[논문 리뷰] Acoustic Weyl nodes from stacking dimerized chains

Zhaoju Yang, Baile Zhang|arXiv (Cornell University)|2016. 01. 29.

Topological Materials and Phenomena인용 수 6

한 줄 요약

이 논문은 1차원 다이머화된 음향 공진기 체인을 적층하여 2차원 및 3차원 시스템에서 음향 Weyl 반도체 상을 실현하는 새로운 방법을 제안한다. 비대칭 힘 모듈레이션을 통한 반사 대칭성의 위반으로, 적층된 시스템은 반대되는 편극성을 가진 Weyl 노드를 지녀, 비영인 감쇠 및 Chern 수로 특징지어지며, 위상적으로 보호된 Fermi 호 표면 상태를 지닌다. 이는 낮은 차원의 구성 요소로부터 음향 위상상에 이르는 확장 가능한 길을 제시한다.

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

연구 동기 및 목표

• 1D 위상상 구성 요소를 사용하여 고차원에서 음향 위상 반도체 상을 확장 가능한 방법으로 구성하는 것.
• 반사 대칭성의 위반을 통한 힘 모듈레이션으로 2D 및 3D 음향 격자에서 반대편극성을 가진 Weyl 노드를 실현하는 것.
• 3D 슬립 구조에서 위상적으로 보호된 측면 상태와 Fermi 호 유사 궤적을 관찰하는 것.
• 음향 시스템에서 1D 위상상 불변량(Zak 위상)과 고차원 위상상 불변량(Chern 수) 간의 관계를 설정하는 것.
• 복잡한 3D 격자 제작이 필요 없이도, Weyl 반도체 물리학을 실현할 수 있는 실용적인 음향 플랫폼을 제공하는 것.

제안 방법

• 교대되는 결합 파이프 반지름을 가진 음향 공진기를 사용하여 Su-Schrieffer-Heeger(SSH) 모델을 모방하는 1D 다이머화된 체인을 구성하는 것.
• 위치에 따라 변하는 힘 강도를 포함한 타이트버딩 해밀토니안을 사용하여 1D, 2D, 3D 시스템을 모델링하고, 반사 대칭성 위반 모듈레이션을 통합하는 것.
• 1D 체인에서 Zak 위상을 계산하여 위상상 전이와 인터페이스 상태를 확인하는 것.
• 추가 차원에 따라 다이머화된 체인을 적층하여 2D 및 3D 격자를 확장하고, 힘의 이방성을 제어하는 것.
• 수치 시뮬레이션과 밴드 구조 계산을 통해 운동량 공간에서 Weyl 노드와 표면 상태를 식별하는 것.
• 대칭 조건 하에서 Weyl 점 근처의 밴드 degeneracy와 분산을 분석하여 위상적 내성의 타당성을 검증하는 것.

실험 결과

연구 질문

• RQ1 1D 위상상 다이머화된 체인을 2D 및 3D 시스템에서 적층함으로써 음향 Weyl 반도체 상을 구성할 수 있는가?
• RQ2 힘 모듈레이션에서의 반사 대칭성 위반이 고차원에서 반대편극성을 가진 Weyl 노드를 어떻게 유도하는가?
• RQ3 2D 및 3D 적층 시스템은 위상적으로 보호된 모서리 및 표면 상태, 예를 들어 Fermi 호를 지닌다?
• RQ4 음향 시스템에서 1D Zak 위상과 2D/3D 위상상 불변량(예: Chern 수) 간의 관계는 무엇인가?
• RQ5 제안된 음향 플랫폼은 복잡한 3D 광학적 또는 결정 격자 구조가 필요 없이 강건한 Weyl 노드를 실현할 수 있는가?

주요 결과

• 2D 적층 시스템은 첫 번째 브릴루앙 영역 내에서 (kx, ky) = (0, ±19.23)에 위치한 두 개의 고립된 Weyl 노드를 나타내며, 주파수는 718.05 Hz로 밴드 구조 계산에 의해 확인되었다.
• 2D Weyl 노드는 편미분에 의한 변화에 강건하며, 시간역전 대칭성과 P-대칭성 위반에 의해 보호되며, 비특이점 근처에서 선형 분산을 보인다.
• 3D 슬립 구조에서는 위상적으로 보호된 측면 상태가 나타나며, Weyl 노드 에너지에서 주파수를 고정하면 Fermi 호 유사 궤적을 형성한다.
• 1D 다이머화된 체인은 위상적으로 다를 수 있는 구성 간의 경계에서 인터페이스 상태를 지녀, Zak 위상 ±π/2로 위상적 차이를 확인한다.
• 3D 시스템에서는 Chern 수가 비영이므로 비자명한 위상상이 존재함을 확인하였고, 2D 시스템은 비영인 감쇠 수로 특징지어진다.
• 적층 방법은 1D 위상상 구성 요소로부터 3D Weyl 반도체 상을 구성할 수 있게 하여, 실험적으로 실현 가능한 확장 가능한 길을 제공한다.