High-temperature cluster expansion for classical and quantum spin lattice systems with multi-body interactions

TL;DR

This paper introduces a new cluster expansion method for classical and quantum spin lattice systems with multi-body interactions, enabling explicit calculations of free energy and correlations at small inverse temperature.

Contribution

It develops a novel cluster expansion using decoupling parameters and a tree-diagram summation scheme, improving bounds on the analyticity radius for quantum systems.

Findings

Explicit expansion in a $eta$-dependent effective fugacity

Larger lower bound of the $eta$-analyticity radius than previous methods for quantum two-body interactions

Applicable to both classical and quantum multi-body spin systems

Abstract

We develop a novel cluster expansion for finite-spin lattice systems subject to multi-body quantum -- and, in particular, classical -- interactions. Our approach is based on the use of ``decoupling parameters", advocated by Park [34], which relates partition functions with successive additional interaction terms. Our treatment, however, leads to an explicit expansion in a $\beta$-dependent effective fugacity that permits an explicit evaluation of free energy and correlation functions at small $\beta$. To determine its convergence region we adopt a relatively recent cluster summation scheme that replaces the traditional use of Kikwood-Salzburg-like integral equations by more precise sums in terms of particular tree-diagrams [2]. As an application we show that our lower bound of the radius of $\beta$-analyticity is larger than Park's for quantum systems two-body interactions.