Weyl points and topological surface states in a three-dimensional elastic lattice

Sai Sanjit Ganti, Ting-Wei Liu, and Fabio Semperlotti∗Ray W. Herrick Laboratories, School of Mechanical Engineering,

Purdue University, West Lafayette, Indiana 47907, USA.

Following the realization of Weyl semimetals in quantum electronic materials, classical wave analogues ofWeyl materials have also been theorized and experimentally demonstrated in photonics and acoustics. Weylpoints in elastic systems, however, have been a much more recent discovery. In this study, we report on thedesign of an elastic fully-continuum three-dimensional material that, while offering structural and load-bearingfunctionalities, is also capable of Weyl degeneracies and surface topologically-protected modes in a way com-pletely analogous to the quantum mechanical counterpart. The topological characteristics of the lattice areobtained by ab initio numerical calculations without employing any further simplifications. The results clearlycharacterize the topological structure of the Weyl points and are in full agreement with the expectations of sur-face topological modes. Finally, full field numerical simulations are used to confirm the existence of surfacestates and to illustrate their extreme robustness towards lattice disorder and defects.

I. INTRODUCTION

The search for novel topological states of matter has re-cently seen many exciting breakthroughs either in quantumor classical wave physics1–5. Phenomena such as the quan-tum Hall effect (QHE), the quantum spin Hall effect (QSHE),and the quantum valley Hall effect (QVHE) have been stud-ied extensively and their nontrivial topological characteristicshave been clearly connected to their ability to support back-scattering-immune topological edge states1,2. Although theconcept of topological material was discovered and developedin the field of condensed matter physics, the many conceptualand mathematical similarities between different fields of wavephysics have led to the development of analogue concepts oftopological materials in electromagnetic, acoustic, and elas-tic systems3–25. Recent studies have identified the existenceof the so-called Weyl semimetals as three dimensional non-trivial topological materials capable of unidirectional back-scattering-protected surface states26.

Weyl semimetals are quantum materials whose behavioris characterized in terms of the Weyl Hamiltonian H(k) =νxkxσx + νykyσy + νzkzσz, where νi, ki, and σi represent thecomponents of the group velocity, momentum, and Pauli ma-trix, respectively. These materials can be thought of as a three-dimensional extension of two-dimensional nontrivial topolog-ical materials characterized by Dirac degeneracies such as, forexample, systems exhibiting QHE, QSHE, and QVHE. Simi-lar to 2D Dirac materials, Weyl semimetals possess a degen-erate nodal point formed by linear intersection of two bandsin the three coordinate directions of the three dimensional re-ciprocal space. This degenerate point, called the Weyl point,carries a nonzero topological charge which confers the ma-terial nontrivial topological properties. The charge (i.e. theintegral of the Berry curvature flux in a 2D manifold enclos-ing the Weyl point) can have either a positive or negative sign,hence determining if the degeneracy acts as a quantized sourceor a sink of berry curvature flux. The total topological chargeof a Brillouin zone (BZ) in reciprocal space should always bezero27 and therefore Weyl points always occur in pairs of op-posite charge. Nontrivial topological surface states connect-ing the Weyl points of opposite charge exist on the boundaryof the system. Apart from topological surface states, Weyl

points have also been linked to other unusual effects such aschiral anomaly28 and quantum anomalous Hall effect29.

Weyl points can be conceptually interpreted as the exten-sion of Dirac points to three-dimensional lattices. However,some fundamental differences exist that have important impli-cations on the robustness of these degeneracies. The existenceof Dirac points can be guaranteed based on symmetry argu-ments (e.g. graphene-like lattices), provided that both parity-inversion (P) and time-reversal (T ) symmetries are preserved.Breaking either P or T -symmetry would lift the degeneracyleaving behind a bandgap with topological significance. Un-fortunately, the existence of Weyl points is much more am-biguous and it cannot be guaranteed a priori. However, someuseful indications can come from the analysis of symmetries.In the case of Weyl points, P and T -symmetries impose con-tradicting requirements. In fact, considering a lattice hav-ing a Weyl point at the k high-symmetry point in reciprocalspace, T -symmetry would require the topological charges atk and −k to be of the same sign while P-symmetry wouldrequire them to be opposite. Due to these contradicting re-quirements, it follows that Weyl points can only be obtainedin systems with broken PT-symmetry. Thus, unlike for Diracpoints in which symmetry breaking is a necessary condition tolift the degeneracy, for Weyl points symmetry breaking is a re-quirement for the existence of the isolated degeneracy. Thesemore stringent requirements on symmetry conditions are atthe foundation of the higher robustness of Weyl points againstexternal perturbations. In fact, previous studies showed thatthey can only be annihilated by combining pair of points withopposite charges. Also, following the requirements dictatedby both P and T -symmetries, the minimum number of Weylpoints in a BZ is 4 (2 pairs) when P-symmetry is broken and2 (1 pair) when T -symmetry is broken3.

