therefore, it is possible to realize a pure 0-3 operation without 2-3 rotation. However, current amplitude necessary for this operation might be very high and the operation is not realistic from the view point of heating. Another QST method based on sequences of the two-photon pulses $X_{02}$, $X_{13}$, $Y_{02}$ and $Y_{13}$ would also be experimentally feasible. But usually rotations between the closest levels are much faster and easier to perform.
Here, we give a simple solution to omit rotations requiring three-photon transitions in, e.g., the rotations $R^{\rm
temp}_{\rm offdiag}$ is to express them as combinations of three one-photon and two-photon rotations as described above. However, we find that combinations of only two rotations are usually sufficient. Thus, we suggest the following three-photon rotations $Y_{03}\rightarrow Y_{01} S_{13}$ and $X_{03}\rightarrow X_{01}
S_{13}$. Note that the new operations do not require the rotation 0-3. We mention that the two-photon transitions can also be replaced by single-photon transitions with the help of the sequences of rotations given by Eq. (\[N28\]).
Conclusions
===========
We described various methods for implementing quantum state tomography for systems of quadrupolar nuclei with spin-3/2 (equivalent to quartit) in an unconventional approach to NMR, which is based on the measurement of longitudinal magnetization $M_z$ instead of the standard measurement of the transverse magnetization $M_{xy}$ [@SlichterBook].
This work has been motivated by the demonstration of high-precision $M_z$-based NMR techniques of coherent manipulation of nuclear spins $I=3/2$ ($^{69}$Ga, $^{71}$Ga, and $^{75}$As) in a GaAs quantum-well device based on an the fractional quantum Hall effect [@Yusa05]. The device, exhibiting extremely low decoherence [@decoherence], offers new possibilities to study interactions in semiconductors but also enables the realization of single- and two-qubit quantum gates [@Hirayama06] and, possibly, testing simple quantum-information processing algorithms.
Although our presentation of the protocols of QST of large-nuclear systems was focused on the nanoscale semiconductor device of Ref. [@Yusa05], it should be stressed that these protocols can also be readily applied to large-nuclear spins in other systems.
We proposed methods with optimized sets of rotations. The optimization was applied in order to improve the robustness against errors, as quantified by condition numbers.
Some of the proposed QST methods for a quartit system require the three-photon transitions (between levels $|0\rangle$ and $|3\rangle$), which are induced by relatively strong pulses. Unfortunately, such pulses can simultaneously induce transitions between levels $|1\rangle$ and $|2\rangle$. Thus, from a practical point of view, it is desired to apply only weak pulses selectively inducing single-photon transitions. We showed how the rotations requiring multiphoton transitions can be replaced by combinations of rotations based only on single-photon transitions.
By applying the condition number based on the spectral norm [@HighamBook], we compared robustness against errors in the measured data for all the described tomographic methods. We have assumed three observation approaches corresponding to: (i) an ideal $M_z$ detection, where all the diagonal elements $\rho_{nn}$ ($n=0,...,3$) of a density matrix can be directly accessed; (ii) an ideal experimental $M_z$ detection, where the population differences ($\rho_{11}-\rho_{00}$, $\rho_{22}-\rho_{11}$, and $\rho_{33}-\rho_{22}$) can be estimated from the amplitude of the signals by integrating the area of the peaks centered at $\omega_{01}$, $\omega_{12}$, and $\omega_{23}$ (see Fig. 2), respectively; and (iii) the non-ideal (“noisy”) experimental detection based on the CYCLOPS method, where the information gathered from the $M_z$-spectra corresponds to some linear functions of the diagonal elements $\rho_{nn}$, as given by Eq. (\[OA3\]).
For the QST methods for a quartit (i.e., two virtual qubits) using the experimental approaches (including the CYCLOPS method), the condition number $\kappa$ is either exactly equal to 1 or very close to 1. This means that the proposed methods are optimally robust against errors.
Let us now compare the error robustness of the discussed NMR QST methods with some known optical QST methods (see, e.g., Ref. [@Miran14] for a review) for two physically-distinct qubits: The well-known QST protocol of James [@James01], which is solely based on local measurements, yields the condition number $\kappa=60.1$. The QST of Refs. [@Altepeter05; @Burgh08] is based on the standard separable basis composed of all of the 36 two-qubit eigenstates of the tensor products of the Pauli operators. This often-applied QST yields $\kappa=9$. Another QST, which is based on local measurements of the 16 tensor products of the Pauli operators, yields $\kappa=2$. In contrast to these optical methods, only the recently-proposed QST of Ref. [@Miran14], which was also experimentally demonstrated [@Bartkiewicz15], is optimal since it yields the condition number $\kappa=1$. This tomography of two optical qubits is based on local and global measurements of generalized Pauli operators. It is worth noting that our optimal NMR tomography is based on a smaller set of measurements in comparison to that for the optimal optical tomography [@Miran14; @Bartkiewicz15].
We also described sequences of NMR pulses to perform various quantum tomography methods and arbitrary gates (including single virtual qubit rotations) with nuclear spins-3/2. This enables a simple translation of arbitrary quantum algorithms from systems of spins-1/2 to higher-number spins.
Finally, we express our hope that this comparative study of various NMR tomographic methods will draw attention to the issue of how such methods are robust against errors and, thus, to the question about the reliability of the reconstructed density matrices.
The authors are grateful to T. Ota, Z. Fojud, K. Hashimoto, N. Kumada, S. Miyashita, T. Saku, and K. Takashina for discussions. A.M. acknowledges a long-term fellowship from the Japan Society for the Promotion of Science (JSPS). A.M. was supported by the Polish National Science Centre under Grants No. DEC-2011/03/B/ST2/01903 and No. DEC-2011/02/A/ST2/00305. J.B. was supported by the Palacký University under Project No. IGA-PřF-2014-014. N.I. was supported by JSPS Grant-in-Aid for Scientific Research(A) (Grant No. No. 25247068). G.Y. was supported by a Grant-in-Aid for Scientific Research (Grant No. 24241039) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan and by the Mitsubishi Foundation. F.N. was partially supported by the RIKEN iTHES Project, the MURI Center for Dynamic Magneto-Optics via the AFOSR Grant No. FA9550-14-1-0040, the IMPACT program of JST, and a Grant-in-Aid for Scientific Research (A).
Selective rotations
===================
Selective rotations in quadrupolar nuclei with large spins are a simple generalization of the standard rotations in a spin-1/2 system: $$\begin{aligned}
{\cal X}(\theta )&\equiv& R^{x}(\theta )={{\left[\begin{array}{cc} \cos
\frac{\theta}{2} & -i \sin \frac{\theta}{2} \\ -i \sin
\frac{\theta}{2} & \cos \frac{\theta}{2} \end{array} \right]}} , \label{A1}
\\ {\cal Y}(\theta
)&\equiv& R^{y}(\theta )={{\left[\begin{array}{cc} \cos \frac{\theta}{2} & - \sin
\frac{\theta}{2} \\ \sin \frac{\theta}{2} & \cos \frac{\theta}{2}
\end{array} \right]}} , \label{A2} \\
{\cal Z}(\theta )&\equiv & R^{z}(\theta)={{\left[\begin{array}{cc} e^{-i \theta/2} &
0 \\ 0 & e^{ i\theta/2} \end{array} \right]}}. \label{A3}\end{aligned}$$ If a two-level rotation is $R^{(i)}(\theta)={{\left[\begin{array}{cc} a\;b \\ c\;d \end{array} \right]}}$ (with $i=X,Y,Z$), then the corresponding selective rotation between levels $m<n$ in a $N$-level system is given by $$\begin{aligned}
R^{(i)}_{mn}(\theta) &=&
a{\mbox{$|m\rangle$}}{\mbox{$\langlem|$}}+b{\mbox{$|m\rangle$}}{\mbox{$\langlen|$}} +c{\mbox{$|n\rangle$}}{\mbox{$\langlem|$}}\nonumber\\
&& +d{\mbox{$|n\rangle$}}{\mbox{$\langlen|$}} +\sum_{k\neq n,m}{\mbox{$|k\rangle$}}{\mbox{$\langlek|$}}.
\label{A4}\end{aligned}$$ For example, the matrix representation of the rotation ${\cal
X}_{02}({{\textstyle{\frac{\pi}{2}}}})$ in a spin-3/2 system reads as: $$\begin{aligned}
{\cal X}_{02}({{\textstyle{\frac{\pi}{2}}}}) = \frac{1}{\sqrt{2}}
\left(
\begin{array}{cccc}
1 & 0 &-i & 0 \\
0 & \sqrt{2} & 0 & 0 \\
-i & 0 & 1 & 0 \\
0 & 0 & 0 & \sqrt{2}
\end{array}
\right). \label{N35}\end{aligned}$$ Note that the rotations calculated by $\exp(-i{\cal H}_{\rm rot}
t_p/\hbar)$ are, in general, not exactly corresponding to Eq. (\[A4\]), because these depend on the quadrupolar frequency $\omega_Q$, even if the conditions $\hbar\omega_{_{\rm RF}}^{(k)}
=\epsilon_{m}-\epsilon_{n}$ and $|\omega_k| \ll |\omega_Q| \ll
|\omega_{0}|$ are satisfied [@ErnstBook]. Nevertheless, these rotations can be effectively reduced to Eq. (\[A4\]) if the pulse duration $t_p$ is equal to $2\pi/\omega_Q$ or its multiple. To fulfill this condition experimentally, the line intensities of spectra can be monitored as a function of the pulse duration (see, e.g., Ref. [@Bonk04]).
$M_{xy}$ detection vs $M_{z}$ detection
=======================================
The $M_{xy}$ detection of a spin-3/2 system provides directly the following off-diagonal elements (as marked in boxes) of the corresponding density matrix $\rho$: $$\begin{aligned}
\rho &=& {{\left[\begin{array}{cccc} \rho_{00} & \frame{$\rho_{01}$} & \rho_{02} & \rho_{03} \\ \frame{$\rho_{10}$} & \rho_{11} & \frame{$\rho_{12}$} & \rho_{13} \\ \rho_{20} & \frame{$\rho_{21}$} & \rho_{22} & \frame{$\rho_{23}$} \\ \rho_{30} & \rho_{31} & \frame{$\rho_{32}$} &
\rho_{33} \end{array} \right]}}.
\label{N04}\end{aligned}$$ This is because the NMR signals obtained by the $M_{xy}$ detection can be proportional to [@SlichterBook] $$\begin{aligned}
M^{\pm}_{xy} &\equiv & M_x \pm \I M_y \;\propto\; \tr[\rho I_{\pm}],
\label{N05}\end{aligned}$$ as given in terms of the total angular momentum operator $I_{\pm} =
I_x {\pm} i I_y$ for spin $I=3/2$, where $$I_x ={{\left[\begin{array}{cccc} 0 & a & 0 & 0 \\ a & 0 & 1 & 0 \\ 0 & 1 & 0 & a \\ 0 & 0 & a & 0 \end{array} \right]}},
\quad I_y =i {{\left[\begin{array}{cccc} 0 & -a & 0 & 0 \\ a & 0 & -1 & 0 \\ 0 & 1 & 0 & -a \\ 0 & 0 & a & 0 \end{array} \right]}},
\label{N06}$$ with $a=\sqrt{3}/2$. In contrast to this, the $M_{z}$ detection of a spin-3/2 system and two spin-1/2 systems enables the determination of only the diagonal elements $\rho_{ii}$ ($i=0,...,3$). This is because the NMR signals obtained by the $M_z$ detection of a spin-3/2 system are given by Eqs. (\[OA2\]), (\[OA2extra\]), or (\[OA3\]).
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
[Fang Duan$^{a, b}$, Qiongxiang Huang$^a$[^1], Xueyi Huang$^a$]{}\
$^{a}$College of Mathematics and Systems Science, Xinjiang University, Urumqi, Xinjiang 830046, P.R. China\
$^b$School of Mathematics Science, Xinjiang normal University, Urumqi, Xinjiang 830054, P.R.China
title: 'On graphs with exactly two positive eigenvalues[^2]'
---
The inertia of a graph $G$ is defined to be the triplet $In(G) = (p(G), n(G), $ $\eta(G))$, where $p(G)$, $n(G)$ and $\eta(G)$ are the numbers of positive, negative and zero eigenvalues (including multiplicities) of the adjacency matrix $A(G)$, respectively. Traditionally $p(G)$ (resp. $n(G)$) is called the positive (resp. negative) inertia index of $G$. In this paper, we introduce three types of congruent transformations for graphs that keep the positive inertia index and negative inertia index. By using these congruent transformations, we determine all graphs with exactly two positive eigenvalues and one zero eigenvalue.
**Keywords:** congruent transformation; positive (negative) inertia index; nullity.
**AMS Classification:**
Introduction
============
All graphs considered here are undirected and simple. For a graph $G$, let $V(G)$ and $E(G)$ denote the vertex set and edge set of $G$, respectively. The order of $G$ is the number of vertices of $G$, denoted by $|G|$. For $v\in V(G)$, we denote by $N_G(v)=\{u\in V(G)\mid uv \in E(G)\}$ the *neighborhood* of $v$, $N_G[v]=N_G(v)\cup\{v\}$ the *closed neighborhood* of $v$ and $d(v)=|N_G(v)|$ the degree of $v$. A vertex of $G$ is said to be *pendant* if it has degree $1$. By $\delta(G)$ we mean the minimum degree of vertices of $G$. As usual, we denote by $G+H$ the disjoint union of two graphs $G$ and $H$, $K_{n_1, \ldots, n_l}$ the complete multipartite graph with $l$ parts of sizes $n_1, \ldots, n_l$, and $K_n$, $C_n$, $P_n$ the complete graph, cycle, path on $n$ vertices, respectively.
The *adjacency matrix* of $G$, denoted by $A(G)=(a_{ij})$, is the square matrix with $a_{ij}=1$ if $v_i$ and $v_j$ are adjacent, and $a_{ij}=0$ otherwise. Clearly, $A(G)$ is a symmetric matrix with zeros on the diagonal, and thus all the eigenvalues of $A(G)$ are real, which are defined to be the *eigenvalues* of $G$. The multiset consisting of eigenvalues along with their multiplicities is called the *spectrum* of $G$ denoted by $Spec(G)$. To characterize graphs in terms of their eigenvalues has always been of the great interests for researchers, for instance to see [@C.Godsil], [@M.R.Oboudi1], [@M.R.Oboudi2], [@M.Petrovi'c], [@J.H.Smith] and references therein.
The *inertia* of a graph $G$ is defined as the triplet $In(G) = (p(G), n(G), \eta(G))$, where $p(G)$, $n(G)$ and $ \eta(G)$ are the numbers of positive, negative and zero eigenvalues (including multiplicities) of $G$, respectively. Traditionally $p(G)$ (resp. $n(G)$) is called the *positive* (resp. *negative*) *inertia index* of $G$ and $\eta(G)$ is called the *nullity* of $G$. Obviously, $p(G)+n(G)=r(G)=n-\eta(G)$ if $G$ has $n$ vertices, where $r(G)$ is the rank of $A(G)$. Let $B$ and $D$ be two real symmetric matrices of order $n$. Then $D$ is called *congruent* to $B$ if there is an real invertible matrix $C$ such that $D=C^TBC$. Traditionally we say that $D$ is obtained from $B$ by congruent transformation. The famous Sylvester’s law of inertia states that the inertia of two matrices is unchanged by congruent transformation.
Since the adjacency matrix $A(G)$ of $G$ has zero diagonal, we have $p(G)\geq 1$ if $G$ has at least one edge. One of the attractive problems is to characterize those graphs with a few positive eigenvalues. In [@J.H.Smith] Smith characterized all graphs with exactly one positive eigenvalue. Recently, Oboudi [@M.R.Oboudi3] completely determined the graphs with exactly two non-negative eigenvalues, i.e., those graphs satisfying $p(G)=1$ and $\eta(G)=1$ or $p(G)=2$ and $\eta(G)=0$.
In this paper, we introduce three types congruent transformations for graphs. By using these congruent transformations and Oboudi’s results in [@M.R.Oboudi3], we completely characterize the graphs satisfying $p(G)=2$ and $\eta(G)=1$.
Preliminaries
=============
In this section, we will introduce some notions and lemmas for the latter use.
\[thm-2-0\] (Interlacing theorem) Let $G$ be a graph of order $n$ and $H$ be an induced subgraph of $G$ with order $m$. Suppose that $\lambda_1(G)\geq
\ldots \geq \lambda_n(G)$ and $\lambda_1(H)\geq \ldots \geq \lambda_m(H)$ are the eigenvalues of $G$ and $H$, respectively. Then for every $1\leq
i\leq m$, $\lambda_i(G)\geq \lambda_i(H)\geq \lambda_{n-m+i}(G)$.
\[lem-2-2\] Let $H$ be an induced subgraph of graph $G$. Then $p(H)\leq p(G)$.
\[lem-2-1\] Let $G$ be a graph containing a pendant vertex, and let $H$ be the induced subgraph of $G$ obtained by deleting the pendant vertex together with the vertex adjacent to it. Then $p(G)=p(H)+1$, $n(G)=n(H)+1$ and $\eta(G)=\eta(H)$.
\[lem-2-3\] (Sylvester’s law of inertia) If two real symmetric matrices $A$ and $B$ are congruent, then they have the same positive (resp., negative) inertia index, the same nullity.
\[thm-3-3\] A graph has exactly one positive eigenvalue if and only if its non-isolated vertices form a complete multipartite graph.
Let $G_1$ be a graph containing a vertex $u$ and $G_2$ be a graph of order $n$ that is disjoint from $G_1$. For $1\leq k \leq n$, the *$k$-joining* graph of $G_1$ and $G_2$ with respect to $u$, denoted by $G_1(u)\odot^k G_2$, is a graph obtained from $G_1\cup G_2$ by joining $u$ to arbitrary $k$ vertices of $G_2$. By using the notion of $k$-joining graph, Yu et al. [@G.H.Yu1] completely determined the connected graphs with at least one pendant vertex that have positive inertia index $2$.
\[thm-2-1\] Let $G$ be a connected graph with pendant vertices. Then $p(G)=2$ if and only if $G\cong K_{1, r}(u)\odot^k K_{n_1, \ldots, n_l}$, where $u$ is the center of $K_{1, r}$ and $1\leq k\leq n_1+\cdots +n_l$.
\[thm-3-1\] Let $G$ be a graph of order $n\geq 2$ with eigenvalues $\lambda_1(G)\geq \ldots\geq \lambda_n(G)$. Assume that $\lambda_3(G)<0$, then the following hold:\
(1) If $\lambda_1(G)>0$ and $\lambda_2(G)=0$, then $G\cong K_1 + K_{n-1}$ or $G\cong K_{n} \setminus e$ for $e\in E(K_{n})$;\
(2) If $\lambda_1(G)>0$ and $\lambda_2(G)<0$, then $G\cong K_{n}$.
Let $\mathcal{H}$ be set of all graphs satisfying $\lambda_2(G)>0$ and $\lambda_3(G)<0$ (in other words, $p(G)=2$ and $\eta(G)=0$). Oboudi [@M.R.Oboudi3] determined all the graphs of $\mathcal{H}$. To give a clear description of this characterization, we introduce the class of graphs $G_n$ defined in [@M.R.Oboudi3].
For every integer $n\geq 2$, let $K_{\lceil\frac{n}{2}\rceil}$ and $K_{\lfloor\frac{n}{2}\rfloor}$ be two disjoint complete graphs with vertex set $V=\{v_1, \ldots , v_{\lceil\frac{n}{2}\rceil}\}$ and $W=\{w_1 ,\ldots,w_{\lfloor\frac{n}{2}\rfloor} \}$. $G_n$ is defined to be the graph obtained from $K_{\lceil\frac{n}{2}\rceil}$ and $K_{\lfloor\frac{n}{2}\rfloor}$ by adding some edges distinguishing whether $n$ is even or not below:
\(1) If $n$ is even, then add some new edges to $K_{\frac{n}{2}}+K_{\frac{n}{2}}$ satisfying $$\begin{array}{ll}
\emptyset=N_W(v_1)\subset N_W(v_2)=\{w_{\frac{n}{2}}\}\subset N_W(v_3)=\{w_{\frac{n}{2}}, w_{\frac{n}{2}-1}\}\subset\cdots\subset N_W(v_{\frac{n}{2}-1})\\=\{w_{\frac{n}{2}}, \ldots, w_{3} \}\subset N_W(v_{\frac{n}{2}})=\{w_{\frac{n}{2}}, \ldots, w_{2} \}
\end{array}$$
\(2) If $n$ is odd, then add some new edges to $K_{\frac{n+1}{2}}+K_{\frac{n-1}{2}}$ satisfying $$\begin{array}{ll}
\emptyset=N_W(v_1)\subset N_W(v_2)=\{w_{\frac{n-1}{2}}\}\subset N_W(v_3)=\{w_{\frac{n-1}{2}}, w_{\frac{n-1}{2}-1}\}\subset\cdots\subset N_W(v_{\frac{n+1}{2}-1})\\=\{w_{\frac{n-1}{2}}, \ldots, w_{2} \}\subset N_W(v_{\frac{n+1}{2}})=\{w_{\frac{n-1}{2}}, \ldots, w_{1} \}
\end{array}$$ By deleting the maximum (resp. minimum) degree vertex from $G_{n+1}$ if $n$ is an even (resp. odd), we obtain $G_n$. It follows the result below.
\[re-0\] $G_n$ is an induced subgraph of $G_{n+1}$ for every $n\geq 2$.
For example, $G_2\cong 2K_1$, $G_3\cong P_3$ and $G_4\cong P_4$. The graphs $G_5$ and $G_6$ are shown in Fig. \[fig-0\]. In general, $G_t$ and $G_{t+1}$ are also shown in Fig. \[fig-0\] for an even number $t$.
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(80.7,20.859)(0,2) (28.225,15.053) (28.314,7.275) (28.374,14.937)[(0,-1)[7.69]{}]{} (25.22,18.589) (25.104,18.737)(.033619565,-.038423913)[92]{}[(0,-1)[.038423913]{}]{} (21.772,14.965) (21.684,7.363) (25.103,18.737)(-.033466019,-.034320388)[103]{}[(0,-1)[.034320388]{}]{} (21.568,15.025)[(0,-1)[7.601]{}]{} (21.568,7.424)[(1,0)[6.718]{}]{} (21.656,15.025)(.033649746,-.039035533)[197]{}[(0,-1)[.039035533]{}]{} (21.656,15.025)[(1,0)[6.629]{}]{} (25.131,3.74) (21.567,7.336)[(1,-1)[3.536]{}]{} (28.462,7.336)(-.033721649,-.036453608)[97]{}[(0,-1)[.036453608]{}]{} (25.103,5.215)[(0,0)\[cc\][$w_1$]{}]{} (20.507,5.922)[(0,0)\[cc\][$w_2$]{}]{} (28.727,5.657)[(0,0)\[cc\][$w_3$]{}]{} (25.279,.088)[(0,0)\[cc\][$G_6$]{}]{} (24.838,16.793)[(0,0)\[cc\][$v_1$]{}]{} (24.97,16.749)[(12.286,5.392)\[\]]{} (24.97,5.701)[(12.286,5.392)\[\]]{} (9.31,15.054) (9.398,7.276) (9.457,14.938)[(0,-1)[7.69]{}]{} (6.303,18.59) (6.187,18.738)(.033619565,-.038423913)[92]{}[(0,-1)[.038423913]{}]{} (2.856,14.966) (2.767,7.364) (6.186,18.738)(-.033466019,-.034320388)[103]{}[(0,-1)[.034320388]{}]{} (2.652,15.026)[(0,-1)[7.601]{}]{} (2.652,7.425)[(1,0)[6.718]{}]{} (2.739,15.026)(.033654822,-.039030457)[197]{}[(0,-1)[.039030457]{}]{} (2.739,15.026)[(1,0)[6.629]{}]{} (6.143,16.661)[(12.286,5.392)\[\]]{} (6.231,7.336)[(12.109,3.182)\[\]]{} (6.276,.442)[(0,0)\[cc\][$G_5$]{}]{} (6.099,17.059)[(0,0)\[cc\][$v_1$]{}]{} (47.774,18.164)[(15.645,5.215)\[\]]{} (47.951,5.083)[(15.645,5.215)\[\]]{} (42.013,18.236) (52.089,18.413) (44.311,18.324) (42.013,4.801) (47.943,18.208) (45.645,4.773) (48.51,4.721) (50.233,4.801) (52.061,18.562)(-.03372807,-.060092105)[228]{}[(0,-1)[.060092105]{}]{} (46.344,18.324) (49.181,18.156) (52.089,4.801) (50.33,18.156) (47.06,4.721) (52.061,18.473)[(0,-1)[13.612]{}]{} (52.061,4.95)(-.033672619,.078916667)[168]{}[(0,1)[.078916667]{}]{} (50.205,4.861)(-.033711864,.113855932)[118]{}[(0,1)[.113855932]{}]{} (44.194,18.385)(.033619658,-.05817094)[234]{}[(0,-1)[.05817094]{}]{} (51.973,18.473)(-.03314286,-.23991071)[56]{}[(0,-1)[.23991071]{}]{} (41.72,3.624)[(0,0)\[cc\][$w_1$]{}]{} (44.371,3.624)[(0,0)\[cc\][$w_2$]{}]{} (44.283,19.357)[(0,0)\[cc\][$v_2$]{}]{} (41.808,19.269)[(0,0)\[cc\][$v_1$]{}]{} (44.577,4.801) (37.831,17.236)[(0,0)\[cc\][$K_{\frac{t}{2}}$]{}]{} (38.096,4.508)[(0,0)\[cc\][$K_{\frac{t}{2}}$]{}]{} (47.819,0)[(0,0)\[cc\][$G_t$]{}]{} (73.097,0)[(0,0)\[cc\][$G_{t+1}$]{}]{} (72.7,18.251)[(15.645,5.215)\[\]]{} (72.877,5.171)[(15.645,5.215)\[\]]{} (66.939,18.324) (77.015,18.501) (69.237,18.409) (66.939,4.889) (72.869,18.296) (70.571,4.86) (73.436,4.808) (75.159,4.889) (76.987,18.65)(-.03372807,-.060087719)[228]{}[(0,-1)[.060087719]{}]{} (71.27,18.409) (74.107,18.244) (77.015,4.889) (75.256,18.244) (71.986,4.808) (76.987,18.561)[(0,-1)[13.612]{}]{} (76.987,5.037)(-.033672619,.078922619)[168]{}[(0,1)[.078922619]{}]{} (75.131,4.95)(-.033711864,.113864407)[118]{}[(0,1)[.113864407]{}]{} (69.12,18.473)(.033619658,-.058175214)[234]{}[(0,-1)[.058175214]{}]{} (76.899,18.561)(-.03314286,-.23989286)[56]{}[(0,-1)[.23989286]{}]{} (66.646,3.712)[(0,0)\[cc\][$w_1$]{}]{} (69.297,3.712)[(0,0)\[cc\][$w_2$]{}]{} (69.209,19.443)[(0,0)\[cc\][$v_2$]{}]{} (66.734,19.357)[(0,0)\[cc\][$v_1$]{}]{} (69.502,4.889) (76.898,18.65)(-.0336296296,-.0461279461)[297]{}[(0,-1)[.0461279461]{}]{} (62.933,17.501)[(0,0)\[cc\][$K_{\frac{t+1}{2}}$]{}]{} (63.11,4.42)[(0,0)\[cc\][$K_{\frac{t-1}{2}}$]{}]{} (2.033,16.086)[(0,0)\[cc\][$v_3$]{}]{} (10.077,16.175)[(0,0)\[cc\][$v_2$]{}]{} (1.68,6.364)[(0,0)\[cc\][$w_1$]{}]{} (10.342,6.275)[(0,0)\[cc\][$w_2$]{}]{} (20.683,15.998)[(0,0)\[cc\][$v_3$]{}]{} (28.815,16.175)[(0,0)\[cc\][$v_2$]{}]{} (53.829,18.119)[(0,0)\[cc\][$v_{\frac{t}{2}}$]{}]{} (53.741,3.889)[(0,0)\[cc\][$w_{\frac{t}{2}}$]{}]{} (78.931,18.384)[(0,0)\[cc\][$v_{\frac{t+1}{2}}$]{}]{} (79.02,4.684)[(0,0)\[cc\][$w_{\frac{t-1}{2}}$]{}]{}
Let $G$ be a graph with vertex set $\{v_1, \ldots, v_n\}$. By $G[K_{t_1}, \ldots, K_{t_n}]$ we mean the *generalized lexicographic product* of $G$ (by $K_{t_1}$, $K_{t_2}$,…,$K_{t_n}$), which is the graph obtained from $G$ by replacing the vertex $v_j$ with $K_{t_j}$ and connecting each vertex of $K_{t_i}$ to each vertex of $K_{t_j}$ if $v_i$ is adjacent to $v_j$ in $G$.
\[thm-2-4\] Let $G\in \mathcal{H}$ of order $n\geq 4$ with eigenvalues $\lambda_1(G)\geq \cdots \geq \lambda_n(G)$.\
(1) If $G$ is disconnected, then $G\cong K_p + K_q$ for some integers $p, q\geq 2$;\
(2) If $G$ is connected, there exist some positive integers $s$ and $t_1, \ldots, t_s$ such that $G\cong G_s[K_{t_1}, \ldots, K_{t_s}]$ where $3\leq s\leq 12$ and $t_1 + \cdots + t_s=n$.
Furthermore, Oboudi gave all the positive integers $t_1, \ldots, t_s$ such that $G_s[K_{t_1},\ldots, K_{t_s}]\in \mathcal{H}$ in Theorems 3.4–3.14 of [@M.R.Oboudi3].
Let $\mathcal{G}$ be the set of all graphs with positive inertia index $p(G)=2$ and nullity $\eta(G)=1$. In next section, we introduce some new congruent transformations for graph that keep to the positive inertia index. By using such congruent transformations we characterize those graphs in $\mathcal{G}$ based on $\mathcal{H}$.
Three congruent transformations of graphs
=========================================
In this section, we introduce three types of congruent transformations for graphs.
\[lem-2-4\] Let $u, v$ be two non-adjacent vertices of a graph $G$. If $u$ and $v$ have the same neighborhood, then $p(G)=p(G-u)$, $n(G)=n(G-u)$ and $\eta(G)=\eta(G-u)+1$.
\[re-1\] *Two non-adjacent vertices $u$ and $v$ are said to be *congruent vertices of I-type* if they have the same neighbors. Lemma \[lem-2-4\] implies that if one of congruent vertices of I-type is deleted from a graph then the positive and negative inertia indices left unchanged, but the nullity reduces just one. Conversely, if we add a new vertex that joins all the neighbors of some vertex in a graph (briefly we refer to add a vertex of I-type in what follows) then the positive and negative inertia indices left unchanged, but the nullity adds just one. The graph transformation of deleting or adding vertices of I-type is called the (graph) *transformation of I-type*.*
Since $Spec(K_s)=[(s-1)^1,(-1)^{s-1}]$. By applying the transformation of I-type, we can simply find the inertia of $K_{n_1,n_2,\ldots,n_s}$.
\[inertia-cor-1\] Let $G=K_{n_1,n_2,\ldots,n_s}$ be a multi-complete graph where $n_1\ge n_2\ge\cdots\ge n_s$ and $i_0=\min\{1\le i\le s\mid n_i\ge 2\}$. Then $G$ has the inertia index: $In(G)=(p(G),\eta(G),n(G))=(1, n_{i_0}+n_{i_0+1}+\cdots+n_s-s+i_0-1, s-1)$.
The following transformation was mentioned in [@M.R.Oboudi1], but the author didn’t prove the result. For the completeness we give a proof below.
\[lem-2-5\] Let $\{u, v, w\}$ be an independent set of a graph $G$. If $N(u)$ is a disjoint union of $N(v)$ and $N(w)$, then $p(G)=p(G-u)$, $n(G)=n(G-u)$ and $\eta(G)=\eta(G-u)+1$.
Since $u, v, w$ are not adjacent to each other, we may assume that $(0, 0, 0, \alpha^T)$, $(0, 0, 0, \beta^T)$ and $(0, 0, 0, \gamma^T)$ are the row vectors of $A(G)$ corresponding to the vertices $u, v, w$, respectively. Thus $A(G)$ can be written as $$A(G)=\left(
\begin{array}{cccc}
0 & 0 & 0 & \alpha^T \\
0 & 0 & 0 & \beta^T \\
0 & 0 & 0 & \gamma^T \\
\alpha & \beta & \gamma & A(G-u-v-w) \\
\end{array}
\right)$$ Since $N(u)=N(v)\cup N(w)$ and $N(v)\cap N(w)=\emptyset$, we have $\alpha=\beta+\gamma$. By letting the $u$-th row (resp. $u$-th column) minus the sum of the $v$-th and $w$-th rows (resp. the sum of the $v$-th and $w$-th columns) of $A(G)$, we get that $A(G)$ is congruent to $$\left(
\begin{array}{cccc}
0 & 0 & 0 & \textbf{0}^T \\
0 & 0 & 0 & \beta^T \\
0 & 0 & 0 & \gamma^T \\
\textbf{0} & \beta & \gamma & A(G-u-v-w) \\
\end{array}
\right)=
\left( \begin{array}{cc}
0 & \textbf{0}^T\\
\textbf{0} & A(G-u) \\
\end{array}
\right)
.$$ Thus $p(G)=p(G-u)$, $n(G)=n(G-u)$ and $\eta(G)=\eta(G-u)+1$ by Lemma \[lem-2-3\].
\[re-2\] *The vertex $u$ is said to be a *congruent vertex of II-type* if there exist two non-adjacent vertices $v$ and $w$ such that $N(u)$ is a disjoint union of $N(v)$ and $N(w)$. Lemma \[lem-2-5\] implies that if one congruent vertex of II-type is deleted from a graph then the positive and negative indices left unchanged, but the nullity reduces just one. Conversely, if there exist two non-adjacent vertices $v$ and $w$ such that $N(v)$ and $N(w)$ are disjoint, we can add a new vertex $u$ that joins all the vertices in $N(v)\cup N(w)$ (briefly we refer to add a vertex of II-type in what follows), then the positive and negative inertia indices left unchanged, but the nullity adds just one. The graph transformation of deleting or adding vertices of II-type is called the (graph) *transformation of II-type*.*
An induced quadrangle $C_4=uvxy$ of $G$ is called *congruent* if there exists a pair of independent edges, say $uv$ and $xy$ in $C_4$, such that $N(u)\backslash \{v, y\}=N(v)\backslash \{u, x\}$ and $N(x)\backslash \{y, v\}=N(y)\backslash \{x, u\}$, where $uv$ and $xy$ are called a pair of *congruent edges* of $C_4$. We call the vertices in a congruent quadrangle the *congruent vertices of III-type*.
\[lem-2-6\] Let $u$ be a congruent vertex of III-type in a graph $G$. Then $p(G)=p(G-u)$, $n(G)=n(G-u)$ and $\eta(G)=\eta(G-u)+1$.
Let $C_4=uvxy$ be the congruent quadrangle of $G$ containing the congruent vertex $u$. Then $(0, 1, 0, 1, \alpha^T)$, $(1, 0, 1, 0, \alpha^T)$, $(0, 1, 0, 1, \beta^T)$, $(1, 0, 1, 0, \beta^T)$ are the row vectors of $A(G)$ corresponding to the vertices $u$, $v$, $x$ and $y$, respectively. Thus $A(G)$ can be presented by $$A(G)=\left(
\begin{array}{ccccc}
0 & 1 & 0& 1 & \alpha^T \\
1 & 0 & 1& 0 & \alpha^T \\
0 & 1 & 0 & 1 & \beta^T \\
1 &0 & 1 & 0 & \beta^T \\
\alpha & \alpha & \beta & \beta & A(G-u-v-x-y) \\
\end{array}
\right)$$
By letting the $u$-th row (resp. $u$-th column) minus the $x$-th row (resp. $x$-th column) of $A(G)$, and letting the $v$-th row (resp. $v$-th column) minus the $y$-th row (resp. $y$-th column) of $A(G)$, we obtain that $A(G)$ is congruent to $$B=\left(
\begin{array}{ccccc}
0 & 0 & 0 & 0 & \alpha^T-\beta^T\\
0 & 0 & 0 & 0 & \alpha^T-\beta^T \\
0 & 0 & 0 & 1 & \beta^T \\
0 & 0 & 1 & 0 & \beta^T \\
\alpha -\beta & \alpha -\beta & \beta & \beta & A(G-u-v-x-y) \\
\end{array}
\right).$$
Again, by letting the $u$-th row (resp. $u$-th column) minus the $v$-th row (resp. $v$-th column) of $B$, and adding the $y$-th row (resp. $y$-th column) to the $v$-th row (resp. $v$-th column) of $B$, we obtain that $B$ is congruent to $$\left(
\begin{array}{ccccc}
0 & 0 & 0 & 0 & \textbf{0}^T\\
0 & 0 & 1 & 0 & \alpha^T \\
0 & 1 & 0 & 1 & \beta^T \\
0 & 0 & 1 & 0 & \beta^T \\
\textbf{0} & \alpha & \beta & \beta & A(G-u-v-x-y) \\
\end{array}
\right)=\left(
\begin{array}{cc}
0 & \textbf{0}^T\\
\textbf{0} & A(G-u) \\
\end{array}
\right).$$ Thus $p(G)=p(G-u)$, $n(G)=n(G-u)$ and $\eta(G)=\eta(G-u)+1$ by Lemma \[lem-2-3\].
\[re-3\] *The Lemma \[lem-2-6\] confirms that if a congruent vertex of III-type is deleted from a graph then the positive and negative inertia indices left unchanged, but the nullity reduces just one. Conversely, if we add a new vertex to a graph that consists of a congruent quadrangle with some other three vertices in this graph (briefly we refer to add a vertex of III-type in what follows) then the positive and negative inertia indices left unchanged, but the nullity adds just one. The graph transformation of deleting or adding vertices of III-type is called the (graph) *transformation of III-type*.*
Remark \[re-1\], Remark \[re-2\] and Remark \[re-3\] provide us three transformations of graphs that keep the positive and negative inertia indices and change the nullity just one. By applying these transformations we will construct the graphs in $\mathcal{G}$. Let $\mathcal{G}_1$ be the set of connected graphs each of them is obtained from some $H\in \mathcal{H}$ by adding one vertex of I-type, $\mathcal{G}_2$ be the set of connected graphs each of them is obtained from some $H\in \mathcal{H}$ by adding one vertex of II-type and $\mathcal{G}_3$ be the set of connected graphs each of them is obtained from some $H\in \mathcal{H}$ by adding one vertex of III-type. At the end of this section, we would like to give an example to illustrate the constructions of the graphs in $\mathcal{G}_i$ ($i=1,2,3$).
*We know the path $P_4$, with spectrum $Sepc(P_4)=\{1.6180, 0.6180, -0.6180, $ $-1.6180\}$, is a graph belonging to $\mathcal{H}$. By adding a vertex $u$ of I-type to $P_4$ we obtain $H_1\in \mathcal{G}_1$ (see Fig. \[fig-1\]) where $Spec(H_1)=\{1.8478, 0.7654, 0, -0.7654, -1.8478\}$, adding a vertex $u$ of II-type to $P_4$ we obtain $H_2\in \mathcal{G}_2$ where $Spec(H_2)=\{2.3028, 0.6180, 0, -1.3028, $ $-1.6180\}$. Finally, by adding a vertex $u$ of III-type to $P_4$ we obtain $H_3\in \mathcal{G}_3$, where $Spec(H_3)=\{2.4812, 0.6889, 0, -1.1701, -2\}$. In fact, $uv$ and $xy$ is a pair of independent edges in $H_3$. Clearly, $N(u)\backslash \{v, y\}=N(v)\backslash \{u, x\}=\{w\}$ and $N(x)\backslash \{y, v\}=N(y)\backslash \{x, u\}=\emptyset$. Thus $C_4=uvxy$ is a congruent quadrangle of $H_3$.*
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(92.077,16.104)(0,0) (81.261,15.568) (90.195,15.568) (81.261,4.636) (90.195,4.636) (81.145,15.662)[(1,0)[9.145]{}]{} (81.145,4.731)[(1,0)[9.04]{}]{} (90.185,15.662)[(0,-1)[10.932]{}]{} (85.571,10.522) (85.666,10.511)(-.033731343,-.044708955)[134]{}[(0,-1)[.044708955]{}]{} (81.251,15.452)(.03370229,-.037717557)[131]{}[(0,-1)[.037717557]{}]{} (85.666,10.511)(.033507246,-.042652174)[138]{}[(0,-1)[.042652174]{}]{} (79.254,15.557)[(0,0)\[cc\][$y$]{}]{} (92.077,15.557)[(0,0)\[cc\][$x$]{}]{} (87.137,10.722)[(0,0)\[cc\][$u$]{}]{} (91.762,4.625)[(0,0)\[cc\][$v$]{}]{} (86.191,.105)[(0,0)\[cc\][$H_3$]{}]{} (55.719,15.568) (64.653,15.568) (55.719,4.636) (64.653,4.636) (55.603,15.662)[(1,0)[9.145]{}]{} (55.603,4.731)[(1,0)[9.04]{}]{} (64.643,15.662)[(0,-1)[10.932]{}]{} (53.932,10.102) (53.816,10.196)(.068754717,.033716981)[159]{}[(1,0)[.068754717]{}]{} (53.711,10.302)(.064686391,-.033585799)[169]{}[(1,0)[.064686391]{}]{} (53.607,15.872)[(0,0)\[cc\][$v$]{}]{} (52.135,10.301)[(0,0)\[cc\][$u$]{}]{} (53.712,4.1)[(0,0)\[cc\][$w$]{}]{} (59.703,.105)[(0,0)\[cc\][$H_2$]{}]{} (.536,15.462) (9.47,15.462) (.536,4.53) (9.47,4.53) (.42,15.557)[(1,0)[9.145]{}]{} (.42,4.625)[(1,0)[9.04]{}]{} (9.46,15.557)[(0,-1)[10.932]{}]{} (5.045,.21)[(0,0)\[cc\][$P_4$]{}]{} (26.918,15.463) (35.852,15.463) (26.918,4.531) (35.852,4.531) (26.802,15.557)[(1,0)[9.145]{}]{} (26.802,4.626)[(1,0)[9.04]{}]{} (35.842,15.557)[(0,-1)[10.932]{}]{} (26.918,11.468) (26.908,11.352)(.074090164,.033606557)[122]{}[(1,0)[.074090164]{}]{} (25.121,10.722)[(0,0)\[cc\][$u$]{}]{} (31.533,0)[(0,0)\[cc\][$H_1$]{}]{} (79.254,4.52)[(0,0)\[cc\][$w$]{}]{}
Clearly, $G=K_{1,2}\cup P_2$ is a non-connected graph in $\mathcal{G}$, and all such graphs we collect in $\mathcal{G}^-=\{G\in \mathcal{G}\mid\mbox{$G$ is disconnected}\}$. Additionally, $H_1$ and $H_2$ shown in Fig. \[fig-1\] are graphs with pendant vertex belonging to $\mathcal{G}$, and all such graphs we collect in $\mathcal{G}^+=\{G\in \mathcal{G}\mid\mbox{$G$ is connected with a pendant vertex} \}$. In next section, we firstly determine the graphs in $\mathcal{G}^-$ and $\mathcal{G}^+$.
The characterization of graphs in $\mathcal{G}^-$ and $\mathcal{G}^+$
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The following result completely characterizes the disconnected graphs of $\mathcal{G}$.
\[thm-3-2\] Let $G$ be a graph of order $n\geq 5$. Then $G\in \mathcal{G}^-$ if and only if $G\cong K_s+K_t+K_1,
H+K_1$ or $K_s+K_{n-s}\setminus e$ for $e\in E(K_{n-s})$, where $H\in \mathcal{H}$ is connected and $s+t=n-1$, $s,t\ge 2$.
All the graphs displayed in Theorem \[thm-3-2\] have two positive and one zero eigenvalues by simple observation. Now we prove the necessity.
Let $G\in \mathcal{G}^-$, and $H_1, H_2,\ldots, H_k$ ($k\geq 2$) the components of $G$. Since $\lambda_1(H_i)\geq 0$ for $i=1,2,\ldots,k$ and $\lambda_4(G)<0$, $G$ has two or three components and so $k\le 3$.
First assume that $G=H_1+H_2+H_3$. It is easy to see that $G$ has exactly one isolated vertex due to $\eta(G)=1$ and $p(G)=2$. Without loss of generality, let $H_3\cong K_1$. Since $\lambda_3(G)=0$ and $\lambda_1(H_i)>0$ ($i=1,2$), we have $\lambda_2(H_1)<0$ and $\lambda_2(H_2)<0$. By Theorem \[thm-3-1\](2), $G\cong K_s+K_t+K_1$ as desired, where $s+t=n-1$ and $s, t\geq 2$.
Next assume that $G=H_1+H_2$. If $H_1\cong K_1$, then $$\lambda_1(G)=\lambda_1(H_2)\ge \lambda_2(G)=\lambda_2(H_2)>\lambda_3(G)=0=\lambda_1(H_1)>\lambda_4(G)=\lambda_3(H_2)<0.$$ Thus $H_2\cong H\in \mathcal{H}$, and so $G\cong
H+K_1$ as desired. If $|H_i|\geq 2$ for $i=1, 2$, then one of $\lambda_2(H_1)$ and $\lambda_2(H_2)$ is equal to zero and another is less than zero because $\lambda_3(G)=0$ and $\lambda_4(G)<0$. Without loss of generality, let $\lambda_2(H_1)<0$ and $\lambda_2(H_2)=0$. We have $\lambda_3(H_1)\leq \lambda_2(H_1)<0$, in addition, $\lambda_3(H_2)<0$ since $\eta(G)=1$. By Theorem \[thm-3-1\](2), $H_1\cong K_s$ for some $s\ge 2$ and by Theorem \[thm-3-1\](1), $H_2\cong K_{n-s}\setminus e$.
We complete this proof.
In terms of Theorem \[thm-2-1\], we will determine all connected graphs with a pendant vertex satisfying $p(G)=2$ and $\eta(G)=d$ for any positive integer $d$.
\[cor-3-0\] Let $G$ be a connected graph of order $n$ with a pendant vertex. Then $p(G)=2$ and $\eta(G)=d\ge1$ if and only if $G\cong K_{1, r}(u)\odot^k K_{n_1, \ldots, n_l}$, where $r+n_1+n_2+\cdots+n_l-(l+1)=d$.
Let $G=K_{1, r}(u)\odot^k K_{n_1, \ldots, n_l}$ and $vu$ is a pendant edge of $G$. By deleting $v$ and $u$ from $G$ we obtain $H=G-\{u,v\}=(r-1)K_1\cup K_{n_1, \ldots, n_l}$. It is well known that $p(K_{n_1, \ldots, n_l})=1$ and $\eta(K_{n_1, \ldots, n_l})=n_1+\cdots+n_l-l$. From Lemma \[lem-2-1\], we have $$p(G)=p(H)+1=p(K_{n_1, \ldots, n_l})+1=2 \mbox{ and } \eta(G)=\eta(H)=(r-1)+ (n_1+\cdots+n_l-l)=d.$$ Conversely, let $G$ be a graph with a pendant vertex and $p(G)=2$. By Theorem \[thm-2-1\], we have $G\cong K_{1, r}(u)\odot^k K_{n_1, \ldots, n_l}$. According to the arguments above, we know that $\eta(G)=r+n_1+n_2+\cdots+n_l-(l+1)=d$.
From Theorem \[cor-3-0\], it immediately follows the result that completely characterizes the graphs in $\mathcal{G}^+$.
\[cor-3-1\] A connected graph $G\in \mathcal{G}^+$ if and only if $G\cong K_{1, 2}(u)\odot^k K_{n-3}$ or $G\cong K_{1, 1}(u)\odot^k K_{n-2}\setminus e$ for $e\in E(K_{n-2})$.
By Theorem \[cor-3-0\], we have $G\in \mathcal{G}^+$ if and only if $G\cong K_{1, r}(u)\odot^k K_{n_1, \ldots, n_l}$, where $r+n_1+n_2+\cdots+n_l-(l+1)=1$ and $r,l, n_1,\ldots,n_l\ge1$. It gives two solutions: one is $r=2$, $n_1=n_2=\cdots=n_{l}=1$ and $l=n-3$ which leads to $G\cong K_{1, 2}(u)\odot^k K_{n-3}$; another is $r=1$, $n_1=2$, $n_2=\cdots=n_{l}=1$ and $l=n-2$ which leads to $G\cong K_{1, 1}(u)\odot^k K_{n-2}\setminus e$ for $e\in E(K_{n-2})$.
Let $\mathcal{G}^*$ denote the set of all connected graphs in $\mathcal{G}$ without pendant vertices. Then $\mathcal{G}=\mathcal{G}^-\cup \mathcal{G}^+\cup \mathcal{G}^*$. Therefore, in order to characterize $\mathcal{G}$, it remains to consider those graphs in $\mathcal{G}^*$.
The characterization of graphs in $\mathcal{G}^\ast$
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First we introduce some symbols which will be persisted in this section. Let $G\in \mathcal{G}^*$. The eigenvalues of $G$ can be arranged as: $$\lambda_1(G)\geq\lambda_2(G)>\lambda_3(G)=0>\lambda_4(G)\geq\cdots\geq \lambda_n(G).$$ We choose $v^*\in V(G)$ such that $d_G(v^*)=\delta(G)=t$, and denote by $X=N_G(v^*)$ and $Y=V(G)-N_G[v^*]$. Then $t=|X|\geq 2$ since $G$ has no pendant vertices. In addition, $|Y|>0$ since otherwise $G$ would be a complete graph. First we characterize the induced subgraph $G[Y]$ in the following result.
\[lem-3-1\] $G[Y]\cong K_{n-t-1}\setminus e, K_1+K_{n-t-2}$ or $K_{n-t-1}$.
First we suppose that $Y$ is an independent set. If $|Y|\geq 3$, then $\lambda_4(G)\geq \lambda_4(G[Y\cup \{v^*\}])=0$ by Theorem \[thm-2-0\], a contradiction. Hence $|Y|\leq 2$, and so $G[Y]\cong K_1$ or $G[Y]\cong K_2\setminus e= 2K_1$.
Next we suppose that $G[Y]$ contains some edges. We distinguish the following three situations.
If $\lambda_2(G[Y])>0$, we have $p(G[Y])\geq 2$. For any $x\in X$, the induced subgraph $G[\{v^*, x\}\cup Y]$ has a pendant vertex $v^*$ by our assumption. By Lemma \[lem-2-2\] and Lemma \[lem-2-1\], we have $p(G)\geq p(G[\{v^*, x\}\cup Y])=p(G[Y])+1\geq 3$, a contradiction.
If $\lambda_2(G[Y])<0$, by Theorem \[thm-3-1\] (2) we have $G[Y]\cong K_{n-t-1}$ as desired.
At last assume that $\lambda_2(G[Y])=0$. If $\lambda_3(G[Y])<0$, by Theorem \[thm-3-1\] (1), we have $G[Y]\cong K_{n-t-1}\setminus e, K_1+K_{n-t-2}$ as desired. If $\lambda_3(G[Y])=0$, by Lemma \[lem-2-1\] we have $p(G[\{v^*, x\}\cup Y])=p(G[Y])+1=2$ and $\eta(G[\{v^*, x\}\cup Y])=\eta(G[Y])\geq 2$, which implies that $\lambda_4(G)\geq \lambda_4(G[\{v^*, x\}\cup Y])=0$, a contradiction.
We complete this proof.
First assume that $Y=\{y_1\}$. If $G[X]=K_t$, then $G=K_n\backslash v^*y_1$. However $K_n\backslash v^*y_1\not\in \mathcal{G}^*$ since $p(K_n\backslash v^*y_1)=1$. Thus there exist $x_1\not\sim x_2$ in $X$. Then $N_G(x_1)=N_G(x_2)$ and $N_G(v^*)=N_G(y_1)$. It follows that $\eta(G)\ge2$ by Lemma \[lem-2-4\]. Next assume that $Y=\{y_1, y\}$ is an independent set. We have $N_G(v^*)=N_G(y_1)=N_G(y)$ since $d_G(y_1), d_G(y)\ge d_G(v^*)=\delta(G)$. Thus, by Lemma \[lem-2-4\] we have $\eta(G)=\eta(G-y_1)+1 =\eta(G-y_1-y)+2\geq 2$. Thus we only need to consider the case that $G[Y]$ contains at least one edge. Concretely, we distinguish three situations in accordance with the proof of Lemma \[lem-3-1\]:\
(a) $G[Y]\cong K_{n-t-2}+K_1$ in case of $\lambda_2(G[Y])=0$ and $\lambda_3(G[Y])<0$, where $n-t-2\ge 2$;\
(b) $G[Y]\cong K_{n-t-1}\setminus e$ in case of $\lambda_2(G[Y])=0$ and $\lambda_3(G[Y])<0$, where $|Y|=n-t-1\ge 3$;\
(c) $G[Y]\cong K_{n-t-1}$ in case of $\lambda_2(G[Y])<0$, where $|Y|=n-t-1\ge 2$.
In the following, we deal with situation (a) in Lemma \[lem-3-3\], (b) in Lemma \[lem-3-2\] and (c) in Lemma \[lem-3-4\], \[X-lem-1\] and Lemma \[thm-6-3\]. We will see that the graph $G\in \mathcal{G}^*$ illustrated in (a) and (b) can be constructed from some $H\in \mathcal{H}$ by the graph transformations of I-, II- and III-type, but (c) can not.
\[lem-3-3\] If $G[Y]\cong K_{n-t-2}+K_1$, where $n-t-2\geq 2$, then $G\in \mathcal{G}_1$.
Since $G[Y]$ is isomorphic to $K_{n-t-2}+K_1(n-t-2\geq 2)$, $Y$ exactly contains one isolated vertex of $G[Y]$, say $y$. We have $N_G(v^*)=N_G(y)$ and thus $y$ is a congruent vertex of I-type. By Lemma \[lem-2-4\], we have $p(G)=p(G-y)$ and $\eta(G)=\eta(G-y)+1$. Notice that $G-y$ is connected, we have $G-y\in \mathcal{H}$, and so $G\in \mathcal{G}_1$. Such a graph $G$, displayed in Fig. \[fig-2\](1), we call the *$v^\ast$-graph of I-type*.
In Fig. \[fig-2\] and Fig. \[fig-4\], two ellipses joining with one full line denote some edges between them. A vertex and an ellipse joining with one full line denote some edges between them, and with two full lines denote that this vertex joins all vertices in the ellipse. Two vertices join with same location of an ellipse denote that they have same neighbours in this ellipse.
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(216.125,33.75)(-10,0) (216.125,33.75)[(0,0)\[cc\][$v^\ast$]{}]{} (12.818,23.652)[(-1,0)[.088]{}]{} (12.404,24.122) (4.803,24.034) (8.929,23.962)[(15.026,4.861)\[\]]{} (8.692,30.486) (8.625,11.688)[(17.25,6.375)\[\]]{} (12.25,24.125)[(0,1)[.25]{}]{} (12.345,11.72) (13.375,10.5)[(0,0)\[cc\][$y$]{}]{} (8.5,4.75)[(0,0)\[cc\][$v^\ast$-graph of I-type]{}]{} (12.25,11.75)[(0,1)[10]{}]{} (12.25,11.875)(-.033505155,.103092784)[97]{}[(0,1)[.103092784]{}]{} (8.25,.375)[(0,0)\[cc\][(1)]{}]{} (9.5,32.375)[(0,0)\[cc\][$v^\ast$]{}]{} (8.75,30.5)(-.033505155,-.042525773)[97]{}[(0,-1)[.042525773]{}]{} (8.75,30.5)(.03353659,-.05030488)[82]{}[(0,-1)[.05030488]{}]{} (7.75,23.75) (8.634,23.75) (9.518,23.75) (22.375,16.125)[(0,0)\[cc\][$G[Y]\cong K_{n-t-2}+K_1$]{}]{} (14,27.875)[(0,0)\[cc\][$X$]{}]{} (46.179,24.009)[(15.026,4.861)\[\]]{} (45.942,30.533) (45,4.375)[(0,0)\[cc\][$Y$-graph of I-type]{}]{} (46.625,32.25)[(0,0)\[cc\][$v^\ast$]{}]{} (44.875,0)[(0,0)\[cc\][(2)]{}]{} (46,30.75)(-.033505155,-.045103093)[97]{}[(0,-1)[.045103093]{}]{} (45.875,30.5)(.03343023,-.0494186)[86]{}[(0,-1)[.0494186]{}]{} (50.11,24.088)[(-1,0)[.088]{}]{} (49.696,24.558) (42.095,24.47) (49.542,24.561)[(0,1)[.25]{}]{} (45.042,24.186) (45.926,24.186) (46.81,24.186) (59.875,15.625)[(0,0)\[cc\][$G[Y]\cong K_{n-t-1}\setminus yy'$]{}]{} (52,27.75)[(0,0)\[cc\][$X$]{}]{} (83.929,24.088)[(15.026,4.861)\[\]]{} (83.692,30.612) (83.625,3.875)[(0,0)\[cc\][$(v^\ast,Y)$-graph of II-type]{}]{} (84,11.563)[(17.25,6.375)\[\]]{} (84.5,32.625)[(0,0)\[cc\][$v^\ast$]{}]{} (83.75,30.625)(-.03370787,-.04634831)[89]{}[(0,-1)[.04634831]{}]{} (83.875,30.5)(.03365385,-.05128205)[78]{}[(0,-1)[.05128205]{}]{} (87.86,24.213)[(-1,0)[.088]{}]{} (87.446,24.683) (79.845,24.595) (87.292,24.686)[(0,1)[.25]{}]{} (82.792,24.311) (83.676,24.311) (84.56,24.311) (99.375,16.125)[(0,0)\[cc\][$G[Y]\cong K_{n-t-1}\setminus yy'$]{}]{} (89.375,27.375)[(0,0)\[cc\][$X$]{}]{} (84.5,0)[(0,0)\[cc\][(3)]{}]{} (87.68,12.305)[(0,1)[.0625]{}]{} (46,11.313)[(17.25,6.375)\[\]]{} (7.468,11.581) (4.416,11.581) (5.332,11.551) (5.858,11.551) (6.383,11.551) (6,21.75)[(0,-1)[8.875]{}]{} (6.063,11.375)[(8.125,3.5)\[\]]{} (50.843,9.72) (52.406,9.868)[(0,0)\[cc\][$y$]{}]{} (50.873,12.625)[(-1,0)[.375]{}]{} (45.218,11.081) (42.165,11.081) (43.081,11.051) (43.608,11.051) (44.132,11.051) (43.813,11)[(8.125,3.5)\[\]]{} (50.845,12.095) (52.497,12.773)[(0,0)\[cc\][$y'$]{}]{} (43.25,21.5)[(0,-1)[8.875]{}]{} (50.875,12.125)(-.033653846,.074038462)[130]{}[(0,1)[.074038462]{}]{} (46.5,21.75)(.033730159,-.095238095)[126]{}[(0,-1)[.095238095]{}]{} (50.875,12.25)[(-1,0)[3.75]{}]{} (47.125,12.25)(.04647436,-.03365385)[78]{}[(1,0)[.04647436]{}]{} (50.75,9.625)[(-1,0)[4]{}]{} (46.75,9.625)(.05809859,.0334507)[71]{}[(1,0)[.05809859]{}]{} (89.218,13.095) (90.87,12.772)[(0,0)\[cc\][$y'$]{}]{} (89.218,10.345) (90.778,10.491)[(0,0)\[cc\][$y$]{}]{} (82.063,11.752)[(8.125,3.5)\[\]]{} (83.394,11.472) (80.343,11.472) (81.26,11.442) (81.784,11.442) (82.311,11.442) (81.125,21.875)[(0,-1)[8.5]{}]{} (89.125,13)[(0,1)[9]{}]{} (89.25,13.125)(-.033653846,.068269231)[130]{}[(0,1)[.068269231]{}]{} (89.25,13.125)[(-1,0)[4.25]{}]{} (85,13.125)(.05087209,-.03343023)[86]{}[(1,0)[.05087209]{}]{} (89.375,10.25)[(-1,0)[4.5]{}]{} (84.875,10.25)(.05769231,.03365385)[78]{}[(1,0)[.05769231]{}]{}
It needs to mention that the $v^\ast$-graph of I-type characterized in Lemma \[lem-3-3\], is a graph obtained from $H\in \mathcal{H}$ by adding a new vertex joining the neighbors of a minimum degree vertex of $H$.
For $S\subseteq V(G)$ and $u\in V(G)$, let $N_S(u)=N_G(u)\cap S$ and $N_S[u]=N_G[u]\cap S$.
\[lem-3-2\] Let $G[Y]\cong K_{n-t-1}\setminus e$, where $n-t-1\geq 3$ and $e=yy'$. Then $G\in \mathcal{G}_1$ if $N_X(y)=N_X(y')$ and $G\in \mathcal{G}_2$ otherwise.
Since $n-t-1\geq 3$, there is $y^\ast\in Y$ other than $y$ and $y'$. It is clear that $N_G(y)=N_X(y)\cup (Y\setminus\{y, y'\})$ and $N_G(y')=N_X(y')\cup (Y\setminus\{y, y'\})$, and thus $N_G(y)=N_G(y')$ if and only if $N_X(y)=N_X(y')$. We consider the following cases.
[**Case 1.**]{} $N_X(y)=N_X(y')$.
By assumption, $N_G(y)=N_G(y')$, thus $y$ and $y'$ are congruent vertices of I-type. By Lemma \[lem-2-4\], we have $p(G)=p(G-y)$ and $\eta(G)=\eta(G-y)+1$. Since $G-y$ is connected, we have $G-y\in \mathcal{H}$ and so $G\in \mathcal{G}_1$. Such a $G$, displayed in Fig. \[fig-2\](2), we call the *$Y$-graph of I-type*.
[**Case 2.**]{} $N_X(y)\not=N_X(y')$.
First suppose that exactly one of $N_X(y)$ and $N_X(y')$ is empty, say $N_X(y)=\emptyset$ and $N_X(y')\not=\emptyset$. Then $yy^\ast$ is a pendant edge of the induced subgraph $G[X\cup\{y, y', y^\ast, v^*\}]$. By Lemma \[lem-2-2\] and Lemma \[lem-2-1\], we have $$2=p(G)\geq p(G[X\cup \{y, y', y^\ast, v^*\}])=p(G[X\cup \{y', v^*\}])+1\ge2.$$ Thus $p(G[X \cup \{y, y', y^\ast, v^*\}])=2$ and $p(G[X\cup \{y', v^*\}])=1$. We see that $\lambda_2(G[X\cup \{y', v^*\}])=0$ (since otherwise $\lambda_2(G[X\cup \{y', v^*\}])<0$ and then $G[X\cup \{y', v^*\}]$ is a complete graph, but $y'\not\sim v^*$). If $\lambda_3(G[X\cup\{y', v^*\}])=0$, we have $\eta(G[X\cup \{y, y', y^\ast, v^*\}])=\eta(G[ X\cup \{y', v^*\}])\ge2$, which implies $\lambda_4(G)\geq \lambda_4(G[X\cup \{y, y', y^\ast, v^*\}])=0$ , a contradiction. If $\lambda_3(G[X\cup\{y', v^*\}])<0$, then $G[X\cup \{y', v^*\}]\cong K_{t+2}\setminus e$ or $K_{t+1}+K_1$ by Theorem \[thm-3-1\] (1). Notice that $G[X\cup \{y', v^*\}]$ is connected, we get $G[X\cup \{y', v^*\}]\cong K_{t+2}\setminus e$ where $e=v^*y'$. Thus $N_X(y')=X$ and so $N_G(y')=X\cup (Y\setminus\{y, y'\})=N_G(v^*)\cup N_G(y)$ is a disjoint union. Additionally, $\{y', v^*, y\}$ is an independent set in $G$, we see that $y'$ is a congruent vertex of II-type. Thus $p(G)=p(G-y')$ and $\eta(G)=\eta(G-y')+1$ by Lemma \[lem-2-5\]. This implies that $G-y'\in \mathcal{H}$, and so $G\in \mathcal{G}_2$. Such a $G$, displayed in Fig. \[fig-2\](3), we call the *$(v^\ast, Y)$-graph of II-type*.
1.15mm
(119.375,54)(0,2) (59.141,52.276)(-.035057851,-.033603306)[121]{}[(-1,0)[.035057851]{}]{} (59.141,52.188)[(1,-1)[3.889]{}]{} (63.119,48.21)[(0,-1)[4.95]{}]{} (63.03,43.26)[(-1,-1)[3.977]{}]{} (54.987,48.299)(.054078947,-.033730263)[152]{}[(1,0)[.054078947]{}]{} (55.076,48.387)[(1,0)[8.132]{}]{} (59.275,52.053) (62.991,48.039) (63.14,43.431) (59.424,39.418) (55.113,48.039) (55.113,43.363) (55.125,48)[(0,-1)[4.75]{}]{} (37.25,0)[(0,0)\[cc\]]{} (23.971,26.238)[(1,-1)[3.889]{}]{} (27.949,22.26)[(0,-1)[4.95]{}]{} (27.86,17.31)[(-1,-1)[3.977]{}]{} (19.729,17.399)(.035793388,-.033603306)[121]{}[(1,0)[.035793388]{}]{} (19.817,22.349)(.054078947,-.033730263)[152]{}[(1,0)[.054078947]{}]{} (19.905,22.172)(.033603306,-.07377686)[121]{}[(0,-1)[.07377686]{}]{} (24.325,26.149)[(-1,0)[.088]{}]{} (20.347,22.702)[(1,0)[.088]{}]{} (20.436,22.702)(-.0295,-.058833)[6]{}[(0,-1)[.058833]{}]{} (20.259,22.349)(-.088375,-.03325)[8]{}[(-1,0)[.088375]{}]{} (24.15,26.042) (28.164,22.028) (28.164,17.272) (24.002,13.407) (19.988,22.028) (19.988,17.272) (19.999,22.388)[(0,-1)[.25]{}]{} (19.749,22.138)[(1,0)[8.375]{}]{} (112.958,26.805)(-.035057851,-.033603306)[121]{}[(-1,0)[.035057851]{}]{} (112.958,26.717)[(1,-1)[3.889]{}]{} (108.716,22.828)[(0,-1)[4.861]{}]{} (116.936,22.739)[(0,-1)[4.95]{}]{} (113.049,26.553) (117.062,17.485) (117.062,22.539) (108.738,22.391) (108.887,17.337) (113.049,13.62) (108.594,22.375)[(1,0)[8.375]{}]{} (112.969,13.5)(.035869565,.033695652)[115]{}[(1,0)[.035869565]{}]{} (108.719,17.25)(.040509259,-.033564815)[108]{}[(1,0)[.040509259]{}]{} (108.625,17.125)[(1,0)[8.375]{}]{} (113,26.5)[(0,-1)[12.75]{}]{} (19.875,22.25)[(0,-1)[5.125]{}]{} (6.226,26.091)[(1,-1)[3.889]{}]{} (10.204,22.113)[(0,-1)[4.95]{}]{} (1.984,17.252)(.035793388,-.033603306)[121]{}[(1,0)[.035793388]{}]{} (2.072,22.202)(.054078947,-.033730263)[152]{}[(1,0)[.054078947]{}]{} (6.58,26.002)[(-1,0)[.088]{}]{} (2.602,22.555)[(1,0)[.088]{}]{} (2.514,22.202)(-.088375,-.03325)[8]{}[(-1,0)[.088375]{}]{} (6.103,25.82) (10.265,21.658) (10.265,16.752) (2.238,21.658) (2.387,16.901) (6.252,13.333) (2.3,21.75)(.033695652,.038043478)[115]{}[(0,1)[.038043478]{}]{} (6.175,13.375)(.033730159,.066468254)[126]{}[(0,1)[.066468254]{}]{} (2.3,21.75)(.033613445,-.070378151)[119]{}[(0,-1)[.070378151]{}]{} (2.175,21.75)[(0,-1)[4.625]{}]{} (5.958,52.717)[(1,-1)[3.889]{}]{} (1.716,48.828)[(0,-1)[4.861]{}]{} (9.936,48.739)[(0,-1)[4.95]{}]{} (6.049,52.553) (10.062,43.485) (10.062,48.539) (1.738,48.391) (1.887,43.337) (6.049,39.62) (1.594,48.375)[(1,0)[8.375]{}]{} (1.719,43.25)(.040509259,-.033564815)[108]{}[(1,0)[.040509259]{}]{} (23.862,52.568)[(1,-1)[3.889]{}]{} (27.84,48.59)[(0,-1)[4.95]{}]{} (27.751,43.64)[(-1,-1)[3.977]{}]{} (23.924,52.255) (27.789,48.39) (27.937,43.485) (23.924,39.769) (19.762,48.688) (19.738,43.363) (42.03,52.291)(-.035057851,-.033603306)[121]{}[(-1,0)[.035057851]{}]{} (42.03,52.203)[(1,-1)[3.889]{}]{} (37.788,48.314)[(0,-1)[4.861]{}]{} (46.008,48.225)[(0,-1)[4.95]{}]{} (42.073,51.951) (41.924,39.613) (45.937,43.032) (37.613,43.181) (37.762,47.937) (37.719,47.969)[(1,0)[8.25]{}]{} (45.957,47.832) (37.594,47.969)(.033653846,-.064423077)[130]{}[(0,-1)[.064423077]{}]{} (41.969,39.719)(.042525773,.033505155)[97]{}[(1,0)[.042525773]{}]{} (76.871,52.578)(-.035057851,-.033603306)[121]{}[(-1,0)[.035057851]{}]{} (76.871,52.49)[(1,-1)[3.889]{}]{} (80.849,48.512)[(0,-1)[4.95]{}]{} (80.76,43.562)[(-1,-1)[3.977]{}]{} (72.629,43.651)(.035793388,-.033603306)[121]{}[(1,0)[.035793388]{}]{} (77.225,52.401)[(-1,0)[.088]{}]{} (76.897,52.307) (80.91,43.239) (81.059,48.591) (72.883,48.442) (77.045,39.672) (72.75,48.5)[(1,0)[.25]{}]{} (72.863,43.238) (72.75,48.5)[(0,-1)[5.375]{}]{} (72.75,48.375)(.054276316,-.033717105)[152]{}[(1,0)[.054276316]{}]{} (76.875,39.625)(.033536585,.075203252)[123]{}[(0,1)[.075203252]{}]{} (112.708,52.68)(-.035057851,-.033603306)[121]{}[(-1,0)[.035057851]{}]{} (112.708,52.592)[(1,-1)[3.889]{}]{} (108.466,48.703)[(0,-1)[4.861]{}]{} (116.686,48.614)[(0,-1)[4.95]{}]{} (112.799,52.428) (116.812,43.36) (116.812,48.414) (108.488,48.266) (108.637,43.212) (112.799,39.495) (108.344,48.25)[(1,0)[8.375]{}]{} (108.469,48.25)(.033653846,-.069230769)[130]{}[(0,-1)[.069230769]{}]{} (112.594,39.625)(.033730159,.070436508)[126]{}[(0,1)[.070436508]{}]{} (112.719,39.375)(.035869565,.033695652)[115]{}[(1,0)[.035869565]{}]{} (108.469,43.125)(.040509259,-.033564815)[108]{}[(1,0)[.040509259]{}]{} (95.265,52.53)(-.035057851,-.033603306)[121]{}[(-1,0)[.035057851]{}]{} (95.265,52.442)[(1,-1)[3.889]{}]{} (99.243,48.464)[(0,-1)[4.95]{}]{} (91.023,43.603)(.035793388,-.033603306)[121]{}[(1,0)[.035793388]{}]{} (91.111,48.553)(.054078947,-.033730263)[152]{}[(1,0)[.054078947]{}]{} (95.276,52.199) (99.14,48.334) (99.289,43.132) (95.276,39.713) (91.262,48.334) (95.125,39.75)(.040865385,.033653846)[104]{}[(1,0)[.040865385]{}]{} (91,48.5)[(1,0)[8.125]{}]{} (91.125,48.25)[(1,-2)[4.25]{}]{} (91.238,43.363) (91.25,43.375)[(0,1)[5.125]{}]{} (7,32.375)[(0,0)\[cc\][$\Gamma_1$]{}]{} (42.75,32.375)[(0,0)\[cc\][$\Gamma_3$]{}]{} (24.75,32.5)[(0,0)\[cc\][$\Gamma_2$]{}]{} (60.75,32.625)[(0,0)\[cc\][$\Gamma_4$]{}]{} (78,32.5)[(0,0)\[cc\][$\Gamma_5$]{}]{} (96.25,32.5)[(0,0)\[cc\][$\Gamma_6$]{}]{} (113.75,32.625)[(0,0)\[cc\][$\Gamma_7$]{}]{} (59.522,26.565)(-.035057851,-.033603306)[121]{}[(-1,0)[.035057851]{}]{} (63.411,17.549)[(-1,-1)[3.977]{}]{} (59.876,26.388)[(-1,0)[.088]{}]{} (55.367,22.676)[(1,0)[8.22]{}]{} (55.367,17.726)[(1,0)[8.22]{}]{} (59.55,26.278) (63.563,22.71) (63.712,17.507) (59.55,13.791) (55.685,22.71) (55.707,17.555) (55.594,17.567)(.034722222,-.033564815)[108]{}[(1,0)[.034722222]{}]{} (59.469,26.442)(.038194444,-.033564815)[108]{}[(1,0)[.038194444]{}]{} (55.719,22.817)(.05,-.03359375)[160]{}[(1,0)[.05]{}]{} (63.594,22.817)[(0,-1)[5.125]{}]{} (55.719,22.942)(.033695652,-.080434783)[115]{}[(0,-1)[.080434783]{}]{} (59.594,13.692)(.033613445,.077731092)[119]{}[(0,1)[.077731092]{}]{} (59.594,26.317)(-.033695652,-.076086957)[115]{}[(0,-1)[.076086957]{}]{} (41.794,26.336)(-.035057851,-.033603306)[121]{}[(-1,0)[.035057851]{}]{} (37.552,17.409)(.035793388,-.033603306)[121]{}[(1,0)[.035793388]{}]{} (45.772,22.27)(-.056082759,-.033524138)[145]{}[(-1,0)[.056082759]{}]{} (37.64,22.359)(.054078947,-.033730263)[152]{}[(1,0)[.054078947]{}]{} (37.728,22.271)[(0,-1)[4.685]{}]{} (45.771,22.359)(-.033711864,-.072652542)[118]{}[(0,-1)[.072652542]{}]{} (45.772,22.36)[(0,-1)[4.861]{}]{} (45.595,17.675)(-.033637168,-.039106195)[113]{}[(0,-1)[.039106195]{}]{} (41.752,25.961) (45.616,21.947) (45.616,17.042) (41.9,13.326) (37.738,17.042) (37.738,21.947) (37.571,17.023)[(1,0)[8.027]{}]{} (41.637,26.016)(.034782609,-.033695652)[115]{}[(1,0)[.034782609]{}]{} (77.188,26.708)(-.035057851,-.033603306)[121]{}[(-1,0)[.035057851]{}]{} (77.188,26.62)[(1,-1)[3.889]{}]{} (72.946,22.731)[(1,0)[8.22]{}]{} (72.946,22.731)[(0,-1)[4.861]{}]{} (81.166,22.642)[(0,-1)[4.95]{}]{} (72.946,17.781)(.035793388,-.033603306)[121]{}[(1,0)[.035793388]{}]{} (72.945,22.731)[(1,0)[8.22]{}]{} (73.034,17.781)[(1,0)[8.132]{}]{} (81.254,22.731)(-.033603306,-.07377686)[121]{}[(0,-1)[.07377686]{}]{} (77.174,26.361) (81.336,22.496) (81.187,17.442) (72.863,17.442) (73.012,22.496) (77.028,13.951) (77.04,14.088)(.045103093,.033505155)[97]{}[(1,0)[.045103093]{}]{} (95.212,26.77)(-.035057851,-.033603306)[121]{}[(-1,0)[.035057851]{}]{} (95.212,26.682)[(1,-1)[3.889]{}]{} (90.97,22.793)[(1,0)[8.22]{}]{} (90.97,22.793)[(0,-1)[4.861]{}]{} (99.19,22.704)[(0,-1)[4.95]{}]{} (99.101,17.754)[(-1,-1)[3.977]{}]{} (90.97,17.843)(.035793388,-.033603306)[121]{}[(1,0)[.035793388]{}]{} (91.057,17.843)[(1,0)[8.132]{}]{} (95.211,26.771)[(0,-1)[13.081]{}]{} (99.189,22.793)(-.033525862,-.075431034)[116]{}[(0,-1)[.075431034]{}]{} (95.15,26.422) (99.015,22.409) (99.015,17.652) (90.988,22.409) (90.988,17.652) (95.299,13.936) (6.125,54.75)[(0,0)\[cc\][$v^\ast$]{}]{} (24,54.625)[(0,0)\[cc\][$v^\ast$]{}]{} (42.25,54.625)[(0,0)\[cc\][$v^\ast$]{}]{} (77,54.5)[(0,0)\[cc\][$v^\ast$]{}]{} (95.125,54.375)[(0,0)\[cc\][$v^\ast$]{}]{} (112.25,54.5)[(0,0)\[cc\][$v^\ast$]{}]{} (6.25,27.875)[(0,0)\[cc\][$v^\ast$]{}]{} (24.25,27.75)[(0,0)\[cc\][$v^\ast$]{}]{} (71.125,48.375)[(0,0)\[cc\][$x$]{}]{} (71.125,43.375)[(0,0)\[cc\][$y$]{}]{} (89.5,48.375)[(0,0)\[cc\][$x$]{}]{} (89.5,43.375)[(0,0)\[cc\][$y$]{}]{} (106.75,47.875)[(0,0)\[cc\][$x$]{}]{} (106.75,42.875)[(0,0)\[cc\][$y$]{}]{} (23.875,13.25)(.033730159,.070436508)[126]{}[(0,1)[.070436508]{}]{} (18.25,22.375)[(0,0)\[cc\][$x$]{}]{} (18.125,16.75)[(0,0)\[cc\][$y$]{}]{} (59.75,27.75)[(0,0)\[cc\][$v^\ast$]{}]{} (54.25,17)[(0,0)\[cc\][$x$]{}]{} (61.25,13.125)[(0,0)\[cc\][$y$]{}]{} (113.125,28.375)[(0,0)\[cc\][$x^\ast$]{}]{} (107.25,16.625)[(0,0)\[cc\][$x$]{}]{} (114.5,12.625)[(0,0)\[cc\][$y$]{}]{} (89.375,22.375)[(0,0)\[cc\][$x$]{}]{} (5.875,39.625)(.036956522,.033695652)[115]{}[(1,0)[.036956522]{}]{} (1.625,48.5)(.035714286,.033730159)[126]{}[(1,0)[.035714286]{}]{} (11.625,43.25)[(0,0)\[cc\][$y^\prime$]{}]{} (11.5,48.75)[(0,0)\[cc\][$x^\prime$]{}]{} (8.75,38.625)[(0,0)\[cc\][$y^\ast$]{}]{} (29.5,43)[(0,0)\[cc\][$y^\prime$]{}]{} (29.375,48.5)[(0,0)\[cc\][$x^\prime$]{}]{} (26.5,38.75)[(0,0)\[cc\][$y^\ast$]{}]{} (48.125,42.625)[(0,0)\[cc\][$y^\prime$]{}]{} (48,48.125)[(0,0)\[cc\][$x^\prime$]{}]{} (44.125,38.5)[(0,0)\[cc\][$y^\ast$]{}]{} (6.25,35.375)[(0,0)\[cc\][$\lambda_3(\Gamma_1)=0.6180$]{}]{} (24.25,35.375)[(0,0)\[cc\][$\lambda_3(\Gamma_2)=0.4142$]{}]{} (42.5,35.5)[(0,0)\[cc\][$\lambda_3(\Gamma_3)=0.5293$]{}]{} (59.25,35.625)[(0,0)\[cc\][$\lambda_3(\Gamma_4)=0.1830$]{}]{} (19.625,48.75)(.039351852,.033564815)[108]{}[(1,0)[.039351852]{}]{} (19.625,48.75)[(0,-1)[5.625]{}]{} (19.625,48.75)(.052884615,-.033653846)[156]{}[(1,0)[.052884615]{}]{} (19.5,43.375)(.040509259,-.033564815)[108]{}[(1,0)[.040509259]{}]{} (37.625,43.25)(.0390625,-.033482143)[112]{}[(1,0)[.0390625]{}]{} (55,43.25)(.036764706,-.033613445)[119]{}[(1,0)[.036764706]{}]{} (61.25,54.375)[(0,0)\[cc\][$v^\ast (y^\prime)$]{}]{} (63.25,38.375)[(0,0)\[cc\][$y^\ast (v^\ast)$]{}]{} (83,48.75)[(0,0)\[cc\][$x^\prime$]{}]{} (83.125,43)[(0,0)\[cc\][$y^\prime$]{}]{} (79.875,38.5)[(0,0)\[cc\][$y^\ast$]{}]{} (77.625,35.625)[(0,0)\[cc\][$\lambda_3(\Gamma_5)=0.6180$]{}]{} (101,48.75)[(0,0)\[cc\][$x^\prime$]{}]{} (101.125,43)[(0,0)\[cc\][$y^\prime$]{}]{} (97.875,38.875)[(0,0)\[cc\][$y^\ast$]{}]{} (95.75,35.375)[(0,0)\[cc\][$\lambda_3(\Gamma_6)=0.1124$]{}]{} (118.5,48.875)[(0,0)\[cc\][$x^\prime$]{}]{} (118.625,43.125)[(0,0)\[cc\][$y^\prime$]{}]{} (115,38.125)[(0,0)\[cc\][$y^\ast$]{}]{} (113.125,35.375)[(0,0)\[cc\][$\lambda_3(\Gamma_7)=0.6180$]{}]{} (11.75,21.5)[(0,0)\[cc\][$x^\prime$]{}]{} (11.875,15.75)[(0,0)\[cc\][$y^\prime$]{}]{} (25.125,4.875)[(0,0)\[cc\][$\Gamma_9$]{}]{} (24.75,8.25)[(0,0)\[cc\][$\lambda_3(\Gamma_9)=0.1589$]{}]{} (114.5,5)[(0,0)\[cc\][$\Gamma_{14}$]{}]{} (113.875,8.375)[(0,0)\[cc\][$\lambda_4(\Gamma_{14})=0$]{}]{} (6.875,4.875)[(0,0)\[cc\][$\Gamma_8$]{}]{} (43.25,4.875)[(0,0)\[cc\][$\Gamma_{10}$]{}]{} (78.875,4.875)[(0,0)\[cc\][$\Gamma_{12}$]{}]{} (61,4.75)[(0,0)\[cc\][$\Gamma_{11}$]{}]{} (96.75,5)[(0,0)\[cc\][$\Gamma_{13}$]{}]{} (6.125,13.375)(.040841584,.033415842)[101]{}[(1,0)[.040841584]{}]{} (6.625,8.375)[(0,0)\[cc\][$\lambda_3(\Gamma_8)=0.2798$]{}]{} (19.75,22.125)(.036585366,.033536585)[123]{}[(1,0)[.036585366]{}]{} (29.625,22.125)[(0,0)\[cc\][$x^\prime$]{}]{} (29.625,17.125)[(0,0)\[cc\][$y^\prime$]{}]{} (41.875,8.5)[(0,0)\[cc\][$\lambda_3(\Gamma_{10})=0.1505$]{}]{} (44.25,27.75)[(0,0)\[cc\][$v^\ast (y^\prime)$]{}]{} (35.875,22.625)[(0,0)\[cc\][$x (y)$]{}]{} (35.75,17.125)[(0,0)\[cc\][$y (x^\ast)$]{}]{} (47.375,22.5)[(0,0)\[cc\][$x^\prime$]{}]{} (65.5,18.375)[(0,0)\[cc\][$y^\prime$]{}]{} (65.25,23.25)[(0,0)\[cc\][$x^\ast$]{}]{} (60,8.375)[(0,0)\[cc\][$\lambda_3(\Gamma_{11})=0.2679$]{}]{} (77.75,8.5)[(0,0)\[cc\][$\lambda_3(\Gamma_{12})=0.1096$]{}]{} (96.5,28.125)[(0,0)\[cc\][$y^\prime (v^\ast)$]{}]{} (97.875,13.625)[(0,0)\[cc\][$x^\prime$]{}]{} (96.5,8.5)[(0,0)\[cc\][$\lambda_3(\Gamma_{13})=0.1873$]{}]{} (107.625,22.625)[(0,0)\[cc\][$v^\ast$]{}]{} (119.125,18.125)[(0,0)\[cc\][$y^\prime$]{}]{} (119.375,23.375)[(0,0)\[cc\][$x^\prime$]{}]{} (.125,48.25)[(0,0)\[cc\][$x$]{}]{} (0,42.625)[(0,0)\[cc\][$y$]{}]{} (18.25,48.75)[(0,0)\[cc\][$x$]{}]{} (18.125,43.125)[(0,0)\[cc\][$y$]{}]{} (35.875,47.875)[(0,0)\[cc\][$x$]{}]{} (35.875,42.875)[(0,0)\[cc\][$y$]{}]{} (53.625,48.25)[(0,0)\[cc\][$x (y)$]{}]{} (53.5,43.5)[(0,0)\[cc\][$y (x)$]{}]{} (.375,21.875)[(0,0)\[cc\][$x$]{}]{} (.25,16.25)[(0,0)\[cc\][$y$]{}]{} (8.125,12)[(0,0)\[cc\][$y^\ast$]{}]{} (26.25,12.125)[(0,0)\[cc\][$y^\ast$]{}]{} (54.25,22.75)[(0,0)\[cc\][$x^\prime$]{}]{} (77.375,27.625)[(0,0)\[cc\][$y^\prime (y^\prime, v^\ast)$]{}]{} (70.5,23)[(0,0)\[cc\][$x (y, x')$]{}]{} (79.375,12.25)[(0,0)\[cc\][$x^\prime (v^\ast, y^\ast)$]{}]{} (102.125,22.75)[(0,0)\[cc\][$y (x^\ast)$]{}]{} (65.875,48.375)[(0,0)\[cc\][$x^\prime (y^\ast)$]{}]{} (66,44)[(0,0)\[cc\][$y^\prime (x^\prime)$]{}]{} (44.875,12.375)[(0,0)\[cc\][$y^\ast (v^\ast)$]{}]{} (48.875,17.625)[(0,0)\[cc\][$y^\prime (x)$]{}]{} (85,23.375)[(0,0)\[cc\][$y (x^\prime, x)$]{}]{} (90.625,16)[(0,0)\[cc\][$v^\ast (y)$]{}]{} (102.375,18.25)[(0,0)\[cc\][$x^\ast (y^\prime)$]{}]{} (69.875,18.5)[(0,0)\[cc\][$v^\ast (x, y')$]{}]{} (85.625,18.375)[(0,0)\[cc\][$x^\ast (x^\ast, y)$]{}]{}
Next suppose that $N_X(y),N_X(y')\neq \emptyset$, without loss of generality, assume that $N_X(y')\backslash$ $N_X(y)$ $\not= \emptyset$. Then there exists $x'\in N_X(y')\backslash N_X(y)$. Thus $x'\sim y'$ and $x'\not\sim y$. Now by taking some $x\in N_X(y)$, we see that $C_6=v^*xyy^\ast y'x'$ is a $6$-cycle in $G$. Note that $x$ may joins each vertex in $\{x', y',y^\ast\}$ and $x'$ may joins $y^\ast$. By distinguishing different situations in according with the number of edges we have $$G[v^\ast, x, y, y^\ast, y', x']\cong\left\{
\begin{array}{ll}
C_6 & \hbox{no edge;}\\
\Gamma_1\ \hbox{or}\ \Gamma_2 & \hbox{one edges;} \\
\Gamma_3, \Gamma_4\ \hbox{or}\ \Gamma_5 & \hbox{two edges;} \\
\Gamma_6,\Gamma_7\ \hbox{or}\ \Gamma_8 & \hbox{three edges;} \\
\Gamma_9 & \hbox{four edges.}
\end{array}
\right.$$ However $C_6$ and $\Gamma_1,\ldots, \Gamma_8$ and $\Gamma_9$ are all forbidden subgraphs of $G$ (see Fig. \[fig-3\]).
We complete this proof.
It remains to characterize the graph $G\in \mathcal{G}^*$ satisfying $G[Y]\cong K_{n-t-1}$. Such a graph $G$ we call $X$-*complete* if $G[X]$ is also complete graph, and $X$-*imcomplete* otherwise. The following result characterizes the $X$-imcomplete graphs.
\[lem-3-4\] Let $G[Y]\cong K_{n-t-1}$, where $n-t-1\geq 2$, and $G$ is $X$-imcomplete. Then $G\in \mathcal{G}_1$ if there exist two vertices $x_1\nsim x_2$ in $G[X]$ such that $N_Y(x_1)=N_Y(x_2)$ and $G\in \mathcal{G}_3$ otherwise.
Let $X=\{x_1,x_2,\ldots,x_t\}$ and $Y=\{y_1,y_2,\ldots,y_{n-t-1}\}$. Then $V(G)=\{v^\ast\}\cup X\cup Y$ and $Y$ induces $K_{n-t-1}$. Let $x$ and $x'$ be two non-adjacent vertices in $X$. Since $d_G(x)\ge d_G(v^*)$ and $n-t-1\geq 2$, we have $|N_Y(x)|\ge1$ and $|Y|\ge 2$, respectively. First we give some claims.
\[claim-1\] If $x\not\sim x'$ in $G[X]$ then one of $N_{Y}(x)$ and $ N_{Y}(x')$ includes another. If $N_{Y}(x)\subsetneqq N_{Y}(x')$ then $|N_Y(x)|=1$ and $N_Y(x')=Y$.
On the contrary, let $y\in N_Y(x)\setminus N_Y(x')$ and $y'\in N_Y(x')\setminus N_Y(x)$, then $G[v^*, x, x', y,$ $ y']$ $\cong C_5$. Thus one of $N_{Y}(x)$ and $ N_{Y}(x')$ includes another. Now assume that $N_{Y}(x)\subsetneqq N_{Y}(x')$. If $|N_Y(x)|\geq 2$, say $\{y, y'\}\subseteq N_Y(x)$, then $x'\sim y, y'$ and exists $y^\ast\in N_{Y}(x')\setminus N_{Y}(x)$. Thus $G[v^*, x, x', y, y', y^\ast]\cong \Gamma_{10}$ (see Fig. \[fig-3\]). However $p(\Gamma_{10})=3$. Hence $|N_Y(x)|=1$, and we may assume that $N_Y(x)=\{y\}$. If $N_Y(x')\neq Y$, then there exists $y'\in Y\backslash N_Y(x')$. Also, there exists $y^\ast\in N_{Y}(x')\setminus N_{Y}(x)$. We have $G[v^*, x, x', y, y', y^\ast]\cong \Gamma_4$ (see the labels in the parentheses of Fig. \[fig-3\]), but $p(\Gamma_{4})=3$. Thus $N_Y(x')=Y$.
\[claim-2\] If $x\not\sim x'$ in $G[X]$ then $N_{X}(x)= N_{X}(x')$.
On the contrary, we may assume that $x^\ast\in N_X(x')\backslash N_X(x)$. Then $x^\ast\sim x'$ and $x^\ast\not\sim x$, thus $|N_Y(x)|\geq 2$ since $|N_G(x)|\geq t$. By Claim \[claim-1\], we have $N_Y(x^\ast), N_Y(x')\subseteq N_Y(x)$. Then either $N_Y(x^\ast)=N_Y(x')=N_Y(x)$ or one of $N_Y(x^\ast)$ and $N_Y(x')$ is a proper subset of $N_Y(x)$ (without loss of generality, assume that $N_Y(x^\ast)\subsetneqq N_Y(x)$, and then $|N_Y(x^\ast)|=1$ and $N_Y(x)=Y$ by Claim \[claim-1\]).
Suppose that $N_Y(x)=N_Y(x^\ast)=N_Y(x')$. Take $y, y'\in N_Y(x)$, we see that $G[v^*, x, x^\ast, x',$ $ y, y']$ $\cong \Gamma_{11}$ (see Fig. \[fig-3\]). However $p(\Gamma_{11})=3$.
Suppose that $|N_Y(x^\ast)|=1$ and $N_Y(x)=Y$. Let $N_Y(x^\ast)=\{y\}$ and there exists another $y'\in Y$. Then $G[v^*, x, x^\ast, x', y, y']$ is isomorphic $\Gamma_{13}$ (see Fig. \[fig-3\]) if $x'\sim y, y'$, or isomorphic to $\Gamma_{12}$ (see Fig. \[fig-3\]) if $x'\sim y$ and $x'\nsim y'$, or isomorphic to $\Gamma_{14}$ (see Fig. \[fig-3\]) if $x'\nsim y$ and $x'\sim y'$. However $p(\Gamma_{12})=p(\Gamma_{13})=3$ and $\lambda_4(\Gamma_{14})=0$. We are done.
Now we distinguish the following cases to prove our result. [**Case 1.**]{} There exist $x_1\not\sim x_2$ such that $N_Y(x_1)=N_Y(x_2)$.
Since $x_1\not\sim x_2$, we have $N_{X}(x_1)= N_{X}(x_2)$ by Claim \[claim-2\], so $N_G(x_1)=N_G(x_2)$. Thus $x_1$ and $x_2$ are congruent vertices of I-type. By Lemma \[lem-2-4\], $p(G)=p(G-x_1)$ and $\eta(G)=\eta(G-x_1)+1$. Thus $G-x_1\in \mathcal{H}$ and so $G\in \mathcal{G}_1$. Such a $G$, displayed in Fig. \[fig-4\] (1), we call the $X$-graph of I-type.
[**Case 2.**]{} For each pair of $x\not\sim x'\in X$, $N_Y(x)\not=N_Y(x')$.
By Claim 1, without loss of generality, assume that $N_Y(x)\subsetneqq N_Y(x')$ and then $N_Y(x)=\{y\}$ and $N_Y(x')=Y$. Thus $y\sim x, x'$ and furthermore we will show that $X\subseteq N_G(y)$. In fact, let $x^\ast\in X\setminus \{x, x'\}$( if any ), if $x \not\sim x^\ast$, we have $N_Y(x^\ast)\supseteq N_Y(x)=\{y\}$ by Claim 1. Thus $y\sim x^\ast$. Otherwise, $x\sim x^\ast$ and thus $x'\sim x^\ast$ since $N_X(x)=N_X(x')$ by Claim 2. Now take $y'\in Y\backslash \{y\}$. If $y\nsim x^\ast$, then $G[v^\ast, x, x', x^\ast, y, y']$ is isomorphic to $\Gamma_{12}$ (see the first labels in the parentheses of Fig. \[fig-3\]) while $x^\ast\nsim y'$, or isomorphic to $\Gamma_{13}$ (see the labels in the parentheses of Fig. \[fig-3\]) while $x^\ast\sim y'$, but $p(\Gamma_{12})=p(\Gamma_{13})=3$. It follows that $N_G(y)=X\cup (Y\setminus \{y\})$ since $Y$ induces a clique.
On the other hand, since $d_G(x)\ge |X|=t$, $x\not\sim x'$ and $N_Y(x)=\{y\}$, we have $N_X(x)=X\backslash\{x,x'\}$ and so $N_X(x')=X\backslash\{x,x'\}$ by Claim 2. Thus $N_G(x)=(X\setminus \{x, x'\})\cup \{v^*,y\}$ and $N_G(x')=(X\setminus \{x, x'\})\cup Y\cup \{v^*\}$. Hence the quadrangle $C_4=xv^*x'y$ is congruent, where $xv^*$ and $x'y$ is a pair of congruent edges of $C_4$. It gives that $x, v^*, x', y$ are congruent vertices of III-type. By Lemma \[lem-2-6\], we have $p(G)=p(G-x)$ and $\eta(G)=\eta(G-x)+1$ thus $G-x\in \mathcal{H}$, and so $G\in \mathcal{G}_3$. Such a $G$, displayed in Fig. \[fig-4\] (2), we call the $(v^\ast, X, Y)$-graph of III-type.
We complete this proof.
1.3mm
(83.544,30.661)(0,0) (41.514,28.533) (42.197,30.25)[(0,0)\[cc\][$v^\ast$]{}]{} (44.164,9.622) (39.016,9.622) (41.359,9.487) (40.563,9.487) (42.35,9.487) (41.539,9.94)[(11.247,5.15)\[\]]{} (41.598,20.959)[(17.25,6.375)\[\]]{} (34.234,25.234)[(0,0)\[cc\][$X$]{}]{} (37.702,10.623)[(0,0)\[cc\][$y$]{}]{} (41.572,28.625)(-.03343023,-.05813953)[86]{}[(0,-1)[.05813953]{}]{} (41.572,28.625)(.03365385,-.05769231)[78]{}[(0,-1)[.05769231]{}]{} (42.239,0)[(0,0)\[cc\][(2)]{}]{} (41.697,3.469)[(0,0)\[cc\][$(v^\ast, X, Y)$-graph of III-type]{}]{} (46.292,22.72) (46.292,19.97) (47.322,22.875)[(0,0)\[cc\][$x$]{}]{} (47.322,20.125)[(0,0)\[cc\][$x'$]{}]{} (46.197,20)[(0,-1)[8.125]{}]{} (46.322,20)(-.03358209,-.1119403)[67]{}[(0,-1)[.1119403]{}]{} (38.96,21.482)[(8.125,3.5)\[\]]{} (40.522,21.568) (37.47,21.568) (38.386,21.538) (38.912,21.538) (39.437,21.538) (46.25,22.885)[(-1,0)[4.204]{}]{} (42.045,22.885)(.04947059,-.03338824)[85]{}[(1,0)[.04947059]{}]{} (46.25,20.047)[(0,1)[0]{}]{} (46.25,20.047)[(-1,0)[4.31]{}]{} (41.94,20.047)(.05384146,.03332927)[82]{}[(1,0)[.05384146]{}]{} (8.916,28.377) (9.724,30.266)[(0,0)\[cc\][$v^\ast$]{}]{} (8.451,3.396)[(0,0)\[cc\][$X$-graph of I-type]{}]{} (8.098,.391)[(0,0)\[cc\][(1)]{}]{} (11.191,9.615) (6.043,9.615) (8.386,9.48) (7.59,9.48) (9.377,9.48) (8.566,9.933)[(11.247,5.15)\[\]]{} (8.625,20.952)[(17.25,6.375)\[\]]{} (1.261,25.227)[(0,0)\[cc\][$X$]{}]{} (8.849,28.391)(-.033602151,-.048387097)[93]{}[(0,-1)[.048387097]{}]{} (9.099,28.391)(.0334507,-.06161972)[71]{}[(0,-1)[.06161972]{}]{} (0,13.891)[(0,0)\[cc\][$G[Y]\cong K_{n-t-1}$]{}]{} (12.694,22.736) (12.694,19.986) (14.224,19.766)[(0,0)\[cc\][$x_2$]{}]{} (14.349,22.266)[(0,0)\[cc\][$x_1$]{}]{} (12.724,22.641)(-.033536585,-.06402439)[164]{}[(0,-1)[.06402439]{}]{} (7.224,12.141)(.033536585,.048018293)[164]{}[(0,1)[.048018293]{}]{} (6.493,21.355) (3.441,21.355) (4.357,21.325) (4.883,21.325) (5.408,21.325) (4.826,21.373)[(8.125,3.5)\[\]]{} (4.968,19.623)[(0,-1)[7.358]{}]{} (12.746,22.776)(-.09077273,-.03343182)[44]{}[(-1,0)[.09077273]{}]{} (8.752,21.305)(.10234211,-.03321053)[38]{}[(1,0)[.10234211]{}]{} (74.776,28.648) (75.584,30.661)[(0,0)\[cc\][$v^\ast$]{}]{} (73.097,.286)[(0,0)\[cc\][(3)]{}]{} (73.362,3.38)[(0,0)\[cc\][$(X, Y)$-graph of III-type]{}]{} (78.776,22.308) (78.776,19.558) (74.919,9.97)[(17.25,6.375)\[\]]{} (64.959,26.123)[(0,0)\[cc\][$G[X]\cong K_t$]{}]{} (70.685,8.303) (70.685,10.931) (78.729,22.339)[(0,-1)[2.838]{}]{} (70.635,10.987)[(0,-1)[2.628]{}]{} (78.729,22.234)(-.033708861,-.048345992)[237]{}[(0,-1)[.048345992]{}]{} (78.834,19.606)(-.033725,-.0477375)[240]{}[(0,-1)[.0477375]{}]{} (80.411,22.864)[(0,0)\[cc\][$x$]{}]{} (68.953,11.512)[(0,0)\[cc\][$y$]{}]{} (74.74,20.902)[(17.25,6.375)\[\]]{} (74.709,28.536)(-.033415842,-.045792079)[101]{}[(0,-1)[.045792079]{}]{} (74.709,28.786)(.033653846,-.048076923)[104]{}[(0,-1)[.048076923]{}]{} (80.709,20.286)[(0,0)\[cc\][$x^\prime$]{}]{} (73.243,21.289) (70.191,21.289) (71.107,21.259) (71.633,21.259) (72.158,21.259) (80.496,9.622) (77.444,9.622) (78.36,9.592) (78.886,9.592) (79.411,9.592) (71.718,19.978)(.033716981,-.043132075)[212]{}[(0,-1)[.043132075]{}]{} (70.562,10.939)(.03369231,.11723077)[78]{}[(0,1)[.11723077]{}]{} (73.19,20.083)(-.03358333,-.1635)[72]{}[(0,-1)[.1635]{}]{} (78.656,19.663)(.03350725,-.12795652)[69]{}[(0,-1)[.12795652]{}]{} (80.968,10.834)(-.03336508,.18352381)[63]{}[(0,1)[.18352381]{}]{} (71.682,21.728)[(8.125,3.5)\[\]]{} (78.724,9.325)[(8.125,3.5)\[\]]{} (30.426,13.636)[(0,0)\[cc\][$G[Y]\cong K_{n-t-1}$]{}]{} (63.221,14.266)[(0,0)\[cc\][$G[Y]\cong K_{n-t-1}$]{}]{} (69.059,8.906)[(0,0)\[cc\][$y^\prime$]{}]{} (38.892,9.536)[(0,1)[8.304]{}]{} (38.892,9.641)[(1,2)[4.099]{}]{}
At last we focus on characterizing $X$-complete graph $G\in \mathcal{G}^*$, i.e., $G[X]\cong K_t$ and $G[Y]\cong K_{n-t-1}$. A $X$-complete graph $G\in \mathcal{G}^*$ is called *reduced* if one of $N_Y(x_i)$ and $N_Y(x_j)$ is a subset of another for any $x_i\not=x_j\in X$ and *non-reduced* otherwise. Thus the $X$-complete graphs are partitioned into a disjoint union of the reduced and non-reduced $X$-complete graphs. Concretely, for a reduced $X$-complete graph $G\in \mathcal{G}^*$, we may assume that $\emptyset=N_Y(v^\ast)\subseteq N_Y(x_1)\subseteq N_Y(x_2)\subseteq\cdots\subseteq N_Y(x_t)$; for a non-reduced $(X, Y)$-complete graph $G\in \mathcal{G}^*$, there exist some $x\not=x'\in X$ such that $N_Y(x)\backslash N_Y(x')\not=\emptyset$ and $N_Y(x')\backslash N_Y(x)\not=\emptyset$. Such vertices $x$ and $x'$ are called *non-reduced vertices*. It remains to characterize the reduced and non-reduced $X$-complete graphs in what follows.
\[X-lem-1\] Let $G\in \mathcal{G}^*$ be a non-reduced $X$-complete graph and $x, x'$ be non-reduced vertices. Then $G\in \mathcal{G}_3$.
Since $x, x'$ are non-reduced vertices, there exist $y\in N_Y(x)\setminus N_Y(x')$ and $y'\in N_Y(x')\setminus N_Y(x)$. Then $x,x',y',y$ induces $C_4$ (see Fig. \[fig-4\](3)). It suffices to verify that $C_4$ is congruent. Clearly, $N_G(x)\supset (X\backslash \{x\})\cup \{v^\ast\}$ and $N_G(x') \supset (X\backslash \{x'\})\cup \{v^\ast\}$. If there exists $y^*\in N_Y(x) \setminus N_Y(x')$ other than $y$, then $G[v^\ast, x, x', y', y, y^*] \cong \Gamma_{12}$ (see the second labels in the parentheses of Fig. \[fig-3\]), however $\Gamma_{12}$ is a forbidden subgraph of $G$. Hence $N_Y(x)\setminus N_Y(x')=\{y\}$. Similarly, $N_Y(x')\setminus N_Y(x)=\{y'\}$. On the other aspect, $x\in N_X(y)\setminus N_X(y')$ and $x'\in N_X(y')\setminus N_X(y)$. If there exists $x^\ast\in N_X(y)\setminus N_X(y')$ other than $x$, then $G[v^\ast, x, x', x^\ast, y, y']\cong \Gamma_{10}$ (see the labels in the parentheses of Fig. \[fig-3\]), however $\Gamma_{10}$ is a forbidden subgraph of $G$. Hence $N_X(y)\setminus N_X(y')=\{x\}$. Similarly, $N_X(y')\setminus N_X(y)=\{x'\}$. Hence $N_X(y)\setminus \{x\}=N_X(y')\setminus\{x'\}$. Note that $N_G(y)\supset Y\backslash \{y\}$ and $N_G(y')\supset Y\backslash \{y'\}$, we have $N_G(y)\setminus\{y', x\}=(Y\backslash \{y, y'\})\cup (N_X(y)\setminus\{x\})=N_G(y')\setminus \{x', y\}$. Hence the quadrangle $C_4=xx'y'y$ is congruent, where $xx'$ and $y'y$ is a pair of congruent edges. It follows that $x, x', y', y$ are congruent vertices of III-type. By Lemma \[lem-2-6\], we have $p(G)=p(G-x)$ and $\eta(G)=\eta(G-x)+1$. Thus $G-x\in \mathcal{H}$, and so $G\in \mathcal{G}_3$. Such a $G$, displayed in Fig. \[fig-4\](3), we call the $(X, Y)$-graph of III-type.
We complete this proof.
To characterize the reduced $X$-complete graph, we need the notion of canonical graph which is introduced in [@Per]. For a graph $G$, a relation $\rho$ on $V(G)$ we mean that $u\rho v$ iff $u\sim v$ and $N_G(u)\backslash v=N_G(v)\backslash u$. Clearly, $\rho$ is symmetric and transitive. In accordance with $\rho$, the vertex set is decomposed into classes: $$\label{de-eq-1}V(G)=V_1\cup V_2\cup\cdots\cup V_k$$ where $v_i\in V_i$ and $V_i=\{x\in V(G)\mid x\rho v_i\}$. By definition of $\rho$, $V_i$ induces a clique $K_{n_i}$ where $n_1+n_2+ \cdots+ n_k=n=|V(G)|$, and vertices of $V_i$ join that of $V_j$ iff $v_i\sim v_j$ in $G$. We call the induced subgraph $G[\{v_1,v_2,\ldots,v_k\}]$ as the *canonical graph* of $G$, denoted by $G_c$. Thus $G=G_c[K_{n_1}, K_{n_2},\ldots,K_{n_k}]$ is a *generalized lexicographic product* of $G_c$ (by $K_{n_1}$, $K_{n_2}$,…,$K_{n_k}$).
Let $G$ be a reduced $X$-complete graph. From (\[de-eq-1\]) we have $G=G_c[K_{n_1}, K_{n_2},\ldots,$ $K_{n_k}]$, where $G_c=G[\{v_1, v_2, \ldots,v_k\}]$ and $V_i=\{x\in V(G)\mid x\rho v_i\}$ induces clique $K_{n_i}$. Without loss of generality, assume $v_1=v^*$. Let $X_c=N_{G_c}(v_1)$ and $Y_c=\{v_2,v_3,\ldots,v_k\}\backslash X_c$. Clearly, $G_c[X_c]$ is a clique since $X_c$ is a subset of $X$ and $X$ induces a clique in $G$. Furthermore, $G_c[Y_c]$ is a clique since $Y_c$ is a subset of $Y$ and $Y$ induces a clique in $G$. Thus $G_c$ is also a $X_c$-complete graph. Additionally, since $G$ is reduced, $G_c$ is also reduced. Let $t_c=d_{G_c}(v_1)$ and $X_c=\{x_1,x_2,\ldots,x_{t_c}\}$, $Y_c=\{y_1,y_2, \ldots,y_{k-t_c-1}\}$. We may assume $N_{Y_c}(v_1)\subset N_{Y_c}(x_1)\subset \cdots\subset N_{Y_c}(x_{t_c})$ and $N_{X_c}(y_1)\subset \cdots
\subset N_{X_c}(y_{k-t_c-1})$. Therefore, $$\label{de-eq-2}
0=|N_{Y_c}(v_1)|<|N_{Y_c}(x_1)|< \cdots <|N_{Y_c}(x_{t_c})|\leq |Y_c|=k-t_c-1,$$ and $$\label{de-eq-3}
0\leq |N_{X_c}(y_1)|<|N_{X_c}(y_2)|<\cdots < |N_{X_c}(y_{k-t_c-1})|\leq |X_c|= t_c.$$ From Eq.(\[de-eq-2\]), we have $t_c\leq k-t_c-1$. Similarly, $k-t_c-2\leq t_c$ from Eq.(\[de-eq-3\]). Thus $k-2\leq 2t_c\leq k-1$, and so $t_c =\lceil\frac{k}{2}\rceil-1$.
If $k$ is even, then $t_c=\frac{k}{2}-1$. From Eq. (\[de-eq-2\]), we have $|N_{Y_c}(x_i)|=i$ for $i=1,2,\ldots,t_c$. Thus we may assume that $$\begin{array}{ll}
N_{Y_c}(v_1)=\emptyset, N_{Y_c}(x_1)=\{y_{\frac{k}{2}}\}, \ldots, N_{Y_c}(x_{\frac{n}{2}-2})=\{y_{\frac{k}{2}}, \ldots, y_{3}\}, N_{Y_c}(x_{\frac{n}{2}-1})=\{y_{\frac{k}{2}}, \ldots, y_{2} \}.
\end{array}$$ This implies that $G\cong G_k$ where $G_k$ is defined in section 2. Similarly, $G\cong G_k$ if $k$ is odd. Thus we obtain the following result.
\[lem-X-1\] Let $G$ be a reduced $X$-complete graph. Then $G_c\cong G_k$ where $k\geq 2$ is determined in (\[de-eq-1\]).
Let $G\in \mathcal{G^\ast}$ be a reduced $X$-complete graph. The following lemma gives a characterization for $G$. First we cite a result due to Oboudi in [@M.R.Oboudi2].
\[[@M.R.Oboudi2]\]\[lem-5-1-1\] Let $G=G_3[K_{n_1}, K_{n_2}, K_{n_3}]$, where $n_1, n_2, n_3$ are some positive integers. Then the following hold:\
(1) If $n_1=n_2=n_3=1$, that is $G\cong P_3$, then $\lambda_3(G)=-\sqrt{2}$;\
(2) If $n_1=n_2=1$ and $n_3\geq 2$, then $\lambda_3(G)=-1$;\
(3) If $n_1n_2> 1$, then $\lambda_3(G)=-1$.
We know that any graph $G$ is a generalized lexicographic product of its canonical graph, i.e., $G=G_c[K_{n_1},K_{n_2},\ldots,K_{n_k}]$. We also have $G_c=G_k$ if $G$ is reduced $X$-complete by Lemma \[lem-X-1\]. Furthermore, the following result prove that $4\leq k\leq 13$.
\[thm 4-1\] Let $G\in \mathcal{G^\ast}$ be a reduced $X$-complete graph. Then there exists $4\leq k\leq 13$ such that $G=G_k[K_{n_1},K_{n_2}, \ldots, K_{n_k}]$.
By Lemma \[lem-X-1\], $G=G_k[K_{n_1},K_{n_2}, \ldots, K_{n_k}]$ for some $k$. If $k=1$ or $2$ then $G\cong K_n\not\in \mathcal{G}^*$, and so $k\ge 3$. If $k=3$, then $G=G_3[K_{n_1}, K_{n_2}, K_{n_3}]$. Thus $\lambda_3(G)<0$ by Lemma \[lem-5-1-1\], a contradiction. Hence $k\geq 4$. On the other hand, since $G_c=G_k$ is an induced subgraph of $G$, we have $\lambda_4(G_k)\leq \lambda_4(G)<0$ by Theorem \[thm-2-0\]. Note that $G_{14}$ is an induced subgraph of $G_k$ (by Remark \[re-0\]) for $k\geq 15$, we have $\lambda_4(G_k)\geq \lambda_4(G_{14})
=0$. It implies that $k\leq 13$.
Next we consider the converse of Lemma \[thm 4-1\]. In other words, we will try to find the values of $n_1,\ldots,n_k$ such that $p(G_k[K_{n_1},
\ldots, K_{n_k}])=2$ and $\eta(G_k[K_{n_1}, \ldots, K_{n_k}])=1$, where $4\leq k\leq 13$ and $n=n_1+n_2+\cdots+n_k$. For the simplicity, we use notation in [@M.Petrovi'c] to denote $G_{2s}[K_{n_1}, \ldots, K_{n_{2s}}]=B_{2s}(n_1, \ldots, $ $n_s; n_{s+1}, \ldots, n_{2s})$ and $G_{2s+1}[K_{n_1}, \ldots, K_{n_{2s+1}}]=B_{2s+1}(n_1, \ldots, n_s; n_{s+1}, \ldots, n_{2s};n_{2s+1})$. By Remark 3.2 in [@M.R.Oboudi3], we know $$H_0=B_{2s}(n_1, \ldots,n_s; n_{s+1}, \ldots, n_{2s})\cong B_{2s}(n_{s+1}\ldots,n_{2s};n_1, \ldots, n_s)=H_0'\ \ \mbox{ and }$$ $$H_1=B_{2s+1}(n_1, \ldots, n_s; n_{s+1}, \ldots, n_{2s}; n_{2s+1})\cong B_{2s+1}(n_{s+1}, \ldots, n_{2s}; n_1, \ldots, n_s; n_{2s+1})=H_1'.$$ In what follows, we always take $H_0$ and $H_1$, in which $(n_1$ $,\ldots, n_s)$ is prior to $(n_{s+1}, \ldots, n_{2s})$ in dictionary ordering, instead of $H_0'$ and $H_1'$. For example we use $B_6(4,3,2;$ $ 4,3,1)$ instead of $B_6(4,3,1; 4,3,2)$ and $B_7(5,3,2; 5,2,4; 8)$ instead of $B_7(5,2,4; 5,3,2; 8)$.
For $4\leq k\leq 13$, let $\mathcal{B}_k(n)=\{G=B_k(n_1,\ldots, n_k)\mid n=n_1+\cdots +n_k, n_i\geq 1 \}$. Let $\mathcal{B}_k^+(n)$, $\mathcal{B}_k^{00}(n)$, $\mathcal{B}_k^0(n)$ and $\mathcal{B}_k^-(n)$ denote the set of graphs in $\mathcal{B}_k(n)$ satisfying $\lambda_3(G)>0$ for $G\in\mathcal{B}_k^+(n)$, $\lambda_4(G)=\lambda_3(G)=0$ for $G\in\mathcal{B}_k^{00}(n)$, $\lambda_4(G)<\lambda_3(G)=0$ for $G\in\mathcal{B}_k^0(n)$ and $\lambda_3(G)<0$ for $G\in\mathcal{B}_k^-(n)$, respectively. Clearly, $\mathcal{B}_k(n)=\mathcal{B}_k^+(n)\cup \mathcal{B}_k^{00}(n)\cup \mathcal{B}_k^0(n)\cup \mathcal{B}_k^-(n)$ is disjoint union and $G=G_k[K_{n_1},K_{n_2}, \ldots, K_{n_k}]\in\mathcal{B}^{0}_k(n)$ if $G\in \mathcal{G}^\ast$ is a reduced $X$-complete graph by Lemma \[thm 4-1\]. In what follows, we further show that $n\le 13$. First, one can verify the following result by using computer.
\[lem-6-1\] $\mathcal{B}_k^{0}(14)=\emptyset$ for $4 \le k\le 13$ (it means that there are no reduced $X$-complete graphs of order $14$).
For $4 \le k\le 13$, the $k$-partition of $14$ gives a solution $(n_1,n_2,\ldots,n_k)$ of the equation $n_1+n_2+\cdots+n_k=14$ that corresponds a graph $G=B_k(n_1,n_2,\ldots,n_k)\in \mathcal{B}_k(14)$. By using computer, we exhaust all the graphs of $\mathcal{B}_k(14)$ to find that there is no any graph $G\in \mathcal{B}_k(14)$ with $\lambda_4(G)<\lambda_3(G)=0$. It implies that $\mathcal{B}_k^{0}(14)=\emptyset$.
In [@M.R.Oboudi3], Oboudi gave all the integers $n_1, \ldots, n_k$ satisfying $\lambda_2(B_k(n_1, \ldots, n_k))>0$ and $\lambda_3(B_k(n_1,
\ldots, n_k))<0$ for $4\leq k\leq 9$. For simplicity, we only cite this result for $k=5$ and the others are listed in Appendix $B$.
\[thm-6-1\] Let $G=B_5(n_1, n_2; n_3, n_4; n_5)$, where $n_1, n_2, n_3, n_4, n_5$ are some positive integers. Then $\lambda_2(G)>0$ and $\lambda_3(G)<0$ if and only if $G$ is isomorphic to one of the following graphs:\
(1) $B_5(a,w;1,1;1)$; (2) $B_5(a,x;1,d;1)$; (3) $B_5(a,x;1,y;z)$; (4) $B_5(a,x;1,1;e)$;\
(5) $B_5(a,1;c,1;e)$; (6) $B_5(a,1;x,w;1)$; (7) $B_5(a,1;x,y;e)$; (8) $B_5(a,1;1,d;e)$;\
(9) $B_5(w,x;y,1;e)$; (10) $B_5(x,b;1,1;1)$; (11) $B_5(x,w;1,d;1)$; (12) $B_5(x,w;1,1;e)$;\
(13) $B_5(1,b;1,d;1)$; (14) $B_5(1,b;1,x;y)$; (15) $B_5(1,x;1,y;e)$;\
(16) 63 specific graphs: 13 graphs of order 10, 25 graphs of order 11, and 25 graphs of order 12, where $a,b,c,d,e,x,y,z,w$ are some positive integers such that $x\leq 2$, $y\leq 2$, $z\leq 2$ and $w\leq 3$.
\[lem-6-5\] Let $G\in \mathcal{B}_k(n)$, where $4 \le k\le 9$ and $n\geq 14$. If $G\notin \mathcal{B}_k^-(n)$, then $G$ has an induced subgraph $\Gamma\in \mathcal{B}_k(14)\setminus \mathcal{B}_k^-(14)$.
We prove this Lemma by induction on $n$. If $n=14$, since $G\in\mathcal{B}_k(14)\setminus \mathcal{B}_k^-(14)$, our result is obviously true by taking $\Gamma=G$. Let $n\geq 15$ and $G'\in \mathcal{B}_k(n-1)$ be an induced subgraph of $G$. If $G'\notin \mathcal{B}_k^-(n-1)$, then $G'$ has an induced subgraph $\Gamma\in \mathcal{B}_k(14)\setminus \mathcal{B}_k^-(14)$ by induction hypothesis, and so does $G$. Hence it suffices to prove that $G$ contains an induced subgraph $G'\in \mathcal{B}_k(n-1)\setminus \mathcal{B}_k^-(n-1)$ for $n\geq 15$ in the following. We will prove that there exists $G'\in \mathcal{B}_5(n-1)\setminus \mathcal{B}_5^-(n-1)$ for $n\geq 15$, and it can be similarly proved for the other $k$ which we keep in the Appendix $B$.
Let $G=B_5(n_1, n_2; n_3, n_4; n_5)\in \mathcal{B}_5(n)$. Then one of $H_1=B_5(n_1-1, n_2; n_3, n_4; n_5)$, $H_2=B_5(n_1, n_2-1; n_3, n_4; n_5)$, $H_3=B_5(n_1, n_2;$ $n_3-1, n_4; n_5)$, $H_4=B_5(n_1, n_2; n_3, n_4-1; n_5)$ and $H_5=B_5(n_1, n_2; n_3, n_4; n_5-1)$ must belong to $\mathcal{B}_5(n-1)$. On the contrary, assume that $H_i\in \mathcal{B}_5^-(n-1)$ for $i=1,2,\ldots,5$. Then $H_i$ is a graph belonging to (1)–(15) in Theorem \[thm-6-1\] since $|H_i|=n-1\geq 14$.
First we consider $H_1$. If $H_1$ is a graph belonging to (1) of Theorem \[thm-6-1\], then $H_1=B_5(a,w;1,1;1)$ where $n_1-1=a$, $n_2=w$, $n_3=n_4=n_5=1$, and hence $G=B_5(a+1,w;1,1;1)\in \mathcal{B}_5^-(n)$, a contradiction. Similarly, $H_1$ cannot belong to (2)–(8) of Theorem \[thm-6-1\]. If $H_1$ is a graph belonging to (9) of Theorem \[thm-6-1\], then $H_1=B_5(w,x;y,1;e)$ where $n_1-1=w$, $n_2=x$, $n_3=y$, $n_4=1$, $n_5=e$. Since $w\le 3$, we have $n_1\leq 4$. If $n_1<4$ then $w+1\leq 3$ and $G=B_5(w+1,x;y,1;e)\in \mathcal{B}_5^-(n)$, a contradiction. Now assume that $n_1=4$. Then $H_1=B_5(3,x;y,1;e)$. Since $x,y\in \{1, 2\}$, we have $G\in \{B_5(4, 1; 1,1; e)$, $B_5(4, 2; 1,1; e)$, $B_5(4, 1; 2,$ $1; e)$, $B_5(4, 2; 2,1; e)\}$. However $B_5(4, 1; 1,1; e)$, $B_5(4, 2; 1,1; $ $e)$, $B_5(4, 1; 2,$ $1; e)$ belong to (4), (5) of Theorem \[thm-6-1\] which contradicts our assumption. Thus $G=B_5(4, 2; 2,1; e)$. By Theorem \[thm-6-1\], $G=B_5(4, 2; 2,1; e)\not\in \mathcal{B}_5^-(n)$, and also its induced subgraph $B_5(4, 2; 2,1; e-1)\notin \mathcal{B}_5^-(n-1)$, a contradiction. Hence $H_1$ belongs to (10)–(15) of Theorem \[thm-6-1\], from which we see that $n_1-1$ is either $x$ or 1. Thus $n_1\leq 3$ due to $x\leq 2$.
By the same method, we can verify that $n_2\leq 3$ if $H_2\in \mathcal{B}_5^-(n-1)$; $n_3\leq 3$ if $H_3\in \mathcal{B}_5^-(n-1)$; $n_4\leq 3$ if $H_4\in \mathcal{B}_5^-(n-1)$ and $n_5\leq 2$ if $H_5\in \mathcal{B}_5^-(n-1)$. Hence $n=n_1+\cdots+n_5\leq 14$, a contradiction. We are done.
\[lem-5-7\] If $n\geq 14$, then $\mathcal{B}_k^-(n)=\emptyset$ for $10\leq k\leq 13$.
\[thm-6-3\] Given $4\le k\le13$, $\mathcal{B}_k^0(n)=\emptyset$ for $n\geq 14$ (it means that there are no reduced $X$-complete graphs of order $n\ge14$).
Let $G\in \mathcal{B}^0_k(n)$ and $n\geq 14$. Then $\lambda_4(G)<\lambda_3(G)=0$. First we assume that $4\le k\le 9$. Since $G\notin \mathcal{B}_k^-(n)$, $G$ has an induced subgraphs $\Gamma\in \mathcal{B}_k(14)\setminus \mathcal{B}_k^-(14)$ by Lemma \[lem-6-5\]. Thus $\lambda_3(\Gamma)\ge 0$. Furthermore, we have $\lambda_3(\Gamma)=0$ since otherwise $0<\lambda_3(\Gamma)\le \lambda_3(G)$. Additionally, $\lambda_4(\Gamma)\le\lambda_4(G)<0$, we have $\Gamma\in \mathcal{B}_k^{0}(14)$, contrary to Lemma \[lem-6-1\]. Next we assume that $10\le k\le 13$. By deleting $n-14$ vertices from $G$, we may obtain an induced subgraph $\Gamma\in \mathcal{B}_{k}(14)$. By Lemma \[lem-5-7\], we have $\lambda_3(\Gamma)\geq 0$, and then $\lambda_3(\Gamma)=0$ by the arguments above. Additionally, $\lambda_4(\Gamma)\le\lambda_4(G)<0$, we have $\Gamma\in \mathcal{B}_k^{0}(14)$ which also contradicts Lemma \[lem-6-1\].
By Lemma \[thm-6-3\], we know that, for any reduced $X$-complete graph $G\in \mathcal{G}^\ast$, there exists $4\le k\le 13$ and $n\le 13$ such that $G\in\mathcal{B}_k^0(n)$. Let $$\mathcal{B}^\ast=\{G=B_k(n_1,n_2, \ldots,n_k)\in \mathcal{B}_k^0(n)\mid 4\le k\le 13 \mbox{ and } n\le 13\}.$$ Thus $G\in \mathcal{G}^\ast$ is a reduced $X$-complete graph if and only if $G\in \mathcal{B}^\ast$.
[r|c|l]{} $k$ & $\mathcal{B}^\ast$ & Number\
$k=4$&
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$B_4(3,2; 3,2)$; $B_4(4,3; 2,2)$, $B_4(4,3; 3,1)$; $B_4(5,4; 2,1)$, $B_4(5,2; 2,3)$, $B_4(3,4; 2,3)$, $B_4(4,1; 3,4)$,
$B_4(5,2; 4,1)$; $B_4(7,3; 2,1)$, $B_4(4,6; 2,1)$, $B_4(7,2; 2,2)$, $B_4(3,6; 2,2)$, $B_4(4,2; 2,5)$, $B_4(3,3; 2,5)$,
$B_4(7,2; 3,1)$, $B_4(3,6; 3,1)$, $B_4(6,1; 3,3)$, $B_4(6,1; 4,2)$.
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: All graphs of $\mathcal{B}^\ast$.[]{data-label="tab-1"}
& 18\
$k=5$ &
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$B_5(2,2; 2,2; 1)$; $B_5(2,3; 1,2; 2)$, $B_5(3,3; 2,1; 1)$; $B_5(3,4; 1,1; 2)$, $B_5(3,4; 1,2; 1)$, $B_5(1,3; 1,3; 3)$,
$B_5(2,2; 1,3; 3)$, $B_5(2,4; 2,1; 2)$, $B_5(4,2; 3,1; 1)$; $B_5(4,5; 1,1; 1)$, $B_5(2,5; 1,1; 3)$, $B_5(4,3; 1,1; 3)$,
$B_5(1,4; 1,2; 4)$, $B_5(3,2; 1,2; 4)$, $B_5(2,5; 1,3; 1)$, $B_5(4,3; 1,3; 1)$, $B_5(1,4; 1,4; 2)$, $B_5(3,2; 1,4; 2)$,
$B_5(5,2; 2,1; 2)$, $B_5(3,1; 2,3; 3)$, $B_5(3,1; 2,5; 1)$, $B_5(4,1; 3,2; 2)$; $B_5(3,7; 1,1; 1)$, $B_5(6,4; 1,1; 1)$,
$B_5(2,7; 1,1; 2)$, $B_5(6,3; 1,1; 2)$, $B_5(2,4; 1,1; 5)$, $B_5(3,3; 1,1; 5)$, $B_5(2,7; 1,2; 1)$, $B_5(6,3; 1,2; 1)$,
$B_5(1,6; 1,2; 3)$, $B_5(5,2; 1,2; 3)$, $B_5(1,3; 1,2; 6)$, $B_5(2,2; 1,2; 6)$, $B_5(1,6; 1,3; 2)$, $B_5(5,2; 1,3; 2)$,
$B_5(2,4; 1,5; 1)$, $B_5(3,3; 1,5; 1)$, $B_5(2,2; 1,6; 2)$, $B_5(2,7; 2,1; 1)$, $B_5(7,2; 2,1; 1)$, $B_5(4,2; 2,1; 4)$,
$B_5(2,3; 2,1; 5)$, $B_5(5,1; 2,3; 2)$, $B_5(5,1; 2,4; 1)$, $B_5(3,2; 3,1; 4)$, $B_5(6,1; 3,2; 1)$.
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: All graphs of $\mathcal{B}^\ast$.[]{data-label="tab-1"}
& 47\
$k=6$ & See Table 2 of Appendix A& 138\
$k=7$ & See Table 3 of Appendix A& 161\
$k=8$ & See Table 4 of Appendix A& 205\
$k=9$ & See Table 5 of Appendix A& 124\
$k=10$ & See Table 6 of Appendix A& 78\
$k=11$&
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$B_{11}(1,1,1,2,1; 1,1,1,1,1;1)$, $B_{11}(2,1,1,1,1; 1,1,1,1,1;1)$; $B_{11}(1,1,1,1,3; 1,1,1,1,1;1)$,
$B_{11}(1,1,1,2,2; 1,1,1,1,1;1)$, $B_{11}(1,1,2,1,2; 1,1,1,1,1;1)$, $B_{11}(1,1,2,2,1; 1,1,1,1,1;1)$,
$B_{11}(1,1,3,1,1; 1,1,1,1,1;1)$, $B_{11}(1,2,1,1,2; 1,1,1,1,1;1)$, $B_{11}(1,2,2,1,1; 1,1,1,1,1;1)$,
$B_{11}(1,3,1,1,1; 1,1,1,1,1;1)$, $B_{11}(2,1,1,1,2; 1,1,1,1,1;1)$, $B_{11}(2,2,1,1,1; 1,1,1,1,1;1)$,
$B_{11}(1,1,1,1,2; 1,1,1,1,1;2)$, $B_{11}(1,1,2,1,1; 1,1,1,1,1;2)$, $B_{11}(1,2,1,1,1; 1,1,1,1,1;2)$,
$B_{11}(1,1,1,1,1; 1,1,1,1,1;3)$, $B_{11}(1,1,1,1,2; 1,1,1,1,2;1)$, $B_{11}(1,1,1,2,1; 1,1,1,1,2;1)$,
$B_{11}(1,1,2,1,1; 1,1,1,1,2;1)$, $B_{11}(1,2,1,1,1; 1,1,1,1,2;1)$, $B_{11}(1,1,2,1,1; 1,1,2,1,1;1)$,
$B_{11}(1,2,1,1,1; 1,1,2,1,1;1)$, $B_{11}(2,1,1,1,1; 1,1,2,1,1;1)$, $B_{11}(1,2,1,1,1; 1,2,1,1,1;1)$.
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: All graphs of $\mathcal{B}^\ast$.[]{data-label="tab-1"}
& 24\
$k=12$ &
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$B_{12}(1,1,1,1,1,2;1,1,1,1,1,1)$, $B_{12}(1,1,1,1,2,1;1,1,1,1,1,1)$, $B_{12}(1,1,1,2,1,1;1,1,1,1,1,1)$,
$B_{12}(1,1,2,1,1,1;1,1,1,1,1,1)$, $B_{12}(1,2,1,1,1,1;1,1,1,1,1,1)$, $B_{12}(2,1,1,1,1,1;1,1,1,1,1,1)$.
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: All graphs of $\mathcal{B}^\ast$.[]{data-label="tab-1"}
& 6\
$k=13$ & $B_{13}(1,1,1,1,1,1;1,1,1,1,1,1;1)$.& 1\
\[re-5-1\] Clearly, $\mathcal{B}=\cup_{4\le k\le 13, n\le 13}\mathcal{B}_k(n)$ contains finite graphs. By using computer we can exhaust all the graphs of $\mathcal{B}$ to find out the graphs in $\mathcal{B}^*$. We list them in Tab. \[tab-1\].
Recall that $\mathcal{G}_1$, $\mathcal{G}_2$ and $\mathcal{G}_3$ are the set of connected graphs each of them is obtained from some $H\in \mathcal{H}$ by adding one vertex of I, II, III-type, respectively. Summarizing Lemmas \[lem-3-3\], \[lem-3-2\], \[lem-3-4\], \[thm-6-3\] and Theorems \[cor-3-0\], finally we give the characterization of the connected graphs in $\mathcal{G}$.
\[thm 5-1\] Let $G$ be a connected graph of order $n\ge 5$. Then $G\in \mathcal{G}$ if and only if $G$ is isomorphic to one of the following graphs listed in (1), (2) and (3):\
(1) $K_{1, 2}(u)\odot^k K_{n-3}$ or $K_{1, 1}(u)\odot^k K_{n-2}\setminus e$ for $e\in E(K_{n-2})$;\
(2) the graphs belonging to $\mathcal{G}_1, \mathcal{G}_2$ or $\mathcal{G}_3$;\
(3) the 802 specific graphs belonging to $\mathcal{B}^\ast$ some of which we list in Tab. \[tab-1\].
If $G^\ast$ is obtained from $G\in \mathcal{G}$ by adding one vertex of I, II or III-type, then the positive and negative indices of $G^\ast$ left unchanged, but the nullity adds just one. Repeating this process, we can get a class of graphs which has two positive eigenvalues and $s$ zero eigenvalues, where $s\ge 2$ is any integer. However, by using the I, II and III-type (graph) transformations, we can not get all such graphs. For example, $H=B_{10}(1,1,2,3,2; 1,1,1,1,1)$ is a graph satisfying $p(H)=2$ and $\eta(H)=2$ that can not be constructed by above (graph) transformation. Hence the characterization of graphs with $p(H)=2$ and $\eta(H)=s$ (especially $\eta(H)=2$) is also an attractive problem.
Five tables
===========
Appendix A contains 5 tables, in which there are 706 specific graphs: 4 graphs of order 10, 32 graphs of order 11, 150 graphs of order 12, 520 graphs of order 13.
[l@l@]{} $10$&$B_6(1,2,2; 1,2,2)$, $B_6(2,2,1; 1,2,2)$;\
$11$ &$B_6(1,3,3; 1,1,2)$, $B_6(2,3,2; 1,1,2)$, $B_6(3,3,1; 1,1,2)$, $B_6(1,3,3; 1,2,1)$, $B_6(2,3,2; 1,2,1)$, $B_6(3,3,1; 1,2,1)$,\
$11$ &$B_6(2,1,1; 1,3,3)$, $B_6(3,2,1; 2,1,2)$, $B_6(2,2,2; 2,2,1)$, $B_6(3,1,2; 3,1,1)$;\
$12$ &$B_6(1,4,4; 1,1,1)$, $B_6(2,4,3; 1,1,1)$, $B_6(3,4,2; 1,1,1)$, $B_6(4,4,1; 1,1,1)$, $B_6(1,2,4; 1,1,3)$, $B_6(1,4,2; 1,1,3)$,\
$12$ &$B_6(2,2,3; 1,1,3)$, $B_6(2,4,1; 1,1,3)$, $B_6(3,2,2; 1,1,3)$, $B_6(4,2,1; 1,1,3)$, $B_6(1,3,1; 1,2,4)$, $B_6(2,1,2; 1,2,4)$,\
$12$ &$B_6(3,1,1; 1,2,4)$, $B_6(1,4,2; 1,3,1)$, $B_6(2,2,3; 1,3,1)$, $B_6(2,4,1; 1,3,1)$, $B_6(3,2,2; 1,3,1)$, $B_6(4,2,1; 1,3,1)$,\
$12$ &$B_6(2,1,2; 1,4,2)$, $B_6(3,1,1; 1,4,2)$, $B_6(2,3,3; 2,1,1)$, $B_6(4,1,3; 2,1,1)$, $B_6(4,3,1; 2,1,1)$, $B_6(2,3,2; 2,1,2)$,\
$12$ &$B_6(3,2,2; 2,1,2)$, $B_6(4,1,2; 2,1,2)$, $B_6(2,3,1; 2,1,3)$, $B_6(4,1,1; 2,1,3)$, $B_6(3,1,3; 2,2,1)$, $B_6(3,2,2; 2,2,1)$,\
$12$ &$B_6(3,3,1; 2,2,1)$, $B_6(3,1,1; 2,2,3)$, $B_6(2,3,1; 2,3,1)$, $B_6(3,2,2; 3,1,1)$, $B_6(4,2,1; 3,1,1)$, $B_6(4,1,1; 4,1,1)$;\
$13$ &$B_6(1,3,6; 1,1,1)$, $B_6(1,6,3; 1,1,1)$, $B_6(2,3,5; 1,1,1)$, $B_6(2,6,2; 1,1,1)$, $B_6(3,3,4; 1,1,1)$, $B_6(3,6,1; 1,1,1)$,\
$13$ &$B_6(4,3,3; 1,1,1)$, $B_6(5,3,2; 1,1,1)$, $B_6(6,3,1; 1,1,1)$, $B_6(1,2,6; 1,1,2)$, $B_6(1,6,2; 1,1,2)$, $B_6(2,2,5; 1,1,2)$,\
$13$ &$B_6(2,6,1; 1,1,2)$, $B_6(3,2,4; 1,1,2)$, $B_6(4,2,3; 1,1,2)$, $B_6(5,2,2; 1,1,2)$, $B_6(6,2,1; 1,1,2)$, $B_6(1,2,3; 1,1,5)$,\
$13$ &$B_6(1,3,2; 1,1,5)$, $B_6(2,2,2; 1,1,5)$, $B_6(2,3,1; 1,1,5)$, $B_6(3,2,1; 1,1,5)$, $B_6(1,2,6; 1,2,1)$, $B_6(1,6,2; 1,2,1)$,\
$13$ &$B_6(2,2,5; 1,2,1)$, $B_6(2,6,1; 1,2,1)$, $B_6(3,2,4; 1,2,1)$, $B_6(4,2,3; 1,2,1)$, $B_6(5,2,2; 1,2,1)$, $B_6(6,2,1; 1,2,1)$,\
$13$ &$B_6(1,5,1; 1,2,3)$, $B_6(2,1,4; 1,2,3)$, $B_6(3,1,3; 1,2,3)$, $B_6(4,1,2; 1,2,3)$, $B_6(5,1,1; 1,2,3)$, $B_6(2,1,1; 1,2,6)$,\
$13$ &$B_6(1,5,1; 1,3,2)$, $B_6(2,1,4; 1,3,2)$, $B_6(3,1,3; 1,3,2)$, $B_6(4,1,2; 1,3,2)$, $B_6(5,1,1; 1,3,2)$, $B_6(2,2,2; 1,5,1)$,\
$13$ &$B_6(2,3,1; 1,5,1)$, $B_6(3,2,1; 1,5,1)$, $B_6(2,1,1; 1,6,2)$, $B_6(2,2,5; 2,1,1)$, $B_6(2,5,2; 2,1,1)$, $B_6(3,1,5; 2,1,1)$,\
$13$ &$B_6(3,2,4; 2,1,1)$, $B_6(3,3,3; 2,1,1)$, $B_6(3,4,2; 2,1,1)$, $B_6(3,5,1; 2,1,1)$, $B_6(4,2,3; 2,1,1)$, $B_6(4,3,2; 2,1,1)$,\
$13$ &$B_6(5,2,2; 2,1,1)$, $B_6(6,1,2; 2,1,1)$, $B_6(6,2,1; 2,1,1)$, $B_6(2,2,4; 2,1,2)$, $B_6(2,5,1; 2,1,2)$, $B_6(3,1,4; 2,1,2)$,\
$13$ &$B_6(3,2,3; 2,1,2)$, $B_6(6,1,1; 2,1,2)$, $B_6(2,2,3; 2,1,3)$, $B_6(3,1,3; 2,1,3)$, $B_6(2,2,2; 2,1,4)$, $B_6(3,1,2; 2,1,4)$,\
$13$ &$B_6(2,2,1; 2,1,5)$, $B_6(3,1,1; 2,1,5)$, $B_6(2,5,1; 2,2,1)$, $B_6(4,2,2; 2,2,1)$, $B_6(5,1,2; 2,2,1)$, $B_6(5,2,1; 2,2,1)$,\
$13$ &$B_6(3,1,3; 2,2,2)$, $B_6(4,1,2; 2,2,2)$, $B_6(5,1,1; 2,2,2)$, $B_6(3,1,2; 2,2,3)$, $B_6(4,1,2; 2,3,1)$, $B_6(4,2,1; 2,3,1)$,\
$13$ &$B_6(3,1,2; 2,3,2)$, $B_6(4,1,1; 2,3,2)$, $B_6(3,1,2; 2,4,1)$, $B_6(3,2,1; 2,4,1)$, $B_6(3,1,1; 2,4,2)$, $B_6(3,3,2; 3,1,1)$,\
$13$ &$B_6(3,4,1; 3,1,1)$, $B_6(6,1,1; 3,1,1)$, $B_6(3,3,1; 3,2,1)$, $B_6(4,2,1; 3,2,1)$, $B_6(5,1,1; 3,2,1)$, $B_6(4,1,1; 3,3,1)$.\
[l@l@]{} $10$ & $B_7(2,2,1; 1,1,2;1)$, $B_7(2,1,2; 2,1,1;1)$;\
$11$ &$B_7(3,3,1; 1,1,1;1)$, $B_7(2,1,3; 1,1,1;2)$, $B_7(2,2,2; 1,1,2;1)$, $B_7(2,1,2; 1,1,2;2)$, $B_7(1,2,1; 1,1,3;2)$, $B_7(2,1,1; 1,1,3;2)$,\
$11$ &$B_7(1,2,3; 1,2,1;1)$, $B_7(1,2,2; 1,2,2;1)$, $B_7(2,1,1; 1,2,3;1)$, $B_7(2,2,2; 2,1,1;1)$, $B_7(3,2,1; 2,1,1;1)$, $B_7(3,1,1; 3,1,1;1)$;\
$12$ &$B_7(1,3,4; 1,1,1;1)$, $B_7(3,1,4; 1,1,1;1)$, $B_7(3,3,2; 1,1,1;1)$, $B_7(1,2,4; 1,1,1;2)$, $B_7(2,2,3; 1,1,1;2)$, $B_7(2,4,1; 1,1,1;2)$,\
$12$ &$B_7(3,2,2; 1,1,1;2)$, $B_7(4,2,1; 1,1,1;2)$, $B_7(1,1,4; 1,1,1;3)$, $B_7(3,1,2; 1,1,1;3)$, $B_7(1,3,3; 1,1,2;1)$, $B_7(2,2,3; 1,1,2;1)$,\
$12$ &$B_7(3,1,3; 1,1,2;1)$, $B_7(1,1,3; 1,1,2;3)$, $B_7(1,3,1; 1,1,2;3)$, $B_7(3,1,1; 1,1,2;3)$, $B_7(1,3,2; 1,1,3;1)$, $B_7(3,1,2; 1,1,3;1)$,\
$12$ &$B_7(1,2,2; 1,1,3;2)$, $B_7(1,3,1; 1,1,4;1)$, $B_7(3,1,1; 1,1,4;1)$, $B_7(2,1,4; 1,2,1;1)$, $B_7(2,2,3; 1,2,1;1)$, $B_7(2,3,2; 1,2,1;1)$,\
$12$ &$B_7(2,4,1; 1,2,1;1)$, $B_7(4,2,1; 1,2,1;1)$, $B_7(2,1,2; 1,2,1;3)$, $B_7(1,2,2; 1,2,2;2)$, $B_7(1,3,1; 1,2,2;2)$, $B_7(2,1,2; 1,2,2;2)$,\
$12$ &$B_7(3,1,1; 1,2,2;2)$, $B_7(2,1,2; 1,2,3;1)$, $B_7(1,3,2; 1,3,1;1)$, $B_7(3,1,1; 1,3,2;1)$, $B_7(2,3,2; 2,1,1;1)$, $B_7(2,3,1; 2,1,1;2)$,\
$12$ &$B_7(4,1,1; 2,1,1;2)$, $B_7(3,2,1; 2,2,1;1)$, $B_7(3,1,1; 2,2,1;2)$;\
$13$ &$B_7(1,2,6; 1,1,1;1)$, $B_7(1,5,3; 1,1,1;1)$, $B_7(2,1,6; 1,1,1;1)$, $B_7(2,2,5; 1,1,1;1)$, $B_7(2,3,4; 1,1,1;1)$, $B_7(2,4,3; 1,1,1;1)$,\
$13$ &$B_7(2,5,2; 1,1,1;1)$, $B_7(2,6,1; 1,1,1;1)$, $B_7(3,2,4; 1,1,1;1)$, $B_7(3,3,3; 1,1,1;1)$, $B_7(4,2,3; 1,1,1;1)$, $B_7(5,1,3; 1,1,1;1)$,\
$13$ &$B_7(5,2,2; 1,1,1;1)$, $B_7(6,2,1; 1,1,1;1)$, $B_7(1,1,6; 1,1,1;2)$, $B_7(1,4,3; 1,1,1;2)$, $B_7(2,3,3; 1,1,1;2)$, $B_7(2,4,2; 1,1,1;2)$,\
$13$ &$B_7(5,1,2; 1,1,1;2)$, $B_7(1,3,3; 1,1,1;3)$, $B_7(2,3,2; 1,1,1;3)$, $B_7(1,2,3; 1,1,1;4)$, $B_7(2,2,2; 1,1,1;4)$, $B_7(2,3,1; 1,1,1;4)$,\
$13$ &$B_7(3,2,1; 1,1,1;4)$, $B_7(1,1,3; 1,1,1;5)$, $B_7(2,1,2; 1,1,1;5)$, $B_7(1,2,5; 1,1,2;1)$, $B_7(1,5,2; 1,1,2;1)$, $B_7(2,1,5; 1,1,2;1)$,\
$13$ &$B_7(2,2,4; 1,1,2;1)$, $B_7(5,1,2; 1,1,2;1)$, $B_7(1,1,5; 1,1,2;2)$, $B_7(1,2,4; 1,1,2;2)$, $B_7(1,3,3; 1,1,2;2)$, $B_7(1,4,2; 1,1,2;2)$,\
$13$ &$B_7(1,5,1; 1,1,2;2)$, $B_7(5,1,1; 1,1,2;2)$, $B_7(1,2,3; 1,1,2;3)$, $B_7(1,3,2; 1,1,2;3)$, $B_7(1,2,2; 1,1,2;4)$, $B_7(1,1,2; 1,1,2;5)$,\
$13$ &$B_7(1,2,1; 1,1,2;5)$, $B_7(2,1,1; 1,1,2;5)$, $B_7(1,2,4; 1,1,3;1)$, $B_7(1,5,1; 1,1,3;1)$, $B_7(2,1,4; 1,1,3;1)$, $B_7(5,1,1; 1,1,3;1)$,\
$13$ &$B_7(1,1,4; 1,1,3;2)$, $B_7(1,2,3; 1,1,3;2)$, $B_7(1,2,3; 1,1,4;1)$, $B_7(2,1,3; 1,1,4;1)$, $B_7(1,2,2; 1,1,5;1)$, $B_7(2,1,2; 1,1,5;1)$,\
$13$ &$B_7(1,2,1; 1,1,6;1)$, $B_7(2,1,1; 1,1,6;1)$, $B_7(1,5,2; 1,2,1;1)$, $B_7(3,2,3; 1,2,1;1)$, $B_7(4,1,3; 1,2,1;1)$, $B_7(4,2,2; 1,2,1;1)$,\
$13$ &$B_7(1,4,2; 1,2,1;2)$, $B_7(2,3,2; 1,2,1;2)$, $B_7(3,2,2; 1,2,1;2)$, $B_7(4,1,2; 1,2,1;2)$, $B_7(1,3,2; 1,2,1;3)$, $B_7(2,2,2; 1,2,1;3)$,\
$13$ &$B_7(2,3,1; 1,2,1;3)$, $B_7(3,2,1; 1,2,1;3)$, $B_7(1,2,2; 1,2,1;4)$, $B_7(1,5,1; 1,2,2;1)$, $B_7(2,1,4; 1,2,2;1)$, $B_7(3,1,3; 1,2,2;1)$,\
$13$ &$B_7(4,1,2; 1,2,2;1)$, $B_7(5,1,1; 1,2,2;1)$, $B_7(1,2,2; 1,2,2;3)$, $B_7(2,1,1; 1,2,2;4)$, $B_7(2,1,3; 1,2,3;1)$, $B_7(3,1,3; 1,3,1;1)$,\
$13$ &$B_7(3,2,2; 1,3,1;1)$, $B_7(2,2,2; 1,3,1;2)$, $B_7(2,3,1; 1,3,1;2)$, $B_7(3,1,2; 1,3,1;2)$, $B_7(3,2,1; 1,3,1;2)$, $B_7(2,1,3; 1,3,2;1)$,\
$13$ &$B_7(3,1,2; 1,3,2;1)$, $B_7(2,1,2; 1,3,2;2)$, $B_7(2,1,1; 1,3,2;3)$, $B_7(2,1,3; 1,4,1;1)$, $B_7(2,2,2; 1,4,1;1)$, $B_7(2,3,1; 1,4,1;1)$,\
$13$ &$B_7(3,2,1; 1,4,1;1)$, $B_7(2,1,2; 1,4,1;2)$, $B_7(2,1,2; 1,4,2;1)$, $B_7(2,1,1; 1,4,2;2)$, $B_7(2,1,1; 1,5,2;1)$, $B_7(2,4,2; 2,1,1;1)$,\
$13$ &$B_7(2,5,1; 2,1,1;1)$, $B_7(6,1,1; 2,1,1;1)$, $B_7(2,2,1; 2,1,1;4)$, $B_7(3,1,1; 2,1,1;4)$, $B_7(2,4,1; 2,2,1;1)$, $B_7(5,1,1; 2,2,1;1)$,\
$13$ &$B_7(2,3,1; 2,2,1;2)$, $B_7(2,2,1; 2,2,1;3)$, $B_7(2,3,1; 2,3,1;1)$, $B_7(3,2,1; 2,3,1;1)$, $B_7(4,1,1; 2,3,1;1)$, $B_7(3,1,1; 2,4,1;1)$.\
[l@l@]{} $11$ & $B_8(1,2,1,2; 1,1,1,2)$, $B_8(2,2,1,1; 1,1,1,2)$, $B_8(1,2,2,1; 1,1,2,1)$, $B_8(1,2,1,1; 1,1,2,2)$, $B_8(2,1,2,1; 1,2,1,1)$,\
$11$ &$B_8(2,1,1,1; 1,2,1,2)$;\
$12$ &$B_8(1,1,3,3; 1,1,1,1)$, $B_8(1,3,1,3; 1,1,1,1)$, $B_8(1,3,3,1; 1,1,1,1)$, $B_8(2,1,3,2; 1,1,1,1)$, $B_8(2,3,1,2; 1,1,1,1)$,\
$12$ &$B_8(3,1,3,1; 1,1,1,1)$, $B_8(3,3,1,1; 1,1,1,1)$, $B_8(1,1,2,3; 1,1,1,2)$, $B_8(1,2,2,2; 1,1,1,2)$, $B_8(1,3,2,1; 1,1,1,2)$,\
$12$ &$B_8(2,1,2,2; 1,1,1,2)$, $B_8(2,2,2,1; 1,1,1,2)$, $B_8(3,1,2,1; 1,1,1,2)$, $B_8(1,1,1,3; 1,1,1,3)$, $B_8(1,3,1,1; 1,1,1,3)$,\
$12$ &$B_8(2,1,1,2; 1,1,1,3)$, $B_8(3,1,1,1; 1,1,1,3)$, $B_8(1,1,3,2; 1,1,2,1)$, $B_8(1,2,2,2; 1,1,2,1)$, $B_8(1,3,1,2; 1,1,2,1)$,\
$12$ &$B_8(2,1,3,1; 1,1,2,1)$, $B_8(2,2,2,1; 1,1,2,1)$, $B_8(2,3,1,1; 1,1,2,1)$, $B_8(2,1,1,1; 1,1,2,3)$, $B_8(1,1,3,1; 1,1,3,1)$,\
$12$ &$B_8(1,3,1,1; 1,1,3,1)$, $B_8(1,2,1,3; 1,2,1,1)$, $B_8(1,2,2,2; 1,2,1,1)$, $B_8(1,2,3,1; 1,2,1,1)$, $B_8(2,2,1,2; 1,2,1,1)$,\
$12$ &$B_8(2,2,2,1; 1,2,1,1)$, $B_8(3,2,1,1; 1,2,1,1)$, $B_8(2,1,1,1; 1,2,2,2)$, $B_8(2,1,1,2; 1,3,1,1)$, $B_8(3,1,1,1; 1,3,1,1)$,\
$12$ &$B_8(2,1,1,1; 1,3,2,1)$, $B_8(2,2,1,2; 2,1,1,1)$, $B_8(3,1,2,1; 2,1,1,1)$, $B_8(2,2,1,1; 2,1,1,2)$, $B_8(3,1,1,1; 2,1,1,2)$,\
$12$ &$B_8(2,1,2,1; 2,1,2,1)$, $B_8(2,2,1,1; 2,1,2,1)$, $B_8(3,1,1,1; 3,1,1,1)$;\
$13$ &$B_8(1,1,2,5; 1,1,1,1)$, $B_8(1,1,5,2; 1,1,1,1)$, $B_8(1,2,1,5; 1,1,1,1)$, $B_8(1,2,2,4; 1,1,1,1)$, $B_8(1,2,3,3; 1,1,1,1)$,\
$13$ &$B_8(1,2,4,2; 1,1,1,1)$, $B_8(1,2,5,1; 1,1,1,1)$, $B_8(1,3,2,3; 1,1,1,1)$, $B_8(1,3,3,2; 1,1,1,1)$, $B_8(1,4,2,2; 1,1,1,1)$,\
$13$ &$B_8(1,5,1,2; 1,1,1,1)$, $B_8(1,5,2,1; 1,1,1,1)$, $B_8(2,1,2,4; 1,1,1,1)$, $B_8(2,1,5,1; 1,1,1,1)$, $B_8(2,2,1,4; 1,1,1,1)$,\
$13$ &$B_8(2,2,2,3; 1,1,1,1)$, $B_8(2,2,3,2; 1,1,1,1)$, $B_8(2,2,4,1; 1,1,1,1)$, $B_8(2,3,2,2; 1,1,1,1)$, $B_8(2,3,3,1; 1,1,1,1)$,\
$13$ &$B_8(2,4,2,1; 1,1,1,1)$, $B_8(2,5,1,1; 1,1,1,1)$, $B_8(3,1,2,3; 1,1,1,1)$, $B_8(3,2,1,3; 1,1,1,1)$, $B_8(3,2,2,2; 1,1,1,1)$,\
$13$ &$B_8(3,2,3,1; 1,1,1,1)$, $B_8(3,3,2,1; 1,1,1,1)$, $B_8(4,1,2,2; 1,1,1,1)$, $B_8(4,2,1,2; 1,1,1,1)$, $B_8(4,2,2,1; 1,1,1,1)$,\
$13$ &$B_8(5,1,2,1; 1,1,1,1)$, $B_8(5,2,1,1; 1,1,1,1)$, $B_8(1,1,1,5; 1,1,1,2)$, $B_8(1,1,4,2; 1,1,1,2)$, $B_8(1,2,3,2; 1,1,1,2)$,\
$13$ &$B_8(1,2,4,1; 1,1,1,2)$, $B_8(1,5,1,1; 1,1,1,2)$, $B_8(2,1,1,4; 1,1,1,2)$, $B_8(2,1,4,1; 1,1,1,2)$, $B_8(2,2,3,1; 1,1,1,2)$,\
$13$ &$B_8(3,1,1,3; 1,1,1,2)$, $B_8(4,1,1,2; 1,1,1,2)$, $B_8(5,1,1,1; 1,1,1,2)$, $B_8(1,1,3,2; 1,1,1,3)$, $B_8(1,2,3,1; 1,1,1,3)$,\
$13$ &$B_8(2,1,3,1; 1,1,1,3)$, $B_8(1,1,2,2; 1,1,1,4)$, $B_8(1,2,2,1; 1,1,1,4)$, $B_8(2,1,2,1; 1,1,1,4)$, $B_8(1,2,1,1; 1,1,1,5)$,\
$13$ &$B_8(2,1,1,1; 1,1,1,5)$, $B_8(1,1,2,4; 1,1,2,1)$, $B_8(1,1,5,1; 1,1,2,1)$, $B_8(1,2,1,4; 1,1,2,1)$, $B_8(1,2,2,3; 1,1,2,1)$,\
$13$ &$B_8(1,5,1,1; 1,1,2,1)$, $B_8(2,1,2,3; 1,1,2,1)$, $B_8(2,2,1,3; 1,1,2,1)$, $B_8(2,2,2,2; 1,1,2,1)$, $B_8(3,1,2,2; 1,1,2,1)$,\
$13$ &$B_8(3,2,1,2; 1,1,2,1)$, $B_8(3,2,2,1; 1,1,2,1)$, $B_8(4,1,2,1; 1,1,2,1)$, $B_8(4,2,1,1; 1,1,2,1)$, $B_8(1,1,2,3; 1,1,2,2)$,\
$13$ &$B_8(1,1,3,2; 1,1,2,2)$, $B_8(1,1,4,1; 1,1,2,2)$, $B_8(2,1,1,3; 1,1,2,2)$, $B_8(2,1,2,2; 1,1,2,2)$, $B_8(2,1,3,1; 1,1,2,2)$,\
$13$ &$B_8(3,1,1,2; 1,1,2,2)$, $B_8(3,1,2,1; 1,1,2,2)$, $B_8(4,1,1,1; 1,1,2,2)$, $B_8(1,1,3,1; 1,1,2,3)$, $B_8(2,1,2,1; 1,1,2,3)$,\
$13$ &$B_8(1,2,1,3; 1,1,3,1)$, $B_8(2,1,2,2; 1,1,3,1)$, $B_8(2,2,1,2; 1,1,3,1)$, $B_8(3,1,2,1; 1,1,3,1)$, $B_8(3,2,1,1; 1,1,3,1)$,\
$13$ &$B_8(2,1,1,2; 1,1,3,2)$, $B_8(2,1,2,1; 1,1,3,2)$, $B_8(3,1,1,1; 1,1,3,2)$, $B_8(1,2,1,2; 1,1,4,1)$, $B_8(2,1,2,1; 1,1,4,1)$,\
$13$ &$B_8(2,2,1,1; 1,1,4,1)$, $B_8(2,1,1,1; 1,1,4,2)$, $B_8(1,2,1,1; 1,1,5,1)$, $B_8(1,3,2,2; 1,2,1,1)$, $B_8(1,4,1,2; 1,2,1,1)$,\
$13$ &$B_8(1,4,2,1; 1,2,1,1)$, $B_8(2,1,1,4; 1,2,1,1)$, $B_8(2,3,2,1; 1,2,1,1)$, $B_8(2,4,1,1; 1,2,1,1)$, $B_8(3,1,1,3; 1,2,1,1)$,\
$13$ &$B_8(4,1,1,2; 1,2,1,1)$, $B_8(5,1,1,1; 1,2,1,1)$, $B_8(1,2,3,1; 1,2,1,2)$, $B_8(1,3,2,1; 1,2,1,2)$, $B_8(1,4,1,1; 1,2,1,2)$,\
$13$ &$B_8(1,2,2,1; 1,2,1,3)$, $B_8(1,3,1,2; 1,2,2,1)$, $B_8(1,4,1,1; 1,2,2,1)$, $B_8(2,1,1,3; 1,2,2,1)$, $B_8(2,2,1,2; 1,2,2,1)$,\
$13$ &$B_8(2,3,1,1; 1,2,2,1)$, $B_8(3,1,1,2; 1,2,2,1)$, $B_8(3,2,1,1; 1,2,2,1)$, $B_8(4,1,1,1; 1,2,2,1)$, $B_8(2,1,1,2; 1,2,3,1)$,\
$13$ &$B_8(2,2,1,1; 1,2,3,1)$, $B_8(3,1,1,1; 1,2,3,1)$, $B_8(2,1,1,1; 1,2,3,2)$, $B_8(2,1,1,1; 1,2,4,1)$, $B_8(1,3,1,2; 1,3,1,1)$,\
$13$ &$B_8(1,3,2,1; 1,3,1,1)$, $B_8(2,3,1,1; 1,3,1,1)$, $B_8(2,2,1,1; 1,3,2,1)$, $B_8(2,2,1,1; 1,4,1,1)$, $B_8(2,1,1,1; 1,5,1,1)$,\
$13$ &$B_8(2,1,1,4; 2,1,1,1)$, $B_8(2,1,2,3; 2,1,1,1)$, $B_8(2,1,3,2; 2,1,1,1)$, $B_8(2,1,4,1; 2,1,1,1)$, $B_8(2,2,2,2; 2,1,1,1)$,\
$13$ &$B_8(2,2,3,1; 2,1,1,1)$, $B_8(2,3,2,1; 2,1,1,1)$, $B_8(2,4,1,1; 2,1,1,1)$, $B_8(3,1,1,3; 2,1,1,1)$, $B_8(3,1,2,2; 2,1,1,1)$,\
$13$ &$B_8(3,2,1,2; 2,1,1,1)$, $B_8(3,2,2,1; 2,1,1,1)$, $B_8(3,3,1,1; 2,1,1,1)$, $B_8(4,1,1,2; 2,1,1,1)$, $B_8(4,2,1,1; 2,1,1,1)$,\
$13$ &$B_8(5,1,1,1; 2,1,1,1)$, $B_8(2,1,1,3; 2,1,1,2)$, $B_8(2,1,2,2; 2,1,1,2)$, $B_8(2,1,3,1; 2,1,1,2)$, $B_8(2,2,2,1; 2,1,1,2)$,\
$13$ &$B_8(3,1,1,2; 2,1,1,2)$, $B_8(2,1,2,1; 2,1,1,3)$, $B_8(2,1,2,2; 2,1,2,1)$, $B_8(3,1,1,2; 2,1,2,1)$, $B_8(3,2,1,1; 2,1,2,1)$,\
$13$ &$B_8(4,1,1,1; 2,1,2,1)$, $B_8(3,1,1,1; 2,1,2,2)$, $B_8(3,1,1,1; 2,1,3,1)$, $B_8(2,2,2,1; 2,2,1,1)$, $B_8(2,3,1,1; 2,2,1,1)$,\
$13$ &$B_8(3,1,1,2; 2,2,1,1)$, $B_8(3,2,1,1; 2,2,1,1)$, $B_8(4,1,1,1; 2,2,1,1)$, $B_8(3,1,1,1; 2,2,2,1)$, $B_8(3,1,1,1; 2,3,1,1)$,\
$13$ &$B_8(3,2,1,1; 3,1,1,1)$.\
[l@l@]{} $11$ &$B_9(2,1,2,1; 1,1,1,1; 1)$, $B_9(2,1,1,1; 1,1,1,2; 1)$, $B_9(1,1,2,1; 1,1,2,1; 1)$, $B_9(2,1,1,1; 2,1,1,1; 1)$;\
$12$ &$B_9(1,2,1,3; 1,1,1,1; 1)$, $B_9(1,2,3,1; 1,1,1,1; 1)$, $B_9(2,1,2,2; 1,1,1,1; 1)$, $B_9(2,2,2,1; 1,1,1,1; 1)$, $B_9(3,2,1,1; 1,1,1,1; 1)$,\
$12$ &$B_9(1,1,1,3; 1,1,1,1; 2)$, $B_9(1,1,3,1; 1,1,1,1; 2)$, $B_9(2,1,1,2; 1,1,1,1; 2)$, $B_9(3,1,1,1; 1,1,1,1; 2)$, $B_9(1,2,1,2; 1,1,1,2; 1)$,\
$12$ &$B_9(2,1,1,2; 1,1,1,2; 1)$, $B_9(1,1,2,1; 1,1,1,2; 2)$, $B_9(1,2,1,1; 1,1,1,3; 1)$, $B_9(1,1,2,2; 1,1,2,1; 1)$, $B_9(1,2,1,2; 1,1,2,1; 1)$,\
$12$ &$B_9(2,1,1,1; 1,1,2,1; 2)$, $B_9(1,2,1,1; 1,1,2,2; 1)$, $B_9(2,1,1,1; 1,1,2,2; 1)$, $B_9(1,2,1,1; 1,1,3,1; 1)$, $B_9(3,1,1,1; 1,2,1,1; 1)$,\
$12$ &$B_9(2,1,1,1; 1,2,2,1; 1)$, $B_9(2,2,1,1; 2,1,1,1; 1)$;\
$13$ &$B_9(1,1,1,5; 1,1,1,1; 1)$, $B_9(1,1,2,4; 1,1,1,1; 1)$, $B_9(1,1,3,3; 1,1,1,1; 1)$, $B_9(1,1,4,2; 1,1,1,1; 1)$, $B_9(1,1,5,1; 1,1,1,1; 1)$,\
$13$ &$B_9(1,2,2,3; 1,1,1,1; 1)$, $B_9(1,2,3,2; 1,1,1,1; 1)$, $B_9(1,3,2,2; 1,1,1,1; 1)$, $B_9(1,4,1,2; 1,1,1,1; 1)$, $B_9(1,4,2,1; 1,1,1,1; 1)$,\
$13$ &$B_9(2,1,1,4; 1,1,1,1; 1)$, $B_9(2,1,2,3; 1,1,1,1; 1)$, $B_9(2,2,1,3; 1,1,1,1; 1)$, $B_9(2,2,2,2; 1,1,1,1; 1)$, $B_9(2,3,1,2; 1,1,1,1; 1)$,\
$13$ &$B_9(2,3,2,1; 1,1,1,1; 1)$, $B_9(2,4,1,1; 1,1,1,1; 1)$, $B_9(3,1,1,3; 1,1,1,1; 1)$, $B_9(3,2,1,2; 1,1,1,1; 1)$, $B_9(4,1,1,2; 1,1,1,1; 1)$,\
$13$ &$B_9(5,1,1,1; 1,1,1,1; 1)$, $B_9(1,1,2,3; 1,1,1,1; 2)$, $B_9(1,1,3,2; 1,1,1,1; 2)$, $B_9(1,2,2,2; 1,1,1,1; 2)$, $B_9(1,3,1,2; 1,1,1,1; 2)$,\
$13$ &$B_9(1,3,2,1; 1,1,1,1; 2)$, $B_9(2,2,1,2; 1,1,1,1; 2)$, $B_9(2,3,1,1; 1,1,1,1; 2)$, $B_9(1,1,2,2; 1,1,1,1; 3)$, $B_9(1,2,1,2; 1,1,1,1; 3)$,\
$13$ &$B_9(1,2,2,1; 1,1,1,1; 3)$, $B_9(2,2,1,1; 1,1,1,1; 3)$, $B_9(1,1,1,2; 1,1,1,1; 4)$, $B_9(1,1,2,1; 1,1,1,1; 4)$, $B_9(2,1,1,1; 1,1,1,1; 4)$,\
$13$ &$B_9(1,1,1,4; 1,1,1,2; 1)$, $B_9(1,1,2,3; 1,1,1,2; 1)$, $B_9(1,1,3,2; 1,1,1,2; 1)$, $B_9(1,1,4,1; 1,1,1,2; 1)$, $B_9(1,2,2,2; 1,1,1,2; 1)$,\
$13$ &$B_9(1,2,3,1; 1,1,1,2; 1)$, $B_9(1,3,2,1; 1,1,1,2; 1)$, $B_9(1,4,1,1; 1,1,1,2; 1)$, $B_9(2,1,1,3; 1,1,1,2; 1)$, $B_9(1,1,1,3; 1,1,1,2; 2)$,\
$13$ &$B_9(1,1,2,2; 1,1,1,2; 2)$, $B_9(1,2,1,2; 1,1,1,2; 2)$, $B_9(1,2,2,1; 1,1,1,2; 2)$, $B_9(1,3,1,1; 1,1,1,2; 2)$, $B_9(1,1,1,2; 1,1,1,2; 3)$,\
$13$ &$B_9(1,2,1,1; 1,1,1,2; 3)$, $B_9(1,1,1,3; 1,1,1,3; 1)$, $B_9(1,1,2,2; 1,1,1,3; 1)$, $B_9(1,1,3,1; 1,1,1,3; 1)$, $B_9(1,2,2,1; 1,1,1,3; 1)$,\
$13$ &$B_9(1,1,2,1; 1,1,1,4; 1)$, $B_9(1,1,2,3; 1,1,2,1; 1)$, $B_9(1,4,1,1; 1,1,2,1; 1)$, $B_9(2,1,1,3; 1,1,2,1; 1)$, $B_9(2,2,1,2; 1,1,2,1; 1)$,\
$13$ &$B_9(2,3,1,1; 1,1,2,1; 1)$, $B_9(3,1,1,2; 1,1,2,1; 1)$, $B_9(3,2,1,1; 1,1,2,1; 1)$, $B_9(4,1,1,1; 1,1,2,1; 1)$, $B_9(1,3,1,1; 1,1,2,1; 2)$,\
$13$ &$B_9(2,2,1,1; 1,1,2,1; 2)$, $B_9(1,2,1,1; 1,1,2,1; 3)$, $B_9(1,1,2,2; 1,1,2,2; 1)$, $B_9(2,1,1,2; 1,1,2,2; 1)$, $B_9(1,2,1,1; 1,1,2,2; 2)$,\
$13$ &$B_9(2,1,1,2; 1,1,3,1; 1)$, $B_9(2,2,1,1; 1,1,3,1; 1)$, $B_9(3,1,1,1; 1,1,3,1; 1)$, $B_9(2,1,1,1; 1,1,3,2; 1)$, $B_9(2,1,1,1; 1,1,4,1; 1)$,\
$13$ &$B_9(1,2,2,2; 1,2,1,1; 1)$, $B_9(1,3,1,2; 1,2,1,1; 1)$, $B_9(1,3,2,1; 1,2,1,1; 1)$, $B_9(2,1,1,3; 1,2,1,1; 1)$, $B_9(2,2,1,2; 1,2,1,1; 1)$,\
$13$ &$B_9(2,3,1,1; 1,2,1,1; 1)$, $B_9(3,1,1,2; 1,2,1,1; 1)$, $B_9(1,2,1,2; 1,2,1,1; 2)$, $B_9(1,2,2,1; 1,2,1,1; 2)$, $B_9(2,1,1,2; 1,2,1,1; 2)$,\
$13$ &$B_9(2,2,1,1; 1,2,1,1; 2)$, $B_9(2,1,1,1; 1,2,1,1; 3)$, $B_9(1,2,2,1; 1,2,1,2; 1)$, $B_9(1,3,1,1; 1,2,1,2; 1)$, $B_9(1,3,1,1; 1,2,2,1; 1)$,\
$13$ &$B_9(2,1,1,2; 1,2,2,1; 1)$, $B_9(2,2,1,1; 1,2,2,1; 1)$, $B_9(2,1,1,2; 1,3,1,1; 1)$, $B_9(2,2,1,1; 1,3,1,1; 1)$, $B_9(2,1,1,1; 1,3,1,1; 2)$,\
$13$ &$B_9(2,1,1,1; 1,4,1,1; 1)$, $B_9(2,3,1,1; 2,1,1,1; 1)$, $B_9(2,2,1,1; 2,2,1,1; 1)$.\
[l@l@]{} $12$ &$B_{10}(1,1,2,1,2; 1,1,1,1,1)$, $B_{10}(1,2,1,2,1; 1,1,1,1,1)$, $B_{10}(2,1,2,1,1; 1,1,1,1,1)$, $B_{10}(1,1,1,1,2; 1,1,1,1,2)$,\
$12$ &$B_{10}(1,2,1,1,1; 1,1,1,1,2)$, $B_{10}(2,1,1,1,1; 1,1,1,1,2)$, $B_{10}(1,1,2,1,1; 1,1,1,2,1)$, $B_{10}(1,2,1,1,1; 1,1,1,2,1)$,\
$12$ &$B_{10}(1,1,2,1,1; 1,1,2,1,1)$, $B_{10}(2,1,1,1,1; 1,2,1,1,1)$;\
$13$ &$B_{10}(1,1,1,1,4; 1,1,1,1,1)$, $B_{10}(1,1,1,2,3; 1,1,1,1,1)$, $B_{10}(1,1,1,3,2; 1,1,1,1,1)$, $B_{10}(1,1,1,4,1; 1,1,1,1,1)$,\
$13$ &$B_{10}(1,1,2,2,2; 1,1,1,1,1)$, $B_{10}(1,1,2,3,1; 1,1,1,1,1)$, $B_{10}(1,1,3,2,1; 1,1,1,1,1)$, $B_{10}(1,1,4,1,1; 1,1,1,1,1)$,\
$13$ &$B_{10}(1,2,1,1,3; 1,1,1,1,1)$, $B_{10}(1,2,1,2,2; 1,1,1,1,1)$, $B_{10}(1,2,2,1,2; 1,1,1,1,1)$, $B_{10}(1,2,2,2,1; 1,1,1,1,1)$,\
$13$ &$B_{10}(1,2,3,1,1; 1,1,1,1,1)$, $B_{10}(1,3,1,1,2; 1,1,1,1,1)$, $B_{10}(1,3,2,1,1; 1,1,1,1,1)$, $B_{10}(1,4,1,1,1; 1,1,1,1,1)$,\
$13$ &$B_{10}(2,1,1,1,3; 1,1,1,1,1)$, $B_{10}(2,1,1,2,2; 1,1,1,1,1)$, $B_{10}(2,1,1,3,1; 1,1,1,1,1)$, $B_{10}(2,1,2,2,1; 1,1,1,1,1)$,\
$13$ &$B_{10}(2,2,1,1,2; 1,1,1,1,1)$, $B_{10}(2,2,1,2,1; 1,1,1,1,1)$, $B_{10}(2,2,2,1,1; 1,1,1,1,1)$, $B_{10}(2,3,1,1,1; 1,1,1,1,1)$,\
$13$ &$B_{10}(3,1,1,1,2; 1,1,1,1,1)$, $B_{10}(3,1,1,2,1; 1,1,1,1,1)$, $B_{10}(3,2,1,1,1; 1,1,1,1,1)$, $B_{10}(4,1,1,1,1; 1,1,1,1,1)$,\
$13$ &$B_{10}(1,1,1,2,2; 1,1,1,1,2)$, $B_{10}(1,1,1,3,1; 1,1,1,1,2)$, $B_{10}(1,1,2,2,1; 1,1,1,1,2)$, $B_{10}(1,1,3,1,1; 1,1,1,1,2)$,\
$13$ &$B_{10}(1,2,2,1,1; 1,1,1,1,2)$, $B_{10}(2,1,1,2,1; 1,1,1,1,2)$, $B_{10}(1,1,1,2,1; 1,1,1,1,3)$, $B_{10}(1,1,2,1,1; 1,1,1,1,3)$,\
$13$ &$B_{10}(1,1,1,2,2; 1,1,1,2,1)$, $B_{10}(1,1,1,3,1; 1,1,1,2,1)$, $B_{10}(1,1,2,2,1; 1,1,1,2,1)$, $B_{10}(1,2,1,1,2; 1,1,1,2,1)$,\
$13$ &$B_{10}(2,1,1,1,2; 1,1,1,2,1)$, $B_{10}(2,1,1,2,1; 1,1,1,2,1)$, $B_{10}(2,2,1,1,1; 1,1,1,2,1)$, $B_{10}(3,1,1,1,1; 1,1,1,2,1)$,\
$13$ &$B_{10}(1,1,2,1,1; 1,1,1,2,2)$, $B_{10}(2,1,1,1,1; 1,1,1,2,2)$, $B_{10}(2,1,1,1,1; 1,1,1,3,1)$, $B_{10}(1,2,1,1,2; 1,1,2,1,1)$,\
$13$ &$B_{10}(1,2,2,1,1; 1,1,2,1,1)$, $B_{10}(1,3,1,1,1; 1,1,2,1,1)$, $B_{10}(2,1,1,1,2; 1,1,2,1,1)$, $B_{10}(2,1,1,2,1; 1,1,2,1,1)$,\
$13$ &$B_{10}(2,2,1,1,1; 1,1,2,1,1)$, $B_{10}(3,1,1,1,1; 1,1,2,1,1)$, $B_{10}(1,2,1,1,1; 1,1,2,2,1)$, $B_{10}(2,1,1,1,1; 1,1,2,2,1)$,\
$13$ &$B_{10}(1,2,1,1,1; 1,1,3,1,1)$, $B_{10}(2,1,1,1,1; 1,1,3,1,1)$, $B_{10}(1,2,1,1,2; 1,2,1,1,1)$, $B_{10}(1,2,2,1,1; 1,2,1,1,1)$,\
$13$ &$B_{10}(1,3,1,1,1; 1,2,1,1,1)$, $B_{10}(2,2,1,1,1; 1,2,1,1,1)$, $B_{10}(2,1,1,1,1; 1,2,2,1,1)$, $B_{10}(2,1,1,1,2; 2,1,1,1,1)$,\
$13$ &$B_{10}(2,1,1,2,1; 2,1,1,1,1)$, $B_{10}(2,1,2,1,1; 2,1,1,1,1)$, $B_{10}(2,2,1,1,1; 2,1,1,1,1)$, $B_{10}(3,1,1,1,1; 2,1,1,1,1)$.\
Some theorems and lemmas
========================
\[thm-7-1\] Let $G=B_4(a_1,a_2; a_3, a_4)$, where $a_1, a_2, a_3, a_4$ are some positive integers. Then $\lambda_2(G)>0$ and $\lambda_3(G)<0$ if and only if $G$ is isomorphic to one of the following graphs:\
(1) $B_4(a, b; 1, d)$; (2) $B_4(a, x; y, 1)$; (3) $B_4(a, 1; c, 1)$; (4) $B_4(a, 1; w, x)$;\
(5) $B_4(a, 1; x, d)$; (6) $B_4(w, b; x, 1)$; (7) $B_4(w, x; y, d)$; (8) $B_4(x, b; y, d)$;\
(9) 25 specific graphs: 5 graphs of order 10, 10 graphs of order 11, and 10 graphs of order 12, where $a, b, c, d, x, y,w $ are some positive integers such that $x\leq 2, y\leq 2$ and $w\leq 3$.
\[lem-6-2\] Let $G\in \mathcal{B}_4(n)$, where $n\geq 14$. If $G\notin \mathcal{B}_4^-(n)$, then $G$ has an induced subgraph $\Gamma \in \mathcal{B}_4(14)\setminus \mathcal{B}_4^-(14)$.
