# Finite-representation approximation of lattice gauge theories at the continuum limit with tensor networks

## TL;DR

This paper investigates the effects of truncating the infinite gauge field representations in tensor network simulations of lattice gauge theories, specifically in the Schwinger model, and finds that the truncation is justified due to exponential decay of representation weights.

## Contribution

It provides a quantitative analysis of the impact of Hilbert space truncation in tensor network approaches to lattice gauge theories, validating the approximation method.

## Key findings

- Representation weights decay exponentially with quadratic Casimir invariant.
- Hilbert space truncation is justified for the Schwinger model.
- Single-particle spectrum varies with electric background field.

## Abstract

It has been established that Matrix Product States can be used to compute the ground state and single-particle excitations and their properties of lattice gauge theories at the continuum limit. However, by construction, in this formalism the Hilbert space of the gauge fields is truncated to a finite number of irreducible representations of the gauge group. We investigate quantitatively the influence of the truncation of the infinite number of representations in the Schwinger model, one-flavour QED$_2$, with a uniform electric background field. We compute the two-site reduced density matrix of the ground state and the weight of each of the representations. We find that this weight decays exponentially with the quadratic Casimir invariant of the representation which justifies the approach of truncating the Hilbert space of the gauge fields. Finally, we compute the single-particle spectrum of the model as a function of the electric background field.