Full Eigenstate Thermalization in Integrable Spin Systems

TL;DR

The paper investigates whether the full ETH, which extends ETH to higher-order matrix-element correlations, holds in integrable quantum systems by studying OTOCs in integrable Ising and XXZ spin chains and contrasting with chaotic variants. Using exact diagonalization, it tests the prediction that multi-point correlators decompose into thermal free cumulants, with the OTOC obeying $\text{OTOC}(t) = 2 k_2(t)^2 + k_4(t)$ and the dominance of non-crossing (cactus) diagrams over crossing diagrams. It finds that the relation holds in integrable models, but late-time OTOC behavior—such as nonzero saturation or persistent oscillations—differs from chaotic systems, and the Hilbert-space scaling of crossing contributions is weaker (e.g., $D^{-0.5}$ vs $D^{-1}$). In XXZ, the observed behavior is operator-dependent, revealing nuanced distinctions between integrable and chaotic dynamics. Overall, the work demonstrates that full ETH-like structures can persist in integrable systems for certain operators, while finite-size scaling and long-time dynamics encode the signature of integrability.

Abstract

The Eigenstate Thermalization Hypothesis(ETH) is a standard tool to understand the thermalization properties of an isolated quantum system. Its generalization to higher order correlations of matrix elements of local operators, dubbed the full ETH, predicts the decomposition of higher-order correlation function into thermal free cumulants. In this work, we numerically test these predictions of full ETH using exact diagonalization of two spin models: the Ising and the XXZ Heisenberg models. The differences from the behavior of full ETH prediction in chaotic systems are highlighted and contrasted along the way. We also show that although in these integrable spin models the dynamics of the four-time correlators, specifically the out-of-time-ordered correlator (OTOC), is encoded in the fourth order free cumulant, it exhibits late-time dynamics that is different from nonintegrable systems.

Figures (10)

• Figure 1: (a)The time evolution OTOC compared with $2k_{2}(t)^{2}$ and $k_{4}(t)$ along with the sum of the two, for integrable Ising model (denoted by I); $L = 16$. (b) Variation of $|\langle r \rangle -1|$ for integrable and chaotic (denoted by C) Ising model with $L=16$. Also shown are the $D^{-1}$ scaling line to be compared with the chaotic case and the $D^{-0.5}$ scaling line to be compared with the integrable case. Factorization of cactus diagram in time domain and crossing contribution as a function of time (c) For integrable Ising chain and (d) chaotic Ising chain with $L=16$. The solid lines are guides to the eyes.
• Figure 2: The time evolution OTOC, compared with $2k_{2}(t)^{2}$ and $k_{4}(t)$ for XXZ model; $L=18$. Also shown is the sum of the contribution of two moments for (a) integrable; $d=0$ (c) chaotic; $d=1$ for operator $\hat{O}_1$ and (b) integrable; $d=0$ (d) chaotic; $d=1$ for operator $\hat{O}_2$. Scaling of $|\braket{r}-1|$, and crossing contribution with system size $L$ (f) for operator $\hat{O}_{1}$ (e) for operator $\hat{O}_{2}$.
• Figure 3: The time evolution OTOC, compared with $2k_{2}(t)^{2}$ and $k_{4}(t)$ for XXZ model; $L=18$. Also shown is the sum of the contribution of two moments.(a)For $d=0$ and operator $\hat{O}_1$. (b)For $d=0$ and operator $\hat{O}_2$. Scaling of $|\braket{r}-1|$, with system size $L$. (c) for operator $\hat{O}_{1}$ and (d) for operator $\hat{O}_{2}$.
• Figure 4: Factorization of cactus diagram in the time domain for XXZ model. (First row) Operator $\hat{O}_{1}$, for integrable ($d=0$) and chaotic ($d=1$) model respectively. (Second row) Operator $\hat{O}_{2}$, and integrable ($d=0$) and chaotic ($d=0$) model respectively. (Third row) Operator $\hat{O}_{2}$, and integrable ($d=0$) and chaotic ($d=0$) model respectively for the diffusive case $\Delta=1.1$. Also shown, in each plot, is the corresponding crossing contribution as a function of time (green markers).
• Figure 5: The time evolution OTOC, $2k_{2}(t)^{2}$ and $k_{4}(t)$ for Ising model with chaotic parameters and system sizes $L =10, 12, 14, 16$. Also shown is the sum $2k_{2}(t)^{2}+k_{4}(t)$.