Coupled-wire construction of static and Floquet second-order topological insulators

Second-order topological insulators (SOTI) exhibit protected gapless boundary states at their hinges or corners. In this paper, we propose a generic means to construct SOTIs in static and Floquet systems by coupling one-dimensional topological insulator wires along a second dimension through dimerized hopping amplitudes. The Hamiltonian of such SOTIs admits a Kronecker sum structure, making it possible for obtaining its features by analyzing two constituent one-dimensional lattice Hamiltonians defined separately in two orthogonal dimensions. The resulting topological corner states do not rely on any delicate spatial symmetries, but are solely protected by the chiral symmetry of the system. We further utilize our idea to construct Floquet SOTIs, whose number of topological corner states is arbitrarily tunable via changing the hopping amplitudes of the system. Finally, we propose to detect the topological invariants of static and Floquet SOTIs constructed with our approach in experiments by measuring the mean chiral displacements of wavepackets.

Figure 1

Constructing SOTI by stacking an array of 1D topological insulators (such as SSH model as shown in each longitudinal box). Each circle represents a lattice site. Red solid and green dotted lines denote different coupling strength between pairs of lattice sites within each array, whereas blue dashed and cyan dash-dotted lines depict different interarray coupling strength.

Figure 2

(a) Topological invariant of $h_{S,k}$ as a function of ${\delta }_S$ (symmetry breaking perturbation strength) under two different set of parameter values, where red crosses (blue dots) correspond to $J_S=0.75$ ($J_S=1.475$) and $\delta J_S=0.25$ ($\delta J_S=1.375$). (b) Probability distribution of each corner mode (marked with different colors) of $\mathcal{H}$ obtained in panel (d). For clarity, only $|{\psi }_c{(i,j)|}^2>0.005$ are shown. (c) Energy level distribution of $\mathcal{H}$ when $h_{x,k}$ is topologically trivial, i.e., $J_x=0.75$, $J_y=1.475$, $\delta J_x=-0.25$, $\delta J_y=1.375$, ${\delta }_x=0.2$, and ${\delta }_y=0.15$. (d) Same as panel (c) but with $\delta J_x=0.25$, so that both $h_{x,k}$ and $h_{y,k}$ are topologically nontrivial. Panel (e) highlights the region near $E=0$ in panel (d), with corner modes highlighted in green.

Figure 3

(a)–(c) Energy band structure of Eq. (1) under OBC in the $y$-direction and PBC in the $x$-direction with (a) $\delta J_x=0.25$ and $\delta J_y=1.375$, (b) $\delta J_x=-0.25$ and $\delta J_y=1.375$, (c) $\delta J_x=-0.25$ and $\delta J_y=-1.375$. (d)–(f) Same as panels (a)–(c) but with OBC in the $x$-direction and PBC in the $y$-direction. Other parameters are the same in all panels with $J_x=0.75$, $J_y=1.475$, symmetry breaking perturbation strengths ${\delta }_x={\delta }_y=0.1$, and Kronecker sum breaking perturbation strengths ${\delta }_{xy,1}={\delta }_{xy,2}=0.1$.

Figure 4

Panel (a) [panel (b)] shows the probability distribution of four (20) corner modes at quasienergy zero of $U_{\mathcal{H}}$ shown in panel (d) [panel (e)]. Panel (f) shows the probability distribution of 16 corner modes at quasienergy $\frac{\pi }{T}$ of $U_{\mathcal{H}}$ shown in panel (f). (g) Full quasienergy level distribution of $U_{\mathcal{H}}$ with $J_2=1$ and $\delta J_y=0.6875$, with corner modes highlighted in panel (d). (h) Full quasienergy level distribution of $U_{\mathcal{H}}$ with $J_2=14.8$ and $\delta J_y=0.6875$, with corner modes highlighted in panels (e) and (f). (i) Same as (h) but with $\delta J_y=-0.6875$, so that no corner modes is present. Other parameters are the same in all panels with $J_1=1$, $J_y=0.7375$, symmetry breaking perturbation strengths ${\delta }_x={\delta }_y=0.1$, and Kronecker sum breaking perturbation strengths ${\delta }_{xy,1}={\delta }_{xy,2}=0.1$.

Figure 5

The Floquet spectrum $\varepsilon$ of $U_{\mathcal{H}}$ versus the hopping amplitude $J_1$. The size of the 2D lattice is $N_x=N_y=50$. Other system parameters are chosen as $J_y=\delta J_y=\pi /40$ and $J_2=\pi /2$. Dashed lines represent boundaries separating different Floquet SOTI phases in the parameter space. $n_0\left(n_{\pi }\right)$ denotes the number of Floquet topological corner states at quasienergy zero ($\frac{\pi }{T}$).

Figure 6

(a)–(c) Quasienergy band structure of Eq. (10) under OBC in the $y$-direction and PBC in the $x$-direction with (a) $J_1=1$, $J_2=1$, and $\delta J_y=0.6875$ (b) $J_1=1$, $J_2=14.8$ and $\delta J_y=0.6875$, (c) $J_1=1$, $J_2=14.8$ and $\delta J_y=-0.6875$. (d)–(f) Same as panels (a)–(c) but with OBC in the $x$-direction and PBC in the $y$-direction. Other parameters are the same in all panels with $J_y=0.7375$, symmetry breaking perturbation strengths ${\delta }_x={\delta }_y=0.1$, and Kronecker sum breaking perturbation strengths ${\delta }_{xy,1}={\delta }_{xy,2}=0.1$.

Figure 7

Winding numbers ${\nu }_0$ (red solid lines) and ${\nu }_{\pi }$ (green dashed lines) as a function of (a) $J_1$, (b) $J_2$, (c) ${\delta }_x$.

Figure 8

MCDs and winding numbers versus the hopping amplitude $J_1$. Numerical results of $2\overline{C_1\left(t\right)}$ and $2\overline{C_2\left(t\right)}$, both averaged over $t=20$ driving periods, are shown by the blue stars and black circles. Theoretical values of winding numbers ${\nu }_0$ and ${\nu }_{\pi }$ are denoted by the red solid and green dashed lines, respectively. Other system parameters are chosen as $J_y=\delta J_y=\pi /40$ and $J_2=\pi /2$. Topological phase transitions happen at $J_1=\pi ,2\pi ,3\pi$, away from which the relations in Eq. (38) are verified.

Figure 9

MCDs of the static SOTI model [defined in Eq. (3)] averaged over a long time duration $\tau =400$ (triangles). Solid lines are theoretical values of ${\nu }_x{\nu }_y/4$, with the two winding numbers ${\nu }_x,{\nu }_y$ defined in Eq. (4). Other system parameters are chosen as $J_x+\delta J_x=1/2,J_y-\delta J_y=0,J_y+\delta J_y=1/2$ in panel (a), and $J_x-\delta J_x=1/2,J_y-\delta J_y=0,J_y+\delta J_y=1/2$ in panel (b).