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

Abstract

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 liquidcrystalline phases. This state can viewed as an array of domain and anti-domain walls of the px + ipy order parameter. We show that the nodes of the PDW order parameter, the location of the domain walls (and anti-domain walls) where the order parameter changes sign, support a pair of symmetry-protected counter-propagating 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 E pdw . The analysis of this interplay yields a rich set of topological states: (1) In the weak coupling regime (EF > E pdw ), 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 striped phase with two co-propagating Majorana modes at the boundary. (3) An 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 < E pdw , 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 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.CONTENTS * These authors contributed equally to the work.

Antichiral and nematicity-wave superconductivity

“…Note added in proof: Recently, there appeared an interesting study that considers a different generalization of the LOFF state for systems to the p x +ip y case [28]. In contrast to our work, there the system has nonalternating chirality but forms inhomogeneous states of stripes of the alternating overall sign of the order parameter with a given chirality.…”

Intertwined Order in Fractional Chern Insulators from Finite-Momentum Pairing of Composite Fermions

“…The distinguishing feature between the physics we present and that of, for instance, Ref. [56] is that we present FCIs as a platform in which to study intertwining of TO and BSO, in that they do not compete with nor are even independent of one another, but rather arise from a common microscopic origin, namely the interplay between the pairing of composite fermions and the commensurability of the lattice and magnetic length scales.…”

“…Related phenomena have been exhibited in recent experiments [53][54][55], which revealed a competition between pairing and nematicity in continuum Landau levels. A subsequent theoretical study [56] proposed a p x + ip y PDW state of composite fermions as a possible explanation for this observation. The distinguishing feature between the physics we present and that of, for instance, Ref.…”

Exploring Quantum Hall Physics at Ultra-Low Temperatures and at High Pressures

“…However, phase transitions between the two distinct classes of phases, such as the transition from a fractional quantum Hall state to the nematic, remain rare and present an opportunity for further theoretical development. [216][217][218][219][220][221] One may gain further insight into the transition by investigating it in the upper spin branch of the second Landau level. Such studies confirmed a qualitatively similar phase transition at filling factor ν = 7/2.…”