Valley Hall phases in Kagome lattices
Natalia Lera, Daniel Torrent, Pablo San-Jose, Johan Christensen, Jose Vicente Alvarez
VValley Hall phases in Kagome lattices
Natalia Lera, Daniel Torrent, P. San-Jose, J. Christensen, and J.V. Alvarez Departamento de F´ısica de la Materia Condensada,Universidad Aut´onoma de Madrid, Madrid 28049, Spain,Condensed Matter Physics Center (IFIMAC) and Instituto Nicol´as Cabrera GROC, Institut de Noves Tecnologies de la Imatge (INIT), Universitat Jaume I, Castellon 12071, (Spain) Materials Science Factory, Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC),Sor Juana Ines de la Cruz 3, 28049 Madrid, Spain Department of Physics, Universidad Carlos III de Madrid, Leganes 28916, Madrid, Spain
We report the finding of the analogous valley Hall effect in phononic systems arising from mirrorsymmetry breaking, in addition to spatial inversion symmetry breaking. We study topological phasesof plates and spring-mass models in Kagome and modified Kagome arrangements. By breaking theinversion symmetry it is well known that a defined valley Chern number arises. We also show thateffectively, breaking the mirror symmetry leads to the same topological invariant. Based on thebulk-edge correspondence principle, protected edge states appear at interfaces between two latticesof different valley Chern numbers. By means a plane wave expansion method and the multiplescattering theory for periodic and finite systems respectively, we computed the Berry curvature,the band inversion, mode shapes and edge modes in plate systems. We also find that appropriatemulti-point excitations in finite system gives rise to propagating waves along a one-way path only.
I. INTRODUCTION
The unusual properties of fabricated metamaterials originate from their designed patterns and geometry as opposedto their chemical composition. Specifically, when created with periodic structures, the study of wave propagation canbe treated similar to electrons in periodic potentials . In this way, topological properties studied in electronic bandstructures can be transferred to classical metamaterials. Inspired by topological electronic systems, the search forprotected modes in classical wave phenomena has been active in recent years in areas such as photonics , acoustics and elastic media . The bulk-boundary correspondence principle has been proved to hold also in these areas byshowing how topological protected waves arise at the edge of systems containing topologically inequivalent phases. Inparticular, mechanical metamaterials present several advantages: 1) the flexibility to create patterns and to modifyband structures in metamaterials is much richer than in real solids . 2.) In electronic systems topological featuresare easier to detect when they occur close to the Fermi energy, which is hard to shift and control. On the otherhand, mechanical systems can be excited in a wide range of frequencies, and the excitation can be easily tuned to thefrequency of the topological mode.We consider mechanical metamaterials with time reversal symmetry, establishing analogy with the quantum valleyHall effect . This approach has been successfully achieved in spring-mass models and plate topology as wellas in photonics or acoustics by breaking the spatial inversion symmetry. The existence of topologicalmodes have been shown experimentally , along with unusual properties in the absence of backscattering . Incontinuous systems like plates, wave guiding through edge modes could have applications for mechanically isolatingstructures or transferring energy and information through elastic waves.In this article we focus on the Kagome lattice, which has a graphene-like structure with degenerate Dirac cones atinequivalent points of the Brillouin Zone. Recent interest in metamaterials based on Kagome arrangement suggestfuture applications . The wide range of crystalline symmetries and the underlying C symmetry of this sys-tem provides a playground to test mechanical topology as well as distinguishing basic features that are relevant totopological mechanics.We study discrete spring-mass models in the linear regime in addition to continuum systems such as plates. Theformer systems allow analytic computations which capture the essentials of topology in easy models with couplingsbetween few neighbors. In plates long ranged waves need to be taken into account. The understanding of topologicalmodes could lead to relevant engineering applications, in particular, efficient and controlled wave guiding. Plateswill be described in the linear regime by Kirchhoff-Love theory. To endow the plate with a crystalline structure, weattach a lattice of resonators on top. Modifications of the unit cell properties might open gaps in the phononic bandstructure with non-trivial topology. Remarkably, the methodology used in this paper to describe flexural waves inplates is not based on commercial software but on the Multiple Scattering Theory (MST) developed in Ref. .The structure of the paper is as follows, in sec II we describe briefly the methodology for studying flexural waves inplates. In section III, we describe the distorted Kagome lattice, its symmetries and the parameter space used in thispaper. In section IV, topology arising from spatial inversion symmetry is deduced from the spring-mass model andexplained via plate physics, we employ ribbons to create topological protected edge states and design finite systems a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec with interesting properties, like one-way wave propagation. In section V, we study the effects of mirror symmetrybreaking in a Kagome lattice. In section VI, we conclude this paper. II. PLATE PHYSICS AND METHODOLOGY