Note also that, although in the current study only Weylpoints with unit charge are considered, other studies30–38 havereported systems possessing Weyl points having higher chargetypically caused by the overlapping of multiple Weyl pointsof unit charge. Such points can be readily identified fromthe band structure because the modes do not intersect linearlyat these points in the kx-ky plane. However, the intersectionalong kz would still be linear. Recent studies have also re-ported a new kind of Weyl point labeled type-II both in pho-

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tonic and phononic systems39–44. The degeneracies discussedin the present study are of type-I and represent the more di-rect analogue to the degenaracies originally studied by Weylin quantum mechanical systems45.

While first theorized as the massless solution of theDirac equation45, Weyl fermions could be realized andexperimentally demonstrated in Weyl semimetals onlyrecently30–33,46–50. The possibility of identifying classical me-chanical systems serving as analogue to the Weyl semimetalshas opened an intriguing and fertile topic of research span-ning many areas of wave physics. As a result, non-quantum-mechanical analogues of the Weyl points were formulatedboth in photonic34,36,40–42,51–57 and acoustic37,38,43,57–60 meta-material systems. Weyl points were realized and experimen-tally demonstrated in photonics using double-gyroid struc-ture characterized by broken P-symmetry51,52. Although suc-cessful in realizing Weyl points, the double-gyroid structureinvolves a complex fabrication. Simpler designs based, asan example, on woodpile photonic crystals36,56 or on rotatedstacked rods in acoustic systems37,38 were also explored. Aplanar fabrication methodology, which outlines a step by stepapproach to obtain Weyl points, was also presented34,55.

Despite the successful implementation of Weyl points inphotonic and acoustic systems, the realization of Weyl pointsin elastic systems has proven to be more challenging thanother classical analogues. This increased complexity stemmedfrom wave coupling and mode conversion between differentpolarizations as well as from the difficulty in fabricating thenecessary 3D structures. To-date, successful realizations werereported only in a few studies35,61. One study made use ofbeams and thin plates with a particular geometry35 purposelyselected to be manufactured by additive manufacturing. An-other study utilized a truss-like construction made of beamelements61. Although elegant and simple to fabricate, thesedesigns are discontinuous in nature therefore not well suitedfor applications where the load-bearing capabilities play acritical role (e.g. applications to structural materials). Also,in both the above mentioned studies, the topological charac-teristics were obtained based on a tight-binding (TB) formula-tion of the Hamiltonian. This approach works well only whenthe fundamental lattice is composed of weakly coupled res-onant elements. In addition, the hopping parameters used inthe TB model usually cannot be clearly connected to the ac-tual geometric and design parameters. These limitations raisethe more important question of how accurately TB models candescribe continuous elastic lattices. In this regard, we proposean elastic analogue of a Weyl semimetal material based on afully continuous design resulting in a solid load-bearing struc-ture. Also, due to the strongly coupled nature of the elemntsin this design, we present a detailed study of the topologicalcharacteristics of such medium based on ab initio calculations,hence bypassing the limitations of TB approaches.

This paper is organized as follows: Sec. II will introduce theproposed design and the corresponding dispersion propertiesin connection with Weyl points. Sec. III will present ab initiocalculations of the topological invariant. Sec. IV will focus onthe dispersion behavior of a supercell and the correspondingdispersion structure, in order to determine the occurrence of

surface states. Finally, Sec. V will show full field simulationsof the elastic Weyl material in order to illustrate the one-way,backscattering-immune nature of the surface states.

II. SYNTHESIS OF THE UNIT CELL AND DISPERSIONPROPERTIES

From a general perspective, in order to develop the funda-mental unit cell, we select a 2D triangular lattice (assumed inthe xy-plane) and build the 3D geometry by periodically re-peating this unit along the z-direction. The final result is alayered 3D structure having intact P and T -symmetries. Notethat, as previously mentioned, starting from a 2D triangu-lar lattice guarantees (due to symmetry arguments) the exis-tence of Dirac degeneracy at the high symmetry points in thekx-ky momentum space. Periodically repeating this unit in thez-direction extends the Dirac degeneracy at virtually any kzvalue, thereby generating a line node degeneracy3,55 along kz.In order to lift this three-dimensional degeneracy, P-symmetrycan be broken by the proper introduction of additional struc-tural element that do not respect inversion symmetry. Forthis purpose, we select slanted beam elements connecting thevertices of the hexagonal structure on two adjacent layers asshown in Fig. 3. The result is the introduction of a chiral cou-pling between the layers that breaks P-symmetry and all exist-ing mirror symmetry. The most direct consequence is that theline degeneracy along kz is reduced to a point, hence givingrise to the Weyl point.

