Valley Hall phases in kagome lattices

We report the finding of the analogous valley Hall effect in phononic systems arising from mirror symmetry breaking, in addition to spatial inversion symmetry breaking. We study topological phases of plates and spring-mass models in kagome and modified kagome arrangements. By breaking the inversion symmetry it is well known that a defined valley Chern number arises. We also show that effectively, breaking the mirror symmetry leads to the same topological invariant. Based on the bulk-edge correspondence principle, protected edge states appear at interfaces between two lattices with different valley Chern numbers. By means of a plane wave expansion method and the multiple scattering theory for periodic and finite systems, respectively, we computed the Berry curvature, the band inversion, mode shapes, and edge modes in plate systems. We also find that appropriate multipoint excitations in finite system gives rise to propagating waves along a one-sided path only.

Figure 1

Schematic representation of a ribbon in an infinite plate. $r_1$ and $r_2$ are a basis of lattice. The two red parallel lines delimit one supercell, the supercell is infinite in size. The unit cell is presented with two different topological phases as we will see later in the text.

Figure 2

(a) Undistorted kagome lattice. The unit cell is indicated in a green box of side $a$. The unit cell contains three resonators forming an equilateral triangles. (b) Parameters used in the paper for deformations of kagome lattice and its effect in the unit cell. They are characterized by an angle $\alpha$ and a uniform expansion factor $f$. (c) Brillouin zone.

Figure 3

Real-space arrangement of resonators for several deformation parameters. The unit cell is highlighted. Notice (a) and (c) look similar but the chosen cell is different. Notice the breaking of spatial inversion symmetry is the three cases.

Figure 4

Gap closings in parameter space at $K$ points. Solid red lines are the closing of the first gap. Dashed black lines are the closings of the second gap. The second gap is a partial gap in $k$ space. Topological transitions studied in this paper are marked with a five- and a four-pointed stars. The driving parameters are $f$ and $\alpha$, respectively, and the symmetry breaking is spatial inversion and mirror symmetry, respectively.

Figure 5

(a) Frequency level crossing at $K$ point for model in Eq. (22) as a function of $\beta$. (b) Representation of $M$ operator in Eq. (33). The arrows indicate how each component transforms, the color indicates different signs.

Figure 6

Band structure of flexural waves plotted along a the hexagonal Brillouin zone for several $f$ values and $\alpha =0$.

Figure 7

In color scale, the Berry curvature of the lower band over the first Brillouin zone. The Berry curvature is localized at $K$ and $K^{\prime }$ points with different signs for different phases. Blue is negative and yellow is positive.

Figure 8

Mode shapes. Real part of $w\left(\vec{r}\right)$ for different bands and phases. Notice the analogy of the first row with the eigenvalue of the spring model $u_{C_3}^0=\frac{1}{\sqrt{3}}{(1,1,1)}^t$ or in the second row with the real part $\text{Re}\left\{u_{C_3}^{+}\right\}=\frac{1}{\sqrt{3}}{(0.5,0.5,-1)}^t$. Notice the band inversion. Mode shapes are not periodic due to the phase $e^{-i\vec{K}·\vec{r}}$ in Eq. (3).

Figure 9

Ribbon of resonators over an infinite plate. The system contains a domain wall between two phases with opposite valley Chern numbers. At the top left, real-space ribbon representation. The horizontal line separates the two phases and black arrows indicate that the ribbon is infinite in horizontal direction. In red, resonators in one supercell. At the top right there is the band structure of the finite system, neglecting nonbulk modes, i.e., modes in the interior of the free dispersion curve $\mathrm{\Omega }a={\left(\frac{k_x}{\pi }\right)}^2$. Two mid gap bands appear. At the bottom, mode shapes or, in other words, real-space displacement field along the supercell sites $w\left({\vec{R}}_{\alpha }\right)$ for different frequencies and momenta as indicated with colored dots on the band structure.

Figure 10

Ribbon of resonators over an infinite plate. The system contains a domain wall between two phases with opposite valley Chern numbers (see Fig. 7). At the top left, real-space ribbon representation. The horizontal line separates the two phases and black arrows indicate that the ribbon is infinite in horizontal direction. In red, resonators in one supercell. At the top right there is the band structure of the finite system, neglecting nonbulk modes, i.e., modes in the interior of the free dispersion curve $\mathrm{\Omega }a={\left(\frac{k_x}{\pi }\right)}^2$. One mid gap band appears. At the bottom, mode shapes or, in other words, real-space displacement field along the supercell sites $w\left({\vec{R}}_{\alpha }\right)$ for different frequencies and momenta as indicated with colored dots on the band structure.

