Non-Abelian interlayer coherent fractional quantum Hall states

We study non-Abelian fractional quantum Hall state in double layer systems at total filling factor $1/2$. Recent progresses in two-dimensional van der Waals materials made it possible to explore the regime with very small interlayer distance. Numerical calculations suggests interlayer phase coherence can develop between the layers such that the electrons may redistribute between them without changing the Hall response. It corresponds to spontaneous breaking of the U(1) symmetry associated with the particle number difference in the layers. This state manifests itself as superfluid in counterflow measurement and has characteristic Hall response when current is passed through one layer and voltages in both layers are measured. As the interlayer distance increases, a phase transition to the Halperin 331 state occurs. We also discuss similar physics for bosonic systems with specially designed interactions.