In the following section, we first describe the geometry andthe dynamic properties of the parity-preserving fundamentalunit. Then, we introduce the chiral coupling and investigateits effect compared with the P-symmetric unit.

A. 3D lattice with intact P-symmetry: line node degeneracy

Fig. 1 (a) and (b) show the fundamental unit cell used tocreate the lattice. The cell is made of a vertical cylinder in-cluded between two homogeneous, hexagonal-shaped, thinplates. The cylinder has its longitudinal axis aligned withthe z-direction. The cylinder is made of iron while the platesare made of aluminum. The selection of these materials wasmotivated by the intent of obtaining marked degeneracies atthe high symmetry points in the momentum space. Clearly,several other combinations of structural materials leading tosimilar conditions could potentially be identified. The dimen-sions of the unit cell are: a = 30 mm, h0 = 2 mm, r1 = 5 mm,h1 = 20 mm, where a is the lattice constant, h0 is the thick-ness of each homogeneous flat plate, r1 and h1 are the radiusand height of the cylinder, respectively. A 2D lattice in thex-y plane can be assembled by periodically repeating the unitcell. The final result is a lattice of cylinders in a hexagonalconfiguration sandwiched between two thin plates. This 2Dlattice can be periodically repeated in the z-direction to formthe final 3D lattice.

The analysis of the band structure of the 3D material can beconveniently performed by applying periodic boundary con-

3

ditions to the fundamental unit cell. More specifically, peri-odic boundary conditions can be applied along the sidewardfaces of the plate in order to create the 2D lattice in the xy-plane, while periodic boundary conditions applied on the topand bottom faces of the top and bottom plates will yield thecomplete 3D lattice. The resulting system is characterized byintact P and T -symmetries and the corresponding 3D Bril-louin zone is shown in Fig. 1 (c). For a fixed value of themomentum component kz, the 3D BZ simplifies into a 2D BZtypical of a triangular lattice.

The dispersion curves along the boundary of the 2D BZin the kx-ky plane at kz = 0, and in the kx-kz plane at ky = 0have been calculated using the commercial finite element (FE)package COMSOL Multiphysics and are shown in Fig. 2. Asvisible in Fig. 2 (a), four modes intersect linearly at a fre-quency of approximately 40 kHz and give rise to two degen-erate points at K. These degenerate points are indicated by thered and green boxes in the inset of Fig. 2 (a).

The resulting degenerate nodes are Dirac points which areprotected by the lattice symmetry of the hexagonal (graphene-like) arrangement of vertical cylinders. As previously men-tioned, this degeneracy exists at all kz, hence resulting in theline degeneracies marked in red and green colors in Fig. 2(b). This behavior is a direct consequence of both P and T -symmetries being preserved. Breaking P-symmetry would re-sult in lifting the line degeneracy along the kz direction henceresulting in the formation of a Weyl points.

B. P-symmetry breaking and Weyl points

In order to break P-symmetry, a chiral coupling can be in-troduced by means of straight slanted cylinders connecting topand bottom layers in proximity of the vertices of the hexag-onal layer. The resulting unit cell is shown in Fig. 3 whileits 3D BZ remains the same as the one shown in Fig. 1 (c).The slanted cylinders are made of aluminum (as the plates)and have a radius of r2 = 2 mm. The dispersion curves alongthe boundary of 2D BZ in the kx-ky plane at kz = 0 and inthe kx-kz plane at ky = 0 are calculated again and shown inFig. 4. The degenerate points resulting from the linear inter-section of two modes in Fig. 2 (a) are also present in Fig. 4(a) (marked by the same red and green boxes). However, theline degeneracies previously observed in Fig. 2 (b) are nowlifted, due to P-breaking, and reduced to nodal degeneraciesat the high-symmetry points K and H (see red and green cir-cles Fig. 4 (b)). These degenaracies are the Weyl points thatare formed by the linear intersection of the two (initially de-generate) modes along all three directions. Fig. 4 (a) showsthe linear intersection in the kx-ky plane while Fig. 4 (b) showsthe linear intersection in kx-kz plane.

As a result, the degeneracy marked in Fig. 4 (a) is lifted fornonzero values of kz (with kz 6= nπ/az for integer n) hence giv-ing rise to a topological bandgap as shown in Fig. 4 (c). Theband structure shown in Fig. 4 (c) corresponds to kz =± 0.1π

az,

where az = h1 + 2h0 = 24 mm is the total height of unit cell.The subscripts ± of the high symmetry points’ labels indicatekz = ± 0.1π

az, as also depicted in Fig. 1 (c). The bandgap size

varies with kz as shown in Fig. 4 (b). Thus, kz can be treatedas a parameter that controls either the opening or closing ofthe topologically nontrivial bandgap, provided that the systemremains periodic along the z-direction.

