Eigenstate thermalization and rotational invariance in ergodic quantum systems

TL;DR

This paper investigates whether a specific relation observed in rotationally invariant random matrix models also holds in ergodic quantum systems, providing a new test for the Eigenstate Thermalization Hypothesis across different models.

Contribution

It introduces a novel relation as a test for ETH in non-localized quantum systems, extending beyond Gaussian ensembles.

Findings

Relation holds in disordered spin chain

Relation holds in SYK model

Relation holds in Floquet system

Abstract

Generic rotationally invariant random matrix models satisfy a simple relation: the probability distribution of off-diagonal elements and the one of half the difference between any two diagonal elements coincide. In the spirit of the Eigenstate Thermalization Hypothesis (ETH), we test the hypothesis that the same relation holds in quantum systems that are non-localized, when one considers small energy differences. The relation provides a stringent test of ETH beyond the Gaussian ensemble. We apply it to a disordered spin chain, the SYK model and a Floquet system.

Equations

$$A_{ij}\sim \mathcal{A}(E){\delta }_{ij}+e^{-S(E)/2}f(E,\omega )R_{ij}$$

$$P(A)=P(U^{†}AU)$$

$$\hat{P}(\hat{A})=\int \mathcal{D}AP(A)\prod _{k\le l=1}^2\delta (A_{{\alpha }_k{\alpha }_l}-{\hat{A}}_{kl})$$

$$\hat{P}(T^{†}\hat{A}T)=\int \mathcal{D}AP(A)\prod _{k\le l=1}^2\delta (A_{{\alpha }_k{\alpha }_l}-T_{vk}{T}_{ul}{\hat{A}}_{vu})$$

$$=\int \mathcal{D}AP(A)\prod _{u\le v=1}^2\delta (T_{uk}{T}_{vl}{A}_{{\alpha }_k{\alpha }_l}-{\hat{A}}_{uv})$$

$$\hat{P}(\hat{A})=\hat{P}(T^{†}\hat{A}T)$$

$$P(\hat{A})=F\left({\hat{A}}_{+}+\sqrt{{\hat{A}}_{-}^2+{\hat{A}}_{ij}^2},{\hat{A}}_{+}-\sqrt{{\hat{A}}_{-}^2+{\hat{A}}_{ij}^2}\right)$$

$$P_d(A_{ii})\simeq e^{-\frac{N}{2}{\text{extr}}_s\left(s{A}_{ii}-{\int }_0^{s}R(x)\mathrm{d}x\right)}$$

$$P_o(A_{ij})\simeq 2\int \mathrm{d}A{P}_d(A)P_d(A-2A_{ij})$$

$$H=\sum _{0\le \alpha <\beta <\gamma <\delta \le L}{J}_{\alpha \beta \gamma \delta }{\psi }_{\alpha }{\psi }_{\beta }{\psi }_{\gamma }{\psi }_{\delta }$$

$$H=\sum _{i=1}^L\left[J{S}_i\cdot S_{i+1}+h_i{S}_i^z\right]$$

Full text

Introduction. ETH has become one of the most accepted elements for our understanding of the dynamical and thermalization properties of quantum systems Srednicki (1994, 1999); D’Alessio et al. (2016). It relies on the idea that the eigenvectors of the Hamiltonian of a chaotic quantum system are affected by a degree of randomness and aims to quantify its statistical properties. The initial assumption, based on work of Berry Berry (1977), was that in certain simple systems like billiards, wavefunctions could be considered as a random superposition of plane waves. Subsequent works by Deutsch and Srednicki Deutsch (1991); Srednicki (1999) extended this idea in order to characterize the fluctuations of matrix elements of physical operators, from element to element and under different realizations of the Hamiltonian D’Alessio et al. (2016). Altogether these assumptions allow us to explain why locally a closed quantum system evolving from some excited state approaches equilibrium and to estimate dynamical correlation functions Srednicki (1994, 1999); D’Alessio et al. (2016). ETH leads to an ansatz for the elements of a generic local observable in the energy eigenbasis given by:

