  We consider a gas of neutral fermions trapped in a specific optical trap that
provides a tight confinement of a Fermi gas in a torus with a potential
periodic along the azimuthal direction. The effective model is interacting
fermions moving in a periodic potential along the ring. We show that the model
demonstrates a novel type of superfluidity different from the BCS mechanism.
This pure 1D quantum gas in a ring can also form crystalline structures with
both ferro- and antiferro-magnetic type orders. Possibilities of realization of
quantum crystals and their application for quantum computing are discussed.
