Higher-order topological phases in a spring-mass model on a breathing kagome lattice

We propose a realization of higher-order topological phases in a spring-mass model with a breathing kagome structure. To demonstrate the existence of the higher-order topological phases, we characterize the topological properties and show that the corner states appear under the fixed boundary condition. To characterize the topological properties, we introduce a formula for the ${\mathbb{Z}}_3$ Berry phases in the Brillouin zone. From the numerical result of this ${\mathbb{Z}}_3$ Berry phase, we have elucidated that coupling between the longitudinal and transverse modes yields a state characterized by the Berry phase $\frac{2\pi }{3}$ for our mechanical breathing kagome model. In addition, we suggest that the corner states can be detected experimentally through a forced vibration.

Figure 1

A spring-mass model in a breathing kagome structure.

Figure 2

(a) The paths in momentum space. The coordinates of ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_2$, and ${\mathrm{\Gamma }}_3$ are (0,0), $\frac{2\pi }{R_a+R_b}(1,\frac{1}{\sqrt{3}})$, and $\frac{2\pi }{R_a+R_b}(0,\frac{2}{\sqrt{3}})$, respectively. The $G$ point is $\frac{2\pi }{R_a+R_b}(\frac{1}{3},\frac{1}{\sqrt{3}})$. The path $L_i$ is ${\mathrm{\Gamma }}_i\to G\to {\mathrm{\Gamma }}_{i+1}({\mathrm{\Gamma }}_4={\mathrm{\Gamma }}_1)$. (b) The triangle area whose vertices are ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_2$, ${\mathrm{\Gamma }}_3$. (c) and (d) Schematic figures for the operations which keep the Hamiltonian invariant. (c) The ${120}^{\circ }$ rotation of the triangle area in momentum space. (d) The translation of the triangle area in momentum space. Combining these two operations, the ${\mathrm{\Gamma }}_i$ point is transformed to ${\mathrm{\Gamma }}_{i+1}$ with ${\mathrm{\Gamma }}_4:={\mathrm{\Gamma }}_1$. Consequently, the path $L_i$ is transformed to $L_{i+1}$ with $L_4=L_1$.

Figure 3

Bulk properties of the spring-mass model. (a) The first Brillouin zone of the spring-mass model. The $\mathrm{\Gamma }$, $K$, and $M$ points are located at $(k_x,k_y)=(0,0)$, $\frac{2\pi }{R_a+R_b}(\frac{2}{3},0)$, and $\frac{2\pi }{R_a+R_b}(\frac{1}{2},\frac{1}{2\sqrt{3}})$, respectively. (b)–(e) The band structure along the line shown in (a) for several parameter sets $({\eta }_{\alpha },t_a/t_b)$. (b), (c), (d), and (e) show the data for $({\eta }_{\alpha },t_a/t_b)=(0,0.4)$, (0.1,0.4), (0.1,1), and (0.1,1.5), respectively.

Figure 4

(a) The Berry phase for the lowest two bands. Schematic figures for two limits with (b) $t_a=0$ and (c) $t_b=0$.

Figure 5

Schematic figures for the spring-mass model in a triangle arrangement for (a) $t_a\ne 0,t_b\ne 0$, (b) $t_a=0,t_b\ne 0$, and (c) $t_a\ne 0,t_b=0$. Black lines represent walls. (d) Eigenfrequencies as a function of $t_a/t_b$ for ${\eta }_a=0.1$ with a triangle arrangement. (e) Kinetic-energy distribution in the real space of the corner states for $t_a/t_b=0.4$. We write the number of small triangles along the edge as $L$; the total number of masses is $3L(L+1)/2$.

Figure 6

(a) Schematic figure for the experimental setup for the forced vibration. The force $f\left(t\right)$ is added to the mass point of the top corner of the triangle (encircled by a green circle). (b) ${\omega }_0$ is near a corner mode, and (c) ${\omega }_0$ is away from a corner mode. In (b) and (c), the amplitudes $A_{\vec{R},p}$ at $t=1000$ are plotted.

Figure 7

(a) The bands structure for ${\eta }_a=0.9$, $t_a/t_b=0.1$. (b) The energy spectrum for ${\eta }_a=0.9$. The corner modes are between the third and fourth bands. (c) The ${\mathbb{Z}}_3$ Berry phase for the lowest fifth bands.

Figure 8

The dispersion relations in the cylinder for (a) ${\eta }_a=0$, $t_a/t_b=0.4$ and (b) ${\eta }_a=0.1$, $t_a/t_b=0.4$. There exist edge modes between the bulk continua (encircled by red ellipses).

Figure 9

(a) and (b) Color map of IPR in the figure of eigenfrequencies. (a) and (b) show the data for ${\eta }_a=0.1$ and 0.9, respectively. Kinetic-energy distribution in the real space of the corner states for (c) $t_a/t_b=0.2$, ${\eta }_a=0.9$, $\omega =1.3954$ and (d) $t_a/t_b=0.2$, ${\eta }_a=0.9$, $\omega =0.854085$.

Figure 10

Schematic figures for the spring-mass model in a parallelogram arrangement with (a) $t_a\ne 0,t_b\ne 0$, (b) $t_a\ne 0,t_b=0$, and (c) $t_a=0,t_b\ne 0$. Black lines represent walls. (d) Eigenfrequencies as a function of $t_a/t_b$ for ${\eta }_a=0.1$ with a parallelogram arrangement. (e) Kinetic-energy distribution in the real space of the corner states for $t_a/t_b=0.4$. The corner mode for $\gamma =0$ arises from the fact that we have cut the unit cells which make the bulk-edge correspondence ubiquitous.