Peierls-Nabarro barrier effect in nonlinear Floquet topological insulators

The Peierls-Nabarro barrier is a discrete effect that frequently occurs in discrete nonlinear systems. A signature of the barrier is the slowing and eventual stopping of discrete solitary waves. This work examines intense electromagnetic waves propagating through a periodic honeycomb lattice of helically driven waveguides, which serves as a paradigmatic Floquet topological insulator. Here it is shown that discrete topologically protected edge modes do not suffer from the typical slowdown associated with the Peierls-Nabarro barrier. Instead, as a result of their topological nature, the modes always move forward and redistribute their energy: a narrow (discrete) mode transforms into a wide effectively continuous mode. On the other hand, a discrete edge mode that is not topologically protected does eventually slow down and stop propagating. Topological modes that are initially narrow naturally tend to wide envelope states that are described by a generalized nonlinear Schrödinger equation. These results provide insight into the nature of nonlinear topological insulators and their application.

Figure 1

Continuous kink solution (4) evaluated on a discrete grid with $v=1/2$ and $h=1$. (a) “On-site” mode centered at grid point $n=0$ where $x_0=0$; (b) “off-site” mode ($x_0=1/2$), not centered at any grid point.

Figure 2

Initial solitons (10) evaluated on a discrete grid with $\xi =-1$, $\eta =1,$ and $h=1$. (a) “On-site” mode ($x_0=0$) centered at grid point $n=0$; (b) “off-site” mode ($x_0=1/2$) not centered at any grid point.

Figure 3

Location of the discrete soliton peak as a function of time for the DNLS Eq. (5) with $h=1$. Initial conditions are taken from (10) with $\xi =-1$, $x_0=-250$, and different values of $\eta$. At fixed $t$, curves appear top-to-bottom in increasing values of $\eta$.

Figure 4

Honeycomb lattice with lattice vectors ${\mathbf{v}}_1,{\mathbf{v}}_2$ defined in (11). The zigzag (armchair) edge is parallel to the $y\left(x\right)$ axis.

Figure 5

Bulk dispersion surfaces (18) computed from (15) with $\kappa =1.36$ and $\epsilon =0.245$ at (a) $\rho =0.7$, (b) $\rho =0.45$, and (c) $\rho =0.2$. Included are the associated Chern numbers, where applicable, for the bands.

Figure 6

Linear edge Floquet bands computed from (21) and (22) for (a) $\rho =0.7$, (b) $\rho =0.45$, and (c) $\rho =0.2$. The blue (red) bands correspond to edge modes localized along the left (right) domain wall. Black regions correspond to bulk modes. A green dot denotes the edge mode localized along the left edge with mode number $k_y=4/\sqrt{3}$.

Figure 7

Dispersion curves corresponding to edge modes in Fig. 6 localized along the left boundary. (left column) $\rho =0.7$, topological; (right column) $\rho =0.2,$ nontopological. The green, black, and red dots denote the location of the $k_y=\pi /\sqrt{3},2.18,$ and $4/\sqrt{3}$ modes, respectively.

Figure 8

Location of a soliton peak as a function of $z$ for the nonlinear, topological case. Balanced initial data (30) with different values of $\nu$ are studied. The thick black line is a trajectory with group velocity ${\alpha }^{\prime }=-0.48$ corresponding to the mode $k_y=2.18$ in Fig. 7. At fixed $z$, curves appear top-to-bottom in increasing values of $\nu$. Inset: peak location for $0\le z\le 250$.

Figure 9

Location of a soliton peak as a function of $z$ for the nonlinear, nontopological case. Balanced initial data (30) with different values of $\nu$ are studied. At fixed $z$, curves appear top-to-bottom in decreasing values of $\nu$.

Figure 10

Location of a soliton peak as a function of $z$ for the linear, nontopological case. The initial conditions are the same as in Fig. 9. At fixed $z$, curves appear top-to-bottom in increasing values of $\nu$. Inset: peak location for $750\le z\le 800$.

