Abstract
The properties of the isotropic incompressible $\nu=5/2$ fractional quantum Hall (FQH) state are described by a paired state of composite fermions in zero (effective) magnetic field, with a uniform $p_x+ip_y$ 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 $\nu=5/2$. This finding motivates us to consider an inhomogeneous paired state - a $p_x+ip_y$ 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 anti-domain 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 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 $E_{F}$ and the PDW pairing energy $E_{\textrm{pdw}}$. The analysis of this interplay yields a rich set of topological states. 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.
Highlights
Fractional quantum Hall (FQH) states are the quintessential example of topological electronic systems
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
We have studied the properties of a 2D pairdensity-wave state with a px þ ipy chiral order parameter, which is periodically varying along one direction
Summary
Fractional quantum Hall (FQH) states are the quintessential example of topological electronic systems. Coupling to an underlying lattice ( weakly, and provided the stripe order is incommensurate) makes the energy of a dislocation logarithmically divergent, leading to a Kosterlitz-Thouless transition where the translation symmetry is fully restored In this scenario, the orientational order is more robust since the associated symmetry is the point group symmetry of the lattice, a square lattice in the case of GaAs 2DEGs. On the other hand, disorder has much more serious effects on states that break translation invariance, such as the px þ ipy PDW paired state. To avoid confusion, we do not refer to the zero modes of the BdG Hamiltonian for domain walls as Majorana zero modes
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