Propagating edge states in strained honeycomb lattices

Abstract

We investigate the helically-propagating edge states associated with pseudo-Landau levels in strained honeycomb lattices. We exploit chiral symmetry to derive a general criterion for the existence of these propagating edge states in the presence of only nearest-neighbour hoppings and we verify our criterion using numerical simulations of both uni-axially and trigonally strained honeycomb lattices. We show that the propagation of the helical edge state can be controlled by engineering the shape of the edges. Sensitivity to chiral-symmetry-breaking next-nearest-neighbour hoppings is assessed. Our result opens up an avenue toward the precise control of edge modes through manipulation of the edge shape.

Direct observation of photonic Landau levels and helical edge states in strained honeycomb lattices

“…For this reason, propagating edge states associated with the n = 0 Landau level appear only at the right edge for the sign of the hopping gradient that we consider. Note that by choosing other terminations, it is possible to engineer propagating edge states only at the left edge, propagating edge states at both edges simultaneously, or no propagating edge states at all 22,47 . This behaviour is a consequence of the sublattice symmetry breaking of the n = 0 Landau level emerging from the presence of the pseudomagnetic field and is very different from the unidirectional chiral edge states that appear in graphene under an external magnetic field, with a propagation direction independent of the type of edge and robust to local disorder.…”

Observation of Photonic Landau Levels in Strained Honeycomb Lattices

“…Since time-reversal symmetry is preserved, edge states in different valleys propagate in opposite di-rections, resulting in helical transport. Bearded terminations (not shown here) result in edge states located on the left edge 43 . The goal of this article is to report on the experimental implementation of these photonic Landau levels and their associated helical edge states.…”

“…To understand why only the right zigzag edge supports propagation of the zeroth Landau level, one has to consider the chiral symmetry of our system and the peculiar form of the zeroth Landau level wavefunction emerging from the pseudomagnetic field 43 . As the zeroth Landau level wavefunction is entirely localised on the B sublattice, its energy is pinned to E p 0 (the onsite energy of the isolated p orbitals) by the chiral symmetry.…”

“…For this reason, for the sign of the hopping gradient we are considering, propagating edge states associated to the n = 0 Landau level only appear at the right edge. Note that by choosing other terminations it is possible to engineer propagating edge states only at the left edge, at both edges simultaneously, or no propagating edge state at all 43,45 . This behaviour is a consequence of the sublattice symmetry breaking of the n = 0 Landau level emerging from the presence of the pseudomagnetic field, and it is very different from the unidirectional chiral edge states that appear in graphene under an external magnetic field, with a propagation direction independent of the type of edge and robust to local disorder.…”

Observation of supersymmetric pseudo-Landau levels in strained microwave graphene

“…In particular, the signatures of photonic pseudo-Landau levels have been detected by probing the edges of a honeycomb array of optical waveguides 18 . A second example is a class of helical edge states in reciprocal systems, as observed, e.g., in zigzag terminated graphene 21,22 . While these bulk and edge phenomena do not naturally fall into the scope of standard topological band structure theory 1 , they are still intimately linked to wavefunctions with a characteristic sublattice polarisation.…”