# Tensor network simulation of QED on infinite lattices: learning from (1+1)d, and prospects for (2+1)d

## TL;DR

This paper advances tensor network methods for simulating lattice gauge theories, specifically QED, in the thermodynamic limit, starting from 1+1 dimensions and proposing a framework for 2+1 dimensions with gauge-invariant PEPS.

## Contribution

It demonstrates gauge-invariant tensor network simulations of the Schwinger model and proposes a novel PEPS ansatz for (2+1)d QED incorporating gauge symmetry and matter fields.

## Key findings

- Benchmarking of Schwinger model with good agreement
- Intuitive insights for (2+1)d simulation strategies
- Proposed PEPS ansatz includes gauge symmetry and matter fields

## Abstract

The simulation of lattice gauge theories with tensor network (TN) methods is becoming increasingly fruitful. The vision is that such methods will, eventually, be used to simulate theories in $(3+1)$ dimensions in regimes difficult for other methods. So far, however, TN methods have mostly simulated lattice gauge theories in $(1+1)$ dimensions. The aim of this paper is to explore the simulation of quantum electrodynamics (QED) on infinite lattices with TNs, i.e., fermionic matter fields coupled to a $U(1)$ gauge field, directly in the thermodynamic limit. With this idea in mind we first consider a gauge-invariant iDMRG simulation of the Schwinger model -i.e., QED in $(1+1)d$-. After giving a precise description of the numerical method, we benchmark our simulations by computing the substracted chiral condensate in the continuum, in good agreement with other approaches. Our simulations of the Schwinger model allow us to build intuition about how a simulation should proceed in $(2+1)$ dimensions. Based on this, we propose a variational ansatz using infinite Projected Entangled Pair States (PEPS) to describe the ground state of $(2+1)d$ QED. The ansatz includes $U(1)$ gauge symmetry at the level of the tensors, as well as fermionic (matter) and bosonic (gauge) degrees of freedom both at the physical and virtual levels. We argue that all the necessary ingredients for the simulation of $(2+1)d$ QED are, a priori, already in place, paving the way for future upcoming results.