By the proof of Lemma \[lem-6-5\], it suffices to prove that $G$ contains an induced subgraph $G'\in \mathcal{B}_4(n-1)\setminus \mathcal{B}_4^-(n-1)$ for $n\geq 15$ in the following.
Let $G=B_4(n_1, n_2; n_3, n_4)\in \mathcal{B}_4(n)$. Then one of $H_1=B_4(n_1-1, n_2; n_3, n_4)$, $H_2=B_4(n_1, n_2-1; n_3, n_4)$, $H_3=B_4(n_1, n_2; n_3-1, n_4)$ and $H_4=B_4(n_1, n_2; n_3, n_4-1)$ must belong to $\mathcal{B}_4(n-1)$. On the contrary, assume that $H_i\in \mathcal{B}_4^-(n-1) (i=1,2,3,4)$. Then $H_i$ is a graph belonging to (1)–(8) in Theorem \[thm-7-1\] since $n\geq 15$.
First we consider $H_1$. If $H_1$ is a graph belonging to (1) of Theorem \[thm-7-1\], then $H_1=B_4(a,b;1,d)$ where $n_1-1=a$, $n_2=b$, $n_3=1$ and $n_4=d$, hence $G=B_4(a+1,b;1,d)\in \mathcal{B}_4^-(n)$, a contradiction. Similarly, $H_1$ cannot belong to (2)–(5) of Theorem \[thm-7-1\]. Hence $H_1$ is belong to (6)–(8) of Theorem \[thm-7-1\] from which we see that $n_1-1$ is either $w$ or $x$. Thus $n_1\leq 4$ due to $w\leq 3$ and $x\leq 2$.
By the same method, we can verify that $n_2\leq 3$ if $H_2\in \mathcal{B}_4^-(n-1)$; $n_3\leq 4$ if $H_3\in \mathcal{B}_4^-(n-1)$ and $n_4\leq 3$ if $H_4\in \mathcal{B}_4^-(n-1)$. Hence $n=n_1+\cdots+n_4\leq 14$, a contradiction. We are done.
\[thm-7-2\] Let $G=B_6(a_1, a_2, a_3; a_4, a_5, a_6)$, where $a_1, \ldots, a_6$ are some positive integers. Then $\lambda_2(G)>0$ and $\lambda_3(G)<0$ if and only if $G$ is isomorphic to one of the following graphs:\
(1) $B_6(a, x, c; 1, 1, 1)$; (2) $B_6(a, 1, c; 1, e, 1)$; (3) $B_6(a, 1, c; 1, x, y)$;\
(4) $B_6(a, 1, c; 1, 1, f)$; (5) $B_6(a, 1, 1; x, e, 1)$; (6) $B_6(x, b, 1; y, 1, 1)$;\
(7) $B_6(x, y, 1; 1, e, 1)$; (8) $B_6(x, y, 1; 1, 1, f)$; (9) $B_6(x, 1, c; y, 1, f)$;\
(10) $B_6(1, b, x; 1, 1, 1)$; (11) $B_6(1, b, 1; 1, e, 1)$; (12) $B_6(1, b, 1; 1, x, y)$;\
(13) $B_6(1, x, y; 1, 1, f)$; (14) 145 specific graphs: 22 graphs of order 10, 54 graphs of order 11, and 69 graphs of order 12, where $a, b, c, d, e, f, x, y$ are some positive integers such that $x\leq 2$ and $y\leq 2$.
\[lem-6-2\] Let $G\in \mathcal{B}_6(n)$, where $n\geq 14$. If $G\notin \mathcal{B}_6^-(n)$, then $G$ has an induced subgraph $\Gamma\in \mathcal{B}_6(14) \setminus \mathcal{B}_6^-(14)$.
By the proof of Lemma \[lem-6-5\], it suffices to prove that $G$ contains an induced subgraph $G'\in \mathcal{B}_6(n-1)\setminus \mathcal{B}_6^-(n-1)$ for $n\geq 15$ in the following.
Let $G=B_6(n_1, n_2, n_3; n_4, n_5, n_6)\in \mathcal{B}_6(n)$. Then one of $H_1=B_6(n_1-1, n_2, n_3; n_4, n_5,$ $ n_6)$, $H_2=B_6(n_1, n_2-1, n_3; n_4, n_5, n_6)$, $H_3=B_6(n_1, n_2, n_3-1; n_4, n_5, n_6)$, $H_4=B_6(n_1, n_2, n_3; n_4-1, n_5, n_6)$, $H_5=B_6(n_1, n_2, n_3; n_4, n_5-1, n_6)$ and $H_6=B_6(n_1,
n_2, n_3; n_4, n_5, n_6-1)$ must belong to $\mathcal{B}_6(n-1)$. On the contrary, assume that $H_i\in \mathcal{B}_6^-(n-1) (i=1,2,\ldots,6)$. Then $H_i$ is a graph belonging to (1)–(13) in Theorem \[thm-7-2\] since $n\geq 15$.
Let us consider $H_3$. If $H_3$ is a graph belonging to (1) of Theorem \[thm-7-2\], then $H_3=B_6(a, x, c; 1, $ $1, 1)$ where $n_1=a$, $n_2=x$, $n_3-1=c$, $n_4=n_5=n_6=1$, hence $G=B_6(a, x, c+1; 1, 1, 1)\in \mathcal{B}_6^-(n)$, a contradiction. Similarly, $H_3$ cannot belong to (2)–(4) and (9) of Theorem \[thm-7-2\]. If $H_3$ is a graph belonging to (10) of Theorem \[thm-7-2\], then $H_3=B_6(1, b, x; 1, 1, 1)$, where $n_1=1$, $n_2=b$, $n_3-1=x$, $n_4=n_5=n_6=1$. Since $x\le 2$, we have $n_3\leq 3$. If $n_3<3$ then $x+1\leq 2$ and $G=B_6(1, b, x+1; 1, 1, 1)\in \mathcal{B}_6^-(n)$, a contradiction. Now assume that $n_3=3$. Then $H_3=B_6(1, b, 2; 1, 1, 1)$, and so $G=B_6(1, b, 3; 1, 1, 1)$. By Theorem \[thm-7-2\], $G\not\in \mathcal{B}_6^-(n)$, and also its induced subgraph $B_6(1, b-1, 3; 1, 1, 1)\notin \mathcal{B}_6^-(n-1)$, a contradiction. Similarly, $H_3$ cannot belong to (13) of Theorem \[thm-7-2\]. Hence $H_3$ is belong to (5)–(8) and (11)–(12) of Theorem \[thm-7-2\] from which we see that $n_3-1\leq 1$. Thus $n_3\leq 2$.
By the same method, we can verify that $n_1\leq 3$ if $H_1\in \mathcal{B}_6^-(n-1)$; $n_2\leq 3$ if $H_2\in \mathcal{B}_6^-(n-1)$; $n_4\leq 2$ if $H_4\in \mathcal{B}_6^-(n-1)$; $n_5\leq 2$ if $H_5\in \mathcal{B}_6^-(n-1)$ and $n_6\leq 2$ if $H_6\in \mathcal{B}_6^-(n-1)$. Hence $n=n_1+\cdots+n_6\leq 14$, a contradiction. We are done.
\[lem-7-3\] Let $G=B_7(a_1, a_2, a_3; a_4, a_5, a_6; a_7)$, where $a_1, \ldots, a_7$ are some positive integers. Then $\lambda_2(G)>0$ and $\lambda_3(G)<0$ if and only if $G$ is isomorphic to one of the following graphs:\
(1) $B_7(a, 1, x; 1, e, 1; 1)$; (2) $B_7(a, 1, 1; 1, e, 1; g)$; (3) $B_7(a, 1, 1; 1, 1, x; 1)$;\
(4) $B_7(x, y, 1; 1, e, 1; g)$; (5) $B_7(x, 1, 1; y, 1, 1; g)$; (6) $B_7(1, b, x; 1, 1, 1; g)$;\
(7) $B_7(1, b, 1; 1, e, 1; g)$; (8) $B_7(1, 1, c; 1, 1, f; 1)$;\
(9) 143 specific graphs: 18 graphs of order 10, 52 graphs of order 11, and 73 graphs of order 12, where $a, b, c, d, e, f, g, x, y$ are some positive integers such that $x\leq 2$ and $y\leq 2$.
\[lem-6-3\] Let $G\in \mathcal{B}_7(n)$, where $n\geq 14$. If $G\notin \mathcal{B}_7^-(n)$, then $G$ has an induced subgraph $\Gamma\in \mathcal{B}_7(14) \setminus \mathcal{B}_7^-(14)$.
By the proof of Lemma \[lem-6-5\], it suffices to prove that $G$ contains an induced subgraph $G'\in \mathcal{B}_7(n-1)\setminus \mathcal{B}_7^-(n-1)$ for $n\geq 15$ in the following.
Let $G=B_7(n_1, n_2, n_3; n_4, n_5, n_6; n_7)\in \mathcal{B}_7(n)$. Then one of $H_1=B_7(n_1-1, n_2, n_3; n_4,$ $ n_5, n_6; n_7)$, $H_2=B_7(n_1, n_2-1, n_3; n_4, n_5, n_6;n_7)$, $H_3=B_7(n_1, n_2, n_3-1; n_4, n_5, n_6; n_7)$, $H_4=B_7(n_1, n_2, n_3; $ $n_4-1, n_5, n_6; n_7)$, $H_5=B_7(n_1, n_2, n_3; n_4, n_5-1, n_6; n_7)$; $H_6=B_7(n_1, n_2, n_3; n_4,$ $n_5, n_6-1; n_7)$ and $H_7=B_7(n_1, n_2, n_3; n_4, n_5, n_6; n_7-1)$ must belong to $\mathcal{B}_7(n-1)$. On the contrary, assume that $H_i\in \mathcal{B}_7^-(n-1) (i=1,2,\ldots,7)$. Then $H_i$ is a graph belonging to (1)–(8) in Theorem \[lem-7-3\] since $n\geq 15$.
Let us consider $H_1$. If $H_1$ is a graph belonging to (1) of Theorem \[lem-7-3\], then $H_1=B_7(a, 1, x; 1, e,$ $ 1; 1)$ where $n_1-1=a$, $n_2=1$, $n_3=x$, $n_4=1$, $n_5=e$, $n_6=n_7=1$, hence $G=B_7(a+1, 1, x; 1, e, 1; 1)\in \mathcal{B}_7^-(n)$, a contradiction. Similarly, $H_1$ cannot belong to (2)–(3) of Theorem \[lem-7-3\]. If $H_1$ is a graph belonging to (4) of Theorem \[lem-7-3\], then $H_1=B_7(x, y, 1; 1, e, 1; g)$, where $n_1-1=x$, $n_2=y$, $n_3=n_4=1$, $n_5=e$, $n_6=1$ and $n_7=g$. Since $x\le 2$, we have $n_1\leq 3$. If $n_1<3$ then $x+1\leq 2$ and $G=B_7(x+1, y, 1; 1, e, 1; g)\in \mathcal{B}_7^-(n)$, a contradiction. Now assume that $n_1=3$. Then $H_1=B_7(2, y, 1; 1, e, 1; g)$, and so $G=B_7(3, y, 1; 1, e, 1; g)$. Since $y\in \{1, 2\}$, we have $G\in \{B_7(3, 1, 1; 1, e, 1; g), B_7(3, 2, 1; 1, e, 1; g)\}$. However $B_7(3, 1, 1; 1, e, 1; g)$ belongs to (2) of Theorem \[lem-7-3\] which contradicts our assumption. Thus $G=B_7(3, 2, 1; 1, e, 1; g)$. By Theorem \[lem-7-3\], $G\not\in \mathcal{B}_7^-(n)$, and also its induced subgraph $B_7(3, 2, 1; 1, e-1, 1; g)$ or $B_7(3, 2, 1; 1, e, 1; g-1)$ is not in $\mathcal{B}_7^-(n-1)$, a contradiction. Similarly, $H_1$ cannot belong to (5) of Theorem \[lem-7-3\]. Hence $H_1$ belongs to (6)–(8) of Theorem \[lem-7-3\] from which we see that $n_1-1\leq 1$. Thus $n_1\leq2$.
By the same method, we can verify that $n_2\leq 2$ if $H_2\in \mathcal{B}_7^-(n-1)$; $n_3\leq 2$ if $H_3\in \mathcal{B}_7^-(n-1)$; $n_4\leq 2$ if $H_4\in \mathcal{B}_7^-(n-1)$; $n_5\leq 2$ if $H_5\in \mathcal{B}_7^-(n-1)$, $n_6\leq 2$ if $H_6\in \mathcal{B}_7^-(n-1)$ and $n_7\leq 2$ if $H_7\in
\mathcal{B}_7^-(n-1)$. Hence $n=n_1+\cdots+n_7\leq 14$, a contradiction. We are done.
\[[@M.R.Oboudi3]\] \[lem-7-4\] Let $G=B_8(a_1, a_2, a_3, a_4; a_5, a_6, a_7, a_8)$, where $a_1, \ldots, a_8$ are some positive integers. Then $\lambda_2(G)>0$ and $\lambda_3(G)<0$ if and only if $G$ is isomorphic to one of the following graphs:\
(1) $B_8(a, 1, 1, d; 1, 1, g, 1)$; (2) $B_8(1, b, 1, 1; 1, f, 1, 1)$;\
(3) 134 specific graphs: 12 graphs of order 10, 42 graphs of order 11, and 80 graphs of order 12, where $a, b, d, f, g$ are some positive integers.
\[lem-6-4\] Let $G\in \mathcal{B}_8(n)$, where $n\geq 14$. If $G\notin \mathcal{B}_8^-(n)$, then $G$ has an induced subgraph $\Gamma\in\mathcal{B}_8(14) \setminus \mathcal{B}_8^-(14)$.
By the proof of Lemma \[lem-6-5\], it suffices to prove that $G$ contains an induced subgraph $G'\in \mathcal{B}_8(n-1)\setminus \mathcal{B}_8^-(n-1)$ for $n\geq 15$ in the following.
Let $G=B_8(n_1, n_2, n_3, n_4; n_5, n_6, n_7, n_8)\in \mathcal{B}_8(n)$ and $H_1=B_8(n_1-1, n_2, n_3, n_4; n_5, n_6,$ $n_7, n_8)$, $H_2=B_8(n_1, n_2-1, n_3, n_4; n_5, n_6, n_7, n_8)$, $H_3=B_8(n_1, n_2, n_3-1, n_4; n_5, n_6,$ $ n_7, n_8)$, $H_4=B_8(n_1, n_2, n_3, n_4$ $-1; n_5, n_6, n_7, n_8)$, $H_5=B_8(n_1, n_2, n_3, n_4; n_5-1, n_6, n_7, n_8)$, $H_6=B_8(n_1, n_2, $ $n_3, n_4; n_5, n_6-1, n_7, n_8)$, $H_7=B_8(n_1, n_2, n_3, n_4; n_5, n_6, n_7-1, n_8)$ and $H_8=B_8(n_1, n_2, n_3, n_4;$ $ n_5, n_6, n_7, n_8-1)$.
If $n_3\geq 3$, then $H_3\in \mathcal{B}_8(n-1)\setminus \mathcal{B}_8^-(n-1)$ by Theorem \[lem-7-4\] as desired. If $n_3=2$, then at least one of $n_1$, $n_2$, $n_4$, $n_5$, $n_6$, $n_7$, $n_8$ is greater than 1 since $n\geq 15$, say $n_2$. Thus $H_2\in \mathcal{B}_8(n-1)\setminus\mathcal{B}_8^-(n-1)$ by Theorem \[lem-7-4\] as desired. Hence let $n_3=1$. Similarly, let $n_5=n_8=1$. Thus one of $H_1, H_2, H_4, H_6, H_7$ must belong to $\mathcal{B}_8(n-1)$. On the contrary, assume that $H_i\in \mathcal{B}_8^-(n-1) (i=1,2,4,6,7)$. Then $H_i$ is a graph belonging to (1)–(2) in Theorem \[lem-7-4\] since $n\geq 15$.
Let us consider $H_1$. If $H_1$ is a graph belonging to (1) of Theorem \[lem-7-4\], then $H_1=B_8(a, 1, 1, d; 1,$ $ 1, g, 1)$; where $n_1-1=a$, $n_2=n_3=1$, $n_4=d$, $n_5=n_6=1$, $n_7=g$ and $n_8=1$, hence $G=B_8(a+1, 1, 1, d; 1, 1, g, 1)\in \mathcal{B}_8^-(n)$, a contradiction. Hence $H_1$ belongs to (2) of Theorem \[lem-7-4\] from which we see that $n_1=2$ due to $n_1-1=1$.
By the same method, we can verify that $n_i=2$ if $H_i\in \mathcal{B}_8^-(n-1)$ for $i=2,4,6,7$. Hence $n=n_1+\cdots+n_8\leq13$, a contradiction. We are done.
\[thm-7-5\] Let $G=B_9(a_1, a_2, a_3, a_4; a_5, a_6, a_7, a_8; a_9)$, where $a_1, \ldots, a_9$ are some positive integers. Then $\lambda_2(G)>0$ and $\lambda_3(G)<0$ if and only if $G$ is isomorphic to one of the following graphs:\
(1) $B_9(1, b, 1, 1; 1, f, 1, 1; k)$; (2) 59 specific graphs: 3 graphs of order 10, 17 graphs of order 11, and 39 graphs of order 12, where $b, f, k$ are some positive integers.
\[lem-6-4\] Let $G\in \mathcal{B}_9(n)$, where $n\geq 14$. If $G\notin \mathcal{B}_9^-(n)$, then $G$ has an induced subgraph $\Gamma\in\mathcal{B}_9(14)\setminus \mathcal{B}_9^-(14)$.
By the proof of Lemma \[lem-6-5\], it suffices to prove that $G$ contains an induced subgraph $G'\in \mathcal{B}_9(n-1)\setminus \mathcal{B}_9^-(n-1)$ for $n\geq 15$ in the following.
Let $G=B_9(n_1, n_2, n_3, n_4; n_5, n_6, n_7, n_8; n_9)\in \mathcal{B}_9(n)$. On the contrary, suppose that every induced subgraphs $G'\in \mathcal{B}_9(n-1)$ of $G$ belongs to $\mathcal{B}_9^-(n-1)$. If $n_1\geq 3$, then $H_1=B_9(n_1-1, n_2, n_3, n_4; n_5, n_6, n_7, n_8; n_9)\notin
\mathcal{B}_9^-(n-1)$ by Theorem \[thm-7-5\], a contradiction. If $n_1=2$, then at least one of $n_2$, $n_3$, $n_4$, $n_5$, $n_6$, $n_7$, $n_8$, $n_9$ is greater than 1 since $n\geq 15$, say $n_2$. Thus $H_2=B_9(n_1, n_2-1, n_3, n_4; n_5, n_6, n_7, n_8; n_9)\notin \mathcal{B}_9^-(n-1)$ by Theorem \[thm-7-5\], a contradiction. Hence $n_1=1$. Similarly, $n_3=n_4=n_5=n_7=n_8=1$. But now $G=B_9(1, n_2, 1, 1; 1, n_6, 1, 1; n_9)\in \mathcal{B}_9^-(n)$ by Theorem \[thm-7-5\], a contradiction. We are done.
[90]{} D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs: Theory and Application, Academic Press, New York, 1980. C.D. Godsil, G. Royle, Algebraic Graph Theory, in: Graduate Texts in Mathematics, vol. 207, Springer, New York, 2001. H.C. Ma, W.H. Yang, S.G. Li, Positive and negative inertia index of a graph, Linear Algebra Appl. 438 (2013) 331–341. M.R. Oboudi, Bipartite graphs with at most six non-zero eigenvalues, Ars Math. Contemp. 11 (2016) 315–325. M.R. Oboudi, On the third largest eigenvalue of graphs, Linear Algebra Appl. 503 (2016) 164–179. M.R. Oboudi, Characterization of graphs with exactly two non-negative eigenvalues, Ars Math. Contemp. 12 (2017) 271–286. M. Petrović, On graphs with exactly one eigenvalue less than $-1$, J. Combin. Theory 52(1) (1991) 102–112. M. Petrović, Graphs with a small number of nonnegative eigenvalues, Graphs Combin. 15 (1999) 221–232. J.H. Smith, Symmetry and multiple eigenvalues of graphs, Glasnik Mat. Ser. III 12(1) (1977) 3–8. A. Torgašev, On graphs with a fixed number of negative eigenvalues, Discrete Math. 57 (1985) 311–317. G.H. Yu, L.H. Feng, H. Qu, Signed graphs with small positive index of inertia, Electron. J. Linear Algebra. 31 (2016) 232–243.
[^1]: Corresponding author. Email: [email protected]
[^2]: Supported by the National Natural Science Foundation of China (Grant nos. 11671344, 11531011).
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{
"pile_set_name": "ArXiv"
}
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abstract: 'We analyze the data from the 6 gravitational waves signals detected by LIGO through the lens of multifractal formalism using the MFDMA method, as well as shuffled and surrogate procedures. We identified two regimes of multifractality in the strain measure of the time series by examining long memory and the presence of nonlinearities. The moment used to divide the series into two parts separates these two regimes and can be interpreted as the moment of collision between the black holes. An empirical relationship between the variation in left side diversity and the chirp mass of each event was also determined.'
---
Introduction
============
Since the first detection, five more signals have been confirmed as GWs: GW151226 ([@GW2andLVT]), GW170104 ([@GW3]), GW170608 ([@GW170608]), GW170814 ([@GW170814]) and GW170817 ([@GW170817]) (the only one coming from a system of coalescing neutron stars), and one signal remains as a suspected GW (LVT151012 [@GW2andLVT]). The GW data used here are within the range of 32 seconds around the event and have a measurement frequency of 4096Hz. We will assume that these GWs, denoted by $y(t)$, are linear combinations of a deterministic signal, $d(t)$, and background noise, $n(t)$. In this context, the present analysis deals with observations that are collected over evenly spaced and discrete time intervals. In this Letter, we reports an analysis of a search for traces of multifractality in GW150914, a fact that may have strong consequences for our understanding of different characteristics of GW. A general discussion of all the GW signals detected to date (with the exception of GW170817) will be also presented. The signal of GW170817 (from coalescence of binary neutron stars) was removed from the sample since it differs in number of data from the signals produced by coalescence of black holes.
Multifractal analysis {#sec:style}
=====================
In monofractal series, one exponent (the Hurst exponent [@Hurst]) is sufficient to characterize the behavior of the series at various scales. $H$ values of $0<H<0.5$ and $0.5<H<1$ indicate persistence and anti-persistence, respectively, while $H=0.5$ indicates that the time series is uncorrelated. In multifractal time series, a range of values for this exponent is calculated. Thus, multifractal analysis consists of studying the scaling behavior in the time series $y(t)$. First, in accordance with the MultiFractal Detrending Moving Average (MFDMA) procedure, we calculated the mean-square function $F^2_\nu(n)$ for a $\nu$ segment of size [*n*]{}:
$$\label{fluctuMS}
F^2_\nu(n)=\frac{1}{n}\sum_{i=1}^{n}[e_\nu(i)]^2,$$
where $e_\nu(i)=y(i)-\tilde{y}(i)$ is the residual series in the segment $\nu$ and $\tilde{y}(i)$ is the moving average function. However, some authors have shown that semi-sinusoidal and power-law trends in multifractal approaches, including Multifractal Detrended Fluctuation Analysis (MFDFA) and MFDMA, are not efficiently removed ([@egh]). In our study, we did not encounter this problem. We then calculated the $q_{th}$ order overall fluctuation function $F_q(n)$, which is given by
$$\label{fluctu}
F_q(n)=\left\lbrace \frac{1}{N_n} \sum_{\nu=1}^{N_n} F^q_\nu(n) \right\rbrace^{1/q} \textrm{for}\; q\neq0$$
and, for $q=0$, $$\label{fluctuzero}
\ln\left[F_{0}(n)\right]=\frac{1}{N_{n}}\sum^{N_{n}}_{\nu=1}\ln [F_{\nu}(n)],$$ where $N_n$ is the number of segments non-overlaping. For larger values of $n$, the fluctuation function follows a power-law given by
$$\label{Fqxn}
F_q(n) \sim n^{h(q)}.$$
The generalized Hurst exponent **$h(q)$** is related to standard multifractal analysis parameters such as the Renyi scaling exponent ($\tau$), which is given by
$$\label{tau}
\tau(q)=qh(q)-1,$$
**when $q = 2$, we return to using monofractal analysis, i.e., $h(2) = H$ is the Hurst exponent.**
Two other important parameters are obtained using a Legendre transform, defined as $$\label{alpha}
\alpha=\frac{d\tau(q)}{dq}, \quad \alpha\in[\alpha_{min},\alpha_{max}]$$ and $$\label{falpha}
f(\alpha)=q\alpha-\tau(q),$$ which are the Hölder exponent and singularity spectrum, respectively.
One way to measure the degree of multifractality $(\Delta\alpha)$ in a series is by using the width of the multifractal singularity spectrum, which [@W2] and [@W1] defined as the difference between the maximum and minimum values of the Hölder exponent, i.e., $\Delta\alpha=\alpha_{max} - \alpha_{min}$.
![\[H1data\] Multifractal analysis of the full data from GW150914 (`H1data`). The top-left panel shows the fluctuation function versus the multi-scale behavior in a log-log diagram. The original series is in red, the shuffled series is in green, and the upper and lower limits correspond to $q=5$ and $q=-5$, respectively, while the bold in the middle corresponds to $q=0$. Dependences on the $q_{th}$ moment of the generalized Hurst exponent, $h(q)$, and the multifractal scaling exponent, $\tau(q)$, are shown in the top-right and bottom-left panels, respectively. The multifractal spectrum is shown in the bottom-right panel.](f1.pdf)
![\[cumulative\] Point-to-point multifractal analysis for the GW150914 time series from Livingston (bottom panel) and Hanford (top panel; shifted and inverted [@GW]), illustrated in green. Red circles represent the degree of multifractality ($\Delta\alpha$) calculated in the time series up to that point; likewise, blue and black circles represent the left side diversity $f(\alpha)_{max}-f(\alpha)^{left}_{min}$ and right side diversity $f(\alpha)_{max}-f(\alpha)^{right}_{min}$, respectively. The vertical lines represent $t = -0.06s$, the time point at which the time series are divided.](f2.pdf)
![\[DeltaDLMc\]The correlation between left side diversity variation $(\Delta D_L)$ and chirp mass for each GW (circles in blue) and LVT151012 (circle in green). The dashed line indicates a quadratic fit adjustment without the LVT signal, and the solid line is the same fit when considering this signal.](f4.pdf)
Results and discussion
======================
**We analyze the data from the 6 gravitational waves signals detected by LIGO identified as GW151226 , GW170104, GW170608, GW170814, GW170817 and LVT151012. All of data were extracted from LIGO. Data were analyzed using the multifractal formalism. Our aim is to study the possible sources of multifractality and to extract a set of multifractality indexes.**
To investigate the source of multifractality, we applied the shuffled method to the original series (the green curves in Figure \[H1data\]). This method destroys the memory signature, but preserves the distribution of the data with $h(q)=0.5$, if the source of multifractality in time series only presents long-range correlations ([@deFreitas2]). We realized that the multifractal behavior remains but with lowered strength. Similarly, the surrogate method (the blue curves in Figure \[H1data\]) also could not eliminate the multifractality in the original series. Already, this method destroys effects of non-linearity of the original series by randomizing the Fourier phases. These results indicate that the source of the multifractality is not only related to long-range correlations but also linked to the existence of non-linear terms that produce a heavy-tailed probability density function (PDF). The same analysis described in the previous three paragraphs was applied to the other three waves and indicated similar behavior both for the Hanford and Livingston detector data.
To study the evolution of the parameters related to the multifractal singularity spectrum throughout the time series, we constructed Fig. \[cumulative\], with the original time series shown in green, for the Hanford data in the top panel and Livingston in the bottom panel. The parameter values at one point in the time series data reflect the values calculated up to that point in a 50-point data window. As seen in Fig. \[cumulative\], the left side diversity $(\Delta f_{L}(\alpha))$ of the multifractal singularity spectrum, defined as $1-f(\alpha)^{left}_{min}$, is shown in blue and is associated with the sensitivity of the series to small-scale fluctuations with large magnitudes. In the same Figure, the right side diversity $(\Delta f_{L}(\alpha))$ of the the multifractal singularity spectrum, denoted by $1-f(\alpha)^{right}_{min}$, is shown in black and is linked to the sensitivity to fluctuations in the series with small magnitudes [@W2]. Furthermore, the parameters $(\Delta f_{L}(\alpha))$ and $(\Delta f_{R}(\alpha))$ indicate either a left or right truncation of the multifractal spectrum, respectively For this analysis, the parameters were calculated for the signal in the interval between 1 second before the event (for GW150914, this time is 1126259462.44s) and 0.05s after the event. We can observe a slight increase in the left side diversity at $t=-0.06s$, which indicates the presence of a strong small-scale fluctuation. This behavior appears in the data analysis of the two advanced LIGO detectors, H1 and L1. Using this time point, we divided the original series into two parts, wherein the first is identified as `H1data1` with 3581 measurements, while the second part comprises 720 measurements and is identified as `H1data2` data.
Using the same procedure as that for the entire time series, we performed multifractal analyses on both the `H1data1` and `H1data2` data. The shuffled method has eliminated the multifractality contained in the `H1data1`, shown in the right pane, and for the shuffled series, $h(2) = 0.5429$ and $\Delta\alpha = 0.0288$. These results indicate that the multifractality present in `H1data1` is due only to long-term correlations and thus does not provide non-linear terms. These correlations can be understood as stemming from the periodic orbital motion of the black holes. As for `H1data1`, multifractality is still present for the original time series, but neither the shuffled nor surrogate methods could eliminate the multifractal behavior; i.e., the multifractal behavior is due to two possible sources, i.e., memory and non-linearity.
These results have two consequences: first, the entire contribution of non-linearity in the analysis of the complete time series occurs in the second part of the series; second, as the periodic movement continues, even in the ringdown phase, the terms associated with long-term correlations continue to appear in the series. The enlargement of the PDF is because the amplitude of the strain grows somewhat in the second part of the series. Given that the strain amplitude is linked to the orbital velocity and mass that generated the gravitational wave, these nonlinear terms are caused by the collision of the black holes. In short, the contribution of long-term temporal correlation is due to the periodic motion of the orbiting black holes, and nonlinear terms occur due to the increase in the strain amplitude.
The difference, presented in the Figure \[cumulative\], between the maximum and minimum value of the left side diversity in the GW amplitude region of increase can be considered as the variation in left side diversity $(\Delta f_{L}(\alpha))$, as indicated in Figure \[cumulative\] for GW150914. We find an empirical correlation between this parameter and the chirp masses of each signal. Figure \[DeltaDLMc\] illustrates these parameters in blue circles for GWs and green circles for the LVT. A quadratic fit with (solid line) and without (dashed line) the LVT151012 signal is also shown in the same Figure. The overlap of these lines indicates that the analysis of the LVT signal falls within the expected behavior according to this correlation. Since we associate the variation in left side diversity with the amplitude increase in the signal, which in turn is related to chirp mass, we are led to conclude that this is an expected correlation. The detection of new GWs can serve as a good test for the correlation found here as well as a check for the chirp mass value of the detected signal.
Conclusions
===========
The statistical approach proposed in this study highlights the scenario opened by detection of the first GWs. We summarize the main results in three points: i) characterize the fractal dynamics of the signals, identifying their multifractal sources; ii) find the moment of the beginning of merger phase in black hole coalescence system, and; iii) determine the empirical relationship between the variation in left side diversity and chirp mass as an additional way for estimating this latter parameter. The methodology applied here may serve as a standard procedure for future analyses of gravitational waves. The prospect of new gravitational wave observatories, both on the ground and in space, provides more opportunities for the field of astronomy to employ the statistical tools already widely used in other areas of knowledge.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own whithin applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.'
address:
- '$^1$M.S. P365, Los Alamos National Laboratory, Los Alamos, NM 87505'
- '$^2$Mathematics Department, UCSB, Santa Barbara, CA 93106'
- '$^3$Theoretical Division and the Center for Nonlinear Studies, Los Alamos, NM 87505'
author:
- 'Mark Mineev$^1$, Mihai Putinar$^2$ and Razvan Teodorescu$^3$'
title: 'Random Matrices in 2D, Laplacian Growth and Operator Theory'
---
Introduction {#sec:intro}
============
During the second half of last century and continuing through the present, random matrix theory has grown from a special method of theoretical physics, meant to approximate energy levels of complex nuclei [@Wigner1; @Wigner2; @Wigner3; @Dyson1; @Dyson2; @Dyson3; @Dyson4], into a vast mathematical theory with many different application in physics, computer and electrical engineering. Simply describing all the developments and methods currently employed in this context would result in a monography much more extensive than this review. Therefore, we will only briefly mention topics which are themselves very interesting, but lie beyond the scope of this work.
The applications of random matrix theory (RMT) into physics have been extended from the original subject, spectra of heavy nuclei, to descriptions of large $N$ $SU(N)$ gauge theory [@tHoft; @IZPB], critical statistical models in two dimensions [@Kazakov; @Kostov; @Serban] disordered electronic systems [@Pastur; @Wegner; @Efetov; @Zirnbauer; @Fyodorov; @Altshuler], quantum chromo-dynamics (QCD) [@Verbaarschot; @Akemann], to name only a few. Non-physics applications range from communication theory [@MIMO] to stochastic processes out of equilibrium [@Schutz1; @Schutz2] and even more exotic topics [@Cuernavaca].
A number of important results, both at theoretical and applied levels, were obtained from the connection between random matrices and orthogonal polynomials, especially in their weighted limit [@Szego; @Saff-Totik; @IZPB; @BEH1; @BEH2]. These works explored the relationship between the branch cuts of spectral (Riemann) curves of systems of differential equations and the support of limit measures for weighted orthogonal polynomials. Yet another interesting connection stemming from this approach is with the general (matrix) version of the Riemann-Hilbert problem with finite support [@Its; @Deift].
In [@WZ03; @Teodorescu04], it was showed that such relationships also hold for the class of normal random matrices. Unlike in previous works, for this ensemble, the support of the equilibrium distribution for the eigenvalues of matrices in the infinite-size limit, is two-dimensional, which allows to interpret it as a growing cluster in the plane. Thus, a direct relation to the class of models known as Laplacian Growth (both in the deterministic and stochastic formulations), was derived, with important consequences. In particular, this approach allowed to study formation of singularities in models of two-dimensional growth. Moreover, these results allowed to define a proper way of continuing the solution for singular Laplacian Growth, beyond the critical point.
From the point of view of the dimensionality of the support for random matrix eigenvalues, it is possible to distinguish between 1-dimensional situations (which characterize 1 and 2-matrix models), and 2-dimensional situations, like in the case of normal random matrix theory. In fact, very recent results point to intermediate cases, where the support is a set of dimensional between 1 and 2. This situation is very similar to the description of disordered, interacting electrons in the plane, in the vicinity of the critical point which separates localized from de-localized behavior [@Efetov]. It is from the perspective of the dimensionality of support for equilibrium measure that we have organized this review.
The paper is structured in the following way: after a brief summary of the main concepts in Section \[first\], we explain the structure of normal random matrices in the limit of infinite size, in Section \[second\]. This allows to connect with planar growth models, of which Laplacian (or harmonic) Growth is a main representative. The following two sections give a solid description of the physical (Section \[third\]) and mathematical (Section \[fourth\]) structure of harmonic growth. The discretized (or quantized) version of this problem is precisely given by normal random matrices, as we indicate in these sections. Next we present a general scheme for encoding shade functions in the plane into linear data, specifically into a linear bounded Hilbert space operator $T$ with rank one self-commutator ${\rm rank} [T^\ast,T] = 1$. This line of research goes back to the perturbation and scattering theory of symmetric operators (M. G. Krein’s phase shift function) and to studies related to singular integral operators with a Cauchy kernel type singularity. Multivariate refinements of the “quantization scheme” we outline in Chapter 5 lie at the foundations of both cyclic (co)homology of operator algebras and of free probability theory. In view of the scope and length of the present survey, we confine ourselves to only outline the surprising link between quadrature domains and such Hilbert space objects.
We conclude with an application of the operator formalism to the description of boundary singular points that are characteristic to Laplacian growth evolution, and a brief overview of other related topics.
Random Matrix Theory in 1D
==========================
The symmetry group ensembles and their physical realisations
------------------------------------------------------------
Following [@Mehta], we reproduce the standard introduction of the symmetry-groups ensemble of random matrices. The traditional ensembles (orthogonal, unitary and symplectic) were introduced mainly because of their significance with respect to symmetries of hamiltonian operators in physical theories: time-reversal and rotational invariance corresponds to the orthogonal ensemble (which, for Gaussian measures, is naturally abbreviated GOE), while time-reversal alone and rotational invariance alone correspond to the symplectic and unitary ensembles, respectively (GSE and GUE for Gaussian measures).
An invariant measure is defined for each of these ensembles, in the form d (M) P(M) d (M) Z\^[-1]{} e\^[-[[Tr]{}]{}\[W(M)\]]{} d (M), where $M$ is a matrix from the ensemble, $Z$ is a normalization factor (partition function), Tr$[W(M)]$ is invariant under the symmetries on the ensemble, and $d \mu(M)$ is the appropriate flat measure for that ensemble: $\prod_{i \le j} d M_{ij}$ for orthogonal, $\prod_{i \le j}d {\mbox{\rm Re }} M_{ij} \prod_{i < j} d {\mbox{\rm Im }}M_{ij}$ for unitary, and $\prod_{i \le j}d M^{(0)}_{ij}
\prod_{k=1}^3 \prod_{i < j} dM^{(k)}_{ij}$ for symplectic (where each matrix element is an element of the real Klein group, $M_{ij} = M^{(0)}_{ij} \cdot 1 + \sum_{k=1}^3 M^{(k)}_{ij} \cdot \mathbb{\sigma}_k $). Correspondingly, to each of these ensembles, a parameter $\beta$, indicating the number of independent real parameters necessary to describe the pair of values $M_{ij}, M_{ji}$, is introduced, with values $\beta = 1, 2, 4$ for orthogonal, unitary and symplectic ensembles, respectively.
The invariance under transformations from the appropriate symmetry group leads to the following simplification of the measure: for any of these ensembles, the generic matrix $M$ can be diagonalized by a transformation $M = U^{-1} \Lambda U$, with $U$ from the same group, and $\Lambda = {\rm{diag}}(\lambda_1, \ldots, \lambda_N)$. The Jacobian of the transformation $M \to \Lambda, U$ (where $U$ is said to carry the “angular" degrees of freedom of $M$) is $J = \prod_{i < j}|\lambda_i - \lambda_j|^{\beta}= |\Delta(\Lambda)|^{\beta}$, with $\Delta$ the Vandermonde determinant. The angular degrees of freedom can be integrated out (a trivial redefinition of the normalization factor), giving the simplified measure (\_1, …, \_N)\_[i=1]{}\^N d\_i = Z\^[-1]{} e\^[[[Tr]{}]{}\[W()\]]{} |()|\^ \_[i=1]{}\^N d\_i For example, in the case of Gaussian measure $W(M) = -M^2$, the joint probability distribution function of eigenvalues, $\rho$, becames (up to normalization) (\_1, …, \_N) = Clearly, this procedure is useful only if we are interested in computing expectation values of quantities which depend only of the distribution of eigenvalues, and not of the angular degrees of freedom. This is indeed the case for all situations of interest.
The next standard transformation (which we discuss for the case of unitary ensemble, $\beta=2$) that is performed on the measure uses the well-known property of Vandermonde determinant $\Delta(\Lambda) = \det [\lambda_i^{j-1}]_{1 \le i, j \le N}$. Because of standard determinantal identities, this is equivalent with replacing each monomial $\lambda_i^{j-1}$ by a [*monic*]{} polynomial of the same order, $P_{j-1}(\lambda_i) = \lambda_{i}^{j-1} + \ldots$. Finally, these polynomials may be chosen to be orthogonal with respect to the measure $e^{W(\lambda)}$, giving for the p.d.f. of eigenvalues the expression (\_1, …, \_N) = ||\^2, which is simply the absolute value-squared of the wavefunction of the ground state for $N$ electrons in the external potential $W$. As we shall see, this kind o physical interpretation may be generalized to the case of matrix ensembles with two-dimensional support of eigenvalues.
#### Generalizations of group ensembles {#generalizations-of-group-ensembles .unnumbered}
Recently, various generalizations were proposed in order to extend the theory for ensembles of matrices which are not associated with symmetry groups. In particular, ensembles of matrices which may be reduced to a tridiagonal form (instead of standard diagonal) by a transformation which eliminates “angular" degrees of freedom, were introduced in [@Dumitriu]. As an interesting consequence, many results carry over to this case, while the parameter $\beta$ is allowed to take any positive real value.
Critical ensembles
------------------
In this section we explain how, using properly chosen non-Gaussian measures, it is possible to construct ensembles of hermitian matrices (corresponding again to the unitary symmetry) which are in a sense, critical, i.e. for which a continuum limit ($N \to\infty$) may be defined. The discussion relies on the formulation based on orthogonal polynomials indicated above, and it follows (at a more elementary level) the general theory of Saff and Totik [@Saff-Totik].
### General formalism
Let $d \mu(x) = e^{W(x)}d x $ be a well-defined measure on the real axis, $W(x) \to - \infty$ as $|x| \to \infty$, and $P^{(1)}_n(x)$ the corresponding family of orthogonal polynomials \_[-]{}\^P\^[(1)]{}\_n(x) P\^[(1)]{}\_m(x) d (x) = \_[nm]{}. Orthonormal functions are obtained through $\psi_n(x) = P_n(x) e^{W(x)/2}$, which are orthogonal with respect to the flat measure on $\mathbb{R}$. We consider a deformation of this ensemble through a positive real parameter $ \lambda \ge 1$, so that $d \mu_{\lambda}(x) = e^{\lambda W(x)} d x$ and \_[-]{}\^P\^[()]{}\_n(x) P\^[()]{}\_m(x) d \_ (x) = \_[nm]{}. Clearly, if $W(x) $ is a monomial of degree $k$, the deformation amounts to a simple rescaling P\^\_n(x) = \^[1/2k]{} P\^[(1)]{}\_n(\^[1/k]{}x) . The first non-trivial example is a quartic polynomial of the type W(x) = -(x\^2 + g x\^4), g> 0, for which the deformation in not a simple rescaling. In this case, it is possible to consider a special limit $n \to \infty, \lambda \to \infty, \lambda \to n r_c$, where $r_c$ is a constant. As we will see, for a specific value of $r_c$, this limit yields a special asymptotic behavior of the orthonormal functions $\psi_n(x)$. However, even for the simplest, trivial monomial (a Gaussian), which yields the Hermite polynomials, the asymptotic behavior of the orthogonal functions is non-trivial, in the sense that there are no known good approximations for the case $r_c = O(1)$.
Generically, in this large $n, \lambda$ limit, we can ask where the wavefunction $\psi_n(x)$ will reach its maximum value, in the saddle point approximation: \_[|x|]{}\_x |\_n(x)| = 0, giving \_x = 0, so that \[saddle\_point\] -r\_c W’(x) = \_[i=1]{}\^n , where $\xi_i, i = 1, \ldots, n$ are the roots of the $n^{\rm th}$ polynomial.
Let \[cauchy\] (z) = \_[i=1]{}\^n , multiply (\[saddle\_point\]) by $(\xi_i - z)^{-1 }$ and sum over $i$, and obtain \[polynomialst\] \^2(z) - r\_c W’(z) (z) = - \_[i=1]{}\^n . Equation (\[polynomialst\]) can be solved in the large $n$ limit by assuming that the roots will be distributed with density $\rho(\xi)$ on some compact (possibly disconnected) set $I \subset \mathbb{R}$. Defining \[definition\] R(z) = - \_I () d , we obtain \[large\] \^2(z) - r\_c W’(z) (z) + ( )\^2 R(z) = 0. The proper solution of (\[polynomialst\]) (considering the behavior at $\infty$ of the function $\omega(z)$), is \[solution\] (z) = , and (since the function $\omega(z)$ is the Cauchy transform of the density $\rho(x)$), it gives us the asymptotic distribution of zeros as (x) = \[(x+ i 0) - (x-i0)\]. Finally, to obtain the asymptotic form of wave functions $\psi_n(x)$, we can write \[asymptotic\] n\^[-1]{} \_n(x) () (x- )d - +W(x).
### Continuum limit and integrable equations
There are two related problems for the large $n$ limit of deformed ensembles described in the previous section. The first is determination of the support of zeros $I$; the second is the scaling behavior of the orthogonal functions $\psi_n(x)$. In general, the limiting support $I$ may consist of several disconnected segments $I_k$, $I = \cup_{k=1}^{k=d} I_k$. In the simplest case, it is just one interval $I = [a, b]\subset \mathbb{R}$. In this section we indicate how to determine this support as well as the density $\rho(x)$, and what this yields for the orthogonal functions.
Let the function $W(x)$ be a polynomial of even degree $d$. From (\[definition\]) we see that $R(z)$ is a polynomial of degree $d-2$, and therefore solution (\[solution\]) has generically $2(d-1)$ branch points. Thus, the function $\omega(z)$ typically has $d-1$ branch cuts, which constitute the disconnected support of distribution $\rho(z)$.
We are interested in a special case, when $d-2$ of these cuts degenerate into double points, and there is a single interval $[a, b]$ which is the support of $\rho(z)$. This special case is called $critical$ and it provides new asymptotic limits for the orthogonal functions. We will also refer to this solution as the “single-cut" solution.
From the equation (x+ i0) - (x-i0) = r\_c W’(x), we obtain for the single-cut solution $$\omega(z) = -\frac{r_c \sqrt{(z-a)(z-b)}}{2 \pi} \int_a^b \frac{W'(\xi)}{\sqrt{(b-\xi)(\xi-a)}} \frac{d \xi}{\xi -z}.$$ The large $|z|$ behavior of this function is known from the continuum limit of (\[cauchy\]), and it implies the absence of regular terms in the Laurent expansion: (z) = - + O(z\^[-2]{}), so that we impose the conditions \[cond1\] 0 = \_a\^b d , \[cond2\] 2 = - r\_c \_a\^b d .
#### Gaussian measure and the Hermite polynomials {#gaussian-measure-and-the-hermite-polynomials .unnumbered}
Let $d=2$ and $-W(x) = ax^2, a > 0$. Then conditions (\[cond1\],\[cond2\]) give a symmetric support $[-b, b]$ where $b^2 = 2/(a r_c)$.
More generally, using the saddle point equation for $|\psi_n(x)|$ at $x = a, b$ and (\[asymptotic\]), we conclude that = C\_b - \_0\^ d Since the integrand behaves like $\eta^{d-3/2}$, we obtain \_n(b+) = \_n(b) . We immediately conclude that for $d=2$ (Hermite polynomials), the asymptotic behavior is given by the Airy function, $\exp z^{3/2}$. The full scaling is achieved by considering the region around the end-point $b$, of order $\zeta = O(n^{-2/(2d-1)})$.Then we obtain \_n(b+n\^[-]{}) \~.
### Scaled limits of orthogonal polynomials and equilibrium measures
The distribution of eigenvalues investigated in the previous sections illustrates the general approach developed by Saff and Totik [@Saff-Totik] for holomorphic polynomials orthogonal on curves in the complex plane. We sketch here the more general result because of its relevance to the main topic of this review.
Given a set $\Sigma \in \mathbb{C}$ and a properly-defined measure on it $w(z)=e^{-Q(z)}$, we construct the holomorphic orthogonal polynomials $P_n(z)$, with respect to $w$. We then pose the question of finding the “extremal" measure (its support $S_w$ and density $\mu_w$), such that the $F$-functional $F(K) \equiv \log {\rm{cap}} (K) - \int Q d \omega_K$, with ${\rm{cap}}(K)$ and $\omega_K$ the capacity, respectively the equilibrium measure of the set $K$, is maximized by $S_w$. Furthermore, $\mu_w$ satisfies energy and capacity constraints on $S_w$.
The remarkable fact noticed in [@Saff-Totik] is that if the extremal value $F(S_w)$ is approximated by the weighted $monic$ polynomials $\tilde{P}_n(z)$ as $(||w^n \tilde{P}_n||_\Sigma^*)^{1/n} \to \exp (-F_w)$ (where we use the weak star norm), then the asymptotic zero distribution of $\tilde{P}_n$ gives the support $S_w$. Hence, (\[ro\]) may be interpreted as giving both the support of the extremal measure (labeled $\rho$ in this formula), as well as its actual density.
The extremal measure has the physical interpretation of the “smallest" equilibrium measure which gives a prescribed logarithmic potential at infinity. According to the concept of “sweeping" (or “balayage", see [@Saff-Totik]), the extremal measure is obtained as a limit of the process, under the constraints imposed on the total mass and energy of the measure. As we have shown in this chapter, for the case of 1D measures, this extreme case is given by weighted limits of orthogonal polynomials.
Random Matrix Theory in higher dimensions
=========================================
In this chapter, we show how to generalize the concepts of equilibrium measure, extremal measure, and their relations to orthogonal polynomials and ensembles of random matrices, in the case of two-dimensional support. The applications of this theory to planar growth processes will be discussed in the following two chapters.
The Ginibre-Girko ensemble
--------------------------
We begin with a brief discussion on the oldest and simplest ensemble of random matrices with planar support. The ensemble of complex, $N \times N$ random matrices with identical, independent, zero-mean Gaussian-distributed entries, was first studied by J. Ginibre in 1965 [@Ginibre], and then it was generalized for non-zero mean Gaussian by Girko in 1985 [@Girko]. Consider $N \times N$ random matrices with eigenvalues $z_k \in \mathbb{C}$, and joint p.d.f. dP\_N \~\_[1 i < j N]{} |z\_i - z\_j|\^2 \_[1 k N]{} \_N(z\_k), where $\mu_N(z_k) = e^{-N|z_k|^2} d {\rm {Re}} z_k d {\rm {Im}} z_k$. Then, in the large $N$ limit, the measure $
\frac{1}{N}\sum_k \delta(z-z_k)
$ converges weakly to the uniform measure on the unit disk. This is known as the Circular Law. If the exponent of the pure Gaussian is perturbed by a quadratic term, the result holds for a corresponding elliptical domain, giving the Elliptical Law. The same limiting curves (circular and elliptical) describe the graph of the distribution of $real$ eigenvalues for Hermitian ensembles, with pure and perturbed Gaussian measures. In that case, the laws are known as Wigner-Dyson [@Wigner1; @Dyson1] and Marchenko-Pastur, respectively (although the last one was originally derived for covariance matrices built from sparse regression matrices [@MarPas]).
Extensions and exceptions from the circular and elliptical laws were found by relaxing the conditions of the theorems. In particular, deviations from uniformity for angular statistics in the case of Gaussian measure were derived in [@Rider], while the case of heavy-tails distributions was investigated in [@Sosh; @Zakh] and subsequent publications.
Normal matrix ensembles
-----------------------
A special case of matrices with complex eigenvalues is given by $normal$ matrices. A matrix $M$ is called normal if it commutes with its Hermitian conjugate: $[M, M^{\dag}]=0$, so that both $M$ and $M^{\dag}$ can be diagonalized simultaneously. The statistical weight of the normal matrix ensemble is given through a general potential $W(M,M^{\dag})$ [@Zaboronsky]: \[ZN\] e\^[ W(M, M\^)]{} [[[d]{}]{}]{}(M). Here $\hbar$ is a parameter, and the measure of integration over normal matrices is induced by the flat metric on the space of all complex matrices ${{{{d}}}}_C M$, where ${{{{d}}}}_C M = \prod_{ij}{{{d}}}\, {\rm Re} \, M_{ij} {{{d}}}\, {\rm Im} \, M_{ij}$. Using a standard procedure, one passes to the joint probability distribution of eigenvalues of normal matrices $z_1,\dots,z_N$, where $N$ is the size of the matrix: \[mean\] |\_N (z)|\^2 \_[j=1]{}\^N e\^[W(z\_j,|z\_j)]{} [[[[d]{}]{}]{}]{}\^2 z\_j Here ${{{d}}}^2 z_j \equiv {{{d}}}x_j \, {{{d}}}y_j$ for $z_j =x_j +iy_j$, $\Delta_N(z)=\det (z_{j}^{i-1})_{1\leq i,j\leq N}=
\prod_{i>j}^{N}(z_i -z_j)$ is the Vandermonde determinant, and \[tau\] \_N = |\_N (z)|\^2 \_[j=1]{}\^[N]{}e\^[ W(z\_j,|z\_j)]{} d\^2z\_j is a normalization factor, the partition function of the matrix model (a $\tau$-function).
A particularly important special case arises if the potential $W$ has the form \[potential\] W=-|z|\^2+V(z)+, where $V(z)$ is a holomorphic function in a domain which includes the support of eigenvalues (see also a comment in the end of Section \[F1\] about a proper definition of the ensemble with this potential). In this case, a normal matrix ensemble gives the same distribution as a general complex matrix ensemble. A general complex matrix can be decomposed as $M=U(Z+R)U^\dagger$, where $U$ and $Z$ are unitary and diagonal matrices, respectively, and $R$ is an upper triangular matrix. The distribution (\[mean\]) holds for the elements of the diagonal matrix $Z$ which are eigenvalues of $M$. Here we mostly focus on the special potential (\[potential\]), and also assume that the field A(z)=\_z V(z) is a globally defined meromorphic function.
Droplets of eigenvalues
-----------------------
In the large $N$ limit ($\hbar\to 0$, $N\hbar$ fixed), the eigenvalues of matrices from the ensemble densely occupy a connected domain $D$ in the complex plane, or, in general, several disconnected domains. This set (called the support of eigenvalues) has sharp edges (Figure \[droplets\]). We refer to the connected components $D_\alpha$ of the domain $D$ as [*droplets*]{}.
![A support of eigenvalues consisting of four disconnected components (left). The distribution of eigenvalues for potential $V(z) = - \alpha \log (1- z/\beta) - \gamma z $. (right) []{data-label="droplets"}](dropletsa.pdf "fig:"){width="5cm"} ![A support of eigenvalues consisting of four disconnected components (left). The distribution of eigenvalues for potential $V(z) = - \alpha \log (1- z/\beta) - \gamma z $. (right) []{data-label="droplets"}](splitjuk.pdf "fig:"){width="5cm"}
For algebraic domains (the definition follows) the eigenvalues are distributed with the density $\rho=-\frac{1}{4 \pi}\Delta W$, where $\Delta =4\p_z \p_{\bar z}$ is the 2-D Laplace operator [@WZ03]. For the potential (\[potential\]) the density is uniform. The shape of the support of eigenvalues is the main subject of this chapter. For example, if the potential is Gaussian [@Ginibre], A(z)=2t\_2 z, the domain is an ellipse. If $A$ has one simple pole, A(z)=--the droplet (under certain conditions discussed below) has the profile of an aircraft wing given by the Joukowsky map (Figure \[droplets\]). If $A$ has one double pole (say, at infinity), A(z)=3t\_3 z\^2, the droplet is a hypotrochoid. If $A$ has two or more simple poles, there may be more than one droplet. This support and density represent the equilibrium solution to an electrostatic problem, as we will indicate in a later section.
Orthogonal polynomials and distribution of eigenvalues
------------------------------------------------------
Define the exact $N$-particle wave function (up to a phase), by \[psi\] \_N(z\_1,…,z\_N)= \_N (z) e\^[\_[j=1]{}\^[N]{} W(z\_j, |z\_j)]{}. The joint probability distribution (\[mean\]) is then equal to $|\Psi(z_1,\dots,z_N)|^2$.
Let the number of eigenvalues (particles) increase while the potential stays fixed. If the support of eigenvalues is simply-connected, its area grows as $\hbar N$. One can describe the evolution of the domain through the density of particles $$\la{51}
\rho_N(z)=N \int|\Psi_{N}(z,z_1,z_2,\dots,z_{N-1})|^2
d^2z_1\dots d^2 z_{N-1},
\ee
where $\Psi_N$ is given by (\ref{psi}).
We introduce a set of orthonormal one-particle functions
on the complex plane as matrix elements of transitions
between $N$ and $(N+1)$-particle states:
\begin{equation}\la{5}
\frac{\psi_N(z)}{\sqrt{N+1}}= \int\Psi_{N+1} (z,z_1,z_2,\dots,z_N)
\overline{\Psi_{N}(z_1,z_2,\dots,z_N)} d^2z_1\dots d^2 z_N$$ Then the rate of the density change is $$\la{1}
\rho_{N+1}(z)-\rho_{N}(z)=|\psi_N(z)|^2.$$ The proof of this formula is based on the representation of the $\psi_n$ through holomorphic biorthogonal polynomials $P_n(z)$. Up to a phase $$\label{O3}
\psi_n (z)=
e^{\frac{1}{2\hbar}W(z, \bar z )}P_n (z),\quad
\quad P_n (z)=\sqrt{\frac{\tau_n}{\tau_{n+1}}} z^n +\ldots$$ The polynomials $P_n(z)$ are biorthogonal on the complex plane with the weight $e^{W/\hbar}$: $$\la{29}
\int e^{W/\hbar}P_n(z)\overline{P_m(z)}d^2 z=
\delta_{mn}.$$ The proof of these formulae is standard in the theory of orthogonal polynomials. Extension to the biorthogonal case adds no difficulties.
We note that, with the choice of potential (\[potential\]), the integral representation (\[29\]) has only a formal meaning, since the integral diverges unless the potential is Gaussian. A proper definition of the wave functions goes through recursive relations (\[L1\], \[M\]) which follow from the integral representation. The same comment applies to the $\tau$-function (\[tau\]). The wave function is not normalized everywhere in the complex plane. It may diverge at the poles of the vector potential field.
Wavefunctions, recursions and integrable hierarchies
----------------------------------------------------
In order to illustrate the mathematical connection between this theory and equivalent formulations which we present in Chapter 5, it is necessary to make a digression through the formalism of infinite, integrable hierarchies. In particular, we choose the case of the Kadomtsev-Petviashvilii (KP) hierarchy, and follow the notations in [@Dikey].
### Pseudo-differential operators
We denote by $\mathcal{A}$ the algebra constructed from differential polynomials of the type $P = \p^n + u_{n-2}\p^{n-2} + \ldots + u_1 \p + u_0$, where $\p = \p / \p z$ is a differential symbol with respect to some (complex) variable $z$, and $u_0, u_1, \ldots, u_{n-2}$ (note: $u_{n-1}$ can be always set to zero) are generically smooth functions in $z$ and (if necessary) other variables $t_1, t_2, \ldots $. On this algebra, we define the ring of pseudo-differential operators $\mathcal R$, consisting of (formal) operators defined by the infinite series L = \_[-]{}\^n c\_k \^k, \^[-1]{} dz, where coefficients are again smooth functions, and the negative powers in the expansions contain $integral$ operators. For any such operator, we denote by $L_+$ the purely differential part and by $L_-$ the remainder of the series: L\_+ \_[0]{}\^n c\_k \^k, L = L\_+ + L\_-.
Let L = + u\_0 \^[-1]{} + u\_1\^[-2]{} + …be a pseudo-differential operator such that $\mathcal{L}_+ = \p$. Then, introducing the infinite set of $times$ ${\bm {t}} = t_1, t_2, \ldots ,$ such that all coefficients $u_k, c_k $ above are generically functions of $\bm t$, the KP hierarchy has the form = \[\^k\_+, \], k = 1, 2, …More explicitly, we note that the hierarchy consists of the differential equations satisfied by the [*coefficients*]{} of the operator $\mathcal L$. As a consequence of the compatibility of all the equations in the hierarchy, we have the [*zero-curvature equations*]{} = 0, t\_k, t\_p.
### Level reductions
The KP hierarchy contains many other known integrable hierarchies, particularly the KdV hierarchy, as reductions to a certain $level$ $n$ in the hierarchy. For example, assume that the operator $\mathcal L$ satisfies the constraint \^2\_- = 0, i.e. it is the square root of a [*differential*]{} operator $L$ of order 2: = L\^[1/2]{}, L = \^2 + 2u\_0. Then it follows that for all even powers $n=2m$, $\mathcal{L}^n_{+} = \mathcal{L}^n$, so that $[\mathcal{L}^n_{+}, \mathcal{L}] = 0$, so there is $no$ dependence on the even times $t_2, t_4, \ldots$. This sub-hierarchy is called $level-2$ KdV, because the first non-trivial zero-curvature equation of the hierarchy is the famous Korteweg-de Vries equation: \^3\_+ P = \^3 + , = \[P, L\] u\_[t\_3]{} = 6 uu\_z + u\_[zzz]{}, where we have used $u_0 = u$ for clarity.
This formulation of the KdV equation makes use of the notion of $Lax$ pair $L, P$, which is central to the [*inverse scattering method*]{} for solving nonlinear integrable differential equations The idea is quite physical: assume that the operators $L, P$ act on a wavefunction $\psi(x,t)$ such that L= , = P , where eigenvalues $\lambda$ form the spectrum of $L$. Then applying the Lax pair equation to the eigenvalue equation, we obtain $\p \lambda / \p t = 0$, i.e. the evolution under these equations leaves the spectrum invariant. This allows to construct the initial state from the final state, hence the inverse scattering appellation.
### Tau functions and Baker-Akhiezer function
At the level of systems of PDE, the $\tau-$function and the Baker-Akhiezer function are introduced, by analogy with the Lax par formulation indicated above, in the following way:
#### Baker-Akhiezer function {#baker-akhiezer-function .unnumbered}
Consider the function $\psi(z,t_1, t_2, \ldots)$ satisfying = z , = \^k\_+ , k 1. This is the Baker-Akhiezer function of the KP hierarchy.
#### Fundamental property of the Baker-Akhiezer function {#fundamental-property-of-the-baker-akhiezer-function .unnumbered}
> Let $\phi = 1 + \sum_0^\infty k_i \p^{-i-1}$ be the “dressing" operator defined such that $\mathcal{L} = \phi \p \phi^{-1}$. Also, introduce the function $g(z, t_1, \ldots) = \exp [\sum_{1}^\infty t_k z^k]$. Then the Baker-Akhiezer function satisfies: $$\psi = \hat{k}(z) g(z, t_1, \ldots), \quad \hat{k}(z) = 1 + \sum_{0}^\infty k_i z^{-i-1},$$ where $\hat{k}$ is the “scalar" analog of the dressing operator $\phi$.
#### Tau function {#tau-function .unnumbered}
Using the notation introduced above, we have the following property:
> There exists a function $\tau(z, t_1, \ldots)$ such that $$\psi(z, t_1, \ldots) = g \cdot \frac{\tau(z, t_1-\frac{1}{z}, t_2 - \frac{1}{2z^2}, \ldots)}{\tau(z, t_1, t_2, \ldots)}
> = g \cdot \frac{\exp \left [\sum_1^\infty -\frac{1}{kz^k}\frac{\p }{\p t_k} \right ] \tau(z, t_1, \ldots)}{\tau(z, t_1, \ldots)}$$
Now let us consider the generalized overlap function $$\psi_N(z,\bar w) = \tau_{N}^{-1}
\int\Psi_{N+1} (z,z_1,z_2,\dots,z_N)
\overline{\Psi_{N+1}(w,z_1,z_2,\dots,z_N)} d^2z_1\dots d^2 z_N,$$ and expand for $|z|, |w| \to \infty$. We obtain \_N(z, |w) = \_N, where $a_{kp}$ is the corresponding interior bi-harmonic moment. Therefore, we may regard the $\tau$-function and the scaled wavefunction introduced earlier as canonical objects describing an integrable hierarchy. This fact will be illustrated in more detail in the next section.
Equations for the wave functions and the spectral curve
-------------------------------------------------------
In this section we specify the potential to be of the form (\[potential\]). It is convenient to modify the exponential factor of the wave function. Namely, we define \_n (z)= e\^[-+V(z)]{}P\_n (z),\_n (z)= e\^[V(z)]{}P\_n (z), where the holomorphic functions $\chi_n(z)$ are orthonormal in the complex plane with the weight $e^{-|z|^2/\hbar}$. Like traditional orthogonal polynomials, the biorthogonal polynomials $P_n$ (and the corresponding wave functions) obey a set of differential equations with respect to the argument $z$, and recurrence relations with respect to the degree $n$. Similar equations for two-matrix models are discussed in numerous papers (see, e.g., [@Aratyn]).
We introduce the $L$-operator (the Lax operator) as multiplication by $z$ in the basis $\chi_n$: $$\la{L1}
L_{nm}\chi_m(z)=z\chi_n(z)$$ (summation over repeated indices is implied). Obviously, $L$ is a lower triangular matrix with one adjacent upper diagonal, $L_{nm}=0$ as $m>n+1$. Similarly, the differentiation $\p_z$ is represented by an upper triangular matrix with one adjacent lower diagonal. Integrating by parts the matrix elements of the $\p_z$, one finds: $$\la{M}
(L^{\dag})_{nm}\chi_m =
\hbar\p_z\chi_n,$$ where $L^{\dag}$ is the Hermitian conjugate operator.
The matrix elements of $L^{\dag}$ are $(L^{\dag})_{nm}=\bar L_{mn}=A(L_{nm})+
\int e^{\frac{1}{\hbar}W}\bar P_m(\bar z)\p_z P_n(z) d^2z$, where the last term is a lower triangular matrix. The latter can be written through negative powers of the Lax operator. Writing $\p_z\log P_n(z)=\frac{n}{z}+\sum_{k>1}v_k(n)z^{-k}$, one represents $L^{\dag}$ in the form $$\la{M1}
L^{\dag}=A(L)+(\hbar n) L^{-1}+\sum_{k>1}v^{(k)}L^{-k},$$ where $v^{(k)}$ and $(\hbar n)$ are diagonal matrices with elements $v_n^{(k)}$ and $(\hbar n)$. The coefficients $v_{n}^{(k)}$ are determined by the condition that lower triangular matrix elements of $A(L_{nm})$ are cancelled.
In order to emphasize the structure of the operator $L$, we write it in the basis of the shift operator [^1] $\hat w$ such that $\hat w f_n =f_{n+1}\hat w$ for any sequence $f_n$. Acting on the wave function, we have: $$\hat w
\chi_n=\chi_{n+1}.$$ In the $n$-representation, the operators $L$, $L^{\dag}$ acquire the form $$\la{M11}
L=r_n \hat w+\sum_{k\geq 0} u_{n}^{(k)} \hat w^{-k},\quad
L^{\dag} = \hat w^{-1} r_n+
\sum_{k\geq 0} \hat w^{k} \bar u_{n}^{(k)}.$$ Clearly, acting on $\chi_n$, we have the commutation relation (“the string equation") $$\la{string}
[L, \, L^{\dag}]=\hbar.$$ This is the compatibility condition of Eqs. (\[L1\]) and (\[M\]).
Equations (\[M11\]) and (\[string\]) completely determine the coefficients $v_n^{(k)}$, $r_n$ and $u_n^{(k)}$. The first one connects the coefficients to the parameters of the potential. The second equation is used to determine how the coefficients $v_n^{(k)}$, $r_n$ and $u_n^{(k)}$ evolve with $n$. In particular, the diagonal part of it reads n=r\_n\^2-\_[k1]{}\_[p=1]{}\^k|u\_[n+p]{}\^[(k)]{}|\^2. Moreover, we note that all the coefficients can be expressed through the $\tau$-function (\[tau\]) and its derivatives with respect to parameters of the potential. This representation is particularly simple for $r_n$: $
r_n^2 =\tau_n \tau_{n+1}^{-2}\tau_{n+2}
$.
### Finite dimensional reductions
If the vector potential $A(z)$ is a rational function, the coefficients $u_{n}^{(k)}$ are not all independent. The number of independent coefficients equals the number of independent parameters of the potential. For example, if the holomorphic part of the potential, $V(z)$, is a polynomial of degree $d$, the series (\[M11\]) are truncated at $k= d-1$.
In this case the semi-infinite system of linear equations (\[M\]) and the recurrence relations (\[L1\]) can be cast in the form of a set of finite dimensional equations whose coefficients are rational functions of $z$, one system for every $n>0$. The system of differential equations generalizes the Cristoffel-Daurboux second order differential equation valid for orthogonal polynomials. This fact has been observed in recent papers [@BEHdual; @Eynard03] for biorthogonal polynomials emerging in the Hermitian two-matrix model with a polynomial potential. It is applicable to our case (holomorphic biorthogonal polynomials) as well.
In a more general case, when $A(z)$ is a general rational function with $d-1$ poles (counting multiplicities), the series (\[M11\]) is not truncated. However, $L$ can be represented as a “ratio", \[LKK\] L=K\_[1]{}\^[-1]{}K\_[2]{}=M\_2 M\_[1]{}\^[-1]{}, where the operators $K_{1,2}$, $M_{1,2}$ are polynomials in $\hat w$: \[KK\] K\_1 =w\^[d-1]{}+\_[j=0]{}\^[d-2]{}A\_[n]{}\^[(j)]{} w\^j, K\_2 =r\_[n + d - 1]{}w\^[d]{}+\_[j=0]{}\^[d-1]{}B\_[n]{}\^[(j)]{} w\^j \[MM\] M\_1 =w\^[d-1]{}+\_[j=0]{}\^[d-2]{}C\_[n]{}\^[(j)]{} w\^j, M\_2 =r\_[n]{}w\^[d]{}+\_[j=0]{}\^[d-1]{}D\_[n]{}\^[(j)]{} w\^j These operators obey the relation \[KMKM\] K\_1 M\_2 =K\_2 M\_1. It can be proven that the pair of operators $M_{1,2}$ is uniquely determined by $K_{1,2}$ and vice versa. We note that the reduction (\[LKK\]) is a difference analog of the “rational" reductions of the Kadomtsev-Petviashvili integrable hierarchy considered in [@Krichev-red].
The linear problems (\[L1\]), (\[M\]) acquire the form \[L1a\] (K\_2)\_n =z (K\_1 )\_n, (M\_[2]{}\^ )\_n =\_z (M\_[1]{}\^) \_n. These equations are of [*finite order*]{} (namely, of order $d$), i.e., they connect values of $\chi_n$ on $d+1$ subsequent sites of the lattice.
The semi-infinite set $\{\chi_0,\chi_1,\dots\}$ is then a “bundle" of $d$-dimensional vectors $${\underline\chi}(n)= (\chi_n,\chi_{n+1},\dots,
\chi_{n+d-1})^{{\rm t}}$$ (the index ${\rm t}$ means transposition, so ${\underline\chi}$ is a column vector). The dimension of the vector is the number of poles of $A(z)$ plus one. Each vector obeys a closed $d$-dimensional linear differential equation $$\la{M2}
\hbar\p_z{\underline\chi}(n)={\mathcal L}_n (z){\underline\chi}(n),$$ where the $d\times d$ matrix ${\mathcal L}_n$ is a “projection" of the operator $L^{\dag}$ onto the $n$-th $d$-dimensional space. Matrix elements of the ${\mathcal L}_n$ are rational functions of $z$ having the same poles as $A(z)$ and also a pole at the point $\overline{A(\infty )}$. (If $A(z)$ is a polynomial, all these poles accumulate to a multiple pole at infinity).
We briefly describe the procedure of constructing the finite dimensional matrix differential equation. We use the first linear problem in (\[L1a\]) to represent the shift operator as a $d\times d$ matrix ${\mathcal W}_n (z)$ with $z$-dependent coefficients: \_n(z)(n)=(n + 1). This is nothing else than rewriting the scalar linear problem in the matrix form. Then the matrix ${\mathcal W}_n (z)$ is to be substituted into the second equation of (\[L1a\]) to determine ${\mathcal L}_n(z)$ (examples follow). The entries of ${\mathcal W}_n(z)$ and ${\mathcal L}_n (z)$ obey the Schlesinger equation, which follows from compatibility of (\[M2\]) and (\[W\]): \_z [W]{}\_n= [L]{}\_[n+1]{} [W]{}\_n -[W]{}\_n [L]{}\_n.
This procedure has been realized explicitly for polynomial potentials in recent papers [@BEHdual; @Eynard03]. We will work it out in detail for our three examples: ${\underline\chi}(n)=(\chi_n,\,\chi_{n+1})^{{\rm t}}$ for the ellipse (\[e\]) and the aircraft wing (\[J1\]) and ${\underline\chi}(n)=(\chi_n,\,\chi_{n+1},\chi_{n+2})^{{\rm t}}$ for the hypotrochoid (\[hyp\]).
Spectral curve
--------------
According to the general theory of linear differential equations, the semiclassical (WKB) asymptotics of solutions to Eq. (\[M2\]), as $\hbar \to 0$, is found by solving the eigenvalue problem for the matrix ${\mathcal L}_n (z)$ [@Wasow]. More precisely, the basic object of the WKB approach is the spectral curve [@Wasow] of the matrix ${\mathcal L}_n$, which is defined, for every integer $n>0$, by the secular equation $\det ({\mathcal L}_n (z)-\tilde z) = 0$ (here $\tilde z$ means $\tilde z \cdot {\bf 1}$, where ${\bf 1}$ is the unit $d\times d$ matrix). It is clear that the left hand side of the secular equation is a polynomial in $\tilde z$ of degree $d$. We define the spectral curve by an equivalent equation $$\la{qc}
f_n (z,\tilde z)=a(z)\det ({\mathcal L}_n (z)-\tilde z) = 0,$$ where the factor $a(z)$ is added to make $f_n(z,\tilde z)$ a polynomial in $z$ as well. The factor $a(z)$ then has zeros at the points where poles of the matrix function ${\mathcal L}(z)$ are located. It does not depend on $n$. We will soon see that the degree of the polynomial $a(z)$ is equal to $d$. Assume that all poles of $A(z)$ are simple, then zeros of the $a(z)$ are just the $d-1$ poles of $A(z)$ and another simple zero at the point $\overline{A(\infty)}$. Therefore, we conclude that the matrix ${\mathcal L}_n (z)$ is rather special. For a general $d\times d$ matrix function with the same $d$ poles, the factor $a(z)$ would be of degree $d^2$.
Note that the matrix ${\mathcal L}_n (z) -\bar z$ enters the differential equation $$\la{M2'}
\hbar\p_{z}|{\underline\psi}(n)|^2=\bar {\underline\psi}(n)
({\mathcal L}_n (z)- \bar z)
{\underline\psi}(n)$$ for the squared amplitude $|{\underline \psi}(n) |^2={\underline \psi}^{\dag}(n)
{\underline \psi}(n) =
e^{-\frac{|z|^2}{\hbar}} |{\underline \chi}(n)|^2$ of the vectors ${\underline \psi}(n)$ built from the orthonormal wave functions (\[O3\]).
The equation of the curve can be interpreted as a “resultant" of the non-commutative polynomials $K_2 -zK_1$ and $M_{2}^{\dag}-\tilde z M_{1}^{\dag}$ (cf. [@BEHdual]). Indeed, the point $(z, \tilde z)$ belongs to the curve if and only if the linear system \[linsys\] {
[l]{} (K\_2 c)\_k =z (K\_1 c)\_k n-dk n-1\
\
(M\_[2]{}\^ c)\_k =z (M\_[1]{}\^ c)\_k nk n+d-1
. has non-trivial solutions. The system contains $2d$ equations for $2d$ variables $c_{n-d}\, , \ldots , c_{n+d -1}$. Vanishing of the $2d \, \times \, 2d$ determinant yields the equation of the spectral curve. Below we use this method to find the equation of the curve in the examples. It appears to be much easier than the determination of the matrix ${\mathcal L}_n(z)$.
The spectral curve (\[qc\]) possesses an important property: it admits an antiholomorphic involution. In the coordinates $z, \tilde z$ the involution reads $(z, \tilde z)\mapsto (\overline{\tilde z}, \bar z)$. This simply means that the secular equation $\det (\bar {\mathcal L}_n (\tilde z)-z) = 0$ for the matrix $\bar {\mathcal L}_n (\tilde z)\equiv
\overline{{\mathcal L}_n (\overline{\tilde z})}$ defines the same curve. Therefore, the polynomial $f_n$ takes real values for $\tilde z =\bar z$: f\_n(z,|z)=. Points of the real section of the curve ($\tilde z =\bar z$) are fixed points of the involution.
The curve (\[qc\]) was discussed in recent papers [@BEHdual; @Eynard03] in the context of Hermitian two-matrix models with polynomial potentials. The dual realizations of the curve pointed out in [@BEHdual] correspond to the antiholomorphic involution in our case. The involution can be proven along the lines of these works. The proof is rather technical and we omit it, restricting ourselves to the examples below. We simply note that the involution relies on the fact that the squared modulus of the wave function is real.
We will give a concrete example for the construction of the spectral curve, after a brief but necessary detour through the continuum limit of this problem.
### Schwarz function
The polynomial $f_n(z, \bar z)$ can be factorized in two ways: f\_n(z,|z)=a(z)(|z-S\_n\^[(1)]{}(z)) …(|z-S\_n\^[(d)]{}(z)), where $S_n^{(i)}(z)$ are eigenvalues of the matrix ${\mathcal L}_n (z)$, or f\_n(z,|z)=( z-|S\_n\^[(1)]{}(|z))…( z-|S\_n\^[(d)]{}(|z)), where $\bar S_n^{(i)}(\bar z)$ are eigenvalues of the matrix $\bar {\mathcal L}_n (\bar z)$. One may understand them as different branches of a multivalued function $S(z)$ (respectively, $\bar S(z)$) on the plane (here we do not indicate the dependence on $n$, for simplicity of the notation). It then follows that $S(z)$ and $\bar S(z)$ are mutually inverse functions: $$\la{anti11}
\bar S(S(z))=z.$$
An algebraic function with this property is called [*the Schwarz function*]{}. By the equation $f(z, S(z))=0$, it defines a complex curve with an antiholomorphic involution. An upper bound for genus of this curve is $g=(d-1)^2$, where $d$ is the number of branches of the Schwarz function. The real section of this curve is a set of all fixed points of the involution. It consists of a number of contours on the plane (and possibly a number of isolated points, if the curve is not smooth). The structure of this set is known to be complicated. Depending on coefficients of the polynomial, the number of disconnected contours in the real section may vary from $0$ to $g+1$. If the contours divide the complex curve into two disconnected “halves", or sides (related by the involution), then the curve can be realized as the [*Schottky double*]{} [@Gustafsson90] of one of these sides. Each side is a Riemann surface with a boundary.
Let us come back to equation (\[M2\]). It has $d$ independent solutions. They are functions on the spectral curve. One of them is a physical solution corresponding to biorthogonal polynomials. The physical solution defines the “physical sheet" of the curve.
The Schwarz function on the physical sheet is a particular root, say $S^{(1)}_n(z)$, of the polynomial $f_n(z, \tilde z)$ (see (\[h1\])). It follows from (\[M1\]) that this root is selected by the requirement that it has the same poles and residues as the potential $A$.
### The Schottky double {#S}
The Schwarz function describes more than just the boundary of clusters of eigenvalues. Together with other sheets it defines a Riemann surface. If the potential $A(z)$ is meromorphic, the Schwarz function is an algebraic function. It satisfies a polynomial equation $f(z, S(z))=0$.
![\[Schottky\] The Schottky double. A Riemann surface with boundaries along the droplets (a front side) is glued to its mirror image (a back side).](double.pdf){width="5cm"}
The function $f(z,\tilde z)$, where $z$ and $\tilde z$ are treated as two independent complex arguments, defines a Riemann surface with antiholomorphic involution (\[anti1\]). If the involution divides the surface into two disconnected parts, as explained above, the Riemann surface is the [*Schottky double*]{} [@Gustafsson90] of one of these parts.
There are two complementary ways to describe this surface. One is through the algebraic covering (\[h1\], \[ah1\]). Among $d$ sheets we distinguish a [*physical*]{} sheet. The physical sheet is selected by the condition that the differential $S(z)dz$ has the same poles and residues as the differential of the potential $A(z)dz$. It may happen that the condition $\bar z=S^{(i)}(z)$ defines a planar curve (or several curves, or a set of isolated points) for branches other than the physical one. We refer to the interior of these planar curves as [*virtual*]{} (or unphysical) droplets situated on sheets other than physical.
Another way emphasizes the antiholomorphic involution. Consider a meromorphic function $h(z)$ defined on a Riemann surface with boundaries. We call this surface the front side. The Schwarz reflection principle extends any meromorphic function on the front side to a meromorphic function on the Riemann surface without boundaries. This is done by adding another copy of the Riemann surface with boundaries (a back side), glued to the front side along the boundaries, Figure \[Schottky\]. The value of the function $h$ on the mirror point on the back side is $h(\overline{S(z)})$. The copies are glued along the boundaries: $h(z)=h(\overline{S(z)})$ if the point $z$ belongs to the boundary. The same extension rule applies to differentials. Having a meromorphic differential $h(z)dz$ on the front side, one extends it to a meromorphic differential $h(\overline{S( z)})d\overline{S(z)}$ on the back side.
This definition can be applied to the Schwarz function itself. We say that the Schwarz function on the double is $S(z)$ if the point is on the front side, and $\bar z$ if the point belongs to the back side (here we understand $S(z)$ as a function defined on the complex curve, not just on the physical sheet).
The number of sheets of the curve is the number of poles (counted with their multiplicity) of the function $A(z)$ plus one. Indeed, poles of $A$ are poles of the Schwarz function on the front side of the double. On the back side, there is also a pole at infinity. Since $S(z=\infty)=A(\infty)$, we have $\bar S(\bar z=A(\infty))=\infty$. Therefore, the factor $a(z)$ is a polynomial with zeros at the poles of $A(z)$ and at $\overline{A(\infty)}$, and $$d\equiv\mbox{number of sheets} =
\mbox{number of poles of $A$ + 1}.$$ The front and back sides meet at planar curves $\bar z=S(z)$. These curves are boundaries of the droplets. We repeat that not all droplets are physical. Some of them may belong to unphysical sheets, Figure \[torus\].
Boundaries of droplets, physical and virtual, form a subset of the $\bf a$-cycles on the curve. Their number cannot exceed the genus of the curve plus one: $$\mbox{number of droplets} \le g+1.$$
The sheets meet along cuts located inside droplets. The cuts that belong to physical droplets show up on unphysical sheets. On the other hand, some cuts show up on the physical sheet (Figure \[torus\]). They correspond to droplets situated on unphysical sheets.
![\[torus\]Physical and unphysical droplets on a torus. The physical sheet (shaded) meets the unphysical sheet along the cuts. The cut situated inside the unphysical droplet appears on the physical sheet. The boundaries of the droplets (physical and virtual) belong to different sheets. This torus is the Riemann surface corresponding to the ensemble with the potential $V(z) = - \alpha \log (1- z/ \beta) - \gamma z $.](torus.pdf){width="5cm"}
The Riemann-Hurwitz theorem computes the genus of the curve as $$g=\mbox{half
the number of
branching points} - d+1.$$
With the help of the Stokes formula, the numbers $\{ \nu_{\alpha} \}$ are identified with areas of the droplets: $|\nu_a|=\frac{1}{2\pi \hbar}\int_{D_a} d^2z$. For a nondegenerate curve, these numbers are not necessarily positive. Negative numbers correspond to droplets located on unphysical sheets. In this case, $\{\nu_a\}$ do not correspond to the number of eigenvalues located inside each droplet, as it is the case for algebraic domains, when all filling numbers are positive.
### Degeneration of the spectral curve
Degeneration of the complex curve gives the most interesting physical aspects of growth. There are several levels of degeneration. We briefly discuss them below.
#### Algebraic domains and double points {#algebraic-domains-and-double-points .unnumbered}
A special case occurs when the Schwarz function on the physical sheet is meromorphic. It has no other singularities than poles of $A$. This is the case of algebraic domains . They appear in the semiclassical case. This situation occurs if cuts on the physical sheet, situated outside physical droplets, shrink to points, i.e., two or more branching points merge. Then the physical sheet meets other sheets along cuts situated inside physical droplets only and also at some points on their exterior ([*double points*]{}). In this case the Riemann surface degenerates. The genus is given by the number of physical droplets only. The filling factors are all positive.
![\[degtorus\]Degenerate torus corresponds to the algebraic domain for the Joukowsky map.](degtorus.pdf){width="5cm"}
In the case of algebraic domains, the physical branch of the Schwarz function is a well-defined meromorphic function. Analytic continuations of $\bar z$ from different disconnected parts of the boundary give the same result. In this case, the Schwarz function can be written through the Cauchy transform of the physical droplets: $$\label{28}
S(z)=A(z)+\frac{1}{\pi}\int_{D}\frac{d^2 \zeta}{z-\zeta}.$$
Although algebraic domains occur in physical problems such as Laplacian growth, their semiclassical evolution is limited. Almost all algebraic domains will be broken in a growth process. Within a finite time (the area of the domain) they degenerate further into critical curves. The Gaussian potential (the Ginibre-Girko ensemble), which leads to a single droplet of the form of an ellipse is a known exception.
#### Critical degenerate curves {#critical-degenerate-curves .unnumbered}
Algebraic domains appear as a result of merging of simple branching points on the physical sheet. The double points are located outside physical droplets. Remaining branching points belong to the interior of physical droplets. Initially, they survive in the degeneration process. However, as known in the theory of Laplacian growth, the process necessarily leads to a further degeneration. Sooner or later, at least one of the interior branching points merges with one of the double points in the exterior. Curves degenerated in this manner are called [*critical*]{}. For the genus one and three this degeneration is discussed below.
Since interior branching points can only merge with exterior branching points on the boundary of the droplet, the boundary develops a cusp, characterized by a pair $p, q$ of mutually prime integers. In local coordinates around such a cusp, the curve looks like $x^p\sim y^q$. The fact that the growth of algebraic domains always leads to critical curves is known in the theory of Laplacian growth as finite time singularities.
The degeneration process seems to be a feature of the semiclassical approximation. Curves treated beyond this approximation never degenerate.
### Example: genus one curve
The potential is $V(z) = - \alpha \log (1- z/\beta) -
\gamma z,\quad A(z)=-\frac{\alpha}{z-\beta}-\gamma$. There is one pole at $z=\beta$ on the first (physical) sheet. At $z=\infty$ on the first sheet $S(z)\to-\gamma+\frac{n\hbar-\alpha}{z}$. Therefore, the Schwarz function has another pole at the point $-\bar\gamma$ on another sheet. All the poles are simple. According to the general arguments of Sec. \[S\], the number of sheets is 2, the number of branching points is 4. The genus is 1. The curve has the form $$f(z, \bar z) = z^2 \bar z^2 +k_1
z^2\bar z +\bar k_1 z\bar z^2 + k_2 z^2 + \bar k_2 \bar z^2 +
k_3 z\bar z
+k_4z+
\bar k_4\bar z +h =0.$$ The points at infinity and $-\bar\gamma$ belong to the second sheet of the algebraic covering. Summing up, $$S(z)=
\left \{\begin{array}{rll}
-\frac{\alpha}{z-\beta}& \mbox{as} & z\to\beta_1, \\
(-\gamma+\frac{n\hbar-\alpha}{z} ) & \mbox{as} & z\to\infty_1, \\
\frac{n\hbar-\bar\alpha}{z+\bar\gamma}& \mbox{as}
& z\to -\bar\gamma_2, \\
(\bar\beta-\frac{\bar\alpha}{z})& \mbox{as} & z\to\infty_2.
\end{array}
\right.$$ where, by 1 and 2 we indicate the sheets.
Poles and residues of the Schwarz function determine all the coefficients of the curve $f(z,\bar z)=a(z) (\bar z-S^{(1)}(z))(\bar z-S^{(2)}(z))=
\overline{a(z)} (z-\bar S^{(1)}(\bar z))(z-\bar
S^{(2)}(\bar z))$ except one. The behavior at $\infty$ of $z, \bar z$ gives $k_1=\gamma -\bar \beta, \, k_2=-\gamma \bar \beta$. Hereafter we choose the origin by setting $\gamma=0$. The equation of the curve then reads $f_n(z, \bar z) = 0$, where $f_n(z, \bar z) $ is given by z\^2 |z\^2 - z\^2|z |-z|z\^2+ ( |||\^2 ++|-n) z|z +z|(n-)+ |z(n-|) +h\_n The free term $h_n$ is to be determined by filling factors of the two droplets $\nu_1$ and $\nu_2=n-\nu_1$. A detailed analysis shows that the droplets belong to different sheets (Figure \[torus\]). Therefore, $\nu_2$ is negative.
A boundary of a physical droplet is given by the equation $\bar z=S^{(1)}(z)$ (Figure \[droplets\]). The second droplet belongs to the unphysical sheet. Its boundary is given by $\bar z=S^{(2)}(z)$. The explicit form of both branches is $$S^{(1,2)}=\frac{1}{2}\bar\beta-
\frac{\beta(n\hbar-\bar\alpha)+(\alpha+\bar\alpha-n\hbar)z\mp
\sqrt{(z-z_1)(z-z_2)(z-z_3)(z-z_4)}}{2(z-\beta)z},$$ where the branching points $z_i$ depend on $h_n$.
If the filling factor of the physical droplet is equal to $n$, the cut inside the unphysical droplet is of the order of $\sqrt{\hbar}$. Although it never vanishes, it shrinks to a double point $z_3=z_4=z_*$ in a semiclassical limit. The sheets meet at the double point $z_*$ rather than along the cut: $\sqrt{(z-z_1)(z-z_2)(z-z_3)(z-z_4)}\to
(z-z_*)\sqrt{(z-z_1)(z-z_2)}$. In this case, genus of the curve reduces to zero and the exterior of the physical droplet becomes an algebraic domain. This condition determines $h$, and also the position of the double point (Figure \[degtorus\]). The double point is a saddle point for the level curves of $f(z,\bar z)$. If all the parameters are real, the double point is stable in $x$-direction and unstable in $y$-direction.
If this solution is chosen, the exterior of the physical droplet can be mapped to the exterior of the unit disk by the Joukowsky map z(w)=rw+u\_0 + ,|w|>1,|a|<1. The inverse map is given by the branch $w_1(z)$ (such that $w_1\to\infty$ as $z\to\infty$) of the double valued function $$w_{1,2}(z) = \frac{1}{2r} \left [
z-u_0 + ar \pm \sqrt{(z -z_1)(z-z_2) } \right ],\quad
z_{1,2}=u_0+ar\mp 2\sqrt{r(u+au_0)}.$$
The function |z(w\^[-1]{})=rw\^[-1]{}+|u\_0 + is a meromorphic function of $w$ with two simple poles at $w=0$ and $w=\bar a^{-1}$. Treated as a function of $z$, it covers the $z$-plane twice. Two branches of the Schwarz function are $S^{(1,2)}(z)=\bar z(w_{1,2}^{-1}(z))$. On the physical sheet, $S^{(1)}(z)=\bar z(w_1(z))$ is the analytic continuation of $\bar z$ away from the boundary. This function is meromorphic outside the droplet. Apart from a cut between the branching points $z_{1,2}$, the sheets also meet at the double point $z_*=-\bar\gamma+a^{-1}re^{2i\phi}$, where $S^{(1)}(z_*)=S^{(2)}(z_*), \phi = \arg (ar+\frac{u\bar a}{1-|a|^2})$.
Analyzing singularities of the Schwarz function, one connects parameters of the conformal map with the deformation parameters: {
[rcl]{}
& = & - |u\_0,\
n-|& = & r\^2 -,\
& = & + u\_0 +
. $$\mbox{Area of the droplet}\sim n\hbar=r^2-\frac{|u|^2}{(1-|a|^2)^2}.$$
A critical degeneration occurs when the double point merges with a branching point located inside the droplet ($z_*=z_2$) to form a triple point $z_{**}$. This may happen on the boundary only. At this point, the boundary has a $(2,\,3)$ cusp. In local coordinates, it is $x^2\sim y^3$. This is a critical point of the conformal map: $w'(z_{**})=\infty$. A critical point inevitably results from the evolution at some finite critical area.
A direct way to obtain the complex curve from the conformal map is the following. First, rewrite (\[222\]) and (\[23\]) as {
[lcl]{} z-u\_0+ar & = & rw+a(z+|)w\^[-1]{}\
|z-|u\_0+|a r & = & rw\^[-1]{}+|a(|z+)w,
. and treat $w$ and $1/w$ as independent variables. Then impose the condition $w\cdot w^{-1} = 1$. One obtains $$\left |\det
\left [
\begin{array}{cc}
z- u_0 + ar & a(z+\gamma) \\
\bar z -\bar u_0 + \bar a r & r
\end{array}
\right ]\right |^2
=\left (\det
\left [
\begin{array}{cc}
r & a(z+\bar\gamma) \\
\bar a( \bar z+\gamma) & r
\end{array}
\right ]\right ) ^2.$$ This gives the equation of the curve and in particular $h$, in terms of $u,\, u_0,\,r,\,a$ and eventually through the deformation parameters $\alpha,\,\beta,\,\gamma$ and $t$.
The semiclassical analysis gives a guidance for the form of the recurrence relations. Let us use an ansatz for the $L$-operator, which resembles the conformal map (\[222\]): $$L=r_n \hat w +u_{n}^{(0)} + (\hat w -a_n )^{-1} u_n,$$ so that $$\begin{aligned}
(\hat w -a_n ) L = (\hat w -a_n ) r_n \hat w +
(\hat w - a_n ) u^{(0)}_n + u_n, \la{701} \\
L^{\dag}(\hat w^{-1} -\bar a_n ) =
\hat w^{-1} r_n (\hat w^{-1} -\bar a_n ) +
\bar u^{(0)}_n (\hat w^{-1} -\bar a_n ) + \bar u_n \, \la{7111},\end{aligned}$$ where $\hat w$ is the shift operator $n\to n+1$.
Now we follow the procedure of the previous section. Since the potential has only one pole, ${\mathcal L}_n$ can be cast into $2\times 2$ matrix form. Let us apply the lines (\[701\], \[7111\]) to an eigenvector $(c_n, c_{n+1})$ of a yet unknown operator ${\mathcal L}_n$, and set the eigenvalue to be $\tilde z$: {
[ccc]{} (z+r\_[n-1]{} a\_[n-1]{}-u\^[(0)]{}\_[n]{})c\_[n]{} & = & r\_[n]{} c\_[n+1]{} + a\_[n-1]{}(z+|\_[n-1]{})c\_[n-1]{}\
(z + r\_[n]{}|a\_[n]{} - |u\^[(0)]{}\_[n+1]{} )c\_[n]{} & = & |a\_[n+1]{}(z + \_[n+1]{}) c\_[n+1]{} + r\_[n]{} c\_[n-1]{}.
. We have defined $\bar \gamma_n = \frac{u_n}{a_n} - u^{0}_n$. The equations are compatible if $c_{n-1}$ and $c_{n+1}$ found through $c_n$ differ by the shift $n\to n+2$. We have c\_[n+1]{} = |
[cc]{} z+r\_[n-1]{} a\_[n-1]{} -u\^[(0)]{}\_[n]{}& a\_[n-1]{}(z+|\_[n-1]{})\
z + r\_n|a\_[n]{} - |u\^[(0)]{}\_[n+1]{} & r\_[n]{}
| = c\_n , c\_[n-1]{} = |
[cc]{} r\_[n]{} & z + r\_[n-1]{} a\_[n-1]{} -u\^[(0)]{}\_[n]{}\
|a\_[n+1]{}( z + \_[n+1]{}) & z + r\_n|a\_[n]{} - |u\^[(0)]{}\_[n+1]{}
| = c\_n , where d\_n = |
[cc]{} r\_[n]{} & a\_[n-1]{}(z+|\_[n-1]{})\
|a\_[n+1]{} (|z +\_[n+1]{}) & r\_[n]{}.
| This yields the curve \_n\_[n+1]{}=d\_nd\_[n+1]{}. Comparing the two forms of the curve (\[8\]) and (\[811\]), we obtain the conservation laws of growth: =\_n = -|u\^[0]{}\_n, = + u\^[(0)]{}\_[n+1]{} + , n -|= r\_n r\_[n+1]{} - . They are the quantum version of (\[27\]).
Continuum limit and conformal maps
----------------------------------
The geometrical meaning of the complex curve (\[qc\]) is straightforward: at fixed shape parameters $t_k$ and area parameter $\hbar$, increasing $n$ yields growing domains that represent the support of the corresponding $n \times n$ model. A remarkable feature of this process is that it preserves the external harmonic moments of the domain $\mathbb{D}_n$, t\_k(n) = t\_k(n-1), t\_k(n) = -\_[ \_n]{} , k 1. The only harmonic moment which changes in this process is the normalized area $t_0 = \frac{1}{\pi} \int d^2 z,$ and it increases in increments of $\hbar$ (hence the meaning of $\hbar$ as quantum of area). We may say that the growth of the NRM ensemble consists of increasing the area of the domain by multiples of $\hbar$, while preserving all the other external harmonic moments. The continuum version of this process, known as $Laplacian$ $Growth$, is a famous problem of complex analysis. It arises in the two-dimensional hydrodynamics of two non-mixing fluids, one inviscid and the other viscous, upon neglecting the effects of surface tension, where it is known as the Hele-Shaw problem. The following chapters discuss this classical problem in great detail.
As we will see, Laplacian growth can be restated simply as a problem of finding the uniform equilibrium measure, subject to constraints on the total mass, and the asymptotic expansion of the logarithmic potential at infinity. As long as a classical solution exists, the machinery of NRM does not seem necessary. However, Laplacian growth (as a class of processes), is characterized by finite-time singularities. In that case, the only way to reformulate the problem is similar to the Saff-Totik approach to the extremal measure, and is deeply related to weighted limits of orthogonal polynomials in the complex plane.
Laplacian Growth
================
Introduction {#introduction}
------------
Laplacian growth (LG) is defined as the motion of a planar domain, whose boundary velocity is a gradient of the Green function of the same domain (also called a [*harmonic measure*]{}). This deceivingly simple process appears to be connected to an impressive number of non-trivial physical and mathematical problems [@Oxf98; @MWZ]. As a highly unstable, dissipative, non-equilibrium, and nonlinear phenomenon, it is famous for producing different universal patterns [@ST; @Ristroph].
Numerous non-equilibrium physical processes of apparently different nature are examples of Laplacian growth: viscous fingering [@ST], slow freezing of fluids (Stefan problem) [@Pelce], growth of snowflakes [@Nakaya], crystal growth, amorphous solidification [@Langer], electrodeposition [@Gollub], bacterial colony growth [@BenJacob], diffusion-limited aggregation (DLA) [@DLA81], motion of a charged surface in liquid Helium [@Zubarev], and secondary petroleum production [@Bear], to name just a few.
A major consequence from the current development of the subject is a discovery of a new and unexpectedly fruitful mathematical structure, which is capable to predict and explain key physical observations in regimes, totally inaccessible by any other available mathematical method.
The first section of this chapter is a brief history of physics covered by the Laplacian growth. The second section addresses in detail the exact time-dependent solutions of the Laplacian Growth Equation, and the last is a detailed presentation of the analytic and algebraic-geometric structure of Laplacian growth.
Physical background
-------------------
#### Darcy’s law {#darcys-law .unnumbered}
In 1856, while completing a hydrological study for the city of Dijon, H. Darcy noticed that the rate of flow (volume per unit time) $Q$ through a given cross-section, $t_0$, is (a) proportional to $t_0$, (b) inversely proportional to the length, $L$, taken between positions of efflux and influx, and (c) linearly proportional to pressure difference, $\Delta p$, taken between the same two levels. In short, $$\la{Darcy}
Q = - \frac{k\,t_0}{L}\,\Delta p ,$$ where $k$ is a positive constant. As one can see, Darcy’s observation coincides with Ohm’s law, upon identifying $Q,\,\Delta p$, and $k$ as the total current through the cross-section $t_0$, the electric potential difference, and the electrical conductivity, respectively. Rewriting (\[Darcy\]) in a differential form, as for Ohm’s law, we obtain $$\la{d'arcy}
{\bf v} = -k \nabla p,$$ where ${\bf v}$ is the velocity vector field of fluid particles, properly coarse-grained to assure its smoothness over infinitesimally small volumes. Here the kinetic coefficient, $k$, (the same as in (\[d’arcy\])), is called a (hydraulic) conductivity and can depend on position. The equation (\[d’arcy\]) constitutes the Darcy’s law in a differential form. For homogenous $k$, $$\la{DARCY}
{\bf v} = \nabla (- k p),$$ Darcy’s law merely states that a flow through uniform porous media (sand in Darcy’s experiments) is purely potential (no vortices), where the pressure field, $p$, is a velocity potential up to a constant factor. Assuming constant $k$ and the fluid incompressible, $\nabla \, {\bf \cdot
\, \bf v} = 0$, we find that pressure $p$ is a harmonic function, $$\la{LaplaceEq}
\nabla^2 p = 0$$ As seen from purely dimensional considerations, the conductivity $k$ equals $$\la{conductivity}
k = C\frac{d^2}{\mu},$$ where $d$ is the average linear size of a pore in cross-section, $\mu$ is the dynamical viscosity of the fluid under consideration, and the dimensionless coefficient, $C$, is usually small and media-dependent. (It is of the order of the density of voids in a given porous medium).
It follows from (\[d’arcy\]) and (\[conductivity\]), that if $\mu$ is negligibly small (an almost inviscid liquid), pressure gradients are also negligibly small, regardless of how fast fluid moves (but still much slower than the velocity of sound in this liquid in order to assure incompressibility assumed earlier).
#### Laplacian growth in porous media {#laplacian-growth-in-porous-media .unnumbered}
Assume that a fluid with a viscosity $\mu_1$ occupying a domain $D_1(t)$ at the moment $t$ pushes another fluid with a viscosity $\mu_2$ occupying the domain $D_2(t)$ at the same time $t$ through a uniform porous media. Then the Laplace equation will hold for both pressures $p_1$ and $p_2$ corresponding to domains $D_1$ and $D_2$ respectively: $$\la{LMuskat}
\nabla^2 p_i = 0 \qquad {\rm{in}} \,\,D_i(t),$$ where $i = 1,2$. At the interface $\Gamma(t)$, where two fluids meet (but do not mix), their normal velocities coincide because of continuity and equal to the normal component $V_n$ of the velocity of the boundary, $\Gamma(t) = \partial D_1 = - \partial D_2$: $$\la{vMuskat}
{\bf v_1}|_n = {\bf v_2}|_n = V_n \qquad \rm{at} \,\,\Gamma(t).$$ The pressure field $p$ at the interface $\Gamma(t)$ (by the Laplace law) has a jump equal to the mean local curvature $\kappa$ multiplied by the [*[surface tension]{}*]{} $\sigma$: $$\la{pMuskat}
p_1 - p_2 = \sigma \kappa \qquad \rm{at} \,\,\Gamma(t).$$ Unless the local curvature is very high, this surface tension correction is usually very small, and so is often neglected. If to supplement the last three equations by boundary conditions at external walls or/and at infinity (they may include sources/sinks of fluids either extended or point-like), then the free boundary problem of finding $\Gamma(t)$ by initially given $D_1$ and $D_2$ is completely formulated.
The process described by (\[LMuskat\], \[vMuskat\], \[pMuskat\]) is typical for various geophysical systems, for instance for petroleum production, where a less viscous fluid (usually water) pushes a much more viscous one (oil) toward production wells. This process is very unstable and most initially smooth water/oil fronts will quickly break down and become fragmented.
#### The Hele-Shaw cell {#the-hele-shaw-cell .unnumbered}
In 1898, H.S. Hele-Shaw proposed an interesting way to observe and study two-dimensional fluid flows by using two closely-placed parallel glass plates with a gap between them occupied by the fluid under consideration [@Hele-Shaw]. This simple device appears to be very useful in various investigations and is now called a Hele-Shaw cell after its inventor. Remarkably, a viscous fluid, governed in 3D by the Stokes law, $$\la{Stokes}
\mu \nabla^2 {\bf v} = \nabla p,$$ after being trapped in a gap of a width $b$, between the plates of a Hele-Shaw cell, obeys Darcy’s law (\[d’arcy\]) with a conductivity equal to $k = b^2/(12\mu)$. The derivation of the formula $$\la{HSdarcy}
{\bf v} = -\frac{b^2}{12\mu}\nabla p,$$ which is to be understood as a 2D vector field in a plane parallel to the Hele-Shaw cell plates, is rather trivial and results from the averaging of (\[Stokes\]) over the dimension perpendicular to the plates [@Lamb; @LL]). Thus, displacement of viscous fluid by the (almost) inviscid one in a Hele-Shaw cell became a major experimental tool to investigate a 2D Laplacian growth. Various versions of 2D Laplacian growth in a Hele-Shaw cell corresponding to different geometries are shown in Figure \[fig:LG\].
![\[fig:LG\] Laplacian Growth in a Hele-Shaw cell for the radial [*a*]{}), channel [*b*]{}), and wedge [*c*]{}) geometries. ](HeleShaw.pdf){width="12cm"}
#### Idealized Laplacian growth {#idealized-laplacian-growth .unnumbered}
In 1945, Polubarinova-Kochina [@PK] and Galin [@Galin] simultaneously, but independently, derived a nonlinear integro-differential equation for an oil/water interface in 2D Laplacian growth, after neglecting surface tension, $\sigma$, and water viscosity, $\mu_{water} = 0$. Assuming for simplicity a singly connected oil bubble, occupying a domain $D(t)$ surrounded by water and having a sink at the origin, $0\in D(t)$, we will obtain this equation starting from the system $$\begin{aligned}
\la{darcy}
\left\{
\begin{array}{l}
\nabla^2 p = \rho\quad {\rm in} \, D(t),\\
p = 0 \quad {\rm at~ the~interface}, \Gamma(t) = \partial D(t)
\label{eq2}
\\
V_n =- \partial_n p \quad \rm{at~the~interface}, \Gamma(t),
\end{array}
\right.\end{aligned}$$ where $\rho$ and $\partial_n$ are density of sources and the normal derivative respectively. This system is a reduction of (\[LMuskat\], \[vMuskat\]) after simplifications mentioned above and using the fact that the normal boundary velocity, $V_n$, equals to the normal components of the fluid velocity at the boundary, which is $- \partial_n p$ by virtue of the Darcy law (\[d’arcy\]). Here and below the conductivity $k$ is scaled to one. The density of sources, $\rho$, in this case equals $\rho(z) = -\delta^2(z)$, which corresponds to a sink of unit strength located at the origin.
#### The Laplacian growth equation {#the-laplacian-growth-equation .unnumbered}
Coming back to the derivation, we apply the conformal map from the unit disc in the complex plane $w = \exp(-p + i\phi)$, where the (stream) function $\phi(x,y)$ is harmonically conjugate to $p(x,y)$, into the domain $D(t)$ in the “physical” complex plane $z = x+iy$, and zero maps to zero with a positive coefficient. Denoting the moving boundary as $z(t,l)$, where $l$ is the arclength along the interface, one obtains V\_n = [Im]{}(|z\_t z\_l) = -\_n p = \_l , It is trivial to see that the chain of three equalities in (\[LG\]) represent respectively the definition of $V_n$ in terms of a moving complex boundary, $z(t,l)$ (the first equality), the kinematic identity expressed by the last equation in the system (\[darcy\]) (the second one), and the Cauchy-Riemann relation (the last one) between $p$ and $\phi$. After reparamerization, $l \to \phi$, we arrive to the equation $$\la{LGE}
{\rm Im}(\bar z_t z_{\phi}) = 1.$$ which possesses many remarkable properties, as will be seen below. The equation (\[LGE\]) is usually referred as the Laplacian growth equation (LGE) or the Polubarinova-Galin equation. In [@PK; @Galin] it was noticed a fully unexpected feature of the equation (\[LGE\]): the boundary, $z(t,\phi)$, taken initially as a polynomial of $w = \exp(i\phi)$, will remain a polynomial of the same degree with time-dependent coefficients during the course of evolution, so new degrees of freedom, describing the moving boundary, will not appear.
An even more remarkable observation concerning the equation (\[LGE\]), belongs to Kufarev [@Kuf], who found that a boundary taken as a rational function with respect to $w = \exp(i\phi)$ will stay as such during the evolution. Moreover, he managed to integrate this dynamical system explicitly, and found first integrals of motion associated with moving poles and residues of the conformal map, $z(t, \exp(i\phi))$, describing the boundary. The authors [@PK; @Galin; @Kuf] have however noticed that all the solutions obtained are short-lived, both because of instability and due to the finite volume of $D(t)$, which is destined to shrink, because of a sink(s) located inside. We will address these interesting observations in detail in the second section of this chapter.
#### LGE in the evolutionary form {#lge-in-the-evolutionary-form .unnumbered}
It is of help to present (\[LGE\]) in the evolutionary form, defined as the dynamical system, where the time derivative constitutes the LHS and does enter the RHS. For this purpose we rewrite (\[LGE\]) as $${\bar z}_t z_{\phi} = i + t_0,$$ where $t_0$ is real. Dividing both sides by $|z_{\phi}|^2$, we will obtain $$\frac{\bar z_t}{\bar z_{\phi}} = \frac{i+t_0}{|z_{\phi}|^2}$$ Taking conjugate form both sides and multiplying by $i$, we will have $$i\frac{z_t}{z_{\phi}} = \frac{1+it_0}{|z_{\phi}|^2}$$ The LHS is the analytic function outside the unit disk in the $w$-plane. In accordance with the last equation, the real part of this analytic finction along the unit circumference equals $|z_{\phi}|^{-2}$. To recover the analytic function from the boundary value of its real part at the unit circle is a well-known procedure involving either the Hilbert transform or the Schwarz integral. The result is i z\_t = - z\_\_0\^[2]{} , where an infinitesimally small positive $\epsilon$ indicates correct limiting value of the integral while approaching the unit circumference. This useful formula was obtained by Shraiman and Bensimon in 1984 [@bs84]. This expression for (\[LGE\]) in the evolutionary form reveals the nonlocal nature of Laplacian growth due to the integral in the RHS.
The equation (\[eLGE\]) helps to prove a beautiful statement, that every singularity of the function $z(t,w)$ moves toward the unit circle from inside, or in other words the radial component of the 2D velocity of any singularity of the conformal map is positive. To prove the claim, we replace $\phi$ in (\[eLGE\]) by $W$, defined earlier as $W = -p + i\phi$. Then after we notice that $$-\frac{z_t(t, e^W)}{z_{W}(t, e^W)} = \left [ \frac{dW}{dt}\right ]_{z = const}$$ and that near a singular point $w=a$ we can replace $W = \log(w) =
\log(a)$, we can rewrite the real part of (\[eLGE\]) as $$\la{out0}
\frac{d\,\log\,|a|}{d\,t} = \left [ \frac{1}{|z_w|^2} \right ]_{w = a} > 0.$$ Thus, we proved that each singular point of the conformal map moves toward the unit circle from inside, so the origin is a repellor for this dynamical system, and the unit circumference is an attractor.
#### Diffusion limited aggregation {#diffusion-limited-aggregation .unnumbered}
The physics section of the survey cannot be completed without mentioning a fascinating discovery by T.A. Witten and L.M. Sander, who observed [@DLA81] in 1981 that a cluster on a 2D square lattice, grown by subsequent attaching to it a Brownian diffusive particles, eventually becomes a self-similar fractal (see Figure \[dla\]) with a robust universal fractal (Haussdorff) dimension given by $$D_0 = \lim_{\epsilon \to 0}\,\frac{\log
N(\epsilon)}{\log(1/\epsilon)} = 1.71 \pm .01,$$ where $1/\epsilon$ is a linear size of a cluster measured by a small “yard stick”, $\epsilon$, and $N(\epsilon)$ is a minimal number of (small) boxes with a side $\epsilon$, which covers the cluster.
![\[dla\] A DLA cluster, $n=100 000$.](DLA.pdf){width="5cm"}
Remarkably, this fractal appeared to be self-similar after appropriate statistical averaging. This means that its higher multi-fractal dimensions, $D_q$, defined as $$D_q = \lim_{\epsilon \to 0}\,\frac{\log( \sum_{i}^N
p_i^q)}{\log(1/\epsilon)},$$ where $p_i$ stands for a portion of a tiny box of a size of $\epsilon$, covered by the cluster under consideration, appear to be equal to each other, and to $D_0$, which is 1.71, as indicated above. Later, these findings were significantly clarified and refined in many respects, but the major challenge: how to calculate the universal dimension defined above still is an open question (see a relatively recent review [@Halsey] and references therein).
This problem is tightly connected to the Laplacian growth. Until very recently there were numerous claims that the DLA process is drastically different from the Laplacian growth, and even statements appeared that the DLA and fractals grown in Laplacian growth belong to different universality classes [@Procaccia]. However, the recent experiments by Praud and Swinney [@HLS-1] made crystal clear that the multi-fractal spectrum of a cluster grown in a viscous fingering process in a Hele-Shaw cell (that is a Laplacian growth) coincides with the DLA spectrum up to the margin accuracy of $1\%$, which is the maximal accuracy in these measurements. Thus, despite of its discrete and a stochastic nature, DLA can be understood by a continuum and deterministic Laplacian growth (\[darcy\]).
#### Related problems {#related-problems .unnumbered}
Below is a list of physical problems connected with Laplacian growth.
First, there is the so-called “singular” Laplacian growth, where a growing domain consists of needles with zero areas and divergent curvature at the tip. The mathematical description for this dynamics should be reformulated, since the gradient of pressure $p$ diverges near moving needle tips, so the boundary velocity should be replaced by an appropriately regularized law. Interesting works by Derrida and Hakim [@DH], and by Peterson [@Peterson] in this direction deserve special attention.
There is also a considerable amount of works in so-called nonlinear mean-field dynamics, where a phase field is involved, which gradually changes from unity in one of moving phases toward zero inside the second one [@CahnHill; @Gollub-Langer]. Many of these processes, including dynamics of miscrostructure [@Khachaturyan] in materials, growth of bacterial colonies in nutritional environment [@Matsushita], and spinodal decomposition [@SpinDec], governed by the time-dependent Ginzburg-Landau and the Cahn-Hilliard equations, are reduced to the Laplacian growth interface dynamics in a special singular limit, when the phase field degenerates to a step-function, thus becoming a characteristic function of a moving domain with a well defined boundary [@caginalp; @Langer-1]. This is certainly worth to mention, both because it significantly enriches a physical process by introducing an additional field (the phase field) and since this is conceptually related to a random matrix approach to Laplacian growth, addressed in the survey, and where a distribution of eigenvalue support will play a role of a mean-field phase, introduced in this paragraph.
Let us also mention several more “selection puzzles", which belong to the Laplacian growth in various settings: selection of a shape of a separated inviscid bubble, observed by Taylor and Saffman in a viscous flow in a rectangular Hele-Shaw cell [@TS] from a continuous family of possible solutions (not to be confused with the Saffman-Taylor fingers family described in [@ST]); selection of a so-called “skinny” finger in a Hele-Shaw cell accelerated by a tiny inviscid bubble near the nose of a finger [@Libch]; and prediction of the periodicity for the so-called side-branching structure in dendritic growth [@Glicksman]. These phenomena have the same (or almost the same) mathematical description.
Another important comment about physics of Laplacian growth is that Darcy’s law (\[d’arcy\]) is invalid near walls of a Hele-Shaw cell, including proximity to both parallel plates. This is because averaging of the Stokes flow, $\mu \nabla^2 {\bf v} = \nabla p$, given by (\[Stokes\]) will no longer bring us to (\[HSdarcy\]), due to boundary layer effects. This apparent difficulty gives rise to the study of an interface dynamics with a Stokes flow, which is an extension of the Hele-Shaw (Darcy’s) flow. The Stokes flow also contains remarkable physics and beautiful mathematics [@Hopper; @Rich; @Crowdy], which is still yet to be fully understood.
Exact solutions
---------------
#### Cardioid {#cardioid .unnumbered}
Consider the equation of motion for the droplet boundary under Laplacian growth $$\label{eq5} {\rm Im}(\bar z_t z_{\phi}) = Q,$$ where $2\pi Q$ stands for a rate of a source (sink). Here, $z(t,e^{i\phi})$ is conformal inside the unit circle, $|w|<1$, $0
\to 0$, and $w = e^{i\phi}$ in the equation. When one tries to solve (\[eq5\]), the solution, $$\la{eq6}
z = r(t)e^{i\phi},$$ comes to mind first, as the simplest one. It describes initially circular droplet centered at the origin, which uniformly grows (shrinks) while continuing to be a circle. Indeed, substituting (\[eq6\]) into (\[eq5\]) one obtains $$\la{eq7}
r(t) = \sqrt{2(|Q|T + Q t)},$$ where a constant of integration, $T$ stands for an initial time. When $Q<0$ (suction), the circle shrinks to a point at $t = T$, and the solution (\[eq6\]) ceases to exist after $T$. Could one find any other exact solutions, less trivial than given by (\[eq6\])?
Remarkably, the answer is yes, despite of nonlinearity of the Laplacian growth equation (\[eq5\]). Let’s add to (\[eq6\]) an initially small quadratic correction, $$\la{eq8}
z = r(t)e^{i\phi} + a(t)e^{2i\phi}.$$ The domain bounded by the curve described by (\[eq8\]), named [*a cardiod*]{}, is connected if $|a| < r/2$. Substituting (\[eq8\]) into (\[eq5\]) one obtains two coupled nonlinear first order ODEs w.r.t. $r$ and $a$: $$\begin{aligned}
\left\{
\begin{array}{l}
r\,\dot r + 2\,a\,\dot a = Q\\
\dot a\,r + 2\,a\,\dot r= 0,
\end{array}
\right.\end{aligned}$$ with an easily found solution $$\begin{aligned}
\left\{
\begin{array}{l}
a r^2 = a_0\\
r^2 + 2 a^2 = 2(|Q|t_0 + Q t)
\end{array} \right.\end{aligned}$$ with $a_0$ and $t_0$ as constants of integration. If $Q>0$ (injection), the cardiod will grow becoming more and more like a circle during the evolution. If instead $Q<0$ (suction) the cardioid (\[eq8\]) shrinks, deforms and ceases to exist after $t^* =
t_0 + 3 a_0^{2/3}/( \sqrt[3]{16}Q)$. This happens when the critical point of the conformal map given by (\[eq8\]) reaches the unit circle from outside. Then the cardioid ceases to be analytic and earns a needle-like cusp (a point of return with infinite curvature). This cusp is called type 3/2 (alternatively (2,3)-cusp) because in local Cartesian coordinates it is described by the equation $y^2 \sim x^3$. We will see later that this kind of cusps is typical for those solutions of Laplacian growth which cease to exist in finite time.
#### Polynomials {#polynomials .unnumbered}
As a generalization, we are going to prove now that all polynomials of $w$, which describe boundaries of analytic domains when $|w|=1$, are solutions of (\[eq5\]). Assume a droplet is initially described by a trigonometric polynomial (with all critical points lying outside the unit disk, because its interior conformally maps onto a droplet): $$\la{eq10}
z = \sum_{k=1}^N\,a_k e^{ik\phi}.$$ Substituting (\[eq10\]) into (\[eq5\]), one obtains $N$ coupled ODE’s for time-dependent coefficients $a_k$, and remarkably there are no other degrees of freedom which appear during the evolution. In other words, the evolving droplet will continue to be described by the polynomial (\[eq10\]), with coefficients, $a_k$, changing in time in accordance with these ODE’s: $$\la{eq11}
\sum_{k=1}^{N-n}\,[k\,a_k\,{\dot{\bar a}}_{k+n} + (k+n)\dot a_k
\,{\bar a}_{k+n}] = Q\delta_{n,0}\qquad n = 0, 1, \ldots ,N-1.$$ Moreover, (\[eq11\]) can be integrated explicitly. Indeed, we notice first that the equation for $k=N-1$, namely $$a_1\,{\dot{\bar a}}_N + N\dot a_1 \,{\bar a}_N = 0,$$ is trivially solved with the answer $$\la{N}
{\bar a}_N\,a_1^N = C_N,$$ where $C_N$ is the constant of integration. Substituting (\[N\]) into the $(N-2)^{nd}$ equation, which has a form $$\la{N-1}
a_1\,{\dot{\bar a}}_{N-1} + (N-1){\dot a_1} \,{\bar a}_{N-1} +
a_2\,{\dot{\bar a}}_N + N{\dot a_2} \,{\bar a}_N = 0,$$ we notice that the LHS of (\[N-1\]) is proportional to a full derivative from the expression $$a_1^{N-1}\,{\bar a_{N-1}} + N\,C_N\frac{a_2}{a_1^2}$$ and is zero in accordance with the RHS of (\[N-1\]). Thus we obtain $$a_1^{N+1}\,{\bar a_{N-1}} + N\,C_N\,a_2= C_{N-1}\,a_1^2,$$ where $C_{N-1}$ is a constant of integration. Knowing $a_{N-1}$ and $a_N$ in terms of $a_1$ and $a_2$ we can easily integrate the third equation from the end of the system (\[eq11\]), namely the $(N-3)^{rd}$ equation. The result is
$$a_1^{N+2}\bar a_{N-2} + (N-1)C_{N-1}a_2a_1^2 + NC_Na_3a_1 -
N(N+1)\frac{C_Na_2^2}{2} = C_{N-2}a_1^4.$$
Continuing in this way, we obtain an explicit dependence of ${\bar
a_k}$ as a linear combination of constants of motion, $C_k$, with coefficients which are polynomial forms w.r.t. $a_1, a_2, \ldots$. The equation for $n=0$ from (\[eq11\]) already constitutes the full derivative and, as such, is trivially integrated: $$\sum_{k=1}^N k|a_k|^2 = 2(C_0 + Qt),$$ where $C_0$ is a constant of integration. Here the LHS is a (scaled) area of the droplet, and the equation states that the area changes linearly in time. In other words, we integrated the system (\[eq11\]), and the solutions are polynomial forms with respect to $a_k$, linear w.r.t. integrals of motion, $C_k$, explicitly obtained.
As in the case of cardioid, in case $Q>0$ the dynamics is stable and the droplet becomes eventually more and more round since all $a_k$ decay in time, but $a_1$ in contrary, grows, as one can easily verify by looking to the system (\[eq11\]). If $Q<0$, then the droplet shrinks and the solution ceases to exist in finite time. This happens because a critical point(s) hits a unit circle from outside manifesting a break of analyticity by making a cusp (of a 3/2 kind in general case). Except such rare cases as a circle centered at the location of sink, the solution stops to exist prior to the formation of a cusp, because of the droplet being completely sucked by the sink.
The fact that a finite time singularity (a cusp) is generic follows directly from (\[N\]): since the conformal radius, $a_1$, should decrease as the area shrinks, then the coefficient, $a_N$, grows in time by virtue of (\[N\]), eventually bringing the system to a cusp.
Now consider the external Laplacian growth, where an inviscid bubble, surrounded by a viscous fluid grows (shrinks) because of a source (sink) at infinity. Then we map conformally the exterior of the unit disk in the $w$-plane to the exterior of a bubble (viscous region) in the physical $z$-plane with a simple pole and positive residue (which is a conformal radius) at infinity.
Here an analogy of the polynomial ansatz (\[eq11\]) will be the formula $$\la{neg}
z = \sum_{k=-1}^N\,a_k e^{-ik\phi},$$ where $a_{-1} = r$ is the conformal radius, that is the radius of a circle perturbed by the rest of $a_k$’s. This case is also integrable in a way, very similar to the interior case shown above [@Mineev90]. One can also see that for an unstable LG, that is a growing bubble in the exterior problem, a finite time cusp is unavoidable. Indeed, the system of ODEs for the ansatz (\[neg\]) will look the same as (\[eq11\]), but with values of $n$ extended from $-1$ to $N+1$. Thus one can easily see that $a_N = C_N r^{N-1}$. This means that the highest harmonic will grow faster than a conformal radius, which should eventually break domain’s analyticity through a cusp [@Howison91].
The area, $t_0$, of the growing bubble in this case equals t\_0 = |r|\^2 - \_[k > 0]{} k |a\_k|\^2.
Consider the simplest non-trivial example for (\[neg\]), which describes a shape with three-fold symmetry, z(w) = rw + , we have a = 3C r\^2, where $C$ is a constant of integration, and the scaled area of the droplet identified with time $t$ in this case, is: t\_0=t = |r|\^2 - 18|C|\^2 |r|\^4. Clearly, this polynomial in $|r|^2$ has a global maximum at $r_c$ solving $36 |r_c|^2 |C|^2 = 1$. We call the corresponding value of the area $t_c = |r_c|^2/2$ [*critical*]{}, and conclude that the dynamics will lead to finite-time singularities for any initial condition $t_3 \ne 0$ Figure \[critical\].
![A Hele-Shaw droplet approaching the critical area.[]{data-label="critical"}](p9n2.pdf){width="7cm"}
In summary, we have shown that the Laplacian growth is integrable for polynomial (time-dependent) conformal mappings both in interior and exterior problem, and in both cases (growing of a bubble in the exterior and suction of a droplet for the interior problem) finite time singularities in the form of the 3/2-cusps are unavoidable. This is caused by an ill-posedness of the Laplacian growth without regularized factors, such as surface tension.
#### Rational functions {#rational-functions .unnumbered}
Kufarev [@Kuf] found a class of rational solutions of the equation (\[eq5\]) with simple poles. Let us show that all rational conformal maps from exterior (interior) of the unit disk to the exterior (interior) of a domain D are solutions of the LGE (\[eq5\]). We will include in our proof multiple poles for the sake of generality. Specifically, we claim that the expression $$\la{rat}
z = rw +
\sum_{k=1}^N\sum_{l=1}^{P_k}\,\frac{A_{kl}}{(w-a_k)^{p_{kl}}},$$ where $p_{kl} \in \mathbb{N}$, solves (\[eq5\]). Indeed, after substitution of (\[rat\]) into (\[eq5\]), putting $w=e^{i\phi}$, we obtain a double sum, which we can decompose to elementary fractions with respect to $(e^{i\phi}-a_k)^{p_{kl}}$ by using repeatedly the identity $$\frac{1}{(e^{i\phi}-a_k)(e^{-i\phi}-{\bar a_l})} =
\frac{1}{1-a_k\,{\bar a_l}}\, \left (1 + \frac{a_k}{e^{i\phi} - a_k} -
\frac{{\bar a_l}}{e^{-i\phi}- {\bar a_l}} \right )$$ Equating coefficients prior to all independent modes to zero and sum of all constants (the zeroth mode) to $Q$ in accordance with (\[eq5\]), we see, after some algebra, that all the expressions are full derivatives, and after integration we obtain the following equations: $$\begin{aligned}
\la{rat.sol}
\left\{
\begin{array}{l}
r^2-\sum_{k=1}^N\sum_{l=1}^{P_k}\,A_{kl}[{\bar z}(1/{\bar
a_k})]^{(p_{kl}-1)}/(p_{kl}-1)! = C + Qt \\
z(1/{\bar a_k}) = \beta_k\qquad k=1,2,\ldots,N\\
A_{kl} = \alpha_{kl}[z_w(1/w)]^{p_{kl}}|_{w={\bar a_k}}\qquad
k=1,2,\ldots,N; \quad l=1, \ldots,P_k,
\end{array}
\right.\end{aligned}$$ where $C$, $\alpha_{kl}$, and $\beta_k$ are constants of integration. It is possible to show that in unstable LG, that is a an exterior problem with growth or an interior problem with shrinking, all the solutions (\[rat\]) blow up in finite time by forming cusps, generally of the 3/2 kind.
Another interesting class of rational solutions was also found by Kufarev [@Kuf] in case there are several sources instead of one, located at $z_k$ with rates $Q_k$, and $k=1,2,\ldots,N$. In this case the velocity potential (scaled pressure) diverges near $z_k$ logarithmically with coefficients $Q_k$: $$-p = Q_k\,\log(z-z_k) + {\rm regular\,\,terms} \qquad ({\rm
when}\,\,z \to z_k).$$ In this case the Laplacian growth equation has a form $${\rm Im}(\bar z_t z_{\phi}) = {\rm Re}\sum_{k=1}^M
\frac{Q_k}{1-b_k(t)e^{i\phi}},$$ where $b_k(t)$ are time dependent inverse conformal pre-images of sources locations, $z_k$, so that $$\la{A}
z_k = z({\bar b_k}^{-1}) \qquad k=1, \ldots , M.$$ In this case, the most general rational solution has a form $$\la{ist}
z = rw +
\sum_{k=1}^N\sum_{l=1}^{P_k}\,\frac{A_{kl}}{(w-a_k)^{p_{kl}}} +
\sum_{k=1}^M\frac{B_k}{w-b_k}$$ The result of integration is then given by (\[rat.sol\]), where summations incorporate the last sum in (\[ist\]), the equation (\[A\]), and $$B_k = C_k \,t\, [z_w(1/w)]|_{w={\bar b_k}} \qquad k=1,2,\ldots,M,$$ where $C_k$ are additional constants of motion. It is worth to mention that even if the initial configuration $z(0,e^{i\phi})$ does not include poles at $b_k$ (knows nothing about sources $Q_k$ at $z_k$), the solution earns terms with simple poles at $b_k$ immediately from the start, as one can see from the last equation.
Thus, the singularities of any solution can be split into those imposed by the source location ($b_k$ in our case) and those determined by initial configuration, that are $a_k$.
One should also beware that the interface can reach sources during evolution, thus breaking analyticity by forming a cusp, and after this moment a solution ceases to exist.
#### Logarithms {#logarithms .unnumbered}
In the paper [@KufVinog] Kufarev and Vinogradov have found a logarithmic class of solutions of (\[LGE\]), which was later rediscovered and studied in detail by several authors [@S59; @BP; @H86; @ms; @sm]. This class appeared to be particularly fruitful from both mathematical and physical points of view: besides providing a significant extension from rational solutions, the logarithmic ones are often free of finite time singularities for an unstable exterior problem, which is the most important for physics. Existence of these solutions for all times allows to study the long time asymptotics, which is perhaps the major goal of this research. These so-called multi-logarithmic solutions have a form $$\la{logr}
z = rw + \sum_{k=1}^N \alpha_k \log \left ( \frac{w}{a_k}-1 \right ),$$ where $r\,\alpha_k,\,a_k$ are parameters (some of them are time dependent), and $|a_k|<1$ for a conformal mapping from the exterior of the unit disk. Using the method outlined above for rational solutions one could easily figure out that (\[logr\]) satisfies the LGE (\[LGE\]) with all $\alpha_k$ to be constants in time and the following time dependence of $r$ and $a_k$: $$\begin{aligned}
\left\{
\begin{array}{l}
\la{eq12}
r^2+\sum_{k=1}^N\sum_{l=1}^N\,\alpha_k\,{\bar
\alpha_l}\log(1-a_k\,{\bar a_l}) = C +
Qt\\
\\
r/{\bar a_k} + \sum_{l=1}^N\,\alpha_l\,\log(\frac{1}{a_l{\bar
a_k}}-1) = \beta_k; \qquad k=1,2,\ldots,N,
\end{array}
\right.\end{aligned}$$ where $C$ and $\beta_k$ are constants of motion. It is less trivial to show that the solutions (\[logr\]) may be free of finite time singularities, but the following example illustrates it well: let’s impose a $\mathbb{Z}_N$ symmetry over the system (\[logr\]) by setting $\alpha_k = \alpha\,\exp{(2\pi i k /N)}$ and $a_k = a\,\exp{(2\pi
i k /N)}$, with positive $a$ and $\alpha$. Then (\[eq12\]) looks significantly simpler: $$\begin{aligned}
\left\{
\begin{array}{l}
\la{eq13}
r^2+N\alpha^2\sum_{k=1}^N\,\gamma_k\log(1-a^2\,\gamma_k) = C + Qt\\
\\
r/a +
\alpha\sum_{k=1}^N\,\gamma_k\,\log(\frac{1}{a^2\,\gamma_k}-1) =
\beta,
\end{array}
\right.\end{aligned}$$ where $\gamma_k = e^{2\pi i k/N}$. Equating the derivative of (\[logr\]) to zero, we find critical points, $b_k$. As expected, $b_k = b\,\gamma_k$ and $$b^N = a^N - \frac{\alpha N a^{N-1}}{r}.$$ Assuming the initial $b$ to be positive, we see that ${\dot b}$ is always positive, if $\dot a$ is, which is always the case, since as follows from the second equation in (\[eq13\]) $${\dot r}=\left (\frac{r}{a}+2\alpha N\frac{a^{2N-2}}{1-a^{2N}} \right ){\dot a},$$ and therefore ${\dot a}$ is positive, since ${\dot r}$ is. Let’s also notice that $a$ cannot reach $1$ since this would make the RHS of the second equation in (\[eq13\]) infinite, which would contradict to the fact that it is a finite constant. Thus, from $$0<b<a<1, \qquad {\dot a}>0, \qquad {\rm and} \qquad {\dot b}>0$$ it follows that critical points and singularities of the conformal map (\[logr\]) will always stay inside the unit circle, which guarantees existence of the solution (\[logr\]) for all times. This simple example illustrates the fact that many of these solutions are free of finite time-singularities (see details in [@ms; @sm]), but the interesting problem of comprehensive classification of initial data for the solutions (\[logr\]) which do not blow up in finite time still is an open question.
Mathematical structure of Laplacian growth
------------------------------------------
### Conservation of harmonic moments
The following remarkable property of the Laplacian growth was found by S. Richardson in 1972 [@Richardson72] for a point-like source Q at the origin, for which
$$\nabla^2 \phi = \frac{Q}{2\pi}\,\delta^2(z).$$
He showed that all positive harmonic moments of the viscous domain, $D(t)$, $$C_k = \int_{D(t)}\,z^k\,dx\,dy, \qquad k=1,2,\ldots$$ do not change in time, while the zeroth moment, which is the area of the growing bubble, changes linearly in time: $$\frac{d C_k}{d t} = \oint_{\partial D(t)}\,z^k\,V_n\,\frac{dl}{\pi} =
\oint_{\partial D(t)}\,(p\,\partial_n z^k - z^k\,\partial_n p)\,
\frac{dl}{\pi},$$ ($dl$ is an element of arclength) because $V_n = -\partial_n\,p$ and $p=0$ along the boundary $\partial D(t)$. By virtue of Gauss’ theorem, it equals $$\int_{D(t)} \nabla(p \nabla z^k - z^k \nabla
p)\,\frac{dx\,dy}{\pi} = Q\,\delta_{k,0}.$$ This property may be used as the definition of the idealized Laplacian growth problem, namely to find an evolution of the domain whose area increases in time, while all positive harmonic moments do not change.
### LG and the Inverse Potential Problem
One can easily notice that the harmonic moments are the coefficients of the (negative) power expansion of the so called Cauchy transform ${\cal C}_{D}(z)$ of the domain $D$, namely $$\la{CT}
{\cal C}_{D}(z) = \frac{1}{\pi}\int_{D}\,\frac{dx'\,dy'}{z -
z'} = \sum_{k=0}^\infty\,\frac{C_k}{z^{k+1}}.$$ Since the Cauchy transform, ${\cal C}_{D}(z)$, is the derivative of the Newtonian potential $\Phi(z)$ created by matter occupied the domain $D$ with a unit density, $$\la{IPP}
\Phi(z) = \int_{D}\,\log|z-z'|\,\frac{\,dx'\,dy'}{\pi},$$ we see a deep connection between the Laplacian growth with the so-called inverse potential problem, asking to find a domain $D$ occupied uniformly by matter which produces a given far field Newtonian potential. The harmonic moments in this context are multipole moments of this potential. If the domain $D(t)$ grows in accordance with the idealized Laplacian growth, then the potential $\Phi(z)$ changes linearly in time, so (up to a constant): $$\Phi(t,z)= \frac{Q t}{2 \pi}\,\log|z|,$$ which is a potential of a point-like (increasing in time) mass at the origin.
### Laplacian growth in terms of the Schwarz function
Let $F(x,y)=0$ define an analytic closed Jordanian contour $\Gamma$ on the plane. Replacing the cartesian coordinates, $x$ and $y$, by complex ones, $z = x+iy$ and ${\bar z} = x-iy$, one obtains a description of $\Gamma$ as $$F \left ( \frac{z+{\bar z}}{2}, \frac{z-{\bar z}}{2i} \right ) = G(z,{\bar z}) =
0.$$ Solving the last equation with respect to ${\bar z}$ one obtains: $${\bar z} = {S}(z),$$ when $z \in \Gamma$. The function ${S}(z)$ is called the Schwarz function of the curve $\Gamma$ [@D]. It is the same mathematical object we encountered in the previous chapter. This function plays an outstanding role in the theory of quadrature domains (see next chapter). It has the following Laurent expansion, valid at least in a strip around the curve $\Gamma$: $$\la{SE}
{S}(z) = \sum_{k=0}^\infty\,\frac{C_k}{z^{k+1}}
+ \sum_{k=0}^\infty \, kt_k z^{k-1},$$ where $t_k$ are the external harmonic moments defined as $$t_k = \frac{1}{\pi k}\int_{D_-} \, \frac{dx\,dy}{z^k}, \qquad k =
1, 2, \ldots,$$ where $D_-$ is the domain complimentary to the domain $D$. From (\[CT\]) and (\[SE\]) we obtain the connection between the Cauchy transform of a domain with the Schwarz function of its boundary:
\_D(z) = \_[D]{} .
Rewriting the Laplacian growth dynamics in terms of the Schwarz function, ${S}(z)$ [@Howison91] one obtains $$\la{lgs}
\partial_t\,{S}(z) = 2\,\partial_z W,$$ where $W = -p + i\phi = \log w$ is the complex potential defined earlier. This last form of the Laplacian growth is very instructive. In particular, it helps to understand the origin of constants of integration in all exact solutions of the Laplacian growth equation presented above as a result of direct integrating efforts. Indeed, the RHS in the last equation is analytic in the viscous domain $D(t)$ except a simple pole at the origin (we consider an internal LG problem with a source at the origin). In order for the LHS to satisfy this condition, all the singularities of ${S}(t,z)$ outside the interface should be constants of motion. At zero the Schwarz function should have a simple pole with a residue (which is the area of the domain $D(t)$) linearly changing in time. This observation can be easily seen as an alternative proof of the Richardson theorem, stated above.
### The correspondence of singularities
The Schwarz function is connected to a conformal map $z = f(w)$ from the unit circle to the domain $D$ through the following formula [@D] $${S}(z) = {\bar f} \left (\frac{1}{f^{-1}(z)} \right ),$$ where $f^{-1}(z)$ is the inverse of the conformal map $w = f^{-1}(z)$. This formula helps to derive a one-to-one correspondence between singularities of ${S}(z)$ inside $D$ and $f(w)$ inside the unit circle: if near a singular point $a$ the conformal map $f(w)$ diverges as $$f(w) = \frac{A}{(w-a)^p},$$ (here by convention $p=0$ stands for a logarithmic divergence), then the Schwarz function ${S}(z)$ diverges near a point $b =
f(1/{\bar a})$ with the same power, $p$, as $${S}(z) = \frac{B}{(z-b)^p},$$ where $$A = \left [ \frac{\bar B}{(-a^2 {\bar f}')^p} \right ]_{w = 1/{\bar a}}.$$ $B$ and $b$ are constants of motions as showed above, thus the last formula together with the relation $b=f(1/{\bar a})$ and the area linearly changing in time and expressed in terms of the parameters of $f(w)$ constitute the whole time dynamics of singularities of $f(t,w)$ [@Richardson72; @Etingof]. The reader can see the equivalence of these formulae with constants of integration obtained earlier when various classes of exact solutions were derived by direct integration.
### A first classification of singularities
As mentioned in the previous sections, existence of the singular limit was established at the same time with the model [@PK; @Galin]. It became a fertile field of study in itself, and led to further developments of the problem [@ST; @Tanveer]. In a series of papers [@S3; @S4; @S5; @Howison86; @Howison85; @Hohlov-Howison94], the possible boundary singularities were studied, as well as the problem of continuing the solutions for certain classes. It was found that, in the free-space set-up, the generic critical boundary features a cusp at $(x_0, y_0)$, with local geometry of the type (x-x\_0)\^q \~(y-y\_0)\^p, (p, q) The most common cusp is characterized by $q=2, p=3$, but $q=2, p=5$ can also be obtained fairly easy by choosing proper initial conditions. Very special situations, where a finite-angle geometry is assumed as initial condition were also considered [@King].
It was shown be several methods that dynamics can be continued through a cusp of type $(2, 4k+1), k > 0$ [@Howison86; @BAZW05].
