Radial lattice quantization of 3D ϕ 4 field theory

The quantum extension of classical finite elements, referred to as quantum finite elements (QFE) [R. C. Brower et al., Lattice ${\ensuremath{\phi}}^{4}$ field theory on Riemann manifolds: Numerical tests for the 2-d Ising CFT on ${\mathbb{S}}^{2}$, Phys. Rev. D 98, 014502 (2018). and R. C. Brower et al., Lattice dirac fermions on a simplicial Riemannian manifold, Phys. Rev. D 95, 114510 (2017).], is applied to the radial quantization of 3D ${\ensuremath{\phi}}^{4}$ theory on a simplicial lattice for the $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{2}$ manifold. Explicit counterterms to cancel the one- and two-loop ultraviolet defects are implemented to reach the quantum continuum theory. Using the Brower-Tamayo [Embedded Dynamics for ${\ensuremath{\phi}}^{4}$ Theory, Phys. Rev. Lett. 62, 1087 (1989).] 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 $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{2}$ 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 operator product expansion couplings, and the central charge.