[论文解读] Acoustic Weyl nodes from stacking dimerized chains

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

Topological Materials and Phenomena被引用 6

一句话总结

该论文提出了一种新颖方法，通过堆叠一维二聚化声学谐振器链，在二维和三维系统中实现声学外尔半金属态。通过非对称跃迁调制打破宇称对称性，堆叠系统表现出具有相反手性的外尔节点，其特征为非零绕数和陈数，并支持拓扑保护的费米弧表面态，展示了从低维构建模块实现声学拓扑态的可扩展路径。

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

研究动机与目标

• 展示一种利用一维拓扑构建模块在更高维度中构建声学拓扑半金属态的可扩展方法。
• 通过宇称对称性破缺，在二维和三维声学晶格中实现具有相反手性的外尔节点。
• 在三维板状结构中观察到拓扑保护的手性表面态和费米弧状轨迹。
• 在声学系统中建立一维拓扑不变量（Zak相位）与高维拓扑不变量（陈数）之间的联系。
• 提供一种实用的声学平台，实现外尔半金属物理，而无需复杂的三维晶格制造。

提出的方法

• 使用具有交替耦合波导半径的声学谐振器构建一维二聚化链，以模拟苏-施里弗-海格模型（Su-Schrieffer-Heeger, SSH）模型。
• 使用具有位置依赖跃迁强度的紧束缚哈密顿量，对一维、二维和三维系统进行建模，包含宇称对称性破缺调制。
• 通过计算一维链中的Zak相位，确认拓扑相变和界面态。
• 通过在额外维度上堆叠二聚化链，将一维模型扩展至二维和三维晶格，实现受控的跃迁各向异性。
• 采用数值模拟和能带结构计算，在动量空间中识别外尔节点和表面态。
• 通过分析在对称性条件下外尔点附近的能带简并和色散关系，验证拓扑鲁棒性。

实验结果

研究问题

• RQ1 能否通过在二维和三维系统中堆叠一维拓扑二聚化链来构建声学外尔半金属态？
• RQ2 跃迁调制中的宇称对称性破缺如何在高维中导致具有相反手性的外尔节点？
• RQ3 二维和三维堆叠系统是否支持拓扑保护的边缘和表面态，如费米弧？
• RQ4 在一维Zak相位与二维/三维拓扑不变量（如陈数）之间存在何种关系？
• RQ5 所提出的声学平台能否在无需复杂三维光子或晶体结构的情况下实现鲁棒的外尔节点？

主要发现

• 二维堆叠系统在第一布里渊区内表现出两个孤立的外尔节点，位置为(kx, ky) = (0, ±19.23)，频率为718.05 Hz，经能带结构计算确认。
• 二维外尔节点在扰动下具有鲁棒性，受时间反演对称性和P对称性破缺保护，且在简并点附近呈现线性色散关系。
• 在三维板状结构中，拓扑保护的手性表面态出现，当频率固定在外尔节点能量时，形成费米弧状轨迹。
• 一维二聚化链在拓扑性质不同的构型边界处支持界面态，Zak相位为±π/2，证实了其拓扑差异。
• 三维系统中陈数非零，证实了非平庸拓扑相的存在；而二维系统则由非零绕数表征。
• 堆叠方法使得能够从一维拓扑构建模块构建三维外尔半金属态，为实现声学拓扑材料提供了可扩展且实验可行的路径。