Subsystem eigenstate thermalization hypothesis for translation invariant systems

TL;DR

This paper proves the subsystem eigenstate thermalization hypothesis for translation invariant quantum spin systems without relying on random matrix theory, using quantum variance and relative entropy bounds.

Contribution

It introduces a new elementary proof of the subsystem ETH for translation invariant systems based on quantum variance and relative entropy, applicable to many lattice models.

Findings

Proves subsystem ETH with algebraic convergence speed

Establishes bounds on quantum variance and relative entropy

Applicable to models with exponential or algebraic decay of correlations

Abstract

The eigenstate thermalization hypothesis for translation invariant quantum spin systems has been proved recently by using random matrices. In this paper, we study the subsystem version of eigenstate thermalization hypothesis for translation invariant quantum systems without referring to random matrices. We first find a relation between the quantum variance and the Belavkin-Staszewski relative entropy. Then, by showing the small upper bounds on the quantum variance and the Belavkin-Staszewski relative entropy, we prove the subsystem eigenstate thermalization hypothesis for translation invariant quantum systems with an algebraic speed of convergence in an elementary way. The proof holds for most of the translation invariant quantum lattice models with exponential or algebraic decays of correlations.