Berry phase and topological spin transport in the chiral d-density wave state

In this paper we demonstrate the possibility of dissipationless spin transport in the chiral d-density wave state, by the sole application of a uniform Zeeman field gradient. The occurrence of these spontaneous spin currents is attributed to the parity (${\cal P}$) and time-reversal (${\cal T}$) violation induced by the $d_{xy}+id_{x^2-y^2}$ density wave order parameter. We calculate the spin Hall conductance and reveal its intimate relation to the Berry phase which is generated when the Zeeman field is applied adiabatically. Finally, we demonstrate that in the zero temperature and doping case, the spin Hall conductance is quantized as it becomes a topological invariant.