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.