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

TL;DR

This paper applies character expansion within the real space renormalization group to analyze the U(1) lattice gauge theory with a theta term in 2D, revealing Gaussian charge distribution, elliptic theta function form of the partition function, and phase transition at theta equals pi.

Contribution

It introduces an analytical approach using character expansion to study the phase structure and zeros of the partition function in U(1) gauge theory with a theta term, complemented by Monte Carlo simulations.

Findings

Charge distribution is Gaussian at all couplings.

Partition function at large volume is an elliptic theta function.

Partition function zeros indicate a phase transition at theta=pi.

Abstract

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".