Lattice ${\varphi }^4$ Field Theory on Riemann Manifolds: Numerical Tests for the 2-d Ising CFT on ${\mathbb{S}}^2$

Abstract

We present a method for defining a lattice realization of the ${\varphi }^4$ quantum field theory on a simplicial complex in order to enable numerical computation on a general Riemann manifold. The procedure begins with adopting methods from traditional Regge Calculus (RC) and finite element methods (FEM) plus the addition of ultraviolet counter terms required to reach the renormalized field theory in the continuum limit. The construction is tested numerically for the two-dimensional ${\varphi }^4$ scalar field theory on the Riemann two-sphere, ${\mathbb{S}}^2$, in comparison with the exact solutions to the two-dimensional Ising conformal field theory (CFT). Numerical results for the Binder cumulants (up to 12th order) and the two- and four-point correlation functions are in agreement with the exact $c=1/2$ CFT solutions.

arXiv comment52 pages, 27 figures

References in corpus (11)

• Properties and uses of the Wilson flow in lattice QCD
• Lecture Notes on General Relativity
• EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions
• Ising Universality in Three Dimensions: A Monte Carlo Study
• Strongly interacting dynamics and the search for new physics at the LHC
• Discrete Exterior Calculus
• A geometric discretisation scheme applied to the Abelian Chern-Simons theory
• Lattice Radial Quantization: 3D Ising
• Lattice Dirac Fermions on a Simplicial Riemannian Manifold
• Simplicial Ricci Flow
• Lattice Gauge Theories at the Energy Frontier

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