Corner states in second-order mechanical topological insulator
Chun-Wei Chen, Rajesh Chaunsali, Johan Christensen, Georgios Theocharis, Jinkyu Yang
CCorner states in second-order mechanical topological insulator
Chun-Wei Chen, Rajesh Chaunsali, Johan Christensen, Georgios Theocharis, and Jinkyu Yang ∗ Aeronautics and Astronautics, University of Washington, Seattle, WA 98195-2400, USA LAUM, CNRS, Le Mans Universit, Avenue Olivier Messiaen, 72085 Le Mans, France Department of Physics, Universidad Carlos III de Madrid, ES-28916 Legan`es, Madrid, Spain (Dated: September 9, 2020)We numerically and experimentally study corner states in a continuous elastic plate with em-bedded bolts in a hexagonal pattern. While preserving C crystalline symmetry, the system cantransition from a topologically trivial to a non-trivial configuration. We create interfacial corners of60 ° and 120 ° by adjoining trivial and non-trivial topological configurations. Due to the rich interac-tion between the bolts and the continuous elastic plate, we find a variety of corner states with andwithout topological origin. Notably, some of the corner states are highly localized and tunable. Bytaking advantage of this property, we experimentally demonstrate one-way corner localization in aZ-shaped domain wall. Introduction .—Topological insulators provide re-searchers with efficient ways to tailor and control theenergy flow. These topologically non-trivial phaseshave drawn growing attentions, since the immunityto back-scattering – the key feature of topologicalprotection – can help overcome defects and sharpbends during energy transfer. Realization of thesetopological boundary states have been demonstratedin classical systems, such as acoustics and mechanics,through mimicking the quantum Hall effect [1, 2], thequantum spin Hall effect [3, 4] or the quantum valleyHall effect [5–7]. Recently, higher-order topologicalinsulators with multiple moments have been predictedtheoretically [8–10], and parallel experimental works ofthe second-order topological quadrupole insulators havebeen demonstrated in mechanics [11], microwave circuits[12], electrical circuits [13], photonics [14], and acoustics[15].To create a topological quadrupole insulator, negativehopping parameter is a requisite ingredient. In practice,however, the negative coupling needs much effort to de-sign. Therefore, another way based on bulk dipole mo-ments has been proposed to form a second-order topo-logical insulator. By leveraging the protection of crys-talline symmetry, it shows the in-gap corner states dueto the “filling anomaly” [16]. The corner states of suchsecond-order topological crystalline insulators with van-ishing quadrupole moment have been studied primarilyin square [17–24] and Kagome lattices [25–29]. Recentstudies in photonics [30, 31] have shown that hexagonallattices can also support corner modes with interestingproperties, but their mechanical analogue has been lim-ited to discrete mechanical structures which lack the en-gineering potential and practicality [32].In this Letter, we propose a bolted plate structureas a platform of a continuous second-order mechani-cal topological insulator. The plate is decorated withbolts, which act as resonators, arranged in C -symmetrichexagonal lattice. For the topological characterization,we approximate our system with a lumped-mass model [33]. Then, the topological indices are determined basedon the rotational symmetries of eigenmodes at the high-symmetry points in the Brillouin zone (BZ). By joiningtwo topologically-distinct configurations, we report thegeneration of two different types of corner modes: onewith topological origin and the other without it. Inter-estingly, we find that the one without topological originexhibits higher localization and tunability than the topo-logical one. By leveraging these characteristics, we ex-perimentally demonstrate a one-way corner localizationof mechanical waves in a Z-shaped domain wall. Thisasymmetric wave localization mechanism can be used foradvanced control of energy flow. System and unit-cell dispersion .—In Fig. 1(a), steelbolts are mounted hexagonally in an aluminum plate.See Fig. 1(b) for the graphical illustration of the unitcell. The lattice constant is a = 45 mm. R is the circum-ferential radius of six bolts, which is a tuning parameterof creating the trivial and non-trivial band gap. The en-larged unit cell in Fig. 1(c) shows that as R increases, thesix bolts gradually expand until reaching the limit of theunit cell’s boundary as represented in yellow; whereas,as the R decreases, the six bolts gradually merge intothe center as indicated in red. The band structure of R = 1 . a/ R = 0 . a/
