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

We make an analysis of the two-dimensional U(1) lattice gauge theory with a $θ$ 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 $θ=π$ 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 $θ=π$ is the first order.

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