Monte Carlo Study of a $U(1)\times U(1)$ Loop Model with Modular Invariance

TL;DR

This paper investigates a (2+1)-dimensional $U(1) imes U(1)$ system with long-range interactions, utilizing modular invariance to propose and confirm a phase diagram, and analyzing phase transitions and critical behavior through Monte Carlo simulations.

Contribution

The study introduces a sign-free reformulation of the model and leverages modular invariance to analytically determine physical properties across phases, including at fixed points.

Findings

Confirmed the proposed phase diagram via Monte Carlo simulations.

Identified segments of second-order phase transitions.

Measured critical exponents at second-order transitions.

Abstract

We study a $U(1)\times U(1)$ system in (2+1)-dimensions with long-range interactions and mutual statistics. The model has the same form after the application of operations from the modular group, a property which we call modular invariance. Using the modular invariance of the model, we propose a possible phase diagram. We obtain a sign-free reformulation of the model and study it in Monte Carlo. This study confirms our proposed phase diagram. We use the modular invariance to analytically determine the current-current correlation functions and conductivities in all the phases in the diagram, as well as at special "fixed" points which are unchanged by an operation from the modular group. We numerically determine the order of the phase transitions, and find segments of second-order transitions. For the statistical interaction parameter $\theta=\pi$, these second-order transitions are evidence of a critical loop phase obtained when both loops are trying to condense simulataneously. We also measure the critical exponents of the second-order transitions.