Nonequilibrium steady state in Lindblad dynamics for infinite quantum spin systems

TL;DR

This paper investigates the properties of nonequilibrium steady states in infinite quantum spin systems under Lindblad dynamics, establishing conditions for their equivalence with finite-system limits using $C^*$-algebraic formalism.

Contribution

It introduces a sufficient condition involving a condition number and spectral gaps that ensures the equivalence of steady states in infinite and finite systems.

Findings

The NESS on an infinite system may differ from the thermodynamic limit of finite-system NESSs.

A condition number quantifies the normality of a Liouvillian affecting steady state equivalence.

An example shows spectral gaps alone do not guarantee the commutation of limits.

Abstract

We consider Lindblad dynamics of quantum spin systems on infinite lattices and define a nonequilibrium steady state (NESS) and a time-averaged nonequilibrium steady state (TANESS) on the basis of $C^*$-algebraic formalism. Generically, the NESS on an infinite system does not equal the thermodynamic limit of NESSs on finite systems. We give a sufficient condition that they coincide with each other, in terms of both a condition number, which quantifies the normality of a Liouvillian, and some spectral gaps on finite subsystems. To appreciate the importance of the condition number, we provide an example in which the spectral gaps have nonzero lower bounds uniformly for any finite subsystems but a thermodynamic limit and a long-time limit (or a long-time average) do not commute with each other.