Anticommuting Integrals and Fermionic Field Theories for Two-Dimensional Ising Models

TL;DR

This paper reviews how anticommuting Grassmann integrals facilitate exact solutions and field-theoretical descriptions of 2D Ising models across various lattices, highlighting the transition from lattice models to continuum fermionic field theories.

Contribution

It introduces a unified fermionic integral approach for 2D Ising models on different lattices and derives the corresponding continuum Majorana and Dirac field theories.

Findings

Exact fermionic Gaussian integral representation of the partition function.

Universal form of the integral across various lattice geometries.

Field theory captures critical behavior near the transition temperature.

Abstract

We review the applications of the integral over anticommuting Grassmann variables (nonquantum fermionic fields) to the analytic solutions and the field-theoretical formulations for the 2D Ising models. The 2D Ising model partition function $Q$ is presentable as the fermionic Gaussian integral. The use of the spin-polynomial interpretation of the 2D Ising problem is stressed, in particular. Starting with the spin-polynomial interpretation of the local Boltzmann weights, the Gaussian integral for $Q$ appears in the universal form for a variety of lattices, including the standard rectangular, triangular, and hexagonal lattices, and with the minimal number of fermionic variables (two per site). The analytic solutions for the correspondent 2D Ising models then follow by passing to the momentum space on a lattice. The symmetries and the question on the location of critical point have an interesting interpretation within this spin-polynomial formulation of the problem. From the exact lattice theory we then pass to the continuum-limit field-theoretical interpretation of the 2D Ising models. The continuum theory captures all relevant features of the original models near $T_c$. The continuum limit corresponds to the low-momentum sector of the exact theory responsible for the critical-point singularities and the large-distance behaviour of correlations. The resulting field theory is the massive two-component Majorana theory, with mass vanishing at $T_c$. By doubling of fermions in the Majorana representation, we obtain as well the 2D Dirac field theory of charged fermions for 2D Ising models. The differences between particular 2D Ising lattices are merely adsorbed, in the field-theoretical formulation, in the definition of the effective mass.