Lattice Dirac Fermions on a Simplicial Riemannian Manifold

TL;DR

This paper develops a lattice formulation of the Dirac equation on simplicial complexes approximating smooth Riemannian manifolds, demonstrating numerical convergence and comparison with continuum conformal field theory results.

Contribution

It introduces a general lattice Dirac operator on simplicial manifolds with a lattice vierbein and spin connection, applicable to any smooth Riemannian manifold with a spin structure.

Findings

Eigenvalues and eigenvectors converge rapidly to continuum results.

Numerical tests on the projective sphere show accurate spectral convergence.

Correlation functions match continuum conformal field theory predictions.

Abstract

The lattice Dirac equation is formulated on a simplicial complex which approximates a smooth Riemann manifold by introducing a lattice vierbein on each site and a lattice spin connection on each link. Care is taken so the construction applies to any smooth D-dimensional Riemannian manifold that permits a spin connection. It is tested numerically in 2D for the projective sphere ${\mathbb S}^2$ in the limit of an increasingly refined sequence of triangles. The eigenspectrum and eigenvectors are shown to converge rapidly to the exact result in the continuum limit. In addition comparison is made with the continuum Ising conformal field theory on ${\mathbb S}^2$. Convergence is tested for the two point, $\langle \epsilon(x_1) \epsilon(x_2) \rangle$, and the four point, $\langle \sigma(x_1) \epsilon(x_2) \epsilon(x_3 )\sigma(x_4) \rangle $, correlators for the energy, $\epsilon(x) = i \bar \psi(x)\psi(x)$, and twist operators, $\sigma(x)$, respectively.