Eigenstate Thermalization and Spectral Imprints of the Hamiltonian in Local Observables

TL;DR

This paper explores how local observables in quantum systems reflect the transition from integrability to chaos, revealing that spectral correlations of the Hamiltonian are encoded locally within the observables' matrix structure, even in partially ergodic regimes.

Contribution

It introduces a submatrix-based framework to analyze local observables in the energy eigenbasis, linking spectral features to ergodicity breaking and chaos onset.

Findings

Local spectral features mirror Hamiltonian correlations.

Small matrix blocks capture global spectral properties.

Chaos signatures are present even in partial ergodicity regimes.

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.