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https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.123.034301
DOI: 10.1103/PhysRevLett.123.034301
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DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo
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Edge States and Topological Pumping in Spatially Modulated Elastic Lattices
Matheus I. N. Rosa,1Raj Kumar Pal,2Jos´e R. F. Arruda,3 and Massimo Ruzzene1,2
1School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA 2School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA 3School of Mechanical Engineering, State University of Campinas, Campinas, São Paulo 13083-970, Brazil
(Received 28 November 2018; revised manuscript received 26 April 2019; published 17 July 2019) Spatial stiffness modulations defined by the sampling of a two-dimensional surface provide one-dimensional elastic lattices with topological properties that are usually attributed to two-one-dimensional crystals. The cyclic modulation of the stiffness defines a family of lattices whose Bloch eigenmodes accumulate a phase quantified by integer valued Chern numbers. Nontrivial gaps are spanned by edge modes in finite lattices whose location is determined by the phase of the stiffness modulation. These observations drive the implementation of a topological pump in the form of an array of continuous elastic beams coupled through a distributed stiffness. Adiabatic stiffness modulations along the beams’ length lead to the transition of localized states from one boundary, to the bulk and, finally, to the opposite boundary. The first demonstration of topological pumping in a continuous elastic system opens new possibilities for its implementation on elastic substrates supporting surface acoustic waves, or to structural components designed to steer waves or isolate vibrations.
DOI:10.1103/PhysRevLett.123.034301
The investigation of topologically protected modes for robust waveguiding along boundaries or interfaces has attracted significant interest across different physical realms, including quantum[1], electromagnetic[2], acoustic[3,4], and elastic[5]media. In mechanics, topologically protected modes have been successfully observed by establishing quantum Hall effect and quantum spin Hall effect analogues through time-reversal symmetry [6–11] and spatial sym-metry[5,12–16]breaking.
Recently, topological waveguiding has been pursued by accessing higher-dimensional topological effects in lower-dimensional systems [17,18]. This approach has been exploited to induce adiabatic pumping of waves in one-dimensional (1D) quasiperiodic photonic waveguides
[19,20], and to implement a four-dimensional (4D) quan-tum Hall system through two-dimensional (2D) periodic arrays[21]. Also, localized modes in a chain of mechanical spinners arranged in patterns obtained by projections from manifolds have been studied in [22]. In this Letter, we investigate a family of 1D elastic lattices where nearest-neighbor interactions are defined by the sampling of a 2D surface, in a configuration inspired by the Aubry-Andr´e-Haper model [23,24]. In line with previous work in photonics and quantum mechanics [19,25], we show that the nontrivial topological properties of the lattice family are related to the existence of edge states for individual, finite 1D lattices. We then consider an array of continuous beams elastically coupled by a distributed stiffness modulated along the beams’ length. Such a system, although con-tinuous in nature, is governed by an eigenvalue problem of the same form as the discrete lattice family, and, therefore,
has similar topological properties. This array supports a topological pump, whereby the edge modes propagate along the beams’ length, and undergo an adiabatic edge-to-edge transition produced by varying the phase of the coupling stiffness modulation. This first demonstration of a topological pump in a continuous elastic system opens new avenues for the physical implementation of topology-based wave guiding in mechanics, which has potential technological relevance for guiding of surface, guided, and bulk waves in acoustic devices, ultrasonic imaging, and nondestructive evaluation.
We consider a 1D lattice [Fig.1(a)] of equal masses m, equally spaced by a unit distance, and connected by springs whose constant kn is defined by sampling the 2D surface
Sðx; ϕÞ ¼ cos ð2πτx þ ϕÞ at xn ¼ n [Fig. 1(b)]. Its
topo-logical properties are defined by specific choices and variations of fτ; ϕg, and lead to localized states at the boundaries of finite lattices, for both rational and irrational τ values [19,25]. Herein, we consider τ ¼ p=q, where integers p and q are co-prime defining a periodic lattice with q masses per unit cell. Accordingly, the spring constant kn is defined as kn¼ k0 1 þ α cos 2πnp qþ ϕ : ð1Þ
A family of lattices is associated with different values ofϕ, which is regarded as an additional dimension remnant of the projection from Sðx; ϕÞ. The cross sections (blue lines) in Fig.1(b)show the spring constant variation along the lattice for τ ¼ p=q ¼ 1=3 and ϕr¼ ð2πr=3Þ (r ∈ ½0; 3), which
illustrates that knðϕ þ 2π=qÞ ¼ knþsðϕÞ, where s obeys the
algebraic congruence relation
ps≡ 1 ðmod qÞ: ð2Þ Equation (2) has two solutions in the range jsj < q [26], which define two shifts of stiffness in opposite directions along the lattice. For p=q¼ 1=3 [Fig.1(b)], the solutions s1;2¼ f1; −2g correspond, respectively, to a shift of one position to the left and of two positions to the right.
