Studying $\textrm{QED}_3$ with radial quantization on the lattice: Free limit

TL;DR

This work develops a lattice formulation of QED$_3$ in radial quantization on $S^2\times\mathbb{R}$ using an icosahedral discretization, focusing on the free limit to validate the approach. It implements both Wilson and overlap fermions and a Gaussian non-compact $U(1)$ gauge action, deriving analytic continuum correlators on $S^2\times\mathbb{R}$ and verifying them against lattice results without Monte Carlo sampling; the results exhibit correct normalization and $O(a^2)$ scaling toward the continuum. Crucially, the overlap fermion preserves the essential lattice symmetries via the Ginsparg-Wilson relation, enabling faithful reproduction of low-lying operator dimensions on reasonably coarse lattices. These findings establish a solid baseline for extending to interacting theories with Hybrid Monte Carlo for multiple flavors and exploring conformal dynamics, while highlighting directions such as improved spherical symmetry, domain-wall formulations, and comparisons with fuzzy-sphere methods.

Abstract

To investigate the three-dimensional quantum electrodynamics in the radial quantization on the lattice, the lattice action is constructed and the free limit is studied on $S^2 \times \mathbb{R}$. With the overlap fermion, it is numerically verified that the important symmetries of the theory can be realized on the lattice. The analytic correlators are derived and compared to the lattice results, which agree including the overall normalization. The $O(a^2)$-scaling is confirmed toward the analytic value in the continuum limit, and the number of reproduced excited states is estimated heuristically for the first few refinement levels. Our study helps us identify the features of the theory that we can study on the icosahedral lattice without fine-tuning.

Figures (14)

• Figure 1: The tessellated $S^2$ with the refinement level $L=2$.
• Figure 2: The area factors for (Left) the site $y$, $A_y$, and (Right) the link $y_1y_2$, $A_{y_1y_2}$.
• Figure 3: (Left) A triangular prism and (Right) the base triangle. The dual point of a spatial plaquette is defined by projecting the circumcenter onto the sphere.
• Figure 4: Spectrum of the free Wilson-Dirac operator $D_W$ with $T=4$. (Left) With various refinement levels $L=1,2,4$ for a fixed $L_t=24$. (Right) With various temporal extent $L_t=16, 24, 48$ for a fixed $L=2$.
• Figure 5: The flat spectrum \ref{['eq:flat_spectrum']} of the Wilson fermion on the equilateral triangular lattice with radial extension, which we use as an approximation of the spectrum on $S^2\times\mathbb{R}$. The couplings are set from $\bar{a}_s$ and $a_t$ of the lattices in Fig. \ref{['fig:eig_W']}.