Bulk-Edge Dualities in Topological Matter

Asorey, M.

doi:10.1007/978-3-030-24748-5_2

Cited by 2 publications

(2 citation statements)

References 18 publications

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“…The analysis carried out in this paper also applies to some of these boundary conditions. Although the continuum effective quantum theory of graphene is given by a Dirac operator, the spectral behavior can be obtained from the square of that operator, which is given, up to a spectral shift, by the Hamiltonian (3) with appropriate boundary conditions [73]. A detailed characterization of the latter aspect will be the object of future research.…”

Section: Discussionmentioning

confidence: 99%

Boundary conditions for the quantum Hall effect

Angelone¹,

Asorey²,

Facchi³

et al. 2023

J. Phys. A: Math. Theor.

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We formulate a self-consistent model of the integer quantum Hall effect on an infinite strip, using boundary conditions to investigate the influence of finite-size effects on the Hall conductivity. By exploiting the translation symmetry along the strip, we determine both the general spectral properties of the system for a large class of boundary conditions respecting such symmetry, and the full spectrum for (fibered) Robin boundary conditions. In particular, we find that the latter introduce a new kind of states with no classical analogues, and add a finer structure to the quantization pattern of the Hall conductivity. Moreover, our model also predicts the breakdown of the quantum Hall effect at high values of the applied electric field.

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Section: Discussionmentioning

confidence: 99%

Boundary conditions for the quantum Hall effect

Angelone¹,

Asorey²,

Facchi³

et al. 2023

J. Phys. A: Math. Theor.

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“…These results apply to both bulk and boundary topological indices and a consequence (using [73,24]) is an alternative proof of a bulk boundary correspondence (BEC) for continuum crystals in a magnetic field; see Figure 1. The first proof of the BEC in the continuum setting for Hamiltonians with a spectral gap appeared in [51], and for Hamiltonians with a mobility gap in [78]; see also [15,23,43,16,7,34].…”

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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Bulk-Edge Dualities in Topological Matter

Cited by 2 publications

References 18 publications

Boundary conditions for the quantum Hall effect

Boundary conditions for the quantum Hall effect

Tight-Binding Reduction and Topological Equivalence in Strong Magnetic Fields

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