Fermions as generalized Ising models

TL;DR

This paper develops a general mapping between fermionic Grassmann functionals and Ising spin models, enabling the representation of fermionic quantum field theories as classical spin systems across various dimensions.

Contribution

It introduces a novel, dimension-independent map that preserves locality, linking fermionic and Ising models, and demonstrates this with a 2D Dirac fermion example.

Findings

Established a general map between Grassmann functionals and Ising models

Preserved locality properties in the mapping

Represented a 2D Dirac fermion as an Ising model

Abstract

We establish a general map between Grassmann functionals for fermions and probability or weight distributions for Ising spins. The equivalence between the two formulations is based on identical transfer matrices and expectation values of products of observables. The map preserves locality properties and can be realized for arbitrary dimensions. We present a simple example where a quantum field theory for free massless Dirac fermions in two-dimensional Minkowski space is represented by an asymmetric Ising model on a euclidean square lattice.