Berry phase and spin quantum Hall effect in the vortex state of superfluid He3 in two dimensions

We show that the spin quantum Hall effect in the vortex state of two-dimensional rotating superfluid He3 can be described as an adiabatic spin transport of Bloch quasiparticles. We show that the spin Hall conductivity is written by the Berry phase as well as the Chern number. The results have similarity to the adiabatic pumping of Bloch electrons and the spontaneous polarization in crystalline dielectrics.