Pseudo-magnetism in a strained discrete honeycomb lattice

TL;DR

This paper derives a continuum Dirac Hamiltonian from a discrete honeycomb lattice under strain, demonstrating the existence of localized pseudo-magnetic states and analyzing their spectral properties.

Contribution

It provides a rigorous derivation of effective magnetic Dirac equations for strained honeycomb media and characterizes localized wave states under specific deformation conditions.

Findings

Localized pseudo-magnetic states exist under unidirectional strain with bounded gradient.

Deformations inducing a perpendicular pseudo-magnetic field produce nearly flat Landau levels.

No localization occurs for deformations preserving zigzag translation symmetry.

Abstract

Slowly varying nonuniform strains of non-magnetic wave propagating media with honeycomb symmetry induce an effective- or pseudo-magnetic field, a phenomenon observed first in graphene, and later in photonic crystals and other physical settings. Starting with a discrete nearest-neighbor tight-binding model of a non-uniformly strained honeycomb medium, we derive the continuum effective magnetic Dirac Hamiltonian governing the envelope dynamics of wave packets, which are spectrally localized near a Dirac point (conical band degeneracy) of the unperturbed honeycomb. For unidirectional deformations of bounded gradient, which preserve translation invariance along the ''armchair'' direction, we prove the existence of time-harmonic states which are plane-wave like (pseudo-periodic) along the armchair direction and exponentially localized transverse to it. We also obtain the leading order multi-scale structure of such modes for small deformation gradients. Their transverse localization are determined by the eigenstates of a one dimensional effective Dirac Hamiltonian. Our rigorous results apply to deformations which induce an approximate perpendicular constant pseudo-magnetic field (Landau gauge), and yield states with nearly flat band (Landau level) spectrum and hence very high density of states. In contrast, the analogous deformation which preserves translations in the zigzag direction induces no such localization. Corroborating numerical simulations for the different deformation types are presented.