Self-ordering induces multiple topological transitions for in-plane bulk waves in solid phononic crystals

Topological defects with symmetry-breaking phase transitions have captured much attention. A vortex generated by topological defects exhibits exotic properties, and its flow direction can be switched by altering the spin configurations. Contrary to electromagnetic and acoustic domains, the topological transport of in-plane bulk waves in periodic structures with topological defects is not well explored due to the mode conversion between longitudinal and transverse modes. Here, we propose an elastic topological insulator with spontaneously broken symmetry based on the topological theory of defects and homotopy theory. Multiple topological transitions for in-plane bulk waves are achieved by topologically modifying the ellipse orientation in a triangular lattice of elliptical cylinders. The solid system, independent of the number of molecules in order-parameter space, breaks through the limit of point-group symmetry to emulate elastic pseudospin-orbit coupling. The transport robustness of the edge states is experimentally demonstrated. Our approach provides new possibilities for controlling and transporting in-plane bulk waves.

Figure 1

(a) Schematic illustration of a triangular lattice of hexagonal clusters composed of six elliptical cylinders embedded in an epoxy resin. (b) Unit-cell diagram and order-parameter space of the PCs. (c) Eigenfrequencies of the p- and d-type states at the Γ point evolving for a sweep of the initial phase $C$. (d) Calculated band structure in which $C=47.5^{\circ }$ and $132.5^{\circ }$, showing the fourfold-degenerate double Dirac cone at the Γ point. (Inset: Brillouin zone of triangular lattice.)

Figure 2

Band structures of the solid PCs with different initial phase. Unit cells are depicted at the bottom right corner in each subfigure. Panel (a) represents the radical configuration with the initial phase $C=0^{\circ }$ and the associated displacement field distributions for the degenerated eigenstates at the Γ point. Panel (b) represents the azimuthal configuration with the initial phase $C={90}^{\circ }$; the $p$- and $d$-type states are inverted with respect to the band gap, which indicates the occurrence of band inversion. The band gap (gray shaded region) in (a) is topologically nontrivial, whereas the one in (b) corresponds to a topologically trivial regime. The time-averaged mechanical energy flux (indicated by the blue arrows) associated with the pseudospin-up and pseudospin-down state around the Γ point above the band gap in the nontrivial regime (c) and the trivial regime (d).

Figure 3

(a) Schematic of the elastic topologically protected domain wall, in which the radical configuration is replicated at the upper domain wall, and the azimuthal configuration is replicated at the lower domain wall, respectively. The domain wall is marked by the dashed red line. (b) Calculated band structure for a 1×20 supercell with a domain wall at the center. The blue and red lines represent edge states of opposite group velocities, while the shaded regions indicate the bulk states. (c) The corresponding displacement field distribution of $A$ and $B$ states. Black arrows indicate the time-averaged mechanical energy flux.

Figure 4

(a)–(c) Band structures of the solid PCs with different initial phases with $C=0^{\circ }, {22}^{\circ },$ and ${45}^{\circ }$, respectively. Unit cells are depicted at the bottom left corner in each subfigure. (d) Calculated band structure of a supercell consisting of a topological nontrivial crystal $(C={45}^{\circ })$ stacked with a topological trivial crystal $(C=0^{\circ })$. The blue and red lines represent an elastic spin+ and spin− edge state, respectively. The inset shows the corresponding energy flux profiles for the edge modes, indicated by the blue arrows. The shaded regions indicate the bulk states. (e) Spatial amplitude profile of the displacement fields localized at the domain wall.

Figure 5

Band inversion process and topological protected edge states of the elastic PCs constructed by elliptical steel cylinders embedded in an epoxy resin. (a)–(c) Band structure of the elastic PCs with different initial phases with $C=0^{\circ }, 49.5^{\circ },$ and ${90}^{\circ }$, respectively. Unit cells are depicted at the bottom right corner in each subfigure. The nontrivial crystal in (a) undergoes a topological phase transition, as the band gap closes in (b) and a fourfold-degenerate double Dirac cone is formed at the Γ point, then reopens with an inverted band structure in (c). (d) Eigenfrequencies of the p- and d-type states at the Γ point evolving for a sweep of the initial phase $C$. The band inversion effect is clearly observed. (e) Calculated band structure of a supercell consisting of a topological nontrivial crystal $(C=0^{\circ })$ stacked with a topological trivial crystal $(C={90}^{\circ })$. The red and blue lines represent an elastic spin+ and spin− edge state, respectively. The shaded regions indicate the bulk states.

