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

TL;DR

This paper analyzes the spectral properties of elastic beam lattices modeled by a fourth order Schrödinger operator on periodic hexagonal structures, revealing Dirac points and spectral features similar to quantum graphs, with extensions to perturbed geometries.

Contribution

It provides a full spectral description of elastic beam lattices on hexagonal graphs, including Dirac points and effects of geometric perturbations, extending quantum graph results to elastic structures.

Findings

Dispersion relation similar to quantum graphs for graphene.

Existence of singular Dirac points in the spectrum.

Spectral properties extend to angle-perturbed lattices.

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.