Do Linear Dispersions of Classical Waves Mean Dirac Cones?

TL;DR

This paper develops a first-principles  method to analyze linear dispersions in phononic and photonic crystals, clarifying when these dispersions resemble Dirac cones and their associated Berry phases.

Contribution

It introduces a theory that predicts Dirac-like dispersions from degeneracy types and symmetry, accurately determining their slopes regardless of frequency or lattice structure.

Findings

Linear dispersions from doubly-degenerate states can be described by Dirac Hamiltonians.

Triply-degenerate states produce Dirac-like cones with spin-1 wavefunctions and zero Berry phase.

The theory accurately predicts slopes of Dirac/Dirac-like cones at various symmetry points.

Abstract

By using the \vec{k}\cdot\vec{p} method, we propose a first-principles theory to study the linear dispersions in phononic and photonic crystals. The theory reveals that only those linear dispersions created by doubly-degenerate states can be described by a reduced Hamiltonian that can be mapped into the Dirac Hamiltonian and possess a Berry phase of -\pi. Triply-degenerate states can also generate Dirac-like cone dispersions, but the wavefunctions transform like a spin-1 particle and the Berry phase is zero. Our theory is capable of predicting accurately the linear slopes of Dirac/Dirac-like cones at various symmetry points in a Brilliouin zone, independent of frequency and lattice structure.