Combinatorial and topological approach to the 3D Ising model

TL;DR

This paper generalizes the Pfaffian formalism for the Ising model to complex 3D lattices with high topological genus, providing a new topological and combinatorial framework for evaluating partition functions.

Contribution

It introduces a novel topological approach to the 3D Ising model using Pfaffians classified by homology cycles, extending previous planar methods.

Findings

Partition function expressed as 2^{2g} Pfaffians

Inclusion of a signature term for correct counting

Formalism applicable to perfect matching and matrix permanent problems

Abstract

We extend the planar Pfaffian formalism for the evaluation of the Ising partition function to lattices of high topological genus g. The 3D Ising model on a cubic lattice, where g is proportional to the number of sites, is discussed in detail. The expansion of the partition function is given in terms of 2^{2 g} Pfaffians classified by the oriented homology cycles of the lattice, i.e. by its spin-structures. Correct counting is guaranteed by a signature term which depends on the topological intersection of the oriented cycles through a simple bilinear formula. The role of a gauge symmetry arising in the above expansion is discussed. The same formalism can be applied to the counting problem of perfect matchings over general lattices and provides a determinant expansion of the permanent of 0-1 matrices.