Abstract
We present a method for defining a lattice realization of the $\phi^4$ 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 $\phi^4$ 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.
Highlights
Lattice field theory (LFT) has proven to be a powerful nonperturbative approach to quantum field theory [1]
We have set up the basic formalism for defining a scalar field theory on nontrivial Riemann manifolds and tested it numerically in the limited context of the 2-d φ4 theory by comparing it against the c 1⁄4 1=2 Ising conformal field theory (CFT) at the WilsonFisher fixed point. This test is at a sufficient accuracy compared with the exact result to encourage us that we are able to define this strongly coupled quantum field theory on a curved manifold, in this case S2
While this may be pursued in future works, our goal here was to sketch a framework for lattice quantum field theories on a nontrivial Riemann manifold as a quantum extension of finite element methods (FEM) which we refer to as quantum finite elements (QFE)
Summary
Lattice field theory (LFT) has proven to be a powerful nonperturbative approach to quantum field theory [1]. Before attempting a nonperturbative lattice construction, one should ask if a particular renormalizable field theory in flat space is even perturbatively renormalizable on a general smooth Riemann manifold. This question was addressed with an avalanche of important research [6,7,8,9] in the 1970s and 1980s. Unlike the hypercubic lattice with toroidal boundary condition, the largest discrete subgroup of the isometries of a sphere is the icosahedron in 2d and the hexacosichoron, or 600 cell, in 3d This greatly complicates constructing a suitable bare lattice Lagrangian that smoothly approaches the continuum limit of the renormalizable quantum field theory when the UV cut-off is removed. We fit the operator product expansion (OPE) as a test case of how to extract the central charge, OPE couplings, and operator dimensions
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