Fig. 1. (a) Spin-momentum locking in elastic wave systems^[49]; (b) pseudo-spin-orbital locking in elastic wave system^[50] (One-direction channels are established). (a)弹性波的内禀自旋轨道锁定^[49]; (b)弹性波的赝自旋轨道锁定^[50](由此都可以得到响应模式的单向通道)

Fig. 2. (a) Valley topological insulator for elastic wave with different valleys supported on each edge; (b) bulk and edge band in a finite ribbon; the color bar implies the value of vertical coordinate, with pure blue (red) representing the upper (lower) edge^[51]; (c) the band inversion during the topological phase transition^[53]; (d) the topological edge state transmission is robust against disorder^[51](a)弹性波的谷拓扑绝缘体, 它的每个边界上都可以支持不同谷的边界态; (b)有限带状材料中弹性波体态和边界态的能带, 其中色阶表示(a)中不同的纵向坐标, 蓝色和红色分别表示上下边界^[51]; (c)拓扑相变过程中的能带反转^[53]; (d)受拓扑保护的边界传输^[51]

Fig. 3. (a) Schematic spring-mass model of a finite 1D system connected by mass-negative and coupling-negative to achieve the topological interface state, the middle of the figure is the transmittance spectrum and the lower is the spatial profile of the eigenmodes of the interface state; (b) the dispersion relation of the system by changing the parameter of the system and the curve of the Zak phase for the lower two band with respect to the system parameter^[56]. (a)由有效负质量和有效负模量系统连接而成的有限长系统以及其界面态, 中间为系统的透射谱, 下面为带隙处共振峰频率下的振动模式; (b)不同参数情况下的系统色散关系以及较低两条能带的Zak 相位随着系统参数的变化^[56]

Fig. 4. (a) Inversion of energy band is broken by spatiotemporal modulation; (b) edge states in temporally varying energy band; the edge-bulk correspondence is present^[16]; (c) the modulated middle part contains disorder and sandwiched between two static parts (S and D); (d) the transmission from S to D. M is the size of disorder, while v is the velocity of moving modulation. The non-reciprocity of the transmission is topologically protected from the disorder^[64]. (a)因时空调制, 空间周期势下的能带(左)对称性打破(右); (b)能带随时间变化中的体带和边界态; 体态和边界态具有体边对应^[16]; (c)包含杂质的受驱动系统夹在两静态系统(S和D)中间; (d)从S到D传输弹性波的透射率, M是杂质大小, 而v是含时行波驱动的移动方向, 透射率的非互易性受到拓扑保护, 不受可能存在的杂质的影响^[64]

Fig. 5. Non-trivial topology induced by broken time reversal symmetry with gyroscopic degrees of freedom: (a) Lattice model of the phononic crystal, in which the black and red lines represent springs of different stiffness; (b) the mechanical unit set on each site; (c) the bulk energy bands with corresponding Chern numbers; (d) eigenstates of a finite system, with uni-directional edge states denoted by red (black) curves. The edge states are uni-directional^[48]. 通过引入陀螺仪打破时间反演实现的非平凡拓扑　(a)声子晶体的格点模型, 其中黑色和红色的线代表不同劲度系数的弹簧; (b)单个格点上的力学单元; (c)系统的能带, 能带上所标的数字代表该带的陈数; (d)有限系统的能态, 红线和蓝线分别为上下边界上的态, 可以看到, 单个边界上的态具有单向性^[48]

Fig. 6. Elastic wave quantum Hall effect of phononic systems in non-inertial reference frame: (a) Diagram of the system, with rotation acting like an effective magnetic field in electronic systems; (b) energy band of a nano-ribbon in the middle panel, with available left (right) edge states on the left (right) panel; (c) using the uni-directionality of edge states to manipulate the transmission in band gap. Remarkably, a perfect wave diode is obtained with the three-terminal system^[34]. 将声子系统放入非惯性系引入的量子霍尔效应　(a)系统示意图, 声子系统的旋转类似对电子系统施加的磁场; (b)中间是准一维纳米带的能带, 左右两图分别为系统左右边界上存在的传输模式; (c)利用边界态单向性来调控频率处于体带带隙内的弹性波的传输, 尤为重要的是, 使用右边的三端口器件, 可以实现完美的弹性波非互易传输, 这是应用非平凡拓扑实现完美二极管的一个例子^[34]

Fig. 7. (a), (b), (c) Heat pump effect in nonlinear quantum molecular junction. The state jumping rate of the junction is dependent on the temperature of two reservoirs, which can then be utilized to modulated temporally to achieve directional heat transfer. (a) The demonstrative diagram of the system; (b) berry curvature in the parameter (the temperature of reservoirs) space; (c) average heat transferring rate proportional to driving frequency, showing the pure geometric origin of pumped heat^[17]; (d), (e) the eigenvalues around EP of the two state system’s effective optomechanical non-Hermitian Hamiltonian; (f), (g) the transient behavior of system’s energy on the two of states. The adiabatic driving in (f) does not circle around EP while that in (g) does. Whether circling around EP determines the energy transfer direction topologically.^[72](a), (b), (c)非线性分子结中的热抽运, 中间系统的跃迁速率和所接的温度相关, 因而可以通过含时调制左右热库的温度, 实现热流的定向输运;　(a)系统示意图; (b)参数(左右两端所接温度)空间内的Berry曲率; (c)在慢驱动极限下, 平均热流大小和驱动频率成正比; 因而每次回路驱动产生的输运热量大小相同, 这种情况下热流是纯几何量; ^[17] (d), (e)非厄密两态光力系统等效哈密顿量在EP点附近的本征值; (f), (g)在t < 0时向系统馈入能量, 在灰色区域对系统进行绝热含时驱动, 其中(f)的驱动不绕过EP点, 而(g)的驱动绕过EP点, 是否绕过EP拓扑性地决定了两模式之间的能量交换方向 ^[72]

Fig. 8. (a) A schematic figure of the experimental system and the time modulation protocol; (b) the stiffness of springs is modulated spatiotemporally to achieve the non-reciprocal energy band. Periodical modulation brings mode coupling in the Bloch band, opening gaps in the position satisfying modified Bragg condition^[76] . (a)系统示意图与驱动方式; (b)对弹簧劲度系数进行时空调制, 实现的能带非互易, 驱动带来附加的准能量和准动量使得能带内出现模式耦合, 使能带在满足广义Bragg条件的位置打开带隙^[76]

Fig. 9. (a) A schematic static thermal diode obtained by manipulation of the spectrum overlap between the two nonlinear parts^[1]; (b) non-reciprocal temperature distribution. The moving direction of the wave-like modulation is opposite between the upper and lower panels, while direction of temperature gradient is not altered. Gamma implies the velocity of modulation motion, which is zero when the modulation is independent of time. The degree of non-reciprocity first increases and then decreases when gamma monotonically grows^[19] . (a)静态热二极管的原理示意图^[1], 关键在于, 调控热传导相关主要模式声子的能谱交叠; (b)含时驱动带来的温度分布非互易性, 上下两图中驱动方向相反, 而左右所接的温度不变. 其中Γ代表了时空调制移动速率, Γ= 0即为无时间调制; 非互易效果随调制移动速度先变大再变小^[19]