The effect of these stiffness shifts on the dispersion topology is investigated by expressing the equation for longitudinal motion of mass n at frequency ω as
mω2un¼ ðknþ kn−1Þun− knunþ1− kn−1un−1: ð3Þ
Imposing Bloch periodicity conditions unþq¼ eiμun,
where μ is the nondimensional wave number, leads to the eigenvalue problem
Kðμ; ϕÞu ¼ Ω2u; ð4Þ
where Kðμ; ϕÞ is a stiffness matrix and Ω ¼ ω=ω0 is a nondimensional frequency, with ω0¼pffiffiffiffiffiffiffiffiffiffiffik0=m. Solution forμ ∈ ½0; 2π gives a set of eigenvalues and eigenvectors that are defined byϕ, whose smooth cyclic variation, i.e., ϕ → ϕ þ 2π, leads to a phase accumulation [see details in the Supplemental Material (SM) [27]]
topology of the Bloch eigenmodesuðμ; ϕÞ and is quantified by the Chern number of this vector field in the ðμ; ϕÞ ∈ T2¼ ½0; 2π × ½0; 2π space[7,30]:
C¼ 1 2πi
Z
D∇ × AdD; ð6Þ
where D ¼ T2, ∇ ¼ ð∂=∂μÞeμþ ð∂=∂ϕÞeϕ, and A ¼ u· ∇u, with ðÞ denoting a complex conjugate. Analytical
evaluation of the Chern number according to the procedure detailed in the SM[27]gives C¼ s, where s is one of the solutions of Eq.(2). Among the two solutions, the Chern number is numerically evaluated using the procedure described in [31], which yields the results displayed in Fig.2(a). A label for gap r is given by the algebraic sum of the Chern numbers of the bands below it, i.e., CðrÞg ¼
Pr
n¼1Cn [32]. This gives Cð1Þg ¼ 1 and Cð2Þg ¼ −1 for the
lattice considered herein [Fig. 2(a)]. Nonzero gap labels signal the presence of topological edge modes spanning the band gaps as the phase parameterϕ varies[25,33]. This is illustrated for a free-free lattice with N¼ 60 masses, p=q¼1=3, N=q ¼ 20 unit cells, and α ¼ 0.6. Figure2(b)
shows the eigenfrequencies (black lines), superimposed to the bulk bands depicted by the shaded gray areas. The presence of two additional modes (red lines) spanning the band gaps is a notable feature of the finite system spectrum, where solid and dashed lines indicate modes localized at the right and left boundary, respectively. The localized states of the finite system change their localization edge as their branches touch the boundaries of the corresponding gaps. An example of such a transition for the edge mode in the second gap is shown in Fig. 2(c). The transitions are characterized by the gap label, which measures singularities (zero components) of the Bloch eigenvectors along the branch defining the lower boundary of the gap [33]. Its absolute valuejCgj equals the number of times the left (or
right) edge state traverses the gap forϕ ∈ ½0; 2π. In Fig.2, both gap labels are jCgj ¼ 1, which indicates that the associated edge states traverse the gap once. Furthermore, the gap label sign defines the direction of the transition: Cg >0 defines a right-to-left transition for increasing ϕ,
whereas Cg<0 signals an opposite transition. For Cg >0, the transitions occur when the branches for both modes merge with the lower boundary of their gap atϕ ¼ 0; 2π, while an opposite occurs when the modes touch the upper boundary of the gap atϕ ¼ π. (See SM[27]for an additional example for a p=q¼ 1=5 lattice.)
The transition of the edge states is employed for a topological pump that exploits the adiabatic variation of ϕ along a second dimension[19]. We illustrate the concept in an array of continuous waveguides (beams) aligned along the y direction, and coupled by distributed springs of
FIG. 1. (a) Spatially modulated 1D lattices with modulated stiffness kn¼k0½1þαcosð2πnp=qþϕÞ. (b) Surface Sðx; ϕÞ ¼
cosð2πτx þ ϕÞ generating the stiffness constants by sampling at xn¼ n. Black lines: stiffness variation with ϕ at a lattice site; blue
lines: cross sections atϕr¼ ð2πr=3Þ (r ∈ ½0; 3) with the red dots
showing the sampled stiffness values.
constantγn along x [Fig. 3(a)]. The springs react propor-tionally to the relative transverse displacement (also along x) of neighboring beams, so that the governing equation for the nth beam reads
EI∂ 4w n ∂y4 þ ρA ∂2w n ∂t2 þ ðγn−1þ γnÞwn − γn−1wn−1− γnwnþ1¼ 0; ð7Þ
where wnis the displacement of the nth beam, whileρ, E,
A, and I denote, respectively, mass density, Young’s modulus, cross-sectional area, and second moment of area of each beam. Imposing plane wave harmonic motion along y, i.e., wnðy; tÞ ¼ wne−iðωt−κyyÞ where κy is the wave
number, gives
ðρAω2− EIκ4 yÞwn
¼ ðγnþ γn−1Þwn− γnwnþ1− γn−1wn−1: ð8Þ
Equation (8)is of the same form as the lattice equation [Eq. (3)]. Thus, similar topological considerations can be made for modulations of the coupling stiffness in the form γn¼ γ0f1 þ α cos½2πnðp=qÞ þ ϕg. Considering again p,
q as co-primes, e.g., p=q¼ 1=3, and enforcing Bloch conditions wnþq ¼ eiμxw
n leads to an eigenvalue problem
in the formKðμx;ϕÞw ¼ λ2w, where K is the same stiffness matrix that appears in Eq.(4). The only differing term is the expression of the eigenvalue λ2¼ ðΩ2− β4yÞ, where Ω ¼
ω=ω0, with ω20¼ ρA=γ0, and βy¼ κyðEI=γ0Þ1=4. For
arrays with a finite number N of beams, letting βy ¼ 0
(a) (b) (c)
FIG. 2. Dispersion properties for p=q¼ 1=3, α ¼ 0.6 lattice. (a) Dispersion surfaces as a function of μ and ϕ showing three bulk bands and two band gaps. Corresponding Chern numbers and gap labels are added for convenience. (b) Natural frequencies of a commensurate finite chain of 60 masses and their variation in terms ofϕ (black lines) superimposed to the bulk bands (shaded gray regions), where edge modes (red lines) span the gaps. (c) Magnitude of the topological mode in the second gap and its the transition from right to left localization resulting from the variation ofϕ.
FIG. 3. Dispersion properties for system of coupled beams. (a) Schematic of the lattice of coupled beams: thick black lines show the beams in a notional deformed configuration, while the red lines correspond to the undeformed state. The thin black horizontal lines depict the distributed springs of constantγncoupling the beams. (b) DispersionΩðϕ; βyÞ for array comprising N ¼ 15 identical beams
with stiffness parameters p=q¼ 1=3, α ¼ 0.6. (c) Wave number βyas function ofϕ for fixed frequency Ω ¼ 2.1, corresponding to the
cross section illustrated by the blue plane in (b). The contours represent the Fourier decomposition of the displacement fieldj ˆWðτ; βyÞj,
and ϕ ∈ ½0; 2π produces a diagram identical to that of Fig.2(b), with edge states spanning the bulk gaps. Variation ofβyshifts the entire spectrum, including the edge states, as
shown for a system of N¼ 15 beams in Fig.3(b), where frequenciesΩðϕ; βyÞ of the edge states are represented by
red surfaces, while the bulk bands are the shaded gray volumes. Motivated by the topological pumping process described next, the dispersion surfaces are displayed in the range ϕ ∈ ½1.75π; 2.25π, which corresponds to ϕ ∈ ½−0.25; 0.25π due to the periodicity of γnðϕÞ. As in 1D
lattices, variations ofϕ lead to transitions of the edge states. The dashed and solid red lines at βy¼ 0.8 in Fig. 3(b),
respectively, denote left-localized and right-localized modes, and illustrate the transition at ϕ ¼ 2π that occurs along the entire surface of the edge states.
To implement the pump, we first observe a cross section of the dispersion diagram of Fig.3(b)at a given frequency [blue plane at Ω ¼ 2.1 in Fig. 3(b)], which is shown in Fig.3(c). The bulk bands (shaded gray areas) are populated by the modes of the finite system (solid black lines), while the edge states in the gaps are again highlighted by the red lines (solid and dashed). By exciting the system of beams at this frequency, and applying a slow modulation of ϕ along y, topological pumping can be induced through the adiabatic[34]evolution of one of the edge states. In contrast to time-modulated materials, the pumping in spatially modulated lattices occurs at a single frequency along the wave number branch of the edge state. To demonstrate this, we consider a system of N¼ 15 coupled beams that are infinite alongþy and are excited at y ¼ 0 by an imposed displacement in the form of a 50-cycle sine burst signal of center frequency Ω ¼ 2.1, and x-wise spatial distribution corresponding to the localized mode in the first gap of Fig.3(c). The parameterϕ is varied linearly along y from ϕ1¼ 1.75π to ϕ2¼ 2.25π to induce the mode transition
from left localized to right localized. The response of the lattice is evaluated using the frequency domain formulation of the equations of motion, as detailed in the SM[27], which
allows for a solution that is computationally efficient and free from potential inaccuracies caused by spatial polynomial discretization. The simulation results of Fig. 4 show the lattice deformed configuration at three subsequent normal-ized time instantsτ ¼ ω0t. The figures illustrate the tran-sition from left-localized wave (left), to a bulk wave (middle), and finally to a right-localized wave (right) that characterizes the pump. The video animation of the full transient response is provided in the SM[27].
To illustrate the adiabatic nature of the pump, we perform a time-frequency analysis to represent the dis-placement field in the reciprocal space ˆWðτ; Ω; μx;βyÞ as a
function of time. For visualization purposes, the L2norm along the fΩ; μxg dimensions is then taken so that a function of only time and normalized wave number βy
remains, i.e., ˆWðτ; βyÞ. The results are displayed as contour plots in Fig.3(c), where the color represents the normalized magnitude of the decomposed displacement fieldj ˆWðτ; βyÞj. The procedure reveals the evolution experienced by the waves: their amplitude is initially distributed around wave numbers βy corresponding to the excited left-localized
topological mode, and, as time elapses, it follows the evolution of that mode. At the end of the process, most of the energy is concentrated on the right-localized mode, while some energy is scattered to a neighboring bulk mode. Under the conditions stated by the adiabatic theorem [34], such scattering is expected to occur near the transition point ϕ ¼ 2π, where the eigenvalue of the topological mode approaches that of the neighboring bulk mode.
We investigate the implementation of a topological pump through spatial modulation of the properties of elastic lattices. The presented results open avenues for the imple-mentation of adiabatic topological pumping in continuous elastic media where material properties and/or geometry can be designed to produce spatial stiffness or inertia modulations. The proposed framework allows the study of systems with properties that are generally modulated along a second dimension, both spatial and temporal.
FIG. 4. Topological pumping through adiabatic evolution of the edge state. Snapshots of three time instants illustrate the transition from left-localized, to bulk and finally to right-localized mode as the wave propagates along the y direction. Deformed configurations show the absolute value of beam deflectionsjwnðy; tÞj, represented along the vertical direction and also quantified by the associated
color map for ease of visualization.
In future work the role of nonlinear interactions may be explored in the context of amplitude-dependent behavior
[35], robustness of the edge states[36], and as a gap opening mechanism in analogy with Mott insulators[37]. The results provide guidelines for future designs of structural compo-nents or acoustic waveguides capable of selectively guiding waves along desired paths, and of localizing perturbation in predefined regions of the domains.
The authors acknowledge the support from the National Science Foundation (EFRI 1741685 grant) and of the Brazilian agency FAPESP through Project No. 2016/ 22432-2.
[1] M. Zahid Hasan and C. L. Kane, Colloquium: Topological insulators,Rev. Mod. Phys. 82, 3045 (2010).
[2] L. Lu, J. D. Joannopoulos, and M. Soljačić, Topological photonics,Nat. Photonics 8, 821 (2014).
[3] Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, Topological Acoustics,Phys. Rev. Lett. 114, 114301 (2015).
[4] J. Lu, C. Qiu, L. Ye, X. Fan, M. Ke, F. Zhang, and Z. Liu, Observation of topological valley transport of sound in sonic crystals,Nat. Phys. 13, 369 (2017).
[5] S. H. Mousavi, A. B. Khanikaev, and Z. Wang, Topologi-cally protected elastic waves in phononic metamaterials,
Nat. Commun. 6, 8682 (2015).
[6] K. v. Klitzing, G. Dorda, and M. Pepper, New Method for High-Accuracy Determination of the Fine-Structure Con-stant Based on Quantized Hall Resistance,Phys. Rev. Lett. 45, 494 (1980).
[7] D. J. Thouless, M. Kohmoto, M. Peter Nightingale, and M. den Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405 (1982).