Figure 11

Ribbon of resonators over an infinite plate. The system contains a domain wall between two phases with opposite valley Chern numbers (see Fig. 7). On the left, real-space ribbon representation. The vertical line separates the two phases and black arrows indicate that the ribbon is infinite in vertical direction. The interface is armchairlike. The band structure does not show localized modes within gap frequencies, bands that appear isolated at gap frequencies are bulk modes.

Figure 12

Schematic representation of a cluster of resonators on top of an infinite plate. The cluster is designed with a Z-shaped interface.

Figure 13

MST simulations of an arrangement of resonators with two phases separated by a zigzag domain wall. The frequency is tuned so that the mode is in a gap and corresponds to topological edge states.

Figure 14

In color scale, the absolute value of the place displacement. MST simulations of an arrangement of resonators with two phases separated by a zigzag domain wall (no mixing valleys) in Z shape. The frequency is tuned so the modes are topological edge states. The red dots correspond to the two excitation points ${\vec{x}}_1=(-1,0)a$ and ${\vec{x}}_2=(1,0)a$. The temporal dephasing is $\phi =0$ (the two points are excited simultaneously) and $\phi =\pi$ (antiphase excitation), respectively.

Figure 15

Distorted kagome lattice for two $f$ values and $\alpha =\pi /6$. Notice that spatial inversion is not a symmetry of the system. Increasing slightly $\alpha$ shortens green links and enlarges blue links. The spring system models changes in distance with appropriate changes in $\beta$.

Figure 16

Frequency levels at $K$ point for the model in Eq. (30) as a function of $\beta$ and for ${\gamma }^{\prime }=1$.

Figure 17

Band structure of deformed kagome lattice for several $\alpha$ values and $f=1.5$.

Figure 18

Berry curvature of the lower band over the first Brillouin zone. Berry curvature is localized at $K$ and $K^{\prime }$ points with different signs for different phases. Blue is negative and yellow is positive.

Figure 19

Ribbon of resonators over an infinite plate. The system contains a domain wall between two phases with opposite valley Chern numbers (see Fig. 7). At the top left, real-space ribbon representation. The horizontal line separates the two phases and the UC is highlighted in red. Black arrows indicate that the ribbon is infinite in horizontal direction. In red, resonators in one supercell. At the top right there is the band structure of the finite system, neglecting nonbulk modes, i.e., modes in the interior of the free dispersion curve $\mathrm{\Omega }a={\left(\frac{k_x}{\pi }\right)}^2$. Two crossing bands appear in the gap, they have different symmetry under domain wall spatial inversion as seen at the bottom. At the bottom, mode shapes, i.e., real-space displacement field along the supercell sites $w\left({\vec{R}}_{\alpha }\right)$ for different frequencies and momenta as indicated with colored dots on the band structure.

Figure 20

Ribbon of resonators over an infinite plate. The system contains a domain wall between two phases with opposite valley Chern numbers (see Fig. 7). On the left, real-space ribbon representation. The vertical line separates the two phases whose interface is armchair type and the UC is highlighted in red. Black arrows indicate that the ribbon is infinite in the vertical direction. The band structure does not show localized modes within gap frequencies, bands that appear isolated at gap frequencies are bulk modes.

Figure 21

MST simulations of an arrangement of resonators with two phases separated by a zigzag domain wall in Z shape. The frequency is tuned so the mode is in a gap and correspond to topological edge states. The red dots correspond to the two excitation points. The dephasing is $\phi =\pi$ on the left and $\phi =-0.36\pi$ on the right.

Figure 22

Schematic of hexagonal arrangement of resonators having two different masses (filled or empty circles represent heavy and light masses) with soft (light-light) and hard (heavy-heavy) interfaces. The solid black line is the domain wall. In green star marker, the inversion symmetry centers of the ribbon. The arrows represent the infinity of the ribbon in horizontal direction.

Figure 23

Ribbon of resonators over an infinite plate. The unit cell highlighted in red contains a domain wall between two phases with different valley Chern numbers. (Top) Band structure of the finite system, neglecting nonbulk modes, i.e., modes in the interior of the free dispersion curve $\mathrm{\Omega }a={\left(\frac{k_x}{\pi }\right)}^2$. (Bottom) Real-space displacement field along the supercell sites for different frequencies and momenta (eigenvectors) as indicated with colored dots on the band structure. The two lines represent sites A and B.

Figure 24

Ribbon of resonators over an infinite plate. The unit cell highlighted in red contains a domain wall between two phases with different valley Chern numbers. (Top) Band structure of the finite system, neglecting nonbulk modes, i.e., modes in the interior of the free dispersion curve $\mathrm{\Omega }a={\left(\frac{k_x}{\pi }\right)}^2$. (Bottom) Real-space displacement field along the supercell sites for different frequencies and momenta (eigenvectors) as indicated with colored dots on the band structure. The two lines represent sites A and B.