The phase transition of the quantum Ising model is sharp

Details The phase transition of the quantum Ising model is sharp The phase transition of the quantum Ising model is sharp

2009-01-03

arxiv.org/abs/0901.0328v2

Physics

Mathematical Physics

Small changes. To appear in the Journal of Statistical Physics

Scientific paper

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.

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