### Hydrodynamics of LG and the singularities of Schwarz function
As indicated above, the Schwarz function encodes information about the conserved moments $\{ t_k \}$, through its expansion at infinity [@MWZ]: S(z) = + \_[k > 0]{}t\_k z\^[k-1]{} + O(z\^[-2]{}). This function is useful when computing averages of integrable analytic functions $f(z)$ over the domain $D_{+}$ (an interior domain): \_[D\_[+]{}]{} f(z) [[[d]{}]{}]{}x [[[d]{}]{}]{}y = \_[k=1]{}\^N\_[i=1]{}\^[n\_k]{} c\_[ik]{}f\^[(i)]{}(z\_k) + \_[m=1]{}\^M \_[\_m]{} h\_m(z)f(z) [[[d]{}]{}]{}z, if the function $S$ has poles of order $n_k$ at $z=z_k$ and branch cuts $\gamma_m$ with jump functions $h_m(z)$. Applying formula (\[average\]) for the characteristic function of the domain $f(z) = \chi_{D_{+}}(z)$ and taking a derivative with respect to $t_0$, we obtain the relation = 1, which shows that the singularity data of the Schwarz function in $D_+$ can be interpreted as giving the location and strength of fluid sources (isolated or line-distributed) [@Richardson72]. Identifying the 2D uniform measure with another, singular (point or line-distributed) distribution, is referred to as [*sweeping*]{} of a measure. We will repeatedly encounter this process in the next chapter. In the case when the Schwarz function is meromorphic in $D_+$ (it has only isolated poles as singular points), (\[average\]) becomes \_[D\_[+]{}]{} f(z) [[[d]{}]{}]{}x [[[d]{}]{}]{}y = \_[k=1]{}\^N\_[i=1]{}\^[n\_k]{} c\_[ik]{}f\^[(i)]{}(z\_k), and the domain is called a [*quadrature domain*]{} [@Bell03; @Bell04; @G1; @G2; @GSh]. Generically, the Schwarz function may have branch cuts in $D_+$, in which case $D_+$ is called a [*generalized quadrature domain*]{} [@Sh]. This is the typical scenario for our problem. The rigorous theory of quadrature domains is outlined in the next chapter.
The hydrodynamic interpretation of the Schwarz function arises from(\[lgs\]), which is worthwhile to rewrite here $$\la{darcy2}
\partial_t\,{S}(z) = \,\partial_z W,$$ after rescaling by 2. Let $C$ be some closed contour, boundary of a domain $B$, and integrate equation (\[darcy2\]) over it. We obtain \_t \_C S(z) [[[d]{}]{}]{}z = \_B [[[d]{}]{}]{}x [[[d]{}]{}]{}y - [[[i]{}]{}]{} \_B v [[[d]{}]{}]{}x [[[d]{}]{}]{}y, where $\omega = \p_y v_x - \p_x v_y$ is the vorticity field, and $\vec \nabla \vec v
= \p_x v_x + \p_y v_y$ is the divergence of velocity field. The real part of this identity shows if the flow has zero vorticity, we have \_t S(z) [[[d]{}]{}]{}z = 0. The imaginary part of (\[re-im\]) illustrates again the interpretation of singularity set of $S(z)$ as sources of [*water*]{} (which occupies $D_+$ in a canonical Laplacian growth formulation, while the exterior domain, $D_-$, is occupied by a viscous fluid, which we call [*oil*]{} [@MWZ]): assume that the contour $C$ in (\[re-im\]) encircles the droplet without crossing any other branch cuts, then the contour integral may be performed using Cauchy’s theorem, giving the total flux of water: \_B v [[[d]{}]{}]{}x [[[d]{}]{}]{}y = Q = 1.
We note here that equation (\[darcy2\]) implies existence of a closed form [[[d]{}]{}]{}= S [[[d]{}]{}]{}z + W [[[d]{}]{}]{}t, whose primitive $\Omega$ has for real part the [*Baiocchi transform*]{} of $p$: = - \_0\^t p(z, ) [[[d]{}]{}]{}. One can see that ${\rm {Re }}\,\, \Omega$ coincides with the potential $\Phi$ introduced earlier. From the continuity equation for water $\dot \rho + \vec \nabla \vec v = 0$ and the Darcy law for water (opposite to oil) $\vec v = \nabla p$, we obtain for the time evolution of water density at a given point $z$, = -p (z,t) = (z,0) - \_0\^t p(z,) [[[d]{}]{}]{}. Equation (\[char-funct\]) may be immediately generalized in a weak sense, replacing the water density by the characteristic function of the domain $D_+, \rho \to \chi_{D_+}$, which shows that the Baiocchi transform Re $\Omega$ may be interpreted as the electrostatic potential giving the growth of the water domain.
Similarly, applying an antiholomorphic derivative to (\[darcy2\]), we obtain v + [[[i]{}]{}]{}= -p + [[[i]{}]{}]{}, so that the imaginary part of the form $\Omega$ can be considered an electrostatic potential for the time integral of vorticity at a given point $z$: [[Im]{}]{} (z, t) = \_0\^t (z, ) [[[d]{}]{}]{}.
### Variational formulation of Hele-Shaw dynamics
Formula (\[average\]) has another physical interpretation, which we explore in this section. Besides hydrodynamics, it also allows to describe the droplet through a variational (minimization) formulation, which will become very relevant when considering the singular limit.
Consider the case when the Schwarz function has only simple poles $\{ z_k \}$ and cuts at $\{ \gamma_m \}$, with residues Res $S(z_k)$ and jump functions $h_m(z)$, inside the droplet. A simple calculation shows that these singular points constitute electrostatic sources for the potential Re $\Omega$: [[Re]{}]{} (z) = \_k [[Res]{}]{} S(z\_k) (z-z\_k) + \_m \_[\_m]{} h\_m() (z-) [[[d]{}]{}]{}. If we apply (\[average\]) to all positive powers $z^k, k \ge 0$, we conclude that the singular distribution $\{ z_k \}, \{ \gamma_m \}$ and the uniform distribution $\rho(z)
= \chi_{D_+}(z)$ have the same interior harmonic moments $v_k = \langle z^k \rangle,
k \ge 0$. Thus, they create the same electrostatic potential outside the droplet. It is therefore possible to substitute the actual singular distribution $\{ z_k \}, \{ \gamma_m \}$ with the smooth, uniform distribution $\rho(z)$ in calculations related to the exterior potential. Beyond the mathematical equivalence, however, this fact has an important physical interpretation, whose full meaning will become apparent in the critical limit: when one more quantum of water is pumped into the droplet, it first appears as a new singular point of the Schwarz function (a $\delta$-function singularity). After a certain time, though, the droplet adjusts to the new area (subject to the constraints given by the fixed exterior harmonic moments), and reaches its new shape (with uniform density of water inside). Therefore, we can say that the singular distribution $\{ z_k \}, \{ \gamma_m \}$ represents the fast-time distribution of sources of water, while the uniform distribution $\rho(z)$ is the long-time, equilibrium distribution of the same amount of water. When the dynamics becomes fully non-equilibrium (after the cusp formation), this equivalence breaks down, and the correct distribution to work with is the set of poles (cuts) of the Schwarz function. In that case, the issue becomes solving the Poisson problem $\Delta \, {\rm{Re}} \,
\Omega = \sum_k {\rm{Res}} S(z_k) \delta(z-z_k)$, and finding the actual (time-dependent) location of the distribution of charges $z_k(t)$, subject to usual conditions for the electrostatic potential $\Delta \, {\rm{Re}} \, \Omega $.
In the equilibrium case, however, it is appropriate to work with the smooth distribution $\rho(z)$. Since the actual electrostatic potential $\Delta \, {\rm{Re}} \, \Omega $ contains the regular expansion $V(z) = \sum_k t_k z^k$, we also add it to the contribution due to the distribution $\rho(z)$. We obtain for the total potential: (z, |z) = \_[D\_+]{}()|z-|\^2 [[[d]{}]{}]{}\^2 + V(z) + . Inside the droplet, this potential solves the Poisson problem $\Delta \, \Phi(z, \bar z) = \rho(z) = 1$, and on the boundary it creates the electric field $E(z) = \bar{\p} \Phi = z$. This means that inside the droplet, this potential is actually equal to $|z|^2$. Therefore, the problem of finding the actual shape of a droplet of area $t_0$ and harmonic moments $\{ t_k \}$ can be stated as:
> *Find the domain $D_+$ of area $t_0$ such that $\Phi(z,\bar z) = |z|^2$ on $D_+$.*
Since $\rho(z)$ is the characteristic function of $D_+$, we may also write this problem in the variational form: $$\frac{\delta \,\,\,\,\,\,\,\,}{\delta \rho(z)} \int_{D_+}
\rho(z) \left [ |z|^2 - V(z) - \overline{V(z)} - \int_{D_+}\rho(\zeta) \log |z-\zeta|^2 {{{d}}}^2 \zeta \right ]
{{{d}}}^2 z = 0.$$ This equation is simply the minimization condition for the total energy of a distribution of charges $\rho(z)$, in the external potential $W(z, \bar z) = -|z|^2
+ V(z) + \overline{V(z)}$. Therefore, the equilibrium (long-time limit) distribution of water has the usual interpretation of minimizing the total electrostatic energy of the system. However, when the system is not in equilibrium, this criterion cannot be used to select the solution.
Quadrature Domains
==================
We have seen in the previous sections that polynomial or rational conformal mappings from the disk have as images planar domains which are relevant for the Laplacian growth (with finitely many sources). The domains in question were previously and independently studied by mathematicians, for at least two separate motivations. First they have appeared in the work of Aharonov and Shapiro, on extremal problems of univalent function theory [@AS]. About the same time, these domains have been isolated by Makoto Sakai in his potential theoretic work [@S1]. These domains, known today as [*quadrature domains*]{}, carry Gaussian type quadrature formulas which are valid for several classes of functions, like integrable analytic, harmonic, and sub-harmonic functions. The geometric structure of their boundary, qualitative properties of their boundary defining function, and dynamics under the Laplacian growth law are well understood. The reader can consult the recent collection of articles [@qd] and the survey [@GP07]. The present section contains a general view of the theory of quadrature domains, with special emphasis of a matrix model realization of their defining function.
This chapter is organized in the following way: after presenting the theory of quadrature domains for subharmonic and analytic functions, we give an overview of the (inverse) Markov problem of moments, followed by its analogue in two dimensions, which is based on the notion of exponential transform in the complex plane. The following sections illustrate the reconstruction algorithm for the shape of a droplet, and point to a few essential properties of the problem for signed measures.
Quadrature domains for subharmonic functions
--------------------------------------------
Let $\varphi$ be a subharmonic function defined on an open subset of the complex plane, that is $\Delta\varphi \geq 0$, in the sense of distributions, or the submeanvalue property $$\varphi (a) \leq \frac{1}{|B(a,r)|}\int_{B(a,r)} \varphi\,{\rm dA}$$ holds for any disc centered at $a$, of radius $r$, $B(a,r)$ contained in the domain of definition of $\varphi$. Henceforth $dA$ denotes Lebesgue planar measure. Thus, with $\Omega=B(a,r)$, $c=|B(a,r)|=
\pi r^2$ and $\mu =c \delta_a$ there holds $$\label{subharmqd}
\int \varphi \, d\mu \leq \int_{\Omega} \varphi\,{\rm dA}$$ for all subharmonic functions $\varphi$ in $\Omega$. This set of inequalities is encoded in the definition that $\Omega$ is a [*quadrature domain for subharmonic functions*]{} with respect to $\mu$ [@S1], and it expresses that $\Omega=B(a,r)$ is a [*swept out*]{} version of the measure $\mu=c\delta_a$. If $c$ increases the corresponding expansion of $\Omega$ is a simple example of Hele-Shaw evolution, or Laplacian growth, as we have seen in the previous section.
The above can be repeated with finitely many points, i.e., with $\mu$ of the form $$\label{mu}
\mu=c_1 \delta_{a_1} +\dots + c_n \delta_{a_n},$$ $a_j\in\mathbb C$, $c_j>0$: there always exists a unique (up to nullsets) open set $\Omega\subset \mathbb C$ such that (\[subharmqd\]) holds for all $\varphi$ subharmonic and integrable in $\Omega$. One can think of it as the union $\bigcup_{j=1}^n B(a_j, r_j)$, $r_j =\sqrt{c_j/\pi}$, with all multiple coverings smashed out to a singly covered set, $\Omega$. In particular, $\bigcup_{j=1}^n
B(a_j, r_j)\subset\Omega$.
The above sweeping process, $\mu\mapsto\Omega$, or better $\mu\mapsto\chi_\Omega\cdot \rm{(dA)}$, called [*partial balayage*]{} [@S1], [@Gustafsson-Sakai94], [@G4], applies to quite general measures $\mu\geq 0$ and can be defined in terms of a natural energy minimization: given $\mu$, $\nu =\chi_\Omega\cdot \rm{(dA)} $ will be the unique solution of $${\rm Minimize}_\nu \, ||\mu -\nu||_e^2 \quad {\rm{s.t.}} \quad \nu\leq \rm{dA},
\int\,d\nu =\int\,d\mu.$$ Here $|| \cdot ||_e$ is the energy norm: $$|| \mu ||_e^2=(\mu,\mu)_e, \quad {\rm with}\quad
(\mu, \nu)_e = \frac{1}{2\pi}\int \log\frac{1}{|z-\zeta|}\, d\mu
(z) d\nu (\zeta).$$ If $\mu$ has infinite energy, like in (\[mu\]), one minimizes $-2(\mu,\nu)+||\nu||_e^2 $ instead of $||\mu -\nu||_e^2$, which can always be given a meaning [@Saff-Totik].
By choosing $$\varphi (\zeta) = \pm \log |z-\zeta|$$ in equation (\[subharmqd\]), the plus sign allowed for all $z\in\mathbb C$, the minus sign allowed only for $z\notin\Omega$, one gets the following statements for potentials: $$\label{potentials}
\left\{\begin{array}{l} U^\mu \geq U^\Omega\quad \rm{in\ all}\quad
\mathbb C,\\
U^\mu = U^\Omega\quad \rm{outside}\,\, \Omega.
\end{array}\right.$$ Here $$U^\mu (z)=\frac{1}{2\pi}\int \log\frac{1}{|z-\zeta|}\, d\mu
(\zeta)$$ denotes the logarithmic potential of the measure $\mu$, and $U^\Omega=U^{\chi_\Omega\cdot {\rm dA}}$. In particular, the measures $\mu$ and $\chi_\Omega\cdot \rm{(dA)}$ are gravi- equivalent outside $\Omega$. By an approximation argument, (\[potentials\]) is actually equivalent to (\[subharmqd\]).
Let us consider now an integrable harmonic function $h$, defined in the domain $\Omega$. Since both $\varphi = \pm h$ are subharmonic functions, we find $$\int_\Omega h dA = \int h d\mu = \sum_{j=1}^n c_j h(a_j).$$ That is, a Gaussian type quadrature formula, with nodes $\{ a_j\}$ and weights $\{ c_j\}$ holds. We say in this case that $\Omega$ is a [*quadrature domain for harmonic functions*]{}. Similarly, one defines a quadrature domain for complex analytic functions, and it is worth mentioning that the inclusions $\{$QD for subharmonic functions$\}
\subset $ $\{$QD for harmonic functions$\} \ \subset $ $\{$QD for analytic functions$\}$ are strict, see for details [@S1].
Recall that for a given positive measure $\sigma$ on the line, rapidly decreasing at infinity, the zeros of the $N$-th orthogonal polynomial are the nodes of a Gauss quadrature formula, valid only for polynomials of degree $2N-1$. The difference above is that the same finite quadrature formula is valid, in the plane, for an infinite dimensional space of functions. A common feature of the two scenarios, which will be clarified in the sequel, is the link between quadrature formulas (on the line or in the plane) and spectral decompositions (of Jacobi matrices, respectively hyponormal operators).
Let $K={\rm conv\,}{\rm supp\,}\mu$ be the convex hull of the support of $\mu$, i.e., the convex hull of the points ${a_1,\dots,a_n}$. As mentioned, $\Omega$ can be thought of as smashed out version of $\bigcup_{j=1}^n B(a_j, r_j)$. The geometry of $\Omega$ which this enforces is expressed in the following sharp result ([@Gustafsson-Sakai94], [@Gustafsson-Sakai02a], [@Gustafsson-Sakai02b]): assume that $\Omega$ satisfies (\[subharmqd\]) for a measure $\mu\geq 0$ of the form (\[mu\]). Then:
1. $\partial\Omega$ may have singular points (cusps, double points, isolated points), but they are all located inside $K$. Outside $K$, $\partial\Omega$ is smooth algebraic.
For $z\in\partial\Omega\setminus K$, let $N_z$ denote the inward normal of $\partial\Omega$ at $z$ (well defined by (i)).
1. For each $z\in\partial\Omega\setminus K$, $N_z$ intersects $K$.
2. For $z,w\in \partial\Omega\setminus K$, $z\ne w$, $N_z$ and $N_w$ do not intersect each other before they reach $K$. Thus $\Omega\setminus K$ is the disjoint union of the inward normals from $\partial\Omega\setminus K$.
3. There exist $r(z)>0$ for $z\in K \cap\Omega$ such that $$\Omega = \bigcup_{z\in K \cap\Omega} B(z,r(z)).$$
(Statement (iv) is actually a consequence of (iii).)\
To better connect our discussion with the moving boundaries encountered in the first part of this survey, we add the following remarks. Since $\Omega$ is uniquely determined by $(a_j,c_j)$ one can steer $\Omega$ by changing the $c_j$ (or $a_j$). Such deformations are of Hele-Shaw type, as can be seen by the following computation, which applies in more general situations: Hele-Shaw evolution $\Omega (t)$ corresponding to a point source at $a\in\mathbb{C}$ (“injection of fluid” at $a$) means that $\Omega(t)$ changes by $\partial\Omega (t)$ moving in the outward normal direction with speed $$-\frac{\partial G_{\Omega (t)}(\cdot, a)}{\partial n}.$$ Here $G_\Omega(z,a)$ denotes the Green function of the domain $\Omega$. If $\varphi$ is subharmonic in a neighborhood of $\overline{\Omega(t)}$ then, as a consequence of $G_\Omega(\cdot,
a)\geq 0$, $G_\Omega(\cdot, a)=0$ on $\partial\Omega$ and $-\Delta
G_\Omega(\cdot, a)=\delta_a$, $$\frac{d}{dt}\int_{\Omega(t)} \varphi \,{\rm dA} =
\int_{\partial \Omega(t)} ({\rm speed\ of\ } \partial\Omega(t) \,
{\rm in\ normal\ direction})\, \varphi\,ds$$ $$=-\int_{\partial\Omega(t)} \frac{\partial G_{\Omega (t)}(\cdot,
a)}{\partial n}\, \varphi\,ds =-\int_{\partial\Omega(t)}
\frac{\partial \varphi}{\partial n}\, G_{\Omega (t)}(\cdot, a)\,ds$$ $$-\int_{\Omega(t)}\varphi \,\Delta G_{\Omega (t)}(\cdot, a)\,{\rm
dA}
+\int_{\Omega(t)}G_{\Omega (t)}(\cdot, a)\, \Delta \varphi\,{\rm
dA} \geq \varphi (a).$$
Hence, integrating from $t=0$ to an arbitrary $t>0$, $$\int_{\Omega(t)} \varphi\, {\rm dA} \geq \int_{\Omega(0)}
\varphi\, {\rm dA} +t\varphi (a),$$ telling that if $\Omega(0)$ is a quadrature domain for $\mu$ then $\Omega(t)$ is a quadrature domain for $\mu + t\delta_a$.
We remark that quadrature domains for subharmonic functions can be defined in any number of variables, but then much less of their qualitative properties are known, see for instance [@qd].
Quadrature domains for analytic functions
-----------------------------------------
Critical for our study is the regularity and algebraicity of the boundary of quadrature domains for analytic functions. This was conjectured in the early works of Aharonov and Shapiro, and proved in full generality by Gustafsson [@G1]. A description of the possible singular points in the boundary of a quadrature domain was completed by Sakai [@S3; @S4; @S5].
Assume that the quadrature domain for analytic functions $\Omega$ has a sufficiently smooth boundary $\Gamma$. Let us consider the Cauchy transform of the area mass, uniformly distributed on $\Omega$: $$C(z) = \frac{-1}{\pi} \int_\Omega \frac{d A(w)}{w-z}.$$ This is an analytic function on the complement of $\overline{\Omega}$, which is continuous (due to the Lebesgue integrability of the kernel) on the whole complex plane. In addition, the quadrature identity implies $$C(z) = \sum_{j=1}^n \frac{c_j}{\pi(z-a_j)}, \ \ z \in {\mathbb C}
\setminus \overline{\Omega}.$$
From the Stokes formula, $$C(z) = \frac{-1}{2 \pi i} \int_\Gamma \frac{ \overline{w}
dw}{w-z}, \ \ z \in {\mathbb C} \setminus \overline{\Omega}.$$ Therefore, by standard arguments in function theory one proves that the continuous function $w \mapsto \overline{w}$ extends meromorphically from $\Gamma$ to $\Omega$. The poles of this meromorphic extension coincide with the quadrature nodes.
The converse also holds, in virtue of Cauchy’s formula: if $f$ is an integrable analytic function in $\Omega$, then $$\int_\Omega f dA = \int_\Gamma f(w) \overline{w} d w =
\sum_{j=1}^n c_j f(a_j).$$ Thus, we recover the following fundamental observation: [ *if $\Omega$ is a bounded planar domain with sufficiently smooth boundary $\Gamma$, then $\Omega$ is a quadrature domain for analytic functions if and only if the function $w \mapsto \overline{w}$ extends meromorphically from $\Gamma$ to $\Omega$.*]{}
Note that above, and elsewhere henceforth, we do not assume that the weights in the quadrature formula for analytic functions are positive. In this way we recover the fact (already noted in the previous chapters) that quadrature domains for analytic functions are characterized by a meromorphic Schwarz function, usually denoted $S(z)$. A second departure from the quadrature domains for subharmonic functions is that the quadrature data $(a_j,c_j)$ [*do not determine*]{} the quadrature domain for analytic functions. Indeed, consider the annulus $A_{r,R} = \{ z, \ r<|z|<R\}.$ Then $$\int_{A_{r,R}} f dA = \pi(R^2-r^2) f(0),$$ for all analytic, integrable functions $f$ in $A_{r,R}$.
The question how weak the smoothness assumption on the boundary $\Gamma$ can be to insure the use of the above arguments has a long history by itself, and we do not enter into its details. Simply the existence of the quadrature formula and the fact that the boundary is a mere continuum implies, via quite sophisticated techniques, the regularity of $\Gamma$. See for instance [@S3; @GP98].
The Schwarz function is a central character in our story. It can also be related to the logarithmic potentials introduced in the previous subsection. More specifically, given any measure $\mu$ as in (\[mu\]) and any open set $\Omega$ containing ${\rm
supp\,}\mu$, define (as distributions in all $\mathbb{C}$) $$u=U^\mu-U^\Omega, \quad
S(z)=\overline{z} -4\frac{\partial u}{\partial z}.$$ Then $$\Delta u = \chi_\Omega- \mu, \quad \frac{\partial S}{\partial
\overline{z}}= 1- \chi_\Omega +\mu.$$ Note that with $\mu$ of the form (\[mu\]) $w$ is harmonic in $\Omega$ except for poles at the points $a_j$ and that in particular, $S(z)$ is meromorphic in $\Omega$.
It is clear from (\[potentials\]) that $\Omega$ is a subharmonic quadrature domain for $\mu$ if and only if $u\geq 0$ everywhere and $u=0$ outside $\Omega$. Then also $\nabla u=0$ outside $\Omega$. Similarly, the criterion for $\Omega$ being a quadrature domain for harmonic functions is that merely $u = \nabla u =0$ on $\mathbb C\setminus \Omega$. (The vanishing of the gradient is a consequence of the vanishing of $u$, except at certain singular points on the boundary.) To be a quadrature domain for analytic functions it is enough that just the gradient vanishes, or better in the complex-valued case, that $\frac{\partial u}{\partial z}
=0$ on $\mathbb C\setminus \Omega$ (or just on $\partial\Omega$).
Gustafsson’s innovative idea, to use the Schottky double of the domain, can be summarized as follows. Let $\Omega$ be a bounded quadrature domain for analytic functions, with boundary $\Gamma$. We consider a second copy $\tilde{\Omega}$ of $\Omega$, endowed with the anti-conformal structure, and “glue" them into a compact Riemann surface $$X = \Omega \cup \Gamma \cup \tilde{\Omega}.$$ This (connected) Riemann surface carries two meromorphic functions: $$f(z) = \left\{ \begin{array}{cc}
S(z), & z \in \Omega \\
\overline{z}, & z \in \tilde{\Omega}\\
\end{array}\right. , \ \ \
g(z) = \left\{ \begin{array}{cc}
z, & z \in \Omega \\
\overline{S(z)}, & z \in \tilde{\Omega}\\
\end{array}\right. .$$ Any pair of meromorphic functions on $X$ is algebraically dependent, that is, there exists a polynomial $Q(z,w)$ with the property $ Q(g,f)=0$, and in particular $$Q(z, S(z)) = Q(z, \overline{z}) = 0, \ \ z \in \Gamma.$$ The involution (flip from one side to its mirror symmetric) on $X$ yields the Hermitian structure of $Q$: $$Q(z,w) = \sum_{i,j}^n a_{ij} z^i w^j,\ \ a_{ij} =
\overline{a_{ji}}.$$ One also proves by elementary means of Riemann surface theory that $Q$ is irreducible, and moreover, its leading part is controlled by the quadrature identity data: $$Q(z,\overline{z}) - |P(z)|^2 = O(z^{n-1},
\overline{z}^{n-1}),$$ where $$P(z) = (z-a_1)(z-a_2)...(z-a_n).$$ This Riemann surface is the continuum limit of the spectral curve (\[qc\]), for $N \to \infty$. Following Gustafsson ([@G1]), we note a surprising result:
*a) The boundary of a quadrature domain for analytic functions is a real algebraic, irreducible curve.*
b\) In every conformal class of finitely connected planar domains there exists a quadrature domain.
c\) Every bounded planar domain can be approximated in the Haudorff distance by a sequence of quadrature domains.
The last two assertions are proven in Gustafsson’s influential thesis [@G1]. Recently, considerable progress was made in the construction of multiply connected quadrature domains, see [@qd; @Crowdy-Marshall03; @Crowdy-Marshall03].
It is important to point out that not every domain bounded by an algebraic curve is an algebraic domain in the above sense. In general, if a domain $\Omega\subset \mathbb C$ is bounded by an algebraic curve $Q(z,\overline{z})=0$ ($Q$ a polynomial with Hermitian symmetry), then one can associate two compact symmetric Riemann surfaces to it: one is the Schottky double of $\Omega$ and the other is the Riemann surface classically associated to the complex curve $Q(z,w)=0$. For the latter the involution is given by $(z,w)\mapsto (\overline{w},\overline{z})$. In the case of [*algebraic domains*]{} (this is another circulating name for quadrature domains for analytic functions), and only in that case, the two Riemann surfaces canonically coincide: the lifting $$z\mapsto (z,S(z))$$ from $\Omega$ to the locus of $Q(z,w)=0$ extends to the Schottky double of $\Omega$ and then gives an isomorphism, respecting the symmetries, between the two Riemann surfaces.
As a simple example, the Schottky double of the simply connected domain $$\Omega=\{ z=x+iy \in {\mathbb{C}} : x^4+ y^4 <1 \}$$ has genus zero, while the Riemann surface associated to the curve $x^{4}+ y^{4} =1$ has genus $3$. Hence they cannot be identified, and in fact $\Omega$ is not an algebraic domain.
Other ways of characterizing algebraic domains, by means of rational embeddings into $n$ dimensional projective space, are discussed in [@GP00].
Markov’s moment problem
-----------------------
We pause for a while the main line of our story, to connect the described phenomenology with a classical, beautiful mathematical construct due to A. A. Markov, all gravitating around moment problems for bounded functions.
The classical [*$L$-problem of moments*]{} (also known as [*Markov’s moment problem*]{}) offers a good theoretical framework for reconstructing extremal measures $\mu$ from their moments, or equivalently, from the germ at infinity of some of their integral transforms. The material below is classical and can be found in the monographs [@AK; @KN]. We present only a simplified version of the abstract $L$-problem, well adapted to the main themes of this survey.
Let $K$ be a compact subset of ${\mathbb{R}}^n$ with interior points and let $A \subset {\mathbb{N}}^n$ be a finite subset of multi-indices. We are interested in the set $\Sigma_A$ of moment sequences $a(f)
= (a_\sigma(f))_{\sigma \in A}$: $$a_\sigma (f) = \int_K x^\sigma f(x) dx, \ \ \sigma \in A,$$ of all measurable functions $f: K \longrightarrow [0,1]$. Regarded as a subset of ${\mathbb{R}}^{|A|}$, $\Sigma_A$ is a compact convex set. An $L^1-L^\infty$ duality argument (known as the abstract $L$-problem of moments) shows that every extremal point of $\Sigma_A$ is a characteristic function of the form $\chi_{ \{ p < \gamma\} },$ where we denote: $$\{p < \gamma \} = \{ x \in K;\ p(x)< \gamma \}.$$ Above $\gamma$ is a real constant and $p$ is an $A$-polynomial with real coefficients, that is $p(x) = \sum_{\sigma \in A}
c_\sigma x^\sigma$. Indeed, to find the special form of the extremal functions $f$, one has to analyze when the inequality $$\int_K p(x) f(x) dx \leq \| p \|_1 \| f \|_\infty = \int_K
|p(x)| dx$$ is an equality. For a complete proof the reader can consult Krein and Nudelman’s monograph [@KN].
As a consequence, the above description of the extremal points in the moment set $\Sigma_A$ implies the following remarkable uniqueness theorem due to Akhiezer and Krein:
[*For each characteristic function $\chi$ of a level set in $K$ of an $A$-polynomial there exists exactly one class of functions $f$ in $L^\infty(K)$ satisfying $a(f) = a(\chi)$. For a non-extremal point $a(f) \in
\Sigma_A$ there are infinitely many non-equivalent classes in $L^\infty(K)$ having the same $A$-moments.*]{}\
Let us consider a simple example: $$K = \{ (x,y); \ x^2 + y^2 \leq 1 \} \subset {\mathbb{R}}^2,$$ and $$\Omega_+ = \{ (x,y) \in K; \ x>0, \ y>0 \}, \ \ \Omega_- = \{ (x,y) \in K; \ x<0, \ y<0 \}.$$ The reader can prove by elementary means that the sets $\Omega_\pm$ cannot be defined in the unit ball $K$ by a single polynomial inequality. On the other hand, the set $$\Omega = \Omega_+ \cup \Omega_- = \{ (x,y); \ xy >0 \},$$ is defined by a single equation of degree two.
Thus, no matter how the finite set of indices $A \subset {\mathbb
N}^2$ is chosen, there is a continuum $f_s, \ s \in {\mathbb{R}},$ of essentially distinct measurable functions $f_s : K \longrightarrow
[0,1]$ possessing the same $A$-moments: $$\int_K x^{\sigma_1} y^{\sigma_2} f_s(x,y) dx dy = \int_{\Omega_+}
x^{\sigma_1} y^{\sigma_2} dx dy, \ \ s \in {\mathbb{R}}, \ \sigma \in A.$$
On the contrary, if the set of indices $A$ contains $(0,0)$ and $(1,1)$, then for every measurable function $f: K \longrightarrow
[0,1]$ satisfying $$\int_K x^{\sigma_1} y^{\sigma_2} f(x,y) dx dy = \int_{\Omega}
x^{\sigma_1} y^{\sigma_2} dx dy, \ \ \sigma \in A,$$ we infer by Akhiezer and Krein’s Theorem that $f = \chi_\Omega,$ almost everywhere.
On a more theoretical side, we can interpret Akhiezer and Krein’s Theorem in terms of geometric tomography, see [@Ga]. Fix a unit vector $\omega \in {\mathbb{R}}^n, \ \| \omega \| =1,$ and let us consider the parallel Radon transform of a function $f:K
\longrightarrow [0,1]$, along the direction $\omega$: $$(Rf)(\omega, s) = \int_{ \langle x, \omega \rangle = s} f(x) dx.$$ Accordingly, the $k$-th moment in the variable $s$ of the Radon transform is, for a sufficiently large constant $M$: $$\int_{-M}^M (Rf)(\omega,s ) s^k ds = \int_K \langle x, \omega
\rangle^k f(x) dx =$$ $$\la{33}
\sum_{|\sigma| = k} \frac{|\sigma|!}{\sigma !} \int_K x^\sigma
\omega^\sigma f(x) dx = \sum_{|\sigma| = k}
\frac{|\sigma|!}{\sigma !} \omega^\sigma a_\sigma(f).$$
Since there are $N(n,d) = C_{n+d}^n$ linearly independent polynomials in $n$ variables of degree less than or equal to $d$, a Vandermonde determinant argument shows, via the above formula, that the same number of different parallel projections of the “shade” function $f: K \longrightarrow [0,1]$, determine, via a matrix inversion, all moments: $$a_\sigma(f), \ \ |\sigma| \leq d.$$ The converse also holds, by formula (\[33\]). These transformations are known and currently used in image processing, see for instance [@GHMP] and the references cited there.
In conclusion, Akhiezer and Krein’s Theorem asserts then that in the measurement process $$f \mapsto ((Rf)(\omega_j,s))_{j=1}^{N(n,d)} \mapsto (a_\sigma(f))_{|\sigma| \leq d}$$ only black and white pictures, delimited by a single algebraic equation of degree less than or equal to $d$, can be exactly reconstructed. Even when these uniqueness conditions are met, the details of the reconstruction from moments are delicate. We shall see some examples in the next sections.
### Markov’s extremal problem and the phase shift
By going back to the source and dropping a few levels of generality, we recall Markov’s original moment problem and some of its modern interpretations. Highly relevant for our “quatization” approach to moving boundaries of planar domains is the matrix interpretation we will describe for Markov’s moment. Again, this material is well exposed in the monograph by Krein and Nudelman [@KN].
Let us consider, for a fixed positive integer $n$, the $L$-moment problem on the line: $$a_k = a_k(f) = \int_{\mathbb{R}} t^k f(t) dt, \ \ \ 0 \leq k \leq 2n,$$ where the unknown function $f$ is measurable, admits all moments up to degree $2n$ and satisfies:$$0 \leq f \leq L, \ {\rm
a.e.}.$$
As noted by Markov, the next formal series transform is quite useful for solving this question: = 1 + + + …. Remark that, although the series under the exponential is finite, the resulting one might be infinite.
The following result is classical, see for instance [@AK] pp. 77-82. Its present form was refined by Akhiezer and Krein; partial similar attempts are due, among others, to Boas, Ghizzetti, Hausdorff, Kantorovich, Verblunsky and Widder, see [@AK; @KN].
[*(Markov) Let $a_0, a_1, \ldots, a_{2n}$ be a sequence of real numbers and let $b_0, b_1, \ldots$ be its exponential $L$-transform. Then there is an integrable function $f, \ 0 \leq f \leq L,$ possessing the moments $a_k(f) = a_k , \ 0
\leq k \leq 2n,$ if and only if the Hankel matrix $(b_{k+l})_{k,l=0}^n$ is non-negative definite. Moreover, the solution $f$ is unique if and only if ${\rm det}
(b_{k+l})_{k,l=0}^n = 0.$ In this case the function $f/L$ is the characteristic function of a union of at most $n$ bounded intervals.*]{}
The reader will recognize above a concrete validation of the abstract moment problem discussed in the previous section.
In order to better understand the nature of the $L$-problem, we interpret below the exponential transform from two different and complementary points of view. For simplicity we take the constant $L$ to be equal to $1$ and consider only compactly supported originals $f$, due to the fact that the extremal solutions have anyway compact support. Let $\mu$ be a positive Borel measure on ${\mathbb{R}}$, with compact support. Its Cauchy transform $$F(z) = 1- \int_{\mathbb{R}} \frac{d\mu(t)}{t-z},$$ provides an analytic function on ${\mathbb{C}} \setminus {\mathbb{R}}$ which is also regular at infinity, and has the normalizing value $1$ there. The power expansion, for large values of $|z|$, yields the generating moment series of the measure $\mu$: $$F(z) = 1 + \frac{b_0(\mu)}{z} + \frac{b_1(\mu)}{z^2} +
\frac{b_2(\mu)}{z^3} + \ldots.$$
On the other hand, $${\rm Im} F(z) = - {\rm Im} z \int \frac{ d\mu(t)}{|t-z|^2},$$ whence $${\rm Im} F(z)\ {\rm Im} z < 0, \ \ z \in {\mathbb C} \setminus {\mathbb{R}}.$$ Thus the main branch of the logarithm ${\rm log} F(z)$ exists in the upper half-plane and its imaginary part, equal to the argument of $F(z)$, is bounded from below by $-\pi$ and from above by $0$. According to Fatou’s theorem, the non-tangential boundary limits $$f(t) = \lim_{\epsilon \rightarrow 0} \frac{-1}{\pi} {\rm Im}\ {\rm log} F(t+i\epsilon),$$ exist and produce a measurable function with values in the interval $[0,1]$. According to Riesz-Herglotz formula for the upper-half plane, we obtain: $${\rm log} F(z) = - \int_{\mathbb{R}} \frac{f(t)dt}{t-z}, \ \ z \in
{\mathbb C} \setminus {\mathbb{R}}.$$ Or equivalently, $$F(z) = {\rm exp} \left [ - \int_{\mathbb{R}} \frac{f(t)dt}{t-z} \right ].$$
One step further, let us consider the Lebesgue space $L^2(\mu)$ and the bounded self-adjoint operator $A = M_t$ of multiplication by the real variable. The vector $\xi = {\bf 1}$ corresponding to the constant function $1$ is $A$-cyclic, and according to the spectral theorem: $$\int_{\mathbb{R}} \frac{d\mu(t)}{t-z} = \langle (A-z)^{-1} \xi, \xi
\rangle, \ \ z \in {\mathbb C} \setminus {\mathbb{R}}.$$
As a matter of fact an arbitrary function $F$ which is analytic on the Riemann sphere minus a compact real segment, and which maps the upper/lower half-plane into the opposite half-plane has one of the above forms. These functions are known in rational approximation theory as [*Markov functions*]{}.
In short, putting together the above comments we can state the following result: the canonical representations: $$F(z) = 1- \int_{\mathbb{R}} \frac{d\mu(t)}{t-z} =
{\rm exp}( - \int_{\mathbb{R}} \frac{f(t)dt}{t-z}) = 1- \langle (A-z)^{-1} \xi, \xi \rangle$$ establish constructive equivalences between the following classes:
a) Markov’s functions F(z);
b) Positive Borel measures $\mu$ of compact support on ${\mathbb{R}}$;
c) Functions $f \in L^\infty_{\rm comp} ({\bf
R})$ of compact support, $0 \leq f \leq 1;$
d) Pairs $(A,\xi)$ of bounded self-adjoint operators with a cyclic vector $\xi$. The extremal solutions correspond, in each case exactly, to: a) Rational Markov functions $F$;
b) Finitely many point masses $\mu$;
c) Characteristic functions $f$ of finitely many intervals;
d) Pairs $(A,\xi)$ acting on a finite dimensional Hilbert space.
For a complete proof see for instance Chapter VIII of [@MP] and the references cited there. The above dictionary is remarkable in many ways. Each of its terms has intrinsic values. They were long ago recognized in moment problems, rational approximation theory or perturbation theory of self-adjoint operators.
For instance, when studying the change of the spectrum under a rank-one perturbation $A \mapsto B= A - \xi\langle \cdot, \xi
\rangle$ one encounters the [*perturbation determinant*]{}: $$\Delta_{A,B}(z) = {\rm det} [ (A - \xi\langle \cdot, \xi \rangle - z)(A-z)^{-1}] =
1- \langle (A-z)^{-1} \xi, \xi \rangle.$$ The above exponential representation leads to the [*phase-shift*]{} function $f_{A,B}(t)
= f(t)$: $$\Delta_{A,B}(z) = {\rm exp}\left [ - \int_{\mathbb{R}} \frac{f_{A,B}(t)dt}{t-z} \right ].$$ The phase shift of, in general, a trace-class perturbation of a self-adjoint operator has certain invariance properties; it reflects by fine qualitative properties the nature of change in the spectrum. The theory of perturbation determinants and of the phase shift is nowadays well developed, mainly for its applications to quantum physics, see [@Krein1953; @Simon].
The reader will recognize above an analytic continuation in the complex plane of the real exponential transform $$F(x) = E_f(x) = {\rm exp} \left [ - \int_{\mathbb{R}} \frac{f(t)dt}{|t-x|} \right ],$$ assuming for instance that $x < M$ and the function $f$ is supported by $[M,\infty)$.
To give the simplest, yet essential, example, we consider a positive number $r$ and the various representations of the function: $$F(z) = 1 + \frac{r}{z} = \frac{z+r}{z} = 1 - \int_{\mathbb{R}} \frac{ r d \delta_0(t)}{t-z} =
{\rm exp} \left [ - \int_{-r}^0 \frac{dt}{t-z} \right ] = {\rm det} [(-r -z)(-z)^{-1}].$$ In this case the underlying Hilbert space has dimension one and the two self-adjoint operators are $A = 0$ and $A - \xi\langle
\cdot, \xi \rangle = -r$.
### The reconstruction algorithm in one real variable
Returning to our main theme, and as a direct continuation of the previous section, we are interested in the exact reconstruction of the original $f:{\mathbb{R}} \longrightarrow [0,1]$ from a finite set of its moments, or equivalently, from a Taylor polynomial of $E_f$ at infinity. The algorithm described in this section is the diagonal Padé approximation of the exponential transform of the moment sequence. Its convergence, even beyond the real axis, is assured by a famous result discovered by A. A. Markov.
Let $a_0, a_1, \ldots, a_{2n}$ be a sequence of real numbers with the property that its exponential transform: $${\rm exp} \left [\frac{1}{L} \left (\frac{a_0}{z} + \frac{a_1}{z^2} +
\ldots \frac{a_{2n}}{z^{2n+1}} \right ) \right ] =
1 + \frac{b_0}{z} + \frac{b_1}{z^2} + \ldots,$$ produces a non-negative Hankel matrix $(b_{k+l})_{k,l=0}^n$.
According to Markov’s Theorem, there exists at least one bounded self-adjoint operator $A \in L(H)$, with a cyclic vector $\xi$, such that: $${\rm exp} \left [\frac{1}{L} \left (\frac{a_0}{z} + \frac{a_1}{z^2} + \ldots
\frac{a_{2n}}{z^{2n+1}} \right ) \right ] = 1 + \frac{\langle \xi, \xi
\rangle}{z} + \frac{\langle A\xi, \xi \rangle}{z^2} + \ldots
\frac{\langle A^{2n} \xi, \xi \rangle}{z^{2n+1}} +
O(\frac{1}{z^{2n+2}}).$$
Let $k <n$ and $H_k$ be the Hilbert subspace spanned by the vectors $\xi, A\xi, \ldots, A^{k-1}\xi$. Suppose that ${\rm dim}
H_k =k$, which is equivalent to saying that ${\rm det}
(b_{i+j})_{i,j=0}^{k-1} \neq 0$. Let $\pi_k$ be the orthogonal projection of $H$ onto $H_k$ and let $A_k = \pi_k A \pi_k$. Then $$\langle A_k^{i+j} \xi, \xi \rangle = \langle A_k^i \xi, A_k^j \xi \rangle =\langle A^i \xi, A^j \xi \rangle = \langle A^{i+j} \xi, \xi \rangle,$$ whenever $0 \leq i, j \leq k-1.$ In other terms, for large values of $|z|$: $$\langle (A-z)^{-1} \xi, \xi \rangle = \langle (A_k-z)^{-1} \xi, \xi \rangle + O(\frac{1}{z^{2k+1}}).$$
By construction, the vector $\xi$ remains cyclic for the matrix $A_k \in L(H_k)$. Let $q_k(z)$ be the minimal polynomial of $A_k$, that is the monic polynomial of degree $k$ which annihilates $A_k$. In particular, $$q_k(z) \langle (A_k-z)^{-1} \xi, \xi \rangle =
\langle (q_k(z) - q_k(A_k)) (A_k-z)^{-1} \xi, \xi \rangle = p_{k-1}(z)$$ is a polynomial of degree $k-1$.
The two observations yield: $$q_k(z) \langle (A-z)^{-1} \xi, \xi \rangle =
q_k(z) \langle (A_k-z)^{-1} \xi, \xi \rangle +
O(\frac{1}{z^{k+1}}) = p_{k-1}(z) + O(\frac{1}{z^{k+1}}).$$
The resulting rational function $R_k(z) =
\frac{p_{k-1}(z)}{q_k(z)}$ is characterized by the property: $$1 + \frac{b_0}{z} + \frac{b_1}{z^2} + \ldots = 1+ R_k(z) + O(\frac{1}{z^{2k+1}});$$ it is known as the [*Padé approximation*]{} of order $(k-1,k)$, of the given series.
A basic observation is now in order: since $b_0, b_1, \ldots,
b_{2k+1}$ is the power moment sequence of a positive measure, $q_k$ is the associated orthogonal polynomial of degree $k$ and $p_k$ is a second order orthogonal polynomial of degree $k-1$. In particular their roots are simple and interlaced. We prove only the first assertion, the second one being of a similar nature. Indeed, let $\mu$ be the spectral measure of $A$ localized at the vector $\xi$. Then, for $j<k$, $$\int_{\mathbb{R}} t^j q_k(t) dt = \langle A^j \xi, q_k(A) \xi
\rangle = \langle A_k^j \xi, q_k(A_k) \xi \rangle = 0.$$
Assume now that we are in the extremal case ${\rm det}
(b_{i+j})_{i,j=0}^n = 0$ and that $n$ is the smallest integer with this property, that is ${\rm det} (b_{i+j})_{i,j=0}^{n-1} \neq 0$. Since $$b_{i+j} = \langle A^i \xi, A^j \xi \rangle,$$ this means that the vectors $\xi, A\xi, \ldots, A^n\xi$ are linearly dependent. Or equivalently that $H_n = H$ and consequently $A_n = A$.
According to the dictionary established above, this is another proof that the extremal case of the truncated moment 1-problem with data $a_0, a_1, \ldots, a_{2n}$ admits a single solution. The unique function $f : {\mathbb{R}} \longrightarrow [0,1]$ with this string of moments will then satisfy: $${\rm exp} \left [ - \int_{\mathbb{R}} \frac{f(t)dt}{t-z} \right ] = 1 + R_n(z) = 1 - \sum_{i=1}^n \frac{r_i}{a_i - z} ={\rm det} [(A- \xi\langle \cdot, \xi \rangle - z)(A-z)^{-1}] =$$ $$\prod_{i=1}^n \frac{b_i-z}{a_i-z},$$ where the spectrum of the matrix $A$ is $\{a_1, \ldots, a_n\},$ that of the perturbed matrix $B = A- \xi\langle \cdot, \xi
\rangle$ is $b_1, \ldots, b_n$ and $r_i$ are positive numbers. Again, one can easily prove that $b_1 < a_1 < b_2 < a_2 < \ldots <
b_n < a_n$. By the last example considered, we infer: $$f = \sum_{i=1}^n \chi_{[b_i,a_i]},$$ or equivalently $$f = \frac{1}{2} \left [1- {\rm sign} \frac{p_{k-1} + q_k}{q_k} \right ].$$
The above computations can therefore be put into a (robust) reconstruction algorithm of all extremal functions $f$. The Hilbert space method outlined above has other benefits, too. We illustrate them with a proof of another celebrated result due to A. A. Markov, and related to the convergence of the mentioned algorithm, in the case of non-extremal functions.
[*Let $\mu$ be a positive measure, compactly supported on the real line and let $F(z) = \int_{\mathbb{R}}
(t-z)^{-1} d\mu(t)$ be its Cauchy transform. Then the diagonal Padé approximation $R_n(z) = p_{n-1}(z)/q_n(z)$ converges to $F(z)$ uniformly on compact subsets of ${\mathbb C} \setminus {\mathbb
R}$.*]{}
This is the basic argument proving the statement: let $A$ be the multiplication operator with the real variable on the Lebesgue space $H= L^2(\mu)$ and let $\xi = {\bf
1}$ be its cyclic vector. The subspace generated by $\xi,
A\xi,...,A^{n-1}\xi$ will be denoted as before by $H_n$ and the corresponding compression of $A$ by $A_n = \pi_n A \pi_n$.
If there exists an integer $n$ such that $H=H_n,$ then the discussion preceding the theorem shows that $F=R_n$ and we have nothing else to prove. Assume the contrary, that is the measure $\mu$ is not finite atomic.
Let $p(t)$ be a polynomial function, regarded as an element of $H$. Then $$(A-A_n)p(t) = tp(t) - (\pi_n A \pi_n) p(t) = tp(t) -
tp(t) = 0$$ provided that ${\rm deg} (p) < n$. Since $\|A_n\| \leq
\|A\|$ for all $n$, and by Weierstrass Theorem, the polynomials are dense in $H$, we deduce: $$\lim_{n \rightarrow \infty} \| (A-A_n) h \| =
0, \ \ h \in H.$$
Fix a point $a \in {\mathbb C} \setminus {\mathbb{R}}$ and a vector $h \in
H$. Then $$\lim_{n \rightarrow \infty} \| [(A-a)^{-1} - (A_n-a)^{-1}]h \| = \lim_{n \rightarrow \infty}
\| (A_n-a)^{-1}(A-A_n)(A-a)^{-1} h \| \leq$$ $$\lim_{n \rightarrow \infty} \frac{1}{| {\rm Im}\ a|} \| (A-A_n)(A-a)^{-1} h \| = 0.$$ A repeated use of the same argument shows that, for every $k \geq
0$, $$\lim_{n \rightarrow \infty} \| [(A-a)^{-k} - (A_n-a)^{-k}]h \| = 0.$$
Choose a radius $r < |{\rm Im}\ a| \leq \| (A_n -a)^{-1} \|^{-1}
$, so that the Neumann series $$(A_n -z)^{-1} = (A_n - a - (z-a))^{-1} = \sum_{k=0}^\infty (z-a)^k (A_n - a)^{-k-1}$$ converges uniformly and absolutely, in $n$ and $z$, in the disk $|z-a| \leq r$. Consequently, for a fixed vector $h \in H$, $$\lim_{n \rightarrow \infty} \| (A_n -z)^{-1} h - (A-z)^{-1} h \| = 0,$$ uniformly in $z,\ |z-a|\leq r$. In particular, $$\lim_{n \rightarrow \infty}
R_n(z) = \langle (A_n-z)^{-1} \xi, \xi \rangle = \lim_{n
\rightarrow \infty} \langle (A_n-z)^{-1} \xi, \xi \rangle =$$ $$\langle (A-z)^{-1} \xi, \xi \rangle = F(z),$$ uniformly in $z, \
|z-a| \leq r$.
Details and a generalization of the above operator theory approach to Markov theorem can be found in [@Put02].
The exponential transform in two dimensions
-------------------------------------------
We return now to two real dimensions, and establish an analog of the matrix model for Markov’s moment problem. Fortunately this is possible due to the import of some key results in the theory of semi-normal operators. We expose first the analog of Markov’s exponential transform, and second, we will make a digression into semi-normal operator theory, with the aim at realizing the exponential transform in terms of (infinite) matrices, and ultimately of reconstructing planar shapes from their moments.
The case of two real variables is special, partly due to the existence of a complex variable in ${\mathbb{R}}^2$ . Let $g : {\mathbb C} \longrightarrow
[0,1]$ be a measurable function and let $dA(\zeta)$ stand for the Lebesgue area measure. The [*exponential transform*]{} of $g$, is by definition the transform: $$E_g(z) = {\rm exp}
(-\frac{1}{\pi} \int_{\mathbb C} \frac{ g(\zeta) dA(\zeta)} {|\zeta -
z|^2}),\ \ z \in {\mathbb C} \setminus {\rm supp}( g).$$ This expression invites to consider a polarization in $z$: $$E_g(z, w) = {\rm exp} (-\frac{1}{\pi} \int_{\mathbb{C}} \frac{ g(\zeta) dA(\zeta)}
{(\zeta - z)(\overline{\zeta}-\overline{w})}),\ \ z, w \in {\mathbb
C} \setminus {\rm supp} (g). \eqno{(6.1)}$$ The resulting function $E_g(z,w)$ is analytic in $z$ and antianalytic in $w$, outside the support of the function $g$. Note that the integral converges for every pair $(z,w) \in {\mathbb{C}}^2$ except the diagonal $z=w$. Moreover, assuming by convention ${\rm exp} (- \infty) =0$, a simple application of Fatou’s Theorem reveals that the function $E_g(z,w)$ extends to the whole ${\mathbb{C}}^2$ and it is separately continuous there. Details about these and other similar computations are contained in [@MP].
As before, the exponential transform contains, in its power expansion at infinity, the moments $$a_{mn}= a_{mn}(g) = \int_{\mathbb{C}} z^m \overline{z}^n g(z) dA(z), \
\ m,n \geq 0.$$ According to Riesz Theorem these data determine $g$. We will denote the resulting series by: $${\rm exp}\left [ \frac{-1}{\pi} \sum_{m,n=0}^\infty \frac{a_{mn}}{z^{n+1}
\overline{w}^{m+1}} \right ] =
1- \sum_{m,n=0}^\infty \frac{b_{mn}}{z^{n+1} \overline{w}^{m+1}}.$$
The exponential transform of a uniformly distributed mass on a disk is simple, and in some sense special, this being the building block for more complicated domains. A direct elementary computation leads to the following formulas for the unit disk ${\bf D}$, cf. [@GP98]: $$E_{\bf D}(z,w) = \cases {
1- \frac{1}{z \overline{w}}, \ \ z,w \in \overline{\bf D}^c,\cr
1- \frac{\overline{z}}{\overline{w}}, \ \ z \in {\bf D},\ w \in \overline{\bf D}^c, \cr
1- \frac{{w}}{{z}}, \ \ w \in {\bf D},\ z \in \overline{\bf D}^c,\cr
\frac{|z-w|^2}{1-z\overline{w}}, \ \ z,w \in {\bf D}.\cr}$$ Remark that $E_{\bf D}(z) = E_{\bf D}(z,z)$ is a rational function and its value for $|z|>1$ is $1- \frac{1}{|z|^2}$. The coefficients $b_{mn}$ of the exponential transform are in this case particularly simple: $b_{00} = 1$ and all other values are zero.
Once more, an additional structure of the exponential transform in two variables comes from operator theory. More specifically, for every measurable function $g: {\mathbb{C}} \to [0,1]$ of compact support there exists a unique irreducible, linear bounded operator $T \in L(H)$ acting on a Hilbert space $H$, with rank-one self-commutator $[T^\ast, T] = \xi \otimes \xi = \xi \langle
\cdot, \xi \rangle$, which factors $E_g$ as follows: $$\label{factorE} E_g(z,w) = 1 - \langle (T^\ast -
\overline{w})^{-1}\xi, (T^\ast - \overline{z})^{-1}\xi \rangle, \
\ z,w \in {\rm supp} (g)^c.$$ As a matter of fact, with a proper extension of the definition of localized resolvent $(T^\ast - \overline{w})^{-1}\xi$ the above formula makes sense on the whole ${\mathbb{C}}^2$. The function $g$ is called the [*principal function*]{} of the operator $T$. The next section will contain a brief incursion into this territory of operator theory.
Let $g : {\mathbb{C}} \to [0,1]$ be a measurable function and let $E_g(z,w)$ be its polarized exponential transform. We retain from the above discussion the fact that the kernel: $$1- E_g(z,w), \ \ \ z,w \in {\mathbb{C}},$$ is positive definite. Therefore the distribution $H_g(z,w) = -
\frac{\partial}{\partial \overline{z}} \frac{\partial}{\partial
{w}} E_g(z,w)$ has compact support and it is positive definite, in the sense: $$\int_{{\mathbb{C}}^2} H_g(z,w) \phi(z) \overline{\phi(w)} dA(z) dA(w) \geq 0, \ \ \phi \in C^\infty({\mathbb{C}}).$$ If $g$ is the characteristic function of a bounded domain $\Omega
\subset {\mathbb{C}}$, then it is elementary to see that the distribution $H_\Omega(z,w) = H_g(z,w)$ is given on $\Omega
\times \Omega$ by a smooth, jointly integrable function which is analytic in $z \in \Omega$ and antianalytic in $w \in \Omega$, see [@GP98].
In particular, this gives the useful representation: $$E_\Omega(z,w) = 1- \frac{1}{\pi^2} \int_{\Omega^2}
\frac{H_\Omega(u,v) dA(u) dA(v)}{(u-z)(\overline{v} -
\overline{w})}, \ \ z,w \in \overline{\Omega}^c,$$ where the kernel $H_\Omega$ is positive definite in $\Omega \times \Omega$.
The example of the disk considered in this section suggests that the exterior exponential transform of a bounded domain $E_\Omega(z,w)$ may extend analytically in each variable inside $\Omega$. This is true whenever $\partial \Omega$ is real analytic smooth. In this case there exists an analytic function $S$ defined in a neighborhood of $\partial \Omega$, with the property: $$S(z) = \overline{z}, \ \ z \in \partial \Omega.$$ The anticonformal local reflection with respect to $\partial
\Omega$ is then the map $z \mapsto \overline{S(z)}$; for this reason $S(z)$ is called the [*Schwarz function*]{} of the real analytic curve $\partial \Omega$, introduced earlier in this text. Let $\omega$ be a relatively compact subdomain of $\Omega$, with smooth boundary, too, and such that the Schwarz function $S(z)$ is defined on a neighborhood of $\Omega \setminus \omega$. A formal use of Stokes’ Theorem yields: $$1- E_\Omega(z,w) = \frac{1}{4 \pi^2} \int_{\partial \Omega}
\int_{\partial \Omega} H_\Omega(u,v) \frac{\overline{u}du}{u-z}
\frac{v d \overline{v}}{\overline{v}-\overline{w}} =$$ $$\frac{1}{4 \pi^2} \int_{\partial \omega}
\int_{\partial \omega} H_\Omega(u,v) \frac{\overline{u}du}{u-z}
\frac{v d \overline{v}}{\overline{v}-\overline{w}}.$$
But the latter integral is analytic/antianalytic for $z,w \in
\overline{\omega}^c$. A little more work with the above Cauchy integrals leads to the following remarkable formula for the analytic extension of $E_\Omega(z,w)$ from $z,w \in
\overline{\Omega}^c$ to $z,w \in \overline{\omega}^c$: $$F(z,w) = \cases{ E(z,w), \ \ z,w \in \Omega^c, \cr
(z-\overline{S(w)})(S(z) - \overline{w})
H_\Omega(z,w), \ \ z,w \in \Omega \setminus
\overline{\omega}. \cr}$$ The study outlined above of the analytic continuation phenomenon of the exponential transform $E_\Omega(z,w)$ led to a proof of a priori regularity of boundaries of domains which admit analytic continuation of their Cauchy transform. The most general result of this type was obtained by different means by Sakai. We simply state the result, giving in this way a little more insight into the proof of the regularity of the boundaries of quadrature domains.
[*[Let $\Omega$ be a bounded planar domain with the property that its Cauchy transform $$\hat{\chi}_\Omega
(z) = \frac{-1}{\pi} \int_\Omega \frac{dA(w)}{w-z}, \ \ z \in
\overline{\Omega}^c$$ extends analytically across $\partial
\Omega$. Then the boundary $\partial \Omega$ is real analytic.]{}*]{}
Moreover, Sakai has classified the possible singular points of the boundary of such a domain. For instance angles not equal to $0$ or $\pi$ cannot occur on the boundary.
Semi-normal operators
---------------------
A normal operator is modelled via the spectral theorem as multiplication by the complex variable on a vector valued Lebesgue $L^2$-space. The interplay between measure theory and the structure of normal operators is well known and widely used in applications. One step further, there are by now well understood functional models, and a complete classification for classes of close to normal operators. We record below a few aspects of the theory of semi-normal operators with trace class self-commutators. They will be serve as Hilbert space counterparts for the study of moving boundaries in two dimensions. The reader is advised to consult the monographs [@MP; @Xia] for full details.
Let $H$ be a separable, complex Hilbert space and let $T \in
\mathcal L(H)$ be a linear bounded operator. We assume that the self-commutator $[T^\ast , T] = T^\ast T - T T^\ast$ is trace-class, and call $T$ semi-normal. If $[T^\ast , T] \geq 0,$ then T is called [*hypo-normal*]{}. For a pair of polynomials $p(z,\overline{z}), q(z,\overline{z})$ one can choose (at random) an ordering in the functional calculus $p(T,T^\ast), q(T,T^\ast)$, for instance putting all adjoins to the left of all other monomials. The functional $$(p,q) \rightarrow {\rm trace} [p(T,T^\ast), q(T,T^\ast)]$$ is then well-defined, independent of the ordering in the functional calculus, and possesses the algebraic identities of the Jacobian $\frac{ \partial(p,q)}{\partial(\overline{z}, z)}$. A direct (algebraic) reasoning will imply the existence of a distribution $u_T \in \mathcal D'(\mathbb C) $ satisfying $${\rm trace} [p(T,T^\ast), q(T,T^\ast)] = u_T \left [\frac{
\partial(p,q)}{\partial(\overline{z}, z)} \right ],$$ see [@HH]. The distribution $u_T$ exists in any number of variables (that is for tuples of self-adjoint operators subject to a trace class multi-commutator condition) and it is known as the [*Helton-Howe functional*]{}.
Dimension two is special because of a theorem of J. D. Pincus which asserts that $u_T = \frac{1}{\pi} g_T \,{\rm dA}$, that is $u_T$ is given by an integrable function function $g_T$, called the [*principal function*]{} of the operator $T$, see [@Pincus; @CareyPincus].
The analogy between the principal function and the phase shift (the density of the measure appearing in Markov’s moment problem in one variable) is worth mentioning in more detail. More precisely, if $B = A- K$ is a trace-class, self-adjoint perturbation of a bounded self-adjoint operator $A \in L(H)$, then for every polynomial $p(z)$, Krein’s [*trace formula*]{} holds: $${\rm tr}[ p(B) - p(A)] = \int_{\mathbb{R}} p'(t) f_{A,B}(t) dt,$$ where $f_{A,B}$ is the corresponding phase-shift function, [@Krein1953]. It is exactly this link between Hilbert space operations and functional expressions which bring the two scenarios very close. Taking one step further, exactly as in the one variable case, the moments of the principal function can be interpreted in terms of the Hilbert space realization, as follows: $$m k \int z^{m-1} \overline{z}^{k-1} g_T(z) {\rm dA} =$$ $${\rm
trace} [T^{\ast k}, T^m], \ \ k,m \geq 1.$$ In general, the principal function can be regarded as a generalized Fredholm index of $T$, that is, when the left hand side below is well defined, we have $${\rm ind} (T-\lambda) = - g_T(\lambda).$$ Moreover $g_T$ enjoys the functoriality properties of the index, and it is obviously invariant under trace class perturbations of $T$. Moreover, in the case of a fully non-normal operator $T$, $${\rm supp} g_T = \sigma (T),$$ and various parts of the spectrum $\sigma(T)$ can be interpreted in terms of the behavior of $g_T$, see for details [@MP].
To give a simple, yet non-trivial, example we proceed as follows. Let $\Omega$ be a planar domain bounded by a smooth Jordan curve $\Gamma$. Let $H^2(\Gamma)$ be the closure of complex polynomials in the space $L^2(\Gamma, ds)$, where $ds$ stands for the arc length measure along $\Gamma$ (the so-called [*Hardy space*]{} attached to $\Gamma$). The elements of $H^2(\Gamma)$ extend analytically to $\Omega$. The multiplication operator by the complex variable, $T_z f = z f, \ \ f \in H^2(\Gamma),$ is obviously linear and bounded. The regularity assumption on $\Gamma$ implies that the commutator $[T_z, T_z^\ast]$ is trace class. Moreover, the associated principal function is the characteristic function of $\Omega$, so that the trace formula above becomes: $${\rm trace} [p(T_z,T_z^\ast), q(T_z,T_z^\ast)] =
\frac{1}{\pi} \int_\Omega \frac{
\partial(p,q)}{\partial(\overline{z}, z)} {\rm dA}, \ \ p,q \in
\mathbb C[z,\overline{z}].$$ See for details [@MP; @Xia].
A second, more interesting (generic example this time) can be constructed as follows. Let $u(t), v(t)$ be real valued, bounded continuous functions on the interval $[0,1]$. Consider the singular integral operator, acting on the Lebesgue space $L^2([0,1],dt)$ by the formula: $$(Tf)(t) = tf(t) +i[u(t)f(t) + \frac{1}{\pi} \int_0^1 \frac{
v(t)v(s) f(s) ds}{s-t}.$$ Then it is easy to see that the self-commutator $[T^\ast,T]$ is rank one. The principal function $g_T$ will be in this case the characteristic function of the closure of the domain $G$ given by the constraints $$G = \{ (x,y) \in {\mathbb R}^2; \ \ |y-u(x)| \leq v(x)^2, \ x\in
[0,1]\}.$$ Based on a refinement of this example, in general every [*hyponormal operator*]{} with trace class self-commutator can be represented by such a singular integral model, with matrix valued functions $u,v$, acting on a direct integral of Hilbert spaces over $[0,1]$; in which case the principal function relates directly to Krein’s phase shift, by the following remarkable formula due to Pincus [@Pincus]: $$g_T(x,y) = f_{u(x)-v(x)^\ast v(x), u(x)+v(x)^\ast v(x)}(y).$$
The case of rank-one self-commutators is singled out in the following key classification result:
[*There exists a bijective correspondence $T
\mapsto g_T$ between irreducible hyponormal operators $T$, with rank-one self-commutator, and bounded measurable functions with compact support in the complex plane.*]{}
An invariant formula, relating the moments of the principal function $g$ to the Hilbert space operator $T, \ [T^\ast,T] = \xi
\langle \cdot, \xi \rangle, $, satisfying $g_T = g, a.e.$ is furnished by the determinantal formula: $${\rm exp} (-\frac{1}{\pi} \int_{\mathbb{C}} \frac{ g(\zeta) dA(\zeta)}
{(\zeta - z)(\overline{\zeta}-\overline{w})}) = {\rm det} [
(T^\ast - \overline{w})^{-1}(T-z)(T^\ast -
\overline{w})(T-z)^{-1}]=$$ $$1 - \langle (T^\ast -
\overline{w})^{-1}\xi, (T^\ast - \overline{z})^{-1}\xi \rangle, \
\ z,w \in {\rm supp} (g)^c.$$ This formula explains the positivity property of the exponential transform, alluded to in the previous section.
The bijective correspondence between classes $g \in L^\infty_{\rm
comp} ({\mathbb{C}}), \ 0 \leq g \leq 1$ and irreducible operators $T$ with rank-one self-commutator was exploited in [@Put96; @Put98] for solving the $L$-problem of moments in two variables. The theory of the principal function has inspired and played a basic role in the foundations of modern non-commutative geometry (specifically the cyclic cohomology of operator algebras) and non-commutative probability.
We have to stress the fact that the above bijective correspondence between“shade functions" $g_T$ and irreducible hyponormal operators $T$ with rank-one self-commutator can in principle transfer [*any*]{} dynamic $g(t)$ into a Hilbert space operator dynamic $T(t)$. However, the details of the evolution law of $T(t)$ even in the case of elliptic growth are not trivial, nor make the integration simpler. We will see some relevant low degree examples in the next section.
### Applications: Laplacian growth
To give a single abstract illustration, consider a growing family of bounded planar domains $D(t)$ with smooth boundary: $$D(t) \subset D(s),\ \ {\rm whenever}\ t<s.$$ The evolution of the exponential transforms $$E_{D(t)}(z,w) = \exp \left [\frac{-1}{\pi} \int_{D(t)}
\frac{dA(\zeta)}{(\zeta-z)(\bar \zeta -\bar w)} \right ] ,$$ is governed by the differential equation (in the standard vector calculus notation) $$\frac{d}{dt} E_{D(t)}(z,w) = \frac{-1}{\pi} E_{D(t)}(z,w) \int_{\partial D(t)} \frac{V_n
d\ell(\zeta)}{(\zeta-z)(\overline{\zeta}-\overline{w})}.$$ Any evolution law at the level of the pair $(T(t),\xi(t))$ will have the form $$\frac{d}{dt} E_{D(t)}(z,w) = \langle (T^\ast(t) - \overline{w})^{-1})
T^\ast (t) (T^\ast(t) - \overline{w})^{-1}) \xi(t),
(T^\ast(t) - \overline{z})^{-1}\xi(t)\rangle -$$ $$\langle (T^\ast(t) - \overline{w})^{-1}\xi'(t),(T^\ast(t) -
\overline{z})^{-1}\xi(t)\rangle +$$ $$\langle (T^\ast(t) - \overline{w})^{-1}\xi(t), (T^\ast(t) -
\overline{z})^{-1} T^\ast (t) (T^\ast(t) -
\overline{z})^{-1}\xi(t)\rangle -$$ $$\langle (T^\ast(t) - \overline{w})^{-1})\xi(t),
(T^\ast(t) - \overline{z})^{-1})\xi'(t)\rangle.$$ A series of simplification in the case of elliptic growth are immediate: for instance $\| \xi(t)\|$ is proportional to the area of $D(t)$, whence we can choose the vector of the form $$\xi(t) = t \xi(0).$$ Second, the higher harmonic moments are preserved by the evolution, whence the Cauchy transform/resolvent $$\frac{d}{dt} \pi \langle \xi(t), (T^\ast(t) -
\overline{z})^{-1}\xi(t)\rangle = \frac{d}{dt} \int_{D(t)}
\frac{dA(\zeta)}
{\zeta - z} = - \frac{c}{z},$$ gives full information about the first row and first column in the matrix representation of $T^\ast(t)$ in the basis obtained by orthonormalizing the sequence $\xi(t), T^\ast(t)\xi(t), T^{\ast 2}\xi(t),...$. The reader can consult the article \[122\] for more details about computations related to the above ones.
Linear analysis of quadrature domains
-------------------------------------
If we would infer from the one-variable picture a good class of extremal domains for Markov’s $L$-problem in two variables we would choose the disjoint unions of disks, as immediate analogs of disjoint unions of intervals. In reality, the nature of the complex plane is much more complicated, but again, fortunately for our survey, the class of quadrature domains plays the role of extremal solutions in two real dimensions.
Recall from our previous sections that a bounded domain $\Omega$ of the complex plane is called a [*quadrature domain*]{} (always henceforth for analytic functions) if there exists a finite set of points $a_1, a_2, \ldots, a_d \in \Omega$, and real weights $c_1, c_2, \ldots, c_d$, with the property: $$\int_\Omega f(z) dA(z) = c_1 f(a_1) + c_2 f(a_2) + \ldots +
c_d f(a_d),\ \ f \in AL^1(\Omega)$$ where the latter denotes the space of all integrable analytic functions in $\Omega$. In case some of the above points coincide, a derivative of $f$ can correspondingly be evaluated.
Let $\Omega$ be a bounded planar domain with moments $$a_{mn} = a_{mn}(\Omega) = \int_\Omega z^m \overline{z}^n dA(z),
\ \ m,n \geq 0.$$ The exponential transform produces the sequence of numbers $b_{mn} = b_{mn}(\Omega), \ \ m,n \geq 0.$ Let $T$ denote the irreducible hyponormal operator with rank-one self-commutator $[T^\ast,T] = \xi \langle \cdot, \xi \rangle.$ In virtue of the factorization (\[factorE\]), $$b_{mn} = \langle T^{\ast m}\xi, T^{\ast n}\xi \rangle, \ \ m,n
\geq 0.$$ Hence the matrix $(b_{mn})_{m,n=0}^\infty$ turns out to be non-negative definite. The following result identifies a part of the extremal solutions of the $L$-problem of moments as the class of quadrature domains:
[*A bounded planar domain $\Omega$ is a quadrature domain if and only if there exists a positive integer $d \geq 1$ with the property*]{} $ {\rm det} (b_{mn}(\Omega))_{m,n=0}^d = 0.$
For a proof see [@Put96]. The vanishing condition in the statement is equivalent to the fact that the span $H_d$ of the vectors $\xi, T^\ast \xi, T^{\ast 2}\xi, \ldots$ is finite dimensional (in the Hilbert space where the associated hyponormal operator $T$ acts). Thus, if $\Omega$ is a quadrature domain with corresponding hyponormal operator $T$, and $T_d$ is the compression of $T$ to the $d$-dimensional subspace $H_d$, then: $$E_\Omega(z,w) = 1 - \langle (T_d^\ast - \overline{w})^{-1}\xi, (T_d^\ast - \overline{z})^{-1}\xi
\rangle, \ \ z,w \in \overline{\Omega}^c.$$ In particular this proves that the exponential transform of a quadrature domain is a rational function. As a matter of fact a more precise statement can easily be deduced:
[*Let $\Omega$ be the quadrature domain defined above. Then $$E_\Omega(z,w) = \frac{Q(z,w)}{P(z) \overline{P(w)}}, \ \ z,w
\in \overline{\Omega}^c.$$*]{}
This result offers an efficient characterization of quadrature domains in terms of a finite set of their moments (see the reconstruction section below) and it opens a natural correspondence between quadrature domains and certain classes of finite rank matrices. We only describe a few results in this direction. For more details see [@GP98; @GP00; @Put96].
In the conditions of the above result, let $\Omega$ be a quadrature domain with associated hyponormal operator $T$; let $H_0 = \bigvee_{k\geq 0} T^{\ast k} \xi$ and let $p$ denote the orthogonal projection of the Hilbert space $H$ (where $T$ acts) onto $H_0$. Denote $C_0 = p T p$ (the compression of $T$ to the $d$-dimensional space $H_0$) and $D_0^2 = [T^\ast, T]$. Then the operator $T$ has a two block-diagonal structure: $$T= \left( \begin{array} {ccccc}
{C_0}&0&0&0&\ldots\\
{D_1}&{C_1}&0&0&\ldots\\
0&{D_2}&{C_2}&0&\ldots\\
0&0&{D_3}&{C_3}&\ldots\\
\vdots& & \vdots& & \ddots\\
\end{array} \right),$$ where the entries are all $d \times d$ matrices, recurrently defined by the system of equations: $$\left\{ \begin{array}{c}
[{{C_{k}}^\ast}, {C_{k}}]
+{{D_{k+1}}^\ast}{D_{k+1}}= {D_k}{{D_k}^\ast}\\
{{C_{k+1}}^\ast}{D_{k+1}}
= {D_{k+1}}{{C_k}^\ast},\hspace{.2in} k \geq 0.\\
\end{array} \right.$$ Note that $D_k >0$ for all $k$. This decomposition has an array of consequences:
1. The spectrum of $C_0$ coincides with the quadrature nodes of $\Omega$;
2. $\Omega = \{ z; \| (C_0^\ast - \overline{z})^{-1}\xi \| >1\}
$ (up to a finite set);
3. The quadrature identity becomes $$\int_\Omega f(z) {\rm dA}(z) = \pi \langle f(C_0)\xi, \xi\rangle,$$ for $f$ analytic in a neighborhood of $\overline{\Omega}$;
4. The Schwarz function of $\Omega$ is $$S(z) = \overline{z} - \langle \xi, (C_0^\ast -
\overline{z})^{-1}\xi\rangle + \langle \xi, (T^\ast -
\overline{z})^{-1}\xi \rangle ,$$ where $ z \in \Omega$.
To give the simplest and most important example, let $\Omega =
\mathbf D$ be the unit disk (which is a quadrature domain of order one) . Then the associated operator is the unilateral shift $T = T_z$ acting on the Hardy space $H^2(\partial \mathbf D)$. Denoting by $z^n$ the orthonormal basis of this space we have $T
z^n = z^{n+1}, \ \ n \geq 0,$ and $[T^\ast , T] = 1\langle \cdot,
1\rangle$ is the projection onto the first coordinate $1 = z^0$. The space $H_0$ is one dimensional and $C_0 = 0$. This will propagate to $C_k =0$ and $D_k =1$ for all $k$. Thus the matricial decomposition of $T$ becomes the familiar realization of the shift as an infinite Jordan block.
In view of the linear algebra realization outlined in the preceding section we obtain more information about the defining equation of the quadrature domain. For instance: $$\frac{Q(z,\overline{w})}{P(z)\overline{P(w)}} = 1 - \langle
(C_0^\ast - \overline{w})^{-1}\xi, (C_0^\ast -
\overline{z})^{-1}\xi \rangle,$$ which yields $$Q(z,{z}) = |P(z)|^2 - \sum_{k=0}^{d-1} |Q_k(z)|^2,$$ where $Q_k$ is a polynomial of degree $k$ in $z$, see [@GP00].
Thus the exponential transform of a quadrature domain contains explicitly the irreducible polynomial $Q$ which defines the boundary and the polynomial $P$ which vanishes at the quadrature nodes. By putting together all these remarks we obtain a strikingly similar picture to that of a single variable. More specifically, if $\Omega$ is a quadrature domain with $d$ nodes, as given above, and associated hyponormal operator $T$, then: $$E_\Omega(z,w) = \frac{Q(z,w)}{P(z) \overline{P(w)}} = 1 - \langle (T_d^\ast - \overline{w})^{-1}\xi, (T_d^\ast - \overline{z})^{-1}\xi
\rangle =$$ $$\frac{1}{\pi^2} \sum_{i,j=1}^d
H_\Omega(a_i,a_j) \frac{c_i}{a_i-z} \frac{
\overline{c_j}}{\overline{a_j}-\overline{w}}, \ \ z,w \in
\overline{\Omega}^c.$$
In particular we infer, assuming that all nodes are simple: $$-\pi^2 \frac{Q(a_i,a_j)}{P'(a_i) \overline{P'(a_j)}} = c_i
\overline{c_j} H_\Omega(a_i, \overline{a_j}), \ \ 1 \leq i,j \leq
d.$$ For details see [@Put96; @GP00].
The interplay between these additive, multiplicative and Hilbert space decompositions of the exponential transform gives an exact reconstruction algorithm of a quadrature domain from its moments. The next section will be devoted to this algorithm.
Before ending the present section we consider an illustration of the above formulas. Let $\Omega = \cup_{i=1}^d D(a_i, r_i)$ be a union of $d$ pairwise disjoint disks. This is a quadrature domain with data: $$P(z) = (z-a_1) \ldots (z-a_d),$$ $$Q(z,w) = [(z-a_1)(\overline{w} - \overline{a_1}) - r_1^2]
\ldots [(z-a_d)(\overline{w} - \overline{a_d}) - r_d^2].$$ The associated matrix $T_d$ is also computable, involving a sequence of square roots of matrices, but we do not need here its precise form. Whence the exponential transform is, for large values of $|z|, |w|$: $$E_\Omega(z,w) = \prod_{i=1}^d [1- \frac{r_i^2}{(z-a_i)(\overline{w} -
\overline{a_i})}] = 1+ \sum_{i,j=1}^d \frac{Q(a_i,\overline{a_j})}{P'(a_i) \overline{P'(a_j)}}
\frac{r_i}{a_i-z} \frac{ r_j}{\overline{a_j}-\overline{w}}.$$
The essential positive definiteness of the exponential transform of an arbitrary domain can be deduced, via an approximation argument, from the positivity of the matrix $(-Q(a_i,
\overline{a_j}))_{i,j=1}^d$, where $Q$ is the defining equation of a disjoint union of disks. We note that $(-Q(a_i,
\overline{a_j}))_{i,j=1}^d \geq 0$ is only a necessary condition for the disks $D(a_i, r_i), \ 1 \leq i \leq d,$ to be disjoint. Exact computations for $d=2$ immediately show that this matrix can remain positive definite even the two disks overlap a little. However, if two disks overlap, then, by adding an external disk, even far away, this prevents the new $3 \times 3$ matrix to be positive definite.
We end this section with two examples, covering the totality of quadrature domains of order two.
[**Quadrature domains with a double node.**]{} Let $z={w^2}+bw$ be the conformal mapping of the disk $|w| <1$, where $b \geq 2$. Then $z$ describes a quadrature domain $\Omega$ of order $2$, whose boundary has the equation:$$Q(z,\overline{z})=|z{|^4}-(2+{b^2})|z{|^2}-{b^2}z-{b^2}\overline{z}+1-{b^2}=0.$$
The Schwarz function of $\Omega$ has a double pole at $z=0$, whence the associated $2 \times 2$-matrix $C_0$ is nilpotent. Moreover, we know that:$$|z{|^4} \|
({C_0^\ast}-\overline{z}{)^{-1}}\xi {\|^2} =
|z{|^4}-P(z,\overline{z}).$$ Therefore $$\|
({C_0^\ast}+\overline{z}) \xi{\|^2}=
(2+{b^2})|z{|^2}+{b^2}z+{b^2}\overline{z} +{b^2}-1,$$ or equivalently: $\|\xi{\|^2}=2+{b^2}, \langle {C_0^\ast}\xi, \xi
\rangle= {b^2}$ and $\| {C_0^\ast}\xi {\|^2}={b^2}-1.$
Consequently the linear data of the quadrature domain $\Omega$ are:$${C_0^\ast}=\left( \begin{array}{cc}
0& \frac{{b^2}-1}{({b^2}-2{)^{1/2}}}\\
0&0
\end{array}\right) ,\hspace{.3in}
\xi=\left( \begin{array}{c}
\frac{b^2}{({b^2}-1{)^{1/2}}}\\
(\frac{{b^2}-2}{{b^2}-1}{)^{1/2}}
\end{array} \right) .$$
[**Quadrature domains with two distinct nodes.**]{} Assume that the nodes are fixed at $\pm 1$. Hence $P(z) = z^2 -1$. The defining equation of the quadrature domain $\Omega$ of order two with these nodes is: $$Q(z, \overline{z}) = (|z+1|^2 - r^2)(|z-1|^2 -r^2)
-c,$$ where $r$ is a positive constant and $c \geq 0$ is chosen so that either $\Omega$ is a union of two disjoint open disks (in which case $c=0$), or $Q(0,0)=0$, see [@G2]. A short computation yields: $$Q(z,\overline{z}) = z^2
\overline{z}^2 - 2r z \overline{z} -z^2 - \overline{z}^2
+\alpha(r),$$ where $$\alpha(r) = \left\{ \begin{array}{lcc}
(1-r^2)^2 , & & r<1\\
0 ,& & r \geq 1.
\end{array}
\right.$$
One step further, we can identify the linear data from the identity: $$|P(z)|^2 (1- \| (C_0^\ast -\overline{z})^{-1} \xi \|^2) =
Q(z,\overline{z}).$$ Consequently, $$\xi = \left( \begin{array}{c}
\sqrt{2} r \\
0
\end{array} \right) ,
\
C_0^\ast = \left( \begin{array}{cc}
0 & \frac{\sqrt{2} r}{\sqrt{1-\alpha(r)}} \\
\frac{\sqrt{1-\alpha(r)}}{\sqrt{2}r} & 0
\end{array}
\right) .$$
This simple computation illustrates the fact that, although the process is affine in $r$, the linear data of the growing domains have discontinuous derivatives at the exact moment when the connectivity changes.
Signed measures, instability, uniqueness
----------------------------------------
Contrary to the uniqueness of a quadrature domain for subharmonic functions with a prescribed quadrature measure, quadrature domains for harmonic or analytic functions are not determined by the quadrature nodes and weights. This is an intriguing global phenomenon which has haunted mathematicians for many decades. we briefly record below some significant discoveries in this direction.
Consider quadrature domains for harmonic test functions and real-valued measures (\[mu\]). As to the relationship between the geometry of $\Omega$ and the location of ${\rm supp\,}\mu$ there are then drastic differences between the cases of having all $c_j>0$ respectively having no restrictions on the signs of $c_j$. This is clearly demonstrated in the following theorem due to M. Sakai [@Sakai98], [@Sakai99a]. The second part of the theorem is discussed (and proved) in some other forms also in [@G1], [@Gustafsson96a], [@Bell03], [@Bell04], [@Zabrodin], for example:
*Let $r$ and $R$ be positive numbers, $R\geq
2r$. Consider measures $\mu$ of the form (\[mu\]) with $c_j$ real and related to $r$ and $R$ by $$\label{supp}
{\rm supp\,}\mu \subset B(0,r),$$ $$\label{R}
\sum_{j=1}^n c_j = \pi R^2.$$*
1. If $\mu\geq 0$, then any quadrature domain $\Omega$ for harmonic functions for $\mu$ is also a quadrature domain for subharmonic functions. Hence the previous result applies, and in addition $$B(0,R-r)\subset \Omega\subset B(0,R+r).$$
2. With $\mu$ not necessarily $\geq 0$, and with no restrictions on $\sum_{j=1}^n |c_j|$ and $n$, any bounded domain containing $B(0,r)$ and having area $\pi R^2$ can be uniformly approximated by quadrature domains for harmonic functions for measures $\mu$ satisfying (\[supp\]), (\[R\]).
With $\mu$ a signed measure of the form (\[mu\]) we still have $\sum_{j=1}^n c_j =|\Omega|$, but $\sum_{j=1}^n |c_j|$ may be much larger. In view of the theorem, the ratio $$\rho=\frac{\sum_{j=1}^n c_j}{\sum_{j=1}^n |c_j|} =\frac{\int
d\mu}{\int|d\mu|}$$ ($0<\rho \leq 1$) might give an indication of how strong is the coupling between the geometry of ${\rm supp\,}\mu$ and the geometry of $\Omega$.
As mentioned, a quadrature domain for harmonic functions is not always uniquely determined by its measure $\mu$. Still there is uniqueness at the infinitesimal level: if $$\label{qi}
\sum_{j=1}^n c_j \varphi (a_j) = \int_\Omega \varphi\, {\rm dA}$$ and (for example) the $a_j$ are kept fixed, then one can always increase the $c_j$ (indefinitely) and get a unique evolution of $\Omega$ (Hele-Shaw evolution). If $\partial\Omega$ has no singularities then one can also decrease the $c_j$ slightly and have a unique evolution (backward Hele-Shaw, which is ill-posed). Thus it makes sense to write $$\Omega = \Omega (c_1, \dots, c_n)$$ for $c_j$ in some interval around the original values. Note however that decreasing the $c_j$ makes the ratio $\rho$ decrease, indicating a loss of control or stability.
In the simply connected case, $\Omega$ will be the image of the unit disc ${\mathbf D}$ under a rational conformal map $f=f_{(c_1,
\dots ,c_n)}: {\mathbf D}\to \Omega(c_1, \dots ,c_n)$. This rational function is simply the conformal pull-back of the meromorphic function $(z,S(z))$ on the Schottky double of $\Omega$ to the Schottky double of ${\mathbf D}$, the latter being identified with the Riemann sphere. It follows that the poles of $f$ are the mirror points (with respect to the unit circle) of the points $f^{-1} (a_j)$. When the $c_j$ increase then the $|f^{-1}
(a_j)|$ decrease (this follows by an application of Schwarz’ lemma to $f^{-1}_{{\rm larger\,} c_j} \circ f_{{\rm original\,} c_j}$), hence the poles of $f$ move away from the unit circle. Conversely, the poles of $f$ approach the unit circle as the $c_j$ decrease, also indicating a loss of stability.
For decreasing $c_j$ the evolution $\Omega(c_1, \dots ,c_n)$ always breaks down by singularity development of $\partial\Omega$ or $\partial\Omega$ reaching some of the points $a_j$ (see e.g. [@Hohlov-Howison94], [@Gustafsson-Vasiliev06]) before $\Omega$ is empty, except in the case that $\Omega(c_1, \dots
,c_n)$ is a quadrature domain for subharmonic functions. In the latter case the $c_j$ (necessarily positive) can be decreased down to zero, and $\Omega$ will be empty in the limit $c_1=\dots
=c_n=0$. However, it may happen that $\Omega(c_1, \dots ,c_n)$ breaks up into components under the evolution.
Assume now that $\Omega$ is simply connected. Then the analytic and harmonic functions are equivalent as test classes for (\[qi\]). In the limit case that all the points $a_j$ coincide, say $a_1=\dots=a_n=0$, then (\[qi\]) corresponds to $$\label{qi0}
\sum_{j=1}^n c_j\varphi^{(j-1)} (0) =\int_\Omega \varphi\, {\rm
dA}$$ for $\varphi$ analytic. The $c_j$ (allowed to be complex) now have a slightly different meaning than before. In fact, they are essentially the analytic moments of $\Omega$: $$c_j =\frac{1}{(j-1) !} \int_\Omega z^{j-1} dA
\quad(j= 1,\dots , n).$$ The higher order moments vanish, and the conformal map $f=f_{(c_1,
\dots ,c_n)}: {\mathbf D}\to \Omega(c_1, \dots ,c_n)$ (normalized by $f(0)=0$, $f'(0)>0$) is a polynomial of degree $n$. A precise form of the local bijectivity of the map $(c_1, \dots, c_n)\mapsto
\Omega (c_1, \dots, c_n)$ has been established by O. Kouznetsova and V. Tkachev [@Kouznetsova-Tkachev2004], [@Tkachev2005], who proved an explicit formula for the (nonzero) Jacobi determinant of the map from the coefficients of $f$ to the moments $(c_1,\dots, c_n))$. This formula was conjectured (and proved in some special cases) by C. Ullemar [@Ullemar80].
On the global level, it does not seem to be known whether (\[qi0\]), or (\[qi\]), with a given left member, can hold for two different simply connected domains and all analytic $\varphi$. Leaving the realm of quadrature domains, an explicit example of two different simply connected domains having the same analytic moments has been given by M. Sakai [@Sakai78]. The idea of the example is that a disc and a concentric annulus of the same area have equal moments. If the disc and annulus are not concentric, then the union of them (if disjoint) will have the same moments as the domain obtained by interchanging their roles. Arranging everything carefully, with removing and adding some common parts, two different Jordan domains having equal analytic moments can be obtained. Similar examples were known earlier by A. Celmin[s]{} [@Celmins57], and probably even by P. S. Novikov. On the positive side, a classical theorem of Novikov [@Novikoff38] asserts that domains which are starshaped with respect to one and the same point are uniquely determined by their moments. See [@Zalcman87] for further discussions.
Returning now to quadrature domains, there is definitely no uniqueness for harmonic and analytic test classes if multiply connected domains are allowed. If $\Omega$ has connectivity $m+1$ ($m\geq 1$), i.e., has $m$ “holes”, then there is generically an $m$-parameter family $\Omega (t_1,\dots,t_m)$ of domains such that $\Omega (0,\dots,0)=\Omega$ and $$\frac{\partial}{\partial t_j} \int_{\Omega (t_1,\dots,t_m)}
\varphi\,{\rm dA} =0 \quad (j=1,\dots ,m)$$ for every $\varphi$ analytic in a neighborhood of the domains. These deformations are Hele-Shaw evolutions, driven not by Green functions but by “harmonic measures”, i.e., regular harmonic functions which take (different) constant boundary values on the components of $\partial\Omega$.
It follows that multiply connected quadrature domains for analytic functions for a given $\mu$ occur in continuous families. It even turns out [@Gustafsson90], [@Sjodin04] that [*any two*]{} algebraic domains for the same $\mu$ can be deformed into each other through families as above. Thus there is a kind of uniqueness at a higher level: given any $\mu$ there is at most one connected family of algebraic domains belonging to it.
For harmonic quadrature domains there are no such continuous families (choosing $\varphi (z) =\log |z-a|$ in (\[qi\]) with $a\in \mathbb{C}\setminus\overline{\Omega}$ in the holes stops them), but one can still construct examples with a discrete set of different domains for the same $\mu$. It is for example possible to imitate the example with a disc and an annulus with quadrature domains for measures $\mu$ of the form (\[mu\]), with $a_j=e^{2\pi j/n}$ ($n\geq 3$) and $c_1=\dots =c_n=c>0$ suitably chosen. However, it seems very difficult to imitate the full Sakai construction, with “removing and adding some common parts”, in the context of quadrature domains. Therefore it is not at all easy to construct different simply connected quadrature domains for the same $\mu$.
We end this section with the simplest example of a continuous class of quadrature domains with the same quadrature data.
[**Three points, non-simply connected quadrature domains and the non-uniqueness phenomenon.**]{} Quadrature domains (for analytic functions) with at most two nodes, as in the above examples, are uniquely determined by their quadrature data and are simply connected. For three nodes and more it is no longer so. The following example, taken from [@G2], with three nodes and symmetry under rotations by $2\pi/3$, illustrates the general situation quite well. More details on the present example are given in [@G2], and similar examples with more nodes are studied in [@Crowdy-Marshall03].
Let the quadrature nodes and weights be $a_j=\omega^j$ and $c_j=\pi r^2$ respectively ($j=1,2,3$), where $\omega=e^{2\pi
i/3}$ and where $r>0$ is a parameter. Considering first the strongest form of quadrature property, namely for subharmonic functions, as in (\[subharmqd\]), (\[mu\]), the situation is in principle easy: $\Omega$ is for any given $r>0$ uniquely determined up to nullsets and can be viewed as a swept out version of the quadrature measure $\mu=\sum_{j=1}^3 c_j \delta_{a_j}$ or as the union of the discs $B(a_j,r)$ with (possible) multiple coverings smashed out.
For $0<r\leq \frac{\sqrt{3}}{2}$ the above discs are disjoint, hence $\Omega =\cup_{j=1}^3 B(a_j,r)$. For $r$ larger than $\frac{\sqrt{3}}{2}$ but smaller than a certain critical value $r_0$ (which seems to be difficult to determine explicitly) $\Omega$ is doubly connected with a hole containing the origin, while for $r\geq r_0$ the hole will be filled in so that $\Omega$ is a simply connected domain. The above quadrature domains (or open sets) are actually uniquely determined even within nullsets, except in the case $r=r_0$ when both $\Omega$ and $\Omega\setminus
\{ 0\}$ satisfy (\[subharmqd\]).
Consider next the general class of quadrature domains for analytic functions (algebraic domains). For $0<r\leq \frac{\sqrt{3}}{2}$ only the disjoint discs qualify, as before. However, for any $r>\frac{\sqrt{3}}{2}$ there is a whole one-parameter family of domains $\Omega$ satisfying the quadrature identity for analytic $\varphi$. These are defined by the polynomials $$\label{polynomial}
Q(z,\overline{z}) = z^3 \overline{z}^3 - z^3 -\overline{z}^3
-3r^2z^2\overline{z}^2-$$ $$3\tau (\tau^3 - 2r^2\tau +1)z\overline{z} +\tau^3 (2\tau^3 -
3r^2\tau +1),$$ where $\tau>0$ is a free parameter, independent of the quadrature data. When completed as to nullsets, the quadrature domains in question are more precisely $$\Omega (r,\tau)= {\rm int clos}\{z\in \mathbb{C}:
Q(z,\overline{z})<0\}.$$
The interpretation of the parameter $\tau$ is that on each radius $\{z=t\omega^{j+\frac{1}{2}}: t>0 \}$, $j=1,2,3$, there is exactly one singular point of the algebraic curve $Q(z,\overline{z})=0$, and $\tau=|z|$ for that point. This singular point is either a cusp on $\partial\Omega$ or an isolated point of $Q(z,\overline{z})=0$, a so-called [*special point*]{}. Special points are those points $a\in\Omega$ for which the quadrature identity admits the (integrable) meromorphic function $\varphi (z)=\frac{1}{z-a}$. Equivalently, $\Omega\setminus\{a\}$ remains to be a quadrature domain for integrable analytic functions.
For $\frac{\sqrt{3}}{2}<r<2^{-\frac{1}{6}}$ the quadrature domains for analytic functions are exactly the domains $\Omega(r,\tau)$ (with possible removal of special points) for $\tau$ in an interval $\tau_1(r)\leq \tau\leq\tau_2(r)$, where $\tau_1(r)$, $\tau_2(r)$ satisfy $0<\tau_1(r)<\frac{1}{2}<\tau_2(r)$, and more precisely can be defined as the positive zeros of the polynomial $4\tau^3-4r^2\tau +1$. (see [@G2] for further explanations and proofs). The domains $\Omega(r,\tau)$ are doubly connected with a hole containing the origin. When $\tau$ increases the hole shrinks and both boundary components move towards the origin. For $\tau=\tau_2(r)$ there are three cusps on the outer boundary component which stop further shrinking of the hole, and for $\tau=\tau_1(r)$ there are three cusps on the inner boundary component which stop the expansion of the hole.
For exactly one parameter value, $\tau=\tau_{\rm subh}(r)$, $\Omega(r,\tau)$ is a quadrature domain for subharmonic functions (and so also for harmonic functions). This $\tau_{\rm subh}(r)$ can be determined implicitly by evaluating the quadrature identity for $\varphi(z)=\log |z|$, which gives the equation $$\int_{\Omega(r,\tau_{\rm subh}(r))} \log |z| \,{\rm dA}(z)=0.$$
For $r=\frac{\sqrt{3}}{2}$, $\tau_1 (r)=\tau_{2}(r)=\frac{1}{2}$, and as $r$ increases, $\tau_1(r)$ decreases and $\tau_2(r)$ increases. What happens when $r= 2^{-\frac{1}{6}}$ is that for $\Omega(r,\tau_2(r))$, i.e., for the domain with cusps on the outer component, the hole has shrunk to a point (the origin). Hence, for $r=2^{-\frac{1}{6}}$, $\Omega(r,\tau_2(r))$ is simply connected, while $\Omega(r,\tau)$ for $\tau_1(r)\leq
\tau<\tau_2(r)$ remain doubly connected.
For all $\frac{\sqrt{3}}{2}<r\leq 2^{-\frac{1}{6}}$, $\tau_1(r)<\tau_{\rm subh}(r)<\tau_2(r)$ because a subharmonic quadrature domain cannot have the type of cusps which appear for $\tau=\tau_1(r), \tau_2(r)$ (see [@S4], [@S5]). It follows that the critical value $r=r_0$, when $\Omega(r,\tau_{\rm
subh}(r))$ becomes simply connected, is larger that $2^{-\frac{1}{6}}$.
For $r\geq 2^{-\frac{1}{6}}$ the quadrature domains for analytic functions are the domains $\Omega(r,\tau)$ (with possible deletion of special points), with $\tau$ in an interval $\tau_1(r)\leq\tau\leq\tau_3(r)$. Here $\tau_1(r)$ is the same as before (i.e., corresponds to cusps on the inner boundary), while $\tau_3(r)$ is the value of $\tau$ for which the hole at the origin degenerates to just the origin itself (which for $r>
2^{-\frac{1}{6}}$ occurs before cusps have developed on the outer boundary). The origin then is a special point, and one concludes from (\[polynomial\]) that $\tau=\tau_3(r)$ is the smallest positive zero of the polynomial $2\tau^3 -3r^2\tau+1$. For $r=2^{-\frac{1}{6}}$, $\tau_3(r)=\tau_2(r)= 2^{-\frac{2}{3}}$.
For $2^{-\frac{1}{6}}\leq r<r_0$ we have $\tau_1(r)<\tau_{\rm
subh}(r)<\tau_3(r)$, while for $r\geq r_0$, $\tau_{\rm
subh}(r)=\tau_3(r)$. Since $\Omega(r,\tau_3(r))$ is simply connected and is a quadrature domain for analytic functions it is also a quadrature domain for harmonic functions. It follows that in the interval $2^{-\frac{1}{6}}\leq r<r_0$ there are (for each $r$) two different quadrature domains for harmonic functions, namely $\Omega(r,\tau_{\rm subh}(r))$ and $\Omega(r,\tau_{3}(r))$ (doubly respectively simply connected).
In summary, we have for each $r>\frac{\sqrt{3}}{2}$ a one-parameter family of algebraic domains $\Omega(r,\tau)$, for exactly one parameter value ($\tau=\tau_{\rm subh}(r)$) this is a quadrature domain for subharmonic functions, and for each $r$ in a certain interval ($2^{-\frac{1}{6}}\leq r<r_0$) there are two different quadrature domains for harmonic functions ($\Omega(r,\tau_{\rm subh}(r))$ and $\Omega(r,\tau_{3}(r))$).
Other physical applications of the operator theory formulation
==============================================================
The preceding chapters provide a review of the relationships between the theory of normal random matrices, where evolution is defined by increasing the size of the matrix (a discrete time), its continuum (or infinite size) limit - Laplacian growth - and the general theory of semi-normal operators whose spectrum approximates generic domains. The exposition reflects, to some extent, the parallel historical development of the two non-commutative generalizations of Laplacian growth (random matrix theory and semi-normal operator theory). It is quite natural, at this point, to investigate the direct relationships between these two theories. However, this is a task of a magnitude which would require a separate review at the very least. We will therefore contend ourselves with exposing only a few of these relations, via their applications to physical problems.
The first application has to do with refined asymptotic expansions which characterize Laplacian growth in the critical case, before formation of a $(2, 3)$ cusp. As we will see, to obtain this limit, one must take a “double-scaling limit" by fine-tuning two parameters of the random matrix ensemble. Alternatively, this procedure is equivalent to a special choice of Padé approximants in the operator theory approach.
The second application described in this section is a very brief introduction of the notion of free, non-commutative random variables, and its relevance in open problems of strongly interacting quantum models, particularly in the 2D metal-insulator transition and the determination of ground state for 2D spin models. The review concludes with this cursory exposition.
Cusps in Laplacian growth: Painlevé equations
---------------------------------------------
In this section, we exploit the formalism built up to now, in order to address a problem of great significance both at the mathematical and physical levels: what happens when a planar domain evolving under Laplacian growth approaches a generic (2, 3) cusp? We have already seen that a $classical$ solution does not exist, in that no singly-connected domain with uniform density would satisfy the conditions of the problem. However, since we now have alternative formulations of Laplacian growth via the $balayage$ of the uniform measure, we may generalize the problem and ask whether there is $any$ equilibrium measure, dropping the uniformity (and indeed, the two-dimensional support) of the classical solution. By analogy with the 1D situation, we seek a solution in the sense of Saff and Totik, where the support and density of the equilibrium measure are given by the proper weighted limit of orthogonal polynomials. In order to obtain this limit, we must organize the evolution equations of the wavefunction such as to extract the correct scaling limit, for $N \to \infty$.
### Universality in the scaling region at critical points – a conjecture
Detailed analysis of critical Hermitian ensembles indicates that the behavior of orthogonal polynomials in a specific region including the critical point (the [*[scaling region]{}*]{}), upon appropriate scaling of the degree $n$, is essentially independent of the bulk features of the ensemble. This $universality$ property (a common working hypothesis in the physics of critical phenomena) is expected to occur for critical NRM ensembles as well – and is indeed easy to verify in critical Gaussian models, $2|t_2|=1$. Analytically, it means that by suitable scaling of the variables $z, n$: $$n \to \infty, \,\,\, \hbar \to 0, \,\,\, n\hbar = t_0, \,\,\,
t_0 = t_c - \hbar^{\delta} \nu, \,\,\, z = z_c + \hbar^{\epsilon} \zeta,$$ where $z_c$ is the location of the critical point and $t_c$ the critical area, the wave function $\Psi_n(z)$ will reveal a universal part $\phi(\nu,\zeta)$ which depends exclusively on the local singular geometry $x^p \sim y^q$ ($p, q$ mutual primes) of the complex curve at the critical point. This conjecture is a subject of active research. Its main consequence is that in order to describe the scaling behavior for a certain choice of $p, q$, it is possible to replace a given ensemble with another which leads to the same type of critical point, though they may be very different at other length scales.
### Scaling at critical points of normal matrix ensembles {#painl}
In the remainder of the section we analyze the regularization of Laplacian Growth for a critical point of type $p=3, q=2$, by discretization of the conformal map as described in the previous paragraph. For simplicity, we start from the conformal map corresponding to the potential $V(z) = t_3z^3$, which is the simplest model leading to the specified type of cusp. It should be noted that the analysis will be identical for any monomial potential $V(z) = t_nz^n, n \ge 3$; for every such map, $n$ singular points of type $p=3, q=2$ will form simultaneously on the boundary. The critical boundary corresponding to $n=3$ is shown in Figure \[critical\].
#### The scaling limit from the string equation
We start from the Lax pair corresponding to the potential $V(z)=t_3z^3$: L\_n = r\_n \_[n+1]{} + u\_n \_[n-2]{}, L\^\_n = r\_[n-1]{}\_[n-1]{} + |u\_[n+2]{}\_[n+2]{}. The string equation (\[string\]) $[L^{\dag}, \, L] = \hbar$ translates into $$\begin{aligned}
\nonumber
(r^2_n +|u|_n|^2 - r^2_{n-1} - |u|_{n+2}^2) \psi_n + (r_n\bar u_{n+3} - r_{n+2}\bar u_{n+2}) \psi_{n+3} \\
+(r_{n-3}u_n - r_{n-1}u_{n-1}) \psi_{n-3} = \hbar \psi_n.
\end{aligned}$$ Identifying the coefficients gives ( r\_n\^2 - |u|\^2\_[n+2]{} - |u|\^2\_[n+1]{} ) - ( r\_[n-1]{}\^2 - |u|\^2\_[n+1]{} - |u|\^2\_[n]{} ) = , and = = 3t\_3. Equation (\[ar\]) gives the quantum area formula r\_n\^2 - (|u|\_[n+2]{}\^2 + |u|\_[n+1]{}\^2) = n, which together with the conservation law (\[co\]) leads to the discrete Painlevé equation r\_n\^2 = n.
In the continuum limit, the equation becomes r\^2 - 18|t\_3|\^2 r\^4 = t\_0. The critical (maximal) area is given by = 0, 36|t\_3|\^2r\^2\_c = 1. Choosing $r_c =1$ gives $6|t_3| = 1$ and $t_c = \frac{1}{2}$. It also follows that u\_n = , z\_c = . Introduce the notations N= t\_c, n= t\_0 = t\_c +\^[4a]{} , r\_n\^2 = 1 - \^[2a]{}u() , z = + \^[2a]{}, where $a=\frac{1}{5}.$ We get $\p_n = \hbar^a \p_\nu$ and r\^2\_[n+k]{} = 1 - \^[2a]{}u - k \^[3a]{}u() - \^[4a]{}, where dot signifies derivative with respect to $\nu$. The scaling limit of the quantum area formula becomes (1-\^[2a]{}u)= + \^[4a]{}, giving at order $\hbar^{4a}$ the Painlevé I equation u - 2 u\^2 = 4 . Rescaling $u \to c_2 u$, $\nu \to c_1 \nu$ gives the standard form u - 3u\^2 = , for $c_2 = 4c_1^3, 8c_1^5 = 3$.
#### Painlevé I as compatibility equation
Inspired by the Saff-Totik approach, we construct the wavefunctions based on monic polynomials, ($\mbox{Pol}$ is the polynomial part) \_n = \_[i=0]{}\^[n-1]{}r\_i \_n, \_n(z) = z\^n + O(z\^[n-1]{}), and rewrite the equations for the Lax pair as L \_n = \_[n+1]{} + \_[n-2]{}, L\^ \_n = r\^2\_[n-1]{}\_[n-1]{} + . Notice that using the shift operator $\mathcal{W}$, the system can also be written L = + ( r\^2\_[n-1]{}\^[-1]{} )\^2, L\^ = r\^2\_[n-1]{}\^[-1]{} + \^2. Introduce the scaling $\psi$ function through \_n(z) = e\^ (, ). The action of Lax operators on $\psi$ gives the representation L = + \^[2a]{}, L\^ = z + \_= + \^[2a]{} + \^[3a]{} \_. Therefore, the action of $\zeta$ is given by the sum of equations at order $\hbar^{2a}$: 3 + 2 \^[2a]{} = + \^2 + r\^2\_[n-1]{}\^[-1]{} + ( r\^2\_[n-1]{}\^[-1]{} )\^2, and the action of $\p_\zeta$ by their difference: \^[3a]{}\_= - + \^2 + r\^2\_[n-1]{}\^[-1]{} - ( r\^2\_[n-1]{}\^[-1]{} )\^2. Equivalently, we can write \^[2a]{}= - 4, \^[3a]{}\_= . Expanding the shift operator in $\hbar$ leads to = 1 + \^[a]{} \_+ \^[2a]{} + \^[3a]{}, and r\^2\_[n-1]{}\^[-1]{} = 1 - \^[a]{}\_+ \^[2a]{}(-u ) + \^[3a]{}( - + u \_+ u ). Substituting into the equations for $\zeta, \p_\zeta$ gives the system of equations = , ’ = + , where primed variables are differentiated with respect to $\zeta$. The equations can be written in matrix form as ’ = , = Q , = (
[c]{}\
), where = (
[cc]{} &\
+ & -
), Q = (
[cc]{} 0 & 1\
& 0
). The compatibility equations - Q’ = \[Q, \] yield the Painlevé equation derived in the previous section: = (
[cc]{} & -\
+ & -
), Q’ = (
[cc]{} 0 & 0\
& 0
), and = (
[cc]{} & -\
& -
). Thus, 0 = - Q’ - \[Q, \] = (
[cc]{} 0 & 0\
- - & 0
). The only non-trivial element of the matrix gives -4uu - 4 = 0, $i.e.$ the Painlevé equation derived in the previous section.
### Conclusions
The derivations presented above indicate that, in the vicinity of a (2, 3) cusp, the refined asymptotics for Laplacian growth are based on the behavior of the Baker-Akhiezer function for the Painlevé I equation. This fact allows to properly define the evolution of the domain beyond the critical time, by identifying the support of the measure with the support of the zeros of this function. This is a work in progress which will be reported elsewhere.
It is also interesting to note that the double-scaling limit required to derive the refined asymptote mirrors an earlier result, due in its original form to Stahl [@Stahl], and related to orthogonal polynomials in [@Saf]. It describes an approximation of the Cauchy transform of a planar domain via a special sequence of Padé approximants (in the spirit of section 5.3.2), which by exponentiation would translate into the double-scaling limit presented in this section.
Non-commutative probability theory and 2D quantum models
--------------------------------------------------------
We conclude this review with a brief presentation of outstanding problems in two-dimensional quantum models, where the use of random matrix theory led to important results, and (perhaps most importantly) pointed out to the need for a probability theory for [*non-commutative*]{} random variables. In turn, such a theory is intimately related to the semi-normal operator approach presented in the previous chapter.
### Metal-insulator transition in two dimensions
The details of the transition from conductive to insulating behavior for a system of interacting 2D electrons, in the presence of disorder, referred to as metal-insulator transition (or MIT) are not well understood, despite decades of research. Here we give a very sketchy description of this problem, in order to illustrate the mathematical essence of the model and of the difficulties, and we refer the reader to one of the several excellent monographs on the subject [@RevModPhys]. The fact that a system of electrons may “jam", i.e. behave like an insulator, because of either strong interactions (Mott transition) or strong disorder (Anderson transition) has been known for roughly half a century. However, creating a theoretical model which could incorporate both interactions and disorder in a proper fashion, was difficult to achieve. The foundation for our current formulation of this problem was laid by Wegner [@Wegner], and later improved by Efetov [@Efetov]. A very clear exposition of this formulation can be found in the synopsis [@Zirnbauer].
In its simplest formulation, the model consists of a lattice in $d-$dimensions (which may be taken to be $\mathbb{Z}^d$), where to each vertex corresponds an $n-$dimensional vector space of states (also called orbitals), and with hamiltonian $$H = H_0 + H_d, \quad H_0 = \sum_{n, \langle x, y \rangle} t_{x,y} |x, n \rangle \langle y, n|,
\quad H_d = \sum_{x, i, j} f^{ij} | x, i\rangle \langle x, j |,$$ where the state $|n, x\rangle$ depends on position $x$ and orbital $n$, $H_0$ refers to the interaction between adjacent vertices and $H_d$ implements the disorder component, via the random matrix $f^{ij}$, which can be Hermitian, Orthogonal, Symplectic, etc. based on symmetries of the system. Efetov’s idea was to use supersymmetry to incorporate interactions and disorder on the same footing; the method was later extended to implement the “Hermitization" of non-Hermitian random matrices with non-Gaussian weights, appearing in the same physical context [@Feinberg]. For rotationally invariant measures, the authors showed that the distribution of eigenvalues can be either a disc or an annulus, and that there is a phase transition between the two, as a function of model parameters.
The difficulties related to this formulation of the problem are due to the fact that the transition cannot be described within the established models of phase transitions. In all these models, the state of the system is obtained by minimization of a proper thermodynamic potential (for instance, free energy), or equivalently, finding the points of extrema of action in a path-integral approach (via a saddle-point condition). “Proper" phase transitions are characterized by potentials that are globally convex, so that the minimization problem is well-defined. However, the supersymmetric formulation of MIT does not lead to a true extremum, but rather a saddle-point, due to the non-compact, hyperbolic geometry structure of the effective theory ($SU(1,1)$ in the simplest case). The interested reader can find a detailed exposition of this phenomenon in [@Spencer], for example. In the case where the system has a finite scale, it can be shown, following Efetov, that only the zero modes of the theory are important, which leads to an effective simplification in computing the multipoint correlation functions. However, the full model is still not solved for the 2D case, due to the difficulties pointed out above.
In a nutshell, we may summarize the problem as non-tractable using the standard statistical physics formulation of phase transitions. In that sense, the situation is similar to another famous unsolved physical model, the disordered spin problem in the presence of magnetic field, where determination of the ground state is a task of exponential complexity (with respect to the size of the system). The phase transition where the system goes from an ordered state to a state with local order but no long-range order (a spin glass) is equally intractable as MIT, for the reasons explained.
Interestingly enough, both problems may be [*approximately*]{} studied using a physicist’s approach notorious for its lack of control: the replica-symmetry breaking (RSB) [@Parisi]. We mention it here mainly because of its statistical interpretation.
Starting from the elementary observation $\log Z = \lim_{n \to 0}(Z^n - 1)/n$, it is tempting to replace averages (over disorder) computed from the thermodynamic potential $\langle \log Z \rangle$, with averages computed with $\langle Z^n \rangle$, because of the implicit assumption that repeated products of the random variable $Z$ will [*self-average*]{} (an implicit application of the Central Limit Theorem). By extension, one may assume that averages of products of operators, projected on special states, $\langle 0 |\phi_1 \phi_2 \ldots \phi_k | 0 \rangle$ (correlation functions), may also be computed by the same argument.
In this (statistical inference) approach, the failure of standard descriptions of phase transitions is related to reducing correlation functions of products of operators, to their [*projections*]{} onto selected states. At the critical point, such projections do not have the expected convergence properties. It is therefore natural to ask whether one may use other (weaker) criteria to determine the critical point. In particular, is it possible to define statistical inference for the operators themselves, rather than special projections?
The answer is affirmative, and such a theory was constructed almost in parallel with the MIT and spin glass models described above.
### Non-commutative probability theory and free random variables
The basic elements in the probability theory for non-commutative operators [@Voiculescu] are the following: $\mathcal{A}$, a non-commutative (operator) algebra over $\mathbb{C}$, $1 \in \mathcal{A}$; a functional $\phi: \mathcal{A} \to \mathbb{C}, \phi(1) = 1$, called expectation functional.
Quantum mechanics offers specific examples:
- Example 1: $\mathcal{A}=$ bounded operators over Hilbert space of states $\mathcal{H}$, $\xi \in \mathcal{H}, ||\xi||=1$, the ground state, and $$\phi(A) = \langle \xi | A |\xi \rangle.$$
- Example 2: $\mathcal{A}=$ von Neumann algebra over $\mathcal{H}$, and functional $\phi = $ Tr.
In order to develop inference methods within this theory, it is necessary to define the equivalent of independent variables in commutative probability. Such variables are called [*free*]{}, and satisfy the following property: $A_1, A_2, \ldots, A_k$ are free if $\phi(A_i) = 0,$ and $$\phi(A_{i_1} A_{i_2} \ldots A_{i_k}) = 0, \quad A_{i_j} \ne A_{i_{j+1}}.$$
Using these tools, generalizations of standard results in large sample theory are possible. We mention a few:
- The “Gaussian" distribution (limit distribution for Central Limit Theorem) in free probability theory is given by operators with eigenvalues obeying the semi-circle distribution (Wigner-Dyson) $\rho(\lambda) = \sqrt{a^2-\lambda^2}$;
- Similarly, the Poisson distribution has as free correspondent the operators with eigenvalues distribution according to the Marchenko-Pastur (elliptical law), $\rho(\lambda) = \sqrt{(\lambda -a )(b-\lambda)}$;
- The free Cauchy distribution is the Cauchy distribution itself.
Likewise, there is a notion of free Fisher entropy, Cramér-Rao bound, etc.
As announced earlier, the relation between this theory, random matrices and operator theory for 2D spectral support is two-fold: on one hand, we have the important result that random matrices, in the large size limit, become free non-commutative random variables. Thus, inference in free non-commutative probability may be approximated using ensembles of random matrices, which explains the success of this concept in the physics of disordered quantum systems.
On the other hand, the limit distributions specified above (via Wigner-Dyson, Marchenko-Pastur laws, and their 2D counterparts), are described through spectral data. Taking this as a starting point, it is relevant to construct sequences of operators which approximate the spectrum, which points directly to the methods of section 5.
As a last remark, an early attempt to employ non-commutative probability theory in MIT was reported in [@Speicher]. It is likely that the application of this generalized inference method will help elucidate open questions like the ones discussed in this section.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors wish to thank P Wiegmann, K Efetov and E Saff for useful discussions regarding parts of this project, and A Zabrodin for comments on the final manuscript. Research of M.M. and R. T. was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE C52-06NA25396. M.M and M.P. were supported by the LANL LDRD project 20070483ER. M.P. was partially supported by the Natl. Sci. Foundation grant DMS-0701094. R.T. acknowledges support from the LANL LDRD Directed Research grant on [*Physics of Algorithms*]{}.
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[^1]: The shift operator $\hat w$ has no inverse. Below $\hat w^{-1}$ is understood as a shift to the left defined as $\hat w^{-1}\hat w=1$. Same is applied to the operator $L^{-1}$. To avoid a possible confusion, we emphasize that although $\chi_n$ is a right-hand eigenvector of $L$, it is not a right-hand eigenvector of $L^{-1}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain $C^k$ lower bounds on the regularity of the embedding. In the resonant case we verify the existence of tori which are only $C^0$ and neither star-shaped nor Lipschitz.'
bibliography:
- 'refs.bib'
---
<span style="font-variant:small-caps;">Maciej J. Capiński$^*$</span>
<span style="font-variant:small-caps;">Emmanuel Fleurantin and J.D. Mireles James</span>
(Communicated by the associate editor name)
Introduction {#sec:intro}
============
Questions about the existence, topology, and regularity of invariant sets have organized the qualitative theory of nonlinear dynamics since the foundational work of Poincaré at the end of the Nineteenth Century. In modern times numerical simulations play a crucial role in this theory, providing deeper insights into the fine structure of phase space than can be obtained by any other means. The digital computer has emerged as a kind of dynamical systems laboratory, where one runs experiments on nonlinear systems far from a trivial solution or other perturbative regime.
In response to this development last four decades have seen a number of researchers put tremendous energy into developing and deploying computer assisted methods of proof which bridge the gap between numerical conjecture and mathematically rigorous theorems. The work of Lanford, Eckman, and Collet on the computer assisted proof of the Feigenbaum Conjectures [@lanford_CAP; @compProofFeig], and the resolution of Smale’s 14th problem by Tucker [@tucker1; @tucker2] provide excellent examples of this trend. The recent review articles [@reviewCAP; @MR3990999] provide historical context and more complete discussion of the literature.
The present work focuses on computer assisted methods of proof for attracting invariant tori in dissipative vector fields. Invariant tori typically appear in systems where there are two or more competing natural frequencies. Two common mechanisms are periodic/quasi-periodic perturbations of a system with an attracting periodic orbit, and when a periodic orbit with complex conjugate Floquet multipliers loses stability – triggering a [Neimark-Sacker]{} bifurcation in a Poincaré section [@MR0132256; @MR2615427]. Both situations are treated in the present work. Some classic references on dissipative dynamical systems having invariant tori are [@MR1005055; @MR1115870; @MR709899; @MR880159; @MR1488520; @MR3435117; @MR3279518; @MR1093209; @Langford], and we refer also to the works of [@Haro1; @Haro2; @MR2299977; @MR3713932; @MR3713933; @MR3309008] for a functional analytic approach to this topic. Of course the study of robust invariant manifolds, or *normally hyperbolic invariant manifolds* (NHIMs) goes back to the classic works of Fenichel [@MR287106], and of Hirsch, Pugh, and Shub [@MR0271991; @MR292101; @MR0501173]. See also the works of [@MR754826; @MR791842; @MR1391508; @MR1460262; @MR2136745; @MR3751167] and the references therein for some numerical investigations of NHIMs.
Related techniques for computer assisted proof of invariant tori are found in the works of [@celettiCAP1; @cellettiCAP2; @rana1; @rana2; @CAPmodernKAM; @Haro_aPosKAM], and the references therein. It should be remarked that the works just cited deal with analytic invariant KAM tori in symplectic/Hamiltonian systems, where the torus cannot be attracting and the dynamics on the torus are conjugate to a Diophantine irrational rotation. The present work deals with attracting invariant tori in dissipative systems. These objects are necessarily of lower regularity [@Llave] – $C^k$ sometimes with $0 \leq k < \infty$ – and computer assisted existence proofs require different strategies.
Our analysis is formulated in terms of topological and geometric hypotheses which we check using mathematically rigorous computational techniques for numerical integration of vector fields and their variational equations. To make the presentation as self contained as possible we focus on the case of 3D fields and include elementary proofs of our arguments. We implement our method in two illustrative examples. The first example is a periodic perturbation of a planar vector field where the unperturbed system has an attracting periodic orbit. Here we prove the existence of $C^k$ invariant tori with rotational dynamics. The second example is a is an autonomous vector field where resonant invariant tori appear naturally after a [Neimark-Sacker bifurcation]{}.
The two situations require different analysis. In the rotational case we develop an *outer approximation* of the torus via coverings by polygons and cone conditions. The union of the polygons is eventually shown to contain a torus. In the case of the autonomous vector field, where the invariant torus appears in a [Neimark-Sacker bifurcation]{}, the tori we consider are resonant. This means that they can be decomposed into attracting and saddle periodic orbits, where the unstable manifold of the saddle is absorbed completely into a trapping neighborhood of the attracting orbit. In the resonant case we provide an *inner approximation*, in the sense that we build the torus out of invariant pieces whose union is shown to be the desired torus. In both the rotational and resonant cases we study the tori away from the perturbative case. We remark that, because our theoretical arguments are formulated for maps (in our case Poincaré maps), our implementations rely heavily on the validated $C^k$ integrators developed over the last decade by Wilczak and Zgliczyński [@cnLohner; @c1Lohner].
A technical remark is that our analysis of the rotational tori is formulated for a star-shaped region in an appropriate surface of section. The star-shaped hypothesis is an implementation detail which allows us to proceed without making a technical digression into the setting of vector bundles. Nevertheless, we indicate in Section \[sec:generalization\] how to proceed more generally. [ A second technical remark is that the torus may be smoother than we are actually able to prove. Put another way, we prove that the rotational torus is *at least* $C^k$ though it may in fact be smoother. We do not claim that our regularity results are sharp.]{} On the other hand the resonant tori we study are globally only $C^0$, and in this case the regularity is sharp as the tori are not globally Lipschitz.
The remainder of the paper is organized as follows. In Section \[sec:prelims\] we review some preliminary notions and definitions from dynamics and validated numerics. In Section \[sec:contractiing-maps\] we state and prove our main theorem on the existence of attracting invariant Lipschitz curves, and investigate conditions which imply their differentiability. Section \[sec:homoclinic-maps\] treats computer assisted methods of validation for the existence of homoclinic/heteroclinic orbits for planar maps. More explicitly we develop techniques for proving the existence of attracting fixed points and obtaining lower bounds on the size of the basin of attraction. Then we recall some tools for validating bounds on the local stable/unstable manifolds attached to saddle fixed points from [@Cap-Lyap; @Zgliczynski-cone-cond]. Finally we prove the existence of heteroclinic connections from the saddle to the attractor. When these techniques are applied in a Poincaré section for an ODE we obtain connections between periodic orbits of the differential equation.
In Section \[sec:tori-3d\] we show how to apply the methods of Section \[sec:homoclinic-maps\] to prove the existence of invariant tori for ODEs. The main idea is to propagate an invariant circle from a Poincaré section by the flow of the ODE. Section \[sec:examples\] is devoted to the implementation [of]{} our methods in two example applications. We consider a periodically forced Van der Pol equation where the natural attracting periodic orbit in the unforced system gives an attracting invariant torus after the application of the forcing. Here we prove the existence of $C^k$ invariant tori. We also consider an autonomous differential equation with an attracting resonant tori. There is an attracting periodic orbit in the invariant the torus which has complex conjugate multipliers, hence the torus is only $C^0$.
Preliminaries {#sec:prelims}
=============
For a set $A$ we write $\overline{A}$ to denote its closure, $\mathrm{int}A$ to denote its interior, and write $\mathbb{S}^{1}$ for a one dimensional circle. [Throughout the paper, for $x\in \mathbb{R}^n$ we shall use $\| x \|$ to stand for the Euclidean norm.]{} Let $B(p,r)$ denote the open ball of radius $r$ centered at $p$.
Suppose that $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is a diffeomorphism and let $p^{\ast }\in \mathbb{R}^{n}$ be a hyperbolic fixed point of $f$ (i.e. the eigenvalues of $Df(p^{\ast })$ are [not on]{} the unit circle). We shall use the notation $W^{u}(p^{\ast })$ and $W^{s}(p^{\ast })$ to stand for the unstable and the stable manifold of $p^{\ast }$, respectively, i.e. $$\begin{aligned}
W^{u}\left( p^{\ast }\right) &=&\left\{ p:\left\Vert f^{n}(p)-p^{\ast
}\right\Vert \rightarrow 0\text{ as }n\rightarrow -\infty \right\} , \\
W^{s}\left( p^{\ast }\right) &=&\left\{ p:\left\Vert f^{n}(p)-p^{\ast
}\right\Vert \rightarrow 0\text{ as }n\rightarrow \infty \right\} .\end{aligned}$$
For $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and $B\subset\mathbb{R}^{n}$ define $\left[ Df\left( B\right) \right] \subset\mathbb{R}^{n}\times\mathbb{R}^{n}$ as$$\left[ Df\left( B\right) \right] :=\left\{ \left( a_{ij}\right)
_{i,j=1}^{n}:a_{ij}\in\left[ \inf_{p\in B}\frac{\partial f_{i}}{\partial
x_{j}}\left( p\right) ,\sup_{p\in B}\frac{\partial f_{i}}{\partial x_{j}}\left( p\right) \right] \text{for }i,j=1,\ldots,n\right\} .$$ We refer to $[ Df\left( B\right) ] $ as the interval enclosure of the derivative of $f$ on $B$, and write $Id$ for the identity matrix.
For an interval matrix $\mathbf{A}$, i.e. a set $\mathbf{A}\subset$ [$\mathbb{R}^{n \times n}$]{}, we will write$$\left\Vert \mathbf{A}\right\Vert :=\sup\left\{ \left\Vert Ax\right\Vert
:\left\Vert x\right\Vert =1,A\in\mathbf{A}\right\} .$$ We say that $\mathbf{A}$ is invertible if each $A\in \mathbf{A}$ is invertible. We define $\mathbf{A}^{-1}:=\{A^{-1}:A\in \mathbf{A}\}\subset
\mathbb{R}^{n}\times\mathbb{R}^{n}$.
The following lemma is a version of the mean value theorem, which is useful in a number of places throughout the paper.
\[lem:Df-difference\]Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be $C^{1}$, let $B\subset\mathbb{R}^{n}$ be a cartesian product of closed intervals in $\mathbb{R}^{n}$ and let $p_{1},p_{2}\in B$, then we can choose an $n\times n$ matrix $A\in\left[ Df\left( B\right) \right] $ for which we will have $$f\left( p_{1}\right) -f\left( p_{2}\right) =A\left( p_{1}-p_{2}\right) .$$ with [ $$A=\int_{0}^{1}Df\left( p_{2}+t\left( p_{1}-p_{2}\right) \right) dt.$$]{}
We use the following classical result.
\[th:interval-Newton\][@Al] (Interval Newton method) Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a $C^{1}$ function and $B$ be a cartesian product of closed intervals in $\mathbb{R}^{n}$. If $[Df(B)]$ is invertible and there exists an $x_{0}$ in $B$ such that$$N(x_{0},B):=x_{0}-\left[ Df(B)\right] ^{-1}f(x_{0})\subset B,$$ then there exists a unique point $x^{\ast}\in B$ such that $f(x^{\ast})=0.$
![On the left we have a cone attached at the point $\protect\gamma_U^{-1}(q)$ in the case when $k=2$ and $n=3$. Note that the cone is *not* the blue (cone shaped) set. The cone $\mathbf{Q}(\protect\gamma_U^{-1}(q))$ is the complement of the blue set in $\mathbb{R}^3$; i.e. the white region outside of the blue set. On the right we have an example of a Lipschitz manifold. []{data-label="fig:Lip-cone"}](fig_Lip-cone.pdf){height="5cm"}
The notion of a Lipschitz manifold requires us to define certain cone conditions. [Fix $1 \leq k \leq n$ and for a point $x=\left( x_{1},\ldots ,x_{n}\right) \in \mathbb{R}^{n}$ write $\pi_{x_1, \ldots, x_k}(x):=\left( x_1,\ldots ,x_{k}\right)$ and $\pi_{x_{k+1},\ldots ,x_{n}}(x) = (x_{k+1}, \ldots, x_n)$. For a point $p\in \mathbb{R}^{n}$ we define the cone attached to $p$ as (see Figure \[fig:Lip-cone\])$$\mathbf{Q}_{k} \left( p\right) :=\left\{ x\in \mathbb{R}^{n}:a\left\Vert
\pi_{x_1, \ldots, x_k}\left( p-x\right) \right\Vert \geq \left\Vert \pi_{x_{k+1}, \ldots, x_n} \left(
p-x\right) \right\Vert \right\},$$ where $0<a \in \mathbb{R}$ is a fixed constant. We suppress the $k$ subscript and simply write $\mathbf{Q}(p)$ when $k$ is clear from context. ]{}
\[def:lip-manifold\] [ Let $\mathcal{M} \subset \mathbb{R}^n$ be a $k$-dimensional compact topological manifold. We say that $\mathcal{M}\subset \mathbb{R}^{n}$ is Lipschitz, if it satisfies cone conditions in the following sense:]{} any point $p\in \mathcal{M}$ there exists an open neighborhood $U $ of $p$ in $\mathbb{R}^{n}$, an open set $B\subset \mathbb{R}^{n}$, and a $C^{1}$ diffeomorphism $\gamma
_{U}:B\rightarrow U$ such that for any $q\in \mathcal{M} \cap U$ [(see Figures \[fig:Lip-cone\], \[fig:curve\])]{} $$\mathcal{M}\cap U\subset \gamma _{U}\left( \mathbf{Q}\left( \gamma
_{U}^{-1}\left( q\right) \right) \cap B\right) .$$
Attracting invariant circles for maps on $\mathbb{R}^{2}$[sec:contractiing-maps]{}
==================================================================================
In this Section we discuss how to establish the existence of attracting invariant curves for planar maps. The methodology is based on taking a neighborhood of the curve and validating that this neighborhood maps into itself. This on its own ensures only existence of an invariant set, and not that the set is a curve. We therefore consider two additional conditions. The first is that we have a ‘well aligned cone field’ which also maps into itself, and the second is that we have uniform contraction inside of the considered set.
The proposed method is similar in spirit to [@CZ0; @CZ]. The main difference is that the papers just mentioned work with a vector bundle around the manifold. In the present work we formulate our results in local coordinates that roughly cover (see Figure \[fig:curve\]) the investigated invariant curve, removing the need for vector bundle coordinates. [ This simplifies the implementation of the method. ]{}
[We give our proof in the setting of closed star-shaped invariant curves. Our results can be directly generalized to the setting where we haver a vector bundle based on a closed curve (not necessarily star-shaped) in $\mathbb{R}^2$. We present this generalization in Section \[sec:generalization\]. We give our proofs in the star-shaped setting due to the simplicity of the setup. The arguments for the more general case of vector bundles are analogous.]{}
In Section \[sec:Lip-curves\] we present the method which ensures the existence of Lipschitz invariant curves, in Section \[sec:Ck\] we add conditions which ensure the $C^{k}$ smoothness, and in Section \[sec:Lip-curves-validation\] we discuss how the assumptions are validated in practice.
Establishing closed attracting star-shaped curves[sec:Lip-curves]{}
-------------------------------------------------------------------
Let$$f:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$$ be a $C^{1}$ diffeomorphism[^1]. Assume that $B_{1},B_{2}\subset\mathbb{R}^{2}$ are homeomorphic to two dimensional open balls in $\mathbb{R}^{2}$ and that $\overline{B_{1}}\subset B_{2}$. Let $U:=\overline{B_{2}}\setminus B_{1}$ and assume that $$U=\bigcup_{i=1}^{N}U_{i}=\bigcup_{i=1}^{N}\gamma_{i}\left( M_{i}\right) , \label{eq:U-set}$$ [ where for each $i=1,\ldots,N$, the $M_{i}=\left[ -R_{i},R_{i}\right] \times\left[ -r_{i},r_{i}\right] $ for some fixed sequence of constants $0<r_{i},$ $R_{i}\in\mathbb{R}$, and $\gamma_i \colon M_i \rightarrow \mathbb{R}^2$ are diffeomorphisms onto their image]{} (See Figure \[fig:curve\]). We think of $\gamma_{i}$ as local coordinates on the set $U$.
![The set $U$ is a collection of boxes, and we prove the existence of a star-shaped invariant closed curve around $q^{\ast}$ which satisfies the cone conditions.[]{data-label="fig:curve"}](fig_curve.pdf){height="4.5cm"}
Our objective is to provide conditions ensuring the existence of a star-shaped Lipschitz closed curve in $U$ homeomorphic to a circle and invariant under $f$.
We equip each box $M_{i}$ with cones as follows. For $p\in\mathbb{R}^{2}$ define$$\begin{aligned}
Q_{i}\left( p\right) & =\left\{ \left( x,y\right) :\left\vert y-\pi
_{y}p\right\vert \leq a_{i}\left\vert x-\pi_{x}p\right\vert \right\} ,
\label{eq:cones-Lip-torus} \\
Q_{i}^{r}\left( p\right) & =Q_{i}\left( p\right) \cap\left\{ \left(
x,y\right) :x>\pi_{x}p\right\} , \notag \\
Q_{i}^{l}\left( p\right) & =Q_{i}\left( p\right) \cap\left\{ \left(
x,y\right) :x<\pi_{x}p\right\} , \notag\end{aligned}$$ where $0<a_{1},\ldots,a_{N}\in\mathbb{R}$ are fixed constants. The superscripts $r$ and $l$ stand for ‘right’ and ‘left’, respectively. Define (see Figure \[fig:curve\]) $$\begin{aligned}
\tilde{Q}_{i}(q) & :=\gamma_{i}\left( Q_{i}\left( \gamma_{i}^{-1}(q)\right)
\right) , \\
\tilde{Q}_{i}^{\kappa}(q) & :=\gamma_{i}\left( Q_{i}^{\kappa}\left(
\gamma_{i}^{-1}(q)\right) \right) ,\qquad\text{for }\kappa\in\left\{
r,l\right\},\end{aligned}$$ and choose $q^{\ast}\in B_{1}\subset\mathbb{R}^{2}$. (From now on the $q^{\ast}$ will remain fixed.) We define the half line emanating from $q^{\ast}$ at an angle $\theta$ as$$l_{\theta}:=\left\{ p\in\mathbb{R}^{2}:p=q^{\ast}+t\left( \cos\theta
,\sin\theta\right) \text{ for }t>0\right\} .$$
\[def:well-aligned-cones\] We say that the cones $Q_{i}$ are well aligned if for any $\theta\in
\lbrack0,2\pi),$ $i\in\left\{ 1,\ldots,N\right\} $ and $p\in\gamma_{i}^{-1}\left( U_{i}\cap l_{\theta}\right) $ we have$$Q_{i}\left( p\right) \cap\gamma_{i}^{-1}(l_{\theta})=p,$$ and $\gamma_{i}^{-1}(l_{\theta})$ intersects $\left\{ y-\pi_{y}p=a_{i}\left(
x-\pi_{x}p\right) \right\} $ and $\left\{ y-\pi_{y}p=-a_{i}\left(
x-\pi_{x}p\right) \right\} $ transversally. (See Figure [fig:well-aligned-cone]{}.)
![A well aligned cone.[]{data-label="fig:well-aligned-cone"}](fig_cone.pdf){height="2cm"}
Let $h:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ be a continuous function. We say that $h$ is a star-shaped closed curve around $q^{\ast}$ if $$h\left( \mathbb{S}^{1}\right) \cap l_{\theta}=h\left( \theta\right),$$ [for all $\theta \in \mathbb{S}^1$.]{}
\[def:closed-curve-cone-cond\] We say that $h:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ is a star-shaped closed curve which satisfies cone conditions, if it is a closed curve around $q^{\ast}$, and for any $\theta$ there exists an $i\in\left\{ 1,\ldots,N\right\} $ and $r>0$ such that (see Figure \[fig:curve\])$$h\left( \mathbb{S}^{1}\right) \cap B\left( h\left( \theta\right) ,r\right)
\subset\tilde{Q}_{i}\left( h\left( \theta\right) \right) . \label{eq:h-cc}$$
We say that $f$ satisfies cone conditions if for any $i\in\left\{
1,\ldots,N\right\} $ and any $p\in\mathrm{int}U_{i}$ there exists an $r>0$ and $j\in\left\{ 1,\ldots,N\right\} $ such that $$f\left( \tilde{Q}_{i}^{r}(p)\cap B(p,r)\right) \subset\tilde{Q}_{j}^{r}\left( f(p)\right) \quad\text{and\quad}f\left( \tilde{Q}_{i}^{l}(p)\cap B(p,r)\right) \subset\tilde{Q}_{j}^{l}\left( f(p)\right)
\label{eq:f-cc1}$$ or$$f\left( \tilde{Q}_{i}^{l}(p)\cap B(p,r)\right) \subset\tilde{Q}_{j}^{r}\left( f(p)\right) \quad\text{and\quad}f\left( \tilde{Q}_{i}^{r}(p)\cap B(p,r)\right) \subset\tilde{Q}_{j}^{l}\left( f(p)\right) .
\label{eq:f-cc2}$$
Let $\mu$ be the Lebesgue measure on $\mathbb{R}^{2}$. We shall say that $f$ is uniformly attracting on $U$, if there exists a constant [$0 \leq \lambda < 1$]{} such that for any Borel set $A\subset U$ we have $\mu\left( f(A)\right)$ [$\leq$]{} $\lambda\mu\left( A\right) $.
We now formulate our main result with which we establish the existence of attracting invariant curves.
\[th:lip-curve\]Assume that the cones $Q_{i}$ are well aligned. Assume also that there exists a sequence of points in $U$, such that the piecewise affine circle which results from joining these points is a closed curve which satisfies cone conditions around $q^{\ast}$. If $f$ is uniformly contracting on $U$, and if $f$ satisfies cone conditions, and if $f\left( U\right)
\subset\mathrm{int}U$, then there exists a closed curve $h^{\ast}$ around $q^*$, which satisfies cone conditions, such that $f\left( h^{\ast}\right) =h^{\ast}$. [ Moreover, for any $p\in U$, the orbit $\{f^{n}(p)\}_{n = 0}^\infty$ accumulates on the curve $h^{\ast}$. That is, the $\omega$-limit set of the orbit is contained in $h^{\ast}$.]{}
The proof is based on the following graph transform type argument. Let $h$ be a closed curve around $q^{\ast}$, which satisfies cone conditions. We show that $f\left( h\left( \mathbb{S}^{1}\right)
\right) $ is the image of another closed curve around $q^{\ast} $, which satisfies cone conditions. Then we show iterates of $h$ converge to $h^*$. Below we provide the details.
Since $f$ is a homeomorphism $f(h(\mathbb{S}^{1}))$ is a circle. We claim that $$f(h(\mathbb{S}^{1}))\cap l_{\theta}\neq\emptyset\text{ \qquad for any }\theta\in\mathbb{S}^{1}. \label{eq:proof-tmp-1}$$ To see this, let $g:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be defined as $$g\left( \theta\right) =\left\{
\begin{array}{lll}
1 & & \text{if }f(h(\mathbb{S}^{1}))\cap l_{\theta}\neq\emptyset, \\
0 & & \text{otherwise.}\end{array}
\right.$$ Once we show that $g$ is continuous, this will prove (\[eq:proof-tmp-1\]). This is because from $f(h(\mathbb{S}^{1}))\subset
f(U)\subset U$ and $U\cap q^{\ast}=\emptyset$ it follows that [$q^{\ast}\notin f(h(\mathbb{S}^{1}))$]{} so for at least one $\theta\in\mathbb{S}^{1}$ we must have $g\left( \theta\right) =1$; then, by continuity, we will have $g\equiv1$. Since $g\equiv1$, this circle intersects $l_{\theta}$ for every $\theta \in \mathbb{S}^2$, which implies (\[eq:proof-tmp-1\]).
To establish the continuity we start by showing that if $g\left( \theta\right) =0$, then for $\beta$ sufficiently close to $\theta$ we will have $g\left( \beta\right) =0.$ Suppose that $g\left( \theta\right) =0$. Since $f(h(\mathbb{S}^{1}))\subset
f(U)\subset U$ we see that $f(h(\mathbb{S}^{1}))$ and $l_{\theta}\cap U$ are disjoint compact sets. This means that we can find their open neighborhoods, which will also be disjoint. Therefore $f(h(\mathbb{S}^{1}))\cap l_{\beta
}=\emptyset$ for $\beta$ close to $\theta$, hence $g\left( \beta\right) =0$, as required.
![Construction of $\mathcal{G}(h)$. []{data-label="fig:graph-transform"}](fig_graph-transform.pdf){height="3.8cm"}
We now show that if $g\left( \theta\right) =1$, then for $\beta$ sufficiently close to $\theta$ we will have $g\left( \beta\right) =1.$ Let $p\in f(h(\mathbb{S}^{1}))\cap l_{\theta}$. There existes a $\bar{\theta}$ such that $p=f\left( h\left( \bar{\theta}\right) \right) $. Take $i$ such that $h\left( \bar{\theta}\right) \in U_{i}.$ From (\[eq:h-cc\]), for $\bar{\beta}$ close to $\bar{\theta}$$$h\left( \bar{\beta}\right) \in\tilde{Q}_{i}\left( h\left( \bar{\theta }\right) \right) .$$ Since $h$ is a closed curve around $q^{\ast}$ we see that we have either $$\bar{\beta}<\bar{\theta}\implies h\left( \bar{\beta}\right) \in\tilde{Q}_{i}^{l}\left( h\left( \bar{\theta}\right) \right) \quad\text{and\quad }\bar{\beta}>\bar{\theta}\implies h\left( \bar{\beta}\right) \in\tilde {Q}_{i}^{r}\left( h\left( \bar{\theta}\right) \right) ,
\label{eq:alignment-1-tmp}$$ or$$\bar{\beta}<\bar{\theta}\implies h\left( \bar{\beta}\right) \in\tilde{Q}_{i}^{r}\left( h\left( \bar{\theta}\right) \right) \quad\text{and\quad }\bar{\beta}>\bar{\theta}\implies h\left( \bar{\beta}\right) \in\tilde {Q}_{i}^{l}\left( h\left( \bar{\theta}\right) \right) .
\label{eq:alignment-2-tmp}$$ Without loss of generality let us assume that we have ([eq:alignment-1-tmp]{}). (If we have (\[eq:alignment-2-tmp\]) then the proof follows from mirror arguments.) We know that $f$ satisfies cone conditions. Let us therefore assume that we have (\[eq:f-cc1\]), from which it follows that [for some $j$ $$\begin{aligned}
f\left( h\left( \bar{\beta}\right) \right) & \in \tilde{Q}_{j}^{r}\left(
f(h\left( \bar{\theta}\right) )\right) =\tilde{Q}_{j}^{r}\left( p\right)
\qquad\text{for }\bar{\beta}>\bar{\theta}, \label{eq:cc-proof-1} \\
f\left( h\left( \bar{\beta}\right) \right) & \in \tilde{Q}_{j}^{l}\left(
f(h\left( \bar{\theta}\right) )\right) =\tilde{Q}_{j}^{l}\left( p\right)
\qquad\text{for }\bar{\beta}<\bar{\theta}. \label{eq:cc-proof-2}\end{aligned}$$]{}(If we have (\[eq:f-cc2\]) then the proof will follow from mirror arguments.) For any $\beta$ sufficiently close to $\theta$ there will therefore exist a $\bar{\beta}$ such that $f\left( h\left( \bar{\beta }\right) \right) \in l_{\beta}$; see Figure \[fig:graph-transform\]. Since $f\left( h\left( \bar{\beta}\right) \right) \in f\left( h\left( \mathbb{S}^{1}\right) \right) $ we see that for $\beta$ sufficiently close $g\left(
\beta\right) =1$, as required.
We have established (\[eq:proof-tmp-1\]). [We will now define a function $\mathcal{G}(h):\mathbb{S}^{1}\rightarrow U$, which we will prove to be a closed curve around $q^*$ which has the same graph as $f(h(\mathbb{S}^1))$. (We use the notation $\mathcal{G}(h)$ since the function follows from a graph transform" type construction.) We start by taking $\theta=0$ and defining $\mathcal{G}(h)(0)$ to be any point from $f(h(\mathbb{S}^1))\cap l_{\theta}$. At this stage we do not know if such point is unique, so we choose an arbitrary point from the intersection. Take $j_0$ such that $\mathcal{G}(h)(0)\in U_{j_0}$. From (\[eq:cc-proof-1\]–\[eq:cc-proof-2\]) we see that for $\theta>0$ close to zero, we can extend $\mathcal{G}(h)$ to obtain a curve by defining $\mathcal{G}(h)(\theta)=f(h(\mathbb{S}^1))\cap l_{\theta} \cap \tilde Q_{j_0} (\mathcal{G}(h)(0))$, as long as $\mathcal{G}(h)(\theta)$ remains in $U_{j_0}$. Let $\theta_1>0$ be some angle such that $\mathcal{G}(h)(\theta_1)\in U_{j_0}\cap U_{j_1}$, for some index $j_1$. We can then continue our construction for $\theta>\theta_1$ as $\mathcal{G}(h)(\theta)=f(h(\mathbb{S}^1))\cap l_{\theta} \cap \tilde Q_{j_1} (\mathcal{G}(h)(\theta_1))$. Continuing in this manner, we can reach $\theta=2\pi$. We are sure that at $\theta =2\pi$ we return to $\mathcal{G}(h)(0)$, since if this were not the case then we could continue with the construction and $f(h(\mathbb{S}^1))$ would contain an infinite spiral. This is not possible since $f(h(\mathbb{S}^1))$ is homeomorphic to a circe. ]{}From ([eq:cc-proof-1]{}–\[eq:cc-proof-2\]) we see that $\mathcal{G}(h)$ satisfies cone conditions, which also implies that it is continuous.
We now show that by starting with the closed curve $h$ which connects the points from the assumption of the theorem, then as we iterate the above defined graph transform we shall converge to the curve we seek; i.e. $$\lim_{n\rightarrow\infty}\mathcal{G}^{n}(h)=h^{\ast}.$$ Convergence follows from the assumption that $f$ is uniformly contracting on $U$.
![Since $\mathcal{G}^{n}(h)$ and $\mathcal{G}^{n-1}(h)$ satisfy cone conditions, we can find an angle $\protect\alpha$, such that the isosceles triangle, with base joining $q_{n}$ and $q_{n-1}$, as in above plot, will fit between $\mathcal{G}^{n}(h)$ and $\mathcal{G}^{n-1}(h)$. By compactness of $U$ and the fact that we have a finite number of $C^{1}$ local maps $\protect\gamma_{i}$, the $\protect\alpha$ can be chosen independently of $n,\protect\theta,q_{n}$ and of $q_{n-1}$. This means that the area between $\mathcal{G}^{n}(h)$ and $\mathcal{G}^{n-1}(h)$ is bounded from below by $C\Vert q_{n}-q_{n-1}\Vert^{2}$, where $C>0$ is some constant independent from $n$ and $\protect\theta$. []{data-label="fig:area"}](fig_area-between-curves.pdf){height="4cm"}
For $n=1,2,\ldots $ let $A_{n}$ be the area between the curves $\mathcal{G}^{n}(h)$ and $\mathcal{G}^{n-1}(h)$. Since $f$ is uniformly contracting, $A_{n}\leq \lambda ^{n-1}A_{1}$. Let us consider two points, $q_{n}\in
\mathcal{G}^{n}(h)\cap l_{\theta }$ and $q_{n-1}\in \mathcal{G}^{n-1}(h)\cap
l_{\theta }$, for some $\theta \in \mathbb{S}^{1}$. Since the curves $\mathcal{G}^{n}(h)$ and $\mathcal{G}^{n-1}(h)$ satisfy cone conditions the area between them has to be at least $C\left\Vert q_{n}-q_{n-1}\right\Vert
^{2}$, where $C>0$ is a constant independent of the choice of $\theta $ and $n$. See Figure \[fig:area\] and the caption below it. This gives us$$C\left\Vert q_{n}-q_{n-1}\right\Vert ^{2}\leq A_{n}\leq \lambda ^{n-1}A_{1}$$so$$\left\Vert q_{n}-q_{n-1}\right\Vert \leq \left( \sqrt{\lambda }\right) ^{n-1}\sqrt{\frac{A_{1}}{C}},$$from which, by the fact that $\sqrt{\lambda }<1$, it follows that the sequence $q_{n}$ is convergent. This means that we can define [$h^{\ast
}\left( \theta \right) :=\lim_{n\to \infty}\mathcal{G}^{n}(h)\left( \theta \right) $]{}. All $\mathcal{G}^{n}(h)$ are closed curves around $q^{\ast }$, which satisfy cone conditions. This property is preserved when passing to the limit, which concludes the proof.
Smoothness\[sec:Ck\]
--------------------
In this Section we discuss how to establish that the invariant curve $h^{\ast }$ from Theorem \[th:lip-curve\] is smooth. We first need to introduce some notation.
Consider local maps $f_{ji}:\left[ -R_{i},R_{i}\right] \times \left[
-r_{i},r_{i}\right] \supset $domain$f_{ji}\rightarrow \mathbb{R}^{2}$ defined as$$f_{ji}:=\gamma _{j}^{-1}\circ f\circ \gamma _{i}.$$(The domain of $f_{ji}$ can be empty.) We now define the following constants $$\begin{aligned}
\xi &:=&\inf \left\{ \frac{\partial \left( \pi _{x}f_{ji}\right) }{\partial x}\left( q\right) -a_{j}\left\vert \frac{\partial \left( \pi
_{x}f_{ji}\right) }{\partial y}\left( q\right) \right\vert :i,j=1,\ldots
,N,~q\in \text{domain}f_{ij}\right\} , \\
\mu &:=&\sup \left\{ \left\vert \frac{\partial \left( \pi _{y}f_{ji}\right)
}{\partial y}\left( q\right) \right\vert +a_{j}\left\vert \frac{\partial
\left( \pi _{x}f_{ji}\right) }{\partial y}\left( q\right) \right\vert
:i,j=1,\ldots ,N,~q\in \text{domain}f_{ij}\right\} .\end{aligned}$$
\[def:rate-cond\] We say that $f$ satisfies rate conditions of order $k$ if $\xi >0$ and for any $j\in \left\{ 1,\ldots ,k\right\} $$$\frac{\mu }{\xi ^{j+1}}<1. \label{eq:rate-cond}$$
The definition is a simplified version of the rate condition considered in [@CZ]. There are two differences. The first is that in [@CZ] three coordinates are considered: the unstable, the stable and central coordinate. Here we only have two: the central coordinate $x$ and the stable coordinate $y$. The second difference is that the rate conditions considered in [@CZ] include also bounds needed to establish the existence of the invariant manifold. Here we do this using the slightly modified method from Theorem \[th:lip-curve\]. Because of these two differences, the nine inequalities for rate conditions from [@CZ] are reduced to the single inequality ([eq:rate-cond]{}). (The condition (\[eq:rate-cond\]) corresponds to the first inequality in equation (4) from [@CZ].)
Below theorem is a reformulation of the smoothness result from [@CZ], adapted to our simplified setting.
\[th:Ck\]If in addition to all assumptions of Theorem \[th:lip-curve\] the map $f$ satisfies rate conditions of order $k,$ then $h^{\ast }$ established in Theorem \[th:lip-curve\] is $C^{k}$.
The result follows from Lemma 48 in [@CZ]. In our setting the only needed condition to apply Lemma 48 from [@CZ] is the condition ([eq:rate-cond]{}); see the remark in the third bullet list item on page 6226 in [@CZ]. Lemma 48 from [@CZ] ensures that when we iterate the graph transform from the proof of Theorem \[th:lip-curve\] starting with a $C^{k}$ curve $h^{0},$ then the $C^{k}$ smoothness persists as we pass to the limit. In the proof of Theorem \[th:lip-curve\] we take $h^{0}$ to be a piecewise affine circle. We can smooth out the corners of such curve to make it $C^{k}$, so the fact that the smoothness is preserved in the limit ensures the claim.
![Generalization to a vector bundle setting. The vector bundle $E$ is in grey, its base is the curve $p^*$, which is in black, with the fibers $E_{\theta}$ represented as the grey lines. The set $U$ consists of the union of the small rectangles. Note that in this picture $p^*$ is not the invariant curve, rather it is the base of the vector bundle.[]{data-label="fig:vector-bundle"}](fig-vect-bundle.pdf){height="4cm"}
Generalization to the setting of vector bundles {#sec:generalization}
-----------------------------------------------
In this section we present how the results from Sections \[sec:Lip-curves\] and \[sec:Ck\] can be generalized. Let us consider a closed curve $p^{\ast}\subset\mathbb{R}^{2}$, parameterized by $\theta\in\mathbb{S}^{1}$, i.e. $p^{\ast}:\mathbb{S}^{1}\rightarrow
\mathbb{R}^{2}$. Consider a vector bundle $E$ in $\mathbb{R}^{2}$ with $p^{\ast}(\mathbb{S}^{1})$ as its base and with fibers $E_{\theta}$ at $p^{\ast}\left( \theta\right) $, for $\theta\in\mathbb{S}^{1}$. Assume that the set $U\subset\mathbb{R}^{2}$ of the form (\[eq:U-set\]) is a subset of $E$ (see Figure \[fig:vector-bundle\]). We will say that the family of cones $Q_{i}$ introduced in the Section \[sec:Lip-curves\] is well aligned with $E_{\theta}$, if it satisfies the assumptions of Definition \[def:well-aligned-cones\], with $l_{\theta}$ changed to $E_{\theta}$. We will say that $h:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ is a closed curve around $p^{\ast}$ iff $h\left(
\mathbb{S}^{1}\right) \cap E_{\theta}=h\left( \theta\right) $ for all $\theta\in \mathbb{S}^1$. With such modiffications Theorems \[th:lip-curve\] and \[th:Ck\] remain true. Their proofs in this more general setting are identical, with the only difference that $l_{\theta}$ needs to be changed to $E_{\theta}$ and $q^{\ast}$ needs to be changed to $p^{\ast}$ throughout the arguments.
[The main difficulty in this setting is actually constructing the needed vector bundles in particular examples, and this is the technicality overcome using the star-shaped assumption in our earlier arguments. The interested reader is referred to [@CZ0; @CZ] for more general discussion. ]{}
Validation of assumptions\[sec:Lip-curves-validation\]
------------------------------------------------------
We finish this Section by describing how the assumptions of Theorems [th:lip-curve]{}, \[th:Ck\] are validated in practice. For the applications we have in mind we take $\gamma _{i}$ to be affine maps, so checking that $Q_{i}$ are well aligned is a simple linear algebra exercise of checking that $Q_{i}\left( p\right) $ and $\gamma _{i}^{-1}(l_{\theta })\ $ intersect at one and only one point. For instance, when $\gamma _{i}\left( q\right)
=A_{i}q+q_{i}$ (with a matrix $A_{i}$ and point $q_{i}\in \mathbb{R}^{2}$) one checks that for any $\theta $ such that $U_{i}\cap l_{\theta }\neq
\emptyset $ for $v_{\theta }=A_{i}^{-1}\left( \left( \cos \theta ,\sin
\theta \right) \right) $ we have $\left\vert \pi _{y}v_{\theta }\right\vert
>a_{i}\left\vert \pi _{x}v_{\theta }\right\vert $. This condition is checked with computer assistance.
Next we take a sequence of points $q_{1}\ldots,q_{N}$ around $q^{\ast}$ (in our application we take the points $q_i$ to be the same as those used to define the affine maps $\gamma_{i}$) and validate that lines joining $q_{i}$ with $q_{i+1}$ (and the line joining $q_{N}$ with $q_{1}$) lie inside the cones $\tilde{Q}_{i}(q_{i})$ and $\tilde{Q}_{i+1}(q_{i+1})$, respectively (and the cones $\tilde{Q}_{N}\left( q_{N}\right) $ and $\tilde{Q}_{1}(q_{1})$, respectively). This is also done with computer assistance.
To check that $f$ is uniformly contracting in $U$ we chevk that for any $i\in \{1,\ldots,N\}$ and any matrix $A\in\left[ Df\left( U_{i}\right) \right] $ we have $\left\vert \det\left( A\right) \right\vert <1.$ This is particularly simple to do in our case, as the computation of determinants of $2\times2$ matrices is straightforward. Once again, this is done with computer assistance.
The condition that $f\left( U\right) $ is a subset of $U$ can be done directly by taking $U_{i}=\bigcup_{k=1}^{m}U_{i,k}$ for some chosen $U_{i,k}$ and checking that for any $i\in \left\{ 1,\ldots ,N\right\} $ and $k\in
\left\{ 1,\ldots ,m\right\} $ there exists a $j\in \left\{ 1,\ldots
,N\right\} $ such that $f\left( U_{i,k}\right) \subset U_{j}$. In our computer assisted approach we do this by taking $$M_{i,k}:=\left[ -R_{i}+(k-1)\frac{2R_{i}}{m},-R_{i}+k\frac{2R_{i}}{m}\right]
\times \left[ -r_{i},r_{i}\right] ,$$$U_{i,k}:=\gamma _{i}\left( M_{i,k}\right) $ and checking that for some $j\in \left\{ 1,\ldots ,N\right\} $ $$\gamma _{j}^{-1}\circ f\circ \gamma _{i}\left( M_{i,k}\right) \subset M_{j}.
\label{eq:fragment-contraction}$$ The computation just described, which is done in the local coordinates, is well suited to interval arithmetic implementation provided the maps $\gamma _{i}$ are well aligned with the invariant curve we wish to established. When $\gamma _{l}$ are affine maps, i.e. $\gamma _{l}\left( q\right) =A_{l}q+q_{l}$ condition ([eq:fragment-contraction]{}) is validated by using Lemma [lem:Df-difference]{} by taking a point $v_{i,k}\in M_{i,k}$ and checking that $$\gamma _{j}^{-1}\circ f\circ \gamma _{i}\left( M_{i,k}\right) \subset \gamma
_{j}^{-1}\circ f\circ \gamma _{i}\left( v_{i,k}\right) +\left( A_{j}^{-1}
\left[ Df\left( \gamma _{i}\left( M_{i,k}\right) \right) \right]
A_{i}\right) \left( M_{i,k}-v_{i,k}\right) .
\label{eq:contraction-lip-curve-validation}$$With a good choice of the matrices $A_{l}$ the matrix $A_{j}^{-1}\left[
Df\left( \gamma _{i}\left( M_{i,k}\right) \right) \right] A_{i}$ can be made close to diagonal, which helps to reduce the wrapping effect of interval arithmetic computations. In this case the derivatives of the local maps are bounded by$$Df_{ji}(q)\in A_{j}^{-1}\left[ Df\left( \gamma _{i}\left( M_{i,k}\right)
\right) \right] A_{i}\qquad \text{for }q\in M_{i,k},
\label{eq:local-map-derivative}$$and these can be used to compute the coefficients $\mu ,\xi $ needed for the rate conditions.
