Grassmannization of classical models

TL;DR

This paper introduces a Grassmann integral reformulation of classical lattice models' high-temperature expansions, enabling diagrammatic analysis and potentially advancing the study of lattice gauge theories with mixed degrees of freedom.

Contribution

It presents a novel method to reformulate classical models using Grassmann variables, overcoming previous limitations for diagrammatic expansions.

Findings

Successfully applied to the 2D Ising model

Allows diagrammatic perturbation around a Gaussian point

Facilitates unified treatment of bosonic and fermionic degrees of freedom

Abstract

Applying Feynman diagrammatics to non-fermionic strongly correlated models with local constraints might seem generically impossible for two separate reasons: (i) the necessity to have a Gaussian (non-interacting) limit on top of which the perturbative diagrammatic expansion is generated by Wick's theorem, and (ii) the Dyson's collapse argument implying that the expansion in powers of coupling constant is divergent. We show that for arbitrary classical lattice models both problems can be solved/circumvented by reformulating the high-temperature expansion (more generally, any discrete representation of the model) in terms of Grassmann integrals. Discrete variables residing on either links, plaquettes, or sites of the lattice are associated with the Grassmann variables in such a way that the partition function (and correlations) of the original system and its Grassmann-field counterpart are identical. The expansion of the latter around its Gaussian point generates Feynman diagrams. A proof-of-principle implementation is presented for the classical 2D Ising model. Our work paves the way for studying lattice gauge theories by treating bosonic and fermionic degrees of freedom on equal footing.