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Layered XY-Models, Anyon Superconductors, and Spin-Liquids

The partition function of the double-layer $XY$ model in the (dual) Villain form is computed exactly in the limit of weak coupling between layers. Both layers are found to be locked together through the Berezinskii-Kosterlitz-Thouless transition, while they become decoupled well inside the normal phase. These results are recovered in the general case of a finite number of such layers. When re-interpreted in terms of the dual problems of lattice anyon superconductivity and of spin-liquids, they also indicate that the essential nature of the transition into the normal state found in two dimensions persists in the case of a finite number of weakly coupled layers.