[TABLE]

where $E=(e_{i}+e_{j})/2$ and $\omega=(e_{i}-e_{j})/2$ and $f$ and ${\cal{A}}$ are smooth functions of $E/N$ and $\omega$ and $S(E)$ is the entropy. The $R_{ij}$ is a Hermitean matrix with essentially random elements. To a first approximation the matrix elements $R_{ij}$ are assumed to be Gaussian and uncorrelated random variables, but subsequent work has shown that, strictly speaking, they are neither: recently Foini and Kurchan (2019), we argued that the matrix entries $A_{ij}$ cannot be in general independent and that the products of certain off-diagonal matrix elements should have small but relevant expectations which enter in the computation of higher order correlation functions. This, together with the fact that the probability distributions of single elements are measurably non-Gaussian Luitz and Lev (2016); Luitz (2016), calls for a better understanding of the link between ETH and random matrix theory. In the second half of this paper we shall assess this connection numerically for several standard quantum models.

The relation between ETH (for small $\omega$) and random matrix ensembles is based upon the argument Deutsch (1991) according to which eigenvectors of a non-localized Hamiltonian with near energies mix with essentially random phases when a small perturbation is applied. This naturally leads one to assume that matrix elements close to the diagonal (where the mixing is maximal) are well represented by matrices such that the joint probability density of their elements $P(A)\equiv P(\{A_{ij}\})$ is left invariant by change of basis, i.e.

[TABLE]

where $U$ may be a orthogonal, unitary or symplectic matrix Mehta (2004). A subclass of well-studied models that enjoys this property has elements of the form Di Francesco et al. (1995) $P(A)\propto\exp\left(-\frac{1}{2}N\text{Tr}V(A)\right)$ where $V(A)$ is some generic function. The Gaussian ensemble $V(A)=A^{2}/2$ is the only one which has both rotational symmetry and independence of matrix entries Livan et al. (2018). The rotational invariance (2) is a weaker condition than gaussianity, but implies testable relations between the joint distributions of elements Foini and Kurchan that, in particular, do not involve determining the distribution itself. These relations are easy to prove for a random ensemble, but for a quantum system remain an assumption, a check of the randomness of the diagonalizing basis below the Thouless energy scale D’Alessio et al. (2016). In this paper we explore the simplest one, relating the distribution of diagonal and off-diagonal matrix elements. We shall see that the validity of the relation in a quantum model is a stringent test of eigenvector thermalization, and may be used together with the standard study of level-spacings.

Ensembles of Matrices: For the purposes of computing large deviations, for example of diagonal matrix elements, we could envisage using the elements of a single matrix. The problem is that there are not enough of those if we wish to compute the tails of the distribution: we are hence forced to consider a distribution of matrices. Furthermore, when we will apply our conclusions to a quantum system, unlike the case of random matrices, we shall not be free to rotate the eigenvectors as we please without destroying the structure of the matrix, so we shall need to clarify from which ensemble the matrices are chosen. Following Deutsch Deutsch (1991), we may consider the original extensive Hamiltonian modified by an ensemble of random perturbations that are also local and have the same symmetries, but are small: sub-extensive in the system’s size. Averaging over this ensemble, we may restrict ourselves (at least in principle) to computing the elements $A_{ii}$, $A_{jj}$ and $A_{ij}$ for any fixed pair $(i,j)$ with $\omega=(e_{i}-e_{j})$ within the Thouless scale.

A simple relation. In order to understand what may be the relation between the probability distributions of diagonal and off-diagonal elements of an operator, we may get inspiration in the corresponding relation for a generic rotationally invariant ensemble as in Eq (2). Let us then consider an $N\times N$ matrix $A$ with probability density $P(A)$, and focus on the (marginal) joint probability distribution $\hat{P}(\hat{A})$ of the elements of a chosen $2\times 2$ submatrix $\hat{A}$ such that $A_{\alpha_{1}\alpha_{1}}=\hat{A}_{11}$, $A_{\alpha_{2}\alpha_{2}}=\hat{A}_{22}$ and $A_{\alpha_{1}\alpha_{2}}=\hat{A}_{12}$ with $\alpha_{1}=i$ and $\alpha_{2}=j$. Restricting for simplicity to the orthogonal ensemble (the generalization to unitary is straightforward) we have:

[TABLE]

with $\mathcal{D}A=\prod_{i=1}^{N}{\rm d}A_{ii}\prod_{i>j}{\rm d}A_{ij}$. Let us now evaluate $\hat{P}(T^{\dagger}\hat{A}T)$ where $T$ is a $2\times 2$ orthogonal matrix:

