Theory of half-integer fractional quantum spin Hall edges

We study the edges of fractional quantum spin Hall insulators (FQSH) with half-integer spin Hall conductance. These states can be viewed as symmetric combinations of a spin-up and spin-down half-integer fractional quantum Hall state (FQH) that conserve the z-component of spin. We consider the states based on the non-Abelian Pfaffian, anti-Pfaffian, PH-Pfaffian, and 221 FQHs, and generic Abelian FQHs. For strong enough spin-conserving interactions, we find that all the non-Abelian and Abelian edges flow to the same fixed point that consists of a single pair of charged counter-propagating bosonic modes. If spin-conservation is broken, but time-reversal symmetry is preserved, the Abelian edge can be fully gapped in a time-reversal symmetric fashion. The non-Abelian edge with broken spin-conservation remains gapless due to time-reversal symmetry, and can flow to a new fixed point with a helical gapless pair of Majorana fermions. From this, we conclude that these non-Abelian FQH bilayers realize half-integer fractional topological insulators, protected by time-reversal symmetry. We discuss the possible relevance of our results to the recent observation of a half-integer edge conductance in twisted MoTe2.