The major finding of this Letter is that a one-dimensional spin-polarized gas composed of an even number of fermionic atoms interacting via attractive p-wave interactions and confined to a mesoscopic ring has a degenerate pair of ground states that are oppositely rotating. In any realization the gas will be measured to rotate one way or the other in spite of the fact that there is no external rotation or bias fields. Our goal is to show that this counterintuitive finding is a natural consequence of the combined effects of quantum statistics, ring topology, and exchange interactions.
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