Character Expansion, Zeroes of Partition Function and  -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 θ-term in 2 dimensions. Topological charge distribution P (Q) is shown to be of Gaussian form at any β(inverse coupling constant). The partition function Z(θ) 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 ζ = e iθ (θ = theta parameter). These partition function zeros lead to the phase transition at θ = π. Analytical results will be compared with the MC simulation results. In MC simulation, we adopt (i)"set method" and (ii)"trial function method".

Monte Carlo studies of two-dimensional systems with aθterm

“…[47,48] There is a simple physical picture of why a limiting θ b arises. Current methods for simulating a system with θ = 0 are done using the θ = 0 system.…”

Dominance of a single topological sector in gauge theory on non-commutative geometry

“…In the commutative case, analytical and numerical studies show that the distribution of the topological charge is gaussian with a finite width, and the width diverges in the infinite-volume limit [11,12]. Thus the situation in the NC case differs drastically from the commutative case.…”

“…It takes a particularly simple form in the 2d U(1) case, which is used in ref. [11,12]. In 4d theories, on the other hand, one usually uses a naively defined topological charge, but one can obtain a distribution peaked at integer values by using some techniques like cooling or renormalization.…”

Two-Dimensional CP2 Model with -Term and Topological Charge Distributions

“…We analyzed a two-dimensional U (1) system with a θ-term 11) using the group character expansion method. 12) The topological charge distribution P (Q) is found to be given by a Gaussian function exp(−κ V Q 2 )) for all β. This leads to the result that the partition function is given by the third elliptic theta function ϑ 3 (ν, τ ).…”