Discrete approximation of topologically protected modes in magneto-optical media

Ablowitz, Mark J.; Cole, Justin T.

doi:10.1103/physreva.101.023811

Cited by 5 publications

(1 citation statement)

References 28 publications

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“…See also [69]. A numerical implementation of this idea appears in [1]. We are interested in the strong binding regime, which we interpret as λ sufficiently large.…”

Section: Introduction and Outlinementioning

confidence: 99%

Tight-Binding Reduction and Topological Equivalence in Strong Magnetic Fields

Shapiro¹,

Weinstein²

2020

Preprint

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We study a class of continuum Schroedinger operators, H λ , on L 2 (R 2 ) which model non-interacting electrons in a two-dimensional crystal subject to a perpendicular constant magnetic field. Such Hamiltonians play an important role in the field of topological insulators. The crystal is modeled by a potential consisting of identical potential wells, centered on an infinite discrete set of atomic centers, G, with strictly positive minimal pairwise distance. The set G, and therefore the crystal potential, is not assumed to be translation invariant. We consider the limit where both magnetic field strength and depths of the array of atomic potential wells are sufficiently large, λ ≫ 1. The topological properties of the crystal are encoded in topological indices derived from H λ . We prove norm resolvent convergence, under an appropriate scaling, of the continuum Hamiltonian to a scale-free tight-binding (discrete) Hamiltonian, H TB , on l 2 (G). For the case G = Z 2 , the H TB is the well-known Harper model. Our analysis makes use of a recent lower bound for the double-well tunneling probability (hopping coefficient) in strongly magnetic systems [30]. We then apply our results on resolvent convergence to prove that the topological invariants associated with H λ , λ ≫ 1 and H TB are equal. These results apply to both bulk and edge invariants. Thus our theorems provide justification for the ubiquitous use of tight-binding models in studies of topological insulators.

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“…See also [69]. A numerical implementation of this idea appears in [1]. We are interested in the strong binding regime, which we interpret as λ sufficiently large.…”

Section: Introduction and Outlinementioning

confidence: 99%