Competing orders in one-dimensional half-integer fermionic cold atoms: A conformal field theory approach

The physical properties of arbitrary half-integer spins F=N−1/2$F=N-1/2$ fermionic cold atoms loaded into a one-dimensional optical lattice are investigated by means of a conformal field theory approach. We show that for attractive interactions two different superfluid phases emerge for F?3/2$F?3/2$: A BCS pairing phase, and a molecular superfluid phase which is formed from bound-states made of 2N   fermions. In the low-energy approach, the competition between these instabilities and charge-density waves is described in terms of ZN${\mathbb{Z}}_N$ parafermionic degrees of freedom. The quantum phase transition for F=3/2,5/2$F=3/2,5/2$ is universal and shown to belong to the Ising and three-state Potts universality classes respectively. In contrast, for F?7/2$F?7/2$, the transition is non-universal. For a filling of one atom per site, a Mott transition occurs and the nature of the possible Mott-insulating phases are determined.