In this section, to present the system and derive the notation, we briefly introduce the classical theory of flexuralwaves for thin plates and describe the methodology, following the approach taken by Torrent et al. and Chaunsali et al. . We consider a thin plate coupled to a lattice of resonators. The equation of motion for the deformation field, w is a fourth order derivative in real space and we look for solutions harmonic in time: w ( (cid:126)r, t ) = w ( (cid:126)r ) e iωt . (cid:0) D ∇ − ω ρh (cid:1) w ( (cid:126)r ) = − (cid:88) (cid:126)R α κ (cid:126)R α (cid:16) w ( (cid:126)R α ) − z ( (cid:126)Rκ α ) (cid:17) δ ( (cid:126)r − (cid:126)R α ) (1)where D = Eh − ν ) is the plate stiffness, ρ is the volume mass density of the plate, h is thickness and the sum runsover all resonator sites (cid:126)R α within the unit cell. Resonators masses and spring constants are respectively m α and κ α and their displacements are z ( (cid:126)R α ), in the direction perpendicular to the plate. The equation for each resonator is, ω m α z ( (cid:126)R α ) = − κ (cid:126)R α (cid:16) w ( (cid:126)R α ) − z ( (cid:126)R α ) (cid:17) (2) A. Plane Wave Expansion
In the Plane Wave Expansion method (PWE), the lattice is infinite in two dimensions and the displacement fieldcan be written in terms of Bloch waves, w ( (cid:126)r ) = (cid:88) (cid:126)G W ( (cid:126)G ) e − i ( (cid:126)G + (cid:126)k ) · (cid:126)r (3)where (cid:126)G = n (cid:126)g + n (cid:126)g are the reciprocal lattice vectors, n , are integers and (cid:126)g j are the basis of vectors fulfilling (cid:126)a i · (cid:126)g j = 2 πδ ij , with (cid:126)a i being the lattice vectors. The result is either a search for zeros of a complex function asdescribed in Ref. or a generalized eigenvalue problem as described in Ref. . For completeness, we highlight somesteps of the derivation. Method 1:
Substituting the resonator equation, Eq. 2 into the plate equation, Eq. 1, we get (cid:18) ∇ − ω ρhD (cid:19) w ( (cid:126)r ) = − (cid:88) (cid:126)R α m α D ω α ω ω α − ω w ( (cid:126)R α ) δ ( (cid:126)r − (cid:126)R α ) (4)where ω α ( ω ) = κ α /m α and t α = m α D ω α ω ω α − ω . Due to the system’s periodicity we omit the (cid:126)R dependence in masses andspring constants. Substituting the Bloch Ansatz Eq. 3 into the previous equation, deriving each independent term inthe Fourier summation and integrating over the unit cell we obtain, (cid:18)(cid:12)(cid:12)(cid:12) (cid:126)k + (cid:126)G (cid:12)(cid:12)(cid:12) − ω ρhD (cid:19) W (cid:126)G = (cid:88) (cid:126)G (cid:48) ,α t α A c e i ( (cid:126)G (cid:48) − (cid:126)G ) · (cid:126)R α W (cid:126)G (cid:48) (5)where a is the lattice parameter and A c is the area of the unit cell. We have used the following identities, (cid:82) UC e − i ( (cid:126)G (cid:48) − (cid:126)G ) · (cid:126)r d(cid:126)r = A c δ ( (cid:126)G (cid:48) − (cid:126)G ); (cid:82) UC f ( (cid:126)r ) δ ( (cid:126)r − (cid:126)R α ) d(cid:126)r = f ( (cid:126)R α ) (6)Now, we write the expected solution expanded on a the Fourier basis, W β = (cid:88) (cid:126)G (cid:48) W (cid:126)G (cid:48) e i (cid:126)G (cid:48) · (cid:126)R β (7)and substitute W (cid:126)G from Eq. 5, W β = (cid:88) (cid:126)G (cid:12)(cid:12)(cid:12) (cid:126)k + (cid:126)G (cid:12)(cid:12)(cid:12) − ω ρhD A c (cid:88) α e i (cid:126)G · ( (cid:126)R α − (cid:126)R β ) t α W α . (8)Therefore a set of N equations with N unknowns can be written, where N is the number of resonators per unit cell.We find solutions of this system as the zeros of the determinant of the following matrix, A αβ ( (cid:126)k ) = δ α,β − γ β Ω a − Ω / Ω α (cid:88) (cid:126)G e − i (cid:126)G · ( (cid:126)R α − (cid:126)R β ) (cid:12)(cid:12)(cid:12) (cid:126)k + (cid:126)G (cid:12)(cid:12)(cid:12) a − Ω a (9)where we have introduced the dimensionless variables Ω = ω ρa h/D and γ α = m α ρa h . We evaluate for each (cid:126)k anddeduce its Ω( (cid:126)k ) solutions. The null space of the A matrix correspond to mode shapes at the resonator points. Method 2:
We substitute Bloch waves from Eq. 3 in the plate Eq. 1. Equating for each mode and integrating overthe unit cell we get, A c (cid:18) D (cid:12)(cid:12)(cid:12) (cid:126)k + (cid:126)G (cid:12)(cid:12)(cid:12) − ω ρh (cid:19) W (cid:126)G = (cid:88) α κ α z ( (cid:126)R α ) − (cid:88) (cid:126)G (cid:48) W (cid:126)G (cid:48) e − i ( (cid:126)G (cid:48) + (cid:126)k ) · (cid:126)R α e i ( (cid:126)G + (cid:126)k ) · (cid:126)R α (10)Using Bloch’s theorem for the resonators we can refer all resonator displacements to the ones of the one unit cell, z ( (cid:126)R α ) = z ( (cid:126)R α ) e − i(cid:126)k · (cid:126)R α . We substitute in previous equation, (cid:18)(cid:12)(cid:12)(cid:12) (cid:126)k + (cid:126)G (cid:12)(cid:12)(cid:12) a − Ω (cid:19) W (cid:126)G = (cid:88) α γ α Ω α e i (cid:126)G · (cid:126)R α z ( (cid:126)R α ) − (cid:88) (cid:126)G (cid:48) W (cid:126)G (cid:48) e − i (cid:126)G (cid:48) · (cid:126)R α (11)and in resonator equation Eq. 2, − Ω z ( (cid:126)R α ) = Ω α (cid:88) (cid:126)G W (cid:126)G e − i (cid:126)G · (cid:126)R α − z ( (cid:126)R α ) (12)Where we have used the same dimensionless variables Ω and γ than in method 1. Now Eq. 11 and 12 are rewritten inmatrix form of dimension N G + N where N G is the number of reciprocal vectors taken for the computation (calculationsin this paper are made with N G = 49) and N is the number of resonators per unit cell. (cid:18) P P P P (cid:19) (cid:18) W (cid:126)G z ( (cid:126)R ,α ) (cid:19) = Ω (cid:18) Q Q (cid:19) (cid:18) W (cid:126)G z ( (cid:126)R ,α ) (cid:19) (13)where P ,ij = a (cid:12)(cid:12)(cid:12) (cid:126)k + (cid:126)G i (cid:12)(cid:12)(cid:12) δ i,j + (cid:80) α γ α Ω α e i ( (cid:126)G j − (cid:126)G i ) · (cid:126)R α P ,iα = − γ α Ω α e i (cid:126)G i · (cid:126)R α = P ∗ ,αi P ,αβ = γ α Ω α δ α,β Q ,ij = δ i,j Q ,αβ = γ α δ α,β . (14)In. Eq (14) we use i, j indices for the N G reciprocal vectors and α, β for the N resonators of the unit cell.The generalized eigenvalue problem gives us the band structure, Ω( (cid:126)k ), and the mode shape by substituting W (cid:126)G into Eq. 3. B. Edge states in ribbons
We consider ribbons of resonators arranged periodically in the (cid:126)r -direction. However, the plate is still infinite, sothe unit cell in direction (cid:126)r is infinite, where (cid:126)r i form a basis in 2D. The unit cell is infinite in size but with finitenumber of resonators present in the supercell, see Fig. 1. Unlike electronic systems where wave functions decayexponentially in space, flexural waves decay slowly in the plate and an infinite large unit cell will account for longrange waves along the (cid:126)r direction. The discrete summation over n (cid:126)g in Eq. 3 transforms into an integral.1 A c (cid:88) G → πa (cid:90) ∞−∞ dg (15)FIG. 1: (Color online) Schematic representation of a ribbon in an infinite plate. r and r are a basis of lattice. Thetwo red parallel lines delimit one supercell, the supercell is infinite in size. The unit cell is presented with twodifferent topological phases as we will see later in the text.Applying this transformation to Eq. 9, A α,β matrix simplifies to depend only on k . The governing equations aredescribed in Ref. . Our main interest creating ribbons consist of studying boundary states between two phases. Theinterface is contained in the supercell. Bands are computed from the zeros of the A ( (cid:126)k ) matrix determinant and itsnull space contains the eigenmodes, i.e. the w ( (cid:126)R α ) weight over the supercell resonators. C. Multiple Scattering Method
For finite clusters in an infinite plate we use Multiple Scattering Theory (MST). The governing equations are Eq.1-2 where the number of (cid:126)R α is finite. The Green’s function of the plate equation without resonators, G ( (cid:126)r ), is usedas a basis to expand the solution of the resulting wave. A system of self-consistent equations lead to the solution ofthe field w ( (cid:126)r ) under some harmonic incident field ψ ( (cid:126)r, t ) = ψ ( (cid:126)r ) e iωt + ϕ w ( (cid:126)r ) = ψ ( (cid:126)r ) + (cid:88) α T α ψ e ( (cid:126)R α ) G ( (cid:126)r − (cid:126)R α ) (16) ψ e is the incident field at scatterer α which allows to deduce the value of T α = t α − it α / (8 k ) . ψ e ( (cid:126)R α ) can be solvedfrom the system of equations, ψ e ( (cid:126)R α ) = ψ ( (cid:126)R α ) + (cid:88) β (1 − δ α,β ) T β G ( (cid:126)R α − (cid:126)R β ) ψ e ( (cid:126)R β ) (17)We compute the resulting field w ( (cid:126)r ) by substituting the solution of ψ e back into Eq. 16. The incident field is theexternal excitation of the system and is taken as a point source ψ ( (cid:126)R α ) = G ( (cid:126)R α − (cid:126)x ), we also consider multipointdephased excitations ψ ( (cid:126)R α ) = (cid:80) j G ( (cid:126)R α − (cid:126)x j ) e iϕ j and solutions without input field ψ ( (cid:126)R α ) = 0 that we call naturalexcitations of the system. III. KAGOME LATTICE, DISTORTIONS AND SYMMETRIES
The standard Kagome lattice consists of three sets of straight parallel lines intersecting at lattice sites as shown inFig. 2. This figure also shows the unit cell chosen in this article as a parallelogram with lattice vectors (cid:126)a = a (1 , (cid:126)a = a (cos( π ) , sin( π )) (18)The normalized masses and resonator frequencies are γ α = 10 and Ω α = 4 π respectively for the three resonators ofthe unit cell. The lattice sites in the unit cell form an equilateral triangle of side a/
2. In this paper we considerdistortions of the standard Kagome lattice with two parameters: f that controls the size of the triangle respect tothe lattice parameter which will remain unchanged, and α the rotation angle of the equilateral triangle respect to itscenter. See Fig. 2.The positions of the three sites in the unit cell are, (cid:126)R n = f · b (cos (Θ n + α ) , sin (Θ n + α )) (19)FIG. 2: (Color online) a) Undistorted Kagome lattice. The unit cell is indicated in a green box of side a . The unitcell contains three resonators forming equilateral triangles. b) parameters used in the paper for deformations ofKagome lattice and its effect in the unit cell. They are characterized by an angle α and a uniform expansion factor f . c) Brillouin ZoneFIG. 3: (Color online) Real space arrangement of resonators for several deformation parameters. The unit cell ishighlighted. Notice a) and c) look similar but the chosen cell is different. Notice the breaking of spatial inversionsymmetry is the three cases.where b = a √ , Θ n = n π − π and n labels the lattice sites n = { , , } . The undistorted Kagome lattice is definedfor f = 1 and α = 0.Kagome lattice in our parameter space have several symmetries. For a constant f , there are three equivalent latticesfor every α corresponding to { α, α + π , α − π } , meaning all systems in this parameter space have C symmetry.For a given angle and f <
1, the lattice with f (cid:48) = 2 − f is equivalent as well. However, lattices with f < > f > f = 1 and α = 0 have C symmetry, inversion symmetry both with centers in the middle of hexagons, C symmetry with center in the middleof triangles and three mirror symmetries. The elastic systems have time reversal symmetry as well. The interrelationof all these symmetries give many interesting features and we will explore some of them.Due to the symmetries of the lattice, some qualitative band features are independent of the system (springs orplates). For instance, the gap closings at K point of the Brillouin Zone will be relevant through the article and theyare represented in Fig. 4 in parameter space. Each red and dashed line correspond to a gap closing in cone-like shape.At momentum K there are Dirac points, and opening the gap gives rise to interesting phenomena.Spring-mass model approach is being used in Kagome lattice to explain band inversion topology and they constitutea first step towards topology in plates. IV. INVERSION SYMMETRY BREAKING AND TOPOLOGYA. Spring-mass model
We design a spring-mass model where masses are located at sites of the Kagome lattice, i.e circles in Fig. 2 andeach blue line connecting neighboring masses are springs. The masses have only one degree of freedom, they movein the direction perpendicular to the plane. The three springs inside the unit cell have spring constant κ and theFIG. 4: (Color online) Gap closings in parameter space at K points. Red full lines are the closing of the first gap.Dashed black lines are the closings of the second gap. The second gap is a partial gap in k-space. Topologicaltransitions studied in this paper are marked with a five and four-pointed stars. The driving parameters are f and α respectively and the symmetry breaking is spatial inversion and mirror symmetry respectively.springs connecting neighboring unit cells have constant κ . The equations of motion read, m ¨ u = − κ ( u − u ) − κ ( u − u ) − κ ( u − u e − i(cid:126)k · (cid:126)a ) − κ ( u − u e − i(cid:126)k (cid:126)a ) m ¨ u = − κ ( u − u ) − κ ( u − u ) − κ ( u − u e i(cid:126)k (cid:126)a ) − κ ( u − u e − i(cid:126)k (cid:126)a ) m ¨ u = − κ ( u − u ) − κ ( u − u ) − κ ( u − u e i(cid:126)k (cid:126)a ) − κ ( u − u e i(cid:126)k (cid:126)a ) (20)where (cid:126)a = (cid:126)a − (cid:126)a . Solving the temporal part as a harmonic function u ( t ) = u e iωt and introducing the dimensionless β = κ − κ κ + κ which plays a role analogous to the distortion f of the preceding section, and Ω = 2 mω / ( κ + κ ) theequation of motion reads: − Ω u u u = − e − i(cid:126)k (cid:126)a e − i(cid:126)k (cid:126)a e i(cid:126)k (cid:126)a − e − i(cid:126)k (cid:126)a e i(cid:126)k (cid:126)a e i(cid:126)k (cid:126)a − + β − e − i(cid:126)k (cid:126)a − e − i(cid:126)k (cid:126)a − e i(cid:126)k (cid:126)a − e − i(cid:126)k (cid:126)a − e i(cid:126)k (cid:126)a − e i(cid:126)k (cid:126)a u u u (21)For β = 0, we recover the dispersion relation of the undistorted Kagome lattice (analogous to f = 0) with Diraccones at K and K (cid:48) points of the Brillouin Zone. Two bands cross linearly at Dirac frequency and the third band haslarger energy. For β (cid:54) = 0 the gap opens up at K and K (cid:48) points, gapping the system. Because C is a symmetry ofthe lattice, its eigenvectors are eigenvectors of the system. C rotation center located in the middle of the triangle ofthe unit cell gives the following matrix form for C symmetry,ˆ C = (22)Thus, the eigenvalues are { , e i π , e − i π } and its corresponding eigenvectors, u C = √ , u + C = √ e − i π e i π − , u − C = √ e i π e − i π − (23)These eigenvectors diagonalize the dynamical matrix (which plays the same role than a Hamiltonian) for β (cid:54) = 0 at K and K (cid:48) points. For β = 0, the two states degenerate at K ( K (cid:48) ) point are u C and u + C ( u − C ). For β >
0, i.e κ > κ , u + K = u + C and u − K = u C and reversed for β < u + K = u C and u − K = u + C . Due to the three mirror symmetries, each K point is related to K (cid:48) = − K , and its eigenvectors are the mirror symmetric of u K . Notice that the superindexFIG. 5: (Color online) a) Energy level crossing at K point for model in Eq. 21 as a function of β . b) Representationof M operator in Eq. 32. The arrows indicates how each component transforms, the color indicates different signs.indicates different things depending on the subindex. When the subindex makes reference to K point, the plus andminus signs correspond to the bands above and below the Dirac energy. The subindex C refers to the symmetry andthe plus minus or zero superindex correspond to its eigenvalues.We see there is a crossing of eigenvectors at β = 0 (the gap must close at the transition), see Fig. 5 a). To capturethe basic topology of this system, let’s derive an effective model near each valley. The effective model can be writtenin the basis of the crossing eigenvectors as follows, D K,ij = (cid:104) u iK | H | u jK (cid:105) (24)where i, j = { + , −} , we expand the dynamical matrix near each valley K and K’ the result is, D η = (cid:18) . − β ) v D ( − ηk x + ik y ) v D ( − ηk x − ik y ) 1 . β ) (cid:19) (25)where v D = √ a and η = ± ± K and (cid:126)k is measured from each valley (cid:126)k = ( ± π a + k x , k y ). This is a well known modelfor graphene with a staggered potential . The whole system has time reversal symmetry, one valley is transformedinto the other with a time reversal transformation. However, each valley is independent from one another, since thereare not direct scattering terms coupling them. Separately each valley dynamical matrix effectively behaves as a Cherninsulator with broken time reversal symmetry where β is a symmetry breaking term responsible for the topologicalgap, and analogous to the magnetic fiel in quantum Hall phases. Opposite Chern numbers are computed near eachvalley when β (cid:54) = 0. Since the valleys are disconnected, a well defined valley Chern number arises. The total systemis still time reversal symmetric and therefore total Chern number is zero.Notice that the eigenvalues of the dynamical matrix in Eq. 24-25 are the square of the actual normalized frequenciesΩ as in the Hamiltoninan in Eq. 21. In any case, the eigenvectors (or normal modes) and conclusions about topologyhold.We can compute the subspace generated by the two Dirac crossing vectors, M = | u C (cid:105) (cid:104) u C | − | u + C (cid:105) (cid:104) u + C | = 12 − i √ − − − (26)This matrix corresponds to the gap-opening operator in the low energy model and is proportional to the linear termin the perturbation evaluated at K point and its imaginary part is schematically represented in Fig.5 b). It gives thespatial inversion symmetry breaking term in the full spring-mass model.This result is relevant for plates with attached resonators. The strength of springs is modeled by the distancebetween resonators. In our case, β > κ is stronger and in a plate system is analogous to a contractionof the sites’ distance in the unit cell, i.e, f <
1. In the same way, β < f > B. Plate model and valley Chern number
To reproduce previous results from spring-mass systems, we study plates with a Kagome arrangement of resonatorsand model spring strength with distance between resonators. Fixing α = 0 and varying f around 1 let us model , FIG. 6: (Color online) Band structure in a path of the hexagonal Brillouin Zone for several f -values and α = 0.FIG. 7: (Color online) Berry curvature of the lower band over the first Brillouin zone. Berry curvature is localized at K and K (cid:48) points with different signs for different phases. Blue is negative and yellow is positive.the variation of spring constants within the unit cell ( κ ) respect to the springs connecting different cells ( κ ). Thecorresponding band structures are in Fig. 6. At both sides of the transition the band structure is the same (see theirsimilar spatial distribution in Fig. 3 a) and c)). However, topology encoded in eigenvectors is inverted as we will see.At the transition point f = 1, the two bands form Dirac cones at first order in momentum around K and K (cid:48) . TheDirac energy is Ω D a = 2 . f (cid:54) = 1 spatial inversion symmetry is broken, while the remaining symmetries are still present (See Fig. 3). Thebroken inversion symmetry allows us to define a valley Chern number, as previously stated in the spring-mass model.In Fig. 7 the computed Berry curvature of first band is plotted. The Berry Curvature in 2D k -space is, B = − i (cid:104) ∂ x u k | ∂ y u k (cid:105) + i (cid:104) ∂ y u k | ∂ x u k (cid:105) (27)where u k is the eigenvector of one band at momentum k . The eigenvector is computed from the PWE method as thenull space of A matrix in Eq. 9. We observe that the Berry curvature is localized near K and K (cid:48) with opposite signand it changes at the transition.For further analogy with the spring system, we compute the mode shapes in real space at the K point for the twolower bands in Fig. 8 which closely resemble the eigenvectors involved in the transition u ± K . Moreover, band inversionis clearly seen. The mode shapes switch energies at both sides of the transition in the same way than eigenvectors inthe spring-mass model (Fig. 5 a). C. Edge states in ribbons
In this section we study the interface states appearing between two lattices with distinct valley Chern numbers,which are topologically protected i. e. with zig-zag interfaces. For analogy with graphene-like lattices, we callzig-zag edges those that go along directions (cid:126)a , (cid:126)a or (cid:126)a − (cid:126)a . We call armchair interface in Kagome lattice to the onealong vertical direction (cid:126)a − (cid:126)a in our definition of the unit cell. We create ribbons in a supercell along (cid:126)a directionand periodic in (cid:126)a direction. Even ribbons with valley topological phases in electronic system don’t have gapless edgesstates, because valleys are not well defined in vacuum, unless the boundary is with another topological phase withFIG. 8: (Color online) Mode shapes. Real part of w ( (cid:126)r ) for different bands and phases. Notice the analogy of thefirst row with the eigenvalue of the spring model u C = √ (1 , , t or in the second row with the real partRe { u + C } = √ (0 . , . , − t . Notice the band inversion. Mode shapes are not periodic due to the phase e − i (cid:126)K · (cid:126)r inEq. 3.opposite valley Chern number . The same reasoning is true for plates. Therefore, boundary states appear at theinterface between two phases with opposite signed topological invariants. Such interface is contained in the supercellof the ribbons as shown in Fig 1. Two types of interfaces can be made, which are depicted in Fig. 9 and 10. Schematicreal space supercell is highlighted, a black full line separates two topological phases distinguished by opposite valleyChern numbers. The bands are limited by the free-wave dispersion relation, outside that region there are not bulksolutions of the plate equation. The two types of interfaces exhibit a band of boundary states localized at the domainwall. In Fig. 9 a second band appears containing edge states at the top of the ribbon which are non-topological. Ananalogous band is present in Fig. 10 with edge states at the bottom of the ribbon as can be seen in the mode shapes.The topological edge modes are robust against certain types of perturbations that do not mix valleys. We haveconfirmed this fact by corroborating that these states are not removed away by the addition of general perturbationsto the boundary. However, there are perturbations mixing valley degrees of freedom such an armchair boundary that will destroy the protection as can be seen in Fig. 11. Notice the change in the unit cell parameter, now in thedirection of periodicity it is a (cid:48) = √ a . D. Finite systems
Now, we study a finite cluster of resonators on top of an infinite plane where multiple scattering theory describedin section II and developed in Ref. applies. The cluster of resonators contain two phases separated by a zig-zaginterface with Z-shape, Fig. 12. Topological protected state appears at mid gap energy. Notice that the horizontalinterface is equivalent to the domain wall in Fig. 9, thus the frequency is tuned to find topological edge modes, in thiscase Ω a = 2 .
51. Fig. 13 shows an edge state without backscattering, this mode is being computed without externalinput field, i.e. ψ = 0. The vector of coefficients ψ e ( (cid:126)R β ) in Eq.17 is the right-singular vector whose single value iszero. This method computes natural excitations of the system at a given frequency.Moreover, in the same cluster we find appropriate multipoint excitation with dephasing in time. A two-point0FIG. 9: (Color online) Ribbon of resonators over an infinite plate. The system contains a domain wall between twophases with opposite valley Chern numbers. At the top left, real space ribbon representation. The horizontal lineseparates the two phases and black arrows indicate that the ribbon is infinite in horizontal direction. In red,resonators in one supercell. At the top right there is the band structure of the finite system, neglecting non-bulkmodes, i. e. modes in the interior of the free dispersion curve Ω a = (cid:0) k x π (cid:1) . Two mid gap bands appear. At thebottom, mode shapes or in other words, real space displacement field along the supercell sites w ( (cid:126)R α ) for differentfrequencies and momenta as indicated with colored dots on the band structure.FIG. 10: (Color online) Ribbon of resonators over an infinite plate. The system contains a domain wall between twophases with opposite valley Chern numbers (See Fig. 7). At the top left, real space ribbon representation. Thehorizontal line separates the two phases and black arrows indicate that the ribbon is infinite in horizontal direction.In red, resonators in one supercell. At the top right there is the band structure of the finite system, neglectingnon-bulk modes, i. e. modes in the interior of the free dispersion curve Ω a = (cid:0) k x π (cid:1) . One mid gap band appears. Atthe bottom, mode shapes or in other words, real space displacement field along the supercell sites w ( (cid:126)R α ) fordifferent frequencies and momenta as indicated with colored dots on the band structure.1FIG. 11: (Color online) Ribbon of resonators over an infinite plate. The system contains a domain wall between twophases with opposite valley Chern numbers (See Fig. 7). On the left, real space ribbon representation. The verticalline separates the two phases and black arrows indicate that the ribbon is infinite in vertical direction. The interfaceis armchair-like. The band structure does not show localized modes within gap frequencies, bands that appearisolated at gap frequencies are bulk modes.FIG. 12: (Color online) Schematic representation of a cluster of resonators on top of an infinite plate. The cluster isdesigned with a Z-shaped interface.excitation ψ ( (cid:126)R α ) = G ( (cid:126)R α − (cid:126)x ) + G ( (cid:126)R α − (cid:126)x ) e iϕ where point sources are located at the horizontal domain wall, (cid:126)x = ( − , a and (cid:126)x = (1 , a . The dephasing ϕ is varied until propagating waves in one direction only are tuned.The results are shown in Fig. 14 and are similar to those presented in Ref. .FIG. 13: (Color online) MST simulations of an arrangement of resonators with two phases separated by a domainwall in zig-zag. The frequency is tuned so the mode is in a gap and correspond to topological edge states.2FIG. 14: (Color online) MST simulations of an arrangement of resonators with two phases separated by a zig-zagdomain wall (no mixing valleys) in Z-shape. The frequency is tuned so the modes are topological edge states. Thered dots correspond to the two excitation points (cid:126)x = ( − , a and (cid:126)x = (1 , a . The temporal dephasing is ϕ = 0(the two points are excited simultaneously) and ϕ = π (anti-phase excitation) respectively. V. MIRROR SYMMETRY BREAKING AND TOPOLOGYA. Spring-mass model
Now we consider a model with mirror symmetry at α = π and consider two continuous deformations that breakmirror symmetry. Changing α towards one side or the other will give two phases differentiated by different eigenvectorsof C symmetry. The spring-mass model is constructed by changing the relative spring constant between green andblue springs as indicated in Fig. 15. The equation of motion read, m ¨ u = − γ ( u − u ) − γ ( u − u ) − κ ( u − u e − i(cid:126)k (cid:126)a ) − κ ( u − u e − i(cid:126)k (cid:126)a ) − κ ( u − u e − i(cid:126)k (cid:126)a ) − κ ( u − u e − i(cid:126)k (cid:126)a ) m ¨ u = − γ ( u − u ) − γ ( u − u ) − κ ( u − u e i(cid:126)k (cid:126)a ) − κ ( u − u e i(cid:126)k (cid:126)a ) − κ ( u − u e i(cid:126)k (cid:126)a ) − κ ( u − u e − i(cid:126)k (cid:126)a ) m ¨ u = − γ ( u − u ) − γ ( u − u ) − κ ( u − u e − i(cid:126)k (cid:126)a ) − κ ( u − u e i(cid:126)k (cid:126)a ) − κ ( u − u e i(cid:126)k (cid:126)a ) − κ ( u − u e i(cid:126)k (cid:126)a )(28)introducing the relative difference β = κ − κ κ + κ we rewrite the system of equations in matrix form, − Ω u u u = γ (cid:48) − − − u u u + − e − i(cid:126)k (cid:126)a + e − i(cid:126)k (cid:126)a e − i(cid:126)k (cid:126)a + e − i(cid:126)k (cid:126)a e i(cid:126)k (cid:126)a + e i(cid:126)k (cid:126)a − e i(cid:126)k (cid:126)a + e − i(cid:126)k (cid:126)a e i(cid:126)k (cid:126)a + e i(cid:126)k (cid:126)a e − i(cid:126)k (cid:126)a + e i(cid:126)k (cid:126)a − u u u + β e − i(cid:126)k (cid:126)a − e − i(cid:126)k (cid:126)a e − i(cid:126)k (cid:126)a − e − i(cid:126)k (cid:126)a e i(cid:126)k (cid:126)a − e i(cid:126)k (cid:126)a e i(cid:126)k (cid:126)a − e − i(cid:126)k (cid:126)a e i(cid:126)k (cid:126)a − e i(cid:126)k (cid:126)a e − i(cid:126)k (cid:126)a − e i(cid:126)k (cid:126)a u u u (29)where Ω = 2 mω / ( κ + κ ) and γ (cid:48) = 2 γ/ ( κ + κ ) The eigenvectors at K point are the same eigenvectors of C symmetry, but its energy order is different from previous section. Now, u ± K = u ± C , i. e. gap closes at K point forminga the Dirac cone at α = π . Notice the closing occurs on first or second gap depending on f (See Fig. 4). In anycase, the Dirac cones are made of states with complex conjugate eigenvalues of C symmetry. Moreover, they areinterchanged at the transition: u ± K = u ± C for β > u ± K = u ∓ C for β < K (cid:48) .We compute the effective model for this band crossing system as in Eq. 24. The result is, D η = (cid:18) γ (cid:48) + 1 . − β ) ηv D e i π/ ( k x + ik y ) v D ηe − i π/ ( k x − ik y ) 3 γ (cid:48) + 1 . β ) (cid:19) (30)where v D = √ a . By rotating (cid:126)k = ( k x , k y )-reference system by π/
3, the dynamical matrix can be written with thesame structure than Eq. 25, D η = (cid:18) γ (cid:48) + 1 . − β ) v D ( ηk (cid:48) x + ik (cid:48) y ) v D ( ηk (cid:48) x − ik (cid:48) y ) 3 γ (cid:48) + 1 . β ) (cid:19) (31)3FIG. 15: (Color online) Distorted Kagome lattice for two f values and α = π/
6. Notice that spatial inversion is nota symmetry of the system. Increasing slightly α shortens green links and enlarges blue links. The spring modelmodels changes in distance with appropriate changes in β .FIG. 16: (Color online) Energy levels at K point for model in Eq. 29 as a function of β for γ (cid:48) = 1.This result illustrates that the mirror symmetry breaking in the original model is analogous to an inversion symmetryin graphene-like systems where β is the pseudo-magnetic field in quantum valley Hall effect. Instead of inducingnonequivalent sublattice potential, here the potential is between eigenstates of the system and C .The subspace generated by the two Dirac eigenstates crossing at K differentiates between states rotating in differentdirections M = i ( ˆ C − ˆ C t ), M = | u + C (cid:105) (cid:104) u + C | − | u − C (cid:105) (cid:104) u − C | = i √ − − − (32)This matrix is proportional to the linear term in β at K point and it is schematically represented in Fig.5 b). Thisgives us the mirror symmetry breaking effect in real space lattice vectors. B. Plate model and valley Chern number
In this section we plot several band structures around α = π/
6. Notice that in Fig. 4 the gap closes for all f at α = π/ K point. For f < √ , the second and third bands are degenerate at K point. For f > √ the firstand second bands form the Dirac cone. The two transitions have equivalent topology. In the spring-mass model this4FIG. 17: (Color online) Band structure of deformed Kagome lattice for several α -values and f = 1 . K and K (cid:48) points with different signs for different phases. Blue is negative and yellow is positive.corresponds to varying the value of γ that tunes the energy of u C but does not affect the other two crossing states.However, the gap opening at K when α (cid:54) = π/ f . For large f , K point is not the minimumof the second band, although topological states come from what happens at K point, the gap is complete and weshow the results of for f = 1 .
5. The band structures of plates with different arrangements of resonators are plottedin Fig. 17. At equidistant points in parameter space from the transition points the band structures are the same,however their topology is not. At the transition point, a Dirac cone at K point is formed which opens upon breakingmirror symmetry. The Dirac energy is Ω a = 2 . α (cid:54) = π mirror symmetry is broken and as we show in the effective model we can define a Berry curvature asin Eq. 27. The result is shown in Fig. 18. The eigenvectors in the Brillouin Zone are computed from PWE methodas the null space of A matrix in Eq. 9 at the appropriate frequency as described in Ref. . We observe that theBerry curvature is localized near K and K (cid:48) with opposite sign and it changes at the transition, consistently with theeffective spring-mass model. C. Edge states in ribbons
We compute the edge states of a ribbon with an interface and find two crossing bands in the middle of the gap.The crossing indicates that the two bands have different symmetry. In Fig. 19, edge states appear in the boundaryof the two phases, due to the different valley Chern numbers. In this transition there are two crossing bands withdifferent symmetries that are topologically protected. The different symmetries can be observed in the modes inFig. 19. They are symmetric or anti-symmetric respect to the domain wall. Notice site two maps onto itself underinversion at the domain wall and site three and one maps onto one another. This symmetry in the eigenvectors reflectthe inversion symmetry present in real space in the ribbon due to the fact that phases are equidistant in real spacefrom the transition point in parameter space. In other words, the two phases are characterized by α = π/ ± φ , where α = π/ φ = 0 .
