Ising Model on the Affine Plane

TL;DR

This paper establishes a precise correspondence between the Ising model on a triangular lattice with three couplings and a conformal field theory, incorporating lattice geometry to restore full conformal invariance in the continuum limit.

Contribution

It introduces a novel lattice formulation linking the Ising model with conformal field theory through affine transformations and geometric metrics, extending the understanding of CFT on curved manifolds.

Findings

Exact lattice formulation of Ising CFT on a torus as a function of modular parameter

Restoration of full conformal invariance via metric introduced through lattice couplings

Connection between simplicial geometry, projective geometry, and CFT on curved spaces

Abstract

We demonstrate that the Ising model on a general triangular graph with 3 distinct couplings $K_1,K_2,K_3$ corresponds to an affine transformed conformal field theory (CFT). Full conformal invariance of the $c= 1/2$ minimal CFT is restored by introducing a metric on the lattice through the map $\sinh(2K_i) = \ell^*_i/ \ell_i$ which relates critical couplings to the ratio of the dual hexagonal and triangular edge lengths. Applied to a 2d toroidal lattice, this provides an exact lattice formulation in the continuum limit to the Ising CFT as a function of the modular parameter. This example can be viewed as a quantum generalization of the finite element method (FEM) applied to the strong coupling CFT at a Wilson-Fisher IR fixed point and suggests a new approach to conformal field theory on curved manifolds based on a synthesis of simplicial geometry and projective geometry on the tangent planes.