$(l,q)$-Deformed Grassmann Field and the Two-dimensional Ising Model

TL;DR

This paper develops an exact fermionic representation of the 2D Ising model and related vertex models using deformed Grassmann fields, extending fermionization techniques with $q$-deformations and functional integrals.

Contribution

It introduces a novel $(l,q)$-deformed Grassmann field framework for representing the Ising model and vertex models, generalizing fermionization methods with $q$-deformations.

Findings

Exact fermionic representation of the Ising partition function.

Representation of the eight-vertex model in external field via fermionic functional integral.

Connection of deformed Grassmann integrals to Pfaffians and determinants.

Abstract

In this paper we construct the exact representation of the Ising partition function in the form of the $ SL_q(2,R)$-invariant functional integral for the lattice free $(l,q)$-fermion field theory ($l=q=-1$). It is shown that the $(l,q)$-fermionization allows one to re-express the partition function of the eight-vertex model in external field through functional integral with four-fermion interaction. To construct these representations, we define a lattice $(l,q,s)$-deformed Grassmann bispinor field and extend the Berezin integration rules to this field. At $l=q=-1, s=1$ we obtain the lattice $(l,q)$-fermion field which allows us to fermionize the two-dimensional Ising model. We show that the Gaussian integral over $(q,s)$-Grassmann variables is expressed through the $(q,s)$-deformed Pfaffian which is equal to square root of the determinant of some matrix at $q=\pm 1, s=\pm 1$.