The fermionic observable in the Ising model and the inverse Kac-Ward operator

TL;DR

This paper links the fermionic observable in the Ising model to the inverse Kac-Ward operator on isoradial graphs, revealing deep connections between operator theory and statistical physics.

Contribution

It identifies the fermionic observable as the inverse of the Kac-Ward operator and introduces a fermionic generating function for planar graphs.

Findings

Inverse Kac-Ward operator acts as an s-holomorphicity operator.

Fermionic observable is the inverse of the Kac-Ward operator.

Provides bounds for spectral radius and operator norm of the transition matrix.

Abstract

We show that the critical Kac-Ward operator on isoradial graphs acts in a certain sense as the operator of s-holomorphicity, and we identify the fermionic observable for the spin Ising model as the inverse of this operator. This result is partially a consequence of a more general observation that the inverse Kac-Ward operator for any planar graph is given by what we call a fermionic generating function. Furthermore, using bounds for the spectral radius and operator norm of the Kac-Ward transition matrix, we provide a general picture of the non-backtracking walk representation of the critical and supercritical inverse Kac-Ward operators on isoradial graphs.