Topological characterization of modified Kane-Mele-Rashba models via local spin Chern marker

In this work, we use the local spin Chern marker (LSCM) recently introduced by Ba\`{u} and Marrazzo [Phys. Rev. B 110, 054203 (2024)] to analyze the topology of the ground-state electronic wave functions in a finite honeycomb lattice flake described by three distinct models. The models considered here are characterized by strong Rashba spin-orbit interaction, which leads to non-conservation of the spin operator, i.e., $[\mathcal{H},\hat{s}_z]\neq 0$. The three spin-orbit couplings associated with the topological aspects of the models are: 1) Standard Kane-Mele coupling, 2) Sublattice-dependent Kane-Mele coupling, and 3) In-plane ($\hat{s}_y$) polarized Kane-Mele coupling. These couplings occur in graphene grown on suitable substrates and are relevant for modeling its van der Waals heterostructures. A particular topological phase diagram characterizes each of these spin-orbit interactions, and our calculations of LSCM successfully capture its general features. We also performed a detailed analysis of the spectral properties of the energy and valence-projected spin matrix eigenvalues, which shows that both exhibit a gap that protects the marker. Our results expand the applicability of the spin Chern number method to a class of Hamiltonians with experimental relevance and may contribute to future research on the real-space topology of realistic materials.