Entanglement renormalization and gauge symmetry

TL;DR

This paper introduces a numerical coarse-graining method that preserves gauge symmetry in lattice models, enabling efficient and accurate analysis of lattice gauge theories and their phase diagrams.

Contribution

It proposes a variational ansatz that maintains local gauge symmetry during coarse-graining, reducing computational costs for lattice gauge theories.

Findings

Successfully applied to Z2 lattice gauge theory with large lattices

Reproduced known phase diagram and quantum phase transition

Accurately estimated energy gaps, fidelities, and Wilson loops

Abstract

A lattice gauge theory is described by a redundantly large vector space that is subject to local constraints, and can be regarded as the low energy limit of an extended lattice model with a local symmetry. We propose a numerical coarse-graining scheme to produce low energy, effective descriptions of lattice models with a local symmetry, such that the local symmetry is exactly preserved during coarse-graining. Our approach results in a variational ansatz for the ground state(s) and low energy excitations of such models and, by extension, of lattice gauge theories. This ansatz incorporates the local symmetry in its structure, and exploits it to obtain a significant reduction of computational costs. We test the approach in the context of the toric code with a magnetic field, equivalent to Z2 lattice gauge theory, for lattices with up to 16 x 16 sites (16^2 x 2 = 512 spins) on a torus. We reproduce the well-known ground state phase diagram of the model, consisting of a deconfined and spin polarized phases separated by a continuous quantum phase transition, and obtain accurate estimates of energy gaps, ground state fidelities, Wilson loops, and several other quantities.