The phase transition of the quantum Ising model is sharp

TL;DR

This paper rigorously proves that the quantum Ising model with a transverse field exhibits a sharp phase transition, providing a precise critical point in one dimension and analyzing the transition's nature in higher dimensions.

Contribution

It introduces a novel 'random-parity' representation for the classical Ising model in d+1 dimensions and establishes the sharpness of the phase transition rigorously.

Findings

Confirmed the sharpness of the phase transition in the quantum Ising model.

Calculated the critical point exactly in one dimension.

Derived differential inequalities related to the model's behavior.

Abstract

An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d+1 dimensions. A so-called `random-parity' representation is developed for the latter model, similar to the random-current representation for the classical Ising model on a discrete lattice. Certain differential inequalities are proved. Integration of these inequalities yields the sharpness of the phase transition, and also a number of other facts concerning the critical and near-critical behaviour of the model under study.