Equilibrium Properties of Quantum Spin Systems with Non-additive Long-Range Interactions

TL;DR

This paper investigates the equilibrium properties of quantum spin systems with long-range interactions, demonstrating ensemble equivalence conditions and analyzing the validity of mean-field approximations, with implications for experimental systems.

Contribution

It extends classical results on free energy minimization to quantum systems and clarifies when mean-field models accurately represent long-range quantum spin interactions.

Findings

Ensemble equivalence depends on parameter regions.

Replacing long-range interactions with mean-field models is justified when ensembles are equivalent.

The Heisenberg XXZ model exemplifies these theoretical insights.

Abstract

We study equilibrium states of quantum spin systems with non-additive long-range interactions by adopting an appropriate scaling of the interaction strength, i.e., the so called Kac prescription. In classical spin systems, it is known that the equilibrium free energy is obtained by minimizing the free energy functional over the coarse-grained magnetization. Here we show that it is also true for quantum spin systems. From this observation, it is found that when the canonical ensemble and the microcanonical ensemble are not equivalent in some parameter region, it is not necessarily justified to replace the actual long-range interaction by the infinite-range interaction (Curie-Weiss type interaction). On the other hand, in the parameter region where the two ensembles are equivalent, this replacement is always justified. We examine the Heisenberg XXZ model as an illustrative example, and discuss the relation to experiments.