Tuning between a fractional topological insulator and competing phases at νT=2/3

We study a spinful, time-reversal symmetric lowest Landau level model for a flatband quantum spin Hall system at total filling fraction νT=2/3. Such models are relevant, e.g. for spin-valley locked moiré transition metal dichalcogenides. The opposite Chern number of the two spins hinders the formation of a quantum Hall ferromagnet, instead favouring other phases. We study the phase diagram in dependence on different short-range Haldane pseudopotentials Vₘ and uncover several phases: A fractional topological insulator, a phase separated state, a spin-polarized fractional quantum Hall state, and the partially particle-hole transformed Halperin (111) state. The effect of the pseudopotentials Vₘ depends on the parity of m, the relative angular momentum.