The next lemma is used to validate cone conditions. We express it in a more general setting where the map is defined on $\mathbb{R}^{n}$, as this setting is needed in Section \[sec:homoclinic-maps\]. Below we state the result using the notations $f$ and $M$, but for our purposes here one would apply it for a local map $f_{ji}$ on a set $M_{i,k}$, with the bound on the derivative from ([eq:local-map-derivative]{}). (We remove the subscripts to simplify the statement and to make it more compatible with the story from Section [sec:homoclinic-maps]{}.)
\[lem:cone-cond\]Let $f:\mathbb{R}\times \mathbb{R}^{n-1}\rightarrow
\mathbb{R}\times \mathbb{R}^{n-1}$ and let $M\subset \mathbb{R}^{n}$. Let $a_{1},a_{2}>0$ and $$\begin{aligned}
Q_{1}\left( p\right) &=&\left\{ \left( x,y\right) \in \mathbb{R}\times
\mathbb{R}^{n-1}:a_{1}\left\vert x-\pi _{x}p\right\vert \geq \left\Vert
y-\pi _{y}p\right\Vert \right\} , \\
Q_{2}\left( p\right) &=&\left\{ \left( x,y\right) \in \mathbb{R}\times
\mathbb{R}^{n-1}:a_{2}\left\vert x-\pi _{x}p\right\vert \geq \left\Vert
y-\pi _{y}p\right\Vert \right\} .\end{aligned}$$If for any $v=\left( 1,v_{y}\right) \in Q_{1}\left( 0\right) $ and any $A\in \left[ Df\left( M\right) \right] $ we have $Av\in Q_{2}\left( 0\right) $ then for any $p_{1},p_{2}\in M$ such that $p_{2}\in Q_{1}\left( p_{1}\right)
$ we have $f\left( p_{2}\right) \in Q_{2}\left( f(p_{1})\right) $.
If $p_{1}=p_{2}$ the result is automatic. Assume that $p_{1}\neq p_{2}$. Since $p_{2}\in Q_{1}\left( p_{1}\right) $ we see that $\pi _{x}\left(
p_{1}-p_{2}\right) \neq 0$. By Lemma \[lem:Df-difference\] for some $A\in \left[ Df\left( B\right) \right] $ we have $f\left( p_{1}\right) -f\left(
p_{2}\right) =A\left( p_{1}-p_{2}\right) $. Take $v=\frac{p_{1}-p_{2}}{\left\vert \pi _{x}\left( p_{1}-p_{2}\right) \right\vert }$. Then since $Av\in Q_{2}\left( 0\right) $ we have $\left\Vert \pi _{y}Av\right\Vert \leq
a_{2}\left\vert \pi _{x}Av\right\vert $ so in turn $$\begin{aligned}
\left\Vert \pi _{y}(f\left( p_{1}\right) -f\left( p_{2}\right) )\right\Vert
& =\left\vert \pi _{x}\left( p_{1}-p_{2}\right) \right\vert \left\Vert \pi
_{y}Av\right\Vert \\
& \leq \left\vert \pi _{x}\left( p_{1}-p_{2}\right) \right\vert
a_{2}\left\vert \pi _{x}Av\right\vert \\
& =a_{2}\left\vert \pi _{x}\left(
f\left( p_{1}\right) -f\left( p_{2}\right) \right) \right\vert ,\end{aligned}$$as required.
Heteroclinic connections between fixed points of maps[sec:homoclinic-maps]{}
============================================================================
In this Section we discuss how to prove the existence of heteroclinic orbits between two fixed points of a map. We are interested in the case when one of the fixed points is hyperbolic, and the other fixed point is a stable focus. The heteroclinic orbits will be found in three steps. The first is to establish an attracting neighborhood for which all trajectories converge to the stable focus. This is discussed in Section \[sec:focus\]. The second step is to establish a bound on the unstable manifold of the hyperbolic fixed point. This is described in Section \[sec:Wu\]. Lastly we propagate the unstable manifold by our map. If it reaches the attracting neighborhood of the stable focus then we have established a heteroclinic orbit. This is discussed in Section \[sec:heteroclinic-subsection\].
Establishing attracting fixed points\[sec:focus\]
-------------------------------------------------
In this Section we show how one can obtain the existence of an attracting fixed point within a prescribed neighborhood. We start with a technical lemma.
\[lem:contraction\]Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be $C^{1}$, and $\lambda>0$ be a fixed constant. Let $B$ be a cartesian product of closed intervals in $\mathbb{R}^{n}$ (an $n$-dimensional cube). If for any $A\in\left[ Df\left( B\right) \right] $ the matrix $\lambda Id-A^{\top}A$ is strictly positive definite, then for any $p_{1},p_{2}\in B$ $$\left\Vert f\left( p_{1}\right) -f\left( p_{2}\right) \right\Vert
^{2}<\lambda\left\Vert p_{1}-p_{2}\right\Vert ^{2}.$$
By Lemma \[lem:Df-difference\] we can choose an $A\in\left[ Df\left(
B\right) \right] $ such that $f\left( p_{1}\right) -f\left( p_{2}\right)
=A\left( p_{1}-p_{2}\right) $. (The choice of $A$ depends on $p_{1}$ and $p_{2}$.) We therefore have$$\begin{aligned}
& \lambda\left\Vert p_{1}-p_{2}\right\Vert ^{2}-\left\Vert f\left(
p_{1}\right) -f\left( p_{2}\right) \right\Vert ^{2} \\
& =\lambda\left( p_{1}-p_{2}\right) ^{\top}\left( p_{1}-p_{2}\right) -\left(
f\left( p_{1}\right) -f\left( p_{2}\right) \right) ^{\top }\left( f\left(
p_{1}\right) -f\left( p_{2}\right) \right) \\
& =\lambda\left( p_{1}-p_{2}\right) ^{\top}\left( p_{1}-p_{2}\right) -\left(
p_{1}-p_{2}\right) ^{\top}A^{\top}A\left( p_{1}-p_{2}\right) \\
& =\left( p_{1}-p_{2}\right) ^{\top}\left( \lambda Id-A^{\top}A\right)
\left( p_{1}-p_{2}\right) \\
& >0,\end{aligned}$$ where the last line follows from the fact that $\lambda Id-A^{\top}A$ is strictly positive definite.
To check that a matrix $2\times2$ matrix $C$ is strictly positive definite, it is enough to establish that$$\det\left( C\right) >0\qquad\text{and\qquad trace}\left( C\right) >0.$$
The next lemma establishes that we have an attracting fixed point within a prescribed neighborhood.
\[lem:contraction-main\]Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be $C^{1}$. Let $\lambda\in\left( 0,1\right) $ be a fixed constant. Let $B$ be a cartesian product of closed intervals in $\mathbb{R}^{n}$ (an $n$-dimensional cube). If $f\left( B\right) \subset B$ and for any $A\in\left[
Df\left( B\right) \right] $ the matrix $\lambda Id-A^{\top}A$ is strictly positive definite, then there exists an attracting fixed point of $f$ in $B$. (By attracting" we mean that for any $p\in B$, $f^{k}(p)
$ will converge to the fixed point as $k$ tends to infinity.)
[Since $\lambda\in (0,1)$, by Lemma \[lem:contraction\] we see that $f$ is contracting, so the result follows from the Banach fixed point theorem.]{}
Establishing unstable manifolds of hyperbolic fixed points[sec:Wu]{}
--------------------------------------------------------------------
We now give a method for establishing mathematically rigorous bounds for a local unstable manifold of a hyperbolic fixed point. We restrict to the case where the unstable manifold is of dimension $1$ as this is the case seen in the applications. Our method is based on [@Cap-Lyap], and a more general procedure is found in [@Zgliczynski-cone-cond].
Let $p^{\ast}$ be a hyperbolic fixed point of a $C^{1}$ map $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$. Assume that the unstable eigenspace of $p^{\ast}$ is of dimension $u=1$. Assume that the unstable eigenvalue of $Df(p^{\ast})$ is $\lambda$, with $|\lambda|>1$.
Let $B_{u}$ be a closed interval and let $B_{s}$ be an $s:=n-u=n-1$ dimensional product of closed intervals (a closed cube in $\mathbb{R}^{s}$). Let $B=B_{u}\times B_{s}$ and assume that $p^{\ast}\subset\mathrm{int}B.$ For any point $p\in\mathbb{R}^{n}=\mathbb{R}^{u}\times\mathbb{R}^{s}$ we shall write $p=\left( p_{u},p_{s}\right) $. The subscripts $u$ and $s$ stand of unstable“ and stable”, respectively. This notation is chosen since in our approach these coordinates will be roughly aligned with the unstable/stable eigenspaces of $p^{\ast}$. We will use the notation $\pi_{u}p=p_{u}$ and $\pi_{s}p=p_{s}$ for the projections.
Let $L>0$ be a fixed constant. For any $p=(p_u,p_s)\in\mathbb{R}^{u}\times\mathbb{R}^{s}$ we define a cone centered at $p$ as$$Q\left( p\right) :=\left\{ q=\left( q_{u},q_{s}\right) :\left\Vert
q_{s}-p_{s}\right\Vert \leq L\left\vert q_{u}-p_{u}\right\vert \right\} .
\label{eq:cone-def}$$
We say that $h:B_{u}\rightarrow B_{u}\times B_{s}$ is a horizontal disc in $B $ if it is continuous, if for any $x\in B_{u}$, $\pi_{u}h\left( x\right) =x$ and if $h\left( B_{u}\right) \subset Q\left( h(x)\right) $.
In other words, horizontal discs are one dimensional curves in $B_{u}\times
B_{s}$, which are graphs of Lipschitz functions with the Lipschitz constant $L.$
The next lemma is our main tool for establishing bounds on the unstable manifold of $p^*$.
\[lem:wu\][@Cap-Lyap]Assume that for any $p_{1},p_{2}\in B$ such that $p_{2}\in Q\left( p_{1}\right) $ we have $$f(p_{2})\in Q\left( f\left( p_{1}\right) \right) . \label{eq:cone-cond-hyp}$$ Let $m\in\left( 1,\left\vert \lambda\right\vert \right) $ be a fixed number. Assume that for any $p\in(Q(p^{\ast})\cap B)\setminus\{p^{\ast}\}$ we have$$\left\Vert f\left( p\right) -p^{\ast}\right\Vert >m\left\Vert p-p^{\ast
}\right\Vert . \label{eq:expansion-hyp}$$ Then the unstable manifold of $p^{\ast}$ is contained in $Q\left( p^{\ast
}\right) $. Moreover, there exists a horizontal disc $w^{u}:B_{u}\rightarrow
B$ in $B$ such that the unstable manifold of $p^{\ast}$ is the graph of $w^{u}.$
We now discuss validation of the assumptions (\[eq:cone-cond-hyp\]), (\[eq:expansion-hyp\]) of Lemma \[lem:wu\]. To verify ([eq:cone-cond-hyp]{}) we use Lemma \[lem:cone-cond\] (taking $M=B$ and $a_{1}=a_{2}=L$). To verify (\[eq:expansion-hyp\]) we use the following lemma.
[@Cap-Lyap]Assume that$$\left[ Df\left( B\right) \right] =\left(
\begin{array}{ll}
\mathbf{A} & \mathbf{C}_{12} \\
\mathbf{C}_{21} & \mathbf{B}\end{array}\right) ,$$where $\mathbf{A}=\left[ a_{1},a_{2}\right] $ is a closed interval, $\mathbf{B}$, $\mathbf{C}_{12}$ and $\mathbf{C}_{21}$ are $s\times s$, $1\times s$ and $s\times 1$ interval matrices, respectively. If $$a_{1}-\left\Vert \mathbf{C}_{12}\right\Vert L>m\sqrt{1+L^{2}}$$then for any $p\in (Q(p^{\ast })\cap B)\setminus \{p^{\ast }\}$ we have ([eq:expansion-hyp]{}).
Establishing heteroclinic connections[sec:heteroclinic-subsection]{}
--------------------------------------------------------------------
In this Section we combine the results of Sections \[sec:focus\], \[sec:Wu\] to obtain a heteroclinic orbit between two fixed points of a map $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ in the special case that one of the fixed points is attracting. The existence of the attracting fixed point is established using the tools from Section \[sec:focus\]. The other fixed point is hyperbolic, and has a one dimensional unstable manifold, as in Section \[sec:Wu\]. The next theorem is used to establish homoclinic connections between such two points. [Computer assisted methods of proof for more general configurations are discussed in ]{} [@MR3461310; @MR3281845; @MR3068557; @MR3207723].
\[th:connecting-curve\] Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be $C^{1}$ and $B^{1},B^{2}\subset\mathbb{R}^{n}$ be two sets which are cartesian products of closed intervals in $\mathbb{R}^n$. Assume that the set $B^{1}$ satisfies the assumptions of Lemma \[lem:contraction-main\]. (that is, the assumptions hold for $B=B^{1}$.)
Assume also that $p_{2}^{\ast }\in
B^{2}=B_{u}\times B_{s}$ is a hyperbolic fixed point and that the assumptions of Lemma \[lem:wu\] are satisfied.
1. If there exists an $n\geq 0$ and $\bar{x}\in B_{u}$ such that $f^{n}(Q\left( p_{2}^{\ast }\right) \cap \left\{ p:\pi _{u}p=\bar{x}\right\}
)\subset B^{1}$, then there exists an attracting fixed point $p_{1}^{\ast
}\in B^{1}$ and a homoclinic orbit from $p_{1}^{\ast }$ to $p_{2}^{\ast }$.
2. If there exists an $n\geq 0$, an interval $I\subset B_{u}$, and an $\bar{x}\in I$ such that $\pi _{u}f\left( Q\left( p_{2}^{\ast }\right) \cap
\left\{ p:\pi _{u}p=\bar{x}\right\} \right) \subset I$ and $f^{n}(\left\{
p\in Q\left( p_{2}^{\ast }\right) :\pi _{u}p\in I\right\} )\subset B^{1}$, then there exists a $C^{0}$ curve, invariant under $f$, which joins $p_{1}^{\ast }$ and $p_{2}^{\ast }$.
We start by proving the first claim. We have $B_{u}\times B_{s}=B^{2}$ and by Lemma \[lem:wu\] there exists the function $w^{u}:B_{u}\rightarrow B^{2}
$ which parameterizes the unstable manifold of $p_{2}^{\ast }$. Since $w^{u}$ is a horizontal disc, it has the properties that $\pi _{u}w^{u}(x)=x$ and that for any $x\in B_{u}$, $w^{u}\left( B_{u}\right) \subset Q\left(
w^{u}(x)\right) $; in particular $w^{u}\left( B_{u}\right) \subset Q\left(
p_{2}^{\ast }\right) $. This means that $w^{u}\left( \bar{x}\right) \in
Q\left( p_{2}^{\ast }\right) \cap \left\{ p:\pi _{u}p=\bar{x}\right\} $. This by the assumption of our lemma implies that $$f^{n}(w^{u}\left( \bar{x}\right) )\subset f^{n}\left( Q\left( p_{2}^{\ast
}\right) \cap \left\{ p:\pi _{u}p=\bar{x}\right\} \right) \subset B^{1}.$$ By Lemma \[lem:contraction-main\] the point $p_{1}^{\ast }$ is attracting in $B^{1}$, which means that $$\lim_{k\rightarrow +\infty}f^{k}(w^{u}\left( \bar{x}\right) )=p_{1}^{\ast }.$$ Since $w^{u}$ parameterizes the unstable manifold of $p_{2}^{\ast }$ we also have $$\lim_{k\rightarrow -\infty }f^{k}(w^{u}\left( \bar{x}\right) )=p_{2}^{\ast },$$ which concludes the proof of the first claim.
To prove the second claim observe that $w^{u}\left( \bar{x}\right) \in
Q\left( p_{2}^{\ast }\right) \cap \left\{ p:\pi _{u}p=\bar{x}\right\} $, so $$\pi _{u}f\left( w^{u}\left( \bar{x}\right) \right) \subset \pi _{u}f\left(
Q\left( p_{2}^{\ast }\right) \cap \left\{ p:\pi _{u}p=\bar{x}\right\}
\right) \subset I.$$Let $x_{1}:=\bar{x}$ and $x_{2}:=\pi _{u}f\left( w^{u}\left( \bar{x}\right)
\right) $. The curve $w^{u}\left( \left[ x_{1},x_{2}\right] \right) $ is a fragment of the unstable manifold, which joins the point $w^{u}\left( \bar{x}\right) $ with the point $f\left( w^{u}\left( \bar{x}\right) \right) $. This means that for any $N\in \mathbb{N}$ we can define a continuous curve $$\gamma _{N}:=w^{u}\left( B_{u}\right) \cup
\bigcup_{k=1}^{N}f^{k}(w^{u}\left( \left[ x_{1},x_{2}\right] \right) ),$$which coincides with a fragment of the unstable manifold. (The larger the $N$ the larger the fragment). From our assumption$$f^{n}(w^{u}\left( \left[ x_{1},x_{2}\right] \right) \subset f^{n}(\left\{
p\in Q\left( p_{2}^{\ast }\right) :\pi _{u}p\in I\right\} )\subset B^{1}.$$ Since $f$ is contracting on $B^{1}$, $f^{k}(w^{u}\left( \left[ x_{1},x_{2}\right] \right) $ converge to $p_{1}^{\ast }$ as $k$ tends to infinity. This means that $$\gamma :=\bigcup_{N=1}^{\infty }\gamma _{N}\cup \left\{ p_{1}^{\ast
}\right\}$$is a continuous curve joining $p_{1}^{\ast }$ and $p_{2}^{\ast }$, as required.
Attracting invariant tori of ODEs in $\mathbb{R}^{3}$[sec:tori-3d]{}
====================================================================
Consider a $C^{l}$, $l\ge 1$ vector field $F:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$. We are interested in the dynamics of the ODE$$x^{\prime }=F\left( x\right) . \label{eq:ode-3d}$$Our goal is to establish two types of invariant tori for the flow of (\[eq:ode-3d\]). First, an attracting torus which is either $C^{k}$ smooth, with $k\le l$, or Lipschitz. The second is a torus that results from homoclinic connections of stable/unstable manifolds of periodic orbits.
Both types of tori are established by considering a section $\Sigma \subset \mathbb{R}^{3}$ and a section to section map $P:\Sigma \rightarrow
\Sigma $ induced by the flow of the ODE. The first type of torus follows from the construction of invariant curves by taking $f=P$ and using the tools from Section [sec:contractiing-maps]{}. The second type follows from homoclinic connections between $m$-periodic orbits of $P$, which are established by taking $f=P^{m}$ and using the methods of Section \[sec:homoclinic-maps\].
The following theorem ensures that the invariant circle established using tools from Section \[sec:contractiing-maps\] leads to an invariant Lipschitz torus. Let $\Phi _{t}$ denote the flow induced by (\[eq:ode-3d\]).
\[th:Lip-torus\] Assume that $h^{\ast }:\mathbb{S}^{1}\rightarrow \Sigma
$ is a closed invariant curve (invariant for $f=P$). Let $$\mathcal{T}:=\left\{ \Phi _{t}(v):v\in h^{\ast }(\mathbb{S}^{1}),t\in
\mathbb{R}\right\} .$$
1. If $h^{\ast }$ satisfies cone conditions (in the sense of Definition \[def:closed-curve-cone-cond\]) then $\mathcal{T}$ is a (two dimensional) Lipschitz invariant torus for (\[eq:ode-3d\]).
2. If $h^{\ast }$ is $C^{k}$ then $\mathcal{T}$ is a $C^{k}$ invariant torus for (\[eq:ode-3d\]).
The set $\mathcal{T}$ is a torus by construction. So we need to show that it is Lipschitz in the sense of Definition \[def:lip-manifold\].
Take $p=\Phi _{t}\left( v\right) \in \mathcal{T}$. Assume that $v\in U_{i}$, meaning that $v\in \Sigma $ is in the local coordinates given by $\gamma
_{i} $ on $\Sigma $. (Throughout the reminder of the proof the $p,v$ and $t$ shall remain fixed.) We extend $\gamma _{i}$ to a neighborhood of $p$ by defining $$\tilde{\gamma}_{i}\left( x_{1},x_{2},x_{3}\right) :=\Phi _{t+x_{2}}\left(
\gamma _{i}\left( a_{i}x_{1},x_{3}\right) \right) .$$(Note that in $\tilde{\gamma}_{i}$ we have added a rescaling on the coordinate $x_{1}$. The $a_i$ used for the rescaling are the parameters from the cones $Q_i$ in (\[eq:cones-Lip-torus\]).) Take a small ball $B$ around $\tilde{\gamma}_{i}^{-1}\left( p\right) $ and define $U:=\tilde{\gamma}_{i}\left( B\right) $ and $$\gamma _{U}:=\tilde{\gamma}_{i}|_{B}.$$ Above $U$ and $\gamma_U$ are those needed for Definition \[def:lip-manifold\].
We need that $$U\cap \mathcal{T}\subset \gamma _{U}\left( \mathbf{Q}\left( \gamma
_{U}^{-1}\left( \bar{q}_{1}\right) \right) \cap B\right) .$$This is equivalent to the condition that for any $q_{1},q_{2}\in U\cap \mathcal{T}
$$$\left\Vert \pi _{x_{1},x_{2}}\left( \gamma _{U}^{-1}\left( q_{1}\right)
-\gamma _{U}^{-1}\left( q_{2}\right) \right) \right\Vert \geq \left\Vert \pi
_{x_{3}}\left( \gamma _{U}^{-1}\left( q_{1}\right) -\gamma _{U}^{-1}\left(
q_{2}\right) \right) \right\Vert , \label{eq:lip-temp-prf}$$which is what we show below.
Before proving (\[eq:lip-temp-prf\]) we make the following auxiliary observation. Consider first $\bar{q}_{1},\bar{q}_{2}\in \mathcal{T}\cap \Sigma $. (Here we do not need $\bar{q}_{1},\bar{q}_{2}$ to be in $U$. In fact, if $p$ is far from $\Sigma $ such $\bar{q}_{1},\bar{q}_{2}$ will not be in $U$.) Since $\bar{q}_{1},\bar{q}_{2}\in h^{\ast }(\mathbb{S}^{1})$ from the fact that $h^{\ast }$ satisfies cone conditions it follows that $q_{2}\notin \tilde{Q}_{i}\left( q_{1}\right) $ hence $$\bar{q}_{2}\notin \tilde{Q}_{i}\left( q_{1}\right) =\gamma _{i}\left(
Q_{i}\left( \gamma _{i}^{-1}\left( \bar{q}_{1}\right) \right) \right) =\tilde{\gamma}_{i}\left( \mathbf{Q}\left( \tilde{\gamma}_{i}^{-1}\left( \bar{q}_{1}\right) \right) \right) \cap \Sigma =\gamma _{U}\left( \mathbf{Q}\left( \gamma _{U}^{-1}\left( \bar{q}_{1}\right) \right) \right) \cap \Sigma
. \label{eq:cone-bar-alignment}$$Since $\bar{q}_{1},\bar{q}_{2}\in \Sigma $ $$\pi _{x_{2}}\tilde{\gamma}_{i}^{-1}\left( \bar{q}_{1}\right) =\pi _{x_{2}}\tilde{\gamma}_{i}^{-1}\left( \bar{q}_{2}\right) =-t,$$so (\[eq:cone-bar-alignment\]) implies$$\left\Vert \pi _{x_{1},x_{2}}\left( \gamma _{U}^{-1}\left( \bar{q}_{1}\right) -\gamma _{U}^{-1}\left( \bar{q}_{2}\right) \right) \right\Vert
\geq \left\Vert \pi _{x_{3}}\left( \gamma _{U}^{-1}\left( \bar{q}_{1}\right)
-\gamma _{U}^{-1}\left( \bar{q}_{2}\right) \right) \right\Vert .
\label{eq:Lip-on-Sigma}$$
Then we are ready to show (\[eq:lip-temp-prf\]). Take $q_{1},q_{2}\in U\cap \mathcal{T}$ where $q_{1}=\Phi _{t_{1}}\left(
\bar{q}_{1}\right) ,$ $q_{2}=\Phi _{t_{2}}\left( \bar{q}_{2}\right) \ $for $\bar{q}_{1},\bar{q}_{2}\in \mathcal{T}\cap \Sigma $ and some $t_{1},t_{2}\in
\mathbb{R}$. By (\[eq:Lip-on-Sigma\]) we obtain$$\begin{aligned}
\left\Vert \pi _{x_{1},x_{2}}\left( \gamma _{U}^{-1}\left( q_{1}\right)
-\gamma _{U}^{-1}\left( q_{2}\right) \right) \right\Vert &\geq &\left\Vert
\pi _{x_{1}}\left( \gamma _{U}^{-1}\left( q_{1}\right) -\gamma
_{U}^{-1}\left( q_{2}\right) \right) \right\Vert \\
&=&\left\Vert \pi _{x_{1}}\left( \gamma _{U}^{-1}\left( \bar{q}_{1}\right)
-\gamma _{U}^{-1}\left( \bar{q}_{2}\right) \right) \right\Vert \\
&=&\left\Vert \pi _{x_{1},x_{2}}\left( \gamma _{U}^{-1}\left( \bar{q}_{1}\right) -\gamma _{U}^{-1}\left( \bar{q}_{2}\right) \right) \right\Vert
\\
&\geq &\left\Vert \pi _{x_{3}}\left( \gamma _{U}^{-1}\left( \bar{q}_{1}\right) -\gamma _{U}^{-1}\left( \bar{q}_{2}\right) \right) \right\Vert
\\
&=&\left\Vert \pi _{x_{3}}\left( \gamma _{U}^{-1}\left( q_{1}\right) -\gamma
_{U}^{-1}\left( q_{2}\right) \right) \right\Vert ,\end{aligned}$$as required.
The second claim follows directly from the fact that $(t,x)\rightarrow \Phi
_{t}(x)$ is $C^{k}$.
The next result ensures that a homoclinic connection established using the tools from Section \[sec:homoclinic-maps\] gives an invariant torus for the ODE.
\[lem:resonant-torus\] Assume that $p_{1}^{\ast }$ is a point on a contracting $k$-periodic orbit of the Poincare map $P$ and $p_{2}^{\ast }$ is a point on a hyperbolic $k$-periodic orbit of $P$. If there exists a curve $\gamma $, invariant under $P^{k}$ (i.e. $P^{k}\left( \gamma \right) =\gamma $), which joins $p_{1}^{\ast }$ with $p_{2}^{\ast }$, [ such that $\bigcup_{i=1}^{k} P^i(\gamma)$ is a closed curve, ]{}then $$\mathcal{T}:=\left\{ \Phi _{t}(v):v\in \gamma ,t\in \mathbb{R}\right\}$$is a two dimensional torus invariant under the flow of (\[eq:ode-3d\]).
Continuing the curve $\bigcup_{i=1}^{k} P^i(\gamma)$ along the flow gives a two dimensional torus, as required.
Applications {#sec:examples}
============
In this Section we apply our methods to two explicit examples. The first is the Van der Pol system with periodic external forcing, where we prove the existence of smooth Lipschitz tori by means of the tools from Section \[sec:contractiing-maps\]. The second example is an autonomous vector field introduced by Langford [@Langford] which exhibits a [Neimark-Sacker bifurcation]{}. For this system we establish the existence of $C^0$ tori by means of the tools from Section \[sec:homoclinic-maps\]. An interesting aspect of the second example is that the tori are neither differentiable nor Lipschitz, so that $C^0$ is in fact the most that can be established. [In all our computer assisted proofs we have used the CAPD[^2] library.]{}
Regular tori for the time dependent Van der Pol system
------------------------------------------------------
In this Section we apply the methods from Sections [sec:contractiing-maps]{}, \[sec:tori-3d\] to establish the existence of $C^k$ and Lipschitz tori in a periodically forced nonlinear oscillator. For our example application we consider the Van der Pol [equation]{} with periodic forcing$$x^{\prime \prime }-v(1-x^{2})x^{\prime }+x-\varepsilon \cos \left( t\right)
=0.$$ The system is a canonical example in dynamical systems theory going back to its introduction by Balthasar van der Pol in 1920 as a mathematical model for an electrical circuit containing a vacuum tube [@vanDerPol_1]. For almost a century the system has been studied as a simple and physically relevant example of a differential equation exhibiting spontaneous nonlinear oscillations. Later van der Pol himself considered the circuit when driven by a periodic external forcing [@vanDerPolNature_forced], and saw what would today be called an attracting invariant torus. For a much more complete theoretical discussion of the dynamics of the forced Van der Pol system, as well as a thorough review of the literature, we refer to the classic study of [@gukenheimerVanDerPol]. The interested reader is referred also to the works of [@deClassified; @vanDerPolPlasma] for interesting applications of the forced system.
We prove the existence of a smooth and attracting invariant torus for the following pairs of parameters$$\left( v,\varepsilon \right) \in \left\{ \left( 0.1,0.002\right) ,\left(
0.2,0.005\right) ,\left( 0.3,0.01\right) ,\left( 0.4,0.015\right) ,\left(
0.5,0.05\right) ,\left( 1,0.1\right) \right\} . \label{eq:VanDerPol-params}$$To do so we consider the system in the extended phase space$$\begin{aligned}
x^{\prime } &=&y, \notag \\
y^{\prime } &=&v(1-x^{2})y-x+\varepsilon \cos \left( t\right) ,
\label{eq:Van-der-Pol-extended} \\
t^{\prime } &=&1, \notag\end{aligned}$$and take the time section $\Sigma =\left\{ t=0\right\} $. We consider the time shift map $f^{v,\varepsilon }:\Sigma \rightarrow \Sigma $ defined as $f^{v,\varepsilon }\left( x,y\right) =\Phi _{2\pi }^{v,\varepsilon
}\left( x,y,0\right) $, where $\Phi _{s}^{v,\varepsilon }$ is the flow induced by (\[eq:Van-der-Pol-extended\]).
For each pair $\left( v,\varepsilon \right) $ of parameters we take a sequence of points $\left\{ p_{i}^{v,\varepsilon }\right\}
_{i=1}^{N^{v,\varepsilon }}\subset \Sigma $, which lie approximately on the intersection of the torus with $\Sigma $. We draw these points in Figure [fig:VenDerPol]{}. (These points are computed numerically. We choose them so that they are roughly uniformly spread along the curves.)
We then choose local coordinates $\gamma _{i}^{v,\varepsilon }:\left[
-R_{i}^{v,\varepsilon },R_{i}^{v,\varepsilon }\right] \times \left[
-r^{v,\varepsilon },r^{v,\varepsilon }\right] \rightarrow \mathbb{R}^{2}$ as $$\gamma _{i}^{v,\varepsilon }\left( x,y\right) =q_{i}^{v,\varepsilon
}+A_{i}^{v,\varepsilon }\left(
\begin{array}{c}
x \\
y\end{array}\right) \qquad \text{for }i=1,\ldots ,N^{v,\varepsilon },$$with$$\begin{aligned}
q_{1}^{v,\varepsilon } &=&\frac{1}{2}\left( p_{2}^{v,\varepsilon
}+p_{N}^{v,\varepsilon }\right) , \\
q_{N}^{v,\varepsilon } &=&\frac{1}{2}\left( p_{1}^{v,\varepsilon
}+p_{N-1}^{v,\varepsilon }\right) , \\
q_{i}^{v,\varepsilon } &=&\frac{1}{2}\left( p_{i+1}^{v,\varepsilon
}+p_{i-1}^{v,\varepsilon }\right) ,\qquad \text{for }i=2,\ldots
,N^{v,\varepsilon }-1,\end{aligned}$$[ $$\begin{aligned}
A_{1}^{v,\varepsilon } &=&\frac{1}{R_1^{v,\varepsilon }}\left(
\begin{array}{ll}
\pi _{x}\frac{1}{2}\left( p_{2}^{v,\varepsilon }-p_{N}^{v,\varepsilon
}\right) & -\pi _{y}\frac{1}{2}\left( p_{2}^{v,\varepsilon
}-p_{N}^{v,\varepsilon }\right) \\
\pi _{y}\frac{1}{2}\left( p_{2}^{v,\varepsilon }-p_{N}^{v,\varepsilon
}\right) & \pi _{x}\frac{1}{2}\left( p_{2}^{v,\varepsilon
}-p_{N}^{v,\varepsilon }\right)\end{array}\right) , \\
A_{N}^{v,\varepsilon } &=&\frac{1}{R_{N}^{v,\varepsilon }}\left(
\begin{array}{ll}
\pi _{x}\frac{1}{2}\left( p_{1}^{v,\varepsilon }-p_{N-1}^{v,\varepsilon
}\right) & -\pi _{y}\frac{1}{2}\left( p_{1}^{v,\varepsilon
}-p_{N-1}^{v,\varepsilon }\right) \\
\pi _{y}\frac{1}{2}\left( p_{1}^{v,\varepsilon }-p_{N-1}^{v,\varepsilon
}\right) & \pi _{x}\frac{1}{2}\left( p_{1}^{v,\varepsilon
}-p_{N-1}^{v,\varepsilon }\right)\end{array}\right) , \\
A_{i}^{v,\varepsilon } &=&\frac{1}{R_{i}^{v,\varepsilon }}\left(
\begin{array}{ll}
\pi _{x}\frac{1}{2}\left( p_{i+1}^{v,\varepsilon }-p_{i-1}^{v,\varepsilon
}\right) & - \pi _{y}\frac{1}{2}\left( p_{i+1}^{v,\varepsilon
}-p_{i-1}^{v,\varepsilon }\right) \\
\pi _{y}\frac{1}{2}\left( p_{i+1}^{v,\varepsilon }-p_{i-1}^{v,\varepsilon
}\right) & \pi _{x}\frac{1}{2}\left( p_{i+1}^{v,\varepsilon
}-p_{i-1}^{v,\varepsilon }\right)\end{array}\right) ,\end{aligned}$$]{} for $ i=2,\ldots ,N^{v,\varepsilon }-1,$ and$$\begin{aligned}
R_{1}^{v,\varepsilon } &=&\left\Vert \frac{1}{2}\left( p_{2}^{v,\varepsilon
}-p_{N}^{v,\varepsilon }\right) \right\Vert , \\
R_{N}^{v,\varepsilon } &=&\left\Vert \frac{1}{2}\left( p_{1}^{v,\varepsilon
}-p_{N-1}^{v,\varepsilon }\right) \right\Vert , \\
R_{i}^{v,\varepsilon } &=&\left\Vert \frac{1}{2}\left(
p_{i+1}^{v,\varepsilon }-p_{i-1}^{v,\varepsilon }\right) \right\Vert ,\qquad
\text{for }i=2,\ldots ,N^{v,\varepsilon }-1.\end{aligned}$$The choice is motivated by the fact that the set $\gamma _{i}\left( \left[
-R_{i}^{v,\varepsilon },R_{i}^{v,\varepsilon }\right] \times \left\{
0\right\} \right) $ is a line connecting the points $p_{i-1}^{v,\varepsilon
} $ and $p_{i+1}^{v,\varepsilon }$. As a result we obtain overlapping sets $U_{i}^{v,\varepsilon }:=\gamma _{i}\left( \left[ -R_{i}^{v,\varepsilon
},R_{i}^{v,\varepsilon }\right] \times \left[ -r^{v,\varepsilon
},r^{v,\varepsilon }\right] \right) $ which cover the true invariant curve for $f^{v,\varepsilon }$. We establish the existence of the curve using the method outlined in Section \[sec:Lip-curves-validation\]. We outline some of the aspects of our computer assisted proof below.
For the first five parameters from (\[eq:VanDerPol-params\]) it turned out to be enough to consider $N^{v,\varepsilon }=1000$. For these five parameters we have chosen $r^{v,\varepsilon }=5\cdot 10^{-4}$, and we have chosen the slope of the cones (\[eq:cones-Lip-torus\]) as $a_{i}=0.3$. Each set $U_i$ was additionally subdivided into $m=6$ parts for the validation of condition (\[eq:fragment-contraction\]). This condition was the most time consuming part of the proof. The computer assisted proof for each of the four tori took under a minute and a half on a single 3 GHz Intel i7 Core processor. [(The parameters $r_i,a_i$ and $m$ were chosen by trial and error.)]{}
The validation of (\[eq:fragment-contraction\]) was based on ([eq:contraction-lip-curve-validation]{}) so as a [byproduct]{} from ([eq:local-map-derivative]{}) we obtained bounds on the derivatives of the local maps, which allows us to compute the bounds $\mu ,\xi $ needed for the rate conditions (see Definition \[def:rate-cond\]). By using Theorem \[th:Ck\] we validate the $C^{k}$ regularity for the torus at parameter pairs $\left( v,\varepsilon \right) =\left( 0.1,0.002\right) ,\left(
0.2,0.005\right) ,\left( 0.3,0.01\right) $ as $k=9,5,2$, respectively. For the remaining parameters from (\[eq:VanDerPol-params\]) we only obtained that the tori are Lipschitz. This is due to the fact that the higher the $v$ the less ‘uniform’ the dynamics on the torus. What we mean by this is that there are regions on the torus in which the dynamics restricted to the torus is expanding or contracting (the torus does not behave uniformly as a central coordinate). This affects the bounds on parameters $\mu ,\xi $ (the second parameter in particular) which results in weaker regularity bounds obtained from our method.
Non-uniformity of the dynamics on the torus for higher $v$ has also made the proof for the parameters $\left( v,\varepsilon \right)
=\left( 1,0.1\right) $ more computationally demanding. We take $N^{v,\varepsilon }=5000$ and $m=20$, covering the curve with a larger number of fragments. We also take $r^{v,\varepsilon }=2\cdot
10^{-5} $, and the slope of the cones (\[eq:cones-Lip-torus\]) were taken as $a_{i}=0.1$. With $a_i$ a larger number, but using smaller sets $U_{i}$ allows us to validate the needed conditions in this example. The computer assisted proof for this parameter pair took 31 minutes on a single 3GHz Intel i7 Core processor.
This demonstrates the following weakness of our method. It performs well if the dynamics on the torus is uniform. If it is not, then proofs require many subdivisions. In the next Section we consider another example in which this problem is even more visible.
In the computer assisted proof we can use a small interval of parameters instead of a single parameter value. By invoking parallel computations on a cluster, one could use our approach to cover whole parameter ranges.
We finish with the comment that by Theorem \[th:Lip-torus\] the invariant curves established for the map $f^{v,\varepsilon }$ lead to two dimensional invariant tori of ([eq:Van-der-Pol-extended]{}).
![Intersections of the invariant Lipschitz tori for the Van der Pol system with the $t = 0$ section for each parameter from (\[eq:VanDerPol-params\]). The smaller the $\protect\mu $ the more circular/smooth the curve.[]{data-label="fig:VenDerPol"}](fig_VenDerPol.pdf){height="5.5cm"}
Resonant tori in the Langford system
------------------------------------
Consider the autonomous vector field $F:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ given by the formula $$F(x,y,z)=\begin{pmatrix}
(z-\beta )x-\delta y \\
\delta x+(z-\beta )y \\
\gamma +\alpha z-\frac{z^{3}}{3}-(x^{2}+y^{2})(1+\epsilon z)+\zeta zx^{3}\end{pmatrix},$$where $\epsilon =0.25$, $\gamma =0.6$, $\delta =3.5$, $\beta =0.7$, $\zeta
=0.1$, and $\alpha =0.95$ are the ‘classical’ parameter values. This system is a toy model for dissipative vortex dynamics, or for a rotating viscus fluid, and was first studied by Langford in [@Langford]. We define a section $\Sigma =\left\{ x=0\right\} $, and the first return time section to section map $P:\Sigma \rightarrow \Sigma $. We treat the parameter $\alpha $ as our bifurcation parameter. For all $\alpha \in \lbrack 0,0.95]$ there exists a fixed point in $\Sigma $ of $P^{2}$ which corresponds to a periodic orbit $\tau $ of the ODE. The periodic orbit has complex conjugate Floquet multipliers which are stable for small $\alpha $ but which later cross the unit circle, loosing stability in a [Neimark-Sacker bifurcation]{} [@Aizawa], which occurs at $\alpha \approx 0.69714$ and gives birth to a $C^{k}$ torus. We give a plot of such torus for one of the parameters in Figure \[fig:Aizawa075\].
![At $\protect\alpha =0.75$ we have an attracting limit cycle of $P^2$ on $\Sigma $ (figure on the left) which is the intersection of the two dimensional $C^k$ torus of the ODE with $\Sigma \cap \{y>0\}$. On the right we plot half of the torus. In black we have both components of the torus intersection with $\Sigma$; one for $y<0$ and the other for $y>0$.[]{data-label="fig:Aizawa075"}](fig_aizawa_075.pdf){width="100.00000%" height="4cm"}
Additionally there exist two period six orbits of $P$ in $\Sigma .$ One is a saddle periodic orbit which we denote as $c_{h}$, and the other is a stable focus periodic orbit, which we denote as $c_{s}$. (We use the subscript $h$ to stand for ‘hyperbolic’ and $s$ to stand for ‘stable’.) We found that one branch of $W^{u}(c_{h})$ wraps around the torus, while the other reaches $c_{s}$; see left plot in Figure \[fig:Aizawa4\]. This happens right until $\alpha \approx 0.822$.
As we increase further our bifurcation parameter $\alpha $, our invariant two-dimensional $C^{k}$ torus bifurcates to a $C^{0}$ torus. This bifurcation happens by $c_{h},c_{s}$ colliding with the torus. Another way of interpreting this is by looking at what happens with $W^{u}(c_{h})$ and $W^{s}(c_{h})$. Before the bifurcation one branch of $W^{u}(c_{h})$ goes inside, wrapping around the torus; see left plot from Figure \[fig:Aizawa3\]. After the bifurcation both branches of $W^{u}(c_{h})$ lead to $c_{s}$; see right plot from Figure \[fig:Aizawa3\] and also Figure \[fig:Aizawa5\]. Since after the bifurcation the tori include the periodic orbits, we refer to them as *resonant tori*.
For parameters between the case of $C^{k}$ tori and the case of resonant tori we have transverse intersections of $W^{u}(c_{h})$ and $W^{s}(c_{h})$ as seen in Figure \[fig:Aizawa4\]. This transverse intersection leads to the presence of Smale horseshoes and thus chaotic dynamics. These transverse intersections are born and terminated at parameters for which we have tangential intersections of $W^{s}(c_{h})$ with $W^{u}(c_{h})$.
Our objective is to apply the tools from Section [sec:heteroclinic-subsection]{} and prove the existence of a resonant tori. Below we describe the proof when $\alpha =0.85$. The resonant torus in our proof is depicted in Figure \[fig:Aizawa5\].
We emphasize that the resonant torus from Figure \[fig:Aizawa5\] is continuous, but *not* $C^{k}$ for $k>0$. In fact, it is not even Lipschitz due to the rotation around the attractinmg periodic orbit.
We now outline the details of the proof. Consider the map $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ $$f:=P^{6}.$$First we find bounds on two fixed points $p_{1}^{\ast },p_{2}^{\ast }\in \Sigma $ of $f$, for which $p_{1}^{\ast }\in
c_{s}$ and $p_{2}^{\ast }\in c_{h}$. We do this using the following shooting method. Consider$$G:\underset{6}{\underbrace{\mathbb{R}^{2}\times \ldots \times \mathbb{R}^{2}}}\rightarrow \mathbb{R}^{12},$$defined as$$G\left( q_{1},\ldots ,q_{6}\right) :=\left( P\left( q_{6}\right)
-q_{1},P\left( q_{1}\right) -q_{2},P\left( q_{2}\right) -q_{3},\ldots
,P\left( q_{5}\right) -q_{6}\right) .$$(Note that $q_{i}=\left( y_{i},z_{i}\right) \in \Sigma $, for $i=1,\ldots 6$.) Establishing that$$G\left( q_{1},\ldots ,q_{6}\right) =0,$$gives points $q_{1},\ldots ,q_{6}$ on a period $6$ orbit of $P$.
We using the interval Newton method (Theorem \[th:interval-Newton\]) to validate that the point $\left( q_{1},\ldots ,q_{6}\right) $ is in $\prod_{i=1}^{12}I_{i}$, for some closed intervals $I_{i}$, for $i=1,\ldots
,12$. Once this is done, the two dimensional box $I_{1}\times I_{2}$ is an enclosure of $q_{1}$ which is a fixed point of $P^{6}.$ We find that $$\begin{aligned}
p_{1}^{\ast } &\in &\left(
\begin{array}{r}
0.611160359286522+4.6\cdot 10^{-13}\cdot \left[ -1,1\right] \\
-0.104496536895459+8.8\cdot 10^{-13}\cdot \left[ -1,1\right]\end{array}\right) , \\
p_{2}^{\ast } &\in &\left(
\begin{array}{c}
0.413216691560642+2.5\cdot 10^{-13}\cdot \left[ -1,1\right] \\
0.150271844775546+9.4\cdot 10^{-13}\cdot \left[ -1,1\right]\end{array}\right) .\end{aligned}$$
Now take the two $2\times 2$ matrices $$\begin{aligned}
A_{1} &:&=\left(
\begin{array}{ll}
0.953174 & -0.0468255 \\
0.169128 & 0.2639\end{array}\right) , \\
A_{2} &:&=\left(
\begin{array}{ll}
0.0138304 & -1 \\
-1 & 0.674926\end{array}\right),\end{aligned}$$and define the local maps $f_{1}$ and $f_{2}$ around $p_{1}^{\ast }$ and $p_{2}^{\ast }$, respectively, as$$f_{i}\left( q\right) :=A_{i}^{-1}\left( f\left( A_{i}q+p_{i}^{\ast }\right)
-p_{i}^{\ast }\right) \qquad \text{for }i=1,2.$$Note that $p_{1}^{\ast }$ and $p_{2}^{\ast }$ are shifted to zero in their respective local coordinates. The choices of $A_{1}$ and $A_{2}$ are such that they put the derivatives of $f_{1}$ and $f_{2},$ respectively, at zero approximately in Jordan form. Such $A_{1}$ and $A_{2}$ are computed using standard numerics (we do not need interval arithmetic validation at this stage). We consider the two cubes $B^{1}$ and $B^{2}$ in $\mathbb{R}^{2}$ defined by $$\begin{aligned}
B^{1} &:=&[-0.0005,0.0005]\times \lbrack -0.0005,0.0005], \\
B^{2} &:=&[-0.0001,0.0001]\times \lbrack -2\cdot 10^{-8},2\cdot 10^{-8}].\end{aligned}$$With computer assistance we established that zero is an attracting fixed point of $f_{1}$ in $B^{1}$. This was done in interval arithmetic by using Lemma \[lem:contraction-main\]. We also established that the unstable manifold of zero for the map $f_{2}$ is contained in the cone $Q(0)$ of the form (\[eq:cone-def\]) with $L=2\cdot 10^{-4}$. We did this by using Lemma \[lem:wu\]. The validation of the assumptions of Lemmas [lem:contraction-main]{} and \[lem:wu\] was based on interval arithmetic bounds on the derivative of the map. Here we write out the bounds we have obtained: $$\begin{aligned}
\left[ Df_{1}\left( B^{1}\right) \right] &=&\left(
\begin{array}{ll}
\lbrack 0.150243,0.220614] & [-0.561824,-0.521109] \\
\lbrack 0.41934,0.663593] & [0.10723,0.263629]\end{array}\right) , \\
\left[ Df_{2}\left( B^{2}\right) \right] &=&\left(
\begin{array}{ll}
\lbrack 2.16813,2.16975] & [-0.000485,0.000485] \\
\lbrack -0.000352,0.000351] & [0.195584,0.195806]\end{array}\right) .\end{aligned}$$
We now consider$$\bar{x}=4.5\cdot 10^{-5}.$$With computer assistance we have validated that $\pi _{u}f_{2}\left(
Q(0) \cap \{p:\pi_x p=\bar{x}\}\right) \subset I:=[\bar{x},0.0001]$ and that for $n=25$ $$A_{1}^{-1}\left( f^{n}\left( A_{2} \left(
Q(0) \cap \{p:\pi_x p\in I\}\right)
+p_{2}^{\ast }\right) -p_{1}^{\ast }\right) \subset B^{1}.
\label{eq:fundamental-domain-condition}$$This by Theorem \[th:connecting-curve\] establishes the existence of an invariant curve for $f$, which joins $p_{1}^{\ast }$ and $p_{2}^{\ast }$. The resonant torus from Figure \[fig:Aizawa5\] follows from Lemma \[lem:resonant-torus\].
The condition (\[eq:fundamental-domain-condition\]) required $n=25$ iterates of the map $f$, which is $6n=150$ iterates of the map $P$; this requires a long integration time of the ODE. This was the most time consuming part of the computer assisted proof, since it required a subdivision of $Q(0) \cap \{p:\pi_x p\in I\}$ into $200$ fragments and checking (\[eq:fundamental-domain-condition\]) for each of them separately.
The computer assisted proof of the resonant torus for $\alpha =0.85$ took under $6$ minutes on a single 3GHz Intel i7 Core processor.
There is nothing particularly special about the parameter $\alpha =0.85$. Using the same techniques, we have obtained proofs of resonant tori for other parameters, including $\alpha =0.835$ for which we have the plot of the torus in the right hand plot from Figure \[fig:Aizawa3\].
We finish this Section by commenting on difficulties we have encountered when trying to validate the $C^{k}$ tori for smaller parameters $\alpha $. We ran into these when considering for instance $\alpha =0.75$ for which the torus is plotted in Figure \[fig:Aizawa075\]. Judging by the shape of the torus it would seem to be well suited for the validation methods of Section \[sec:contractiing-maps\]. Our problem in this particular example is that the dynamics near the torus are not uniformly contracting. There are some regions of expansion, and other regions of strong contraction. In total the torus is an attractor, but it is not a uniform one and the methods of Section \[sec:contractiing-maps\] do not apply. When $\alpha =0.75$ such uniform contraction is achieved for $f=P^{16}$. We have been able to enclose the curve in a set $U$ which consists of $10000$ cubes and validate that $f$ is contracting in $U$ and that $f(U)\subset U$. (Such validation has been very time consuming and took 5 hours and 27 minutes on a single 3GHz Intel i7 Core processor.) This establishes the existence of an invariant set in $U$, but does not prove that this set is a torus. Using the results from [@CK] one obtains that this invariant set projects surjectively onto a torus, but other than this we do not get any information about its topology.
To prove that the invariant set is a torus we would need to also validate cone conditions. The fact that $f$ consists of $16$ iterates of $P$ leads to long integration of the ODE. This resulted in insufficiently sharp estimates on the derivative of $f$ and we were unable to validate cone conditions. An additional difficulty we have encountered is that in the neighborhood of the invariant curve of $f$ we do not have ‘vertical’ contraction towards the curve, but also strong twist dynamics. This makes the validation of cone conditions even harder, since the angle between the center and the stable bundles becomes very small.
[ To be more precise, if on the section $\Sigma$ we choose the tangent vector and the normal vector to the invariant circle as the basis for our coordinates, then for a point $p$ from the invariant circle the derivative of $f=P^{16}$ is of the form $$Df(p)=\left(
\begin{array}
[c]{cc}1 & \delta\\
0 & \lambda
\end{array}
\right),$$ with $\left\vert \lambda\right\vert <1$, but $\left\vert \lambda\right\vert
\approx1$, and $\left\vert \delta\right\vert \gg1.$ This means that in order to validate cone conditions, the fact that $(x,y)\in Q(0)$, i.e. $\left\vert
y\right\vert <a\left\vert x\right\vert $, should imply that $Df(p)(x,y)\in
Q(0)$, i.e. $\left\vert \lambda y\right\vert < a \left\vert x+y\delta\right\vert
$. The choice of $y=-\delta^{-1}x$ will result in zero on the right hand side, which means that a necessary condition is to have $a\leq\left\vert
\delta\right\vert ^{-1}$. This means that in the case when $\delta$ is a large number (we have a strong twist), we have to choose small $a$, which means that we need to use sharp cones. The smaller the $a$ the more difficult is the validation of cone conditions. On top of that, we need a large number of iterates of $P$ to compute $f$, which leads to long integration times, resulting in insufficiently accurate bounds on the derivatives of $f$ in order to validate the cone conditions. ]{}
[ We encounter exactly the same problem when the parameters of the system are close to the Neimark-Sacker bifurcation. In such setting the torus is not strongly attracting, meaning that a large number of iterates of $P$ is needed for the contraction to be strong enough so that we can validate $f(U)\subset U$. This results in the appearance of a large twist parameter $\delta$ in derivatives of $f$, and we run into identical problems as those described above. ]{}
This demonstrates that our method has limitations in the presence of twist and nonuniform contraction of the invariant tori. Developing a computer assisted proof strategy which overcomes these difficulties would be an interesting future project. Another interesting project would be to formulate functional analytic methods for studying rotational invariant tori which could possibly lead to sharp or sharper regularity bounds.
Acknowledgements
================
We would like to thank the anonymous Reviewers for their comments, suggestions and corrections, which helped us improve our paper.
Received xxxx 20xx; revised xxxx 20xx.
[^1]: For the purposes of this Section we could assume that $f$ is a homeomorphism, however the validation of the required assumptions is easier using the derivative of $f$. This is why we assume $C^{1}$ smoothness straightaway.
[^2]: Computer Assisted Proofs in Dynamics: http://capd.ii.uj.edu.pl/
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A detailed level collisional-radiative model of the E1 transition spectrum of Ca-like W$^{54+}$ ion has been constructed. All the necessary atomic data has been calculated by relativistic configuration interaction (RCI) method with the implementation of Flexible Atomic Code (FAC). The results are in reasonable agreement with the available experimental and previous theoretical data. The synthetic spectrum has explained the EBIT spectrum in 29.5-32.5 Å, while several new strong transitions has been proposed to be observed in 18.5-19.6 Å for the future EBIT experiment with electron density $n_e$ = $10^{12}$ cm$^{-3}$ and electron beam energy $E_e$ = 18.2 keV.'
address:
- 'Key Laboratory of Atomic and Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China'
- 'Department of physics, Sophia University, Tokyo 102-8554, Japan'
- 'National Institute for Fusion Science, Toki, Gifu 509-5292, Japan'
- 'Institute for Laser Science, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan'
author:
- Xiaobin Ding
- Jiaoxia Yang
- Fumihiro Koike
- Izumi Murakami
- Daiji Kato
- Hiroyuki A Sakaue
- Nobuyuki Nakamura
- Chenzhong Dong
bibliography:
- 'abc.bib'
title: 'Theoretical investigation on the soft X-ray spectrum of the highly-charged W$^{54+}$ ions'
---
Collisional-radiative model,Ca-like Tungsten,Relativistic configuration interaction
Introduction
============
Tungsten (W) and its alloy had been used as the armor plate material for the plasma facing component in the divertor region of ITER (International Thermonuclear Experimental Reactor Tokamak) and the other magnetic confinement fusion reactors because of their high melting point, low sputtering yield, and low tritium retention rate[@0029-5515-45-3-007; @Matthews2009934]. However, tungsten impurity ions might be produced during the plasma-wall interaction in the edge and then they might be transported to the high-temperature core plasma region. In the core plasma, these tungsten impurities can be ionized further to highly charged ions and radiate high energy photons. Therefore, the large radiation loss can be caused by these highly charged tungsten impurity ions, which will lead to the plasma disruption if the relative concentration of W ions in the core plasma is higher than about 10$^{-5}$[@Radtke2001]. Monitoring and controlling the flux of the highly charged W impurity ions are crucial to the success of the fusion[@1402-4896-2009-T134-014022]. Thus a thorough knowledge of the atomic properties of tungsten ions were strongly needed by the magnetic confinement fusion research. Furthermore, the spectra of tungsten impurity ions observed from the fusion plasma also provide plenty of important information about the fusion plasma parameters such as electron density, electron temperature, and ion temperature. Therefore, it can also be used to diagnose the plasma.
There are many research works related to the energy levels and transition properties of tungsten in various ionization stage in the past several decades[@Fei2014; @Kramida2006; @Kramida2009; @Kramida2011; @20072_S1060; @0953-4075-45-3-035003], where only a few studies have focused on W$^{54+}$ ion[@0953-4075-50-4-045004; @Dipti2015Electron; @PhysRevA.87.062505; @Ralchenko2008; @0953-4075-43-7-074026; @0953-4075-44-19-195007; @PhysRevA.83.032517]. Y. Ralchenko *et al.* used the electron beam ion trap (EBIT) to observe the M1 spectrum from 3d$^n$(n=1-9) ground state fine structure multiplets of Co-like W$^{47+}$ to K-like W$^{55+}$ ions and a non-Maxwellian collisional-radiative model (CRM) was used to analyze the observed spectrum[@PhysRevA.83.032517]. U. I. Safronova and A. S. Safronova calculated the wavelength and transition rates of the magnetic dipole (M1) and electric quadrupole (E2) transitions between the multiplets of the ground state configuration (\[Ne\]3s$^2$3p$^6$3d$^2$) of W$^{54+}$ ion by using relativistic many-body perturbation theory (RMBPT)[@0953-4075-43-7-074026]. P. Quinet used a full relativistic Dirac-Fock method to calculate the line wavelengthes and transition rates of the forbidden transitions within the 3p$^k$(k=1-5) and 3d$^n$(n=1-9) ground state configuration multiplets from Al-like W$^{61+}$ to Co-like W$^{47+}$ ions[@0953-4075-44-19-195007]. Dipti *et al.* calculated the excitation energies and the electron collisional excitation cross sections of Ca-like W$^{54+}$ ion by relativistic distorted wave theory[@Dipti2015Electron]. C. F. Fischer *et al.* found the core correlation is very important for the transition energies of 3d$^k$ configuration in tungsten ions[@Atomcs].
T. Lennartsson *et al.* observed the electric dipole (E1) transitions from the excited states \[Ne\]3s$^2$3p$^5$3d$^3$ to the ground state \[Ne\]3s$^2$3p$^6$3d$^2$ of W$^{54+}$ ion in the wavelength range of 26.3-43.5Å in the Lawrence Livermore National Laboratory (LLNL) EBIT at the electron beam energy of 18.2 keV, and a CRM was applied to explain the observed spectrum[@PhysRevA.87.062505]. Ding and his collaborators have performed a calculation on the E1, E2, M1, M2 transitions between \[Ne\]3s$^2$3p$^5$3d$^3$ and \[Ne\]3s$^2$3p$^6$3d$^2$ levels of W$^{54+}$ ion by using an MCDF method with the restricted active space method; the relativistic and electron correlation effects as well as the Breit interaction and some Quantum Electrodynamic effects have been taken into account[@0953-4075-50-4-045004; @1701-05504; @Ding2017]. The results are in reasonable agreement with available experimental data for both M1 and E1 transition lines. Several strong E1 transitions have been predicted. Some of these strong transitions are in good agreement with the observation from the EBIT, while others were not observed even in the similar wavelength range with similar transition rates. And some transitions with wavelength in 18.5-19.6 Å are suggested to be observed in the future experiment.
The present paper focuses on the explanation of the E1 transition spectrum of W$^{54+}$ ion from the EBIT with the electron density $n_e = 10^{12}$ cm$^{-3}$ and the energy of electron beam $E_e$ = 18.2 keV by CRM. The present paper is constructed as follows. In section 2, the theory of CRM and the calculation of the necessary atomic data are described. The result of the present calculation and the discussion are given in Section 3. Finally, the conclusion is presented in section 4.
Theoretical method
==================
The CRM is one of the most useful simulation methods for the spectrum from the optically thin and isotropic plasma. It has been widely used to study the spectrum of highly charged ions in X-ray, VUV and visible region[@0953-4075-48-14-144028; @Ding2012Collisional; @0953-4075-47-17-175002]. In order to carry out the analysis of the fine structure of the spectrum, a detailed-level CRM should be used[@Ding2016Collisional; @Ding2012Collisional; @0953-4075-48-14-144029; @0953-4075-47-17-175002].
The spectral intensity $I_{i,j}( \lambda)$ of a transition from the upper level $i$ to the lower level $j$ with wavelength $\lambda$ can be defined as: $$\begin{aligned}
\label{eq1}
I_{i,j}(\lambda) \propto n(i) A(i,j)\phi(\lambda).\end{aligned}$$ where $A(i,j)$ is the transition rate for the transition from the energy level $i$ to $j$, which can be obtained by experimental observations or by accurate theoretical calculations. The function $\phi(\lambda)$ is the normalized line profile, which was taken as a Gaussian profile to include the effect of Doppler, natural, collisional and instrumental broadenings in the present work. The notation $n(i)$ is the population of the ions in the excited upper level $i$, which was determined by the detailed atomic physics processes in the plasma, e.g., spontaneous radiative transitions, collisional excitation and deexcitation, collisional ionization, radiative recombination and three-body recombination etc.. These atomic processes can be calculated by using an appropriate theory.
For the plasma in the EBIT, which is in low electron density and the energy distribution of the free electron is almost mono-energy, the radiative recombination, three-body recombination, electron collisional ionization, and dielectronic recombination processes are expected to be negligible in this situation. Only the electron collisional excitations, deexcitations, and spontaneous radiative transitions processes will determine the population $n(i)$ of the excited upper level $i$. For a specifically excited level $i$, the temporal development of the population $n(i)$ can be obtained by solving the collisional-radiative rate equations: $$\begin{aligned}
\label{eq2}\nonumber
\frac{{\rm d}}{{\rm d} t}n(i)&=\sum_{j<i}C(j,i)n_en(j) \\
&-[{\sum_{j<i}F(i,j)n_e+A(i,j)}+\sum_{j>i}C(i,j)n_e]n(i)\\\nonumber
&+\sum_{j>i}[F(j,i)n_e+A(j,i)]n(j).
\end{aligned}$$ where ${n_e}$ is the electron density of the plasma, $C(i,j)$ and $F(j,i)$ are collisional excitation and deexcitation rates coefficient from the level $j$ to $i$, respectively. These rate coefficient can be calculated by convoluting the cross section of the collisional (de) excitation processes with the free electron energy distribution function, which can be assumed as Maxwellian distribution, for the plasma in local thermodynamic equilibrium (LTE) with the electron temperature $T_e$. However, the electron energy distribution in the electron beam of the EBIT is more like mono-energy distribution function instead of Maxwellian distribution. Thus, the rate coefficient for the atomic processes in the EBIT may be calculated by taking the electron energy distribution as the $\delta$ function. These rate equations are solved under the Quasi-Steady-State (QSS) approximation, i.e., $ {\rm d}n(i)/{{\rm d} t} = 0$. Finally, the intensity $I_{i,j}(\lambda)$ of the specific transition can be calculated when the population $n(i)$ of the upper level $i$ of the transition is obtained.
Because the W$^{54+}$ ion is a heavy highly charged ion, the relativistic effects will play an important role in the energy level structure and transition properties. The ground state of W$^{54+}$ ion is \[Ne\]3s$^{2}$3p$^6$3d$^2$, which have two electrons in the open $3d$ subshells. Therefore, the relativistic and electron correlation effects should both be taken into account in the theoretical calculation. The present calculation is performed by using the relativistic configuration interaction (RCI) method with the implementation of FAC packages[@Gu2008The]. The atomic data including the energy levels, radiative transition rates, and cross sections of collisional (de) excitation are calculated. The collisional deexcitation process is the inverse of collisional excitation process. Therefore, the collisional deexcitation rate coefficient can be calculated by the principle of detailed balance from the corresponding electron collisional excitation process. In the present calculation, most of the important configuration interaction are included by single and double electron substitution from n=3 shells to n=4 subshells (nSD). For instance, the configuration interaction from configuration 3s$^2$3p$^6$3dnl(n=4), 3s$^2$3p$^5$3d$^2$nl (n=3, 4), 3s$^1$3p$^6$3d$^2$nl (n=3, 4), 3s$^2$3p$^6$nln’l’ (n, n’=4), 3s$^2$3p$^4$3d$^2$nln’l’ (n, n’=3, 4), 3p$^6$3d$^2$nln’l’ (n, n’=3, 4), 3s$^2$3p$^5$3dnln’l’ (n, n’=3, 4), 3s$^1$3p$^6$3dnln’l’ (n, n’=3, 4) were taken into account in the present calculation. The E1 radiative transitions, electron collisional excitation processees between the levels of the configuration 3p$^6$3d$^2$, 3p$^5$3d$^3$, 3p$^6$3d4l are included to simulate the spectrum of W$^{54+}$ ion.
Result and discussion
=====================
The level energies (in eV) of the ground state 3p$^6$3d$^2$ and the first excited state 3p$^5$3d$^3$ multiplets of W$^{54+}$ ion were presented in Table \[Tab1\]. The levels are sorted by their excitation energies. There are 9 levels in the ground state 3p$^6$3d$^2$ and 110 levels in the first excited state 3p$^5$3d$^3$. The $jj$ coupling labels are provided to designate the levels. The level energies calculated from different electron correlation models (nSD) are given to show the configuration interaction effects on the excitation energies. It can be inferred from the table that the level energies converge along with the increase of configurations interaction. The result of the present calculation agrees well with previous MCDF calculations[@Ding2017]. The discrepancy was found to be about 0.12%. For the levels of ground configuration, the results from the NIST database[@nist] were compared with the present calculation. The discrepancy was about 0.19%. A good agreement between the present result and previous result was found indicating that the most of the important configuration interaction effects has been included in the present work.
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Index Levels DF cal(3SD) cal(4SD) Theo.$^a$ NIST
\[3pt\]
Index Levels DF cal(3SD) cal(4SD) Theo.$^a$ NIST
\[3pt\]
$Ground state 3s^{2}3p^{6}3d^{2}$
1 \[3p$^{6}$3d$_{3/2}^{2}$\]$_{2}$ 0 0 0 0 0
2 \[3p$^{6}$3d$_{3/2}^{2}$\]$_{0}$ 24.317 24.173 23.248 23.123 23.309
3 \[3p$^{6}$3d$_{3/2}$3d$_{5/2}$\]$_{3}$ 71.992 72.023 72.321 72.456 72.59
4 \[3p$^{6}$3d$_{3/2}$3d$_{5/2}$\]$_{2}$ 83.069 83.067 82.736 82.805 82.882
5 \[3p$^{6}$3d$_{3/2}$3d$_{5/2}$\]$_{4}$ 86.125 86.118 86.300 86.359 86.4
6 \[3p$^{6}$3d$_{3/2}$3d$_{5/2}$\]$_{1}$ 88.161 88.138 87.597 87.613 87.962
7 \[3p$^{6}$3d$_{5/2}^{2}$\]$_{4}$ 152.557 152.596 152.933 153.158 153
8 \[3p$^{6}$3d$_{5/2}^{2}$\]$_{2}$ 161.371 161.384 160.966 161.160 161.1
9 \[3p$^{6}$3d$_{5/2}^{2}$\]$_{0}$ 186.754 186.482 185.041 185.140 185.5
$Excited state 3s^{2}3p^{5}3d^{3}$
10 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{3/2}$\]$_{2}$ 271.864 267.172 273.656 273.769
11 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{3}$)$_{3/2}$\]$_{1}$ 276.776 272.177 278.597 278.693
12 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{3}$)$_{3/2}$\]$_{0}$ 281.019 276.325 282.713 282.774
13 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{3/2}$\]$_{3}$ 282.739 278.073 284.507 284.582
14 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{3/2}$3d$_{5/2}$\]$_{3}$ 334.793 330.688 337.025 337.262
15 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{4}$ 340.543 336.665 342.952 343.159
16 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{3/2}$3d$_{5/2}$\]$_{2}$ 342.013 337.558 343.879 344.043
17 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{1/2}$3d$_{5/2}$\]$_{2}$ 344.868 340.627 346.909 347.135
18 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{3/2}$3d$_{5/2}$\]$_{1}$ 346.876 342.447 348.711 348.901
19 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{5}$ 348.958 345.129 351.385 351.541
20 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{7/2}$3d$_{5/2}$\]$_{3}$ 349.878 345.632 351.879 352.048
21 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{0}$ 349.968 345.632 351.926 352.089
22 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{7/2}$3d$_{5/2}$\]$_{6}$ 350.377 346.507 352.741 352.878
23 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{3/2}$3d$_{5/2}$\]$_{4}$ 350.991 346.859 353.116 353.284
24 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{7/2}$3d$_{5/2}$\]$_{4}$ 357.312 353.222 359.492 359.671
25 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{2}$ 359.202 354.601 360.928 361.129
26 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{3}$ 359.647 355.507 361.754 361.952
27 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{1}$ 359.997 355.537 361.840 362.029
28 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{7/2}$3d$_{5/2}$\]$_{5}$ 365.560 361.917 368.102 368.256
29 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{0}$)$_{3/2}$3d$_{5/2}$\]$_{4}$ 377.203 372.566 378.857 378.929
30 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{0}$)$_{3/2}$3d$_{5/2}$\]$_{2}$ 381.494 376.402 382.637 382.657
31 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{7/2}$3d$_{5/2}$\]$_{1}$ 390.190 383.853 390.051 390.056
32 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{0}$)$_{3/2}$3d$_{5/2}$\]$_{3}$ 390.357 384.849 390.889 390.731
33 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{2}$ 393.665 387.333 393.325 393.157
34 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{0}$)$_{3/2}$3d$_{5/2}$\]$_{3}$ 395.220 388.874 394.857 394.733
35 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{0}$(3d$_{5/2}^{2}$)$_{4}$\]$_{4}$ 410.421 406.464 412.697 413.038
36 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{4}$\]$_{5}$ 414.800 411.268 417.424 417.736
37 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{0}$(3d$_{5/2}^{2}$)$_{2}$\]$_{2}$ 416.667 412.128 418.373 418.689
38 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{3/2}^{2}$)$_{0}$)$_{3/2}$3d$_{5/2}$\]$_{1}$ 418.451 414.824 420.451 419.866
39 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{4}$\]$_{6}$ 420.957 415.297 421.437 421.736
40 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{4}$\]$_{3}$ 425.424 421.069 427.235 427.500
41 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{2}$\]$_{2}$ 425.944 422.356 428.549 428.802
42 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{4}$\]$_{5}$ 425.991 422.467 428.574 428.881
43 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{2}$\]$_{1}$ 426.923 422.600 428.819 429.044
44 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{4}$\]$_{7}$ 427.623 423.133 429.239 429.467
45 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{4}$\]$_{4}$ 427.672 423.334 429.534 429.789
46 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{2}$\]$_{3}$ 432.445 427.718 433.929 434.203
47 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{4}$\]$_{4}$ 437.602 433.178 439.202 439.326
48 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{4}$\]$_{3}$ 438.543 434.006 440.144 440.334
49 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{2}$\]$_{1}$ 438.641 434.784 441.037 441.338
50 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{4}$\]$_{6}$ 439.931 435.500 441.552 441.773
51 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{2}$\]$_{4}$ 440.325 435.762 441.869 442.007
52 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{2}$\]$_{5}$ 440.336 436.009 442.079 442.165
53 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{0}$(3d$_{5/2}^{2}$)$_{0}$\]$_{0}$ 444.847 439.450 445.855 446.163
54 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{2}$\]$_{2}$ 447.059 442.116 448.282 448.481
55 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{4}$\]$_{3}$ 452.440 447.295 453.385 453.580
56 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{4}$\]$_{4}$ 456.750 451.980 458.032 458.198
57 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{4}$\]$_{2}$ 460.988 455.504 461.538 461.647
58 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{4}$\]$_{5}$ 464.235 459.539 465.368 465.389
59 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{0}$\]$_{1}$ 465.621 460.007 466.285 466.474
60 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{0}$\]$_{3}$ 470.088 464.555 470.752 470.828
61 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{4}$\]$_{2}$ 471.224 466.182 471.920 471.719
62 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{4}$\]$_{1}$ 474.370 467.146 473.333 473.503
63 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{0}$\]$_{2}$ 474.387 468.779 474.956 475.067
64 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{2}$\]$_{3}$ 476.453 469.685 475.569 475.534
65 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{2}$\]$_{4}$ 477.385 470.400 476.323 476.313
66 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{0}$\]$_{2}$ 481.909 475.078 480.941 480.731
67 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{2}$\]$_{1}$ 486.173 479.787 485.476 485.082
68 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{2}$\]$_{0}$ 486.665 481.009 486.558 486.092
69 \[((3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$3d$_{3/2}$)$_{3}$(3d$_{5/2}^{2}$)$_{2}$\]$_{3}$ 487.142 482.322 487.985 487.630
70 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{9/2}$\]$_{6}$ 493.589 490.274 496.363 496.802
71 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{3/2}$\]$_{0}$ 504.879 500.470 506.656 507.002
72 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{9/2}$\]$_{5}$ 505.289 500.800 506.733 507.038
73 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{3/2}$\]$_{3}$ 506.786 502.182 508.242 508.543
74 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{3/2}$\]$_{2}$ 523.308 517.979 524.093 524.365
75 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{5/2}$\]$_{4}$ 524.286 519.340 525.606 525.942
76 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{5/2}$\]$_{2}$ 535.660 530.254 536.143 536.137
77 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{9/2}$\]$_{4}$ 540.624 534.476 540.343 540.525
78 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{5/2}$\]$_{3}$ 544.344 538.414 544.495 544.801
79 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{5/2}$\]$_{3}$ 553.354 548.107 553.875 553.794
80 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{3/2}$\]$_{1}$ 556.678 550.699 556.634 556.773
81 \[(3p$_{1/2}^{2}$3p$_{3/2}^{3}$)$_{3/2}$(3d$_{5/2}^{3}$)$_{5/2}$\]$_{1}$ 567.604 558.998 565.219 565.442
82 \[(3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{3}$)$_{3/2}$\]$_{2}$ 632.534 626.627 632.604 632.492
83 \[(3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{3}$)$_{3/2}$\]$_{1}$ 669.418 660.920 666.964 666.827
84 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{3}$ 685.896 680.855 686.909 687.221
85 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{2}$ 686.471 681.280 687.301 687.555
86 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{1}$ 691.022 685.843 691.842 692.014
87 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{4}$ 692.811 687.884 693.899 694.156
88 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{0}$ 693.116 688.045 694.040 694.211
89 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{5/2}$3d$_{5/2}$\]$_{5}$ 694.625 690.205 696.166 696.384
90 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{0}$)$_{1/2}$3d$_{5/2}$\]$_{2}$ 726.627 720.330 726.189 726.150
91 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{3/2}$3d$_{5/2}$\]$_{3}$ 731.048 724.068 729.856 729.825
92 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{3/2}$3d$_{5/2}$\]$_{4}$ 738.006 730.188 735.993 735.966
93 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{3/2}$3d$_{5/2}$\]$_{2}$ 739.656 732.355 738.170 738.129
94 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{2}$)$_{3/2}$3d$_{5/2}$\]$_{1}$ 743.215 734.282 740.378 740.497
95 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{2}$)$_{0}$)$_{1/2}$3d$_{5/2}$\]$_{3}$ 744.170 738.029 743.776 743.576
96 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{4}$\]$_{4}$ 752.738 748.314 754.373 754.976
97 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{4}$\]$_{3}$ 756.827 751.744 757.770 758.279
98 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{4}$\]$_{5}$ 759.325 755.182 761.205 761.763
99 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{2}$\]$_{0}$ 763.576 758.015 764.190 764.767
100 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{2}$\]$_{1}$ 763.984 758.436 764.604 765.173
101 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{4}$\]$_{2}$ 764.909 759.220 765.349 765.853
102 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{2}$\]$_{2}$ 766.354 760.652 766.765 767.260
103 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{4}$\]$_{6}$ 769.565 765.786 771.778 772.272
104 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{2}$\]$_{4}$ 773.439 768.403 774.429 774.884
105 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{2}$\]$_{3}$ 776.986 771.433 777.559 778.043
106 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{2}$(3d$_{5/2}^{2}$)$_{0}$\]$_{2}$ 796.183 789.842 796.082 796.470
107 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{4}$\]$_{5}$ 797.381 791.151 796.687 796.760
108 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{4}$\]$_{4}$ 798.948 792.159 797.999 798.247
109 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{2}$\]$_{3}$ 803.296 796.136 802.009 802.234
110 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{2}$\]$_{1}$ 806.282 798.409 804.420 804.706
111 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{4}$\]$_{3}$ 811.530 805.282 810.977 810.831
112 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{2}$\]$_{2}$ 812.190 805.501 811.002 811.046
113 \[((3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}$)$_{3/2}$)$_{1}$(3d$_{5/2}^{2}$)$_{0}$\]$_{1}$ 831.396 823.862 829.747 829.814
114 \[(3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{5/2}^{3}$)$_{9/2}$\]$_{4}$ 831.475 826.499 832.469 833.226
115 \[(3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{5/2}^{3}$)$_{9/2}$\]$_{5}$ 840.088 835.570 841.539 842.263
116 \[(3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{5/2}^{3}$)$_{3/2}$\]$_{1}$ 851.857 845.181 851.236 851.798
117 \[(3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{5/2}^{3}$)$_{3/2}$\]$_{2}$ 852.971 847.109 853.197 853.855
118 \[(3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{5/2}^{3}$)$_{5/2}$\]$_{2}$ 859.833 853.175 859.378 860.010
119 \[(3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{5/2}^{3}$)$_{5/2}$\]$_{3}$ 864.667 858.388 864.592 865.208
------------------------------------ ------------------------------------------------------------------------------------------------ --------- ---------- ---------- ----------- --------
: The level energies (in eV) of the ground state 3p$^6$3d$^2$ and the first excited state 3p$^5$3d$^3$ of W$^{54+}$ ion. The column DF is the level energy calculated with Dirac-Hartree-Fock approximation. The fourth and fifth column labeled as cal(3SD) and cal(4SD) represent the level energy obtained by considering the configuration interaction from single and double substitution in n=3 and n=4 subshells. The column ’Theo.’ and ’NIST’ represent the previous available data obtained by MCDF calcualtion and the NIST database[@nist] for comparison.\[Tab1\]
$^{\rm a}$From Ding et al by MCDF method[@Ding2017].
According to the present calculation, the wavelength of E1 transition from the first excited state 3s$^2$3p$^5$3d$^3$ to the ground configuration state 3s$^2$3p$^6$3d$^2$ covered the range of 18.5-32.5 Å. The strong transition with large transition rates was concentrated in two wavelength range 29.5-32.5 Å and 18.5-19.6 Å.
The transition wavelength $\lambda$ (in Å), transition rate A (in s$^{-1}$), population $n(i)$ and intensity $I_{ij}(\lambda)$ of E1 transition from 3s$^2$3p$^5$3d$^3$ to 3s$^2$3p$^6$3d$^2$ of W$^{54+}$ ion are presented for a wavelength range from 29.5-32.5 Å in Table \[Tab2\]. The experimental value observed by EBIT and the theoretical values from FAC[@Gu2008The] and MCDF[@0953-4075-50-4-045004; @Dipti2015Electron] are also included for comparison. The present calculation agrees quite well with the experimental data and other theoretical values. The wavelength discrepancy between T. Lennartsson *et al.* [@PhysRevA.87.062505] by experiment, Dipti *et al.* [@Dipti2015Electron] and Ding *et al.* [@0953-4075-50-4-045004] by MCDF method, T. Lennartsson *et al.* [@PhysRevA.87.062505] by FAC calculations and the present calculation are about 0.25%, 0.14%, 0.11% and 0.07%, respectively. All the observed transition lines from the EBIT experiment were identified in the present calculation. It was found that the observed transition lines in the EBIT experiment have large transition rates. However, a few transitions in this range with large transition rates have not been observed in the previous EBIT experiment, such as transitions with the key 7, 8 and 9. The reason is the population (Pop) of the excited upper levels of these unobserved transitions is extremely small. The results show that the intensity (Int) of the unobserved transitions are generally smaller by four orders of magnitude than the intensity which could be observed. The intensity might be changed with the plasma conditions. It can be expected that these unobserved strong transitions might be observed by the appropriate plasma condition.
Lower Upper $\lambda(\AA)$ $\lambda_{others}(\AA)$ $A(/s^{-1}$) $A^{c}(/s^{-1}$) Pop Int Key
------- ------- ---------------- --------------------------------------------- -------------- ------------------ ----------- ---------- -----
1 30 32.403 32.264$^a$ 32.502$^b$ 32.401$^c$ 32.416$^d$ 8.30(10) 8.50(10) 6.92(-14) 5.74(-3) 1
1 31 31.787 31.811$^a$ 31.783$^b$ 31.787$^c$ 31.786$^d$ 7.49(11) 7.44(11) 3.63(-14) 2.72(-2) 2
1 32 31.719 31.776$^a$ 31.765$^b$ 31.732$^c$ 31.711$^d$ 5.43(11) 5.91(11) 9.05(-14) 4.92(-2) 3
1 33 31.522 31.563$^a$ 31.503$^b$ 31.536$^c$ 31.505$^d$ 9.60(11) 9.43(11) 6.35(-14) 6.09(-2) 4
1 34 31.400 31.430$^a$ 31.378$^b$ 31.410$^c$ 31.386$^d$ 5.89(11) 5.18(11) 8.80(-14) 5.18(-2) 5
2 38 31.215 31.245$^a$ 31.236$^b$ 31.251$^c$ 31.155$^d$ 9.39(11) 9.33(11) 7.75(-15) 7.28(-3) 6
6 68 31.077 31.115$^c$ 1.18(12) 1.16(12) 8.68(-20) 1.03(-7) 7
7 79 30.924 30.948$^c$ 1.19(12) 1.17(12) 1.88(-20) 2.23(-8) 8
5 69 30.866 30.898$^c$ 1.19(12) 1.16(12) 3.26(-16) 3.86(-4) 9
1 38 29.489 29.560$^a$ 29.456$^b$ 29.530$^c$ 29.452$^d$ 3.08(11) 2.96(11) 7.75(-15) 2.39(-3) 10
: The calculated transition wavelength lambda (Å), transition rate A(in $s^{-1}$), population (Pop), intensity (Int) and available experimental and other theoretical values for the strong E1 transition of W$^{54+}$ ion. The column ’Key’ correspond to the label in figure \[fig1\]. Notion $a(b)$ for transition probabilities A means a$\times$10$^b$.[]{data-label="Tab2"}
$^{\rm a}$ From T. Lennartsson by EBIT experiment[@PhysRevA.87.062505].
$^{\rm b}$ From Dipti et al by MCDF method[@Dipti2015Electron].
$^{\rm c}$ From Ding et al by MCDF method[@0953-4075-50-4-045004].
$^{\rm d}$ From T. Lennartsson by collisional-radiative model[@PhysRevA.87.062505]\
The synthetic spectrum of W$^{54+}$ ion in the wavelength 29.5-32.5 Å is shown in Fig. \[fig1\]. Each individual transition was assumed to have the Gaussian profile with full width at half maximum (FWHF) 0.09 Å. The upper part Fig. \[fig1\](a) is the spectrum obtained by convoluting the transition rate, while the middle part Fig. \[fig1\](b) is the spectrum by considering the population of the upper level in the EBIT case with the electron density $n_e$ = 10$^{12}$ cm$^{-3}$ and the energy of electron beam $E_e$ = 18.2 keV. All 7 peaks observed by the experimental observation[@PhysRevA.87.062505] can be reproduced by the synthetic spectrum. The lower part Fig. \[fig1\](c) is the spectrum by considering the excited upper levels in the LTE plasma with the electron density $n_e$ = 10$^{15} $ cm$^{-3}$ and the electron temperature $T_e$ = 18.2 keV, and the electron energy distribution is Maxwellian. The results indicate that all the strong transition lines are observable under this plasma condition.
The large difference between Fig. \[fig1\] (b) and (c) is caused by the dependence of population mechanism on the free electron energy distribution function in a different plasma environment. The intensity for a specific transition line is proportion to the population of the excited upper level $i$ which will be populated by the collisional excitation processes from other lower energy levels and collisional deexcitation processes from other higher energy levels (referenced as population flux), and will be depopulated by the collisional excitation processes to other higher energy levels and collisional deexcitation processes to other lower energy levels (referenced as depopulation flux). Collisional excitation and deexcitation rates coefficient are obtained by convoluting the cross section of the corresponding collisional (de)excitation processes with the free electron energy distribution function (EEDF) in the plasma. In the present work, the EEDF in the EBIT plasma and LTE plasma are taken as the $\delta$ function and the Maxwellian distribution function, respectively. In the EBIT plasma, for example, the strong transition line have been observed in the previous EBIT experiment with the key 4, the ratio of the population flux and depopulation flux of the excited upper level is 1.12, while the weak transition line which have not been observed in the previous EBIT experiment with the key 9, the ratio of the population flux and depopulation flux is 0.15. This means the excited upper level of the weak peak have a small population compare with the strong peak. In the LTE plasma, for all excited upper levels, the ratio of the population flux and depopulation flux about are 0.07, which means the population for the excited upper levels almost same. Thus the intensity only proportional to the transition rates. As a result, the transition 7, 8 and 9 are large enough to be observed.
![The synthetic spectrum of W$^{54+}$ ion in wavelength 29.5-32.5 Å. (a).convoluting the transition rate with the Gaussian profile; (b).the spectrum for the EBIT case with $n_e$ = 10$^{12}$ cm$^{-3}$ and $E_e$ = 18.2 keV; (c).the spectrum for the LTE plasma with $n_e$ = 10$^{15}$ cm$^{-3}$ and $T_e$ = 18.2 keV.[]{data-label="fig1"}](Fig1.eps)
The wavelength $\lambda$ (in Å), transition rate A (in s$^{-1}$), population $n(i)$ and intensity $I_{ij}(\lambda)$ of E1 transition 3s$^2$3p$^5$3d$^3$ to 3s$^2$3p$^6$3d$^2$ in the wavelength range 18.5-19.6 Å are presented in Table \[Tab3\]. The theoretical values from MCDF calculation[@0953-4075-50-4-045004] are also included for comparison. The present calculation values generally made a good agreement with the previous data. The wavelength discrepancy was found to be about 0.04%. According to the present calculation, some strong transitions may be observed in the wavelength range of 18.5-19.6 Å. The synthetic spectrum of W$^{54+}$ ion in this wavelength range was shown in Fig. \[Fig2\]. All the peaks were obtained with FWHM = 0.05 Å for each individually transition to make the spectrum clear. The upper part Fig. \[Fig2\](a) is the spectrum obtained by convoluting the transition rate with the Gaussian profile. The middle part Fig. \[Fig2\](b) is the spectrum by considering the population of the upper levels in the EBIT case with $n_e$ = 10$^{12}$ cm$^{-3}$ and $E_e$ = 18.2 keV. According to the synthetic spectrum there are only 3 peaks that would be observed in the EBIT experiment with this condition, namely, the transition lines with key 1, 5, and 20 in Fig. \[Fig2\](b), These 3 peaks are regarded as the E1 transition [\[(3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{3}$)$_{3/2}$\]$_{2}$]{} $\rightarrow$ [\[3p$^{6}$3d$_{3/2}^{2}$\]$_{2}$]{}, [\[(3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{3}$)$_{3/2}$\]$_{1}$]{} $\rightarrow$ [\[3p$^{6}$3d$_{3/2}^{2}$\]$_{0}$]{}, [\[(3p$_{1/2}$3p$_{3/2}^{4}$)$_{1/2}$(3d$_{3/2}^{3}$)$_{3/2}$\]$_{1}$]{} $\rightarrow$ [\[3p$^{6}$3d$_{3/2}^{2}$\]$_{2}$]{} with transition wavelength 19.599 Å, 19.260 Å and 18.590 Å, respectively. The lower part Fig. \[Fig2\](c) is the spectrum by considering the population of the upper levels in the LTE plasma with $n_e$ = 10$^{15}$ cm$^{-3}$ and $T_e$ = 18.2 keV, and the electron energy distribution is Maxwellian. The results indicate that all transition lines are observable in this condition. The difference between Fig. \[Fig2\] (b) and (c) are caused by the same reason as in Fig. \[fig1\].
Lower Upper $\lambda(\AA)$ $\lambda^{a}(\AA)$ $A(/s^{-1}$) $A^{a}(/s^{-1})$ Pop Int Key
------- ------- ---------------- -------------------- -------------- ------------------ ----------- ---------- -----
1 82 19.599 19.603 2.95(12) 2.88(12) 1.31(-14) 3.85(-2) 1
8 109 19.341 19.340 1.66(12) 1.61(12) 1.32(-19) 2.19(-7) 2
8 110 19.269 19.266 4.09(12) 4.03(12) 2.58(-18) 1.06(-5) 3
7 107 19.261 19.264 2.85(12) 2.79(12) 9.40(-20) 2.68(-7) 4
2 83 19.260 19.261 1.78(12) 1.76(12) 4.86(-15) 8.64(-3) 5
9 113 19.231 19.232 2.81(12) 2.79(12) 1.84(-17) 5.15(-5) 6
7 108 19.221 19.220 3.36(12) 3.30(12) 5.78(-19) 1.94(-6) 7
4 91 19.160 19.162 1.96(12) 1.93(12) 3.71(-17) 7.26(-5) 8
7 109 19.102 19.102 2.14(12) 2.13(12) 1.32(-19) 2.82(-7) 9
5 92 19.084 19.086 4.59(12) 4.52(12) 2.44(-17) 1.12(-4) 10
8 111 19.074 19.084 1.46(12) 1.45(12) 6.80(-19) 9.90(-7) 11
8 112 19.074 19.078 3.10(12) 3.00(12) 1.06(-17) 3.29(-5) 12
6 93 19.058 19.060 2.63(12) 2.60(12) 2.11(-17) 5.55(-5) 13
6 94 18.993 18.991 1.66(12) 1.64(12) 1.66(-17) 2.74(-5) 14
3 90 18.962 18.967 4.08(12) 3.97(12) 4.48(-17) 1.83(-4) 15
5 95 18.858 18.865 4.35(12) 4.24(12) 1.86(-17) 8.11(-5) 16
3 91 18.856 18.861 1.99(12) 1.96(12) 3.71(-17) 7.40(-5) 17
4 94 18.853 18.852 4.55(12) 4.48(12) 1.66(-17) 7.55(-5) 18
7 111 18.842 18.852 2.25(12) 2.15(12) 6.80(-19) 1.53(-6) 19
1 83 18.590 18.594 5.18(12) 5.09(12) 4.86(-15) 2.52(-2) 20
: The calculated transition wavelength lambda (Å), transition rate A(in $s^{-1}$), population (Pop), intensity (Int) and the theoretical values from MCDF calculation for the strong E1 transition of W$^{54+}$ ion. The column ’Key’ correspond to the label in figure \[Fig2\]. Notion $a(b)$ for transition probabilities A means a$\times$10$^b$.[]{data-label="Tab3"}
$^{\rm a}$From Ding et al by MCDF method[@0953-4075-50-4-045004].
![The synthetic spectrum of W$^{54+}$ ion in wavelength 18.5-19.6 Å. (a).convoluting the transition rate with the Gaussian profile; (b).the spectrum for the EBIT case with $n_e$ = 10$^{12}$ cm$^{-3}$ and $E_e$ = 18.2 keV; (c).the spectrum for the LTE plasma with $n_e$ = 10$^{15}$ cm$^{-3}$ and $T_e$ = 18.2 keV.[]{data-label="Fig2"}](Fig2.eps "fig:")\
Conclusion
==========
The energy level, E1 transition rate, and electron collisional excitation of the ground state 3s$^2$3p$^6$3d$^2$ and the first excited state 3s$^2$3p$^5$3d$^3$ of W$^{54+}$ ion were calculated by relativistic configuration interaction method. A collisional-radiative model (CRM) was constructed to simulate the E1 transition spectrum for the EBIT and the LTE plasma. All the necessary atomic data for constructing the CRM was calculated by FAC packages. The most important configuration interaction effects were taken into account. The energy levels and transition rates made a reasonable agreement with the EBIT experimental observation and the previous theoretical values. The synthetic spectrum from the CRM explained the EBIT observation in the 29.5-32.5 Å. Furthermore, some possible transitions were proposed to be observed in 18.5-19.6 Å of the future the EBIT observations. Finally, the difference of the spectrum in different plasma condition was observed and explained.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by National Key Research and Development Program of China, Grant No:NYK0123, National Nature Science Foundation of China, Grant No: 11264035, Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP), Grant No: 20126203120004, International Scientific and Technological Cooperative Project of Gansu Province of China (Grant No. 1104WCGA186), JSPS-NRF-NSFC A3 Foresight Program in the field of Plasma Physics (NSFC: No. 11261140328, NRF: 2012K2A2A6000443).
References {#references .unnumbered}
==========
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We consider the learning of algorithmic tasks by mere observation of input-output pairs. Rather than studying this as a black-box discrete regression problem with no assumption whatsoever on the input-output mapping, we concentrate on tasks that are amenable to the principle of *divide and conquer*, and study what are its implications in terms of learning.
This principle creates a powerful inductive bias that we leverage with neural architectures that are defined recursively and dynamically, by learning two scale-invariant atomic operations: how to *split* a given input into smaller sets, and how to *merge* two partially solved tasks into a larger partial solution. Our model can be trained in weakly supervised environments, namely by just observing input-output pairs, and in even weaker environments, using a non-differentiable reward signal. Moreover, thanks to the dynamic aspect of our architecture, we can incorporate the computational complexity as a regularization term that can be optimized by backpropagation. We demonstrate the flexibility and efficiency of the Divide-and-Conquer Network on several combinatorial and geometric tasks: convex hull, clustering, knapsack and euclidean TSP. Thanks to the dynamic programming nature of our model, we show significant improvements in terms of generalization error and computational complexity.
author:
- |
Alex Nowak\
Courant Institute of Mathematical Sciences\
Center for Data Science\
New York University\
New York, NY 10012, USA\
`[email protected]`\
David Folqué\
Courant Institute of Mathematical Sciences\
Center for Data Science\
New York University\
New York, NY 10012, USA\
`[email protected]`\
Joan Bruna\
Courant Institute of Mathematical Sciences\
Center for Data Science\
New York, NY 10012, USA\
`[email protected]`\
bibliography:
- 'mainbib.bib'
title: Divide and Conquer Networks
---
### Acknowledgments {#acknowledgments .unnumbered}
This work was partly supported by Samsung Electronics (Improving Deep Learning using Latent Structure)
|
{
"pile_set_name": "ArXiv"
}
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---
address: |
$^{a}$Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853-2501\
$^{b}$Lyman Laboratory of Physics, Harvard University, Cambridge MA 02138\
\
author:
- 'P. W. Brouwer$^{a}$, X. Waintal$^{a}$, and B. I. Halperin$^{b}$'
title: 'Fluctuating spin $g$-tensor in small metal grains'
---
With the advance of nanoparticle technology, it has become possible to resolve individual energy levels for electrons in ultrasmall metal grains. Recent experiments addressed their Zeeman splitting under the application of a magnetic field $\vec B$ [@Ralph; @Davidovic; @Salinas]. The splitting of a level $\varepsilon_{\mu}$ is described by a $g$-factor, $\delta
\varepsilon_{\mu} = \pm \case{1}{2} \mu_B g B_z$, where $\mu_B$ is the Bohr magneton. A free electron has $g = 2$, but in small metal grains the effective $g$-factor may be reduced as a result of spin-orbit scattering [@Halperin]. In order to study this reduction, Salinas [et al.]{} [@Salinas] have doped Al grains (which do not have significant spin-orbit scattering) with Au (which has). For small concentrations of Au, the effective $g$-factor was seen to drop from 2 to around 0.7. Even lower values $g \sim 0.3$ were reported in experiments on Au grains [@Davidovic].
For disordered systems with spin-orbit scattering, the splitting of a level $\varepsilon_{\mu}$ does not only depend on the magnitude of the magnetic field $\vec B$, but also on its direction. Hence, an analysis in terms of a “$g$-tensor” is more appropriate [@Slichter]. To be precise, the Zeeman field splits the Kramers’ doublet $\varepsilon_{\mu} \to
\varepsilon_{\mu} \pm \delta \varepsilon_{\mu}$ with $$\begin{aligned}
\delta \varepsilon_{\mu}^2 = (\mu_B/2)^2 \vec B \cdot {\cal G_{\mu}}
\cdot \vec B,
\label{eq:deltaE}\end{aligned}$$ where ${\cal G}_{\mu}$ is a $3 \times 3$ tensor. In the absence of spin-orbit scattering, the tensor ${\cal G}_{\mu}$ is isotropic, $({\cal G_{\mu}})_{ij} = 4 \delta_{ij}$. The effect of spin-orbit scattering on ${\cal G}_{\mu}$ is threefold: It leads to a decrease of the typical magnitude of ${\cal G}_{\mu}$, it makes the tensor structure of ${\cal G}_{\mu}$ important (i.e., it introduces an anisotropic response to the magnetic field $\vec B$), and it causes ${\cal G}_{\mu}$ to be different for each level $\varepsilon_{\mu}$. Hence ${\cal G}_{\mu}$ becomes a fluctuating quantity, and it is important to know its statistical distribution. The latter problem was addressed in a recent paper by Matveev et al. [@Matveev], however without considering the tensor structure of ${\cal
G}_{\mu}$. The anisotropy of the $g$-tensor is a measurable quantity and we here consider the distribution of the entire tensor ${\cal G}_{\mu}$. The distribution $P({\cal G}_{\mu})$ is defined with respect to an ensemble of small metal grains of roughly equal size. The same distribution applies to the fluctuations of ${\cal G}_{\mu}$ as a function of the level $\varepsilon_{\mu}$ in the same grain.
In general, ${\cal G}_{\mu}$ has a contribution ${\cal
G}_{\mu}^{\rm spin}$ from the magnetic moment of electron spins, and a contribution ${\cal G}_{\mu}^{\rm orb}$ for the orbital angular moment of the state $|\psi_{\mu}\rangle$. In Ref. , the typical sizes of both contributions were estimated as ${\cal G}^{\rm
spin} \sim \tau_{\rm so} \Delta$ and ${\cal G}^{\rm orb} \sim \ell/L$, where $\tau_{\rm so}$ is the mean spin-orbit scattering time, $L$ is the grain size, $\Delta \propto L^{-3}$ is the mean level spacing, and $\ell \ll L$ is the elastic mean free path. We restrict ourselves to the spin contribution ${\cal G}^{\rm spin}$, which should be dominant for small grain sizes [@Matveev], provided $\tau_{\rm so}$ does not depend on system size, as should be the case for the experiments of Ref. . When orbital contributions are important, the anisotropy of ${\cal G}$ will be affected by the shape of the grain. In that case, our main conclusions apply only to a roughly spherical grain. As the typical magnitude of ${\cal G}$ (we drop the superscript “spin” and the subscript $\mu$ if there is no ambiguity) depends on the microscopic parameters $\tau_{\rm so}$ and $\Delta$, which are in most cases not known accurately, we choose to have the typical magnitude of ${\cal G}$ serve as an external parameter in our theory.
We first present our main results. With a suitable choice of the coordinate axes (“principal axes”), the tensor ${\cal G}$ can be diagonalized. Writing its eigenvalues as $g_j^2$ and denoting the components of the magnetic field along the principal axes by $B_j$, $j=1,2,3$, Eq. (\[eq:deltaE\]) takes a particularly simple form, $$\delta \varepsilon_{\mu}^2 = \case{1}{4} \mu_B^2 (
g_1^2 B_1^2 + g_2^2 B_2^2 + g_3^3 B_3^2). \label{eq:deltaE2}$$ We refer to the numbers $g_1$, $g_2$, and $g_3$ as principal $g$-factors. For a generic metal grain of a cubic material, rotational symmetry implies that, for a given level $\varepsilon_{\mu}$, the positioning of the principal axes is entirely random in space, as long as they are mutually orthogonal. Hence, it remains to study the distribution $P(g_1,g_2,g_3)$ of the principal $g$-factors $g_1$, $g_2$, and $g_3$ for the level $\varepsilon_{\mu}$. Our main result is, that for sufficiently strong spin-orbit scattering, $P(g_1,g_2,g_3)$ is given by the distribution $$P(g_1,g_2,g_3) \propto \prod_{i<j} |g_i^2 - g_j^2| \prod_{i}
e^{-3 g_i^2/2 \langle g^2 \rangle}, \label{eq:PgGSE0}$$ where $g^2 = \case{1}{3}(g_1^2 + g_2^2 + g_3^2)$ is the average of $(2 \delta \varepsilon_{\mu}/ \mu_B B)^2$ over all directions of $\vec B$ and $\langle g^2 \rangle$ is its average over the ensemble of grains. In random matrix theory [@Mehta], this distribution is known as the Laguerre ensemble. Without loss of generality we may assume that $g_1^2 \le g_2^2 \le g_3^2$. Figure \[fig:3\] shows the averages $\langle g_j^2 \rangle$ and a realization of the principal $g$-factors $g_1$, $g_2$, and $g_3$ for a specific sample, as a function of a parameter $\lambda \sim (\tau_{\rm so} \Delta)^{-1/2}$ measuring the strength of the spin-orbit scattering. (A formal definition of $\lambda$ in a random-matrix model will be given below.) From the figure, one readily observes that, typically, the three principal $g$-factors differ by a factor $2$–$3$. This implies that, in spite of the average rotational symmetry of the grains, the response of a given level $\varepsilon_{\mu}$ to an applied magnetic field is highly anisotropic because of mesoscopic fluctuations. The mathematical origin of this effect is the “level repulsion” factor $|g_i^2 - g_j^2|$ in the probability distribution (\[eq:PgGSE0\]), which signifies that, to a certain extent, ${\cal G}_{\mu}$ can be viewed a as a “random matrix”.
-0.5cm =0.99
Let us now turn to a more detailed discussion of our results. Without magnetic field, the Hamiltonian ${\cal H}$ of the grain is invariant under time-reversal, so that all eigenstates come in doublets $|\psi_{\mu}\rangle$ and $|{\cal T} \psi_{\mu}\rangle$, where ${\cal
T} \psi = i \sigma_2 \psi^{*}$ is the time-reversal operator. To study the splitting of the doublets by a magnetic field, we add a term $
\mu_B \vec B \cdot \vec \sigma$ to ${\cal H}$, $\vec \sigma =
(\sigma_1,\sigma_2,\sigma_3)$ being the vector of Pauli matrices. From degenerate perturbation theory we find that a level $\varepsilon_{\mu}$ is split into $\varepsilon_{\mu} \pm \delta
\varepsilon_{\mu}$, with $\delta \varepsilon_{\mu}$ of the form (\[eq:deltaE\]). For the real symmetric $3 \times 3$ matrix ${\cal G}_{\mu}$ one has $${\cal G}_{\mu} = G_{\mu}^{\rm T} G_{\mu},$$ where $G_{\mu}$ is a real $3 \times 3$ matrix with elements $$\begin{aligned}
(G_{\mu})_{1j} + i (G_{\mu})_{2j} &=&
- 2 \langle {\cal T} \psi_{\mu} | \sigma_j | \psi_{\mu} \rangle
\nonumber \\
(G_{\mu})_{3j} &=&
2\langle \psi_{\mu} | \sigma_{j} | \psi_{\mu} \rangle,
\label{eq:gpsi} \end{aligned}$$ We use random-matrix theory (RMT) to compute the distribution of ${\cal G}_{\mu}$. In RMT, the microscopic Hamiltonian ${\cal H}$ is replaced by a $2N \times 2N$ random hermitian matrix $H$, where at the end of the calculation the limit $N \to \infty$ is taken. (The factor $2$ accounts for spin.) The wavefunction $\psi_{\mu}(\vec r)$ is replaced by an $N$-component spinor eigenvector $\psi_{\mu n}$ of $H$, where $n$ is a vector index. To study the effect of spin-orbit scattering, we take $H$ of the form
\[eq:HSA\] $$H(\lambda) = S \otimes \openone_2 + i {\lambda \over \sqrt{4N}}
\sum_{j} A_j \otimes \sigma_j,$$ where $S$ ($A_j$) is a real symmetric (antisymmetric) $N \times N$ matrix with the Gaussian distribution $$\begin{aligned}
P(S) &\propto& e^{- (\pi^2/4 N \Delta^2)\, {\rm tr}\, S^{\rm T} S},
\label{eq:distr}\\
P(A_j) &\propto& e^{- (\pi^2/4 N \Delta^2)\, {\rm tr}\, A_j^{\rm T} A_j},\
\ j=1,2,3. \nonumber\end{aligned}$$
The Hamiltonian $H(\lambda)$ is similar to the Pandey-Mehta Hamiltonian used to describe the effect of time-reversal symmetry breaking in a system of spinless particles [@Pandey]. In Eq.(\[eq:distr\]), $\Delta$ is the average spacing between the Kramers doublets near $\varepsilon=0$. The amount of spin-orbit scattering is measured by the parameter $\lambda \sim (\tau_{\rm so} \Delta)^{-1/2}$ [@Halperin]. The case $\lambda=0$ corresponds to the absence of spin-orbit scattering, when $H = S$ is a member of the Gaussian Orthogonal Ensemble (GOE) of random matrix theory. The case $\lambda=(4N)^{1/2}$ corresponds to the case of strong spin-orbit scattering, when $H$ is a member of the Gaussian Symplectic Ensemble (GSE). The ensemble of Hamiltonians $H(\lambda)$ corresponds to a crossover from the GOE to the GSE. Similar crossovers were studied previously in the literature, in particular for the cases GOE–GUE and GSE–GUE (GUE is Gaussian Unitary Ensemble) [@Pandey; @French; @Sommers; @Falko; @VanLangen].
The distribution of the tensor ${\cal G}_{\mu}$ for an eigenvalue $\varepsilon_{\mu}$ of the matrix $H(\lambda)$ is related to the statistics of eigenvectors of $H(\lambda)$ in this crossover ensemble. To deal with the twofold degeneracy of the eigenvalue $\varepsilon_{\mu}$, we combine the two $N$-component spinor eigenvectors $\psi_{\mu}$ and ${\cal T}\psi_{\mu}$ into a single $N$-component vector of quaternions $\bar \psi = (\psi,{\cal T}\psi)$ [@Mehta; @quaternion]. The quaternion vector $\bar \psi$ can be parameterized as, $$\bar \psi = \sum_{k=0}^{3} \alpha_k u_k \otimes \phi_k, \label{eq:barpsiphi}$$ where the $u_k$ are quaternion numbers with $\mbox{tr}\, u_k^{\dagger}
u_l = 2 \delta_{kl}$ (“quaternion phase factors”), the $\phi_k$ are $N$-component real orthonormal vectors, and the $\alpha_k$ are positive numbers such that $\sum_{k} \alpha_k^2 = 1$ ($k,l=0,1,2,3$). A eigenvector in the GOE corresponds to $\alpha_0 = 1$, $\alpha_1 = \alpha_2 = \alpha_3 = 0$, while an eigenvector in the GSE has typically $\alpha_0 \approx \alpha_1 \approx \alpha_2 \approx \alpha_3
\approx \case{1}{2}$. A similar parameterization has been applied to the GOE–GUE crossover [@French]. Orthogonal invariance of the distributions of $S$ and $A_j$, together with the freedom to choose the overall quaternion phase of $\bar \psi$, give a distribution of the $u_k$ and $\phi_k$ that is as random as possible, provided the above mentioned orthogonality constraints are obeyed. Hence, all nontrivial information about the eigenvector statistics is encoded in the numbers $\alpha_k$. Substitution of the parameterization (\[eq:barpsiphi\]) into Eq. (\[eq:gpsi\]) yields $$\begin{aligned}
% g_{j} &=& 2(\alpha_0^2 - \alpha_1^2 - \alpha_2^2 - \alpha_3^2) +
% 4 \alpha_j^2, \ \ j=1,2,3. \label{eq:ga}
g_{1} &=& 2(\alpha_0^2 + \alpha_1^2 - \alpha_2^2 - \alpha_3^2), \nonumber \\
g_{2} &=& 2(\alpha_0^2 - \alpha_1^2 + \alpha_2^2 - \alpha_3^2),
\label{eq:ga}\\
g_{3} &=& 2(\alpha_0^2 - \alpha_1^2 - \alpha_2^2 + \alpha_3^2). \nonumber\end{aligned}$$ While the squares $\alpha_k^2$ ($k=0,1,2,3$) are all positive, the principal $g$-factors as given by Eq. (\[eq:ga\]) can also be negative. Permutations of the $\alpha_k$ alter the signs of the individual $g_j$, but not of their product $g_1 g_2 g_3$. \[The product $g_1 g_2 g_3 = \det G$ also follows from Eq. (\[eq:gpsi\]); one verifies that it does not change when $|\psi\rangle$ is replaced by a linear combination of $|\psi\rangle$ and $|{\cal
T}\psi\rangle$.\] Without loss of generality, we may assume that $g_1^2 \le g_2^2 \le g_3^2$, and that $g_2$ and $g_3$ are positive. Then equation (\[eq:ga\]) provides the constraint $g_2 + g_3 \le 2 + g_1$, which poses a bound on the occurrence of negative values for the product $g_1 g_2 g_3$. We conclude that all information on the eigenvector statistics in the GOE–GSE crossover is encoded in the magnitudes of $g_1$, $g_2$, and $g_3$ and the sign of their product. Since for the level splitting $\delta \varepsilon_{\mu}(\vec B)$ only the squares $g_j^2$ are of relevance, we disregard the sign of $g_1 g_2 g_3$ in the remainder of the paper. The sign of $g_1 g_2 g_3$ may be determined in principle, however, by a spin-resonance experiment [@spinres].
In order to calculate the distribution $P(g_1,g_2,g_3)$ one has, in principle, to carry out the same program as was done in Refs. for the GOE–GUE crossover. However, it turns out that in the present case the calculation is considerably more complicated. This can already be seen from the mere observation that the wavefunction statistics in the GOE–GSE crossover is governed by three variables $g_1$, $g_2$, and $g_3$, whereas in the case of the GOE–GUE crossover only one variable was needed [@Sommers; @Falko; @VanLangen]. In the field-theoretic language of Ref. , one has to use a nonlinear sigma model of $16 \times 16$ supermatrices, instead of the usual $8 \times 8$ for the GOE–GUE crossover [@Klaus]. Here we refrain from such a truly heroic enterprise. Instead we focus on the regimes of strong and weak spin-orbit coupling, and study the intermediate regime by means of numerical simulations of the model (\[eq:HSA\]).
Before we address the case of strong spin-orbit scattering $\lambda \gg 1$ in the crossover Hamiltonian, we first consider the GSE, corresponding to $\lambda^2 = 4N$. In the GSE, the wavefunction $\psi$ is a vector of independently Gaussian distributed complex numbers. Then, one easily verifies that, for large $N$, the elements of the matrix $G$ of Eq. (\[eq:gpsi\]) are real random variables, independently distributed, with a Gaussian distribution of zero mean and variance $2/N$. Hence $G$ is a random real matrix with distribution $$P(G) \propto \exp(-N {\rm tr}\, G^{\rm T} G/4). \label{eq:GProb}$$ The principal $g$-factors are the eigenvalues $g_{j}^2$ of the product ${\cal G} = G^{\rm T} G$. The distribution of the eigenvalues of such a matrix product is known in literature [@Brezin]. It is given by Eq. (\[eq:PgGSE0\]) with $\langle g^2 \rangle = 6/N$.
Let us now turn to the Hamiltonian $H(\lambda)$ for large $\lambda \gg 1$, but still $\lambda \ll N^{1/2}$. In that case, spin-rotation invariance is broken globally (so that a wavefunction as a whole does not have a well-defined spin), but not locally; on short length scales, the particle keeps a well-defined spin. We then argue that, in the random matrix language, one may think of the quaternion wavevector $\bar \psi$ as consisting of $\sim \lambda^2 \gg 1$ components, each with a well-defined spin (or “quaternion phase”), but with uncorrelated spins for each component. The distribution of ${\cal G}$ is then given by the distribution for the GSE with $N$ replaced by a number $\sim \lambda^2$ [@phaserigidity]. We have found that the precise correspondence is $N \to 2 \lambda^2$, by estimating the exponential term in the exact distribution, along the lines of Ref.. In order to verify this statement we have numerically generated random matrices of the form (\[eq:HSA\]). The comparison with the GSE distribution with $N$ replaced by $2 \lambda^2$ is excellent, see Fig. \[fig:1\].
=0.89
In order to further analyze $P({\cal G})$ for strong spin-orbit scattering, we introduce the orientationally averaged $g$-factor, $$g^2 = \case{1}{3}(g_1^2 + g_2^2 + g_3^2)
= \left \langle { (2\delta \varepsilon_{\mu} / \mu_B |B|)^2}
\right \rangle_{\Omega},$$ where the brackets $\langle \ldots \rangle_{\Omega}$ indicate an average over all directions of the magnetic field. Further, we introduce the ratios $r_{12} = |g_1/g_2|$ and $r_{23} = |g_2/g_3|$ to characterize the anisotropy of ${\cal G}$. Changing variables in Eq. (\[eq:PgGSE0\]), we find that $P(g,r_{12},r_{23})$ reads $$\begin{aligned}
P
&\propto& % \nonumber
{r_{23}^3 (1-r_{23}^2) (1-r_{23}^2 r_{12}^2)
(1-r_{12}^2) \over (1 + r_{23}^2 + r_{23}^2 r_{12}^2)^{9/2}}\,
g^8 e^{-9 g^2/2 \langle g^2 \rangle}.\! \label{eq:Pr12}\end{aligned}$$ Note that the distribution of $r_{12}$ and $r_{23}$ does not depend on $\langle g^2 \rangle$ (provided the spin-orbit scattering is sufficiently strong). The “$g$-factor” $g_z$ for a magnetic field in the $z$-direction (which is a random direction with respect to the principal axes) is given by $g_z = ({\cal G}_{zz})^{1/2}$. Its distribution follows from Eq. (\[eq:GProb\]) as $P(g_z) \propto
g_z^2 \exp(-3 g_z^2/2\langle g^2 \rangle)$, in agreement with Ref..
The case of weak spin-orbit scattering can be addressed by treating the terms proportional to $\lambda$ in Eq. (\[eq:HSA\]) as a small perturbation. To second order in $\lambda$ we find, $${\cal G} =
4 - {4 \lambda^2} \sum_{\nu \neq \mu}
a_{\mu\nu}^{\rm T} a_{\mu\nu}^{\vphantom{no superscript {\rm T}}}
{1 \over (\varepsilon_{\nu} - \varepsilon_{\mu})^2},
\label{eq:pert}$$ where $\Delta$ is the mean level spacing and $a_{\mu\nu}$ is an antisymmetric $3 \times 3$ matrix proportional to the matrix elements of the perturbation in the eigenbasis $\{ |\psi_{\nu} \rangle \}$ of $H(0) = S$, $(a_{\mu\nu})_{ij} = N^{-1/2} \langle
\psi_{\mu} | A_k | \psi_{\nu} \rangle \varepsilon_{kij}$, where $\varepsilon_{kij}$ is the antisymmetric tensor. We first consider the change in the principal $g$-factors due to the matrix element $a_{\mu\nu}$ coupling the level $\varepsilon_{\mu}$ to a close neighboring level $\varepsilon_{\nu}$ where $\nu = \mu + 1$ or $\mu - 1$. (Level repulsion rules out the possibility that both levels $\varepsilon_{\mu \pm 1}$ are very close.) In view of the energy denominators in Eq.(\[eq:pert\]), we may expect that this contribution is dominant. Taking only the relevant matrix element $a_{\mu \nu}$ into account, we find $$g_3 = 2,\ \ g_1 = g_2 = 2 - {\case{1}{2} \lambda^2}
{(\varepsilon_{\mu} - \varepsilon_{\nu})^{-2}}
{\rm tr}\, a_{\mu\nu}^{\rm T}
a_{\mu\nu}^{\vphantom{no superscript {\rm T}}},$$ where $\nu = \mu \pm 1$. Since the spacing distribution $P(|\varepsilon_{\mu}
- \varepsilon_{\nu}|) \approx \pi \Delta^{-2}|\varepsilon_{\mu} -
\varepsilon_{\nu}|$ for small $\varepsilon_{\mu} - \varepsilon_{\nu}$ [@Mehta], we find that the distribution $P(g)$ of both $g_1$ and $g_2$ has tails $P(g) = (3 \lambda^2 / 2 \pi) (2 - g)^{-2}$ for $2-g
\gg \lambda^2$. The main effect of contributions from the other energy levels in Eq. (\[eq:pert\]) is a reduction of $g_3$ below $2$, and a separation of $g_1$ and $g_2$. This is illustrated in Fig. \[fig:3\]. The three regimes of weak, intermediate, and strong spin-orbit scattering are compared in Fig. \[fig:2\], using a numerical evaluation of the distributions of the three principal $g$-values.
=0.99
We gratefully acknowledge discussions with T. A. Arias, D. Davidovic, K. M. Frahm, Y. Oreg, D. C. Ralph, and M. Tinkham. Upon completion of this project, we learned of Ref. , which contains some overlap with our work. This work was supported in part by the NSF through the Harvard MRSEC (grant DMR 98-09363), and by grant DMR 99-81283.
D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. Lett. [**74**]{}, 3241 (1995); [*ibid*]{} [**78**]{}, 4087 (1997); D. Davidovic and M. Tinkham, Phys. Rev. Lett. [**83**]{}, 1644 (1999); cond-mat/9910396.
D. G. Salinas, S. Guéron, D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. B [**60**]{}, 6137 (1999).
W. P. Halperin, Rev. Mod. Phys. [**58**]{}, 533 (1986).
C. P. Slichter, [*Principles of Magnetic Resonance*]{} (Springer, Berlin, 1980).
K. A. Matveev, L. I. Glazman, and A. I. Larkin, cond-mat/0001431.
M. L. Mehta, [*Random Matrices*]{} (Academic, New York, 1991).
A. Pandey and M. L. Mehta, Commun. Math. Phys. [**87**]{}, 449 (1983).
J. B. French, V. K. B. Kota, A. Pandey, and S. Tomsovic, Ann. Phys. (N. Y.) [**181**]{}, 198 (1988).
H.-J. Sommers and S. Iida, Phys. Rev. E [**49**]{}, 2513 (1994).
V. I. Fal’ko and K. B. Efetov, Phys. Rev. B [**50**]{}, 11267 (1994); Phys. Rev. Lett. [**77**]{}, 912 (1996).
S. A. van Langen, P. W. Brouwer, and C. W. J. Beenakker, Phys. Rev. E [**55**]{}, 1 (1997).
A quaternion is a $2 \times 2$ matrix $q$ of the form $
q = q_0 \openone + i \sum_{j} q_j \sigma_j,
$ where the $q_j$ are real numbers ($j=0,1,2,3$).
For example, if the principal axes of ${\cal G}$ are labeled $\hat e_1$, $\hat e_2$, and $\hat e_3 = \hat e_1 \times \hat e_2$, and we apply a static field $B \hat e_3$, then a resonant AC field $\vec b \propto \mbox{Re}\, [(g_2 \hat e_1 + i \eta
\hat e_2 g_1) e^{-i \omega t}]$, with $\omega = g_3 |\mu_B| B/\hbar > 0$, will produce spin flips for $\eta=1$ but not for $\eta=-1$.
K. M. Frahm, private communication.
E. Brézin, S. Hikami, and A. Zee, Nucl. Phys. B [**464**]{}, 411 (1996).
The same is true for the GOE–GUE crossover, where the relevant quantity is the “phase-rigidity” $\rho = |\langle {\cal T} \psi |
\psi \rangle|^2$. The distribution $P(\rho)$ for large magnetic fields equals $P(\rho)$ in the GUE, with $N$ replaced by $2 \alpha^2$, where $\alpha$ is a crossover parameter analogous to $\lambda$, see Ref..
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $E,F$ be Banach spaces. In the case that $F$ is reflexive we give a description for the solutions $(f,g)$ of the Banach-orthogonality equation $$\sl f(x),g(\alpha)\sr=\sl x,\alpha\sr\hspace{10mm}\forall x\in E,\forall \alpha\in E^*,$$ where $f:E\to F,g:E^*\to F^*$ are two maps. Our result generalizes the recent result of [Ł]{}ukasik and Wójcik in the case that $E$ and $F$ are Hilbert spaces.'
address: |
Department of Mathematics\
Institute for Advanced Studies in Basic Sciences\
P.O. Box 45195-1159, Zanjan 45137-66731, Iran
author:
- Maysam Maysami Sadr
title: Decomposition of functions between Banach spaces in the orthogonality equation
---
Introduction
============
The topological dual of a Banach space $E$ is denoted by $E^*$, and $\sl\cdot,\cdot\sr$ denotes the usual pairing between elements of $E$ and $E^*$. Let $E,F$ be Banach spaces and $f:E\to F$, $g:E^*\to F^*$ be two maps. Consider the following functional equation, which we call *Banach-orthogonality equation*, for unknown maps $f,g$: $$\label{e1}
\sl f(x),g(\alpha)\sr=\sl x,\alpha\sr\hspace{10mm}\forall x\in E,\forall \alpha\in E^*.$$ The *generalized orthogonality equation* introduced in [@LukasikWojcik1] and [@Chmielinski1] is a special case of (\[e1\]) where $E$ and $F$ are Hilbert spaces, $E^*$ and $F^*$ are identical with $E$ and $F$ via the canonical (conjugate) linear isometric isomorphisms induced by the inner products, and $\sl\cdot,\cdot\sr$ denotes the inner products. [Ł]{}ukasik and Wójcik [@LukasikWojcik1] have recently characterized the solutions $(f,g)$ of the generalized orthogonality equation. (See also [@Lukasik1] in the case that $E,F$ are pre-Hilbert spaces.) In this paper we give a characterization of the solutions $(f,g)$ of (\[e1\]) in the case that $F$ is a reflexive Banach space, via decompositions of $f,g$ by invertible linear and arbitrary nonlinear parts. Then the result of [@LukasikWojcik1] is seen as a special case of our main result. In the remainder of this section we consider two preliminary lemmas and in the next section we give the main result. The proof of the following lemma is similar to the proofs of Lemmas 1 and 2 of [@LukasikWojcik1]. For the sake of completeness we add the proof.
\[l1\] Let $C,D$ be Banach spaces and $S:C\to D,T:C^*\to D^*$ be two maps satisfying $$\sl S(x),T(\alpha)\sr=\sl x,\alpha\sr\hspace{10mm}\forall x\in C,\forall \alpha\in C^*.$$
1. If $\overline{\r{Lin}S(C)}=D$ then $T$ is a bounded linear operator.
2. If $\overline{\r{Lin}T(C^*)}=D^*$ then $S$ is a bounded linear operator.
\(i) For every $\alpha,\alpha',x$ we have $$\sl S(x),T(\alpha+\alpha')\sr=\sl x,\alpha+\alpha'\sr=\sl S(x),T(\alpha)+T(\alpha')\sr,$$ and thus $T(\alpha+\alpha')|_{S(C)}=(T(\alpha)+T(\alpha'))|_{S(C)}$. Since $T(\alpha+\alpha'),T(\alpha)+T(\alpha')$ are bounded linear functionals and $\overline{\r{Lin}S(C)}=D$ we have $T(\alpha+\alpha')=T(\alpha)+T(\alpha')$. Similarly, for every scalar $r$ we have $T(r\alpha)=rT(\alpha)$. So it was proved that $T$ is linear. Suppose that $\alpha_n\to\alpha$ and $T(\alpha_n)\to\beta$. We have $\sl S(x),T(\alpha_n)\sr=\sl x,\alpha_n\sr\to\sl x,\alpha\sr=\sl S(x),T(\alpha)\sr$. On the other hand, $\sl S(x),T(\alpha_n)\sr\to\sl S(x),\beta\sr$. Thus, $\sl S(x),T(\alpha)\sr=\sl S(x),\beta\sr$ and hence $T(\alpha)|_{S(C)}=\beta|_{S(C)}$. It follows that $T(\alpha)=\beta$. Now since $C^*,D^*$ are Banach spaces Closed Graph Theorem implies that $T$ is continuous. The proof of (ii) is similar.
Let $C$ be a Banach space. For subsets $U\subseteq C$ and $V\subset C^*$ we let $$U^\bot:=\{\alpha\in C^*: \sl x,\alpha\sr=0, \forall x\in U\},\hspace{5mm} V^\bot:=\{x\in C: \sl x,\alpha\sr=0, \forall \alpha\in V\}.$$ It is easily checked that $U^\bot$ and $V^\bot$ are closed linear subspaces. Let $K$ be a closed linear subspace of $C$. Then the quotient vector space $C/K$ is a Banach space with the quotient norm defined by $\|x+K\|:=\inf_{x'\in K}\|x+x'\|$. It is clear that $(C/K)^*$ embeds canonically in $C^*$ via the map $(C/K)^*\xrightarrow{I}C^*$ defined by $\alpha\mapsto \alpha P$ ($\alpha\in(C/K)^*$) where $C\xrightarrow{P}C/K$ denotes the canonical projection. We also call the map $C^*\xrightarrow{R}K^*$, defined by $\alpha\mapsto\alpha|_K$, the canonical restriction.
\[l2\] Let $C$ be a reflexive Banach space and $W$ be a closed linear subspace of $C^*$. Then $W=W^{\bot\bot}$. It follows that $(C/W^\bot)^*$ coincides with $W$ via the embedding $I$ defined as above with $K:=W^\bot$.
It is clear that $W\subseteq W^{\bot\bot}$. We must show the reverse inclusion. Assume, to reach a contradiction, that there exists an $\alpha$ in $W^{\bot\bot}\setminus W$. By geometric version of the Hahn-Banach Theorem and reflexivity of $C$ there is a $x\in C$ with $\sl x,\alpha\sr\ne0$ and $\sl x,\alpha'\sr=0$ for every $\alpha'\in W$. This contradicts $\alpha\in W^{\bot\bot}$. Thus $W=W^{\bot\bot}$. For the proof of the second part, let $P$ be as above with $K:=W^\bot$. Let $\beta\in(C/W^\bot)^*$. We must show that $\beta P\in W$. But it is obvious because $\beta P\in W^{\bot\bot}=W$.
The Main Result
===============
Let $S:C\to D$ be a bounded linear operator. Then its adjoint $S^*:D^*\to C^*$ is a bounded linear operator defined by $\beta\mapsto \beta S$ for every $\beta\in D^*$. Thus we have the following identity. $$\sl Sx,\beta\sr=\sl x,S^*\beta\sr\hspace{10mm}\forall x\in C,\forall \beta\in D^*.$$ It is easily checked that $S$ is invertible if and only if $S^*$ is invertible. Also if $S^*$ is surjective then $S$ is injective. Now we are ready to state our main result.
\[t1\] Let $E,F$ be Banach spaces such that $F$ is reflexive. Let $f:E\to F$, $g:E^*\to F^*$ be two maps. Then the pair $(f,g)$ satisfies (\[e1\]) if and only if there are
1. closed linear subspaces $M\subseteq L\subseteq F$,
2. an invertible bounded linear operator $A:E\to L/M\subseteq F/M$,
3. a (not necessarily linear) right inverse $\varphi$ of the canonical projection $L\xrightarrow{P}L/M$, i.e. $P\varphi=\r{id}_{L/M}$, and
4. a (not necessarily linear) right inverse $\psi$ of the canonical restriction $F^*\xrightarrow{R}L^*$, i.e. $R\psi=\r{id}_{L^*}$,
such that $$f=\varphi A,\hspace{10mm}g=\psi I(A^*)^{-1},$$ where $I$ denotes the canonical injection $(L/M)^*\xrightarrow{I}L^*$.
The “if” part of the theorem is easily checked. We only prove the “only if” part.
Let $L:=\overline{\r{Lin}f(E)}\subseteq F$. Let $F^*\xrightarrow{R}L^*$ denote the canonical restriction and let $Q_0:=Rg:E^*\to L^*$. We have $\sl f(x),Q_0(\alpha)\sr=\sl x,\alpha\sr$, $\forall x\in E,\forall \alpha\in E^*$. Applying Lemma \[l1\](i) with $C:=E,D:=L,S:=f,T:=Q_0$, we find that $Q_0$ is a bounded linear operator. Note that there is a right inverse $\psi$ of $R$ such that $g=\psi Q_0$.
Let $M:=Q_0(E^*)^\bot=\overline{Q_0(E^*)}^\bot\subseteq L$. Thus $Q_0(\alpha)|_M=0$ for every $\alpha\in E^*$. This shows that $Q_0$ actually takes the elements of $E^*$ to the space $(L/M)^*$ that is considered as a subspace of $L^*$ via the canonical injection $(L/M)^*\xrightarrow{I}L^*$. We let $\hat{Q}_0$ denote the same operator $Q_0$ but with the new codomain $(L/M)^*$. Thus we have $Q_0=I\hat{Q}_0$ and also $g=\psi I\hat{Q}_0$.
Since $F$ is reflexive, $L$ is reflexive and thus it follows from Lemma \[l2\] that $\overline{\hat{Q}_0(E^*)}=\overline{Q_0(E^*)}=(L/M)^*\subseteq L^*$. So $\hat{Q}_0$ is a bounded linear operator from $E^*$ to $(L/M)^*$ with dense range.
Let $Q_1:=Pf$ where $L\xrightarrow{P}L/M$ denotes the canonical projection. Thus there is a right inverse $\varphi$ of $P$ such that $f=\varphi Q_1$. We also have $\sl Q_1(x),\hat{Q}_0(\alpha)\sr=\sl x,\alpha\sr$, $\forall x\in E,\forall \alpha\in E^*$. Applying Lemma \[l1\](ii) with $C:=E,D:=L/M,S:=Q_1,T:=\hat{Q}_0$, we find that $Q_1$ is a bounded linear operator.
Let $x$ be an arbitrary element of $E$. Let $\alpha\in E^*$ be such that $\|\alpha\|=1$ and $\sl x,\alpha\sr=\|x\|$. (The existence of such a functional is a consequence of the Hahn-Banach theorem.) We have $$\|x\|=|\sl x,\alpha\sr|=|\sl Q_1(x),\hat{Q}_0(\alpha)\sr|\leq\|\hat{Q}_0\|\|Q_1(x)\|.$$ This implies that for every $x\in E$, $\frac{\|x\|}{\|\hat{Q}_0\|}\leq \|Q_1(x)\|$. It follows that the range of $Q_1$ is a closed linear subspace of $L/M$. On the other hand, since $\overline{\r{Lin}f(E)}=L$, we have $\overline{Q_1(E)}=L/M$. Thus $Q_1$ is a surjective operator.
For every $x\in E$ and $\alpha\in E^*$ we have $$\sl x,\alpha\sr=\sl Q_1(x),\hat{Q}_0(\alpha)\sr=\sl x,Q_1^*\hat{Q}_0(\alpha)\sr.$$ This implies that $Q_1^*\hat{Q}_0=\r{id}_{E^*}$. Thus $Q_1^*$ is surjective and $Q_1$ is injective. It follows that $Q_1$ is invertible and $\hat{Q}_0=(Q_1^*)^{-1}$. Now, we let $A:=Q_1$. The proof is complete.
The following corollary of Theorem \[t1\] is the main result of [@LukasikWojcik1].
Let $E$,$F$ be Hilbert spaces and $f,g:E\to F$ be two maps. Suppose that $f$,$g$ satisfy the generalized orthogonality equation: $$\sl f(x),g(y)\sr=\sl x,y\sr,\hspace{10mm}\forall x,y\in E,$$ where $\sl\cdot,\cdot\sr$ denotes the inner products. Then there exist
1. pairwise orthogonal closed linear subspaces $F_1,F_2,F_3$ of $F$ such that $$F=F_1\oplus F_2\oplus F_3,$$
2. an invertible bounded linear operator $B:E\to F_1$,
3. a (not necessarily linear) map $\mu:E\to F_2$, and
4. a (not necessarily linear) map $\nu:E\to F_3$,
such that $$f=B+\mu\hspace{10mm}g=(B^*)^{-1}+\nu.$$
We identify $E^*$ and $F^*$ with $E$ and $F$ via the canonical (conjugate) linear isomorphisms induced by inner products. Let $L,M,A,\varphi,\psi$ be as in the formulation of Theorem \[t1\]. Let $F_3$ be the orthogonal complement of $L$ in $F$, let $F_2:=M$, and let $F_1$ be the orthogonal complement of $M$ in $L$. Thus $L/M$ is canonically identified with $F_1$ and $L=F_1\oplus F_2$. We let $B$ be the same operator $A$ but with codomain $F_1\cong L/M$. Let $F_1\xrightarrow{\hat{\varphi}}F_2\oplus F_1$ denote the same map $L/M\xrightarrow{\varphi}L$. Thus there is a unique map $F_1\xrightarrow{\tilde{\varphi}}F_2$ such that $\hat{\varphi}=\tilde{\varphi}+\r{id}_{F_1}$. We let $\mu:=\tilde{\varphi}B$. Let $L\xrightarrow{\hat{\psi}}L\oplus F_3$ denote the same map $L^*\xrightarrow{\psi}F^*$. Thus there is a unique map $L\xrightarrow{\tilde{\psi}}F_3$ such that $\hat{\psi}=\r{id}_L+\tilde{\psi}$. We let $\nu:=\tilde{\psi}(B^*)^{-1}$. This completes the proof.
Characterize the solutions of (\[e1\]) in the case that $F$ is not reflexive.
A natural generalization of the problem considered in this paper is as follows.
Let $(E_1,E_2)$ be a pair of Banach spaces together with a pairing $\sl\cdot,\cdot\sr$ that is a (non degenerate) bilinear functional on $E_1\times E_2$. Let $(F_1,F_2)$ be another pair of Banach spaces with the pairing $[\cdot,\cdot]$. Characterize the solutions $(E_1\xrightarrow{f_1}F_1,E_2\xrightarrow{f_2}F_2)$ of the following functional equation. $$[f_1(x_1),f_2(x_2)]=\sl x_1,x_2\sr\hspace{10mm}\forall x_1\in E_1,\forall x_2\in E_2.$$
Another possible extension is related to the theory of Hilbert C\*-modules [@Lance1]:
Characterize the solutions of equation (\[e1\]) in the case that $E$ and $F$ are Hilbert C\*-modules over a C\*-algebra $\mathcal{A}$ and $\sl\cdot,\cdot\sr$ denotes the $\mathcal{A}$-valued inner products.
[10]{} J. Chmieliński, *Orthogonality equation with two unknown functions*, Aequationes mathematicae 90, no. 1 (2016): 11–23. C. Lance, *Hilbert C\*-modules: a toolkit for operator algebraists*, London Math. Soc. Lecture Notes Series, vol. 210, Cambridge University Press, Cambridge, 1994. R. [Ł]{}ukasik, *A note on the orthogonality equation with two functions*, Aequationes mathematicae 90, no. 5 (2016): 961–965. R. [Ł]{}ukasik, P. Wójcik, *Decomposition of two functions in the orthogonality equation*, Aequationes mathematicae 90, no. 3 (2016): 495–499.
|
{
"pile_set_name": "ArXiv"
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---
abstract: 'Determining the binary fraction for a population of asteroids, particularly as a function of separation between the two components, helps describe the dynamical environment at the time the binaries formed, which in turn offers constraints on the dynamical evolution of the solar system. We searched the [*NEOWISE*]{} archival dataset for close and contact binary Trojans and Hildas via their diagnostically large lightcurve amplitudes. We present 48 out of 554 Hilda and 34 out of 953 Trojan binary candidates in need of follow-up to confirm their large lightcurve amplitudes and subsequently constrain the binary orbit and component sizes. From these candidates, we calculate a preliminary estimate of the binary fraction without confirmation or debiasing of $14-23$% for Trojans larger than $\sim 12$ km and $30-51$% for Hildas larger than $\sim 4$ km. Once the binary candidates have been confirmed, it should be possible to infer the underlying, debiased binary fraction through estimation of survey biases.'
author:
- 'S. Sonnett, A. Mainzer, T. Grav, J. Masiero, J. Bauer'
title: |
Binary Candidates in the Jovian Trojan and Hilda\
Populations from [*NEOWISE*]{} Lightcurves
---
Introduction
============
Trojan asteroids lie in stable orbits at the L4 (leading) and L5 (trailing) Lagrange points of a planet. There are currently $\sim 5,500$ Jovian Trojan asteroids known, making them the most numerous known Trojan population and thus one of the most useful for constraining the dynamical processes that shaped their orbits and physical states (size, structure, etc.). Just inward of the Jovian Trojans are the Hildas in 3:2 orbital resonance with Jupiter. In early solar system formation models of minimal planetary migration, Jovian Trojans (hereafter, Trojans) and Hildas were captured relatively gently in situ .
The Nice model instead proposes that when Jupiter and Saturn reached 2:1 orbital resonance, a violent scattering episode was ignited, with Neptune moving into the Trans-Neptunian region and chaotically scattering planetesimals [[[*e.g.*]{}]{}, @2005Natur.435..462M; @2005Natur.435..459T; @2005Natur.435..466G; @2011AJ....142..152L]. The Nice model thus predicts that Trojans were captured from the trans-Neptunian region and experienced a turbulent dynamical environment relative to previous formation models. A later version of the Nice model suggests that if one of the ice giants traversed one of the Trojan clouds during migration, the clouds would undergo asymmetric depletion, producing the difference in population ratio observed today [$N_{L4} : N_{L5} = 1.4 \pm 0.2$; @2011ApJ...742...40G; @2013ApJ...768...45N]. Combined with the Grand Tack model of inner solar system mixing through Jupiter’s migration, the Nice model also predicts Hildas to have similar origins as Trojans [@2009Natur.460..364L; @2011Natur.475..206W].
In order to help discern the Trojans’ formation location, their present dynamical state should be well-characterized and compared with those of other small body populations like the Hildas. Determining the fraction of Trojans and Hildas in binary or multiple systems is one of the fundamental modes of constraining dynamical and collisional history. For example, a turbulent environment like the one described in the Nice model might imply more interaction between small bodies and consequently a higher probability of either disrupting more weakly bound wide binaries or causing wide binaries to spiral inward, in which case we should see a low wide binary fraction but perhaps a high tight binary fraction [separations less than five times the Hill radius of the primary @2011ApJ...727L...3P]. Several binary formation models exist, each of which make a set of predictions about the binary’s synodic orbit and sometimes its physical properties (mass ratio, similarity between component surfaces, etc.). For example, dynamical friction tends to produce tight binaries while exchange reactions and three-body interactions increases mutual separation, favoring production of wide binaries [@2002Natur.420..643G; @2002Icar..160..212W; @2004Natur.427..518F; @2005MNRAS.360..401A].
Some tight binaries can be identified by their lightcurves. If the binary components are near-fluid rubble piles and not monoliths, they become tidally elongated toward each other, distorting into Jacobi ellipsoid shapes stretched along the semi major axis of the system [@1969efe..book.....C]. The lightcurve of a binary made of two elongated components can have an amplitude so high that it cannot be explained by a singular equilibrium rubble pile. Very large amplitude lightcurves therefore offer a means of identifying candidate rubble pile binaries [$\Delta m > 0.9$ magnitudes compared to an average lightcurve amplitude of 0.3 magnitudes for Trojans; @2004AJ....127.3023S; @2009Icar..202..134W]. This technique of identifying binary candidates is limited to systems oriented such that the variation can be observed and with mass ratios high enough ($\geq 0.6$ mags) to cause sufficiently diagnostic elongation of the components . Still, [@2007AJ....134.1133M] successfully used it to identify two L5 Trojan binaries (17365 and 29314 Eurydamas).
Apart from 17365 (1978 VF$_{11}$) and 29314 Eurydamas, two other Trojans binaries are known: 617 Patroclus-Menoetius and 624 Hektor [@2001IAUC.7741....2M; @2006DPS....38.6507M]. 624 Hektor is in fact a triple system possibly formed through a low-velocity collision, with a bilobate primary and a moderately separated satellite [@2014ApJ...783L..37M]. 617 Patroclus-Menoetius is a moderately separated system consisting of two spheroids nearly equal in size and with a low bulk density [@2006Natur.439..565M].
In addition to the [@2007AJ....134.1133M] survey for tight binaries, three other dedicated observational surveys for wide Trojan binaries have been conducted. [@2006DPS....38.6507M] used high-resolution direct imaging to search for L4 binaries, detecting one (624 Hektor) out of 55 objects observed. [@2007DPS....39.6009M] also used direct imaging on a sample of 35 Trojans, finding no binaries. Lastly, [@2014LPI....45.1703N] directly imaged 8 Trojans, finding no binaries. Surveys that could inadvertently detect tight binaries by being aimed at determining rotation periods and amplitudes of Trojans have mostly sampled only large objects ($\gtrsim 30$ km) and found no bound pairs [[[*e.g.*]{}]{}, @1992Icar...95..222B; @2011AJ....141..170M]. The Hildas have never been explicitly searched for binaries, though several have well-constrained lightcurves, some of which exhibit large amplitudes typical of contact binaries [@1988Icar...73..487H; @1998Icar..133..247D; @1999Icar..138..259D].
In this work, we seek to more fully explore the tight binary fraction in an effort to understand how Trojans are dynamically linked to other small body populations. To that end, we harvested Trojan and Hilda lightcurves identified by the solar system data processing portion of the Wide-field Infrared Survey Explorer mission [[*WISE*]{}; @2010AJ....140.1868W], known as NEOWISE [@2011ApJ...731...53M]. Here, we present the candidates identified by our binary search algorithm for objects within our sensitivity range in the 12 $\mu$m band (roughly corresponding to diameters $\gtrsim 12$ km). Follow-up is needed on each of these 29 candidates previously not known to be binary in order to: (i) reduce the uncertainty in their photometric ranges, which in some cases is needed to confirm their high amplitudes; and (ii) enable characterization of the system through detailed Roche binary modeling of the component sizes and orientations. In an upcoming publication, we will report the binary fraction that can be extrapolated from these candidates as a function of dynamical class (Trojans vs. Hildas), Trojan cloud designation, taxonomic type, and separation between components.
Observations
============
Lightcurve data were taken by the [*WISE*]{} spacecraft, which conducted a space-based all-sky survey that operated in four bandpasses simultaneously: 3.4, 4.6, 12, and 22 $\mu$m [denoted W1, W2, W3, and W4; @2010AJ....140.1868W]. The [*WISE*]{} observing cadence typically provided 12 observations per object per bandpass spanning $\sim 36$ hours. Several Trojans and Hildas were also observed at multiple epochs [@2012ApJ...759...49G]. Profile-fitting photometry was done using the [*WISE*]{} science data processing pipeline described in [@2012wise.rept....1C].
The [*WISE*]{} cadence with 3 hour spacing covering a 1.5 day span cannot be repeated from ground-based observatories unless telescopes at multiple longitudes are coordinated, so NEOWISE lightcurves offer a nearly unique advantage in sampling periodicities on the order of $\sim1-2$ days . Also, the W3 and W4 bandpasses contain almost purely thermal emission from Trojans and Hildas, somewhat isolating shape as the cause of brightness variations (Fig. \[Fig:THSED\]). Moreover, the peak of the Trojan and Hilda black bodies lie between W3 and W4, making them relatively bright at those wavelengths. Of these two bandpasses, W3 is more sensitive, making it an ideal filter choice for calculating the lightcurve amplitude.
Sample and Analysis\[Sec:Sample\]
=================================
In total, [*WISE*]{} observed $\sim 1800$ Trojans and $\sim 1100$ Hildas, with diameters between $4-150$ km and $1-220$ km, respectively [@2011ApJ...742...40G; @2012ApJ...744..197G]. We chose to limit our sample to objects that could have binary lightcurve minima with a signal-to-noise (S/N) greater than five ([[*i.e.*]{}]{}, magnitude uncertainty $\leq 0.2$ magnitudes). We explored the relationship between W3 magnitude of our sample and its uncertainty by fitting polynomials with various numbers of terms (Fig. \[Fig:W3magVsW3sigma\]). We found that a six-term polynomial produced the best quality fit (lowest $\chi_{\nu}^{2}$). The typical W3 magnitude corresponding to a W3 uncertainty of 0.2 was 10.3 mag, which roughly corresponds to $\sim12$ km for the Trojans and $\sim 4$ km for the Hildas. Therefore, we limited our sample to only objects with mean W3 magnitudes such that their binary lightcurve minima would be brighter than 10.3 magnitudes in W3, meaning that at no point will a typical large-amplitude binary have S/N $< 5$. We were left with 953 Trojans and 554 Hildas in our sample that met these criteria.
Choosing a sample with this physical size range meant including objects with a range of possible structures, which could affect the applicability of our search technique. This method of identifying binary candidates through large lightcurve amplitudes relies on the components being near-fluid rubble piles that tidally distort as opposed to rigid monoliths. Shape and structure of an asteroid is thought to correlate with its size, with larger objects being massive enough to remain gravitationally bound after successive impacts and smaller objects likely being monolithic collisional remnants [[[*e.g.*]{}]{}, @1982Icar...52..409F]. The diameter below which a small body population starts to become strength-dominated monoliths is not well known. Observational studies found that sub-kilometer sized near-Earth asteroids contain a large fraction of objects spinning faster than the critical period (below which rubble piles spin apart), suggesting that asteroids start to become strength-dominated at $\lesssim 1$km [[[*e.g.*]{}]{}, @2000Icar..148...12P; @2013Icar..225..141S]. In between rubble piles and monoliths, an object can be fractured but with a lower porosity than pure rubble piles, allowing them to take on more elongated shapes than rubble piles, which have a limiting axis ratio of $a:b \sim 2.3$ . The non-binary near-Earth asteroid 433 Eros is an example of such a fractured body elongated beyond the rubble pile limit [$a\sim 31$ km, $b \sim 14$ km; @2000Sci...289.2088V; @2001LPI....32.1721W]. Eros’ large size violates the suggestion that objects larger than the sub-kilometer range cannot have $a:b > 2.3$. These results’ relevance to the Trojan region, which may have formed in the more ice-rich outer solar system, has not been developed. Tidal disruption caused by close flybys with planets might also change the shape of an asteroid [@1999AJ....117.1921B]. This distortion mechanism has been studied for near-Earth asteroids through simulations by [@2014Icar..239..118B], but it has not been explored in the context of Trojans or Hildas. It is therefore possible that some of the high-amplitude candidates presented here may in fact be monolithic shards or elongated low-porosity fractured bodies.
For each object, we excluded measurements with field sources within a radius of $10\arcsec$ of their centroid (the beam size being $\sim 6\arcsec$ in W1-W3 and $\sim 12\arcsec$ in W4) by comparing the single-frame extracted source lists with the AllWISE Catalog [@2013yCat.2328....0C], which coadds together all exposures available at each point on the sky. We estimate that most sources outside this radius would not affect profile-fitting photometry. However, a very extended source could still significantly affect the background determination, effectively raising the background threshold and underestimating the target flux. Source or background flux can also be affected by observations taken inside the South Atlantic Anomaly, which would mean a significant increase in cosmic ray hits, or by observations taken close to the Moon, which would introduce a significant (usually non-linear) gradient to the background. To guard against these contaminants and also against bad pixels affecting the lightcurve, the images for every candidate were visually inspected. We constrained the W3 lightcurve amplitude by calculating the photometric range for each epoch and each object observed from the maximum and minimum usable data points. Corresponding range uncertainties are the sum in quadrature of the maximum’s and minimum’s uncertainties. The values reported are lower limits to the amplitudes since the observations may have missed the intrinsic lightcurve extrema.
Results and Discussion
======================
The Trojan and Hilda binary candidates identified after performing the analysis described in §\[Sec:Sample\] are reported in Tables \[Table:TrojanCandidates\] and \[Table:HildaCandidates\], respectively. We found that 38 of the 953 Trojans in our sample had photometric ranges larger than 0.9 mag, 35 of which were not known binaries, making them new candidate binary objects (Fig. \[Fig:TrojanCandidates\]). The three known binaries flagged by our binary search technique were 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas. As described in the introduction, observations of 624 Hektor are consistent with a triple system, where a small satellite is moderately separated from the large, bilobate primary [@2014ApJ...783L..37M]. Our technique is only sensitive to detecting the bilobate primary (itself being a binary), since the satellite is too small to produce significant effect on the lightcurve. WISE obtained two epochs of data on 624 Hektor. Using the ephemeris generator from the Institut de Mécanique Céleste et de Calcul des Éphémérides (IMCCE), we determined that 624 Hektor was only oriented such that the primary would produce a large lightcurve amplitude during one of our epochs of coverage, which was indeed flagged as a binary in our search. Binary Trojans 17365 and 29314 Eurydamas are consistent with contact binaries, though their binary orbital elements are not as well known, preventing us from checking their orientation at the time of our coverage epochs [@2007AJ....134.1133M].
The only other known Trojan binary, 617 Patroclus-Menoetius, is moderately separated with roughly spherical components, consequently giving it a lower thermal lightcurve amplitude than would be detected by our algorithm. We used the IMCCE ephemeris generator again to determine the configuration of the Patroclus-Menoetius system and found that is was neither fully nor partially eclipsing during either of our epochs of coverage, giving it a very low WISE thermal lightcurve amplitude of $0.18 \pm 0.02$ magnitudes. Of the 554 Hildas explored here, 48 had photometric ranges larger than the binary candidate limit (Fig. \[Fig:HildaCandidates\]). There are 503 L4 Trojans in the sample, 21 of which are binary candidates, compared to 16 of the 446 L5 Trojans being binary candidates. We note that the sum of the L4 and L5 sample does not equal the complete Trojan sample explored because 4 Trojans do not yet have cloud designations due to their unusual orbits ([[*i.e.*]{}]{}, spending much of their time opposite Jupiter or traversing both clouds equally).
We can estimate the observed binary fraction before debiasing (assuming all candidates are true binaries) by dividing the number of candidates by the total number of objects in that sample, then dividing by the probability that the system will be oriented such that a large amplitude is projected back to the observer [$17-29$% depending on the angularity, or “boxiness” of the components’ shapes; [[*e.g.*]{}]{}, @2007AJ....134.1133M]. This approach gives us first-order observed binary fractions of $13-23$% for all Trojans, $14-25$% for L4 Trojans, $12-21$% for L5 Trojans, and $30-51$% for Hildas. However, proper debiasing for incompleteness of the survey, Poisson statistics, and consideration for the probability of detecting a large amplitude given the observing cadence, uncertainties, and theoretical distribution of rotation periods and amplitudes is needed to determine the true binary fraction. This debiasing is the subject of an upcoming publication by the authors.
Compared to other large-sample lightcurve surveys of Trojans and Hildas, we found a greater number of photometric ranges indicative of possible binarity. Of the 47 Hildas whose lightcurves were constrained by [@1998Icar..133..247D], one showed a very large lightcurve amplitude – 3923 Radzievskij – an object also flagged as having a large amplitude in our results. [@2007AJ....134.1133M] found that 2 of their 114 Trojans had large amplitudes, giving a binary fraction estimate of $6-10$% when computed as described in the preceding paragraph. The differences between their observing cadence and ours could explain the discrepancy in the fraction of Trojans observed to have large amplitudes since this factor was not accounted for in the binary fraction estimate. They sampled each object’s lightcurve five times whereas we observed each object an average of 12 times per epoch, affording us more opportunities to catch the lightcurve extrema.
Another possible contributor to the discrepancy in estimated binary fractions is the physical size range of the survey sample. [@2007AJ....134.1133M] were able to extend their search down to $\sim 20$ km objects, whereas our Trojan sample reached $\sim 12$ km, introducing a possible bias in our results toward detecting smaller highly elongated bodies with a fractured structure instead of a rubble pile (see §1 for discussion of asteroid structure versus size). Lastly, if only half of our binary candidates are true binaries and not highly elongated fractured bodies like 433 Eros, then these fractions will decrease by a factor of two, making our results consistent with the [@2007AJ....134.1133M] survey.
[@2011AJ....141..170M] densely sampled the lightcurves of 80 Trojans with diameters ranging $\sim 60-150$ km, finding none with large amplitudes. Their cadence and sample diameter range may have similarly affected their null detection, especially after noting that only one of our 34 Trojans with high lightcurve amplitudes (the bilobate primary of the 624 Hektor system) has a diameter in the range they explored [@2014ApJ...783L..37M]. However, another explanation for our high observed binary fraction could be that the binary fraction amongst smaller Trojans is higher than larger Trojans. Debiasing and follow-up of our candidates is needed to confirm that possibility. We therefore encourage the observing community to obtain densely sampled lightcurves of the binary candidates presented here in order to confirm their nature, allowing tight constraints to be set on the binary fraction as a function of orbital elements, size, taxonomy, dynamical classification, and Trojan cloud designation.
