Radial Lattice Quantization of 3D $\phi^4$ Field Theory

TL;DR

This paper applies quantum finite elements to radial quantization of 3D $$ theory on a lattice, demonstrating that the continuum conformal field theory can be accurately reached and analyzed with improved Monte Carlo methods.

Contribution

It introduces a quantum finite element approach to radial quantization of 3D $$ theory, including counter terms and curvature effects, enabling high-precision CFT data extraction.

Findings

Quantum finite elements successfully reach the continuum CFT.

Counter terms cancel ultraviolet defects effectively.

Ricci curvature improves convergence in simulations.

Abstract

The quantum extension of classical finite elements, referred to as quantum finite elements ({\bf QFE})~\cite{Brower:2018szu,Brower:2016vsl}, is applied to the radial quantization of 3d $\phi^4$ theory on a simplicial lattice for the $\mathbb R \times \mathbb S^2$ manifold. Explicit counter terms to cancel the one- and two-loop ultraviolet defects are implemented to reach the quantum continuum theory. Using the Brower-Tamayo~\cite{Brower:1989mt} cluster Monte Carlo algorithm, numerical results support the QFE ansatz that the critical conformal field theory (CFT) is reached in the continuum with the full isometries of $\mathbb R \times \mathbb S^2$ restored. The Ricci curvature term, while technically irrelevant in the quantum theory, is shown to dramatically improve the convergence opening, the way for high precision Monte Carlo simulation to determine the CFT data: operator dimensions, trilinear OPE couplings and the central charge.