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.

Small changes. To appear in the Journal of Statistical Physics

References in corpus (10)

• The Random-Cluster Model
• Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
• The Phase Diagram of the Quantum Curie-Weiss Model
• Entanglement in the quantum Ising model
• Correlation inequalities for the Potts model
• On the Critical Behavior at the Lower Phase Transition of the Contact Process
• Critical Value of the Quantum Ising Model on Star-Like Graphs
• Graphical representations of Ising and Potts models
• Random Current Representation for Transverse Field Ising Models
• Rigidity of the interface for percolation and random-cluster models

Cited by in corpus (16)

• Universality for the random-cluster model on isoradial graphs
• The free energy in a class of quantum spin systems and interchange processes
• Correlation inequalities for the Potts model
• Quantum Griffiths inequalities
• Vanishing critical magnetization in the quantum Ising model
• Decay of transverse correlations in quantum Heisenberg models
• Critical Value of the Quantum Ising Model on Star-Like Graphs
• Infrared bound and mean-field behaviour in the quantum Ising model
• Bounded entanglement entropy in the quantum Ising model
• Graphical representations of Ising and Potts models
• Random Current Representation for Transverse Field Ising Models
• Fermionic observables in the transverse Ising chain
• The Interface of the FK-representation of the Quantum Ising Model Converges to the $SL{E}_{16/3}$
• Geometric analysis of Ising models, Part III
• Multidimensional Contours à la Fröhlich-Spencer and Boundary Conditions for Quantum Spin Systems
• Kac-Ward solution of the 2D classical and 1D quantum Ising models