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Keywords: Topological edge state Reconﬁgurability Lamb wave Pseudospin–orbit coupling Robustness

We propose two-dimensional reconﬁgurable phononic crystal slabs that support topologically protected edge states for Lamb waves. It consists of a triangular lattice of triangular-like air holes perforated in a slab and two sandwich cylinders are placed in the matrix of each unit cell. The pseudospin–orbit coupling is achieved by shifting the sandwich cylinders up or down. The robust transport of Lamb waves along reconﬁgurable interfaces is further demonstrated. The dynamically reconﬁgurable design provides an excellent platform for tuning the frequency range of topological edge states and steering eﬃcient transport of Lamb waves along arbitrary pathways, which can be employed for elastic-wave communications, signal processing, and sensing. © 2018 Published by Elsevier B.V.

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Article history: Received 27 February 2018 Received in revised form 7 June 2018 Accepted 16 June 2018 Available online xxxx Communicated by R. Wu

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1. Introduction

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The discoveries of the quantum Hall effect, the quantum spin Hall effect, and topological insulators have revolutionized our scientiﬁc cognition of condensed matter physics. Since the gapless edge states in graphene and HgTe quantum wells were discovered [1,2], more studies on analogous states in structures supporting classical waves have been conducted. Topologically protected edge states have been demonstrated in the topological photonic crystals [3–13], which is of great importance in both fundamental science and the practical application. In parallel, these concepts have also inspired a novel ﬁeld of topological acoustics [14–27]. Achieving topologically protected acoustic-wave propagation depends mainly on three categories of mechanisms. The ﬁrst is to realize one-way edge states similar to the quantum Hall effect. By introducing a circulating ﬂuid ﬂow to break the time-reversal symmetry [14–16], the topologically protected unidirectional edge states with strong robustness were demonstrated in nonreciprocal systems. The second mechanism exhibits the pairwise topological edge states in analogy with the quantum spin Hall effect. By constructing a pair of pseudospins [19–22], the robust pseudospin-dependent transport has been successfully demonstrated in systems with pseudo-time-reversal symmetry. The third is to achieve valley-polarized topological edge

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states by emulating the valley Hall effect. The topological valley transport has been successfully observed by breaking the spatialreversal symmetry [23–26]. Different from the previous studies on acoustics in the ﬂuid domain where acoustic waves are purely longitudinal, the research of topological physics in the solid slabs is extremely challenging due to the mixture of all three polarizations of elastic waves. Recently, topologically protected edge states for Lamb waves have been achieved by using a dual-scale topological phononic crystal (TPC) [19]. However, its topological bandgaps and transport pathways are limited to speciﬁc situations that cannot be tuned after machining. Most recently, many studies on the reconﬁgurability of the TPCs have been performed in the ﬂuid domain due to the merits of tuning the topological bandgaps and transport pathways [22,23], which can enrich the design and use of the TPCs. So far, there is few investigation on the reconﬁgurability of the TPCs in the solid domain owing to the limitation of the movement of the scatterers. Thus, it is meaningful and challenging to investigate the reconﬁgurable TPCs in the solid domain. The purpose of this letter is to realize the reconﬁgurable phononic crystal (PC) slabs supporting topological edge states for Lamb waves. It consists of a triangular lattice of triangular-like air holes perforated in a slab and two sandwich cylinders are inserted in the matrix material of each unit cell. The inversion symmetry breaking caused by moving the sandwich cylinders up or down leads to the pseudospin–orbit coupling. We explicitly demonstrate that the topologically protected transport of Lamb waves is robust against various kinds of defects along reconﬁgurable interfaces. The tunability of the designed TPC slabs makes it possible to dynamically control the frequency range of topological edge states and

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Lu-yang Feng, Hong-bo Huang, Jian-cheng Zhang, Xiao-ping Xie ∗ , Jiu-jiu Chen ∗

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Reconﬁgurable topological phononic crystal slabs

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Corresponding authors. E-mail addresses: [email protected] (X.-p. Xie), [email protected] (J.-j. Chen). https://doi.org/10.1016/j.physleta.2018.06.029 0375-9601/© 2018 Published by Elsevier B.V.

