Discrete Riemann Surfaces and the Ising model - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Communications in Mathematical Physics Année : 2001

Discrete Riemann Surfaces and the Ising model

Résumé

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.
Fichier principal
Vignette du fichier
DiscRiem.pdf (479.53 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00418532 , version 1 (19-09-2009)

Identifiants

Citer

Christian Mercat. Discrete Riemann Surfaces and the Ising model. Communications in Mathematical Physics, 2001, 218 (1), pp.177-216. ⟨10.1007/s002200000348⟩. ⟨hal-00418532⟩
145 Consultations
582 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More