Estimating bulk and edge topological indices in finite open chiral chains

We develop a formalism to estimate, simultaneously, the usual bulk and edge indices from topological insulators in case of finite sample with open boundary conditions provide physical interpretation these quantities. then show that they converge exponentially fast an integer value when we increase system size also index estimates coincide at size. The theorem applies any non-homogeneous system, such as disordered or defect configurations. focus on one-dimensional chains chiral symmetry, Su–Schrieffer–Heeger model, but proof actually only requires Hamiltonian be short range spectral gap bulk. definition relies finite-size version switch-function where Fermi projector is smoothed energy using carefully chosen regularization parameter.