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Fig. 1. (a) Perspective view (left panel) of a unit cell and top view (right panel) of the PC slabs formed by a triangular array of triangular-like air holes in a slab. The host material of each unit cell is inserted by two movable sandwich cylinders. (b) Sandwich cylinders at different positions in the slab: from left to right, the sandwich cylinders are shifted from the lower to upper position, leading to a topological transition as indicated by the mass term m. The sandwich cylinder consists of one cylindrical core (blue) in the center and two cylindrical face sheets (green) on both ends. The black and green dotted lines represent the position of the neutral planes of the slab and the sandwich cylinders, respectively. (For interpretation of the colors in the ﬁgure(s), the reader is referred to the web version of this article.)

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reconﬁgure arbitrary transport pathways for Lamb waves without obvious backscattering.

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As shown in Figs. 1(a) and (b), the designed PC slabs are composed of a triangular array of triangular-like air holes perforated in an elastic slab and the host material of each unit cell is inserted by two movable sandwich cylinders, each of which consists of one cylindrical core (blue) in the center and two cylindrical face sheets (green) on both ends. The lattice constant is a = 40 mm and the slab thickness is t = 10.5 mm. The ﬁllet radius of the triangular-like hole is r = 8.3 mm and the distance from center to side of the triangular-like hole is l = 14.85 mm. The diameter of the cylinder, the length of the cylindrical core, and the thickness of the cylindrical face sheet are d = 6.4 mm, h = 10.5 mm, and b = 0.64 mm, respectively; the displacement of the cylinders is denoted as s (suppose that s = 0 mm when the neutral plane of the sandwich cylinders overlaps with that of the slab [middle panel in Fig. 1(b)]). The material of the cylindrical core of the sandwich cylinder is polyetheretherketone (PEEK) which is a common engineering plastics with excellent friction and wear properties as well as mechanical properties, and the material of the remaining structures is steel. The material parameters are chosen as follows: the density ρ = 7800 kg/m3 , the Young modulus E = 206 GPa, and the Poisson’s ratio σ = 0.3 for steel; ρ = 1285 kg/m3 , E = 4.21 GPa, and σ = 0.388 for PEEK. The propagation of Lamb waves is governed by the wave equilibrium equation ρ u¨ = ∇(λ + 2μ)(∇ · u ) − ∇ × (μ∇ × u ), where ρ is the material density, u is the displacement vector, and both λ = E σ /[(1 + σ )(1 − 2σ )] and μ = E /[2(1 + σ )] are the Lame’s coeﬃcients. The wave equilibrium equation can be transferred into the eigenvalue problem (K − ω2 (k)M)U = 0 [28], where K and M are the stiffness matrix and mass matrix, respectively, and k

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ˆ = ν D (τˆz sˆ 0 σˆ x δkx + τˆ0 sˆ 0 σˆ y δk y ) + mτˆz sˆ z σˆ z , H

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represents the wave vector in the irreducible Brillouin zone. The eigenvalue problem is solved by using eigenfrequency analysis of solid mechanics in the commercial ﬁnite element software COMSOL Multiphysics. Periodic boundary conditions are imposed on the four external cross section of the unit cell, and the stress-free boundary conditions are set on the other surfaces. When the wave vector k varies along the boundary of the irreducible Brillouin zone (–K(K )–M–) [inset of Fig. 2(a)], the Lamb-wave band structures ω(k) are obtained, as shown in Fig. 1(a), (c), and (d). For a phononic system, it is necessary to increase the degrees of freedom to two-fold states to emulate two pseudospin states [20,22]. Consequently, a four-fold Dirac degeneracy in the phononic band structure is required. For s = 0 mm [middle panel of Fig. 1(b)], both the in-plane C 3v and the z-direction symmetries of the PC slabs are maintained, resulting in the four-fold Dirac degeneracy at point K [Fig. 2(a)]. The degenerate Lamb-wave modes are classiﬁed by their displacement ﬁelds as the symmetric (S) modes (blue lines) and the anti-symmetric ( A) modes (red lines). The frequency and group velocity of these two families of modes are matched around the Dirac point, which allows one to use unitary transform of the orthogonal basis (S and A modes) as pseudospin states [19,20]. We then introduce pseudospin–orbit coupling to induce topological phase translation, accomplished by the opening of a complete topological bandgap. When the sandwich cylinders are moved up or down from the equilibrium position [left or right panel of Fig. 1(b)], the z-direction inversion symmetry of the PC slabs is broken, leading to the formation of a complete topological bandgap and two two-fold degeneracies: the lower (upper) A and upper (lower) S modes [Fig. 2(c)]. For all frequencies in the proximity of the original Dirac points where the frequency and group velocity degeneracy are maintained, √ the A and S modes of the PC slabs become hybridized ( A ± S )/ 2 [19], which can be used as two pseudospin states [20]. As shown in Fig. 2(b), the evolution of displacement ﬁelds of the unit cells from s = 0 mm to s = 0 mm also conﬁrms the pairwise hybridizations of the A and S modes. In practice, the z-direction symmetry breaking caused by moving the sandwich cylinders produces an effect analogous to spin–orbit coupling in graphene. To reveal the pseudospin–orbit effect, the effective Hamiltonian for the present PC slabs can be obtained by the perturbation theory as [10,19]:

(1)

where σˆ i , τˆi , and sˆ i (i = x, y , z) are the Pauli matrices acting on orbit, valley, and pseudospin state vectors, respectively; τˆ0 and sˆ 0 are the identity matrix, v D is the group velocity near the Dirac point, and m (|m| = ωgap /2, where ωgap is the topological bandgap) is the effective mass induced by the z-direction symmetry breaking. The last term mτˆz sˆ z σˆ z [Eq. (1)] caused by the z-direction symmetry breaking is identical to the spin–orbit coupling of the Kane– Mele Hamiltonian [1], which demonstrates that the z-direction symmetry breaking leads to the pseudospin–orbit coupling in the present PC slabs. The occurrence of the pseudospin–orbit coupling can induce a topological phase transition. In order to reveal the topological phase transition [29,30], the pseudospin Chern numbers are introduced to characterize various topological phases. For the Hamiltonian given in Eq. (1), the pseudospin Chern numbers of each band can be evaluated as C = ±sgn(m) [10,19]. For s = 0 mm [middle panel of Fig. 1(b)], the pseudospin Chern numbers C are equal to zero due to m = 0, meaning trivial topological phase; while for s = 0 mm, the pseudospin Chern numbers C are nonzero due to m = 0, corresponding to the nontrivial topological phase. Obviously, by moving the sandwich cylinders up or down, the PC slabs experience a topological phase transition. Note

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Fig. 2. (a) Band structure for s = 0 mm. A four-fold Dirac degeneracy is formed at point K. The red and blue lines indicate the anti-symmetric ( A) modes and the symmetric (S) modes, respectively. (b) The evolution of displacement ﬁelds of the unit cells around the point K from s = 0 mm to s = 0 mm. The four displacement ﬁelds from bottom to top on the right or left side correspond to four bands of interest from lower to upper, respectively. (c) Band structure for |s| = 0.64 mm. A complete topological bandgap is induced by shifting sandwich cylinders up or down. The parameters C are the pseudospin Chern numbers of the PC slabs with the sandwich cylinders shifted up. (d) The evolution of the topological bandgap as a function of the displacement |s| of the sandwich cylinders.

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that although the band structures are identical in the PC slabs with the sandwich cylinders shifted up and down, the sign of the effective mass m is opposite [left or right panel of Fig. 1(b)], indicating the opposite pseudospin Chern numbers and different topological properties. To illustrate this further, we calculate numerically the pseudospin Chern numbers by integrating the Berry curvature (k) = ∇k × A (k) of the same pseudospins throughout the ﬁrst Brillouin zone, where A (k) is the Berry connection and ∇k = (∂kx , ∂ky , ∂kz ) [19,20]. The Berry connection of the nth band is deﬁned as A n (k) = Im unit cell un (k)|∇k |un (k)dV , where un (k) is the normalized Bloch wave function that can be obtained through the COMSOL simulation. In this way, we ﬁnd that the pseudospin Chern numbers for four bands from lower to upper reverse from C = −1, +1, −1, +1 to C = +1, −1, +1, −1, as the sandwich cylinders are moved from up to down. The opposite pseudospin Chern numbers reveal that the two PC slab structures are topologically distinct, which is consistent with the previous discussion. Therefore, the ability to independently shift the sandwich cylinders makes it possible to conveniently control the topological properties of the PC slabs. Next, we investigate the inﬂuence of the displacement s of the sandwich cylinders on the topological bandgap. As shown in Fig. 2(d), the topological bandgap enlarges with the increase of |s|. Because the increase of the displacement s leads to the bigger differences of the height of the stubbed surface on both sides of the