3, and R = 1 . a/ z ) direction, whichis defined as Π z = (cid:82) V | w | dV (cid:82) V ( | u | + | v | + | w | ) dV , where V is thevolume of the plate of a unit cell and u , v and w arethe displacement components in x , y and z axes. When Π z = 1, it means that the eigenmode is completely domi-nated by the out-of-plane motion; whereas, when Π z = 0,the eigenmode is entirely dominated by the in-plane mo-tion. From Fig. 1(d), we see that there is a double Diraccone at the Γ point at 7.27 kHz. Once the radius R = a/ a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p (a)(d) (b) a a R (e) (f) R = 0.8 a /3 R = 1.1 a /3R R = 1.0 a /3 π z M K MM K M2468101214 F r e qu e n c y ( k H z ) M K M
Trivial BG
Non-trivial BG (c) M1 Γ Γ (+) (+)(-) (+) XYZ (-) Γ Γ M2 (-)(-)(-) M1 M b b Γ K a FIG. 1. (a) Continuous plate structure with hexagonally ar-ranged bolts (blue dots). The translation vectors (cid:126)a and (cid:126)a and the corresponding reciprocal vectors (cid:126)b and (cid:126)b in the firstBrillouin zone (inset). (b) A graphical illustration of the unitcell. (c) An enlarged plot of the unit cell with expanded( R > a/
3, yellow) and shrunk (
R < a/
3, red) arrangementsof the mounted bolts. (d) R = a/ Γ point.Colorbar represents the level of the out-of-plane motion. (e) R = 0 . a/ M and Γ points. (f) R = 1 . a/ Topological characterization .—We perform the topo-logical characterization of the band gaps based on thepseudospins of eigenmodes at the Γ point [35]. This ana-logue of the quantum spin-Hall effect helps us predict theexistence of chiral edge states at the interface betweentwo domains made of shrunk ( R < a/
3) and expanded(
R > a/
3) unit cells. In the present study, however,we are interested in the corner states, and therefore, wecharacterize the band gaps based on quadrupole [36] orrotation invariants [16, 30]. These rely on the parity (theeigenvalue of π rotation over the z axis) of eigenmodes atthe Γ and M points of the BZ for every band below theband gap. In the insets of Figs. 1(e) and 1(f), we ploteigenmodes for the first two bands immediately belowthe band gap at the Γ and M points (light green stars)and calculate their parity. For the shrunk configuration,the two bands have − Γ and M points.However, for the expanded configuration, the two bandshave +1 parity at the Γ point, but the opposite parityat the M point.Ideally, this characterization process needs to be re-peated for all bands below the band gaps. This is acomplicated task due to the existence of the numerousdispersion curves [see Figs. 1(e) and 1(f)], which resultfrom the multi degrees of freedom and coupling between - /a 0 /a67891011 F r e qu e n c y ( k H z ) (a) (b) BG1BG2
Topologicalinterface R = 1.1 a /3 R = 0.8 a /3 FIG. 2. (a) A supercell made by placing six non-trivial( R = 1 . a/
3) and six trivial ( R = 0 . a/
3) cells adjacently.(b) Eigenfrequencies of the supercell as a function of wavenumbers in the periodic direction. Bulk bands are in black.There are two edge modes with opposite pseudospins (purpleand yellow) inside the bulk band gap, where two mini gaps(BG1 and BG2) are generated. the bolt and the plate. To simplify this, we approximatethe system into a lumped-mass model, in which the boltsare modeled as point masses connected to the plate withtransverse springs [33] (see Supplemental Material [34]for more details on this model and its validity). As aresult, we can calculate the parity for all the bands be-low the band gap, and obtain quadrupole 0 and 1/2, androtation invariants [0 ,
0] and [2 ,
0] for the shrunk (trivial)and expanded (nontrivial) unit cells, respectively. Thisestablishes topological distinction of the two band gapsshown in Figs. 1(e) and 1(f).
