Anomalous thermalization in ergodic systems

TL;DR

This paper demonstrates that some quantum systems can thermalize anomalously, exhibiting subdiffusive behavior and violating traditional assumptions of the eigenstate thermalization hypothesis, with implications for understanding quantum transport.

Contribution

It introduces a modified ETH framework for subdiffusive systems and links the scaling of matrix element variances to dynamical exponents, supported by numerical analysis.

Findings

Subdiffusive thermalization systems satisfy a modified ETH.

Variance of off-diagonal matrix elements scales more slowly in subdiffusive systems.

Eigenfunction distributions are non-Gaussian, violating Berry's conjecture.

Abstract

It is commonly believed that quantum isolated systems satisfying the eigenstate thermalization hypothesis (ETH) are diffusive. We show that this assumption is too restrictive, since there are systems that are asymptotically in a thermal state, yet exhibit anomalous, subdiffusive thermalization. We show that such systems satisfy a modified version of the ETH ansatz and derive a general connection between the scaling of the variance of the offdiagonal matrix elements of local operators, written in the eigenbasis of the Hamiltonian, and the dynamical exponent. We find that for subdiffusively thermalizing systems the variance scales more slowly with system size than expected for diffusive systems. We corroborate our findings by numerically studying the distribution of the coefficients of the eigenfunctions and the offdiagonal matrix elements of local operators of the random field Heisenberg chain, which has anomalous transport in its thermal phase. Surprisingly, this system also has non-Gaussian distributions of the eigenfunctions, thus directly violating Berry's conjecture.