Probes of Full Eigenstate Thermalization in Ergodicity-Breaking Quantum Circuits

TL;DR

The paper investigates whether full ETH, which encodes higher-order correlations via FP, persists in ergodicity-breaking quantum circuits. It analyzes a dual-unitary, interacting XXZ circuit to obtain exact eigenstates in a soliton basis and uses FP diagrams to describe cactus (non-crossing) and crossing contributions to higher-order matrix-element correlations. In the soliton basis, crossing contributions scale as $C_m \sim 1/L$ with a sector-averaged result $C = \frac{2}{L}-\frac{4}{L^2}+\frac{2}{L^3}$, while in a generic basis the suppression can be quasi-exponential, $\sim L^{3/2}/2^L$, due to degenerate subspaces; away from DU the scaling remains, yet off-DU matrix elements show a log-normal distribution and the ETH-like structure changes. Upon breaking integrability, the crossing diagrams revert to chaotic-system scaling $C\sim 1/2^L$, and frequency-domain features of ETH become broad, signaling a transition toward ergodic behavior. Overall, the results demonstrate that FP-based descriptions of higher-order ETH extend to interacting nonergodic systems and elucidate how soliton dynamics and degeneracies shape ETH diagnostics.

Abstract

The eigenstate thermalization hypothesis (ETH) is the leading interpretation in our current understanding of quantum thermalization. Recent results uncovered strong connections between quantum correlations in thermalizing systems and the structure of free probability theory, leading to the notion of full ETH. However, most studies have been performed for ergodic systems and it is still unclear whether or how full ETH manifests in ergodicity-breaking models. We fill this gap by studying standard probes of full ETH in ergodicity-breaking quantum circuits, presenting numerical and analytical results for interacting integrable systems. These probes can display distinct behavior and undergo a different scaling than the ones observed in ergodic systems. For the analytical results we consider an interacting integrable dual-unitary model and present the exact eigenstates, allowing us to analytically express common probes for full ETH. We discuss the underlying mechanisms responsible for these differences and show how the presence of solitons dictates the behavior of ETH-related quantities in the dual-unitary model. We show numerical evidence that this behavior is sufficiently generic away from dual-unitarity when restricted to the appropriate symmetry sectors.

Figures (12)

• Figure 1: According to the eigenstate thermalization hypothesis (ETH), eigenstates of quantum many-body systems appear thermal whenever local observables are probed. For such ergodic systems, the dynamics of out-of-time order correlators can be decomposed in the sum of thermal cumulants -- an ETH-based analogue of free cumulants. This decomposition is based on the combinatorics of non-crossing partitions and correctly captures the long-time behavior of OTOCs.
• Figure 2: (a) Example of cactus and non-cactus diagrams depicting the generalized ETH averages of Eq. \ref{['eq:general_eth']}. The cacti in the ETH picture correspond to non-crossing partitions as understood by free probability theory. The correspondence between non-cactus diagrams and crossing partitions is analogous. (b) Illustration of the factorization of the cactus diagrams in terms of its 'leaves'. That is, complicated averages corresponding to cacti can be factorized into simple averages of lower order, as written in Eq. \ref{['eq:eth_factorization']}.
• Figure 3: (a) Action of the DU XXZ gate on solitons. The darker green shade represents the conjugate phase in the gate $U^\dagger$. (b) Illustration of the action of the SWAP circuit on the soliton $Z_{10100}$ for $L = 5$.
• Figure 4: Illustration of how two spins interact twice along the orbit of $\ket{\sigma^t(m)}$, as highlighted by the red marks. The two spins will scatter each other whenever one of the is right moving and the other one is left moving. Here they meet at $t = 2$ and $t = 6$. For visual clarity, we omit the controlled-phase gates whenever the control qubit is zero.
• Figure 5: Distribution of eigenvalues for the one-body charge (magnetization) and two-body charges for $L = 17$. The former is binomially distributed as ${L \choose l}$ with $l = 0, 1, ..., L - 1, L$, while the latter follows a modified binomial distribution $2{L \choose l}$ with $l = 1, 3, ..., L - 2, L$.