Edge states and the Valley Hall Effect

Abstract

We study energy propagation along line-defects (edges) in 2D 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: 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 which are localized near and propagating along the line-defect. 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 provide a rigorous explanation of the numerical observations [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.