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

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.