Unconventional edge states in a two-leg ladder

TL;DR

This paper demonstrates the emergence of unconventional edge states in a simple two-leg ladder of coupled harmonic oscillators, influenced by next-nearest neighbour interactions, with potential applications in waveguiding and quantum transport.

Contribution

It reveals how edge states can form in a two-leg ladder model through specific interactions beyond nearest neighbors, expanding understanding of localized states in lattice systems.

Findings

Edge states depend on the interplay of rung and next-nearest neighbour couplings.

Edge states correspond to stationary points in the continuum bandstructure.

Results are applicable to classical and quantum lattice model simulators.

Abstract

Some popular mechanisms for restricting the diffusion of waves include introducing disorder (to provoke Anderson localization) and engineering topologically non-trivial phases (to allow for topological edge states to form). However, other methods for inducing somewhat localized states in elementary lattice models have been historically much less studied. Here we show how edge states can emerge within a simple two-leg ladder of coupled harmonic oscillators, where it is important to include interactions beyond those at the nearest neighbour range. Remarkably, depending upon the interplay between the coupling strength along the rungs of the ladder and the next-nearest neighbour coupling strength along one side of the ladder, edge states can indeed appear at particular energies. In a wonderful manifestation of a type of bulk-edge correspondence, these edge state energies correspond to the quantum number for which additional stationary points appear in the continuum bandstructure of the equivalent problem studied with periodic boundary conditions. Our theoretical results are relevant to a swathe of classical or quantum lattice model simulators, such that the proposed edge states may be useful for applications including waveguiding in metamaterials and quantum transport.