Non-Hermitian Hamiltonians Violate the Eigenstate Thermalization Hypothesis

TL;DR

This paper demonstrates that non-Hermitian quantum systems do not satisfy the Eigenstate Thermalization Hypothesis, showing that eigenstate fluctuations are comparable to averages, thus preventing thermalization.

Contribution

It provides the first rigorous analysis and numerical evidence that non-Hermitian systems violate ETH, revealing a fundamental difference from Hermitian systems in thermalization behavior.

Findings

Eigenstate fluctuations are of the same order as the average in non-Hermitian systems.

Non-Hermitian systems do not exhibit thermalization according to ETH.

Results are supported by rigorous proofs and numerical simulations across ensembles.

Abstract

The Eigenstate Thermalization Hypothesis (ETH) represents a cornerstone in the theoretical understanding of the emergence of thermal behavior in closed quantum systems. The ETH asserts that expectation values of simple observables in energy eigenstates are accurately described by smooth functions of the thermodynamic parameters, with fluctuations and off-diagonal matrix elements exponentially suppressed in the entropy. We investigate to what extent the ETH holds in non-Hermitian many-body systems and come to the surprising conclusion that the fluctuations between eigenstates is of equal order to the average, indicating no thermalization. We support this conclusion with mathematically rigorous results in the Ginibre ensemble and numerical results in other ensembles, including the non-Hermitian Sachdev-Ye-Kitaev model, indicating universality in chaotic non-Hermitian quantum systems.