Single Scale Cluster Expansions with Applications to Many Boson and Unbounded Spin Systems

TL;DR

This paper introduces a novel single-scale cluster expansion technique that demonstrates exponential decay of correlations in various spin systems, with applications to Bose gases and unbounded spin models.

Contribution

It develops a new cluster expansion method using a single application of the BKAR interpolation formula for general single-scale spin systems.

Findings

Proves exponential decay of correlations in lattice quantum field models.

Applies the method to Bose gas correlation functions at negative chemical potential.

Extends to unbounded spin systems with complex covariances.

Abstract

We develop a cluster expansion to show exponential decay of correlations for quite general single scale spin systems, as they arise in lattice quantum field theory and discretized functional integral representations for observables of quantum statistical mechanics. We apply our results to: the small field approximation to the coherent state correlation functions of the grand canonical Bose gas at negative chemical potential, constructed by Balaban, Feldman, Kn\"orrer and Trubowitz (2010); and to N component unbounded spin systems with repulsive two body interaction and massive, possibly complex, covariance. Our cluster expansion is derived by a single application of the BKAR interpolation formula.