III. AB INITIO CALCULATIONS OF THETOPOLOGICAL PROPERTIES

A. Topological charge

In order to assess the topological significance of the degen-eracies identified in the chiral 3D lattice and, consequently,the existence of Weyl points, this section presents a numeri-cal investigation into the calculation of the topological charge.The charge can be calculated by integrating the Berry curva-ture flux on a 2D manifold S enclosing the Weyl point in thereciprocal space, that is

C =1

2π

∮S

ΩΩΩ(k) ·dS (1)

where C is the topological monopole charge, i.e., the Chernnumber, dS is the vector surface element aligned with its localnormal direction, and ΩΩΩ is the Berry curvature (vector field)given by

ΩΩΩ(k) = ∇k×〈u(k)|i∇k|u(k)〉 (2)

where k is the wavevector, and un(k) is the displacementeigenstate as a function of k. Consider the Weyl point at K,at a frequency of approximately 40 kHz, as indicated by thered dashed box in Fig. 4 (a). This point corresponds to theintersection of the 16th and the 17th bands. To calculate thecorresponding topological charge, we can integrate the Berrycurvature flux threading the outer surface area (on the top, bot-tom, and side faces) of a fictitious prismatic manifold enclos-ing the K point (see Fig. 5). We anticipate, and show later, thatthe flux on the side faces is zero if a certain triangular prismis chosen as prototypical 2D manifold.

Without loss of generality, we can select kz =± 0.1π

azas the

top and bottom faces of the prism. At these planes, we haveK+ and K− with the same (kx,ky) coordinate as the K point.The z-component of the Berry curvature on the top plane isthen calculated using Eq. 2 (see Fig. 6 (a) and (b)) for the 16th

and 17th bands, respectively. Note that, at the two distinctpoints (i.e. they cannot be transformed into one another bytranslation symmetry operations), K+ and K′+ points exhibitthe same Berry curvature pattern. This is a direct result ofthe C6 symmetry of the lattice. Therefore, if the 2D manifoldis chosen to be the triangular prism with the right red dashedtriangle in Fig. 6 (a) as its top face, we can assure that the fluxintegral on the side faces of the prism is zero due to symmetryin the reciprocal space. In fact, the C6 symmetry guaranteesthat the outward flux on the side face ¬ equals the inward fluxon the side face ­; the same cyclic symmetry along with thetranslation symmetry (from the left to the right, for example)forces the flux integral on the entire side surface ® to vanish.

It follows that we only need to integrate the Berry curva-ture flux on the top and bottom faces. The bottom face which

4

kxky

kz

K+

K−

H

Γ

K

M

A L

K'+

K'−K'

r1

+

a

h1

h0

(a) (b) (c)

FIG. 1. (a) Schematics of the unit cell with intact P- and T -symmetry. Some characteristic dimensions are shown. The top and bottom platesare made of aluminum (transparent white) while the cylinder is made of iron (black). Periodic boundary conditions are used on the boundariesto create a 3D lattice. (b) Rendered view of a bulk piece of the 3D lattice. (c) Brillouin zone of the resulting 3D lattice in reciprocal space.Orange dashed lines mark the irreducible part of the Brillouin zone (IBZ).

FIG. 2. (a) Dispersion curves of the 3D lattice based on the unit cell shown in Fig. 1. (a) the dispersion along the 2D BZ in the kx-ky planeat kz = 0. The region marked in blue is magnified in the inset to show the presence of two Dirac points (marked by the red and green boxes).These degeneracies are protected by the symmetry of the triangular lattice in the x-y plane. (b) Dispersion curves plotted along the 2D BZ inthe kx-kz plane at ky = 0. The nodal degeneracies at the corner of the BZ (marked in Fig. 2 (a)) exist for all values of kz, thereby resulting inline degeneracies along the K-H directions indicated by the red and green lines.

(a) (b)

+

r2

FIG. 3. (a) Schematics of the unit cell with slanted cylinders. Thecylinders have radius r2 and are made of aluminum. The resultingunit cell does not preserve P-symmetry and mirror symmetry. Peri-odic boundary conditions are used on the boundaries to create a 3Dlattice. (b) Rendered view of a bulk piece of the 3D lattice.

is around the K− point, is actually the time reversed counter-part of the triangular face around the K′+ point (inversion of

