Estimating bulk and edge topological indices in finite open chiral chains

Abstract

We develop a formalism to extend, simultaneously, the usual definition of bulk and edge indices from topological insulators to 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 that bulk and edge quantities 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 short-range and with a spectral gap in the bulk. The definition of bulk and edge indices 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.

arXiv comment19 pages

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• Local Chern Marker for Periodic Systems
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• Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
• Interacting Local Topological Markers: A one-particle density matrix approach for characterizing the topology of interacting and disordered states
• Edge-dependent anomalous topology in synthetic photonic lattices subject to discrete step walks
• Topological zero-modes of the spectral localizer of trivial metals
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- part II: Higher-dimensional spectral invariants
• Fluctuations and Correlations of Local Topological Order Parameters in Disordered Two-dimensional Topological Insulators
• Real-space formulation of the Chern invariant and topological phases in a disordered Chern insulator