[論文レビュー] Acoustic Weyl nodes from stacking dimerized chains

Zhaoju Yang, Baile Zhang|arXiv (Cornell University)|Jan 29, 2016

Topological Materials and Phenomena被引用数 6

ひとこと要約

本論文は、1次元のデュアリズド音響共鳴器鎖を積み重ねることで、2次元および3次元系において音響ウェイル半金属相を実現する新規な手法を提案する。非対称なホッピング調制によるパリティ対称性の破壊により、相反するヘリシティを有するウェイルノードを有する積み重ね系が得られ、非ゼロの巻き数およびチーン数によって特徴付けられ、トポロジカル的に保護されたフェルミアーク表面状態を支持する。これは、低次元の構築ブロックから音響トポロジカル相をスケーラブルに実現する道筋を示している。

ABSTRACT

The discovery of three-dimensional (3D) Weyl semimetals hosting isolated Weyl nodes has drawn remarkable attention in condensed matter physics as well as in photonics. In acoustics, 3D Weyl nodes were proposed theoretically with coupling manipulations in a graphite structure. Here, we propose an approach of constructing acoustic topological semimetal phases in 2D and 3D systems by stacking one-dimensional dimerized chains as building blocks. These 2D and 3D acoustic systems exhibit Weyl nodes with opposite chirality, which can be characterized by nonzero winding number and Chern number, respectively. The stacked 2D ribbon structures possess nearly flat edge states. For stacked 3D slab structures, topologically protected chiral surface states localize at the boundaries, which, by fixing the frequency at the Weyl node, can trace out the trajectories of “Fermi Arcs”, similar to previous observations in condensed matter Weyl semimetals. Weyl semimetals [1] that host isolated Weyl nodes in three-dimensional (3D) momentum space have recently been discovered in material TaAs [2,3] and a double-gyroid photonic crystal [4], as a new topological phase of matter beyond topological insulators. In classical acoustics, topological concepts have gradually been introduced with many new phenomena predicted [5-8]. A recent theoretical proposal shows that, by applying on-site coupling difference and chiral coupling in a 3D graphite structure, Weyl nodes [8] can be constructed for acoustic waves. Yet it remains unclear if acoustic Weyl semimetal phase can be constructed from lower-dimensional topological phases, although it is known in condensed matter physics that 3D integer quantum Hall states [9] and 3D weak topological insulators [10,11] can be formed by stacking layers of two-dimensional (2D) quantum Hall states and quantum spin Hall insulators. In fact, one-dimensional (1D) systems can also exhibit rich topological physics. The Zak phase [12], as predicated in the 1D dimerized chains of polyacetylene [13] or linearly conjugated diatomic polymers [14], has been experimentally measured in 1D optical lattice with ultracold atoms [15], and then in an acoustic system [16] with periodic tubes. In the following, by stacking 1D dimerized chains of acoustic resonators, we construct acoustic topological semimetal phases in 2D and 3D systems. These 1D dimerized chains can be described by Su-Schrieffer-Heeger (SSH) model [13] with equivalent on-site energies of two neighboring lattice sites. These 2D and 3D acoustic systems exhibit Weyl nodes with opposite chirality, characterized by nonzero winding number and Chern number, respectively. The adopted principle of constructing Weyl nodes with opposite chirality in higher dimensions is parity (P) symmetry breaking. The idea of constructing higher-dimensional topological phases from stacking 1D chains may offer new platforms for exploring topological physics with acoustic waves. The schematic of the 1D dimerized chain is shown in upper panel of Fig. 1(a). The filled (open) circle indicates A (B) type atom. The left and right nearest-neighbor (NN) hopping strengths of A type resonator are t t   and t t   , respectively. By setting zero energy offset between two sites, we can arrive at the SSH model and obtain the Bloch Hamiltonian H(k) for the 1D system: 1( ) 2 cos( ) 2 sin( ) x x x y H k t k a t k a      . (1) This Hamiltonian can be implemented in an acoustic dimerized chain. One unit cell of the dimerized chain consists of two resonators, connected by two coupling waveguides with different radii, as shown in the lower part of Fig. 1(a). The periodic boundary condition is applied to the left and right surfaces. Other surfaces (marked with blue color) of the unit cell are treated as hard boundaries for sound. The distance between two nearest resonators is 0.1 a  m. The radius and height of the cylinder (resonator) is 0.4 r a  and 0.8 h a  . For dimerization, we apply modulation of 0.3 w w   to the original radius of coupling waveguide 0.26 w r  . We thus have w w   for one radius of the coupling waveguide, and