(kx,ky,kz)) as shown in the left red dashed triangle in Fig. 6(a). T -symmetry requires that ΩΩΩ(−k) = −ΩΩΩ(k) which re-sults in the z-component of the Berry curvature on the bottomface to have the same strength but opposite sign. Also, giventhat the dS element on the bottom face is oriented in the −kzdirection, it still contributes a positive outward flux. As a re-sult, the topological charge is the sum of the integrals of theBerry curvature calculated over the two red dashed trianglesin Fig. 6 (a). While solving for the Bloch eigenmodes, we per-formed a parametric analysis across the kx and ky grid pointsby using an adaptive resolution (finer near K′+ point). TheBerry curvature was calculated based on Eq. 2 using a finitedifference formulation on the two triangular regions, and theresults are shown in Fig. 6 (c) and (d). The numerical inte-grals yield 3.1561 and 3.1479, respectively, hence confirmingthe monopole charge of +1 (with only +0.3% error). Usinga very similar numerical approach, the topological charge atthe H point can be found to be −1. This result is in agree-ment with the T -symmetric nature of the Weyl points whichrequires the appearance of the degeneracies in pairs of points

5

± ± ± ±

FIG. 4. Dispersion curves for the 3D lattice with broken P-symmetry. (a) Dispersion along the 2D BZ in the kx-ky plane at kz = 0. The nodaldegeneracies from Fig. 2 (a) are still present, despite a slight shift in frequency. (b) Dispersion curves of the same 3D lattice along the 2D BZin kx-kz plane at ky = 0. The line degeneracies in Fig. 2 (b) are lifted (magnified image of the green modes is shown in the inset). The resultingnodal degeneracies at K and H are Weyl points, marked by red and green circles. (c) Dispersion curves along the 2D BZ in the kx-ky planewhen kz =

0.1π

az. The degeneracy indicated by the red box in Fig. 4 (a) is now lifted, due to the nonzero value of kz, hence leaving behind a

topological bandgap. The degeneracy marked by the green box in Fig. 4 (a) is also lifted due to the nonzero value of kz, which opens a second(although narrower) bandgap.

KK+

Γ

ky

kz

Ωz

K−kx

FIG. 5. The topological charge at the K point can be calculatedby integrating the outward Berry curvature flux of a 2D manifoldenclosing the K point. In this case, the surfaces of a triangular prismare chosen.

with opposite charge. Fig. 7 shows all the Weyl points withinthe half BZ (in the kz direction) and their corresponding topo-logical monopole nature (either sink or source). On the kz = 0plane, there are two Weyl points (K and K′) both with charge+1 (that is sources of Berry curvature), while on the kz = π/azplane, each of the two Weyl points (H and H′) are of charge−1 (that is sinks of Berry curvature).

B. Bandgap-Chern number

An alternative way to obtain the topological charge of theWeyl point is to use a topological invariant called the bandgap-Chern number Cg. This invariant is obtained by summing theChern numbers of all the bands below the gap of interest. Inthe following, we consider kz as a parameter and focus on thebandgap that forms in the kx-ky plane. Cg remains constant

(b) 17th band

K+

K'+

K'+

K'+

K+

K+

(a) 16th band

K'+

(c)

K+

(d)

z

a2

π2

1

2

3

1

2

3

FIG. 6. (a,b) The z-Berry curvature distribution of the 16th and 17th

bands, respectively, on the kz = 0.1π/az plane. The two red dashedtriangles in (a) indicate the top faces of the chosen prismatic man-ifolds enclosing the Weyl points, K or K′ points, respectively. Thetotal integral of the Berry curvature flux on the vertical side walls ¬,­ and ® of the prism vanishes due to the C6 and translation sym-metry. (c,d) Zoomed-in view of the Berry curvature correspondingto the two triangular areas in (a). Each yields the outward flux inte-gral of π , which shows the Weyl points K and K′ are of +1 Chernnumber.

with varying kz except at K and H where the bandgap closesand reopens and Cg experiences a sudden change in its value.This change is equal to the topological charge of the Weylpoints on that plane because Weyl points act as either sources

6

K

K'

K'K'

K

KH

H'H

H

H'

H'

FIG. 7. Schematic illustration of the Weyl points within the kz > 0half BZ and their corresponding topological monopole nature (eitheras a sink or a source).

or sinks of Berry curvature flux56. This principle can be usedto calculate the charge of the Weyl points in the band struc-ture. Particularly, we focus on the degeneracies occurring atapproximately 40 kHz. It is found that the Berry curvatureintegrals of the first 15 bands cancel each others, therefore Cgreduces to the integral of Berry curvature in the 2D BZ for the16th mode only.

