Combinatorial and topological approach to the 3D Ising model

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 gis proportional to the number of sites, is discussed in detail. The expansion of the partition function is given in terms of 22g 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.