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

TL;DR

The paper analyzes two-dimensional fermion pairing with broken parity and time-reversal symmetry, focusing on complex p-wave and d-wave pairings to reveal a topological weak- vs strong-pairing dichotomy and associated Majorana edge and vortex modes. It connects these superconducting-like phases to fractional quantum Hall states via the composite-fermion framework, identifying Moore–Read-type nonabelian statistics for spinless p-wave weak-pairing and abelian analogs in other channels, with the Haldane–Rezayi state arising at the critical point. The work develops a Bogoliubov–de Gennes, conformal-field-theory, and Chern–Simons perspective to derive ground-state degeneracies, edge theories, and spin-Hall/Chern-Simons responses, and analyzes disorder effects within symmetry-class frameworks. The findings illuminate how topology, geometry, and symmetry dictate quasiparticle content, edge excitations, and transport in paired fermion systems, with implications for FQHE realizations and topological quantum computation. The study underscores the need for comprehensive effective-field theories that encapsulate disorder, transitions, and nonabelian statistics in these 2D paired states.

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 $l=-1$ (p-wave, spinless or spin-triplet) and $l=-2$ (d-wave, spin-singlet). For $l\neq0$, 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 wavefunction 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 {\em 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.

Figures (2)

• Figure 1: Schematic phase diagram for the p-wave phases, as discussed in the text. The A-phase with unbroken U(1) symmetry appears as the vertical axis $t=0$, with the region $\mu>0$ being the 331 phase. Similarly, the Fermi-liquid phase in which pairing disappears is identified with the line $\mu=E_F$, since that is the value of $\mu$ there at fixed density, neglecting Hartree-Fock corrections.
• Figure 2: Proposed renormalization group flow diagrams for (a) the unitary ensemble (IQHE), as in pruisken, and class C, and (b) class D. The dashed curves represent schematically the (nonuniversal) bare values of the coupling parameters. Other features are universal when the renormalized couplings are defined using the as-measured conductivity parameters, as explained in the text, and repeat periodically in the $\theta$ variable.