Subcriticality at High Temperatures in Spin Lattice Systems

TL;DR

This paper establishes new criteria for subcriticality in classical and quantum spin lattice systems at high temperatures, based on the uniqueness of KMS states and a novel analytical approach.

Contribution

It introduces a uniform condition independent of local Hilbert space dimension, enlarging the class of interactions for which subcriticality can be proven.

Findings

Provides sufficient conditions for subcriticality based on KMS state uniqueness.

Enlarges the class of interactions by relying only on the $C^*$-norm estimates.

Offers improved lower bounds on the inverse temperature for subcriticality.

Abstract

We provide new sufficient conditions for subcriticality of classical and quantum spin lattice systems, formulated in terms of the uniqueness of Kubo-Martin-Schwinger (KMS) states. This is achieved by exploiting a non-commutative analog of the Kirkwood-Salzburg equations together with a novel decomposition of local observables. In contrast to standard approaches \cite{Bratteli_Robinson_97,Frohlich_Ueltschi_2015}, our condition is uniform with respect to the dimension of the single-site Hilbert space. Moreover, unlike the results of \cite{Drago_Pettinari_Van_de_Ven_2025}, which required control over the growth of the derivatives of the interaction potentials, our result only involves estimating the natural $C^*$-norm of these potentials. This substantially enlarges the class of interactions for which the theorems apply and provides better lower bounds on the subcritical inverse temperature.