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