Figure 6

(a) Photograph of the experiment setup. The yellow dashed lines indicate the location of the waveguides constructed by stacking a topological trivial crystal $(C={90}^{\circ })$ with a topological nontrivial crystal $(C=0^{\circ })$. (b)–(d) Simulated displacement field distribution of the topologically protected waveguide and the ordinary waveguide at a bulk band-gap frequency (125.1 kHz). The green dashed rectangle denotes the region of the area $125\times 110\mathrm{m}{\mathrm{m}}^2$ measured in experiments. Green and orange regions represent the trivial and nontrivial crystals. (e) Experimentally measured transmission spectra for the elastic topologically protected waveguide and the ordinary waveguide. The shaded region corresponds to the topological band gap.

Figure 7

(a) Calculated band structure in which $C=47.5^{\circ }$. (b) The associated model displacement trajectories (near the Γ point). (c) Simulated transmission spectra in the frequency range of the four degeneracy bands with longitudinal input excitation.

Figure 8

Acoustic pseudospin states and band inversion process in a solid-in-air system, in which the phononic crystal structure consisting of elliptical copper cylinders is embedded in air. (a),(b) Band structures of the PCs with different initial phases with $C=0^{\circ }$ and ${90}^{\circ }$, respectively. Unit cells are depicted at the right corner in each subfigure. The lattice constant $a_s=15\mathrm{mm}$ and the distance between the adjacent cylinders $l=0.3030a_s$. The major axis of the ellipse is $a=2.2481\mathrm{mm}$ and the minor axis is $b=1.6085\mathrm{mm}$. (c) The band inversion underlying the topological transition from nontrivial phases to trivial phases upon changing the initial phase $C$.

Figure 9

Acoustic pseudospin states and band inversion process in an air-in-fluid system, in which the phononic crystal structure consisting of elliptical cylinders filled with air is embedded in water. (a),(b) Band structures of the PCs with different initial phases with $C=0^{\circ }$ and ${90}^{\circ }$, respectively. Unit cells are depicted at the bottom right corner in each subfigure. The lattice constant $a_s=15\mathrm{mm}$ and the distance between the adjacent cylinders $l=a_{\mathrm{s}}/3$. The major axis of the ellipse is $a=2.1\mathrm{mm}$ and the minor axis is $b=1.5\mathrm{mm}$. (c) The band inversion underlying the topological transition from nontrivial phases to trivial phases upon changing the initial phase $C$.

Figure 10

Band structures of the trivial rectilinear modification structures and the associated displacement field distributions for the degenerate eigenstates at the Γ point. Band inversion no longer takes place since there is no reversal between the $p$- and $d$-type states with respect to the band gap. Unit cells are depicted at the bottom right corner in each subfigure. The shaded regions indicate the complete band gap.

Figure 11

Topological mode inversion between the elastic PCs consisting of rectangular tungsten cylinders embedded in an epoxy resin matrix. (a),(b) Band structures of the PCs with different initial phases with $C=0^{\circ }$ and ${90}^{\circ }$, respectively. Unit cells are depicted at the bottom right corner in each subfigure. The lattice constant $a_s=15\mathrm{mm}$ and the distance between the adjacent cylinders $l=a_s/3$. The length and width of the rectangle is $a=3.0314\mathrm{mm}$ and $b=2.3318\mathrm{mm}$, respectively, whose filling ratio is identical with the elliptical tungsten cylinders in the main text. (c) The band inversion underlying the topological transition from nontrivial phases to trivial phases upon changing the initial phase $C$.

Figure 12

Topological phase transition and band inversion process in an air-in-fluid system beyond $C_{6v}$ symmetry, in which the phononic crystal structure consisting of 15 elliptical cylinders filled with air is embedded in water. (a)–(c) Band structures of the PCs with different initial phases with $C=0^{\circ }, {46}^{\circ },$ and ${90}^{\circ },$ respectively. A double Dirac cone is seen at the Γ point. Unit cells are depicted at the bottom right corner in each subfigure. The lattice constant $a_s=15\mathrm{mm}$ and the distance between the adjacent cylinders $l=0.3829a_{\mathrm{s}}$. The major axis of the ellipse is $a=1.06\mathrm{mm}$ and the minor axis is $b=0.53\mathrm{mm}$. The band inversion underlying the topological transition from nontrivial phases to trivial phases upon changing the initial phase $C$.