On the Spectra of Periodic Elastic Beam Lattices: Single-Layer Graph

Abstract

We present full description of spectra for a Hamiltonian defined on periodic hexagonal elastic lattices. These continua are constructed out of Euler-Bernoulli beams, each governed by a scalar-valued self-adjoint operator, which is also known as the fourth order Schr\"{o}dinger operator, equipped with a real periodic symmetric potential. In contrast to the second order Schr\"{o}dinger operator commonly applied in quantum graph literature, here vertex matching conditions encode geometry of the underlying graph by their dependence on angles at which edges are met. We show that for a special equal angle lattice, known as graphene, dispersion relation has a similar structure as reported for the periodic second order Schr\"{o}dinger operator on hexagonal lattices. This property is then further utilized to prove existence of singular Dirac points. We further discuss reducibility of Fermi surface at uncountably many low-energy levels for this special lattice. Applying perturbation analysis, we extend the developed theory to derive dispersion relation for angle-perturbed Hamiltonian of hexagonal lattices in a geometric neighborhood of graphene. In these graphs, unlike graphene, dispersion relation is not splitted into purely energy and quasimomentum dependent terms, however singular Dirac points exist similar to the graphene case.