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We study a $U(1)\times U(1)$ system in (2+1)-dimensions with long-rangeinteractions and mutual statistics. The model has the same form after theapplication of operations from the modular group, a property which we callmodular invariance. Using the modular invariance of the model, we propose apossible phase diagram. We obtain a sign-free reformulation of the model andstudy it in Monte Carlo. This study confirms our proposed phase diagram. We usethe modular invariance to analytically determine the current-currentcorrelation functions and conductivities in all the phases in the diagram, aswell as at special "fixed" points which are unchanged by an operation from themodular group. We numerically determine the order of the phase transitions, andfind segments of second-order transitions. For the statistical interactionparameter $\theta=\pi$, these second-order transitions are evidence of acritical loop phase obtained when both loops are trying to condensesimulataneously. We also measure the critical exponents of the second-ordertransitions.

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