Trial wavefunction for fractional quantum spin Hall insulators

Fermions with opposite spins occupying half-filled conjugate Chern bands exhibit interaction physics distinct from their multi-component Landau-level counterparts with the same chirality. This is largely due to unavoidable inter-species collisions that preclude the Halperin-type wavefunctions available in multi-component Landau levels. In this work, we propose and evaluate a variational wavefunction for a fractional quantum spin Hall state with Z_4 topological order in a pair of conjugate Landau levels. This Z_4 topological order has previously been shown to be the minimal topological order compatible with charge conservation, $S_z$ conservation, time-reversal symmetry, and the fractional spin Hall conductance 1/2 suggested by previous twisted MoTe$_2$ experiments. Our construction is based on the condensation of an anyonic exciton formed by the neutral fermionic excitations in a decoupled pair of Moore-Read Pfaffian state and its conjugate. By coupling the chiral and anti-chiral Ising conformal field theories associated with the two spin species, we introduce a variational mass parameter in the Z_4 trial wavefunction that captures the inter-spin-species s-wave pairing of composite fermions alongside the intra-spin-species p-wave pairing. We assess the energetics of this trial state using Monte Carlo sampling on a spherical geometry. Because the coupled state intrinsically involves Landau-level mixing, we explicitly evaluate the resulting kinetic energy penalty. Our phase diagram reveals that the proposed Z_4 state becomes energetically favorable in a sizable region of parameter space, over both the decoupled pair of conjugate Pfaffian states and an alternative exciton condensate state. These results provide a concrete microscopic wavefunction realization of this Z_4 fractional quantum spin Hall phase, and propose a route to constructing additional families of such states.