Tight-Binding Reduction and Topological Equivalence in Strong Magnetic Fields

Abstract

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.

Fredholm Homotopies for Strongly-Disordered 2D Insulators

“…While we use the above definition in the present work, it is worth noting that in the spectral gap regime there is an alternative, convenient way to encode the fact that an edge system descends from a bulk gapped system (see [29,Definition 3.4]): Definition 2.8 (Edge systems with a bulk spectral gap). Let ∆ ⊆ R be a given interval.…”

Is the continuum SSH model topological?

“…It is natural to ask whether the topological properties of 1D discrete models are present in the continuum models of which these discrete models are approximations. For the case of continuum two-dimensional crystals in a strong constant magnetic field, the setting of the integer quantum Hall effect (IQHE), we recently proved the equality of topological invariants for continuum Hamiltonians -in the strong binding regime -with those of the discrete tight binding limit [13].…”

“…Remark 1.1. The arguments of [13] establish norm resolvent convergence of the continuum Hamiltonian to the tight-binding (discrete) limit, a notion of convergence enabling the control of topological indices for IQHE, which are expressed in terms of spectral (Fermi) projections onto an isolated spectral set. The results of this paper suggest then that there is no continuum index (e.g.…”

“…In more general settings, e.g. 2D models considered in [4,13], lower bounds on these matrix elements are proved. We now introduce a λ− dependent atomic lattice with the choice:…”

“…Since d in < d out , we have from (3.4)-(3.5) that ρ λ 1 > ρ λ 2 . Our procedure allow for the centering of atoms to be dependent on the asymptotic parameter, λ; see [13,Example 2.11].…”