Partial thermalisation of a two-state system coupled to a finite quantum bath

TL;DR

This paper develops an ETH-like framework for a partially thermalising spin-1/2 system coupled to a finite quantum bath, revealing complex distributions and entropic behaviors due to rare many-body resonances.

Contribution

It introduces a novel ETH-like ansatz for weakly coupled finite quantum systems, analyzing distributions of fidelity susceptibilities, entropies, and matrix elements, and exploring their connection to many-body localization.

Findings

Distribution of fidelity susceptibilities is broadly spread.

Spin eigenstate entropies are bi-modal.

Intermediate bath entropic enhancement occurs due to rare resonances.

Abstract

The eigenstate thermalisation hypothesis (ETH) is a statistical characterisation of eigen-energies, eigenstates and matrix elements of local operators in thermalising quantum systems. We develop an ETH-like ansatz of a partially thermalising system composed of a spin-1/2 coupled to a finite quantum bath. The spin-bath coupling is sufficiently weak that ETH does not apply, but sufficiently strong that perturbation theory fails. We calculate (i) the distribution of fidelity susceptibilities, which takes a broadly distributed form, (ii) the distribution of spin eigenstate entropies, which takes a bi-modal form, (iii) infinite time memory of spin observables, (iv) the distribution of matrix elements of local operators on the bath, which is non-Gaussian, and (v) the intermediate entropic enhancement of the bath, which interpolates smoothly between zero and the ETH value of $\log 2$. The enhancement is a consequence of rare many-body resonances, and is asymptotically larger than the typical eigenstate entanglement entropy. We verify these results numerically and discuss their connections to the many-body localisation transition.