We develop a formalism to estimate, simultaneously, the usual bulk and edge indices from topological insulators in the case of a finite sample with open boundary conditions and provide a physical interpretation of these quantities. We then show that they converge exponentially fast to an integer value when we increase the system size and also show that bulk and edge index estimates coincide at finite size. The theorem applies to any non-homogeneous system, such as disordered or defect configurations. We focus on one-dimensional chains with chiral symmetry, such as the Su–Schrieffer–Heeger model, but the proof actually only requires the Hamiltonian to be of short range and with a spectral gap in the bulk. The definition of bulk and edge index estimates relies on a finite-size version of the switch-function formalism where the Fermi projector is smoothed in energy using a carefully chosen regularization parameter.
Estimating bulk and edge topological indices in finite open chiral chains
Note: This paper is part of the Special Topic on Mathematical Aspects of Topological Phases.
Lucien Jezequel, Clément Tauber, Pierre Delplace; Estimating bulk and edge topological indices in finite open chiral chains. J. Math. Phys. 1 December 2022; 63 (12): 121901. https://doi.org/10.1063/5.0096720
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