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Huber reviews how a recently established bridge between the phenomenology of electrons in topological insulators and the world of classical mechanical systems might lead to new design principles for such metamaterials.

A topological insulator, as originally proposed for electrons governed by quantum mechanics, is characterized by a dichotomy between the interior and the edge of a finite system: The bulk has an energy gap, and the edges sustain excitations traversing this gap. However, it has remained an open question whether the same physics can be observed for systems obeying Newton's equations of motion. We conducted experiments to characterize the collective behavior of mechanical oscillators exhibiting the phenomenology of the quantum spin Hall effect. The phononic edge modes are shown to be helical, and we demonstrate their topological protection via the stability of the edge states against imperfections. Our results may enable the design of topological acoustic metamaterials that can capitalize on the stability of the surface phonons as reliable wave guides.

Topological phononic crystals, alike their electronic counterparts, are characterized by a bulk–edge correspondence where the interior of a material dictates the existence of stable surface or boundary modes. In the mechanical setup, such surface modes can be used for various applications such as wave guiding, vibration isolation, or the design of static properties such as stable floppy modes where parts of a system move freely. Here, we provide a classification scheme of topological phonons based on local symmetries. We import and adapt the classification of noninteracting electron systems and embed it into the mechanical setup. Moreover, we provide an extensive set of examples that illustrate our scheme and can be used to generate models in unexplored symmetry classes. Our work unifies the vast recent literature on topological phonons and paves the way to future applications of topological surface modes in mechanical metamaterials.

A neural-network technique can exploit the power of machine learning to mine the exponentially large data sets characterizing the state space of condensed-matter systems. Topological transitions and many-body localization are first on the list. Classifying phases of matter is key to our understanding of many problems in physics. For quantum-mechanical systems in particular, the task can be daunting due to the exponentially large Hilbert space. With modern computing power and access to ever-larger data sets, classification problems are now routinely solved using machine-learning techniques 1 . Here, we propose a neural-network approach to finding phase transitions, based on the performance of a neural network after it is trained with data that are deliberately labelled incorrectly. We demonstrate the success of this method on the topological phase transition in the Kitaev chain 2 , the thermal phase transition in the classical Ising model 3 , and the many-body-localization transition in a disordered quantum spin chain 4 . Our method does not depend on order parameters, knowledge of the topological content of the phases, or any other specifics of the transition at hand. It therefore paves the way to the development of a generic tool for identifying unexplored phase transitions.

A two-dimensional phononic quadrupole topological insulator is demonstrated experimentally using mechanical metamaterials, which has both the one-dimensional edge states and the zero-dimensional corner states predicted by theory. Topological edge and corner states The properties of many materials with topological band structures can be understood in terms of a quantization of the electric polarization. By considering a quantization of higher-order polarizations, a new class of higher-order topological insulators has recently been predicted. Marc Serra-Garcia et al. use a mechanical metamaterial to demonstrate such a system experimentally: a two-dimensional phononic quadrupole topological insulator, which has both one-dimensional states at the edges as well as zero-dimensional states at the corners. As these topological corner states are two dimensions lower than the bulk, they could provide a route to engineering one-dimensional channels along the edges of three-dimensional systems. The modern theory of charge polarization in solids 1 , 2 is based on a generalization of Berry’s phase 3 . The possibility of the quantization of this phase 4 , 5 arising from parallel transport in momentum space is essential to our understanding of systems with topological band structures 6 , 7 , 8 , 9 , 10 . Although based on the concept of charge polarization, this same theory can also be used to characterize the Bloch bands of neutral bosonic systems such as photonic 11 or phononic crystals 12 , 13 . The theory of this quantized polarization has recently been extended from the dipole moment to higher multipole moments 14 . In particular, a two-dimensional quantized quadrupole insulator is predicted to have gapped yet topological one-dimensional edge modes, which stabilize zero-dimensional in-gap corner states 14 . However, such a state of matter has not previously been observed experimentally. Here we report measurements of a phononic quadrupole topological insulator. We experimentally characterize the bulk, edge and corner physics of a mechanical metamaterial (a material with tailored mechanical properties) and find the predicted gapped edge and in-gap corner states. We corroborate our findings by comparing the mechanical properties of a topologically non-trivial system to samples in other phases that are predicted by the quadrupole theory. These topological corner states are an important stepping stone to the experimental realization of topologically protected wave guides 12 , 15 in higher dimensions, and thereby open up a new path for the design of metamaterials 16 , 17 .

