Gauss Law, Minimal Coupling and Fermionic PEPS for Lattice Gauge Theories

TL;DR

This paper reviews recent tensor network approaches to Hamiltonian lattice gauge theories, emphasizing the dual perspectives on Gauss's law and their implications for constructing fermionic PEPS.

Contribution

It introduces a novel perspective on Gauss's law for gauge fields and matter, and explores how these insights facilitate the construction of PEPS for lattice gauge theories.

Findings

Dual viewpoints on Gauss's law enable new gauge fixing techniques

Explicit algebraic form of matter simplifies gauge theory analysis

PEPS construction benefits from the dual perspectives on gauge constraints

Abstract

In these lecture notes, we review some recent works on Hamiltonian lattice gauge theories, that involve, in particular, tensor network methods. The results reviewed here are tailored together in a slightly different way from the one used in the contexts where they were first introduced, by looking at the Gauss law from two different points of view: for the gauge field it is a differential equation, while from the matter point of view, on the other hand, it is a simple, explicit algebraic equation. We will review and discuss what these two points of view allow and do not allow us to do, in terms of unitarily gauging a pure-matter theory and eliminating the matter from a gauge theory, and relate that to the construction of PEPS (Projected Entangled Pair States) for lattice gauge theories.