Random Current Representation for Transverse Field Ising Models

TL;DR

This paper develops a space-time random current representation for transverse field Ising models, proving correlation inequalities and exponential decay of correlations, advancing understanding of phase transitions in quantum spin systems.

Contribution

It formulates and proves space-time versions of the classical switching lemma and correlation inequalities for the transverse field Ising model.

Findings

Proves exponential decay of two-point functions at positive magnetic fields.

Establishes correlation inequalities using the new representation.

Addresses the sharpness of phase transition in the model.

Abstract

Recently, a random current representation for transverse field Ising models has been introduced in \cite{ILN}. This representation is a space-time version of the classical random current representation exploited by Aizenman et. al. %It is a space-time version of the classical random current representation \cite{Ai82, ABF, AF}. In this paper we formulate and prove corresponding space-time versions of the classical switching lemma and show how they generate various correlation inequalities. In particular we prove exponential decay of truncated two-point functions at positive magnetic fields in $\sfz$-direction and address the issue of the sharpness of phase transition.