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The partition function of the double-layer $XY$ model in the (dual) Villainform is computed exactly in the limit of weak coupling between layers. Bothlayers are found to be locked together through theBerezinskii-Kosterlitz-Thouless transition, while they become decoupled wellinside the normal phase. These results are recovered in the general case of afinite number of such layers. When re-interpreted in terms of the dual problemsof lattice anyon superconductivity and of spin-liquids, they also indicate thatthe essential nature of the transition into the normal state found in twodimensions persists in the case of a finite number of weakly coupled layers.Comment: 10 pgs, TeX, LA-UR-94-394

Layered XY Models, Anyon Superconductors, and Spin-Liquids