Condensed-matter and other engineered systems, such as cold atoms1, photonic2 or phononic metamaterials3, have proved to be versatile platforms for the observation of low-energy counterparts of elementary particles from relativistic field theories. These include the celebrated Majorana modes4, as well as Dirac5,6 and Weyl fermions7–9. An intriguing feature of the Weyl equation10 is the chiral symmetry, where the two chiral sectors have an independent gauge freedom. Although this freedom leads to a quantum anomaly11–15, there is no corresponding axial background field coupling differently to opposite chiralities in quantum electrodynamics. Here, we provide the experimental characterization of the effect of such an axial field in an acoustic metamaterial. We implement the axial field through an inhomogeneous potential16 and observe the induced chiral Landau levels. From the metamaterials perspective these chiral channels open the possibility for the observation of non-local Weyl orbits17 and might enable unidirectional bulk transport in a time-reversal-invariant system18.Axial fields couple to the states of different chiralities with opposite signs. In an acoustic Weyl system, the implementation of such fields induces chiral Landau levels, which is now observed experimentally.

Symmetries crucially underlie the classification of topological phases of matter. Most materials, both natural as well as architectured, possess crystalline symmetries. Recent theoretical works unveiled that these crystalline symmetries can stabilize fragile Bloch bands that challenge our very notion of topology: Although answering to the most basic definition of topology, one can trivialize these bands through the addition of trivial Bloch bands. Here, we fully characterize the symmetry properties of the response of an acoustic metamaterial to establish the fragile nature of the low-lying Bloch bands. Additionally, we present a spectral signature in the form of spectral flow under twisted boundary conditions.

Identifying material geometries that lead to metamaterials with desired functionalities presents a challenge for the field. Discrete, or reduced-order, models provide a concise description of complex phenomena, such as negative refraction, or topological surface states; therefore, the combination of geometric building blocks to replicate discrete models presenting the desired features represents a promising approach. However, there is no reliable way to solve such an inverse problem. Here, we introduce ‘perturbative metamaterials’, a class of metamaterials consisting of weakly interacting unit cells. The weak interaction allows us to associate each element of the discrete model with individual geometric features of the metamaterial, thereby enabling a systematic design process. We demonstrate our approach by designing two-dimensional elastic metamaterials that realize Veselago lenses, zero-dispersion bands and topological surface phonons. While our selected examples are within the mechanical domain, the same design principle can be applied to acoustic, thermal and photonic metamaterials composed of weakly interacting unit cells.

Machine-learning driven models have proven to be powerful tools for the identification of phases of matter. In particular, unsupervised methods hold the promise to help discover new phases of matter without the need for any prior theoretical knowledge. While for phases characterized by a broken symmetry, the use of unsupervised methods has proven to be successful, topological phases without a local order parameter seem to be much harder to identify without supervision. Here, we use an unsupervised approach to identify boundaries of the topological phases. We train artificial neural nets to relate configurational data or measurement outcomes to quantities like temperature or tuning parameters in the Hamiltonian. The accuracy of these predictive models can then serve as an indicator for phase transitions. We successfully illustrate this approach on both the classical Ising gauge theory as well as on the quantum ground state of a generalized toric code.

Topologically protected surface modes of classical waves hold the promise to enable a variety of applications ranging from robust transport of energy to reliable information processing networks. However, both the route of implementing an analogue of the quantum Hall effect as well as the quantum spin Hall effect are obstructed for acoustics by the requirement of a magnetic field, or the presence of fermionic quantum statistics, respectively. Here, we construct a two-dimensional topological acoustic crystal induced by the synthetic spin-orbit coupling, a crucial ingredient of topological insulators, with spin non-conservation. Our setup allows us to free ourselves of symmetry constraints as we rely on the concept of a non-vanishing “spin” Chern number. We experimentally characterize the emerging boundary states which we show to be gapless and helical. More importantly, we observe the spin flipping transport in an H-shaped device, demonstrating evidently the spin non-conservation of the boundary states. In acoustic systems, quantum spin Hall physics is rarely observed because the crucial ingredient spin-orbit coupling is missing. Here, the authors construct an acoustic crystal with synthetic spin-orbit coupling and observe gapless helical boundary states as well as spin flipping effect.