Quantum pumping induced by disorder in one dimension

The topological property in one dimension (1D) is protected by symmetry. Based on a concrete model, we show that since a 1D topological model usually contain two of the three Pauli matrix, the left one automatically become the protecting symmetry. We study the effect of disorder preserving or breaking the symmetry and show the nature of symmetry protecting in the 1D topological phase. Based on the 1D topological model, a stable quantum pumping can be constructed, which is topologically nontrivial and can be characterized by the Chern number. By calculating the instantaneous local current we show that an integer charge is pumped across a periodic chain in a cyclic process. Also on an open chain, an edge state can be transferred to the other edge by the quantum pumping. Furthermore we find that not only the quantum pumping is stable to on-site disorder, but also can be induced by it. These results may be realized experimentally using quasicrystals.