Cobordism invariance of topological edge-following states

Abstract

We prove that a spectral gap-filling phenomenon occurs whenever a Hamiltonian operator encounters a coarse index obstruction upon compression to a domain with boundary. Furthermore, the gap-filling spectra contribute to quantised current channels, which follow and are localised at the possibly complicated boundary. This index obstruction is shown to be insensitive to deformations of the domain boundary, so the phenomenon is generic for magnetic Laplacians modelling quantum Hall systems and Chern topological insulators. A key construction is a quasi-equivariant version of Roe's algebra of locally compact finite propagation operators.

Delocalized spectra of Landau operators on helical surfaces

“…, has a more refined interpretation using the methods of [17] Section 6. Technically, this interpretation requires a polynomial growth condition, which is satisfied by helicoids.…”

“…In brief, let X +,↑ c = {(ρ, φ) ∈ X + c : φ ≥ 0} be the upper-half of X + c , and X +,↓ c be the lower-half. Then there is a well-defined integer-valued map, Definition 4.6 of [17],…”

“…The stability of Landau levels (as essential spectra) against perturbations of a constant magnetic field vanishing at infinity, were considered in [10,11]. A recent perspective [16,17] is that the Euclidean/hyperbolic plane Landau level spectral projections, while infinitely-degenerate and thus not characterized by a Fredholm index, actually realize the coarse index [22] of an associated Dirac operator. This coarse geometric perspective, together with supersymmetry techniques, can account for small geometric perturbations very efficiently, see Section 2.3.…”

Topological Lattice Defects by Groupoid Methods and Kasparov's KK-Theory

“…Disordered and quasi-periodic lattice systems can be conveniently analyzed in the framework of crossed-product algebras and the bulk-boundary correspondence principle can be derived from the exact sequence of boundary, half-space and bulk algebras, which in essence is just the Pimsner-Voiculescu exact sequence [22]. Roe algebras [12] have been also successfully used to explore the bulk-boundary correspondence principle in settings with irregular boundaries [30,21] or in hyperbolic geometries [20]. Recently, Roe algebras have been also employed to formulate a bulk-defect correspondence principle for a weak topological insulator [18].…”

Computing spectral properties of topological insulators without artificial truncation or supercell approximation

“…The edge currents of TIs are mediated by electronic states localised at edges known as edge states. The robustness of the edge conductance of a TI can be seen at the level of localised wave-packets formed from these edge states, which snake around corners and defects of the edge, even in the presence of disorder [3,11,17,19,41,46,59,61,66,75,80,81,85,93]. The behaviour of these wave-packets has spurred interest in building photonic and acoustic devices which mimic topological insulators for wave-guiding applications [38,72,86,99,100,105].…”