Supercell analysis .—We return to the FEA approachand first perform a supercell analysis by adjoining sixnon-trivial ( R = 1 . a/
3) and six trivial cells ( R = 0 . a/ gapless edge statesprotected by the time-reversal symmetry [38]. Eventhough there are ways to close these gaps for a broad-band wave propagation at the interface [39], we deliber-ately use them to look for potential corner modes in thisstudy. BG1 is at low frequency and resides within theinterface mode spectrum. This emerges due to the break-age of crystalline C symmetry at the interface and existsas long as there are cells across the interface with dif-ferent R (see the details in Supplemental Material [34]).BG2 is at a higher frequency and lies above this spectrumand below the bulk spectrum (black) and emerges due tothe large mismatch of radii between trivial and nontrivialbolted-plate lattices (Supplemental Material [34]). (a) (b)(c)(d) R = 0.8 a /3 R = 1.1 a /3 Solu � on number F r e qu e n c y ( k H z ) (a)Inverted (a) Corner (60°)Corner (120°) ƒ = 7.41 kHz ƒ = 7.42 kHzƒ = 7.43 kHz ƒ = 7.46 kHz (c) (e)(f)(d) (f) ƒ = 8.48 kHz ƒ = 8.51 kHz (e) ƒ = 8.40 kHz ƒ = 8.42 kHz BG1BG2
Max.0
FIG. 3. (a) A rhombus-shaped structure with an interfacebetween two domains: R = 1 . a/ R = 0 . a/ | w | . Emergence of corner states .—To observe these cor-ner states and investigate their differences systemati-cally, we construct a rhombus-shaped, topologically non-trivial domain ( R = 1 . a/
3) inside the trivial domain( R = 0 . a/
3) [Fig. 3(a)]. This contains two 120 ° cornersand two 60 ° corners. We also consider the “inverted” con-figuration, in which the two domains are interchanged.We then perform the eigenfrequency analysis on bothconfigurations and show the results in Figs. 3(b). We ob-serve the emergence of several corner states marked withthe green and red stars.In Fig. 3(c), we show the low-frequency corner statesfor the rhombus-shaped structure of Fig. 3(a) while in Fig. 3(d) the corner modes of the inverted configuration.These corner states reside in the BG1. Importantly, wefind that only 120 ° corners support corner states. Thesestates are of topological origin. For the verification, weparameterize our system with the unit cells with varyingradii and find that these corner states exist robustly evenfor a small difference in radii between the trivial and non-trivial cells (see Supplemental Material [34] for details).Moreover, a corner state of the topological origin shouldpersist when two domains are interchanged, since theirexistence is based on the difference in the topological na-ture of the two adjacent domains. This is exactly whatwe observe in Figs. 3(c) and 3(d). Upon closely exam-ining the mode shapes of these states in Fig. 3(c), wefind that the peak displacement within the non-trivialunit cell, located at the corner, occurs at the two boltsadjacent to the most cornered bolt, which shows the min-imum displacement. This is consistent with the shapesof the topologically-nontrivial modes reported in photon-ics recently [30]. The inversion of domains in Fig. 3(d),however, changes the mode shape, and now the peak dis-placement within the non-trivial unit cell occurs at themost cornered bolt.Interestingly, there are also high-frequency cornerstates, marked by red and green star in Fig. 3(b), whichreside within BG2. We observe that both ° and 120 ° corners support these states as shown in Figs. 3(e) and3(f). However, these exist only in the regular configura-tion shown in Fig. 3(a), but not in the inverted configura-tion. This hints at their nontopological origin, which weverify by performing a parametric study again with vary-ing radii of unit cells (see Supplemental Material [34] fordetails). We find that these states exist only for a largedifference in radii between the trivial and non-trivial unitcells and are not predicted by our simplified lumped-massmodel, as opposed to the corner states within BG1. Thissuggests that they appear due to the complex interactionof bolt-plate assembly that the lumped-mass model failsto capture. Their mode shapes also differ from the onesin BG1. For example, the states shown in Fig. 3(e) havethe peak displacement occurring at the most corneredbolt within the non-trivial unit cell, while the adjacenttwo bolts have also nonzero displacements. This is alsoconsistent with the topologically-trivial corner modes re-ported in Ref.[30].What makes the corner states observed in BG2 uniqueis their tunability when domains are interchanged, andalso their spatial localization which is much higher com-pared to the corner states in BG1 [compare the localiza-tion lengths in Figs. 3(e) and 3(f) to those in Figs. 3(c)and 3(d)]. These characteristics will be leveraged toachieve one-way energy localization as will be describedbelow. Experimental demonstration .—We build a Z-shapeddomain wall that includes two different types of 60 ° cor-ners in one setup (Fig. 4(a), see Supplemental Material R = 0.8 a /3 R = 1.1 a /3 ƒ = 8.49 kHzƒ = 8.49 kHz R = 0.8 a /3 R = 1.1 a /3 (a) Max. (b)(c) (d) R = 0.8 a /3 R = 1.1 a /3 (II)(I) ƒ = 8.52 kHz Sim.Exp.Exp.
