Abstract
We report a new topological phononic crystal in a ring-waveguide acoustic system. In the previous reports on topological phononic crystals, there are two types of topological phases:quantum Hall phase and quantum spin Hall phase. A key point in achieving quantum Hall insulator is to break the time-reversal (TR) symmetry, and for quantum spin Hall insulator, the construction of pseudo-spin is necessary. We build such pseudo-spin states under particular crystalline symmetry (C6v) and then break the degeneracy of the pseudo-spin states by introducing airflow to the ring. We study the topology evolution by changing both the geometric parameters of the unit cell and the strength of the applied airflow. We find that the system exhibits three phases:quantum spin Hall phase, conventional insulator phase and a new quantum anomalous Hall phase.The quantum anomalous Hall phase is first observed in phononics and cannot be simply classified by the Chern number or Z2 index since it results from TR-broken quantum spin Hall phase. We develop a tight-binding model to capture the essential physics of the topological phase transition. The analytical calculation based on the tight-binding model shows that the spin Chern number is a topological invariant to classify the bandgap. The quantum anomalous Hall insulator has a spin Chern number C±=(1,0) indicating the edge state is pseudo-spin orientation dependent and robust against TR-broken impurities.We also perform finite-element numerical simulations to verify the topological differences of the bandgaps. At the interface between a conventional insulator and a quantum anomalous Hall insulator, pseudo-spin dependent one-way propagation interface states are clearly observed, which are strikingly deferent from chiral edge states resulting from quantum Hall insulator and pairs of helical edge states resulting from quantum spin Hall insulator. Moreover, our pseudo-spin dependent edge state is robust against TR-broken impurities, which also sheds lights on spintronic devices.
Highlights
形边界态的 QSH 绝缘体,我们提出的 QAH 绝缘体,它在边界上只有一支自旋锁定的 手性边界态[32]。这支自旋锁定的手性边界态,它无需时间反演对称性保护,因此 对于实现实际系统中的波的无损传播,有很大的应用价值。
图 3(a) 引入气流后, 点的本征态随着外加气流的变化。 (b) 引入气流后,系统的对 称性从 C6v 降到 C6,可以用其矩形波导内部能流的方向类比为自旋,这里我们定义逆时 针能流为自旋向上。 (c) 考虑两种声子晶体系统,随着外加气流变化,其能带演化的示 意图,其中灰色区域代表全带隙。随着气流的增大,我们可以看到能带的闭合与翻转。 (d)系统的拓扑相图,改变参数 R 和外加气流强度 v,系统一共呈现三个相,分别用不同 自旋陈数表示。 Figure 3 (a) The change of eigenfrequencies at the point as functions of the introduced external airflow (b) The illustration of pseudo-spin up component when we introduce the airflow that breaks the TR symmetry and changes the symmetry of the system from C6v to C6 . (C) A schematic of the evolution of the band under an increasing external airflow V
图 5 超原胞边界态(a)用图 2a 中的声子晶体 和图 4b 中的声子晶体 V 构造出的新的声子 晶体系统,共包含 20 个原胞,其投影能带色散关系。图中显示了系统在边界上支持向右 传播的自旋向上的边界态 (b) 系统由图 2b 中的声子晶体 I 和图 4b 中的声子晶体 V 组 成,图中显示了系统在边界上支持向右传播的自旋向吓的边界态。 Fig 5 Projected band structures and edge states. (a) Dispersion relation for a ribbon-shaped 2D topological phononic crystal with 20 unit cells formed by Systems I and IV
Summary
形边界态的 QSH 绝缘体,我们提出的 QAH 绝缘体,它在边界上只有一支自旋锁定的 手性边界态[32]。这支自旋锁定的手性边界态,它无需时间反演对称性保护,因此 对于实现实际系统中的波的无损传播,有很大的应用价值。 记能带。我们可以看到在 a R 3.1 的时候, E1 态的频率低于 E2 态。而增大 R 可以 图 2: 不引入气流下的两种声子晶体(a) 我们取参数 a R 3.1 , E1 态的频率低于 E2 态,系统是个普通绝缘体 (系统 I) (b)我们取参数 a R 2.9 , E1 态的频率高于 E2 态 , 系统是个量子自旋霍尔效应绝缘体(系统 II)。 到[36]: H0(k) A( ˆzkyx kxy ) E0 Dk 2 (M0 Bk 2) ˆz
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