Variational Monte Carlo simulation with tensor networks of a pure $\mathbb{Z}_3$ gauge theory in (2+1)d

TL;DR

This paper demonstrates a variational Monte Carlo method combined with tensor networks to numerically study the ground state of a 2D pure $ ext{Z}_3$ lattice gauge theory, advancing computational techniques in quantum gauge systems.

Contribution

It introduces a novel application of gauged Gaussian PEPS with Monte Carlo to simulate a $ ext{Z}_3$ gauge theory in two dimensions, enabling more efficient and scalable computations.

Findings

Successful variational Monte Carlo simulation of a $ ext{Z}_3$ gauge theory

First proof of principle for this combined tensor network and Monte Carlo approach

Method can be extended to systems with fermions

Abstract

Variational minimization of tensor network states enables the exploration of low energy states of lattice gauge theories. However, the exact numerical evaluation of high-dimensional tensor network states remains challenging in general. In [E. Zohar, J. I. Cirac, Phys. Rev. D 97, 034510 (2018)] it was shown how, by combining gauged Gaussian projected entangled pair states with a variational Monte Carlo procedure, it is possible to efficiently compute physical observables. In this paper we demonstrate how this approach can be used to investigate numerically the ground state of a lattice gauge theory. More concretely, we explicitly carry out the variational Monte Carlo procedure based on such contraction methods for a pure gauge Kogut-Susskind Hamiltonian with a $\mathbb{Z}_3$ gauge field in two spatial dimensions. This is a first proof of principle to the method, which provides an inherent way to increase the number of variational parameters and can be readily extended to systems with physical fermions.