[TABLE]

where the sums over repeated indices are implicit and the equality holds because in the delta functions we make a change of variable with Jacobian equal to one. Let us now consider the $N\times N$ orthogonal transformation $U_{kl}=T_{11}\delta_{ki}\delta_{li}+T_{22}\delta_{kj}\delta_{lj}+T_{12}\delta_{ki}\delta_{lj}+T_{21}\delta_{kj}\delta_{li}+\delta_{kl}(1-\delta_{ki})(1-\delta_{kj})$. In the integral (4) we can make the change of variable $A^{\prime}=UAU^{{\dagger}}$ and exploiting the rotational invariance of the overall probability distribution $P(A)$ and of the measure $\mathcal{D}A$ we see that

[TABLE]

i.e. the submatrix inherits the symmetries of the original matrix. This invariance implies that the joint probability distribution $\hat{P}(\hat{A})$ of all elements of the submatrix $\hat{A}$ depend only on its eigenvalues $\tilde{A}_{a}$.

For a two by two submatrix (with any $(i,j)$), in terms of the difference $\hat{A}_{-}=(\hat{A}_{ii}-\hat{A}_{jj})/2$, the sum $\hat{A}_{+}=(\hat{A}_{ii}+\hat{A}_{jj})/2$ and the off-diagonal element $\hat{A}_{ij}$, we have:

[TABLE]

This function is symmetric with respect to exchange of $\hat{A}_{-}$ and $\hat{A}_{ij}$. This means that their probability marginals are the same function $\bar{F}$ evaluated in $\hat{A}_{-}$ and $\hat{A}_{ij}$, respectively, and hence we conclude that they are equally distributed: $P(\hat{A}_{ij})=\bar{F}(\hat{A}_{ij})$ and $P(\hat{A}_{-})=\bar{F}(\hat{A}_{-})$. The same may be said of the real and imaginary parts of the off-diagonal elements in the complex case.

Large $N$ results. The previous relation holds – in a matrix model with rotational invariance – for any system size. For a distribution $P(A)\propto\exp\left(-\frac{1}{2}N\text{Tr}V(A)\right)$ one can determine the probability distribution of matrix elements (or their real and imaginary part for hermitian operators) in the large $N$ limit, in a general way. They take the form of a large deviation function, for instance for the diagonal matrix elements Guionnet and Maı (2005); Foini and Kurchan :

[TABLE]

where $R$ is a function dependent on the potential $V$, the ‘$R$-transform’ of the spectrum Tulino and Verdú (2004). The distribution of off-diagonal matrix elements can be written at the saddle point level:

[TABLE]

which is the large $N$ limit of our result when the probability of $A_{ii}$ and $A_{jj}$ factorize.

The exponential scaling (7) suggests that the probability distributions of matrix entries of quantum operators are also large deviation functions (with some ‘effective $N$’ related to level-spacing) and the deviations from the Gaussian behavior should be visible only in the tails of those distributions for small system sizes.

Applications to quantum systems.

We now check if in ergodic quantum systems the result we have obtained relating the distribution of diagonal and off-diagonal matrix elements applies. Note that what was a theorem for a matrix model is here a leap of faith. For a true matrix model the relation is valid for any $N$ while for a quantum system one may expect that a random matrix ETH regime sets in for $N$ sufficiently large. The particular case of variances of elements in a Gaussian case was already discussed in Refs. Mondaini and Rigol (2017); Hamazaki and Ueda (2019).

In the following, in the spirit of ETH we look at matrix elements of physical observables in the energy eigenbasis (or in the basis of the evolution operator for Floquet systems) and we order eigenstates according to their energy or their phase.

