Tensor network approach to 2D Yang-Mills theories

TL;DR

This paper introduces a new tensor network method for 2D Yang-Mills theories that directly uses group elements, enabling accurate computation of physical quantities for $SU(2)$ and $SU(3)$.

Contribution

It presents a novel tensor network representation for 2D Yang-Mills theories that bypasses character expansion, with applications to $SU(2)$ and $SU(3)$.

Findings

Accurate evaluation of free energy density and energy density.

Tensor singular value decomposition reveals group-theoretic structure.

Method applicable to arbitrary compact gauge groups.

Abstract

We propose a novel tensor network representation for two-dimensional Yang-Mills theories with arbitrary compact gauge groups. In this method, tensor indices are directly given by group elements with no direct use of the character expansion. We apply the tensor renormalization group method to this tensor network for $SU(2)$ and $SU(3)$, and find that the free energy density and the energy density are accurately evaluated. We also show that the singular value decomposition of a tensor has a group theoretic structure and can be associated with the character expansion.