  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.
