Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect

We analyze pairing of fermions in two dimensions for fully gapped cases wit h broken parity (P) and time reversal (T), especially cases in which the ga p function is an orbital angular momentum (l) eigenstate, in particular l = -1 Gn wave, spinless, or spin triplet) and l = -2 (d wave, spin singlet). For l not equal 0, these fall into two phases, weak and strong pairing, whi ch may be distinguished topologically. In the cases with conserved spin, we derive explicitly the Hall conductivity for spin as the corresponding topo logical invariant. For the spinless p-wave case, the weak-pairing phase has a pair wave function that is asympototically the same as that in the Moore -Read (Pfaffian) quantum Hall state, and we argue that its other properties ledge states, quasihole, and toroidal ground states) are also the same, in dicating that nonabelian statistics is a generic property of such a paired phase. The strong-pairing phase is an abelian state, and the transition bet ween the two phases involves a bulk Majorana fermion, the mass of which cha nges sign at the transition. For the d-wave case, we argue that the Haldane -Rezayi state is not the generic behavior of a phase but describes the asym ptotics at the critical point between weak and strong pairing, and has gapl ess fermion excitations in the bulk. In this case the weak-pairing phase is an abelian phase, which has been considered previously. In the p-wave case with an unbroken U(1) symmetry, which can be applied to the double layer q uantum Hall problem, the weak-pairing phase has the properties of the 331 s tate, and with nonzero tunneling there is a transition to the Moore-Read ph ase. The effects of disorder on noninteracting quasiparticles are considere d. The gapped phases survive, but there is an intermediate thermally conduc ting phase in the spinless p-a ave case, in which the quasiparticles are ex tended.