Self-localized topological states in three dimensions

Three-dimensional (3D) topological materials exhibit much richer phenomena than their lower-dimensional counterparts. Here, we propose self-localized topological states (i.e., topological solitons) in a 3D nonlinear photonic Chern insulator. Despite being in the bulk and self-localized in all 3D, the topological solitons at high-symmetry points $K$ and $K^{\prime }$ rotate in the same direction, due to the underlying topology. Specifically, under the saturable nonlinearity the solitons are stable over a broad frequency range. Our results highlight how topology and nonlinearity interact with each other and can be extended to other 3D topological systems.

Figure 1

(a) A 3D Chern insulator constructed by AA stacking the 2D Haldane honeycomb lattices. The orange and purple spheres denote sublattice sites $A$ and $B$, respectively. In the $xy$ plane, the black solid lines represent the NN hopping $t_1$, and the orange and purple arrows represent the NNN hopping $t_{2}exp(\pm i\varphi )$. The orange and purple lines represent interlayer hoppings $t_A$ and $t_B$, respectively. (b) Brillouin zone of the 3D Chern insulator.

Figure 2

(a1) Two different isosurfaces and (a2) the phase distribution on the isosurface with $0.8\times \text{Max}\left(\text{Mag}\left[{\psi }_A\right]\right)$ of the pseudospin component ${\psi }_A$ at $K$. (b1) Isosurfaces and (b2) the phase distribution of the pseudospin component ${\psi }_B$ at $K$. (c1) Isosurfaces and (c2) the phase distribution of ${\psi }_A$ at $K^{\prime }$. (d1) Isosurfaces and (d2) the phase distribution of ${\psi }_B$ at $K^{\prime }$. The isosurfaces are plotted with $\omega =10$ and parts of the isosurfaces are removed for the sake of clarity.

Figure 3

(a), (b) The power of the two pseudospin components ${\psi }_{A,B}$. The insets are enlarged figures for $P_{A,B}$. (c) The growth rate $\text{Im}\left(\delta \right)$ of the topological solitons. The dashed lines in (a)–(c) denote the linear band edges. (d), (e) The perturbation eigenmode ${\tilde{\varepsilon }}_A$ and ${\tilde{\varepsilon }}_B$ at $\omega =9.953$.

Figure 4

(a) The isosurfaces with $0.8\times \text{Max}\left(\text{Mag}\left[{\psi }_A(t=0)\right]\right)$ of the pseudospin component ${\psi }_A$ of the stable topological soliton with $\omega =10$ at $K$. The four subfigures from left to right correspond to $t=6000$, 6650, 7300, and 7950, respectively. (b) The isosurfaces with $1.0^{*}\text{Max}\left(\text{Mag}\left[{\psi }_A(t=0)\right]\right)$ of ${\psi }_A$ of the unstable topological soliton with $\omega =9.953$.