1. This ribbon symmetry is also present in ribbons with two phases breakinginversion symmetry in honeycomb lattice like in Ref . Since the two phases are equidistant from the transition point,there is an inversion that gives symmetric and anti-symmetric edge modes respect to the domain wall. (See AppendixVII). Unlike honeycomb lattice in Kagome arrangement each number site has its inversion point. In graphene, the5spatial inversion is clearly seen in the eigenvectors u A = (1 , t and u B = (0 , t than transform into one another byappropriate inversion in real space. In our case, Eq. 31, the eigenvectors at a given frequency and at each side of thedomain wall are related by spatial inversion too u β> K = √ e − i π e i π = e i π u + C u β< K = √ e i π e − i π = e i π u − C (33)Site 2 maps into itself, while sites 1 and 3 interchange and appropriate combinations. The result shows symmetricand anti-symmetric modes, as observed in the ribbon eigenvectors (Fig 19). Notice inversion symmetry is not presentin domain walls in ribbons with phases of Kagome lattice with broken inversion symmetry shown in Fig. 9-10.Modes are not symmetric or anti-symmetric and neither the eigenvectors at K involved in the transition ( u C and u + C ) exhibit inversion symmetry, as expected.Valley topology is not protected against perturbations mixing the valleys. For instance, a vertical interface, (arm-chair type) mixes the valleys and the edge states disappear as shown in Fig. 20. The bands displayed in the middleof the gap are also bulk bands. D. Finite systems
We design a finite structure of resonators over an infinite plate and compute the real part of w ( (cid:126)r ). A similar resultoccurs for natural modes of the system, as in Fig. 13. We also find two-point time-dephased excitation at mid gapfrequency, so one-way propagation is achieved. See Fig. 21, the red dots are the points where the external excitationforce is applied, (cid:126)x = ( − ,
0) and (cid:126)x = (1 , ϕ excites different directional waves. VI. CONCLUSION
We have studied two types of topological transitions in mechanical metamaterials based on the distorted Kagomelattice, namely inversion symmetry or mirror symmetry breaking. In spring-mass systems, we derived a dynamicalmatrix for each valley that effectively behaves as a Chern insulator. We have identified, in the microscopic model, theoperator acting as a pseudo-magnetic field which is controlled by relative values of springs’ strengths. We also exploitthis finding for flexural waves in plates coupled to resonators. In this context the ”magnetic field” is controlled bythe distance between resonators. The main manifestation of the valley Hall effect in our system is the presence ofprotected boundary states located at interfaces between domains with opposite signed valley Chern numbers. Theseinterfaces must have appropriate edges as shown in simulations of ribbons and finite clusters of resonators with zig-zag domains. We also illustrated how mixing valleys with armachair-type interfaces produces back-scattering anddestroys the topological modes. However, we also claim that a lattice lacking inversion symmetry at the transitiondespite intact mirror symmetry exhibits the same type of valley topology of broken mirror symmetry. We compute asimilar effective model for springs and find protected edge states with different symmetry. We find simple two-pointexcitation generating one-way flexural waves in finite systems that can propagate through desired bends in 2D space.It is well known that the dynamics of spring-mass systems is dissimilar in several ways to the one of interactingresonators coupled to plates. For instance, interaction between the resonators is long-ranged and the dynamicalmatrix is frequency dependent in the latter. However, throughout this work we have established a common originto their topological properties. We hope all these findings help enlightening the path towards future applications inwave guiding and related fields.
Acknowledgments : N.L and J.V.A. acknowledges financial support from MINECO grant FIS2015-64886-C5-5-P.NL acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through The Mar´ıade Maeztu Programme for Units of Excellence in R&D (MDM-2014-0377), and also hospitality from the UniversitatJaume I in Castellon where part of this work was done. D.T. acknowledges financial support through the “Ram´on yCajal” fellowship under grant number RYC-2016-21188. P.S-J. acknowledges financial support from the Spanish Min-istry of Economy and Competitiveness through Grant No. FIS2015-65706-P (MINECO/FEDER) J. C. acknowledgesthe support from the European Research Council (ERC) through the Starting Grant No. 714577 PHONOMETA andfrom the MINECO through a Ram´on y Cajal grant (Grant No. RYC-2015-17156).6FIG. 19: (Color online) Ribbon of resonators over an infinite plate. The system contains a domain wall between twophases with opposite valley Chern numbers (See Fig. 7). At the top left, real space ribbon representation. Thehorizontal line separates the two phases and black arrows indicate that the ribbon is infinite in horizontal direction.In red, resonators in one supercell. At the top right there is the band structure of the finite system, neglectingnon-bulk modes, i. e. modes in the interior of the free dispersion curve Ω a = (cid:0) k x π (cid:1) . Two crossing bands appear inthe gap, they have different symmetry under domain wall spatial inversion as seen at the bottom. At the bottom,mode shapes, i. e. real space displacement field along the supercell sites w ( (cid:126)R α ) for different frequencies andmomenta as indicated with colored dots on the band structure. VII. APPENDIX: HONEYCOMB RIBBONS WITH BROKEN INVERSION SYMMETRY
As computed in Ref. , the analogous to quantum valley Hall effect guarantees boundary modes localized at theinterface between two phases. Inversion symmetry is broken by different masses of resonators in the two dimensionalunit cell and two types of interface can be created (with zig-zag boundary). In this appendix we examine the symmetryof the boundary modes. As explained in the main text, the ribbon structure has inversion symmetry at the domainwall provided the two phases are equally large and masses are the same. See Fig.22, full circles correspond to γ = 11and empty circles to γ = 9 (the same at each side of the domain wall), all resonators have the same spring constantand their frequency is Ω R = 4 π . Dirac frequency for γ = 10 is Ω D = 2 . k x = π . At each side, the symmetry7FIG. 20: (Color online) Ribbon of resonators over an infinite plate. The system contains a domain wall between twophases with opposite valley Chern numbers (See Fig. 7). On the left, real space ribbon representation. The verticalline separates the two phases whose interface is armchair type and black arrows indicate that the ribbon is infinite inthe vertical direction. The band structure does not show localized modes within gap frequencies, bands that appearisolated at gap frequencies are bulk modes.FIG. 21: (Color online) MST simulations of an arrangement of resonators with two phases separated by a zig-zagdomain wall in Z-shape. The frequency is tuned so the mode is in a gap and correspond to topological edge states.The red dots correspond to the two excitation points. The dephasing is ϕ = π on the left and ϕ = − . π on theright.is different, for k x < π modes are anti-symmetric under inversion symmetry and for k x > π modes are symmetric. R. LAKES, “Foam structures with a negative poisson’s ratio,”
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