w w   for the other, as shown in the lower part of Fig. 1(a). Since there are two atoms in one unit cell, hereafter we only consider the two-band model with two lowest acoustic eigen modes, whose pressure field patterns are single valued in each acoustic resonator. By choosing three values of modulation 0.3 ,0, 0.3 w w w    , we arrive at three band diagrams by solving acoustic wave equation in the first Brillouin zone (BZ) as shown in Fig. 1(b). The closing of bandgap at 0 w   indicates the existence of topological phase transition. For the lower bands of three cases in Fig.1(b), we can characterize their topological properties by calculating the topological invariant—Zak phase [12] /2 /2 | | a Zak k k k a i dk u u        . The results are / 2   , 0 and / 2  for 0 w   , 0 w   and 0 w   , respectively. Note that the Zak phase of each dimerization is a gauge dependent value, but the difference between the Zak phases of two dimerized configurations with 0 w   and 0 w   , which is 2 1 Zak Zak Zak         in our acoustic model, is topologically defined [15]. Because the topological property of a bandgap is determined by the summation of Zak phases of all bands below the gap, the two dimerizations in Fig. 1(b) (red and blue curves) are topologically distinct to each other. The above topologically nontrivial phases in acoustic resonators ensures the existence of interface states between two configurations of dimerized lattices. Figure 1(c) demonstrates the results from numerical simulations. For the left panel, we apply 0.3 w w   and 0.3 w w    on two sides of an interface. For the right panel, 0.3 w w   and 0.1 w w   are applied. There is an interface state, as predicated, locating inside the bandgap in the left panel, as highlighted by the red line. The acoustic pressure field pattern of the interface state is shown in Fig. 1(d). The green arrow points to the interface between two topologically distinct structures. Utilizing these 1D dimerized chains as building blocks, we can extend the acoustic topological nontrivial phase into higher dimensional structures by constructing 2D and 3D dimerized lattices. First, we start from the Bloch Hamiltonian of a 2D dimerized acoustic lattice: 2( ) [2 cos( ) 2 cos( )] 2 sin( ) x x y y x x x y H k t k a t k a t k a       (2) where x t ( y t ) is the hopping strength along x (y) direction, and x t  is the modulation of the hopping strength along x direction. In order to acquire topological semimetal phase with two linear degenerate points in the first BZ, we find a necessary condition of x y t t  , as otherwise there will be a trivial bandgap for x y t t  , or a single degenerate point with a quadratic dispersion in the corners of 2D BZ for x y t t  . Thus in the Hamiltonian Eqn. (2), the T symmetry is preserved and the P symmetry is broken. With parameters 1 x t   , 2 y t   and 0.5 x t    , the band diagram in the 2D momentum space ( , ) x y k k , as illustrated in Fig. 2(b), can be calculated from the Hamiltonian Eqn. (2), as shown in Fig. 2(c). Two isolated degenerate points locate at (0, 2 3 ) a   in the first 2D BZ, enclosed by blue lines in Fig. 2(b). Following the above tight-binding model, we set the unit cell of the acoustic lattice as shown in Fig. 2(a). The right inset is the schematic of 2D lattice whose unit cell is enclosed by green dashed lines. The lattice constant and parameters of the resonator (radius and height) are the same with those in Fig. 1(a). Similar to 1D dimerized chains, the modulation 0.3 x x w w   , where 0.26 x w r  , applies to coupling waveguides along x direction, whose radii are x x w w   , respectively. Coupling waveguides along y direction with radius 2 y x w w  connect these 1D dimerized chains. For this real acoustic structure, the band diagram along high symmetry lines in the first BZ is shown in Fig. 2(d). It can be seen that there are two degenerate points (2D Weyl nodes [17,18]) with frequency 718.05 Hz located at ( , ) (0, 19.23) x y k k   and ( , ) (0, 19.23) x y k k   in high symmetry lines 2 M  and 3 M  . Note that, usually Dirac points are protected by PT symmetry. Here the 2D Weyl nodes [18] are under P symmetry breaking and T symmetry preservation. They are robust against perturbations within terms , x y   and can only be removed through pair annihilation at x y t t  . After expanding the Hamiltonian Eqn. (2) by substituting 0 x x x k k a k a    and 0 y y y k k a k a    around the degenerate points 0 0 ( , ) x y k k and keeping the first order term, we

