Communications in Mathematical Physics

, Volume 218, Issue 1, pp 177–216

Discrete Riemann Surfaces and the Ising Model

  • Christian Mercat

DOI: 10.1007/s002200000348

Cite this article as:
Mercat, C. Commun. Math. Phys. (2001) 218: 177. doi:10.1007/s002200000348


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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Christian Mercat
    • 1
  1. 1.Université Louis Pasteur, Strasbourg, France. E-mail: mercat@math.u-strasbg.frFR