Figure 11

(a) Peak magnitude, (b) participation number, and (c) product of peak magnitude and participation number evolutions for the balanced case. Initial data (30) with $A_0=\nu \sqrt{{\alpha }_{*}^{″}/{\alpha }_{\mathrm{nl}}}$ and $K=4/\sqrt{3}$ are used. At $z=1000$, curves in panel (c) appear top-to-bottom in decreasing values of $\nu$.

Figure 12

Initial condition ($z=0$), numerical solution at given $z$, and secant hyperbolic (sech) fit of edge profile magnitude, $|b_{m0}\left(z\right)|$, for the topologically protected, balanced case. The initial width $\nu$ and $z$ value of the profiles are (a) $(\nu ,z)=(0.25,250)$, (b) $(\nu ,z)=(0.7,250)$, (c) $(\nu ,z)=(0.25,1000)$, and (d) $(\nu ,z)=(0.7,1000)$. Note: each mode shown here has been recentered to $m=0$ to appreciate their structure.

Figure 13

Initial amplitude and width used in initial condition (30). The center green line corresponds to the balanced initial state. The upper red (lower blue) line corresponds to increased amplitude (decreased width) relative to the balanced cases. The off-balanced case is computed by increasing $A_0$ ($\nu$) from the green line while leaving $\nu$ ($A_0$) fixed. Points on the center green, upper red, and lower blue lines correspond to the results shown in Figs. 11, 14, and 16, respectively.

Figure 14

(a) Peak position, (b) peak magnitude, (c) participation number, and (d) product of peak magnitude and participation number evolutions for the unbalanced, increased amplitude case (red line in Fig. 13). Initial data (30) with $A_0=1.1\nu \sqrt{{\alpha }_{*}^{″}/{\alpha }_{\mathrm{nl}}}$ are taken. At $z=1000$, curves in panels (a) and (d) appear top-to-bottom in increasing and decreasing values of $\nu$, respectively.

Figure 15

Initial condition ($z=0$), numerical final condition ($z=1000$), and secant hyperbolic (sech) fit of edge profile magnitudes, $|b_{m0}\left(z\right)|$, for the topologically protected but unbalanced cases. The initial width $\nu$ and amplitude $A_0$ of the profiles are (a) $(\nu ,A_0)=(0.25,0.57)$, (b) $(\nu ,A_0)=(0.7,1.60)$, (c) $(\nu ,A_0)=(0.275,0.52)$, and (d) $(\nu ,A_0)=(0.77,1.46)$. Top row: increased amplitude case; bottom row: decreased width case. Left column: wider mode; right column: narrower mode. Note: each mode shown here has been recentered to $m=0$ to help describe their structure.

Figure 16

(a) Peak position, (b) peak magnitude, (b) participation number, and (c) product of peak magnitude and participation number evolutions for the unbalanced, decreased width case (blue line in Fig. 13). Initial data (30) with $\nu =1.1A_0\sqrt{{\alpha }_{\mathrm{nl}}/{\alpha }_{*}^{″}}$ are taken. At $z=1000$, curves in panels (a) and (d) appear top-to-bottom in increasing and decreasing values of $\nu$, respectively.

Figure 17

Location of a soliton peak as a function of $z$ for the nonlinear, topological case. Initial data (30) with $A_0=1.04$ and different values of $\nu$ are used. All other parameters are the same as Fig. 8. The thick black line is a trajectory with group velocity ${\alpha }^{\prime }=-0.583$ corresponding to the mode $k_x=\pi /\sqrt{3}$ (green dot) in Fig. 7. At fixed $z$, curves appear top-to-bottom in increasing values of $\nu$. Inset: peak location for $380\le z\le 400$.

Figure 18

The average Hamiltonian value (dashed curve, right axis) of a typical wide ($\nu =0.25$) balanced and topologically protected envelope initial profile (30). Also included is a portion of the topologically protected edge dispersion band (solid curve, left axis), originally shown in Fig. 7.