Nonlinear excitations in the honeycomb lattice: Beyond the high-symmetry points of the band structure

The interplay between nonlinearity and the band structure of pristine honeycomb lattices is systematically explored. For that purpose, a theory of collective excitations valid for the first Brillouin zone of the lattice is developed. Closed-form expressions of two-dimensional excitations are derived for Bloch wave numbers beyond the high-symmetry points of the band structure. A description of the regions of validity of different nonlinear excitations in the first-Brillouin zone is given. We find that the unbounded nature of these excitations in nonlinear honeycomb latices is a signature of the strong influence of the Dirac cones in other parts of the band structure.

Figure 1

The $\mathbf{N}=\{l,m\}$ site index distribution in the honeycomb lattice. Here ${\delta }_0$ (black), ${\delta }_1$ (blue), and ${\delta }_2$ (green) are displacement vectors between ${\mathbf{N}}_{\mathbf{A}}$ and ${\mathbf{N}}_{\mathbf{B}}$ sites.

Figure 2

(a) Honeycomb band structure and its projection into the first Brillouin zone. (b) Honeycomb band structure along the symmetry points of the structure.

Figure 3

(a) and (c) Intensity distribution of the kink solutions as a function of the spatial coordinates $x$ and $y$. (b) and (d) Corresponding density plots of (a) and (c), respectively. The directions of the $x$ and $y$ coordinates with respect to the honeycomb lattice are shown in Fig. 1. The arrows indicate the direction of the velocity $\mathit{v}$.

Figure 4

Schematic diagram of the Dirac-like cone form of the band structure in the vicinity of the Dirac points. Circle-number positions in the band structure are related to the solutions plotted in Fig. 5. Here $q>0$ and is defined such that $q/\left|\mathbf{K}\right|\ll 1$.

Figure 5

Density plots of the kink intensity distribution for different points in the vicinity of the Dirac point. The arrows in this plot indicate the velocity direction for each case. The positions of the circle numbers in the band structure of Fig. 4 are related to the density plots as follows: For $U>0$ solutions (a) and (e) correspond to the $\mathrm{①}$ position and solutions (b)–(d) correspond to the $\mathrm{②},\mathrm{③}$, and $\mathrm{④}$ positions, respectively; for $U<0$ solutions (h) and (i) correspond to the $\mathrm{⑦}$ position and solutions (f), (g), and (j) correspond to the $\mathrm{⑤},\mathrm{⑥}$, and $\mathrm{⑧}$ positions, respectively.

Figure 6

Bright soliton solution with $\theta =\pi /4$ and $\beta >0$.

Figure 7

Dark soliton solution with $\theta =\pi /4$ and $\beta >0$.

Figure 8

Regions in the reciprocal space where bright and dark solutions appear. Clear and dark regions correspond to bright and dark soliton solutions, respectively. (a) $\theta =\pi /4$ and $\beta >0$, (b) $\theta =\pi /4$ and $\beta <0$, (c) $\theta =\pi$ and $\beta >0$, and (d) $\theta =\pi$ and $\beta <0$.