Fermionic Structure of Two-Dimensional Ising Model with Quenched Site Dilution

TL;DR

This paper introduces a fermionic path integral approach to analyze the 2D Ising model with quenched site dilution, revealing a complex interaction structure and critical behavior near the transition temperature.

Contribution

It reformulates the diluted 2D Ising model as an interacting fermion theory using a novel anticommuting path integral technique, highlighting unique interaction orders and critical singularities.

Findings

Exact lattice fermionic interaction structure identified

Effective four-fermion interaction near critical temperature

Doubled-logarithmic singularity in specific heat near T_c

Abstract

We apply a new anticommuting path integral technique to clarify the fermionic structure of the 2D Ising model with quenched site dilution. In the $N$-replica scheme, the model is explicitly reformulated as a theory of interacting fermions on a lattice. An unusual feature is that the leading term of interaction in the exact lattice theory is of order $2N$ in fermions, where $N$ is the number of replicas. The continuum-limit approximation near $T_c $ for weak dilution produces, however, an effective four-fermion interaction. In particular, this implies the doubled-logarithmic singularity in the specific heat near $T_c$ for weak site dilution. The exact value of the initial slope is also obtained. Keywords: Ising model, site disorder, anticommuting Grassmann variables