The phase transition of the quantum Ising model is sharp
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.
Cite
@article{arxiv.0901.0328,
title = {The phase transition of the quantum Ising model is sharp},
author = {J. E. Björnberg and G. R. Grimmett},
journal= {arXiv preprint arXiv:0901.0328},
year = {2015}
}
Comments
Small changes. To appear in the Journal of Statistical Physics