Discrete holomorphicity and Ising model operator formalism

TL;DR

This paper connects the transfer matrix formalism with discrete complex analysis in the 2D Ising model, revealing that lattice fermion operators exhibit discrete holomorphicity and relate to parafermionic observables, extending beyond criticality.

Contribution

It introduces a discrete analytic continuation matrix linking transfer matrix formalism with discrete holomorphicity in the Ising model, including off-critical extensions.

Findings

Discrete holomorphicity of lattice fermion operators.

Correlation functions as Ising parafermionic observables.

Critical correlations computable via discrete Cauchy data spaces.

Abstract

We explore the connection between the transfer matrix formalism and discrete complex analysis approach to the two dimensional Ising model. We construct a discrete analytic continuation matrix, analyze its spectrum and establish a direct connection with the critical Ising transfer matrix. We show that the lattice fermion operators of the transfer matrix formalism satisfy, as operators, discrete holomorphicity, and we show that their correlation functions are Ising parafermionic observables. We extend these correspondences also to outside the critical point. We show that critical Ising correlations can be computed with operators on discrete Cauchy data spaces, which encode the geometry and operator insertions in a manner analogous to the quantum states in the transfer matrix formalism.