Magnetic field inversion symmetry in quantum pumps with discrete   symmetries
  We investigate the magnetic field inversion symmetry of the pumped currents in quantum pumps with various discrete symmetries using Floquet scattering matrix approach. We found the pumped currents can have symmetries $I(B,\phi) = -I(-B,-\phi)$ and $I(B,\phi) = I(-B,-\phi)$, where $\phi$ is the phase difference of two time dependent perturbations, depending on the discrete symmetry considered. The results in the adiabatic limit for each discrete symmetry are compared with those of Brouwer's formula. 
Sang Wook Kim