# Topological vacuum structure of the Schwinger model with matrix product states

## TL;DR

This paper uses tensor network methods to study the topological properties of the Schwinger model with a theta-term, overcoming the sign problem faced by traditional lattice simulations, and explores the model's behavior across different fermion masses.

## Contribution

It introduces tensor network techniques to analyze the Schwinger model with a topological term, revealing theta-dependence and mass effects beyond lattice Monte Carlo limitations.

## Key findings

- Continuum limit is theta-independent in the chiral limit.
- Negative fermion masses relate to positive masses via theta shift due to axial anomaly.
- Lattice artifacts depend on theta and distort continuum mappings.

## Abstract

We numerically study the single-flavor Schwinger model with a topological $\theta$-term, which is practically inaccessible by standard lattice Monte Carlo simulations due to the sign problem. By using numerical methods based on tensor networks, especially the one-dimensional matrix product states, we explore the non-trivial $\theta$-dependence of several lattice and continuum quantities in the Hamiltonian formulation. In particular, we compute the ground-state energy, the electric field, the chiral fermion condensate, and the topological vacuum susceptibility for positive, zero, and even negative fermion mass. In the chiral limit, we demonstrate that the continuum model becomes independent of the vacuum angle $\theta$, thus respecting CP invariance, while lattice artifacts still depend on $\theta$. We also confirm that negative masses can be mapped to positive masses by shifting $\theta\rightarrow \theta +\pi$ due to the axial anomaly in the continuum, while lattice artifacts non-trivially distort this mapping. This mass regime is particularly interesting for the (3+1)-dimensional QCD analog of the Schwinger model, the sign problem of which requires the development and testing of new numerical techniques beyond the conventional Monte Carlo approach.