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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 .

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.

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.

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.

The physics of complex systems stands to greatly benefit from the qualitative changes in data availability and advances in data-driven computational methods. Many of these systems can be represented by interacting degrees of freedom on inhomogeneous graphs. However, the lack of translational invariance presents a fundamental challenge to theoretical tools, such as the renormalization group, which were so successful in characterizing the universal physical behaviour in critical phenomena. Here we show that compression theory allows the extraction of relevant degrees of freedom in arbitrary geometries, and the development of efficient numerical tools to build an effective theory from data. We demonstrate our method by applying it to a strongly correlated system on an Ammann-Beenker quasicrystal, where it discovers an exotic critical point with broken conformal symmetry. We also apply it to an antiferromagnetic system on non-bipartite random graphs, where any periodicity is absent. More often seems better when it comes to information, but its excess can obstruct the real relevant features of inhomogeneous systems. Here, the authors use a compression algorithm based on renormalization group to uncover exotic criticality on a quasicrystal by purposefully subtracting information.

Phonon engineering of solids enables the creation of materials with tailored heat-transfer properties, controlled elastic and acoustic vibration propagation, and custom phonon–electron and phonon–photon interactions. These can be leveraged for energy transport, harvesting, or isolation applications and in the creation of novel phonon-based devices, including photoacoustic systems and phonon-communication networks. Here we introduce nanocrystal superlattices as a platform for phonon engineering. Using a combination of inelastic neutron scattering and modeling, we characterize superlattice-phonons in assemblies of colloidal nanocrystals and demonstrate that they can be systematically engineered by tailoring the constituent nanocrystals, their surfaces, and the topology of superlattice. This highlights that phonon engineering can be effectively carried out within nanocrystal-based devices to enhance functionality, and that solution processed nanocrystal assemblies hold promise not only as engineered electronic and optical materials, but also as functional metamaterials with phonon energy and length scales that are unreachable by traditional architectures. Phonon engineering enables tailored heat-transfer properties, controlled elastic and acoustic vibration propagation, and custom phonon-electron and phonon-photon interactions. Here, the authors present an approach for the observation and tunability of vibrational modes of nanocrystal superlattices.