Acknowledgments
===============
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![\[Fig:THSED\] Theoretical spectral energy distributions for Trojans and Hildas with four different albedos. Fluxes are in Jy. Dotted curves show the reflected light component, dashed curves show the emitted light, and solid curves are the combined flux. The vertical black dotted lines show the band centers for the four WISE filters, and the vertical colored bands show the effective wavelength coverage of the W1, W2, W3, and W4 bands [cyan, purple, orange, and red, respectively; @2010AJ....140.1868W]. We used the W3 filter at 12 $\mu$m to identify diagnostically large lightcurve amplitudes in this work. For the nominal range of Trojan and Hilda albedos (0.02, 0.06, 0.10, 0.15 for red, blue, green, magenta, respectively) the W3 filter is dominated by thermal emission.](TrojHilda_sed.pdf){width="6.5in"}
![\[Fig:W3magVsW3sigma\] Six-term polynomial fit (solid red line) to the Trojan and Hilda W3 magnitudes versus their photometric uncertainties (black dots). This relationship was used in helping determine the sensitivity limits of our binary search technique. ](W3mproVsW3sigmpro.pdf){width="6in"}
[lccrrcrcl]{} 624 & 150 $\pm$ 2 & 5.249 & 0.024 & 18.2 & L4 & 11 & 1.33 $\pm$ 0.02 & Known binary Hektor\
9431 & 38 $\pm$ 3 & 5.126 & 0.084 & 21.3 & L4 & 11 & 0.91 $\pm$ 0.31 &\
11429 & 38 $\pm$ 1 & 5.272 & 0.029 & 17.1 & L4 & 13 & 0.96 $\pm$ 0.10 &\
13323 & 23.2 $\pm$ 0.6 & 5.112 & 0.089 & 0.9 & L4 & 12 & 0.92 $\pm$ 0.08 &\
15398 & 36 $\pm$ 1 & 5.128 & 0.027 & 28.5 & L4 & 11 & 0.98 $\pm$ 0.19 &\
16152 & 16.2 $\pm$ 0.6 & 5.125 & 0.096 & 3.5 & L4 & 9 & 0.97 $\pm$ 0.15 &\
17365 & 44.9 $\pm$ 0.5 & 5.268 & 0.079 & 11.6 & L5 & 9 & 1.17 $\pm$ 0.04 & Known binary 1978 VF11\
17414 & 21.6 $\pm$ 0.3 & 5.130 & 0.032 & 16.6 & L5 & 9 & 1.02 $\pm$ 0.12 &\
20428 & 27 $\pm$ 3 & 5.219 & 0.145 & 21.0 & L4 & 11 & 0.94 $\pm$ 0.26 &\
25911 & 18 $\pm$ 1 & 5.226 & 0.044 & 21.4 & L4 & 11 & 1.08 $\pm$ 0.19 &\
29314 & 21.4 $\pm$ 0.8 & 5.280 & 0.073 & 15.2 & L5 & 17 & 0.99 $\pm$ 0.15 & Known binary Eurydamas\
51357 & 19.3 $\pm$ 0.8 & 5.201 & 0.070 & 9.0 & L5 & 8 & 1.23 $\pm$ 0.19 &\
55474 & 21 $\pm$ 1 & 5.204 & 0.095 & 18.0 & L5 & 10 & 0.93 $\pm$ 0.19 &\
63241 & 22.6 $\pm$ 0.9 & 5.250 & 0.049 & 25.8 & L4 & 8 & 1.21 $\pm$ 0.18 &\
64270 & 16.5 $\pm$ 0.7 & 5.159 & 0.095 & 12.9 & L5 & 10 & 1.25 $\pm$ 0.15 &\
65225 & 16.7 $\pm$ 0.2 & 5.287 & 0.081 & 7.0 & L4 & 11 & 1.07 $\pm$ 0.15 &\
76820 & 17.5 $\pm$ 0.6 & 5.164 & 0.097 & 18.4 & L5 & 10 & 0.91 $\pm$ 0.17 &\
76836 & 18.3 $\pm$ 0.6 & 5.242 & 0.099 & 23.8 & L5 & 10 & 0.97 $\pm$ 0.12 &\
114141 & 20.9 $\pm$ 0.6 & 5.131 & 0.069 & 19.7 & L5 & 10 & 1.08 $\pm$ 0.13 &\
129135 & 20.0 $\pm$ 0.7 & 5.302 & 0.038 & 33.1 & L5 & 11 & 1.41 $\pm$ 0.26 &\
130190 & 17.2 $\pm$ 0.7 & 5.238 & 0.044 & 14.7 & L4 & 13 & 1.13 $\pm$ 0.18 &\
155337 & 17.2 $\pm$ 0.8 & 5.230 & 0.089 & 17.0 & L5 & 10 & 1.15 $\pm$ 0.21 &\
160140 & 19.3 $\pm$ 0.6 & 5.200 & 0.057 & 24.5 & L4 & 12 & 1.13 $\pm$ 0.13 &\
161018 & 19.2 $\pm$ 0.7 & 5.098 & 0.054 & 12.0 & L4 & 12 & 1.05 $\pm$ 0.15 &\
182178 & 15.1 $\pm$ 0.7 & 5.200 & 0.115 & 25.5 & L5 & 7 & 1.25 $\pm$ 0.18 &\
182445 & 14.3 $\pm$ 0.7 & 5.150 & 0.059 & 17.3 & L5 & 7 & 1.66 $\pm$ 0.51 &\
192221 & 21.4 $\pm$ 0.7 & 5.188 & 0.045 & 27.3 & L4 & 16 & 1.21 $\pm$ 0.22 &\
192389 & 16.1 $\pm$ 0.8 & 5.253 & 0.013 & 22.8 & L4 & 12 & 0.94 $\pm$ 0.22 &\
222861 & 13.0 $\pm$ 0.9 & 5.167 & 0.100 & 6.7 & L4 & 5 & 0.93 $\pm$ 0.21 &\
228114 & 17.0 $\pm$ 0.9 & 5.132 & 0.018 & 14.0 & L4 & 12 & 1.03 $\pm$ 0.21 &\
231631 & 13.5 $\pm$ 0.9 & 5.102 & 0.059 & 9.5 & L4 & 7 & 1.03 $\pm$ 0.18 &\
246550 & 15.2 $\pm$ 0.5 & 5.146 & 0.223 & 6.7 & L4 & 11 & 1.04 $\pm$ 0.15 &\
247969 & 15 $\pm$ 1 & 5.279 & 0.098 & 17.0 & L5 & 8 & 1.03 $\pm$ 0.26 &\
321611 & 16.0 $\pm$ 0.8 & 5.188 & 0.058 & 26.9 & L4 & 11 & 1.00 $\pm$ 0.20 &\
341880 & 15.6 $\pm$ 0.6 & 5.175 & 0.137 & 35.8 & L5 & 15 & 1.13 $\pm$ 0.18 &\
343993 & 13.4 $\pm$ 0.7 & 5.236 & 0.202 & 19.5 & L5 & 11 & 0.95 $\pm$ 0.13 &\
356261 & 17.8 $\pm$ 0.8 & 5.338 & 0.055 & 22.7 & L4 & 10 & 1.48 $\pm$ 0.26 &\
[lccrrrc]{} 2483 & 35.7 $\pm$ 0.2 & 3.972 & 0.278 & 4.5 & 13 & 1.51 $\pm$ 0.02\
3923 & 29.9 $\pm$ 0.2 & 3.963 & 0.223 & 3.5 & 15 & 0.98 $\pm$ 0.02\
4230 & 28.5 $\pm$ 0.8 & 3.946 & 0.133 & 3.1 & 9 & 1.77 $\pm$ 0.04\
15626 & 18.6 $\pm$ 0.4 & 3.950 & 0.112 & 1.8 & 10 & 1.04 $\pm$ 0.07\
16927 & 22.6 $\pm$ 0.1 & 3.980 & 0.139 & 12.9 & 11 & 0.92 $\pm$ 0.04\
21047 & 16.0 $\pm$ 0.3 & 3.982 & 0.176 & 4.5 & 12 & 1.1 $\pm$ 0.2\
22070 & 18.8 $\pm$ 0.5 & 3.977 & 0.276 & 13.8 & 11 & 1.0 $\pm$ 0.2\
23405 & 14.5 $\pm$ 0.5 & 3.952 & 0.130 & 5.2 & 23 & 1.9 $\pm$ 0.1\
31097 & 15.3 $\pm$ 0.7 & 3.960 & 0.109 & 2.7 & 11 & 1.33 $\pm$ 0.09\
39405 & 14.6 $\pm$ 0.7 & 3.960 & 0.222 & 1.8 & 11 & 1.1 $\pm$ 0.3\
39415 & 9.4 $\pm$ 0.4 & 3.925 & 0.209 & 2.4 & 21 & 1.0 $\pm$ 0.2\
45862 & 8.6 $\pm$ 0.1 & 3.972 & 0.162 & 3.2 & 11 & 1.0 $\pm$ 0.2\
46629 & 15.7 $\pm$ 0.1 & 3.953 & 0.243 & 1.7 & 14 & 1.35 $\pm$ 0.03\
54630 & 16.07$\pm$ 0.09 & 3.981 & 0.139 & 9.0 & 12 & 0.91 $\pm$ 0.04\
60398 & 11.5 $\pm$ 0.2 & 3.931 & 0.146 & 1.8 & 8 & 1.03 $\pm$ 0.09\
64390 & 6.60 $\pm$ 0.06 & 3.936 & 0.250 & 2.5 & 16 & 1.5 $\pm$ 0.2\
65389 & 9.78 $\pm$ 0.02 & 3.948 & 0.260 & 2.3 & 14 & 1.02 $\pm$ 0.04\
83900 & 7.5 $\pm$ 0.5 & 3.935 & 0.101 & 3.4 & 13 & 1.1 $\pm$ 0.2\
88230 & 18.5 $\pm$ 0.4 & 3.974 & 0.150 & 7.4 & 26 & 1.19 $\pm$ 0.04\
94266 & 10.9 $\pm$ 0.5 & 3.933 & 0.100 & 8.6 & 23 & 1.5 $\pm$ 0.1\
112822 & 10.2 $\pm$ 0.3 & 3.935 & 0.185 & 10.5 & 13 & 1.04 $\pm$ 0.08\
121005 & 10.0 $\pm$ 0.2 & 3.932 & 0.171 & 7.9 & 17 & 1.2 $\pm$ 0.2\
132868 & 8.9 $\pm$ 0.3 & 3.959 & 0.242 & 2.0 & 14 & 1.12 $\pm$ 0.06\
141557 & 10.5 $\pm$ 0.4 & 3.971 & 0.117 & 4.1 & 22 & 1.6 $\pm$ 0.1\
186649 & 8.9 $\pm$ 0.2 & 3.973 & 0.289 & 5.4 & 11 & 1.5 $\pm$ 0.2\
193291 & 7.7 $\pm$ 0.5 & 3.960 & 0.307 & 9.3 & 10 & 1.2 $\pm$ 0.2\
197558 & 8.5 $\pm$ 0.5 & 4.000 & 0.084 & 7.7 & 13 & 1.0 $\pm$ 0.1\
209512 & 8.1 $\pm$ 0.6 & 3.969 & 0.076 & 8.0 & 21 & 1.1 $\pm$ 0.3\
222490 & 6.2 $\pm$ 0.3 & 3.975 & 0.269 & 3.5 & 11 & 1.5 $\pm$ 0.2\
233939 & 5.1 $\pm$ 0.1 & 3.960 & 0.185 & 6.8 & 12 & 1.1 $\pm$ 0.2\
241528 & 7.3 $\pm$ 0.7 & 3.953 & 0.138 & 3.6 & 18 & 1.3 $\pm$ 0.2\
241994 & 7.77 $\pm$ 0.01 & 3.937 & 0.289 & 5.5 & 12 & 1.1 $\pm$ 0.1\
247405 & 6.4 $\pm$ 0.3 & 3.977 & 0.236 & 10.0 & 16 & 1.2 $\pm$ 0.2\
249416 & 6.2 $\pm$ 0.3 & 3.938 & 0.115 & 4.2 & 22 & 0.9 $\pm$ 0.2\
250139 & 8.3 $\pm$ 0.1 & 3.933 & 0.178 & 8.0 & 14 & 1.5 $\pm$ 0.1\
251338 & 7.2 $\pm$ 0.4 & 3.921 & 0.114 & 12.3 & 12 & 1.1 $\pm$ 0.2\
263793 & 6.34 $\pm$ 0.04 & 3.939 & 0.202 & 2.6 & 11 & 1.4 $\pm$ 0.2\
288443 & 6.7 $\pm$ 0.2 & 3.955 & 0.133 & 8.6 & 11 & 0.9 $\pm$ 0.2\
307321 & 5.9 $\pm$ 0.4 & 3.923 & 0.167 & 4.2 & 12 & 1.0 $\pm$ 0.2\
310756 & 6.2 $\pm$ 0.1 & 3.949 & 0.266 & 8.3 & 16 & 1.05 $\pm$ 0.09\
317150 & 7.0 $\pm$ 0.4 & 3.962 & 0.216 & 2.8 & 8 & 1.7 $\pm$ 0.3\
368099 & 5.5 $\pm$ 0.3 & 3.933 & 0.205 & 9.7 & 14 & 0.9 $\pm$ 0.1\
2002 RM & 5.3 $\pm$ 0.1 & 3.953 & 0.279 & 3.4 & 16 & 1.5 $\pm$ 0.2\
2008 SE$_{268}$ & 4.7 $\pm$ 0.2 & 3.955 & 0.223 & 3.2 & 13 & 1.3 $\pm$ 0.2\
2010 MJ$_{93}$ & 4 $\pm$ 1 & 3.967 & 0.240 & 7.5 & 12 & 1.1 $\pm$ 0.2\
2010 NO$_{52}$ & 4.4 $\pm$ 0.5 & 3.964 & 0.252 & 4.0 & 16 & 1.4 $\pm$ 0.3\
2010 NB$_{115}$ & 5.5 $\pm$ 0.2 & 3.979 & 0.274 & 4.1 & 16 & 1.3 $\pm$ 0.2\
2010 OS$_{14}$ & 6.1 $\pm$ 0.2 & 3.913 & 0.253 & 4.0 & 18 & 1.5 $\pm$ 0.1\
![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_00624.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_09431.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_11429.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_13323.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_15398_mod.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_16152.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_17365.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_17414.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_20428.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_25911.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_29314.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_51357.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_55474.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_63241.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_64270.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_65225.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_76820.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_76836.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_B4141.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_C9135.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_D0190.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_F5337.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_G0140.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_G1018.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_I2178.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_I2445.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_J2221_mod.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_J2389.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_M2861.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_M8114.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_N1631.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_O6550.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_O7969.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_W1611.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_Y1880.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_Y3993.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:TrojanCandidates\] Candidate binary Trojans from our survey identified by their anomalously high lightcurve photometric ranges, including known binaries 624 Hektor, 17365 (1978 VF$_{11}$), and 29314 Eurydamas.](BinaryCandidateLC_Z6261.pdf){width="3.5in" height="2.6in"}
![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_02483.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_03923.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_04230_mod.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_15626.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_16927.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_21047.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_22070.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_23405.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_31097.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_39405.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_39415_mod.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_45862.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_46629.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_54630.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_60398.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_64390.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_65389.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_83900.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_88230.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_94266.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_B2822.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_C1005.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_D2868.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_E1557.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_I6649.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_J3291.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_J7558.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_K9512.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_M2490.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_N3939.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_O1528.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_O1994.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_O7405.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_O9416.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_P0139.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_P1338.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_Q3793.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_S8443.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_U7321.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_V0756.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_V7150.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_K02R00M.pdf "fig:"){width="3.5in" height="2.6in"}
![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_K08SQ8E.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_K10M93J.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_K10N52O.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_K10NB5B.pdf "fig:"){width="3.5in" height="2.6in"} ![\[Fig:HildaCandidates\] Candidate binary Hildas from our survey identified by their anomalously high lightcurve photometric ranges.](BinaryCandidateLC_K10O14S.pdf "fig:"){width="3.5in" height="2.6in"}
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{
"pile_set_name": "ArXiv"
}
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abstract: 'We present a new type of coordination mechanism among multiple agents for the allocation of a finite resource, such as the allocation of time slots for passing an intersection. We consider the setting where we associate one counter to each agent, which we call *karma value*, and where there is an established mechanism to decide resource allocation based on agents exchanging karma. The idea is that agents might be inclined to pass on using resources today, in exchange for karma, which will make it easier for them to claim the resource use in the future. To understand whether such a system might work robustly, we only design the protocol and not the agents’ policies. We take a game-theoretic perspective and compute policies corresponding to Nash equilibria for the game. We find, surprisingly, that the Nash equilibria for a society of self-interested agents are very close in social welfare to a centralized cooperative solution. These results suggest that many resource allocation problems can have a simple, elegant, and robust solution, assuming the availability of a karma accounting mechanism.'
author:
- 'Andrea Censi, Saverio Bolognani, Julian G. Zilly, Shima Sadat Mousavi, Emilio Frazzoli [^1][^2]'
bibliography:
- 'karmagames.bib'
title: |
***Today Me, Tomorrow Thee*: Efficient Resource Allocation\
in Competitive Settings using Karma Games**
---
(15mm,18mm) ****
Introduction
============
The very survival and success of a society with shared resources depends on the rules and protocols agents use to interact with each other.
In designing the rules of these societies, there is always a trade-off concerning centralization, efficiency, robustness, and resiliency. A centralized system for resource allocation needs more infrastructure and is less robust and resilient, yet it is the most efficient. A distributed system is more resilient and privacy-preserving.
In intelligent transportation systems, we can distinguish the “macro” level of the fleet, and the “micro” level of the vehicles. At the macro level, much research has shown how it is possible to obtain a substantial improvement in the efficiency of a transportation network [@samaranayake2017ridepooling; @ruch2018amodeus] by optimizing resource use through cooperative approaches; that is, one takes the perspective of a single agent which is able to control centrally a fleet of vehicles. At the micro level there are similar resource allocation problems. Because of the advent of self-driving cars to be used in autonomous mobility on demand networks, the ‘micro’ coordination problems become interesting, as we study how the codes, customs, and conventions of human drivers can be generalized to a scenario with both artificial and human agents.
![ We propose an innovative approach to the problem of resource allocation in a competitive setting based on the notion of “karma”, an accounting system that summarizes the agent’s actions in the past. The karma system allows agents to accept to give in at a particular interaction, while receiving a karma compensation. This allows an overall more efficient use of resources. Agents interact by meeting in pairs, e.g. $\{i, j\}$, and bid on the resource by sending messages $\{m_{i}, m_{j}\}$ specifying how much karma they are willing to bid in that particular interaction. The agent with the larger bid wins and gets access to the limited resource which in this case is access to an intersection leading to no delay $\delta$ in travel time for the winner. []{data-label="fig:interaction_oerview"}](figures/interactions-fat.pdf){height="3.5cm"}
The prototypical problem is intersection management. Deciding which car may pass first is a resource allocation problem, in which the resource is the use of the space inside the intersection in a given time interval. Similar resource allocation problems happen also in maneuvers outside of intersections, as drivers compete for the use of space, although the outcome is not as simple as a discrete decision as in intersection management. These interactions happen between independent agents, with competitive goals, and typically are not repeated, as it is rare to encounter the same vehicle again. Therefore, there is little incentive to give in at one interaction; at face value, this appear to be a non-repeated game.
Typical human drivers do not act like self-interested agents. Humans have ways to communicate urgency and politely negotiate maneuvers while they drive. Ultimately this is due to the altruism and pro-sociality bias that evolved in our species [@doi:10.1146/annurev-psych-010814-015355]; the bias makes the single individual intrinsically happy to accommodate somebody who seems to be in a hurry. Our species thrived because individuals are not completely self-interested. When we lived in tribes, deviant antisocial behavior was easily spotted and repressed; now that our social groups are counted in the billions, a set of rules (laws) and corresponding incentives (punishments) help in aligning the individual and societal interests in the handling of common resources [@ostrom2015governing]. When driving, some of our behaviors derive from these incentives (we do not speed because we are afraid of tickets), but many polite behaviors are due to our visceral intrinsic motivation rather than extrinsic rewards/punishment.
How can we ensure that a population of artificial agents, such as self-driving cars, can attain the same efficiency of a pro-social species like humans? In this paper, we consider the problem of resource allocation in a setting that we call *Karma Game*. The idea is that considerable gains can be realized if an agent is inclined to give in at one interaction, if it is compensated with “karma”. Thus, we introduce karma as a way to account for an agent’s past actions. (This concept is closer to how “karma” is used in video game mechanics, rather than to how it is understood in Indian religions.)
We define a karma protocol with which agents can negotiate the use of resources. The protocol describes the exchange of bidding messages and how karma is updated based on the outcome of the interaction. The protocol does not need a third party, and the primitives needed to implement karma accounting and the interaction are those provided by many blockchains, such as Ethereum [@wood2014ethereum].
Having fixed the protocol, we study how a population of self-interested agents will use it, by computing the Nash equilibria for the resulting Karma game. We then compare the Nash equilibria of the distributed system with the baseline of the optimal centralized policies. We observe that the efficiency of the system is remarkably similar. The social welfare is thus closely aligned with the self-interest of the agents, assuming the agents have reasonable discount factors. An agent that does not care about the future and lives for the present will also create an inefficient society.
Related work
============
#### Intersection control
Traditional intersection control strategies have been substantially based on utilizing control devices such as traffic lights, in which an offline optimization based on historical data can be used to provide a control signal [@robertson1969transyt; @sims1980sydney]. The main drawback of this control strategy is that it cannot adapt to changes in request patterns and environment. Improving upon classical control strategies, communication-based schemes [@chen2016cooperative] are based on a competitive scenario, in which different vehicles aim at minimizing their own selfish cost. It is assumed that the urgency $u_i(t)$ is a piece of *private information* of each vehicle $i$, and is therefore not accessible to other vehicles. This kind of scenario is typically tackled via auctions, which can be designed in order to induce selfish agents to disclose their true urgency [@schepperle2007towards; @carlino2013auction; @levin2015intersection; @vasirani2012market; @mashayekhi2015multi; @raphael2017intersection; @isukapati2017accommodating; @sayin2018information]. For example, in [@schepperle2007towards], the earliest time-slot in an intersection is auctioned off by an intersection manager among all vehicles at the front of each lane. In [@carlino2013auction], having an infinite budget, any agent in a lane can participate in a second-price auction to enhance the winning chance of the agent at the front. In [@levin2015intersection], a mechanism based on a first-price auction is proposed for the management of intersections. Two scenarios for single intersection and a network of intersections are considered in [@vasirani2012market], and a policy based on a combinatorial auction for assigning the reservations of time-space slots is presented. However, finding the winner of a combinatorial auction is NP-hard [@nisan2007algorithmic]. Finally, to schedule the intersection usage, [@sayin2018information] proposes a variant of the Vickrey-Clarke-Groove mechanism in which an intersection unit charges each agent at the front of any lane with a time-token based on its impact on others.
We note that our approach departs from the auction-based schemes in the mentioned papers in that to maintain the fairness properties between wealthier drivers and those without many funds, it does not require any monetary transactions, and therefore does not require to attach an objective value to the cost incurred by the vehicles. We will discuss later how this sheds light on the true nature of this coordination problem. Any vehicle is assigned an initial karma level. In light of the budget-balance property of our mechanism, the total amount of karma remains constant over the whole transportation network. Also, unlike the assumption in [@carlino2013auction], every agent is assumed to have a *limited* total karma at any time period, which neither is negative, nor exceeds a maximum value.
Almost all works in the literature which proposed an auction-based approach for the intersection control are static, one-time decision problems. However since the urgencies and the agent’s private information change over time, a sequence of decisions needs to be made, resulting in a dynamic resource allocation and a dynamic bidding process [@bergemann2010dynamic]. Thus, the utility function of each agent along with the social welfare are defined based on the discounted utility over time. We assume that in every interaction, vehicles are allowed to communicate a scalar *message* $m_i(t)$. The karma value of each agent is a public state $k_i(t)$. Both the outcome of the interaction (who goes first) and the update of the public state $k_i$ are determined based on a set of rules which are known and verifiable to all agents (as they only depend on public information: the states $k_i,k_j$ and the messages $m_i,m_j$).
#### Karma-like concepts
A “karma” system was introduced in [@vishnumurthy2003karma] in the context of file-sharing to prevent “freeloading” in peer-to-peer networks. In this framework, karma represents the standing of each agent in the system, that increases when contributing and decreases when consuming a resource, and thereby incentivizes agents to contribute resources [@garcia2004off]. In this and similar systems, the “value” of the karma is fixed—in our approach, the agents are free to assign a value to karma according to their goals and current state.
#### Population games
This competitive scenario can be modeled as a repeated game (interactions) between randomly selected agents in a large population. For the analysis of the resulting game, we adopt the approach that is typically used in the study of *population games* [@Sandholm2010], which has its underpinnings in the following abstractions: 1) populations are continuous rather than discrete; the payoffs to a given strategy therefore depend on society’s aggregate behavior in a continuous fashion; 2) the aggregate behavior in a population game is described by a “social state”, which specifies the empirical distribution of strategy choices (or *types*) in the population; for simplicity, this social state is generally finite-dimensional. The specific application that we are considering has however some peculiarities, compared to standard population games: for example, each agent’s *type* is also determined by an exogenous time-varying signal (their urgency). Moreover, there is no natural *revision protocol* or *adaptation*, and therefore no evolution of the agents. We therefore prefer to present the resulting game in a self-contained and specialized form, without explicitly tapping into that literature for definitions or results. Notice that the game we are formulating is more general than the specific traffic interaction problem, although clearly inspired by that setup.
Resource allocation in a “drive-by” scenario {#section:model}
============================================
In this section we introduce a deliberately simple model for vehicle-to-vehicle interaction at intersections. We strove to simplify the model to its core features, in order to isolate the essential phenomena in this problem. We understand the problem of vehicle-to-vehicle interaction at intersections as an example of a “drive-by” scenario, in which:
1. There is a large number of agents in the systems.
2. Agents interact with a random schedule.
3. Each agent interacts many times with other agents over its lifetime.
4. The value of a resource to an agent varies in time according to an exogenous factor.
For vehicle-to-vehicle (V2V) interactions at intersection:
1. There is a large number of cars on the road.
2. Cars meet randomly at intersections.
3. Each car encounters many intersections over its lifetime.
4. The value of time saved to a car varies in time according to its urgency on that day.
Formalization
-------------
More formally, consider a population $\mathcal N$ of $N$ vehicles. Each vehicle $i \in \mathcal N$ has an associated *urgency* process $u_i(t)$. The urgency $u_i$ at time $t$ indicates the marginal value that agent $i$ gives to a unitary delay in its trip. It is an exogenous process that is not affected by the behavior of the vehicles.
The vehicles interact at intersections. Each interaction at time $t$ involves only a pair of vehicles $\mathcal I(t) =\{ i, j\} \subset \mathcal N$.
Every time two vehicles interact, one of the two vehicles is necessarily delayed by a unitary delay, while the other vehicle does not incur any delay. We therefore have two possible outcomes $o(t)$, that is $o(t) \in \mathcal O := \{i,j\}$. Agent $i$ (and, in a completely symmetric way, agent $j$) incurs a cost $c: {\mathcal{O}}\times {\mathcal{U}}\to {\mathbb{R}_{+}}$ that is a function of the outcome and of its own urgency, and is defined as $$c_i(o,u_i) =
\begin{cases}
u_i, & \text{if }o=i\text{;}\\
0, & \text{otherwise.}
\end{cases}
\label{eq:costfunction}$$
Assumptions
-----------
We propose the following assumptions about the model.
\[assumption:iidinteraction\] The sequence $\mathcal I(t)$ is random and identically distributed at all times $t$ over the set $\{\mathcal I \subset \mathcal N, |\mathcal I| = 2\}$, and each vehicle has the same probability of belonging to $\mathcal I(t)$ at a given $t$.
\[assumption:equalurgency\] The urgency processes $u_i(t)$ are identical for all vehicles $i\in\mathcal N$. The urgency at each time $t$ is independent and identically distributed, and takes values in $\mathcal U := \{0, U\}$.
We defer the discussion on how to relax these assumptions to Section \[section:conclusions\]. For the most part, these assumptions are introduced for technical convenience, as they yield a simpler analysis, a computational advantage (see also Section \[section:computing\]), and a more immediate interpretation of the results.
Performance measures {#section:social}
--------------------
The focus of this paper is on *policies* that allow to decide $o(t)$ optimally, where the notion of optimality is to be defined hereafter.
We define two measures of social cost for the entire population, which are associated to two different interpretations. The first measure simply quantifies the expected aggregate cost for the entire system at each interaction: $$W_1 := \mathbb{E}\big[\, \textstyle{\sum_{\ell\in \mathcal N}}\ c_\ell(o(t),u_\ell(t))\, \big].$$ The second measure quantifies the expected rate at which the variance (across agents) of the accumulated cost grows: $$W_2 = \lim_{t \rightarrow \infty}
{\mathbb{E}\left[\operatorname{var}a(t+1) - \operatorname{var}a(t)\right]}$$ where $$\operatorname{var}a := \frac{1}{N} \textstyle{\sum_{\ell \in \mathcal N}}
\left(
a_\ell - \frac{1}{N} \textstyle{\sum_{k \in \mathcal N}}\, a_k
\right)^2$$ and $a$ denotes the vector of accumulated costs of the agents, defined element-wise as $
a_\ell(t) = \sum_{\tau=0}^t c_\ell(o(t), u_\ell(t)).
$ In these expressions, ${\mathbb{E}\left[\cdot\right]}$ represents the expectation with respect to both the stochastic urgency processes and the interaction selection process (which are independent processes).
Centralized policies
--------------------
In this section, we derive the optimal centralized policies for the simplified intersection management problem that we presented, under the notions of social optimality that we described. These optimal centralized policies will constitute a baseline for the analysis of the policies that emerge in a distributed competitive setting.
In a centralized setting, we are allowed to adopt causal policies of the kind $$o(t) = \Pi
\left(
\mathcal I(t), \left\{u(\tau)\right\}_{\tau=0}^{t}, \left\{o(\tau)\right\}_{\tau=0}^{t-1}
\right),$$ where by $u(t)$, we indicate the past urgencies of all agents.
Under Assumptions \[assumption:iidinteraction\] and \[assumption:equalurgency\], the optimal policies for the two social costs $W_1$ and $W_2$ can be computed explicitly.
\[proposition:optimalpolicies\] The social costs $W_1$ and $W_2$ are minimized, respectively, by the policies $$o_1^*(t) \in \arg\min_{\ell \in \mathcal I(t)} u_\ell(t) \label{eq:centralized-urgency}
$$ and $$o_2^*(t) \in \arg\min_{\ell \in \mathcal I(t)} a_\ell(t-1) + u_\ell(t). \label{eq:centralized-cost}
$$
If the $\arg \min$ operation does not return a singleton, then any of the two choices is optimal. Here and thereafter, we assume that $\arg \min$ ties are resolved via fair coin flipping.
We also define a third centralized policy, which prioritizes the minimization of $W_1$ (therefore obtaining the same value for $W_1$ as $o_1^*$) and, in case of ties between the urgencies $u_i$ and $u_j$ (where $\mathcal I = \{i,j\}$), aims at minimizing the unfairness defined by $W_2$: $$\begin{split}
o_{1,2}^*(t) \in &\arg\min_{\ell \in \mathcal I(t)} u_\ell(t)\\ &\text{and} \\
u_i(t) = u_j(t) \Rightarrow
o_{1,2}^*(t) \in &\arg\min_{\ell \in \mathcal I(t)} a_\ell(t-1) + u_\ell(t).
\end{split}
\label{eq:centralized-urgency-then-cost} $$
Resource allocation using Karma Games
=====================================
In this section, we formulate a mechanism for resource allocation based on the notion of karma. We only design the mechanism and not the agents’ policy, which is going to be found automatically through optimization.
Informal definition of karma interaction mechanism
--------------------------------------------------
We assume that there is an integer counter $k_i(t)$ (karma) associated to each agent bounded by ${k_{\max}}$. The agents exchange one message at each interaction. Each agent $i$ can produce a message $m_i$ which contains a value not to exceed its current karma:$
0 \leq m_i(t) \leq k_i(t).
$
We give this message the semantics of how much karma the agent sees fit to bid on the current interaction. The agent that provides the highest message is allowed to use the resource (go first at the intersection) and must pay the other agent *up to* the karma value that it has bet. The karma transferred is reduced if the transfer would make the other agent overflow ${k_{\max}}$. Suppose that agent $i$ wins betting $m_i$. Then the karma transferred is $\min(m_i, {k_{\max}}- k_j)$.
In this paper we do not delve into the technical implementation of such a scheme, but we would like to remark that it is possible to implement such a scheme, in a completely distributed way, without an arbiter to preside at each interaction, by using some of the cryptographic primitives associated to blockchain technology. The counters are implemented using public addresses. Non-refutable messages are implemented using cryptographic commitments. The resolution and the outcome can be easily implemented using the primitives of, for example, Ethereum’s Solidity language.
Formal definition of Karma Game
-------------------------------
We formalize the discussion so far by defining *Karma Games* in a way that is slightly more general.
\[def:karma-game\] A Karma Game $G$ is a tuple $$G = \langle {\mathcal{K}}, {\mathcal{M}}, {\mathcal{O}}, {\mathcal{U}}, p, c, \alpha, {\gamma}, {\phi}\rangle,$$ where:
- ${\mathcal{K}}$ is a set of possible public states (*karma*) of an agent;
- ${\mathcal{M}}$ is a set of possible *messages* of an agent;
- ${\mathcal{O}}$ is a set of possible *outcomes* of an interaction;
- ${\mathcal{U}}$ is a set of possible *exogenous states* of an agent and $p$ is a probability distribution on ${\mathcal{U}}$;
- $c: {\mathcal{O}}\times {\mathcal{U}}\to {\mathbb{R}_{+}}$ is the instantaneous cost for each agent, which depends on the outcome of the interaction and on the exogenous state of the agent;
- $0\le\alpha<1$ is a discount factor;
- ${\gamma}: {\mathcal{K}}\times {\mathcal{M}}\times {\mathcal{K}}\times {\mathcal{M}}\to {\mathcal{P}}({\mathcal{O}})$ is the *interaction outcome function*, as a probability distribution on ${\mathcal{O}}$;
- ${\phi}: {\mathcal{K}}\times {\mathcal{M}}\times {\mathcal{K}}\times {\mathcal{M}}\times {\mathcal{O}}\to {\mathcal{P}}({\mathcal{K}})$ is the *state transition function*.
The interpretation is as follows. Suppose an agent of karma $k_i(t)$ meets an agent of karma $k_j(t)$, and they exchange messages $m_i(t)$ and $m_j(t)$. The function ${\gamma}$ gives a distribution on the possible outcome $o(t) \in {\mathcal{O}}$ given by $$o(t) \sim {\gamma}(k_i(t), m_i(t), k_j(t), m_j(t)).$$ As for the consequences, ${\phi}$ is the map that specifies the probability distribution of the next value of $k_i$ and $k_j$: $$k_i(t+1) \sim {\phi}(k_i(t), m_i(t), k_j(t), m_j(t), o(t) ).$$ The cost for each agent is given by the following series, where time is to be interpreted as ranging over the instants in which the agent participated in an interaction: $$C = \mathbb{E} \Big[\,
\textstyle{\sum_{t=0}}\, \alpha^t c(o(t), u(t))
\,\Big].
\label{eq:totalcost}$$
Vehicle interaction as a Karma Game
-----------------------------------
We now put in the form of Definition \[def:karma-game\], the model we described so far. ${\mathcal{K}}= {\mathcal{M}}$ is the set of integers up to ${k_{\max}}$: $${\mathcal{K}}= \{0, 1, 2, \dots, {k_{\max}}\}.$$ There are two possible outcomes of each interaction, as explained in Section \[section:model\]: ${\mathcal{O}}= \{ i, j \}.$ For the outcome distribution ${\gamma}(k_i, m_i, k_j, m_j)$, we have $$\mathbb P(o = i) =
\begin{cases}
0, & \text{if $\tilde{m}_i > \tilde{m}_j $}, \\
1, & \text{if $\tilde{m}_i < \tilde{m}_j $}, \\
0.5, & \text{if $\tilde{m}_i = \tilde{m}_j $}, \\
\end{cases}$$ where we defined $\tilde{m}_i = \min(m_i, k_i)$.
For the state transition function ${\phi}(k_i , m_i , k_j , m_j , o)$, we have that with probability 1 $$k_i(t+1) =
\begin{cases}
k_i - \min(\tilde{m}_i, {k_{\max}}- k_j),& \text{if } o(t)=i, \\
\min(k_i + \tilde{m}_j, {k_{\max}}), & \text{if } o(t)=j.
\end{cases}$$ These rules guarantee that
- the total amount of karma is conserved.
- karma is bounded above by ${k_{\max}}$ and below by $0$.
The cost function $c$ is the one already defined in .
Acting rationally in a Karma Game
=================================
We now turn attention to what is the rational behavior of an agent in a Karma Game. An agent’s behavior is completely defined by its policy.
In a Karma Game, the agent’s policy ${\pi}$ is a probability distribution over the possible messages, which varies as a function of the agent’s current urgency $u_i(t)$ and current karma $k_i(t)$: $$m_i(t) \sim {\pi}(u_i(t), k_i(t)).$$
As an agent, we need to decide what message to send for each combination of urgency $u_i\in{\mathcal{U}}$ and karma $k_i\in{\mathcal{K}}$. In game theory jargon, we speak of a set $\mathcal{A}$ of different agent “types”; in this case, $\mathcal{A} \simeq {\mathcal{U}}\times {\mathcal{K}}$. The traditional notion of “agent type” does not fully capture our setting; because following an interaction, the type of an agent changes as they gain/lose karma. Moreover, the urgency is an exogenous variable that nobody can predict. Still, we use “agent type” in the following.
Under our assumptions, it is easy to compute the optimal policy for an agent if the urgency is zero. In that case, the optimal action for the agent is to send a message $m_i(t) = 0$. That is because the agent is indifferent to losing or winning the interaction regarding the cost; and, regarding the karma, the agent prefers to lose the interaction hoping to gain some karma.
If an agent has a nonzero urgency, how much karma should she bid today? This does not have an easy answer, except in special cases. For example, if the discount factor $\alpha$ is zero—the agent does not care about the future, then the optimal policy is to send the maximum message $m_i(t) = k_i(t)$. In all other cases, we need to characterize and compute Nash equilibria for this game. Figure \[fig:definitions-policy\] shows a representation of such an optimal policy obtained as a Nash equilibrium.
\
![Expected message value given a karma level for mixed policies for different $\alpha$ discounting factors of future costs. Strategies with small discounting factor spend almost all available karma on a message whereas strategies with a large discounting factor save karma for the future.[]{data-label="fig:nash_eq_overview"}](figures/anonymous-equilibria.pdf){width="\columnwidth"}
Characterization of Nash equilibria for a Karma Game
----------------------------------------------------
To characterize the equilibrium of the game, we must consider, in addition to the policy, a series of other related quantities. These are:
- ${D}\in {\mathcal{P}}({\mathcal{K}})$ is the stationary distribution of karma values. Figure \[fig:definitions-stationary\] shows a typical stationary distribution.
- ${T}: {\mathcal{K}}\to {\mathcal{P}}({\mathcal{K}})$ is a transition function for the karma levels; Figure \[fig:definitions-transitions\] shows a representation of such a transition function.
- ${\overline{c}}: {\mathcal{K}}\to {\mathbb{R}_{+}}$ is the expected cost of one interaction, as a function of the agent’s karma.
The transition function ${T}$ immediately descends from the composition of $\phi$ with the policy $\pi$ and the outcome distribution $\gamma$, assuming the karma distribution $D$ for the other agents and $p$ for all agents’ urgencies.
To express ${\overline{c}}$, it is convenient to define the function $\rho(u_i,k_i,m_i)$ which gives the expected utility of choosing message $m_i$ for an agent of type $u_i,k_i$.
$$\begin{gathered}
\rho(u_i,k_i,m_i) =
\sum_{k_j \in {\mathcal{K}}} {{{D}}}_{k_j}
\sum_{u_j \in {\mathcal{U}}} p_{u_j} \cdot \\
\sum_{m_j \in {\mathcal{M}}} {{{\pi}}}_{m_j}(u_j, k_j)
\sum_{o \in {\mathcal{O}}}
{\gamma}_o(k_i, m_i, k_j, m_j) \cdot \\
\left[
c(o, u_i) + \alpha \sum_{k'\in{\mathcal{K}}}
{\phi}_{k'}(k_i, m_i, k_j, m_j, o)
{{{\theta}}}(k')
\right].
\label{eq:rho}\end{gathered}$$
Figure \[fig:definitions-utilities\] shows a representation of a typical $\rho$. Based on this definition, the expected cost of an interaction is $${\overline{c}}(k_i) =
\sum_{u_i \in {\mathcal{U}}} p_{u_i}
\sum_{m_i \in {\mathcal{M}}} {\pi}_{m_i}(u_i, k_i)
\rho(u_i,k_i,m_i).
\label{eq:expectedcost}$$
We can now define the notion of Nash equilibrium for a Karma Game.
\[def:nash\_eq\] A policy ${\pi}$ is a Nash equilibrium for the Karma Game $G$ if there exist ${D}, {T}, {\overline{c}}$ that satisfy three properties:
**P1: Stationarity**: ${D}$ is the equilibrium distribution for the transition map ${T}$: $${D}= \textstyle{\sum_{\tau \in {\mathcal{K}}}}\, {D}_\tau {T}(\tau).$$
**P2: Bellman**: There exists a function ${\theta}: {\mathcal{K}}\to {\mathbb{R}_{+}}$, representing the expected total cost for an agent as a function of the present value of the karma, that satisfies the Bellman-like equation $$\label{eq:exputility-fixed-point}
{\theta}(k) = {\overline{c}}(k) + \alpha \sum_{\tau \in {\mathcal{K}}}
{T}_\tau(k) {\theta}(\tau)$$ for the expected interaction cost ${\overline{c}}$ defined in and the discount factor $\alpha$.
**P3: Rationality**: The policy ${\pi}$ must yield the best expected outcome: $$C(\pi) \le C(\pi') \quad \forall \pi',$$ where $C$ was defined in and can be expressed as $$C = \sum_{k_i \in {\mathcal{K}}} {D}_{k_i} {\theta}(k_i).$$
The next section will be devoted to the numerical computation of a Nash equilibrium for the Karma Game of interest and to the interpretation of the resulting policies and outcome.
Computing Nash equilibria of Karma Games {#section:computing}
========================================
In general, Nash equilibria can be computed by iterative algorithms. Starting with an initial policy, one computes the other unknown (stationary distribution, karma utility); then one re-computes the optimal policy. If the recomputed policy is different from the initial one, the delta is a profitable perturbation of the policy. Based on the perturbation, one can make a small update of the policy, and repeat the process until convergence. If this process converges to a distribution, then by definition, we have found a Nash equilibrium as defined above. However, there is in general no guarantee that the iterative process converges.
Fixed point computation {#sec:fixed}
-----------------------
We show here how to rearrange the equations to put them in the form of a fixed point.
Suppose we have a current guess of the policy ${\pi}$, the stationary distribution ${D}$, and the utility ${\theta}$.
**Step 1:** Compute the policy ${\pi}$ from the previous policy, the stationary distribution ${D}$, and the expected utility ${\theta}$. The policy is computed using based on the values of $\rho$ obtained from .
**Step 2:** Compute the transitions ${T}$ from the policy ${\pi}$ and the stationary distribution ${D}$. Given the policy and the stationary distribution, we can compute the transitions of the system. For each type $u_i, k_i$, we know the distribution of the types it will encounter, and we know their policy. Thus, we can compute the outcomes, and the consequences of the outcomes in terms of what will be the next value of $k_i$.
**Step 3:** Compute the stationary distribution ${D}$ from the transitions ${T}$. This is a standard step - given a transition matrix, compute the equilibrium distribution. It can be done by iteration or by solving an eigenvector problem.
**Step 4:** Compute the expected utility ${\theta}$ from ${D}$ and ${\pi}$. We can compute the expected utility using . The expected daily cost $\overline{c}(k)$ is computed by setting $\alpha=0$ in .
Momentum and simulated annealing
--------------------------------
We found two simple devices that make the convergence robust, in the sense that the policy converges to the same solution no matter the initial conditions of the policy, stationary distribution, and karma utility.
### Momentum
In Section \[sec:fixed\], we have defined a way to update the policy ${\pi}$ that we can abstract as a function $\Psi$ such that: $${\pi}_t^{\text{new}} = \Psi({\pi}_t, {D}_t, {\theta}_t).$$ Define the “momentum” $\tau$ as a scalar $0<\tau\leq 1$. Then we update the policy as $${\pi}_{t+1} = \tau {\pi}_t^{\text{new}} +(1-\tau) {\pi}_{t}.$$ For the set of simulations described below the optimization parameters were constant, but we did find in general that for different values of the model properties, the optimization parameters had to be optimized.
### Simulated annealing
Let $T > 0$ be a temperature parameter. Rather than looking for a pure strategy, we set $$\label{eq:policy-soft}
{\pi}(u_i, k_i, m_i) \propto \exp(- \rho_{u_i,k_i}(m_i) / T).$$ For large values of $T$, agents choose a random action. As $T$ decreases, the agents choose more often actions with good rewards. As $T\to 0$, the policy tends to the deterministic policy, where we select the maximum of $\rho_{u_i,k_i}(\cdot)$: $${\pi}(u_i, k_i, m) \stackrel{T\to 0}{\longrightarrow}
\begin{cases}
1,& \text{if $m$ maximizes $\rho_{u_i,k_i}(m)$}, \\
0,& \text{otherwise}.
\end{cases}$$ In the simulations, we gradually decrease the temperature of the system in a series of “eras” (Figure \[fig:policy\_evolution\]).
Equilibria parametrization in $\alpha$
--------------------------------------
The parameter $\alpha$ introduced as a cost discounting factor in determines how much importance an agent assigns to future costs. In the limit $\alpha \ll 1$, the agent is only occupied with minimizing instantaneous costs. When $\alpha$ approaches 1, future costs are deemed almost as important as present costs. To determine the influence this factor has on agent policies, we ran experiments with different $\alpha$ values ranging from $0$ to $1$ in $0.05$ increments. As an overview of the effect, we provide Figure \[fig:discounting\] which depicts the gradual changes in policy as $\alpha$ is increased. Similarly we offer Figure \[fig:nash\_eq\_overview\] as an overview of the effect of time discounting on the best message to send given a karma level.
One caveat that we have is that the Nash equilibria are not well defined when $\alpha = 1$ as some of the series in the formalization do not converge. Still, we also include the results of the algorithm for $\alpha=1$. Similarly, we believe that for $\alpha \rightarrow 1$ there are numerical instabilities, and in fact we find that there are much larger oscillations. Rather than tuning the optimization parameters for each $\alpha$, we keep the same parameters, and we still picture the results for $\alpha=0.9$ and $\alpha=0.95$, without fully believing they are Nash equilibria for the game.
Policy comparison
=================
In this section, we are interested in gaining an empirical understanding of different solutions to the proposed distributed interaction problem.
#### Evaluation protocol
All simulations of interactions follow the same general procedure. As described in Section \[section:model\], agents randomly meet in pairs and bid karma if they are urgent in order to pass first in an intersection. All experiments were conducted with 200 agents and a total of 1000 time periods. On each day, there are an average of 0.1 interactions per agent. Agents are urgent with magnitude $3$ with probability $0.5$ and not urgent (magnitude $0$) again with probability $0.5$. Each agent has an initial karma level uniformly randomly chosen between $0$ and $12$. Agents can, through interactions, attain a minimum karma level of $0$ and a maximum karma level of $12$.
In the following, we compare various policies as well as the underlying parameters influencing the agents’ policies. We consider two performance metrics which are finite-sample proxies for $W_1$ and $W_2$, respectively:
- “Inefficiency”: This is the average cost per interaction attained by the agent at the end of the simulation period. Note that this is not the $\alpha$-discounted factor that each agent is trying to minimize; rather, this is the social welfare—which roughly corresponds to the case $\alpha=1$.
- “Unfairness”: This is the standard deviations of the costs at the end of the simulation period.
![Overview of efficiency and unfairness of random, centralized and karma-based strategies. Random solutions fare the worst in both domains whereas centralized solutions with access to all information are optimal to their respective objectives. Karma-based solutions describe a trend of better efficiency and fairness with increasing discount factor $\alpha$ up to a limit.[]{data-label="fig:results_overview"}](figures/anonymous-plot.pdf)
#### Policies
In addition to the Nash equilibria found for sweeping $\alpha$ between 0 and 1, we consider these other policies, as they are useful reference points:
- : The winner is decided randomly.
- : The agents always bid 1.
- : The agents bid 1 if the urgency is nonzero, and zero otherwise.
- : The policy .
- : The policy .
- : The policy .
#### Results
The overall results are shown in Figure \[fig:results\_overview\].
(top right) obtains the worst results, as one might expect.
(bottom left) obtains the best results for both fairness and efficiency, as expected.
does well in terms of unfairness, as it tries to reduce the spread of the costs, but it is very inefficient.
obtains minimum inefficiency (as predicted), but it does not do anything to reduce the spread of the costs, leading to a relatively high unfairness.
The baselines provide a reference frame to interpret the results for the karma-based policies.
We find many interesting nuggets. For example, is very inefficient, as inefficient as , but it is less unfair. This is because the karma accounting keeps track of previous times when the agent lost, thereby slightly reducing the unfairness even if the policy is trivial.
Next consider the performance of . This corresponds to a mechanism in which the agents use the karma message to reveal their urgency. Notwithstanding the fact that this is not an equilibrium for the game (this can be easily verified by noting that this is not a fixed point of the procedure discussed above), what we found surprising is that the efficiency is *not* as good as some of the Nash equilibria that we find.
Next we consider the performance of the Nash equilibrium as a function of $\alpha$. The sequence draws a hook in the inefficiency/unfairness space. The continuity of this curve also is good evidence that the procedure converged well (as noted before, for $\alpha\geq0.9$ the convergence is not assured).
We find the surprising result that for $\alpha\geq0.4$, the Nash equilibria are better in efficiency than the strategy. The reason is that the agents should bid more or less if their karma levels allow—bidding only 1 is not the best strategy (neither for the agents nor society). For $\alpha<0.4$, the agents do worse.
The $\alpha=0$ “there is no tomorrow” strategy (bid everything if urgent) is particularly bad for society, though not as bad as random: karma still allows some reparations to be made.
We observe that for $\alpha>0.4$, the karma strategies beat the strategy in unfairness. There is a minimum unfairness observed for $\alpha=0.8$—we are not sure how this relates to the parameters of the problem.
In these experiments, for $\alpha=0.85$, the performance is closest to the strategy in both inefficiency and unfairness, in fact surprisingly close.
In conclusion, we obtain the surprising result that, for agents that are reasonably future-conscious, Nash equilibrium strategies beat heuristic solutions in both efficiency and fairness, and their performance is extremely close to the centralized solutions.
Conclusions {#section:conclusions}
===========
We have demonstrated how the efficient use of a shared infrastructure can emerge from simple coordination protocols among competitive agents, without the need of any monetary transaction or complex decision infrastructures, in sharp contrast to most of the literature. The enabler is the notion of *karma*: a public state that links the decision of the same agent at different times (as long as each agent reasonably values its own future cost). A solid understanding of the mechanisms that are necessary and sufficient for fair sharing of an infrastructure has the potential to guide the design of scalable solutions in many applications, and in particular for autonomous mobility.
[^1]: The authors are with the Institute for Dynamic Systems and Control (IDSC) and the Automatic Control Laboratory (IfA) at ETH Zurich, 8092 Zurich, Switzerland. E-mail: [{acensi, bsaverio, jzilly, mousavis, efrazzoli}@ethz.ch]{}
[^2]: 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we study inverse local time at 0 of one-dimensional reflected diffusions on $[0, \infty)$, and establish a comparison principle for inverse local times. Applications to Green function estimates for non-local operators are given.'
author:
- '[Zhen-Qing Chen]{}'
title: 'Inverse local time of one-dimensional diffusions and its comparison theorem'
---
[**AMS 2010 Mathematics Subject Classification**]{}: Primary 60J55, 60J75; Secondary 60J75, 60H10
[**Keywords and phrases**]{}: diffusion, local time, inverse local time, subordinator, Lévy measure, non-local operator, Esscher transform, Girsanov transform, comparison theorem, Green function estimate
Introduction {#sec:intr}
============
It is well known that the trace of Brownian motion in $\R^{d+1}$ (or reflected Brownian motion in the upper half space $\R^{d+1}_+:=\{x=(x_1, \dots, x_d, x_{d+1})\in \R^{d+1}: x_{d+1}>0\}$) on the hyperplane $\{x_{d+1}=0\}$ is a $d$-dimensional Cauchy process. Molchanov and Ostrovski [@MO] in 1969 showed that in fact any rotationally symmetric $\alpha$-stable process on $\R^d$ with $0<\alpha <2$ can be realized as the boundary trace on $\{x_{d+1}=0\}$ of some reflected diffusion in the upper half space $\R_+^{d+1}$. In terms of the generators, Molchanov and Ostrovski’s result says that the fractional Laplacian $\Delta^{\alpha/2}:=-(-\Delta)^{\alpha/2}$ in $\R^{d+1}$ can be realized as the boundary trace of some (degenerate) differential operator in upper half space of one-dimensional higher. The latter fact is rediscovered by Caffarelli and Silvestre [@CS] in 2007 using a purely analytic approach. Realizing non-local operators as boundary trace of some differential operators is a powerful way to study non-local operators from analytic point of view as one can employ many well developed techniques and ideas from partial differential equations (PDE). It is a natural and interesting question to investigate the scope of non-local operators that can be realized as the boundary trace of differential operators.
Note that rotationally symmetric stable process has the same distribution as Brownian motion time-changed by an independent stable subordinator. So the key to Molchanov and Ostrovski’s result is to show that $(\alpha/2)$-stable subordinator can be realzed as the inverse local time at 0 of some one-dimensional reflected diffusion on $[0, \infty)$. Indeed, Molchanov-Ostrovski [@MO] showed that for $0<\alpha<1$, $\alpha$-stable subordinator can be realized as the inverse local time of a reflecting Bessel process on $[0, \infty)$ determined locally by the generator $$\label{e:1.1}
\op^{(\alpha)}=\frac{1}{2}\frac{d^2}{dx^2}+\frac{1-2\alpha}{2x}\frac{d}{dx}.$$
The class of subordinate Brownian motions that can be realized as boundary traces of diffusion processes in upper half space of one-dimensional higher is in one-to-one correspondence with the class of subordinators that can be realized as the inverse local time at 0 of some reflected diffusions on the half line $[0, \infty)$. The latter is exactly a question raised in Itô-McKean [@IM] called Krein representation problem. This problem remains open. However, Knight [@Knight] and Kotani-Watanabe [@KW] showed independently in 1981-1982 that if one relaxes $\R_+$-valued diffusions to gap diffusions (a family of Markov processes having possibly discontinuous trajactories), then the answer is affirmative. It is a general fact that the inverse local time of a Markov process as a point having positive capacity is always a subordinator, that is, a non-decreasing real-valued Lévy process.
Relativistic Cauchy process in $\R^d$ (also called relativistic Brownian motion in the study of relativistic Hamiltonian system in physics [@CMS]) is a subordinate Brownian motion $X_t$ characterized by $$\Ex e^{ i\xi \cdot (X_t-X_0)} = e^{ t \left( \sqrt{m}-\sqrt{ m+|\xi|^2 } \right) }, \quad \xi \in \R^d ,$$ where $m>0$ stands for the mass of the particle. The infinitesimal generator of $X_t$ is $\sqrt{m}-\sqrt{m-\Delta}$. It is not hard to see that the inverse local time at $0$ of the reflected Brownian motion with downward constant drift $\sqrt{2m}$ on $[0, \infty)$ is a subordinator $S_t$ with $S_0=0$ and $$\Ex e ^{-\lambda S_t} = e^{ t \left( \sqrt{m}-\sqrt{m+\lambda } \right)}, \quad \lambda \geq 0.$$ Hence the relativisitc Cauchy process on $\R^d$ can be regarded as the boundary trace on $\{x_{d+1}=0\}$ in the upper half space of $\R^{d+1}$ where the vertical motion in the $x_{d+1}$ direction is a Brownian motion with downward constant drift while the horizontal motion is an independent Brownian motion in $\R^d$. However for general $\alpha \not= 1/2$, relativistic $\alpha$-stable processes can not be simply realized, by analogy with rotationally symmetric stable processes, as a diffusion in the upper half space of one-dimensional high whose vertical motion is a Bessel process with constant drift.
By using Esscher transform, Martin and Yor[@Yor1] showed that relativistic $\alpha$-subordinator for $0<\alpha<1$ is in fact the inverse local time at $0$ of the reflected diffusion on $[0, \infty)$ determined by generator $$\label{e:1.2}
\op^{(\alpha,m)}=\frac{1}{2}\frac{d^2}{dx^2}+\bigg(\frac{1-2\alpha}{2x}+\frac{\wh K'_\alpha(\sqrt{2m}x)}{\wh K_\alpha(\sqrt{2m}x)}\bigg)\frac{d}{dx},$$ where $\wh K(x)=x^\alpha K_{\alpha}(x)$ and $K_\alpha(x)$ is the modified Bessel function of the second kind defined in .
In this paper, we set out a modest goal to investigate properties of the inverse local time at $0$ of reflected diffusions on $[0, \infty)$ with infinitesimal generator of the form $$\label{e:1.3}
\op=\frac{1}{2}\frac{d^2}{d x^2}+\left(\frac{1-2\alpha}{2x}-f(x)\right)\frac{d}{d x},$$ where $f\geq 0$ is a function on $(0, \infty)$, and the corresponding subordinate Brownian motions.
For two functions $f$ and $g$, notation $f\lesssim g$ means there is a constant $c>0$ so that $f\leq c g$. The following are the main results of this paper.
\[T:1.1\] Let $Y_t$ be the reflected diffusion process on $[0,\infty)$ determined by the local generator of the form with $$0\leq f(x)\lesssim (1\wedge x)^{2\alpha -1} \quad \hbox{on } (0, \infty) .$$ Let $S_t$ be the inverse local time of $Y_t$ at $0$. Then there is a constant $m>0$ so that stochastically $S_t^{(\alpha,m)}\leq S_t\leq S^{(\alpha)}_t$ for all $t\geq 0$, where $S^{(\alpha)}_t$ and $S_t^{(\alpha,m)}$ are the $\alpha$-stable subordinator and relativistic $\alpha$-stable subordinator with mass $m$, respectively.
As an application, we have the following Green function estimates for the trace processes of diffusion processes in $\R^{d+1}$ whose vertical $x_{d+1}$-coordinate is a reflected diffusion on $[0, \infty)$ with infinitesimal generator , and the horizontal direction is an independent Brownian motion in $\R^d$.
\[T:1.2\] Under the setting of Theorem \[T:1.1\], let $B_t$ be a $d$-dimensional Brownian motion independent of $Y_t$ with variance $2t$, and $\mu(x)$ be the density of the measure of trace process $B_{S_t}$. Denote the density of the measure of symmetric $2\alpha$-stable process by $\mu^{(\alpha)}(x)$. Then $j(x) :=\mu^{(\alpha)}(x)-\mu(x)\geq 0$, and there exists a constant $C$ such that for $|x|\leq 1$, $$j(x)\leq C|x|^{2-2\alpha -d}.$$ Let $D\subset\R^d$ be a bounded connected Lipschitz open set. Denote Green functions of the trace process $B_{S_t}$ in $D$ by $G_D(x,y)$. Then there exists a constant $C_1=C_1( d, \alpha, D,C)$ such that $$C_1^{-1} G^{(2\alpha)}_D(x,y)\leq G_D(x,y)\leq C_1G^{(2\alpha)}_D(x,y)
\text{ for } x,y\in D,$$ where $G^{(2\alpha)}_D$ is the Green function of rotationally symmetric $(2\alpha)$-stable process, or equivalently of the fractional Laplacian $\Delta^\alpha$, in $D$.
The rest of the paper is organized as follows. Esscher transform (an Economics terminology) and Girsanov transform for reflected diffusions on $[0, \infty)$ are discussed in Section \[sec:EG\]. This extends and refines the corresponding part of Martin-Yor [@Yor1] with a more complete and rigorous proof. In Section \[sec:com\], we present a key comparison result for the inverse local times at $0$ for reflected diffusion processes on $[0, \infty)$ and their corresponding Lévy measures. Hausdorff measure of zero sets and Girsanov transform are the main tools to get this result. Regenerative embedding theory for subordinators are also used. With the comparison result obtained in Section \[sec:com\], Theorems \[T:1.1\] and \[T:1.2\] are established in Section \[S:4\].
Esscher transforms {#sec:EG}
==================
Recall that the Laplace exponent and Lévy measure for $\alpha$-stable subordinator, where $0<\alpha <1$, are $\phi^{(\alpha)}(\lambda)=c_\alpha\lambda^\alpha$ and $\nu^{(\alpha)}(dx):=c_\alpha x^{-1-\alpha} dx$, where $c_\alpha = \alpha/\Gamma (1-\alpha)$; while that for relativisitic $\alpha$-stable subordinator with mass $m>0$ are $\phi^{(\alpha,m)}(\lambda)=c_\alpha[(m +\lambda)^{\alpha}-m^\alpha]$ and $\nu^{(\alpha, m)} (dx):= c_{\alpha }x^{-1-\alpha} e^{-mx} dx$.
Fix $0<\alpha <1$. Let $(\Omega,\F, \mathbb P)$ be a probability space on which a reflected Bessel process $Y_t$ on $[0, \infty)$ with generator is defined. The filtration generated by $Y_t$ will be denoted as $\{\F_t; t\geq 0\}$. Let $L_t$ be the local time of $Y$ at $0$, and $$S_t:= \inf\{s>0: L_s >t\} ,$$ the inverse of $L$, which is a stopping time with respect to the filtration $\{\F_t; t\geq 0\}$. We know that $S_t$ is an $\alpha$-stable subordinator. We can define a new probability measure $\Q$ on $(\Omega,\F)$ by $$\label{e:2.1}
\frac{\Q(dx)}{\P (dx)} = \frac{e^{-mx}}{\Ex_0[\exp(-mS_t ]}
= \exp(tc_\alpha m^\alpha-mx) \quad \hbox{ on } \F_{S_t}.$$ This change of measure is called Esscher transform in literature (see Chapter VII, 3c, [@E]). Note that under the new probability measure $\Q$, $$\begin{aligned}
\Ex^\Q [\exp(-\lambda S_t ]=&\exp(tc_\alpha m^\alpha)\Ex^\P[\exp(-(\lambda+m)S_t ]\notag\\
=&\exp(tc_\alpha m^\alpha-t\phi^{(\alpha)(\lambda+m)})\notag\\
=&\exp(-t\phi^{(\alpha,m)}(\lambda)).\end{aligned}$$ In other words, under $\Q $, $\{S_t ; t\geq 0\}$ is a relativistic $\alpha$-stable subordinator with mass $m$.
We now extend the above Esscher transform to general one-dimensional diffusions.
\[thm:EG\] Suppose that $X_t$ is a reflected diffusion process on $[0, \infty)$ defined on a probability space $(\Omega,\F, \mathbb P)$, determined locally by the generator $$\op=a(x)\frac{d^2}{d x^2}+b(x)\frac{d}{d x}.$$ Let $L_t$ be the local time of $X_t$ at $0$, and $S_t=\inf\{s:L_s>t\}$ its inverse local time at $0$, which is a subordinator. Denote by $\phi(\lambda)$ the Laplace exponent of $\{S_t; t\geq 0\}$. Define $$\label{e:2.3}
\frac{d \Q }{d \P }:=\frac{\exp(-mS_t)}{\Ex [\exp(-mS_t)]} \quad \hbox{on }\F_{S_t}, \ t\geq 0.$$ Then
- defines a new measure $\Q $ on $\F_\infty$ in a consistent way;
- Under $\Q$, the original diffusion $X$, write as $X^{(m)}$ for emphasis, is a reflected diffusion on $[0, \infty)$ having generator $$\op^{(m)}=\op+2a(x)\frac{\rho'_m(x)}{\rho_m(x)}\frac{d}{d x}, \ x>0,$$ where $\rho_m(x):=\Ex_x[\exp(-mT_0)]$ and $T_0$ is the first hitting time of $0$ by the process $X_t$.
- Denote by $S_t^{(m)}$ the inverse local time of $X_t^{(m)}$ at $0$ and $\phi^{(m)}(\lambda)$ the Laplace exponent for subordinator $S_t^{(m)}$. Then we have $$\phi^{(m)}(\lambda)=\phi(\lambda+m)-\phi(m).$$
The definition (\[e:2.3\]) gives $$\label{eqn:gir2}
M_{s,t}:=\frac{d\Q }{d\P }\bigg|_{\F_{s\wedge S_t}}=\frac{\Ex [\exp(-mS_t)|\F_{s\wedge S_t}]}{\Ex [\exp(-mS_t)]}.$$ To see the consistency, we observe for $r\leq t$, since $S_t$ is a subordinator, $$\begin{aligned}
\frac{d\Q }{d\P }\bigg|_{\F_{s\wedge S_r}}=&\frac{d\Q }{d\P }\bigg|_{\F_{s\wedge S_r\wedge S_t}}=\frac{\Ex [\exp(-mS_t)|\F_{s\wedge S_r\wedge S_t}]}{\Ex [\exp(-mS_t)]}\\
=&\frac{\Ex [\exp(-mS_t)|\F_{s\wedge S_r}]}{\Ex [\exp(-mS_t)]}\\
=&\frac{\Ex [\exp(-mS_r)\exp(-m(S_t-S_r))|\F_{s\wedge S_r}]}{\Ex [\exp(-mS_r)\exp(-m(S_t-S_r)]}\\
=&\frac{\Ex [\exp(-m(S_t-S_r))]\Ex [\exp(-mS_r)|\F_{s\wedge S_r}]}{\Ex [\exp(-m(S_t-S_r))]\Ex [\exp(-mS_r)]}\\
=&\frac{\Ex [\exp(-mS_r)|\F_{s\wedge S_r}]}{\Ex [\exp(-mS_r)]}\end{aligned}$$ To see uniform integrability, we first have the decomposition of the inverse local time at $0$, since $S_0=T_0$, $$\begin{aligned}
\label{eqn:inv}
S_t\text{ under }\P =&\inf\{s:L_s>t \text{ with }X_0=x\}\notag\\
=&\inf\{s:s=T_0+r,L_{T_0}+L_r\circ\theta_{T_0}>t\text{ with }X_0=x\}\notag\\
=&T_0+\inf\{r:L_r\circ\theta_{T_0}>t\text{ with }X_0=x\}\notag\\
=&T_0+\inf\{r:L_r>t\text{ with }X_0=0\}\notag\\
=&T_0+S_t\text{ under }\P_0\end{aligned}$$ With the decomposition, $$\begin{aligned}
\label{eqn:exp}
\Ex [\exp(-mS_t)]=&\Ex [e^{-mT_0}]\Ex_0[e^{-mS_t}]\notag\\
=&\rho_m(x)\Ex_0[e^{-mS_t}]\notag\\
=&\rho_m(x)\exp[-t\phi(m)]\end{aligned}$$ Similarly, $$\begin{aligned}
\1_{\{s\leq S_t\}}\Ex [e^{-mS_t}|\F_{s}]=&\1_{\{s\leq S_t\}}\Ex [e^{-mS_t}|\F_s]\\
=&\1_{\{s\leq S_t\}}e^{-ms}\Ex_{X_s}[\exp(-mS_{t-r})]|_{r=L_s}\end{aligned}$$ The last equation holds because on $\{s\leq S_t\}$, $$\begin{aligned}
S_t=&\inf\{r+s:L_{r+s}>t\}=s+\inf\{r:L_s+L_r\circ\theta_s>t\}\\
=&s+\inf\{r:L_r\circ\theta_s>t-L_s\}\\
=&s+S_{t-r}\circ\theta_s|_{r=L_s}.\end{aligned}$$ Now using (\[eqn:exp\]), we have $$\begin{aligned}
\Ex_{X_s}[\exp(-mS_{t-r})]|_{r=L_s}=&\rho_m(X_s)\Ex_0[\exp(-mS_{t-r})]|_{r=L_s}\\
=&\rho_m(X_s)\exp(-(t-r)\phi(m))|_{r=L_s}\\
=&\rho_m(X_s)\exp(-(t-L_s)\phi(m)).\end{aligned}$$ Thus, restricted to $\F_{s\wedge S_t}$, $$\begin{aligned}
M_{s,t}=&\frac{d\Q }{d\P }\bigg|_{\F_{s\wedge S_t}}=\1_{\{s\leq S_t\}} \frac{\Ex [\exp(-mS_t)|\F_{s}]}{\Ex [\exp(-mS_t)]}+\1_{\{s>S_t\}}\frac{\exp(-mS_t)}{\Ex [\exp(-mS_t)]}\\
=&\1_{\{s\leq S_t\}}\frac{e^{-ms}\rho_m(X_s)\exp(-(t-L_s)\phi(m))}{\rho_m(x)\exp(-t\phi(m))}+\1_{\{s>S_t\}}\frac{\exp(-mS_t)}{\Ex [\exp(-mS_t)]}\\
=&\1_{\{s\leq S_t\}}\frac{\rho_m(X_s)}{\rho_m(x)}\exp(-ms+L_s\phi(m))+\1_{\{s>S_t\}}\frac{\exp(-mS_t)}{\Ex [\exp(-mS_t)]}\end{aligned}$$ As $t\to\infty$, $M_{s,t}\to M_s$, a.s. and $$M_s:=\frac{\rho_m(X_s)}{\rho_m(x)}\exp(-ms+L_s\phi(m)).$$ It’s obvious that $M_s\in\F_s$. Also, from the original definition of $M_{s,t}$ in (\[eqn:gir2\]), for any $s,t$, $\Ex M_{s,t}\leq 1$, so by Fatou’s lemma, $$\Ex M_s\leq\liminf\Ex M_{s,t}\leq 1.$$ Thus, $M_s\in L^1$, and on the other hand, with $\F_{s\wedge S_t}\subset\F_s$ $$\begin{aligned}
\Ex [M_s|\F_{s\wedge S_t}]=&\Ex \bigg[\frac{\rho_m(X_s)}{\rho_m(x)}\exp(-ms+L_s\phi(m))\Big|\F_{s\wedge S_t}\bigg]\\
=&\1_{\{s\leq S_t\}}\frac{\rho_m(X_s)}{\rho_m(x)}\exp(-ms+L_s\phi(m))+\1_{\{s>S_t\}}\frac{\rho_m(0)}{\rho_m(x)}\exp(-mS_t+t\phi(m)])\\
=&\1_{\{s\leq S_t\}}\frac{\rho_m(X_s)}{\rho_m(x)}\exp(-ms+L_s\phi(m))+\1_{\{s>S_t\}}\frac{\exp(-mS_t)}{\Ex (\exp(-mS_t))}\\
=&M_{s,t},\end{aligned}$$ the second to the last equality comes from $\rho_m(0)=1$ and (\[eqn:exp\]). Thus $\{M_{s,t}=\Ex [M_s|\F_{s\wedge S_t}]\}_{t\geq0}$ is uniformly integrable. Taking $t\to \infty$ yields $$\label{eqn:gir3}
\frac{d\Q }{d\P }\bigg|_{\F_s}=\frac{\rho_m(X_s)}{\rho_m(x)}\exp(-ms+L_s\phi(m)).$$ The above is a combination of Doob’s $h$-transform and a Feynman-Kac transform by local time $L_t$. It follows that for $x>0$, $$\begin{aligned}
\op^{(m)}f(x)=&\rho_m^{-1}(x)(\op-m)(\rho_m\cdot f)(x)\\
=&\op f(x)+2a(x)\frac{\rho'_m(x)}{\rho_m(x)}f'(x)+\rho_m^{-1}(x)(\op-m)(\rho_m)(x)f(x)\end{aligned}$$ Since $\rho_m(x)$ satisfies $(\op-m)\rho_m(x)=0$, under the new measure $\Q$, the diffusion process $X_t$ is a reflected diffusion on $[0, \infty)$ with generator $$\op^{(m)}=\op+2a(x)\frac{\rho'_m(x)}{\rho_m(x)}\frac{d}{d x}, \text{ for }x>0.$$ By , for every $\lambda >0$, $$\Ex^\Q e^{-\lambda S_t}=\frac{\Ex e^{-(\lambda +m)S_t}}{\exp(-t\phi(m))}=\exp\{-t(\phi(\lambda+m)-\phi(m))\}.$$ This proves that the Laplace exponent of $S^{(m)}_t$ is $\phi_m(\lambda)=\phi(\lambda+m)-\phi(m)$.
\[rmk:EG\] By Feymann-Kac transformation, $\rho_m(x)=\Ex [\exp(-mT_0)]$ is the unique solution to $$\begin{cases}
(\op-m)\rho_m=0;\\
\rho_m(0)=1,\ \rho_m(\infty)=0.
\end{cases}$$
Comparison theorem for inverse local time {#sec:com}
=========================================
Let $X_t$ and $Y_t$ be reflected diffusion processes on $[0, \infty)$ defined on a probability space $(\Omega, \F, \P)$ and driven by a common Brownian motion, whose generators are $$\begin{aligned}
&\op^X=a(x)\frac{d^2}{d x^2}+b(x)\frac{d}{d x};\\
&\op^Y=a(x)\frac{d^2}{d x^2}+B(x)\frac{d}{d x}.\end{aligned}$$ Denote by $Z^X$ and $ Z^Y$ the zero sets for $X$ and $Y$ respectively; that is, $$Z^X:= {\{t\in[0,\infty): X_t=0\}} \quad \hbox{ and } \quad
Z^Y:= {\{t\in[0,\infty): Y_t=0\}}.$$ These are random closed subsets of $[0, \infty)$, and are regenerate (also called Markov) sets in the sense of Maisonneuve (cf [@B]).
\[lem:lap\] Let $S_t^X$ and $ S_t^Y$ be inverse local times at $0$ for $X_t, Y_t$, whose Laplace exponents are denoted by $\phi^X(\lambda)$, $\phi^Y(\lambda)$, respectively. Suppose that $b(x)\leq B(x)$ for all $x$. Then $\phi^Y/\phi^X$ is a completely monotone function.
If $X_0\leq Y_0$, then by the comparison theorem for one-dimensional diffusions (see, e.g., [@Bass Theorem I.6.2]), we have, almost surely, $X_t\leq Y_t$ for all $t\geq 0$. Consequently, $$\label{eqn:z}
Z^Y\subset Z^X, \ \P\text{-a.s.}$$ Note that $S_t^X$ and $ S_t^Y$ are the subordinators associated with the regenerative sets $Z^X$ and $Z^Y$, respectively. It follows from the regenerative embedding theorem due to Bertoin (see [@B Theorem 1]) that $\phi^Y/\phi^X$ is a completely monotone function.
\[ex:stablecom\] Consider two reflected Bessel processes on $[0, \infty)$: $X_t^{(\alpha)}, X_t^{(\beta)}$, determined by local generators: $$\begin{aligned}
&\op^{(\alpha)}=\frac{1}{2}\frac{d^2}{d x^2}+\frac{1-2\alpha}{2x}\frac{d}{d x};\\
&\op^{(\beta)}=\frac{1}{2}\frac{d^2}{d x^2}+\frac{1-2\beta}{2x}\frac{d}{d x},\end{aligned}$$ where $0<\beta<\alpha<1$ and $X_0^{(\alpha)}\leq X_0^{(\beta)}$. Denote by $S_t^{(\alpha)}, S_t^{(\beta)}$ the inverse local times at $0$, and $\phi^{(\alpha)}, \phi^{(\beta)}$ their Laplace exponents.
Since $\frac{1-2\alpha}{2x}<\frac{1-2\beta}{2x}$ for $x>0$, apply the classic comparison theorem for one-dimensional SDE and Lemma \[lem:lap\], $\phi^{(\beta)}/\phi^{(\alpha)}$ is completely monotone.
On the other hand, as we know from [@MO], $S_t^{(\alpha)}, S_t^{(\beta)}$ are $\alpha$- and $\beta$-stable subordinators, respectively. Since $\phi^{(\alpha)}(\lambda)=c_\alpha \lambda^\alpha$, $\phi^{(\beta)}(\lambda)=c_\beta\lambda^\beta$, $\phi^{(\beta)} (\lambda )/\phi^{(\alpha)} (\lambda )
=(c_\beta/c_\alpha) \lambda^{\beta -\alpha}$ is indeed completely monotone in $\lambda$.
In general one can not conclude that the inverse local time at 0 of $Y$ is dominated by that of $X$. Indeed, there is no monotonicity between $\alpha$-stable and $\beta$-stable subordinators, as there is no monotonicity between their Laplace exponents. However we have the following comparison theorem for inverse local times.
\[thm:com\] Suppose $X_t$ and $Y_t$ defined on a probability space $(\Omega,\F, \mathbb P)$ are reflected diffusions on $[0, \infty)$, determined by the local generator $$\begin{aligned}
&\op^X=\frac{1}{2}\frac{d^2}{d x^2}+b(x)\frac{d}{d x};\\
&\op^Y=\frac{1}{2}\frac{d^2}{d x^2}+B(x)\frac{d}{d x}.\end{aligned}$$ Let $S_t^X$ and $S_t^Y$ be the corresponding inverse local times at $0$, respectively. Suppose $f(x)=B(x)-b(x) \geq 0$ satisfies the condition $$\label{con:Gir}
\sup_{x>0}\Ex_x\bigg[\int_0^{T}|f(X_t)|^2d t\bigg]<\infty,\ \text{ for any fixed time }T>0.$$ Then stochastically, $S_t^X\leq S_t^Y$ for all $t\geq 0$.
We first define a Girsanov transform between $X_t$ and $Y_t$, $$\frac{d \Q}{d \P}\bigg|_{\F_t}=\exp\bigg[\int_0^t f(X_s)d B_s-\frac{1}{2}\int_0^tf^2(X_s)d s\bigg].$$ Note that due to condition , by [@Chen Theorem 3.2], the right hand side of the above is a uniformly integrable martingale. Thus with $Z^X$, $Z^Y$ as zero sets, we have the relations $$\label{eqn:zero}
(X_t,\Q)\overset{d}{=}(Y_t,\P)\Rightarrow (Z^X,\Q)\overset{d}{=}(Z^Y,\P).$$ In other words, under $\Q$, $X_t$ can be viewed as $Y_t$. This leads to the same properties for zero sets of $X_t$ and $Y_t$.
Now let $L_t^X$ be a choice of the local time for $X_t$ at $0$ such that $L_t^X$ satisfies (cf. Theorem X.2. in [@M]) $$\label{eqn:lt}
\Ex_x\bigg[\int_t^\infty e^{-s}d L_s^X\bigg |\F_t\bigg]=\Ex_x\big[e^{-T_0\circ\theta_t}|\F_t\big],$$ where $T_0$ is the first hitting time at $0$ for $X_t$. We claim that $M_t\triangleq\{e^{-T_0}\1_{\{T_0\leq t\}}-\int_0^te^{-s}d L_s^X;t\geq 0\}$ is a $\P$-martingale with respect to the filtration $\{\F_t; t\geq 0\}$. This is because for every $t\geq r$, $$\begin{aligned}
\Ex_x[M_t|\F_r]=&M_r+\Ex_x\bigg[e^{-T}\1_{\{r<T\leq t\}}-\int_r^te^{-s}d L_s^X\bigg|\F_r\bigg]\\
=&M_r+\Ex_x\bigg[e^{-T}\1_{\{r<T\leq t\}}-\int_r^te^{-s}d L_s^X\bigg|\F_r\bigg]+\Ex_x\bigg[e^{-T\circ\theta_t}-\int_t^\infty e^{-s}d L_s^X\bigg |\F_r\bigg]\\
=&M_r+\Ex_x\bigg[e^{-T\circ\theta_r}-\int_r^\infty e^{-s}d L_s^X\bigg |\F_r\bigg]\\
=&M_r, \end{aligned}$$ where the last equality is due to . This proves the claim that $\{M\}_t$ is a martingale with respect to $\{\F_t\}_{\{t\geq0\}}$. Clearly, it is purely discontinuous martingale of finite variation.
Applying the same Girsanov transform to $M_t$, $M_t-[M,\int_0^\cdot f(X_s)d B_s]_t$ is a $\Q$-martingale. Since $M_t$ is a purely discontinuous martingale of finite variation and $\int_0^tf(X_s)d B_s$ is continuous, $[M,\int_0^\cdot f(X_s)d B_s]_t=0$, $M_t$ is a $\Q$-martingale as well. Thus, $$\Ex_x^\Q [e^{-T_0}\1_{\{T_0\leq t\}}]=\Ex_x^\Q\bigg[\int_0^te^{-s}d L_s^X\bigg]$$ By letting $t\to\infty$, one can have $$\Ex_x^\Q[e^{-T_0}]=\Ex_x^\Q\bigg[\int_0^\infty e^{-s}d L_s^X\bigg].$$ Thus, we get the relation that $(L_t^X, \mathbb Q)\overset{d}{=}(L_t^Y,\mathbb P)$.
Fristedt-Pruitt showed in [@FP] there exists an increasing function $g$ such that $$g\text{-}m(S^X[0,t])=t,$$ where the left hand side represents the Hausdorff measure of the range of $S^X$ on the time interval $[0,t]$ with respect to the function $g$. Since the closure of the range for $S^X$ is $Z^X$, it follows that $g\text{-}m(Z^X\cap[0,t])=L_t^X$, $\mathbb P\text{-a.s.}$ and $$(g\text{-}m(Z^X\cap[0,t]);\Q)=(L_t^X,\Q)\overset{d}{=}(L_t^Y,\P).$$ Also, by the classic comparison theorem, $X_t\leq Y_t$ almost surely for all $t$, we have $Z^X\supset Z^Y$, $\P$-a.s. Together with (\[eqn:zero\]), $$\begin{aligned}
(L_t^Y,\P)=(g\text{-}m(Z^X\cap[0,t]);\Q)=&(g\text{-}m(Z^Y\cap[0,t]);\P)\\
\leq& (g\text{-}m(Z^X\cap[0,t]);\P)\\
=&(L_t^X,\P)\end{aligned}$$ The conclusion of the theorem now follows.
Denote by $\mu_X$ and $\mu_Y$ the Lévy measure for the subordinators $S^X_t$ and $S^Y_t$, respectively. The following is a comparison theorem on Lévy measures.
\[T:3.4\] Suppose $X_t$ and $Y_t$ are reflected diffusions on $[0, \infty)$ as in Theorem \[thm:com\], $\phi^{X}$ and $\phi^{Y}$ are the Laplace exponents of inverse local times, respectively. Then $\phi^Y-\phi^X \geq 0$ is completely monotone and, consequently, $\mu_X \leq \mu_Y$.
Applying Theorem \[thm:com\], we have $S_t^X\leq S_t^Y$, $\P$-a.s. and so $0\leq\phi^X\leq\phi^Y$.\
On the other hand, since $b(x)\leq B(x)$, by Lemma \[lem:lap\], $\phi^Y/\phi^X$ is completely monotone. Combining the two facts, we see that $$\frac{\phi^Y}{\phi^X}-1\geq 0\text{ is completely monotone.}$$ Since completely monotone relation is preserved under multiplication (check details in Chapter 1, [@SSV]), we would have $$\phi^Y-\phi^X= \left(\frac{\phi^Y}{\phi^X}-1\right)\phi^X\geq0\text{ is completely monotone.}$$ This says that $\phi^Y-\phi^X$ is the Laplace exponent of some subordinator $Z$. Hence $S^Y$ has the same distribution as the independent sum of two subordinators $S^X$ and $Z$. Denote by $\nu$ the Lévy measure for $Z$. It follows then $\mu_Y - \mu_X =\nu\geq 0$.
Properties of non-local operators {#S:4}
=================================
We use the same notations as in Example \[ex:stablecom\]. For $0<\alpha<1$, let $X^{(\alpha)}_t$ is a reflected Bessel process on $[0,\infty)$ with the local generator $$\op^{(\alpha)}=\frac{1}{2}\frac{d^2}{d x^2}+\frac{1-2\alpha}{2x}\frac{d}{d x}.$$ As we noted earlier, the inverse local time $S^{(\alpha)}_t$ is an $\alpha$-stable subordinator, with the Laplace exponent $$\label{eqn:stable}
\phi^{(\alpha)}(\lambda)=c_\alpha\lambda^{\alpha}.$$ We know from Theorem \[thm:EG\] that under the new probability measure $\Q$ defined by , the inverse local time of the Girsanov transformed diffusion, $X_t^{(\alpha,m)}$, is a relativistic $\alpha$-stable subordinator, with the Laplace exponent $$\label{eqn:rela}
\phi^{(\alpha,m)}(\lambda)=\phi^{(\alpha)}(\lambda+m)-\phi^{(\alpha)}(m)=c_\alpha\big((\lambda+m)^{\alpha}-m^\alpha\big).$$ The new reflected diffusion $X_t^{(\alpha,m)}$ on $[0, \infty)$ has generator $$\label{eqn:relabessel}
\op^{(\alpha,m)}=\frac{1}{2}\frac{d^2}{dx^2}+\bigg(\frac{1-2\alpha}{2x}+\frac{\rho'_m(x)}{\rho_m(x)}\bigg)\frac{d}{dx}, \text{ for }x>0,$$ where $\rho_m (x):= \Ex_x \left[ \exp (-m T_0 )\right]$ with $T_0$ being the first hitting time of $0$ by $X^{(\alpha)}$. By Remark \[rmk:EG\], $\rho_m(x)$ is the unique solution to $$\begin{cases}
(\op^{(\alpha)}-m)\rho(x)=0,\\
\rho(0)=1;\ \rho(\infty)=0.
\end{cases}$$ It is know from ODE, $$\rho_m(x)=\wh c_\alpha \wh K_\alpha(\sqrt{2m}x) ,$$ where $\wh c_\alpha$ is a normalizing constant depending on $\alpha$ only, $\wh K_\alpha=x^\alpha K_\alpha$ and $$\label{eqn:Kalpha}
K_\alpha(x)=\frac{\pi}{2}\frac{I_{-\alpha}(x)-I_\alpha(x)}{\sin(\alpha\pi)},$$ where $$I_\alpha(x)=\sum_{n=0}^\infty\frac{1}{n!\Gamma(n+\alpha+1)}\left(\frac{x}{2}\right)^{2n+\alpha}.$$ The function $I_\alpha$ is a solution to the following modified Bessel’s equation $$x^2u''(x)+xu'(x)-(x^2+\alpha^2)u=0.$$ Clearly, $K_\alpha$ also satisfies the above equation, and is called a modified Bessel function of the second kind.
\[ex:Cauchy\] Suppose $\alpha=0.5$. Note that $$\begin{aligned}
K_{0.5}(x)=&\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{(\frac{x}{2})^{2n}}{n!}\bigg[\frac{(\frac{x}{2})^{-0.5}}{\Gamma(0.5+n)}-\frac{(\frac{x}{2})^{0.5}}{\Gamma(1.5+n)}\bigg]\\
=&\frac{\pi}{\sqrt{2x}}\sum_{n=0}^\infty\bigg[\frac{(\frac{x}{2})^{2n}}{n!\Gamma(0.5+n)}-\frac{(\frac{x}{2})^{2n+1}}{n!\Gamma(1.5+n)}\bigg]\\
=&\frac{\pi}{\sqrt{2x}}\sum_{n=0}^\infty\bigg[\frac{x^{2n}}{\Gamma(0.5)(2n)!}-\frac{x^{2n+1}}{\Gamma(0.5)(2n+1)!}\bigg]\\
=&\frac{\pi e^{-x}}{\Gamma(0.5)\sqrt{2x}}.\end{aligned}$$ Thus for $\alpha=0.5$, $$\rho_m(x)=\frac{\pi}{\Gamma(0.5)\sqrt{2}}\exp(-\sqrt{2m}x).$$ Consequently, we have the perturbation part as $$\frac{\rho'_m(x)}{\rho_m(x)}=-\sqrt{2m}.$$ Hence if $X^{(0.5,m)}_t$ is a reflected process on $[0,\infty)$ with the local generator $$\op^{(0.5,m)}=\frac{1}{2}\frac{d^2}{d x^2}-\sqrt{2m}\frac{d}{d x},$$ then its inverse local time at $0$ is a relativistic Cauchy subordinator with the Laplace exponent $$\phi^{(0.5,m)}(x)=c(\sqrt{\lambda+m}-\sqrt{m}).$$
Now if we operate another Girsanov transform on $\op^{(\alpha,m)}$, that is $$\op^{(1)}=\op^{(\alpha,m)}+\frac{q'_n(x)}{q_n(x)}\frac{d}{d x},$$ where $q_n(x)$ is the unique solution to $$\begin{cases}
(\op^{(\alpha,m)}-n)q_n(x)=0;\\
q_n(0)=1,\ q_n(\infty)=0.
\end{cases}$$ Then the inverse local time at $0$, $S_t^{(1)}$, of the new reflecting diffusion generated by the above generator, has the Laplace exponent $$\phi^{(1)}(\lambda)=\phi^{(\alpha,m)}(\lambda+n)-\phi^{(\alpha,m)}(n)=c_\alpha\big((\lambda+m+n)^\alpha-(m+n)^\alpha\big).$$ It can also be viewed as the Laplace exponent of a relativistic $\alpha$-stable subordinator with mass $m+n$, which is obtained as the inverse local time at $0$ for a reflecting diffusion determined locally by $$\op^{(\alpha,m+n)}=\op+\frac{\rho'_{m+n}(x)}{\rho_{m+n}(x)}\frac{d}{d x}.$$ The two generators should be the same, so we get the relation $$\frac{\rho'_m(x)}{\rho_m(x)}+\frac{q'_n(x)}{q_n(x)}=\frac{\rho'_{m+n}(x)}{\rho_{m+n}(x)}$$
S. Watanabe [@W] has defined a conservative diffusion process $\wt X_t$ on $[0,\infty)$, determined by the local generator in the same form $$\wt\op=\frac{1}{2}\frac{d^2}{dx^2}+\left(\frac{1-2\alpha}{2x}+\frac{\rho_c'(x)}{\rho_c(x)}\right)\frac{d}{dx},$$ but with $\rho_c(x)$ as the unique solution to $$\begin{cases}
(\op^{(\alpha)}-c)\rho(x)=0;\\
\rho(0)=1,\ \rho'(0)=0.
\end{cases}$$ Thus, the generator can be written as $$\wt\op u=\frac{1}{\rho_c}(\op^{(\alpha)}-c)(\rho_c\cdot u)(x),$$ and this yiels an explicit expression of the transition density for $X_t^{(\alpha,c)}$ $$\wt p(t,x,y)=\frac{e^{-mt}p^{(\alpha)}(t,x,y)}{\rho_c(x)\rho_c(y)},\ x,y\geq 0,$$ where $p^{(\alpha)}(t,x,y)$ is the transition density of the Bessel process $X_t^{(\alpha)}$ with respect to the measure $m^{(\alpha)}(dx)=x^{1-2\alpha}dx$.
We continue to discuss the reflecting diffusions, $X_t^{(\alpha,m)}$, which is locally determined by the generator (\[eqn:relabessel\]). Consider the drift term $\frac{\rho'_m(x)}{\rho_m(x)}$, because $K'_\alpha(x)=-\frac{\alpha}{x}K_\alpha(x)-K_{\alpha-1}(x)$ $$\begin{aligned}
\label{eqn:rho}
\frac{\rho'_m(x)}{\rho_m(x)}=&\sqrt{2m}\frac{\wh K'_\alpha(\sqrt{2m}x)}{\wh K_\alpha(\sqrt{2m}x)}\notag\\
=&\frac{\alpha}{x}+\sqrt{2m}\frac{K'_\alpha(\sqrt{2m}x)}{K_\alpha(\sqrt{2m}x)}\notag\\
=&\frac{\alpha}{x}+\sqrt{2m}\frac{-\frac{\alpha}{\sqrt{2m}x}K_\alpha(\sqrt{2m}x)-K_{\alpha-1}(\sqrt{2m}x)}{K_\alpha(\sqrt{2m}x)}\notag\\
=&-\sqrt{2m}\frac{K_{\alpha-1}(\sqrt{2m}x)}{K_{\alpha}(\sqrt{2m}x)}.\end{aligned}$$ We will focus on the asymptotic behaviors of $\frac{\rho'_m(x)}{\rho_m(x)}$ near $0$ and $\infty$ and have the following lemma:
For $m\geq 0$, $0<\alpha<1$, $$\label{eqn:asymp}
\frac{\rho'_m(x)}{\rho_m(x)}=-\sqrt{2m}\frac{K_{\alpha-1}(\sqrt{2m}x)}{K_{\alpha}(\sqrt{2m}x)}\sim\begin{cases}
-\frac{m^\alpha\Gamma(1-\alpha)}{2^{\alpha-1}\Gamma(\alpha)}x^{2\alpha-1}\text{ as }x\to 0+;\\
-\sqrt{2m}\text{ as }x\to\infty,
\end{cases}$$ where $\sim$ means the ratio between two sides approaches $1$ as $x$ goes to $0+$ or $\infty$.
When $x\to 0+$ and $\nu\notin\Z$, $K_\nu(x)$ has the following series expansion: $$\begin{aligned}
K_\nu(x)\propto \frac{1}{2}\bigg(\Gamma&(\nu)\Big(\frac{x}{2}\Big)^{-\nu}\Big(1+\frac{x^2}{4(1-\nu)}+\frac{x^4}{32(1-\nu)(2-\nu)}+\cdots\Big)\\
&+\Gamma(-\nu)\Big(\frac{x}{2}\Big)^\nu\Big(1+\frac{x^2}{4(\nu+1)}+\frac{x^4}{32(\nu+1)(\nu+2)}+\cdots\Big)\bigg).\end{aligned}$$ Since $0<\alpha<1$, $\alpha,\alpha-1\notin\Z$ in (\[eqn:rho\]), as $x\to 0+$, $$\begin{aligned}
\label{eqn:asym0}
\frac{\rho'_m(x)}{\rho_m(x)}\sim&-\sqrt{2m} \frac{\Gamma(1-\alpha)\Big(\frac{\sqrt{2m}x}{2}\Big)^{\alpha-1}}{\Gamma(\alpha)\Big(\frac{\sqrt{2m}x}{2}\Big)^{-\alpha}}\notag\\
\sim&-\frac{m^\alpha\Gamma(1-\alpha)}{2^{\alpha-1}\Gamma(\alpha)}x^{2\alpha-1}.\end{aligned}$$ When $x\to\infty$, $K_\nu(x)$ can be described as the following formula: $$K_\nu(x)\propto \sqrt{\frac{\pi}{2}}\frac{e^{-x}}{\sqrt{x}}\bigg(1+O\Big(\frac{1}{x}\Big)\bigg).$$ The asymptotic behavior near $\infty$ is independent of the index $\nu$. Thus, as $x\to\infty$ $$\label{eqn:asyminf}
\frac{\rho'_m(x)}{\rho_m(x)}\sim -\sqrt{2m}.$$
We are now in the position to present the proof for Theorem \[T:1.1\].
**Proof of Theorem \[T:1.1\].** $Y_t$ is a reflecting diffusion process, determined by the local generator $$\op=\frac{1}{2}\frac{d^2}{d x^2}+\left(\frac{1-2\alpha}{2x}-f(x)\right)\frac{d}{d x},$$ and there exists a constant $c_1$ such that $$0\leq f(x)\leq c_1(1\wedge x)^{2\alpha-1}.$$ Now we check the condition (\[con:Gir\]) in Theorem \[thm:com\], $f(x)$ is bounded when $1/2\leq\alpha<1$, so the condition is naturally satisfied. When $0<\alpha< 1/2$, for a fixed $T>0$, with $p^{(\alpha)}(t,x,y)$ representing the transition density of a Bessel process of index $\alpha$ with respect to the measure $m^{(\alpha)}(dx)=2x^{1-2\alpha}dx$, $$\begin{aligned}
\sup_{x>0}\Ex_x\bigg[\int_0^T|f(X^{(\alpha)}_t)|^2\rd t\bigg]\leq&c_1\sup_{x>0}\int_{(0,\infty)}\bigg(\int_0^T p^{(\alpha)}(t,x,y)(1\vee y^{4\alpha-2})d t\bigg)m^{(\alpha)}(dy)\\
\leq&c_1T+c_1\sup_{x>0}\int_0^1\bigg(\int_0^T \frac{x^\alpha y^{3\alpha-1}}{t}\exp\left(-\frac{x^2+y^2}{2t}\right)I_{-\alpha}\Big(\frac{xy}{t}\Big)d t\bigg)dy\\
=&c_1T+c_1\sup_{x>0}\int_0^1\int_{xy/T}^\infty \frac{x^\alpha y^{3\alpha-1}}{s}\exp\left(-\frac{(x^2+y^2)s}{2xy}\right)I_{-\alpha}(s)dsdy\\
\leq&c_1T+c_1\sup_{x>0}\int_0^1x^\alpha y^{3\alpha-1}\bigg(\int_{xy/T}^\infty \frac{e^{-s}}{s}I_{-\alpha}(s)ds\bigg)dy,\end{aligned}$$ where $I_{-\alpha}(s)$ is the modified Bessel function of the first kind. Then $$I_{-\alpha}(s)\propto\begin{cases}
\frac{1}{\Gamma(1-\alpha)}\Big(\frac{s}{2}\Big)^{-\alpha}\bigg(1+\frac{s^2}{4(1-\alpha)}+\frac{s^4}{32(1-\alpha)(2-\alpha)}+\cdots\bigg),\ s\to0;\\
\frac{e^s}{\sqrt{2\pi s}}\bigg(1+O\Big(\frac{1}{s}\Big)\bigg),\ s\to\infty.
\end{cases}$$ Since we have $0<y<1$, by the above asymptotic behavior $$\int_{xy/T}^\infty \frac{e^{-s}}{s}I_{-\alpha}(s)ds\text{ is dominated by } C_\alpha T^\alpha (xy)^{-\alpha} \text{ as }x\to 0+; \text{ by }C_\alpha T^{1/2}(xy)^{-1/2}\text{ as }x\to\infty.$$ We would then get for $0<\alpha<1/2$ $$\sup_{x>0}\Ex_x\bigg[\int_0^T|f(X^{(\alpha)}_t)|^2\rd t\bigg]<\infty.$$ Thus, one can set up a Girsanov transform between $X_t$ and $X_t^{(\alpha)}$, or, $X_t$ and $X_t^{(\alpha)}$ are absolutely continuous to each other. Applying Theorem \[thm:com\], we have $S_t\leq S_t^{(\alpha)}$, $\P$-a.s.
For any $0<\alpha<1$, from the asymptotic behaviors of $\frac{\rho'_m(x)}{\rho_m(x)}$ near $0$ and $\infty$ shown in (\[eqn:asymp\]), we can always choose a proper value of $m$ such that $$c_1\leq\sqrt{2m}\wedge\frac{m^\alpha\Gamma(1-\alpha)}{2^{\alpha-1}\Gamma(\alpha)},$$ consequently, $$0\leq f(x)\leq -\frac{\rho'_m(x)}{\rho_m(x)},\ m\text{ depends on }c_1.$$ Thus, by the classic Comparison theorem, $$X_t^{(\alpha,m)}\leq Y_t\leq X_t^{(\alpha)},\ \P\text{-a.s.}$$ By Theorem \[thm:EG\], $X_t^{(\alpha)}$ and $X_t^{(\alpha,m)}$ are absolutely continuous to each other. Thus, $Y_t$ and $X_t^{(\alpha,m)}$ are absolutely continuous to each other, i.e., there exists a Girsanov transform between them. Applying Theorem \[thm:com\] again, $S_t^{(\alpha,m)}\leq S_t$, $\P$-a.s.
To prove Theorem \[T:1.2\], we first recall the following result from Grzywny-Ryznar [@GR] (with slightly different notation here).
\[thm:GR\] [[@GR Theorem 1.1] ]{} Let $D\subset\R^d$ be a bounded Lipschitz open set. Suppose that $Y_t$ is a symmetric Lévy process on $\R^d$ with Lévy measure $\nu(x) dx$. Denote by $\nu^{(\alpha)}(x)$ the Lévy density for the isotropic $\alpha$-stable process $Z$ on $\R^d$. Denote by $G_D$ and $G^{(\alpha)}_D$ the Green functions of $Y$ and $Z$ in $D$, respectively. Assume that $j(x)=\nu^{(\alpha)}(x)-\nu(x)\geq0$ on $\R^d$, and that $j(x)\leq c|x|^{\rho-d}$ for $|x|\leq 1$, where $c,\rho>0$. Then there exists a constant $C=C(d,\alpha, D, \rho, c)$, such that $$C^{-1} G_D^{(\alpha)}(x,y)\leq G_D(x,y)\leq CG_D^{(\alpha)}(x,y) \quad
\hbox{for all } x,y\in D.$$
**Proof of Theorem \[T:1.2\].** Denote the Laplace exponents of $S_t$, $S_t^{(\alpha)}$, $S_t^{(\alpha,m)}$, by $\phi(\lambda)$, $\phi^{(\alpha)}(\lambda)$, $\phi^{(\alpha,m)}(\lambda)$, respectively. Now applying Theorem \[T:3.4\], $$\phi(\lambda)-\phi^{(\alpha,m)}(\lambda)\text{ and } \phi^{(\alpha)}(\lambda)-\phi(\lambda)\text{ are completely monotone.}$$ Denote $\nu^{(\alpha,m)}, \nu,\nu^{(\alpha)}$ as measures of inverse local times respectively, then $$\nu^{(\alpha)}-\nu\geq 0;\ \nu-\nu^{(\alpha,m)}\geq 0$$ and for any $0<\alpha<1$, $t>0$, $$\label{eqn:mea}
0\leq(\nu^{(\alpha)}-\nu)(t)\leq (\nu^{(\alpha)}-\nu^{(\alpha,m)})(t)\leq c_\alpha\frac{1-e^{-mt}}{t^{\alpha+1}}.$$ Thus, for $|x|\leq1$, the difference between measures of trace processes, $B_{S_t}, B_{S_t^{(\alpha)}}$, would be $$\begin{aligned}
j(x)=&c_\alpha\int_0^\infty (4\pi t)^{-d/2}e^{-\frac{|x|^2}{4t}}(\nu^{(\alpha)}-\nu)(t)dt\\
=&c_\alpha\int_0^\infty (4\pi t)^{-d/2}e^{-\frac{|x|^2}{4t}}\frac{1-e^{-mt}}{t^{\alpha+1}}dt\\
\leq&c_\alpha\int_0^\infty (4\pi t)^{-d/2}e^{-\frac{|x|^2}{4t}}\frac{mt}{t^{\alpha+1}}dt\\
=&c_\alpha\pi m^{-d/2}4^{\alpha-1}|x|^{-d+2-2\alpha}\int_0^\infty s^{d/2+\alpha-2}e^{-s}ds\\
\leq &C|x|^{-d+2-2\alpha},\end{aligned}$$ where $C=C(\alpha,m,d,c_\alpha)$. The second to the last equality is obtained by doing change of variables $s=|x|^2/(4t)$. From the proof of Theorem \[T:1.1\], $m$ is fixed once given a $f(x)$.
Applying Theorem \[thm:GR\] with $\rho=2-2\alpha>0$, we conclude that there exists a constant $C_1=C(d,\alpha, D,C)$ such that $$C_1^{-1}G_D^{(2\alpha)}(x,y)\leq G_D(x,y)\leq C_1G_D^{(2\alpha)}(x,y)$$ for all $x,y\in D$.
[**Acknowledgement.**]{} We thank P. J. Fitzsimmons, M. M. Meerschaert and Z. Vondracek for helpful discussions.
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R. L. Schilling, R. Song and Z. Vondra$\check{c}$ek, Bernstein functions, [*de Gruyter studies in Mathematics **37***]{} (2010).
S. Watanabe, On Time Inversion of One-Dimensional Diffusion Processes, [*Z. Wahrsch. verw. Geb.*]{} [**31**]{} (1975) 115-124.
0.3truein
Department of Mathematics, University of Washington, Seattle, WA 98195, USA
Email: `[email protected]`
Email: `[email protected]`
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $\overline{{\mathcal{M}}}_{g,l}$ be the moduli space of stable algebraic curves of genus $g$ with $l$ marked points. With the operations which relate the different moduli spaces identifying marked points, the family $(\overline{{\mathcal{M}}}_{g,l})_{g,l}$ is a modular operad of projective smooth Deligne-Mumford stacks, $\overline{{\mathcal{M}}}$. In this paper we prove that the modular operad of singular chains $C_*(\overline{{\mathcal{M}}}_{};\mathbb{Q})$ is formal; so it is weakly equivalent to the modular operad of its homology $H_*(\overline{{\mathcal{M}}}_{};\mathbb{Q})$. As a consequence, the up to homotopy" algebras of these two operads are the same. To obtain this result we prove a formality theorem for operads analogous to Deligne-Griffiths-Morgan-Sullivan formality theorem, the existence of minimal models of modular operads, and a characterization of formality for operads which shows that formality is independent of the ground field.'
address:
- |
Departament d’Àlgebra i Geometria\
Universitat de Barcelona\
Gran Via 585, 08007 Barcelona (Spain)
- |
Departament de Matemàtica Aplicada I\
Universitat Politècnica de Catalunya\
Diagonal 647, 08028 Barcelona (Spain).
author:
- 'F. Guill[é]{}n Santos'
- 'V. Navarro'
- 'P. Pascual'
- 'A. Roig'
title: Moduli spaces and formal operads
---
Introduction
============
In recent years, moduli spaces of Riemann surfaces such as the moduli spaces of stable algebraic curves of genus $g$ with $l$ marked points, $\overline{{\mathcal{M}}}_{g,l}$, have played an important role in the mathematical formulation of certain theories inspired by physics, such as the complete cohomological field theories.
In these developments, the operations which relate the different moduli spaces $\overline{{\mathcal{M}}}_{g,l}$ identifying marked points, $ \overline{{\mathcal{M}}}_{g,l} \times
\overline{{\mathcal{M}}}_{h,m} \longrightarrow \overline{{\mathcal{M}}}_{g+h,l+m-2} $ and $ \overline{{\mathcal{M}}}_{g,l}
\longrightarrow \overline{{\mathcal{M}}}_{g+1,l-2} $, have been interpreted in terms of operads. With these operations the spaces $\overline{{\mathcal{M}}}_{0,l}$, $l\geq 3$, form a cyclic operad of projective smooth varieties, $\overline{{\mathcal{M}}}_0$ ([@GeK94]), and the spaces $\overline{{\mathcal{M}}}_{g,l}$, $g,l\geq 0$, $2g-2+l > 0$, form a modular operad of projective smooth Deligne-Mumford stacks, $\overline{{\mathcal{M}}}$ ([@GeK98]). Therefore, the homologies of these operads, $H_*(\overline{{\mathcal{M}}}_{0};\mathbb{Q})$ and $H_*(\overline{{\mathcal{M}}};\mathbb{Q})$, are cyclic and modular operads respectively.
An important result in the algebraic theory of the Gromov-Witten invariants is that, if $X$ is a complex projective manifold and $\Lambda(X)$ is the Novikov ring of $X$, the cohomology $H^*(X;\Lambda(X))$ has a natural structure of an algebra over the modular operad $H_*(\overline{{\mathcal{M}}}_{};\mathbb{Q})$, and so it is a complete cohomological field theory ([@Be], see [@Man]).
But there is another modular operad associated to the geometric operad $\overline{{\mathcal{M}}}_{}$: the modular operad $C_*(\overline{{\mathcal{M}}}_{};\mathbb{Q})$ of singular chains. Algebras over this operad have been studied in [@GeK98], [@KSV] and [@KVZ].
In this paper we prove that the modular operad $C_*(\overline{{\mathcal{M}}}_{};\mathbb{Q})$ is formal; so it is weakly equivalent to the modular operad of its homology $H_*(\overline{{\mathcal{M}}}_{};\mathbb{Q})$. As a consequence, the up to homotopy" algebras of these two operads are the same.
A paradigmatic example of operad is the little $2$-disc operad of Boardman-Vogt, $\mathcal{D}_2(l)$, of configurations of $l$ disjoint discs in the unity disc of $\mathbb{R}^2$. Our result can be seen as the analogue for $\overline{{\mathcal{M}}}_{}$ of the Kontsevich-Tamarkin’s formality theorem of for $C_*(\mathcal{D}_2;\mathbb{Q})$ ([@Ko] and [@T]; moreover [@Ko] also explains the relation between this formality theorem, Deligne’s conjecture in Hochschild cohomology and Kontsevich’s formality theorem in deformation quantization).
Our paper is organized as follows. In section 2, we study symmetric monoidal functors between symmetric monoidal categories, since they induce functors between the categories of their operads. After recalling some definitions and fixing some notations of operads and monoidal categories, we prove a symmetric De Rham theorem. We then introduce the notion of formal symmetric monoidal functor, and we see how this kind of functor produces formal operads.
In section 3, as a consequence of Hodge theory, we prove that the cubic chain functor on the category of compact Kähler manifolds $C_*: {\mathbf{K\ddot{a}h}}\longrightarrow {\mathbf{C}_*(\mathbb{R})}$ is a formal symmetric monoidal functor. It follows that, if $X$ is an operad of compact Kähler manifolds, then the operad of chains $C_*(X;\Bbb R)$ is formal. This is the analogue in the theory of operads of the Deligne-Griffiths-Morgan-Sullivan formality theorem in rational homotopy theory ([@DGMS]).
The goal of sections $4$, $5$ and $6$ is to prove the [ descent]{} of formality from $\mathbb{R}$ to $\mathbb{Q}$. In section $4$ we recall some results due to M. Markl on minimal models of operads in the form that we will use in order to generalize them to cyclic and modular operads.
In section 5, drawing on Deligne’s weight theory for Frobenius endomorphism in étale cohomology, we introduce weights and show the formality of the category of complexes endowed with a pure endomorphism. Next, in th. \[aixeca\], we prove a characterization of formality of an operad in terms of the lifting of automorphisms of the homology of the operad to automorphisms of the operad itself.
The automorphism group of a minimal operad with homology of finite type is a pro-algebraic group. This result allows us to use the descent theory of algebraic groups to prove the independence of formality of the ground field in th. \[descens\].
In section 7 we show how the above results can be extended easily to cyclic operads. In particular we obtain the formality of the cyclic operad $C_*({\overline{\mathcal{M}}_0}; \mathbb{Q}
)$.
In the last section, we go one step further and prove the above results also for modular operads. In particular, we introduce minimal models of modular operads and we prove their existence and lifting properties. Here, we follow Grothendieck’s idea in his “jeu de Légo-Teichmüller" ([@Gro]), in which he builds the complete Teichmüller tower inductively on the modular dimension. Once this is established, the proofs of the previous sections can be transferred to the modular context without difficulty. Finally, we conclude the formality of the modular operad $C_*(\overline{\mathcal M}_{};
\mathbb{Q})$.
Formal operads
==============
Operads
-------
Let us recall some definitions and notations about operads (see [@GK], [@KM], [@MSS]).
###
Let $\Sigma$ be the [*symmetric groupoid*]{}, that is, the category whose objects are the sets $\{1,\dots , n \} $, $n\ge 1$ , and the only morphisms are those of the symmetric groups $\Sigma_n = Aut\{1,\dots , n \} $.
###
Let $\mathcal C$ be a category. The category of contravariant functors from $\Sigma$ to $\mathcal{C}$ is called the category of $\Sigma$-[*modules*]{} and is denoted by ${\mathbf{\Sigma Mod}}_{\mathcal{C}}$, or just ${\mathbf{\Sigma Mod}}$ if $\mathcal C$ is understood. We identify its objects with sequences of objects in $\mathcal{C}$, $E=\left((E(l)\right)_{l\geq 1}$, with a right $\Sigma_l$-action on each $E(l)$. If $e$ is an element of $E(l)$, $l$ is called the [*arity*]{} of $e$. If $E$ and $F$ are $\Sigma$-modules, a [*morphism of $\Sigma$-modules*]{} $f:E \longrightarrow F$ is a sequence of $\Sigma_l$-[ equivariant]{} morphisms $f(l): E(l) \longrightarrow F(l), \
l\geq 1$.
###
Let $(\mathcal{C}, \otimes, \mathbf{1})$ be a symmetric monoidal category. A [*unital $\Sigma$-operad*]{} (an [*operad*]{} for short) in $\mathcal{C}$ is a $\Sigma$-module $ P$ together with a family of [ structure morphisms]{}: [ composition]{} $\gamma_{l;m_1,\dots ,m_l} : P(l)\otimes P(m_1) \otimes \dots
\otimes P(m_l)
\longrightarrow P(m_1 + \cdots + m_l)$, and [ unit]{} $\eta : \mathbf{1} \longrightarrow
P(1)$, satisfying the axioms of equivariance, associativity, and unit. A [*morphism of operads*]{} is a morphism of $\Sigma$-modules compatible with structure morphisms. Let us denote by ${\mathbf{Op}}_{\mathcal{C}}$, or simply ${\mathbf{Op}}$ when $\mathcal
C$ is understood, the category of operads in $\mathcal C$ and its morphisms.
Symmetric monoidal categories and functors
------------------------------------------
In the study of $\Sigma$-operads the commutativity constraint plays an important role. In particular the functors we are interested in are functors between symmetric monoidal categories which are compatible with the associativity, commutativity and unit constraints.
###
The following are some of the symmetric monoidal categories we will deal with in this paper. On the one hand, the geometric ones:
${\mathbf{Top}}$: the category of topological spaces.
${\mathbf{Dif}}$: the category of differentiable manifolds.
${\mathbf{K\ddot{a}h}}$: the category of compact Kähler manifolds.
${\mathbf{V}(\mathbb{C})}$: the category of smooth projective $\mathbb
C$-schemes.
On the other hand, the algebraic categories, which will be subcategories, or variants of
${{\mathbf{C}_*({\mathcal{A}})}}$: the category of complexes with a differential of degree $-1$ of an abelian monoidal symmetric category $({\mathcal{A}}, \otimes , \mathbf{1})$. The morphisms are called chain maps. If ${\mathcal{A}}$ is the category of $R$-modules for some ring $R$, we will denote it by ${\mathbf{C}_*(R)}$. Operads in ${{\mathbf{C}_*({\mathcal{A}})}}$ are also called [*dg operads*]{}.
In a symmetric monoidal category $(\mathcal C,\otimes ,\mathbf
1)$ we usually denote the natural commutativity isomorphism by $\tau_{X,Y}:X\otimes Y\longrightarrow Y\otimes X$. For example, in ${{\mathbf{C}_*({\mathcal{A}})}}$, the natural commutativity isomorphism $$\tau_{X,Y} : X \otimes Y \longrightarrow Y \otimes X$$ includes the signs: $$\tau_{X,Y} (x \otimes y) = (-1)^{\deg (x)\deg (y)} y \otimes x\ .$$
###
As usual, we move from a geometric category to an algebraic one through a functor. Let us recall (see [@KS]) that a [*monoidal functor*]{} $$(F , \kappa ,\eta ) : (\mathcal{C}, \otimes , \mathbf{1})
\longrightarrow (\mathcal{D},
\otimes , \mathbf{1}')$$ between monoidal categories is a functor $F : \mathcal{C}
\longrightarrow \mathcal{D}$ together with a natural morphism of $\mathcal{D}$, $$\kappa_{X,Y} : FX \otimes FY
\longrightarrow F(X \otimes Y),$$ for all objects $X, Y \in \mathcal{C}$, and a morphism of $\mathcal{D}$, $ \eta : \mathbf{1}' \longrightarrow
F\mathbf{1}$, compatibles with the constraints of associativity, and unit. We will refer the [*Künneth morphism*]{} as $\kappa$.
If $\mathcal C$ and $\mathcal D$ are symmetric monoidal categories, a monoidal functor $F :
\mathcal{C} \longrightarrow
\mathcal{D}$ is said to be [*symmetric*]{} if $\kappa$ is compatible with the commutativity constraint.
For example, the [*homology functor*]{} $H_*: {{\mathbf{C}_*({\mathcal{A}})}}\longrightarrow {{\mathbf{C}_*({\mathcal{A}})}}$ is a symmetric monoidal functor, taking the usual K[ü]{}nneth morphism $$H_*(X)\otimes H_*(Y) \longrightarrow H_*(X \otimes Y).$$ as $\kappa$.
Let $F,G:\mathcal{C}\rightrightarrows
\mathcal{D}$ be two monoidal functors. A natural transformation $ \phi : F \Rightarrow G
$ is said to be [*monoidal*]{} if it is compatible with $\kappa$ and $\eta$.
###
Let $F : \mathcal{C} \longrightarrow \mathcal{D}$ be a symmetric monoidal functor. It is easy to prove that, applied componentwise, $F$ induces a functor between $\Sigma$-operads $${\mathbf{Op}}_{F}:{\mathbf{Op}}_{\mathcal{C}} \longrightarrow {\mathbf{Op}}_{\mathcal{D}} \ ,$$ also denoted by $F$.
In particular, for an operad $P\in {\mathbf{Op}}_{{{\mathbf{C}_*({\mathcal{A}})}}}$, its homology is an operad $HP \in
{\mathbf{Op}}_{{{\mathbf{C}_*({\mathcal{A}})}}}$.
In the same way, if $F, G:\mathcal{C} \rightrightarrows
\mathcal{D}$ are two symmetric monoidal functors, a monoidal natural transformation $\phi:F\Rightarrow G$ induces a natural transformation $${\mathbf{Op}}_\phi:{\mathbf{Op}}_{F}\Rightarrow {\mathbf{Op}}_G,$$ also denoted by $\phi$.
Weak equivalences
-----------------
We will use weak equivalences in several contexts.
###
Let $X$ and $Y$ be objects of ${\mathbf{C}}_*({\mathcal{A}})$. A chain map $f:
X \longrightarrow Y$ is said to be a [*weak equivalence of complexes* ]{} if the induced morphism $f_* = Hf : HX
\longrightarrow HY$ is an isomorphism.
###
Let $\mathcal C$ be a category, ${\mathcal{A}}$ an abelian category, and $F,G:\mathcal C\rightrightarrows
\mathbf C_*({\mathcal{A}})$ two functors. A natural transformation $\phi:F\Rightarrow G $ is said to be a [*weak equivalence of functors*]{}, if the morphism $\phi(X):F(X)\rightarrow G(X)$ is a weak equivalence, for every object $X$ in $\mathcal C$.
###
A morphism $\rho : P \longrightarrow Q $ of operads in ${{\mathbf{C}_*({\mathcal{A}})}}$ is said to be a [*weak equivalence of operads* ]{} if $\rho (l) : P(l)
\longrightarrow Q(l)$ is a weak equivalence of chain complexes, for all $l$.
###
Let $\mathcal C$ be a category endowed with a distinguished class of morphism called [*weak equivalences*]{}. We suppose that this is a saturated class of morphisms which contains all isomorphisms. Two objects $X$ and $Y$ of $\mathcal C$ are said to be [*weakly equivalent*]{} if there exists a sequence of morphism of $\mathcal C$ $$X \longleftarrow X_1 \longrightarrow \cdots \longleftarrow
X_{n-1} \longrightarrow Y.$$ which are weak equivalences. If $X$ and $Y$ are weakly equivalent, we say that $Y$ is a [*model*]{} of $X$.
###
The following proposition is an easy consequence of the definitions.
\[Operadsequivalents\] If $F,G:\mathcal C\rightrightarrows \mathcal D$ are two weakly equivalent symmetric monoidal functors, the functors ${\mathbf{Op}}_F$ and ${\mathbf{Op}}_G$ are weakly equivalent. In particular, for every operad $P$ in $\mathcal C$, the operads $F(P)$ and $G(P)$ are weakly equivalent.
Symmetric De Rham’s theorem
---------------------------
In this section we will demonstrate a symmetric version of De Rham’s theorem comparing the complex of singular cubic chains and De Rham’s complex of currents, including its symmetric structure as monoidal functors.
###
It is well known that the functor of [*singular chains* ]{}$$S_*(\phantom{X}
; \mathbb{Z}) : {\mathbf{Top}}\longrightarrow {\mathbf{C}_*(\mathbb{Z})} \ ,$$ together with the [*shuffle product*]{} as the Künneth morphism, is a symmetric monoidal functor.
On the other hand, let $\mathcal D'_*(M)$ be the complex of De Rham’s [*currents*]{} of a differentiable manifold $M$; that is, $\mathcal D'_*(M)$ is the topological dual of the complex $\mathcal D^*(M)$ of differential forms with compact support. Then the functor $$\mathcal{D}'_* : {\mathbf{Dif}}\longrightarrow {\mathbf{C}_*(\mathbb{R})} \ ,$$ is a symmetric monoidal functor with the Künneth morphism $$\kappa_{M,N}: \mathcal{D}'_*(M) \otimes \mathcal{D}'_*(N) \longrightarrow
\mathcal{D}'_*(M \times N)$$ induced by the tensor product of currents. Thereby, if $S \in
\mathcal{D}'_*(M)$, and $T
\in \mathcal{D}'_*(N)$, then $$<\!\kappa(S \otimes T)\,, \,\pi_M^*(\omega ) \wedge \pi_N^*
(\nu )\!> \,=\, <\! S\,,\,\omega \!>\,\cdot \,<\!T\,,\,\nu\! >$$ for all $\omega \in \mathcal{D}^*(M)$ and $\nu \in
\mathcal{D}^*(N)$.
In order to compare the functor of currents with the functor of singular chains on differentiable manifolds, one can consider the complex of chains $S_*^\infty(M)$ generated by the $\mathcal
C^{\infty}$-maps $\Delta^p \longrightarrow M$. The corresponding functor of [*$\mathcal C^{\infty}$-singular chains*]{} $S_*^\infty:{\mathbf{Dif}}\longrightarrow\bold C _*(\mathbb
Z)$ is also a symmetric monoidal functor with the shuffle product.
On the one hand, the natural inclusion of $\mathcal
C^{\infty}$-singular chains into singular ones defines a monoidal natural transformation $S_*^\infty \Rightarrow S_* :{\mathbf{Dif}}\rightrightarrows \bf C_*(\mathbb Z), $ and from the approximation theorem it follows that it is a weak equivalence of functors. On the other hand, by Stokes’ theorem, integration along ${\mathcal{C}}^\infty$-singular simplexes induces a natural transformation $ \int:S_*^\infty
\Rightarrow\mathcal{D}'_*:{\mathbf{Dif}}\rightrightarrows \bold C_*(\mathbb R) $, which is a weak equivalence of functors by De Rham’s theorem.
So the functors of singular chains and currents are weakly equivalent. However, the natural transformation $\int$ is not compatible with the monoidal structures. To overcome this deficiency, we will use the cubic singular chains instead of the simplicial ones.
###
For a topological space $X$, cubic chains are generated by the singular cubes of $X$; that is, the continuous maps $I^p\longrightarrow X$, where $I$ is the unit interval of the real line $\mathbb R$ and $p\in \mathbb
N$, modulo the degenerate ones (see e.g. [@Mas]). The functor of [*cubic*]{} chains $$C_*(\phantom{X} ; \mathbb{Z}) : {\mathbf{Top}}\longrightarrow
{\mathbf{C}_*(\mathbb{Z})} \ ,$$ with the [*cross product*]{} $$\times\;: C_*(X ; \mathbb{Z}) \otimes C_*(Y ; \mathbb{Z})
\longrightarrow C_*(X \times Y ; \mathbb{Z}) \ ,$$ which for singular cubes $c : I^p \longrightarrow X$ and $d :
I^q \longrightarrow Y$ is defined as the cartesian product $$c \times d : I^{p+q} = I^p \times I^q \longrightarrow X \times
Y\, ,$$ is a monoidal functor.
If $M$ is a differentiable manifold, one may consider chains generated by ${\mathcal{C}}^\infty$-maps $ I^p \longrightarrow M$. The corresponding functor of [*${\mathcal{C}}^\infty$-cubic chains*]{} $C_*^\infty(\phantom{X} ; \mathbb{Z}):{\mathbf{Dif}}\longrightarrow\bold C _*(\mathbb Z)$ is also a monoidal functor.
Finally the inclusion $C^{\infty}_*\Rightarrow C_*:{\mathbf{Dif}}\longrightarrow {\mathbf{C}}_*(\mathbb Z)$ and integration $\int:C_*^{\infty}\Rightarrow \mathcal D'_*:{\mathbf{Dif}}\longrightarrow \mathbf
C_*(\mathbb R)$ are monoidal natural transformations, which are weak equivalences of monoidal functors.
In spite of this, in this case another problem arises: the monoidal functor of cubic chains is not symmetric, because the cross product is not commutative.
###
However, over $\mathbb{Q}$, or more generally over a $\mathbb
Q$-algebra, we can symmetrize cross product with the classical alternating operator. For every $n$, define the map $ {\mathrm{Alt}\,}:
C_n(X;\mathbb{Q}) \longrightarrow C_n(X;\mathbb{Q}) $ as $${\mathrm{Alt}\,}(c) = \frac{1}{n!} \sum_{\sigma \in \Sigma_n} (-1)^{|\sigma|} c \circ \sigma \ ,$$ for all singular cubes $c:I^p \longrightarrow X$. The map ${\mathrm{Alt}\,}$ is well defined because, if $c$ is a degenerate singular cube, then ${\mathrm{Alt}\,}(c)$ is a degenerate cubic chain.
The functor of cubic chains $C_*(\phantom{X} ;
\mathbb{Q}):{\mathbf{Top}}\longrightarrow
{\mathbf{C}_*(\mathbb{Q})} $, together with the [ product]{} $$\kappa_{X,Y} : C_*(X;\mathbb{Q}) \otimes C_*(Y;\mathbb{Q})
\longrightarrow C_*(X \times Y;\mathbb{Q})
\,,$$ defined by $\kappa(c\otimes d)={\mathrm{Alt}\,}(c\times d)$, is a symmetric monoidal functor.
First of all, compatibility with units is trivial. Next, compatibility with associativity and commutativity constraints is proved in a similar way to the usual proofs of the associativity and commutativity properties for the wedge product of multi-linear alternating tensors. Finally, the fact that ${\mathrm{Alt}\,}$ is a chain map is possibly a classical result, but we have not found a proof in the literature. So let us see that ${\mathrm{Alt}\,}$ is a chain map. Recall the definition of the differential $d:C_n(X;\mathbb{Q}) \longrightarrow C_{n-1}(X;\mathbb{Q})$ of the cubic chain complex. For $1\le i\le n,\ \epsilon\in\{0,1\}$, let $\delta_i^{\epsilon}:I^{n-1} \longrightarrow I^{n}$ denote the face defined by $ \delta^{\epsilon}_{i}
(t_1,\dots,t_{n-1})=(t_1,\dots,\epsilon,\dots,t_{n-1})\,\,, $ where $\epsilon$ is in the $i$-th place. Now, if $c\in C_n(X,\mathbb Q)$, $d(c)$ is defined by $$d (c)=\sum_{i,\epsilon}(-1)^{i+\epsilon}c\circ
\delta_i^{\epsilon}\,\, ,$$ and we have $$d({\mathrm{Alt}\,}(c)) =\frac{1}{n!}\sum_{i,\epsilon}\sum_{\sigma\in
\Sigma_n}(-1)^{i+\epsilon+ |\sigma|} c \circ \sigma\circ
\delta_i^{\epsilon} \ ,$$ and $${\mathrm{Alt}\,}(d c) =
\frac{1}{(n-1)!}\sum_{\tau\in \Sigma_{n-1}}\sum_{r,\epsilon}(-1)^{|\tau |+r+\epsilon}c
\circ \delta_r^{\epsilon}\circ\tau$$ Then, it is easy to check the following claim
[**Claim:**]{}
*For all $\tau\in \Sigma_{n-1},$ and $r,i\in\{1,\dots,n\}$ let $$\sigma_{\tau,r,i}:=(r,\dots,n)\circ \overline{\tau}\circ (i,\dots,n)^{-1}\,,$$ where $\overline{\tau}\in \Sigma_{n}$ is defined by $
\overline{\tau}(i)= \tau(i)$ for all $ 1\leq i < n $, and $(r,\dots,n)$ denotes the cycle $$(1,2,\dots, r-1,r,r+1,\dots,n)\mapsto (1,2\dots,r-1,r+1,\dots,n,r)\ .$$ Then $\sigma_{\tau,r,i}$ is the only permutation that satisfies $ \sigma_{\tau,r,i}\circ
\delta^\epsilon_i=\delta^\epsilon_r\circ \tau. $*
Moreover, the set map $$\begin{aligned} \Sigma_{n-1}\times
\{1,\dots,n\}\times
\{1,\dots,n\} &\longrightarrow \Sigma_n\times \{1,\dots,n\}\\(\tau,r,i)&\mapsto
(\sigma_{\tau,r,i},i)\end{aligned}$$ is bijective.
To finish checking that $ d\circ {\mathrm{Alt}\,}= {\mathrm{Alt}\,}\circ d \ $, it suffices to note that, from $
(-1)^{\vert\sigma_{\tau,r,i}\vert+i}=(-1)^{|\tau|+r} \ , $ it follows $$\frac{1}{n}\sum_{i=1}^{n}(-1)^{ |\sigma |+i+\epsilon}c \circ \sigma_{\tau,r,i}\circ
\delta_i^{\epsilon}=(-1)^{|\tau | +r+\epsilon}c \circ \delta_r^{\epsilon}\circ\tau$$ for all $\epsilon$, $r$, $\tau$.
The functor of [*$\mathcal C^{\infty}$-cubic chains*]{} $C_*^\infty(\phantom{X};\mathbb Q):{\mathbf{Dif}}\longrightarrow\bold C
_*(\mathbb Q)$ is also a symmetric monoidal functor with the corresponding product.
We will now check that integration is compatible with the monoidal structure.
Integration along $C^\infty$-cubes induces a monoidal natural transformation $$\int:C_*^\infty \Rightarrow\mathcal{D}'_*:{\mathbf{Dif}}\rightrightarrows \bold C_*(\mathbb R) ,$$ which is a weak equivalence of symmetric monoidal functors.
Let $M$ be a differentiable manifold and $c: I^p \longrightarrow
M$ a $\mathcal C^\infty$-singular cube. Integration along $c$, $\int^M_c
\omega=\int_{I^{p}}c^*(\omega)$, defines a chain map $$\int^M :C_*^\infty(M; \mathbb{R}) \longrightarrow \mathcal{D}'_*(M;\mathbb{R})$$ by $c \mapsto \int_c^M $, which is obviously natural in $M$, and a weak equivalence by De Rham’s theorem.
The natural transformation $\int^M$ is monoidal, by Fubini’s theorem and by the change of variables theorems. Indeed, for $c:I^p \longrightarrow M$ and $d: I^q \longrightarrow N$ $\mathcal C^\infty$-singular cubes, and $\omega \in
\mathcal{D}^p(M)$ and $\nu \in
\mathcal{D}^q(N)$, we have $$\begin{aligned}
\int^{M\times
N}_{\kappa (c\otimes d)} \pi^*_M (\omega) \wedge
\pi^*_N(\nu)\notag
&=\frac{1}{(p+q)!} \sum_{\sigma
\in
\Sigma_{p+q}}(-1)^{|\sigma |}
\int_{(c\times d)\circ \sigma}^{M\times N} \pi^*_M (\omega)
\wedge \pi^*_N(\nu) \notag\\
&= \int_{c\times d}^{M\times N}\! \pi^*_M (\omega)
\wedge \pi^*_N(\nu) \notag\\
& = \left(\int_c^M \pi^*_M (\omega ) \right)
\left(\int_{d}^N \pi^*_N (\nu ) \right) \notag\\
& = \left(\kappa \circ \left(\int^M _c\otimes \int^N _d\right)
\right)(\pi^*_M (\omega) \wedge
\pi^*_N(\nu))\ .\notag\end{aligned}$$
To sum up, we can state the following symmetric version of De Rham’s theorem
\[cubicchainswecurrents\] The functors of cubic chains and currents $$C_*,\mathcal
D'_*:{\mathbf{Dif}}\rightrightarrows {\mathbf{C}}_*(\mathbb R)$$ are weakly equivalent symmetric monoidal functors.
A similar result can be obtained with the functor of [*oriented cubic chains* ]{} $C_*^{\mathrm{or}}$, used by Kontsevich (see [@Ko], (2.2)), which, with the cross product, is a monoidal symmetric functor from ${\mathbf{Top}}$ to ${\mathbf{C}_*(\mathbb{Z})}$. It can be proved using the alternating operator that, at least over $\mathbb{Q}$, computes the usual homology of a topological space. This solution is equivalent to the previous one, because the natural projection $
C_*(\phantom{X};\mathbb Q) \longrightarrow
C_*^{\mathrm{or}}(\phantom{X};\mathbb Q)$ is a monoidal natural transformation, which is a weak equivalence of symmetric monoidal functors.
The above results can be seen as analogous in the covariant setting to the following well known statements. The contravariant functor of singular cochains $S^*:{\mathbf{Top}}^{op}\longrightarrow {\mathbf{C}}^*(\mathbb Z)$, together with the Alexander-Whitney product, is a monoidal contravariant functor. Although the functor of ${\mathcal{C}}^\infty$-singular cochains $S^*_\infty:{\mathbf{Dif}}^{op}{\longrightarrow}{\mathbf{C}}^*(\mathbb R)$ is weakly equivalent to the symmetric monoidal functor of differential forms $\mathcal E^*:{\mathbf{Dif}}^{op}\longrightarrow {\mathbf{C}}^*(\mathbb
R)$, the monoidal functor $S^*_\infty$ is symmetric only up to homotopy. $\mathcal E^*$ can be topologically defined as a symmetric monoidal functor using the Sullivan’s $\mathbb
Q$-cdga $Su_\mathbb Q$ of simplicial differential forms, which defines a symmetric monoidal contravariant functor ${\mathbf{Top}}^{op}\longrightarrow {\mathbf{C}}^*(\mathbb Q)$, together with a weak equivalence of symmetric monoidal functors $\mathcal
E^*\Rightarrow Su_\mathbb R:{\mathbf{Dif}}\longrightarrow
{\mathbf{C}}^*(\mathbb R)$.
Formality
---------
###
The notion of formality has attracted interest since Sullivan’s work on rational homotopy theory. In the operadic setting the notion of formality appears in [@Mkl] and [@Ko].
An operad $P$ in ${{\mathbf{C}_*({\mathcal{A}})}}$ is said to be [*formal*]{} if it is weakly equivalent to its homology $HP$.
More generally, we can give the following definition
Let $\mathcal C$ be a category endowed with an idempotent endofunctor $H:\mathcal C{\longrightarrow}\mathcal C$, and take as weak equivalences the morphisms $f:X{\longrightarrow}Y$ such that $H(f)$ is an isomorphism. An object $X$ of $\mathcal C$ is said to be [*formal*]{} if $X$ and $HX$ are weakly equivalent.
###
In particular, a functor $F: \mathcal{C} \longrightarrow
{{\mathbf{C}_*({\mathcal{A}})}}$ is [*formal*]{} if it is weakly equivalent to its homology $HF: \mathcal{C}
\longrightarrow {{\mathbf{C}_*({\mathcal{A}})}}$. However, we will use this notion in the context of symmetric monoidal functors. So, the definition of formality in this case is
Let $\mathcal C$ be a symmetric monoidal category, and $F: \mathcal{C} \longrightarrow
{{\mathbf{C}_*({\mathcal{A}})}}$ a symmetric monoidal functor. It is said that $F$ is a [*formal symmetric monoidal functor*]{} if $F$ and $HF$ are weakly equivalent in the category of symmetric monoidal functors.
If the identity functor of ${{\mathbf{C}_*({\mathcal{A}})}}$ is a formal symmetric functor, ${{\mathbf{C}_*({\mathcal{A}})}}$ is said to be a [*formal symmetric monoidal category*]{}.
Let $\mathcal B$ be a symmetric monoidal subcategory of ${{\mathbf{C}_*({\mathcal{A}})}}$. If the inclusion functor $\mathcal B{\longrightarrow}{{\mathbf{C}_*({\mathcal{A}})}}$ is a formal symmetric monoidal functor, we will say that $\mathcal B$ is a [*formal symmetric monoidal subcategory of ${{\mathbf{C}_*({\mathcal{A}})}}$*]{}.
The properties below follow immediately from the definitions.
Let $\mathcal B$ be a formal symmetric monoidal subcategory of ${{\mathbf{C}_*({\mathcal{A}})}}$. If $F:{\mathcal{C}}\longrightarrow {{\mathbf{C}_*({\mathcal{A}})}}$ is a monoidal functor with values in the subcategory $\mathcal B$, then $F$ is a formal symmetric monoidal functor.
\[formalfunctorscreateformaloperads\] Let $F: \mathcal{C}
\longrightarrow {{\mathbf{C}_*({\mathcal{A}})}}$ be a functor. If $F$ is a formal symmetric monoidal functor, then $$F: {\mathbf{Op}}_{\mathcal{C}} \longrightarrow {\mathbf{Op}}_{{{\mathbf{C}_*({\mathcal{A}})}}}$$ sends operads in $\mathcal{C}$ to formal operads in ${{\mathbf{C}_*({\mathcal{A}})}}$.
###
Let $R$ be a commutative ring, and $R-\mathbf{cdga}$ the category of differential graded-commutative $R$-algebras, or simply cdg $R$-algebras. It is a symmetric monoidal category. Then, if $\mathcal C$ a symmetric monoidal category and $F: \mathcal{C}
\longrightarrow \mathbf C^*(R)$ is a symmetric monoidal functor, it is a well known fact that $F$ induces a functor from the category of commutative monoids of $(\mathcal
C,\otimes,\mathbf 1)$ to cdg $R$-algebras.
Besides, if $F$ is a formal symmetric monoidal functor, then $F$ sends commutative monoids to formal cdg $R$-algebras.
If $\mathcal C$ is a category with finite products, and a final object $\mathbf 1$, then $(\mathcal C,\times,\mathbf 1)$, is a symmetric monoidal category. In this case every object $X$ of $\mathcal C$ is a comonoid object with the diagonal $X\rightarrow X\times X$ and the unit $X\rightarrow \mathbf 1$. So we have
\[formalidaddealgebras\] Let $\mathcal C$ be a category with finite products, and a final object $\mathbf 1$. Every formal symmetric monoidal contravariant functor $F:\mathcal{C}^{op}
\longrightarrow \mathbf C^*(R) $ sends objects in $\mathcal C$ to formal cdg $R$-algebras.
Hodge theory implies formality
==============================
In [@DGMS], Deligne et al. prove the formality of the De Rham cdg algebra of a compact K[ä]{}hler manifold. In this section we will see how this result can be mimicked for the cubic chain complex of an operad of compact K[ä]{}hler manifolds.
Formality of De Rham’s functor
------------------------------
In [[@DGMS]]{}, the first of the proofs of formality th. 5.22 relies on the [*Hodge decomposition*]{} for the complex of forms and the K[ä]{}hler identities. From them, the $dd^c$-[*lemma*]{} is proved and the existence of a diagram of complexes, called [*$d^c$-diagram*]{}, $$(\mathcal{E}^*(M),d) \longleftarrow \ (^c\mathcal{E}^*(M),d)
\longrightarrow (H^*_{d^c}(M),d) \ ,$$ is deduced. Here $M$ is a compact K[ä]{}hler manifold, $\mathcal{E}^*(M)$ is the real De Rham complex of $M$, $\ ^c\mathcal{E}^*(M)$ the subcomplex of $d^c$-closed forms, and $H_{d^c}^*(M)$ the quotient complex $\
^c\mathcal{E}^*(M) / d^c(\mathcal{E}^*(M))$. In the $d^c$-diagram, both maps are weak equivalences of chain complexes, and the differential induced by $d$ on $H_{d^c}^*(M)$ is zero. Since $\ ^c\mathcal{E}^*$ is a symmetric monoidal subfunctor of $\mathcal{E}^*$ and the morphisms of this diagram are natural, the functor $\mathcal{E}^*$ is formal. So the theorem of formality can also be stated with the previous definitions as follows
The functor of differential forms $\mathcal{E}^* : {\mathbf{K\ddot{a}h}}^{op} \longrightarrow \mathbf
C^*(\mathbb{R})$ is a formal symmetric monoidal functor.
This result, together with prop. \[formalidaddealgebras\], implies the formality theorem for De Rham’s cdg-algebra in its usual formulation: The De Rham functor $\mathcal{E}^*: {\mathbf{K\ddot{a}h}}^{op} \longrightarrow \mathbb R -\mathbf{cdga}$ sends objects in ${\mathbf{K\ddot{a}h}}$ to formal cdg $\mathbb R$-algebras.
Formality of the current complex functor
----------------------------------------
We claim that an analogous theorem of formality is obtained replacing forms with currents.
\[curretsformalfunctor\] The functor of currents $\mathcal{D}'_* : {\mathbf{K\ddot{a}h}}\longrightarrow {\mathbf{C}_*(\mathbb{R})}$ is a formal symmetric monoidal functor.
Let $M$ be a compact Kähler manifold. It is a classical result of Hodge theory (see [@Sch]) that the K[ä]{}hler identities between the operators $d, d^c, \Delta ,\dots $ of the De Rham complex of differential forms are also satisfied by the corresponding dual operators on De Rham complex of currents. Hence we have the following $dd^c$-lemma.
Let $T$ be a $d^c$-closed and $d$-exact current. Then, there exists a current $S$ such that $T = dd^cS$.
From this lemma, we can follow verbatim the first proof of theorem 5.22 ([*the $d^c$-Diagram Method*]{}) in [@DGMS] and we obtain a $d^c$-diagram for currents: $$(\mathcal{D}'_*(M),d) \longleftarrow \ (^c\mathcal{D}'_*(M),d)
\longrightarrow (H^{d^c}_*(M),d) \ .$$ Here $\ ^c\mathcal{D}'_*(M)$ denotes the subcomplex of $\mathcal D'_*(M)$ defined by the $d^c$-closed currents, and $H^{d^c}_*(M)$ is the quotient $\ ^c\mathcal{D}'_*(M) / d^c
(\mathcal{D}'_*(M))$. In this $d^c$-diagram both maps are weak equivalences, and the differential induced by $d$ on the latter is zero. So we have $H_*^{d^c}(M)\cong H_*(\mathcal
D'_*(M)$.
Now, since $d^c$ satisfies the Leibnitz rule, $\
^c\mathcal{D}'_*$ is a symmetric monoidal subfunctor of $\mathcal D'_*$ .
Finally, since the morphisms of the above $d^c$-diagram are natural and compatible with the Künneth morphism, it follows that $\mathcal{D}'_* $ is a formal symmetric monoidal functor.
As a consequence of the formality of the current functor and the symmetric De Rham theorem for currents (th. \[cubicchainswecurrents\]), the formality of the cubic chains functor for compact Kähler manifolds follows.
\[cadenesformals\] The functor of cubic chains $C_*(\phantom{X}; \mathbb R): {\mathbf{K\ddot{a}h}}\longrightarrow
{\mathbf{C}_*(\mathbb{R})}$ is a formal symmetric monoidal functor.
Formality of Kählerian operads {#hodge}
------------------------------
From \[formalfunctorscreateformaloperads\] and \[cadenesformals\] we obtain the operadic version of the formality DGMS theorem.
\[cadenesformalsR\] If $X$ is an operad in ${\mathbf{K\ddot{a}h}}$, then the operad of cubic chains $C_*({X}; \mathbb R)$ is formal.
This result, together with prop. \[formalfunctorscreateformaloperads\] and the descent theorem th. \[descens\] below, implies the formality of the operad of cubic chains with rational coefficients for every operad of compact Kähler manifolds (see cor. \[cadenesformalsQ\] below).
Formality of $DM$-operads
-------------------------
The above results can be easily generalized to the category of [*Deligne-Mumford projective and smooth stacks*]{} over $\mathbb C$, which we will denote by ${\mathbf{DMV}(\mathbb{C})}$.
Indeed, every stack of this kind defines a compact Kähler $V$-manifold and for such $V$-manifolds we have the functors of cubic chains, ${\mathcal{C}}^\infty$-cubic chains and currents, and also Hodge theory (see [@Ba]). This allows us to obtain an analogous result to cor. \[cadenesformals\]:
\[hodgeforVman\] The functor of cubic chains $C_*(\phantom{X};\mathbb R): {\mathbf{DMV}(\mathbb{C})}\longrightarrow {\mathbf{C}_*(\mathbb{R})}$ is a formal symmetric monoidal functor.
And, from \[formalfunctorscreateformaloperads\] follows
\[Coperadformal\] If $X$ is an operad in ${\mathbf{DMV}(\mathbb{C})}$, then the operad of cubic chains $C_*(X;\mathbb{R})$ is formal.
Minimal operads
===============
In this section $\mathbf{k}$ will denote a field of characteristic zero, and an operad will be an operad in the category of dg vector spaces over $\mathbf {k}$, ${\mathbf{C}}_*(\mathbf k)$. The category of these operads is denoted simply by ${\mathbf{Op}}$. It is a complete and cocomplete category (see [@Hi]).
Some preliminaries
------------------
Let us start by recalling some basic results on minimal operads due to M. Markl ([@Mkl], see [@MSS]).
###
A [*minimal operad*]{} is an operad of the form $(\Gamma
(V),d_M)$, where $\Gamma:{\mathbf{\Sigma Mod}}{\longrightarrow}{\mathbf{Op}}$ is the free operad functor, $V$ is a $\Sigma$-module with zero differential with $V(1)=0$, and the differential $d_M$ is [*decomposable*]{}.
The free operad functor $\Gamma:{\mathbf{\Sigma Mod}}{\longrightarrow}{\mathbf{Op}}$ is a right adjoint functor for the forgetful functor $U:{\mathbf{Op}}{\longrightarrow}{\mathbf{\Sigma Mod}}$.
A [*minimal model*]{} of an operad $P$ is a minimal operad $P_\infty$, together with a weak equivalence $P_{\infty}
\longrightarrow P$.
Let $P=\left(P(l)\right)_{l\ge 1}$ be an operad. M. Markl has proved that, if $HP(1) =
\mathbf{k}$, $P$ has a minimal model $P_\infty $ with $P_\infty(1)=\mathbf{k}$ ([@MSS], th. 3.125).
As observed in [@MSS], remark II.1.62, the category of operads $P$ with $P(1) =
\mathbf{k}$ is equivalent to the category of [*pseudo-operads* ]{} $Q$ with $Q(1)=
\mathbf{0}$, the zero dg vector space (see [*op. cit.*]{} def. II.1.16).
[*In the sequel, we will work only with pseudo-operads, with $HP(1)=0$, and we will call them simply operads. We will denote by ${\mathbf{Op}}$ the category of these operads, and $$\circ_i:P(l)\otimes P(m){\longrightarrow}P(l+m-1),\quad 1\le i\le l,$$ their composition operations*]{}.
Truncated operads {#Truncatedoperads}
-----------------
We will now introduce the arity truncation and their right and left adjoints, which enables us to introduce in the operadic setting the analogs of the skeleton and coskeleton functors of simplicial set theory.
Here we establish the results for the arity truncation in a form that can be easily translated to modular operads in §$8$.
### {#propiedadideal}
Let $E=\left(E(l)\right)_{l\ge 1}$ be a $\Sigma$-module, and $n\ge 1$ an integer. The grading of $E$ induces a decreasing filtration $\left(E(\ge \!l)\right)_{l\ge 1}$, by the sub-$\Sigma$-modules $$E(\ge\! l):=\left(E(i)\right)_{i\ge l}\, .$$
Let $P$ be an operad. We will denote by $P\cdot P(n)$ the sub-$\Sigma$-module consisting of elements $\alpha\circ_i\beta$ with $\alpha\in P(l)$ and $\beta\in
P(m)$, such that at least one of $l,m$ is $n$. It follows from the definitions that $$P\cdot P(n)\subset P(\ge\! n).$$ If, moreover, $P(1)=0$, then $$P\cdot P(n)\subset P(\ge\! n\!+\!1\!).$$ The first property implies that $P(\ge \!n)$ is an ideal of $P$, so the quotient $P/P(\ge\! n\!+\!1\!)$ is an operad, which is zero in arities $>n$. This is a so-called $n$-truncated operad. However, we find it more natural to give the following definition of $n$-truncated operad.
A [*$n$-truncated operad*]{} is a finite sequence of objects in ${\mathbf{C}}(\mathbf k)$, $$P=(P(1),\dots,P(n)),$$ with a right $\Sigma_l$-action on each $P(l)$, together with a family of composition operations, satisfying those axioms of composition operations in ${\mathbf{Op}}$ that make sense for truncated operads. A [*morphism of $n$-truncated operads*]{} $f: P
\longrightarrow Q$ is a finite sequence of morphisms of $\Sigma_l$-modules $f(l) : P(l) \longrightarrow Q(l) , \ 1\le
l\le n$, which commute with composition operations.
Let ${{\mathbf{Op}}(\leq\! n)}$ denote the category of $n$-[*truncated operads*]{} of ${\mathbf{C}}_*(\mathbf k)$.
A [*weak equivalence*]{} of $n$-truncated operads is a morphism of $n$-truncated operads $\phi : P \longrightarrow Q$ which induces isomorphisms of graded $\mathbf{k}$-vector spaces, $H\phi (l) : HP(l) \longrightarrow HQ(l)$, for $l = 1,
\dots , n$.
Given an operad $P$, $t_nP:=(P(1),\dots, P(n))$ defines a [*truncation functor*]{} $$t_{n}:{\mathbf{Op}}\longrightarrow {{\mathbf{Op}}(\leq\! n)}\ .$$
### {#section-18}
For a $n$-truncated operad $P$ denote by $t_*P$ the $\Sigma$-module that is $\mathbf 0$ in arities $>n$ and coincides with $P$ in arities $\le n$. Since $P\cdot P(n)\subset
P(\ge\! n)$, $t_*P$ together with the structural morphisms of $P$ trivially extended, is an operad, and the proposition below follows easily from the definitions
\[propiedadest\*\] Let $n\ge 1$ be an integer. Then
1. $t_{*}:{\mathbf{Op}}(\le \! n){\longrightarrow}{\mathbf{Op}}$ is a right adjoint functor for $t_{n}$.
2. There exists a canonical isomorphism $t_{n}\circ t_{*}\cong Id_{{\mathbf{Op}}(\le n)}$.
3. $t_{*}$ is a fully faithful functor.
4. $t_{*}$ preserves limits.
5. For $m\ge n$, there exists a natural morphism $$\psi_{m,n}:t_*t_{m} {\longrightarrow}t_{*}t_{n}$$ such that $\psi_{l,n}=\psi_{m,n}\circ\psi_{l,m}$, for $l\ge
m\ge n$. For an operad $P$, the family $\left(t_{*}t_{
n}P\right)_{n}$, with the morphisms $\psi_{m,n}$, is an inverse system of operads. The family of unit morphisms of the adjunctions $$\psi_n:P{\longrightarrow}t_{*}t_{n}P,$$ induces an isomorphism $\psi:P{\longrightarrow}\lim\limits_{\leftarrow}t_{*}t_{n}P$.
6. Let $P, Q$ be operads. If $n\ge 2$, $t_{n-1}P=0$, and $Q\cong
t_{*}t_{n}Q$, then $${\mathrm{Hom}}_{{\mathbf{Op}}}(P,Q)\cong {\mathrm{Hom}}_{\Sigma_n}(P(n),Q(n)).$$
### {#section-19}
On the other hand, the functor $t_{n}$ also has a left adjoint. For a $n$-truncated operad $P$, denote by $t_{!}P$ the operad obtained freely adding to $P$ the operations generated in arities $> n$, that is, $$t_{!}P=\Gamma(Ut_*P)/J\ ,$$ where $J$ is the ideal in $\Gamma (Ut_*P)$ generated by the kernel of $ t_{
n}\Gamma(Ut_*P){\longrightarrow}P$.
\[propiedadest!\] Let $n\ge 1$ be an integer. Then
1. $t_{!}:{\mathbf{Op}}(\le n){\longrightarrow}{\mathbf{Op}}$ is a left adjoint functor for $t_{n}$.
2. There exists a canonical isomorphism $t_{n}\circ
t_{!}\cong Id_{Op(\le n)}$.
3. $t_{!}$ is a fully faithful functor.
4. $t_{!}$ preserves colimits.
5. For $m\le n$, there exists a natural morphism $$\phi_{m,n}:t_{!}t_{m} {\longrightarrow}t_{!}t_{n}$$ such that $\phi_{l,n}=\phi_{m,n}\circ\phi_{l,m}$, for $l\le m\le n$. For an operad $P$, the family $\left(t_{!}t_{n}P\right)_{n}$, with the morphisms $\phi_{m,n}$, is a directed system of operads. The family of unit morphisms of the adjunctions $$\phi_n:t_{!}t_{n}P{\longrightarrow}P\,,$$ induces an isomorphism $\phi:
\lim\limits_{\rightarrow}t_{!}t_{n}P{\longrightarrow}P$.
6. Let $P, Q$ be operads. If $t_{n-1}Q=0$, $P\cong
t_{!}t_{n}P$, and $P(1)=0$, then $${\mathrm{Hom}}_{{\mathbf{Op}}}(P,Q)\cong {\mathrm{Hom}}_{\Sigma_n }(P(n),Q(n)).$$
Part (1) follows from the definition of $t_!$, and the remaining parts follow from (1) and prop. \[propiedadest\*\].
We will call the direct system of operads given by $$0\rightarrow t_{!}t_{1}P\rightarrow\cdots \rightarrow t_{!}t_{n-1} P \rightarrow
t_{!}t_{n}P\rightarrow\cdots$$ the [*canonical tower*]{} of $P$.
As an easy consequence of the existence of right and left adjoint functors for $t_{n}$ we obtain the following result.
The truncation functors $t_{n}$ preserve limits and colimits. In particular, they commute with homology, send weak equivalences to weak equivalences, and preserve formality.
Principal extensions
--------------------
Next, we recall the definition of a principal extension of operads and show that the canonical tower of a minimal operad is a sequence of principal extensions. This will allow us to extend these notions to the truncated setting.
### {#section-20}
To begin with, we establish some notations on suspension and mapping cones of complexes in an additive category.
If $A$ is a chain complex and $n$ is an integer, we denote by $A[n]$ the complex defined by $A[n]_i = A_{i-n}$ with the differential given by $d_{A[n]}=(-1)^n d_A$.
For a chain map $\eta: B\longrightarrow A$ we will denote by $C\eta$, or by $A\oplus_\eta B[1]$, the mapping cone of $\eta$, that is to say, the complex that in degree $i$ is given by $(C\eta)_i=A_i\oplus B_{i-1}$ with the differential $d(a,b)=(d_Aa +\eta b, -d_Bb)$. Therefore $ C\eta$ comes with a canonical chain map $i_A:A{\longrightarrow}C\eta$ and a canonical homogeneous map of graded objects $j_B:B[1]{\longrightarrow}C\eta$.
For a chain complex $X$, a chain map $\phi :
C\eta\longrightarrow X$ is determined by the chain map $\phi
i_A:A\longrightarrow X$ together with the homogeneous map $\phi j_B :B[1]\longrightarrow X$. Conversely, if $f:A{\longrightarrow}X$ is a chain map and $g:B[1]{\longrightarrow}X$ is a homogeneous map such that $f\eta = d_Xg + gd_B$, that is, $g$ is a homotopy between $f\eta$ and $0$, then there exists a unique chain map $\phi:C\eta{\longrightarrow}X$ such that $\phi i_A=f$ and $\phi j_B=g$. In other words, $C\eta$ represents the functor $h_{\eta}:{\mathbf{C}}_*(\mathbf k){\longrightarrow}\mathbf{Sets}$ defined, for $X\in
{\mathbf{C}}_*(\mathbf k)$, by $$h_{\eta}(X)= \{(f,g); \ f\in {\mathrm{Hom}}_{{\mathbf{C}}_*(\mathbf k)}(A, X)\ ,\ \, g\in {\mathrm{Hom}}_{\mathbf
k}(B, X)_{1}\ , \ d_Xg+gd_B=f\eta\ \},$$ where ${\mathrm{Hom}}_{\mathbf k}(B, X)_1$ denotes the set of homogeneous maps of degree $1$ of graded $\mathbf k$-vectorial spaces.
### {#section-21}
Recall the construction of standard cofibrations introduced in ([@Hi]). Let $P$ be an operad, $V$ a dg $\Sigma$-module and $\xi :V[-1] \longrightarrow P$ a chain map of dg $\Sigma$-modules. The standard cofibration associated to these data, denoted by $P\!\langle\!V,\xi\!\rangle $ in [@Hi], is an operad that represents the functor $h_{\xi}:{\mathbf{Op}}{\longrightarrow}\mathbf{Sets}$ defined, for $Q\in
{\mathbf{Op}}$, by $$h_{\xi}(Q)= \{(f,g);\ f\in {\mathrm{Hom}}_{{\mathbf{Op}}}(P, Q)\ ,\, g\in {\mathrm{Hom}}_{\bf Gr\Sigma Mod}(V,
UQ)_{0}\ , \ d_Qg-gd_V=f\xi\ \},$$ where ${\mathrm{Hom}}_{\bf Gr\Sigma Mod}(V, UQ))_0$ denotes the set of homogeneous maps of degree $0$ of graded $\Sigma$-modules. When $V$ has zero differential, this construction is called a [*principal extension*]{} and denoted by $P*_\xi \Gamma(V)$, see [@MSS]. For reasons that will become clear at once, we will denote it by $ P\sqcup_\xi V$. From the definition it follows that $ P\sqcup_\xi V$ comes with a canonical morphism of operads $i_P:P{\longrightarrow}P\sqcup_\xi V$ and a canonical homogeneous map of degree $0$ of graded $\Sigma$-modules $j_V:V{\longrightarrow}P\sqcup_\xi V$.
Now, one can express $P\sqcup_\xi V$ as a push-out. Let $C(V[-1])$ be the mapping cone of id$_{V[-1]}$, $S(V)=\Gamma(V[-1])$, $T(V)=\Gamma(C(V[-1]))$ , $i_V:S(V){\longrightarrow}T(V)$ the morphism of operads induced by the canonical chain map $i:V[-1]{\longrightarrow}C(V[-1])$, and $\widetilde\xi:S(V){\longrightarrow}P$ the morphism of operads induced by $\xi$. Then $P\sqcup_\xi V$ is isomorphic to the push-out of the following diagram of operads $$\begin{CD}
S(V) @>\widetilde\xi>> P \\
@Vi_{V}VV@.\\
T(V)\ .@.
\end{CD}$$ If $V$ is concentrated in arity $n$, and its differential is zero, the operad $P\sqcup_\xi V$ is called an [*arity $n$ principal extension*]{}.
Let us explicitly describe it in the case that $n\ge 2$, $P(1)=0$, and $P\cong
t_{!}t_{n}P$. First of all, since, for a truncated operad $Q\in {\mathbf{Op}}(\le n-1)$, there is a chain of isomorphisms $$\begin{aligned}\label{}
{\mathrm{Hom}}(t_{n-1}(P\sqcup_\xi V),Q)&\cong {\mathrm{Hom}}(P\sqcup_\xi V, t_{*}Q)
\\
&\cong {\mathrm{Hom}}(P,t_{*}Q)
\\
&\cong {\mathrm{Hom}}(t_{n-1}P,Q),
\end{aligned}$$ we have $t_{n-1}(P\sqcup_\xi V)\cong t_{n-1}P$. Next, let $X$ be the $n$-truncated operad extending $t_{n}P$ defined by $$X(i)=\left\{
\aligned \label{}&P(i),\hskip 13mm \text{ if } i<n,\\
&P( n)\oplus_\xi V,\hskip 3mm \text{ if } i=n,\endaligned
\right.$$ the composition operations involving $V$ being trivial, because $P(1)=0$. Then, it is clear that $X$ represents the functor $h_{\xi}$ restricted to the category ${\mathbf{Op}}(\le n)$, so $t_{n}\left(P\sqcup_\xi V\right)\cong X$. Finally, it is easy to check that $t_{!}X$ satisfies the universal property of $P\sqcup_\xi V.$ Summing up, we have proven:
\[estructuradeunaextensionprincipal\]Let $n\ge 2$ an integer. Let $P$ be an operad such that $P(1)=0$ and $t_!t_{n}P\cong P$, $V$ a dg $\Sigma$-module concentrated in arity $n$ with zero differential, and $\xi:V[-1]{\longrightarrow}P(n)$ a chain map of $\Sigma_n$-modules. The principal extension $P\sqcup_\xi V$ satisfies:
(1)$t_{n-1}(P\sqcup_\xi V)\cong t_{n-1}P$.
(2)$(P\sqcup_\xi V)(n)\cong C(\xi),$ in particular, there exists an exact sequence of complexes $$0{\longrightarrow}P(n){\longrightarrow}(P\sqcup_\xi V)(n){\longrightarrow}V{\longrightarrow}0.$$ (3) $P\sqcup_\xi V\cong t_{!}t_{n}(P\sqcup_\xi V) $.
\(4) A morphism of operads $\phi:P\sqcup_\xi V{\longrightarrow}Q$ is determined by a morphism of $n$-truncated operads $f:t_nP{\longrightarrow}t_nQ$, and a homogeneous map of $\Sigma_n$-modules $g:V{\longrightarrow}Q(n)$, such that $f\xi =dg$.
These results extend trivially to truncated operads.
Minimal objects
---------------
Now we can translate the definition of minimality of operads of dg modules in terms of the canonical tower:
An operad $M$ is minimal if, and only if, $M(1)=0$ and the canonical tower of $M$ $$0=t_{!}t_{1}M\rightarrow\cdots \rightarrow t_{!}t_{n-1}M
\rightarrow t_{!}t_{n}M\rightarrow\cdots$$ is a sequence of principal extensions.
Let $M = (\Gamma (V),d_M)$ be a minimal operad. Then (see [@MSS], formula II.(3.89)) $$t_{!}t_{n}M\cong(\Gamma(V(\le n)),\partial_n),$$ and $t_{!}t_{n}M$ is an arity $n$ principal extension of $t_{!}t_{n-1}M$ defined by $\partial_n:V(n){\longrightarrow}\left(
t_{!}t_{n-1}M\right)(n)$.
Conversely, let us suppose $M(1)=0$, and that $t_{!}t_{n-1}M{\longrightarrow}t_{!}t_{n}M$ is an arity $n$ principal extension defined by a $\Sigma$-module $V(n)$ concentrated in arity $n$ and zero differential, for each $n$. Then, $M=\Gamma(\bigoplus_{n\ge\! 2}V( n))$ and its differential is decomposable, because $M(1)=0$. So $M$ is a minimal operad.
### {#section-22}
We now give the definition of minimality for truncated operads.
For $m\leq n$ we have an obvious truncation functor $$t_{m}:{{\mathbf{Op}}(\leq\! n)} \longrightarrow
{{\mathbf{Op}}(\leq\! m)},$$ which has a right adjoint $t_{*}$ and a left adjoint $t_{!}$.
A $n$-truncated operad $M$ is said to be [*minimal*]{} if $M(1)=0$ and the canonical tower $$0=t_!t_{1}M{\longrightarrow}t_!t_{2}M{\longrightarrow}\cdots {\longrightarrow}t_!t_{n-1}M{\longrightarrow}M$$ is a sequence of ($n$-truncated) principal extensions.
An operad $M$ is said to be [*$n$-minimal*]{} if the truncation $t_{n}M$ is minimal.
It follows from the definitions that an operad $M$ is $n$-minimal if, and only if, $t_{!}t_{n}M$ is minimal. It is clear that an operad $M$ is minimal if and only if it is $n$-minimal for every $n$, and that theorems 3.120, 3.123 and 3.125 of [@MSS] remain true in ${{\mathbf{Op}}(\leq\! n)}$, merely replacing minimal“ by $n$-minimal”.
### {#section-23}
The category ${\mathbf{Op}}$ has a natural structure of closed model category ([@Hi]). For our present purposes, we will not need all the model structure, only a small piece: the notion of homotopy between morphisms of operads and the fact that minimal operads are cofibrant objects in ${\mathbf{Op}}$; this can be developed independently, as in [@MSS], II.3.10. From these results the next one follows easily.
\[almendruco\] Let $M$ be a minimal operad and $P$ a suboperad. If the inclusion $P
\hookrightarrow M$ is a weak equivalence, then $P = M$.
Let us call $i : P \hookrightarrow M$ the inclusion. By [@MSS], th. II.3.123, we can lift, up to homotopy, the identity of $M$ in the diagram below $$\begin{CD}
@. P \\
@. @VV i V \\
M @>\mathrm{id}>> M \ .
\end{CD}$$ So we obtain a morphism of operads $f : M \longrightarrow P $ such that $i f$ is homotopic to $\mathrm{id}$. Hence $if$ is a weak equivalence and, by [@MSS], prop. II.3.120, it is an isomorphism. Therefore $i$ is an isomorphism too.
Automorphisms of a formal minimal operad
----------------------------------------
For an operad $P$, let ${\mathrm{Aut}}(P)$ denote the group of its automorphisms. The following lifting property from automorphisms of the homology of the operad to automorphisms of the operad itself is the first part of the characterization of formality that we will establish in th. \[aixeca\].
\[ooth\] Let $M$ be a minimal operad. If $M$ is formal, then the map $H: {\mathrm{Aut}}(M) \longrightarrow {\mathrm{Aut}}(HM)$ is surjective.
Because $M$ is a formal operad, we have a sequence of weak equivalences $$M \longleftarrow X_1{\longrightarrow}X_2{\longleftarrow}\cdots {\longrightarrow}X_{n-1}{\longleftarrow}X_{n}{\longrightarrow}HM \, .$$ By the lifting property of minimal operads ([@MSS], th. II.3.123) there exists a weak equivalence $$\rho: M \longrightarrow HM \ .$$ Let $\phi\in {\mathrm{Aut}}(HM)$. Again by the lifting property of minimal operads, given the diagram
$$\begin{CD}
@. M \\
@. @VV \rho V \\
M @>(H\rho) \phi (H\rho)^{-1} \rho>> HM \ .
\end{CD}$$ there exists a morphism $f : M \longrightarrow M$ such that $\rho f$ is homotopic to $(H\rho) \phi (H\rho)^{-1}\rho$. Since homotopic maps induce the same morphism in homology, it turns out that $f$ is a weak equivalence and, by ([@MSS] II.th.3.120), it is also an isomorphism, because $M$ is minimal. Finally, from $(H\rho)(H f )= (H\rho)\phi
(H\rho)^{-1}(H\rho)$, $Hf= \phi$ follows.
