Figure 3: Emergence of chiral spin superfluid and the spin Hall effect.

(a) The energy difference between chiral charge and spin superfluid states with varying inter-spin interaction V and fixed intra-species interaction U. The red ‘+’ symbols are results calculated by solving the Bogoliubov spectra numerically, and the blue solid line is from a renormalized perturbation theory. See the Methods section for details of the tight-binding Hamiltonian we solve. The intra-spin interaction strength is chosen to be U/t_eff=1 here. For interaction strength V>V_c≈U/3 (not shown here), the system is in a spin polarized state (see Fig. 4). (b) The dynamics of the charge centres, r_↑↓ , of the two spin components with a force F applied in the r_‖ direction. The two spins move oppositely in the transverse (r_⊥) direction, signifying a spin Hall effect. Here we choose Δ_D/t=2 and F/t=0.5 (t and Δ_D are nearest neighbour tunnelling and Dirac gap respectively, in the hexagonal lattice.).