Square-root higher-order topological insulator on a decorated honeycomb lattice

Square-root topological insulators are recently proposed intriguing topological insulators, where the topologically nontrivial nature of Bloch wave functions is inherited from the square of the Hamiltonian. In this paper, we propose that higher-order topological insulators can also have their square-root descendants, which we term square-root higher-order topological insulators. There, emergence of in-gap corner states is inherited from the squared Hamiltonian which hosts higher-order topology. As an example of such systems, we investigate the tight-binding model on a decorated honeycomb lattice, whose squared Hamiltonian includes a breathing kagome-lattice model, a well-known example of higher-order topological insulators. We show that the in-gap corner states appear at finite energies, which coincides with the nontrivial bulk polarization. We further show that the existence of in-gap corner states results in characteristic single-particle dynamics; namely, setting the initial state to be localized at the corner, the particle stays at the corner even after a long time. Such characteristic dynamics may experimentally be detectable in photonic crystals.

Figure 1

(a) Model considered in this paper. Gray arrows represent the lattice vectors: ${\mathit{a}}_1=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$ and ${\mathit{a}}_2=\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. (b) A schematic figure of the relation between the original Hamiltonian and its square. The squared Hamiltonian is the direct sum of the honeycomb-lattice model with sublattice-dependent on-site potentials and the breathing kagome-lattice model.

Figure 2

Band structures of the Hamiltonian ${\mathcal{H}}_{\mathit{k}}$ for (a) $t_1=1$, $t_2=1$, and (b) $t_1=1$, $t_2=1.2$. $\mathrm{\Gamma }=(0,0)$, $K=\left(\frac{4\pi }{3},0\right)$, and $M=\left(\pi ,\frac{\pi }{\sqrt{3}}\right)$ are high-symmetry points in the Brillouin zone.

Figure 3

(a) Finite system under the open boundary conditions, consisting of 45 black and 36 white sites. (b) The energy spectrum as a function of $t_1/t_2$. The red dashed lines correspond to the in-gap corner states. (c) The energy spectrum for $t_1=0.3$, $t_2=1$. The in-gap corner states are encircled by cyan (lower) and magenta (upper) ellipses. The charge density of (d) the lower three in-gap states and (e) the higher three in-gap states. The radii of yellow shaded dots represent the probability density.

Figure 4

Polarization of Eq. (18) as a function of $t_1/t_2$. Blue dots and orange triangles are for the first and the fifth bands of the decorated honeycomb-lattice model, respectively. Purple squares are for the upper dispersive band of the breaking kagome-lattice model $h_{\mathit{k}}^{\left(\mathrm{K}\right)}$ and the green diamonds are for the upper dispersive band of the honeycomb-lattice model $h_{\mathit{k}}^{\left(\mathrm{H}\right)}$.

Figure 5

Charge density $N_{\ell }\left(t\right)$ of Eq. (23) at $t=500$ represented by the radii of green shaded dots. The panels (a) and (b) are for $t_1=1$, $t_2=0.3$ and (c) and (d) are for $t_1=0.3$, $t_2=1$. Red circles represent the initial position of the particle. For (a) and (b), the green dots are smaller than black and white symbols representing the sites.