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

Abstract

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

Comments: 14 pages, plaintex, 4 figures