Magnetic slowdown of topological edge states

Abstract

We study the propagation of wavepackets along curved interfaces between topological, magnetic materials. Our Hamiltonian is a massive Dirac operator with a magnetic potential. We construct semiclassical wavepackets propagating along the curved interface as adiabatic modulations of straight edge states under constant magnetic fields. While in the magnetic-free case, the wavepackets propagate coherently at speed one, here they experience slowdown, dispersion, and Aharonov-Bohm effects. Several numerical simulations illustrate our results.

Semiclassical propagation along curved domain walls

“…In cases where (7) holds for all branches m and ξ ∈ Ξ independent of m with realvalued phases G m (see assumptions (vii)-(viii) below), we show that any sufficiently smooth initial condition for u concentrated in the vicinity of (x 0 , y 0 ) ∈ Γ can indeed be represented by a superposition over the branches of continuous spectrum of objects of the form (6); see Theorem 2.4.…”

“…Wavepackets such as (2) encode this asymmetry; see [7, §1.4]. For recent analyses of relativistic modes propagating along curved domain walls, see [6,7,24]. Note that in the presence of (sufficiently highly oscillatory) perturbations of L D , the aforementioned asymmetric transport is encoded by a perturbation-dependent dispersive mode rather than the relativistic mode (2); see [8].…”

“…where A m and B m are appropriate functions with the main constraint that ∂ x A m (x 0 , ξ; x 0 ) = ξ as expected so that the ansatz (6) approximately corresponds to a semiclassical Fourier transform at time t = 0.…”

Traveling edge states in massive Dirac equations along slowly varying edges

“…Among the general edge-admissible mass terms, we asymptotically propose two typical edge states where the edges can be locally treated as straight lines. Here, we exploit the explicit asymptotic behaviors that cling to the edges locally other than solving the wave guidance derived from ODE [10,11]. In the next two subsections, one edge is a circular ring with the sufficiently large radius, and another is a slowly varying curve which is generated by adding a small perturbation to the straight line on the normal direction.…”

“…These physical phenomena stimulate the interests of wave propagation along the nontrivial edges. Recently, a class of Dirac equations with a small semi-classical parameter described the wave packets which propagate along the curved edge for long times [10,11]. However, these effective models depend on the small parameter in the semi-classical equation and the curvature of nearly straight edge provides the limited effect to the time validity.…”