Eigenstate Thermalization and Spectral Imprints of the Hamiltonian in Local Observables

TL;DR

This paper addresses how ergodicity and thermalization emerge across the integrability–chaos crossover in a many-body quantum system. It introduces a local perturbation in the spin-$\tfrac{1}{2}$ XXZ chain and a submatrix framework to extract spectral correlations directly from local observables expressed in the Hamiltonian eigenbasis. The authors show that diagonal ETH fluctuations diminish with chaos, off-diagonal elements become Gaussian with an exponential decay in energy difference that weakens as chaos strengthens, and submatrix blocks reproduce GOE-level statistics via NNSD, SFF, and entanglement Page curves, even in partially ergodic regimes. The results imply that chaos signatures are imprinted locally within operator structure, enabling diagnostics of ergodicity and thermalization from small blocks and offering insights into dynamics and potential extensions to other many-body systems.

Abstract

The Eigenstate Thermalization Hypothesis explains thermalization in isolated quantum systems through the statistical properties of observables in the energy eigenbasis. We investigate the crossover from integrability to chaos in the spin-$1/2$ XXZ chain, establishing a direct correspondence between the spectral correlations of the Hamiltonian and local observables expressed in the energy eigenbasis as a signature of ergodicity breaking. By introducing a local perturbation that drives the system from integrability to chaos, we track the standard ETH indicators and the eigenstate entanglement entropy. We introduce a submatrix-based framework for analyzing local observables in the energy eigenbasis. By extracting real-symmetric blocks along the diagonal of the local observables represented in eigenbasis, we show that these submatrices exhibit both the short-range and long-range spectral features of the Hamiltonian. Remarkably, this correspondence persists even in a partially ergodic regime, indicating that the emergence of chaos is already encoded locally within the observables' matrix structure and that small blocks are sufficient to capture the underlying spectral correlations.

Figures (13)

• Figure 1: (a) Schematic representation of submatrices of a local observable in the eigenbasis of the Hamiltonian, where each block corresponds to an energy-resolved window and captures correlations among matrix elements within that localized sector. (b) Density plot of a representative block of the operator $O$. Qualitatively similar structure is observed for blocks taken across the diagonal for both operators [see Eq. (\ref{['eq:local_op1']}) and (\ref{['eq:local_op2']}) ].
• Figure 2: (Color online) (a) Nearest-neighbor spacing distribution and (b) level number variance of the XXZ Hamiltonian [see Eq. (\ref{['eq:perturbation']})] across the integrability chaos crossover.
• Figure 3: (Color online) [(a),(c)] Diagonal matrix elements of the observables $T$ and $O$ with normalized energy across the crossover. The black solid line represents the microcanonical average, computed by averaging over states within $[\epsilon_n - \delta \epsilon_n, \epsilon_n + \delta \epsilon_n]$ with $\delta \epsilon_n = 0.02$. [(b),(d)] Deviation $\Delta_{\mathrm{mic}} Z$ versus normalized energy quantifying fluctuations from the microcanonical average, computed over the same energy window.
• Figure 4: (Color online) Distribution of off-diagonal matrix elements $T_{\alpha\beta}$ [panels (a)--(f)] and $O_{\alpha\beta}$ ($\alpha \neq \beta$) [panels (g)--(l)] for different field strengths. The dashed green line represents a double-Gaussian fit, while the solid red line corresponds to a single-Gaussian fit.
• Figure 5: (Color online) Scaled variance of off-diagonal matrix elements of observables $T$ Eq. (\ref{['eq:local_op1']}) [(a)--(f)] and $O$ Eq. (\ref{['eq:local_op2']}) [(g)--(l)] versus frequency $\omega$ across the crossover. The matrix elements are computed for pairs of eigenstates whose average energy $\bar{E} \in [-0.5, 0.5]$ and averaged over frequency bins of $\delta \omega = 0.05$. Fitting is done in the shaded region