Single-point spin Chern number in a supercell framework

We present an approach for the calculation of the Z₂ topological invariant in non-crystalline two-dimensional quantum spin Hall insulators. While topological invariants were originally mathematically introduced for crystalline periodic systems, and crucially hinge on tracking the evolution of occupied states through the Brillouin zone, the introduction of disorder or dynamical effects can break the translational symmetry and imply the use of larger simulation cells, where the k-point sampling is typically reduced to the single Γ-point. Here, we introduce a single-point formula for the spin Chern number that enables to adopt the supercell framework, where a single Hamiltonian diagonalisation is performed. Our single-point approach allows to calculate the spin Chern number even when the spin operator does not commute with the Hamiltonian, as in the presence of Rashba spin-orbit coupling. This archive entry contains the results of the single-point calculations of the topological invariant on the tight-binding Kane-Mele model for large supercells (up to 7200 sites). Convergence tests as function of the supercell size are reported, both in the pristine case and in presence of Anderson disorder. Single-point calculations of the spin Chern number are carried out over the entire phase diagram of the Kane-Mele model. The study of topological phase transitions due to disorder is performed in terms of single-point spin Chern number by averaging over several realisations of the Anderson disorder in the supercell.