Coupling of Edge States and Topological Bragg Solitons

The existence of the edge states at the interface between two media with different topological properties is protected by symmetry, which makes such states robust against structural defects or disorder. We show that, if a system supports more than one topological edge state at the interface, even a weak periodic deformation may scatter one edge state into another without coupling to bulk modes. This is the Bragg scattering of the edge modes, which in a topological system is highly selective, with closed bulk and backward scattering channels, even when conditions for resonant scattering are not satisfied. When such a system bears nonlinearity, Bragg scattering enables the formation of a new type of soliton—topological Bragg solitons. We report them in a spin-orbit-coupled (SOC) Bose-Einstein condensate in a homogeneous honeycomb Zeeman lattice. An interface supporting two edge states is created by two different SOCs, with the $y$ component of the synthetic magnetic field having opposite directions at different sides of the interface. The reported Bragg solitons are found to be stable.

Figure 1

(a) Two lowest bands in the spectrum of the ZL with $R(\mathit{r})=-\rho {\sum }_{m,n}\exp [-(\mathit{r}-{\mathit{r}}_{mn}{)}^2/d^2]$ shown in (b) for one component. The amplitude of the lattice is $\rho =8$, the characteristic width of the potential maxima and minima centered at the nodes ${\mathit{r}}_{mn}=(x_m,y_n)$ of the honeycomb grid is $d=0.5$, the distance between neighboring sites is $a=1.4$. The dashed line in (b) indicates the interface between domains with Rashba SOC (left) and Dresselhaus SOC (right) with equal amplitudes $\beta =1.5$. Moduli of the components of the edge state 1 at $k_1=0.375K$ (c),(d) and of the edge state 2 at $k_2=0.575K$ (e),(f), where $K=2\pi /\sqrt{3}a$. Both states, marked by the blue dots in (a), have the energy ${ϵ}_0\approx -3.376$.

Figure 2

Linear evolution of ${\nu }_{1,2}$ in the exact Bragg resonance ${\kappa }_2={\kappa }_1+0.2K$ (a) and near Bragg resonance ${\kappa }_2={\kappa }_1+0.225K$, i.e., $\mathrm{\Delta }k=0.025K$ corresponding to $\mathrm{\Delta }ϵ=0.0077$ (b). The dots and solid lines show, respectively, the results of direct solution of Eq. (1) and predictions of the coupled-mode theory. In both cases the modulation depth is $\delta =0.004$, while at $t=0$, ${\nu }_1=1$, and ${\nu }_2=0$.

Figure 3

Resonant scattering of the initial edge state prepared at $k_1=0.375K$ for $\delta =0.004$. Real-space $|{\mathrm{\Psi }}^{(1)}|$ distribution within $x,y\in [-16L,16L]$ window (top row), and corresponding momentum-space $|{\phi }^{(1)}|$ distribution within $k_x$, $k_y\in [-1.5K,1.5K]$ window (bottom row) are shown at approximately a quarter-period ($t=320$), half-period ($t=640$), and one period ($t=1280$).

Figure 4

(a) Energy mismatch $\mathrm{\Delta }ϵ$, coupling constant $c$, and (b) maximal density ${\nu }_2^{\max }$ versus $k_2-k_1$ at $k_1=0.375K$ and $\delta =0.004$. Dots in (a) are guides for the eye. Dots in (b) show the results of direct simulations of Eq. (1) while the solid line shows predictions of the coupled-mode theory. The exact Bragg resonance occurs at $\kappa =0.2K$.

Figure 5

$|{\mathrm{\Psi }}^{(1)}|$ distributions of a wave packet at $g=0$ (a)–(c) and a topological Bragg soliton at $g=-1$ (d)–(f), obtained by the direct simulation of (1) are shown at instants within the $x\in [-16L,16L]$, $y\in [-60L,60L]$ window. Initial distributions were constructed using (3) with $\tau =1$, $\sigma =0.44\pi$. The modulation amplitude $\delta =0.008$ corresponds to $c=0.00484$. The coupled modes have group velocities $v_1=0.2543$ and $v_2=0.1215$. We mention high accuracy of the definition of the soliton velocity, which is 0.1828 in the shown numerical results, as compared to 0.1879 predicted by (3). The nonlinear coefficients are ${\chi }_1=0.0739$, ${\chi }_2=0.1515$, and $\tilde{\chi }=0.1965$.