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

摘要

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 p x + i p y 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 p x + i p y 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 p x + i p y 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 E F and the PDW pairing energy E P D W . The analysis of this interplay yields a rich set of topological states: (1)?In the weak-coupling regime ( E F E P D W ), 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, E F E P D W , 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.