Character Expansion, Zeros of Partition Function and $theta$-term in U(1) Gauge Theory

Character expansion developed in real space renormalization group (RSRG) approach is applied to U(1) lattice gauge theory with $th$-term in 2 dimensions. Topological charge distribution $P(Q)$ is shown to be of Gaussian form at any $b$(inverse coupling constant). The partition function $Z(th)$ at large volume is shown to be given by the elliptic theta function. It provides the information of the zeros of partition function as an analytic function of $ze= e^{i th}$ ($th$ = theta parameter). These partition function zeros lead to the phase transition at $th=pi$. Analytical results will be compared with the MC simulation results. In MC simulation, we adopt (i)``set method and (ii)``trial function method.