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

TLDR

In two‑dimensional fermion systems with nonzero angular momentum pairing, weak and strong pairing phases exist that are topologically distinct, with the weak‑pairing phase being abelian in the d‑wave case. The authors analyze fully gapped 2D fermion pairing with broken parity and time‑reversal symmetry for l = −1 (p‑wave) and l = −2 (d‑wave) states, derive the spin Hall conductivity as a topological invariant in spin‑conserving cases, and study disorder effects on quasiparticles. The study shows that the weak‑pairing spinless p‑wave phase realizes the Moore‑Read Pfaffian with nonabelian statistics, the strong‑pairing phase is abelian and the transition involves a sign change of a bulk Majorana fermion, the d‑wave case is governed by the Haldane‑Rezayi critical point with gapless bulk excitations, the U(1)‑symmetric p‑wave weak phase corresponds to the 331 state with a tunneling‑induced transition to Moore‑Read, and disorder preserves the gapped phases but introduces an intermediate thermally conducting phase with extended quasiparticles in the spinless p‑wave case.

Abstract

We analyze pairing of fermions in two dimensions for fully gapped cases with broken parity (P) and time reversal (T), especially cases in which the gap function is an orbital angular momentum (l) eigenstate, in particular $l=\ensuremath{-}1$ (p wave, spinless, or spin triplet) and $l=\ensuremath{-}2$ (d wave, spin singlet). For $l\ensuremath{\ne}0,$ these fall into two phases, weak and strong pairing, which may be distinguished topologically. In the cases with conserved spin, we derive explicitly the Hall conductivity for spin as the corresponding topological 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 (edge states, quasihole, and toroidal ground states) are also the same, indicating that nonabelian statistics is a generic property of such a paired phase. The strong-pairing phase is an abelian state, and the transition between the two phases involves a bulk Majorana fermion, the mass of which changes 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 asymptotics at the critical point between weak and strong pairing, and has gapless 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 quantum Hall problem, the weak-pairing phase has the properties of the 331 state, and with nonzero tunneling there is a transition to the Moore-Read phase. The effects of disorder on noninteracting quasiparticles are considered. The gapped phases survive, but there is an intermediate thermally conducting phase in the spinless p-wave case, in which the quasiparticles are extended.