[8] P. Wang, L. Lu, and K. Bertoldi, Topological Phononic Crystals with One-Way Elastic Edge Waves, Phys. Rev. Lett. 115, 104302 (2015).
[9] L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, Topological mechanics of gyroscopic metamaterials, Proc. Natl. Acad. Sci. U.S.A. 112, 14495 (2015).
[10] A. Souslov, B. C. van Zuiden, D. Bartolo, and V. Vitelli, Topological sound in active-liquid metamaterials,Nat. Phys. 13, 1091 (2017).
[11] N. P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine, Amorphous topological insulators con-structed from random point sets,Nat. Phys. 14, 380 (2018). [12] R. Süsstrunk and S. D. Huber, Observation of phononic helical edge states in a mechanical topological insulator,
Science 349, 47 (2015).
[13] R. K. Pal, M. Schaeffer, and M. Ruzzene, Helical edge states and topological phase transitions in phononic systems using bi-layered lattices,J. Appl. Phys. 119, 084305 (2016). [14] M. Miniaci, R. K. Pal, B. Morvan, and M. Ruzzene,
Experimental Observation of Topologically Protected Hel-ical Edge Modes in Patterned Elastic Plates,Phys. Rev. X 8, 031074 (2018).
[15] J. Chen, H. Huang, S. Huo, Z. Tan, X. Xie, J. Cheng, and G.-l. Huang, Self-ordering induces multiple topological transitions for in-plane bulk waves in solid phononic crystals, Phys. Rev. B 98, 014302 (2018).
[16] R. Chaunsali, C.-W. Chen, and J. Yang, Subwavelength and directional control of flexural waves in zone-folding induced topological plates, Phys. Rev. B 97, 054307 (2018).
[17] T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics,Phys. Rev. A 93, 043827 (2016).
[18] Y. E. Kraus and O. Zilberberg, Quasiperiodicity and top-ology transcend dimensions,Nat. Phys. 12, 624 (2016). [19] Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O.
Zilberberg, Topological States and Adiabatic Pumping in Quasicrystals, Phys. Rev. Lett. 109, 106402 (2012). [20] M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y.
Silberberg, Observation of Topological Phase Transitions in Photonic Quasicrystals,Phys. Rev. Lett. 110, 076403 (2013). [21] O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, Photonic topo-logical boundary pumping as a probe of 4D quantum Hall physics,Nature (London) 553, 59 (2018).
[22] D. J. Apigo, K. Qian, C. Prodan, and E. Prodan, Topological edge modes by smart patterning, Phys. Rev. Mater. 2, 124203 (2018).
[23] P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field,Proc. Phys. Soc. London Sect. A 68, 874 (1955).
[24] S. Aubry and G. Andr´e, Analyticity breaking and anderson localization in incommensurate lattices, Ann. Isr. Phys. Soc. 3, 18 (1980).
[25] V. M. Alvarez and M. D. Coutinho-Filho, Edge states in trimer lattices,Phys. Rev. A 99, 013833 (2019).
[26] D. M. Burton, Elementary Number Theory (Tata McGraw-Hill Education, New York, 2006).
[27] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.123.034301 for more
details, which includes Refs. [28,29], and for a video animation of the transient simulation.
[28] J. F. Doyle, Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms, Mechanical Engineering Series (Springer, New York, 1997). [29] J. Bellissard and B. Simon, Cantor spectrum for the almost
Mathieu equation,J. Funct. Anal. 48, 408 (1982). [30] Y. Hatsugai, Chern Number and Edge States in the Integer
Quantum Hall Effect, Phys. Rev. Lett. 71, 3697 (1993). [31] T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in
discretized Brillouin zone: Efficient method of computing (spin) Hall conductances, J. Phys. Soc. Jpn. 74, 1674 (2005).
[32] Y. Hatsugai, Topological aspects of the quantum Hall effect,
J. Phys. Condens. Matter 9, 2507 (1997).
[33] Y. Hatsugai, Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function,Phys. Rev. B 48, 11851 (1993).
[34] H. Nassar, H. Chen, A. N. Norris, and G. L. Huang, Quantization of band tilting in modulated phononic crystals,
phononic lattices,Phys. Rev. E 97, 032209 (2018). [36] J. Vila, G. H. Paulino, and M. Ruzzene, Role of
non-linearities in topological protection: Testing magnetically
[37] M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, Exploring 4D quantum Hall physics with a 2D topological charge pump,Nature (London) 553, 55 (2018).