It is clear that prop. \[ooth\] remains true in ${{\mathbf{Op}}(\leq\! n)}$, merely replacing minimal“ and formal” by $n$-minimal“ and $n$-formal”, respectively.
Finiteness of the minimal model
-------------------------------
In this section we will show that we can transfer the finiteness conditions of the homology of an operad to the finiteness conditions of its minimal model.
A $\Sigma$-module $V$ is said to be of [*finite type*]{} if, for every $l$, $V(l)$ is a finite dimensional $\mathbf{k}$-vector space. An operad $P$ is said to be [*of finite type*]{} if the underlying $\Sigma$-module, $UP$, is of finite type.
If $V$ is a $\Sigma$-module of finite type such that $V(1)=0$, then the free operad $\Gamma(V)$ is of finite type, because $$\Gamma(V)(n)\cong\bigoplus_{T\in \mathcal{T}{ree}(n)} V(T),$$ where $ \mathcal{T}{ree}(n)$ is the finite set of isomorphism classes of $n$-labelled reduced trees, and $V(T)=\bigotimes_{v\in\text {Vert}(T)}V(\text{In}(v))$, for every $n$-labelled tree $T$, (see [@MSS], II.1.84). In particular, if $P$ is a $n$-truncated operad of finite type, then $t_{!}P\cong\Gamma(P)/J$ (see prop. \[propiedadest!\]) is of finite type as well.
\[finitudob\] Let $P$ be an operad. If the homology of $P$ is of finite type, then every minimal model $P_{\infty}$ of $P$ is of finite type.
Let $M$ be a minimal operad such that $HM$ is of finite type. Since $$M(n)=(t_{!}t_{n}M)(n),$$ it suffices to check that $t_{!}t_{n}M$ is of finite type. We proceed by induction. The first step of the induction is trivial because $M(1)=0$. Then, $t_{!}t_{n}M$ is an arity $n$ principal extension of the operad $t_{!}t_{n-1} M$ by the vector space $$V(n)=
HC\left(\left(t_{!}t_{n-1}M\right)(n)\rightarrow
M(n)\right),$$ thus $t_{n}\left(t_{!}t_{n}M\right)$ is finite dimensional, by the induction hypothesis. Therefore $t_{!}t_{n}M=t_{!}\left(t_{n}t_!t_{n}M\right)$ is also of finite type, by the previous example.
Weight theory implies formality
===============================
In analogy to the formality theorem of [@DGMS] for the rational homotopy type of a compact Kähler manifold, Deligne ([@D]) proved formality of the “${\overline{\mathbb Q_{\l}}}$-homotopy type" of a smooth projective variety defined over a finite field using the weights of the Frobenius action in the $\l$-adic cohomology and his solution of the Riemann hypothesis. In this section we follow this approach and introduce weights to establish a criterion of formality for operads based on the formality of the category of pure complexes, defined below.
In this section $\mathbf{k}$ will denote a field of characteristic zero, and an operad will be an operad in ${\mathbf{C}}_*(\mathbf k)$.
Weights
-------
A [*weight function*]{} on $\mathbf k$ is a group morphism $w:\Gamma\longrightarrow
\mathbb Z$ defined on a subgroup $\Gamma$ of the multiplicative group $\overline
{\mathbf k}^*$ of an algebraic closure $\overline {\mathbf k}$ of $\mathbf k$.
An element $\lambda\in \overline{ \mathbf k} $ is said to be [*pure of weight*]{} $n$ if $\lambda\in \Gamma$ and $w(\lambda)=n$. A polynomial $q(t)\in \mathbf k[t]$ is said to be [*pure of weight $n$*]{} if all the roots of $q(t)$ in $
\overline{\mathbf k}$ are pure of weight $n$. We will write $w(q)=n$ in this case.
Let $f$ be an endomorphism of a $\mathbf k$-vector space $V$. If $V$ is of finite dimension, we will say that $f$ is [*pure of weight $n$*]{} if its characteristic polynomial is pure of weight $n$. When $f$ is understood, we will say that $V$ is pure of weight $n$.
Let $f$ be an endomorphism of a finite dimensional $\mathbf
k$-vector space $V$. If $q(t)\in \mathbf k[t]$ is an irreducible polynomial, we will denote by $\ker q(f)^\infty$ the primary component corresponding to the irreducible polynomial $q(t)$, that is, the union of the subspaces $\ker
q(f)^n$, $n\ge 1$. The space $V$ decomposes as a direct sum of primary components $V=\bigoplus \ker q(f)^\infty$, where $q(t)$ runs through the set of all irreducible factors of the minimal polynomial of $f$. The sum of the primary components corresponding to the pure polynomials of weight $n$ will be denoted by $V^n$, that is $V^n=\bigoplus_{w(q)=n}\ker
q(f)^\infty$. Hence we have a decomposition $$V=\bigoplus_n V^n\oplus C,$$ where $C$ is the sum of the primary components corresponding to the polynomials which are not pure. This decomposition will be called the [*weight decomposition*]{} of $V$.
This weight decomposition is obviously functorial on the category of pairs $(V,f)$.
Let $P$ be a complex of $\mathbf k$-vector spaces such that $HP$ is of finite type, that is, $H_iP$ is finite dimensional, for all $i$. An endomorphism $f$ of $P$ is said to be [*pure of weight $n$*]{} if $H_i(f)$ is pure of weight $n+i$, for all $i$. In that case, we will say that $(P,f)$, or simply $P$, is [*pure of weight*]{} $n$, if the endomorphism $f$ is understood. Obviously, if $(P,f)$ is pure of weight $n$, so do is $(HP,Hf)$.
The following example will be useful in the sequel. Take $\alpha\in \mathbf k^*$ that is not a root of unity and define $w:\{\alpha^{n};\ n\in \mathbb Z\ \}{\longrightarrow}\mathbb Z $ by $w(\alpha^n)=n$. Let $P$ be a finite type complex with zero differential. Then, the [*grading automorphism*]{} $\phi_\alpha$ of $P$, defined by $\phi_\alpha=\alpha^{i}\cdot
\mathrm{id}$ on $P_i$, for all $i\in \mathbb Z$, is pure of weight $0$.
Let $\mathbb F$ be a finite field of characteristic $p$ and $q$ elements, $\l$ a prime $\not=p$. In [@D], Deligne defined a weight function $w$ on the field $\mathbb Q_l$ as follows. If $\iota: {{\overline{\mathbb Q_{\l}}}} {\longrightarrow}\mathbb C$ is an embedding, and $$\Gamma=\{\alpha\in {\overline{\mathbb Q_{\l}}}\,;\, \exists n\in \mathbb Z \text { such
that } |\iota \alpha|=q^{\frac{n}{2}}\},$$ then $w:\Gamma {\longrightarrow}\mathbb Z$ is defined by $w(\alpha)=n$, if $ |\iota
\alpha|=q^{\frac{n}{2}}$. The Riemann hypothesis, proved by Deligne, asserts that the Frobenius action is a pure endomorphism (of weight $0$) of the étale cohomology $H^*(X,\mathbb Q_{\l})$ of every smooth projective $\mathbb
F$-scheme $X$.
Formality criterion
-------------------
Let $w$ be a weight function on $\mathbf k$ and denote by ${\mathbf{C}}^w_*(\mathbf k)$ the category of couples $(P,f)$ where $P$ is a finite type complex and $f$ is an endomorphism of $P$ which is pure of weight $0$.
\[categoriadepesosformal\] ${\mathbf{C}}^w_*(\mathbf k)$ is a formal symmetric monoidal category.
First of all, $\mathbf 1$, with the identity, is pure of weight $0$. Next, by the Künneth theorem and elementary linear algebra, if $(P,f)$ and $(Q,g)$ are pure complexes of weights $n$ and $m$ respectively, then $(P\otimes Q, f\otimes g)$ is pure of weight $n+m$. Then it is easy to check that ${\mathbf{C}}^w_*(\mathbf
k)$ has a structure of symmetric monoidal category such that the assignment $${\mathbf{C}}^w_*({\mathbf{k}}){\longrightarrow}{\mathbf{C}}_*({\mathbf{k}}), \quad (P,f)\mapsto P$$ is a symmetric monoidal functor, and that the functor of homology $H:{\mathbf{C}}^w_*(\mathbf k){\longrightarrow}{\mathbf{C}}^w_*(\mathbf k)$ is a symmetric monoidal functor as well.
To prove that ${\mathbf{C}}^w_*(\mathbf k)$ is formal we will use the weight function $w$ to define a symmetric monoidal functor $T$ and weak equivalences $$\mbox{id}_{{\mathbf{C}}^w_*(\mathbf k)}\longleftarrow T \longrightarrow H.$$
Let $(P,f)$ an object ${\mathbf{C}}^w_*(\mathbf k)$. Since each $P_i$ is finite dimensional, $P_i$ has a weight decomposition $
P_i=C_i\oplus \bigoplus_n P_i^n. $ Then, the components of weight $n$, $$P^n:=\bigoplus_i P_i^n\ ,$$ form a subcomplex of $P$, and the same is true for $C:=\bigoplus C_i$. So we have a weight decomposition of $P$ as a direct sum of complexes $$P=C\oplus \bigoplus_n P^n.$$ Taking homology we obtain $$HP=HC\oplus\bigoplus_n HP^n.$$ Obviously, this decomposition is exactly the weight decomposition of $HP$. Purity of $f$ implies $HC=0$ and $H(P^n)=H_n(P)$, for all $n\in \mathbb Z$. Hence the inclusion $
\bigoplus_{n} P^n\rightarrow P$ is a weak equivalence.
Next, for every $n\in \mathbb Z$, the homology of the complex $P^n$ is concentrated in degree $n$. So there is a natural way to define a weak equivalence between the complex $P^n$ and its homology $H(P^n)$. Let $\tau_{\ge n}P^n$ be the canonical truncation in degree $n$ of $P^n$, $$\tau_{\ge n}P^n := Z_n P^n\oplus \bigoplus_{i>n} P^n_{i}\ .$$ This is a subcomplex of $P^n$, and the inclusion $\tau_{\ge
n}P^n\rightarrow P^n$ is a weak equivalence. Since $\tau_{\ge
n}P^n$ is non trivial only in degrees $\ge n$, and its homology is concentrated in degree $n$, the canonical projection $\tau_{\ge n}P^n
\rightarrow H(P^n)$ is a chain map, which is a weak equivalence.
Define $T$ by $$TP: = \bigoplus_{n}\tau_{\ge n}P^n \ .$$ Obviously $T$ is an additive functor, and $TP=\bigoplus_nT(P^n)$. Moreover, $T$ is a subfunctor of the identity functor of ${\mathbf{C}}^w(\mathbf k)$, and the canonical projection $TP{\longrightarrow}HP$ is a weak equivalence.
We prove now that $T$ is a symmetric monoidal subfunctor of the identity. Let $P$ and $Q$ be pure complexes of weight $0$. Since $T$ is additive, and $ \sum_{i+j=n}P^i\otimes
Q^j\subset(P\otimes Q)^n$, it suffices to show that $T(P^i)\otimes T(Q^j)\subset T(P^i\otimes Q^j)$. By the Leibnitz rule, we have an inclusion in degree $i+j$: $$Z_iP^i\otimes Z_jQ^j \subset Z_{i+j}(P^i\otimes Q^j).$$ In the other degrees the inclusion is trivially true. Hence, $T$ being stable by products, it is a symmetric monoidal subfunctor of the identity.
Finally, the projection on the homology $TP\rightarrow HP$ is well defined and obviously compatible with the Künneth morphism, so the canonical projection $T\rightarrow H$ is a monoidal natural transformation. Therefore $C^w_*(\mathbf k) $ is a formal symmetric monoidal category.
The formality theorem for the current complex, \[curretsformalfunctor\], could be obtained as a corollary from the formality of the full subcategory of ${\mathbf{C}}_*(\mathbb R)$ whose objects are the double complexes that satisfy the $dd^c$-lemma.
\[pesosimplicaformal\] Let $P$ be an operad with homology of finite type. If $P$ has a pure endomorphism (with respect to some weight function $w$), then $P$ is a formal operad.
If $P_\infty\rightarrow P$ is a minimal model of $P$, then $P_\infty$ is an operad of finite type by \[finitudob\]. From the lifting property ([@MSS], 3.123), there exists an induced pure endomorphism $f$ on $P_\infty$. Thus $(P_\infty,f)$ is an operad of ${\mathbf{C}}^w_*({\mathbf{k}})$, and the corollary follows from th. \[categoriadepesosformal\] and prop. \[formalfunctorscreateformaloperads\].
Let $P$ be an operad. Since $\Sigma$-actions and compositions $\circ_i$ are homogeneous maps of degree $0$, every grading automorphism, with respect to a non root of unit $\alpha$, is a pure endomorphism of the operad $HP$.
\[aixeca\] Let $\mathbf k$ a field of characteristic zero, and $P$ an operad with homology of finite type. The following statements are equivalent:
1. $P$ is formal.
2. There exists a model $P'$ of $P$ such that $H: {\mathrm{Aut}}(P') \longrightarrow {\mathrm{Aut}}(HP)$ is surjective.
3. There exists a model $P'$ of $P$ and $f\in Aut(P')$ such that $H(f)=\phi_{\alpha}$, for some $\alpha \in \mathbf k^*$ non root of unity.
4. There exists a pure endomorphism $f$ in a model $P'$ of $P$.
$(1)\Rightarrow (2)$ is prop. \[ooth\]. $(2) \Rightarrow
(3)$ and $(3) \Rightarrow (4)$ are obvious. Finally, $(4)\Rightarrow (1)$ is cor. \[pesosimplicaformal\].
Descent of formality
====================
In this section $\mathbf{k}$ will denote a field of characteristic zero, and operad will means an operad in the category ${\mathbf{C}}_*(\mathbf k)$, unless another category was mentioned. Using the characterization of formality of th. \[aixeca\], we will prove now that formality does not depend on the ground field, if it has zero characteristic.
Automorphism group of a finite type operad
------------------------------------------
### {#section-24}
Let $P$ be an operad. Restricting the automorphism we have an inverse system of groups $
\left({\mathrm{Aut}}(t_nP) \right)_n
$ and a morphism of groups ${\mathrm{Aut}}(P) \longrightarrow {\underset{\leftarrow}{\lim}\,}{\mathrm{Aut}}(t_nP)$. Because $P\cong
\lim\limits_{\leftarrow}t_{*}t_nP$, the following lemma is clear.
\[limitauts\] The morphisms of restriction induce a canonical isomorphism of groups $${\mathrm{Aut}}(P) {\longrightarrow}{\underset{\leftarrow}{\lim}\,}{\mathrm{Aut}}(t_nP).$$
### {#section-25}
In order to prove that the group of automorphisms of a finite type operad is an algebraic group, we start by fixing some notations about group schemes. Let $\mathbf k{\longrightarrow}R$ be a commutative $\mathbf k$-algebra. If $P$ is an operad, its extension of scalars $P\otimes_{\mathbf k} R$ is an operad in ${\mathbf{C}}_*(R)$, and the correspondence $$R \mapsto {\mathbf{Aut} (P)\,}(R) = {\mathrm{Aut}}_R (P \otimes_{\mathbf{k}} R) \ ,$$ where ${\mathrm{Aut}}_R $ means the set of automorphisms of operads in ${\mathbf{C}}_*(R)$, defines a functor $${\mathbf{Aut} (P)\,} : \mathbf{k}-\mathbf{alg} \longrightarrow \mathbf{Gr},$$ from the category $\mathbf{k}-\mathbf{alg} $ of commutative $\mathbf k$-algebras, to the category $\mathbf{Gr}$ of groups. It is clear that $${\mathbf{Aut} (P)\,} (\mathbf{k}) = {\mathrm{Aut}}(P) \ .$$
We will denote by $\mathbb{G}_m$ the [*multiplicative group scheme*]{} defined over the ground field $\mathbf{k}$.
\[algebraicos\] Let $P$ be a truncated operad. If $P$ is of finite type, then
1. ${\mathrm{Aut}}(P)$ is an algebraic matrix group over $\mathbf k$.
2. ${\mathbf{Aut} (P)\,}$ is an algebraic affine group scheme over $\mathbf
k$, represented by the algebraic matrix group ${\mathrm{Aut}}(P)$.
3. Homology defines a morphism $\mathbf{H} :
{\mathbf{Aut} (P)\,} \longrightarrow {\mathbf{Aut} (HP)\,}$ of algebraic affine group schemes.
Let $P$ a finite type $n$-truncated operad. The sum $M = \sum_{l \leq n} \mathrm{dim}\, P(l)$ is finite, hence ${\mathrm{Aut}}(P)$ is the closed subgroup of ${\mathbf{GL}_{M}({\mathbf{k}})}$ defined by the polynomial equations that express the compatibility with the $\Sigma$-action, the differential, and the bilinear compositions $\circ_i$. Thus ${\mathrm{Aut}}(P)$ is an algebraic matrix group. Moreover, ${\mathbf{Aut} (P)\,}$ is obviously the algebraic affine group scheme represented by the matrix group ${\mathrm{Aut}}(P)$.
Next, for every commutative $\mathbf k$-algebra $R$, the map $${\mathbf{Aut} (P)\,}(R) = {\mathrm{Aut}}_R (P \otimes_{\mathbf{k}}R) \longrightarrow
{\mathrm{Aut}}_R (HP \otimes_{\mathbf{k}}R) = {\mathbf{Aut} (HP)\,} (R)$$ is a morphism of groups and it is natural in $R$; thus (3) follows.
\[grupsalgebraics\] Let $\mathbf k$ be a field of characteristic zero, and $P$ a finite type truncated operad. If $P$ is minimal, then $$\mathbf{N} = \ker \left( \mathbf{H} :{\mathbf{Aut} (P)\,} \longrightarrow {\mathbf{Aut} (HP)\,} \right)$$ is a unipotent algebraic affine group scheme over $\mathbf k$ .
Since $\mathbf{k}$ has zero characteristic, and ${\mathbf{Aut} (P)\,}$, ${\mathbf{Aut} (HP)\,}$ are algebraic by prop. \[algebraicos\], $\mathbf{N}$ is represented by an algebraic matrix group defined over $\mathbf{k}$ (see [@Bo]). So it suffices to verify that all elements in $\mathbf{N}(\mathbf{k})$ are unipotent.
Given $f \in \mathbf{N}(\mathbf{k})$, let $P^1=\ker
(f-\mathrm{id})^{\infty}$ be the primary component of $P$ corresponding to the eigenvalue $1$ (see \[weights\]). Then $P^1$ is a suboperad of $P$, and the inclusion $P^1
\hookrightarrow P$ is a weak equivalence. Since $P$ is minimal, it follows from prop. \[almendruco\] that $P =
P^1$, thus $f$ is unipotent.
A descent theorem
-----------------
After these preliminaries, let us prove the descent theorem of the formality for operads. In rational homotopy theory, this corresponds to the descent theorem of formality for cdg algebras of Sullivan and Halperin-Stasheff ([@Su] and [@HS], see also [@Mor] and [@R])
\[descens\] Let $\mathbf{k}$ be a field of characteristic zero, and $\mathbf{k}
\subset \mathbf{K}$ a field extension. If $P$ is an operad in ${\mathbf{C}}_*(\mathbf k)$ with homology of finite type, then the following statements are equivalent:
1. $P$ is formal.
2. $P\otimes \mathbf{K}$ is a formal operad in ${\mathbf{C}}_*(\mathbf K)$.
3. For every $n$, $t_nP$ is formal.
Because the statements of the theorem only depend on the homotopy type of the operad, we can assume $P$ to be minimal and, by th. \[finitudob\], of finite type. Moreover, minimality of $P$ is equivalent to the minimality of all its truncations, $t_nP$.
Let us consider the following additional statement:
$(2\frac{1}{2})$ For every $n$, $t_nP\otimes\mathbf{K}$ is formal.
We will prove the following sequence of implications $$(1)\Rightarrow (2)\Rightarrow (2\tfrac{1}{2})\Rightarrow
(3)\Rightarrow (1),$$
$(1)$ implies $(2)$ because $\_
\otimes_{\mathbf{k}}\mathbf{K} $ is an exact functor.
If $P\otimes \mathbf{K}$ is formal, then so are all of its truncations $t_n (P\otimes \mathbf K) \cong t_nP \otimes
\mathbf{K}$, because truncation functors are exact, so $(2)$ implies $(2\frac{1}{2})$.
Let us see that $(2\frac{1}{2})$ implies $(3)$. From the implication $(1)\Rightarrow (2)$, already proven, it is clear that we may assume $\mathbf{K}$ to be algebraically closed. So, let $\mathbf{K}$ be an algebraically closed field, $n$ an integer, and $P$ a finite type minimal operad such that $t_nP
\otimes \mathbf{K}$ is formal. Since $${\mathbf{Aut} (t_nP)\,}(\mathbf{K}) \longrightarrow {\mathbf{Aut} (Ht_nP)\,}(\mathbf{K})$$ is a surjective map, by th. \[aixeca\], it results that $${\mathbf{Aut} (t_nP)\,} \longrightarrow {\mathbf{Aut} (Ht_nP)\,}$$ is a quotient map. Thus, by ([@Wa], 18.1), we have an exact sequence of groups $$1 \longrightarrow \mathbf{N}(\mathbf{k}) \longrightarrow
{\mathbf{Aut} (t_nP)\,}(\mathbf{k})
\longrightarrow {\mathbf{Aut} (Ht_nP)\,} (\mathbf{k}) \longrightarrow H^1(\mathbf{K}/\mathbf{k},
\mathbf{N}) \longrightarrow \dots$$ Since $\mathbf{N}$ is unipotent by th. \[grupsalgebraics\], and $\mathbf k$ has zero characteristic, it follows that $H^1(\mathbf{K}/\mathbf{k}, \mathbf{N})$ is trivial ([*op. cit.*]{}, 18.2.e). So we have an exact sequence of groups $$1 \rightarrow \mathbf{N}(\mathbf{k}) \longrightarrow {\mathrm{Aut}}(t_nP) \longrightarrow {\mathrm{Aut}}(Ht_nP ) \longrightarrow 1$$ In particular, ${\mathrm{Aut}}(t_nP) \longrightarrow {\mathrm{Aut}}(Ht_nP)$ is surjective. Hence, again by th. \[aixeca\], $t_nP$ is a formal operad.
Let us see finally that $(3)$ implies $(1)$. By th. \[aixeca\], it suffices to prove that all the grading automorphisms have a lift. Let $\phi:\mathbb
G_m{\longrightarrow}{\mathbf{Aut} (Ht_nP)\,} $ the grading representation that sends $\alpha\in \mathbb G_m$ to the grading automorphism $\phi_\alpha$ defined in \[weights\]. For every $n$, form the pull-back of algebraic affine group schemes: $$\begin{CD}
{\mathbf{F}_{n}} @>>> \mathbb{G}_m \\
@VVV @VV\phi V \\
{\mathbf{Aut} (t_nP)\,} @>\mathbf H>> {\mathbf{Aut} (Ht_nP)\,}
\end{CD}$$ That is to say, for every commutative $\mathbf k$-algebra $R$, $${\mathbf{F}_{n}}(R) = \mathrm{}\left\{ (f,\alpha ) \in {\mathbf{Aut} (t_nP)\,} (R) \times \mathbb{G}_m (R) \ ; \
Hf = \phi_{\alpha} \right\}\ .$$ By \[limitauts\], we have a commutative diagram $$\begin{CD}
\lim\limits_\leftarrow\mathbf F_n(\mathbf k)@>>>\mathbb
G_m(\mathbf k)\\
@VVV@VV\phi V\\
{\mathrm{Aut}}(P)\cong\lim\limits_\leftarrow {\mathrm{Aut}}(t_nP)@>H>>{\mathrm{Aut}}(HP)\cong\lim\limits_\leftarrow
{\mathrm{Aut}}(Ht_nP)
\end{CD}$$ so, to lift grading automorphisms, it suffices to verify that the map $\lim\limits_\leftarrow\mathbf F_n(\mathbf
k)\longrightarrow\mathbb G_m(\mathbf k)$ is surjective. In order to prove this surjectivity, first we will replace the inverse system $\left(\mathbf F_n\right(\mathbf k))_n$ by an inverse system $\left(\mathbf F'_n\right(\mathbf k))_n$ whose transition maps are surjective. Indeed, for all $p \geq n$, the restriction $\mathbf{\varrho}_{p,n} : {\mathbf{F}_{p}}
\longrightarrow {\mathbf{F}_{n}}$ is a morphism of algebraic affine group schemes which are represented by algebraic matrix groups, so, by ([@Wa], 15.1), it factors as a quotient map and a closed embedding: $${\mathbf{F}_{p}} \longrightarrow \mathrm{im}\mathbf{\varrho}_{p,n} \longrightarrow {\mathbf{F}_{n}} \ .$$ Denote ${\mathbf{F}'_{n}} :=\bigcap_{p \geq n}
\mathrm{im}\mathbf{\varrho}_{p,n}$. Since $\{
\mathrm{im}\mathbf{\varrho}_{p,n} \}_{p \geq n}$ is a descending chain of closed subschemes of the noetherian scheme ${\mathbf{F}_{n}}$, there exists an integer $N(n) \geq n$ such that $${\mathbf{F}'_{n}} = \mathrm{im} \mathbf{\varrho}_{N(n),n} \ ,$$ thus the restrictions $\mathbf{\varrho}_{n+1,n}$ induce quotient maps $\mathbf{\varrho}_{n+1,n} : {\mathbf{F}'_{n+1}} \longrightarrow {\mathbf{F}'_{n}}$. So, applying again ([@Wa], 18.1), we have an exact sequence of groups $$1 \longrightarrow \mathbf{N}'(\mathbf{k}) \longrightarrow
{\mathbf{F}'_{n+1}}(\mathbf{k})
\longrightarrow {\mathbf{F}'_{n}}(\mathbf{k}) \longrightarrow H^1({\overline{\mathbf{k}}}/\mathbf{k},
\mathbf{N}) \longrightarrow \dots$$ Here, $\mathbf{N}'(\mathbf{k})$ is a closed subscheme of $\mathbf{N}(\mathbf{k})$ because, for every $(f, \alpha) \in \mathbf{N}'(\mathbf{k})$ we have $\alpha = 1$ and so $Hf = 1$ in $Ht^*_{\le n+1}P$, which means that $f \in \mathbf{N}(\mathbf{k})$. By th. \[grupsalgebraics\], $\mathbf{N}'(\mathbf{k})$ is unipotent, thus, as in the previous implication, it follows that ${\mathbf{F}'_{n+1}}(\mathbf{k})
\longrightarrow {\mathbf{F}'_{n}} (\mathbf{k})$ is surjective for all $n \geq 2$.
Since in the inverse system $\left(\mathbf F'_n\right(\mathbf
k))_n$ all the transition maps are surjective, the map $${\underset{\leftarrow}{\lim}\,}{\mathbf{F}'_{p}}(\mathbf{k} )\longrightarrow {\mathbf{F}'_{2}} (\mathbf{k} ) \ .$$ is surjective as well. Moreover, ${\mathbf{F}'_{2}}(\mathbf{k})
\longrightarrow
\mathbb{G}_m(\mathbf k)$ is also surjective. Indeed, given $\alpha \in
\mathbb{G}_m(\mathbf k)$, since $t_{ N(2)}{P}$ is formal by hypothesis, by th. \[aixeca\] we can lift the grading automorphism $\phi_{\alpha} \in {\mathrm{Aut}}(Ht_{ N(2)}P)$ to an automorphism $f\in {\mathrm{Aut}}(t_{ N(2)}{P})$. So we have an element $(f, \alpha) \in
{\mathbf{F}_{N(2)}}(\mathbf{k})$, whose image in ${\mathbf{F}_{2}}(\mathbf{k})$ will be an element of ${\mathbf{F}'_{2}} ( \mathbf{k})$ which will project onto $\alpha$.
We conclude that ${\underset{\leftarrow}{\lim}\,}{\mathbf{F}'_{p}}(\mathbf{k} )
\longrightarrow \mathbb{G}_m(\mathbf k) $ is surjective, hence $P$ is formal.
Applications
------------
As an immediate consequence of th. \[descens\], the previous theorems \[cadenesformalsR\], and \[Coperadformal\] of formality over $\mathbb R$ imply, respectively, the following corollaries
\[cadenesformalsQ\] If $X$ is an operad in ${\mathbf{K\ddot{a}h}}$, then the operad of cubic chains $C_*({X}; \mathbb Q)$ is formal.
If $X$ is an operad in ${\mathbf{DMV}(\mathbb{C})}$, then the operad of cubic chains $C_*(X;\mathbb{Q})$ is formal.
Finally, we can apply th. \[descens\] to the formality of the little $k$-disc operad. Let $\mathcal D_k$ denote the [*little $k$-discs* ]{} operad of Boardman and Vogt. It is the topological operad with $\mathcal D_k (1) = \mathbf{pt} $, and, for $l\geq 2$, $\mathcal D_k (l) $ is the space of configurations of $l$ disjoint discs inside the unity disc of $\mathbb{R}^k$.
M. Kontsevich proved that the operad of cubic chains $C_*(\mathcal D_k,\mathbb R) $ is formal ([@Ko]). Therefore, from th. \[descens\], we obtain
The operad of cubic chains of the little $k$-discs operad $C_*(\mathcal D_k ;\mathbb{Q})$ is formal.
Cyclic operads
==============
Basic results
-------------
Let us recall some definitions from [@GeK94] (see also [@GeK98] and [@MSS]). For all $l\in \mathbb N$, the group $\Sigma^+_l:={\mathrm{Aut}}\{0,1,\dots,l\}$ contains $\Sigma_l$ as a subgroup, and it is generated by $\Sigma_l$ and the cyclic permutation of order $l+1$,$\tau_l:(0,1,\dots,l)\mapsto(1,2,\dots,l,0)$.
Let $\mathcal C$ be a symmetric monoidal category. A [*cyclic*]{} $\Sigma$-module $E$ in $\mathcal{C}$ is a sequence $(E(l))_{l\ge 1}$ of objects of $\mathcal C$ together with an action of $\Sigma^+_l $ on each $E(l)$. Let ${\mathbf{\Sigma^+ Mod}}$ denote the category of cyclic $\Sigma$-modules. Forgetting the action of the cyclic permutation we have a functor $$U^-: {\mathbf{\Sigma^+ Mod}}\longrightarrow {\mathbf{\Sigma Mod}}\, .$$
A [*cyclic operad*]{} is a cyclic $\Sigma$-module $P$ whose underlying $\Sigma$-module $U^-P$ has the structure of an operad compatible with the action of the cyclic permutation (see [*loc. cit.*]{}). Let ${\mathbf{Op^+}}$ denote the category of cyclic operads. We also have an obvious forgetful functor $$U^- : {\mathbf{Op^+}}\longrightarrow {\mathbf{Op}}\ .$$
There are obvious extensions of the notions of free operad, homology, weak equivalence, minimality and formality for cyclic dg operads and all the results in the previous sections can be easily transferred to the cyclic setting. In particular, every cyclic dg operad $P$ with $HP(1)=0$, has a minimal model $P_\infty$. Moreover $U^-(P_\infty)$ is a minimal model of $U^-(P)$. Finally, we can deduce results analogous to the formality criterion (th. \[aixeca\]), and to the descent of formality (th. \[descens\]) for cyclic operads.
Let $\mathcal A$ an abelian category. It is clear that a formal symmetric monoidal functor $F: \mathcal{C} \longrightarrow {{\mathbf{C}_*({\mathcal{A}})}}$ induces a functor of cyclic operads $$F: {\mathbf{Op^+}}\!_{\mathcal{C}} \longrightarrow
{\mathbf{Op^+}}\!_{{{\mathbf{C}_*({\mathcal{A}})}}}$$ which sends cyclic operads in $\mathcal{C}$ to formal cyclic operads in ${{\mathbf{C}_*({\mathcal{A}})}}$. From theorems \[cadenesformals\] and \[descens\] it follows that $C_*(X; \mathbb{Q})$ is a formal cyclic operad, for every cyclic operad $X$ in ${\mathbf{K\ddot{a}h}}$.
Formality of the cyclic operad $C_*(\overline{{\mathcal{M}}}_{0};\mathbb Q)$
-----------------------------------------------------------------------------
Let us apply the previous results to the [ configuration operad]{}.
Let ${\mathcal{M}}_{0,l}$ be the moduli space of $l$ different labelled points on the complex projective line ${\mathbb P}^1$. For $l
\geq 3$, let $\overline{{\mathcal{M}}}_{0,l}$ denote its [*Grothendieck-Knudsen compactification*]{}, that is, the moduli space of stable curves of genus $0$, with $l$ different labelled points.
For $l=1$, put ${\overline{\mathcal{M}}_0}(1) = *$, a point, and for $l \geq
2$, let ${\overline{\mathcal{M}}_0}(l) =
\overline{{\mathcal{M}}}_{0,l+1}$. The family of spaces ${\overline{\mathcal{M}}_0}= \left({\overline{\mathcal{M}}_0}(l)\right)_{l\geq 1}$ is a cyclic operad in ${\mathbf{V}(\mathbb{C})}$ ([@GK], or [@MSS]). Applying the functor of cubic chains componentwise we obtain a dg cyclic operad $C_*({\overline{\mathcal{M}}_0};
\mathbb{Q} )$. So, we have the following results.
\[modulispaceformal\] The cyclic operad of cubic chains $C_*({\overline{\mathcal{M}}_0}; \mathbb{Q})$ is formal.
The categories of strongly homotopy $C_*({\overline{\mathcal{M}}_0}; \mathbb{Q})$ and $H_*({\overline{\mathcal{M}}_0};
\mathbb{Q})$-algebras are equivalent.
Modular operads
===============
Preliminaries
-------------
Let us recall some definitions and notations about modular operads (see [@GeK98], or [@MSS], for details).
### {#section-26}
Let $\mathcal C$ be a symmetric monoidal category. A [*modular $\Sigma$-module*]{} of $\mathcal C$ is a bigraded object of $\mathcal{C}$, $E=\left(E((g,l))\right)_{g,l}$, with $g,l\ge0,\, 2g-2+l>0$, such that $E((g,l))$ has a right $ \Sigma_{l}$-action. Let us denote the category of modular $\Sigma$-modules by ${\mathbf{MMod}}_{\mathcal{C}}$, or just ${\mathbf{MMod}}$ if no confusion can arise.
### {#section-27}
A [*modular operad*]{} is a modular $\Sigma$-module $ P$, together with [*composition*]{} morphisms $$\circ_i : P((g,l)) \otimes P((h,m)) \longrightarrow P((g+h,l+m-2)) \ ,\; 1\le i\le l,$$ and [*contraction*]{} morphisms $$\xi_{ij}: P((g,l)) \longrightarrow P((g+1,l-2)) \ , \;1\le i\not= j\le l,$$ which verify axioms of associativity, commutativity and compatibility (see [@GeK98], [@MSS]). Let us denote the category of modular operads by ${\mathbf{MOp}}_{\mathcal{C}}$, or just ${\mathbf{MOp}}$ if no confusion can arise.
As for operads and cyclic operads, from the definitions it follows that every symmetric monoidal functor $F : \mathcal{C} \longrightarrow \mathcal{D}$ applied componentwise induces a functor $${\mathbf{MOp}}_{F}:{\mathbf{MOp}}_{\mathcal{C}} \longrightarrow {\mathbf{MOp}}_{\mathcal{D}} \ ,$$ and every monoidal natural transformation $\phi:F\Rightarrow
G$ between symmetric monoidal functors induces a natural transformation ${\mathbf{MOp}}_\phi:{\mathbf{MOp}}_{F}\Rightarrow
{\mathbf{MOp}}_G.$
As Getzler and Kapranov proved ([@GeK98]), the family $\overline{{\mathcal{M}}}((g,l)):=\overline{{\mathcal{M}}}_{g,l}$ of Deligne-Knudsen-Mumford moduli spaces of stable genus $g$ algebraic curves with $l$ marked points, with the maps that identify marked points, is a modular operad in the category of projective smooth DM-stacks.
dg modular operads {#functorformalmodular}
------------------
[*From now on, $\mathbf k$ will denote a field of characteristic zero, and modular operads in ${\mathbf{C}}_*(\mathbf k)$ will be called simply dg modular operads.*]{}
Let $V$ be a finite type chain complex of $\mathbf k$-vector spaces, and $B$ an inner product over $V$, that is to say, a non-degenerate graded symmetric bilinear form $B:V\otimes
V{\longrightarrow}\mathbf k$ of degree $0$. It is shown in [@GeK98] that there exists a dg modular operad $\mathcal E[V]$ such that $$\mathcal E[V]((g,l))=V^{\otimes l},$$ with the obvious structure morphisms.
An [*ideal*]{} of a dg modular operad $P$ is a modular $\Sigma$-submodule $I$ of $P$, such that $P\cdot I\subset I$, $I\cdot P\subset I$, and $I$ is closed under the contractions $\xi_{ij}$.
For any dg modular operad $P$ and any ideal $I $ of $P$, the quotient $P/I$, inherits a natural structure of dg modular operad and the projection $P \longrightarrow P/I$ is a morphism of dg modular operads.
If $P$ is a dg modular operad its [*homology*]{} $HP$, defined by $(HP)((g,l))=H(P((g,l)))$, is also a dg modular operad. A morphism $\rho:P{\longrightarrow}Q$ of dg modular operads is said to be a [*weak equivalence*]{} if $\rho((g,l)):P((g,l)){\longrightarrow}Q((g,l))$ is a weak equivalence for all $(g,l)$.
The localization of ${\mathbf{MOp}}_{{\mathbf{C}}_*(\mathbf k)}$ with respect to the weak equivalences is denoted by $\mathrm{ Ho}
{\mathbf{MOp}}_{{\mathbf{C}}_*(\mathbf k)}$.
A dg modular operad $P$ is said to be [*formal*]{} if $P$ is weakly equivalent to its homology $HP$.
Clearly, for a formal symmetric monoidal functor $F : \mathcal{C} \longrightarrow
{\mathbf{C}}_*(\mathbf k)$, the induced functor $$F: {\mathbf{MOp}}_{\mathcal{C}} \longrightarrow {\mathbf{MOp}}_{{\mathbf{C}}_*(\mathbf k)}$$ transforms modular operads in $\mathcal{C}$ to formal modular operads in ${\mathbf{C}}_*(\mathbf k)$.
Modular dimension
-----------------
In order to study the homotopy properties of dg modular operads we will replace the arity truncation with the truncation with respect to the modular dimension.
Let $\mathcal C$ be a symmetric monoidal category.
Recall that, the dimension as algebraic variety of the moduli space $\overline{{\mathcal{M}}}_{g,l}$ is $3g-3+l$. So the following definition is a natural one. The function $d:\mathbb Z^2{\longrightarrow}\mathbb Z$, given by $d(g,l)=3g-3+l$, will be called the [*modular dimension*]{} function.
Let $E$ be a modular $\Sigma$-module in $\mathcal C$. The modular dimension function induces a graduation $\left( E_n\right)_{n\ge 0}$ on $E$ by $E_n=\left(E((g,l))\right)_{d(g,l)=n}$, and a decreasing filtration $\left(E_{\ge n}\right)_n$ of $E$ by $$E_{\ge n}:=\left(E((g,l))\right)_{d(g,l)\ge n} \ .$$
The following properties are easily checked.
\[propiedaddimensionmodular\] Let $P$ be a modular operad in $\mathcal C$. The modular dimension grading satisfies $$P\cdot P_n\subset P_{\ge n+1},$$ where $P\cdot P_n$ is the set of meaningful products $\alpha\circ_i\beta$, with $\alpha\in P_m$, $\beta\in P_l$ and at least one of $l,\,m$ is $n$. On the other hand, the contraction maps satisfies $$\xi_{ij}: P_n \longrightarrow P_{\ge n+1} \ ,$$ for all $i,j$.
Truncation of modular operads
-----------------------------
A [*$n$-truncated modular operad*]{} in a symmetric monoidal category $\mathcal C$ is a modular operad defined only up to modular dimension $n$, that is, a family of $\mathcal{C}$, $\{P((g,l));\,g,l\ge 0, \; 2g-2+l>0,\, d(g,l)\le n\,\}$, such that $P((g,l))$ has a right $
\Sigma_{l}$-action, with morphisms $$\circ_i : P((g,l)) \otimes P((h,m)) \longrightarrow P((g+h,l+m-2)) \ ,\; 1\le i\le l,$$ and [*contractions*]{} $$\xi_{ij}: P((g,l)) \longrightarrow P((g+1,l-2)) \ , \;1\le i\not= j\le l,$$ satisfying those axioms in ${\mathbf{MOp}}$ that make sense.
If ${\mathbf{MOp}}_{\le n}$ denotes the category of $n$-[*truncated dg modular operads*]{}, we have a [*modular dimension truncation*]{} functor, $$t_{ n}:{\mathbf{MOp}}\longrightarrow {\mathbf{MOp}}_{\le n}\,,$$ defined by $t_n(P)=(P((g,l)))_{d(g,l)\le n}$.
Since the obvious forgetful functor $$U: {\mathbf{MOp}}\longrightarrow {\mathbf{MMod}}$$ has a left adjoint, the [*free modular operad functor*]{} $$\mathbb{M} : {\mathbf{MMod}}\longrightarrow {\mathbf{MOp}}\ ,$$ (see[@GeK98], 2.18), by prop. \[propiedaddimensionmodular\] we can translate the truncation formalism developed in \[Truncatedoperads\] to the setting of dg modular operads. So we have a sequence of adjunctions $t_{!}\dashv t_{n}\dashv t_{*}$, and the propositions \[propiedadest\*\] and \[propiedadest!\] are still true, merely replacing “operad" with “modular operad", and “arity" with “modular dimension" shifted by $+2$. For instance, the arity truncation begins with $t_2$, whereas the modular dimension truncation begins with $t_0$.
If $P$ is a dg modular operad, the direct system of dg modular operads given by $$0{\longrightarrow}t_{!}t_{0}P\rightarrow\cdots \rightarrow t_{!}t_{n-1} P
\rightarrow t_{!}t_{n}P\rightarrow\cdots$$ is called the [*canonical tower*]{} of $P$.
Principal extensions
--------------------
Let us explicitly describe the construction of a principal extension in the context of modular operads. Let $P$ be a dg modular operad, $V$ a dg modular $\Sigma$-module with zero differential, concentrated in modular dimension $n\ge 0$, and $\xi:V[-1]{\longrightarrow}P_n$ a chain map. Then the principal extension of $P$ by $\xi$, $P\sqcup_\xi V$, is defined by a universal property as in \[estructuradeunaextensionprincipal\], $${\mathrm{Hom}}_{{\mathbf{MOp}}}(P\sqcup_\xi V,Q)= \{(f,g);\ f\in {\mathrm{Hom}}_{{\mathbf{MOp}}}(P, Q)\ ,\, g\in {\mathrm{Hom}}_{\bf
GrMMod}(V, UQ)_{0}\ , \ d_Qg-gd_V=f\xi\ \}\, .$$
In particular we have $$(P\sqcup_\xi
V)_i=\left\{
\aligned \label{}&P_i,\hskip 3mm \text{ if } i<n,\\
&P_n\oplus_\xi V,\hskip 3mm \text{ if } i=n,\endaligned
\right.$$ because in $t_{ n}(P\sqcup_\xi
V)$ all the structural morphisms involving $V$ are trivial, by prop. \[propiedaddimensionmodular\].
Furthermore, the following property, analogous to prop. \[estructuradeunaextensionprincipal\], is satisfied.
\[estructuradeunaextensionprincipaldemodularoperads\] Let $n\ge 0$ an integer. Let $P$ be a dg modular operad such that $t_!t_nP\cong P$, $V$ a dg modular $\Sigma$-module concentrated in modular dimension $n$, with zero differential, and $\xi:V[-1]{\longrightarrow}P_n$ a morphism of dg modular $\Sigma_n$-modules. The principal extension $P\sqcup_\xi V$ satisfy
(1)$t_{n-1}(P\sqcup_\xi V)\cong t_{n-1}P$.
(2)$(P\sqcup_\xi V)_n\cong C\xi,$ in particular, there exists an exact sequence of complexes $$0{\longrightarrow}P_n{\longrightarrow}(P\sqcup_\xi V)_n{\longrightarrow}V{\longrightarrow}0.$$ (3) $P\sqcup_\xi V\cong t_{!}t_n(P\sqcup_\xi V) $.
\(4) A morphism of dg modular operads $\phi:P\sqcup_\xi V{\longrightarrow}Q$ is determined by a morphism of $n$-truncated dg modular operads $f:t_nP{\longrightarrow}t_nQ$, and a homogeneous map $g:V{\longrightarrow}Q_n$ of modular $\Sigma$-modules, such that $f\xi =dg$.
Minimal models
--------------
### Minimal objects
A dg modular operad $M$ is said to be [*minimal*]{} if the canonical tower $$0\longrightarrow t_{!}t_{0}M\longrightarrow \dots
\longrightarrow t_{!}t_{ n-1}M
\longrightarrow t_!t_{ n}M \longrightarrow \dots$$ is a sequence of principal extensions.
A [*minimal model*]{} of a dg modular operad $P$ is a minimal dg modular operad $P_\infty$ together with a weak equivalence $P_\infty
\longrightarrow P$ in ${\mathbf{MOp}}$.
From prop. \[estructuradeunaextensionprincipaldemodularoperads\], it follows, by induction on the modular dimension, that:
\[quis=isomodular\] Let $M$, $N$ be minimal dg modular operads. If $\rho:M
\longrightarrow N$ is a weak equivalence of dg modular operads, then $\rho$ is an isomorphism.
### Existence of minimal models
\[recetapaella\] Let $\mathbf k$ be a field of characteristic zero. Every modular operad $P$ in ${\mathbf{C}}_*(\mathbf k)$ has a minimal model.
We start in modular dimension $0$. Let $M^{0}=\mathbb{M}HP_0$, and $s:HP_0{\longrightarrow}ZP_0$ a section of the canonical projection. Then $s$ induces a morphism of modular operads $$\rho^{0}: M^{0} \longrightarrow P$$ which is a weak equivalence of modular operads up to modular dimension $0$, because $M^{0}_0=HP_0$.
For $n\ge 1$, assume that we have already constructed a morphism of modular operads $$\rho^{n-1} : M^{n-1} \longrightarrow P$$ such that
1. $M^{n-1}\cong t_{!}t_{n-1}M^{n-1}$ is a minimal modular operad, and
2. $t_{ n-1}(\rho^{n-1})$ is a weak equivalence.
To define the next step of the induction we will use the following statement, which contains the main homological part of the inductive construction of minimal models.
Let $$\begin{CD}
B@>\eta>>A@.@.\\
@V\lambda VV@VV\mu V @.@.\\
(C\zeta)[-1]@>-p_Y>>Y@>\zeta>>X
\end{CD}$$ be a commutative diagram of complexes of an additive category, then there exists a chain map $\nu:C\eta{\longrightarrow}X$ such that in the diagram $$\begin{CD}
@.@. B @>\eta>> A @>>> C\eta @>>> B[1] \\
(8.6.3.1)\hskip 30mm @.@. @V\lambda VV
@VV\mu V@VV\nu V @VV\lambda[1] V\\
@.@.(C\zeta)[-1]@>-p_Y>>Y @>\zeta>>X@>>>C\zeta
\end{CD}\hskip 50mm$$ the central square is commutative, and the right hand side square is homotopy commutative. Moreover, the rows of $(8.6.3.1)$ are distinguished triangles, and the vertical maps define a morphism of triangles in the derived category.
We have $C\eta=A\oplus_\eta B[1]$, and $C\zeta=X\oplus_\zeta
Y[1]$. Let $(\lambda_X,\lambda_Y)$ be the components of $\lambda$, then one can check that $
\nu(a,b)=\lambda_X(b)+\zeta\mu(a), $ with the homotopy $
h(a,b)=(0,\mu (a))$, satisfies the conditions of the statement.
The upper row of the diagram $(8.6.3.1)$ is obviously a distinguished triangle. By axiom $(TR2)$ of a triangulated category, turning the distinguished triangle $$\begin{CD}
Y@>\zeta>>X@>>>C\zeta@>p_Y>>Y[1]
\end{CD}$$ one step to the left we obtain that the lower row of the diagram (8.6.3.1) is also a distinguished triangle.
Now we return to the proof of the theorem. Since $\mathbf k$ is a field of zero characteristic, the category of modular $\Sigma$-modules is semisimple, and $C\rho^{n-1}_n$ is a formal complex of modular $\Sigma$-modules. Therefore, if $V=
HC\rho^{n-1}_n$, with the zero differential, there exists a weak equivalence $$s:V{\longrightarrow}C\rho^{n-1}_n\,.$$ In fact, $s$ can be obtained from a $\Sigma$-equivariant section of the canonical projection from cycles to homology.
Let $\xi$ be the composition $$\begin{CD}
{V[-1]}@>s[-1]>> \left(C\rho^{n-1}_n\right)[-1]
@>-p>>M^{n-1}_n,
\end{CD}$$ where the second arrow is the opposite of the canonical projection. We have a commutative diagram of complexes $$\begin{CD}
V[-1]@>\xi>>M^{n-1}_{n}@.\\
@V s [-1]VV@VV \mathrm{id} V\\
\left(C\rho^{n-1}_n\right)[-1]@>-p>> M^{n-1}_n@>\rho^{n-1}_n>> P_n
.\\
\end{CD}$$ By the previous lemma, there exists a chain map $$\nu:C\xi{\longrightarrow}P_n$$ that completes the previous diagram in a diagram $$\begin{CD}
@.@. V[-1]@>\xi>>M^{n-1}_n@>>>C\xi@>>>
V@.@.\\
(8.6.3.2)\hskip 30mm@.@.@Vs[-1]VV@V\mathrm{id}VV@V\nu VV@VVsV@.@.\hskip 20mm
\\
@.@.\left(C\rho^{n-1}_n\right)[-1]@>-p>> M^{n-1}_n
@>\rho^{n-1}_n>> P_n@>>> C\rho^{n-1}_n.@.@.\\
\end{CD}\hskip 20mm$$ where the rows are distinguished triangles in the category of complexes, the central square is commutative, and the vertical maps define a morphism of triangles in the derived category.
The step $M^{n}$ is defined as the principal extension of $M^{n-1}$ by the attachment map $\xi:V[-1]{\longrightarrow}M^{n-1}_n$, $$M^{n}:=M^{n-1}\sqcup_{\xi} V\,.$$ Let $\nu_V:V{\longrightarrow}P$ be the graded map $$\begin{CD} V@>>> C\xi@>\nu>>P_n\end{CD}$$ where the first map is the canonical inclusion. Since $\rho^{n-1}\xi=d \nu_{V}$ , the maps $\rho^{n-1}$ and $\nu_{V}$ define, according to the universal property of $M^{n}=M^{n-1}\sqcup_{\xi} V$, a morphism of modular operads $$\rho^{n}:M^{n}{\longrightarrow}P$$ such that $t_{ n-1}\rho^{n}=t_{ n-1}\rho^{n-1}$ and $\rho_n^{n}=\nu$. By the inductive hypothesis, $\rho^{n}$ is a weak equivalence in modular dimensions $<n$. Finally, in the diagram $(8.6.3.2)$ $s$ is a weak equivalence, hence $\nu$ is a weak equivalence as well. It follows that $t_{n}\rho^{n}$ is a weak equivalence, which finishes the induction. Therefore, $\lim\limits_{\rightarrow}M^{n}$ is a minimal model of $P$.
### Finiteness of minimal models
A modular $\Sigma$-module $V$ is said to be [*of finite type*]{} if, for every $(g,l)$, $V((g,l))$ is a finite dimensional $\mathbf{k}$-vector space. A dg modular operad $P$ is said to be of [*finite type*]{} if $UP$ is of finite type.
Obviously, for every integer $n\ge 0$, there are only a finite number of pairs $(g,l)$ such that $g,l,2g-2+l>0$ and $d(g,l)=n$, thus a modular $\Sigma$-module $V$ is of finite type if, and only if, $V_n$ is finite dimensional, for every $n\ge 0$.
\[finitudmodular\] If $V$ is a modular $\Sigma$-module of finite type, then $\mathbb M(V)$ is of finite type.
Indeed, for every pair $(g,l)$, there is an isomorphism $$\mathbb M(V)((g,l))\cong \bigoplus_{\gamma\in \{\mathbf\Gamma((g,l))\}
}V((\gamma))_{\text{Aut}(\gamma)}$$ where $\{\mathbf\Gamma((g,l))\} $ denotes the set of equivalence classes of isomorphisms of stable $l$-labelled graphs of genus $g$, the subscript $\text{Aut}(\gamma)$ denotes the space of coinvariants, and $$V((\gamma))=\bigotimes_{v\in
\text{Vert}(\gamma)}V((g(v),\text{Leg }{(v)})).$$ By [@GeK98] lemma 2.16, the set $\{\mathbf\Gamma((g,l))\}
$ is finite, for every pair $(g,l)$. Therefore the free modular operad $\mathbb M(V)$ is of finite type.
As a consequence of prop. \[finitudmodular\] we obtain the finiteness result analogous to \[finitudob\].
Let $P$ be a dg modular operad. If $HP$ is of finite type, then every minimal model of $P$ is of finite type.
Lifting properties
------------------
Analogous to def. II.3.121 of [@MSS], there exists a similarly defined path object and a notion of homotopy in the category of dg modular operads.
### Homotopy
Let $\mathbf I:=\mathbf k[t,\delta t]$ be the differential graded commutative $\mathbf k$-algebra generated by a generator $t$ in degree $0$ an its differential $\delta t$ in degree $-1$. For every dg modular operad $P$, the path object of $P$ is the dg modular operad $P\otimes\mathbf I $, obtained by extension of scalars.
The evaluations at $0$ and $1$ define two morphisms of modular operads $\rho_0,\rho_1:P\otimes\mathbf I\rightrightarrows P$ which are weak equivalences. An [*elementary homotopy*]{} between two morphisms of dg modular operads $f_0,f_1:P\rightrightarrows Q$ is a morphism $H:P{\longrightarrow}Q\otimes\mathbf I$ of dg modular operads such that $\rho_i H=f_i$, for $i=0,1$. Elementary homotopy is a reflexive and symmetric relation, and the [*homotopy relation*]{} between morphisms is the equivalence relation generated by elementary homotopy. Homotopic morphisms induce the same morphism in $\mathrm{ Ho }{\mathbf{MOp}}$.
### Lifting properties of minimal objects
Obstruction theory, that is, lemma II.3.139 [*op. cit.*]{}, and its consequences: the homotopy properties of the minimal objects (theorems II.3.120 and II.3.123 [*op. cit.*]{}), is easily established in the context of modular operads. So we have
\[liftingproperty\] Let $\rho:Q{\longrightarrow}R$ be a weak equivalence of dg modular operads, and $\iota : P
\longrightarrow P \sqcup_\xi V$ a principal extension. For every homotopy commutative diagram in ${\mathbf{MOp}}$ $$\begin{CD}
P @>\phi >> Q \\
@V\iota VV @V\rho VV \\
P \sqcup_\xi V @>\psi >> R
\end{CD}$$ there exists an extension $\overline{\phi} : P \sqcup_\xi V
\longrightarrow Q$ of $\phi$ such that $\rho \overline{\phi}$ is homotopic to $\psi$. Moreover, $\overline{\phi}$ is unique up to homotopy.
From this lemma the lifting property of minimal modular operads follows by induction:
\[liftingpropertyforminimalsmodular\] Let $\rho : Q \longrightarrow R$ be a weak equivalence of dg modular operads, and $M$ a minimal modular operad. For every morphism $\psi : M \longrightarrow R$, there exists a morphism $\widetilde{\psi} : M
\longrightarrow Q$ such that $\rho\widetilde{{\psi}}$ is homotopic to $\psi$. Moreover, $\widetilde{\psi}$ is unique up to homotopy.
### Uniqueness of minimal models
From th. \[liftingpropertyforminimalsmodular\] and prop. \[quis=isomodular\], we obtain
\[unicidadmmmodular\] Two minimal models of a modular operad are isomorphic.
The modular analogue of prop. \[almendruco\] follows in the same way.
\[almendrucomodular\] Let $M$ be a minimal dg modular operad and $P$ a subobject of $M$. If the inclusion $P \hookrightarrow M$ is a weak equivalence, then $P = M$.
Formality
---------
From theorems \[liftingpropertyforminimalsmodular\] and \[quis=isomodular\], the modular analogue of prop. \[ooth\] follows easily.
\[oothmodular\] Let $M$ be a minimal dg modular operad. If $M$ is formal, then the map $H:\nolinebreak{\mathrm{Aut}}(M)\longrightarrow{\mathrm{Aut}}(HM)$ is surjective.
Now, from th. \[pesosimplicaformal\] and prop. \[oothmodular\], the formality criterion for modular operads follows with the same proof as th. \[aixeca\].
\[aixecamodular\] Let $\mathbf k$ be a field of characteristic zero, and $P$ a dg modular operad with homology of finite type. The following statements are equivalent:
1. $P$ is formal.
2. There exists a model $P'$ of $P$ such that $H: {\mathrm{Aut}}(P')
\longrightarrow {\mathrm{Aut}}(HP)$ is surjective.
3. There exists a model $P'$ of $P$, and $f\in {\mathrm{Aut}}(P')$ such that $Hf=\phi_{\alpha}$, for some $\alpha\in \mathbf{k^*} $ non root of unity.
4. There exists a pure endomorphism $f$ in a model $P'$ of $P$.
Then, using this result, the descent of formality for modular operads follows as th. \[descens\].
\[descensmodular\] Let $\mathbf{k}$ be a field of characteristic zero, and $\mathbf{k} \subset \mathbf{K}$ a field extension. If $P$ is a modular operad in ${\mathbf{C}}_*(\mathbf k)$ with homology of finite type, then $P$ is formal if, and only if, $P\otimes \mathbf{K}$ is a formal modular operad in ${\mathbf{C}}_*(\mathbf K)$.
Finally, the result below follows from \[functorformalmodular\], and theorems \[hodgeforVman\], \[descensmodular\].
\[Coperadmodformal\] Let $X$ be a modular operad in ${\mathbf{DMV}(\mathbb{C})}$. Then $C_*(X; \mathbb{Q})$ is a formal modular operad.
Strongly homotopy algebras over a modular operad
------------------------------------------------
Let $P$ be a dg modular operad. Recall ([@GeK98]) that a $P$-algebra is a finite type chain complex $V$ with an inner product $B$, together with a morphism of modular operads $P
\longrightarrow \mathcal E[V]$.
We give the following definition. A [*strongly homotopy $P$-algebra*]{}, or [*sh $P$-algebra*]{}, is a finite type chain complex $V$ with an inner product $B$, together with a morphism $P
\longrightarrow \mathcal E[V]$ in $\mathrm {Ho}{\mathbf{MOp}}$. By \[liftingpropertyforminimalsmodular\], this is equivalent to giving a homotopy class of morphisms $P_\infty{\longrightarrow}\mathcal E[V]$.
Let $(V,B)$, and $(W,B)$ be sh $P$-algebras. A [*morphism of sh $P$-algebras*]{} $f$ is a chain map $f:V{\longrightarrow}W$ compatible with the inner products and such that the following diagram $$\begin{array}{ccc}
& & \mathcal E[V] \\
& \nearrow & \\
P & & \downarrow \; f_* \\
& \searrow & \\
& & \mathcal E[W]
\end{array}$$ commutes in $\mathrm{Ho }{\mathbf{MOp}}$.
The homotopical invariance is an immediate consequence of the above definitions:
Let $(V,B)$ be a finite type chain complex with an inner product, $(W,B)$ a sh $P$-algebra, and $f: (V,B)
\longrightarrow (W,B)$ a chain map compatible with $B$ and such that $f:V{\longrightarrow}W$ is a homotopy equivalence. Then $(V,B)$ has a unique structure of sh $P$-algebra such that $f$ becomes a morphism of sh $P$-algebras.
Application to moduli spaces
----------------------------
Let us apply these results to the modular operad of moduli spaces $\overline{{\mathcal{M}}}$.
From th. \[Coperadmodformal\] it follows
$C_*(\overline{\mathcal M} ; \mathbb{Q})$ is a formal modular operad.
So, we obtain
Every structure of $H_*(\overline{\mathcal M} ; \mathbb{Q})$-algebra lifts to a structure of sh $C_*(\overline{\mathcal M} ; \mathbb{Q})$-algebra.
In conclusion, we see that the minimal model $H_*(\overline{\mathcal M} ;
\mathbb{Q})_{\infty}$ of the modular operad $H_*(\overline{\mathcal M} ; \mathbb{Q})$ plays an important role in the description of the sh $H_*(\overline{\mathcal M} ;
\mathbb{Q})$-algebras and therefore of the sh $C_*(\overline{\mathcal M};\mathbb
Q)$-algebras. The explicit construction of the modular operad $H_*(\overline{\mathcal M}
; \mathbb{Q})_{\infty}$, as in the proof of th. \[recetapaella\], would require the knowledge of the homology of the moduli spaces and all its relations. We think that a motivic formulation of this minimal modular operad (see [@BM]) and the determination of the basic pieces for this building (for instance, a minimal tensor generating family of simple motives for the smaller abelian tensor subcategory where this operad lives) would be a nice variant of Grothendieck’s “Lego-Teichmüller game".
[BE]{}
W.L. Baily, Jr., *The decomposition theorem for $V$-manifolds*, Amer. J. Math. [**78**]{}(1956), 862–888. K. Behrend, *Gromov-Witten invariants in algebraic geometry*, Invent. Math. [**127**]{} (1997), 601–617. K. Behrend, Y. Manin, *Stacks of stable maps and Gromov-Witten invariants*, Duke Math. J. [**85** ]{} (1996), 1–60. A. Borel, *Linear algebraic groups*, Second enlarged edition, Springer GTM [**126**]{} (1991) P. Deligne, *Conjecture de Weil. II*, Publ. Math. IHES, [**52**]{} (1980), 137–252. P. Deligne, P. Griffiths, J.W. Morgan and D. Sullivan, *Real homotopy theory of K[ä]{}hler manifolds*, Invent. Math. [**29**]{} (1975), 245–274. E. Getzler, *Operads and moduli spaces of genus 0 Riemann surfaces*, The moduli space of curves (Texel Island, 1994), 199–230, Progr. Math., 129, Birkhäuser Boston, Boston, MA, 1995. E. Getzler and M. Kapranov, *Cyclic operads and cyclic homology*, in *Geometry, Topology and Physics for Raoul Bott*, 167–201, Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA, 1995. E. Getzler and M. Kapranov, *Modular operads*, Compositio Math. [**110**]{} (1998), 65–126. V. Ginzburg and M. Kapranov, *Koszul duality for operads*, Duke Math. J. [**76**]{} (1994), 203–272. A. Grothendieck, *Esquisse d’un programme*, manuscript 1984. Published in *Geometric Galois actions 1*, 5–48. London Math. Soc. Lecture Note Ser., 242. Cambridge Univ. Press, Cambridge, 1997. V. Hinich, *Homological algebra of homotopy algebras*, Comm. Algebra [**25**]{} (1997), 3291–3323. S. Halperin and J. Stasheff, *Obstructions to homotopy equivalences*, Adv. Math. [**32**]{} (1979), 233–279. I. Kriz and J.P. May *Operads, algebras, modules and motives*, Asterisque [**133**]{}, SMF (1995). M. Kontsevich, *Operads and motives in deformation quantization*, Lett. Math. Physics [**48**]{} (1999), 35-72. G.M. Kelly and R. Street *Review of the elements of 2-categories*, in *Category Seminar, Sydney 1972/73* Springer LNM [**420**]{} (1974), 75–103. T. Kimura, J. Stasheff and A.A. Voronov, *Homology of moduli of curves and commutative homotopy algebras*, in *The Gelfand Mathematical Seminars, 1993–1995*, Gelfand Math. Sem., Birkh[ä]{}user (1996), 151–170. T. Kimura, A.A. Voronov and G.J. Zuckerman, *Homotopy Gerstenaber algebras and topological field theory*, in *Operads: Proceedings of Renaissance Conferences*, Contemp. Math. [**202**]{} (1997), 305–333. Y.I. Manin *Frobenius manifolds, quantum cohomology and moduli spaces*, Coll. Publ. AMS [**47**]{} (1999). W.S. Massey *Singular homology theory*, Springer GTM [**70**]{} (1980). M. Markl, *Models for operads*, Comm. Algebra [**24**]{} (1996), 1471–1500. J.W. Morgan, *The algebraic topology of smooth algebraic varieties*, Publ. Maht. IHES [**48**]{} (1978), 137–204. M. Markl, S. Shnider and J. Stasheff, *Operads in Algebra, Topology and Physics*, Surv. and Monog. AMS [**96**]{} (2002). A. Roig, *Formalizability of DG algebras and morphisms of CDG algebras*, Ill. J. of Math. [**38**]{} (1994), 434–451. L. Schwartz, *Lectures on complex analytic manifolds*, Tata Institute (1955). D. Sullivan, *Infinitesimal computations in topology*, Publ. Math. IHES [**47**]{} (1977), 269–331. D.E. Tamarkin, *Formality of chain operad of small squares*, Preprint [math.QA/9809164]{}, September 1998. W.C. Waterhouse, *Introduction to affine group schemes*, Springer GTM [**66**]{} (1979)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We briefly review the Hopf algebra structure arising in the renormalization of quantum field theories. We construct the Hopf algebra explicitly for a simple toy model and show how renormalization is achieved for this particular model.'
author:
- |
Usman Naseer\
*Center for Theoretical Physics*\
*Massachusetts Institute of Technology*\
title: |
------------------------------------------------------------------------
\
Hopf algebra and renormalization: A brief review\
------------------------------------------------------------------------
\
---
Introduction
============
The underlying Hopf algebraic strucuture of the process of renormalization was discovered first by Kreimer in [@Kreimer1]. Further progress was made in formulating the renormalization procedure in the language of Hopf algebra and doing explicit computations using this algebraic structure, in [@Kreimer2; @Kreimer3; @Kreimer4; @Kreimer5]. The purpose of this article is to briefly review this algebraic structure. For simplicity, we avoid many of the technicalities of the quantum field theory by considering a simple toy model containing only nested divergences. Issues related to overlapping divergences and more realistic field theoretic models have been discussed in literature( see refs. [@Kreimer6] and [@QED]). For a more detailed review of this subject see [@review].
This article closely follows the conventions and notation used in [@knots]. For a detailed treatment of renormalization procedure, see [@collins]. For a mathematically rigorous introduction to Hopf algebra, see [@qgroup].
The article is organized as follows. In section \[sec:pre\], we describe necessary notation and conventions. In section \[sec:halgebra\] we explicitly construct the Hopf algebra structure. For clarity of our arguments and construction, most of the proofs have been relegated to appendices. The article is concluded with a very simple example in appendix \[app:example\]
Preliminaries {#sec:pre}
=============
The forest formula
------------------
A brief summary of the BPHZ renormalization procedure and the derivation of the forest formula is given in the appendix \[app:bphz\]. The key result of BPHZ renormalization is an iterative formula (forest formula) which gives a renormalized Feynman graph in terms of the divergent graph, its subgraphs and the corresponding counter terms. Forest formula can be written in a schematic form as follows: $$\begin{aligned}
\Gamma_{r}&=&\overline{\Gamma} + Z_{\Gamma},\label{eq:for1}\\
\overline{\Gamma}&=&\Gamma +\sum_{\gamma \subset \Gamma} Z_{\gamma} \left(\Gamma/\gamma\right),\label{eq:for2}\\
Z_{\Gamma} &=&-t_{\Gamma} \overline{\Gamma},\label{eq:for3}\end{aligned}$$ where $\Gamma$ and $\Gamma_r$ are bare and renormalized graphs respectivley. $\overline{\Gamma}$ is the graph with all the subdivergences removed. The sum is over all non-empty proper forests of $\Gamma$. $Z_{\gamma}$ and $Z_{\Gamma}$ are counter terms. $t_{\Gamma}$ is a renormalization scheme dependent operator, which removes the overall divergence associated with graph $\Gamma$. To make the notion of forest precise, let $H_{1},\cdots,H_m$ be all 1PI, non overlapping divergent subgraphs of $\Gamma$, then a proper forest of $\Gamma$ is any subset of the following set: $$\begin{aligned}
\lbrace H_1,\cdots,H_m\rbrace.\end{aligned}$$
Representing the graph
----------------------
We would like to represent Feynman graphs in a more algebraic fashion such that their forest structure and subdivergences become manifest. This would be done by representing them as ‘parenthesized words’. Parentheses encode information about the nestedness or the disjointness of the subdivergences and letters appearing in these words correspond to graphs without subdivergences. Parenthesized words can be assigned to a graph by the following procedure:
- For every forest we write down a pair of brackets respecting the forest structure, i.e., if a forest $A$ is inside a forest $B$ then the pair of brackets corresponding to the forest $A$ are contained inside the pair of brackets corresponding to $B$.
- Consider a given pair of brackets, if we shrink all the brackets/forests inside it to a point the remainder is a graph $\gamma_{i}$ without any subdivergences. We write the letter corresponding to $\gamma_{i}$ next to the right closing bracket of the pair of brackets under consideration.
- Rest of what is contained in the pair under consideration is written to the left of this letter.
For an example, consider the diagram in figure \[fig:TwoDiv\]. It has two disjoint subdivergences and is overall divergent when the subdivergences are shrunk to a point. The two subdivergences are contained in rectangular boxes. These subdivergences themselves are both 1PI and do not contain any subdivergences. In the figure we have also shown the letters corresponding to these subdivergences. It is easy to see that, using our rules above, this diagram corresponds to the parenthesized word $\left(\left(x_1\right)\left(x_2\right)x_1\right)$.
![A divergent diagram with two disjoint subdivergences[]{data-label="fig:TwoDiv"}](hopf1)
Important features of this construction are following.
- Disjoint forests and configurations inside disjoint pair of brackets commute in this construction. i.e., $$\begin{aligned}
\left(\left(x_1\right)\left(x_2\right)x_1\right)=\left(\left(x_2\right)\left(x_1\right)x_1\right).\end{aligned}$$
- Only the forest structure of the graph is made manifest in this construction and we lose information about to which propagator or to which vertex of a graph $\gamma_j$ another graph $\gamma_i$ is attached. Several different attachment can yield the same forest structure. Hence any Feynman diagram belongs to a class given by a Parenthesized word. For example, the two diagrams in figure \[fig:samePW\] belong to the class represented by the parenthesized word $\left(\left(x_2\right)\left(x_2\right)x_1\right)$.
![These two Feynman diagrams corresponding to the same parenthesized word ${\left({\left(x_2\right)}{\left(x_2\right)}x_1\right)}$[]{data-label="fig:samePW"}](hopf2)
- A letter $x_i$ has one and only one closing bracket on its right side while it can have more than one opening brackets.
- We include the empty graph as $\left(\right)$ which would act as a unit element (not to be confused with the unit map) in the construction of the Hopf Algebra.
- An important characteristic of a parenthesized word is its length, which is simply the total number of letters $x_i$ appearing in it. For example, in collection (\[coll\]), the parentheized words have lengths $0,1,2,2,3,\cdots$ respectively.
- In general we will have a class of Feynman graphs represented by the notion of parenthesized words constructed out of letters $x_i$. Some examples are: $$\begin{aligned}
\left(\right),\ \ \left(x_i\right),\ \ \left(\left(x_i\right)x_j\right),\ \ \left(x_i\right)\left(x_j\right),\ \ \left(\left(x_i\right)\left(x_j\right)x_k\right),\cdots\label{coll}\end{aligned}$$
- A parenthesized word, whose left most bracket is matched with its right most bracket is called an irreducible parenthesized word and corresponds to a 1PI Feynman graph. Examples are: $$\begin{aligned}
{\left(x_i\right)},\ \ \ {\left({\left(x_i\right)}x_j\right)},\ \ \ {\left({\left({\left(x_i\right)}x_j\right)}x_k\right)},\cdots .\end{aligned}$$ An arbitrary irreducible parenthesized word can be represented as ${\left(Xx_i\right)}$, where $X$ is an any parenthesized word.
- A parenthesized word, whose left most and the right most brackets do not match with each other is called a reducible parenthesized word and can be written as product of irreducible parenthesized words. For example $$\begin{aligned}
{\left({\left(x_i\right)}x_j\right)}{\left(x_k\right)},\end{aligned}$$ is a reducible parenthesized word and is written as a product of two irreducible parenthesized words ${\left({\left(x_i\right)}x_j\right)}$ and ${\left(x_k\right)}$.
Hopf algebra
------------
[r]{}[0.4]{}
HH & & & &HH&\
m &\
H && K & & H&\
m &\
HH & & \_[S]{}& &HH &\
A detailed discussion of the mathematical properties of Hopf algebra will lead us off topic. In this subsection, we will give the formal definition of a Hopf algebra and different elements appearing in the definition. We will also give a rough sketch of how the procedure of renomalization can be described by an underlying Hopf algebra structure. These notions will be made more precise in the next section.
Formally a Hopf algebra is defined as following.
\[defHopf\] A Hopf algebra is an associative and co-associative bialbegra $H$ over a field $K$ with a K-linear map $S:\ H\rightarrow H$, called antipode such that the diagram \[fig:hopf\] commutes. $E, \overline{e},m, \Delta $ are called unit, co-unit, product and co-product maps respectively. The condition for the commutativity of the diagram can be written algebraically as: $$\begin{aligned}
m\left[\left(S\otimes {\mathbf{1_d}}\right)\Delta\left[X\right]\right]=m\left[\left({\mathbf{1_d}}\otimes S\right)\Delta\left[X\right]\right]=E\circ \overline{e}\left[X\right]\label{hopfCom}\end{aligned}$$ where $X$ is an element of Hopf algebra and ${\mathbf{1_d}}$ is the identity map.
Now we will give a brief overview of how renormalization would turn out to be related to the Hopf algebra structure.
- Basic objects of the Hopf algebra are Feynman graphs $\Gamma$ which will be represented by the corresponding parenthesized word $X_{\Gamma}$. Representatives of the overall divergent graphs without subdivergences will be identified as the primitive elements of the Hopf algebra. All other elements $X_{\Gamma}$ can be built out of these primitive elements.
- The co-product resolves the graph into its forests. $$\begin{aligned}
\Delta\left[X_{\Gamma}\right]=\sum_{\text{all forests }\gamma}\ X_{\gamma}\otimes X_{\Gamma/\gamma}.\end{aligned}$$
- We have a renormalization map $R$, which extracts the divergent parts of a graph (depending on the renormalization scheme).
- The antipode $S$ gives the counter term $Z_{\Gamma}$ through the renormalization map. $$\begin{aligned}
S_{R}\left[X_{\Gamma}\right]=-R\left[X_{\Gamma}\right]-\ \sum_{\text{all non-empty proper forests }\gamma}R\left[S_R\left[X_{\gamma}\right]X_{\Gamma/\gamma}\right].\end{aligned}$$
- The renormalized Feynman graph will related to the term $m\left[\left(S\otimes {\mathbf{1_d}}\right)\Delta\left[X\right]\right]$, appearing in the condition of the commutativity. We would indeed see that $m\left[\left(S\otimes {\mathbf{1_d}}\right)\Delta\left[X\right]\right]=0$, expressing the fact that the we get a finite result.
Construction of Hopf Algebra {#sec:halgebra}
============================
In this section we will construct the Hopf algebra related to renormalization. This will be done by explicitly defining the all the maps and elements appearing in the definition (\[defHopf\]). We will proceed in several steps, establishing algebra, co-algebra, bialgebra and finally Hopf algebra structure.
The algebra structure
---------------------
As discussed in the previous section, we will represent Feynman diagrams by parenthesized words. We will arrange these parenthesized words into an algebra structure here. Let $\mathcal{A}$ be the set of all parenthesized words. We regard this as a $\mathbb{Q}$ vector space. It is easy to see that $\mathcal{A}$ is a vector space over $\mathbb{Q}$. Now, we introduce a bilinear product map as follows: $$\begin{aligned}
&m&:\ \ \ \ \ \ \ \mathcal{A}\otimes \mathcal{A}\ \rightarrow \ \mathcal A,\\
&m& \left[X\otimes Y\right]\ \equiv \ \ XY \equiv YX,\ \ \ \forall\ X,Y \in \mathcal{A}.\label{defProd}\end{aligned}$$ Also we have an identity element $e=\left(\right)$ which satisfies: $$\begin{aligned}
eX=Xe=X\ \ \ \forall \ X\in \mathcal{A}.\end{aligned}$$ To understand the product (\[defProd\]) consider the example with $X=\left(\left(x\right)x\right)$ and $Y=\left(y\right)$ then $XY$ is a well defined product given by $\left(\left(x\right)x\right){\left(y\right)}$, i.e., the product of two parenthesized words give a reducible parenthesized word. By introducing the product we have furnished $\mathcal{A}$ with an algebra structure.
Now we define a homomorphism (the unit map) from $\mathbb{Q}$ to the set $\mathcal{A}$ as follows: $$\begin{aligned}
&E&:\ \ \ \ \ \ \ \mathbb{Q}\ \to \mathcal{A},\\
&E\left[q\right]&\ \equiv e,\ \ \ \forall\ \text{rational numbers }q.\label{defunit}\end{aligned}$$ Now, by definition, the bilinear product $m$ is associative, our algebra $\mathcal{A}$ has an identitiy element $e$ and we have constructed a homomorphism from the field of rational numbers $\mathbb{Q}$ to algebra $\mathcal{A}$, this means that the set $\mathcal{A}$ is a unital associative algebra.
The coalgebra structure
-----------------------
In this subsection, we furnish $\mathcal{A}$ with the structure of a coalgebra. Let us first give the formal definition of a coalgebra.
\[defCoalg\] A coalgebra, $C$ over a field $K$ is a vector space $C$ over $K$ together with linear maps $\overline{e}:\ C\to K$ (counit) and $\Delta:\ C\to C\otimes C$ (coproduct) such that $$\begin{aligned}
{\left({\mathbf{1_d}}\otimes\overline{e}\right)}\Delta&=&{\left(\overline{e}\otimes{\mathbf{1_d}}\right)}\Delta,\label{coalg1}\\
{\left(\Delta\otimes{\mathbf{1_d}}\right)}\Delta&=&{\left({\mathbf{1_d}}\otimes\Delta\right)}\Delta.\label{coalg2}\end{aligned}$$ where ${\mathbf{1_d}}$ is the identity map on $C$, or quivalently, the two diagrams in figure \[fig:coalg\] commute. In the second diagram, we have identified the naturally isomorphic spaces $C$, $C\otimes K$,$K\otimes C$. The second equation above is also called the coassociativity condition for the coproduct $\Delta$.
C &\^&CC & & &C&\^& &CC\
& & & &\
CC&\_&CCC & & &CC &\_& &KCCCK
Now, we will define the counit and the coproduct maps for the set $\mathcal{A}$ under consideration.
### The counit {#the-counit .unnumbered}
We define a counit by: $$\begin{aligned}
&\overline{e}&:\ \ \ \ \ \mathcal{A}\to \mathbb{Q},\\
&\overline{e}\left[e\right]&\equiv \ 1,\\
&\overline{e}\left[X\right]&\equiv \ 0,\ \ \forall\ X\neq e, \in \mathcal{A}.\ \ \ \ \ \ \ \ \ \ \ \label{defcounit}\end{aligned}$$ This definition is motivated by the fact that there is no rational number which should be assigned naturally to an arbitrary parenthesized word and thus the counit annihilates Feynman graphs. On the other hand we assign the rational number $1$ to the empty graph $e$.
### The coproduct {#the-coproduct .unnumbered}
The definition of the coproduct is more involved as compared to the elements defined so far. Roughly speaking, coproduct yields a sum of terms $\sum_{i}X_i\otimes Y_i$, where the first terms, $X_i$, are to be identified with divergent subgraphs and the second terms, $Y_{i}$, correspond to the remainder of the graph obtained by reducing $X_{i}$ to a point.
To give a rigorous definition of the coproduct, it will be useful to define a projection map $P$ as follows: $$\begin{aligned}
&P&:\ \ \ \ \ \mathcal{A}\otimes\mathcal{A}\ \to \ \mathcal{A}\otimes \mathcal{A},\\
&P&\ \equiv \left({\mathbf{1_d}}-E\circ\overline{e}\right)\otimes {\mathbf{1_d}}.\end{aligned}$$ It is easy to confirm the following properties of the map $P$ by explicit computation. $$\begin{aligned}
P\left[e\otimes X\right]&=&0,\ \ \forall\ X\in \mathcal{A},\\
P\left[X\otimes Y\right]&=&X\otimes Y,\ \ \ \ \forall\ X\neq e,\ Y, \in \mathcal{A},\\
P^2&=&P.\end{aligned}$$
We also define a useful endomorphism $B_{\left(x_{i}\right)}$, which is parametrized by a single letter $x_i$, corresponding to a primitive graph. $$\begin{aligned}
&B_{\left(x_i\right)}&:\ \ \ \ \ \mathcal{A}\to \mathcal{A},\\
&B_{\left(x_i\right)}\left[X\right]&\equiv \left(Xx_i\right).\end{aligned}$$ For example, $B_{\left(x_1\right)}\left[\left(x_2\right)\right]=\left(\left(x_2\right)x_1\right)$. With the help of the maps $P$ and $B$, we are now in a position to define the coproduct as follows. $$\begin{aligned}
&\Delta&:\ \ \ \ \ \ \mathcal{A}\to \mathcal{A}\otimes \mathcal{A} ,\\
&\Delta\left[e\right]&\ \equiv\ e\otimes e,\\
&\Delta\left[\left(Xx_i\right)\right]&\ \equiv\ \left(Xx_i\right)\otimes e+e\otimes\left(Xx_i\right)+\left({\mathbf{1_d}}\otimes B_{\left(x_i\right)}\right)\left[P\left[\Delta\left[X\right]\right]\right].\label{coprod}\end{aligned}$$ This definition of the coproduct is complete. It is easy to use the above definition to show an important property of the coproduct. $$\begin{aligned}
&\Delta\left[\left(x_i\right)\right]&\ \equiv\ \left(x_i\right)\otimes e + e\otimes\left(x_i\right).\label{coprodPrim}\end{aligned}$$ Another important property of the coproduct is: $$\begin{aligned}
&\Delta\left[XY\right]&\ \equiv \ \Delta\left[X\right]\Delta\left[Y\right].\label{compatcoprod}\end{aligned}$$ This can also be shown by using the definition (\[coprod\]), however the proof is a bit involved. The proof is based on the standard induction argument on the length of the words $X$ and $Y$.
Another way to write the coproduct is by using the Sweedler’s notation, $\Delta\left[X\right]=\sum_X X_1\otimes X_2$, where the sum is over the subwords $X_1$ of $X$ and $X_2=X/X_1$. Proof of this assertion is given in appendix \[app:Sweed\]. Using this notation and the properties of the map $P$, we can write the equation (\[coprod\]) of the coproduct as: $$\begin{aligned}
\Delta\left[\left(Xx\right)\right]=\left(Xx\right)\otimes e +\left({\mathbf{1_d}}\otimes B_{\left(x\right)}\right)\left[\sum_X X_1\otimes X_2\right]\label{coprodSweed}.\end{aligned}$$
Let us now consider a few example to explain how the coproduct acts on the elements of the set $\mathcal{A}$.
1. $$\begin{aligned}
\Delta{\left[{\left({\left(x_i\right)}x_j\right)}\right]}&=&{\left({\left(x_i\right)}x_j\right)}\otimes e + e\otimes {\left({\left(x_i\right)}x_j\right)}+{\left({\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}P{\left(\Delta{\left[{\left(x_i\right)}\right]}\right)},\\
&=&{\left({\left(x_i\right)}x_j\right)}\otimes e + e\otimes {\left({\left(x_i\right)}x_j\right)}+{\left({\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}{\left[{\left(x_i\right)}\otimes e\right]},\\
&=&{\left({\left(x_i\right)}x_j\right)}\otimes e + e\otimes {\left({\left(x_i\right)}x_j\right)}+{\left(x_i\right)}\otimes {\left(x_j\right)}.\label{eq:exampleDel1}\end{aligned}$$
2. Using the similar method (but after more tedious algebra) we can also compute: $$\begin{aligned}
\Delta{\left[{\left({\left(x_i\right)}{\left(x_j\right)}x_k\right)}\right]}=&&{\left({\left(x_i\right)}{\left(x_j\right)}x_k\right)}\otimes e + e\otimes {\left({\left(x_i\right)}{\left(x_j\right)}x_k\right)}+{\left(x_i\right)}\otimes {\left({\left(x_j\right)}x_k\right)}\nonumber \\ &&+{\left(x_j\right)}\otimes {\left({\left(x_i\right)}x_k\right)}+{\left(x_i\right)}{\left(x_j\right)}\otimes {\left(x_k\right)}.\label{eq:exampleDel2}\end{aligned}$$
### Coalgebra check {#coalgebra-check .unnumbered}
We have defined the counit and the coproduct maps for $\mathcal{A}$, but in order to furnish the coalgebra structure on $\mathcal{A}$ we need to show that these maps satisfy the equations (\[coalg1\]) and (\[coalg2\]). The first of these relations is trivial to show due to the definition of the counit as : $$\begin{aligned}
{\left({\mathbf{1_d}}\otimes\overline{e}\right)}\Delta{\left[X\right]}=X={\left(\overline{e}\otimes{\mathbf{1_d}}\right)}\Delta{\left[X\right]}.\end{aligned}$$
Next, we want to show the equation (\[coalg2\]) holds. This can be proved using induction on the length of the words. A detailed proof is given in the appendix \[app:coalg\].
After successfully defining a counit and a coproduct on $\mathcal{A}$, we have completed the construction of the coalgebra structure on $\mathcal{A}$. We have already established the fact that $\mathcal{A}$ is a unital coassociative algebra. The property (\[compatcoprod\]) ensures that the algebra and the coalgebra structures are compatible. This implies that $\mathcal{A}$ is actually a bialgebra.
The antipode
------------
To complete the construction of the Hopf algebra, what remains to find is an antipode. It turns out that antipode is actually the object which achieves the renormalization, it combines the terms generated by the coproduct and combines them in a way which is similar to the forest formula. We define the antipode as follows: $$\begin{aligned}
S:&&\ \ \ \ \ \ \mathcal{A}\to \mathcal{A},\\
S{\left[e\right]}&=&e,\\
S{\left[{\left(x_i\right)}\right]}&=&-{\left(x_i\right)},\\
S{\left[XY\right]}&=&S{\left[Y\right]}S{\left[X\right]},\\
S{\left[{\left(Xx_i\right)}\right]}&=&-{\left(Xx_i\right)}-m{\left[{\left(S\otimes{\mathbf{1_d}}\right)}P_2{\left(\Delta{\left[{\left(Xx_i\right)}\right]}\right)}\right]},\label{eq:Sdef1}\\
S{\left[{\left(Xx_i\right)}\right]}&=&-{\left(Xx_i\right)}-m{\left[{\left({\mathbf{1_d}}\otimes S\right)}P_2{\left(\Delta{\left[{\left(Xx_i\right)}\right]}\right)}\right]},\label{eq:Sdef2}\\
P_2&\equiv& {\left({\mathbf{1_d}}-E\circ\overline{e}\right)}\otimes {\left({\mathbf{1_d}}-E\circ\overline{e}\right)}\equiv P_1\otimes P_1.\end{aligned}$$ This completely defines the antipode. However, we need to show that this antipode is actually well defined and induces a Hopf algebra structure. This amounts to showing that equations (\[eq:Sdef1\]) and (\[eq:Sdef2\]) are equivalent[^1] and also the condition (\[hopfCom\]) is satisfied. Equivalence of the two definitions follow from the associativity of the product $m$ and the coassociativity of the coproduct $\Delta$. The detailed proof is given in appendix \[app:antipodeDef\]. The proof that the condition (\[hopfCom\]) is satisfied, is given in appendix \[app:hopfcheck\]. We have now completely furnished the set of all Feynman diagrams, $\mathcal{A}$ with the structure of a Hopf algebra. We have not yet discussed precisely how the renomalization is achieved by this structure. This will be the subject of the next section.
From Hopf algebra to the forest formula
---------------------------------------
In this section, we describe how the Hopf algebra constructed above produces the forest formula, generates counter terms and the renormalized Feynman graphs. We will see that an important ingredient in this regard is the renormalization map, $R$, which is renormalization scheme dependent.
Given a Feynman graph $\Gamma$, we associate a parenthesized word $X_{\Gamma}$ to it. Using the Feynman rules we obtain an integral expression associated with the graph $\Gamma$, denote it by $\phi{\left(X_{\Gamma}\right)}$ $\in V$, where $V$ is a vector space, endowed with suitable structure which is not important for our considerations. For example, it could be the space of Laurent polynomials in the regularization parameter. These Feynman integrals are subject to some renormalization conditions which are described by renormalization map $R:\ \ V\to V$. The renormalization map depends on the renormalization scheme, for example, in the case of minimal subtraction, $R$ picks out the only the divergent part of $\phi{\left(X_{\Gamma}\right)}$. The map $\phi$, the renormalization map $R$ and the antipode of the Hopf algebra $S$ give rise to a map $S_{R}$ at the level of the Feynman integrals, which is written as: $$\begin{aligned}
S_{R}{\left[{\left(Xx\right)}\right]}=-R{\left[\phi{\left({\left(Xx\right)}\right)}\right]}-R{\left[m{\left[{\left(S_{R}\otimes\phi\right)}P_2{\left(\Delta{\left[{\left(Xx\right)}\right]}\right)}\right]}\right]}.\label{eq:antipode2}\end{aligned}$$ with $S_{R}{\left[e\right]}=e$. This map $S_{R}$ gives the counter terms for a given graph depending on the particular renormalization scheme $R$. Consider the following examples where, for simplicity, we omit writing $\phi$ explicitly:
1. $$\begin{aligned}
S_{R}{\left[{\left({\left(x_i\right)}x_j\right)}\right]}=-R{\left[{\left({\left(x_i\right)}x_j\right)}\right]}-R{\left[m{\left[{\left(S_R\otimes {\mathbf{1_d}}\right)}P_2{\left(\Delta{\left[{\left({\left(x_i\right)}x_j\right)}\right]}\right)}\right]}\right]}.\end{aligned}$$
We can use $\Delta{\left[{\left({\left(x_i\right)}x_j\right)}\right]}$ as computed in equation (\[eq:exampleDel1\]). Since, $P_1$ annihilates $e$, we finally find that: $$\begin{aligned}
S_{R}{\left[{\left({\left(x_i\right)}x_j\right)}\right]}=-R{\left[{\left({\left(x_i\right)}x_j\right)}\right]}+R{\left[R{\left[{\left(x_i\right)}\right]}{\left(x_j\right)}\right]}.\end{aligned}$$
2. Similarly, after a straightforward but tedious computation one can find that: $$\begin{aligned}
S_{R}{\left[{\left({\left(x_i\right)}{\left(x_j\right)}x_k\right)}\right]}&=&-R{\left[{\left({\left(x_i\right)}{\left(x_j\right)}x_k\right)}\right]}+R{\left[R{\left[{\left(x_i\right)}\right]}{\left({\left(x_j\right)}x_k\right)}\right]}+R{\left[R{\left[{\left(x_j\right)}\right]}{\left({\left(x_i\right)}x_k\right)}\right]}\nonumber \\&&-R{\left[R{\left[{\left(x_i\right)}\right]}R{\left[{\left(x_j\right)}\right]}{\left(x_k\right)}\right]}.\label{eq:example}\end{aligned}$$
Let us now proceed further to show that the forest structure in equations (\[eq:for1\],\[eq:for2\],\[eq:for3\]) emerges from the Hopf the algebra structure.
Let $U$ be a subword of $X$, then by using the representation of the coproduct in Sweedler’s notation and the fact that $P_1$ annihilates $e$, the antipode can be written as: $$\begin{aligned}
S{\left[X\right]}=- X - \sum_{U\neq e, X} S{\left[U\right]}{\left(X/U\right)}.\end{aligned}$$ If the parenthesized word $X$ is associated to a Feynman graph $\Gamma$ then the subwords $U\neq e, X$ are associated to the proper forest $\gamma$ of the graph $\Gamma$. Using this fact, we can now write the map $S_{R}$ in the following way: $$\begin{aligned}
S_{R}{\left[\Gamma\right]}&=&-R{\left[\Gamma\right]}-\sum_{\text{proper forest }\gamma\subset \Gamma} R{\left[S_{R}{\left[\gamma\right]}\Gamma/\gamma\right]},\\
&=&-R{\left[\Gamma+\sum_{\text{proper forest }\gamma\subset \Gamma} S_{R}{\left[\gamma\right]}\Gamma/\gamma\right]}.\end{aligned}$$ Now, if we identify $S_{R}{\left[\gamma\right]}$ with counter term associated to the subgraph $\gamma$, then the argument of the map $R$ in the above equation is just $\overline{\Gamma}$, the graph $\Gamma$ with all its subdivergences renormalized as defined in equation (\[eq:for2\]). We can also identify the renormalization map $R$ with the operator $t_{\Gamma}$, both are renormalizatio scheme dependent operators and picks out just the divergent part of a Feynman integral in MS scheme. With this identification, we see that $S_{R}{\left[\Gamma\right]}$ just gives the counter term $Z_{\Gamma}$ and we recover the forest structure of equations (\[eq:for1\],\[eq:for2\],\[eq:for3\]). The renormalized Feynman graph $\Gamma_{ren}$ is obtained as follows. Let $X$ be the parenthesized word associated with the graph $\Gamma$ (We will use the parenthesized word $X$, the corresponding graph $\Gamma$ and the corresponding Feynman integral $\phi{\left(\Gamma\right)}$ interchangeably) then: $$\begin{aligned}
m{\left[S_R\otimes \phi \right]}\Delta{\left[X\right]}&=&m{\left[S_{R}\otimes \phi\right]}{\left(e\otimes X + X\otimes e+ \sum_{\text{subwords } U\neq e,X}U\otimes {\left(X/U\right)}\right)},\\
&=&\phi{\left[X\right]}+S_R{\left[X\right]}+\sum_{\text{subwords }U\neq e, X } S_{R}{\left[U\right]}\phi{\left[X/U\right]},\end{aligned}$$ in the last equation, the first term is just the Feynman integral associated with graph $\Gamma$, the second term is the counter term $Z_{\Gamma}$ and the last term just removes the subdivergences as we have seen earlier. Now, we omit writing $\phi$ and replace parenthesized words with the respecting graphs to find: $$\begin{aligned}
m{\left[S_R\otimes \phi \right]}\Delta{\left[X\right]}=\Gamma +Z_{\Gamma}+\sum_{\text{proper forests }\gamma\subset \Gamma}Z_{\gamma}\Gamma/\gamma=\overline{\Gamma}+Z_{\Gamma}=\Gamma_{ren}.\end{aligned}$$ Earlier, we showed that at the Hopf algebra level, the operator $m{\left[{\left(S\otimes {\mathbf{1_d}}\right)}\Delta\right]}$ annihilates any parenthesized word other than the unit $e$. This expresses the fact that at the level of the Feynman integrals we will get essentially a finite result.
Summary {#sec:sum}
=======
In this section we will briefly summarize the key results of this article. By representing the Feynman diagrams as parenthesized words, we furnished them into a set $\mathcal{A}$. We also included the empty graph, represented by the unit element $e$, in that set. Then we introduced an algebra structure on $\mathcal{A}$ by defining a bilinear product $m:\mathcal{A}\to \mathcal{A}$. We also defined a unit map $E:\mathbb{Q}\to \mathcal{A}$, furnishing $\mathcal{A}$ into a unital associative algebra. Next, we introduced the coalgebra structure on $\mathcal{A}$ by defining the counit map and the coproduct map. The coproduct was defined in such a way that it was compatible with the product $m$ and hence we obtained a bialgebra structure on $\mathcal{A}$. To complete the construction of the Hopf algebra, we defined an antipode map $S:\mathcal{A}\to\mathcal{A}$. We also showed that the struture of the forest formula is recovered if we identify the antipode with the counter term of a specific graph. To make this notion precise, we defined a map $\phi:\mathcal{A}\to V$, which assigns a parenthesized word an analytic expression (Feynman integral) using the Feynman rules. We defined the renormalization map $R$ which gives the divergent part of a Feynman integral. It turned out antipode $S$ induced the counter term for a graph via $R$.
The most important result we obtained is the equivalence of the antipode and the forest formula. This equivalence followed by making a set of identifications between the elements of the Hopf algebra and the objects of the standard renormalization theory. We list these identifications here.
- 1PI Feynman graph $\Gamma$ with subdivergences are identified with irreducible parenthesized word $(Xx)$ whose bracket structure matches the forest structure of $\Gamma$, and the letters label the components of $\Gamma$ obtained after reducing the subdivergences to a point.
- The counter term $Z_{\Gamma}$ is identified with $S_{R}{\left[\Gamma\right]}$.
- The Feynman graph, with all its subdivergences renormalized, $\overline{\Gamma}$ is identified with the object: $$\begin{aligned}
m{\left[{\left(S_R\otimes \phi\right)}P_{R}\Delta{\left[{\left(Xx\right)}\right]}\right]},\ \ \text{where}\ \ \ P_R={\mathbf{1_d}}\otimes{\left({\mathbf{1_d}}-E\circ\overline{e}\right)}.\end{aligned}$$
- The renormalized Feynman graph $\Gamma_{ren}=\overline{\Gamma}+Z_{\Gamma}$ is identified with: $$\begin{aligned}
m{\left[{\left(S_{R}\otimes\phi\right)}\Delta{\left[{\left(Xx\right)}\right]}\right]}.\end{aligned}$$
BPHZ Renormalization {#app:bphz}
====================
Consider a Feynman graph $\Gamma$. By using Feynman rules we can obtain the corresponding analytic expression $F_{\Gamma}$. In general this expression can be written as a Laurent series in the regularization parameter $\epsilon$. If we consider $\phi$-cubed theory in $6$ spacetime dimensions and use dimensional regularization then $$\begin{aligned}
F_{\Gamma}\ \equiv\ \sum_{n=-N}^{\infty} a_{n}\epsilon^{n} \label{Fgamma},\end{aligned}$$ where $a_{n}$ are some coefficients and the integer $N$ is bounded above by the number of loops in the graph $\Gamma$, which can be shown explicitly. We stress here that in the general argument for the BPHZ renormalization nothing depends crucially on the particular toy model chosen here. Let us now define a ‘subtraction’ operator associated with the graph $\Gamma$ as follows $$\begin{aligned}
t_{\Gamma} F_{\Gamma}\ \equiv\ \sum_{n=-N}^{-1} a_{n}\epsilon^{n} \label{SubOp},\end{aligned}$$ i.e., it picks out the divergent part of $F_{\Gamma}$. In general, the subtraction operator is renormalization scheme dependent, here we have chosen the minimal subtraction scheme. The finite part of the graph can now be written as: $$\begin{aligned}
F_{\Gamma}^{r}\ =\ \left(1-t_{\Gamma}\right) F_{\Gamma}.\end{aligned}$$ So, we see that the term ‘$-t_{\Gamma}F_{\Gamma}$’ provides the counter term for the graph $\Gamma$ and $1-t_{\Gamma}$ removes the divergence associated with graph $\Gamma$ and makes it finite in the $\epsilon\rightarrow 0$ limit.
Now, consider the graph $\Gamma$ to have proper 1PI subgraphs $H_i,\ i=1,\cdots, m.$ For simplicity, we assume that all these subgraphs are overall divergent, if they are not divergent, there is no need for renormalization. We order these graphs such that if $H_{i}\subset H_{j}$ then $i<j$. Now we define the following: $$\begin{aligned}
\overline{R}_{\Gamma} F_{\Gamma}\ \equiv \left(1-t_{H_m}\right)\cdots \left(1-t_{H_1}\right)=\left(\prod_{H_i\subset G} \left(1-t_{H_i}\right)\right) F_{\Gamma},\label{SubDivR}\end{aligned}$$ where the product in the second equality needs to be ordered. Since the operator ‘$1-t_{H_i}$’ removes the divergence associated with the subgraph $H_i$, we see that equation (\[SubDivR\]) is nothing but the graph $\Gamma$ with all its subdivergences renormalized. Now we define the ‘Bogoliubov $R$ operator’ which removes the over all divergence associated with $\Gamma$ and renders it finite: $$\begin{aligned}
R_{\Gamma} &\equiv& \left(1-t_{\Gamma}\right)\overline{R}_{\Gamma},\\
\Rightarrow R_{\Gamma} F_{\Gamma}&=&\left(1-t_{\Gamma}\right)\ \left(\prod_{H_i\subset G } \left(1-t_{H_i}\right)\right) F_{\Gamma}. \label{RenGam}\end{aligned}$$ Let us now define a restricted graph $\Gamma /H$ as the graph obtained by reducing $H$ to a point inside $\Gamma$, then it is easy to see that $-t_H\ F_{\Gamma}=\left(-t_HF_H\right)F_{\Gamma/H}$, i.e., we can replace the subgraph $H$ in $\Gamma$ by $\Gamma/H$ and multiply by the counter term which makes $H$ finite. We can write equation (\[RenGam\]) as: $$\begin{aligned}
R_{\Gamma} F_{\Gamma}&=&\left(1-t_{\Gamma}\right)\ \left( F_{\Gamma}+\sum_{\phi}\left(\prod_{H\in \phi } \left(-t_{H}\right)\right) F_{\Gamma}\right),\label{RenF2}\end{aligned}$$ where the sum is taken over all subgraphs of $\Gamma$ (i.e., all non empty subsets (denoted by $\phi$) of the set $\lbrace H_1,\cdots, H_m\rbrace$). We will also need the following theorem due to Hepp [@hepp1966], which we state here without proof.
\[hepp\] Let $H_1,\cdots,H_j$ be overlapping 1PI subgraphs of $\Gamma$. Then consider a subgraph $H_{12\cdots j}$ such that $H_{i}\subset H_{12\cdots j}, \forall i=1,\cdots,j$ then $$\begin{aligned}
\left(1-t_{H_{12\cdots j}}\right) t_{H_1}\cdots t_{H_j}\ =\ 0,\end{aligned}$$ i.e., the finite part of the graph left after replacing the overlapping subdivergences is zero.
Courtesy this theorem we can restrict the $\phi$ in equation (\[RenF2\]) to be the subset of non overlapping 1PI divergences. Since $\left(\prod_{H\in \phi } \left(-t_{H}\right)\right) F_{\Gamma}$ provides the counter term associated with the subgraph $\phi$ we can write: $$\begin{aligned}
R_{\Gamma} F_{\Gamma}&=&\left(1-t_{\Gamma}\right)\ \left( F_{\Gamma}+\sum_{\phi} Z_{\phi} F_{\Gamma/\phi}\right),\label{RenF3}\end{aligned}$$ where $Z_{\phi}$ is the counter term which makes the subgraph $\phi$ finite. The subgraph $\phi$ is formally defined as: $$\begin{aligned}
\phi=\lbrace H_{i}|H_{i}\subset G, H_{i} \text{ are non overlaping, 1PI} \rbrace,\end{aligned}$$ and is called a ‘forest’ of graph $\Gamma$. In the above expression the term inside second set of parenthesis is the graph $\Gamma$ with all non-overlapping subdivergences renormalized. The remaining divergence is then removed by the operator $\left(1-t_{\Gamma}\right)$. Equation (\[RenF3\]) is called ‘Zimmermann’s Forest Formula’. We can write the forest formula in a schematic fashion as follows: $$\begin{aligned}
\Gamma_{r}&=&\overline{\Gamma} + Z_{\Gamma},\label{eq:for1A}\\
\overline{\Gamma}&=&\Gamma +\sum_{\gamma \subset \Gamma} Z_{\gamma} \left(\Gamma/\gamma\right),\label{eq:for2A}\\
Z_{\Gamma} &=&-t_{\Gamma} \overline{\Gamma},\label{eq:for3A}\end{aligned}$$ where $\Gamma$ and $\Gamma_r$ are bare and renormalized graphs respectivley. $\overline{\Gamma}$ is the graph with all the subdivergences removed. $\gamma$ denotes all proper forests of $\Gamma$. $Z_{\gamma}$ and $Z_{\Gamma}$ are the counter terms.
### Example {#example .unnumbered}
We now consider an example which explains some important aspects of the forest structure of a Feynman graph and the application of the forest formula. Let us look at the diagram in figure \[fig:TwoDiv\]. This graph (say $\Gamma_{1}$) has only two non overlapping 1PI subgraphs, say $H_1$ and $H_2$, as labeled and boxed in the diagram. The corresponding proper forests are: $$\begin{aligned}
\gamma_{1}=\lbrace H_1\rbrace,\ \ \gamma_{2}=\lbrace H_2\rbrace,\ \ \gamma_{3}=\lbrace H_1,H_2 \rbrace.\end{aligned}$$ So we find that: $$\begin{aligned}
\overline{\Gamma_1}&=&\Gamma_1+Z_{\gamma_1} \Gamma_1/\gamma_1 + Z_{\gamma_2}\Gamma_1/\gamma_2+Z_{\gamma_3}\Gamma_1/\gamma_3,\\
Z_{\gamma_1}&=&-t_{\gamma_1}\gamma_1,\ \ Z_{\gamma_2}=-t_{\gamma_2}\gamma_2,\ \ Z_{\gamma_3}=Z_{\gamma_1}Z_{\gamma_2},\\
Z_{\Gamma_1}&=&-t_{\Gamma_1}\overline{\Gamma_1},\\
\Gamma_{1r}&=&\overline{\Gamma_1}+Z_{\Gamma_1}.\end{aligned}$$ We shoowed earlier that this diagram corresponds a parenthesized word ${\left({\left(x_1\right)}{\left(x_2\right)}x_1\right)}$. If we compare the structure of the counter term $Z_{\Gamma}$ obtained here with equation (\[eq:example\]) (which computes $S_{R}{\left[{\left({\left(x_1\right)}{\left(x_2\right)}x_1\right)}\right]}$), we see that the two objects have exactly the same structure after the identifications described in the section \[sec:sum\].
Proofs
======
Sweedler’s Notation {#app:Sweed}
-------------------
Let $U$ be any subword of a parenthesized word $X$, then our coproduct is defined in such a way that: $$\begin{aligned}
\Delta[X]=\sum_{ U }U\otimes {\left(X/U\right)}.\end{aligned}$$ This assertion is easy to prove using the induction on length of the words. It is obviously true for words of length $1$. Assume that it is true for word $X$ of length $n$ and then induce. Let us consider an irreducible parenthesized word ${\left(Xx\right)}$ of length $n+1$. $$\begin{aligned}
\Delta{\left[{\left(Xx\right)}\right]}&=&{\left(Xx\right)}\otimes e +e\otimes {\left(Xx\right)}+{\left({\mathbf{1_d}}\otimes B_{{\left(x\right)}}\right)}P\Delta{\left[X\right]},\\
&=&{\left(Xx\right)}\otimes e +e\otimes {\left(Xx\right)}+{\left({\mathbf{1_d}}\otimes B_{{\left(x\right)}}\right)}{\left(\sum_{\text{all subwords } U\neq e \text{ of X}}U\otimes {\left(X/U\right)}\right)},\\
&=&{\left(Xx\right)}\otimes e +e\otimes {\left(Xx\right)}+{\left(\sum_{\text{all subwords } U\neq e \text{ of X}}U\otimes {\left(X/U x\right)}\right)} ,\\
&=&{\left(Xx\right)}\otimes e +{\left(\sum_{\text{all subwords } U \text{ of X}}U\otimes {\left(X/U x\right)}\right)} ,\\
&=&\sum_{\text{all subwords } U \text{ of }{\left(Xx\right)}}U\otimes {\left(Xx\right)}/U \end{aligned}$$ which proves our assertion for irreducible word of length $n+1$. For an arbitrary word $XY$ of length $n+1$, the assertion follows by using the induction assumption and the fact that $\Delta{\left[XY\right]}=\Delta{\left[X\right]}\Delta{\left[Y\right]}$. This completes our proof.
Coassociativity of the coproduct {#app:coalg}
---------------------------------
Here we prove that the coproduct defined in equation (\[coprod\]) is coassociative and satisfies the following condition: $$\begin{aligned}
{\left(\Delta\otimes{\mathbf{1_d}}\right)}\Delta{\left[X\right]}&=&{\left({\mathbf{1_d}}\otimes\Delta\right)}\Delta{\left[X\right]},\ \ \ \forall X\in\mathcal{A}.\label{coass}\end{aligned}$$
We will prove this using induction on the length of the parenthesized words. It is trivial to see that $\Delta$ is coassociative when acting on the words of length $1$. For the induction we assume that it is coassociative acting on words of length $n$. First, we show that it is coassociative on irreducible parenthesized words of length $n+1$ and then we prove the assertion for arbitrary parenthesized words. We use the Sweedler’s notation and also drop the summation sign $\sum$ to simplify the notation further. Let $X$ be a parenthesized word of length $n$ then: $$\begin{aligned}
\Delta{\left[X\right]}&=&X_1\otimes X_2,\label{eq:Dx}\\
{\left(\Delta\otimes{\mathbf{1_d}}\right)}\Delta{\left[X\right]}&=&{\left({\mathbf{1_d}}\otimes\Delta\right)}\Delta{\left[X\right]}\label{eq:IndAss},\end{aligned}$$ where equation (\[eq:Dx\]) is just the simplified Sweedler’s notation and equation (\[eq:IndAss\]) is the induction assumption. Now, consider the parenthesized word $\left(Xx_j\right)$ of length $n+1$. A straightforward computation gives: $$\begin{aligned}
{\left(\Delta\otimes{\mathbf{1_d}}\right)}\Delta{\left[Xx_j\right]}&=&{\left(\Delta\otimes{\mathbf{1_d}}\right)}{\left({\left(Xx_j\right)}\otimes e+{\left({\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}{\left(X_1\otimes X_2\right)}\right)},\\
&=&\Delta{\left[Xx_j\right]}\otimes e + \Delta{\left[X_1\right]}\otimes {\left(X_2x_j\right)},\\
&=&{\left(Xx_j\right)}\otimes e\otimes e + X_1\otimes {\left(X_2x_j\right)}\otimes e + \Delta{\left[X_1\right]}\otimes {\left(X_2x_j\right)}.\label{eq:pr1}\end{aligned}$$ Now, let us compute the RHS of equation (\[coass\]). By using the definition (\[coprodSweed\]), we get. $$\begin{aligned}
{\left({\mathbf{1_d}}\otimes\Delta\right)}\Delta{\left[{\left(Xx_j\right)}\right]}&=&{\left(Xx_j\right)}\otimes e\otimes e +X_1\otimes {\left(X_2x_j\right)}\otimes e + X_1\otimes{\left[{\left({\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}\Delta{\left[X_2\right]}\right]}.\ \ \ \label{eq:pr2}\end{aligned}$$ First two terms in the above equation are the same as in equaton (\[eq:pr1\]). Let’s focus on the third term. An important result in this regard is the following. $$\begin{aligned}
{\left({\mathbf{1_d}}\otimes{\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}{\left({\mathbf{1_d}}\otimes\Delta\right)}\Delta{\left[X\right]} &=& {\left({\mathbf{1_d}}\otimes{\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}{\left({\mathbf{1_d}}\otimes\Delta\right)}{\left(X_1\otimes X_2\right)},\\
&=&{\left({\mathbf{1_d}}\otimes{\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}{\left(X_1\otimes \Delta{\left[X_2\right]}\right)},\\
&=& X_1\otimes {\left[{\left({\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}\Delta{\left[X_2\right]}\right]}.\end{aligned}$$ Using this we can write: $$\begin{aligned}
X_1\otimes {\left[{\left({\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}\Delta{\left[X_2\right]}\right]}&=&{\left({\mathbf{1_d}}\otimes{\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}{\left({\mathbf{1_d}}\otimes\Delta\right)}\Delta{\left[X\right]},\\
&=&{\left({\mathbf{1_d}}\otimes{\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}{\left(\Delta\otimes{\mathbf{1_d}}\right)}\Delta{\left[X\right]}, \\
&=&{\left({\mathbf{1_d}}\otimes{\mathbf{1_d}}\otimes B_{{\left(x_j\right)}}\right)}{\left(\Delta\otimes{\mathbf{1_d}}\right)}{\left[X_1\otimes X_2\right]},\\
&=&\Delta{\left[X_1\right]}\otimes {\left(X_2x_j\right)},\end{aligned}$$ where, the second equality just follows from the induction assumption (\[eq:IndAss\]). This is precisely the third term in equation (\[eq:pr1\]) and this complete the proof of coassociativity for parenthesized words of length $n+1$ an of the form ${\left(Xx_j\right)}$. Now, for a general parenthesized word $XY$ of length $n+1$, we use the property of the coproduct (\[compatcoprod\]) to get: $$\begin{aligned}
{\left({\mathbf{1_d}}\otimes\Delta\right)}\Delta{\left[XY\right]}&=&{\left({\mathbf{1_d}}\otimes\Delta\right)}{\left(\Delta{\left[X\right]}\Delta{\left[Y\right]}\right)},\\
&=&{\left({\left({\mathbf{1_d}}\otimes\Delta\right)}\Delta{\left[X\right]}\right)}{\left({\left({\mathbf{1_d}}\otimes\Delta\right)}\Delta{\left[Y\right]}\right)},\\
&=&{\left({\left(\Delta\otimes{\mathbf{1_d}}\right)}\Delta{\left[X\right]}\right)}{\left({\left(\Delta\otimes{\mathbf{1_d}}\right)}\Delta{\left[Y\right]}\right)},\\
&=&{\left(\Delta\otimes{\mathbf{1_d}}\right)}{\left(\Delta{\left[X\right]}\Delta{\left[Y\right]}\right)},\\
&=&{\left(\Delta\otimes{\mathbf{1_d}}\right)}\Delta{\left[XY\right]},\end{aligned}$$ where the first and second lines follow from property (\[compatcoprod\]), third equality follows fromt the induction assumption. Fourth and fifth lines again follow from (\[compatcoprod\]). This complete the proof of coassociativity for the coproduct.
Equivalence of definitions of antipode {#app:antipodeDef}
--------------------------------------
In the definition of the antipode, two definitions, (\[eq:Sdef1\]) and (\[eq:Sdef2\]), were given. For the antipode to be well defined, these two definitions should be equivalent. We prove this equivalence in the following.
We can strip off the parenthesized word ${\left(Xx_i\right)}$ from the argument of the antipode in equations (\[eq:Sdef1\]) and (\[eq:Sdef2\]) and represent antipode as an operator acting on $\mathcal{A}$. Then, we need to show that: $$\begin{aligned}
-{\mathbf{1_d}}-m{\left[{\left(S\otimes{\mathbf{1_d}}\right)}P_2 \Delta\right]}\ &=&-{\mathbf{1_d}}-\ m{\left[{\left({\mathbf{1_d}}\otimes S\right)}P_2\Delta\right]},\\
-{\mathbf{1_d}}-m{\left[{\left(S P_1\otimes P_1\right)} \Delta\right]}\ &=&-{\mathbf{1_d}}-\ m{\left[{\left(P_1\otimes SP_1\right)}\Delta\right]}.\label{eq:claim1}\end{aligned}$$ Both sides still involve the antipode $S$, let us do one more iteration on the both sides. For the left hand side we get: $$\begin{aligned}
LHS&=& -{\mathbf{1_d}}-m{\left[{\left({\left(-{\mathbf{1_d}}-m{\left[{\left(S P_1\otimes P_1\right)} \Delta\right]}\right)}P_1\otimes P_1\right)} \Delta\right]},\\
&=&-{\mathbf{1_d}}+m{\left[{\left(P_1\otimes P_1\right)} \Delta\right]}+m{\left[{\left({\left(m{\left[{\left(S P_1\otimes P_1\right)} \Delta\right]}\right)}P_1\otimes P_1\right)} \Delta\right]},\\
&=&-{\mathbf{1_d}}+m{\left[{\left(P_1\otimes P_1\right)} \Delta\right]}\nonumber \\ &&+m{\left[{\left(m\otimes{\mathbf{1_d}}\right)}{\left(S\otimes{\mathbf{1_d}}\otimes{\mathbf{1_d}}\right)}{\left(P_1\otimes P_1\otimes{\mathbf{1_d}}\right)}{\left(\Delta\otimes{\mathbf{1_d}}\right)}{\left(P_1\otimes P_1\right)}\Delta\right]},\\
&=&-{\mathbf{1_d}}+m{\left[{\left(P_1\otimes P_1\right)} \Delta\right]}\nonumber \\ &&+m{\left[{\left(m\otimes{\mathbf{1_d}}\right)}{\left(S\otimes{\mathbf{1_d}}\otimes{\mathbf{1_d}}\right)}{\left(P_1\otimes P_1\otimes P_1\right)}{\left(\Delta\otimes{\mathbf{1_d}}\right)}\Delta\right]},\label{LHS1}\end{aligned}$$ where the last equality follows because of the fact that $P_1\otimes P_1 \Delta P_1 = P_1\otimes P_1\Delta$, which is easy to confirm. For the right hand side, a similar computation yields: $$\begin{aligned}
RHS&=&-{\mathbf{1_d}}+m{\left[{\left(P_1\otimes P_1\right)} \Delta\right]}\nonumber \\ &&+m{\left[{\left({\mathbf{1_d}}\otimes m\right)}{\left({\mathbf{1_d}}\otimes{\mathbf{1_d}}\otimes S\right)}{\left(P_1\otimes P_1\otimes P_1\right)}{\left({\mathbf{1_d}}\otimes \Delta\right)}\Delta\right]}.\label{RHS1}\end{aligned}$$ From equations (\[LHS1\]) and (\[RHS1\]), we deduce that, to show the equivalence of the two definitions we need to prove the following: $$\begin{aligned}
&&m{\left[{\left({\mathbf{1_d}}\otimes m\right)}{\left({\mathbf{1_d}}\otimes{\mathbf{1_d}}\otimes S\right)}{\left(P_1\otimes P_1\otimes P_1\right)}{\left({\mathbf{1_d}}\otimes \Delta\right)}\Delta\right]}\nonumber \\ &=&
m{\left[{\left(m\otimes{\mathbf{1_d}}\right)}{\left(S\otimes{\mathbf{1_d}}\otimes{\mathbf{1_d}}\right)}{\left(P_1\otimes P_1\otimes P_1\right)}{\left(\Delta\otimes{\mathbf{1_d}}\right)}\Delta\right]}\end{aligned}$$ This is very easy to show using the previously established properties of the coproduct $\Delta$ and the product $m$. Using the coassociativity ${\left(\Delta\otimes {\mathbf{1_d}}\right)}\Delta={\left({\mathbf{1_d}}\otimes\Delta\right)}\Delta$, we can freely make the following change in the left side of the above equation: $$\begin{aligned}
{\mathbf{1_d}}\otimes{\mathbf{1_d}}\otimes S \to {\mathbf{1_d}}\otimes S \otimes {\mathbf{1_d}}.\end{aligned}$$ Similarly, now we make use of the associativity of the product, this implies that $m{\left({\mathbf{1_d}}\otimes m\right)}=m{\left(m\otimes {\mathbf{1_d}}\right)}$. Using this, we can again move the last two operators in the direct product to the first two places, yielding: $$\begin{aligned}
{\mathbf{1_d}}\otimes S \otimes {\mathbf{1_d}}\to S\otimes{\mathbf{1_d}}\otimes{\mathbf{1_d}}.\end{aligned}$$ This completes the proof for the equivalence of the two definitions.
Hopf algebra check {#app:hopfcheck}
------------------
Here, we show that the antipode defined earlier in this article actually satisfies the condition (\[hopfCom\]).
We will do this using induction. For a parenthesized word of length 1, $(x)$, it is easy to see that: $$\begin{aligned}
E\circ \overline{e}{\left[{\left(x\right)}\right]}=0,\end{aligned}$$ and $$\begin{aligned}
m{\left[{\left(S\otimes {\mathbf{1_d}}\right)}\Delta{\left[{\left(x\right)}\right]}\right]}&=&m{\left[{\left(S\otimes{\mathbf{1_d}}\right)}{\left({\left(x\right)}\otimes e + e\otimes {\left(x\right)}\right)}\right]},\\
&=&m{\left[-{\left(x\right)}\otimes e + e\otimes {\left(x\right)}\right]}=0.\end{aligned}$$ A similar computation yields $$\begin{aligned}
m{\left[{\left({\mathbf{1_d}}\otimes S\right)}\Delta{\left[{\left(x\right)}\right]}\right]}=0.\end{aligned}$$ Let us now assume that the assertion holds for parenthesized words of length $n$, consider an irreducible parenthesized word ${\left(Xx\right)}$ of length $n+1$. Since the map $P_1$ annihilates $e$, using the Sweedler’s notation we can write the antipode of ${\left(Xx\right)}$ as follows: $$\begin{aligned}
S{\left[{\left(Xx\right)}\right]}=-{\left(Xx\right)}-\sum_{X_{1}\neq e}S{\left[X_1\right]}{\left(X_{2}\ x\right)}.\end{aligned}$$ Now, $$\begin{aligned}
\Delta{\left[{\left(Xx\right)}\right]}&=&{\left(Xx\right)}\otimes e +\sum_{X_1}X_{1}\otimes {\left(X_2\ x\right)},\\
{\left(S\otimes {\mathbf{1_d}}\right)}\Delta{\left[{\left(Xx\right)}\right]}&=& S{\left[{\left(Xx\right)}\right]}\otimes e + \sum_{X_1}S{\left[X_{1}\right]}\otimes {\left(X_2\ x\right)},\\
&=&-{\left(Xx\right)}\otimes e-\sum_{X_{1}\neq e}S{\left[X_1\right]}{\left(X_{2}\ x\right)}\otimes e+ \sum_{X_1}S{\left[X_{1}\right]}\otimes {\left(X_2\ x\right)},\\
m{\left(S\otimes {\mathbf{1_d}}\right)}\Delta{\left[{\left(Xx\right)}\right]}&=&-{\left(Xx\right)}-\sum_{X_{1}\neq e}S{\left[X_1\right]}{\left(X_{2}\ x\right)}+ \sum_{X_1}S{\left[X_{1}\right]} {\left(X_2\ x\right)},\\
&=&-{\left(Xx\right)}+S{\left[e\right]}{\left(Xx\right)}=0,\end{aligned}$$ where the last line follows from the fact that then when $X_1=e$, $X_2=X$. Now, let us consider the case for $m{\left[{\left({\mathbf{1_d}}\otimes S\right)}\Delta{\left[{\left(Xx\right)}\right]}\right]}$. Due to the equivalence of two definitions (\[eq:Sdef1\]) and (\[eq:Sdef2\]), and the properties of $P_2$ we have the following identity: $$\begin{aligned}
{\left(S\otimes {\mathbf{1_d}}\right)}\sum_{X_1\neq e}X_1\otimes {\left(X_2 x\right)}= {\left({\mathbf{1_d}}\otimes S\right)}\sum_{X_1\neq e}X_1\otimes {\left(X_2 x\right)}.\end{aligned}$$ Using this, we find that: $$\begin{aligned}
{\left({\mathbf{1_d}}\otimes S\right)}\Delta{\left[{\left(Xx\right)}\right]}&=&{\left(Xx\right)}\otimes e + {\left({\mathbf{1_d}}\otimes S\right)}\sum_{X_1}X_1\otimes {\left(X_2 x\right)},
\\
&=& {\left(Xx\right)}\otimes e + {\left(S\otimes {\mathbf{1_d}}\right)}\sum_{X_1\neq e}X_1\otimes {\left(X_2 x\right)}+{\left({\mathbf{1_d}}\otimes S\right)}e\otimes {\left(Xx\right)},
\\
&=&{\left(Xx\right)}\otimes e + \sum_{X_1\neq e}S{\left[X_1\right]}\otimes {\left(X_2 x\right)}+{\left({\mathbf{1_d}}\otimes S\right)}{\left[e\otimes {\left(Xx\right)}\right]},\\
&=& {\left(Xx\right)}\otimes e + \sum_{X_1\neq e}S{\left[X_1\right]}\otimes {\left(X_2 x\right)}-e\otimes {\left(Xx\right)}\nonumber \\ &&-\sum_{X_{1}\neq e}e\otimes S{\left[X_1\right]}{\left(X_2x\right)},\end{aligned}$$ which implies $$\begin{aligned}
m{\left[{\left({\mathbf{1_d}}\otimes S\right)}\Delta{\left[{\left(Xx\right)}\right]}\right]}&=& 0.\end{aligned}$$ Since the counit annihilates any parenthesized word we finally conclude that: $$\begin{aligned}
m{\left[{\left(S\otimes {\mathbf{1_d}}\right)}\Delta{\left[{\left(Xx\right)}\right]}\right]}=0=E\circ\overline{e}{\left[{\left(Xx\right)}\right]}.\end{aligned}$$ For an arbitrary parenthesized word $XY$, due to the induction assumption and the property (\[compatcoprod\]) of the coproduct, the assertion holds trivially. This completes our proof.
Example {#app:example}
=======
Here, we will work out an elementary example which elucidates how all the different elements of the Hopf algebra fit together to give a finite result for a divergent integral. We will use a very simple toy model, defined below: $$\begin{aligned}
{\left(x_{j}\right)}{\left[c\right]}&\equiv& \int_{c}^{\infty}dy y^{-1-j\epsilon}\equiv I_j,\\
{\left(Xx_{j}\right)}{\left[c\right]}&\equiv& \int_{c}^{\infty}dy y^{-1-j\epsilon}X{\left[y\right]},\\
{\left(x_j\right)}{\left(x_k\right)}{\left[c\right]}&=& I_jI_k,\\
R{\left[X{\left[c\right]}\right]}&\equiv& X{\left[1\right]}.\end{aligned}$$ It is easy to see that $I_{j}$ is divergent as $\frac{1}{j\epsilon}$. We call the subscript $j$ in ${\left(x_j\right)}$, the loop order of ${\left(x_j\right)}$. This toy model is the simplest realization of our Hopf algebra. Let us consider the divergent graph $X={\left({\left(x_1\right)}{\left(x_2\right)}x_1\right)}$. Our claim is that the expression $X_r\equiv m{\left[{\left(S_R\otimes {\mathbf{1_d}}\right)}\Delta{\left[X{\left[c\right]}\right]}\right]}$ is a finite integral as expected from our Hopf algebra construction. By making use of the already worked out examples for $\Delta{\left[{\left({\left(x_i\right)}{\left(x_j\right)}x_k\right)}\right]}$, $S_R{\left[{\left({\left(x_i\right)}x_j\right)}\right]}$, $S_{R}{\left[{\left({\left(x_i\right)}{\left(x_j\right)}x_k\right)}\right]}$ and the fact $S_{R}{\left[XY\right]}=S_R{\left[X\right]}S_R{\left[Y\right]}$ we find that: $$\begin{aligned}
X_r&=&X{\left[c\right]}-{\left(x_1\right)}{\left[1\right]}{\left({\left(x_2\right)}x_1\right)}{\left[c\right]}-{\left(x_2\right)}{\left[1\right]}{\left({\left(x_1\right)}x_1\right)}{\left[c\right]}+{\left(x_1\right)}{\left[1\right]}{\left(x_2\right)}{\left[1\right]}{\left(x_1\right)}{\left[c\right]}\nonumber \\ &&-{\left(\text{first four terms with }c\text{ replaced by }1\right)}.\label{eq:Xr}\end{aligned}$$ Now, $$\begin{aligned}
T_1\equiv X{\left[c\right]}&=&\int_{c}^{\infty}dx x^{-1-\epsilon}\int_{x}^{\infty}dy y^{-1-2\epsilon}\int_{x}^{\infty}dz z^{-1-\epsilon},\\
T_2\equiv - {\left(x_1\right)}{\left[1\right]}{\left({\left(x_2\right)}x_1\right)}{\left[c\right]}&=&- \int_{c}^{\infty}dx x^{-1-\epsilon}\int_{x}^{\infty}dy y^{-1-2\epsilon}\int_{1}^{\infty}dz z^{-1-\epsilon},\\
T_3\equiv - {\left(x_2\right)}{\left[1\right]}{\left({\left(x_1\right)}x_1\right)}{\left[c\right]}&=&-\int_{c}^{\infty}dx x^{-1-\epsilon}\int_{1}^{\infty}dy y^{-1-2\epsilon}\int_{x}^{\infty}dz z^{-1-\epsilon},\\
T_4\equiv {\left(x_1\right)}{\left[1\right]}{\left(x_2\right)}{\left[1\right]}{\left(x_1\right)}{\left[c\right]}&=&\int_{c}^{\infty}dx x^{-1-\epsilon}\int_{1}^{\infty}dy y^{-1-2\epsilon}\int_{1}^{\infty}dz z^{-1-\epsilon}.\end{aligned}$$ The first two terms can be combined to get: $$\begin{aligned}
T_1+T_2=-\int_{c}^{\infty}dx x^{-1-\epsilon}\int_{x}^{\infty}dy y^{-1-2\epsilon}\int_{1}^{x}dz z^{-1-\epsilon}.\end{aligned}$$ The third term can be written as $$\begin{aligned}
T_3&=&-\int_{c}^{\infty}dx x^{-1-\epsilon}\int_{1}^{\infty}dy y^{-1-2\epsilon}{\left(\int_{1}^{\infty}dz z^{-1-\epsilon}-\int_{1}^{x}dz z^{-1-\epsilon}\right)},\\
T_3 &=&-T_4+\int_{c}^{\infty}dx x^{-1-\epsilon}\int_{1}^{\infty}dy y^{-1-2\epsilon}\int_{1}^{x}dz z^{-1-\epsilon}.\end{aligned}$$ So that the sum of the four terms is: $$\begin{aligned}
T_1+T_2+T_3+T_4=\int_{c}^{\infty}dx x^{-1-\epsilon}\int_{1}^{x}dy y^{-1-2\epsilon}\int_{1}^{x}dz z^{-1-\epsilon}.\end{aligned}$$ Plug this in equation (\[eq:Xr\]) we finally obtain the expression: $$\begin{aligned}
X_{r}=-\int_{1}^{c}dx x^{-1-\epsilon}\int_{1}^{x}dy y^{-1-2\epsilon}\int_{1}^{x}dz z^{-1-\epsilon},\end{aligned}$$ which is clearly well defined and finite in the $\epsilon \to 0$ limit. Although this was a very simple example, there should be no hinderance in generalizing this to more realistic QFT examples. If we consider some realistic Feynman graph, our Hopf algebra will renormalize it with the same ease by applying the operator $m{\left[{\left(S\otimes {\mathbf{1_d}}\right)}\Delta\right]}$.
[100]{}
D. Kreimer, *On the Hopf algebra structure of perturbative quantum field theories*, [arXiv:q-alg/9707029](http://arxiv.org/pdf/q-alg/9707029v4.pdf). A. Connes and D. Kreimer, *Hopf Algebra, Renormalization and Non Commutative Geometry*, [arXiv:hep-th/9808042](http://arxiv.org/pdf/hep-th/9808042.pdf). D. J. Broadhurst, D. Kreimer, *Renormalization automated by Hopf algebra*, [arXiv:hep-th/9810087](http://arxiv.org/pdf/hep-th/9810087.pdf). D. Kreimer, R. Delbourgo, *Using the Hopf algebra structure of QFT in calculations*, [arXiv:hep-th/9903249](http://arxiv.org/pdf/hep-th/9903249.pdf). A. Connes, D. Kreimer, *Lessons from Quantum Field Theory*, [hep-th/9904044](http://arxiv.org/pdf/hep-th/9904044.pdf). D. Kreimer, *On Overlapping Divergences*, [arXiv:hep-th/9810022](http://arxiv.org/abs/hep-th/9810022). W. v. Suijlekom, *The Hopf algebra of Feynman graphs in QED*, [hep-th/0602126](http://arxiv.org/abs/hep-th/0602126) E. Panzer, *Hopf Algebraic Renormalization of Kreimer’s Toy Model*, [arXiv:1202.3552\[math-ph\]](http://arxiv.org/abs/1202.3552). D. Kreimer, *Knots and Feynman Diagrams*, 2000, Cambrige University Press. J. C. Collins, *Renormalization*, 1986, Cambridge University Press. S. Majid, *Foundations of Quantum Group Theory*, 1996, Cambridge University Press. K. Hepp, *Proof of the Bogoliubov-Parasiuk theorem on renormalization*, Comm. Math. Phys., 1966, Springer.
[^1]: The two definitions correspond to recursive and non-recursive form of the forest formula
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the property for an input-output system to map unimodal inputs to unimodal outputs. As a first step, we analyse this property for linear time-invariant (LTI) systems, static nonlinearities, and interconnections of those. In particular, we show how unimodality is closely related to the concepts of positivity, monotonicity, and total positivity.'
author:
- 'Christian Grussler and Rodolphe Sepulchre [^1][^2]'
bibliography:
- 'refkpos.bib'
- 'refopt.bib'
- 'refpos.bib'
- 'science.bib'
title: Strongly unimodal systems
---
Strong unimodality, logarithmic concavity, external positivity, positive systems, damping, neural networks, total positivity.
Introduction
============
System analysis via the concept of [*positivity*]{} has gained considerable popularity in the recent years [@farina2011positive; @rantzer2015scalable; @son1996robust; @tanaka2011bounded; @sootla2012scalable]. From a modelling viewpoint, the value of devoting a special treatment to dynamical models that manipulate positive variables (states, inputs, or outputs) was recognised early by Luenberger [@luenberger1979introduction], as this situation frequently arises in networks, economics, biology, transport, etc. From an analysis viewpoint, the increasing use of convexity analysis in system theory led a number of authors to revisit the classical linear-quadratic theory of linear-time invariant (LTI) systems in the presence of positive constraints. Positivity was shown to be a source of numerical tractability and simplicity even in the standard context of Lyapunov analysis [@rantzer2015scalable], optimal control design [@tanaka2011bounded; @ebihara2012optimal], or system gain computation [@farina2011positive]. It is surprising that such properties have only started to gain widespread interest. Positivity concepts have also proven useful beyond LTI systems. Positivity is central to consensus and distributed system analysis, which involves linear time-varying models [@Sepulchre2010]. It is also central to the theory of monotone systems [@Hirsch2006; @Angeli2003] and to the recent development of differential positivity analysis [@Forni2016; @Mostajeran2018a].
The present paper focuses on the input-output (or external) concept of positivity: a system is called (externally) positive if it maps positive inputs to positive outputs. For linear systems, this property is equivalent to (external) monotonicity: input signals with a time-derivative that has no sign variation are mapped to output signals with the same property. In a similar spirit, we aim at characterizing systems that map unimodal inputs to unimodal outputs : input signals with a time-derivative that has at most [*one*]{} sign variation are mapped to output signals with the same property. We call such systems [*strongly unimodal*]{} because a well-known result in probability theory: strongly unimodal densities are precisely those that map unimodal densities to unimodal densities by convolution [@ibragimov1956composition].
While strong unimodality has been extensively studied in statistics and interpolation theory, it does not seem to have received much attention in system theory. Our motivation is that it is nevertheless a natural property to expect in the context of mean-field models. Classical examples include amplifier modelling in electronics, conductance modelling in neurophysiology, or reaction rate modelling in biochemistry. In first approximation, it is natural in all those examples to expect that a unimodal input is mapped to a unimodal output (see e.g. [@Sepulchre2018] for details). This motivation makes direct contact with the questions that have motivated the development of total positivity theory in interpolation theory and in statistics [@karlin1968total; @dharmadhikari1988unimodality; @Schoenberg1988polyI]. Starting with the early work of Schoenberg [@Schoenberg1930vari], the entire theory has been motivated by a characterization of maps with [*variation-diminishing*]{} properties. We believe that such properties could play an important role as well in system analysis of models grounded in mean-field principles.
As a first step towards a more general theory, the main focus of the present paper is the simple class of LTI systems. Our main result is to show that strong unimodality of a LTI state-space model can be studied as the external positivity of a [*compound*]{} LTI state-space model. This methodology is central to the theory of total positivity: properties of unimodal systems are studied via the properties of a compound [*externally positive*]{} system. As an elementary application, we single out the main difference between positive and unimodal LTI systems: externally positive systems have one dominant real pole, whereas unimodal systems have two dominant real poles. Using properties of log-concave functions, we derive a number of properties for the interconnections of LTI unimodal systems and monotone functions. While elementary, those preliminary results suggest a strong potential of total positivity theory in system analysis.
The remainder of the paper is organized as follows. After some preliminaries in , the theory of external positive systems is briefly reviewed in . Then introduces the analog concept of strong unimodality. Our preliminary results are presented in . introduces elementary interconnection properties of unimodal systems. The papers ends with concluding remarks in . Proofs are given in the appendix.
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Preliminaries {#sec:prelim}
=============
Notations
---------
For real valued matrices $X = (x_{ij}) \in {\mathbb{R}^{n \times m}}$, including vectors $x = (x_i) \in \mathbb{R}^n$, we use the following notation. $X$ is called *nonnegative*, $X \in {\mathbb{R}^{n \times m}}_{\geq 0}$, if all its entries $x_{ij}$ are nonnegative. If $X \in {\mathbb{R}^{n \times n}}$, then $\sigma(X) = \{\lambda_1(X),\dots,\lambda_n(X)\}$ denotes its *spectrum*, where we order the eigenvalues by descending real part, i.e., $\lambda_1(X)$ is the eigenvalue with the largest real part, counting multiplicity. In case that the real part of two eigenvalues is equal, we subsort them by ascending imaginary part. In case that the real part of some eigenvalues is equal, we subsort them by ascending modulus. Further, $X$ is said to be *positive semidefinite*, $X \succeq 0$, if $X = X^{\mathsf{T}}$ and $\sigma(X) \subset \mathbb{R}_{\geq 0}$. Analogously, we define positive and positive definite matrices, $X \in \mathbb{R}_{>0}$ and $X \succ 0$, respectively. Letting $I_n$ denote the identity matrix in $\mathbb{R}^{n \times n}$, the *Kronecker sum* of two matrices $X \in\mathbb{R}^{n \times n}$ and $Y \in \mathbb{R}^{m \times m}$ is given by $X \oplus Y := (X \otimes I_m) + (I_n \otimes Y)$, where $\otimes$ stands for the Kronecker product.
For a real valued function $g: \mathbb{R} \to \mathbb{R} \cup \{ -\infty \}$, we say that it is *concave* if $g(\lambda x + (1-\lambda)y) \geq \lambda g(x) + (1-\lambda) g(y)$ for all $0 \leq \lambda \leq 1$. The set of all concave functions will be denoted by $\mathcal{S}_{\text{c}}$, the set of all *nonnegative functions* is given by $\mathcal{S}_{\geq 0} := \{g: \mathbb{R} \to \mathbb{R}_{\geq0}\}$. Further, the set of all integrable functions will be denoted by $L_1$. Then, the *convolution* of two real-valued functions $g$ and $u$ is defined as $(g \ast u)(t) = \int_{-\infty}^\infty f(t-\tau) g(\tau) d \tau$ and for $\mathcal{S} \subset \mathbb{R}$, the (1-0) indicator function is defined as $$\begin{aligned}
\mathds{1}_{\mathcal{S}}(x) = \begin{cases}
1 & x \in \mathcal{S}\\
0 & x \notin \mathcal{S}
\end{cases}
\end{aligned}$$ Finally, we define $\dot{g}:= \dfrac{d}{dt}g $, $\ddot{g} = \dfrac{d^2}{dt^2}g$, $\dddot{g} = \dfrac{d^3}{dt^3}g$ to be the first, second and third derivative of a real valued function $g$.
State-space realizations
------------------------
A LTI state-space model $$\label{eq:SISO}
\begin{aligned}
\dot{x} &= Ax + bu\\
y &= cx
\end{aligned}$$ where $A \in \mathbb{R}^{n\times n}$, $b,c^{\mathsf{T}}\in \mathbb{R}^n$ defines a unique causal LTI system with impulse response $g(t) = ce^{At}b \mathds{1}_{[0,\infty)}$. The triple $(A,b,c)$ is called a *realization* of this impulse response. Further, we also refer to $(A,b,c)$ as an LTI system and mean \[eq:SISO\]. The following proposition is crucial for the derivation of our results.
\[thm:impprod\] For $A_1 \in \mathbb{R}^{n_1 \times n_1}$, $b_1,c_1^{\mathsf{T}}\in \mathbb{R}^{n_1}$, $A_2 \in \mathbb{R}^{n_2 \times n_2}$ and $b_2,c_2^{\mathsf{T}}\in \mathbb{R}^{n_2}$, let $g_1(t) = c_1e^{A_1t}b_1 \mathds{1}_{[0,\infty)}$ and $g_2(t) = c_2e^{A_2t}b_2 \mathds{1}_{[0,\infty)}$. Then $g_1(t)g_2(t) \mathds{1}_{[0,\infty)} = \bar{c} e^{\bar{A}t}\bar{b} \mathds{1}_{[0,\infty)}$, where $$\begin{aligned}
\bar{A} = A_1 \oplus A_2, \quad \bar{b} = b_1 \otimes b_2, \quad \bar{c} = c_1 \otimes c_2.
\end{aligned}$$
For the ease of exposition, we only consider [*causal*]{} LTI systems in the remainder of this paper, that is, we assume $g(t) = \mathds{1}_{[0,\infty)}g(t)$.
Externally positive LTI systems {#sec:ex_pos}
===============================
Next we review the concept of external positivity.
An LTI system with impulse response $g \mathds{1}_{[0,\infty)}$ is called *externally positive* if $$\forall u \in \mathcal{S}_{\geq 0}: \ g \ast u \in \mathcal{S}_{\geq 0}$$
The set of all externally positive system defines a convex cone. It is closed under parallel as well as serial interconnection. We review two classical equivalent definitions of external positivity.
\[lem:nonneg\_imp\] An LTI system is externally positive if and only if its impulse response is nonnegative.
\[lem:ex\_mono\] An LTI system with impulse response $g $ is externally positive if and only if for all monotonically increasing $u \in \mathcal{S}_{\geq 0}$ it holds that $y= g \ast u \in \mathcal{S}_{\geq 0}$ is monotonically increasing.
For completeness, a proof of \[lem:ex\_mono\] is given in . We also recall the following important consequence of positivity.
\[prop:dominantpole\] If $(A,b,c)$ is an externally positive LTI system, then $\lambda_1(A) \in \mathbb{R}$.
This proposition links external positivity to the internal positivity property of mapping a cone to a cone in the state-space. Perron-Frobenius theory shows that matrices that contract a cone also have a dominant eigenvalue and an eigenvector in the interior of the cone [@luenberger1979introduction].
Verifying external positivity is hard. Nevertheless, there exist several sufficient tests [@anderson1996nonnegative; @grussler2014modified; @altafini2016minimal; @grussler2012symmetry]. The following test from [@grussler2012symmetry] is particularly tractable.
\[prop:ex\_pos\_test\] Let $(A,b,c)$ be an LTI system and assume that there exists $Q = Q^{\mathsf{T}}\in {\mathbb{R}^{n \times n}}$ and $\gamma \in \mathbb{R}$ such that
$$\begin{aligned}
&{A}^{\mathsf{T}}Q + Q {A} + 2\gamma Q \preceq 0\\
&{b}^{\mathsf{T}}Q {b} \leq 0\\
&Q + {c}^{\mathsf{T}}{c} \succ 0\\
&{c}{b} \geq 0\\
&\lambda_{n-1}(Q) > 0 > \lambda_{n}(Q)
\end{aligned}$$
Then $(A,b,c)$ is externally positive, i.e., $\forall t \geq 0 : ce^{At}b \geq 0$.
Strongly unimodal LTI systems {#sec:strong_uni}
=============================
In the following, we introduce the class of strongly unimodal LTI systems.
A function $g: \mathbb{R} \to \mathbb{R}$ is called *unimodal* if one of the following equivalent conditions hold:
1. $g$ has a unique local maximum, i.e. there exists a mode $m \in \mathbb{R}$ such that $f$ is montonotonically increasing on $(-\infty, m]$ and montonically decreasing on $[m,+\infty)$.
2. $g$ is quasi-concave, i.e., $$g(\lambda x + (1-\lambda) y) \geq \min \lbrace g(x), g(y) \rbrace$$ for all $x,y$ and $\lambda \in [0,1]$.
The set of all unimodal functions is denoted by $\mathcal{S}_{\text{qc}}$.
\[def:strong\_unimod\] An LTI system with impulse response $g $ is called *strongly unimodal* if $$\forall u \in \mathcal{S}_{\text{qc}}: g \ast u \in \mathcal{S}_{\text{qc}}$$
The impulse response of a strongly unimodal LTI system is certainly unimodal (approximate the Dirac impulse with the unimodal Dirac sequence, $\delta_\epsilon(t) = \frac{1}{2 \pi \epsilon} e^{-\frac{t^2}{\epsilon}}$ for $\epsilon > 0$ and apply the definition). However, unimodality of the impulse response is not sufficient. This observation was first made by Ibragimov [@ibragimov1956composition], who introduced the terminology of strong unimodality in the context of probability distributions.
$g \in \mathcal{S}_{\geq 0}$ is called *log-concave* if for all $x,y \in \mathbb{R}$ and $\lambda \in [0,1] $: $$g(\lambda x + (1-\lambda) y) \geq g(x)^\lambda g(y)^{1-\lambda}.$$ Equivalently, $g$ is log-concave if and only if $g(x) = e^{\phi(x)}$ for some $\phi \in \mathcal{S}_c$, i.e., $\log(g) \in \mathcal{S}_{\text{c}}$. The set of all log-concave functions is denoted by $\mathcal{S}_{\text{logc}}$
\[prop:log\_conv\_unimod\] $g \in L_1 \cap \mathcal{S}_{\text{logc}}$ if and only if $$\forall u \in \mathcal{S}_{\text{qc}}: g \ast u \in \mathcal{S}_{\text{qc}}.$$
Thus, an LTI system is strongly unimodal if and only if its impulse response is log-concave. This means that strongly unimodal systems are a subset of externally positive systems \[lem:nonneg\_imp\]. In particular, it holds that $$\mathcal{S}_{\text{c}} \cap \mathcal{S}_{\geq 0} \subset \mathcal{S}_{\text{logc}} \subset \mathcal{S}_{\text{qc}} \label{prop:log_conc_unimod}.$$
We note that many unimodal density functions are also log-concave, e.g., for the exponential distribution, normal distribution, Laplace distribution, etc. [@boyd2004convex; @karlin1968total; @dharmadhikari1988unimodality]. In probability theory, log-concave density functions form a set of [*well-behaved*]{} unimodal density functions [@Samworth2017].
The next results reformulate log-concavity as a positivity condition.
\[lem:twice\_diff\] Let $g \in \mathcal{S}_{\geq 0}$ be twice-differentiable and $\mathcal{I} \subset \mathbb{R}$ be an interval. Then $g \in \mathcal{S}_{\text{logc}}$ if and only if $$\forall t \in \mathcal{I}: \ \dot{g}(t)^2 - g(t) \ddot{g}(t) \geq 0.$$
Follows by [@boyd2004convex Sec. 3.5.2] and the fact that if $g \in \mathcal{S}_{\text{logc}}$ is then $g \mathds{1}_{\mathcal{I}} \in \mathcal{S}_{\text{logc}}$.
\[prop:strong\_uni\_sys\] A causal LTI system with impulse response $g \in L_1$ is strongly unimodal if and only if $g \mathds{1}_{[0, \infty)} \in \mathcal{S}_{\geq 0}$ and $$\forall t \geq 0: \dot{g}(t)^2 - g(t) \ddot{g}(t) \geq 0.$$
With this proposition, one immediately verifies that any externally positive first-order system is also strongly unimodal. One also obtains the following test for second-order systems.
\[cor:second\_order\] Let $g$ be the impulse response of a causal stable LTI second-order system. Then the system is strongly unimodal if and only if $g \in \mathcal{S}_{\geq 0}$ and $$\dot{g}(0)^2 - g(0) \ddot{g}(0) \geq 0.$$
A proof to \[cor:second\_order\] is provided in .
\[ex:MSD\] Strong unimodality prevents oscillations in the step response of a system. As an illustration, the classical *mass-spring-damper system* with external force $u$ (see ), is modelled by the differential equation $$\begin{aligned}
\ddot{x} + \frac{\beta}{m} \dot{x} + \frac{k}{m} x = u \label{eq:MSD}
\end{aligned}$$ where $x$ stands for the displacement of the mass and $m,k,\beta >0$ denote the mass, spring and damping coefficients, respectively. Letting, $p := \sqrt{\beta^2 - 4k}$, the (causal) impulse response $g $ of this system is $$\label{eq:MSD_imp}
g(t) = \frac{m}{p }\left(e^{-\frac{(\beta - p)t}{2}}-e^{- \frac{(\beta+p)t}{2}}\right).$$ In the overdamped case, $p \geq 0$, it follows that $g \in \mathcal{S}_{\geq 0}$. Further, since $\dot{g}(0)^2 - g(0) \ddot{g}(0) = 1$, \[cor:second\_order\] implies that the system is also strongly unimodal. An example output for a unimodal input can be found in \[fig:imp\_MSDs\]. Thus, strong unimodality requires the mass-spring-damper system to be overdamped. This will be made even clearer in \[thm:poles\].
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\(M) ++ (0,.75 cm) – ++(0,.5 cm) – ++(1.75,0) node\[near end, above, draw = none\][$x$]{};
(ground) \[ground,anchor=north,yshift=-0.3cm,minimum width=4 cm,xshift = .5 cm\] at (M.south) ; (ground.north east) – (ground.north west);
(M.south west) ++ (0.2cm,-0.15cm) circle (0.15cm) (M.south east) ++ (-0.2cm,-0.15cm) circle (0.15cm);
(wall.157) – ($(M.north west)!(wall.157)!(M.south west)$) node\[above, draw = none, midway,yshift = 1.5 pt\][$k$]{}; (wall.18) – ($(M.north west)!(wall.18)!(M.south west)$) node\[above, draw = none, midway, yshift = 1.5 pt\][$\beta$]{};
(M.west) ++ (1.5 cm,0) – ++(1,0) node\[above,draw = none, midway\] [$u$]{};
State-space characterization of strong unimodality {#sec:state_space}
==================================================
In this section, we present our main results on strongly unimodal systems. Our first result shows that strong unimodality of a state-space model is equivalent to external positivity of a compound state-space model.
\[thm:log\_sys\] Let $(A,b,c)$ be the realization of a causal impulse response $g (t) = ce^{At}b \mathds{1}_{[0,\infty)}(t)$. Then $\dot{g}(t)^2-g(t) \ddot{g}(t) $ is the impulse response of the state-space model $(\bar{A},\bar{b},\bar{c})$, where $$\begin{aligned}
\bar{A} = A \oplus A , \quad \bar{b} =
Ab \otimes Ab - b \otimes A^2b, \quad \bar{c} =
c \otimes c, \label{eq:log_state}
\end{aligned}$$ i.e., $(A,b,c)$ is strongly unimodal if and only if $(A,b,c)$ and $(\bar{A},\bar{b},\bar{c})$ are externally positive systems.
A minimal realization $(\tilde{A},\tilde{b},\tilde{c})$ of $(\bar{A},\bar{b},\bar{c})$ has the following poles: $$\sigma(\tilde{A}) \subset \{\lambda_i(A) + \lambda_j(A): j > i\}. \label{eq:set_poles}$$ Further, if $(A,b,c)$ is minimal, then equality holds in \[eq:set\_poles\].
\[thm:log\_sys\] is proven in . Note that a tractable, sufficient test for external positivity of $(A,b,c)$ and $(\tilde{A},\tilde{b},\tilde{c})$ is given in \[prop:ex\_pos\_test\].
Next we present a key property of strongly unimodal LTI systems.
\[thm:poles\] If $(A,b,c)$ is the minimal realization of a strongly unimodal LTI system, then it has two dominant real poles, that is, $\lambda_1(A) \in \mathbb{R}$ and $ \lambda_2(A)
\in \mathbb{R}$.
This property illustrates how much strong unimodality restricts positivity: externally positive sytems require one dominant real pole, whereas stronly unimodal systems have two dominant real poles. The property also provides a mathematical justification for our damping interpretation in for unimodal LTI systems of arbitrary order. In particular, if the system is of order three, then the three poles are necessarily real.
Consider the mass-spring-damper system \[eq:MSD\] in series with an integrator. The dynamics are described by $$\begin{aligned}
\dddot{x} + \frac{\beta}{m} \ddot{x} + \frac{k}{m} \dot{x} = u \label{eq:MSD_int}
\end{aligned}$$ and the impulse response $g$ is $$\label{eq:MSD_int_imp}
g(t) = \frac{m}{p} \int_{0}^{t} \left(e^{-\frac{(\beta-p)\tau}{2}}-e^{- \frac{(\beta+p)\tau}{2}}\right) d \tau.$$ In the underdamped case, $p < 0$, the integrand undergoes a harmonic damped oscillation with an initial positive displacement (spring extension), which is why the system is externally positive However, due to the negative displacement phases of the integrand (spring contraction), $g$ inherits those oscillations, and is therefore not unimodal. As we noticed earlier, unimodality of the impulse response is necessary for a system to be strongly unimodal. This fact is also visualized by the example input \[subfig:MSD\_input\] with corresponding output \[subfig:output\_MSD\].
file[u\_msd\_gauss.txt]{}; (nl) at (current axis.north); file[y\_msd\_gauss.txt]{}; \[output:MSD\_gauss\_int\] file[y\_msd\_gauss\_over.txt]{}; \[output:MSD\_gaus\_over\] (nl) at (current axis.north);
at (\[yshift= -8 mm\]my plots c1r1.south) ; at (\[yshift=-8 mm\]my plots c1r2.south) ;
Linear and Non-linear interconnections {#sec:inter}
======================================
What makes positivity and unimodality properties attractive is that they are not restricted to linear models. Here we illustrate some interconnection properties that involve LTI models and static nonlinearities. We first recall the following two results.
\[lem:prop\_log\_conc\] Log-concave functions are closed under convolution and multiplication.
In particular, products of log-concave impulse responses lead to strongly unimodal LTI systems.
\[lem:com\_log\_unimod\] If $g \in \mathcal{S}_{\text{qc}}$ and $f:\mathbb{R} \to \mathbb{R}$ is monotonically increasing, then the composition $f \circ g\in \mathcal{S}_{\text{qc}}$.
By adopting the definition of strongly unimodal LTI systems in \[def:strong\_unimod\], provide us with the following interconnection properties.
Serial interconnections of strongly unimodal LTI systems and static monotonically increasing non-linearities are strongly unimodal.
=\[draw, very thick, align = center,circle\]
\(a) \[neu\] [$\frac{w_1}{\tau_1 s+1}$]{}; (b) \[left of=a,node distance=1.2cm, coordinate\]; (b) edge node\[above\][$u$]{} node(u\_inc)\[above\] (a); (end) \[right of=a, node distance=1cm\]; (a) edge node\[above\] (end);
(sig) \[int, at = (end), minimum height = .5 cm, anchor = [west]{}\]
file[sigmoid.txt]{};
;
(mid1) \[right of=sig, node distance = 1.2 cm\]; (sig) edge node\[above\] (mid1);
(mid2) \[right of=mid1, node distance = .12 cm\];
(mid3) \[right of=mid2, node distance = .3 cm\];
(mid2) edge node\[above\] (mid3);
(b1) \[right of=mid3, node distance = .12 cm\];
(sys2) \[right of=b1, node distance = 1.2 cm\];
(a1) \[neu, at = (sys2)\] [$\frac{w_n }{\tau_n s+1}$]{}; (b1) edge node\[above\] (a1);
(end1) \[right of=a1, node distance=1cm\]; (a1) edge node\[above\] (end1);
(sig1) \[int, at = (end1), minimum height = .5 cm, anchor = [west]{}\]
file[sigmoid.txt]{};
; Output Sigmoid 2 (end2) \[right of=sig1, node distance = 1.2 cm\]; (sig1) edge node\[above\][$v$]{} (end2);
An example of such non-linear systems is the serial interconnection of neurons \[fig:neuron\], where the output firing rate $v$ of each neuron is modeled by [@dayan2001theoretical] $$\begin{aligned}
\tau \dot{I_s} &= - I_s + w u\\
v &= F(I_s)
\end{aligned}$$ where $u$ is an input rate, $\tau,w > 0$ are rate and weight coefficients, $I_s$ is the synaptic current and $F$ is a static non-linear activation function, e.g., the sigmoid function $F(x) = \frac{1}{1+e^{-x}}$. By our interconnection rules, the serial interconnection of such systems is strongly unimodal.
A contrario, the next result shows that parallel interconnections of strongly unimodal LTI systems are not necessarily strongly unimodal. This is in contrast to positive systems. The following result shows that this is the case even for first-order models.
\[thm:symm\_sys\] Let $g(t) = \sum_{i=1}^n b_i e^{-\alpha_i t}$ with $b_i, \alpha_i > 0$ and $n \geq 2$ be such that $\alpha_{i} \neq \alpha_{i+1}$ for all $i$. Then, $$\begin{aligned}
\forall t \geq 0: \ \dot{g}(t)^2-g(t) \ddot{g}(t) < 0.
\end{aligned}$$ Therefore, $g \mathds{1}_{[0,\infty)} \in \mathcal{S}_{\geq 0} \setminus \mathcal{S}_{\text{logc}}$, which implies that strongly unimodal LTI systems are not closed under parallel interconnection.
The following result shows that the difference of two positive systems can be strongly unimodal.
\[prop:diff\] Let $g = b_1 e^{-\alpha_1 t} - \sum_{i=2}^n b_i e^{-\alpha_i t}$ with $b_i, \alpha_i > 0$ and $n \geq 1$ be such that $$\forall j \geq 2: \ 2 (\alpha_{1} - \alpha_j)^2 \geq \max_i (\alpha_i -\alpha_j)^2 \label{eq:ass}$$ Then, $g \mathds{1}_{[0,\infty)} \in \mathcal{S}_{\geq 0}$ if and only if $g \mathds{1}_{[0,\infty)} \in \mathcal{S}_{\text{logc}}$.
By \[thm:symm\_sys,prop:diff\], we can see that while the sum of two first order strongly unimodal LTI systems is not strongly unimodal, the difference preserve strong unimodality if it is externally positive.
Finally note that \[lem:prop\_log\_conc\] also gives the following stronger result for log-concave inputs.
Let $g $ be the impulse response to a strongly unimodal LTI system. Then, $$\forall u \in \mathcal{S}_{\text{logc}}: g \ast u \in \mathcal{S}_{\text{logc}}.$$
Concluding remarks {#sec:conc}
==================
This paper has introduced the class of strongly unimodal systems, which is characterized by preserving unimodality from input to output. Our main result is that unimodality of a state-space model is equivalent to positivity of a compound state-space model (\[thm:log\_sys\]). We have also shown that unimodal systems are a subclass of externally positive systems. As a main property, they have [*two*]{} dominant real poles rather than [*one*]{}.
