Pair-Density-Wave Order and Paired Fractional Quantum Hall Fluids

The properties of the isotropic incompressible ν=5/2 fractional quantum Hall (FQH) state are described by a paired state of composite fermions in zero (effective) magnetic field, with a uniform px+ipy pairing order parameter, which is a non-Abelian topological phase with chiral Majorana and charge modes at the boundary. Recent experiments suggest the existence of a proximate nematic phase at ν=5/2. This finding motivates us to consider an inhomogeneous paired state - a px+ipy pair-density wave (PDW) - whose melting could be the origin of the observed liquid-crystalline phases. This state can viewed as an array of domain and antidomain walls of the px+ipy order parameter. We show that the nodes of the PDW order parameter, the location of the domain walls (and antidomain walls) where the order parameter changes sign, support a pair of symmetry-protected counterpropagating Majorana modes. The coupling behavior of the domain-wall Majorana modes crucially depends on the interplay of the Fermi energy EF and the PDW pairing energy EPDW. The analysis of this interplay yields a rich set of topological states: (1) In the weak-coupling regime (EF>EPDW), the hybridization of domain walls leads to a Majorana Fermi surface (MFS), which is protected by inversion and particle-hole symmetries. (2) As the MFS shrinks towards degenerate Dirac points, lattice effects render it unstable towards an Abelian stripe phase with two copropagating Majorana modes at the boundary. (3) A uniform component of the order parameter, which breaks inversion symmetry, gaps the MFS and causes the system to enter a non-Abelian FQH state supporting a chiral Majorana edge state. (4) In the strong-coupling regime, EF<EPDW, the bulk fermionic spectrum becomes gapped; this is a trivial phase with no chiral Majorana edge states, which is in the universality class of an Abelian Halperin paired state. The pair-density-wave order state in a paired FQH system provides a fertile setting to study Abelian and non-Abelian FQH phases - as well as transitions thereof - tuned by the strength of the paired liquid crystalline order.