Determine whether triplet-order fluctuations support fractional quantum spin Hall order

Determine whether extended spin-rotation-symmetric interactions can stabilize a fractional quantum spin Hall phase when the triplet-order orientation is allowed to fluctuate, and determine the resulting skyrmion energy and stability after restoring the dynamics of the order-parameter orientation.

Background

The mixed singlet–triplet d-density-wave parent state selects a triplet orientation, denoted by $\hat{\mathbf n}$, and the numerical calculations fix this orientation rather than allowing it to fluctuate. In the spin-decoupled limit, the authors find a nine-state manifold at filling $\nu=1/3$, while spin-independent interactions generate strong period-three charge correlations. Consequently, the calculations do not determine whether the fractional quantum spin Hall phase survives in a spin-rotation-symmetric setting with dynamical order-parameter orientations.

The paper argues that, if such a phase survives, skyrmion textures of the triplet order could carry fractional charge. Establishing whether suitable extended interactions stabilize this phase and quantifying the energy and stability of its skyrmions therefore remain unresolved.