In future work, we would like to generalize our results with the theory of total positivity [@karlin1968total; @gantmacher1950oszillationsmatrizen]. We anticipate that systems with a fixed number of dominant real poles can be studied via the external positivity of a compound system. This paves the way for novel system analysis tools to characterize input-output properties via the important [*variation diminishing*]{} concept: inputs with a certain number of variations are mapped to outputs with the same (or less) number of variations. The theory of total positivity suggests that this analysis framework is general, with plausible extensions to discrete LTI systems, linear time-space-invariant (LTSI) models, linear time-varying linear systems, and nonlinear systems.
\[sec:app\]
Proofs
------
### Proof to \[lem:ex\_mono\] {#proof:ex_mono}
We want to show that $\dot{y}: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$. By [@flanders1973differentiation], $\frac{d}{dt} y(t) = g(0)u(t) + \int_0^t ce^{A(t-\tau)}b \dot{u}(\tau) d\tau$, where the monotonicity of $u$ implies that $\dot{u}(s)$ exist almost everywhere on $[0,t]$ [@cohn2013measure Theorem 6.3.3]. Hence, since $\dot{u}, u \in \mathcal{S}_{\geq 0}$, applying \[lem:nonneg\_imp\] proves our claim.
### Proof to \[cor:second\_order\] {#proof:second_order}
By \[lem:nonneg\_imp\], it follows that $g(t) = \beta_1 e^{-\lambda_1 t} + \beta_2 e^{-\lambda_2}$, $\lambda_{1}, \lambda_2 > 0$ and $\beta_1,\beta_2 \in \mathbb{R}$. Then, $$\dot{g}(t)^2 - g(t) \ddot{g}(t) = -\beta_1 \beta_2e^{-t(\lambda_1 + \alpha_2)}(\lambda_1-\lambda_2)^2.$$
### Proof to \[thm:log\_sys\] {#proof:log_sys}
The first part is an application of and the fact that $\frac{d^k}{dt^k} g(t) = cA^k e^{At}b$. In order to prove the second part, let $T \in \mathbb{C}^{n \times n}$ be such that $\hat{J} = T^{-1}AT$ is the complex Jordan form of $A$. Then, with $\hat{b} := T^{-1}b$ and $\hat{c} := cT$,
$$\begin{gathered}
\left[\frac{d}{dt} g(t)\right]^2-g(t) \frac{d^2}{dt^2} g(t) \\
= {c}e^{{A}t}\left({A}{b}{b}^{\mathsf{T}}- {b}{b}^{\mathsf{T}}{A}^{\mathsf{T}}\right)e^{{A}^{\mathsf{T}}t}{A}^{\mathsf{T}}{c}^{\mathsf{T}}= \hat{c}K(t)\hat{J}^{\mathsf{T}}\hat{c}^{\mathsf{T}}.
\end{gathered}$$
with $K(t):=e^{\hat{J}t}\left({J}\hat{b}\hat{b}^{\mathsf{T}}- \hat{b}\hat{b}^{\mathsf{T}}\hat{J}^{\mathsf{T}}\right)e^{\hat{J}^{\mathsf{T}}t}$. Since $K(t) = -K(t)^{\mathsf{T}}$ and $\hat{J}$ in Jordan form, we conclude that $\left[\frac{d}{dt} g(t)\right]^2-g(t) \frac{d^2}{dt^2} g(t)$ only depends on exponentials of the form $e^{(\lambda_i(A)+\lambda_j(A))t}$ with $j > i$. Hence, \[eq:set\_poles\] follows by $\sigma(\tilde{A}) \subset(\bar{A})$.
To see the last claim notice that if $(\hat{J},\hat{b},\hat{c})$ is minimal, then the controllability of $(\hat{J},\hat{b})$ implies that $K_{ij}(t)\not\equiv 0$ for $i \neq j$. Thus, $\left[\frac{d}{dt} g(t)\right]^2-g(t) \frac{d^2}{dt^2} g(t)$ does not depend on $e^{(\lambda_i(A)+\lambda_j(A))t}$ for some $i > j$ if and only if for all $t\geq 0$ $$\begin{aligned}
\hat{c}_i (\hat{c}\hat{J})_j K_{ij}(t) + \hat{c}_j (\hat{c}\hat{J})_i K_{ji}(t) = 0
\intertext{which by $K = -K^{\mathsf{T}}$ is equivalent to}
\hat{c}_i (\hat{c}\hat{J})_j = \hat{c}_j (\hat{c}\hat{J})_i .
\end{aligned}$$ However, since $(\hat{J},\hat{c})$ is observable, $\hat{c}$ does not contain any zero entries and therefore in conjunction with the Jordan form of $\hat{J}$, this case cannot occur.
### Proof to \[thm:poles\] {#proof:poles}
The fact that $\lambda_1(A) \in \mathbb{R}$ is inherited from the external positivity (see \[lem:nonneg\_imp\]). Further, with the notation of \[thm:log\_sys\], it follows that $\lambda_1(\tilde{A}) = \lambda_1(A) + \lambda_2(A)$, which by \[lem:nonneg\_imp\] has to be real. The last claim is then a trivial consequence.
### Proof to \[thm:symm\_sys\] {#proof:symm_sys}
Obviously, $g \mathds{1}_{[0,\infty)}$ is nonnegative as the sum of nonnegative functions. Further, for all $t\geq 0$ $$\begin{gathered}
\dot{g}(t)^2-g(t) \ddot{g}(t) = \sum_{i=1}^n \sum_{j=1}^n b_i b_j e^{-(\alpha_i+\alpha_j)t} (\alpha_i \alpha_j-\alpha_j^2) \\
= - \frac{1}{2}\sum_{i=1}^n \sum_{j=1}^n b_i b_j e^{-(\alpha_i+\alpha_j)t} (\alpha_{i} - \alpha_j)^2 < 0
\end{gathered}$$
Therefore, by \[lem:twice\_diff\], $g$ is not log-concave, which by \[prop:strong\_uni\_sys\] implies that the parallel interconnection of first order log-concave systems is not log-concave.
### Proof to \[prop:diff\] {#proof:diff}
We only need to show the case with $g \mathds{1}_{[0,\infty)} \in \mathcal{S}_{\geq 0}$. Then for $t \geq 0$ $$\begin{aligned}
&\dot{g}(t)^2-g(t) \ddot{g}(t) \\
& = \sum_{j=1}^n b_1 b_j e^{-(\alpha_1+\alpha_j)t} (\alpha_{1} - \alpha_j)^2\\
& - \frac{1}{2}\sum_{i=2}^n \sum_{j=2}^n b_i b_j e^{-(\alpha_i+\alpha_j)t} (\alpha_{i} - \alpha_j)^2 \\
& \geq \sum_{i=2}^n b_i \sum_{j=2}^n b_j e^{-(\alpha_1+\alpha_j)t} (\alpha_{1} - \alpha_j)^2\\
& - \frac{1}{2}\sum_{i=2}^n b_i \sum_{j=2}^nb_j e^{-(\alpha_i+\alpha_j)t} (\alpha_{i} - \alpha_j)^2 \geq 0\\
&\geq \sum_{i=2}^n b_i \sum_{j=2}^n b_j e^{-(\alpha_i+\alpha_j)t} \left((\alpha_{1} - \alpha_j)^2 - \frac{1}{2}(\alpha_{i} - \alpha_j)^2\right) \\
& \geq 0,
\end{aligned}$$ where the last inequality follows by \[eq:ass\] and the other inequalities are a consequence of $g \mathds{1}_{[0,\infty)} \in \mathcal{S}_{\geq 0}$, which implies that $b_1 \geq \sum_{i=2}^n b_i$ and $\alpha_1 \leq \alpha_i$ for all $i$.
[^1]: The authors are with the Control Group at the Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, United Kingdom. [{christian.grussler, r.sepulchre}@eng.cam.ac.uk]{}
[^2]: The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n.670645 and from DGAPA-UNAM under the grant PAPIIT RA105518.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: '*Scikit-learn* is a Python module integrating a wide range of state-of-the-art machine learning algorithms for medium-scale supervised and unsupervised problems. This package focuses on bringing machine learning to non-specialists using a general-purpose high-level language. Emphasis is put on ease of use, performance, documentation, and API consistency. It has minimal dependencies and is distributed under the simplified BSD license, encouraging its use in both academic and commercial settings. Source code, binaries, and documentation can be downloaded from <http://scikit-learn.org>.'
author:
- |
Fabian Pedregosa [email protected]\
Gaël Varoquaux [email protected]\
Alexandre Gramfort [email protected]\
Vincent Michel [email protected]\
Bertrand Thirion [email protected]\
Parietal, INRIA Saclay\
Neurospin, Bât 145, CEA Saclay\
91191 Gif sur Yvette – France Olivier Grisel [email protected]\
Nuxeo\
20 rue Soleillet\
75 020 Paris – France Mathieu Blondel [email protected]\
Kobe University\
1-1 Rokkodai, Nada\
Kobe 657-8501 – Japan Peter Prettenhofer [email protected]\
Bauhaus-Universität Weimar\
Bauhausstr. 11\
99421 Weimar – Germany Ron Weiss [email protected]\
Google Inc\
76 Ninth Avenue\
New York, NY 10011 – USA Vincent Dubourg [email protected]\
Clermont Université, IFMA, EA 3867, LaMI\
BP 10448, 63000 Clermont-Ferrand – France Jake Vanderplas [email protected]\
Astronomy Department\
University of Washington, Box 351580\
Seattle, WA 98195 – USA Alexandre Passos [email protected]\
IESL Lab\
UMass Amherst\
Amherst MA 01002 – USA David Cournapeau [email protected]\
Enthought\
21 J.J. Thompson Avenue\
Cambridge, CB3 0FA – UK
- |
Matthieu Brucher [email protected]\
Total SA, CSTJF\
avenue Larribau\
64000 Pau – France Matthieu Perrot [email protected]\
Édouard Duchesnay [email protected]\
LNAO\
Neurospin, Bât 145, CEA Saclay\
91191 Gif sur Yvette – France
bibliography:
- 'scikit.bib'
title: 'Scikit-learn: Machine Learning in Python'
---
Python, supervised learning, unsupervised learning, model selection
Introduction
============
The Python programming language is establishing itself as one of the most popular languages for scientific computing. Thanks to its high-level interactive nature and its maturing ecosystem of scientific libraries, it is an appealing choice for algorithmic development and exploratory data analysis [@cise2007; @cise2011]. Yet, as a general-purpose language, it is increasingly used not only in academic settings but also in industry.
[*Scikit-learn*]{} harnesses this rich environment to provide state-of-the-art implementations of many well known machine learning algorithms, while maintaining an easy-to-use interface tightly integrated with the Python language. This answers the growing need for statistical data analysis by non-specialists in the software and web industries, as well as in fields outside of computer-science, such as biology or physics. *Scikit-learn* differs from other machine learning toolboxes in Python for various reasons: *i)* it is distributed under the BSD license *ii)* it incorporates compiled code for efficiency, unlike MDP [@zito2008] and pybrain [@schaul2010], *iii)* it depends only on numpy and scipy to facilitate easy distribution, unlike pymvpa [@hanke2009] that has optional dependencies such as R and shogun, and *iv)* it focuses on imperative programming, unlike pybrain which uses a data-flow framework. While the package is mostly written in Python, it incorporates the C++ libraries LibSVM [@chang2001] and LibLinear [@fan2008] that provide reference implementations of SVMs and generalized linear models with compatible licenses. Binary packages are available on a rich set of platforms including Windows and any POSIX platforms. Furthermore, thanks to its liberal license, it has been widely distributed as part of major free software distributions such as Ubuntu, Debian, Mandriva, NetBSD and Macports and in commercial distributions such as the “Enthought Python Distribution”.
Project Vision
==============
*Code quality.* Rather than providing as many features as possible, the project’s goal has been to provide solid implementations. Code quality is ensured with unit tests—as of release 0.8, test coverage is 81%—and the use of static analysis tools such as [pyflakes]{} and [pep8]{}. Finally, we strive to use consistent naming for the functions and parameters used throughout a strict adherence to the Python coding guidelines and numpy style documentation.
*BSD licensing.* Most of the Python ecosystem is licensed with non-copyleft licenses. While such policy is beneficial for adoption of these tools by commercial projects, it does impose some restrictions: we are unable to use some existing scientific code, such as the GSL.
*Bare-bone design and API.* To lower the barrier of entry, we avoid framework code and keep the number of different objects to a minimum, relying on numpy arrays for data containers. *Community-driven development.* We base our development on collaborative tools such as git, github and public mailing lists. External contributions are welcome and encouraged. *Documentation.* *Scikit-learn* provides a $\sim$300 page user guide including narrative documentation, class references, a tutorial, installation instructions, as well as more than 60 examples, some featuring real-world applications. We try to minimize the use of machine-learning jargon, while maintaining precision with regards to the algorithms employed.
Underlying Technologies
=======================
*Numpy:* the base data structure used for data and model parameters. Input data is presented as numpy arrays, thus integrating seamlessly with other scientific Python libraries. Numpy’s view-based memory model limits copies, even when binding with compiled code [@Vanderwalt2011]. It also provides basic arithmetic operations.
*Scipy:* efficient algorithms for linear algebra, sparse matrix representation, special functions and basic statistical functions. [*Scipy*]{} has bindings for many Fortran-based standard numerical packages, such as LAPACK. This is important for ease of installation and portability, as providing libraries around Fortran code can prove challenging on various platforms.
*Cython:* a language for combining C in Python. Cython makes it easy to reach the performance of compiled languages with Python-like syntax and high-level operations. It is also used to bind compiled libraries, eliminating the boilerplate code of Python/C extensions.
Code Design
===========
*Objects specified by interface, not by inheritance.* To facilitate the use of external objects with *scikit-learn*, inheritance is not enforced; instead, code conventions provide a consistent interface. The central object is an [estimator]{}, that implements a [fit]{} method, accepting as arguments an input data array and, optionally, an array of labels for supervised problems. Supervised estimators, such as SVM classifiers, can implement a [predict]{} method. Some estimators, that we call [transformers]{}, for example, PCA, implement a [transform]{} method, returning modified input data. Estimators may also provide a [score]{} method, which is an increasing evaluation of goodness of fit: a log-likelihood, or a negated loss function. The other important object is the *cross-validation iterator*, which provides pairs of train and test indices to split input data, for example K-fold, leave one out, or stratified cross-validation.
*Model selection.* *Scikit-learn* can evaluate an estimator’s performance or select parameters using cross-validation, optionally distributing the computation to several cores. This is accomplished by wrapping an estimator in a [GridSearchCV]{} object, where the “CV” stands for “cross-validated”. During the call to [fit]{}, it selects the parameters on a specified parameter grid, maximizing a score (the [score]{} method of the underlying estimator). [predict]{}, [score]{}, or [transform]{} are then delegated to the tuned estimator. This object can therefore be used transparently as any other estimator. Cross validation can be made more efficient for certain estimators by exploiting specific properties, such as warm restarts or regularization paths [@friedman2010]. This is supported through special objects, such as the [LassoCV]{}. Finally, a [Pipeline]{} object can combine several [transformers]{} and an estimator to create a combined estimator to, for example, apply dimension reduction before fitting. It behaves as a standard estimator, and [GridSearchCV]{} therefore tune the parameters of all steps.
High-level yet Efficient: Some Trade Offs
=========================================
While *scikit-learn* focuses on ease of use, and is mostly written in a high level language, care has been taken to maximize computational efficiency. In Table \[tab:comparisons\], we compare computation time for a few algorithms implemented in the major machine learning toolkits accessible in Python. We use the Madelon data set [@Guyon2004], 4400 instances and 500 attributes, The data set is quite large, but small enough for most algorithms to run.
----------------------------------------- -------------- ------- --------- -------------- ------- --------------
scikit-learn mlpy pybrain pymvpa mdp shogun
\[0.5ex\] Support Vector Classification [**5.2**]{} 9.47 17.5 11.52 40.48 5.63
Lasso (LARS) [**1.17**]{} 105.3 - 37.35 - -
Elastic Net [**0.52**]{} 73.7 - 1.44 - -
k-Nearest Neighbors 0.57 1.41 - [**0.56**]{} 0.58 1.36
PCA (9 components) [**0.18**]{} - - 8.93 0.47 0.33
k-Means (9 clusters) 1.34 0.79 $\star$ - 35.75 [**0.68**]{}
License BSD GPL BSD BSD BSD GPL
----------------------------------------- -------------- ------- --------- -------------- ------- --------------
: Time in seconds on the Madelon data set for various machine learning libraries exposed in Python: MLPy [@albanese2008], PyBrain [@schaul2010], pymvpa [@hanke2009], MDP [@zito2008] and Shogun [@sonnenburg2010]. For more benchmarks see [http://github.com/scikit-learn]{}. \[tab:comparisons\]
-: Not implemented. $\star$: Does not converge within 1 hour.
*SVM.* While all of the packages compared call libsvm in the background, the performance of *scikit-learn* can be explained by two factors. First, our bindings avoid memory copies and have up to 40% less overhead than the original libsvm Python bindings. Second, we patch libsvm to improve efficiency on dense data, use a smaller memory footprint, and better use memory alignment and pipelining capabilities of modern processors. This patched version also provides unique features, such as setting weights for individual samples.
*LARS.* Iteratively refining the residuals instead of recomputing them gives performance gains of 2–10 times over the reference R implementation [@LARS]. [*Pymvpa*]{} uses this implementation via the Rpy R bindings and pays a heavy price to memory copies.
*Elastic Net.* We benchmarked the *scikit-learn* coordinate descent implementations of Elastic Net. It achieves the same order of performance as the highly optimized Fortran version *glmnet* [@friedman2010] on medium-scale problems, but performance on very large problems is limited since we do not use the KKT conditions to define an active set.
*kNN.* The k-nearest neighbors classifier implementation constructs a ball tree [@omohundro1989] of the samples, but uses a more efficient brute force search in large dimensions.
*PCA.* For medium to large data sets, *scikit-learn* provides an implementation of a truncated PCA based on random projections [@rokhlin2009]. *k-means.* *scikit-learn*’s k-means algorithm is implemented in pure Python. Its performance is limited by the fact that numpy’s array operations take multiple passes over data.
Conclusion
==========
*Scikit-learn* exposes a wide variety of machine learning algorithms, both supervised and unsupervised, using a consistent, task-oriented interface, thus enabling easy comparison of methods for a given application. Since it relies on the scientific Python ecosystem, it can easily be integrated into applications outside the traditional range of statistical data analysis. Importantly, the algorithms, implemented in a high-level language, can be used as building blocks for approaches specific to a use case, for example, in medical imaging [@Michel2011]. Future work includes *online* learning, to scale to large data sets.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Let $G$ be a finite group of Lie type $E_6$ over ${\mathbb{F}}_q$ (adjoint or simply connected) and $W$ be the Weyl group of $G$. We describe maximal tori $T$ such that $T$ has a complement in its algebraic normalizer $N(G,T)$. It is well known that for each maximal torus $T$ of $G$ there exists an element $w\in W$ such that $N(G,T)/T\simeq C_W(w)$. When $T$ does not have a complement isomorphic to $C_W(w)$, we show that $w$ has a lift in $N(G,T)$ of the same order.'
author:
- 'Alexey Galt and Alexey Staroletov[^1]'
title: 'On splitting of the normalizer of a maximal torus in $E_6(q)$'
---
Introduction
============
Finite groups of Lie type arise from linear algebraic groups as sets of fixed points of a Steinberg endomorphism. Let $\overline{G}$ be a simple connected linear algebraic group over an algebraically closed field $\overline{{\mathbb{F}}}_p$ of positive characteristic $p$. Consider a Steinberg endomorphism $\sigma$ and a maximal $\sigma$-invariant torus $\overline{T}$ of $\overline{G}$. It is well known that all maximal tori are conjugate in $\overline{G}$ and a quotient $N_{\overline{G}}(\overline{T})/\overline{T}$ is isomorphic to the Weyl group $W$ of $\overline{G}$.
The natural question is to describe groups $\overline{G}$, in which $N_{\overline{G}}(\overline{T})$ splits over $\overline{T}$. A similar question can be formulated for finite groups of Lie type. More precisely, let $G$ be a finite group of Lie type, that is $O^{p'}(\overline{G}_{\sigma})\leqslant G\leqslant\overline{G}_{\sigma}$. Let $T=\overline{T}\cap G$ be a maximal torus in $G$ and $N(G,T)=N_{\overline{G}}(\overline{T})\cap G$ be the algebraic normalizer of $T$. Then the question is to describe groups $G$ and their maximal tori $T$ such that $N(G,T)$ splits over $T$.
These questions were stated by J.Tits in [@Tits]. In the case of algebraic groups it was solved independently in [@AdamsHe] and in [@Galt1; @Galt2; @Galt3; @Galt4]. In the case of finite groups the problem was studied for the groups of Lie types $A_n, B_n, C_n$ and $D_n$ in [@Galt2; @Galt3; @Galt4].
In this paper we consider finite groups $G$ of Lie type $E_6$ over a finite field ${\mathbb{F}}_q$ of characteristic $p$. Recall that in this case $G/Z(G)$ is isomorphic to the simple group $E_6(q)$ and either $|Z(G)|=1$ or $|Z(G)|=3$. There are 25 conjugacy classes of maximal tori in $E_6(q)$ and we enumerate them as in [@DerF]. Let $\Delta=\{r_1,r_2,r_3,r_4,r_5,r_6\}$ be a fundamental system of a root system $E_6$ and $r_{14}=r_2+r_4+r_5$, $r_{31}=r_2+r_3+2r_4+2r_5+r_6$, $r_{36}=r_1+2r_2+2r_3+3r_4+2r_5+r_6$. Denote by $w_i$ the elements of the Weyl group $W$ corresponding to reflections in the hyperplanes orthogonal to the roots $r_i$. There is a bijection between conjugacy classes of maximal tori and conjugacy classes of $W$. The main result of the paper is the following
\[th1\] Let $G$ be a finite group of Lie type $E_6$ over ${{\mathbb{F}}}_q$ (adjoint or simply connected) with the Weyl group $W$. Let $T$ be a maximal torus of $G$ corresponding to an element $w$ of $W$ and $N$ its algebraic normalizer. Then $T$ does not have a complement in $N$ if and only if one of the following claims holds:
- $q$ is odd and $w$ is conjugate to one of the following: $1$, $w_1$, $w_1w_2$, $w_2w_3w_5$, $w_1w_3w_4$, $w_1w_4w_6w_{36}$, $w_1w_4w_6w_3$, $w_1w_4w_6w_3w_{36}$;
- $q\equiv3\pmod4$ and $w$ is conjugate to $w_3w_2w_4w_{14}$.
In the end of the paper we illustrate this result in Table \[table\] with some useful information about the maximal tori. In fact, when $T$ has a complement in $N$ we construct it explicitly.
J.Adams and X.He [@AdamsHe] considered a related problem. Namely, it is natural to ask about the orders of lifts of $w\in W$ to $N_{\overline{G}}(\overline{T})$. They noticed that if $d$ is the order of $w$ then the minimal order of a lift of $w$ has order $d$ or $2d$, but it can be a subtle question which holds. Clearly, if $N_{\overline{G}}(\overline{T})$ splits over $\overline{T}$ then the minimal order is equal to $d$. In the case of Lie type $E_6$ it was proved that the normalizer does not split and they showed that the minimal order of lifts of $w$ is $d$ if $w$ belongs to so-called regular or elliptic conjugacy classes. We prove that the minimal order of lifts of $w$ is always equal to the order $w$ in this case. Moreover, we construct these lifts in the corresponding algebraic normalizers.
\[th2\] Let $G$ be a finite group of Lie type $E_6$ over ${{\mathbb{F}}}_q$ (adjoint or simply connected) with the Weyl group $W$. Let $T$ be a maximal torus of $G$ corresponding to an element $w$ of $W$ and $N$ its algebraic normalizer. Then there exists a lift for $w$ in $N$ with the same order.
This paper is organized as follows. In Section 2 we recall notation and basic facts about algebraic groups. In Section 3 we prove auxiliary results and explain how we use MAGMA in the proofs. Section 4 is devoted to the proof of the main results. It consists of two subsections. First, we consider the maximal tori that do not have a complement. In these cases we also prove the existence of the lifts of the corresponding elements of $W$. In the second subsection we construct the complements for the remaining tori.
Notation and preliminary results
================================
By $q$ we always denote some power of a prime $p$, $\overline{{\mathbb{F}}}_p$ is an algebraic closure of a finite field ${\mathbb{F}}_p$. The symmetric group on $n$ elements is denoted by $S_n$. Let $T$ be a normal subgroup of a group $N$. Then [*$T$ has a complement in $N$*]{} (or [*$N$ splits over $T$*]{}) if there exists a subgroup $H$ such that $N=T H$ and $T\cap H=1$.
By $\overline{G}$ we denote a simple connected linear algebraic group over $\overline{{\mathbb{F}}}_p$. A surjective endomorphism $\sigma$ of $\overline{G}$ is called a [*Steinberg endomorphism*]{}, if the set of $\sigma$-stable points $\overline{G}_\sigma$ is finite [@GorLySol Definition 1.15.1]. Any group $G$ satisfying $O^{p'}(\overline{G}_\sigma)\leqslant G\leqslant\overline{G}_\sigma$, is called a [*finite group of Lie type*]{}. It is well known that $\overline{G}$ always has a $\sigma$-stable maximal torus, which is denoted by $\overline{T}$. All maximal tori are conjugate to $\overline{T}$ in $\overline{G}$. If $\overline{T}$ is a $\sigma$-stable maximal torus of $\overline{G}$ then $T=\overline{T}\cap G$ is called a [*maximal torus*]{} of $G$. If $G_2\unlhd G_1\unlhd G$ then the image of $T\cap G_1$ in $G_1/G_2$ is called a [*maximal torus*]{} of $G_1/G_2$. A group $N_{\overline{G}}(\overline{T})\cap G$ is denoted by $N(G,T)$ or just $N$. Notice that $N(G,T)\leqslant N_G(T)$, but the equality is not true in general. For example, let $G=\operatorname{SL}_n(2)$ then the subgroup of diagonal matrices $T$ of $G$ is trivial, hence, $N_G(T)=G$. But $G=(\operatorname{SL}_n(\overline{{\mathbb{F}}}_2))_\sigma$, where $\sigma$ is a Frobenius map $(\sigma : (a_{i,j}) \mapsto (a_{i,j}^2))$. Then $T=\overline{T}_\sigma$, where $\overline{T}$ is the subgroup of diagonal matrices in $\operatorname{SL}_n(\overline{{\mathbb{F}}}_2)$. Thus $N(G,T)$ is the group of monomial matrices in $G$. That is why for the group $N(G,T)$ we use the term [*algebraic normalizer*]{}.
By $\overline{N}$ and $W$ we denote the normalizer $N_{\overline{G}}(\overline{T})$ and the Weyl group $\overline{N}/\overline{T}$, and $\pi$ stands for the natural homomorphism from $\overline{N}$ onto $W$. Define the action of $\sigma$ on $W$ in the natural way. Elements $w_1, w_2\in W$ are called $\sigma$-conjugate if $w_1=(w^{-1})^{\sigma}w_2w$ for some $w\in W$.
[*[@Car Propositions 3.3.1, 3.3.3]*]{}\[torus\]. A torus $\overline{T}^g$ is $\sigma$-stable if and only if $g^{\sigma}g^{-1}\in\overline{N}$. The map $\overline{T}^g\mapsto\pi(g^{\sigma}g^{-1})$ determines a bijection between the $G$-classes of $\sigma$-stable maximal tori of $\overline{G}$ and the $\sigma$-conjugacy classes of $W$.
As follows from Proposition \[torus\] the cyclic structure of a torus $(\overline{T}^g)_{\sigma}$ in $G$ and of the corresponding tori in the sections of $G$ is determined only by a $\sigma$-conjugacy class of the element $\pi(g^{\sigma}g^{-1})$.
[*[@ButGre Lemma 1.2]*]{}\[prop2.5\]. Let $n=g^{\sigma}g^{-1}\in\overline{N}$. Then $(\overline{T}^g)_\sigma=(\overline{T}_{\sigma n})^g$, where $n$ acts on $\overline{T}$ by conjugation.
[*[@Car Proposition 3.3.6]*]{}\[normalizer\]. Let $g^{\sigma}g^{-1}\in\overline{N}$ and $\pi(g^{\sigma}g^{-1})=w$. Then $$(N_{\overline{G}}({\overline{T}}^g))_{\sigma}/({\overline{T}}^g)_{\sigma}\simeq C_{W,\sigma}(w)=\{x\in W~|~(x^{-1})^{\sigma}wx=w\}.$$
From now on, we suppose that $\overline{G}$ has Lie type $E_6$. Then the Weyl group $W$ has order $2^{7}\cdot3^4\cdot5$ and is isomorphic to the group $\operatorname{PSp}_4(3):2$ (in the notation of [@Atlas]). Let $\Phi$ and $\Pi=\{r_1,r_2,r_3,r_4,r_5,r_6\}$ be a set of positive and fundamental roots of a root system $E_6$, respectively. The Dynkin diagram of $E_6$ has the form
(330,50)(-110,-30) (0,0)[(1,0)[50]{}]{} (50,0)[(1,0)[50]{}]{} (100,0)[(1,0)[50]{}]{} (150,0)[(1,0)[50]{}]{} (0,0) (50,0) (100,0) (150,0) (200,0) (100,0)[(0,-1)[20]{}]{} (100,-20) (0,-10)[(0,0)[$r_1$]{}]{} (50,-10)[(0,0)[$r_3$]{}]{} (107,-10)[(0,0)[$r_4$]{}]{} (150,-10)[(0,0)[$r_5$]{}]{} (200,-10)[(0,0)[$r_6$]{}]{} (105,-30)[(0,0)[$r_2$]{}]{}
\
Following [@Car], we write $x^y=yxy^{-1}$ and $[x,y]=y^xy^{-1}$. It is well known [@Car] that
$\overline{T}=\langle h_r(\lambda)~|~r\in\Phi,\lambda\in \overline{{\mathbb{F}}}_p^*\rangle, \quad
\overline{N}=\langle \overline{T},n_r~|~r\in\Phi\rangle$,
where $$n_r(1)=n_r,\quad
n_r^2=h_r(-1),\quad
h_r(\lambda)=n_r(\lambda)n_r(-1).$$ For simplicity of notation, we write $h_r$ for $h_r(-1)$. If $r=r_i$, then $h_i$ stands for $h_{r_i}$ and $n_i$ stands for $n_{r_i}$. Every element $H$ of $\overline{T}$ can be written in the form $H=\prod\limits_{i=1}^6h_{r_i}(\lambda_i)$. Then the element $H$ is uniquely determined by $\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5,\lambda_6$, and we write $H = (\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5,\lambda_6)$.
The group $\mathcal{T}=\langle n_r~|~ r\in\Pi\rangle$ is called [*the Tits group*]{}. Let $\mathcal{H}=\overline{T}\cap\mathcal{T}$. It is known that $\mathcal{H}=\langle h_r~|~r\in\Pi\rangle$ and so $\mathcal{H}$ is an elementary abelian group such that $\mathcal{T}/\mathcal{H}\simeq W$. Observe that if $p=2$ then $h_r=1$ for every $r\in\Pi$, in particular, $\mathcal{H}=1$ and $\mathcal{T}\simeq W$. This implies the assertions of the both theorems, when $q$ is even (for details see Section 4.2).
Let $\xi\in\overline{{\mathbb{F}}}_p$ such that $\xi^3=1$. According to Table $1.12.6$ in [@GorLySol] the centre $Z(E_6(\overline{{\mathbb{F}}}_p))$ of the simply connected group $E_6(\overline{{\mathbb{F}}}_p)$ is generated by the element $z=h_{r_1}(\xi)h_{r_3}(\xi^2)h_{r_5}(\xi)h_{r_6}(\xi^2)$ of order 3.
According [@Car Theorem 7.2.2] we have:
$n_s n_r n_s^{-1}=h_{w_s(r)}(\eta_{s,r})n_{w_s(r)}=
n_{w_s(r)}(\eta_{s,r}),\quad \eta_{s,r}=\pm1,$
$n_s h_r(\lambda)n_s^{-1}=h_{w_s(r)}(\lambda).$
We choose values of $\eta_{r,s}$ as follows. Let $r\in\Phi$ and $r=\sum\limits_{i=1}^6\alpha_i r_i$. The sum of the coefficients $\alpha_1+\alpha_2+\alpha_3+\alpha_4+\alpha_5+\alpha_6$ is called the height of $r$. Following to [@Vavilov], we fix the following total ordering of positive roots: we write $r\prec s$ if either $h(r)<h(s)$ or $h(r)=h(s)$ and the first nonzero coordinate of $s-r$ is positive. The table of positive roots with respect to this ordering can be found in [@Vavilov].
Recall that a pair of positive roots $(r,s)$ is called [*special*]{} if $r+s\in\Phi$ and $r\prec s$. A pair $(r,s)$ is called [*extraspecial*]{} if it is special and for any special pair $(r_1, s_1)$ such that $r+s=r_1+s_1$ one has $r\preccurlyeq s$. Let $N_{r,s}$ be the structure constants of the corresponding simple Lie algebra. Then the values $N_{r,s}$ may be taken arbitrarily at the extraspecial pairs and then all other structure constants are uniquely determined [@Car Proposition 4.2.2]. In our case, we choose $\operatorname{sgn}(N_{r,s})=+$ for all extraspecial pairs $(r,s)$. Values of the structure constants for all pairs can be found in [@Vavilov]. The numbers $\eta_{r,s}$ are uniquely determined by the structure constants [@Car Proposition 6.4.3].
Preliminaries: calculations
===========================
We use MAGMA to calculate products of elements in $\overline{N}$. All calculations can be performed using online Magma Calculator [@MC] as well. At the moment it uses Magma V2.23-10. We use the following preparatory commands:$L:=LieAlgebra("E6", Rationals());$$R:=RootDatum(L);$$B:=ChevalleyBasis(L);$
The following command produces the list of extraspecial pairs and signs of the corresponding structure constants:
Here, for example, $\langle2,4,1\rangle$ means that the pair $(r_2,r_4)$ is extraspecial and $N_{r_2,r_4}=1$. It is straightforward to verify the defined above ordering gives the same set of extraspecial pairs. Thus calculations in MAGMA for $\overline{N}$ correspond to the ordering and structure constants defined in the previous section. The following commands construct elements $n_i$ and $h_i$. $G:=GroupOfLieType(L);$ $n:=[ elt\langle G~|~i\rangle : i\ in\ [1..36]];$ $h:=[TorusTerm(G, i,-1) : i\ in\ [1..36]];$
To obtain the list of matrices of the fundamental reflections one can use the following command:$w:=[Transpose(i)\ :\ i\ in\ ReflectionMatrices(R)];$
\[normalizer\] Let $g\in\overline{G}$ and $n=g^\sigma g^{-1}\in\overline{N}$. Suppose that $H\in \overline{T}$ and $u\in\mathcal{T}$. Then
\(i) $Hu\in\overline{N}_{\sigma n}$ if and only if $H=H^{\sigma n}[n,u];$
\(ii) If $H\in \mathcal{H}$ then $Hu\in\overline{N}_{\sigma n}$ if and only if $[n,Hu]=1$.
\(i) Since $\sigma$ acts trivially on $\mathcal{T}$, we have $Hu=(Hu)^{\sigma n}=H^{\sigma n}u^{\sigma n}=
H^{\sigma n}u^{n}$ and $H=H^{\sigma n}[n,u]$.
\(ii) Since $\sigma$ acts trivially on $\mathcal{H}$, we have $Hu=(Hu)^{\sigma n}=(Hu)^{n}.$
\[conjugation\] Let $n=\prod\limits_{j=1}^{k}n_{r_{i_j}}$ and $w=\prod\limits_{j=1}^{k}w_{r_{i_j}}$, where $i_j\in\{1..36\}$. Suppose that $A=(a_{ij})_{6\times6}$ is the matrix of $w$ in the basis $r_1,r_2,...,r_6$ and $H=(\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5,\lambda_6)$ is an element of $\overline{T}$. Then the following claims hold:
\(i) $H^n=(\lambda_1',\lambda_2',\lambda_3',\lambda_4',\lambda_5',\lambda_6')$, where $\lambda_i'=\lambda_1^{a_{i1}}\lambda_2^{a_{i2}}\lambda_3^{a_{i3}}\lambda_4^{a_{i4}}\lambda_5^{a_{i5}}\lambda_6^{a_{i6}}$ for $1\leqslant{i}\leqslant6$;
\(ii) $(Hn)^m=(\lambda_1',\lambda_2',\lambda_3',\lambda_4',\lambda_5',\lambda_6')n^m$, where $m$ is a positive integer, $\lambda_i'=\lambda_1^{b_{i1}}\lambda_2^{b_{i2}}\lambda_3^{b_{i3}}\lambda_4^{b_{i4}}\lambda_5^{b_{i5}}\lambda_6^{b_{i6}}$ for $1\leqslant{i}\leqslant6$ and $b_{ij}$ are elements of the matrix $\sum\limits_{k=0}^{m-1}A^k$.
Since the matrix of the composition of two linear transformations is the product of their matrices in the reverse order, it is sufficient to proof the lemma for $n=n_r$. By formulas, $H^{n_r}=\prod\limits_{i=1}^6h_{r_i}(\lambda_i)^{n_r}=\prod\limits_{i=1}^6h_{w_r(r_i)}(\lambda_i)=\prod\limits_{i=1}^6(\prod\limits_{j=1}^6h_{r_j}(\lambda_i^{a_{ji}}))=
\prod\limits_{i=1}^6h_{r_i}(\prod\limits_{j=1}^6\lambda_j^{a_{ij}})$.
To prove $(ii)$ observe that $(Hn)^m=H^{n^0}H^{n^1}H^{n^2}..H^{n^{m-1}}n^m$. By $(i)$, we know that the $i$-th row of $A^j$ corresponds to the exponents of $\lambda_1$, $\lambda_2$, $\ldots$ , $\lambda_6$ in $i$-th coordinate of $H^{n^j}$ for every $i,j\in\{1,\ldots,6\}$. To compute the product, we need to sum the exponents up for each coordinate. The lemma is proved.
Since we often use Lemma \[conjugation\], we illustrate its applying with the following example.
[**Example.**]{} Let $w=w_1w_3$ and $n=n_1n_3$. Then it is easy to see that $w(r_1)=r_3$, $w(r_2)=r_2$, $w(r_3)=-r_1-r_3$, $w(r_4)=r_1+r_3+r_4$, $w(r_5)=r_5$ and $w(r_6)=r_6$. Therefore, in this case $$A=\left(\begin{array}{cccccc} 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1
\end{array} \right) .$$ Let $H=(\lambda_1, \lambda_2, \lambda_3, \lambda_4,\lambda_5,\lambda_6)\in\overline{T}$. Then by Lemma \[conjugation\], we can use the rows of $A$ to compute $H^n$, namely $H^{n_1n_3}=(\lambda_3^{-1}\lambda_4,\lambda_2,\lambda_1\lambda_3^{-1}\lambda_4,\lambda_4,\lambda_5,\lambda_6)$. Now let $B=A^0+A+A^2$. Then $$B=\left(\begin{array}{cccccc} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0\\ 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3
\end{array} \right) .$$ This matrix helps us to compute $(Hn)^3$. It is easy to see that $n^3=1$, so $$(Hn)^3=(\lambda_4,\lambda_2^3,\lambda_4^2,\lambda_4^3,\lambda_5^3,\lambda_6^3)n^3=(\lambda_4,\lambda_2^3,\lambda_4^2,\lambda_4^3,\lambda_5^3,\lambda_6^3).$$
The following lemma is clear.
\[commutator\] Let $N_1=H_1u_1, N_2=H_2u_2$, where $H_1, H_2\in T$ and $u_1,u_2\in N$. Then $$N_1N_2=N_2N_1 \text{ if and only if } H_1^{-1}H_1^{u_2}\cdot u_2u_1u_2^{-1}u_1^{-1}=H_2^{-1}H_2^{u_1}.$$ In particular, if $[u_1,u_2]=1$, then $N_1N_2=N_2N_1 \text{ if and only if } H_1^{-1}H_1^{u_2}=H_2^{-1}H_2^{u_1}.$
Proof of the main results
=========================
The proof of Theorem \[th1\] is divided into two subsections. First, we consider maximal tori that do not have a complement in their algebraic normalizer. In these cases we show that the corresponding element of $W$ has preimage in $N$ of the same order. The second subsection is devoted to the construction of the complements for the remaining maximal tori. In these cases the lifts obviously exist in the complements. We use the numeration of the maximal tori as in [@DerF Table I] and Table \[table\]. All calculations in $W$ can be verified in MAGMA [@MAGMA] or GAP [@GAP].
Non-complement cases
--------------------
Our strategy is similar in all cases. Throughout this subsection we suppose that $q$ is odd and $T$ is a maximal torus corresponding to the conjugacy class of $w$ in $W$, where $w$ is one from the assertion of Theorem \[th1\]. We suppose that there exists a complement $K$ in $N$, in particular $K\simeq C_W(w)$. We arrive at a contradiction in each case in the universal group $E_6(q)$. Then we explain that the same contradiction can be obtained in the adjoint group $E_6(q)$. Throughout this subsection $H=(\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5,\lambda_6)$ is an arbitrary element of $T$.
**Torus 1.** In this case $w=1$ and $C_W(w)=W$. It was proved [@Galt1 Corollary] that the corresponding maximal torus in algebraic group does not have a complement in $\overline{N}$. Therefore, $T$ does not have a complement in $N$ as well.
**Tori 2, 3.** In these cases $w=w_1$ or $w=w_1w_2$, respectively. Observe that both centralizers $C_W(w_1)$ and $C_W(w_1w_2)$ contain a subgroup $\langle w_1,w_2,w_5,w_{29}\rangle$ and we deduce a contradiction from it. Let $N_1,N_2,N_3,N_4$ be preimages of $w_1,w_2,w_5,w_{29}$ in $K$, respectively. Then $$N_1=H_1n_1,N_2=H_2n_2,N_3=H_3n_5,N_4=H_4n_{29},$$ where $$H_1=(\mu_1,\mu_2,\mu_3,\mu_4,\mu_5,\mu_6),
H_2=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6),
H_3=(\beta_1,\beta_2,\beta_3,\beta_4,\beta_5,\beta_6),$$ $H_4=(\delta_1,\delta_2,\delta_3,\delta_4,\delta_5,\delta_6)$ are elements of $T$. Since $K\simeq C_W(w)$, we have $N_1^2=1$ and $N_1N_i=N_iN_1$ for $i=2,3,4.$ By Lemma \[conjugation\], we get $$N_1^2=(H_1n_1)^2=H_1H_1^{n_1}h_1=(-\mu_3, \mu_2^2,\mu_3^2,\mu_4^2,\mu_5^2,\mu_6^2)=1,$$ so $1=\mu_2^2=\mu_4^2=\mu_5^2=\mu_6^2=-\mu_3$. If $j\in\{2,5,29\}$ then using MAGMA [@MAGMA] we see that $[n_1,n_j]=1$. It follows from Lemma \[commutator\] that $H_1^{-1}H_1^{n_j}=H_j^{-1}H_j^{n_1}$. By Lemma \[conjugation\] we conclude $$j=2 \Rightarrow (1,\mu_2^{-2}\mu_4,1,1,1,1)=(\alpha_1^{-2}\alpha_3,1,1,1,1,1),
\text{ so } \mu_4=\mu_2^2=1.$$ $$j=5 \Rightarrow (1,1,1,1,\mu_4\mu_5^{-2}\mu_6,1)=(\beta_1^{-2}\beta_3,1,1,1,1,1),
\text{ so } \mu_6=\mu_5^2\mu_4^{-1}=1.$$ $$j=29 \Rightarrow (\mu_3^{-1}\mu_6,\mu_3^{-1}\mu_6,\mu_3^{-2}\mu_6^2,\mu_3^{-2}\mu_6^2,\mu_3^{-1}\mu_6,1)= (\delta_1^{-2}\delta_3,1,1,1,1,1),$$ so $\mu_3=\mu_6=1.$ This contradicts $\mu_3=-1$.
In the adjoint group $E_6(q)$, an element $H=(\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5,\lambda_6)$ is the identity if and only if $\lambda_1=\lambda_5=\xi,\lambda_3=\lambda_6=\xi^2,\lambda_2=\lambda_4=1,$ where $\xi^3=1$. In particular, $\lambda_i^3=1$. Note that $(-1)^3=-1$. Hence, we can consider the elements $\widetilde{\lambda}_i=\lambda_i^3$ for $\lambda_i\in\{\mu_i,\alpha_i,\beta_i,\delta_i\}$ with $1\leqslant i\leqslant6$. Then we obtain the same equalities and the same contradiction.
Now we provide the lifts for $w_1$ and $w_1w_2$ to prove Theorem \[th2\] in these cases. Let $\zeta$ be an element of $\overline{{\mathbb{F}}}_p$ such that $\zeta^{q+1}=-1$. Put $H_1=(\zeta,1,-1,1,1,1)$. We claim that $H_1n_1\in N$ and $(H_1n_1)^2=1$. By Lemma \[conjugation\], we have $H^{n_1}=(\lambda_1^{-1}\lambda_3,\lambda_2,\lambda_3,\lambda_4,\lambda_5,\lambda_6)$. Hence, $H_1^{\sigma n_1}=(-\zeta^{-q},1,-1,1,1,1)=H_1,$ and so $H_1\in T$. Using MAGMA [@MAGMA], $$(Hn_1)^2=(-\lambda_3,\lambda_2^2,\lambda_3^2,\lambda_4^2,\lambda_5^2,\lambda_6^2).$$
Thus $(H_1n_1)^2=1$ and $H_1n_1$ is a required lift for $w_1$.
Similarly, put $H_2=(\zeta,\zeta,-1,-1,1,1)$. By Lemma \[conjugation\], we have $$H^{n_1n_2}=(\lambda_1^{-1}\lambda_3,\lambda_2^{-1}\lambda_4,\lambda_3,\lambda_4,\lambda_5,\lambda_6).$$ Hence, $H_1^{\sigma n_1n_2}=(-\zeta^{-q},-\zeta^{-q},-1,-1,1,1)=H_2,$ and so $H_2\in T$. Using MAGMA [@MAGMA], $$(Hn_1n_2)^2=(-\lambda_3,-\lambda_4,\lambda_3^2,\lambda_4^2,\lambda_5^2,\lambda_6^2).$$
Thus $(H_1n_1n_2)^2=1$ and $H_1n_1n_2$ is a required lift for $w_1w_2$.
**Torus 5.** In this case $w=w_2w_3w_5$ and $$C_W(w)=\langle w\rangle\times \langle w_{24}\rangle\times\langle x,y,z\rangle\simeq
\mathbb{Z}_2\times\mathbb{Z}_2\times S_4,$$ where $$x=w_{17}w_{18},y=w_{20}w_{21},z=w_{16}w_{25}, \text{ and } x^2=y^2=z^2=(yz)^2=(xy)^3=(xz)^3=1.$$ Using MAGMA, we have $[n_{24},n_2n_3n_5]=[n_{17}n_{18},n_2n_3n_5]= [h_4h_6n_{20}n_{21},n_2n_3n_5]=[n_{16}n_{25},n_2n_3n_5]=1.$ Hence, Lemma \[normalizer\] (ii) yields $w_{24},x,y,z$ are images of $n_{24},n_{17}n_{18}, h_4h_6n_{20}n_{21}, n_{16}n_{25}$, respectively. Let $N_1,N_2,N_3,N_4$ be preimages of $w_2w_3w_5,w_{24},w_{20}w_{21},w_{16}w_{25}$ in $K$. Then $$N_1=H_1n_2n_3n_5,N_2=H_2n_{24},N_3=H_3h_4h_6n_{20}n_{21},N_4=H_4n_{16}n_{25},$$ where $$H_1=(\mu_1,\mu_2,\mu_3,\mu_4,\mu_5,\mu_6),
H_2=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6),
H_3=(\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5,\gamma_6),$$ $H_4=(\delta_1,\delta_2,\delta_3,\delta_4,\delta_5,\delta_6).$ Since $K\simeq C_W(w)$, we get $N_2^2=1$. By Lemma \[conjugation\], it follows that $$N_2^2=H_2H_2^{n_{24}}h_2h_3h_5=(\alpha_1^2,-\alpha_1\alpha_2^2\alpha_4^{-1}\alpha_6,-\alpha_1\alpha_3^2\alpha_4^{-1}\alpha_6,
\alpha_1^2\alpha_6^2,-\alpha_1\alpha_4^{-1}\alpha_5^2\alpha_6,\alpha_6^2)=1,$$ so $\alpha_1^2=\alpha_6^2=1, \alpha_2^2=\alpha_3^2=\alpha_5^2=-\alpha_4\alpha_1^{-1}\alpha_6^{-1}$.
By Lemma \[commutator\], $N_2N_1=N_1N_2$ is equivalent to $H_2^{-1}H_2^{n_2n_3n_5}=H_1^{-1}H_1^{n_{24}}$. Then according to Lemma \[conjugation\] we have $$(1,\alpha_2^{-2}\alpha_4,\alpha_1\alpha_3^{-2}\alpha_4,1,\alpha_4\alpha_5^{-2}\alpha_6,1)
=(1,\mu_1\mu_4^{-1}\mu_6,\mu_1\mu_4^{-1}\mu_6,
\mu_1^2\mu_4^{-2}\mu_6^2,\mu_1\mu_4^{-1}\mu_6,1).$$ Then $\mu_1\mu_4^{-1}\mu_6=\alpha_2^{-2}\alpha_4=\alpha_1\alpha_3^{-2}\alpha_4=\alpha_4\alpha_5^{-2}\alpha_6.$ Since $\alpha_2^2=\alpha_3^2=\alpha_5^2$, we obtain $\alpha_1=1$ and $\alpha_6=1$. From $\alpha_2^2=-\alpha_4\alpha_1^{-1}\alpha_6^{-1}$, we get $\alpha_2^2=-\alpha_4.$
Calculations in MAGMA show $[n_{24},h_4h_6n_{20}n_{21}]=1.$ Hence, $N_2N_3=N_3N_2$ implies that $H_2^{-1}H_2^{n_{20}n_{21}}=H_3^{-1}H_3^{n_{24}}$. Then by Lemma \[conjugation\] $$(1,\alpha_2^{-1}\alpha_3\alpha_6^{-1},\alpha_1\alpha_2\alpha_3^{-1}\alpha_6^{-1},
\alpha_1\alpha_6^{-2},\alpha_1\alpha_6^{-2},\alpha_1\alpha_6^{-2})=(1,\gamma_1\gamma_4^{-1}\gamma_6,\gamma_1\gamma_4^{-1}\gamma_6,\gamma_1^2\gamma_4^{-2}\gamma_6^2,\gamma_1\gamma_4^{-1}\gamma_6,1).$$
Since $\alpha_1=\alpha_6=1$, it follows that $\alpha_2\alpha_3^{-1}=\gamma_1\gamma_4^{-1}\gamma_6$ and $1=\gamma_1\gamma_4^{-1}\gamma_6$. Therefore $\alpha_2=\alpha_3.$ Using MAGMA we get $[n_{24},n_{16}n_{25}]=1.$ Hence, $N_2N_4=N_4N_2$ implies $H_2^{-1}H_2^{n_{16}n_{25}}=H_4^{-1}H_4^{n_{24}}$. Then by Lemma \[conjugation\] $$(1,\alpha_1\alpha_2^{-1}\alpha_3^{-1}\alpha_4\alpha_6^{-1},\alpha_1\alpha_2^{-1}\alpha_3^{-1}\alpha_4\alpha_6^{-1},\alpha_1\alpha_6^{-2},\alpha_1\alpha_6^{-2},\alpha_1\alpha_6^{-2})=(1,\delta_1\delta_4^{-1}\delta_6,\delta_1\delta_4^{-1}\delta_6,\delta_1^2\delta_4^{-2}\delta_6^2,\delta_1\delta_4^{-1}\delta_6,1).$$ Since $\alpha_1=\alpha_6=1$, it follows that $\alpha_2^{-1}\alpha_3^{-1}\alpha_4=\delta_1\delta_4^{-1}\delta_6$ and $1=\delta_1\delta_4^{-1}\delta_6$. Therefore, $\alpha_4=\alpha_2\alpha_3=\alpha_2^2;$ a contradiction with $\alpha_4=-\alpha_2^2$.
Now we provide the lift for $w$. Let $\zeta$ be an element of $\overline{{\mathbb{F}}}_p$ such that $\zeta^{q+1}=-1$. Put $H_1=(1,\zeta,\zeta,-1,\zeta,1)$. We claim that $H_1n_2n_3n_5\in N$ and $(H_1n_2n_3n_5)^2=1$. By Lemma \[conjugation\], we have $H^{n_2n_3n_5}=(\lambda_1,\lambda_2^{-1}\lambda_4,\lambda_1\lambda_3^{-1}\lambda_4,
\lambda_4,\lambda_4\lambda_5^{-1}\lambda_6,\lambda_6)$. Hence, $H_1^{\sigma n_2n_3n_5}=(1,-\zeta^{-q},-\zeta^{-q}-1,-\zeta^{-q},1)=H_1$ and so $H_1\in T$. Using MAGMA we get $$(Hn_2n_3n_5)^2=(\lambda_1^2, -\lambda_4, -\lambda_1\lambda_4, \lambda_4^2, -\lambda_4\lambda_6,\lambda_6^2)$$
Thus $(H_1n_2n_3n_5)^2=1$ and $H_1n_2n_3n_5$ is a required lift for $w_2w_3w_5$.
**Torus 7.** In this case $w=w_1w_3w_4$ and $$C_W(w)=~\langle w\rangle\times\langle w_6, w_{19}w_{26}
\rangle\simeq\mathbb{Z}_4 \times D_8.$$ Put $n=n_1n_3n_4$. Let $N_2, N_3$ be preimages of $w_6$ and $w_{19}w_{26}$ in $K$, respectively. Then $$N_2=H_2n_6, N_3=H_3n_{19}n_{26},$$ where $H_2=(\mu_1,\mu_2,\mu_3,\mu_4,\mu_5,\mu_6)$ and $H_3=(\beta_1,\beta_2,\beta_3,\beta_4,\beta_5,\beta_6)$. Since $[n_1n_3n_4, n_6]=[n_1n_3n_4,n_{16}n_{26}]=1$, Lemma \[normalizer\] (ii) yields $H_2, H_3\in T$. Since $w_6^2=1$, we obtain $N_2^2=1$. By Lemma \[conjugation\], $$(Hn_6)^2=(\lambda_1^2,\lambda_2^2,\lambda_3^2,\lambda_4^2,\lambda_5^2,-\lambda_5).$$ Therefore, we obtain $\mu_1^2=\mu_2^2=\mu_3^2=\mu_4^2=1$, $\mu_5=-1$. In particular, $\mu_i^q=\mu_i^{-1}=\mu_i$ for $1\leqslant i\leqslant5$. By Lemma \[conjugation\], $$H^{n}=(\lambda_2\lambda_4^{-1}\lambda_5, \lambda_2, \lambda_1\lambda_2\lambda_4^{-1}\lambda_5, \lambda_2\lambda_3\lambda_4^{-1}\lambda_5, \lambda_5, \lambda_6).$$ Since $H_2^{\sigma n}=H_2$, we have $(\mu_2\mu_4\mu_5,\mu_2,\mu_1\mu_2\mu_4\mu_5,\mu_2\mu_3\mu_4\mu_5,\mu_5,\mu_6^q)=H_2$. Using $\mu_5=-1$, we conclude that $\mu_2\mu_4=-\mu_1$, $\mu_1\mu_2\mu_4=-\mu_3$, $\mu_2\mu_3=-1$. From the last equality we get $\mu_2=-\mu_3$. Therefore $\mu_1=\mu_4$, $\mu_2=-1$ and $\mu_3=1$. Thus $H_2=(\mu_1,-1,1,\mu_1,-1,\mu_6)$.
Since $(w_{19}w_{26})^2=1$, we have $N_3^2=1$. Applying MAGMA, we get $(n_{19}n_{26})^2=h_1h_4$ and $$(Hn_{19}n_{26})^2=(-\lambda_1\lambda_3\lambda_4^{-1}\lambda_6,\lambda_2\lambda_5^{-1}\lambda_6^2,\lambda_2^{-1}\lambda_3^2\lambda_5^{-1}\lambda_6^2,-\lambda_1^{-1}\lambda_2^{-1}\lambda_3\lambda_4\lambda_5^{-1}\lambda_6^3, \lambda_2^{-1}\lambda_5\lambda_6^2,\lambda_6^2).$$ So $\beta_4=-\beta_1\beta_3\beta_6$, $\beta_2\beta_6^2=\beta_5$ and $\beta_6^2=1$. Therefore $\beta_2=\beta_5$.
Now $(w_6w_{19}w_{26})^4=1$ and so $(N_2N_3)^4=1$. Observe that $N_2N_3=H_2n_6H_3n_{19}n_{26}=H_2H_3^{n_6}n_6n_{19}n_{26}$. By Lemma \[conjugation\], $H^{n_6}=(\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5,\lambda_5\lambda_6^{-1}).$ Applying to $H_2H_3^{n_6}$, we obtain $$H_2H_3^{n_6}=(\mu_1,-1,1,\mu_1,-1,\mu_6)(\beta_1,\beta_2,\beta_3,\beta_4,\beta_2,\beta_2\beta_6^{-1})=(\mu_1\beta_1,-\beta_2,\beta_3,\mu_1\beta_4,-\beta_2,\mu_6\beta_2\beta_6^{-1}).$$ Since $(n_6n_{19}n_{26})^4=h_1h_4$, By Lemma \[conjugation\] we have $$(Hn_6n_{19}n_{26})^4=(-\lambda_1^2\lambda_2^{-1}\lambda_3^2\lambda_4^{-2}\lambda_5,\ast).$$ So $1=(N_2N_3)^4=(-\mu_1^2\beta_1^2(-\beta_2^{-1})\beta_3^2\mu_1^{-2}\beta_4^{-2}(-\beta_2),\ast)$. Therefore, $-\beta_1^2\beta_3^2\beta_4^{-2}=1$, which is equivalent to $\beta_4^2=-\beta_1^2\beta_3^2$. As noted above, $\beta_4=-\beta_1\beta_3\beta_6$. Thus $\beta_4^2=\beta_1^2\beta_3^2\beta_6^2=\beta_1^2\beta_3^2$; a contradiction.
Now we provide the lift for $w$. Let $\zeta$ be an element of $\overline{{\mathbb{F}}}_p$ such that $\zeta^{2(q+1)}=-1$. Put $H_1=(-\zeta^{-q},-1,\zeta^{-q^2-q},\zeta,1,1)$. We claim that $H_1n\in N$ and $|H_1n|=4$. Since $\zeta^{q^3+q^2+q+1}=\zeta^{(2q+2)((q^2+1)/2)}=-1$ and $-\zeta^{-q^3-q^2-q}=\zeta$, we have $$H_1^{\sigma n}=(-\zeta^{-q}, -1, \zeta^{-q^2-q}, -\zeta^{-q^3-q^2-q}, 1,1)=H_1.$$ Thus $H_1\in T$. By Lemma \[conjugation\], $$(Hn)^4=(-\lambda_2\lambda_5, \lambda_2^4, (\lambda_2\lambda_5)^2, -(\lambda_2\lambda_5)^3, \lambda_5^4,\lambda_6^4).$$
So $(H_1n)^4=1$ and $H_1n$ is a required lift for $w$.
**Torus 8.** In this case $w=w_1w_4w_6w_{36}$ and $C_W(w)\geqslant\langle w_1,w_4,w_6,w_{36}\rangle \simeq\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$.
Put $n=n_1n_4n_6n_3$. Using MAGMA, we see that $[n,n_1]=[n,n_4]=[n,n_6]=[n,n_{36}]=1$ and hence $n_1,n_4,n_6,n_{36}\in N$. Let $N_1,N_2,N_3,N_4$ be preimages of $w_1,w_4,w_6,w_{36}$ in $K$, respectively. Then $$N_1=H_1n_1,N_2=H_2n_4,N_3=H_3n_6,N_4=H_4n_{36},$$ where $H_1=(\mu_1,\mu_2,\mu_3,\mu_4,\mu_5,\mu_6),$ $H_2=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6),$ $H_3=(\beta_1,\beta_2,\beta_3,\beta_4,\beta_5,\beta_6),$ $H_4=(\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5,\gamma_6).$ Since $K\simeq C_W(w)$, we have $N_3^2=1$ and $N_3N_i=N_iN_3$, where $i=1,2,4.$ By Lemma \[conjugation\], $$N_3^2=H_3H_3^{n_6}h_6=(\beta_1^2, \beta_2^2,\beta_3^2,\beta_4^2,\beta_5^2,-\beta_5)=1.$$ Therefore $1=\beta_1^2=\beta_2^2=\beta_3^2=\beta_4^2=-\beta_5$. Using MAGMA we see that $[n_6,n_j]=1$ for each $j\in\{1,4,36\}$, whence $H_3^{-1}H_3^{n_j}=H_j^{-1}H_j^{n_6}$. By Lemma \[conjugation\], $$j=1 \Rightarrow (\beta_1^{-2}\beta_3,1,1,1,1,1)=(1,1,1,1,1,\mu_6^{-2}\mu_5),
\text{ whence } \beta_3=\beta_1^2=1.$$ $$j=4 \Rightarrow (1,1,1,\beta_2\beta_3\beta_4^{-2}\beta_5,1,1)=(1,1,1,1,1,\alpha_6^{-2}\alpha_5),
\text{ whence } \beta_4^2=\beta_2\beta_3\beta_5.$$ $$j=36 \Rightarrow (\beta_2^{-1},\beta_2^{-2},\beta_2^{-2},\beta_2^{-3},\beta_2^{-2},\beta_2^{-1})= (1,1,1,1,1,\gamma_6^{-2}\gamma_5), \text{ whence } \beta_2=1.$$ We derive a contradiction with $1=\beta_4^2=\beta_2\beta_3\beta_5=-1$.
Calculations in MAGMA show that $(n_1n_4n_6n_{36})^4=1$. Thus $n_1n_4n_6n_{36}$ is a required lift for $w$.
**Torus 11.** In this case $w=w_1w_4w_6w_3$ and $C_W(w)=\langle w, w_6,w_{36}\rangle\simeq\mathbb{Z}_4\times\mathbb{Z}_2\times\mathbb{Z}_2$.
Put $n=n_1n_4n_6n_3$. Let $N_1,N_2,N_3$ be preimages of $w_1w_4w_6w_3,w_6,w_{36}$ in $K$, respectively. Then $$N_1=H_1n,N_2=H_2n_6,N_3=H_3n_{36},$$ where $H_1=(\mu_1,\mu_2,\mu_3,\mu_4,\mu_5,\mu_6),$ $H_2=(\beta_1,\beta_2,\beta_3,\beta_4,\beta_5,\beta_6),$ $H_3=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6).$ Using MAGMA we see that $[n,n_6]=[n,n_{36}]=1$ and hence $H_1, H_2, H_3\in~T$. Since $K\simeq C_W(w)$, we have $N_2^2=1$ and $[N_3,N_1]=[N_3,N_2]=1$. The calculations from the previous case (Torus 8) show that $$N_2^2=1 \text{ implies } 1=\beta_1^2=\beta_2^2=\beta_3^2=\beta_4^2=-\beta_5,$$ and $N_3N_2=N_2N_3$ implies $\beta_2=1.$ By Lemma \[commutator\], we obtain $H_2^{-1}H_2^{n}=H_1^{-1}H_1^{n_{6}}$. It follows from Lemma \[conjugation\] that $$(\beta_1^{-1}\beta_3^{-1}\beta_4,1,\beta_1\beta_3^{-2}\beta_4,
\beta_1\beta_2\beta_3^{-1}\beta_4^{-1}\beta_5,1,\beta_5\beta_6^{-2})=(1,1,1,1,1,\mu_6^{-2}\mu_5).$$ Hence, $\beta_1\beta_4=\beta_3^2=1$ and $\beta_4=\beta_1\beta_3$. Therefore, $1=\beta_1\beta_4=\beta_1\cdot\beta_1\beta_3=\beta_3$ and hence $\beta_4=\beta_1$. Moreover, $1=\beta_1\beta_2\beta_3^{-1}\beta_4^{-1}\beta_5=\beta_1\beta_4^{-1}\beta_5=\beta_5;$ a contradiction with $\beta_5=-1$.
Now we provide the lift for $w$. Let $\zeta$ be an element of $\overline{{\mathbb{F}}}_p$ such that $\zeta^{q^3+q^2+q+1}=~-1$. Put $H_1=(\zeta,-1,\zeta^{q+1},-\zeta^{-q^2},1,1)$. We claim that $H_1n\in N$ and $|H_1n|=4$. By Lemma \[conjugation\], we get $H^{n}=(\lambda_3^{-1}\lambda_4,\lambda_2,\lambda_1\lambda_3^{-1}\lambda_4,
\lambda_1\lambda_2\lambda_3^{-1}\lambda_5,\lambda_5,\lambda_5\lambda_6^{-1})$. Hence, $H_1^{\sigma n}=(-\zeta^{-q^2-q-1},-1,-\zeta^{-q^2-q},-\zeta^{-q},1,1)^q=(-\zeta^{-q^3-q^2-q},-1,-\zeta^{-q^3-q^2},-\zeta^{-q^2},1,1)=H_1.$ Thus $H_1\in T$. Using MAGMA, we see that $$(Hn)^4=(-\lambda_2\lambda_5, \lambda_2^4, (\lambda_2\lambda_5)^2, -(\lambda_2\lambda_5)^3, \lambda_5^4,\lambda_5^2).$$ So $(H_1n)^4=1$ and $H_1n$ is a required lift for $w$.
**Torus 14.** In this case $w=w_3w_2w_4w_{14}$ and $q\equiv-1\pmod{4}$. Observe that $w_6w_{15}w_{20}\in C_W(w)$.
Put $n=n_3n_2n_4n_{14}$. Using MAGMA, we see that $[n,h_6n_6n_{15}n_{20}]=1$, and therefore $h_6n_6n_{15}n_{20}\in N$. Let $N_1$ and $N_2$ be preimages of $w$, $w_6w_{15}w_{20}$ in $K$, respectively. Then $N_1=H_1n$ and $N_2=H_2h_6n_6n_{15}n_{20}$, where $H_1=(\mu_1,\mu_2,\mu_3,\mu_4,\mu_5\,\mu_6)$, $H_2=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6)$ are elements of $T$. Using MAGMA, we get $(h_6n_6n_{15}n_{20})^4=h_2h_3$. Now Lemma \[conjugation\] implies that $$(HN_2)^4=(\lambda_1^4,-\lambda_1\lambda_2^2\lambda_3^2\lambda_5^{-2},-\lambda_1^3\lambda_2^2\lambda_3^2\lambda_5^{-2},\lambda_1^4\lambda_4^4\lambda_5^{-4},\lambda_1^4,\lambda_1^2).$$
It follows that $-\alpha_5^2=\alpha_1\alpha_2^2\alpha_3^2$, $\alpha_1^2=1$. Since $[w,d]=1$, we have $H_1^{-1}{H_1}^{N_2}=H_2^{-1}{H_2}^{N_1}$. By Lemma \[conjugation\], $$H^{n}=(\lambda_1,\lambda_3\lambda_4^{-1}\lambda_5,\lambda_1\lambda_3\lambda_4^{-1}\lambda_6,\lambda_3^2\lambda_4^{-1}\lambda_6, \lambda_2^{-1}\lambda_3\lambda_6,\lambda_6),$$ $$H^{n_6n_{15}n_{20}}=(\lambda_1,\lambda_3\lambda_6^{-1},\lambda_1\lambda_3\lambda_5^{-1},\lambda_1\lambda_2^{-1}\lambda_3\lambda_4\lambda_5^{-1}\lambda_6^{-1},\lambda_1\lambda_2^{-1}\lambda_3\lambda_6^{-1},\lambda_1\lambda_6^{-1}).$$ Whence $$\begin{gathered}
(1,\mu_2^{-1}\mu_3\mu_6^{-1},\mu_1\mu_5^{-1},\mu_1\mu_2^{-1}\mu_3\mu_5^{-1}\mu_6^{-1},\mu_1\mu_2^{-1}\mu_3\mu_5^{-1}\mu_6^{-1},\mu_1\mu_6^{-2})=\\
=(1,\alpha_2^{-1}\alpha_3\alpha_4^{-1}\alpha_5,\alpha_1\alpha_4^{-1}\alpha_6,\alpha_3^2\alpha_4^{-2}\alpha_6,\alpha_2^{-1}\alpha_3\alpha_5^{-1}\alpha_6,1).\end{gathered}$$ On the left-hand side, we see that the product of the second and third coordinates equals the fourth one, hence $(\alpha_2^{-1}\alpha_3\alpha_4^{-1}\alpha_5)(\alpha_1\alpha_4^{-1}\alpha_6)=\alpha_3^2\alpha_4^{-2}\alpha_6$. So we infer that $\alpha_1\alpha_5=\alpha_2\alpha_3$. Since $-\alpha_5^2=\alpha_1\alpha_2^2\alpha_3^2$, we obtain $\alpha_1=-1$ and hence $\alpha_5=-\alpha_2\alpha_3$. Moreover, values of the fourth and fifth coordinates coincide, so $\alpha_3^2\alpha_4^{-2}\alpha_6=\alpha_2^{-1}\alpha_3\alpha_5^{-1}\alpha_6$. Therefore, $\alpha_2\alpha_3\alpha_5=\alpha_4^2$ and hence $\alpha_5^2=-\alpha_4^2$.
Since $H_2$ belongs to the torus, we have $H_2^{\sigma{n}}=H_2$. Therefore $$(\alpha_1^q,(\alpha_3\alpha_4^{-1}\alpha_5)^q,(\alpha_1\alpha_3\alpha_4^{-1}\alpha_6)^q,(\alpha_3^2\alpha_4^{-1}\alpha_6)^q,(\alpha_2^{-1}\alpha_3\alpha_6)^q,\alpha_6^q)=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6).$$ Whence $\alpha_2=(\alpha_3\alpha_4^{-1}\alpha_5)^q$, $\alpha_3=(\alpha_1\alpha_3\alpha_4^{-1}\alpha_6)^q$ and $\alpha_4=(\alpha_3^2\alpha_4^{-1}\alpha_6)^q$. After squaring up the both sides of the equation for $\alpha_2$ and using $\alpha_4^2=-\alpha_5^2$, we have $-\alpha_3^{2q}=\alpha_2^2$. Since $\alpha_1=-1$, it is true that $\alpha_4\alpha_3^{-1}=(\alpha_3^2\alpha_4^{-1}\alpha_6)^q(-\alpha_3\alpha_4^{-1}\alpha_6)^{-q}=-\alpha_3^q$ and hence $\alpha_4=-\alpha_3^{q+1}$. Therefore, $\alpha_3=(\alpha_1\alpha_3\alpha_4^{-1}\alpha_6)^q=-\alpha_3^q\alpha_4^{-q}\alpha_6^q =-\alpha_3^q(-\alpha_3^{-q^2-q})\alpha_6$ and hence $\alpha_6=\alpha_3^{q^2+1}$. On the other hand, we have $\alpha_5=(\alpha_2^{-1}\alpha_3\alpha_6)^q$. Since $\alpha_6^q=\alpha_6$ and $\alpha_5=-\alpha_2\alpha_3$, we obtain $\alpha_6=-\alpha_2^{q+1}\alpha_3^{1-q}$. We know that $\alpha_2^2=-\alpha_3^{2q}$ and hence $\alpha_2^{q+1}=\alpha_3^{q^2+q}$. Thus $\alpha_6=-\alpha_3^{q^2+1}$; a contradiction with $\alpha_6=\alpha_3^{q^2+1}$.
Calculations in MAGMA show that $n^4=1$ and hence $n$ is the required lift for $w$ in this case.
**Torus 16.** In this case $w=w_1w_4w_6w_3w_{36}$ and $C_W(w)=\langle w\rangle\times\langle w_6,w_{27},w_{36}\rangle\simeq \mathbb{Z}_4\times S_4$. Observe that $(w_6w_{27})^3=(w_{36}w_{27})^3=(w_6w_{36})^2=1$ and $w_1w_4w_6w_3\in C_W(w)$. Put $n=n_1n_4n_6n_3n_{36}$. Using MAGMA we see that $[n,n_1n_4n_6n_3]=[n,n_{36}]=[n,n_6]=1.$ Let $N_1,N_2,N_3$ be preimages of $w_1w_4w_6w_3,w_{36},w_{6}$ in $K$, respectively. Then $$N_1=H_1n_1n_4n_6n_3,N_2=H_2n_{36},N_3=H_3n_{6},$$ where $$H_1=(\mu_1,\mu_2,\mu_3,\mu_4,\mu_5,\mu_6),H_2=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6),H_3=(\beta_1,\beta_2,\beta_3,\beta_4,\beta_5,\beta_6)$$ are elements of $T$. Since $H\simeq C_W(w)$, we have $N_3^2=1$ and $[N_3,N_1]=[N_3,N_2]=1$. The calculations for the Torus 8 show that $$N_3^2=1 \text{ implies } 1=\beta_1^2=\beta_2^2=\beta_3^2=\beta_4^2=-\beta_5,$$ and $N_3N_2=N_2N_3$ implies $\beta_2=1.$ By Lemma \[commutator\], the equation $N_3N_1=N_1N_3$ is equivalent to $H_3^{-1}H_3^{n_1n_4n_6n_3}=H_1^{-1}H_1^{n_{6}}$. By Lemma \[conjugation\], the latter equation yields $$(\beta_1^{-1}\beta_3^{-1}\beta_4,1,\beta_1\beta_3^{-2}\beta_4,
\beta_1\beta_2\beta_3^{-1}\beta_4^{-1}\beta_5,1,\beta_5\beta_6^{-2})=(1,1,1,1,1,\mu_6^{-2}\mu_5).$$ Hence, $\beta_1\beta_4=\beta_3^2=1$ and $\beta_4=\beta_1\beta_3$. So $1=\beta_1\beta_4=\beta_1\cdot\beta_1\beta_3=\beta_3$ and $\beta_4=\beta_1$. Finally, $$1=\beta_1\beta_2\beta_3^{-1}\beta_4^{-1}\beta_5=\beta_1\beta_4^{-1}\beta_5=\beta_5;$$ a contradiction with $\beta_5=-1$.
Now we provide the lift for $w$. Let $\zeta$ be an element of $\overline{{\mathbb{F}}}_p$ such that $\zeta^{q^3+q^2+q+1}=-1$. Put $H_1=(\zeta,1,\zeta^{q+1},-\zeta^{-q^2},-1,\zeta^{q^2+1})$. We claim that $H_1n\in N$ and $|H_1n|=4$. By Lemma \[conjugation\] $$H^{n}=(\lambda_2^{-1}\lambda_3^{-1}\lambda_4,\lambda_2^{-1},
\lambda_1\lambda_2^{-2}\lambda_3^{-1}\lambda_4,\lambda_1\lambda_2^{-2}\lambda_3^{-1}\lambda_5,\lambda_2^{-2}\lambda_5,\lambda_2^{-1}\lambda_5\lambda_6^{-1}).$$ Therefore $$H_1^{\sigma{n}}=(-\zeta^{-q^2-q-1},1,-\zeta^{-q^2-q},-\zeta^{-q},-1,-\zeta^{-q^2-1})^q
=(-\zeta^{-q^3-q^2-q},1,-\zeta^{-q^3-q^2},-\zeta^{-q^2},1,-\zeta^{-q^3-q})=H_1.$$ So $H_1\in T$. By Lemma \[conjugation\], $$(Hn_1n_4n_6n_3n_{36})^4=(-\lambda_2^{-1}\lambda_5, 1, \lambda_2^{-2}\lambda_5^2, -\lambda_2^{-3}\lambda_5^3, \lambda_2^{-4}\lambda_5^4, \lambda_2^{-2}\lambda_5^2).$$
Thus $(H_1n)^4=1$ and $H_1n$ is a required lift for $w$.
Complement cases
----------------
Now we will deal with maximal tori of $E_6(q)$ that have a complement in their algebraic normalizer. Throughout this subsection we suppose that $T$ is a maximal torus corresponding to the conjugacy class of $w$ in $W$. We use $w$ as in Table \[table\]. If $w=w_{i_1}w_{i_2}...w_{i_k}$, where $w_{i_j}$ are fundamental reflections, then we put $n=n_{i_1}n_{i_2}...n_{i_k}$. Observe that $n$ is a lift to $N$ for $w$.
As was mentioned in Section 3, if $q$ is even then $\mathcal{T}\simeq W$. So, the orders of $n$ and $w$ are equal. If $K$ is the isomorphic copy of $C_W(w)$ in $\mathcal{T}$ and $x\in K$ then $[x,n]=1$ and hence by Lemma \[normalizer\] we have $x\in N$. Therefore, $K$ is a complement for $T$ in $N$.
Throughout this subsection we assume that $q$ is odd and $H=(\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5,\lambda_6)$ is an arbitrary element of $T$.
**Torus 4.** In this case $w=w_3w_1$ and $C_W(w)=\langle w,w_5,w_6,w_2,
w_{36},v\rangle\simeq
\mathbb{Z}_3\times((S_3\times S_3):\mathbb{Z}_2)$, where $v=w_1w_4w_{14}w_{29}$, $w_5^v=w_2$, $w_6^v=w_{36}$ and $\langle w_5,w_6\rangle\simeq\langle w_2,w_{36}\rangle\simeq S_3$. Using MAGMA we see that $[n,n_1n_4n_{14}n_{29}]=[n,n_5]=[n,n_6]=[n,n_2]=[n,n_{36}]=1$, whence $n_1n_4n_{14}n_{29},n_5,n_6,n_2,n_{36}\in N$. Put $$N_1=n_1n_3, N_2=h_{36}n_2, N_3=h_2n_{36}, N_4=n_1n_4n_{14}n_{29}, N_5=h_5h_6n_5, N_6=h_5n_6.$$ We claim that $K=\langle N_1,N_2,N_3,N_4,N_5,N_6\rangle$ is a complement for $T$. By Lemma \[conjugation\], $$H^{n}=(\lambda_1^{-1}\lambda_3,\lambda_2,
\lambda_1^{-1}\lambda_4,\lambda_4,\lambda_5,\lambda_6).$$ Then $h_{36}^{\sigma{n}}=(-1,1,1,-1,1,-1)^{\sigma{n}}=h_{36}^\sigma=h_{36}.$ Similarly, $h_2^{\sigma{n}}=h_2$, $h_5^{\sigma{n}}=h_5$, $h_6^{\sigma{n}}=h_6$. Therefore $h_{36}, h_2, h_5$ and $h_6$ belong to $T$. Using MAGMA we have $$N_2^2=N_3^2=N_5^2=N_6^2=1,\quad (N_2N_3)^3=1, (N_5N_6)^3=1,$$ that is $\langle N_5,N_6\rangle\simeq\langle N_2,N_3\rangle\simeq S_3.$ Further, using MAGMA we obtain $$N_1^3=1,\quad N_4^2=1,\quad N_1^{N_2}=N_1^{N_3}=N_1^{N_4}=N_1^{N_5}=N_1^{N_6}=N_1.$$ Hence, $K=\langle N_1\rangle\times\langle N_2,N_3,N_4,N_5,N_6\rangle$. Finally, calculations in MAGMA show that $N_5^{N_4}=N_2, N_6^{N_4}=N_3$, and so $K\simeq\mathbb{Z}_3\times((S_3\times S_3):\mathbb{Z}_2)\simeq C_W(w)$, as claimed.
**Torus 6**. In this case $w=w_1w_3w_{5}$ and $C_W(w)=~\langle w\rangle\times\langle w_2, w_{36} \rangle\simeq\mathbb{Z}_6 \times S_3$. Using MAGMA we see that $[n, n_2]=[n, n_{36}]=1$. Therefore $n_2, n_{36}\in N$. Let $\zeta$ be an element of $\overline{{\mathbb{F}}}_p$ such that $|\zeta|=2(q+1)$. Then it is clear that $\zeta^{(q+1)}=-1$. Put $H_1=(1,1,1,1,-\zeta,-1)$, $N_1=H_1n$, $N_2=h_{36}n_2$ and $N_3=h_{2}n_{36}$. We claim that $K=\langle N_1, N_2, N_3\rangle$ is a complement for $T$ in $N$. By Lemma \[conjugation\], $$H^{n}=(\lambda_3^{-1}\lambda_4, \lambda_2, \lambda_1\lambda_3^{-1}\lambda_4, \lambda_4, \lambda_4\lambda_5^{-1}\lambda_6,\lambda_6).$$ Using this equality, we see that $h_2^{n}=h_2$, $h_{36}^{n}=h_{36}$ and hence $h_2, h_{36}\in T$. Furthermore, we have $H_1^{n}=(1,1,1,1,\zeta^{-1},-1)$. So $H_1^{\sigma n}=(1,1,1,1,\zeta^{-q},-1)$. Since $\zeta^{q}=-\zeta^{-1}$, we infer that $H_1^{\sigma{n}}=(1,1,1,1,-\zeta,-1)=H_1$. Calculations in MAGMA show that $(N_2)^2=1$, $(N_3)^2=1$, and $(N_2N_3)^3=1$. It remains to verify that $N_1^6=1$ and $[N_1,N_2]=[N_1,N_3]=1$. Applying MAGMA, we see that $n^6=h_5$, and therefore Lemma \[conjugation\] implies that $$(Hn)^6=(\lambda_4^2, \lambda_2^6, \lambda_4^4, \lambda_4^6, -\lambda_4^3\lambda_6^3, \lambda_6^6).$$ This equality yields $N_1^6=1$. By Lemma \[conjugation\], $$H^{n_2}=(\lambda_1,\lambda_2^{-1}\lambda_4,\lambda_3,\lambda_4,\lambda_5,\lambda_6), H^{n_{36}}=(\lambda_1\lambda_2^{-1}, \lambda_2^{-1}, \lambda_2^{-2}\lambda_3,\lambda_2^{-3}\lambda_4,\lambda_2^{-2}\lambda_5, \lambda_2^{-1}\lambda_6).$$ It follows that $H_1^{-1}H_1^{n_{2}}=(1,1,1,1,1,1)=h_{36}^{-1}h_{36}^{n}$ and $H_1^{-1}H_1^{n_{36}}=(1,1,1,1,1,1)=h_{2}^{-1}h_{2}^{n}$. Therefore $[N_1,N_2]=[N_1,N_3]=1$ by Lemma \[commutator\]. Thus $\langle N_1, N_2, N_3 \rangle\simeq\mathbb{Z}_6\times S_3$, as claimed.
**Torus 9.** In this case $w=w_1w_2w_3w_5$ and $C_W(w_1w_2w_3w_5)=\langle w_1w_3\rangle\times \langle w_2,w_5, v\rangle\simeq
\mathbb{Z}_3\times D_8$, where $v=w_1w_4w_{14}w_{29}, w_2^v=w_5$ and $D_8=(w_2\times w_5)\rtimes v.$ Let $\xi$ be a primitive $(q^2-1)$th root of unity and $\lambda=\xi^{\frac{q-1}{2}}$. Observe that $\lambda^{q+1}=\xi^{(q^2-1)/2}=-1$. Put $$N_1=n_1n_3, N_2=H_2n_2, N_3=H_3n_5, N_4=h_1h_4n_1n_4n_{14}n_{29},$$ where $H_2=(-1,\lambda,1,-1,1,-1), H_3=(1,1,1,1,\lambda^{-1},-1)$.
Using MAGMA, we see that $[n,n_1n_3]=[n,n_2]=[n,n_5]=[n,N_4]=1$, whence $N_1$, $n_2$, $n_5$, $N_4$ belong to $N$. By Lemma \[conjugation\], $$H^n=(\lambda_3^{-1}\lambda_4,\lambda_2^{-1}\lambda_4,\lambda_1\lambda_3^{-1}\lambda_4,
\lambda_4,\lambda_4\lambda_5^{-1}\lambda_6,\lambda_6).$$ Since $\lambda^{q+1}=-1$, we have $H_2^{\sigma{n}}=(-1,-\lambda^{-1},1,-1,1,-1)^\sigma=H_2$ and $H_3^{\sigma{n}}=(1,1,1,1,-\lambda,-1)^\sigma=H_3$, Therefore, $H_2$ and $H_3$ belong to $T$. We claim that $K=\langle N_1,N_2,N_3,N_4\rangle$ is a complement.
Calculations in MAGMA show that $N_1^3=[N_1,N_4]=1$. Now we prove that $[N_1,N_2]=[N_1,N_3]=1$. Since $[n_1n_3,n_2]=[n_1n_3,n_5]=1$, it suffices to verify $H_2^{-1}H_2^{N_1}=H_3^{-1}H_3^{N_1}=1$. By Lemma \[conjugation\], $H^{N_1}=(\lambda_3^{-1}\lambda_4, \lambda_2, \lambda_1\lambda_3^{-1}\lambda_4,\lambda_4,
\lambda_5,\lambda_6)$ and hence $H_2^{N_1}=H_2$ and $H_3^{N_1}=H_3$, as required.
Now we verify that $\langle N_2,N_3,N_4 \rangle\simeq D_8$. Using MAGMA, we see that $N_4^2=1$. By Lemma \[conjugation\], $(Hn_2)^2=(\lambda_1^2,-\lambda_4,\lambda_3^2,\lambda_4^2,\lambda_5^2,\lambda_6^2)$ and $(Hn_5)^2=(\lambda_1^2,\lambda_2^2,\lambda_3^2,\lambda_4^2,-\lambda_4\lambda_6,\lambda_6^2)$. Therefore $N_2^2=N_3^2=1$. It remains to verify the equation $N_2N_4=N_4N_3$. Note that $N_2N_4=H_2n_2N_4$. On the other hand, $N_4N_3=H_3^{N^4}N_4n_5$. By Lemma \[conjugation\], $$H^{N_4}=(\lambda_1^{-1}\lambda_6, \lambda_5^{-1}\lambda_6^2,\lambda_3^{-1}\lambda_6^2,\lambda_3^{-1}\lambda_6^3, \lambda_2^{-1}\lambda_6^2,\lambda_6).$$ Therefore $H_3^{N_4}=(1,-\lambda,1,-1,1,-1)=H_2$. Using MAGMA, we see that $n_2N_4=N_4n_5$ and hence $N_2N_4=N_4N_3$. Thus $K\simeq\langle N_1\rangle\times\langle N_2,N_3,N_4\rangle\simeq \mathbb{Z}_3\times D_8\simeq C_W(w)$.
**Torus 10.** In this case $w=w_1w_5w_3w_6$ and $C_W(w)=\langle w,w_2, w_{36},u,v\rangle
\simeq\mathbb{Z}_3\times S_3\times S_3$, where $u=w_2w_{26}w_{28}w_{34}, v=w_2w_{24}w_{32}w_{33}$ and $\langle w_2,w_{36}\rangle\simeq\langle u,v\rangle\simeq S_3.$ Put $$N_1=n, N_2=h_{36}n_2,N_3=h_2n_{36}, N_4=h_1h_6n_2n_{26}n_{28}n_{34}, N_5=h_1h_3h_6n_2n_{24}n_{32}n_{33}.$$ We claim that $K=\langle N_1,N_2,N_3,N_4,N_5\rangle$ is a complement. Using MAGMA, we see that $[N_1,N_2]=[N_1,N_3]=[N_1,N_4]=[N_1,N_5]=1$ and $N_1^3=1$. Therefore, $N_1, N_2, N_3, N_4, N_5$ belong to $N$ and $K=\langle N_1\rangle\times\langle N_2,N_3,N_4,N_5\rangle$.
Computations in MAGMA show that $N_2^2=N_3^2=N_4^2=N_5^2=1, (N_2N_3)^3=1$ and $(N_4N_5)^3=1$. Whence $\langle N_2,N_3\rangle\simeq\langle N_4,N_5\rangle\simeq S_3.$ Finally, we have $[N_2,N_4]=[N_2,N_5]=[N_3,N_4]=[N_3,N_5]=1$. Thus $K\simeq C_W(w)$, as claimed.
**Torus 12.** In this case $w=w_1w_4w_3w_2$ and $C_W(w)=\langle w,w_{6}\rangle\simeq\mathbb{Z}_5\times\mathbb{Z}_2$. Put $$N_1=n_1n_4n_3n_2, N_2=h_2h_5n_6.$$ Using MAGMA, we see that $[N_1,N_2]=1$, and therefore $N_2$ belongs to $N$. Moreover, we have $N_1^5=N_2^2=1$, so the group $K=\langle N_1,N_2\rangle\simeq\mathbb{Z}_5\times\mathbb{Z}_2$ is a complement for $T$.
**Torus 13.** In this case $w=w_3w_2w_5w_4$ and $C_W(w)=\langle w,w_{17}w_{18},w_{20}w_{21}\rangle\simeq\mathbb{Z}_6\times S_3$. Let $$N_1=n_3n_2n_5n_4, N_2=h_3h_5n_{17}n_{18}, N_3=h_4h_6n_{20}n_{21}.$$ Using MAGMA, we see that $[N_1,N_2]=[N_1,N_3]=1$. Therefore, $N_2$ and $N_3$ belong to $N$. Moreover, we have $N_1^6=1, N_2^2=N_3^2=(N_2N_3)^3=1.$ Thus $K=\langle N_1,N_2,N_3\rangle\simeq\mathbb{Z}_6\times S_3$ is a required complement for $T$.
**Torus 14.** In this case $w=w_3w_2w_4w_{14}$, $q\equiv1\pmod{4}$ and $C_W(w)\simeq SL_2(3):\mathbb{Z}_4$. Moreover, we have $C_W(w)=\langle d,y,c \rangle$, where $d=w_6w_{15}w_{20}$, $y=w_4w_{11}w_{28}$ and $c=w_1w_2w_4w_6w_{31}w_{32}$. Note that $d^4=y^4=c^3=1$, $yd=dy$ and $d^3y^2c=c^2y$. Using GAP, we see that these relations are determined $C_W(w)$ as abstract group generated by three elements. Put $n=n_3n_2n_4n_{14}$, $D=h_6n_6n_{15}n_{20}$, $Y=h_4n_4n_{11}n_{28}$ and $C=h_1h_6n_1n_2n_4n_6n_{31}n_{32}$. Using MAGMA, we see that $[n,D]=[n,Y]=[n,C]=1$ and hence $D$, $Y$, $C$ are elements of the normalizer of $T$. Let $\alpha$ be an element of $\overline{{\mathbb{F}}}_p$ such that $\alpha^2=-1$ and $H_1=(-1,-1,\alpha,1,\alpha,-1)$, $H_2=(-1,\alpha,1,-1,-\alpha,1)$. Put $N_1=H_1D$ and $N_2=H_2Y$. We claim that $K=\langle N_1, N_2, C\rangle$ is a complement for $T$ in $N$. It suffices to verify that $H_1$ and $H_2$ belong to $T$, and $N_1^4=N_2^4=C^3=1$, $N_1N_2=N_2N_1$ and $N_1^3N_2^2C=C^2N_2$.
By Lemma \[conjugation\], we get $H^{n}=(\lambda_1, \lambda_3\lambda_4^{-1}\lambda_5,\lambda_1\lambda_3\lambda_4^{-1}\lambda_6,\lambda_3^2\lambda_4^{-1}\lambda_6,\lambda_2^{-1}\lambda_3\lambda_6,\lambda_6).$
Applying to $H_1$ and $H_2$, we obtain $H_1^n=H_1$ and $H_2^n=H_2$. Since $q\equiv1\pmod{4}$, we have $\alpha^q=\alpha$ and hence $H_1^{\sigma n}=H_1$, $H_2^{\sigma n}=H_2$. So $H_1$ and $H_2$ belong to $T$. Since $D^4=Y^4=h_2h_3$, Lemma \[conjugation\] implies that $$(HD)^4=(\lambda_1^4,-\lambda_1\lambda_2^2\lambda_3^2\lambda_5^{-2},-\lambda_1^3\lambda_2^2\lambda_3^2\lambda_5^{-2},\lambda_1^4\lambda_4^4\lambda_5^{-4},\lambda_1^4,\lambda_1^2)$$ and $$(HY)^4=(\lambda_1^4,-\lambda_1^3\lambda_2^2\lambda_3^{-2}\lambda_5^2\lambda_6^{-2},-\lambda_1^3\lambda_2^{-2}\lambda_3^2\lambda_5^2\lambda_6^{-2},\lambda_1^4\lambda_5^4\lambda_6^{-4},\lambda_1^2\lambda_5^4\lambda_6^{-4},\lambda_1^2).$$ Now it is easy to see that $N_1^4=N_2^4=1$. Using MAGMA, we obtain $C^3=1$.
It follows from Lemma \[commutator\] that the equality $N_1N_2=N_2N_1$ is equivalent to $H_1^{-1}H_1^Y=H_2^{-1}H_2^D[D,Y]$. By Lemma \[conjugation\], we have $$H^{-1}H^D=(1,\lambda_2^{-1}\lambda_3\lambda_6^{-1},\lambda_1\lambda_5^{-1},\lambda_1\lambda_2^{-1}\lambda_3\lambda_5^{-1}\lambda_6^{-1},\lambda_1\lambda_2^{-1}\lambda_3\lambda_5^{-1}\lambda_6^{-1},\lambda_1\lambda_6^{-2})$$ and $$H^{-1}H^Y=(1,\lambda_1\lambda_4^{-1}\lambda_5\lambda_6^{-1},\lambda_1\lambda_4^{-1}\lambda_5\lambda_6^{-1},\lambda_1\lambda_2\lambda_3\lambda_4^{-2}\lambda_5\lambda_6^{-2},\lambda_1\lambda_6^{-2},\lambda_1\lambda_6^{-2}).$$
Therefore, we have $H_1^{-1}H_1^Y=(1,\alpha,\alpha,-1,-1,-1)$ and $H_2^{-1}H_2^D=(1,-\alpha,-\alpha,-1,-1,-1)$. Since $[D,Y]=h_2h_3$, we infer that $N_1N_2=N_2N_1$.
It remains to verify the equality $N_1^3N_2^2C=C^2N_2$. Observe that $N_1^3N_2^2C=(H_1D)^3(H_2Y)^2C=H_1H_1^DH_1^{D^2}(H_2H_2^Y)^{D^3}D^3Y^2C$. By Lemma \[conjugation\], we have $$(HD)^3=(\lambda_1^3,\lambda_2\lambda_3^2\lambda_5^{-1},\lambda_1^2\lambda_2\lambda_3^2\lambda_5^{-2}\lambda_6,\lambda_1^2\lambda_2^{-1}\lambda_3\lambda_4^3\lambda_5^{-3}\lambda_6,\lambda_1^2\lambda_2^{-1}\lambda_3\lambda_6,\lambda_1\lambda_6)D^3,$$ $$(HY)^2=(\lambda_1^2,\lambda_1\lambda_2^2\lambda_4^{-1}\lambda_5\lambda_6^{-1},\lambda_1\lambda_3^2\lambda_4^{-1}\lambda_5\lambda_6^{-1},\lambda_1\lambda_2\lambda_3\lambda_5\lambda_6^{-2},\lambda_1\lambda_5^2\lambda_6^{-2},\lambda_1)Y^2,$$ $$H^{D^3}=(\lambda_1,\lambda_1\lambda_2\lambda_5^{-1},\lambda_1\lambda_2\lambda_6^{-1},\lambda_1^2\lambda_2\lambda_3^{-1}\lambda_4\lambda_5^{-1}\lambda_6^{-1},\lambda_1^2\lambda_2\lambda_3^{-1}\lambda_6^{-1},\lambda_1\lambda_6^{-1}).$$ Therefore, $H_1H_1^DH_1^{D^2}=(-1,-\alpha,1,-1,\alpha,1)$ and $H_2H_2^Y=(1,\alpha,-\alpha,-1,1,-1)$. Finally, $(H_2H_2^Y)^{D^3}=(1,\alpha,-\alpha,-1,1,-1)$. Then $N_1^3N_2^2C=(-1,1,-\alpha,1,\alpha,-1)D^3Y^2C$. On the other hand, we have $C^2N_2=H_2^{C^2}C^2Y$. By Lemma \[conjugation\], we obtain $$H^{C^2}=(\lambda_1^{-1}\lambda_6,\lambda_1^{-1}\lambda_2^{-1}\lambda_4\lambda_5^{-1}\lambda_6,\lambda_1^{-1}\lambda_5^{-1}\lambda_6^2,\lambda_1^{-2}\lambda_2^{-1}\lambda_3\lambda_5^{-1}\lambda_6^2,\lambda_1^{-2}\lambda_3\lambda_5^{-1}\lambda_6,\lambda_1^{-1}).$$
Therefore, $C^2N_2=(-1,1,-\alpha,1,\alpha,-1)C^2Y$. Calculations in MAGMA show that $D^3Y^2C=C^2Y$ and hence $N_1^3N_2^2C=C^2N_2$. Thus $K$ is a complement, as claimed.
**Torus 15**. In this case, $w=w_1w_5w_3w_6w_2$ and $C_W(w)=\langle w\rangle\times\langle w_{24}w_{32}w_{33}, w_{26}w_{28}w_{34}\rangle\simeq\mathbb{Z}_6\times S_3$. Let $\xi, \zeta$ be elements of $\overline{{\mathbb{F}}}_p$ such that $|\xi|=2(q^3-1)$ and $|\zeta|=2(q+1)$. Observe that $\xi^{q^3-1}=-1$, $\zeta^{q+1}=-1$. Put $$H_1=(-1,\zeta,1,-1,-\xi^{q-1},-\xi^{q^2-1}),$$ $$H_2=(\xi^{q^2+q},\xi^{q^2+q+1},-\xi^{2q^2+q+1},-\xi^{2(q^2+q+1)},\xi^{(q+1)^2},\xi^{q^2+q}),$$ $$H_3=(\xi^{q+1},\xi^{q^2+q+1},\xi^{(q+1)^2},-\xi^{2(q^2+q+1)},\xi^{(q+1)^2},\xi^{q^2+q}).$$ For convenience, we denote $n_{24}n_{32}n_{33}$ by $u$ and $n_{26}n_{28}n_{34}$ by $v$. Put $N_1=H_1n$, $N_2=H_2h_1h_3h_6u$ and $N_3=H_3h_1h_6v$. We claim that $K=\langle N_1, N_2, N_3 \rangle$ is a complement for $T$ in $N$. Using MAGMA, we see that $[h_1h_3h_6u,n]=[h_1h_6v,n]=1$, so $h_1h_3h_6u$, $h_1h_6v$ belong to $N$ by Lemma \[normalizer\]. Now we verify that $H_1$, $H_2$ and $H_3$ belong to $T$. By Lemma \[conjugation\], $$H^{n}=(\lambda_3^{-1}\lambda_4,\lambda_2^{-1}\lambda_4,\lambda_1\lambda_3^{-1}\lambda_4,\lambda_4,\lambda_4\lambda_6^{-1},\lambda_5\lambda_6^{-1}).$$ Putting $H=H_1$, we obtain $H_1^{n}=(-1,-\zeta^{-1}, 1, -1, \xi^{1-q^2},\xi^{q-q^2})$. Whence $H_1^{\sigma{n}}=(-1,-\zeta^{-q},1,-1,\xi^{q-q^3},\xi^{q^2-q^3})$. Observe that $\zeta^{-q}=-\zeta$ and $\xi^{q^3}=-\xi$. Therefore, $H_1^{\sigma{n}}=(-1,\zeta,1,-1,-\xi^{q-1},-\xi^{q^2-1})=H_1$. Furthermore, $$H_2^{n}=(\xi^{q+1},-\xi^{q^2+q+1},\xi^{q^2+2q+1},-\xi^{2(q^2+q+1)},-\xi^{q^2+q+2},\xi^{q+1}).$$ Therefore, $H_2^{\sigma{n}}=(\xi^{q^2+q},-\xi^{q^3+q^2+q},\xi^{q^3+2q^2+q},-\xi^{2(q^3+q^2+q)},-\xi^{q^3+q^2+2q},\xi^{q^2+q})$. Since $\xi^{q^3}=-\xi$, we have $H_2^{\sigma{n}}=(\xi^{q^2+q},\xi^{1+q^2+q},-\xi^{1+2q^2+q},-\xi^{2(1+q^2+q)},\xi^{1+q^2+2q},\xi^{q^2+q})=H_2$. Finally, $H_3^n=(-\xi^{q^2+1},-\xi^{q^2+q+1},-\xi^{q^2+q+2},-\xi^{2(q^2+q+1)},-\xi^{q^2+q+2},\xi^{q+1})$, so $H_3^{\sigma{n}}=(-\xi^{q^3+q},-\xi^{q^3+q^2+q},-\xi^{q^3+q^2+2q},-\xi^{2(q^3+q^2+q)},-\xi^{q^3+q^2+2q},\xi^{q^2+q})$. Since $\xi^{q^3}=-\xi$ and $\xi^{2q^3}=\xi^2$, we have $H_3^{\sigma{n}}=(\xi^{1+q},\xi^{1+q^2+q},\xi^{1+q^2+2q},-\xi^{2(1+q^2+q)},\xi^{1+q^2+2q},\xi^{q^2+q})=H_3$. Since $n^6=h_2$, Lemma \[conjugation\] implies that $(H_1n)^6=(\lambda_4^2,-\lambda_4^3,\lambda_4^4,\lambda_4^6,\lambda_4^4,\lambda_4^2)$, whence $|N_1|=6$. Now we prove that $N_2$ and $N_3$ are involutions. By Lemma \[conjugation\], $(Hu)^2=(\lambda_1\lambda_6^{-1},-\lambda_2^2\lambda_4^{-1},\lambda_3\lambda_4^{-1}\lambda_5\lambda_6^{-1},1,\lambda_1^{-1}\lambda_3\lambda_4^{-1}\lambda_5,\lambda_1^{-1}\lambda_6)$. Putting $H=H_2h_1h_3h_6$, we see that $N_2^2=1$. Similarly, we have $$(Hv)^2=(\lambda_1\lambda_5^{-1}\lambda_6,-\lambda_2^2\lambda_4^{-1},\lambda_3\lambda_5^{-1},1,\lambda_3^{-1}\lambda_5,\lambda_1\lambda_3^{-1}\lambda_6).$$ Applying this to $H_3h_1h_6$ we obtain $N_3^2=1$.
Now we prove that $N_1$ commutes with $N_2$ and $N_3$. By Lemma \[commutator\], this is equivalent to $H_1^{-1}H_1^{N_2}=H_2^{-1}H_2^{N_1}$ and $H_1^{-1}H_1^{N_3}=H_3^{-1}H_3^{N_1}$, respectively. Using the equations for $H_2^n$ and $H_3^n$, we have $$H_2^{-1}H_2^{N_1}=(\xi^{1-q^2},-1,-\xi^{q-q^2},1,-\xi^{1-q},\xi^{1-q^2})=H_1^{-1}H_1^{N_2},$$ $$H_3^{-1}H_3^{N_1}=(-\xi^{q^2-q},-1,-\xi^{1-q},1,-\xi^{1-q},\xi^{1-q^2})=H_1^{-1}H_1^{N_3},$$ as required.
It remains to prove that $(N_2N_3)^3=1$. Observe that $$\begin{gathered}
N_2N_3=H_2h_1h_3h_6uH_3h_1h_6v=H_2h_1h_3h_6(H_3h_1h_6)^{u}uv=\\
=(\ast,\xi^{q^2+q+1},\ast,-\xi^{2(q^2+q+1)},\ast,\ast)(\ast,-\xi^{-(q^2+q+1)},\ast,-\xi^{-2(q^2+q+1)},\ast,\ast)uv
=(\ast,-1,\ast,1,\ast,\ast)uv \end{gathered}$$ By Lemma \[conjugation\], $$(Huv)^3=(\lambda_4,-\lambda_2^3,\lambda_4^2,\lambda_4^3,\lambda_4^2,\lambda_4).$$ Thus $(N_2N_3)^3=1$ and $K$ is a required complement.
**Torus 17**. In this case $w=w_1w_4w_5w_3w_{36}$ and $C_W(w)=~\langle w\rangle\simeq\mathbb{Z}_{10}$. Let $\xi$ be an element of $\overline{{\mathbb{F}}}_p$ such that $|\xi|=(q+1)(q^5-1)$. Let $\zeta=\xi^{(q-1)/2}$. Observe that $\xi^{q(q^5-1)}\xi^{(q^5-1)}=1$ and hence $\zeta^{q^6-q}=\zeta^{1-q^5}$. Put $$H_1=(\zeta^{q^6+q^3-q},\zeta^{-q^5+1},\zeta^{-q^5+q^4+q^3+1},\zeta^{-q^5+q^4+q^3+q^2+1},\zeta^{q^4+q^3+q^2+1},\zeta^{q^4+q^3+q^2+q+1}).$$ Now we show that $H_1\in T$. By Lemma \[conjugation\], $$H^n=(\lambda_2^{-1}\lambda_3^{-1}\lambda_4,\lambda_2^{-1},\lambda_1\lambda_2^{-2}\lambda_3^{-1}\lambda_4,\lambda_1\lambda_2^{-2}\lambda_3^{-1}\lambda_4\lambda_5^{-1}\lambda_6,\lambda_2^{-2}\lambda_4\lambda_5^{-1}\lambda_6,
\lambda_2^{-1}\lambda_6).$$ Putting $H=H_1$, we see that $$H_1^{n}=(\zeta^{q^5+q^2-1},\zeta^{q^5-1},\zeta^{q^5+q^3+q^2-1},\zeta^{q^5+q^3+q^2+q-1},\zeta^{q^5+q^4+q^3+q^2+q-1},\zeta^{q^5+q^4+q^3+q^2+q}).$$ Therefore, $$\begin{gathered}
H_1^{\sigma{n}}=(\zeta^{q^6+q^3-q},\zeta^{q^6-q},\zeta^{q^6+q^4+q^3-q},\zeta^{q^6+q^4+q^3+q^2-q},\zeta^{q^6+q^5+q^4+q^3+q^2-q},\zeta^{q^6+q^5+q^4+q^3+q^2})=\\
=(\zeta^{q^6+q^3-q},\zeta^{1-q^5},\zeta^{-q^5+q^4+q^3+1},\zeta^{-q^5+q^4+q^3+q^2+1},\zeta^{q^4+q^3+q^2+1},\zeta^{q^4+q^3+q^2+q+1})=H_1.\end{gathered}$$ Thus $H_1$ belongs to $T$.
Using MAGMA, we see that $n^{10}=h_1h_4h_6$, and therefore Lemma \[conjugation\] implies that $(Hn)^{10}=(-\lambda_2^{-1}\lambda_6^2,1,\lambda_2^{-2}\lambda_6^4,-\lambda_2^{-3}\lambda_6^6,\lambda_2^{-4}\lambda_6^8,-\lambda_2^{-5}\lambda_6^{10})$.
Putting $H=H_1$, we have $\lambda_2^{-1}\lambda_6^2=\zeta^{q^5-1}\zeta^{2q^4+2q^3+2q^2+2q+2}=\zeta^{q^5+2q^4+2q^3+2q^2+2q+1}=\zeta^{(q+1)(q^4+q^3+q^2+q+1)}=\xi^{(q^5-1)(q+1)/2}=-1$. Thus $(H_1n)^{10}=1$ and $\langle H_1n \rangle$ is a complement for $T$ in $N$.
**Torus 18.** In this case $w=w_1w_4w_6w_3w_5$ and $C_W(w)=\langle w,w_{36}\rangle\simeq\mathbb{Z}_6\times\mathbb{Z}_2$. Let $\xi$ and $\zeta$ be elements of $\overline{{\mathbb{F}}}_p$ such that $\xi^{q+1}=-1$ and $\zeta^{q-1}=-1$. Put $H_1=(\xi, -1, -1, \xi^{-1},-1,\xi)$ and $H_2=(\zeta,-\zeta^2,-\zeta^2,\zeta^3,-\zeta^2,\zeta)$. Now we verify that $H_1, H_2\in T$. By Lemma \[conjugation\], $$H^{n}=(\lambda_3^{-1}\lambda_4,\lambda_2,\lambda_1\lambda_3^{-1}\lambda_4,\lambda_1\lambda_2\lambda_3^{-1}\lambda_4\lambda_5^{-1}\lambda_6,\lambda_4\lambda_5^{-1}\lambda_6,\lambda_4\lambda_5^{-1}).$$ So $H_1^{\sigma{n}}=(-\xi^{-q}, (-1)^{q},(-1)^{q}, (-\xi)^q,(-1)^q,(-\xi)^{-q})$. Since $\xi^{q+1}=-1$, we have $-\xi^{-q}=\xi$ and $(-\xi)^q=\xi^{-1}$. Therefore, $H_1^{\sigma{n}}=H_1$ and hence $H_1\in T$. Now $H_2^{\sigma{n}}=(-\zeta^q,-\zeta^{2q},-\zeta^{2q},-\zeta^{3q},-\zeta^{2q},-\zeta^q)$. Since $\zeta^q=-\zeta$, we have $H_2^{\sigma{n}}=(\zeta,-\zeta^2,-\zeta^2,\zeta^3,-\zeta^2,\zeta)=H_2$.
Put $N_1=H_1n_1n_4n_6n_3n_5$ and $N_2=H_2n_{36}$. We claim that $K=\langle N_1, N_2\rangle$ is a complement for $T$ in $N$. It suffices to show that $|N_1|=6$, $|N_2|=2$ and $[N_1,N_2]=1$. Using MAGMA, we see that $n^6=h_1h_4h_6$, and therefore Lemma \[conjugation\] implies that $(Hn)^6=(-\lambda_2^3,\lambda_2^6,\lambda_2^6,-\lambda_2^9,\lambda_2^6,-\lambda_2^3)$. Therefore, we obtain $N_1^6=(H_1n)^6=1$. Similarly, we have $(Hn_{36})^2=(-\lambda_1^2\lambda_2^{-1},1,\lambda_2^{-2}\lambda_3^2,-\lambda_2^{-3}\lambda_4^2,\lambda_2^{-2}\lambda_5^2,-\lambda_2^{-1}\lambda_6^2)$, so $(H_2n_{36})^2=1$. Note that $[n,n_{36}]=1$, so it remains to prove that $H_2^{-1}H_2^{n}=H_1^{-1}H_1^{n_{36}}$ by Lemma \[commutator\]. Using equations for $H^{n}$ and $H^{n_{36}}$, we obtain $$H_1^{-1}H_1^{n_{36}}=(\xi^{-1}, -1, -1, \xi,-1,\xi^{-1})(-\xi, -1, -1, -\xi^{-1},-1,-\xi)=(-1,1,1,-1,1,-1).$$ and $$H_2^{-1}H_2^{n}=(\zeta^{-1},-\zeta^{-2},-\zeta^{-2},\zeta^{-3},-\zeta^{-2},\zeta^{-1})(-\zeta,-\zeta^{2},-\zeta^{2},-\zeta^{3},-\zeta^{2},-\zeta)=(-1,1,1,-1,1,-1).$$ Thus $[N_1,N_2]=1$ and $\langle N_1, N_2\rangle$ is a complement for $T$, as claimed.
**Tori 19, 23, 24.** In these cases $C_W(w)$ is cyclic and generated by $w$. Depending on torus, we have $w=w_2w_5w_3w_4w_6$, $w_1w_4w_6w_3w_2w_5$ or $w_1w_4w_{14}w_3w_2w_6$. Put $N_1=n_2n_5n_3n_4n_6$, $n_1n_4n_6n_3n_2n_5$ or $n_1n_4n_{14}n_3n_2n_6$, respectively. Calculations in MAGMA show that $|N_1|=|w|$ in each case and so $\langle N_1\rangle$ is a complement.
**Torus 20.** In this case $w=w_{20}w_5w_4w_3w_2$ and $C_W(w)=~\langle w\rangle\simeq\mathbb{Z}_{12}$. Put $n=n_{20}n_5n_4n_3n_2$. Let $\xi$ be an element of $\overline{{\mathbb{F}}}_p$ such that $|\xi|=|T|=(q-1)(q^2+1)(q^3+1)$ and $\zeta=\xi^{(q-1)/2}$. $$H_1=(-1,-\zeta^{q},-\zeta^{-q^4},-\zeta^{-q^4-q^3},\zeta^{-q^4-q^3-q^2-1},\zeta^{-q^3-1}).$$ By Lemma \[conjugation\], $$H^n=(\lambda_1, \lambda_1\lambda_3^{-1}\lambda_4\lambda_6^{-1}, \lambda_1\lambda_3^{-1}\lambda_4,\lambda_1^2\lambda_3^{-2}\lambda_4\lambda_5\lambda_6^{-1},\lambda_1^2\lambda_3^{-2}\lambda_4,\lambda_1\lambda_2\lambda_3^{-1}).$$ Therefore, we obtain $H_1^n=(-1,-\zeta,-\zeta^{-q^3},-\zeta^{-q^3-q^2},-\zeta^{q^4-q^3},-\zeta^{q^4+q})$. Hence $H_1^{\sigma n} =(-1,-\zeta^{q},-\zeta^{-q^4},-\zeta^{-q^4-q^3},-\zeta^{q^5-q^4},-\zeta^{q^5+q^2}).$ Observe that $|\xi|=2(q^5+q^3+q^2+1)(q-1)/2$ and hence $\zeta^{q^5+q^3+q^2+1}=-1$. Then $\zeta^{q^5}=-\zeta^{-q^3-q^2-1}$. So $-\zeta^{q^5-q^4}=\zeta^{-q^4-q^3-q^2-1}$ and $-\zeta^{q^5+q^2}=\zeta^{-q^3-q^2-1+q^2}$. Thus $H_1^{\sigma n}=H_1$. Using MAGMA, we have $n^{12}=h_2h_3$, and therefore $(Hn)^{12}=(\lambda_1^{12},-\lambda_1^9,-\lambda_1^{15},\lambda_1^{18},\lambda_1^{12},\lambda_1^6)$. Thus $(H_1n)^{12}=1$ and $\langle H_1n\rangle$ is a complement for $T$ in $N$.
**Torus 21.** In this case $w=w_1w_5w_2w_3w_6w_{36}$ and $C_W(w)\simeq(((\mathbb{Z}_3\times\mathbb{Z}_3):\mathbb{Z}_3):Q_8):\mathbb{Z}_3$. Moreover, we have $C_W(w)=\langle u,v\rangle$, where $u=w_1w_2w_5w_{23}w_{26}w_{31}, v=w_1w_2w_6w_8w_{10}w_{29}$ and $$\langle u,v\rangle\simeq\langle a,b~|~a^{12}=b^6=a^8ba^{-8}b^{-1}=(a^6b^{-1})^3=a^6b^2a^6b^{-2}=ba^8(a^{-1}b)^2a^{-1}=1\rangle$$ Put $N_1=h_1h_2h_5\cdot n_1n_2n_5n_{23}n_{26}n_{31}, N_2=h_1h_5\cdot n_1n_2n_6n_8n_{10}n_{29},$ and $N=n$. Using MAGMA, we see that $[N,N_1]=[N,N_2]=1$. So $N_1$ and $N_2$ belong to $N$ by Lemma \[normalizer\]. It is easy to verify the relations of $C_W(w)$ for $N_1$ and $N_2$ using MAGMA, but we provide arguments that do not involve computations.
The relations of $C_W(w)$ are valid for $u,v$. Substituting $a=N_1$ and $b=N_2$, we obtain each relation up to some element $h\in T$. Since $N_1\in\mathcal{T}$ and $N_2\in\mathcal{T}$, we infer that every such $h\in\mathcal{T}\cap{T}=\mathcal{H}\cap{T}$. Since $T\simeq(q^2+q+1)^3$, every element of $T$ has odd order. On the other hand, $\mathcal{H}$ is an elementary abelain 2-group and hence $\mathcal{H}\cap{T}=1$. Thus all relations of $C_W(w)$ hold true for $K=\langle N_1, N_2\rangle$, and therefore $K$ is a required complement.
**Torus 22.** In this case $w=w_1w_4w_6w_3w_5w_{36}$ and $C_W(w)=\langle w,w_{36},w_{24}\rangle\simeq\mathbb{Z}_6\times S_3$. By Lemma \[conjugation\], $$H^n=
(\lambda_2^{-1}\lambda_3^{-1}\lambda_4,\lambda_2^{-1},\lambda_1\lambda_2^{-2}\lambda_3^{-1}\lambda_4,
\lambda_1\lambda_2^{-2}\lambda_3^{-1}\lambda_4\lambda_5^{-1}\lambda_6,
\lambda_2^{-2}\lambda_4\lambda_5^{-1}\lambda_6,\lambda_2^{-1}\lambda_4\lambda_5^{-1}).$$ Using MAGMA, we see that $[n,n_{36}]=1$ and $[n,n_{24}]=h_2h_3h_5$. Let ${h}=(\alpha^2,\alpha,\alpha,1,\alpha,\alpha^2)$, where $\alpha\in\overline{{\mathbb{F}}}_p$ with $\alpha^{q+1}=-1$. Then ${h}^{\sigma{n}}=(\alpha^{-2},\alpha^{-1},\alpha^{-1},1,\alpha^{-1},\alpha^{-2})^\sigma=
(\alpha^{2},-\alpha,-\alpha,1,-\alpha,\alpha^{2})=h_2h_3h_5\cdot h$. Hence, Lemma \[normalizer\] implies that ${h}n_{24}$ belongs to $N$. Let $\xi$ be a primitive $(2q^3+2)$th root of unity, $\lambda=-\xi^2$ and $\alpha=\xi^{-q^2+q-1}$. Observe that $\lambda^{-q^3}=-\xi^{-2q^3}=-\xi^2=\lambda$. Put $$N_1=H_1n,N_2=H_2{h}n_{24},N_3=H_3n_{36},$$ where $H_1=(\lambda,1,\lambda^{q+1},\lambda^{-q^2+q+1},\lambda^{q+1},\lambda),
H_2=(-\alpha^{-2},1,1,-\alpha^2,1,-\alpha^{-2}),
H_3=h_2h_3h_5.$ We claim that $K=\langle N_1,N_2,N_3\rangle$ is a complement. First, we have $H_1^{\sigma{n}}=
(\lambda^{-q^2},1,\lambda^{-q^2+1},\lambda^{-q^2-q+1},\lambda^{-q^2+1},\lambda^{-q^2})^q=
(\lambda,1,\lambda^{1+q},\lambda^{1-q^2+q},\lambda^{1+q},\lambda)=H_1$. Moreover, $H_{2}^{\sigma{n}}=(-\alpha^2,1,1,-\alpha^{-2},1,-\alpha^2)^q=
(-\alpha^{-2},1,1,-\alpha^2,1,-\alpha^{-2})=H_{2}$ and $H_3^{\sigma{n}}=H_3^q=H_3$. Hence, $H_1,H_2,H_3\in T$. Lemma \[conjugation\] implies that $$H^{n_{24}}=(\lambda_1,\lambda_1\lambda_2\lambda_4^{-1}\lambda_6,\lambda_1\lambda_3\lambda_4^{-1}\lambda_6,
\lambda_1^2\lambda_4^{-1}\lambda_6^2,\lambda_1\lambda_4^{-1}\lambda_5\lambda_6,\lambda_6),$$ $$(Hn_{24})^2=(\lambda_1^2,-\lambda_1\lambda_2^2\lambda_4^{-1}\lambda_6,-\lambda_1\lambda_3^2\lambda_4^{-1}\lambda_6,
\lambda_1^2\lambda_6^2,-\lambda_1\lambda_4^{-1}\lambda_5^2\lambda_6,\lambda_6^2).$$ Therefore, $N_{2}^2=1.$ Using MAGMA, we see that $N_3^2=1$. Observe that $$\begin{gathered}
N_{2}N_{3}=H_{2}{h}n_{24}H_{3}n_{36}=H_{2}{h}H_{3}^{n_{24}}n_{24}n_{36}=\\
=(-1,\alpha,\alpha,-\alpha^2,\alpha,-1)(1,-1,-1,1,-1,1)n_{24}n_{36}=\\
=(-1,-\alpha,-\alpha,-\alpha^2,-\alpha,-1)n_{24}n_{36}.\end{gathered}$$ Calculations in MAGMA show that $(n_{24}n_{36})^3=1$, and therefore Lemma \[conjugation\] yields $$(Hn_{24}n_{36})^3=(\lambda_1^2\lambda_2^{-2}\lambda_4\lambda_6^{-1},1,
\lambda_2^{-3}\lambda_3^3,\lambda_1\lambda_2^{-4}\lambda_4^2\lambda_6,\lambda_2^{-3}\lambda_5^3,
\lambda_1^{-1}\lambda_2^{-2}\lambda_4\lambda_6^2).$$ Hence, $(N_{2}N_{3})^3=1$ and $\langle N_{2},N_{3}\rangle\simeq S_3$.
Furthermore, by Lemma \[commutator\] we have $N_1N_3=N_3N_1$ is equivalent to $H_{3}^{-1}H_{3}^{n}=H_1^{-1}H_1^{n_{36}}$. Lemma \[conjugation\] implies that $H^{n_{36}}=(\lambda_1\lambda_2^{-1},\lambda_2^{-1},\lambda_2^{-2}\lambda_3,
\lambda_2^{-3}\lambda_4,\lambda_2^{-2}\lambda_5,\lambda_2^{-1}\lambda_6).$ Then $H_1^{n_{36}}=H_1$ and $H_{3}^{n}=H_{3}$. Therefore $N_1N_3=N_3N_1.$ By Lemma \[commutator\], we know that $$N_1N_2=N_2N_1\text{ is equivalent to } H_1^{-1}H_1^{n_{24}}\cdot [n_{24},n]=(H_2{h})^{-1}(H_2{h})^n.$$ Lemma \[conjugation\] implies that $H^{-1}H^{n_{24}}=(1,\lambda_1\lambda_4^{-1}\lambda_6,\lambda_1\lambda_4^{-1}\lambda_6,
\lambda_1^2\lambda_4^{-2}\lambda_6^2,\lambda_1\lambda_4^{-1}\lambda_6,1)$. Putting $H=H_1$ and using $\lambda^{q^2-q+1}=-\xi^{2(q^2-q+1)}=-\alpha^{-2}$, we have $H_1^{-1}H_1^{n_{24}}=(1,-\alpha^{-2},-\alpha^{-2},\alpha^{-4},-\alpha^{-2},1)$. On the other hand, $H_2{h}=
(-1,\alpha,\alpha,-\alpha^2,\alpha,-1),$ so $(H_2{h})^n=(-1,\alpha^{-1},\alpha^{-1},-\alpha^{-2},\alpha^{-1},-1).$ Therefore, $$(H_2{h})^{-1}(H_2{h})^n=
(1,\alpha^{-2},\alpha^{-2},\alpha^{-4},\alpha^{-2},1).$$ Since $[n,n_{24}]=h_2h_3h_5$, we get $N_1N_2=N_2N_1$.
Finally, using MAGMA, we see that $(Hn)^6=1$. Thus $N_1^6=1$ and $K\simeq\langle N_1\rangle\times\langle N_2,N_3\rangle\simeq\mathbb{Z}_6\times S_3$, as claimed.
**Torus 25.** In this case $w=w_1w_4w_{14}w_3w_2w_{31}$ and $C_W(w)=\langle w^2 \rangle\times\langle i,j,c \rangle $, where $i=w_3w_6w_{19}w_{26}$, $j=w_3w_6w_{14}w_{30}$, $c=w_1w_4w_6w_{13}w_{20}w_{34}$ and $\langle i,j,c\rangle\simeq SL_2(3)$. Observe that $|T|=(q^2-q+1)(q^4+q^2+1)$ is odd.
Put $N_1=n^2$, $N_2=h_1h_2h_5n_3n_6n_{19}n_{26}$, $N_3=h_2h_3h_4h_5n_3n_6n_{14}n_{30}$, $N_4=h_1h_2h_4h_6n_1n_4n_6n_{13}n_{20}n_{34}$. We claim that $K=\langle N_1, N_2, N_3, N_4 \rangle$ is a complement for $T$ in $N$. Using MAGMA, we see that $[n,N_2]=[n,N_2]=[n,N_3]=1$ and hence $N_2, N_3, N_4\in N$ by Lemma \[normalizer\]. Clearly the image of $\langle N_1, N_2, N_3, N_4 \rangle$ in $W$ is $C_W(w)$. Now we argue as in the case of Torus 21. The group $C_W(w)$ is defined by some relations on $w^2$, $i$, $j$ and $c$. These relations hold true for $N_1$, $N_2$, $N_3$ and $N_4$ up to some elements $h\in T\cap\mathcal{H}$. However, since the order of $T$ is odd, we have $h=1$ for each relation. Thus $K\simeq C_W(w)$ and $K$ is a required complement.
Representative $w$ $|w|$ $|C_W(w)|$ Structure of $C_W(w)$ Cyclic structure of $T$ Splits
---- ---------------------------- ------- ------------ ----------------------------------- ------------------------------------- -----------------------
1 $1$ 1 51840 $O_5(3):Z_2$ $(q-1)^6$ –
2 $w_1$ 2 1440 $S_2\times S_6$ $(q-1)^4\times(q^2-1)$ –
3 $w_1w_2$ 2 192 $D_8\times S_4$ $(q-1)^2\times(q^2-1)^2$ –
4 $w_3w_1$ 3 216 $Z_3\times((S_3\times S_3):Z_2)$ $(q-1)^3\times(q^3-1)$ +
5 $w_2w_3w_5$ 2 96 $Z_2\times Z_2\times S_4$ $(q^2-1)^3$ –
6 $w_1w_3w_5$ 6 36 $Z_6\times S_3$ $(q-1)\times(q^2-1)\times(q^3-1)$ +
7 $w_1w_3w_4$ 4 32 $Z_4\times D_8$ $(q-1)^2\times(q^4-1)$ –
8 $w_1w_4w_6w_{36}$ 2 1152 $Z_2:(((A_4\times A_4):Z_2):Z_2)$ $(q+1)^2\times(q^2-1)^2$ –
9 $w_1w_2w_3w_5$ 6 24 $Z_3\times D_8$ $(q^2-1)\times(q+1)(q^3-1)$ +
10 $w_1w_5w_3w_6$ 3 108 $Z_3\times S_3\times S_3$ $(q-1)\times(q^2+q+1)\times(q^3-1)$ +
11 $w_1w_4w_6w_3$ 4 16 $Z_4\times Z_2\times Z_2$ $(q^2-1)\times(q^4-1)$ –
12 $w_1w_4w_3w_2$ 5 10 $ Z_2\times Z_5$ $(q-1)\times(q^5-1)$ +
13 $w_3w_2w_5w_4$ 6 36 $ Z_6\times S_3$ $(q^2-1)\times(q-1)(q^3+1)$ +
14 $w_3w_2w_4w_{14}$ 4 96 $ SL_2(3): Z_4$ $(q-1)(q^2+1)^2$ + $(q\not\equiv3(4))$
– $(q\equiv3(4))$
15 $w_1w_5w_3w_6w_2$ 6 36 $ Z_6\times S_3$ $(q^2+q+1)\times(q+1)(q^3-1)$ +
16 $w_1w_4w_6w_3w_{36}$ 4 96 $ Z_4\times S_4$ $(q+1)^2\times(q^4-1)$ –
17 $w_1w_4w_5w_3w_{36}$ 10 10 $ Z_{10}$ $(q+1)(q^5-1)$ +
18 $w_1w_4w_6w_3w_5$ 6 12 $ Z_6\times Z_2$ $(q^2+q+1)\times(q-1)(q^3+1)$ +
19 $w_2w_5w_3w_4w_6$ 8 8 $ Z_8$ $(q^2-1)(q^4+1)$ +
20 $w_{20}w_5w_4w_3w_2$ 12 12 $ Z_{12}$ $(q-1)(q^2+1)(q^3+1)$ +
21 $w_1w_5w_2w_3w_6w_{36}$ 3 648 $(((Z_3\times Z_3):Z_3):Q_8):Z_3$ $(q^2+q+1)^3$ +
22 $w_1w_4w_6w_3w_5w_{36}$ 6 36 $ Z_6\times S_3$ $(q+1)\times(q^5+q^4+q^3+q^2+q+1)$ +
23 $w_1w_4w_6w_3w_2w_5$ 12 12 $ Z_{12}$ $(q^2+q+1)(q^4-q^2+1)$ +
24 $w_1w_4w_{14}w_3w_2w_6$ 9 9 $ Z_9$ $(q^6+q^3+1)$ +
25 $w_1w_4w_{14}w_3w_2w_{31}$ 6 72 $ Z_3\times SL_2(3)$ $(q^2-q+1)\times(q^4+q^2+1)$ +
: The maximal tori of simply connected group $E_6(q)$\[table\]
[99]{}
J.Adams, X.He, [*Lifting of elements of Weyl groups*]{}, J. Algebra. V.485 (2017), 142–165.
A.A.Buturlakin, M.A.Grechkoseeva, [*The cyclic structure of maximal tori of the finite classical groups*]{}, Algebra Logic, V.46:2 (2007), 73–89.
R.W.Carter, [*Finite groups of Lie type, Conjugacy classes and complex characters*]{}, John Wiley and Sons, 1985.
D.I.Deriziotis, A.P.Fakiolas, [*The maximal tori in the finite Chevalley groups of type $E_6,E_7$ and $E_8$*]{}, Comm. Algebra, V.19:3 (1991) 889–903.
A.A.Gal$'$t, [*On the splitting of the normalizer of a maximal torus in the exceptional linear algebraic groups*]{}, Izv. Math., V.81:2 (2017), 269–285.
A.A.Galt, [*On splitting of the normalizer of a maximal torus in orthogonal groups*]{}, J. Algebra Appl., V.16:9 (2017) 1750174 (23 pages).
A.A.Galt, [*On splitting of the normalizer of a maximal torus in linear groups*]{}, J. Algebra Appl., V.14:7 (2015) 1550114 (20 pages).
A.A.Gal$'$t, [*On the splitting of the normalizer of a maximal torus in symplectic groups*]{}, Izv. Math., V.78:3, (2014), 443–458.
Gorenstein D., Lyons R., Solomon R., [*The classification of the finite simple groups*]{}. Number 3. Part I. Chapter A. Almost simple $K$-groups. Mathematical Surveys and Monographs, [**40**]{}, N.3, American Mathematical Society, Providence, RI, 1998.
J.H.Ċonway, R.T.Ċurtis, S.P.Norton, R.A.Parker, R.A.Wilson, [*Atlas of Finite Groups*]{}, Clarendon Press, Oxford, 1985.
N.A.Vavilov, [*Do it yourself structure constants for Lie algebras of types $E_l$*]{}, J. Math. Sci. (N.Y.), V.120:4 (2004), 1513–1548.
http://magma.maths.usyd.edu.au/calc/
W.Bosma, J.Cannon, and C.Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., V.24 (1997), 235–265.
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.9.1; 2018. (http://www.gap-system.org)
J.Tits, [*Normalisateurs de tores I. Groupes de Coxeter Étendus*]{}, J. Algebra, V.4 (1966), 96–116.
A. Galt, <span style="font-variant:small-caps;">Sobolev Institute of Mathematics, Novosibirsk, Russia;</span>
*E-mail address:* `[email protected]`
A. Staroletov, <span style="font-variant:small-caps;">Sobolev Institute of Mathematics, Novosibirsk, Russia;</span>
*E-mail address:* `[email protected]`
[^1]: This research was supported by the Russian Science Foundation (project no. 14-21-00065).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
This paper is devoted to the algorithmic development of inverse elastic scattering problems. We focus on reconstructing the locations and shapes of elastic scatterers with known dictionary data for the nearly incompressible materials. The scatterers include non-penetrable rigid obstacles and penetrable mediums, and we use time-harmonic elastic point signals as the incident input waves. The scattered waves are collected in a relatively small backscattering aperture on a bounded surface. A two-stage algorithm is proposed for the reconstruction and only two incident waves of different wavenumbers are required. The unknown scatterer is first approximately located by using the measured data at a small wavenumber, and then the shape of the scatterer is determined by the computed location of the scatterer together with the measured data at a regular wavenumber. The corresponding mathematical principle with rigorous analysis is presented. Numerical tests illustrate the effectiveness and efficiency of the proposed method.
[**Keywords:**]{} inverse scattering, elastic wave propagation, reconstruction scheme, nearly impressible materials
[**2010 Mathematics Subject Classification:**]{} 35R30, 35P25, 78A46
title: |
On an inverse elastic wave imaging scheme for\
nearly incompressible materials
---
Introduction {#sec:intro}
============
Inverse scattering associated with acoustic, electromagnetic and elastic waves are important for various applications including sonar and radar imaging, geophysical exploration, seismology, medical imaging and remote sensing; see [@AA1; @AA2; @AA3; @CK; @Kir1] and the references therein. Inverse scattering problems are concerned with the recovery of unknown scatterers by wave probing. To that end, one sends an incident detecting wave to probe the scatterers, and then measures the scattered wave data away from the scatterers. By using the measurement data, one can infer knowledge about the unknown scatterers including locations, shapes or material properties of the scatterers.
In this paper, we mainly consider the inverse elastic wave imaging. In order to uniquely reconstruct the unknown obstacles or medium, one usually needs scattered data of full aperture for incident elastic plane waves of all directions at least theoretically; see [@PH] for the obstacles case and [@PH0] for the medium case with extra scattering data with multiple frequencies within some positive real interval. However, this causes a lot of challenges for practical applications, and particularly is not efficient for real-time applications. One reason is that no priori information is used. Inspired by the modern technology including machine learning, several imaging models and efficient numerical schemes with a priori information from dictionary data are developed [@AA1; @AA2; @AA3; @LWY]. In [@AA1], an efficient imaging procedure was developed for target identification based on the dictionary matching with precomputed generalized polarization tensors, and in [@LWY], a fast gesture computing scheme with acoustic wave is developed with precomputed scattering data.
To motivate the current study, we briefly discuss the ideas of the design in [@LWY]. Time-harmonic point wave signals are first emitted, and one then collects the scattering wave data within a relatively small backscattering aperture. The reconstruction process is divided into two steps. First, a low-frequency wave signal is emitted and one then uses the collected scattering data to determine the location of the scatterer. Second, a regular-frequency (compared to the size of the scatterers) wave signal is emitted, and one then uses the collected scattering data to determine the shape of the scatterer. Besides, the numerical implementation is totally “direct" without any inversions or iterations, and hence it is very fast and robust.
Now we turn our focus to the inverse elastic problem. The nearly incompressible material is widely studied in engineering [@MDR; @SMG] and mathematical community [@BM; @SBC]. For homogeneous deformations, the Lamé coefficients $\lambda$ and $\mu$ have the following descriptions [@LAU; @SBC], $$\label{eq:strain:stree:rela}
\lambda = \frac{\nu E}{(1+\nu)(1-2\nu)}, \quad \mu = \frac{E}{2(1+\nu)},$$ where $E$ is the *Young’s modulus* and $\nu$ is the *Poisson’s ratio*. By , we conclude the following limiting properties, $$\label{eq:nearly:in:para}
\lambda \rightarrow +\infty \quad \text{while} \quad \nu \rightarrow (1/2)-,$$ where this kind of material with is commonly referred to as *nearly incompressible material* [@SBC]. By , one can also use $\lambda/\mu \gg 1$ or $\lambda \gg \mu$ to characterize this property because of the boundedness of $\mu$ [@BLP]. The nearly incompressible materials are frequently seen in real applications including polymer materials, rubber and biologic tissues [@YT; @MRR; @MDR; @KV]. The numerical computation of static elasticity or elastic wave propagation in the nearly incompressible materials is also very challenging and important, and various finite element or spectral element methods are developed in the last several decades for overcoming the computational difficulty [@SBC; @BLP; @BM; @SMG]. In addition, there is also some related study on the inverse problems for engineering applications [@KV].
In this article, we aim to further develop the imaging technique with a prior dictionary data in a practical setting of using elastic waves. There are several challenges we are confronted with the design of the reconstruction schemes. First, different from [@HLLS], full aperture scattering data is usually neither practical nor feasible in assorted applications for nearly incompressible materials. The measurement information is very limited in our numerical scheme, and the only available information is the backscattering data in a small aperture associated with a few time harmonic signals. Second, the whole reconstruction process should be completed in a timely manner. Third, elastic point signal includes the mixture of the fundamental solutions of two different wave numbers, and is more complicated than the acoustic or electromagnetic point signal which has only one wave number. Moreover, it is difficult to separate the shear wave and the pressure wave from the finite-aperture scattering data with different wave numbers, and meanwhile the separation is very critical for the “direct" imaging method developed in [@LWY].
Nonetheless, we manage to overcome the aforementioned challenges in our study by incorporating the key dictionary ingredient as mentioned before. We assume the shape of the unknown scatterers are all from a *dictionary* that is known a priori, which is reasonable since the shapes of various scatterers can be collected by experience and learning. A few dictionary techniques have been developed for inverse acoustic or electromagnetic scattering problems; see [@AA1; @AA2; @AA3; @LWY] and the references therein. The critical ingredients of a dictionary method are the design of the appropriate dictionary class and the dictionary searching method. These are also the major technical contributions of the current article including the low frequency analysis for scattering by the nearly incompressible materials. One needs first to determine the location of the scatterer. After that, one can use the dictionary matching algorithm to determine the specific shape. However, in the dictionary class, the scattering information of the admissible shapes should be independent of any location requirement. This challenge can be overcome by using the so-called translation relation if incident plane waves are used; see [@AA1; @AA2; @AA3; @LWY; @LLS]. However in the current design, elastic point signals with two different wave numbers are used and the scattering data are collected in a special manner. This requires some special technical analysis and treatments to determine the dominating shear or pressure wave in our study. Moreover, for timely dictionary matching, we propose a fast and robust “direct" method with the dominating wave based on our detailed theoretical analysis.
The rest of the paper is organized as follows. In section \[sec:mathground\], we present the mathematical framework for our imaging scheme with elastic waves. In section \[sec:analysis:lowfre\], we give the necessary low frequency analysis for nearly incompressible materials with preparation for locating scatterers with a low frequency. In section \[sec:two-stage\], we present the two-stage recognition algorithm based on the theoretical analysis. In section \[sec:num\], extensive numerical tests are conducted to verify the effectiveness and efficiency of the proposed algorithm. We conclude our study in Section \[sect:conclusion\] with some relevant discussion.
Mathematical Framework {#sec:mathground}
======================
In this section, we present the general mathematical setting and fundamentals for the proposed reconstruction scheme. The shape of the unknown scatterer is supposed to be a $C^2$ domain $\Omega$, which is assumed to have a connected complement $\Omega^c := \mathbb{R}^{3} \backslash \bar \Omega$. It is assumed that there exists a [*dictionary*]{} of $C^2$ domains, which can be calibrated beforehand, i.e., $$\label{eq:dic:class}
\mathfrak{D} = \{ D_j\}_{j=1}^{N}, \quad N \in \mathbb{N}.$$ Here each $D_j$ is simply connected and contains the origin, such that there exists a translation operator $F: \mathbb{R}^3 \rightarrow \mathbb{R}^3$, $$\label{eq:trasi:domain}
\Omega = F(D) = D + z: = \{ x+z; \ x \in D \}, \quad D \in \mathfrak{D}, \quad z \in \mathbb{R}^3.$$ Our reconstruction strategy with elastic waves is a typical inverse elastic scattering problem. In the setup of the current study, the scatterer $\Omega$ or the dictionary domain $D_j$ shall be assumed to be a non-penetrable rigid obstacles or a penetrable medium scatterer, which covers many practical scenarios of important applications [@AR; @Kup]. The incident elastic point waves were located at a fixed position. With the incident waves, one then measures the scattered wave due to the unknown scatterer $\Omega$ on a measurement surface $\Gamma$ with multiple receivers. In our study, the measurement surface $\Gamma$ contains the location of the incident point waves.
Throughout the rest of the paper, we need the following two assumptions, $$\label{eq:assume:1}
\|D_j\|: = \max_{x \in D} |x| \simeq 1, \quad 1 \leq j \leq N,$$ and $$\label{eq:assume:2}
|z| \gg 1,$$ where $z$ is the location of $\Omega$ as in . Assumption means that the size of the scatterer $\Omega$ can be calibrated such that the regular frequency scale is characterized as $\frac{2 \pi}{k_s} \simeq \|\Omega\|$ and the low frequency of the elastic waves is characterized as $\frac{2 \pi}{k_s} \gg \|\Omega\|$, where $k_s\in\mathbb{R}_+$ signifies the wave number of the shear waves. This is practically feasible, since the frequency band of the elastic waves is of a wide range [@AR]. Throughout the rest of the paper, for exposition convenience, we always assume that $\rho\equiv 1$.
Actually, we would like to emphasize that we only need to collect the scattered field of $\Omega$ in a small aperture scattered by the elastic point source, which is convenient and quite practical. We also point out that this condition is mainly required for the theoretical justification of the proposed algorithm in what follows. Indeed, in our numerical tests, the proposed reconstruction algorithm works effectively and efficiently, as long as the scatterer $\Omega$ is located away from the point sources of a reasonable distance.
For the setup described above, we next introduce the direct elastic wave scattering. The displacement of a time-harmonic elastic wave is governed by the following Navier’s equations, $$\label{eq:elastic:wave}
\mu \Delta u(x) +(\lambda + \mu) \operatorname{grad}\operatorname{div}u(x) + \omega^2 \rho u(x) = 0,$$ where $\lambda$ and $\mu$ are Lamé coefficients, $\rho$ the density, and $\omega$ the frequency. In the sequel, we take one scatterer $\Omega$ for example in the analysis. We mainly consider two cases, i.e., $\Omega$ is a non-penetrable rigid obstacle or $\Omega$ is a penetrable medium.
For the rigid obstacle case, the elastic wave scattering is governed by the following homogeneous PDE system in $\mathbb{R}^3$, i.e., to find $u_{\omega,\Omega}^{s} \in H^{1}_{loc}(\Omega^c)$, such that $$\label{eq:scat:pec}
\begin{cases}
&\mu \Delta u_{\omega,\Omega}^s(x) +(\lambda + \mu) \operatorname{grad}\operatorname{div}u_{\omega,\Omega}^s(x) + \omega^2 u_{\omega,\Omega}^s(x) = 0,
\quad x \in \Omega^c, \\
& u_{\omega,\Omega}^s(x) + u_{\omega,p,ep}^i(x,y) = 0, \quad x \in \partial \Omega, \quad y \in \Omega^{c}, \\
&\displaystyle{\lim_{|x| \rightarrow \infty}|x|( \operatorname{curl}\operatorname{curl}u_{\omega,\Omega}^s \times \hat{x} -i k_{s}u_{\omega,\Omega}^s) = 0}, \\
& \displaystyle{\lim_{|x| \rightarrow \infty}|x|( \hat{x}\cdot \operatorname{grad}\operatorname{div}u_{\omega,\Omega}^s -ik_p \operatorname{div}u_{\omega,\Omega}^s) = 0},
\end{cases}$$ where the last two equations are known as the Kupradze radiation conditions [@Kup]. Moreover, $u_{\omega,p,ep}^i$ the elastic incident point source wave with a polarization $p\in\mathbb{R}^3$ and a source position $y\in\mathbb{R}^3$ [@PHhab], $$\label{eq:point:source}
u_{\omega,p,ep}^{i}(x,y)=\Gamma(x,y)p, \quad y \neq x,$$ where $\Gamma(x,y)$ is the fundamental solution of with $\rho \equiv 1$ (see [@PHhab Chap. 5], [@PH]), $$\begin{aligned}
\Gamma(x,y)p :&= \frac{e^{ik_{s}|x-y|}}{4 \pi \mu |x-y|}p + \frac{1}{\omega^2}\operatorname{grad}_{x} \operatorname{div}_{x} \left[\frac{e^{ik_s|x-y|} -e^{ik_p|x-y|}}{4 \pi |x-y|}p\right] \label{eq:pointsource:1}\\
&= \frac{1}{\omega^2}\operatorname{curl}_{x} \operatorname{curl}_{x} \left[\frac{e^{ik_{s}|x-y|}}{4 \pi |x-y|}p\right] - \frac{1}{\omega^2} \operatorname{grad}_{x} \operatorname{div}_{x} \left[\frac{e^{ik_{p}|x-y|}}{4 \pi |x-y|}p\right],\label{eq:point}\end{aligned}$$ with the wave numbers $k_s, k_p > 0 $ given by $$k_s = \frac{\omega}{\sqrt{\mu}},\ \quad k_p = \frac{\omega}{\sqrt{2\mu + \lambda}}.$$
For the medium case, denoting the mass density as $\rho_{\Omega}(x)$ and $n_{\Omega} =1 -\rho_{\Omega}$, we assume $n_{\Omega}$ has a compact support, and define $\Omega: = \{x \in \mathbb{R}^3: n_{\Omega}(x) \neq 0\}$ as the non-homogeneous medium. The elastic medium scattering problem of an inhomogeneous medium $\Omega$ with the incident elastic point source reads as follows: to find $u_{\omega,\Omega}^s \in H^{1}_{loc}(\mathbb{R}^3)$, such that $$\label{eq:scat:medium}
\begin{cases}
&\mu \Delta u_{\omega,\Omega}(x) +(\lambda + \mu) \operatorname{grad}\operatorname{div}u_{\omega,\Omega}(x) + \omega^2 (1-n_{\Omega}) u_{\omega,\Omega}(x) = 0, \quad x \in \mathbb{R}^3\backslash \{y\}, \\
& u_{\omega,\Omega}(x) = u_{\omega,\Omega}^s(x) + u_{\omega,p,ep}^i(x,y), \quad x \in \mathbb{R}^3\backslash \{y\}, \\
& \displaystyle{\lim_{|x| \rightarrow \infty}|x|( \operatorname{curl}\operatorname{curl}u_{\omega,\Omega}^s \times \hat{x} -i k_{s}u_{\omega,\Omega}^s) = 0}, \\
& \displaystyle{\lim_{|x| \rightarrow \infty}|x|( \hat{x}\cdot \operatorname{grad}\operatorname{div}u_{\omega,\Omega}^s -ik_p \operatorname{div}u_{\omega,\Omega}^s) = 0}.
\end{cases}$$ The well-posedness of the direct scattering problem and are known [@PH; @PH0; @PHhab]. The main focus of this paper is the following inverse problem:
*Given the measured scattering field $u_{\omega,\Omega}^s(x)$ on a bounded surface $\Lambda$, to find the location $z$ and the shape of the scatterer $\Omega$ for the nearly incompressible material.*
As introduced in Section \[sec:intro\], our reconstruction algorithm contains two stages. In the first stage, we locate the scatterer $\Omega$ by the measured scattering field $u_{\omega,\Omega}^s$ in or under low frequency on a bounded surface $\Lambda$. To this end, we develop specially designed indicator functionals, which are originated from the low frequency analysis of the nearly incompressible materials. Once the location is found, we then collect the measured scattered field $u_{\omega,\Omega}^s$ on $\Lambda$ as in or under regular frequency to reconstruct the shape of $\Omega$. In each process, only one incident point source wave is needed. Additionally, for the shape determination, we shall benefit from the scattering field of $D$ scattered by the incident plane wave stored in the precomputed dictionary. With the priori information in the dictionary and the dictionary matching process, we can reconstruct the shape efficiently. In light of these, we next introduce the elastic wave scattering of $D$ due to an incident plane wave, which also includes both the non-penetrable rigid obstacle case and the penetrable medium case.
We first introduce the elastic wave scattering of the rigid scatterer $D$, i.e., to find the radiating field $u^s(D,d,p;x) \in H^{1}_{loc}(D^c)$ satisfying the Kupradze radiation conditions, such that $$\begin{aligned}
&\mu \Delta u^s(D,d,p;x) +(\lambda + \mu) \operatorname{grad}\operatorname{div}u^s(D,d,p;x) + \omega^2 u^s(D,d,p;x) = 0,
\ \ x \in D^c = \mathbb{R}^3\backslash \bar D, \notag \\
& u^s(D,d,p;x) + u^i(x,d,p) = 0, \quad x \in \partial D. \label{eq:scat:pec:plane}
$$
For the elastic medium scattering of $D$, similar to the case of $\Omega$, we need to introduce $\rho_{D}(x)$ and $n_{D} =1 -\rho_{D}$. By the translation relation , we have $$\label{eq:n:medium}
n_{\Omega}(y) = n_{D}(x), \quad y = x+z, \quad x \in \mathbb{R}^3.$$ The scattering of the inhomogeneous elastic medium $D$ by a plane incident wave is to find $u^s(D,d,p;x) \in H^{1}_{loc}(\mathbb{R}^3)$ with the Kupradze radiation conditions, such that $$\begin{aligned}
&\mu \Delta u(D,d,p;x) +(\lambda + \mu) \operatorname{grad}\operatorname{div}u(D,d,p;x) + \omega^2 (1-n_{D}) u(D,d,p;x) = 0, \quad x \in \mathbb{R}^3, \notag \\
& u(D,d,p;x) = u^s(D,d,p;x) + u^i(x,d,p) = 0, \quad x \in \mathbb{R}^3. \label{eq:scat:medium:plane}\end{aligned}$$ $u^i(x,d,p)$ as in and is the elastic plane wave with the polarization $p \in \mathbb{R}^3$ and the travelling direction $d \in \mathbb{R}^3$, i.e., $u^{i}(x,d,p) = u_{p}^{i}(x,d,p) + u_{s}^{i}(x,d,p)$,
\[eq:plane\] $$\begin{aligned}
\quad u_{p}^{i}(x,d,p):&= -\frac{1}{\omega^2}\operatorname{grad}_{x} (\operatorname{div}_{x}[ p e^{ik_p d \cdot x}]) =
\frac{k_{p}^2}{\omega^2} d\cdot p d e^{ik_px\cdot d}, \\
u_{s}^i(x,d,p):&= \frac{1}{\omega^2} \operatorname{curl}_x \operatorname{curl}_x[pe^{ik_sd\cdot x}]= \frac{k_{s}^2}{\omega^2} (d\times p) \times d e^{ik_sx\cdot d}.\end{aligned}$$
Moreover, the following asymptotic relations between the elastic point incident wave and elastic plane incident waves are useful for our subsequent discussion. Denote $\hat{z} = z/|z|$ for $z\neq 0$. Here and also in what follows, we use ${\mathcal{O}}(\cdot)$ or ${
\mathchoice
{{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(\cdot)$ to represent the usual asymptotic behavior of real scalar variables.
\[lem:incident\] As $|z| \rightarrow \infty$, for large $\lambda \gg \mu$, we have $$\begin{aligned}
u_{\omega,p,ep}^{i}(x+z,y) &= \frac{e^{ik_p|z|-ik_p\hat{z} \cdot y}}{4 \pi |z|}u_{p}^{i}(x,\hat{z},p)+
\frac{e^{ik_s|z|-ik_s\hat{z} \cdot y}}{4 \pi |z|}u_{s}^{i}(x,\hat{z},p) + {\mathcal{O}}(\frac{1}{|z|^2})\frac{k_p^2 + k_s^2}{\omega^2} \label{eq:asym:point1}\\
|u_{p}^{i}(x,\hat{z},p)|& = {\mathcal{O}}(\frac{1}{\lambda}), \quad |u_{s}^{i}(x,\hat{z},p)| = {\mathcal{O}}(\frac{1}{\mu}). \label{eq:asym:plane:incident}
$$
First, we have $$\begin{aligned}
\Gamma(x+z,y)p &= \frac{1}{\omega^2}\operatorname{curl}_{x} \operatorname{curl}_{x} \left[\frac{e^{ik_{s}|x+z-y|}}{4 \pi |x+z-y|}p\right] - \frac{1}{\omega^2} \operatorname{grad}_{x} \operatorname{div}_{x} \left[\frac{e^{ik_{p}|x+z-y|}}{4 \pi |x+z-y|}p\right] \notag \\
&=\frac{1}{\omega^2}\operatorname{curl}_{x} \operatorname{curl}_{x} \left[\frac{e^{ik_{s}|x-(y-z)|}}{4 \pi |x-(y-z)|}p\right] - \frac{1}{\omega^2} \operatorname{grad}_{x} \operatorname{div}_{x} \left[\frac{e^{ik_{p}|x-(y-z)|}}{4 \pi |x-(y-z)|}p\right]. \label{point:green:expan}\end{aligned}$$ For the first term of the RHS of , by direct calculation, we have $$\label{up:first:expan}
\begin{aligned}
&\frac{1}{\omega^2}\operatorname{curl}_{x} \operatorname{curl}_{x} \left[\frac{e^{ik_{s}|x-(y-z)|}}{4 \pi |x-(y-z)|}p\right] =
\frac{1}{\omega^2}(-\Delta + \nabla \operatorname{div})\left[\frac{e^{ik_{s}|x-(y-z)|}}{4 \pi |x-(y-z)|}p\right] \\
&=\frac{k_s^2}{\omega^2} \left[\Phi_{k_s}(x,y-z) \left(p-\frac{z-(y-x)}{|z-(y-x)|}\frac{[z-(y-x)]\cdot p}{|z-(y-x)|}\right) + \mathcal{O}(\frac{1}{|z|^2})\right] \\
& = \frac{k_s^2}{\omega^2} \left[ \frac{e^{ik_s|z|}}{4 \pi |z|} e^{ik_s \hat{z} \cdot (x-y)}\{p - \hat{z} \hat{z} \cdot p\} + \mathcal{O}(\frac{1}{|z|^2})\right] \\
& = \frac{k_s^2}{\omega^2} \left[\frac{e^{ik_s|z|-ik_s\hat{z} \cdot y}}{4 \pi |z|} (\hat{z}\times p) \times \hat{z} e^{ik_sx\cdot \hat{z}} + \mathcal{O}(\frac{1}{|z|^2})\right].
\end{aligned}$$ For the second term of the RHS of , by direct calculation, we can obtain $$\label{up:second:expan}
\begin{aligned}
- \frac{1}{\omega^2} \operatorname{grad}_{x} \operatorname{div}_{x} \left[\frac{e^{ik_{p}|x-(y-z)|}}{4 \pi |x-(y-z)|}p\right]
& = \frac{1}{\omega^2}(-\Delta_x - \operatorname{curl}_{x} \operatorname{curl}_{x})\left[\frac{e^{ik_{p}|x-(y-z)|}}{4 \pi |x-(y-z)|}p\right]\\
&=\frac{k_{p}^2}{\omega^2} \left[\frac{e^{ik_p|z|-ik_p\hat{z} \cdot y}}{4 \pi |z|} \hat{z} \hat{z}\cdot p e^{ik_px\cdot \hat{z}}+\mathcal{O}(\frac{1}{|z|^2})\right].
\end{aligned}$$ By and , compared with , we have . Combining , and the assumption that $\lambda \gg \mu$, we finally arrive at .
Henceforth, we use $\nu$ to signify the exterior unit normal of the domain concerned. For a vector $u \in C^{1}(D^c)^3$ (or $u \in C^{1}(\Omega^c)^3$) where $D^c = \mathbb{R}^3 \backslash \bar D$ (or $\Omega^c = \mathbb{R}^3 \backslash \bar \Omega$), we define for $x \in \partial D$ (or $x \in \partial \Omega$ ) $$\label{eq:traction:elastic}
[\mathcal{P}u](x): = (\alpha + \mu)(\nu\cdot \operatorname{grad})u + \beta \nu \operatorname{div}u + \alpha [\nu \times \operatorname{curl}u(x)],$$ which is the traction vector at $x$, with $\alpha$ and $\beta$ defined as follows $$\label{eq:alpha:beta}
\alpha :=\frac{\mu(\lambda + \mu)}{\lambda+3\mu}, \quad \beta: = \frac{(\lambda+\mu)(\lambda+2\mu)}{\lambda+3 \mu}.$$ For $y \in \partial D$, $x\in \mathbb{R}^3$, $x \neq y$, we define $\Pi(x,y) \in \mathbb{C}^{3\times3}$ by $$\Pi(x,y)^{T}P := \mathcal{P}_{y}(\Gamma(x,y)P),$$ where the superscript $T$ denotes the transpose. For $\phi(x) \in C(\partial D)^3$, we introduce the following single and double layers, $$(S_{\omega, D} \phi)(x) : = 2 \int_{\partial D} \Gamma(x,y) \phi(y)ds(y), \quad (K_{\omega, D}\phi)(x): = 2 \int_{\partial D} \Pi(x,y) \phi(y)ds(y).$$ It is well-known [@PH] that $\Pi(x,y)$ is weakly singular with specifically chosen $\alpha$, $\beta$ in , and $K_{\omega, D}: C(\partial D)^3 \rightarrow C^{0, \alpha}(\partial D)^3$ is bounded. For $\varphi(x) \in C(D)^3$, we introduce the volume potential $$(V_{\omega, D}\varphi)(x): = \int_{D}\Gamma(x,y)\varphi(y)dy.$$ In the sequel, we study the translation relation of the scattered field of $\Omega$ due to an incident point source $u_{\omega,p,ep}^i$ and the scattered field of the translated $D$ due to an incident plane wave $u^i(x,d,p)$. For expositional simplification and by normalization, we assume in the following that $\rho\equiv 1$ for the background space $\mathbb{R}^3\backslash \overline{\Omega}$.
Elastic wave scattering of rigid obstacle
-----------------------------------------
In fact, for the scattered elastic waves of system and , we have the following asymptotic relation under the translation condition . For clarity, we assume that the point source in is located in $y_0$ instead of $y$ hereafter.
\[thm:pec\] For a fixed $\omega \in \mathbb{R}_{+}$, we have the following asymptotic expansions for the rigid obstacle scattering problem , $$\begin{aligned}
\label{eq:rr}
u_{\omega,\Omega}^s(x)
& =\frac{e^{ik_p|z|-ik_p\hat{z} \cdot y_0}}{4 \pi |z|}u_{p}^s(D,\hat{z},p;x-z)+ \frac{e^{ik_s|z|-ik_s\hat{z} \cdot y_0}}{4 \pi |z|}u_{s}^s(D,\hat{z},p;x-z)+ \mathcal{O}(|z|^{-2}),
\end{aligned}$$ for any fixed $x \in \Omega^c$ as $|z| \rightarrow \infty$ uniformly for all $\hat{z} \in \mathbb{S}^2$. In , $u_{p}^s(D,\hat{z},p;x)$ and $u_{s}^s(D,\hat{z},p;x)$ are the scattered fields of the rigid obstacle $D$ as in corresponding to the incident plane waves $u_{p}^i(x,\hat{z},p)$ and $u_{s}^i(x,\hat{z},p)$ respectively.
By [@PH], $ u_{\omega,\Omega}^s(x) $ can be represented as the following combined potentials, $$u_{\omega,\Omega}^s(x) = 2\int_{\partial \Omega}\Pi(x,y) \phi(y)ds(y) + 2i \int_{\partial \Omega} \Gamma(x,y) \phi(y)ds(y),$$ where $\phi(y)$ is a vectorial density on $\partial \Omega$. Here and also in what follows, we denote $$\Phi_{\kappa}(x,y) = e^{i \kappa |x-y|}/({4 \pi|x-y|}), \quad \text{with} \ \kappa = k_s \ \text{or} \ \kappa = k_p.$$ By direct calculations, one can verify that for any $f(y) \in \mathbb{C}^3$, $$\begin{aligned}
\Delta_{x}[f(y)\Phi_{\kappa}(x,y)] &= \Delta_{x}\Phi_{\kappa}(x,y) f(y), \\
\operatorname{curl}_{x} \operatorname{curl}_{x} [f(y)\Phi_{\kappa}(x,y)] & = (-\Delta_{x} + \nabla_{x} \operatorname{div}_{x})[f(y)\Phi_{\kappa}(x,y)].
