Experimental verification of acoustic pseudospin multipoles in a symmetry-broken snowflakelike topological insulator

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Experimental verification of acoustic pseudospin multipoles in a symmetry-broken snowflakelike topological insulator

Zhiwang Zhang, Ye Tian, Ying Cheng, Xiaojun Liu, and Johan Christensen

Phys. Rev. B 96, 241306(R) – Published 29 December 2017

Abstract

Topologically protected wave engineering in artificially structured media resides at the frontier of ongoing metamaterials research, which is inspired by quantum mechanics. Acoustic analogs of electronic topological insulators have recently led to a wealth of new opportunities in manipulating sound propagation by means of robust edge mode excitations through analogies drawn to exotic quantum states. A variety of artificial acoustic systems hosting topological edge states have been proposed analogous to the quantum Hall effect, topological insulators, and Floquet topological insulators in electronic systems. However, those systems were characterized by a fixed geometry and a very narrow frequency response, which severely hinders the exploration and design of useful applications. Here we establish acoustic multipolar pseudospin states as an engineering degree of freedom in time-reversal invariant flow-free phononic crystals and develop reconfigurable topological insulators through rotation of their meta-atoms and reshaping of the metamolecules. Specifically, we show how rotation forms man-made snowflakelike molecules, whose topological phase mimics pseudospin-down (pseudospin-up) dipolar and quadrupolar states, which are responsible for a plethora of robust edge confined properties and topological controlled refraction disobeying Snell's law.

Received 23 October 2017

DOI:https://doi.org/10.1103/PhysRevB.96.241306

Physics Subject Headings (PhySH)

Acoustic metamaterials Edge states

Topological insulators

Condensed Matter, Materials & Applied PhysicsFluid DynamicsNonlinear Dynamics

Authors & Affiliations

Zhiwang Zhang¹, Ye Tian¹, Ying Cheng^1,*, Xiaojun Liu^1,†, and Johan Christensen^2,‡

¹Department of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
²Instituto Gregorio Millan Barbany, Universidad Carlos III de Madrid, ES-28916 Leganés, Madrid, Spain

^*chengying@nju.edu.cn
^†liuxiaojun@nju.edu.cn
^‡johan.christensen@uc3m.es

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Issue

Vol. 96, Iss. 24 — 15 December 2017

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Images

Figure 1
(a) Schematic of the snowflakelike metamolecule unit of the phononic triangular lattice composed of TLRs. (b) Dispersion relation of the structure in which $d = 0.578$ cm. (c) Topological modes inversion underlying the transition of pseudospin states. The rainbow color represents the amplitude of total pressure fields. $p_{x}$ and $p_{y}$ show pseudospin dipole modes, while $d_{x y}$ and $d_{x^{2} - y^{2}}$ show pseudospin quadrupole modes. (d) The eigenfrequencies of the double twofold degenerated states are controlled by the length of the arm $d$ at the BZ center. The band inversion effect is clearly observed as indicated in different colors.
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Figure 2
Schematics of two snowflakelike metamolecules: the topologically (a) trivial and (b) nontrivial triangular lattice composed of TLRs with $d = 0.85$ cm. Corresponding dispersion relations of the topologically (c) trivial regime and (d) nontrivial structure. Insets: topological modes inversion underlying the transition between pseudospin states. Sound intensity $I$ associated with pseudospin-down and pseudospin-up modes at the $Γ$ point below the band gap for the (e) trivial and (f) nontrivial regimes. The arrows show the direction and amplitude of the intensity.
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Figure 3
(a) Schematic of a ribbon-shaped supercell composed of 20 for the topologically nontrivial region and two times ten metamolecules for the trivial regions, respectively. (b) Dispersion relation of the ribbon-shaped supercell. Dark-gray regions indicate the bulk states, whereas the doubly degenerated solid red and blue lines represent the edge states of opposite group velocities. (c) Acoustic pressure fields and intensity of the pseudospin-dependent one-way edge modes localized at the interface of the supercell, corresponding to the points $A$ and $B$ indicated in (b).
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Figure 4
(a) Photograph of a fabricated PnC sample (without the top cover). (b) Simulated distributions of the absolute pressure fields along the interfaces between trivial and nontrivial regions (left panel, structure I) and through the trivial bulk region (right panel, structure II). Cyan and yellow regions represent the trivial and nontrivial regimes. (c) Experimentally measured transmission spectra of topological edge states and bulk states. (d) Frequency-dependent spatial profiles of the absolute pressure fields measured in the middle of the crystal [red dashed line in (b)]. The frequency range corresponds to the one in (c).
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Figure 5
Simulated absolute pressure fields (a) at the frequency $f = 8200$ Hz in the TPWG and (b) at the frequency of $f = 8000$ Hz in the TTWG. Green and cyan arrows denote the directions of transmitted and reflected sound waves, respectively. Inset: the partial structures of the uninterrupted interface and the interfaces with bends. (c)–(d) Experimental transmission spectra for the TPWG and the TTWG, respectively. Shaded regions represent the band gap.
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Figure 6
(a) Schematic of the metamolecule with the rotation angle $ϕ$ , which is positive/negative when rotating TLRs anticlockwise/clockwise. Topological birefringence: pressure fields along the interface with (b) $ϕ = 0^{\circ}$ (trivial) above and $ϕ = 30^{\circ}$ (nontrivial) below; (c) $ϕ = 30^{\circ}$ (nontrivial) above and $ϕ = 0^{\circ}$ (trivial) below. (d) Schematic of primitive cell of triangular lattice with lattice constant $a_{0} = 2.17 cm$ . Topological positive/negative refraction: pressure fields along the interface with (b) $α = - 10^{\circ}$ above and $α = 10^{\circ}$ below; (c) $α = 10^{\circ}$ above and $α = - 10^{\circ}$ below.
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