Discrete Riemann Surfaces and the Ising model

TL;DR

This paper develops a new theory of discrete Riemann surfaces, establishing classical complex analysis results in a discrete setting and connecting criticality with the Ising model through a Dirac equation framework.

Contribution

It introduces a novel discrete Riemann surface theory incorporating dual cellular decompositions and links criticality to the existence of Dirac spinors in the Ising model.

Findings

Discrete holomorphy defined via discretised Cauchy-Riemann equations

Classical Riemann surface results have discrete analogues

Existence of Dirac spinors characterizes criticality in the Ising model

Abstract

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy-Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality.