Topological spin-Hall edge states of flexural wave in perforated metamaterial plates
TTopological spin-Hall edge states of flexural wave inperforated metamaterial plates
Linyun Yang, Kaiping Yu ∗ , Ying Wu, Rui Zhao andShuaishuai Liu Department of Astronautic Science and Mechanics, Harbin Institute ofTechnology, Harbin, Heilongjiang 150001, ChinaE-mail: [email protected]
March 2018
Abstract.
This paper investigates the pseudo-spin based edge states for flexuralwaves in a honeycomb perforated phononic plate, which behaves an elasticanalogue of the quantum spin Hall effect. We utilize finite element methodto analyse the dispersion for flexural waves based on Mindlin’s plate theory.Topological transition takes place around a double Dirac cone at Γ point byadjusting the sizes of perforated holes. We develop an effective Hamiltonian todescribe the bands around the two doubly degenerated states and analyse thetopological invariants. This further leads us to observe the topologically protectededge states localized at the interface between two lattices. We demonstrate theunidirectional propagation of the edge waves along topological interface, as wellas their robustness against defects and sharp bends.
Keywords : topological edge state, quantum spin Hall effect, metamaterial plate,flexural wave, unidirectional transport
Submitted to:
J. Phys. D: Appl. Phys.
1. Introduction
The discovery of topological insulators [1–3], which can exhibit topologically protectededge states propagating in a single direction along the sample edges, has opened a newchapter in the research realm of condensed matter physics. The edge states are immuneto backscattering from disorder or sharp bends because of the underlying topology ofthe band structures. Recently these concepts in quantum systems have been extendedto the field of photonic [4–6], acoustic [7–18] and elastic [19–31] systems, due to theirpotential practical applications in wave guiding, isolating, filtering, etc.Topologically protected states in acoustic and mechanical systems become aresearch focus very recently. Breaking the time-reversal symmetry and mimicking thequantum Hall effect (QHE) [7–9, 20, 21] in mechanical and acoustic systems requiresexternal active components, such as gyroscope and circulating fluid flow, which addscomplexity to the systems and remains challenging to practical realization. Anotherscheme to achieve topological states is establishing acoustic/mechanical analogues tothe quantum spin Hall effect (QSHE) [10–13, 24–28], which makes use of the twoirreducible representations of C v point group symmetry to construct (pseudo) spin a r X i v : . [ phy s i c s . a pp - ph ] J un Hall states. Forming a double Dirac cone to increase the degrees of freedom is essentialto realize the acoustic/mechanical analogue of QSHE, since Kramers doublet exists inthe form of two double degenerate states, pseudo spin states can be constructed bythe hybridization of two pairs of degenerate Bloch modes.Different from acoustic waves, elastic waves in plates exhibit much more complexdispersion behaviors due to the existence of both longitudinal and shear waves, andmoreover, the reflections and couplings at the stress-free boundaries. Elastic plateshave become an attractive platform for the study of topological states for bothacademic and practical interests very recently. These investigations mainly focus onthe lamb modes [24–26, 30] and flexural modes [23, 27–29] in plates with perforatedholes [24–26] or resonators [23, 27–30], forming elastic analogues of QSHE [24–28] orquantum valley Hall effect (QVHE) [29, 30]. Recently, Rajesh and co-authors [27]investigated the topological spin Hall effects for flexural waves based on local resonantmetamaterial plate with mass-spring systems attached on one face of the plate. Intheir work, the distance between the resonators and the unit cell center is tuned tobreak the translational symmetry in order to open the double Dirac cone. In thispaper, inspired by zone-folding mechanism, we report the observation of topologicallyprotected edge states for flexural waves in plates with solely honeycomb arranged circular holes . By simply perturbing the holes’ radii, the topology transition froma topological trival band to non-trival one can be realized. Our structural designstrategy is free of fabrication complexity, and provides an ideal platform for practicalrealization for elastic QSHE analogues.The structure of this work is outlined as follows, in section 2, we present thedispersion analysis of flexural waves based on Mindlin’s plate theory using finiteelement method (FEM). In section 3, we show the band inversion process and study thetopology of the band structures. In section 4, we consider a ribbon-shaped supercellcomposed topological trival and non-trival cells, and show the existence of a pair oftopological edge states. This is followed in section 5 by a full wave simulation tocheck the topological protected pseudo-spin dependent flexural wave transport. Abrief summary and discussion of this paper is provided in section 6.
2. Dispersion Analysis for Perforated Phononic Plates
We consider a flat thick plate with hexagonal arranged perforated holes, as figure1 shows. Two types of unit cells are highlighted by red hexagon and blue rhombin figure 1 (a). The hexagonal cell (primitive unit cell) can clearly illustrate thehoneycomb arrangement pattern and the C v symmetry of our system, but the complexboundaries may bring some inconvenience in the following analysis. Therefore, wechoose a bigger rhombic shaped unit cell, shown in figure 1 (b). s and s denotethe two lattice vectors. The plate material is acrylic glass, with Young’s modulus E = 3 . ν = 0 .
33 and mass density ρ = 1062kg / m . a = 0 . a = √ a are the lattice constants of the hexagonal and rhombic unit cells, andthe plate thickness is set to be h = 0 . a .Dispersion analysis for flexural waves propagating in plate structures in this paperis based on Mindlin’s plate theory, which is an extension of classical Kirchhoff platetheory by taking into account the first order shear deformation [32]. We employFEM to calculate the dispersion relation in phononic crystal plate described above.According to the First-Order shear deformation theory for flexural vibration modes (b)(a) s s xyz Figure 1. Schematic view of the phononic plate and unit cells. (a)
Twotypes of unit cells are marked by blue (rhomb) and red (hexagon) lines. (b)
Azoom in view of the rhombic cell, s , s are the two lattice vectors, a denotesthe lattice constant, and r and r are the radii of the smaller- and bigger-sizedperforated holes. in elastic plates, the displacement can be expressed as u ( x, y, z ) = zθ x ( x, y ) v ( x, y, z ) = zθ y ( x, y ) w ( x, y, z ) = w ( x, y ) (1)in which u, v, w are the particle displacement components in x, y and z directions, and θ x , θ y are the rotations of the normal to the mid-plane with respect to axis y and x ,respectively.So the strains can be expressed as ε x = z ∂θ x ∂x , ε y = z ∂θ y ∂y , γ xy = z (cid:18) ∂θ x ∂y + ∂θ y ∂x (cid:19) (2a) γ xz = θ x + ∂w∂x , γ yz = θ y + ∂w∂y (2b)And stresses are given by the stress-strain relation σ f = D f ε f , σ c = D c ε c (3)where σ f = [ σ x , σ y , τ xy ] (cid:62) , ε f = [ ε x , ε y , γ xy ] (cid:62) , and σ c = [ τ xz , τ yz ] (cid:62) , ε c = [ γ xz , γ yz ] (cid:62) . D f and D c are defined as D f = E − ν ν ν − ν , D c = (cid:20) µ µ (cid:21) (4)in which E and µ are Young’s modulus and shear modulus, ν is Poisson’s ratio. Afour-node bi-linear isoparametric element is employed to discretize the unit cell [33],and displacement variables are interpolated by the nodal displacements, w ( x, y ) = n (cid:88) i =1 N i ( x, y ) w i , θ x ( x, y ) = n (cid:88) i =1 N i ( x, y ) θ x,i , θ y ( x, y ) = n (cid:88) i =1 N i ( x, y ) θ y,i (5)where n denote the number of nodes in each element, and w i , θ x,i and θ y,i are thedisplacement components at the i th node. N i ( x, y ) are shape functions that are usedto interpolate the nodal displacements. Inserting equation (5) into equation (2), wehave ε f = z B f u e , ε c = B c u e (6)with u e = [ w , θ x, , θ y, , , . . . , w , θ x, , θ y, ] (cid:62) the element nodal displacement vector. B f and B c are given by B f = N ,x . . . N ,x
00 0 N ,y . . . N ,y N ,y N ,x . . . N ,y N ,x (7a) B c = (cid:20) N ,x N . . . N ,x N N ,y N . . . N ,y N (cid:21) (7b)Here N i,x = ∂N i ∂x and N i,y = ∂N i ∂y . The strain energy and kinetic energy in eachelement can be expressed as U e = 12 (cid:90) V e ε (cid:62) f D f ε f d V + κ (cid:90) V ε (cid:62) c D c ε c d V (8a) T e = 12 (cid:90) V e ˙ u (cid:62) diag( ρ, I, I ) ˙ u d V (8b)in which κ = π /
12 is the correction factor, and ˙ u = ∂u∂t . Thus the Lagrangian of thissystem is given by L = T − U . From the Lagrangian equationdd t ∂ L ∂ ˙ u e − ∂ L ∂ u e = 0 (9)We can obtain that K e u e + M e ¨ u e = 0 (10)where K e , M e are so called element stiffness matrix and element mass matrix, whoseexpressions are given by K e = h (cid:90) A e B (cid:62) f D f B f d A + κh (cid:90) A e B (cid:62) c D c B c d A (11a) M e = (cid:90) A e ρ N (cid:62) diag( h, h / , h / N d A (11b)By assembling all the element stiffness matrices and element mass matrices, wecan obtain the global governing equation K U + M ¨ U = 0 (12)in which K and M denote the global stiffness matrix and mass matrix, respectively.Because of the periodic nature of phononic plate, the studying domain can be limitedwithin a single unit cell on the basis of Floquet-Bloch theorem. Applying the Floquet-Bloch periodic condition in FEM analysis is equivalent to a nodal displacementtransformation U = PU , where P is a k -dependent matrix, as stated in our earlierwork Ref. [34]. The readers can also see Appendix A for more details. Under time-harmonic assumption, we have (cid:16) K ( k ) − ω M ( k ) (cid:17) U = (13)where K = P † K P , M = P † M P are both Hermitian and k -dependent matrices.Here † represents Hermitian transpose. By solving the Hermitian eigenvalue problemequation (13) with k varying along the edges of the first Irreducible Brillouin Zone(IBZ), we can obtain the dispersion relation ω = ω ( k ).
3. Topological Phase Transition
Double Dirac cone plays an important role in imitating the quantum spin Hall effectsin classical periodic systems. The emergence of a double Dirac cone at Γ pointcan be achieved by varying the radii of perforated holes such that r = r (= r )while their locations remain unchanged. When all the holes possess the same size,we observe a double Dirac cone at the BZ center. Figure 2 (a) shows the bandstructure of the presented phononic plate in the case of r = 0 . a . A four-folddegenerated state is observed at the frequency f = 6452Hz at Γ point. Moreover,in the vicinity of Γ, four branches of dispersion curves touch at the degenerate pointlinearly (see Appendix C), in other words, a double Dirac cone has been formed. Theconstruction of the double Dirac cone can be explained by zone-folding mechanism.When r = r , there are 9 uniform holes in the unit cell as shown in figure 1(b). Inthis case, the unit cell is actually a supercell, with 3 × K point (vertex of the BZ of theprimitive cell) in the band structure of the primitive cell. By taking a 3 × K point are folded into Γ point twice.Therefore, a double Dirac cone is constructed at Γ point in the BZ of the supercell. Thefour-fold degeneracy will be split into doubly degenerated dipolar states ( p ) and doublydegenerated quadripolar states ( d ) if the radii of the holes are tuned because of thebrokenness of the translational symmetry. We adjust r in the range of [0 . r , . r ]with a restriction that 2 r + r = 3 r . The results plotted in figure 2 (b) show that thegap between the p and d states closes and reopens as r is tuned from 0 . r to 1 . r .It is worth noting that r = r when r = r under the condition that 2 r + r = 3 r ,therefore, the emergence of a double Dirac cone takes place.As figure 2 (b) shows, adjusting r such that r < r or r > r can both opena band gap. Below we will show band inversion process by perturbing parameters r from less to greater than r , which further leads to the topological transition froma topologically trival state to a non-trival state. Two cases are studied, case (I)for r = 0 . r and case (II) for r = 1 . r , both breaking the double Dirac coneand opening a complete band gap, as shown in figure 2 (c) and (d). Calculationresults show that case (I) opens a gap ranged from 6314 ∼ ∼ w ( x, y ) of theeigenmodes corresponding to the p and d states in figure 2 (c) and (d). For case (I),two modes below the gap behave like dipoles (marked by p x , p y ), and the modesabove the gap behave like quadripolar (marked by d x − y , d xy ). For case (II) theeigenmodes are flipped, that p modes are above the gap and d modes are below thegap. Here p x ( p y ) represents the mirror symmetry of the eigenmodes along x/y axisare even/odd(odd/even), and for d x − y ( d xy ) being odd/odd (even/even). The bandstructure has experienced a process of gap closing to gap reopening, during which the F r equen cy ( k H z ) M K Γ M4.55.05.56.06.57.07.58.0 Wave Vector, k F r equen cy ( k H z ) double dirac cone KM Γ M K Γ M4.55.05.56.06.57.07.58.0 Wave Vector, k F r equen cy ( k H z ) (c) M K Γ M4.55.05.56.06.57.07.58.0 Wave Vector, k F r equen cy ( k H z ) (d)(a) (b) trival nontrival Figure 2. Band structures of the phononic plates. (a)
A double Diraccone emerges at the brillouin zone center when r = r . (b) The upper and lowerbounds of the band gap vary versus r . (c) and (d) show the band structuresand the eigenmodes when r = 0 . r and r = 1 . r , respectively. When r isperturbed, r can be determined from the restriction condition 2 r + r = 3 r . eigenmodes above and below the gap at Γ point have inversed. In this section we demonstrate that band inversion further induces the topologicalphase transition of the band structures. To reveal the topological property of theband gaps in figure 2 (c) and (d), we employ the k · p perturbation method [35]to construct an effective Hamiltonian for the proposed phononic plate, and furthercalculate the topological invariant, i.e., spin Chern number. The stiffness matrix K ( k )in equation (13) can be rewritten as the summation of K ( k ) and a perturbed term K (∆ k ) approximately by using the Taylor series. So equation (13) can be expressedas (cid:0) K ( k ) + K (cid:48) − ω M ( k ) (cid:1) U = (14)in which K (cid:48) ≈ k x K x + k y K y , and K x , K y are constant matrices (that can be calculatedfrom equation (A.1). All the eigenvalues( ω k ,n ) and eigenvectors( U k ,n ) at Γ point( k = ) can be calculated beforehand, so we can expand the eigenvector in equation(13) into linear combination of U k ,n , U = (cid:88) n a n U k ,n (15)where a n are expansion coefficients to be determined. Substituting equation (15) intoequation (13), and making use of the orthogonality relation (cid:104) U k ,m | M | U k ,n (cid:105) = δ mn ,we can obtain that ( H + H (cid:48) − ω I ) { a } = (16)in which H (cid:48) mn = (cid:104) U k ,m | k x K x + k y K y | U k ,n (cid:105) is the first order perturbation term,and H = diag { ω k ,n } is a diagonal matrix. One should note that equation (16) stilldescribes the eigenvalue problem in the entire space. Here we develop an effectiveHamiltonian to describe the dispersion around the two doubly degenerate states ina subspace, i.e., span {| p x (cid:105) , | p y (cid:105) , | d x − y (cid:105) , | d xy (cid:105)} . Following the approach introducedin [6, 10, 11], we can obtain the matrix elements of the 4 × p x , p y , d x − y , d xy ] as H eff mn = H (cid:48) mn + (cid:88) α (cid:54) = m ( n ) H (cid:48) mα H (cid:48) αn ω k ,m ( n ) − ω k ,α (17)where the second term comes from the second order perturbation [6,10,11]. Withthe FEM analysis, H eff mn can be numerically calculated. Rewriting H eff on basis[ p + , d + , p − , d − ] by an unitary transformation, we find that H eff can be expressedas H eff = − M − Bk Ak + A ∗ k − M + Bk − M − Bk Ak − A ∗ k + M + Bk (18)with p ± = ( p x ± ip y ) / √ , d ± = ( d x − y ± id xy ) / √ , k ± = k x ± ik y , and M =( ω k ,d − ω k ,p ) / p and d modes. Coefficients A and B can be determined numerically following the above descriptions. The obtainedeffective Hamiltonian in equation (18) shares a similar form with Bernevig-Hughes-Zhang (BHZ) model for CdTe/HgTe/CdTe quantum well [36], indicating that ourperforated phononic plate can behave a “spin” Hall effect. We can further calculatethe spin Chern number [10, 11] C = ± (cid:0) sgn( M ) + sgn( B ) (cid:1) (19)Since the sign of M changes from positive ( ω k ,p < ω k ,d ) to negative ( ω k ,p >ω k ,d ), and noticing the sign of B , which can be numerically evaluated, is typicallynegative, we can conclude that M B < C ± = 0. Whileafter the band inversion, we have M B > C ± = ±
1, indicating the topologicaltransition from trival to non-trival band structure has taken place.
4. Topological Edge States
To confirm the existence of topologically protected edge states for flexural waves inour system, we consider a ribbon-shaped supercell, constructed by 16 unit cells in s direction (8 topologically trival and 8 non-trival unit cells are arranged adjacentlyalong the s direction, and in s direction the system is still assumed to be periodic.Figure 3 (a) shows the projected band structure of the supercell in Γ K directioncalculated from Mindlin’s plate model. For pure lattice structure I (the topologicaltrival lattice) or structure II (the topological non-trival lattice), there exists a completeband gap and the two gaps share a common frequency range [6318Hz , M and state M ). We plot the square of the amplitude | w ( x, y ) | (representing the vibrationenergy) in figure 3 (b), which unambiguously demonstrates that M and M statesare localized at the lattice-interface since the deformation amplitude decreases rapidlywith the distance away from the lattice interface. Figure 3 (c) supports a magnifiedview for two adjacent unit cells at the interface. We plot the real parts Re( w ), Figure 3. (a)
Projected band structure of a ribbon-shaped supercell. The blueand red lines represent edge states, while the gray dot-solid lines represent bulkstates. The green dots labelled as M and M are two specified edge states,with negative and positive group velocities, respectively. (b) shows the energydistributions for eigenmodes of states M and M . It is evidently that the energyis localized at the interface between the topological trival and non-trival lattices. (c) shows the real parts, imaginary parts and the arguments for states M and M near the interface with magnified views. In the real-part plots, we also presentthe averaged time harmonic Poynting vectors, which represent the energy flowsfor both states. Energy flow for M is anticlockwise, while energy flow for M isclockwise, as shown by the thick blue and red circular arrows. imaginary parts Im( w ) and the arguments Arg( w ) for states M and M , and find thatRe( w M ) = Re( w M ) , Im( w M ) = − Im( w M ) and Arg( w M ) = − Arg( w M ). Thisfinding, especially the argument information, gives us guidance about the selectivelyexcitation of a one-way edge mode with multiple sources, as we will discuss in the nextsection. We also plot the time-averaged Poynting vectors of two adjacent unit cell atthe interface by black arrows. The power flow of the center regions of each unit cellis anti-clockwise for state M , and clockwise for state M , unveiling the pseudo spinup and down characteristics of each state. When we introduce complex number inelastic wave propagation problems, physical quantities like displacements, velocitiesand accelerations, should take only the real (or imaginary) part. Since we observedRe( w M ) = Re( w M ) , Im( w M ) = − Im( w M ), and do not forget eigen-vectors allowa constant scale ratio difference, so whichever part (real or imaginary) we take, weconclude that the eigenvector of state M and M are the “same”. But for physicalquantities like energy and power, we have to take both the real and imaginary partsinto consideration [32]. So states M and M possessing the same real part andopposite imaginary part is the reason why the Poynting vectors (energy flow) forthese two states are pointing to the opposite directions.
5. Robust One-way Edge Wave Propagation
Full wave simulations for flexural waves propagation in a finite lattice (15 × F i = F exp( iϕ i ) are shown in theright panel of figure 4 (a). Note that the time harmonic term exp( − ωt ) is omittedhere, and all the forces possess the same amplitude F but different phases, ϕ = ϕ = 0 , ϕ = ± π/ , ϕ = ∓ π/ ϕ = ϕ = 0 , ϕ = π/ , ϕ = − π/
2, which isconsistent with the argument of state M , as we can see in figure 3 (c). The simulatedwave field With excitation frequency f = 6500Hz, which is exactly the frequency forstate M , is plotted in figure 4 (b). The wave travels only in a single direction withpositive group velocity, being consistent with state M . In this case we successfullyexcited only one-way edge mode. Similarly, when taking the lower sign in equation(20), we would expect another one-way edge mode with negative group velocity beexcited at frequency of 6500Hz. In other words, edge state M is selective excited,which is confirmed by the wave propagation simulation shown in 4 (c).0 Figure 4. Unidirectional propagation of topologically protected edgestates. (a)
Schematic view of the present wave guide. Two types of latticesare separated by a “Z” shape domain wall. Multi-line-excitations are utilizedto selectively excite a particular edge state, with F = F = F , F = F exp( ± iπ/ , F = F exp( ∓ iπ/ (b) up going edge state (positive groupvelocity) and (c) down going edge state (negative group velocity) are selectivelyexcited, by taking the upper/lower sign in the expressions of F and F . The C v symmetry protected flexural wave propagation of our topological waveguide is studied by intentionally introducing different defects. For comparison,wave propagation in conventional wave guides with the same defects is also studied.Topological wave guide studied here is constructed by the topological trival lattice(structure I) and non-trival lattice (structure II), as addressed in section 5.1.Conventional wave guide is realized by simply removing a set of holes along a desiredpath in pure lattice I. The incident waves are generated by a line-excitation fromthe left side of the sample, and the force amplitude along the excitation line is setto be Gaussian distribution, which can be expressed as F = F exp (cid:16) − ( y − y ) b (cid:17) .Here y determines the wave-beam center, and parameter b determines the widthof the wave-beam. The first example is a straight wave guide with additional pointdefects, as figure 5 (a) shows. For topological wave guide, the normally incident wavescan travel through the entire sample, and no backscattering is observed obviouslyat the point defect. As for the conventional wave guide, strong backscattering areobserved so that no output waves can be detected at the right port. The secondexample studies the wave propagating along a path with sharp bends for both typesof structures. Simulation results in figure 5 (b) show that waves in topological lattice1 Figure 5. Wave field of topological (left) and conventional (right) waveguides with different defects.
Excitation frequency is chosen to be 6500Hz. are backscattering free against sharp bends while waves in conventional wave guidesare not.
6. Conclusions
In conclusion, we have proposed a perforated phononic plate and studied the pseudo-spin states for flexural waves propagation based on Mindlin’s plate theory. A doubleDirac cone is observed at Γ by carefully adjusting the size of perforated holes suchthat all the holes are uniform, which can be explained by the zone-folding mechanism.When the radii are perturbed in different directions, the double Dirac cone opensbecause the translational symmetry is broken. We have shown that a topologicalphase transition takes place due to the band inversion by perturbing the radii ofthe holes while preserving the C v point group symmetry. An effective Hamiltonian isdeveloped to describe the topological properties of the band structures and to calculatetopological invariants, i.e., the spin Chern number. Zero/non-zero Chern numbers areobtained before/after the band inversion, which again confirms that the topologicaltransition has taken place. In addition, the projected dispersion of a ribbon-shapedfinite supercell with topologically trival and non-trival cells arranged adjacently isinvestigated. Result demonstrates the existence of a pair of topological edge stateslocated within the bulk gap range, which support two pseudo-spin states. With fullwave simulations, we demonstrate the robust unidirectional transport of topologicallyprotected edge states for flexural waves. Observation in this paper paves a new wayfor the studies and practical applications in wave guiding, vibration isolating, wavefiltering and related fields. Further studies including the experimental verification ofedge wave propagation in the proposed metamaterial plate will be reported in authors’future publications.2 Acknowledgements
We acknowledge Rajesh Chaunsali (University of Washington) gratefully for fruitfuldiscussions.
Appendix A. Applying Bloch Condition in FEM
For the dispersion calculation of periodic systems, we should apply the Bloch-Floquetconditions on the periodic edge pairs of the unit cell. As figure A1 shows, all the finiteelement nodes are grouped into 9 node sets. Node set 1 includes all the nodes locatewithin the inner domain, and node sets 2 – 5 the nodes on the four edges (but notincluding the endpoints of the edges). The rest 4 nodes at the unit cell corners arenode sets 6 – 9. We would like to note that the current mesh shown in Fig. 1A (a)may be a little coarse that the results may not meet the convergence condition, thusa finer mesh as Fig. 1A (b) shows is required to check the convergence of FE analysis.The Bloch-Floquet conditions states that u = exp( i k · a ) u , u = exp( i k · a ) u u = exp( i k · a ) u , u = exp( i k · a ) u u = exp( i k · ( a + a )) u (A.1)in which k is wave vector, a , a are the two lattice vectors. From equation (A.1) wecan reduce the displacement vector U which consists of all of the nodal displacementsto a “reduced” vector U whose components are the nodal displacements of node sets1, 2, 4 and 6, expressed as U = PU (A.2)where U = [ u , u , u , u , u , u , u , u , u ] (cid:62) , U = [ u , u , u , u ] (cid:62) , and matrix P iscalled the transformation matrix, whose components can be obtained from equation(A.1). For the dispersion analysis of periodic structures, by applying Bloch-Floquetconditions equation (A.1) or equation (A.2), one can obtain the eigenvalue problemequation (13). E dg e nod e s e t E dg e nod e s e t E d g e n o d e s e t E d g e n o d e s e t N od e Node 7 N od e Node 9 (a) (b) a a Figure A1.
FE meshes of the unit cell. (a) A coarse mesh to better illustratethe 9 node sets: node sets 2 – 9 are nodes on the boundaries and node set 1 arethose locate in the inner domain. (b) A finer mesh is utilized in the FE analysisto ensure the convergence of results. Appendix B. Comparison Between Plate Model and Solid Model
In this section, we will check the accuracy of the proposed FEM for flexural waveanalysis. Both 2D model based on Mindlin’s plate theory and 3D model based onelastic dynamic equation ( λ + µ ) ∇∇ · u + µ ∇ u = ρ ¨ u (B.1)are utilized to analyse the dispersion of the present plate. The blue solid lines in figureB1 represent the dispersion curves obtained from Mindlin’s plate theory model, andthe green circles represent dispersion curves of elastic solid model. By comparing thesetwo set of results we can find that the plate model can accurately describe the flexuralwave modes for a thick plate and efficiently eliminate other types of wave modes (SHmodes and in-plane extensional modes) which are not our research interests in thispaper. M K Γ M4.55.05.56.06.57.07.58.0 Wave Vector, k F r equen cy ( k H z ) Solid ModelPlate Model in planeextentional modes (a)
M K Γ M4.55.05.56.06.57.07.58.0 Wave Vector, k F r equen cy ( k H z ) Solid ModelPlate Model in planeextentional modes (b)
M K Γ M4.55.05.56.06.57.07.58.0 Wave Vector, k F r equen cy ( k H z ) Solid ModelPlate Model in planeextentional modes (c)
Figure B1. Validity of our FEM for flexural wave modes analysis .Dispersion curves obtained from Mindlin’s plate theory (blue lines) matchwell with those obtained from elastodynamics equation (green circles). (a)-(c)represent the band structures for r /r = 0 . , . . Appendix C. Effective Hamiltonian near The Double Dirac Cone
We employ the degenerate second-order k · p perturbation theory [35] to calculate theeffective Hamiltonian around Γ point. For a fixed plate structure, for example, r = r or r > r or r < r , the eigenvalues ω k ,n and eigenvectors U k ,n at k = 0 arecalculated beforehand. Small wave vector k (around k ) is treated as a perturbation.Our goal is to find the effective Hamiltonian when wave vector takes k , with only ω k ,n and U k ,n are already known. According to the degenerate second-order k · p perturbation theory, the matrix components of the effective Hamiltonian at k point isgiven by (see page 316–319 in Ref. [35] for more details about the derivation) H eff mn = H (cid:48) mn + ∞ (cid:88) α H (cid:48) mα H (cid:48) αn ω k ,m ( n ) − ω k ,α (C.1)in which subscripts m, n denote the degenerate states that we are interestedin, and subscript α denotes all of the rest states. We recall that H (cid:48) mn = (cid:104) U k ,m | k x K x + k y K y | U k ,n (cid:105) is the first order perturbation term and K x and K y are constant matrices.In this section, we consider the case of r = r . Since a double Dirac cone (four-fold degenerate states) emerges, m, n should be taken over the corresponding 4 states4indexes and α the rest. We should also note that the other states ( α ) are far awayfrom the double Dirac cone, so | ω k ,m ( n ) − ω k ,α | is very large that the summationterm over α can be neglected (only when r = r , but not for the case r (cid:54) = r ). Sowe have H eff mn = H (cid:48) mn = (cid:104) U k ,m | k x K x + k y K y | U k ,n (cid:105) (C.2)Eq.(C.2) gives the expressions of each matrix components of H eff . By numericalcalculation, we find that H eff can be expressed as H eff = 1 . − A − B CA − C − BB C A − C B − A (C.3)where A = (0 . k x − . k y ) i, B = (0 . k x + 2 . k y ) i, C = (2 . k x − . k y ) i .Eigenvalues of equation (C.3) can be solved explicitly, ε , = − i (cid:112) A + B + C ≈ +2 . kε , = + i (cid:112) A + B + C ≈ − . k (C.4)in which ε = ω − ω . so equation (C.4) can be rewritten as ω − ω ≈ ω ∆ ω = ± . k −
0) (C.5)so the slopes of the dispersion curves near the double Dirac cone are given by∆ ω ∆ k = ± . ω (C.6)We note that each root is two-fold, indicating the two cones are uniform. AroundΓ point, dispersions obtained from our k · p method show excellent agreement withthose from FEM, as shown in figure C1. Γ K Γ Γ M Wave Vector, k F r equen cy ( k H z ) Figure C1.
Double Dirac Cone near Γ ( r = r = 0 . a ). Solid lines arepredicted from the effective Hamiltonian and the green circles are calculated byFEM. References [1] Klaus von Klitzing. The quantized hall effect.
Rev. Mod. Phys. , 58:519–531, Jul 1986.[2] M. Z. Hasan and C. L. Kane. Colloquium.
Rev. Mod. Phys. , 82:3045–3067, Nov 2010.[3] Xiao-Liang Qi and Shou-Cheng Zhang. Topological insulators and superconductors.
Rev. Mod.Phys. , 83:1057–1110, Oct 2011.[4] F. D. M. Haldane and S. Raghu. Possible realization of directional optical waveguides in photoniccrystals with broken time-reversal symmetry.
Phys. Rev. Lett. , 100:013904, Jan 2008.[5] Ling Lu, John D Joannopoulos, and Marin Soljaˇci´c. Topological photonics.
Nat. Photon. ,8(11):821–829, 2014.[6] Long-Hua Wu and Xiao Hu. Scheme for achieving a topological photonic crystal by usingdielectric material.
Phys. Rev. Lett. , 114:223901, Jun 2015.[7] Alexander B Khanikaev, Romain Fleury, S Hossein Mousavi, and Andrea Al`u. Topologicallyrobust sound propagation in an angular-momentum-biased graphene-like resonator lattice.
Nat. Commun. , 6, 2015.[8] Zhaoju Yang, Fei Gao, Xihang Shi, Xiao Lin, Zhen Gao, Yidong Chong, and Baile Zhang.Topological acoustics.
Phys. Rev. Lett. , 114:114301, Mar 2015.[9] Ze-Guo Chen and Ying Wu. Tunable topological phononic crystals.
Phys. Rev. Applied ,5:054021, May 2016.[10] Cheng He, Xu Ni, Hao Ge, Xiao-Chen Sun, Yan-Bin Chen, Ming-Hui Lu, Xiao-Ping Liu, andYan-Feng Chen. Acoustic topological insulator and robust one-way sound transport.
Nat.Phys. , 12(12):1124–1129, 2016.[11] Jun Mei, Zeguo Chen, and Ying Wu. Pseudo-time-reversal symmetry and topological edge statesin two-dimensional acoustic crystals.
Sci. Rep. , 6, 2016.[12] Bai-Zhan Xia, Ting-Ting Liu, Guo-Liang Huang, Hong-Qing Dai, Jun-Rui Jiao, Xian-GuoZang, De-Jie Yu, Sheng-Jie Zheng, and Jian Liu. Topological phononic insulator with robustpseudospin-dependent transport.
Phys. Rev. B , 96:094106, Sep 2017.[13] Simon Yves, Romain Fleury, Fabrice Lemoult, Mathias Fink, and Geoffroy Lerosey. Topologicalacoustic polaritons: robust sound manipulation at the subwavelength scale.
New J. Phys. ,19(7):075003, 2017.[14] Jiuyang Lu, Chunyin Qiu, Manzhu Ke, and Zhengyou Liu. Valley vortex states in sonic crystals.
Phys. Rev. Lett. , 116:093901, Feb 2016.[15] Romain Fleury, Alexander B Khanikaev, and Andrea Al`u. Floquet topological insulators forsound.
Nat. Commun. , 7, 2016.[16] Jiuyang Lu, Chunyin Qiu, Liping Ye, Xiying Fan, Manzhu Ke, Fan Zhang, and Zhengyou Liu.Observation of topological valley transport of sound in sonic crystals.
Nat. Phys. , 13(4):369–374, 2017.[17] Zhiwang Zhang, Qi Wei, Ying Cheng, Ting Zhang, Dajian Wu, and Xiaojun Liu. Topologicalcreation of acoustic pseudospin multipoles in a flow-free symmetry-broken metamateriallattice.
Phys. Rev. Lett. , 118:084303, Feb 2017.[18] Xiang Ni, Maxim A Gorlach, Andrea Alu, and Alexander B Khanikaev. Topological edge statesin acoustic kagome lattices.
New J. Phys. , 19(5):055002, 2017.[19] Roman S¨usstrunk and Sebastian D. Huber. Observation of phononic helical edge states in amechanical topological insulator.
Science , 349(6243):47–50, 2015.[20] Pai Wang, Ling Lu, and Katia Bertoldi. Topological phononic crystals with one-way elasticedge waves.
Phys. Rev. Lett. , 115:104302, Sep 2015.[21] Lisa M. Nash, Dustin Kleckner, Alismari Read, Vincenzo Vitelli, Ari M. Turner, and WilliamT. M. Irvine. Topological mechanics of gyroscopic metamaterials.
Proc. Natl. Acad. Sci.U.S.A. , 112(47):14495–14500, 2015.[22] Sebastian D Huber. Topological mechanics.
Nat. Phys. , 12(7):621–623, 2016.[23] Daniel Torrent, Didier Mayou, and Jos´e S´anchez-Dehesa. Elastic analog of graphene: Diraccones and edge states for flexural waves in thin plates.
Phys. Rev. B , 87:115143, Mar 2013.[24] S Hossein Mousavi, Alexander B Khanikaev, and Zheng Wang. Topologically protected elasticwaves in phononic metamaterials.
Nat. Commun. , 6, 2015.[25] M. Miniaci, R. K. Pal, B. Morvan, and M. Ruzzene. Observation of topologically protectedhelical edge modes in Kagome elastic plates.
ArXiv e-prints , October 2017.[26] S.-Y. Yu, C. He, Z. Wang, F.-K. Liu, X.-C. Sun, Z. Li, M.-H. Lu, X.-P. Liu, and Y.-F. Chen. AMonolithic Topologically Protected Phononic Circuit.
ArXiv e-prints , July 2017.[27] Rajesh Chaunsali, Chun-Wei Chen, and Jinkyu Yang. Subwavelength and directional control offlexural waves in zone-folding induced topological plates.
Phys. Rev. B , 97:054307, Feb 2018.[28] A. Foehr, O. R. Bilal, S. D. Huber, and C. Daraio. Spiral-based phononic plates: From wave beaming to topological insulators. ArXiv e-prints , December 2017.[29] Raj Kumar Pal and Massimo Ruzzene. Edge waves in plates with resonators: an elastic analogueof the quantum valley hall effect.
New J. Phys. , 19(2):025001, 2017.[30] H. Zhu, T.-W. Liu, and F. Semperlotti. Design and Experimental Observation of Valley-HallEdge States in Diatomic-Graphene-like Elastic Waveguides.
ArXiv e-prints , December 2017.[31] Ying Wu, Rajesh Chaunsali, Hiromi Yasuda, Kaiping Yu, and Jinkyu Yang. Dial-in topologicalmetamaterials based on bistable stewart platform.
Sci. Rep. , 8(1):112, 2018.[32] Jan Achenbach.
Wave propagation in elastic solids , volume 16. Elsevier, 2012.[33]
Analysis of Mindlin plates , pages 161–201. Springer Netherlands, Dordrecht, 2009.[34] Ying Wu, Kaiping Yu, Linyun Yang, Rui Zhao, Xiaotian Shi, and Kuo Tian. Effect of thermalstresses on frequency band structures of elastic metamaterial plates.
J. Sound Vib. , 413:101– 119, 2018.[35] Mildred S Dresselhaus, Gene Dresselhaus, and Ado Jorio.
Group theory: application to thephysics of condensed matter . Springer Science & Business Media, 2007.[36] B Andrei Bernevig, Taylor L Hughes, and Shou-Cheng Zhang. Quantum spin hall effect andtopological phase transition in hgte quantum wells.