LAUSR

We investigate the effect of disorder on a gapped crystalline system by introducing a class of local quantities for an energy band, which is referred as to band correlation function (BCF) and is the sum of correlation functions for all eigenstates of the band. We show that the BCFs are robust in the presence of disorder if the band gap is not collapse. The eigenstate set of an energy band can be almost completely mapped onto the perturbated eigenstate set, referred as to quasi-closed mapping, when it is sufficiently isolated from other bands. Some features relate to translational symmetry may emerge in a randomly perturbed system. We demonstrate this by simulating numerically the pumping process for a 1D Rice-Mele (RM) model with disorders on the hopping strength and on-site potential. It is shown that the quantized pumping charge is robust against with the dynamic disorder. This result indicates the possibility of measuring the topological invariant in experimental system with imperfection.