研究の動機と目的

• 1次元のトポロジカルな構築ブロックを用いて、高次元における音響トポロジカル半金属相をスケーラブルに構築する方法を示すこと。
• パリティ対称性の破壊によって、2次元および3次元の音響格子において相反するヘリシティを有するウェイルノードを実現すること。
• 3次元スラブ構造において、トポロジカル的に保護されたヘリカルな表面状態およびフェルミアークに類似した軌跡を観測すること。
• 音響系において、1次元のトポロジカル不変量（Zak位相）と高次元のトポロジカル不変量（チーン数）との関係を確立すること。
• 複雑な3次元格子のプロセスを必要としない、実用的な音響プラットフォームを提供することにより、ウェイル半金属物理学を実現すること。

提案手法

• 交互に異なる結合波ガイド半径を有する音響共鳴器を用いて、Su-Schrieffer-Heeger（SSH）モデルを模倣する1次元のデュアリズド鎖を構築する。
• 位置に依存するホッピング強度を含むタイトバインディングハミルトニアンを用いて、1次元、2次元、3次元系をモデル化し、パリティ対称性の破壊を組み込む。
• 1次元鎖におけるZak位相の計算により、トポロジカル相転移および界面状態の確認を行う。
• 追加の次元に沿ってデュアリズド鎖を積み重ねることで、ホッピング異方性を制御して1次元モデルを2次元および3次元格子に拡張する。
• 数値シミュレーションおよびバンド構造計算を用いて、運動量空間におけるウェイルノードおよび表面状態を同定する。
• 対称性条件下でのウェイル点近傍のバンド簡約性および分散関係の解析により、トポロジカルな頑健性を検証する。

実験結果

リサーチクエスチョン

• RQ1 2次元および3次元系において、1次元のトポロジカルなデュアリズド鎖を積み重ねることで、音響ウェイル半金属相を構築できるか？
• RQ2 ホッピング調制におけるパリティ対称性の破壊は、高次元において相反するヘリシティを有するウェイルノードをどのようにもたらすか？
• RQ3 2次元および3次元の積み重ね系は、フェルミアークに類似した状態を含む、トポロジカル的に保護された端状態および表面状態を支持するか？
• RQ4 音響系において、1次元のZak位相と2次元／3次元のトポロジカル不変量（例えばチーン数）との関係は何か？
• RQ5 提案された音響プラットフォームは、複雑な3次元フォトニクスや結晶構造を必要とせずに、頑健なウェイルノードを実現できるか？

主な発見

• 2次元の積み重ね系は、1次ブリユアンゾーン内に (kx, ky) = (0, ±19.23) に位置する2つの孤立したウェイルノードを示し、周波数は718.05 Hz であることがバンド構造計算により確認された。
• 2次元のウェイルノードは摂動に対して頑健であり、時間反転対称性およびP対称性の破壊によって保護されており、縮重点近傍で線形分散を示す。
• 3次元スラブ構造では、トポロジカル的に保護されたヘリカルな表面状態が出現し、周波数をウェイルノードエネルギーに固定した場合、フェルミアークに類似した軌跡を形成する。
• 1次元のデュアリズド鎖は、トポロジカルに異なる構成の境界で界面状態を支持し、Zak位相が ±π/2 であることで、トポロジカルな差異が確認された。
• 3次元系ではチーン数が非ゼロであり、非自明なトポロジカル相の存在が確認された。一方、2次元系は非ゼロの巻き数によって特徴付けられる。
• 積み重ねアプローチにより、1次元のトポロジカルな構築ブロックから3次元のウェイル半金属相を構築可能であり、音響トポロジカル材料の実験的かつスケーラブルな実現ルートを提供する。