The Phase Transition of the Quantum Ising Model is Sharp

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...

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Bibliographic Details
Published in:Journal of statistical physics 2009-07, Vol.136 (2), p.231-273
Main Authors: Björnberg, J. E., Grimmett, G. R.
Format: Article
Language:English
Subjects:
Mathematical and Computational Physics
Physical Chemistry
Physics
Physics and Astronomy
Quantum Physics
Statistical Physics and Dynamical Systems
Theoretical
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Summary: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.
ISSN:0022-4715
1572-9613
DOI:10.1007/s10955-009-9788-z