Topological crystalline phases in a disordered inversion-symmetric chain

When translational symmetry is broken by bulk disorder, the topological nature of states in crystalline systems may change depending on type disorder that applied. In this work, we characterize phases a one-dimensional chain with inversion and chiral symmetries, where every configuration symmetric. By using basis-independent formulation for invariant, winding number, polarization, are able to construct phase diagrams these quantities when present. We show unlike number invariant can fluctuate spectral gap closes at strong disorder. Using position-space renormalization group, compare how polarization behave low energies limit. inversion-symmetry-preserving value determined eigenvalues centers, while such as still have contributions from state throughout chain. also it possible alter clean system occupying additional centers keeping fixed. discuss implications our results higher-dimensional electronic systems, potential experimental realizations ultracold-atomic systems.