Robustness of the pumping charge to dynamic disorder

We investigated the effect of disorder in a gapped crystalline system by introducing a class of local quantities for an energy band, which is referred to as a band correlation function (BCF) and is the sum of correlation functions for all eigenstates of the band. We demonstrated that the BCFs are robust in the presence of a disorder if the band gap does not collapse. The eigenstate set of an energy band can be almost completely mapped onto the perturbed eigenstate set, referred to as quasiclosed mapping, when it is sufficiently isolated from other bands. A randomly perturbed system may exhibit some features of translational symmetry, which we demonstrated by numerically simulating the pumping process for a one-dimensional Rice-Mele model with disorders in hopping strength and on-site potential. The quantized pumping charge was robust against dynamic disorder. In addition, we investigated a dimerized one-dimensional Kitaev model to illustrate the universality of our finding. The result indicates the possibility of measuring the topological invariant of an experimental system with imperfections.