Equality of the Bulk and Edge Hall Conductances in a Mobility Gap

Elgart, Alexander; Graf, Gian Michele; Schenker, Jeffrey H.

doi:10.1007/s00220-005-1369-7

Cited by 88 publications

(117 citation statements)

References 32 publications

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“…A similar result was established for discrete Schrödinger operators in Ref. 8. We should mention that certain regularization of the edge Hall conductance was necessary for the case of strong disorder.…”

Section: Introductionsupporting

confidence: 78%

The edge spectrum of Chern insulators with rough boundaries

Prodan

2009

Journal of Mathematical Physics

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Chern insulators are periodic band insulators with the property that their projector onto the occupied bands have non-zero Chern number. Chern insulator with a homogeneous boundary display continuum spectrum that fills the entire insulating gap. The local density of states corresponding to this part of the spectrum is localized near the boundary, hence the terminology edge spectrum. An interesting question arises, namely, if a rough boundary, which can be seen as a strong random potential acting on these quasi 1-dimensional states, would destroy the continuum edge spectrum. This paper shows how such question can be answered via a newly formulated abstract framework in which the expectation value of the current of a general observable is connected to the index of a specific Fredholm operator. For the present application, we will connect the expectation value of the charge edge current with the index of a Fredholm operator that remains invariant under arbitrary deformations of the boundary

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Section: Introductionsupporting

confidence: 78%

The edge spectrum of Chern insulators with rough boundaries

Prodan

2009

Journal of Mathematical Physics

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“…The extensive physics literature on topologically robust edge states goes back to investigations of the quantum Hall effect; see, for example, [30,31,55,58] and the rigorous mathematical articles [12,13,41,54]. In [29,46] a proposal for realizing photonic edge states in periodic electromagnetic structures that exhibit the magneto-optic effect was made.…”

Section: Graphene and Its Artificial Analogues-physical Motivationmentioning

confidence: 99%

Honeycomb Schrödinger Operators in the Strong Binding Regime

Fefferman

Lee-Thorp

Weinstein

2017

Comm Pure Appl Math

107

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In this article, we study the Schrödinger operator for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane. The potentials we consider are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure corresponding to the single electron model of graphene and its artificial analogues. We consider this Schrödinger operator in the regime of strong binding, where the depth of the potential wells is large. Our main result is that for sufficiently deep potentials, the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, converge uniformly to those of the two-band tight-binding model (Wallace, 1947 [56]). Furthermore, we establish as corollaries, in the regime of strong binding, results on (a) the existence of spectral gaps for honeycomb potentials that break PT symmetry and (b) the existence of topologically protected edge states-states that propagate parallel to and are localized transverse to a line defect or "edge"-for a large class of rational edges, and that are robust to a class of large transverse-localized perturbations of the edge. We believe that the ideas of this article may be applicable in other settings for which a tight-binding model emerges in an extreme parameter limit.We begin with a review of Floquet-Bloch theory. See, for example, [11,37,38,49] and [3,23,24,34,53]. Fourier Analysis on L 2 .R=ƒ/Let fv 1 ; v 2 g be a linearly independent set in R 2 , and introduce the following:Lattice: ƒ D Zv 1˚Z v 2 D fm 1 v 1 C m 2 v 2 W m 1 ; m 2 2 ZgI Dual lattice: ƒ D ZK 1˚Z K 2 D fm E K

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“…In applications, edge states are of great interest due to their potential as robust vehicles for channeling energy. The extensive physics literature on topologically robust edge states goes back to investigations of the quantum Hall effect; see, for example, [18,19,39,42] and the rigorous mathematical articles [5,6,28,38]. In [17,32] a proposal for realizing photonic edge states in periodic electromagnetic structures which exhibit the magneto-optic effect was made.…”

Section: Introduction and Outlinementioning

confidence: 99%

Edge States in Honeycomb Structures

2016

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An edge state is a time-harmonic solution of a conservative wave system, e.g. Schrödinger, Maxwell, which is propagating (plane-wave-like) parallel to, and localized transverse to, a line-defect or "edge". Topologically protected edge states are edge states which are stable against spatially localized (even strong) deformations of the edge. First studied in the context of the quantum Hall effect, protected edge states have attracted huge interest due to their role in the field of topological insulators. Theoretical understanding of topological protection has mainly come from discrete (tight-binding) models and direct numerical simulation. In this paper we consider a rich family of continuum PDE models for which we rigorously study regimes where topologically protected edge states exist. Our model is a class of Schrödinger operators on R 2 with a background two-dimensional honeycomb potential perturbed by an "edge-potential". The edge potential is a domain-wall interpolation, transverse to a prescribed "rational" edge, between two distinct periodic structures. General conditions are given for the bifurcation of a branch of topologically protected edge states from Dirac points of the background honeycomb structure. The bifurcation is seeded by the zero mode of a one-dimensional effective Dirac operator. A key condition is a spectral no-fold condition for the prescribed edge. We then use this result to prove the existence of topologically protected edge states along zigzag edges of certain honeycomb structures. Our results are consistent with the physics literature and appear to be the first rigorous results on the existence of topologically protected edge states for continuum 2D PDE systems describing waves in a non-trivial periodic medium. We also show that the family of Hamiltonians we study contains cases where zigzag edge states exist, but which are not topologically protected.

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Equality of the Bulk and Edge Hall Conductances in a Mobility Gap

Cited by 88 publications

References 32 publications

The edge spectrum of Chern insulators with rough boundaries

The edge spectrum of Chern insulators with rough boundaries

Honeycomb Schrödinger Operators in the Strong Binding Regime

Edge States in Honeycomb Structures

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