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The properties of the isotropic incompressible $\nu=5/2$ fractional quantumHall (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 modesat the boundary. Recent experiments suggest the existence of a proximatenematic phase at $\nu=5/2$. This finding motivates us to consider aninhomogeneous paired state - a $p_x+ip_y$ pair-density-wave (PDW) - whosemelting could be the origin of the observed liquid-crystalline phases. Thisstate 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, thelocation of the domain walls (and anti-domain walls) where the order parameterchanges sign, support a pair of symmetry-protected counter-propagating Majoranamodes. The coupling behavior of the domain wall Majorana modes cruciallydepends 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 oftopological states. The pair-density-wave order state in paired FQH systemprovides a fertile setting to study Abelian and non-Abelian FQH phases - aswell as transitions thereof - tuned by the strength of the paired liquidcrystalline order.

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