Pseudo-magnetism in a strained discrete honeycomb lattice

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.

Figures (10)

• Figure 1: The undeformed and deformed honeycomb and their band structures: (a) the undeformed honeycomb with the unit cell highlighted by shading; (b) the band structure of the tight-binding Hamiltonian $H^0$ for the undeformed honeycomb with Dirac points occurring at the intersection of the two dispersion surfaces, at the six vertices of the Brillouin zone $\mathcal{B}$; (c) the deformed honeycomb with a slowly-varying quadratic deformation $\bm u(\delta \bm X) = (0,\delta^2X_1^2)^T$; (d) Landau levels generated by the effect of strain in (c) on the spectrum.
• Figure 3: (a) The honeycomb lattice, $\mathbb Z\bm{v}_1+\mathbb Z\bm{v}_2$, containing two nodes, $A$ in red and $B$ in blue, per unit cell. $\mathbb R^2$ is the union of cells (parallelograms), each containing two nodes (circled pair of dots), labeled with $\mathbb Z^2$ indices. Dark-shaded parallelogram is the $(m,n)$-cell; adjacent lighter-shaded regions are translations by $\pm \bm{v}_1$ or $\pm \bm{v}_2$. (b) nearest neighbors of $A_{m,n}$ and $B_{m,n}$, bond direction vectors $\bm{e}_i$ ($i=1,2,3$); (c) Hexagonal Brillouin zone $\mathcal{B}$ with high symmetry quasimomenta $\bm{K},\bm{K}'$, at the vertices of $\mathcal{B}$. Dual lattice vectors $\bm{a}_1, \bm{a}_2$, such that $\bm a_l\cdot\bm v_j=2\pi\delta_{lj}$.
• Figure 4: The undeformed honeycomb along the AC edge: (a) our numerical model of the honeycomb along the AC edge as a periodic structure in the vertical direction; (b) the numerical eigenvalue curves with truncation size $N_T = 200$ and the Dirac point occurs at $q_\parallel = 0$; (c) a zoomed-in view at $q_\parallel = 0$ with the lowest 40 eigenvalues.
• Figure 5: Numerical eigenvalue curves of the honeycomb lattice under $\bm{u}^{\mathrm{AC}}$ and $\bm{u}^{\mathrm{AC}}_{\mathrm{reg}}$ with different $\delta$ and truncation size $N_T=200$: (a) $\bm{u}^{\mathrm{AC}}$ with $\delta=0.04$; (b) $\bm{u}^{\mathrm{AC}}_{\mathrm{reg}}$ with $\delta=0.04$ and $\alpha_\text{reg} = 0.5$; (c) $\bm{u}^{\mathrm{AC}}$ with $\delta=0.08$; (d)-(f) are zoomed-in numerical eigenvalue curves near $q_\parallel=0$ with the 40 numerical eigenvalue curves of smallest magnitude for (a)-(c), and markers in (d) correspond to eigenstates plotted in Figure \ref{['fig:ac-defmodes']}. The value of $q_\parallel$ is 0.02 for the dot and square in (d).
• Figure 6: Eigenstates of the deformed honeycomb lattice under $\bm{u}^{\mathrm{AC}}$ with $\delta=0.04$: (a) $|\psi_s^B|$ as a function of $s$ of zero eigenstates for $q_\parallel=0$ at the diamond in Figure \ref{['fig:ac-def-middle-band']} and $q_\parallel$ at the circle in Figure \ref{['fig:ac-def-middle-band']}; note that the zero eigenstates vanish on the $A$ nodes; (b)-(c) eigenstates corresponding to the triangle and square markers in Figure \ref{['fig:ac-def-middle-band']}, representing the smallest nonzero eigenvalues in magnitude; (b) shows $|\psi_s^A|$, oscillations on the $A$ nodes, while (c) shows $|\psi_s^B|$, oscillations on the nodes.

Theorems & Definitions (40)

• Proposition 3.1
• Proposition 3.2
• Remark 3.3: Comparing the discrete and continuous case
• Proposition 4.1
• Proposition 4.2
• Lemma 4.3: Spectral properties of $\mathcal{D}(k_\parallel)$ with bounded $\kappa(X_1;k_\parallel)$
• Remark 4.4: The topologically proteced zero eigenstates
• Remark 4.5: The number of nonzero eigenvalues in the spectral gap
• Theorem 5.1
• Remark 5.2: Inverting $\mathcal{D}(k_\parallel) - E_1(k_\parallel)$