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Journal of Mathematical Physics

The use of anticommuting variable integrals in statistical mechanics. I. The computation of partition functions

Stuart Samuel

Published Dec 1, 1980• 145 citations
Publisher

Abstract

Integrals over anticommuting variables are use to rewrite partition functions as fermionic field theories. The method is used to solve the two-dimensional Ising model, the planar close-packed dimer problems, and the free-fermion eight vertex model.
DOI
10.1063/1.524404

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The Ising model in three dimensions is fermionized by using integrals over anticommuting variables. The result is generalized to the Ising model in arbitrary dimensions and in a magnetic field. Approximation methods are developed to attack unsolved statistical mechanics models. Perturbation theory and the Hartree approximation are applied to the unsolved monomer-dimer problems. The result is a numerical solution to this unsolved class of problems. Anticommuting variables appear to be a powerful approach to unsolved problems.
We review some aspects of the fermionic interpretation of the two-dimensional Ising model.The use is made of the notion of the integral over the anticommuting Grassmann variables.For simple and more complicated 2D Ising lattices, the partition function can be expressed as a fermionic Gaussian integral.Equivalently, the 2D Ising model can be reformulated as a free-fermion theory on a lattice.For regular lattices the analytic solution then readily follows by passing to the momentum space for fermions.We also comment on the eective eld{theoretical (continuum{limit) fermionic formulations for the 2D Ising models near the critical point.