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.
Highlights
Numerous important aspects of nonperturbative quantum field theories are best understood on curved space-time manifolds
As an example, when a conformal field theory (CFT) on Euclidean Rd is mapped to the Riemann sphere, Sd, the free energy gives direct access to the central charge [4]
Model building to search for potential new physics in composite Higgs or dark matter scenarios beyond the standard model increasingly focus on small
Summary
Numerous important aspects of nonperturbative quantum field theories are best understood on curved space-time manifolds. One solution is the extension of lattice field theory methods beyond Euclidean flat space Towards this solution, we have developed in previous works a quantum extension of classical finite elements by placing the lattice theory on a simplicial complex with appropriate counterterms, referred to as quantum finite elements (QFE) [1,2]. Explicit QFE counterterms are introduced in order to restore the exact nonperturbative quantum physics as the cutoff is removed To date this QFE method has been tested with numerical simulations for the 2D φ4 theory on S2 and has been found to be in precise agreement with the exact solution of the minimal c 1⁄4 1=2 Ising CFT [1].
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