Observation of nonlinear higher-order topological insulators with unconventional boundary truncations

Abstract

In higher-order topological insulators (HOTIs), topologically nontrivial phases are usually associated with the shift of Wannier centers to topologically nontrivial positions on the edges of the unit cells, and the emergence of fractional spectral charges in the corners of the lattice upon its truncation that keeps the number of its unit cells integer. Here we propose theoretically and illustrate experimentally a different approach to the construction of HOTIs. This approach utilizes lattices with incomplete unit cells and achieves localized modes of topological origin across a broader parameter space. When truncation disrupts translational symmetry by cutting through the interior of multiple unit cells, boundary modes in our system emerge for both trivial and topologically nontrivial positions of the Wannier centers. We link these modes to the appearance of fractional Wannier centers. We also demonstrate that linear boundary states give rise to rich families of stable solitons bifurcating from them in the presence of focusing nonlinearity. Multiple types of thresholdless topological solitons with different internal symmetries are observed in waveguide arrays with triangular configurations featuring incomplete unit cells for any dimerization of waveguide spacings. Our results expand the family of HOTIs and pave the way for the observation of boundary states with different symmetries.

Figures (6)

• Figure 1: Construction of higher-order topological insulators with unconventional boundary truncations. Microphotographs of the triangular femtosecond laser-written waveguide arrays with complete and incomplete honeycomb unit cells (the former are indicated by blue contours superimposed on the photomicrographs) with different degrees of dimerization $\gamma=0.6$ (a), $\gamma=1.0$ (b) and $\gamma=1.6$ (c), where ${\gamma=d_1/d_2}$ defines the ratio between the distance $d_1$ between the nearest waveguides within the unit cell and the distance $d_2$ between the waveguides of neighboring cells (d). The insets schematically show weak and strong couplings in such arrays. (e) Propagation constants $\beta$ of the linear eigenmodes as a function of ${t_2}/{(t_1+t_2)}$, calculated with the tight-binding model. Note the existence of boundary states with $\beta=0$ both at $t_2<t_1$ and $t_2>t_1$. (f), (g) Schematic representation of the concept of fractional Wannier center, density of states, and distributions of mode density and Wannier centers within the array in type-I [$t_2<t_1$] and type-II [$t_2>t_1$] phases of the system, shown with different background colors in (e).
• Figure 2: Linear spectrum of the system and its eigenmodes. Linear spectrum of the array showing the eigenvalues of all linear modes as a function of the dimerization parameter $\gamma$(a) as well as the eigenvalues at $\gamma=0.6$(b) and $\gamma=1.6$ (c), which correspond to the red and blue dashed lines in the $\beta(\gamma)$ dependence, respectively. The top line in (d) shows representative linear modes $\psi$ with eigenvalues that fall into the gap at $\gamma=0.6$ and $1.6$. At $\gamma=0.6$, the modes with indices $n=67\sim69$ correspond to the corner modes, while the modes with $n=70\sim78$ are edge modes. For $\gamma=1.6$, the indices $n=67\sim 69, 74, 77, 78$ correspond to the corner modes (including the modes $n=67$, 68, and 74 with two out-of-phase peaks, while the modes $n=69$, 77 and 78 have two in-phase peaks near the corner), while the indices $n=70\sim73$, 75, and 76 correspond to the edge modes. The bottom row in (d) shows linear modes from two bands with indices $n=22$ and 121, which are localized at $\gamma=1.6$ (i.e., represent bound states in the continuum) and delocalized at $\gamma=0.6$. The field distributions of all linear modes in the spectral gap can be found in Supplementary Figure S1.
• Figure 3: Families of thresholdless topological solitons arising at different dimerization values $\gamma$.Soliton power $U$ versus propagation constant $b$, illustrating soliton families bifurcating from different linear localized modes: (a) the corner mode with $n=69$ at $\gamma=0.6$; (b) the edge mode with $n=70$, which is located in the incomplete edge cell at $\gamma=0.6$; (c) the edge mode with $n=75$, which is located in the complete cell at $\gamma=1.6$; and (d) the out-of-phase mode near the corner with $n=74$ at $\gamma=1.6$. The bottom line shows the intensity distributions $|\psi|^2$ corresponding to the points in the $U(\beta)$ dependencies. Stable branches are shown with solid black lines, while unstable branches are shown with red dashed lines. Gray regions indicate bulk bands, and vertical gray dashed lines represent the eigenvalues of the linear modes.
• Figure 4: Excitation of linear and non-linear states at dimerization parameter$\gamma=1.0$. Comparison of the experimental output intensity distributions [(a)-(d), maroon background] with theoretically calculated output distributions [(e)-(h), white background] for different excitation positions indicated by the arrow and the circles. The contours superimposed on the intensity distributions show only complete (not truncated) unit cells of the lattice. (a,e) Waveguide in the bottom corner; (b,f) the seventh waveguide from the top left corner at the left edge; (c,g) the ninth waveguide from the top left corner at the top edge; and (d,h) two waveguides in the bottom corner, out-of-phase excitation. The total input pulse energies in the experiment and the input powers in the simulations are given on each panel. Here and below, the corresponding $E$ values are typically doubled in the case of two-site excitation to obtain the same nonlinear contribution to the refractive index in each waveguide.
• Figure 5: Excitation of linear and nonlinear states at dimerization parameter$\gamma=0.6$. The arrangement and meaning of the panels are as in Fig. 4. (a) and (e) show the intensity distributions for the single-site excitation of the waveguide in the lower corner (where the mode with index $n=69$ is located); (b) and (f) show the single-site excitation of the seventh waveguide from the upper left corner at the left edge (strong overlap with the edge mode with $n=72$ in the incomplete cell); (c,g) single-site excitation of the ninth waveguide at the upper edge belonging to the complete unit cell; (d,h) out-of-phase excitation of two lower waveguides.