Journal of Mathematical Physics

The use of anticommuting variable integrals in statistical mechanics. III. Unsolved models

Stuart Samuel

Published Dec 1, 1980• 48 citations

Abstract

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.

DOI: 10.1063/1.524406

Related papers

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

Stuart Samuel

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.

Journal of Mathematical PhysicsDec 1, 1980145 citations

Free Fermions and Two-Dimensional Ising Model

V. N. Plechko

The two-dimensional Ising model is representable as a lattice free-fermion field theory in terms of the integral over anticommuting Grassmann variables. The exact solution in a zero magnetic field then follows by evaluating Gaussian fermionic integral for partition function. It is argued that by switching on of a non-zero magnetic field the non-local and non-Gaussian fermionic terms appear in the action, which rather imply approximating methods of the analysis.

arXiv:math-ph/0411084Nov 29, 20042 citations

The use of anticommuting variable integrals in statistical mechanics. II. The computation of correlation functions

Stuart Samuel

By using integrals over anticommuting variables all the correlation functions in the two-dimensional Ising model and free-fermion eight vertex model are computed. The method is quite general and applicable to other solvable systems.

Journal of Mathematical PhysicsDec 1, 198092 citations

Fermions and Correlations in the Two-Dimensional Ising Model

V. N. Plechko

The anticommuting analysis with Grassmann variables is applied to the two-dimensional Ising model in statistical mechanics. The discussion includes the transformation of the partition function into a Gaussian fermionic integral, the momentum-space representation and the spin-fermion correspondence at the level of the correlation functions.

arXiv:hep-th/0512263Dec 20, 20051 citation

Grassmann Variables and Exact Solutions for Two-Dimensional Dimer Models

R. Hayn, V. N. Plechko

We discuss some aspects of a new noncombinatorial fermionic approach to the two-dimensional dimer problem in statistical mechanics based on the integration over anticommuting Grassmann variables and factorization ideas for dimer density matrix. The dimer partition function can be expressed as a Gaussian fermionic integral. For regular lattices, the analytic solution then follows by passing to the momentum space for fermions.

arXiv:cond-mat/9711156PANNov 17, 19972 citations

Fermionic Path Integrals and Analytic Solutions for Two-Dimensional Ising Models

V. N. Plechko

The notion of the integral over the anticommuting Grassmann variables (nonquantum fermionic fields) seems to be the most powerful tool in order to extract the exact analytic solutions for the 2D Ising models on simple and more complicated lattices, which is the subject of a discussion in this report.

arXiv:hep-th/9609044Sep 5, 19960 citations