The importance of the Ising model

Abstract

Understanding the relationship which integrable (solvable) models, all of which possess very special symmetry properties, have with the generic non-integrable models that are used to describe real experiments, which do not have the symmetry properties, is one of the most fundamental open questions in both statistical mechanics and quantum field theory. The importance of the two-dimensional Ising model in a magnetic field is that it is the simplest system where this relationship may be concretely studied. We here review the advances made in this study, and concentrate on the magnetic susceptibility which has revealed an unexpected natural boundary phenomenon. When this is combined with the Fermionic representations of conformal characters, it is suggested that the scaling theory, which smoothly connects the lattice with the correlation length scale, may be incomplete for $H\ne 0$.

arXiv comment33 pages

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• From Network Reliability to the Ising Model: A Parallel Scheme for Estimating the Joint Density of States
• Existence and Uniqueness of Tronquée Solutions of the Third and Fourth Painlevé Equations
• Leading corrections to the scaling function on the diagonal for the two-dimensional Ising model
• Simulation of classical Ising-like magnetism with a Mott insulator of paired atoms
• Conformal Measure Ensembles and Planar Ising Magnetization: A Review
• Factorization of Ising correlations C(M,N) for $\nu =-k$ and M+N odd, $M\le N$, $T<T_c$ and their lambda extensions
• Exact matrix model for generalized Ising model
• A Gaussian process related to the mass spectrum of the near-critical Ising model
• The Gaussian process for particle masses in the near-critical Ising model
• Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman
• Localized dopant motion across the 2D Ising phase transition
• Infinite-temperature thermostats by energy localization in a nonequilibrium setup