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The Phase Transition of the Quantum Ising Model is Sharp

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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.

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  1. Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    J. E. Björnberg & G. R. Grimmett

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  1. J. E. Björnberg
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  2. G. R. Grimmett
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Correspondence to G. R. Grimmett.

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Björnberg, J.E., Grimmett, G.R. The Phase Transition of the Quantum Ising Model is Sharp. J Stat Phys 136, 231–273 (2009). https://doi.org/10.1007/s10955-009-9788-z

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