Locally accurate tensor networks for thermal states and time evolution

TL;DR

This paper introduces locally accurate tensor network methods that efficiently approximate thermal states and time evolution in quantum systems with bond dimensions independent of system size, backed by rigorous mathematical foundations.

Contribution

It constructs PEPOs that approximate thermal expectation values and Heisenberg time evolution with bond dimensions depending only on temperature or time, not system size.

Findings

PEPOs can approximate thermal states with size-independent bond dimension.

PEPOs effectively model Heisenberg time evolution for local observables.

Method enables accurate approximation of thermal correlations and quenches.

Abstract

Tensor network methods are routinely used in approximating various equilibrium and non-equilibrium scenarios, with the algorithms requiring a small bond dimension at low enough time or inverse temperature. These approaches so far lacked a rigorous mathematical justification, since existing approximations to thermal states and time evolution demand a bond dimension growing with system size. To address this problem, we construct PEPOs that approximate, for all local observables, $i)$ their thermal expectation values and $ii)$ their Heisenberg time evolution. The bond dimension required does not depend on system size, but only on the temperature or time. We also show how these can be used to approximate thermal correlation functions and expectation values in quantum quenches.