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slabs, the bigger bandgap can form at crossings of the Lamb-wave band structure [31,32]. Therefore, we can tune the topological bandgap by simply moving the sandwich cylinders. The tunability of the topological bandgap, absence in the conventional ﬁxed TPCs, can dramatically enrich the design and use of the TPC slabs.

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To realize topologically protected edge states for Lamb waves, we construct a ribbon-shaped supercell with one interface which is between two PC slab structures with different pseudospin Chern numbers [Fig. 3(a)]. The sandwich cylinders of the left PC slab structure (type I) are shifted up, while the sandwich cylinders of the right PC slab structure (type II) are shifted down. We calculate the projected band structure by eigenfrequency analysis of solid mechanics in the COMSOL. Periodic (absorbing) boundary conditions are applied to the edges along x ( y) direction. When the wave vector k sweeps from −π /a to π /a along x direction, the projected band structure is obtained, as shown in Fig. 3(b). It is observed that there are two pairs of topological edge states within the topological bandgap: the forward-propagating edge states (red lines) and the backward-propagating edge states (blue lines). Because the pseudospin Chern numbers differ by 2 across the interface, according to the bulk-boundary correspondence [33–35], there are four topologically protected edge states for the TPC slabs

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Fig. 4. Displacement ﬁeld (51.4 kHz) of the TPC slabs with a linear interface. The sign I and II represent the PC slabs with the sandwich cylinders shifted up and down, respectively. Lamb waves are excited by a small displacement on the left end (top panel) and right end (bottom panel) of the interface, respectively. The red arrows indicate the input/output ports.

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Fig. 3. (a) Side view (upper panel) and top view (lower panel) of a ribbon-shaped supercell consisting of 16 unit cells on each side of the interface (light blue dotted lines). The red arrow on each side of the interface indicates the moving directions of the corresponding sandwich cylinders. The PC slabs on both sides of the interface are marked as I and II, corresponding to the mass term m > 0 and m < 0. (b) Projected band structure of the supercell. The red and blue dotted lines indicate the forward-propagating and backward-propagating edge states, respectively. The black dotted lines represent the bulk bands. (c) Displacement ﬁeld and Poynting vector of the supercell at points 1, 2, 3, and 4 in (b). The black and red arrows denote the Poynting vector and the overall energy ﬂow, respectively.

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[10,19]. As shown in Fig. 3(c), the displacement ﬁelds of the supercell display that two pairs of topological edge states are conﬁned at the interface and decay rapidly into the bulk. In order to further understand the topological edge states, we check the real-space = Re[− V ∗ · T ]/2 ( V is the distribution of the Poynting vector P velocity vector and T is the stress tensor) averaged over a period τ = 2π /ω , which describes the energy ﬂow in the TPC slabs

Fig. 5. Displacement ﬁeld (top panel) of the TPC slabs with a zigzag interface including three kinds of defects (bottom panel): sharp bends, cavity and disorder (the disordered sandwich cylinders are marked in red). Lamb waves are excited by a small displacement on the left end of the interface. The red arrows indicate the input/output ports.

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[36]. The energy ﬂow of the forward-propagating edge states [1 and 3 enlarged views in Fig. 3(c)] has the opposite senses of rotation on both sides of the interface, resulting in the directional transport of the Lamb waves [10]. While for backward-propagating edge states, the senses of rotation are reversed [2 and 4 enlarged views in Fig. 3(c)]. It is worth noting that the conjunctive points [green points in Fig. 3(b)] between two topological edge states are transition points where there is no energy ﬂow. Additionally, the frequency range of topological edge states is controllable considering the tunable topological bandgap.