Figure 8 shows the plot of Cg as a function of kz for theWeyl points at 40kHz. It clearly shows that Cg has a value of−1 for kz < 0 and +1 for kz > 0 which results in a net dif-ference ∆Cg = +2. This is consistent with the previous anal-ysis that showed the kz = 0 plane to host two Weyl points ofequal charge at K and K′ due to intact T -symmetry. The twoWeyl points at K and K′ therefore have a topological chargeof +1 each. In the same figure, there is also a change in thevalue of Cg at kz = ± π

az. This indicates that the charge of the

Weyl point at H (H′) is −1. Once again, this is expected andconsistent with the fact that Weyl points always occur in pairswith opposite topological charge. The nonzero value of chargealso confirms that the nodal degenerate points under consider-ation are in fact Weyl points with nontrivial topological sig-nificance. In light of the bulk-surface correspondence2, anonzero Cg indicates that topological surface modes can existwithin the target bandgap while the magnitude of Cg indicateshow many surface modes should be expected. In our specificexample, Cg = ±1 for kz ∈ (0,π/a) or kz ∈ (−π/a,0), re-spectively. Compared with trivial materials (such as vacuum),there is a kz-locked net difference of ±1 in Cg. Therefore, onesurface mode should be expected either in case of positive ornegative kz. The corresponding surface states propagate uni-directionally either in the clockwise or counter-clockwise di-rection along the structure border (depending on the sign ofCg), as shown later on in the numerical simulations. This re-sult further confirms that the selected geometry leads to theformation of Weyl points and it is capable of supporting topo-logical surface states.

We note that the nonzero Chern number and the Berry cur-vature distribution for, as an example, kz = 0.1π/az (Fig. 6) isreminiscent of 2D topological materials manifesting quantumHall effect (QHE)62. In QHE materials, T -symmetry is bro-ken due to the use of an external field. In the present material,the chiral structure breaks the z-mirror symmetry therefore a

-1.0 -0.5 0.0 0.5 1.0-2

-1

0

1

2

kzaz/π

bandgap-Chernnumber

FIG. 8. The bandgap-Chern number Cg associated with the gaparound 40 kHz as a function of kz. As the bandgap closes and reopensat kz = 0, Cg experiences a +2 discontinuous jump in its value, whichindicates the sum of topological monopole charges on the kz = 0plane.

mode propagating in the z−direction in the infinite lattice isexpected to exhibit kz-locked unidirectional surface states. Infact, at steady state, the role of temporal and the spatial (in thiscase z) variables can be interchanged. A similar idea was alsopresented in studies of 3D Floquet topological insulators63.This latter aspect will be further clarified in the next section.

IV. SUPERCELL DISPERSION AND SURFACE MODES

Weyl points with opposite topological charges are con-nected by nontrivial topological surface states that exist onlyon the boundary of the medium. In this section, we considerthe physical response of a periodic supercell made out of the3D unit cell described above. Consider a supercell havinga finite dimension of 38 units in the y direction and infinitedimensions (obtained by means of periodic boundary condi-tions) in the x and z directions. Fig. 9 (a) shows the actualdomain used for the numerical calculations.

We focus the following analysis on the topological bandgapthat develops at the approximate frequency of 40kHz and thatis marked by the red box in Fig. 4 (c). Fig. 9 (b) shows thedispersion curves for the supercell at kz =

0.1π

az. It is evident

from the visual inspection of the dispersion that the supercelladmits two additional modes, the surface modes, that existwithin the topological bandgap. Bulk modes are in grey colorwhile the surface modes are indicated in green and red. Thegreen modes correspond to the lower edge of the supercellwhile red modes correspond to upper edge. This correspon-dence is also reflected in the plot of the eigenvectors on thesupercell shown in Fig. 9 (c) and 9 (d). The dispersion of thesurface modes corresponding to the upper edge has a positiveslope (i.e. positive group velocity), which means that thesemodes are expected to travel in the positive x-direction. Sim-ilarly, the surface modes defined on the lower edge have neg-ative slope and therefore are expected to travel in the negativex-direction.

Further, it can be observed that the supercell in Fig. 9 is

7

FIG. 9. (a) Supercell domain having finite size in the y-direction (38 units) and infinite size (via the application of periodic boundaryconditions) in the x- and z-directions. (b) Dispersion curves corresponding to the supercell at kz =

0.1π

azand in the neighborhood of the bandgap

marked by red box in Fig. 4 (c). Topological surface modes exist in the bandgap and are indicated in red and green color. The bulk modes areindicated in grey color. The green colored modes occur at the lower edge of the supercell while the red colored modes occur at the upper edgeas indicated in Fig. 9 (c) and Fig. 9 (d), respectively. The eigenvectors corresponding to the points indicated by (c) the green star, and (d) thered star in the dispersion curves and plotted on the supercell.

obtained by terminating the triangular lattice along the zigzagedge. However, triangular lattices exhibit also another edgetype named the armchair edge. We explored the possible prop-agation and existence of topological modes also on this edgetype. For this purpose, a finite size supercell terminated alongthe x-direction was developed, as shown in Fig. 10 (a). Thesupercell extended indefinitely in the y and z directions (pe-riodic boundary conditions were used to simulate the infinitesize). The corresponding band structure at kz =

0.1π

azis shown

in Fig. 10 (b). The bulk modes are marked in grey color, thesurface modes corresponding to the right edge of the supercellare indicated in green color, and those corresponding to theleft edge are indicated in red color. The eigenvectors corre-sponding to the surface states on both the right and left edgesare shown in Fig. 10 (c) and 10 (d), respectively. Similar tothe zigzag edge, the dispersion of the surface modes corre-sponding to the left edge has a positive slope (i.e. positivegroup velocity), which means that these modes are expectedto travel in the positive y-direction. The surface modes de-fined on the right edge have negative slope and therefore areexpected to travel in the negative y-direction. Therefore, if weassemble an extended 2D domain in the plane x-y we wouldexpect to see the surface mode to propagate in a clockwisedirection along the boundary of the lattice.

