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We study a $U(1)\times U(1)$ system with two species of loops with mutual$\pi$-statistics in (2+1) dimensions. We are able to reformulate the model in away that can be studied by Monte Carlo and we determine the phase diagram. Inaddition to a phase with no loops, we find two phases with only one species ofloop proliferated. The model has a self-dual line, a segment of which separatesthese two phases. Everywhere on the segment, we find the transition to befirst-order, signifying that the two loop systems behave as immiscible fluidswhen they are both trying to condense. Moving further along the self-dual line,we find a phase where both loops proliferate, but they are only of evenstrength, and therefore avoid the statistical interactions. We study anothermodel which does not have this phase, and also find first-order behavior on theself-dual segment.

Monte Carlo Study of a U(1)xU(1) system with \pi-statistical Interaction