Edge states and the valley Hall effect

Abstract

We study energy propagation along line-defects (edges) in two dimensional continuous, energy preserving periodic media. The unperturbed medium (bulk) is modeled by a honeycomb Schroedinger operator, which is periodic with respect to the triangular lattice, invariant under parity, P, and complex-conjugation, C . A honeycomb operator has Dirac points in its band structure: two dispersion surfaces touch conically at an energy level, E D [25,27]. Periodic perturbations which break P or C open a gap in the essential spectrum about energy E D . Such operators model an insulator near energy E D .Our edge operator is a small perturbation of the bulk and models a transition (via a domain wall) between distinct periodic, P or C breaking perturbations. The edge operator permits energy transport along the line-defect. The associated energy channels are called edge states. They are time-harmonic solutions of the underlying wave equation, which are localized near and propagating along the line-defect. They are of great scientific interest due to their remarkable stability, and are a key property of topological insulators.We completely characterize the edge state spectrum within the bulk spectral gap about E D . At the center of our analysis is an expansion of the edge operator resolvent for energies near E D . The leading term features the resolvent of an effective Dirac operator. Edge state eigenvalues are poles of the resolvent, which bifurcate from the Dirac point. The corresponding eigenstates have the multiscale structure identified in [23].We extend earlier work on zigzag-type edges [14] to all rational edges. We elucidate the role in edge state formation played by the type of symmetry-breaking and the orientation of the edge. We prove the resolvent expansion by a new direct and transparent strategy. Our results also provide a rigorous explanation of the numerical observations in [22,38]; see also the photonic experimental study in [42]. Finally we discuss implications for the Valley Hall Effect, which concerns quantum Hall-like energy transport in honeycomb structures.

Non-reciprocal wave propagation in space-time modulated media

“…By turning on the modulation (Figures 7b and 7c), the Dirac cones open up to local extrema of the band functions. The local extrema are called the valleys [19], or valley degrees of freedom. By breaking reciprocity, we obtain different valleys for K and −K.…”

“…More recently, the field of topological insulators in condensed matter physics has been teeming with intriguing and very exciting discoveries. Notably, the capacity of guiding currents towards specific directions according to the spin of the travelling electrons has a great potential for electronic devices [15,19].…”

Wave packets in the fractional nonlinear Schrödinger equation with a honeycomb potential

“…Examples include formulations of fractional derivatives acting on quasi-periodic functions, asymptotics of the fractional Laplacian acting on a wave packet with highly oscillating Floquet-Bloch modes, homogenization of such modes with degenerate eigenvalues and so on. The results also shed some light on the rigorous analysis of topologically protected wave propagation in honeycomb-based media if additional assumptions are added to the slowly varying modulations [13,16,22,29]. This paper will be organized as follows: In section 2 and 3, we briefly review the Floquet-Bloch theory from honeycomb latticed fractional Schödinger operator and verify the existence of Dirac point.…”

Magnetic slowdown of topological edge states

“…From a physical point of a view, our motivation stems from the ubiquity of / D in the field of topological phases of matter [Wit16,MM21], and in particular one-particle models of topological insulators and topological superconductors [Vol89,Ber13,PSB16,Bal19a,Bal19b], which generically come with conical points [Dro21b]. The domain wall κ(x) models the interface between two topologically distinct insulating phases [FLTW16,Dro19b,DW20]. This in turn generates an asymmetric transport along the interface Γ by a principle called the bulk-interface correspondence; see e.g.…”