Grassmann tensor renormalization group for the massive Schwinger model with a $\theta$ term using staggered fermions

TL;DR

This paper applies the Grassmann tensor renormalization group method to study the $N_f=2$ Schwinger model with a $ heta$ term, revealing phase structure differences between lattice and continuum versions.

Contribution

The study introduces a novel application of the Grassmann tensor renormalization group to the massive Schwinger model with a $ heta$ term, enabling direct analysis of fermions and $ heta$ dependence.

Findings

Results agree with analytical solutions in large mass limit.

Consistent with previous Monte Carlo studies in small mass regime.

Suggests different phase structure on lattice compared to continuum.

Abstract

We use the Grassmann tensor renormalization group method to investigate the $N_f=2$ Schwinger model with the staggered fermions in the presence of a $2\pi$ periodic $\theta$ term in a broad range of mass. The method allows us to deal with the massive staggered fermions straightforwardly and to study the $\theta$ dependence of the free energy and topological charge in the thermodynamic limit. Our calculation provides consistent results with not only the analytical solution in the large mass limit but also the previous Monte Carlo studies in the small mass regime. Our numerical results also suggest that the $N_f=2$ Schwinger model on a lattice has a different phase structure, than the model in the continuum limit.