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arXiv · 2209.15546

Ising Model on the Affine Plane

Richard C. Brower·Evan K. Owen
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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.

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Richard C. Brower, Evan K. Owen. 2022-09-30. Ising Model on the Affine Plane. https://doi.org/10.1103/physrevd.108.014511

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