Ising Model on the Affine Plane

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 $\sinh (2K_i)={\ell }_i^{\ast }/{\ell }_i$ 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.

References in corpus (8)

• Bayesian model averaging for analysis of lattice field theory results
• Lattice Dirac Fermions on a Simplicial Riemannian Manifold
• Lattice ${\varphi }^4$ Field Theory on Riemann Manifolds: Numerical Tests for the 2-d Ising CFT on ${\mathbb{S}}^2$
• Hyperbolic Lattice for Scalar Field Theory in AdS${}_3$
• Modular invariance, lattice field theories and finite size corrections
• Critical Behavior of Ferromagnetic Ising Model on Triangular Lattice
• Euclidean Dynamical Triangulations Revisited
• Ising model as Wilson-Majorana Fermions

Cited by in corpus (1)

• The Operator Product Expansion for Radial Lattice Quantization of 3D ${\varphi }^4$ Theory