Quantum charge pumping and electric polarization in Anderson insulators

We investigate adiabatic charge pumping in the disordered system in one dimension with open and closed boundary conditions. In contrast to the Thouless charge pumping, the system has no gap even though all the states are localized, i.e., strong localization. Charge pumping can be achieved by making a loop adiabatically in the two-dimensional parameter space of the Hamiltonian. It is because there are many delta-function-like fluxes distributed over the parameter space with random strength, in sharp contrast to the single delta-function in the pure case. This provides a more efficient way of charge pumping and polarization.