As a first example we consider a paradigmatic model of chaos, the Sachdev-Ye-Kitaev (SYK) model Maldacena and Stanford (2016). This is defined considering $N$ Majorana fermions $\{\psi_{\alpha},\psi_{\beta}\}=\delta_{\alpha\beta}$ interacting via a disordered multibody term:

[TABLE]

where $J_{\alpha\beta\gamma\delta}$ are Gaussian random variables with zero mean and variance $\langle J_{\alpha\beta\gamma\delta}^{2}\rangle=6/L^{3}$. We consider the observable $A=i\psi_{\alpha}\psi_{\beta}$ for all pairs of $\alpha$ and $\beta$. The Hamiltonian commutes with the parity operator $P=i^{-L/2}\prod_{\alpha=1}^{L}\psi_{\alpha}$ and we restrict ourselves to the sector with eigenvalue $\lambda_{P}=-1$. In Fig. 1 we show the results for $L=10$ where we compare the distribution of off-diagonal matrix elements $P(\text{Re}A_{ij})$ with the difference of two diagonal matrix elements $P((A_{ii}-A_{jj})/2)$ in the middle of the spectrum. We also show the distribution of diagonal matrix elements $P(A_{ii})$ that is far from a Gaussian distribution in the wings, where the agreement between $P(\text{Re}A_{ij})$ and $P((A_{ii}-A_{jj})/2)$ is still good. In the inset we show the collapse of the large deviation function $P(A_{ii})$ for two sizes $L=10$ and $L=14$.

Next we consider a Floquet system as the one implemented in Chan et al. (2018a). We take a one dimensional chain and we consider as evolution operator $U=W_{1}W_{2}$ with $W_{1}=U_{1,2}\times U_{3,4}\times\dots\times U_{L-1,L}$ and $W_{2}=S^{T}(U_{1,2}\times U_{3,4}\times\dots\times U_{L-1,L})S$ where $U_{i,j}$ are random unitary matrix and $S$ is the shift operator which translate the spins of one site implementing periodic boundary conditions. We consider as observable $A=\sigma^{z}_{L/2}$ in the basis of the operator $U$. In Fig. 2 we show the result of the distribution of diagonal and off-diagonal matrix elements for a system of size $L=6$. The deviations from a Gaussian are clear as well as the agreement between $P(\text{Re}A_{ij})$ and $P((A_{ii}-A_{jj})/2)$.

We finally discuss a disordered spin chain:

[TABLE]

where $S_{i}$ is a spin $1/2$, the sum is over periodic boundary conditions and $h_{i}$ are random uniform variables between $[-h,h]$. In Fig. 3 we show the results for $L=12$ and $h=1$ and we choose $S^{z}_{L/2}$. In our procedure to construct the histogram we generate a configuration of fields and then we vary it by a small amount ($\delta h_{i}\in[-0.05,0.05]$), to avoid very rare values of disorder fields dominating the tails – an effect enhanced at small sizes. We obtain a good agreement between $P(A_{ij})$ and $P((A_{ii}-A_{jj})/2)$. The deviations from the Gaussian are not very marked for this particular sample but are inevitable given that $P(A_{ii})$ is clearly not Gaussian. Allowing the fields to vary freely over the disorder gives different distributions (as we show in the inset of Fig. 3), and the symmetry of the distributions is clearly recovered.

Let us note that the distributions of off-diagonal matrix elements in many models (also considered here) are observed to be Gaussian Haque and McClarty (2017); Chan et al. (2018b); Luitz (2016); Luitz and Lev (2016); Beugeling et al. (2015), when relatively small deviations are considered, as is usual in large-deviation theory.

Perhaps the most interesting situation is when the distributions do not coincide. This can be seen for instance increasing the disorder in the model (10). Already for $h=2$ and $L=12$ the distributions $P(A_{ij})$ and $P((A_{ii}-A_{jj})/2)$ differ as shown in Fig. 4. This value of $h$ is expected to be below the MBL transition Pal and Huse (2010); Luitz and Lev (2016); Šuntajs et al. (2019), but at these sizes eigenstate thermalization does not hold. For large $h$, in the MBL phase, the relation is expected to break down for every size.

Conclusions. We have introduced a test of eigenstate thermalization inspired by matrix models that can be used in conjunction with level-statistics to detect subtle localization properties. The test is a direct check on the eigenvectors of the Hamiltonian rather than an indirect test through its spectrum.

The example of the matrix models underlines the fact that there is a close connection between the non-gaussian nature of the distribution and the existence of correlations between matrix elements (which are relevant for the higher-order correlation functions Foini and Kurchan (2019)). This relation does not apply for level-spacing, which is insensitive the existence or not of correlations between matrix elements.

Acknowledgments We thank discussions with J. Chalker, A. De Luca, B. Eynard and P. Vivo. This work is supported by “Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM) (EquiDystant project, L. Foini) J.K. is supported by the Simons Foundation Grant No 454943.