Tensor renormalization group and the volume independence in 2D U($N$) and SU($N$) gauge theories

TL;DR

This paper applies tensor renormalization group methods to 2D U(N) and SU(N) gauge theories, revealing large-N spectral behaviors and a novel volume independence phenomenon beyond traditional Eguchi-Kawai reduction.

Contribution

It introduces a practical strategy for handling higher-rank gauge groups in tensor networks and demonstrates volume independence in 2D U(N) gauge theories with a theta term.

Findings

Spectral profile of the fundamental tensor in large-N limit

Qualitative change in singular-value spectrum at the Gross-Witten-Wadia transition

Discovery of a new volume independence in the strong coupling phase

Abstract

The tensor renormalization group method is a promising approach to lattice field theories, which is free from the sign problem unlike standard Monte Carlo methods. One of the remaining issues is the application to gauge theories, which is so far limited to U(1) and SU(2) gauge groups. In the case of higher rank, it becomes highly nontrivial to restrict the number of representations in the character expansion to be used in constructing the fundamental tensor. We propose a practical strategy to accomplish this and demonstrate it in 2D U($N$) and SU($N$) gauge theories, which are exactly solvable. Using this strategy, we obtain the singular-value spectrum of the fundamental tensor, which turns out to have a definite profile in the large-$N$ limit. For the U($N$) case, in particular, we show that the large-$N$ behavior of the singular-value spectrum changes qualitatively at the critical coupling of the Gross-Witten-Wadia phase transition. As an interesting consequence, we find a new type of volume independence in the large-$N$ limit of the 2D U($N$) gauge theory with the $\theta$ term in the strong coupling phase, which goes beyond the Eguchi-Kawai reduction.