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Classification of Topological Phonons in Linear Mechanical Metamaterials

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Classification of Topological Phonons in Linear Mechanical Metamaterials Classification of topological phonons in linear PNAS PLUS mechanical metamaterials Roman Süsstrunka,1 and Sebastian D. Hubera aInstitute for Theoretical Physics, ETH Zürich, 8093 Zurich, Switzerland Edited by Mordechai (Moti) Segev, Technion–Israel Institute of Technology, Haifa, Israel, and approved June 21, 2016 (received for review April 4, 2016) Topological phononic crystals, alike their electronic counterparts, are states of self-stress (35), where external loads on a material are characterized by a bulk–edge correspondence where the in- can be absorbed in the region of a topological boundary mode. terior of a material dictates the existence of stable surface or Defining such details of the load-bearing properties of a material boundary modes. In the mechanical setup, such surface modes are relevant both for smart adaptive materials (9) as well as for can be used for various applications such as wave guiding, vibra- civil engineering applications. The design of finite-frequency tion isolation, or the design of static properties such as stable properties, on the other hand, constitutes a quite different field floppy modes where parts of a system move freely. Here, we of research. Here, the goals are to control the propagation, re- provide a classification scheme of topological phonons based flection, or absorption of mechanical vibrations. This includes, on local symmetries. We import and adapt the classification of for example, waveguiding, acoustic cloaking, or vibration noninteracting electron systems and embed it into the mechan- isolation ranging from the seismic all of the way to the radi- ical setup. Moreover, we provide an extensive set of examples ofrequency scale. that illustrate our scheme and can be used to generate models in A second important separation into two distinct classes of unexplored symmetry classes. Our work unifies the vast recent materials arises from the presence or absence of nonrecipro- literature on topological phonons and paves the way to fu- cal elements (43). Generically, nondissipative mechanical prop- ture applications of topological surface modes in mechanical erties are invariant under the reversal of the arrow of time. metamaterials. Nonreciprocal elements, however, transmit waves asymmet- rically between different points in space. The absence of time- PHYSICS topological matter | mechanical metamaterials | adaptive materials reversal symmetry allows for a topological invariant, the Chern number, which encodes chiral, or unidirectional wave propaga- echanical metamaterials derive their properties not from tion. We will see that these two attributes: static vs. dynamic Mtheir microscopic composition but rather through a clever and reciprocal vs. nonreciprocal will be key to understand how engineering of their structure at larger scales (1). Various design the electronic classification is naturally modified for mechani- principles have been put forward and successfully applied in the cal systems. past. Examples range from periodic modifications leading to Before we embark on the development of the framework band gaps via Bragg scattering (2) to the use of local resonances needed for our classification, let us state our goals more pre- (3) to achieve subwavelength functionalities. Recently, the con- cisely. Our aim is to import and adapt the classification of cept of “band topology” emerged as a new design principle for noninteracting electron systems according to their local sym- mechanical metamaterials (4–14). Colloquially speaking, a sys- metries T , C, and S = T ○ C, that is, time reversal, charge conju- tem with a topological phonon band structure will possess me- gation, and their combination, respectively (20, 21). Clearly, we chanical modes bound to surfaces or lattice defects that are will have to specify the role of these symmetries in mechanical immune to a large class of perturbations. If the targeted purpose systems. Moreover, we cover only the “strong” indices, which do of a metamaterial is encoded in such a topologically protected not rely on any spatial symmetries. The extension to weak indices, mode, its functioning will be largely independent of production arising from a stacking of lower-dimensional systems carrying strong imperfections or environmental influences. indices, is straightforward (5). Finally, there are many recent de- The introduction of topology to the field of mechanical met- velopments dealing with topological phases stabilized by spatial amaterials was largely motivated by its successful application to – the description of electrons in solids (15) and to photonics (16 Significance 19). One of the key elements in the understanding of the elec- tronic systems was the classification of different topological – A mechanical metamaterial is an engineered material that is phases according to their symmetry properties (20 22). Although characterized by properties that go beyond the properties of over the last years numerous proposals (4, 5, 23–42) and several – its microscopic building blocks. Specifically, in topological experiments (6 14) were put forward promoting mechanical metamaterials, one makes use of surface or boundary modes topological metamaterials, a complete classification of linear that are stable against imperfections or environmental influ- topological phonons is missing to date. In this report, we intend ences and hence constitute reliable building blocks for various to fill in this gap. applications. In our work, we provide a classification scheme of At first sight, the dynamics in classical mechanics seems to be possible topological metamaterials and an extensive number of rather different from quantum-mechanical electron systems. Our examples illustrating this scheme. Our classification can serve approach is therefore to map the first to the second problem as an important blueprint for many future applications that (26). This, in principle, allows us to import the classification (20, target such stable boundary modes for engineering purposes. 21) from the description of electronic systems. However, a bare import of this classification is not doing justice to the rich structure Author contributions: R.S. and S.D.H. designed research, performed research, and wrote mechanical systems possess by themselves. the paper. We can categorize mechanical metamaterials by two inde- The authors declare no conflict of interest. pendent properties. First, the targeted functionality can either be This article is a PNAS Direct Submission. at zero or at finite frequencies. Zero-frequency modes define 1To whom correspondence should be addressed. Email: [email protected]. structural properties such as mechanisms where parts of a ma- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. terial move freely (5, 8). The dual partners of freely moving parts 1073/pnas.1605462113/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1605462113 PNAS | Published online August 1, 2016 | E4767–E4775 Downloaded by guest on September 29, 2021 properties (44–48) such as point group symmetries. Although such of complex eigensolutions always allows to create real solutions spatial symmetries are more easily broken by disorder, the with ~xðtÞ ∈ RN. required ingredients might be very well tailored to the me- Although Eq. 2 contains all of the information about the chanical setup (49, 50). eigensolutions, for the topological classification it is advanta- The remainder of this paper is organized as follows: we start geous to transform it into a Hermitian form. To this end, we by developing a framework to map classical problems to an apply the transformation equation that formally looks like a Schrödinger equation of a pffiffiffiffi quantum-mechanical problem. We then introduce the three D 0 T = [3] symmetries, T , C, and S, and discuss their appearance in me- 0 i chanical problems before we provide the sought classification. Finally, an extensive example section serves two purposes: we to ~y. The square root of the matrix D is defined through its illustrate and apply our approach. Moreover, we show a way how spectral decomposition, where the positive branch of the square to construct new symmetry classes from generic building blocks. root of the eigenvalues is chosen. With this, we arrive at Models and Theoretical Framework pffiffiffiffi d − λ 0 D i ~ψðtÞ = H~ψðtÞ, ~ψðtÞ = e i tT~y, H = pffiffiffiffi . [4] In this manuscript, we aim at characterizing discrete systems of dt D iΓ undamped, linear mechanical oscillators. Although this setup is directly relevant for simple mass-spring systems (11) or mag- As D is symmetric positive-definite and Γ is skew-symmetric, the netically coupled gyroscopes (12), the scope here is actually matrix H is Hermitian and the differential equation for ~ψ has the considerably broader. Any system that can be reliably reduced to sought-after form of a Schrödinger equation. a discrete linear model is amenable to our treatment. This in- The formulation in Eq. 4 is reminiscent of a single-particle cludes 1D (40), 2D (25, 38, 39, 41), or 3D (40) systems made tight-binding problem in quantum mechanics. Therefore, the from continuous media, where a targeted microstructuring en- discussion of topological properties of the eigenvectors ~ψ can be ables the description in terms of a discrete model. Once we deal directly carried over. Remember that the topological classifica- with a discrete model, we have a direct way to import the tion is based on the spatial dimensionality of the problem as well methods known from electronic topological
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