FIG. 4. Experimental verification of the corner state at 60 ° corners. (a) A Z-shaped interface with two different 60 ° cor-ners is created via placing trivial cells ( R = 0 . a/
3) withnon-trivial cells ( R = 1 . a/
3) adjacently. The red stars markthe locations of the piezo-actuators that excite the elasticplate. (b) A simulated eigenmode shows that corner stateappear only at the corner (II) at f = 8 .
52 kHz. (c)-(d) Themeasured wave-field response of the bolted plate when thepiezo-actuator is attached on the corner (I) and corner (II),respectively, and excited at f = 8 .
49 kHz. [34] for the fabrication and measurements detail). Fromthe simulation results shown earlier, we know that thecorner state without the topological origin exists onlyin the case when the non-trivial cells are surroundedby the trivial cells [Figs. 3(e) and 3(f)], i.e., at corner(II) in Fig. 4(a). This is again verified by the eigen-frequency analysis on this particular setup. Fig. 4(b)shows a highly-localized corner state with f = 8 .
52 kHzat the corner (II).We excite the plate using a piezoelectric ceramic actu-ator by placing it at corner (I) and (II) in two separateexperiments. We use a chirp signal with the frequencyrange of 2–40 kHz. A point-by-point measurement is thenconducted by using the laser Doppler vibrometer to de-tect the flexural waves. By gathering and reconstructingmeasured data from all the points, we plot the steadystate wave-field at f = 8 .
49 kHz (Figs. 4(c, d)). Whencorner (I) is excited, there is no evidence of a corner stateapart from the usual exponentially decaying evanescentfield [Fig. 4(c)]. When we excite corner (II), however, weobserve clear confinement of energy due to the presenceof the corner mode [Fig. 4(d)]. In addition, the profileof the corner mode matches closely with the simulationresults in that the last two resonators of the non-trivialunit cell have peak displacements [compare Fig. 4(b) andFig. 4(d)]. It should also be noted that the frequencies ofthe corner state between the experimental and computa-tional results are in excellent agreement.Next, we exploit this selective localization observed inthe previous test to demonstrate a one-way localizationthrough the Z-shaped interface. We excite the plate witha harmonic excitation at f = 8 .
49 kHz in the middle ofthe interface as shown in in Fig. 5. We observe that the flexural waves departing from the point of excitationpropagate towards the right direction only, thereby ex-citing the corner (II) only, whereas the corner (I) beingat the same distance does not see any such energy local-ization. Such asymmetric wave localization is a highlyuseful – yet relatively unexplored – feature that can beexploited to manipulate energy flow at will.
Conclusions .—We propose a ubiquitous design of abolted plate in the hexagonal arrangement to demon-strate in-gap corner states in our C -symmetry-protectedsystem. By changing the radius of the unit cell, we con-struct two configurations that show topologically distinctband gaps. We perform topological characterization ofthe bolted-plate assembly based on a simple lumped-massmodel. When two such topologically distinct bolted-plates are placed adjacently, we conduct full geometrysimulations to show that there are two regions (minigaps) in frequency where different types of corner modescan exist. We find that the low-frequency corner statesare of topological origin and the high-frequency cornerstates are of nontopological origin. While the formercan be predicted by the lumped-mass model, the latercan not. However, the later are highly localized for at both ° and 120 ° corners and can be made to exist ornon-exist based on the inversion of topologically-distinctdomains across the interface. This fact is thus used tocreate a Z-shaped interface between topologically distinctdomains for achieving an asymmetric localization of en-ergy. We expect that these findings will enrich the wave-localization phenomena in mechanics and encourage newapplications in vibration management.C.-W. C. and J. Y. are grateful for the support fromNSF (CAREER1553202 and EFRI-1741685). R. C. andG. T. acknowledge support by the project CS.MICROfunded under the program Etoiles Montantes of the Re-gion Pays de la Loire. J. C. acknowledges the supportfrom the European Research Council (ERC) throughthe Starting Grant 714577 PHONOMETA and from theMINECO through a Ram´on y Cajal grant (Grant No. ƒ = 8.49 kHz Max.0
Piezo-actuator Corner state R = 0.8 a /3 R = 1.1 a /3 Exp.
FIG. 5. Experimental demonstration of asymmetric wave lo-calization when the piezo-actuator is attached in the middleof the Z-shaped interface.
RYC-2015-17156).C.-W. C. and R. C. contributed equally to this work.
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