Topological Classification of Heterostructures from first-principles: Systematic Computation of the Chern Number from Wannier-Based Bogoliubov–de Gennes Hamiltonians for Heterostructures

Reliable computation of the Chern number is essential for identifying topological phases. While first-principles methods for evaluating the Chern number of electronic Hamiltonians are well established, extending them to Bogoliubov–de Gennes (BdG) Hamiltonians, particularly in semiconductor (SM)–superconductor (SC) heterostructures, is both challenging and crucial for designing platforms that can host Majorana zero modes. We present a systematic framework for computing Chern numbers across models of increasing complexity. Starting from a single-orbital BdG model relevant to cold-atom systems, where the bandwidth is comparable to the superconducting pairing scale, we establish convergence criteria. We then extend the approach to SM-SC heterostructures derived from first principle methods where maximally localized Wannier functions yield multiband electronic Hamiltonians truncated to the topologically relevant low-energy sector while preserving orbital character. Incorporating orbitally selective Rashba spin–orbit coupling and various superconducting pairings, we construct the BdG Hamiltonian and its Matsubara Green's function to evaluate the Chern number and test convergence with respect to Brillouin-zone sampling and Matsubara frequencies. This study demonstrates robust Chern number evaluation in realistic heterostructures and identifies when multiband effects are essential for determining the topological phase.