Real-space formulation of topology for disordered Rice-Mele chains without chiral symmetry

In this paper, we derive a real-space topological invariant that involves all energy states in the system. This global invariant, denoted by $Q$, is always quantized to be 0 or 1, independent of symmetries. In terms of $Q$, we numerically investigate topological properties of the nonchiral Rice-Mele model including random onsite potentials to show that nontrivial bulk topology is sustained for weak enough disorder. In this regime, a finite spectral gap persists, and then $Q$ is definitely identified. We also consider sublattice polarization of disorder potentials. In this case, the energy spectrum retains a gap regardless of disorder strength so that $Q$ is unaffected by disorder. This implies that bulk topology remains intact as long as the spectral gap opens.