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Fig. 6. Dynamic reconﬁguration of topological transport pathways and displacement ﬁelds. The green and light blue lines represent two different transport pathways which are successively constructed by shifting the sandwich cylinders up or down in the same TPC slabs. The red and gray arrows indicate the presence and absence of incident or received Lamb waves, respectively.

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4. Conclusion

To sum up, we have presented the PC slabs exhibiting tunable topological edge states and reconﬁgurable transport pathways for Lamb waves. The pseudospin–orbit coupling is achieved by shifting the sandwich cylinders up or down. With numerical simulations, we unambiguously demonstrate topologically protected propagation of Lamb waves along reconﬁgurable interfaces and the robustness of topological edge states against defects. In view of the scale invariance of the elastic wave equation, Lamb waves can be manipulated and controlled over a higher frequency range. The dynamically reconﬁgurable TPC slabs provide an excellent platform for freely steering the topologically protected transport of Lamb waves along arbitrary pathways, which has potential applications in elastic-wave communications, signal processing, and sensing.

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be achieved in the TPC slabs, which could be used for steering efﬁcient transport of Lamb waves along arbitrary pathways in real time.

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To verify topologically protected transmission of edge states, full wave simulation in ﬁnite TPC slabs (16 × 16 unit cells) is carried out via frequency-domain analysis of solid mechanics in the COMSOL. As shown in Fig. 4, this TPC slabs consist of two different PC slab structures (type I and II), separated by a linear interface. Perfectly matched layers are applied around the TPC slabs to suppress the boundary reﬂections. Lamb waves (51.4 KHz) are excited by a small displacement on the left end (top panel) and right end (bottom panel) of the interface, respectively. The Lamb waves propagate along the linear interface without obvious backscattering, which demonstrates eﬃcient transmission of topological edge states for Lamb waves in the TPC slabs. Considering the unidirectional transmission of each kind of topological edge states [21,22], the propagation of Lamb waves in two directions along the interface [Fig. 4] veriﬁes the presence of two types of edge states in the TPC slabs. Topologically protected edge states are robust owing to the locking of the pseudospins to their respective propagation directions. In order to verify the robustness of topologically protected edge states, we intentionally introduce different defects into TPC slabs and investigate their effects. As shown in bottom panel of Fig. 5, there are three types of common defects (two sharp bends, cavity, and disorder) in a zigzag interface of the TPC slabs. It is observed from the top panel of Fig. 5 that Lamb waves excited by a small displacement at the left end maintain a complete transmission along the interface without obvious backscattering. The result explicitly demonstrates the topological edge states are robust against various kinds of defects, which is a universal characteristic caused by the intrinsic topological feature of the TPC slabs and their topological protection. In order to illustrate the reconﬁgurability of the TPC slabs, two different transport pathways are successively constructed by shifting the sandwich cylinders up or down in the same TPC slabs. As shown in Fig. 6, during 0–t 1 period, Lamb waves incident from the port 1 transmit to port 4 along the ﬁrst pathway (green line) with high transmission eﬃciency; during t 1 –t 2 , the second topological transport pathway (light blue line) is reconﬁgured rapidly from the ﬁrst pathway; during t 2 –t 3 period, Lamb waves effectively propagate from port 3 to port 2 along the second pathway. Therefore, by moving the sandwich cylinders, we can realize topologically protected transmission of Lamb waves along arbitrary pathways from arbitrary input ports to output ports at different time. When each sandwich cylinder is equipped with one electronically controlled linear actuator, the reconﬁguration of the topological transport pathways can be completed instantaneously. In other words, the dynamic reconﬁguration of the topological transport pathways can

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Acknowledgements

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The authors gratefully acknowledge ﬁnancial support from National Science Foundation of China under Grant No. 11374093 and Young Scholar fund sponsored by common university and college of the province in Hunan.

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15 16 17

RO

18 19 20 21 22 23

DP

24 25 26 27 28 29 30

TE

33 34 35

EC

36 37 38 39 40

43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

UN

42

CO RR

41

69 70 71 72 73 74 75

78

14

32

68

77

13

31

67

76

OF

1

AID:25181 /SCO Doctopic: Condensed matter

79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

60

126

61

127

62

128

63

129

64

130

65

131

66

132

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