Matrix Product Operator Constructions for Gauge Theories in the Thermodynamic Limit

TL;DR

This paper introduces a general method using infinite matrix product states and operators to simulate 1+1D lattice gauge theories, efficiently handling gauge fields and including background effects, enabling studies of gauge dynamics in the thermodynamic limit.

Contribution

The authors develop a novel MPO construction for gauge theory Hamiltonians directly in the thermodynamic limit, compatible with standard iDMRG algorithms and extendable to quasi-2D geometries.

Findings

Successfully applied to the Schwinger model, reproducing confinement and string breaking.

Provides a flexible framework including background fields and $ heta$-terms.

Enables efficient simulation of gauge theories in the thermodynamic limit.

Abstract

We present a general method for simulating lattice gauge theories in low dimensions using infinite matrix product states (iMPS). A central challenge in Hamiltonian formulations of gauge theories is the unbounded local Hilbert space associated with gauge degrees of freedom. In one spatial dimension, Gauss's law permits these gauge fields to be integrated out, yielding an effective Hamiltonian with long-range interactions among matter fields. We construct efficient matrix product operator (MPO) representations of these Hamiltonians directly in the thermodynamic limit. Our formulation naturally includes background fields and $\theta$-terms, requiring no modifications to the standard iDMRG algorithm. This provides a broadly applicable framework for 1+1D gauge theories and can be extended to quasi-two-dimensional geometries such as infinite cylinders, where tensor-network methods remain tractable. As a benchmark, we apply our construction to the Schwinger model, reproducing expected features including confinement, string breaking, and the critical behavior at finite mass. Because the method alters only the MPO structure, it can be incorporated with little effort into a wide range of iMPS and infinite-boundary-condition algorithms, opening the way to efficient studies of both equilibrium and non-equilibrium gauge dynamics.