Spin-Hall effect in triplet chiral superconductors and graphene

We study spin-Hall effects in time-reversal-symmetry- (TRS-) broken systems such as triplet chiral superconductors and TRS-preserved ones such as graphene. For chiral triplet superconductors, we show that the edge states carry a quantized spin-Hall current in response to an applied Zeeman magnetic field $B$ along the $\mathbf{d}$ vector [A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975)], whereas the edge spin current for $\mathbf{B}\ensuremath{\perp}\mathbf{d}$ is screened by the condensate. We also derive the bulk spin-Hall current for chiral triplet superconductors for arbitrary relative orientation of $\mathbf{B}$ and $\mathbf{d}$ and discuss its relation with the edge spin current. For TRS-invariant system graphene, we show that the bulk effective action, unlike its TRS-broken counterparts, does not support a SU(2) Hopf term but allows a crossed Hopf term in the presence of an external electromagnetic field, which yields a quantized bulk spin-Hall current in response to an electric field. We also present an analytical solution of the edge problem for armchair edges of graphene and contrast the properties of these edge states with their time-reversal-symmetry-broken counterparts in chiral superconductors. We propose possible experiments to test our results.