\end{aligned}$$ It can be shown that $ \Phi_{\kappa}(x+z,y) = \Phi_{\kappa}(x,y-z)$ and $$\quad \frac{\partial \Phi_{\kappa}(\tilde x,y)}{\partial \tilde x_{j}}|_{\tilde{x} = x+z} = \frac{\partial \Phi_{\kappa}( x,y-z)}{\partial x_{j}},\quad \frac{\partial^2 \Phi_{\kappa}(\tilde x,y)}{\partial \tilde x_{j}\partial \tilde x_{i}}|_{\tilde{x} = x+z} = \frac{\partial^2 \Phi_{\kappa}( x,y-z)}{\partial x_{j}\partial x_{j}}.$$ Thus we have $$\label{eq:curl:trans}
\begin{aligned}
\operatorname{grad}_{x}\operatorname{div}_{x}[f(y)\Phi_{\kappa}(x,y)](x+z) &= \operatorname{grad}_{x}\operatorname{div}_{x}[f(y)\Phi_{\kappa}(x,y-z)], \\
\operatorname{curl}_{x} \operatorname{curl}_{x} [f(y)\Phi_{\kappa}(x,y)](x+z) & = \operatorname{curl}_{x} \operatorname{curl}_{x} [f(y)\Phi_{\kappa}(x,y-z)] .
\end{aligned}$$
We are in a position to prove the theorem. First, considering $ u_{\omega,\Omega}^s(x+z)$ with $x\in \Omega^c$, we have $$\begin{aligned}
\operatorname{curl}_{x}[f(y)\Phi_{\kappa}(x,y)] &= \nabla_{x}\Phi_{\kappa}(x,y)\times f(y), \\
\operatorname{curl}_{x} \operatorname{curl}_{x} [f(y)\Phi_{\kappa}(x,y)] & = (-\Delta_{x} + \nabla_{x} \operatorname{div}_{x})[f(y)\Phi_{\kappa}(x,y)].
\end{aligned}$$ It can be shown that $ \Phi_{\kappa}(x+z,y) = \Phi_{\kappa}(x,y-z)$ and $$\quad \frac{\partial \Phi_{\kappa}(\tilde x,y)}{\partial \tilde x_{j}}|_{\tilde{x} = x+z} = \frac{\partial \Phi_{\kappa}( x,y-z)}{\partial x_{j}},\quad \frac{\partial^2 \Phi_{\kappa}(\tilde x,y)}{\partial \tilde x_{j}\partial \tilde x_{i}}|_{\tilde{x} = x+z} = \frac{\partial^2 \Phi_{\kappa}( x,y-z)}{\partial x_{j}\partial x_{j}}.$$ Hence, we have $$\label{eq:curlcurl:trans}
\begin{aligned}
\operatorname{curl}_{x}[f(y)\Phi_{\kappa}(x,y)](x+z) &= \operatorname{curl}_{x}[f(y)\Phi_{\kappa}(x,y-z)], \\
\operatorname{curl}_{x} \operatorname{curl}_{x} [f(y)\Phi_{\kappa}(x,y)](x+z) & = \operatorname{curl}_{x} \operatorname{curl}_{x} [f(y)\Phi_{\kappa}(x,y-z)] .
\end{aligned}$$ By the jump relation of the vector potentials of $K_{\omega, \Omega}$ and $S_{\omega, \Omega}$ [@PH; @PHhab], we have $$\label{eq:density:eq:Omega}
\phi(x) + (K_{\omega,\Omega}\phi)(x) + i (S_{\omega,\Omega} \phi)(x) = - u_{\omega, p,ep}^i(x,y_0), \quad x \in \partial \Omega, \quad y_0 \notin \partial \bar \Omega$$ Then we shall write $ u_{\omega,\Omega}^s(x+z)$ by and as follows, $$\begin{aligned}
u_{\omega,\Omega}^s(x+z)
& = 2\int_{\partial \Omega}\Pi(x+z,y) \phi(y)ds(y) + 2i \int_{\partial \Omega} \Gamma(x+z,y) \phi(y)ds(y),
\label{eq:E:repre:Omega}
\end{aligned}$$ By changing variable with $y = z+t$ in and the assumption , we have $$\begin{aligned}
u_{\omega,\Omega}^s(x+z)
&= 2\int_{\partial D}\Pi(x,t) \phi(z+t)ds(t) + 2i \int_{\partial D} \Gamma(x,t) \phi(z+t)ds(t), \notag \\
& = [(K_{\omega,D}\phi(t+z))(x)+ i(S_{\omega,D}\phi(t+z))(x)].\label{eq:E:representaion:D}
\end{aligned}$$ In the sequel, we turn to solving the density $\phi(z+t)$ with $t \in \partial D$. Letting $x = t+z$ in and denoting $\tilde \phi(y)|_{y\in\partial D} = \phi(t+z)|_{t \in \partial D}$, again by the change of variables and similar arguments as before, we have $$\label{eq:density:sa}
[(I + K_{\omega,D} + iS_{\omega,D})\tilde \phi](t+z)|_{\partial D} = - u_{\omega,p,ep}^i(t+z, y_0), \quad t \in \partial D.$$ By , we have $$\label{eq:a:trans:D:a}
\phi(t+z) = \tilde \phi(y) = (I + K_{\omega,D} + i S_{\omega,D})^{-1}(- u_{\omega,p, ep}^i(\cdot+z, y_0)), \quad t \in \partial D.$$ Substituting $\phi(t+z)$ in into , together with Lemma \[lem:incident\], we have for any $x \in \Omega^c$ $$\begin{aligned}
u_{\omega,\Omega}^s(x+z) &= [K_{\omega,D} + iS_{\omega,D}] (I + K_{\omega,D} + i S_{\omega,D})^{-1}[- u_{\omega,p,ep}^i(\cdot+z,y_0)], \notag \\
&= \frac{e^{ik_p|z|-ik_p\hat{z} \cdot y_0}}{4 \pi |z|} [K_{\omega,D} + iS_{\omega,D}] (I + K_{\omega,D} + i S_{\omega,D})^{-1}[- u_{p}^{i}(t,\hat{z},p)+ \mathcal{O}(|z|^{-1})], \notag \\
& \quad +\frac{e^{ik_s|z|-ik_s\hat{z} \cdot y_0}}{4 \pi |z|} [K_{\omega,D} + iS_{\omega,D}] (I + K_{\omega,D} + i S_{\omega,D})^{-1}[- u_{s}^{i}(t,\hat{z},p)+ \mathcal{O}(|z|^{-1})], \notag \\
&= \frac{e^{ik_p|z|-ik_p\hat{z} \cdot y_0}}{4 \pi |z|}u_{p}^s(D,\hat{z},p;x) + \frac{e^{ik_s|z|-ik_s\hat{z} \cdot y_0}}{4 \pi |z|}u_{s}^s(D,\hat{z},p;x)+ \mathcal{O}(|z|^{-2}). \label{eq:asym:rigid:twp}
\end{aligned}$$ In , $u_{p}^s(D,\hat{z},p;x)$ and $u_{s}^s(D,\hat{z},p;x)$ are the scattering fields corresponding to the incident plane elastic wave $u_{p}^i(x, \hat{z}, p)$ and $u_{s}^i(x,\hat{z},p)$ respectively, i.e., $u_{p}^s(D,\hat{z},p;x)$ $= T_{\omega,D}(- u_{p}^{i}(t,\hat{z},p))(x)$ and $u_{s}^s(D,\hat{z},p;x)$ $= T_{\omega,D}(-u_{s}^{i}(t,\hat{z},p))(x)$ with $T_{\omega,D}$ defined as follows, $$T_{\omega,D} : C(\partial D)^3 \rightarrow C(\partial D)^3, \quad T_{\omega,D} = [K_{\omega,D} + iS_{\omega,D}] (I + K_{\omega,D} + i S_{\omega,D})^{-1}.$$ Therefore, by change of variables, we have $$u_{\omega,\Omega}^s(x) = \frac{e^{ik_p|z|-ik_p\hat{z} \cdot y_0}}{4 \pi |z|}u_{p}^s(D,\hat{z},p;x-z) + \frac{e^{ik_s|z|-ik_s\hat{z} \cdot y_0}}{4 \pi |z|}u_{s}^s(D,\hat{z},p;x-z)+ \mathcal{O}(|z|^{-2}). \label{eq:asym:rigid:twoo}$$ which completes the proof.
Elastic scattering of inhomogeneous medium
------------------------------------------
For the scattered elastic waves of system and of the penetrable medium scattering, we have the following asymptotic relation under the translation condition .
\[thm:medium\] For fixed $k_s, k_p \in \mathbb{R}_{+}$, we have the following asymptotic expansion for the elastic medium scattering problem , $$\begin{aligned}
\label{eq:rrr}
u_{\omega,\Omega}^s(x) & = \frac{e^{ik_p|z|-ik_p\hat{z} \cdot y_0}}{4 \pi |z|}u_{p}^s(D,\hat{z},p;x-z) + \frac{e^{ik_s|z|-ik_s\hat{z} \cdot y_0}}{4 \pi |z|}u_{s}^s(D,\hat{z},p;x-z)+ \mathcal{O}(|z|^{-2}),
\end{aligned}$$ for any fixed $x \in \mathbb{R}^3$ as $|z| \rightarrow \infty$ uniformly for all $\hat{z} \in \mathbb{S}^2$. In , $u_{p}^s(D,\hat{z},p;x)$ and $u_{s}^s(D,\hat{z},p;x)$ are the scattered fields of the penetrable medium $D$ as in corresponding to the incident plane elastic waves $u_{p}^i(x, \hat{z}, p)$ and $u_{s}^i(x, \hat{z},p)$ respectively.
By [@PH0 Lemma 2] and [@PHhab Lemma 5.7 ], the radiating scattering elastic wave field has the following integral representation, $$\label{eq:repre:medium}
u_{\omega,\Omega}^{s}= -\omega^2 \int_{\Omega}\Gamma(x,y)n_{\Omega}(y)u_{\omega,\Omega}(y)dy.$$ In the following, we set $-\omega^2 \int_{\Omega}\Gamma(x,y)n_{\Omega}(y)u_{\omega,\Omega}(y)dy:= \tilde{V}_{\omega,\Omega}u_{\omega,\Omega}$. It is known that the operator $(I -\tilde{V}_{\omega,\Omega} )$ is continuously invertible in $C(\Omega)^3$ [@PHhab]. Thus we have $$\label{eq:eq:elec}
u_{\omega,\Omega} = (I - \tilde{V}_{\omega,\Omega})^{-1}u_{\omega,p,ep}^{i}(\cdot, y_0),\quad x \in \Omega, \quad y_0 \notin \bar \Omega.$$ Considering $u_{\omega,\Omega}^s(x+z)$, by , it can be written as $$u_{\omega,\Omega}^s(x+z)= -\omega^2 \int_{\Omega}\Gamma(x+z,y)n_{\Omega}(y)u_{\omega,\Omega}(y)dy.$$ Setting $t=y-z$ and denoting $u_{\omega,D}(t+z)|_{t \in D}= u_{\omega,\Omega}(y)|_{y \in \Omega}$, by change of variables and noting that the Jacobian matrix of the change of variables is the identity matrix in $\mathbb{R}^3$, together with and , we have $$u_{\omega,\Omega}^s(x+z) = -\omega^2 \int_{D}\Gamma(x,t)n_{D}(t)u_{\omega,D}(t+z)dy.$$ It can be readily verified that $$\label{eq:e:omega:z}
u_{\omega,\Omega}^s(x+z) = (\tilde{V}_{\omega,D} u_{\omega,D}(t+z))(x).$$ Next, we solve for $u_{\omega,D}(t+z)$. Rewriting as $(I - \tilde{V}_{\omega,\Omega})u_{\omega,\Omega}(x) = u_{\omega,p,ep}^{i}(x,y_0)$, we have $$\label{eq:e:omega:medium}
u_{\omega,p,ep}^{i}(x,y_0)=u_{\omega,\Omega}(x) + \omega^2 \int_{\Omega}\Gamma(x,y)n_{\Omega}(y)u_{\omega, \Omega}(y)dy.$$ By change of variables with $x = t+z$, becomes $$\label{eq:mega:2}
u_{\omega,p,ep}^{i}(t+z,y_0)-u_{\omega,D}(t+z)=\omega^2 \int_{\Omega}\Gamma(t+z,y)n_{\Omega}(y)u_{\omega, \Omega}(y)dy.$$ By using change of variables with $\tilde y = y-z$, we see can be written as $$\label{eq:volume:reso:add}
u_{\omega,p,ep}^{i}(t+z,y_0)-u_{\omega,D}(t+z)=\omega^2 \int_{D}\Gamma(t,\tilde y)n_{D}(\tilde y)u_{\omega,D}(\tilde y +z)d\tilde y.$$ With , it can be shown $$u_{\omega,D}(t+z) = (I - \tilde{V}_{\omega,D})^{-1}u_{\omega,p,ep}^{i}(\cdot+z,y_0), \quad t \in D,$$ where $\tilde{V}_{\omega,D}$ is defined as follows, $$\label{eq:volume:p:D}
(\tilde{V}_{\omega,D} u)(x): = -\omega^2\int_{D} \Gamma(x,y) n_{D}(y) u(y)dy, \quad u \in C(D)^3.$$ Substituting it into , we have $$u_{\omega,\Omega}^s(x+z) = [\tilde{V}_{\omega,D} (I - \tilde{V}_{\omega,D})^{-1}u_{\omega,p,ep}^{i}(\cdot+z,y_0)](x).$$ Again by Lemma \[lem:incident\], we have $$\begin{aligned}
u_{\omega,\Omega}^s(x+z) =& \frac{e^{ik_p|z|-ik_p\hat{z} \cdot y_0}}{4 \pi |z|}[\tilde{V}_{\omega,D} (I - \tilde{V}_{\omega,D})^{-1}u_{p}^{i}(x,\hat{z},p)(x) + \mathcal{O}(|z|^{-1})], \label{eq:aym:me:1} \\
& +\frac{e^{ik_s|z|-ik_s\hat{z} \cdot y_0}}{4 \pi |z|}[\tilde{V}_{\omega,D} (I - \tilde{V}_{\omega,D})^{-1}u_{s}^{i}(x,\hat{z},p)(x) + \mathcal{O}(|z|^{-1})]. \label{eq:aym:me:2}\end{aligned}$$ Now we introduce $u_{p}^{s}(D, \hat{z},p;x) := V_{\omega,D}(- u_{p}^{i}(t,\hat{z},p))(x)$ and $u_{s}^{s}(D, \hat{z},p;x) := V_{\omega,D}(-u_{s}^{i}(t,\hat{z},p))(x)$ with $V_{\omega,D}$ defined as follows, $$V_{\omega,D} : C(D)^3 \rightarrow C(D)^3, \quad V_{\omega,D} := \tilde{V}_{\omega,D} (I - \tilde{V}_{\omega,D})^{-1}.$$ In fact, $u_{p}^{s}(D, \hat{z},p;x)$ and $u_{s}^{s}(D, \hat{z},p;x)$ are the scattering fields associated with the incident plane elastic wave $u_{p}^i(x, \hat{z},p)$ and $u_{s}^i(x, \hat{z},p)$, respectively. Again by change of variables, we have $$\label{eq:medium:syste}
u_{\omega,\Omega}^s(x) = \frac{e^{ik_p|z|-ik_p\hat{z} \cdot y_0}}{4 \pi |z|}u_{p}^s(D,\hat{z},p;{x-z})+\frac{e^{ik_s|z|-ik_s\hat{z} \cdot y_0}}{4 \pi |z|}u_{s}^s(D,\hat{z},p;{x-z})+ \mathcal{O}(|z|^{-2}).$$
Low Frequency Asymptotic Analysis {#sec:analysis:lowfre}
=================================
In this section, we discuss the low frequency asymptotic approximations of the nearly incompressible material for our algorithmic development. There are some existing results on low frequency asymptotic analysis for elastic wave scattering [@DR]. However, there are no direct and clear results that could cover the case of the nearly incompressible materials for our purpose, and thus we provide a complete study in what follows. In the sequel, for notational convenience and without loss of generality, we assume that the elastic point source is always located at the origin, i.e., $u_{\omega,p,ep}^{i}(x,0)=\Gamma(x,0)p$.
For the low frequency analysis, we first consider the limiting case with $\omega=0$. We introduce $\Gamma^0(x)$ as in [@PH], which is the fundamental solution of with $\omega=0$, $\rho \equiv 1$ $$\label{eq:gamma0}
\Gamma_{ij}^0(x): = \frac{\delta_{jk}}{4 \pi \mu |x|} - \frac{\lambda + \mu}{8\pi \mu(2\mu + \lambda)} \frac{\partial |x|}{\partial x_j \partial x_k},$$ and define $\Pi^0(x,y)$ as follows, $$\Pi^0(x,y)^{T}d: = \mathcal{P}_y(\Gamma^0(x-y)d).$$ Then the single and double layers associated with $\Gamma_{i,j}^0(x,y)$ can be defined similarly, $$\label{eq:gamma0:layer}
(S^{0}_{D}\phi)(x): = 2\int_{\partial D} \Gamma^{0}(x,y)\phi(y)ds(y), \quad (K^{0}_{D}\phi)(x): = 2\int_{\partial D} \Pi^{0}(x,y)\phi(y)ds(y).$$ In fact, as $\omega \rightarrow 0$, the single layer $S^{0}_{D}$ and the double layer $K^{0}_{D}$ approximate $S_{\omega,D}$ and $K_{\omega,D}$, respectively, as shown in the lemma below.
\[lem:layer:diff\] As the frequency $\omega \rightarrow 0$, we have the following estimates, $$\label{eq:estimates:layers}
||S_{\omega, D} - S_{D}^{0}||_{\mathcal{L}(C(\partial D)^3, C^{0, \alpha}(\partial D)^3)} \sim \mathcal{O}(\omega), \quad ||K_{\omega, D} - K_{D}^{0}||_{\mathcal{L}(C(\partial D), C^{0, \alpha}(\partial D)^3)} \sim \mathcal{O}(\omega).$$
By [@PHhab Lemma 5.1], as $\kappa\rightarrow 0$ with $\kappa = k_{p}$ or $\kappa = k_s$, we have $$\begin{aligned}
\frac{e^{i \kappa x}}{4\pi |x|} &= \frac{cos(\kappa |x|)}{4 \pi |x|} + \frac{sin(\kappa |x|)}{4 \pi |x|} \\
& = \frac{1}{4\pi |x|} - \frac{\kappa^2}{8\pi}|x| + \kappa^4|x|^3f_1(\kappa^2|x|^2) + i\kappa f_2(\kappa^2|x|^2),\end{aligned}$$ where $f_1$ and $f_2$ are entire functions. By direct calculation, as $\omega \rightarrow 0$, we see $$\label{gamma:gamma0:first}
\frac{e^{ik_sx}}{4\pi \mu |x|} - \frac{1}{4 \pi \mu |x|} = \frac{1}{\mu}(- \frac{k_s^2}{8\pi}|x| + k_s^4|x|^3f_1(k_s^2|x|^2) + ik_s f_2(k_s^2|x|^2),$$ and by $(k_p^2-k_s^2)/\omega^2 = (\lambda+ \mu)/[\mu(2\mu + \lambda)]$, we also have $$\begin{aligned}
&\frac{\partial^2}{\partial x_j \partial x_k} \left[ \frac{1}{\omega^2}\frac{e^{ik_sx} -e^{ik_px}}{4 \pi |x|} + \frac{(\lambda+\mu)|x|}{8\pi \mu(2\mu + \lambda)}\right]\\
= & \omega \frac{\partial^2}{\partial x_j \partial x_k}\left[ |x|^3g_{1}(|x|^2)+ig_{2}(|x|^2) +|x|^3h_{1}(|x|^2)+ih_{2}(|x|^2) \right], \label{gamma:gamma0:second}\end{aligned}$$ where $g_1$, $g_2$, $h_1$ and $h_2$ are entire functions whose coefficients only depend on $\mu$, $\lambda$, $\omega^k$ with $k \in \mathbb{N}$.
Hence, for $\omega \ll 1$, and any constant $c_1$, there exist a constant $c_2$, such that for all $|x| \leq c_1$, by and the expressions of $\Gamma$ and $\Gamma^0$, we have $$\label{eq:weak:singular:S}
|\Gamma_{jk}(x)-\Gamma_{jk}^0(x)| \leq c_2 \omega, \quad j,k=1,2,3.$$ By differentiating , and noting for odd integer $l$ the relation $\partial |x|^{l}/\partial x_m=lx_m|x|^{l-2}$, we have $$\label{eq:weak:singular:K}
|\frac{\partial}{\partial x_{l_1}}(\Gamma_{jk}(x)-\Gamma_{jk}^0(x))| \leq \frac{c_2\omega}{|x|} \Rightarrow |(\Pi(x)^{t}-{\Pi^0}^t(x))e_j| \leq \frac{c_2\omega}{|x|}, \quad \quad l_1, l_2, l_3=1,2,3,$$ where $e_j$, $j=1,2,3$, denote the usual Cartesian orthonormal unit basis vector.
By and and Chapter 2.3 of [@CK0], since the integral kernel of $S_{\omega, D} - S_{D}^{0}$ and $K_{\omega, D} - K_{D}^{0}$ are weakly singular, we see the mapping properties of them are at least as good as $S_{\omega, D}$ or $K_{\omega, D}$, which are bounded and linear mappings from $C(\partial D)^3$ to $C^{0,\alpha}(\partial D)^3$ . This leads to the desired estimates and completes the proof.
It is necessary to introduce the far field patterns of any radiating elastic waves $u^{s}(x)$ for convenience [@AK; @PH], $$u^{s}(x) = \frac{e^{ik_{s}|x|}}{|x|}u_{s,\infty}(\hat{x}) + \frac{e^{ik_{p}|x|}}{|x|}u_{p,\infty}(\hat{x})\hat{x} + \mathcal{O}(|x|^{-2}),$$ for $|x| \rightarrow \infty$ where $\hat{x}\cdot u_{s,\infty}(\hat{x})=0$ for all $\hat{x} \in \mathbb{S}^2$ and $u_{p,\infty}(\hat{x})$ is complex valued scalar function.
With Lemma \[lem:layer:diff\] and Theorem \[thm:pec\], we are now in the position to give the theorem on the low frequency asymptotic approximations of the nearly incompressible materials.
\[thm:obstacle:asym\] Assuming $\Omega$ and $D$ be rigid obstacles with the translation and $\lambda \gg \mu$, $ \mu = {\mathcal{O}}(1)$, $|z| \gg 1$, we have the following asymptotic estimate of the scattered fields of as $|x| \rightarrow \infty$, $\omega \rightarrow 0$, $$\begin{aligned}
u_{\omega,\Omega}^s(x) =& \frac{e^{ik_p|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} ({\mathcal{O}}(\frac{1}{\lambda})(I-\hat{x}\hat{x}^T)a_0 + \frac{e^{ik_p|x-z|}}{|x-z|}{\mathcal{O}}(\frac{1}{\lambda^2}) \hat{x}\hat{x}^{T}b_0 \right] + {\mathcal{O}}(\frac{1}{|z|^2}) \label{eq:asym:obs:lambda}\\
&+\frac{e^{ik_s|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} ({\mathcal{O}}(1)(I-\hat{x}\hat{x}^T)c_0 + \frac{e^{ik_p|x-z|}}{|x-z|}
{\mathcal{O}}(\frac{1}{\lambda}) \hat{x}\hat{x}^{T}d_0\right] + \mathcal{O}(\frac{1}{|x|^2}) + {\mathcal{O}}(\omega). \notag\end{aligned}$$ where $a_0$, $b_0$, $c_0$ and $d_0$ are all constant complex vectors that do not depend on $\omega$ and $\lambda$.
Here we assume that $\partial D$ is $C^2$-continuous such that we use the mapping properties of the layer potentials $K_{\omega, D}$ and $S_{\omega, D}$ in [@PH]. We assume $u^s(D,d, p;x)$ has the following combined layer potential representation [@PH] with incident plane wave $u^i(x,d,p)$, $$u^s(D,d, p;x) = [(K_{\omega, D} + i S_{\omega, D})\phi](x).$$ It is well known that $u^s(D,d, p;x)$ has the following asymptotic behavior as $|x|\rightarrow \infty$ [@AK], $$\label{eq:scatter:repre:rigi}
u^s(D,d, p;x)= \frac{e^{ik_s|x|}}{|x|}(\mathcal{F}_{s,\infty}\phi)(\hat{x})+ \frac{e^{ik_p|x|}}{|x|} (\mathcal{F}_{p,\infty}\phi)(\hat{x}) + \mathcal{O}(\frac{1}{|x|^2}),$$ where $(\mathcal{F}_{s,\infty}\phi)(\hat{x})$ is the shear far field pattern that is tangential to $\mathbb{S}^2$, and $(\mathcal{F}_{p,\infty}\phi)(\hat{x})$ is the pressure far field pattern that is normal to $\mathbb{S}^2$. Beside, they have the the following integral representations [@AK], $$\begin{aligned}
(\mathcal{F}_{s,\infty}\phi)(\hat{x})&= \frac{1}{2\pi \mu}\int_{\partial D}\left\{ [I -\hat{x}\hat{x}^{T}]e^{-ik_s\hat{x}\cdot y} + i [\mathcal{P}_{y}(I-\hat{x}\hat{x}^{T})e^{-ik_s\hat{x}\cdot y}]^{T}\right\}\phi(y)ds(y),\label{eq:far:s:asy}\\
(\mathcal{F}_{p,\infty}\phi)(\hat{x})& = \frac{1}{2\pi (2\mu + \lambda)}\int_{\partial D}\left\{ \hat{x}\hat{x}^{T}e^{-ik_p\hat{x}\cdot y} + i [\mathcal{P}_{y}\hat{x}\hat{x}^{T}e^{-ik_p\hat{x}\cdot y}]^{T}\right\}\phi(y)ds(y). \label{eq:far:p:asy}\end{aligned}$$ By , as $\omega \rightarrow 0$, we have $\lim_{\omega \rightarrow 0}u^i(x,d,p) = u^0 = u_{s}^{0} + u_{p}^0$, where $$\label{eq:lowlimit}
u_{s}^{0}= \lim_{\omega \rightarrow 0} u_{s}^i = \frac{1}{\mu} (d\times p)\times d,\ \quad u_{p}^0 = \lim_{\omega \rightarrow 0} u_{p}^i = \frac{1}{2\mu + \lambda}(d\cdot p) d.$$
Next we introduce $\phi^{\omega}(x) : = \phi_{p}^{\omega}(x) + \phi_{s}^{\omega}(x)$, $$\phi_{p}^{\omega}(x) := (I + K_{\omega, D} + i S_{\omega, D})^{-1}(-u_{p}^{i}(\cdot, d,p)), \ \ \phi_{s}^{\omega}(x) := (I + K_{\omega, D} + i S_{\omega, D})^{-1}(-u_{s}^{i}(\cdot, d,p)).$$ By Lemma \[lem:layer:diff\], it can be shown that as $\omega \rightarrow 0$, $$\begin{aligned}
\phi^{\omega}(x) &= (I + K_{\omega, D} + i S_{\omega, D})^{-1}(-u^{i}) = [I + K_{D}^{0} + i S_{D}^{0} + \mathcal{O}(\omega)]^{-1}(-u^0 + \mathcal{O}(\omega)) \notag \\
&= -[I + K_{D}^{0} + i S_{D}^{0}]^{-1}u^0 + \mathcal{O}(\omega). \label{eq:}\end{aligned}$$ Thus we arrive at $$\phi^{0}(x): = \lim_{\omega \rightarrow 0}\phi(x) = [I + K_{D}^{0} + i S_{D}^{0}]^{-1}(-u^0)= \phi_{s}^0+ \phi_{p}^0,$$ where $$\label{eq:phi:u:rela}
\phi_{s}^0: = [I + K_{D}^{0} + i S_{D}^{0}]^{-1}(-u_{s}^0),\quad \quad \phi_{p}^0: = [I + K_{D}^{0} + i S_{D}^{0}]^{-1}(- u_{p}^0).$$ It then follows that $$\begin{aligned}
u_{p}^s(D,\hat{z},p;x) &= [K_{\omega,D} + iS_{\omega,D}] (I + K_{\omega,D} + i S_{\omega,D})^{-1}(- u_{s}^{i}(t,\hat{z},p)) \\
& = [K_{\omega,D} + iS_{\omega,D}]\phi_{s}^0 + {\mathcal{O}}(\omega) \\
&= \frac{e^{ik_s|x|}}{|x|} (\mathcal{F}_{s,\infty}\phi_{s}^0)(\hat{x}) + \frac{e^{ik_p|x|}}{|x|} (\mathcal{F}_{p,\infty}\phi_{s}^0)(\hat{x})+ \mathcal{O}(\frac{1}{|x|^2}) + {\mathcal{O}}(\omega).\end{aligned}$$ Similarly, we have $$u_{p}^s(D,\hat{z},p;x) = \frac{e^{ik_s|x|}}{|x|} (\mathcal{F}_{s,\infty}\phi_{p}^0)(\hat{x}) + \frac{e^{ik_p|x|}}{|x|} (\mathcal{F}_{p,\infty}\phi_{p}^0)(\hat{x})+ \mathcal{O}(\frac{1}{|x|^2}) + {\mathcal{O}}(\omega).$$ By and Theorem \[thm:pec\], it is straightforward to show that $$\begin{aligned}
u_{\omega,\Omega}^s(x+z) =& \frac{e^{ik_p|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x|}}{|x|} (\mathcal{F}_{s,\infty}\phi_{p}^0)(\hat{x}) + \frac{e^{ik_p|x|}}{|x|} (\mathcal{F}_{p,\infty}\phi_{p}^0)(\hat{x})\right] + \mathcal{O}(\frac{1}{|x|^2}) \\
&+\frac{e^{ik_s|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x|}}{|x|} (\mathcal{F}_{s,\infty}\phi_{s}^0)(\hat{x}) + \frac{e^{ik_p|x|}}{|x|} (\mathcal{F}_{p,\infty}\phi_{s}^0)(\hat{x})\right] + {\mathcal{O}}(\frac{1}{|z|^2}) + {\mathcal{O}}(\omega).\end{aligned}$$ Furthermore, we can show that $$\begin{aligned}
u_{\omega,\Omega}^s(x) =& \frac{e^{ik_p|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}_{s,\infty}\phi_{p}^0)(\widehat{x-z}) + \frac{e^{ik_p|x-z|}}{|x-z|} (\mathcal{F}_{p,\infty}\phi_{p}^0)(\widehat{x-z})\right] + {\mathcal{O}}(\frac{1}{|z|^2}) \label{eq:rigid:fullexpan} \\
&+\frac{e^{ik_s|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}_{s,\infty}\phi_{s}^0)(\widehat{x-z}) + \frac{e^{ik_p|x-z|}}{|x-z|} (\mathcal{F}_{p,\infty}\phi_{s}^0)(\widehat{x-z})\right] + \mathcal{O}(\frac{1}{|x|^2}) + {\mathcal{O}}(\omega). \notag\end{aligned}$$ By combining , , and , as $|x| \rightarrow \infty$ and $\omega \rightarrow 0$, we have $$\begin{aligned}
&(\mathcal{F}_{s,\infty}\phi_{p}^0)(\widehat{x-z}) =(I -\hat{x}\hat{x}^T){\mathcal{O}}(\frac{1}{\lambda})a_0 , \quad (\mathcal{F}_{p,\infty}\phi_{p}^0)(\widehat{x-z}) = \hat{x}\hat{x}^T{\mathcal{O}}(\frac{1}{\lambda^2})b_0, \\
&(\mathcal{F}_{s,\infty}\phi_{s}^0)(\widehat{x-z}) =(I- \hat{x}\hat{x}^{T}){\mathcal{O}}(1)c_0, \quad (\mathcal{F}_{p,\infty}\phi_{s}^0)(\widehat{x-z}) =\hat{x}\hat{x}^T {\mathcal{O}}(\frac{1}{\lambda})d_0,\end{aligned}$$ where $a_0$, $b_0$, $c_0$ and $d_0$ are constant vectors that do not depend on $\lambda$ and $\omega$.
The proof is complete.
For the scattering of the rigid obstacle under a regular frequency, we have the following corollary.
\[cor:near:shape:rigid:dete\] Assuming $\Omega$ and $D$ be rigid obstacles with the translation and $\lambda \gg \mu$, $ \mu = {\mathcal{O}}(1)$, $|z| \gg 1$, we have the following asymptotic estimate of the scattered fields of as $|x| \rightarrow \infty$, with the incident plane wave $u^{i}(x,d,p)=u_{p}^i + u_{s}^i$ as in
\[eq:normal:fre:repre\] $$\begin{aligned}
u_{\omega,\Omega}^s(x) =& \frac{e^{ik_s|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}_{s,\infty}\phi_{s}^\omega)(\widehat{x-z}) + \frac{e^{ik_p|x-z|}}{|x-z|}
(\mathcal{F}_{p,\infty}\phi_{s}^\omega)(\widehat{x-z}) \right] + \mathcal{O}(\frac{1}{|x|^2}) \\
&+\frac{e^{ik_p|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}_{s,\infty}\phi_{p}^\omega)(\widehat{x-z}) + \frac{e^{ik_p|x-z|}}{|x-z|} (\mathcal{F}_{p,\infty}\phi_{p}^\omega)(\widehat{x-z})\right] + {\mathcal{O}}(\frac{1}{|z|^2}).\end{aligned}$$
Furthermore, we have $$\begin{aligned}
&|(\mathcal{F}_{s,\infty}\phi_{p}^\omega)(\widehat{x-z})| = {\mathcal{O}}(\frac{1}{\lambda}), \quad |(\mathcal{F}_{p,\infty}\phi_{p}^\omega)(\widehat{x-z})| = {\mathcal{O}}(\frac{1}{\lambda^2}), \\
&|(\mathcal{F}_{s,\infty}\phi_{s}^\omega)(\widehat{x-z})| = {\mathcal{O}}(1), \quad |(\mathcal{F}_{p,\infty}\phi_{s}^\omega)(\widehat{x-z})| = {\mathcal{O}}(\frac{1}{\lambda}).\end{aligned}$$
By and , and the boundedness of the linear operator $(I + K_{\omega,D} + i S_{\omega,D})^{-1}$ in $C(\partial D)^3$ [@KK; @PH], we have $$\begin{aligned}
&| (I + K_{\omega,D} + i S_{\omega,D})^{-1}(- u_{p}^{i}(t,\hat{z},p))| \sim {\mathcal{O}}(\frac{1}{\lambda}), \\
&| (I + K_{\omega,D} + i S_{\omega,D})^{-1}(- u_{s}^{i}(t,\hat{z},p))| \sim {\mathcal{O}}(\frac{1}{\mu}). \label{eq:asym:normal:fre}\end{aligned}$$ By the integral representation $[K_{\omega,D} + iS_{\omega,D}]$ in , and the representations of $\mathcal{F}_{s,\infty}$ in and $\mathcal{F}_{p,\infty}$ in , we can obtain . Furthermore with , , and and , we can obtain their asymptotic magnitudes as $\lambda \rightarrow \infty$.
For the scattering of the penetrable medium case, we have a similar theorem for the low frequency scattering.
\[thm:medium:asym\] Let $\Omega$ and $D$ be medium scatterers as described earlier with the translation . Suppose $\lambda \gg \mu$ and $ \mu = {\mathcal{O}}(1)$. Then as $|z| \gg 1$ and $\omega \ll 1$ and denoting ${\mathcal{O}}(x,z) = {\mathcal{O}}(|z|^{-1}) + {\mathcal{O}}(|x|^{-2})$, we have the following asymptotic estimate of the scattered fields of with $|x| \rightarrow \infty$, $$\begin{aligned}
&u_{\omega,\Omega}^s(x) = -\omega^2\frac{e^{ik_p|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} ({\mathcal{O}}(\frac{1}{\lambda})(I-\hat{x}\hat{x}^T)a^m_0 + \frac{e^{ik_p|x-z|}}{|x-z|}{\mathcal{O}}(\frac{1}{\lambda^2}) \hat{x}\hat{x}^{T}b^m_0 + {\mathcal{O}}(x,z) \right] \notag \\
&-\omega^2\frac{e^{ik_s|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} ({\mathcal{O}}(1)(I-\hat{x}\hat{x}^T)c^m_0 + \frac{e^{ik_p|x-z|}}{|x-z|}
{\mathcal{O}}(\frac{1}{\lambda}) \hat{x}\hat{x}^{T}d^m_0 + {\mathcal{O}}(x,z) \right] + {
\mathchoice
{{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(\omega^2). \notag\end{aligned}$$ where $a^m_0$, $b^m_0$, $c^m_0$ and $d^m_0$ are all constant vectors that do not depend on $\omega$ and $\lambda$.
The proof is similar to those of Lemma \[lem:layer:diff\] and Theorem \[thm:obstacle:asym\]. We introduce the following volume potential $$(\tilde{V}_{D}^{0}\varphi)(x) := - \int_{D}\Gamma^{0}(x,y)n_{D}(y)\varphi(y)dy.$$ By and [@PHhab Chap. 5], we see as $\omega \rightarrow 0$ $$\|\tilde{V}_{\omega,D}-\omega^2\tilde{V}_{D}^0\|_{C(D)\rightarrow C^{1, \gamma}(D)} \sim \|n_{D}\|_{C(D)}\mathcal{O}(\omega^3).$$ For the scattering by plane incident wave of the medium $D$ , as $\omega \rightarrow 0$, we have $$\label{eq:asym:medium:total:1}
u(D,d, p;x) = (I- \tilde{V}_{\omega,D})^{-1}u_{\omega, p,d}^{i} = (I - \omega^2 \tilde{V}_{D}^{0} +\mathcal{O}(\omega) )^{-1}(u^0 + \mathcal{O}(\omega)) = u^0 + \mathcal{O}(\omega^2).$$ In addition, the fundamental solution matrix $\Gamma(x,y)$ has the following asymptotic behavior [@Kup], $$\label{eq:asym:elastic:funda}
\Gamma(x,y) = \frac{e^{ik_p|x|}}{|x|}\frac{\hat{x}\hat{x}^{T}e^{-ik_p \hat{x}\cdot y}}{4\pi(\lambda+2\mu)}
+ \frac{e^{ik_s|x|}}{|x|}\frac{1}{4\pi \mu}(I-\hat{x}\hat{x}^{T})e^{-ik_s\hat{x}\cdot y} + \mathcal{O}(\frac{1}{|x|^2}), \ |x| \rightarrow \infty.$$ By the integral representation of the scattered field of , $ u^s(D,d, p;x) = [\tilde{V}_{\omega,D}u(D,d,p;\cdot)](x)$, together with and , we have $$u^{\infty}(D,d, p;\hat{x}) = -\omega^2 \left[ \frac{e^{ik_p|x|}}{|x|} (\mathcal{F}^{m}_{p,\infty} u(D,d, p;\cdot))(\hat{x}) + \frac{e^{ik_s|x|}}{|x|}(\mathcal{F}^{m}_{s,\infty} u(D,d, p;\cdot))(\hat{x}) \right],$$ where the far field mapping operators $\mathcal{F}^{m}_{s,\infty}$ and $\mathcal{F}^{m}_{p,\infty}$ are defined by $$\begin{aligned}
(\mathcal{F}^{m}_{s,\infty}\phi)(\hat{x})&:= \frac{1}{4\pi \mu}\int_{D} [I -\hat{x}\hat{x}^{T}]e^{-ik_s\hat{x}\cdot y} n_{D}(y) \phi(y) dy,\label{eq:far:s:asy:medium}\\
(\mathcal{F}^{m}_{p,\infty}\phi)(\hat{x})& := \frac{1}{4\pi (2\mu + \lambda)}\int_{D} \hat{x}\hat{x}^{T}e^{-ik_p\hat{x}\cdot y}n_{D}(y) \phi(y) dy. \label{eq:far:p:asy:medium}\end{aligned}$$ Recalling , together with and , similar to the rigid obstacle case , we see $$\begin{aligned}
&u_{\omega,\Omega}^s(x) = -\omega^2\frac{e^{ik_p|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}^m_{s,\infty}u_{p}^0)(\widehat{x-z}) + \frac{e^{ik_p|x-z|}}{|x-z|} (\mathcal{F}^m_{p,\infty}u_{p}^0)(\widehat{x-z}) + {\mathcal{O}}(x,z) \right] \notag \\
&-\omega^2\frac{e^{ik_s|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}^m_{s,\infty}u_{s}^0)(\widehat{x-z}) + \frac{e^{ik_p|x-z|}}{|x-z|} (\mathcal{F}^m_{p,\infty}u_{s}^0)(\widehat{x-z}) + {\mathcal{O}}(x,z)\right] + {
\mathchoice
{{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(\omega^2). \label{eq:medium:fullexpan}\end{aligned}$$ Combining , and , as $|x| \rightarrow \infty$ and $\omega \rightarrow 0$, we have $$\begin{aligned}
&(\mathcal{F}^m_{s,\infty}u_{p}^0)(\widehat{x-z}) = (I-\hat{x}\hat{x}^{T}){\mathcal{O}}(\frac{1}{\lambda})a^m_0 , \quad (\mathcal{F}^m_{p,\infty}u_{p}^0)(\widehat{x-z}) = \hat{x}\hat{x}^{T}{\mathcal{O}}(\frac{1}{\lambda^2})b^m_0, \\
&(\mathcal{F}^m_{s,\infty}u_{s}^0)(\widehat{x-z}) = (I-\hat{x}\hat{x}^{T}){\mathcal{O}}(1)c^m_0, \quad (\mathcal{F}^m_{p,\infty}u_{s}^0)(\widehat{x-z}) = \hat{x}\hat{x}^{T} {\mathcal{O}}(\frac{1}{\lambda})d^m_0,\end{aligned}$$ where $a^m_0$, $b^m_0$, $c^m_0$ and $d^m_0$ are all constant complex vectors that do not depend on $\omega$ and $\lambda$, which leads to this theorem.
For the medium scattering under a regular frequency, we have the following result. We denote $$v_{p}^{\omega}(x) = (I- \tilde{V}_{\omega,D})^{-1}u_{p}^{i}, \ \ v_{s}^{\omega}(x) = (I- \tilde{V}_{\omega,D})^{-1}u_{s}^{i}.$$
\[cor:asym:medium:shape:dete\] Let $\Omega$ and $D$ be medium with the translation . Supposing $\lambda \gg \mu$, $ \mu = {\mathcal{O}}(1)$, $|z| \gg 1$, then we have the following asymptotic estimate of the scattered fields of as $|x| \rightarrow \infty$, with the incident plane wave $u^{i}(x,d,p)=u_{p}^i + u_{s}^i$ $$\begin{aligned}
u_{\omega,\Omega}^s(x) =& -\omega^2\frac{e^{ik_s|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}^m_{s,\infty}v_{s}^\omega)(\widehat{x-z}) + \frac{e^{ik_p|x-z|}}{|x-z|}
(\mathcal{F}^m_{p,\infty}v_{s}^\omega)(\widehat{x-z}) \right] + \mathcal{O}(\frac{1}{|x|^2}) \\
&-\omega^2\frac{e^{ik_p|z|}}{4 \pi |z|} \left[ \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}^m_{s,\infty}v_{p}^\omega)(\widehat{x-z}) + \frac{e^{ik_p|x-z|}}{|x-z|} (\mathcal{F}^m_{p,\infty}v_{p}^\omega)(\widehat{x-z})\right] + {\mathcal{O}}(\frac{1}{|z|^2}). \notag\end{aligned}$$ Additionally, we have $$\begin{aligned}
&|(\mathcal{F}^m_{s,\infty}v_{p}^\omega)(\widehat{x-z})| = {\mathcal{O}}(\frac{1}{\lambda}), \quad |(\mathcal{F}^m_{p,\infty}v_{p}^\omega)(\widehat{x-z})| = {\mathcal{O}}(\frac{1}{\lambda^2}), \\
&|(\mathcal{F}^m_{s,\infty}v_{s}^\omega)(\widehat{x-z})| = {\mathcal{O}}(1), \quad |(\mathcal{F}^m_{p,\infty}v_{s}^\omega)(\widehat{x-z})| = {\mathcal{O}}(\frac{1}{\lambda}).\end{aligned}$$
The proof of Corollary \[cor:asym:medium:shape:dete\] is completely similar to that of Theorem \[thm:medium:asym\] and we skip it here.
A Two-stage Reconstruction Algorithm {#sec:two-stage}
====================================
For the elastic wave scattering, Theorems \[thm:pec\] and \[thm:medium\] indicate that either with the incident plane pressure wave $u_{p}^i$ or with the incident plane shear wave $u_{s}^i$, the scattered field consists of both the shear and pressure waves which are difficult to separate one from the other. However, one can benefit from the orthogonality of the shear and pressure waves with only a single incident plane wave, i.e., with only $u_{p}^i$ or $u_{s}^i$. Similar ideas could be found in [@HLLS] and [@Sini], where the direction and phase information of the far fields can be separated for reconstructions. For the incident point waves, by Lemma \[lem:incident\], two kinds of incident plane waves are mixed with complex phases even in the asymptotic expansion of point source waves. Fortunately, with the help of the low frequency analysis developed in Section \[sec:analysis:lowfre\], we can benefit from the dominating part of the point source wave for the nearly compressible materials. This crucial observation, together with the results derived in Section \[sec:mathground\], enables us to propose a two-stage algorithm for finding both the locations and the shapes of the unknown elastic scatterers from an admissible dictionary. At the first stage, a low frequency is used for locating the scatterers, and at the second stage, a regular frequency is used for determining the shapes of the scatterers from the dictionary.
Locating the scatterers {#subsect:locating}
-----------------------
We let $\Lambda$ denote the bounded measurement surface in $\mathbb{R}^3$, and $\tilde{z}$ be an arbitrary sampling point contained in a bounded sampling region $S \subseteq \mathbb{R}^3$. We also introduce $u_{\omega}^{s}(D,z;x):=u_{\omega,\Omega}^{s}(x)$ and $u_{\omega}^{\infty}(D,z;\hat{x}):=u_{\omega,\Omega}^{\infty}(\hat{x})$ as the corresponding scattered field and far field of $\Omega$ due to the incident point source $u_{\omega, p, ep}^i(x,0)$ as in . These are the measurement data for our reconstruction schemes.
With the transition relations built in Theorems \[thm:pec\] and \[thm:medium\], and the asymptotic estimates given in Theorems \[thm:obstacle:asym\] and \[thm:medium:asym\], we propose the following two-stage algorithm for locating the position and determining the shape of the scatterer.
There are two separate cases in our subsequent discussion, depending on the noise level $\epsilon$ in the measured data. From the low frequency analysis in Theorems \[thm:obstacle:asym\] and \[thm:medium:asym\], it can be seen that if the order of noise level is ${
\mathchoice
{{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(\frac{1}{\lambda})$, then we can extract the $\mathcal{O}(\frac{1}{\lambda})$ order pressure wave alone for the reconstruction. If the noise level is of order $\mathcal{O}(\frac{1}{\lambda})$, the $\mathcal{O}(\frac{1}{\lambda})$ order pressure wave would be polluted. We then extract the dominating $\mathcal{O}(1)$ order shear wave instead for the reconstruction.
Case I: The noise level $\epsilon \sim {
\mathchoice
{{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(\frac{1}{\lambda})$. In this case, we only use the pressure wave field for the reconstruction and introduce the following imaging functional, $$\label{eq:indicator:p}
I_{p}(D,z, \tilde{z}): = \frac{|\langle P_{\hat{x}}[u_{\omega}^s(D,z;\cdot)], \mathring{u}_{p}^s(\tilde{z};\cdot)\rangle_{L^{2}(\Lambda)} |}{\|P_{\hat{x}}[u_{\omega}^s(D,z;\cdot)]\|_{L^2(\Lambda)} \|\mathring{u}_{p}^s(\tilde{z};\cdot)\|_{L^2(\Lambda)} }, \quad \tilde{z} \in S,$$ where $u_{\omega,\Omega}^s(D,z;\cdot) $ is the scattered field produced by the incident point source $u_{\omega, p, ep}^i(x,0)$ as in . $P_{\hat{x}}=\hat{x}\hat{x}^{T}$ is the projection along the direction $\hat{x}$, and the test function $\mathring{u}_{p}^s(\tilde{z};x)$ is defined as $$\mathring{u}_{p}^s(\tilde{z};x): = \frac{e^{ik_s|\tilde{z}|}}{4\pi |\tilde{z}|} \frac{e^{ik_p|x-\tilde{z}|}}{|x-\tilde{z}|}\hat{x}.$$ If the measurement data are phaseless, then the imaging functional is modified as follows, and see [@Klib1; @Klib2] for more reconstruction schemes and algorithms with phaseless data.
$$\label{eq:indicator:p:phaseless}
I_{|p|}(D,z, \tilde{z}): = \frac{|\langle |P_{\hat{x}}[u_{\omega}^s(D,z;\cdot)]|, |\mathring{u}_{p}^s(\tilde{z};\cdot)|\rangle_{L^{2}(\Lambda)} |}{\|P_{\hat{x}}[u_{\omega}^s(D,z;\cdot)]\|_{L^2(\Lambda)} \|\mathring{u}_{p}^s(\tilde{z};\cdot)\|_{L^2(\Lambda)} }, \quad \tilde{z} \in S.$$
Case II: the noise level $ \epsilon \sim {\mathcal{O}}(\frac{1}{\lambda})$. In this case, we will use the shear far field for the reconstruction. According to Theorems \[thm:obstacle:asym\] and \[thm:medium:asym\], although the pressure and shear parts are mixed together in the near scattered field, the shear far field is the dominant part with magnitude ${\mathcal{O}}(\mu)$. We introduce $$\mathring{u}_{k_s, m, H_1}^{\infty}(\tilde{z};x) := \frac{e^{ik_s|\tilde{z}|}}{4\pi |\tilde{z}|} e^{-ik_s \hat{x}\cdot \tilde{z}}U_{1}^{m}(\hat{x}),\quad \mathring{u}_{k_s, m, H_2}^{\infty}(\tilde{z};x) := \frac{e^{ik_s|\tilde{z}|}}{4\pi |\tilde{z}|} e^{-ik_s \hat{x}\cdot \tilde{z}}V_{1}^{m}(\hat{x}), \quad m=-1,0,1,$$ where $U_{1}^m$ and $V_{1}^m$ are the vectorial spherical harmonics (cf. [@CK]), $$U_{1}^m: = \frac{1}{2} \text{Grad}Y_{1}^m(\hat{x}), \quad V_{1}^m: = \frac{1}{2} \hat{x} \times \text{Grad}Y_{1}^m(\hat{x}), \quad m=-1,0,1.$$ We propose the following imaging functional, $$\label{eq:indicator:s}
I_{s}(D,z, \tilde{z}): = \frac{\sqrt{\sum_{j=1}^2\sum_{m=-1}^1|\langle (I-P_{\hat{x}})u_{\omega}^{\infty}(D,z;\cdot), \mathring{u}_{k_s,m, H_j}^{\infty}(\tilde{z};\cdot)\rangle_{T^{2}(\mathbb{S}^2)}|^2}}{\|(I-P_{\hat{x}})u_{\omega}^{\infty}(D,z;\cdot)\|_{T^2(\mathbb{S}^2)} 1/{(4\pi |\tilde{z}|)}}, \quad \tilde{z} \in S,$$ where $T^2(\mathbb{S}^2)$ is the tangential vector space on the unit sphere $\mathbb{S}^2$, and $(I-P_{\hat{x}})$ is the projection to the shear part, $$(I-P_{\hat{x}})u_{\omega}^{\infty}(D,z;\cdot) =u_{\omega}^{\infty}(D,z;\cdot) - \hat{x} \hat{x}^{T}u_{\omega}^{\infty}(D,z;\cdot).$$
Next we show the indicating behaviours of the imaging functionals introduced in , and , for locating the position $z$ of the scatterer $\Omega$.
If $D$ is a rigid scatterer as in Theorem \[thm:obstacle:asym\] (or a penetrable medium given in Theorem \[thm:medium:asym\]) and assume $|a_0|$, $|b_0|$, $|c_0|$, $|d_0|$ (or $|a^m_0|$, $|b^m_0|$, $|c^m_0|$, $|d^m_0|$ for medium case) are positive for all $D \in \mathfrak{D}$. Then we have the following asymptotic expansion $$\label{eq:indica:relation:p}
\lim_{\lambda \rightarrow \infty}\lim_{\omega \rightarrow 0} I_{p}(D,z, \tilde{z}) = \mathring{I}_{p}(z;\tilde{z})[1+ \mathcal{O}(|z|^{-1})], \quad |z|\rightarrow \infty, \quad |x| \rightarrow \infty,$$ uniformly for all $D \in \mathfrak{D}$, $z \in \Lambda$ and $\tilde{z} \in \Lambda$, where for the indicator functional , one has $$\mathring{I}_{p}(z;\tilde{z}) = \frac{|\langle \mathring{u}_{p}^s(z;\cdot), \mathring{u}_{p}^s(\tilde{z};\cdot) \rangle_{L^2(\Lambda)} |}{\|\mathring{u}_{p}^s({z};\cdot)\|_{L^2(\Lambda)} \|\mathring{u}_{p}^s(\tilde{z};\cdot)\|_{L^2(\Lambda)} }, \quad \tilde{z} \in S,$$ and for the phaseless indicator functional , denoting $ \mathring{I}_{|p|}$ as the similar limit of $I_{|p|}$ as in , one has $$\mathring{I}_{|p|}(z;\tilde{z}) = \frac{|\langle |\mathring{u}_{p}^s(z;\cdot)|, |\mathring{u}_{p}^s(\tilde{z};\cdot) | \rangle_{L^2(\Lambda)} |}{\|\mathring{u}_{p}^s({z};\cdot) \|_{L^2(\Lambda)}\|\mathring{u}_{p}^s(\tilde{z};\cdot)\|_{L^2(\Lambda)} }, \quad \tilde{z} \in S.$$ Similarly for the indicator $I_{s}(D,z, \tilde{z})$, we have $$\label{eq:indica:relation:s}
\lim_{\lambda \rightarrow \infty}\lim_{\omega \rightarrow 0} I_{s}(D,z, \tilde{z}) = \mathring{I}_{s}(z;\tilde{z})[1+ \mathcal{O}(|z|^{-1})], \quad |z|\rightarrow \infty, \quad |x| \rightarrow \infty,$$ uniformly for all $D \in \mathfrak{D}$, $\hat{z}\in \mathbb{S}^2$ and $\tilde{z} \in S$, where $$\label{eq:indicator:asym}
\mathring{I}_{s}(z, \tilde{z}): = \frac{\sqrt{\sum_{j=1}^2\sum_{m=-1}^1|\langle \tilde{u}_{k_s}^{\infty}(D,z;\cdot), \mathring{u}_{k_s,m, H_j}^{\infty}(\tilde{z};\cdot)\rangle_{T^{2}(\mathbb{S}^2)}|^2}}{\|\tilde{u}_{k_s}^{\infty}(D,z;\cdot)\|_{T^2(\mathbb{S}^2)} 1/{(4\pi |\tilde{z}|)}}, \quad \tilde{z} \in S,$$ with $$\tilde{u}_{k_s}^{\infty}(D,z;\cdot) = \frac{e^{ik_s|z|}}{4 \pi |z|} e^{-ik_s\hat{x}\cdot z} \begin{cases} (\mathcal{F}_{s,\infty}\phi_{s}^0)(\widehat{x-z}), \quad D \ \text{rigid obstacle}, \ \phi_{s}^0 \ \text{as in} \ \eqref{eq:phi:u:rela}, \ \\
(\mathcal{F}^m_{s,\infty}u_{s}^0)(\widehat{x-z}), \quad D \ \text{medium}, \ u_{s}^0 \ \text{as in}\ \eqref{eq:medium:fullexpan}.
\end{cases}$$
We only prove the case that $D$ is a rigid obstacle with the indicator functional given by $I_{p}(D,z, \tilde{z})$. The other cases can be proved in a similar manner. By Theorems \[thm:pec\] and \[thm:obstacle:asym\], there exists a constant vector $d_0$, such that as $|z|\rightarrow \infty$, we have by , $$\begin{aligned}
\lim_{\omega \rightarrow 0} P_{\hat{x}}[u_{\omega}^s(D,z;x)] &= \mathring{u}_{p}^s(z;x)[\hat{x}^Td_{0}{\mathcal{O}}(\frac{1}{\lambda}) \hat{x} +
\mathcal{O}(|z|^{-1}) + {\mathcal{O}}(\omega)][1+ \mathcal{O}(|z|^{-1})], \label{eq:relation:near:far:p}
$$ uniformly for $\hat{z}\in \mathbb{S}^2$ and $x\in S$. Substituting into , we arrive at .
By the Cauchy-Schwarz inequality, we see $ \mathring{I}_{p}(z;\tilde{z}) \leq 1$ unless $\mathring{u}_{p}^s(z;\cdot)$ and $\mathring{u}_{p}^s(\tilde{z};\cdot)$ are constant multiples of each other, which can only happen when $z=\tilde{z}$. This then helps us to find the location $z$ with the sampling points $\tilde{z}$.
Shape determination
-------------------
Once the location is approximated by $\mathring{z}= \operatorname*{arg\,max}_{\tilde{z}}I_{p}(D,z;\tilde{z})$ or $\mathring{z}= \operatorname*{arg\,max}_{\tilde{z}}I_{s}(D,z;\tilde{z})$ in Section \[subsect:locating\], together with Corollary \[cor:near:shape:rigid:dete\] and \[cor:asym:medium:shape:dete\], we next present the numerical schemes for reconstructing the shapes through dictionary matching under a regular frequency. The corresponding analysis also depends on the noise level $\epsilon$ through a similar strategy as before.
Case I: The noise level $\epsilon \sim {
\mathchoice
{{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(\frac{1}{\lambda})$. In view of Corollaries \[cor:near:shape:rigid:dete\] and \[cor:asym:medium:shape:dete\], we propose the following imaging functionals, $$\label{eq:indicator:p:multi}
J_{p}(D_i, D_j; z, \mathring{z}): = \frac{|\langle P_{\hat{x}}[u_{\omega}^s(D_i,z;\cdot)], P_{\hat{x}}[u_{sp}^s(D_j,\mathring{z};\cdot)]\rangle_{L^{2}(\Lambda)} |}{\|P_{\hat{x}}[u_{\omega}^s(D_i,z;\cdot)]\|_{L^2(\Lambda)} \|P_{\hat{x}}[u_{sp}^s(D_j,\mathring{z};\cdot)]\|_{L^2(\Lambda)} },$$ for $D_i, D_j \in \mathfrak{D}$, where $u_{\omega}^s(D_i,z;\cdot): = u_{\omega, D_i+z}^{s}(\cdot)$, and the test function $$\label{eq:shape:sp:p}
u_{sp}^s(D_j,\mathring{z};x): = \frac{e^{ik_s|\mathring{z}|}}{4\pi |\mathring{z}|} \frac{e^{ik_p|x-\mathring{z}|}}{|x-\mathring{z}|}u_{s}^{\infty}(D_j,\hat{\mathring{z}},p;\widehat{x-\mathring{z}}),$$ with $u_{s}^{\infty}(D_j,\hat{\mathring{z}},p;\widehat{x-\mathring{z}})$ the far field of the scattered wave of $D$ ($D$ is a rigid obstacle or a penetrable medium) due to the incident shear plane wave only, i.e., $u_{s}^i$ in . Case II: The noise level $ \epsilon \sim {\mathcal{O}}(\frac{1}{\lambda})$. We can again use near-field scattered data in the dictionary produced by the shear incident plane wave $u_{s}^i$. We denote it as $u_{ss}^s$, i.e., $$u_{ss}^s(D_j,\mathring{z};x): = \frac{e^{ik_s|\mathring{z}|}}{4\pi |\mathring{z}|} \frac{e^{ik_s|x-\mathring{z}|}}{|x-\mathring{z}|}u_{s}^{\infty}(D_j,\hat{\mathring{z}},p;\widehat{x-\mathring{z}}).$$ Here $u_{s}^{\infty}(D_j,\hat{\mathring{z}},p;\widehat{x-\mathring{z}})$ is the same as in . Then the imaging functional becomes
$$\label{eq:indicator:s:multi}
J_{s}(D_i, D_j;z, \mathring{z}): = \frac{|\langle (I - P_{\hat{x}}) u_{\omega}^{s}(D_i,z;\hat{x}), (I - P_{\hat{x}})\mathring{u}_{ss}^{s}(D_j,\mathring{z};\hat{x})\rangle_{L^2(\Lambda)} | }{\|(I - P_{\hat{x}})u_{\omega}^{s}(D_i,z;\hat{x})\|_{L^2(\Lambda)} \|(I - P_{\hat{x}})\mathring{u}_{ss}^{s}(D_j,\mathring{z};\hat{x})\|_{L^2(\Lambda)} }$$
With these preparations, we can present our second stage algorithm for the shape reconstruction. Through this scheme, we can find the shape of the scatterer $\Omega$ by dictionary searching.
Assume that there exists a constant $c_0>0$ such that $\|P_{\hat{x}}[u_{sp}^s(D_j,\mathring{z};\cdot)]\|_{L^2(\Lambda)} \geq c_0$ (or $\|(I - P_{\hat{x}})\mathring{u}_{ss}^{s}(D_j,\mathring{z};\hat{x})\|_{L^2(\Lambda)} \geq c_0$) for all $D_i \in \mathfrak{D}$. Then for any sufficiently small $\varepsilon>0$ there exist $R_0$ and $\delta>0$ such that if $|z| \geq R_0$ and $|z-\mathring{z}| \leq \delta$, $$\label{eq:shape:appro}
|J_{l_i}(D_i,D_j;z, \mathring{z}) - \hat{J}_{l_i}(D_i,D_j;z)| \leq z, \quad \forall D_i, D_j \in \mathfrak{D},$$ where $$\hat{J}_{l_i}(D_i,D_j;z): = J_{l_i}(D_i,D_j;z,z), \quad l_i = p \ \ \text{or} \ \ l_i=s.$$ Furthermore, as $\mathring{z} \rightarrow z$ and $D_i = D_j$, $J_{l_i}(D_i,D_j;z, \mathring{z})$ obtains its maximal value $1$ approximately.
We first consider the case that $D$ is a rigid obstacle. For the indicator $J_{p}$ with the noise level $\epsilon \sim {
\mathchoice
{{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(\frac{1}{\lambda})$, it suffices to calculate $P_{\hat{x}}[u_{\omega}^s(D_i,z;\cdot)]$ and $P_{\hat{x}}[u_{sp}^s(D_j,\mathring{z};\cdot)]$ as in . According to Corollary \[cor:near:shape:rigid:dete\], we have $$\begin{aligned}
P_{\hat{x}}[u_{\omega}^s(D_i,z;\cdot)] = P_{\hat{x}} [u_{\omega,\Omega}^s(x)] =& \frac{e^{ik_s|z|}}{4 \pi |z|} \frac{e^{ik_p|x-z|}}{|x-z|}
(\mathcal{F}_{p,\infty}\phi_{s}^\omega)(\widehat{x-z}) + {\mathcal{O}}(\frac{1}{|z|^2}) \\
& +\frac{e^{ik_p|z|}}{4 \pi |z|} \frac{e^{ik_p|x-z|}}{|x-z|} (\mathcal{F}_{p,\infty}\phi_{p}^\omega)(\widehat{x-z}) + \mathcal{O}(\frac{1}{|x|^2}).\end{aligned}$$ Since $|(\mathcal{F}_{p,\infty}\phi_{p}^\omega)(\widehat{x-z})| = {\mathcal{O}}(\frac{1}{\lambda^2})$ and $|(\mathcal{F}_{p,\infty}\phi_{s}^\omega)(\widehat{x-z})| = {\mathcal{O}}(\frac{1}{\lambda})$, we have $$\label{eq:shape:unknown:p}
P_{\hat{x}}[u_{\omega}^s(D_i,z;\cdot)] = \frac{e^{ik_s|z|}}{4 \pi |z|} \frac{e^{ik_p|x-z|}}{|x-z|} (\mathcal{F}_{p,\infty}\phi_{s}^\omega)(\widehat{x-z}) + \mathcal{O}(\frac{1}{|x|^2}) + {\mathcal{O}}(\frac{1}{\lambda^2}).$$ Moreover, for the precomputed dictionary data $u_{sp}^s(D_j,\mathring{z};\cdot)$, we have $$\label{eq:shape:known:p}
P_{\hat{x}}[u_{sp}^s(D_j,\mathring{z};\cdot)] = \frac{e^{ik_s|\mathring{z}|}}{4 \pi |\mathring{z}|} \frac{e^{ik_p|x-\mathring{z}|}}{|x-\mathring{z}|} (\mathcal{F}_{p,\infty}\phi_{s}^\omega)(\widehat{x-\mathring{z}}) + \mathcal{O}(\frac{1}{|x|^2}).$$ It can be seen that the dictionary data approximate the measured scattered field under the condition of this theorem, and then the approximation follows. By the Cauchy-Schwarz inequality and comparing the above two formulas, we see that $J_{p}(D_i,D_j;z, \mathring{z})$ achieves its maximal value approximately if and only if $D_i = D_j$.
For the indicator $J_s$ with the noise level $ \epsilon \sim {\mathcal{O}}(\frac{1}{\lambda})$, it is sufficient for us to calculate $(I - P_{\hat{x}}) u_{\omega}^{s}(D,z;\hat{x})$ and $(I - P_{\hat{x}})\mathring{u}_{ss}^{s}(D_j,\mathring{z};\hat{x})$. By Corollary \[cor:near:shape:rigid:dete\], we have $$\begin{aligned}
(I - P_{\hat{x}})u_{\omega}^{s}(D,z;\hat{x}) =& (I - P_{\hat{x}}) u_{\omega,\Omega}^s(x)
= \frac{e^{ik_s|z|}}{4 \pi |z|} \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}_{s,\infty}\phi_{s}^\omega)(\widehat{x-z}) + \mathcal{O}(\frac{1}{|x|^2}) \\
&+\frac{e^{ik_p|z|}}{4 \pi |z|} \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}_{s,\infty}\phi_{p}^\omega)(\widehat{x-z}) + {\mathcal{O}}(\frac{1}{|z|^2}).\end{aligned}$$ Since $|(\mathcal{F}_{s,\infty}\phi_{s}^\omega)(\widehat{x-z}) |= {\mathcal{O}}(1)$ and $|(\mathcal{F}_{s,\infty}\phi_{p}^\omega)(\widehat{x-z})| = {\mathcal{O}}(\frac{1}{\lambda})$, we thus have $$\label{eq:shape:unknown}
(I - P_{\hat{x}})u_{\omega}^{s}(D,z;\hat{x}) = \frac{e^{ik_s|z|}}{4 \pi |z|} \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}_{s,\infty}\phi_{s}^\omega)(\widehat{x-z}) + \mathcal{O}(\frac{1}{|x|^2}) + {\mathcal{O}}(\frac{1}{\lambda}) + \mathcal{O}(\frac{1}{|z|^2}).$$ Similarly, it can be verified that $$\label{eq:shape:known}
(I - P_{\hat{x}})\mathring{u}_{ss}^{s}(D_j,\mathring{z};\hat{x}) = \frac{e^{ik_s|z|}}{4 \pi |z|} \frac{e^{ik_s|x-z|}}{|x-z|} (\mathcal{F}_{s,\infty}\phi_{s}^\omega)(\widehat{x-z}) + \mathcal{O}(\frac{1}{|x|^2}).$$ Hence the dictionary data approximate the measured scattered field under the condition of this theorem, and the approximation also follows. Again by Cauchy-Schwarz inequality and comparing the above two formulas, we conclude that $J_{s}(D_i,D_j;z, \mathring{z})$ achieves its maximal value approximately if and only if $D_i = D_j$.
The other case that $\Omega$ is a penetrable medium follows by completely similar arguments, along with the use of Corollary \[cor:asym:medium:shape:dete\].
Numerical Tests {#sec:num}
===============
In this section, we present numerical tests to illustrate the effectiveness and efficiency of the proposed method. We verify the elastic imaging technique using limited aperture near field data. All the numerical experiments are carried out using MATLAB R2017a on a Lenovo workstation with 2.3GHz Intel Xeon E5-2670 v3 processor and 512GB of RAM.
The experimental setup is as follows. As shown in Figure \[fig:Dictionary\], the admissible dictionary set consists of six referemce domains, $D_i$, $i=1,\ldots,6$, which are composed of a number of unit cubes. The measurement surface $\Lambda$ is set to be a unit square in the $x^2 x^3$-plane and centered at the origin. The scattered elastic near fields on the measurement surface $\Lambda$ of reference domains in the dictionary $\mathfrak{D}$ as in are first collected in advance for incident point signal waves with different locations.
In the following tests, we let the Poisson ratio $\nu=0.475$ and Young’s modulus $E=3$, thus $\mu \sim {\mathcal{O}}(1)$, $\lambda \sim {\mathcal{O}}(10^2)$ and $\lambda \gg \mu$. The detecting wavelength for locating objects is set to be $\omega_{1}:=1$ and the detecting wavelength for shape determination is set to be $\omega_{2}:=20$. Without loss of generality, the target scatterer is given by $\Omega:=D_{i}+z_{0}$. Here $z_{0}$ is fixed as $(40,\,0,\,0)$.
![\[fig:Dictionary\]Dictionary of a priori known scatterers.](gesture1 "fig:"){width="30.00000%"}![\[fig:Dictionary\]Dictionary of a priori known scatterers.](gesture2 "fig:"){width="30.00000%"}![\[fig:Dictionary\]Dictionary of a priori known scatterers.](gesture11 "fig:"){width="30.00000%"}
(a)(b)(c)
![\[fig:Dictionary\]Dictionary of a priori known scatterers.](gesture12 "fig:"){width="30.00000%"}![\[fig:Dictionary\]Dictionary of a priori known scatterers.](gesture13 "fig:"){width="30.00000%"}![\[fig:Dictionary\]Dictionary of a priori known scatterers.](gesture6 "fig:"){width="30.00000%"}
(d)(e)(f)
Rigid body case
---------------
In the first example, we test with six reference domains with the rigid body boundary condition.
In the first stage, the positions are found in the locating stage by finding the maximum value of the indicator functions $I_p$ in and $I_s$ in . In the noise-free case, the coordinates and the distance from the exact location are shown in Table \[tab:location-test-pec\] for each reference domain in the dictionary. One can find that the error of positions deduced from $I_{s}$ is much smaller than those from $I_{p}$. The reason for this phenomenon is due to the fact that $k_{s}\gg k_{p}$, thus the shear wave with wave number $k_{s}$ has a higher resolution. In light of the high resolution of shear waves, it is enough for us to locate and imaging the reference domain only by $I_{s}$ in . Compared with the exact position $z_0$, the difference between the exact and estimated positions using $I_s$ in are always below $0.1\%$ in terms of the Euclidean distance.
Next, we compute indicator function value $J_s$ in using the near field data measured on $\Lambda$ and the approximate position found in Table \[tab:location-test-pec\]. The result of shape determination is shown in Table \[tab:gesture-test-pec\]. The values of the indicator function value have been rescaled between $0$ and $1$ by normalizing with respect to the maximum function value among all six reference gestures in each row of the table to highlight the unique reference domain identified. The same normalization procedure is employed in the sequel. We see from Table \[tab:gesture-test-pec\] that the peak value are always taken in the diagonal line when the measurement data match with the precomputed data of the correct gesture.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
--------------------------------------- ----------- ----------- ----------- ----------- ----------- -----------
$\mathring{z}_{0}^{1}$ $40.0278$ $40.0547$ $40.0965$ $40.0158$ $40.0971$ $40.0958$
$\mathring{z}_{0}^{2}$ $0.0679$ $0.0743$ $0.0655$ $0.0706$ $0.0277$ $0.0097$
$\mathring{z}_{0}^{3}$ $0.0758$ $0.0392$ $0.0171$ $0.0032$ $0.0046$ $0.0823$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.1055$ $0.1003$ $0.1173$ $0.1196$ $0.0322$ $0.1277$
: \[tab:location-test-pec\] Location test using $I_{p}$ and $I_{s}$ for elastic rigid bodies without noise.
$I_{p}$ indicator function.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
--------------------------------------- ----------- ----------- ----------- ----------- ----------- -----------
$\mathring{z}_{0}^{1}$ $39.9901$ $39.9811$ $39.9784$ $39.9846$ $40.0035$ $39.9853$
$\mathring{z}_{0}^{2}$ $0.0100$ $0.0180$ $-0.0056$ $0.0201$ $-0.0251$ $0.0121$
$\mathring{z}_{0}^{3}$ $-0.0101$ $-0.0110$ $-0.0204$ $-0.0256$ $-0.0265$ $-0.0265$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.0173$ $0.0283$ $0.0302$ $0.0360$ $0.0367$ $0.0326$
: \[tab:location-test-pec\] Location test using $I_{p}$ and $I_{s}$ for elastic rigid bodies without noise.
$I_{s}$ indicator function.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
--------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- -----------------------
$D_{1}$ $\boldsymbol{1.0000}$ $0.9234$ $0.9291$ $0.8660$ $0.8249$ $0.9453$
$D_{2}$ $0.9232$ $\boldsymbol{1.0000}$ $0.9245$ $0.9502$ $0.9109$ $0.9146$
$D_{3}$ $0.9280$ $0.9242$ $\boldsymbol{1.0000}$ $0.9040$ $0.9494$ $0.9851$
$D_{4}$ $0.8650$ $0.9510$ $0.9040$ $\boldsymbol{1.0000}$ $0.9301$ $0.9742$
$D_{5}$ $0.8249$ $0.9007$ $0.9592$ $0.9310$ $\boldsymbol{1.0000}$ $0.9169$
$D_{6}$ $0.9451$ $0.9147$ $0.9849$ $0.9732$ $0.9249$ $\boldsymbol{1.0000}$
: \[tab:gesture-test-pec\]Shape determination test using $J_{p}$ and $J_{s}$ for elastic rigid body without noise.
$J_{p} $ indicator function.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
--------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- -----------------------
$D_{1}$ $\boldsymbol{1.0000}$ $0.9134$ $0.8291$ $0.8260$ $0.8249$ $0.9453$
$D_{2}$ $0.9212$ $\boldsymbol{1.0000}$ $0.9245$ $0.9202$ $0.9309$ $0.9146$
$D_{3}$ $0.9260$ $0.9242$ $\boldsymbol{1.0000}$ $0.9240$ $0.9194$ $0.9351$
$D_{4}$ $0.8630$ $0.8510$ $0.9140$ $\boldsymbol{1.0000}$ $0.9101$ $0.9542$
$D_{5}$ $0.8149$ $0.8007$ $0.9492$ $0.9310$ $\boldsymbol{1.0000}$ $0.9269$
$D_{6}$ $0.9461$ $0.8147$ $0.9449$ $0.9532$ $0.9249$ $\boldsymbol{1.0000}$
: \[tab:gesture-test-pec\]Shape determination test using $J_{p}$ and $J_{s}$ for elastic rigid body without noise.
$J_{s}$ indicator function.
In the noisy case with noise level of $5\%$, the positions found in the first stage are shown in Table \[tab:location-test-pec-noise\]. the difference between the exact and estimated positions is still very small. The result of gesture recognition is shown in Table \[tab:gesture-test-pec-noise\], which clearly shows that all the correct pairs matches the best. The test with noisy data shows the robustness with respect to noisy measurement data of both the locating and recognition indicator functions $I_s$ in and $J_s$ in , respectively. This salient robustness is due to the inner product operation, which eliminates implicitly the noisy part in light of the orthogonality.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
--------------------------------------- ----------- ----------- ----------- ----------- ----------- -----------
$\mathring{z}_{0}^{1}$ $39.8738$ $39.7611$ $39.8742$ $39.8632$ $40.1075$ $39.8453$
$\mathring{z}_{0}^{2}$ $0.1307$ $0.3280$ $-0.0896$ $0.1542$ $-0.1451$ $0.1321$
$\mathring{z}_{0}^{3}$ $-0.1731$ $-0.2110$ $-0.1204$ $-0.1456$ $-0.1695$ $-0.1475$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.2509$ $0.4574$ $0.1958$ $0.2524$ $0.2477$ $0.2513$
: \[tab:location-test-pec-noise\]Location test using $I_{s}$ for elastic rigid body with noise $5\%$.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
--------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- -----------------------
$D_{1}$ $\boldsymbol{1.0000}$ $0.9453$ $0.9632$ $0.8213$ $0.9182$ $0.9649$
$D_{2}$ $0.9431$ $\boldsymbol{1.0000}$ $0.9374$ $0.8864$ $0.9205$ $0.9061$
$D_{3}$ $0.9651$ $0.9255$ $\boldsymbol{1.0000}$ $0.9213$ $0.9070$ $0.9124$
$D_{4}$ $0.8268$ $0.8811$ $0.9219$ $\boldsymbol{1.0000}$ $0.9621$ $0.9450$
$D_{5}$ $0.9152$ $0.9213$ $0.9071$ $0.9491$ $\boldsymbol{1.0000}$ $0.9378$
$D_{6}$ $0.9649$ $0.9066$ $0.9123$ $0.9459$ $0.9367$ $\boldsymbol{1.0000}$
: \[tab:gesture-test-pec-noise\]Shape determination test using $J_{s}$ for elastic rigid body with noise $5\%$.
Medium case
-----------
In the second example, we test with an inhomogeneous elastic medium among the six reference gesture domains.
$$n_{k,\,\Omega}=\begin{cases}
1 & x\in\mathbb{R}^{3}\backslash\bar{\Omega}\\
5 & x\in\Omega
\end{cases}.$$
The results of location and shape determination tests are shown, respectively, in Tables \[tab:location-test-medium\] and \[tab:gesture-test-medium\] for the noise-free case, and in Tables \[tab:location-test-medium-noise\] and \[tab:gesture-test-medium-noise\] for the noisy case with $5\%$ noise level. Both noise-free and noisy cases tell us that our locating and shape determination algorithms are very robust with noise and work very well even with data of limited aperture. Besides, the computational efforts is quite less and the shape determination schemes are very efficient only involving with inner product by known data at hand.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
--------------------------------------- ----------- ----------- ----------- ----------- ----------- -----------
$\mathring{z}_{0}^{1}$ $40.0417$ $40.0254$ $39.9576$ $40.0279$ $40.0069$ $39.9837$
$\mathring{z}_{0}^{2}$ $-0.0214$ $-0.0120$ $-0.0446$ $0.0434$ $-0.0031$ $-0.0338$
$\mathring{z}_{0}^{3}$ $0.0257$ $0.0068$ $0.0031$ $-0.0370$ $-0.0488$ $0.0294$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.0535$ $0.0289$ $0.0616$ $0.0635$ $0.0494$ $0.0477$
: \[tab:location-test-medium\]Location test using $I_{s}$ for elastic media without noise.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
--------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- -----------------------
$D_{1}$ $\boldsymbol{1.0000}$ $0.9311$ $0.8848$ $0.9393$ $0.8716$ $0.9418$
$D_{2}$ $0.9312$ $\boldsymbol{1.0000}$ $0.9174$ $0.8838$ $0.9100$ $0.9086$
$D_{3}$ $0.8834$ $0.9179$ $\boldsymbol{1.0000}$ $0.9295$ $0.9740$ $0.8854$
$D_{4}$ $0.9398$ $0.8899$ $0.9004$ $\boldsymbol{1.0000}$ $0.9200$ $0.9864$
$D_{5}$ $0.8737$ $0.9171$ $0.9422$ $0.9183$ $\boldsymbol{1.0000}$ $0.9131$
$D_{6}$ $0.9346$ $0.9087$ $0.8557$ $0.9131$ $0.9089$ $\boldsymbol{1.0000}$
: \[tab:gesture-test-medium\]Shape determination test using $J_{s}$ for elastic media without noise.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
--------------------------------------- ----------- ----------- ----------- ----------- ----------- -----------
$\mathring{z}_{0}^{1}$ $39.9758$ $39.9762$ $39.9722$ $39.9819$ $39.9586$ $39.9529$
$\mathring{z}_{0}^{2}$ $-0.0091$ $0.0103$ $-0.0383$ $-0.0076$ $-0.0238$ $0.0429$
$\mathring{z}_{0}^{3}$ $0.0095$ $0.0211$ $-0.0203$ $0.0008$ $0.0301$ $0.0230$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.0275$ $0.0334$ $0.0515$ $0.0197$ $0.0565$ $0.0677$
: \[tab:location-test-medium-noise\]Location test using $I_{s}$ for elastic media with noise $5\%$.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
--------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- -----------------------
$D_{1}$ $\boldsymbol{1.0000}$ $0.8800$ $0.9109$ $0.9321$ $0.8895$ $0.9280$
$D_{2}$ $0.8738$ $\boldsymbol{1.0000}$ $0.8666$ $0.9520$ $0.9303$ $0.9032$
$D_{3}$ $0.9105$ $0.8656$ $\boldsymbol{1.0000}$ $0.9095$ $0.8817$ $0.9021$
$D_{4}$ $0.9320$ $0.9221$ $0.9802$ $\boldsymbol{1.0000}$ $0.9160$ $0.9287$
$D_{5}$ $0.8863$ $0.9285$ $0.8819$ $0.9162$ $\boldsymbol{1.0000}$ $0.9169$
$D_{6}$ $0.9279$ $0.9030$ $0.9256$ $0.9141$ $0.9171$ $\boldsymbol{1.0000}$
: \[tab:gesture-test-medium-noise\]Shape determination test using $J_{s}$ for elastic media with noise $5\%$.
Conclusion {#sect:conclusion}
==========
We proposed and analyzed a reconstruction scheme with elastic wave detection for the nearly incompressible materials. The inverse scattering theory of elastic wave is of essential importance. We employed the translation relations between the scattering by incident elastic point source and the scattering by incident elastic plane waves. We also analysed the asymptotic amplitude of scattering shear wave and pressure wave. Besides, we presented a detailed low frequency analysis for the nearly incompressible materials. With these theoretical analysis, we proposed a two-stage reconstruction algorithm. In the first stage, we employ low frequency scattering data to locate the positions with special designed test functions, and in the second stage we employ the regular frequency scattering data to determine the shapes. We use a precomputed dictionary to store the data needed for the reconstruction algorithm, and the involved computations are mainly inner products of the corresponding data, which are very robust to noise. Various numerical experiments also show the efficiency of the proposed algorithm.
Acknowledgement {#acknowledgement .unnumbered}
===============
[The work of J. Li was supported by the NSF of China under the grants No. 11571161 and 11731006, the Shenzhen Sci-Tech Fund No. JCYJ20160530184212170 and the SUSTech startup fund. The work of H. Liu was supported by the FRG grants and startup fund from Hong Kong Baptist University, and Hong Kong RGC General Research Funds (12302415 and 12302017). H. Sun acknowledges the support of Fundamental Research Funds for the Central Universities, and the research funds of Renmin University of China (15XNLF20) and NSF of China under grant No. 11701563. ]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Recently, the author and Yamamoto invented a new proof of the duality for multiple zeta values. The technique is applicable in other series identities. In this article, we exhibit such proofs for some series identities.'
address: 'Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-Ku, Sendai, 980-8578, Japan'
author:
- 'Shin-ichiro Seki'
title: Connectors
---
Dynamic (or algorithmic) proofs
===============================
For a given series identity, the strategy to prove it in this article is as follows:
(i) Define the *connected sum* with the *connector* or the *connecting relation*.
(ii) \[sec1:2\] Check its *transport relations*.
(iii) Transport indices, algorithmically.
(iv) \[sec1:4\] Check the *boundary conditions*.
Only these!
Proofs of and are relatively easier than the proof of the original identity itself. We can often prove them by using partial fraction decompositions or telescoping sums and we leave the proofs to the reader. The hardest part in each dynamic proof is how to find a suitable connected sum.
Note that values of connected sums may diverge, however we use only convergent values when we apply them.
Notation
========
$N$ denotes a positive integer. $p$ denotes a prime number. For a positive integer $a$, we set $$J_a\coloneqq\underbrace{{\mathbb{Z}}_{\geq0}\times\cdots\times{\mathbb{Z}}_{\geq0}}_a,\quad I_a\coloneqq\underbrace{{\mathbb{Z}}_{\geq1}\times\cdots\times{\mathbb{Z}}_{\geq1}}_a,$$ $$I'_a\coloneqq\{(k_1,\dots,k_a)\in I_a : k_a\geq 2\}.$$ Further, $J_0=I_0=I'_0\coloneqq\{\varnothing\}$ and $${\mathcal{I}}\coloneqq\bigsqcup_{a=1}^{\infty}I_a,\quad {\mathcal{I}}'\coloneqq\bigsqcup_{a=1}^{\infty}I'_a,\quad{\mathcal{I}}_0\coloneqq\bigsqcup_{a=0}^{\infty}I_a,\quad {\mathcal{I}}_0'\coloneqq\bigsqcup_{a=0}^{\infty}I'_a.$$
Let $a$ be a positive integer and ${\boldsymbol{k}}=(k_1,\dots,k_a)\in I_a$. The weight of ${\boldsymbol{k}}$ is defined as ${\mathrm{wt}}({\boldsymbol{k}})\coloneqq k_1+\cdots+k_a$. The reverse index of ${\boldsymbol{k}}$ is defined as $\overline{{\boldsymbol{k}}}=(k_a,\dots,k_1)$. For $0\leq i\leq a$, ${\boldsymbol{k}}_{(i)}\coloneqq(k_1,\dots,k_i)$, ${\boldsymbol{k}}^{(i)}\coloneqq(k_{i+1},\dots,k_a)$ (${\boldsymbol{k}}_{(0)}={\boldsymbol{k}}^{(a)}=\varnothing$). Further, we define the *arrow-notation* as $$\begin{aligned}
{\boldsymbol{k}}_{\to}&\coloneqq(k_1,\dots,k_a,1),\qquad\qquad{}_{\leftarrow}{\boldsymbol{k}}\coloneqq(1,k_1,\dots,k_a),\\
{\boldsymbol{k}}_{\uparrow}&\coloneqq(k_1,\dots,k_{a-1},k_a+1),\quad{}_{\uparrow}{\boldsymbol{k}}\coloneqq(k_1+1,k_2,\dots,k_a),\\
{\boldsymbol{k}}_{\downarrow}&\coloneqq(k_1,\dots,k_{a-1},k_a-1),\quad{}_{\downarrow}{\boldsymbol{k}}\coloneqq(k_1-1,k_2,\dots,k_a).\end{aligned}$$ $\{\uparrow\}^j$ denotes $\underbrace{\uparrow\cdots\uparrow}_j$ and $\{\downarrow\}^j$ denotes $\underbrace{\downarrow\cdots\downarrow}_j$. By convention, we use ${\mathrm{wt}}(\varnothing)=0$, $\overline{\varnothing}=\varnothing$, $\varnothing_{\to}={}_{\leftarrow}\varnothing=(1)$, and $\varnothing_{\uparrow}={}_{\uparrow}\varnothing=\varnothing_{\downarrow}={}_{\downarrow}\varnothing=\varnothing$.
For ${\boldsymbol{k}}=(k_1,\dots,k_a)\in I_a$, ${\boldsymbol{l}}=(l_1,\dots,l_b)\in I_b$ ($a,b\geq 0$), $({\boldsymbol{k}},{\boldsymbol{l}})$ denotes the concatenation index $(k_1,\dots, k_a, l_1,\dots, l_b)$.
In definitions of connected sums and , we use abbreviated notation ${\boldsymbol{n}},{\boldsymbol{m}},{\boldsymbol{r}}\in{\mathcal{I}}_0$ as $${\boldsymbol{n}}=(n_1,\dots,n_a),\quad {\boldsymbol{m}}=(m_1,\dots,m_b),\quad {\boldsymbol{r}}=(r_1,\dots,r_c),$$ even if there appear extra variables $n_0$, $m_{b+1}$ and so on.
For ${\boldsymbol{k}}=(k_1,\dots,k_a)\in J_a$ and ${\boldsymbol{n}}=(n_1,\dots, n_a)\in I_a$, ${\boldsymbol{n}}^{{\boldsymbol{k}}}\coloneqq\prod_{i=1}^an_i^{k_i}$ (${\boldsymbol{n}}^{{\boldsymbol{k}}}\coloneqq1$ when $a=0$).
For ${\boldsymbol{k}}\in I_a$, we define $\zeta^{}_N({\boldsymbol{k}})$ and $\zeta^{\star}_N({\boldsymbol{k}})$ as $$\label{eq:MHS}
\zeta^{}_N({\boldsymbol{k}})\coloneqq\sum_{0=n_0< n_1<\cdots<n_a\leq N}\frac{1}{{\boldsymbol{n}}^{{\boldsymbol{k}}}},\quad \zeta^{\star}_N({\boldsymbol{k}})\coloneqq\sum_{0=n_0< n_1\leq\cdots\leq n_a\leq N}\frac{1}{{\boldsymbol{n}}^{{\boldsymbol{k}}}}.$$ For ${\boldsymbol{k}}\in {\mathcal{I}}'_0$, $\zeta({\boldsymbol{k}})\coloneqq\lim\limits_{N\to\infty}\zeta^{}_N({\boldsymbol{k}})<+\infty$. For a connected sum $Z_N$, $Z_{\infty}$ means $\lim\limits_{N\to\infty}Z_N$.
We omit to define some standard notions including the shuffle product ${\mathbin{\mathcyr{sh}}}$, the harmonic product $\ast$, the dual index ${\boldsymbol{k}}^{\dagger}$, and the Hoffman dual index ${\boldsymbol{k}}^{\vee}$. Rather, we can define these notions by the following transport relations and algorithms.
Shuffle product formula
=======================
For ${\boldsymbol{k}},{\boldsymbol{l}}\in {\mathcal{I}}'_0$, we have $$\label{eq:shuffle1}
\zeta({\boldsymbol{k}})\zeta({\boldsymbol{l}})=\sum a_{{\boldsymbol{h}}}\zeta({\boldsymbol{h}}),$$ where ${\boldsymbol{h}}\in {\mathcal{I}}'_0$ runs over indices appearing in ${\boldsymbol{k}}{\mathbin{\mathcyr{sh}}}{\boldsymbol{l}}=\sum a_{{\boldsymbol{h}}}{\boldsymbol{h}}$.
For ${\boldsymbol{k}},{\boldsymbol{l}}\in {\mathcal{I}}_0$, we have $$\label{eq:shuffle1'}
(-1)^{{\mathrm{wt}}({\boldsymbol{l}})}\zeta^{}_{p-1}({\boldsymbol{k}},\overline{{\boldsymbol{l}}})\equiv\sum a_{{\boldsymbol{h}}}\zeta^{}_{p-1}({\boldsymbol{h}})\pmod{p},$$ where ${\boldsymbol{h}}\in {\mathcal{I}}_0$ runs over indices appearing in ${\boldsymbol{k}}{\mathbin{\mathcyr{sh}}}{\boldsymbol{l}}=\sum a_{{\boldsymbol{h}}}{\boldsymbol{h}}$.
Connected sum
-------------
For ${\boldsymbol{k}}\in I_a, {\boldsymbol{l}}\in I_b, {\boldsymbol{h}}\in I_c$ ($a,b\in{\mathbb{Z}}_{\geq 0}, c\in {\mathbb{Z}}_{\geq 1}$), the connected sum $Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}})$ is defined by $$Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}})\coloneqq\sum_{\substack{0=n_0<n_1<\cdots<n_a \\ 0=m_0<m_1<\cdots<m_b \\ n_a+m_b=r_1<r_2<\cdots<r_c\leq N}}\frac{1}{{\boldsymbol{n}}^{({\boldsymbol{k}}_{\downarrow})}{\boldsymbol{m}}^{({\boldsymbol{l}}_{\downarrow})}{\boldsymbol{r}}^{({}_{\downarrow}{\boldsymbol{h}})}}.$$ The connecting relation is $n_a+m_b=r_1$. By definition, there is a symmetry $$\label{eq:shuffle2}
Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}})=Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{l}};{\boldsymbol{k}};{\boldsymbol{h}}).$$ This connected sum is essentially defined in the right-hand side of [@KMT (18)].
Transport relations
-------------------
For ${\boldsymbol{k}},{\boldsymbol{l}}\in{\mathcal{I}}_0, {\boldsymbol{h}}\in {\mathcal{I}}$, we have $$\label{eq:shuffle3}
Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}}_{\uparrow};{\boldsymbol{l}}_{\uparrow};{\boldsymbol{h}})=Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}};{\boldsymbol{l}}_{\uparrow};{}_{\uparrow}{\boldsymbol{h}})+Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}}_{\uparrow};{\boldsymbol{l}};{}_{\uparrow}{\boldsymbol{h}})\quad ({\boldsymbol{k}},{\boldsymbol{l}}\neq\varnothing),$$ $$\label{eq:shuffle4}
Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}}_{\to};{\boldsymbol{l}};{}_{\uparrow}{\boldsymbol{h}})=Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}}_{\uparrow};{\boldsymbol{l}};{}_{\leftarrow}{\boldsymbol{h}}).$$
Algorithm {#sh-alg}
---------
For each $Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}})$, we set rules as follows:
(i) \[sec3:1\] If ${\boldsymbol{k}}$ and ${\boldsymbol{l}}\in{\mathcal{I}}'$, then use the transport relation .
(ii) \[sec3:2\] If ${\boldsymbol{l}}\in{\mathcal{I}}_0\setminus{\mathcal{I}}'$, then use the symmetry .
(iii) \[sec3:3\] If ${\boldsymbol{k}}\in{\mathcal{I}}\setminus{\mathcal{I}}'$, then use the transport relation .
(iv) \[sec3:4\] If ${\boldsymbol{k}}=\varnothing$, then stop.
Start from $Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}}_{\uparrow};{\boldsymbol{l}}_{\uparrow};(1))$ (${\boldsymbol{k}},{\boldsymbol{l}}\in{\mathcal{I}}_0$) and transport indices according to the above rules until the algorithm stops for all connected sums. Then, we have an identity which has the following form (${\boldsymbol{h}},{\boldsymbol{h}}'\in{\mathcal{I}}_0$): $$\label{eq:shuffle5}
Z_N^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}}_{\uparrow};{\boldsymbol{l}}_{\uparrow};(1))=\sum Z_N^{{\mathbin{\mathcyr{sh}}}}(\varnothing;{\boldsymbol{h}}_{\uparrow};{}_{\leftarrow}{\boldsymbol{h}}').$$
Boundary conditions
-------------------
For ${\boldsymbol{k}},{\boldsymbol{l}}\in{\mathcal{I}}'_0$, we have $$\label{eq:shuffle6}
Z_{\infty}^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}}_{\uparrow};{\boldsymbol{l}}_{\uparrow};(1))=\zeta({\boldsymbol{k}})\zeta({\boldsymbol{l}})$$ and $$Z_N^{{\mathbin{\mathcyr{sh}}}}(\varnothing;{\boldsymbol{h}}_{\uparrow};{}_{\leftarrow}{\boldsymbol{h}}')=\zeta^{}_N({\boldsymbol{h}},{\boldsymbol{h}}').$$ By these boundary conditions, we see that the case $N\to\infty$ of the identity is nothing but the shuffle product formula .
When ${\boldsymbol{k}},{\boldsymbol{l}}\in{\mathcal{I}}_0$, by using the following congruence instead of using the boundary condition , we can also prove the shuffle relation for finite multiple zeta values : $$Z_{p-1}^{{\mathbin{\mathcyr{sh}}}}({\boldsymbol{k}}_{\uparrow};{\boldsymbol{l}}_{\uparrow};(1))\equiv (-1)^{{\mathrm{wt}}({\boldsymbol{l}})}\zeta^{}_{p-1}({\boldsymbol{k}},\overline{{\boldsymbol{l}}})\pmod{p}.$$
Example
-------
By applying the algorithm in \[sh-alg\] to $Z_N^{{\mathbin{\mathcyr{sh}}}}((1,2);(2);(1))$, we have $$\begin{aligned}
&Z_N^{{\mathbin{\mathcyr{sh}}}}((1,2);(2);(1))\stackrel{\eqref{sec3:1}}{=}Z_N^{{\mathbin{\mathcyr{sh}}}}((1,1);(2);(2))+Z_N^{{\mathbin{\mathcyr{sh}}}}((1,2);(1);(2)),\\
&\qquad Z_N^{{\mathbin{\mathcyr{sh}}}}((1,1);(2);(2))\stackrel{\eqref{sec3:3}}{=}Z_N^{{\mathbin{\mathcyr{sh}}}}((2);(2);(1,1))\stackrel{\eqref{sec3:1}}{=}Z_N^{{\mathbin{\mathcyr{sh}}}}((1);(2);(2,1))+Z_N^{{\mathbin{\mathcyr{sh}}}}((2);(1);(2,1)),\\
&\qquad\qquad Z_N^{{\mathbin{\mathcyr{sh}}}}((2);(1);(2,1))\stackrel{\eqref{sec3:2}}{=}Z_N^{{\mathbin{\mathcyr{sh}}}}((1);(2);(2,1))\stackrel{\eqref{sec3:3}}{=}Z_N^{{\mathbin{\mathcyr{sh}}}}(\varnothing;(2);(1,1,1)),\\
&\qquad Z_N^{{\mathbin{\mathcyr{sh}}}}((1,2);(1);(2))\stackrel{\eqref{sec3:2}}{=}Z_N^{{\mathbin{\mathcyr{sh}}}}((1);(1,2);(2))\stackrel{\eqref{sec3:3}}{=}Z_N^{{\mathbin{\mathcyr{sh}}}}(\varnothing;(1,2);(1,1)).\end{aligned}$$ Thus, $$Z_N^{{\mathbin{\mathcyr{sh}}}}((1,1)_{\uparrow};(1)_{\uparrow};(1))=2Z_N^{{\mathbin{\mathcyr{sh}}}}(\varnothing;(1)_{\uparrow};{}_{\leftarrow}(1,1))+Z_N^{{\mathbin{\mathcyr{sh}}}}(\varnothing;(1,1)_{\uparrow};{}_{\leftarrow}(1))$$ and this corresponds to $$(1,1){\mathbin{\mathcyr{sh}}}(1)=2((1),(1,1))+((1,1),(1))=3(1,1,1).$$
Harmonic product formula
========================
For ${\boldsymbol{k}},{\boldsymbol{l}}\in {\mathcal{I}}_0$, we have $$\label{eq:harmonic1}
\zeta^{}_N({\boldsymbol{k}})\zeta^{}_N({\boldsymbol{l}})=\sum b_{{\boldsymbol{h}}}\zeta^{}_N({\boldsymbol{h}}),$$ where ${\boldsymbol{h}}\in {\mathcal{I}}_0$ runs over indices appearing in ${\boldsymbol{k}}\ast{\boldsymbol{l}}=\sum b_{{\boldsymbol{h}}}{\boldsymbol{h}}$.
Connected sum
-------------
For ${\boldsymbol{k}}\in I_a, {\boldsymbol{l}}\in I_b, {\boldsymbol{h}}\in I_c$ ($a,b\in{\mathbb{Z}}_{\geq 1}, c\in{\mathbb{Z}}_{\geq 0}$), the connected sum $Z_N^{\ast}({\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}})$ is defined by $$Z_N^{\ast}({\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}})\coloneqq\sum_{\substack{1=r_0\leq r_1<\cdots<r_c \\ r_c=n_1<\cdots<n_a\leq N \\ r_c=m_1<\cdots<m_b\leq N}}\frac{1}{{\boldsymbol{n}}^{({}_{\downarrow}{\boldsymbol{k}})}{\boldsymbol{m}}^{({}_{\downarrow}{\boldsymbol{l}})}{\boldsymbol{r}}^{{\boldsymbol{h}}}}.$$ The connecting relation is $n_1=m_1=r_c$. By definition, there is a symmetry $$\label{eq:harmonic2}
Z_N^{\ast}({\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}})=Z_N^{\ast}({\boldsymbol{l}};{\boldsymbol{k}};{\boldsymbol{h}}).$$ The definition of this connected sum is essentially due to Hirose.
Transport relations
-------------------
For ${\boldsymbol{k}},{\boldsymbol{l}}\in{\mathcal{I}}, {\boldsymbol{h}}\in {\mathcal{I}}_0$, we have $$\label{eq:harmonic3}
Z_N^{\ast}({}_{\leftarrow}{\boldsymbol{k}};{}_{\leftarrow}{\boldsymbol{l}};{\boldsymbol{h}})=
Z_N^{\ast}({\boldsymbol{k}};{}_{\leftarrow}{\boldsymbol{l}};{\boldsymbol{h}}_{\to})
+Z_N^{\ast}({}_{\leftarrow}{\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}}_{\to})
+Z_N^{\ast}({\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}}_{\to\uparrow}),$$ $$\label{eq:harmonic4}
Z_N^{\ast}({}_{\uparrow}{\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}})=Z_N^{\ast}({\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}}_{\uparrow}).$$
Algorithm {#har-alg}
---------
For each $Z_N^{\ast}({\boldsymbol{k}};{\boldsymbol{l}};{\boldsymbol{h}})$, we set rules as follows:
(i) \[sec4:1\] If $\overline{{\boldsymbol{k}}}$ and $\overline{{\boldsymbol{l}}}\in{\mathcal{I}}\setminus({\mathcal{I}}'\cup\{(1)\})$, then use the transport relation .
(ii) \[sec4:2\] If $\overline{{\boldsymbol{l}}}\in{\mathcal{I}}'\cup\{( 1)\}$, then use the symmetry .
(iii) \[sec4:3\] If $\overline{{\boldsymbol{k}}}\in{\mathcal{I}}'$, then use the transport relation .
(iv) \[sec4:4\] If ${\boldsymbol{k}}=(1)$, then stop.
Start from $Z_N^{\ast}({}_{\leftarrow}{\boldsymbol{k}};{}_{\leftarrow}{\boldsymbol{l}};\varnothing)$ (${\boldsymbol{k}},{\boldsymbol{l}}\in{\mathcal{I}}_0$) and transport indices according to the above rules until the algorithm stops for all connected sums. Then, we have an identity which has the following form (${\boldsymbol{h}},{\boldsymbol{h}}'\in{\mathcal{I}}_0, h\in{\mathbb{Z}}_{\geq 0}$): $$\label{eq:harmonic5}
Z_N^{\ast}({}_{\leftarrow}{\boldsymbol{k}};{}_{\leftarrow}{\boldsymbol{l}};\varnothing)=\sum Z_N^{\ast}((1);{}_{\{\uparrow\}^h\leftarrow}{\boldsymbol{h}};{\boldsymbol{h}}').$$
Boundary conditions
-------------------
For ${\boldsymbol{k}},{\boldsymbol{l}},{\boldsymbol{h}},{\boldsymbol{h}}'\in{\mathcal{I}}_0, h\in{\mathbb{Z}}_{\geq 0}$, we have $$Z_N^{\ast}({}_{\leftarrow}{\boldsymbol{k}};{}_{\leftarrow}{\boldsymbol{l}};\varnothing)=\zeta^{}_N({\boldsymbol{k}})\zeta^{}_N({\boldsymbol{l}})$$ and $$Z_N^{\ast}((1);{}_{\{\uparrow\}^h\leftarrow}{\boldsymbol{h}};{\boldsymbol{h}}')=\zeta^{}_N({\boldsymbol{h}}'_{\{\uparrow\}^{h}},{\boldsymbol{h}}).$$ By these boundary conditions, we see that the identity is nothing but the harmonic product formula .
Example
-------
By applying the algorithm in \[har-alg\] to $Z_N^{\ast}((1,1);(1,2);\varnothing)$, we have $$\begin{aligned}
&Z_N^{\ast}((1,1);(1,2);\varnothing)\stackrel{\eqref{sec4:1}}{=}Z_N^{\ast}((1);(1,2);(1))+Z_N^{\ast}((1,1);(2);(1))+Z_N^{\ast}((1);(2);(2)),\\
&\qquad Z_N^{\ast}((1,1);(2);(1))\stackrel{\eqref{sec4:2}}{=}Z_N^{\ast}((2);(1,1);(1))\stackrel{\eqref{sec4:3}}{=}Z_N^{\ast}((1);(1,1);(2)).\end{aligned}$$ Thus, $$Z_N^{\ast}({}_{\leftarrow}(1);{}_{\leftarrow}(2);\varnothing)=Z_N^{\ast}((1);{}_{\leftarrow}(2);(1))+Z_N^{\ast}((1);{}_{\leftarrow}(1);(2))+Z_N^{\ast}((1);{}_{\uparrow\leftarrow}\varnothing;(2))$$ and this corresponds to $$(1)\ast(2)=(1,2)+(2,1)+(3).$$
By applying the algorithm in \[har-alg\] to $Z_N^{\ast}((1,1,1);(1,1);\varnothing)$, we have $$\begin{aligned}
&Z_N^{\ast}((1,1,1);(1,1);\varnothing)\stackrel{\eqref{sec4:1}}{=}Z_N^{\ast}((1,1);(1,1);(1))+Z_N^{\ast}((1,1,1);(1);(1))+Z_N^{\ast}((1,1);(1);(2)),\\
&\quad Z_N^{\ast}((1,1);(1,1);(1))\stackrel{\eqref{sec4:1}}{=}Z_N^{\ast}((1);(1,1);(1,1))+Z_N^{\ast}((1,1);(1);(1,1))+Z_N^{\ast}((1);(1);(1,2)),\\
&\quad\quad Z_N^{\ast}((1,1);(1);(1,1))\stackrel{\eqref{sec4:2}}{=}Z_N^{\ast}((1);(1,1);(1,1)),\\
&\quad Z_N^{\ast}((1,1,1);(1);(1))\stackrel{\eqref{sec4:2}}{=}Z_N^{\ast}((1);(1,1,1);(1)),\\
&\quad Z_N^{\ast}((1,1);(1);(2))\stackrel{\eqref{sec4:2}}{=}Z_N^{\ast}((1);(1,1);(2)).\end{aligned}$$ Thus, $$\begin{aligned}
Z_N^{\ast}({}_{\leftarrow}(1,1);{}_{\leftarrow}(1);\varnothing)&=2Z_N^{\ast}((1);{}_{\leftarrow}(1);(1,1))+Z_N^{\ast}((1);{}_{\leftarrow}\varnothing;(1,2))+Z_N^{\ast}((1);{}_{\leftarrow}(1,1);(1))\\
&\qquad+Z_N^{\ast}((1);{}_{\leftarrow}(1);(2))\end{aligned}$$ and this corresponds to $$(1,1)\ast(1)=2((1,1),(1))+(1,2)+((1),(1,1))+(2,1)=3(1,1,1)+(1,2)+(2,1).$$
Duality
=======
Let $n,m$ be non-negative integers. For ${\boldsymbol{k}}\in I'_a$, we define the *multiple zeta value with double tails* $\zeta^{}_{n,m}({\boldsymbol{k}})$ by $$\zeta^{}_{n,m}({\boldsymbol{k}})\coloneqq\sum_{n=n_0< n_1<\cdots<n_a}\frac{1}{{\boldsymbol{n}}^{{\boldsymbol{k}}}}\cdot\frac{1}{\binom{n_a+m}{m}}.$$
For any ${\boldsymbol{k}}\in{\mathcal{I}}'$, $$\label{eq:duality1}
\zeta^{}_{n,m}({\boldsymbol{k}})=\zeta^{}_{m,n}({\boldsymbol{k}}^{\dagger}).$$
The case $n=m=0$ is the usual duality relation for multiple zeta values.
Connected sum
-------------
For ${\boldsymbol{k}}, {\boldsymbol{l}}\in{\mathcal{I}}_0$, the connected sum $Z_{n,m}^{\text{D}}({\boldsymbol{k}};{\boldsymbol{l}})$ ([@SY1]) is defined by $$Z_{n,m}^{\text{D}}({\boldsymbol{k}};{\boldsymbol{l}})\coloneqq\sum_{\substack{n=n_0<n_1<\cdots<n_a \\ m=m_0<m_1<\cdots<m_b}}\frac{1}{{\boldsymbol{n}}^{{\boldsymbol{k}}}}\cdot C^{\text{D}}(n_a,m_b)\cdot\frac{1}{{\boldsymbol{m}}^{{\boldsymbol{l}}}}.$$ Here, the connector is $$\label{con}
C^{\text{D}}(n_a,m_b)\coloneqq\frac{n_a!\cdot m_b!}{(n_a+m_b)!}.$$ By definition, there is a symmetry $$\label{eq:duality2}
Z_{n,m}^{\text{D}}({\boldsymbol{k}};{\boldsymbol{l}})=Z_{m,n}^{\text{D}}({\boldsymbol{l}};{\boldsymbol{k}}).$$
Transport relations
-------------------
For ${\boldsymbol{k}},{\boldsymbol{l}}\in{\mathcal{I}}_0$, we have $$\begin{aligned}
Z_{n,m}^{\text{D}}({\boldsymbol{k}}_{\uparrow};{\boldsymbol{l}})&=Z_{n,m}^{\text{D}}({\boldsymbol{k}};{\boldsymbol{l}}_{\to})\quad ({\boldsymbol{k}}\neq\varnothing),\label{eq:duality3}\\
Z_{n,m}^{\text{D}}({\boldsymbol{k}}_{\to};{\boldsymbol{l}})&=Z_{n,m}^{\text{D}}({\boldsymbol{k}};{\boldsymbol{l}}_{\uparrow})\quad ({\boldsymbol{l}}\neq\varnothing).\label{eq:duality4}\end{aligned}$$
Algorithm {#D-alg}
---------
For each $Z_{n,m}^{\text{D}}({\boldsymbol{k}};{\boldsymbol{l}})$, we set rules as follows:
(i) \[sec5:1\] If ${\boldsymbol{k}}\in{\mathcal{I}}'$, then use the transport relation .
(ii) \[sec5:2\] If ${\boldsymbol{k}}\in{\mathcal{I}}\setminus{\mathcal{I}}'$, then use the transport relation .
(iii) \[sec5:3\] If ${\boldsymbol{k}}=\varnothing$, then stop.
Start from $Z_{n,m}^{\text{D}}({\boldsymbol{k}};\varnothing)$ (${\boldsymbol{k}}\in{\mathcal{I}}'$) and transport indices from left to right according to the above rules until the algorithm stops. Then, we have $$\label{eq:duality5}
Z_{n,m}^{\text{D}}({\boldsymbol{k}};\varnothing)=Z_{n,m}^{\text{D}}(\varnothing;{\boldsymbol{k}}^{\dagger}).$$
Boundary conditions
-------------------
For ${\boldsymbol{k}}\in{\mathcal{I}}'$, $$Z_{n,m}^{\text{D}}({\boldsymbol{k}};\varnothing)=\zeta^{}_{n,m}({\boldsymbol{k}})$$ holds by definition. Therefore, by the symmetry , we see that the identity is nothing but the duality .
Example
-------
By applying the algorithm in \[D-alg\] to $Z_{n,m}^{\text{D}}((3,2);\varnothing)$, we have $$\begin{aligned}
Z_{n,m}^{\text{D}}((3,2);\varnothing)&=Z_{n,m}^{\text{D}}(\varnothing_{\to\uparrow\uparrow\to\uparrow};\varnothing)\\
&\stackrel{\eqref{sec5:1}}{=}Z_{n,m}^{\text{D}}(\varnothing_{\to\uparrow\uparrow\to};\varnothing_{\to}) \ =Z_{n,m}^{\text{D}}((3,1);(1))&\\
&\stackrel{\eqref{sec5:2}}{=}Z_{n,m}^{\text{D}}(\varnothing_{\to\uparrow\uparrow};\varnothing_{\to\uparrow}) \ \, =Z_{n,m}^{\text{D}}((3);(2))&\\
&\stackrel{\eqref{sec5:1}}{=}Z_{n,m}^{\text{D}}(\varnothing_{\to\uparrow};\varnothing_{\to\uparrow\to}) \ =Z_{n,m}^{\text{D}}((2);(2,1))&\\
&\stackrel{\eqref{sec5:1}}{=}Z_{n,m}^{\text{D}}(\varnothing_{\to};\varnothing_{\to\uparrow\to\to}) =Z_{n,m}^{\text{D}}((1);(2,1,1))&\\
&\stackrel{\eqref{sec5:2}}{=}Z_{n,m}^{\text{D}}(\varnothing;\varnothing_{\to\uparrow\to\to\uparrow}) \; =Z_{n,m}^{\text{D}}(\varnothing;(2,1,2))&\end{aligned}$$ and this proves $$\zeta^{}_{n,m}(3,2)=\zeta^{}_{m,n}(2,1,2).$$
Hoffman’s identity
==================
For ${\boldsymbol{k}}\in I_a$ $(a\geq 1)$, $$\label{eq:HD1}
H_N^{\star}({\boldsymbol{k}})\coloneqq\sum_{1\leq n_1\leq\cdots\leq n_a\leq N}\frac{(-1)^{n_a-1}}{{\boldsymbol{n}}^{{\boldsymbol{k}}}}\binom{N}{n_a}=\zeta_N^{\star}({\boldsymbol{k}}^{\vee}).$$
Connected sum
-------------
For ${\boldsymbol{k}}\in{\mathcal{I}}$ and ${\boldsymbol{l}}\in{\mathcal{I}}_0$, the connected sum $Z_N^{\text{HD}}({\boldsymbol{k}};{\boldsymbol{l}})$ ([@SY2]) is defined by $$Z_N^{\text{HD}}({\boldsymbol{k}};{\boldsymbol{l}})\coloneqq\sum_{1\leq n_1\leq\cdots\leq n_a\leq m_1\leq \cdots \leq m_b\leq m_{b+1}=N}\frac{1}{{\boldsymbol{n}}^{({\boldsymbol{k}}_{\downarrow})}}\cdot C^{\text{HD}}(n_a,m_1)\cdot\frac{1}{{\boldsymbol{m}}^{{\boldsymbol{l}}}}.$$ Here, the connector is $$C^{\text{HD}}(n_a,m_1)\coloneqq(-1)^{n_a-1}\binom{m_1}{n_a}$$ and the connecting relation is $n_a\leq m_1$.
Transport relations
-------------------
For ${\boldsymbol{k}}\in{\mathcal{I}}$ and ${\boldsymbol{l}}\in{\mathcal{I}}_0$, we have $$\begin{aligned}
Z_N^{\text{HD}}({\boldsymbol{k}}_{\uparrow};{\boldsymbol{l}})&=Z_N^{\text{HD}}({\boldsymbol{k}};{}_{\leftarrow}{\boldsymbol{l}}),\label{eq:HD2}\\
Z_N^{\text{HD}}({\boldsymbol{k}}_{\to};{\boldsymbol{l}})&=Z_N^{\text{HD}}({\boldsymbol{k}};{}_{\uparrow}{\boldsymbol{l}})\quad ({\boldsymbol{l}}\neq\varnothing).\label{eq:HD3}\end{aligned}$$
Algorithm {#HD-alg}
---------
For each $Z_N^{\text{HD}}({\boldsymbol{k}};{\boldsymbol{l}})$, we set rules as follows:
(i) \[sec6:1\] If ${\boldsymbol{k}}\in{\mathcal{I}}'$, then use the transport relation .
(ii) \[sec6:2\] If ${\boldsymbol{k}}\in{\mathcal{I}}\setminus{\mathcal{I}}'$ and ${\boldsymbol{k}}\neq (1)$, then use the transport relation .
(iii) \[sec6:3\] If ${\boldsymbol{k}}=(1)$, then stop.
Start from $Z_N^{\text{HD}}({\boldsymbol{k}}_{\uparrow};\varnothing)$ (${\boldsymbol{k}}\in{\mathcal{I}}$) and transport indices from left to right according to the above rules until the algorithm stops. Then, we have $$\label{eq:HD4}
Z_N^{\text{HD}}({\boldsymbol{k}}_{\uparrow};\varnothing)=Z_N^{\text{HD}}((1);{\boldsymbol{k}}^{\vee}).$$
Boundary conditions
-------------------
For ${\boldsymbol{k}}\in{\mathcal{I}}$, we have $$Z_N^{\text{HD}}({\boldsymbol{k}}_{\uparrow};\varnothing)=H_N^{\star}({\boldsymbol{k}})$$ and $$Z_N^{\text{HD}}((1);{\boldsymbol{k}})=\zeta^{\star}_N({\boldsymbol{k}}).$$ Therefore, we see that the identity is nothing but Hoffman’s identity .
Example
-------
By applying the algorithm in \[HD-alg\] to $Z_{N}^{\text{HD}}((3,3);\varnothing)$, we have $$\begin{aligned}
Z_{N}^{\text{HD}}((3,2)_{\uparrow};\varnothing)&=Z_{N}^{\text{HD}}((1)_{\uparrow\uparrow\to\uparrow\uparrow};\varnothing)\\
&\stackrel{\eqref{sec6:1}}{=}Z_{N}^{\text{HD}}((1)_{\uparrow\uparrow\to\uparrow};{}_{\leftarrow}\varnothing) \ \, \, =Z_{N}^{\text{HD}}((3,2);(1))\\
&\stackrel{\eqref{sec6:1}}{=}Z_{N}^{\text{HD}}((1)_{\uparrow\uparrow\to};{}_{\leftarrow\leftarrow}\varnothing) \ =Z_{N}^{\text{HD}}((3,1);(1,1))\\
&\stackrel{\eqref{sec6:2}}{=}Z_{N}^{\text{HD}}((1)_{\uparrow\uparrow};{}_{\uparrow\leftarrow\leftarrow}\varnothing) \ \, =Z_{N}^{\text{HD}}((3);(2,1))\\
&\stackrel{\eqref{sec6:1}}{=}Z_{N}^{\text{HD}}((1)_{\uparrow};{}_{\leftarrow\uparrow\leftarrow\leftarrow}\varnothing) \ =Z_{N}^{\text{HD}}((2);(1,2,1))\\
&\stackrel{\eqref{sec6:1}}{=}Z_{N}^{\text{HD}}((1);{}_{\leftarrow\leftarrow\uparrow\leftarrow\leftarrow}\varnothing)=Z_{N}^{\text{HD}}((1);(1,1,2,1))\end{aligned}$$ and this proves $$H_N^{\star}(3,2)=\zeta^{\star}_N(1,1,2,1).$$
Cyclic sum formula {#sec:CS}
==================
For a cyclic equivalent class $\alpha$ of an element of ${\mathcal{I}}'$, we have $$\label{eq:CS1}
\sum_{{\boldsymbol{k}}\in\alpha}\zeta({\boldsymbol{k}}_{\uparrow})=\sum_{{\boldsymbol{k}}\in\alpha}\sum_{j=0}^{k_a-2}\zeta({}_{\{\uparrow\}^j\leftarrow}{\boldsymbol{k}}_{\{\downarrow\}^{j}}),$$ where ${\boldsymbol{k}}=(k_1,\dots,k_a)$.
For a cyclic equivalent class $\alpha$ of an element of ${\mathcal{I}}$, we have $$\label{eq:CS1'}
\sum_{{\boldsymbol{k}}\in\alpha}\left(\zeta^{}_{p-1}({\boldsymbol{k}}_{\uparrow})+\zeta^{}_{p-1}({}_{\uparrow}{\boldsymbol{k}}^{\sigma})+\zeta^{}_{p-1}({}_{\leftarrow}{\boldsymbol{k}}^{\sigma})\right)\equiv \sum_{{\boldsymbol{k}}\in\alpha}\sum_{j=0}^{k_a-2}\zeta^{}_{p-1}({}_{\{\uparrow\}^j\leftarrow}{\boldsymbol{k}}_{\{\downarrow\}^{j}})\pmod{p},$$ where ${\boldsymbol{k}}=(k_1,\dots,k_a)$ and ${\boldsymbol{k}}^{\sigma}=(k_a,k_1,\dots, k_{a-1})$.
Connected sum
-------------
For ${\boldsymbol{k}}\in{\mathcal{I}}$, the connected sum $Z_N^{\text{O}}({\boldsymbol{k}})$ is defined by $$Z_N^{\text{O}}({\boldsymbol{k}})\coloneqq\sum_{0<n_1<\cdots<n_a\leq N}\frac{1}{{\boldsymbol{n}}^{({}_{\downarrow}{\boldsymbol{k}})}}\cdot C^{\text{O}}(n_1,n_a).$$ Here, the connector discovered by Ohno is $$C^{\text{O}}(n_1,n_a)\coloneqq\frac{1}{n_a-n_1}.$$
Transport relations
-------------------
For ${\boldsymbol{k}}\in{\mathcal{I}}$, we have $$\begin{aligned}
Z_N^{\text{O}}({\boldsymbol{k}}_{\uparrow})&=Z_N^{\text{O}}({}_{\uparrow}{\boldsymbol{k}})-\zeta^{}_N({\boldsymbol{k}}_{\uparrow}),\label{eq:CS2}\\
Z_{\infty}^{\text{O}}({\boldsymbol{k}}_{\to})&=Z_{\infty}^{\text{O}}({}_{\leftarrow}{\boldsymbol{k}})+\zeta({\boldsymbol{k}}_{\uparrow}),\label{eq:CS3}\\
Z_{p-1}^{\text{O}}({\boldsymbol{k}}_{\to})&\equiv Z_{p-1}^{\text{O}}({}_{\leftarrow}{\boldsymbol{k}})+\zeta^{}_{p-1}({\boldsymbol{k}}_{\uparrow})+\zeta^{}_{p-1}({}_{\uparrow}{\boldsymbol{k}})+\zeta^{}_{p-1}({}_{\leftarrow}{\boldsymbol{k}})\pmod{p}.\label{eq:CS4}\end{aligned}$$
Algorithm {#C-alg}
---------
For each $Z_{\infty}^{\text{O}}({\boldsymbol{k}})$ with ${\boldsymbol{k}}\neq (1)$, we set rules as follows:
(i) \[sec7:1\] If ${\boldsymbol{k}}\in{\mathcal{I}}'$, then use the case $N\to\infty$ of the transport relation .
(ii) \[sec7:2\] If ${\boldsymbol{k}}\in{\mathcal{I}}\setminus{\mathcal{I}}'$, then use the transport relation .
Start from $Z_{\infty}^{\text{O}}({}_{\leftarrow}{\boldsymbol{k}})$ (${\boldsymbol{k}}\in{\mathcal{I}}'$) and use transport relations according to the above rules until the first value $Z_{\infty}^{\text{O}}({}_{\leftarrow}{\boldsymbol{k}})$ appears again. Then, we have $$\begin{gathered}
Z_{\infty}^{\text{O}}({}_{\leftarrow}{\boldsymbol{k}})=Z_{\infty}^{\text{O}}({}_{\leftarrow}{\boldsymbol{k}})+(\text{L.H.S of \eqref{eq:CS1} for the class $[{\boldsymbol{k}}]$})\\
-(\text{R.H.S of \eqref{eq:CS1} for the class $[{\boldsymbol{k}}]$}).\end{gathered}$$
Boundary conditions
-------------------
We need the fact that the connected sum $Z_{\infty}^{\text{O}}({\boldsymbol{k}})$ for an index ${\boldsymbol{k}}$ one of whose component is greater than $1$ converges instead of boundary conditions (see [@HO Theorem 3.1]).
For ${\boldsymbol{k}}\in{\mathcal{I}}$, we can also prove the cyclic sum formula for finite multiple zeta values by using the transport relation instead of using the transport relation in the above algorithm.
Example
-------
For ${\boldsymbol{k}}=(k_1,\dots,k_a)\in I_a$ $(a\geq 1)$, we set $$S({\boldsymbol{k}})\coloneqq\sum_{j=0}^{k_a-2}\zeta({}_{\{\uparrow\}^j\leftarrow}{\boldsymbol{k}}_{\{\downarrow\}^{j}}).$$ Here, $S({\boldsymbol{k}})=0$ when $k_a=1$. By applying the algorithm in \[C-alg\] to $Z_{\infty}^{\text{O}}(1,2,1,3)$, we have $$\begin{aligned}
&Z_{\infty}^{\text{O}}({}_{\leftarrow}(2,1,3))\\
&\stackrel{\eqref{sec7:1}}{=}Z_{\infty}^{\text{O}}({}_{\uparrow\leftarrow}(2,1,3)_{\downarrow})-\zeta({}_{\leftarrow}(2,1,3))\\
&\stackrel{\eqref{sec7:1}}{=}Z_{\infty}^{\text{O}}({}_{\uparrow\uparrow\leftarrow}(2,1,3)_{\downarrow\downarrow})-\zeta^{}({}_{\uparrow\leftarrow}(2,1,3)_{\downarrow})-\zeta({}_{\leftarrow}(2,1,3))\\
&\stackrel{\eqref{sec7:2}}{=}Z_{\infty}^{\text{O}}({}_{\leftarrow}(3,2,1))+\zeta((3,2,1)_{\uparrow})-S(2,1,3)\\
&\stackrel{\eqref{sec7:2}}{=}Z_{\infty}^{\text{O}}({}_{\leftarrow}(1,3,2))+\zeta((1,3,2)_{\uparrow})+\zeta((3,2,1)_{\uparrow})-S(2,1,3)\\
&\stackrel{\eqref{sec7:1}}{=}Z_{\infty}^{\text{O}}({}_{\uparrow\leftarrow}(1,3,2)_{\downarrow})-\zeta({}_{\leftarrow}(1,3,2))+\zeta((1,3,2)_{\uparrow})+\zeta((3,2,1)_{\uparrow})-S(2,1,3)\\
&\stackrel{\eqref{sec7:2}}{=}Z_{\infty}^{\text{O}}({}_{\leftarrow}(2,1,3))+\zeta((2,1,3)_{\uparrow})-S(1,3,2)+\zeta((1,3,2)_{\uparrow})+\zeta((3,2,1)_{\uparrow})-S(2,1,3).\end{aligned}$$ This proves $$\zeta((2,1,3)_{\uparrow})+\zeta((1,3,2)_{\uparrow})+\zeta((3,2,1)_{\uparrow})=S(2,1,3)+S(1,3,2)+S(3,2,1),$$ that is, $$\zeta(2,1,4)+\zeta(1,3,3)+\zeta(3,2,2)=\zeta(1,2,1,3)+\zeta(2,2,1,2)+\zeta(1,1,3,2).$$
Hoffman’s relation
==================
For ${\boldsymbol{k}}=(k_1,\dots,k_a)\in I'_a$ $(a\geq 1)$, $$\label{eq:H1}
\sum_{i=0}^{a-1}\zeta({\boldsymbol{k}}_{(i)},{}_{\uparrow}{\boldsymbol{k}}^{(i)})=\sum_{i=1}^{a}\sum_{j=1}^{k_i-1}\zeta(({\boldsymbol{k}}_{(i)})_{\{\downarrow\}^j},{}_{\{\uparrow\}^j\leftarrow}{\boldsymbol{k}}^{(i)}).$$
Connected sum
-------------
For ${\boldsymbol{k}}\in I_a$, ${\boldsymbol{l}}\in I_b$ ($a,b\geq 1$), the connected sum $Z^{\text{H}}({\boldsymbol{k}};{\boldsymbol{l}})$ is defined by $$Z^{\text{H}}({\boldsymbol{k}};{\boldsymbol{l}})\coloneqq\sum_{0<n_1<\cdots<n_a<m_1<\cdots<m_b}\frac{1}{{\boldsymbol{n}}^{({\boldsymbol{k}}_{\downarrow})}}\cdot C^{\text{H}}(n_a,m_1)\cdot\frac{1}{{\boldsymbol{m}}^{{\boldsymbol{l}}}}.$$ Here, the connector is $$C^{\text{H}}(n_a,m_1)\coloneqq\frac{1}{m_1-n_a}$$ and the connecting relation is $n_a<m_1$.
Transport relations
-------------------
For ${\boldsymbol{k}}\in{\mathcal{I}}$, ${\boldsymbol{l}}\in{\mathcal{I}}'\cup\{(1)\}$, we have $$\begin{aligned}
Z^{\text{H}}({\boldsymbol{k}}_{\uparrow};{\boldsymbol{l}})&=Z^{\text{H}}({\boldsymbol{k}};{}_{\uparrow}{\boldsymbol{l}})+\zeta({\boldsymbol{k}},{}_{\uparrow}{\boldsymbol{l}}),\label{eq:H2}\\
Z^{\text{H}}({\boldsymbol{k}}_{\to};{\boldsymbol{l}})&=Z^{\text{H}}({\boldsymbol{k}};{}_{\leftarrow}{\boldsymbol{l}})+\zeta({\boldsymbol{k}},{}_{\leftarrow}{\boldsymbol{l}})\quad ({\boldsymbol{l}}\neq (1)).\label{eq:H3}\end{aligned}$$
Algorithm {#H-alg}
---------
For each $Z^{\text{H}}({\boldsymbol{k}};{\boldsymbol{l}})$, we set rules as follows:
(i) \[sec8:1\] If ${\boldsymbol{k}}\in{\mathcal{I}}'$, then use the transport relation .
(ii) \[sec8:2\] If ${\boldsymbol{k}}\in{\mathcal{I}}\setminus{\mathcal{I}}'$ and ${\boldsymbol{k}}\neq(1)$, then use the transport relation .
(iii) \[sec8:3\] If ${\boldsymbol{k}}=(1)$, then stop.
Start from $Z^{\text{H}}({\boldsymbol{k}};(1))$ (${\boldsymbol{k}}\in{\mathcal{I}}'$) and transport indices from left to right according to the above rules until the algorithm stops. Then, we have $$\label{eq:H4}
Z^{\text{H}}({\boldsymbol{k}};(1))=Z^{\text{H}}((1);{\boldsymbol{k}})+\sum_{i=1}^{a}\sum_{j=1}^{k_i-1}\zeta(({\boldsymbol{k}}_{(i)})_{\{\downarrow\}^j},{}_{\{\uparrow\}^j\leftarrow}{\boldsymbol{k}}^{(i)})+\sum_{i=1}^{a-1}\zeta({\boldsymbol{k}}_{(i)},{}_{\leftarrow}{\boldsymbol{k}}^{(i)}).$$
Boundary conditions
-------------------
For ${\boldsymbol{k}}\in{\mathcal{I}}'$, we have $$Z^{\text{H}}({\boldsymbol{k}};(1))=\sum_{i=0}^{a-1}\zeta({\boldsymbol{k}}_{(i)},{}_{\uparrow}{\boldsymbol{k}}^{(i)})+\sum_{i=0}^{a-1}\zeta({\boldsymbol{k}}_{(i)},{}_{\leftarrow}{\boldsymbol{k}}^{(i)})$$ and $$Z^{\text{H}}((1);{\boldsymbol{k}})=\zeta({}_{\leftarrow}{\boldsymbol{k}}).$$ Therefore, we see that the identity is nothing but Hoffman’s relation .
Example
-------
For ${\boldsymbol{k}}=(k_1,\dots,k_a)\in I'_a$ $(a\geq 1)$ and $1\leq i\leq a$, we set $$H_i({\boldsymbol{k}})\coloneqq\sum_{j=1}^{k_i-1}\zeta(({\boldsymbol{k}}_{(i)})_{\{\downarrow\}^j},{}_{\{\uparrow\}^j\leftarrow}{\boldsymbol{k}}^{(i)}).$$ Here, $H_i({\boldsymbol{k}})=0$ when $k_i=1$. By applying the algorithm in \[H-alg\] to $Z^{\text{H}}((2,1,3);(1))$, we have $$\begin{aligned}
&Z^{\text{H}}((2,1,3);(1))\\
&\stackrel{\eqref{sec8:1}}{=}Z^{\text{H}}((2,1,3)_{\downarrow};{}_{\uparrow\leftarrow}\varnothing)+\zeta((2,1,3)_{\downarrow},{}_{\uparrow\leftarrow}\varnothing)\\
&\stackrel{\eqref{sec8:1}}{=}Z^{\text{H}}((2,1,3)_{\downarrow\downarrow};{}_{\uparrow\uparrow\leftarrow}\varnothing)+\zeta((2,1,3)_{\downarrow\downarrow},{}_{\uparrow\uparrow\leftarrow}\varnothing)+\zeta((2,1,3)_{\downarrow},{}_{\uparrow\leftarrow}\varnothing)\\
&\stackrel{\eqref{sec8:2}}{=}Z^{\text{H}}((2,1);{}_{\leftarrow}(3))+\zeta((2,1),{}_{\leftarrow}(3))+H_3(2,1,3)\\
&\stackrel{\eqref{sec8:2}}{=}Z^{\text{H}}((2);{}_{\leftarrow}(1,3))+\zeta((2),{}_{\leftarrow}(1,3))+\zeta((2,1),{}_{\leftarrow}(3))+H_3(2,1,3)\\
&\stackrel{\eqref{sec8:1}}{=}Z^{\text{H}}((2)_{\downarrow};{}_{\uparrow\leftarrow}(1,3))+\zeta((2)_{\downarrow},{}_{\uparrow\leftarrow}(1,3))+\zeta((2),{}_{\leftarrow}(1,3))+\zeta((2,1),{}_{\leftarrow}(3))+H_3(2,1,3)\\
&=Z^{\text{H}}((1);(2,1,3))+H_1(2,1,3)+\zeta((2),{}_{\leftarrow}(1,3))+\zeta((2,1),{}_{\leftarrow}(3))+H_3(2,1,3).\end{aligned}$$ On the other hand, the boundary conditions are $$\begin{aligned}
Z^{\text{H}}((2,1,3);(1))&=\zeta({}_{\uparrow}(2,1,3))+\zeta((2),{}_{\uparrow}(1,3))+\zeta((2,1),{}_{\uparrow}(3))\\
&\qquad +\zeta({}_{\leftarrow}(2,1,3))+\zeta((2),{}_{\leftarrow}(1,3))+\zeta((2,1),{}_{\leftarrow}(3))\end{aligned}$$ and $Z^{\text{H}}((1);(2,1,3))=\zeta({}_{\leftarrow}(2,1,3))$. Therefore, we have $$\zeta({}_{\uparrow}(2,1,3))+\zeta((2),{}_{\uparrow}(1,3))+\zeta((2,1),{}_{\uparrow}(3))=H_1(2,1,3)+H_2(2,1,3)+H_3(2,1,3),$$ that is, $$\zeta(3,1,3)+\zeta(2,2,3)+\zeta(2,1,4)=\zeta(1,2,1,3)+\zeta(2,1,2,2)+\zeta(2,1,1,3).$$
Some remarks
============
It is known that there exist various generalizations of each connected sum. For example, $$\sum_{\substack{0=n_0<n_1<\cdots<n_a \\ 0=m_0<m_1<\cdots<m_b \\ n_a+m_b=r_1<r_2<\cdots<r_c\leq N}}\frac{x_1^{n_1}x_2^{n_2-n_1}\cdots x_a^{n_a-n_{a-1}}\cdot y_1^{m_1}y_2^{m_2-m_1}\cdots y_b^{m_b-m_{b-1}}\cdot z_2^{r_2-r_1}\cdots z_c^{r_c-r_{c-1}}}{{\boldsymbol{n}}^{({\boldsymbol{k}}_{\downarrow})}{\boldsymbol{m}}^{({\boldsymbol{l}}_{\downarrow})}{\boldsymbol{r}}^{({}_{\downarrow}{\boldsymbol{h}})}}$$ and $$\sum_{\substack{1=r_0\leq r_1<\cdots<r_c \\ r_c=n_1<\cdots<n_a\leq N \\ r_c=m_1<\cdots<m_b\leq N}}\frac{x_2^{n_2}\cdots x_a^{n_a}\cdot y_2^{m_2}\cdots y_b^{m_b}\cdot z_1^{r_1}\cdots z_c^{r_c}}{{\boldsymbol{n}}^{({}_{\downarrow}{\boldsymbol{k}})}{\boldsymbol{m}}^{({}_{\downarrow}{\boldsymbol{l}})}{\boldsymbol{r}}^{{\boldsymbol{h}}}}$$ give the shuffle product formula and the harmonic product formula for multiple polylogarithms, respectively. Here, $x_1, \dots, x_a, y_1,\dots, y_b, z_1,\dots, z_c$ are suitable complex variables.
We can also obtain a simple proof of the Ohno relation proved by Ohno in [@Oh] by generalizing the connector as $$\frac{[n_a;x][m_b;x]}{[n_a+m_b;x]} \qquad \left(|x|<1, \ [n;x]\coloneqq \prod_{i=1}^n(i-x)\right).$$ Furthermore, we can get a simple proof of the $q$-Ohno relation proved by Bradley in [@B] by generalizing the above connector as $$\frac{[n_a;x]_q[m_b;x]_q}{[n_a+m_b;x]_q} \qquad \left(0<q<1, \ [n;x]_q\coloneqq \prod_{i=1}^n([i]_q-q^ix), \quad [i]_q=\frac{1-q^i}{1-q}\right).$$ See [@SY1] for details.
We can also prove other series identities by using this dynamic proof method: the cyclic sum formula for (finite) multiple zeta-star values (same as §\[sec:CS\]), Leshchiner’s identity which is a generalization of the Apéry–Markov identity (Ono–Seki; unpublished), Zhao’s binomial identity [@Z Theorem 1.4] which implies the two-one formlua (Yamamoto; unpublished), the duality for MZVs of level 2 (Ono–Seki, Yamamoto; both unpublished), and the double Ohno relation (Hirose–Sato–Seki; [@HSS]), and so on.
Let’s find new connectors!
Acknowledgments {#acknowledgments .unnumbered}
===============
This article is based on the talk by the author at the workshop “Various Aspects of Multiple Zeta Values” held at RIMS, Kyoto Univ. (November 18–22, 2019). The author thanks the organizer Prof. Hidekazu Furusho for the kind invitation. This work was supported by JSPS KAKENHI Grant Number JP18J00151. The author sincerely thanks Dr. Minoru Hirose, Dr. Masataka Ono, Dr. Nobuo Sato, and Prof. Shuji Yamamoto for their helpful comments. He also would like to thank Prof. Masanobu Kaneko and Prof. Yasuo Ohno for reading the manuscript carefully.
[9]{} P. Akhilesh, *Double tails of multiple zeta values*, J. Number Theory **170** (2017), 228–249. D. M. Bradley, *Multiple $q$-zeta values*, J. Algebra **283** (2005), no. 2, 752–798. M. Hirose, N. Sato, S. Seki, *The connector for double Ohno relation*, preprint, arXiv:2006.09036. M. Hoffman, *Multiple harmonic series*, Pacific J. Math., **152** (1992), 275–290. M. Hoffman, *The algebra of multiple harmonic series*, J. Algebra **194** (1997), 477–495. M. Hoffman, *Quasi-symmetric functions and mod $p$ multiple harmonic sums*, Kyushu J. Math., **69** (2015), no. 2, 345–366. M. Hoffman, Y. Ohno, *Relations of multiple zeta values and their algebraic expression*, J. of Algebra, **262** (2003), 332–347. Y. Komori, K. Matsumoto, H. Tsumura, *Shuffle products of multiple zeta values and partial fraction decompositions of zeta-functions of root systems*, Math. Z. **268** (2011), 993–1011. M. Kaneko, D. Zagier, *Finite multiple zeta values*, in preparation. N. Kawasaki, K. Oyama, *Cyclic sums of finite multiple zeta values*, Acta Arith., Advance publication (2020), 8 pages. Y. Ohno, *A generalization of the duality and sum formulas on the multiple zeta values*, J. Number Theory **74** (1999), no. 1, 39–43. M. Ono, *Finite multiple zeta values associated with $2$-colored rooted trees*, J. Number Theory, **181** (2017), 99–116. S. Seki, S. Yamamoto, *A new proof of the duality of multiple zeta values and its generalizations*, Int. J. of Number Theory, Vol. 15, No. 6 (2019), 1261–1265. S. Seki, S. Yamamoto, *Ohno-type identities for multiple harmonic sums*, J. Math. Soc. Japan, Advance publication (2020), 14 pages. J. Zhao, *Identity families of multiple harmonic sums and multiple zeta star values*, J. Math. Soc. Japan **68** (2016), no. 4,1669–1694.
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---
abstract: 'Detecting reliably copy-move forgeries is difficult because images do contain similar objects. The question is : how to discard natural image self-similarities while still detecting copy-moved parts as being “unnaturally similar”? Copy-move may have been performed after a rotation, a change of scale and followed by JPEG compression or the addition of noise. For this reason, we base our method on SIFT, which provides sparse keypoints with scale, rotation and illumination invariant descriptors. To discriminate natural descriptor matches from artificial ones, we introduce an *a contrario* method which gives theoretical guarantees on the number of false alarms. We validate our method on several databases. Being fully unsupervised it can be integrated into any generic automated image tampering detection pipeline.'
address: 'CMLA, ENS Cachan, CNRS, Université Paris-Saclay, 94235 Cachan, France'
bibliography:
- 'main.bib'
title: 'Robust copy-move forgery detection by false alarms control'
---
Sift, copy-move, a-contrario, forgery
Introduction {#sec:intro}
============
Photo and video editing includes the insertion or removal of parts of the image, often performed by internal or external copy-move operations. The Poisson editing technique [@perez2003poisson; @ipol.2016.163] allows for seamless insertions and is now routinely used for special effects in movies, in software like Photoshop or in popular mobile phone applications. Most editing operations are driven by aesthetic goals. Yet their usage can easily become malicious and help forging false evidence, fake news, or alter results in scientific publications [@bik2016prevalence]. It is therefore of primary importance to provide public and professionals with reliable scientific tools detecting traces of any intentional alteration of a photograph. Several different techniques are relevant here: image splicing (internal or external) can be detected through its local alterations of the compression encoding and of the JPEG blocks [@lin2009fast; @cao2012robust; @nikoukhah2018automatic], its inconsistent demosaicking traces [@popescu2005exposing; @ferrara2012image; @bammey2018automatic], or directly [@ng2004blind; @hsu2007image; @huh2018fighting]. Methods tracking other features such as noise inconsistencies, lightning inconsistencies, chromatic aberration inconsistencies, etc. were listed in the broad review [@farid2009image].
Our paper focuses on a specific type of image splicing called “copy-move". As its name indicates it consists in copying a region of the image and pasting it somewhere else. Rotation, scaling, change of contrasts and other manipulations are sometimes applied to the piece being copied before pasting it. The method can be used to replicate objects, but sometimes also to hide an object by a texture borrowed elsewhere in the image. Copy-move detection methods can be divided into two main categories: Block-based and keypoint-based. The block-based approaches try to match regions by blocks. In order to match the blocks more easily and more efficiently it is frequent to represent the block in a compact form by dimensionality reduction, e.g. with PCA, DCT [@cao2012robust] or DWT [@li2007sorted]. The compact representation may also ensure the invariance of the detection to rotations by using Zernike moments [@cozzolino2015efficient; @ipol.2018.213] or a similarity invariance with the Fourier-Mellin transform [@bayram2009efficient; @li2010rotation]. These methods generally manage to detect the forged regions, but are computationally demanding. Instead of directly trying to match blocks, featured-based methods compute sets of keypoints and then match these keypoints. Many of these methods are based on SIFT [@amerini2010geometric; @ardizzone2010detecting] or SURF [@bo2010image]. The descriptors associated to the keypoints are invariant to rotation, scaling and even moderate affine distortions. Yet, precisely out of too much robustness, these methods may cause false detections when similar objects are present in an image. As argued in [@wen2016coverage], most methods therefore suffer from a false positive problem caused by the occurrence of “natural” self-similarity. This is the problem that we attempt to tackle here.
Section \[sec:method\] introduces our method and Section \[sec:experiments\] shows experimental results on different datasets. Finally perspectives are going to be presented in Section \[sec:conclusion\].
Copy-move matching with SIFT-like matching {#sec:method}
==========================================
Like in the SIFT algorithm [@lowe1999object] we start by computing a set of sparse keypoints. These keypoints are usually located in textured regions. Then a descriptor is associated to each of these keypoints. Finally the descriptors are matched to each other to define the detection. These three steps are summarized in the next paragraphs.
Keypoints
---------
The keypoints correspond to the extrema of the normalized Laplacian scale-space. In practice they are computed using differences of Gaussians. The positions of the maxima are then found for each scale. To each keypoint is associated a scale and a principal orientation. A more detailed analysis can be found in [@ipol.2014.82].
Descriptors
-----------
This first, classic, SIFT step gives a list $\mathcal{K} = (k_i)$ of keypoints. From each of these keypoints (consisting of a spatial position, a scale and an orientation), a square patch $p_i$ of size $(N+2) \times (N+2)$ can be sampled. The gradients in both directions are then computed from these patches yielding an $N \times N$ gradient patch $\mathcal{D}^i$ with vector values $(\frac{\partial p_i}{\partial x}, \frac{\partial p_i}{\partial y})$.
Contrary to SIFT, we keep these matrices for the matching step. Indeed, using histograms of gradients (HOGs) to represent the gradient patch would be too robust a representation and lead to the detection of natural repetitions. Hence, following [@von2015contrario] and [@rodriguez2018affine]. we encode the key point $k_i$ by its gradient patch $\mathcal D_i$. This allows for an invariance to uniform illumination changes. In SIFT, the descriptors are computed on a grayscale version of the image for matching applications. In the forgery case, it is interesting to consider all information available. So our gradient descriptors keep three channels, one for each color. For simplicity our matching step will be presented using a grayscale descriptor, but extends immediately to color as well. Color descriptors will be used for the experiments in Section \[sec:experiments\].
Matching {#sec:matching}
--------
Two naturally similar objects are rarely exactly similar. This is because there are always differences in their illumination in a real scene, and physical differences that do not necessarily catch the eye, in addition to the acquisition noise. Our matching process takes advantage of these serious variations to discriminate between similar objects and digital copies. Consider two keypoints $k_i$ and $k_j$ located respectively on the original object and on the forged copy, so that they match with a regular matching method such as SIFT [@lowe1999object] or [@rodriguez2018affine]. In that case the descriptors $\mathcal D^i$ and $\mathcal D^j$ associated to these keypoints should be *exactly* the same, namely $\forall k,l \in \{1,\dots,N\}, \mathcal{D}^i_{k,l} = \mathcal{D}^j_{k,l}$. Of course this perfect quality is not reached in practice. Several copy-move steps could introduce small differences such as: the interpolation due to a rotation or zoom or even a post-processing step such as the addition of noise and/or a compression after forgery. Nevertheless, we can enforce a very close match between each part of the descriptors by an exigence like $\forall k,l \in \{1,\dots,N\}, \|\mathcal{D}^i_{k,l} - \mathcal{D}^j_{k,l}\|^2_2 \leqslant \tau$. For the distance $d_{max}$ defined by $$d_{max}(\mathcal{D}^i, \mathcal{D}^j) = \max_{k,l \in \{1,\dots,N\}} \|\mathcal{D}^i_{k,l} - \mathcal{D}^j_{k,l}\|^2_2,$$ the suspicious match test is simply $d_{max}(\mathcal{D}^i, \mathcal{D}^j) \leqslant \tau$.
![On the left image the two objects are similar but not digital copies of each other. On the right, one is a digital copy of the other. The patches shown below each respective image correspond to the red dots in the images. They show that a difference is visible at this level and therefore the descriptors can be discriminated. This is why the detection method can discard genuinely similar objects.[]{data-label="fig:distinguish"}](images/distinguish/1.png "fig:"){width="0.49\linewidth"} ![On the left image the two objects are similar but not digital copies of each other. On the right, one is a digital copy of the other. The patches shown below each respective image correspond to the red dots in the images. They show that a difference is visible at this level and therefore the descriptors can be discriminated. This is why the detection method can discard genuinely similar objects.[]{data-label="fig:distinguish"}](images/distinguish/1t.png "fig:"){width="0.49\linewidth"} ![On the left image the two objects are similar but not digital copies of each other. On the right, one is a digital copy of the other. The patches shown below each respective image correspond to the red dots in the images. They show that a difference is visible at this level and therefore the descriptors can be discriminated. This is why the detection method can discard genuinely similar objects.[]{data-label="fig:distinguish"}](images/distinguish/patch_similar.png "fig:"){width="0.24\linewidth"} ![On the left image the two objects are similar but not digital copies of each other. On the right, one is a digital copy of the other. The patches shown below each respective image correspond to the red dots in the images. They show that a difference is visible at this level and therefore the descriptors can be discriminated. This is why the detection method can discard genuinely similar objects.[]{data-label="fig:distinguish"}](images/distinguish/patch_orig.png "fig:"){width="0.24\linewidth"} ![On the left image the two objects are similar but not digital copies of each other. On the right, one is a digital copy of the other. The patches shown below each respective image correspond to the red dots in the images. They show that a difference is visible at this level and therefore the descriptors can be discriminated. This is why the detection method can discard genuinely similar objects.[]{data-label="fig:distinguish"}](images/distinguish/patch_copy.png "fig:"){width="0.24\linewidth"} ![On the left image the two objects are similar but not digital copies of each other. On the right, one is a digital copy of the other. The patches shown below each respective image correspond to the red dots in the images. They show that a difference is visible at this level and therefore the descriptors can be discriminated. This is why the detection method can discard genuinely similar objects.[]{data-label="fig:distinguish"}](images/distinguish/patch_orig.png "fig:"){width="0.24\linewidth"}
The key question is to fix the right detection threshold $\tau$, to have a matching criterion that rejects genuinely similar objects while still detecting well copy-move forgeries. This threshold can be computed rigorously using the *a-contrario* theory [@desolneux2007gestalt] which is a probabilistic formalization of the *non-accidentalness* principle [@lowe1985perceptual]. This principle has shown its practical use for detection purposes such as segment detection [@grompone2010lsd], vanishing points detection [@lezama2014finding] and anomaly detection [@davy2018reducing]. The *a-contrario* theory provides a way to compute automatically detection thresholds while having a control on the number of false alarms (NFA). It replaces the usual $p$-value by drawing into account the number of tests and therefore controlling the overall number of false alarms in a given detection task. The method only requires a simple *a contrario* stochastic model on the perturbation. We will consider for now that $\mathcal{D}^i$ and $\mathcal{D}^j$ are derived from the same patch but one of them has been corrupted by Gaussian noise of variance $\sigma^2$ *i.e.* since the descriptors consist of gradients $\forall k,l, \mathcal{D}^j_{k,l} = \mathcal{D}^i_{k,l} + n_{k,l}$ where $n_{k,l} \sim \mathcal{N}(0,2\sigma^2)$ and are independent. Matching both descriptors requires that $\max_{k,l} n_{k,l}^2 \leqslant \tau$. The probability of matching in this case is then $$\mathbb{P}\left(\max_{k,l} n_{k,l}^2 \leqslant \tau\right) = \prod_{k,l} \mathbb{P}\left(n_{k,l}^2 \leqslant \tau\right)
= \mathbb{P}\left(n \leqslant \frac{\tau}{2\sigma^2}\right)^{N^2}$$ where $n$ follows a $\chi^2$ distribution with $1$ degree of freedom. We can therefore control the number of false detections by choosing the proper $\tau$ according to $$\tau = 2\sigma^2 \times chi2inv\left(\sqrt[N^2]{\frac{\epsilon}{N_{tests}}}\right),
\label{eq:threshold}$$ where $\epsilon$ is the number of false alarms per $N_{tests}$ number of tests and $chi2inv$ is inverse of the $\chi^2$ cumulative distribution function. The main point of formula is that it reduces the initial method dependency on many detection parameters to just one, namely $\sigma$. We can argue that this last one is not critical. Indeed, even though the dependency on $\sigma$ is strong, as long as the degradation is not too large, there will be a scale in which $\sigma$ is small enough so the detection will work: indeed $\sigma$ is divided by two at each octave in the SIFT method. For example, this exigent threshold can work for a noise of 4, but requires the tampered area to be four times larger for a detection. It might be objected that zooming down also makes naturally similar objects become more similar. Yet our experiments indicate that this is not the case, indeed their small but significant differences encompass all scales. To summarize, granting that we allow for one false detection on average on a set of images, the method is parameterless as it adapts to the number of tests and to the patch size. Of course it might be coupled with an automatic noise estimator to give an good guess of $\sigma$. Assuming a perturbation noise of variance $\sigma^2=1$ and the use of descriptors of size $4 \times 4 \times 3$ (derived from $6\times6$ color patches), a number of false alarms of $\epsilon=1$ and testing on $100$ images with on average $50$ keypoints (this corresponds to the COVERAGE dataset presented in Section \[sec:experiments\]), Equation gives $\tau=2.9$. The advantage of using color descriptors in this case is either to increase the size of the descriptor (allowing for a larger $\tau$ and detecting more) or to reduce the spatial size for a same size of descriptor (allowing to detect smaller forgeries).
An interesting side effect is that this test is really fast to compute. Indeed to detect forgeries each keypoint must to compared against all others. Since we are comparing keypoints inside a single image this gives $\frac{(K-1)(K-2)}{2}$ pairs to be tested, where $K$ is the number of descriptors. (Of course all descriptor self-matches are discarded). For large images the computation of the distance becomes quickly a bottleneck for distances that are costly. In our case it is not necessary to compute $d_{max}$ before doing the test, the test can be done during the computation of $d_{max}$ which allows for early stopping. Since most keypoints won’t match, the number of operations done per comparison is in practice much smaller than the size $N^2$ of the descriptor. An experimental verification of this fact is made in Section \[sec:experiments\].
Finally we need to take into account all possible flips for the forged regions. While the matching process doesn’t detect flips it is possible to still detect them at the cost of a few more computations. The modified distance to test flips is then $$d_{flip}(\mathcal{D}^i, \mathcal{D}^j) = \max_{k,l} \left(x^i_{k,l} - x^j_{k,l}\right)^2+ \left(y^i_{k,l} + y^j_{N-k+1,l}\right)^2$$ where $D_{k,l} = (x_{k,l},y_{k,l})$. Indeed when flipped, the indexes in one direction are reversed but also the gradients in that direction are opposite. Thanks to the rotation invariance all flips are taken into account by just testing the flip in one direction (in our case in the $y$ direction). In the end, we test each pair of keypoints with both distances to take into account flips. Having to do twice the computation is not a problem in practice as each test is very efficient.
Experiments {#sec:experiments}
===========
Dataset Method
------------ ---------- ------- ------
COVERAGE Proposed 43% 1%
Previous 50.5% -
GRIP Proposed 70% 1.3%
Previous 71% -
IM Proposed 81.2% 0%
Previous 75% -
IM JPEG80 Proposed 64.6% 0%
IM NOISE20 Proposed 79.2% 0%
: Detection statistics on the different datasets compared to the reported results from [@wen2016coverage]. The proposed method achieves similar true positive detections for a very limited number of false detections. The only false detection is shown in \[fig:failure\]. The false detections are computed on the original images (with no forgeries).[]{data-label="tab:detection"}
In this section the images are all shown in grayscale even though they are originally in color. This allows for a better visualization of the matches. Nevertheless, the descriptors were color descriptors. We present results on three different datasets: GRIP [@cozzolino2015efficient], Image Manipulation (IM) [@christlein2012evaluation] and COVERAGE [@wen2016coverage] which is the dataset that inspired this study as it focuses on distinguishing forgeries from similar but genuine objects. All images shown in this section come from these datasets.
We decided to use descriptors of size $3\times8\times8$ for the IM dataset and $3\times4\times4$ otherwise as the images from the IM dataset are much larger than the ones from the other datasets. Indeed the size of the descriptors needs be chosen so to be smaller than the expected size of the forged regions. Each time the threshold was computed using Equation from Section \[sec:matching\]. We also verified that the number of comparisons done to compare two descriptors was much smaller than the size of a descriptor. For example for the image shown in Figure \[fig:distinguish\], of size $424 \times 421$ and containing $207$ descriptors only $1.1$ comparisons were necessary on average for a descriptor size of $3\times6\times6=108$ that is almost the size of the descriptor used for SIFT. Thus the detection is really fast even for large images with a large number of keypoints.
Table \[tab:detection\] shows that while the methods focuses on being robust to similar objects and reduces as much as possible false detections, it is actually competitive with previous keypoint based methods. Moreover, the number of false alarm is definitely under control : only very little false alarms were found in all three datasets. One of these false detection is shown in Figure \[fig:failure\]. We also verified that while robust to similar objects (and therefore very precise) the method still is robust to reasonable noise and compression.
Figure \[fig:success\] shows different examples of successful detections. The method is able to detect well rotation, uniform illumination changes, scaling and compression. As can be seen, the more texture the forged region has the easier it is to detect. This is because a textured region will generate more keypoints and therefore will increase its chances of matching.
Figure \[fig:failure\] shows several failure examples. Most failures come from the fact that the method can’t deal with more severe distortions such as a tilt or non-uniform illumination change. The method also fails to detect flat regions, as no keypoints are computed on these regions. As for the false detection, it does not contradict the a contrario model. We requested at most $\epsilon=1$ false detection per $100$ images with the threshold given in Section \[sec:matching\], and we found one with $200$ images tested for the COVERAGE dataset.
![Examples of forgeries that were successfully detected in the different datasets. From the top to bottom, left to right: an example with a rotation, a change of illumination, a change of scale and with JPEG compression.[]{data-label="fig:success"}](images/example/true_rotation.png "fig:"){width="0.49\linewidth"} ![Examples of forgeries that were successfully detected in the different datasets. From the top to bottom, left to right: an example with a rotation, a change of illumination, a change of scale and with JPEG compression.[]{data-label="fig:success"}](images/example/true_illumination.png "fig:"){width="0.49\linewidth"} ![Examples of forgeries that were successfully detected in the different datasets. From the top to bottom, left to right: an example with a rotation, a change of illumination, a change of scale and with JPEG compression.[]{data-label="fig:success"}](images/example/true_scale.png "fig:"){width="0.49\linewidth"} ![Examples of forgeries that were successfully detected in the different datasets. From the top to bottom, left to right: an example with a rotation, a change of illumination, a change of scale and with JPEG compression.[]{data-label="fig:success"}](images/example/true_jpeg.png "fig:"){width="0.49\linewidth"}
![Examples of forgeries that were not successfully detect in the different datasets. From top to bottom, left to right: an example where the forged region is completely flat, with a non-uniform change of illumination, with a tilt applied and the false detection.[]{data-label="fig:failure"}](images/example/false_flat.png "fig:"){width="0.49\linewidth"} ![Examples of forgeries that were not successfully detect in the different datasets. From top to bottom, left to right: an example where the forged region is completely flat, with a non-uniform change of illumination, with a tilt applied and the false detection.[]{data-label="fig:failure"}](images/example/false_illumination.png "fig:"){width="0.49\linewidth"} ![Examples of forgeries that were not successfully detect in the different datasets. From top to bottom, left to right: an example where the forged region is completely flat, with a non-uniform change of illumination, with a tilt applied and the false detection.[]{data-label="fig:failure"}](images/example/false_tilt.png "fig:"){width="0.49\linewidth"} ![Examples of forgeries that were not successfully detect in the different datasets. From top to bottom, left to right: an example where the forged region is completely flat, with a non-uniform change of illumination, with a tilt applied and the false detection.[]{data-label="fig:failure"}](images/example/false_alarm.png "fig:"){width="0.49\linewidth"}
Perspectives {#sec:conclusion}
============
In this paper we have presented an unsupervised method to detect copy-move forgeries that is not only invariant to rotation, scaling and global change of illumination, but also robust to the presence of similar but genuinely different objects or regions. The method, being parameter-less and very fast, can be included in the necessary long series of tampering tests applied to a suspicious image.
The limits of the method are closely linked to its strength. Because it is robust to the presence of naturally similar objects, it is less reliable in case of large degradation of a copied digital ones. We nevertheless found that the method is robust enough to usual noise and compression levels. An image that has been degraded too much is suspicious anyway, since nowadays the quality of an image taken with a mobile is very good. Thus only images with a good enough quality should be tested. Highly degraded images would be anyway suspicious regardless of any such more sophisticated examination. The second limit is the usage of sparse keypoints. These keypoints are only computed in regions that are contrasted enough (non-flat areas) which means that forgeries in these regions might not detected. Finally matching keypoints give anchor points and do not delimit forged regions precisely. A natural extension of the method would be to extract the forged regions from the anchor points while still keeping a good control over the number of false detections. Finally coupling the method with a noise estimator could arguably make it still more discriminant.
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