Fermionic Integrals and Analytic Solutions for Two-Dimensional Ising Models

TL;DR

This paper reviews the fermionic interpretation of the 2D Ising model, showing how its partition function can be expressed as a fermionic Gaussian integral and reformulated as a free-fermion theory, facilitating analytic solutions.

Contribution

It provides a comprehensive overview of fermionic integral methods and their application to solving the 2D Ising model analytically, including continuum-limit formulations.

Findings

Partition function expressed as fermionic Gaussian integral

Reformulation as free-fermion theory on a lattice

Analytic solutions obtained via momentum space analysis

Abstract

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 effective field-theoretical (continuum-limit) fermionic formulations for the 2D Ising models near the critical point.