For kz = − 0.1π

az, the dispersion curves remain unaltered

but the direction of propagation of the surface modes is re-

versed. The surface mode corresponding to the upper edgeacquires a negative group velocity (propagation in the neg-ative x-direction) while the surface mode corresponding tothe lower edge acquires a positive group velocity (propaga-tion in the positive x-direction). Similarly, the surface modecorresponding to the left edge acquires a negative group ve-locity (propagation in the negative y-direction) while the sur-face mode corresponding to the right edge acquires a positivegroup velocity (propagation in the positive y-direction). Anextended 2D domain in the x-y plane would then show a wavepropagating in the anti-clockwise sense along the boundaries.

The combined behavior illustrated above is well consistentwith those of dynamical systems characterized by Weyl pointsand corresponding topologically nontrivial modes. Resultsalso suggest that, upon injecting a wave into the system, thedirection of propagation can be controlled by properly choos-ing the sign of kz. This latter aspect will be further clarified bymeans of full field simulations in the next section.

Another interesting feature of the proposed geometry is thatthe group velocity of the surface modes can be controlled sim-ply by changing the radius of the center vertical cylinder r1.Fig. 11 shows the dispersion curve of the supercell (a) for var-ious values of r1. The group velocity of the surface modes de-creases with increasing radius r1 of the center cylinder. Thisreduction of the group velocity can be explained by observ-ing that by increasing r1 the homogenized mass density of the

8

FIG. 10. (a) Supercell domain having finite size in the x−direction (40 units) and infinite size (via the application of periodic boundaryconditions) in the y- and z-directions. (b) Dispersion curves corresponding to the supercell at kz =

0.1π

azaround the bandgap marked by red

box in Fig. 4 (c). Topological surface modes exist in the bandgap and are indicated in red and green color, while the bulk modes are indicatedin grey color. The green colored modes occur at the right edge of supercell while red colored modes occur at the left edge of the supercell asindicated in Fig. 10 (c) and Fig. 10 (d) respectively. The eigenvectors corresponding to the points indicated by (c) the green star (d) the red starin the dispersion curves and plotted on the supercell.

lattice increases, but without significantly increasing the rigid-ity. Hence, given that the eigenmode does not involve signif-icant compressive or bending motion of the center cylinder,the surface modes tend to slow down. By tuning the radius r1,the surface states can gradually evolve from the single valleymechanism (similar to the 2D model discussed by Raghu62)to the intervalley mechanisms (similar to the model discussedin Kane64). In either case, the unidirectional gapless surfacestates connecting the lower and upper bulk bands are protectedby the topological nature of the band structure and are guaran-teed to be present. Note that, while a similar variation in thegroup velocity of the topological edge modes was earlier ob-served in 2D chiral materials63, this tuning ability of the sur-face states in 3D lattice was never observed and documentedin earlier studies.

V. FULL FIELD NUMERICAL SIMULATIONS

Based on the 3D unit cell developed above, a complete do-main in the x-y plane was also simulated in order to observeboth the direction of propagation and the scattering behaviorin presence of defects. The domain used is shown in Fig. 12(a). The domain is rectangular and finite along both the x-and y-directions. Both edges were set to free boundaries. Theboundaries normal to the z-axis were assigned periodic bound-ary conditions so to render the lattice infinite in this direction.Perfectly matched layers (PML) were used at the left edge toabsorb the incoming wave and allow a clear assessment of the

direction of propagation of the surface state when perform-ing a steady state analysis (i.e. it avoids the surface mode topropagate all around the boundary and get back to the source).

The domain was excited by a harmonic point force acting inthe z-direction at the location indicated by the red star markerin Fig. 12 (a). A harmonic excitation at a frequency of 40kHz was selected because right within the target topologicalbandgap where surface states exist. At kz =

0.1π

az, the source

excites the surface state with positive group velocity which isobserved propagating in the clockwise direction (see Fig. 12(b)). At kz =− 0.1π

az, the situation is inverted and the source ex-

cites an counter-clockwise propagating surface state (Fig. 12(c)). These results confirm previous observations concerningthe one-way propagation property of the surface states.

