Grassmann Tensor Renormalization Group for two-flavor Schwinger model with a theta term

QCD has been understood through numerical calculations by the Monte Carlo method. However, this method does not work for some parameter regions because of the sign problem. For example, QCD with a theta term has a sign problem, so the nature of QCD with a finite theta parameter is unknown. The theta dependence is also important to axion physics.
To reveal such systems, tensor network methods are powerful tools. Tensor network methods have been developed by condensed matter theorists. Furthermore, recently there have been some attempts to apply them to high energy physics. In particular, the tensor renormalization group (TRG) method is remarkable for its applicability to higher dimensions.
The Schwinger model is known as a two-dimensional toy model of QCD. It has the chiral symmetry and theta term as the same as QCD.

In this study, the free energy of the two-flavor Schwinger model is calculated in a broad range of mass and theta parameters. We use TRG to calculate it, with obvious 2pi periodicity of theta parameter. We check the consistency with analytical values in large and small mass limits.