[Paper Review] Anticommuting Integrals and Fermionic Field Theories for Two-Dimensional Ising Models

V. N. Plechko|ArXiv.org|Jul 7, 1996

Theoretical and Computational Physics3 references3 citations

TL;DR

This paper presents a fermionic field theory formulation of two-dimensional Ising models using Grassmann integrals, showing that the partition function can be expressed as a Gaussian fermionic integral via a spin-polynomial representation. The method yields the massive two-component Majorana theory in the continuum limit, with critical behavior captured by a vanishing mass at T_c, unifying various lattice types under a universal field-theoretic framework.

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.

Motivation & Objective

• To establish a unified field-theoretic formulation of 2D Ising models across different lattices using Grassmann variables.
• To demonstrate that the partition function of the 2D Ising model can be expressed as a Gaussian fermionic integral through a spin-polynomial representation.
• To derive the continuum limit of the lattice theory, identifying the resulting field theory as a massive two-component Majorana theory.
• To show that critical point singularities and long-distance correlations emerge from the low-momentum sector of the exact lattice theory.
• To explore the role of symmetries and lattice-specific differences in the field-theoretic formulation, particularly in the effective mass term.

Proposed method

• Represent the local Boltzmann weights of the Ising model as polynomials in spin variables, enabling a Grassmann integral formulation.
• Express the full partition function Q as a Gaussian integral over anticommuting Grassmann variables, using only two fermionic variables per lattice site.
• Apply a momentum-space transformation on the lattice to analytically solve the model, revealing universal behavior across rectangular, triangular, and hexagonal lattices.
• Identify the continuum limit as the low-momentum sector of the lattice theory, where critical phenomena are governed by the massless limit of a Majorana field.
• Double the fermion degrees of freedom in the Majorana representation to obtain an equivalent 2D Dirac field theory for charged fermions.
• Absorb lattice-specific differences into the definition of the effective mass in the continuum field theory.

Experimental results

Research questions

• RQ1 Can the partition function of the 2D Ising model be universally formulated as a Grassmann Gaussian integral across different lattice types?
• RQ2 How do symmetries and critical point locations manifest in the spin-polynomial formulation of the Ising model?
• RQ3 What is the continuum field theory that captures the critical and long-distance behavior of the 2D Ising model?
• RQ4 How do lattice-specific details (e.g., triangular vs. rectangular) influence the effective field theory in the continuum limit?
• RQ5 What is the relationship between the Majorana and Dirac field theories in the context of 2D Ising models?

Key findings

• The partition function of the 2D Ising model is exactly representable as a Gaussian Grassmann integral using a spin-polynomial formulation.
• The method yields a universal form of the partition function for rectangular, triangular, and hexagonal lattices with minimal fermionic degrees of freedom (two per site).
• Analytic solutions are obtained by transforming to momentum space on the lattice, revealing universal critical behavior.
• The continuum limit of the theory is a massive two-component Majorana field theory, with the mass vanishing at the critical temperature T_c.
• Lattice-specific differences are encoded in the effective mass term of the continuum field theory, not in the underlying field structure.
• By doubling the fermions in the Majorana representation, the theory is equivalent to a 2D Dirac field theory of charged fermions, providing a field-theoretic realization of the Ising model.