These results also allow an additional observation on theback scattering immune properties of the surface states. As thewave propagates beyond the corner of the rectangular domain,the wave amplitude on both sides is exactly comparable. Thissuggests that there are no significant reflections taking placeat the sharp corner and the wave is capable of propagatingunidirectionally.

It should be noted that while the states propagate with thesame intensity along the x- and y-aligned edges, they are asso-ciated with different eigenstates. This can be clearly observedin Fig. 12 (b) and 12 (c), and it is due to the different structureof the edges (i.e. zigzag versus armchair).

The immunity to back scattering was also assessed by in-troducing a defect in the rectangular domain represented by a

9

FIG. 11. Dispersion curves of the supercell shown in Fig. 9(a) for different values of the radius of the center cylinder. Values of r1 equalto (a) 4mm (b) 5mm (c) 6mm (d) 7mm (e) 8mm (e) 9mm are considered. It is seen that the group velocity of the topological surface modesdecreases with increasing r1 and that the surface states evolve from a single-valley to an inter-valley dominated behavior.

cutout on the top edge, as shown in Fig. 13 (a). The sourcewas located on the left edge as indicated by the star marker.The excitation was kept identical to the previous simulationscenario. By exciting at kz =

0.1π

az, a clockwise propagating

surface state is clearly obtained as shown in Fig. 13 (b). Asthe state encounters the defect on the top edge, it propagatesaround its perimeter and it continues along the original direc-tion. Observing that the intensity of the wave amplitudes bothbefore and after the defect are exactly comparable, and thatthere are no waves traveling back towards the source (reachingthe section of the edge before the source), one can concludethat no appreciable back-scattering takes place due to the de-fect. The latter is a clear hallmark of topologically protectedstates.

VI. CONCLUSIONS

Weyl points and topological surface modes in elastic solidmedia are still in their early stages of formulation and design.The very few implementations proposed to-date are based ondiscrete designs that are not conducive to use in practical ap-plications, particularly in those requiring load-bearing mate-

rials. This study presented a fully continuous, load-bearing,elastic system capable of unidirectionally-propagating andtopologically-protected surface states. The design of the 3Dunit cell consisted in a layered prismatic lattice with hexag-onal cross section in which the layers were spaced by solidcylindrical elements and by slanted circular beams connect-ing consecutive faces of the prismatic unit cell. The cylin-ders were used to set the necessary in-plane symmetry condi-tions and to provide high load carrying capacity. The slantedbeams were used to achieve chirality and allowed achievingthe necessary P-symmetry breaking conditions. Note that thelayered structure resulted in a very feasible and practical de-sign that has the potential to greatly facilitate the fabricationphase. The analysis of the lattice dynamics highlighted the ex-istence of Weyl points following the breaking of the z-mirror-symmetry and the P-symmetry of the lattice. To gain insightinto the mechanism leading to the formation of these degen-eracy points, we evaluated the topological invariants using abinitio calculations and without employing any further simpli-fications. Results were very well aligned with the expectedquantized values. The relationship between the surface statesand the topological invariants was also elaborated upon fromdifferent perspectives. kz-locked unidirectional propagating

10

FIG. 12. (a) Domain used for full field simulations. The domain is finite in size along the x- and y-directions but it employs periodic boundaryconditions along the z-direction. A PML boundary was applied along the left edge of the domain to absorb the incoming wave and to facilitatethe visualization of the direction of propagation of the surface mode during steady state analyses. The wave was injected by means of aharmonic point force applied at the location marked by the red star at a frequency of 40 kHz. (b) Eigenstate showing the propagation of thesurface state when excited with kz =

0.1π

az. (c) Eigenstate showing the propagation of the surface state when excited with kz =

−0.1π

az. These

results clearly show the unidirectional nature of the topological surface modes.

FIG. 13. (a) Full field simulations on a domain with an edge defect represented by a cut-out at the top boundary. The domain is finite alongthe x- and y-directions and it has periodic boundary conditions along the z-direction. PMLs were used along the bottom boundary to absorb theincoming wave and facilitate the observation of unidirectionally propagating states. Harmonic point load applied at the location indicated byred star at a frequency of 40 kHz. (b) Response showing the propagation of the surface wave when excited with kz =

0.1π

az. The wave clearly

travels unidirectionally from the source and can efficiently propagate around the defect with no appreciable back-scattering.

surface elastic states were predicted on the external surface ofthe 3D lattice and their existence was further confirmed by fullfield numerical simulations. Numerical results also confirmedthe extreme robustness of these states against strong latticedefects. This study may serve as a basis to develop structureshaving surface-elastic-wave-guiding capabilities for applica-tions in fields such as vibration control, energy harvesting,structural health monitoring, and on-chip telecommunication

signal processing.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support ofthe National Science Foundation under Grant No. 1761423.

11

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