Topological Phase Transition in a Disordered Inversion-Symmetric Chain

Topological crystalline phases are states which are protected by crystalline symmetries. When translational invariance is broken by bulk disorder, the topological nature of these states may change depending on the type of disorder that is applied. In this work, we characterize the phases of a one-dimensional (1D) chain with inversion and chiral symmetries where the disorder preserves the inversion symmetry on every configuration. By using a basis-independent formulation for the inversion invariant and chiral winding number, we are able to construct phase diagrams for both quantities when disorder is present. Unlike the chiral winding number, the inversion invariant is prone to fluctuations past the spectral gap closing at strong disorder. Using the real-space renormalization group, we are able to compare how differently the inversion invariant and chiral winding number behave at low energies when disorder is present.