Tensor renormalization group study of two-dimensional U(1) lattice gauge theory with a $\theta$ term

We make an analysis of the two-dimensional U(1) lattice gauge theory with a $\theta$ term by using the tensor renormalization group. Our numerical result for the free energy shows good consistency with the exact one at finite coupling constant. The topological charge density generates a finite gap at $\theta =\pi$ toward the thermodynamic limit. In addition finite size scaling analysis of the topological susceptibility up to $V=L\times L=1024\times 1024$ allows us to determine the phase transition at $\theta =\pi$ is the first order.