PHYSICS. Observation of phononic helical edge states in a mechanical topological insulator

First- and Second-Order Topological States in Two-Dimensional Noncovalent Molecular Chiral Crystals

Topological band physics has been extensively investigated in inorganic solid-state materials with a bonding structure. While covalent organic frameworks or metal-organic frameworks have garnered significant research interest, the high-order topological states in two-dimensional noncovalent molecular crystals remain largely uncharted. Here we investigated noncovalent molecular chiral crystals assembled from achiral molecules using first-principles calculations and tight-binding model analysis. The rotated achiral molecules introduce chiral enantiomers, promoting topological chiral states in the vicinity of conduction band edges. The structural chirality in the rotated monolayer crystal is described by the rotating Kekulé model, which possesses an opposite Berry curvature. Therefore, we further obtain the topological kink states of parallel propagating channels at the neighboring boundary of the two chiral enantiomers. In addition, this model breaks the chiral symmetry, thereby accounting for the origin of the higher-order topological corner states in the valence band. We simultaneously identify valley topological gapless edge states and second-order topological corner states, representing a significant step toward the development of first- and second-order topological insulators in noncovalent molecular chiral crystals. This work sheds light on intriguing higher-order topological states in noncovalent molecular crystals.

Quantum computing predicts particle trajectories in optical tweezers

A recent study demonstrated advancements in quantum computing by applying it to address a non-Hermitian optical manipulation problem. The emergence of exceptional points and the dynamics of optically trapped single or multiple particles were simulated using a quantum computing approach.

Emulation of Schrödinger dynamics with metamaterials

The exploration of metamaterials with artificial sub-wavelength structures has empowered researchers to engineer the propagation of classical waves, enabling advancements in areas such as imaging, sensing, communication, and energy harvesting. Concurrently, the investigation into topology and symmetry has not only unveiled valuable insights into fundamental physics, but also expanded our ability to manipulate waves effectively. Combined with the remarkable flexibility and diversity of artificial metamaterials, these considerations have sparked a focused research interest. Notably, a class of structures capable of supporting topological propagation modes akin to the Schrödinger equation has been identified. Leveraging metamaterials to emulate Schrödinger dynamics has emerged as a promising avenue for robust wave manipulation and the exploration of quantum phenomena beyond the confines of electronic systems. Despite rapid progress in this burgeoning field, comprehensive summaries are scarce. Thus, this review aims to systematically consolidate recent advancements in classical wave physics based on a Schrödinger equation approach. This discourse initiates with an overview of quantum and classical wave descriptions, subsequently delving into the elucidation of numerous models realized across diverse experimental platforms, including photonic/phononic waveguides, acoustic cavities, and optomechanics. Finally, we address the challenges and prospects associated with emulating Schrödinger dynamics, underscoring the potential for groundbreaking developments in this captivating domain.

Singular topological edge states in locally resonant metamaterials

Band topology has emerged as a novel tool for material design across various domains, including photonic and phononic systems, and metamaterials. A prominent model for band topology is the Su-Schrieffer-Heeger (SSH) chain, which reveals topological in-gap states within Bragg-type gaps (BG) formed by periodic modification. Apart from classical BGs, another mechanism for bandgap formation in metamaterials involves strong coupling between local resonances and propagating waves, resulting in a local resonance-induced bandgap (LRG). Previous studies have shown the challenge of topological edge state emergence within the LRG. Here, we reveal that topological edge states can emerge within an LRG by achieving both topological phase and bandgap transitions simultaneously. We describe this using a model of inversion-symmetric extended SSH chains for locally resonant metamaterials. Notably, this topological state can lead to highly localized modes, comparable to a subwavelength unit cell, when it emerges within the LRG. We experimentally demonstrate distinct differences in topologically protected modes-highlighted by wave localization-between the BG and the LRG using locally resonant granule-based metamaterials. Our findings suggest the scope of topological metamaterials may be extended via their bandgap nature.

Backscattering-free edge states below all bands in two-dimensional auxetic media

Unidirectional and backscattering-free propagation of sound waves is of fundamental interest in physics and highly sought-after in engineering. Current strategies utilize topologically protected chiral edge modes in bandgaps, or complex mechanisms involving active constituents or nonlinearity. Here we propose passive, linear, one-way edge states based on spin-momentum locking of Rayleigh waves in two-dimensional media in the limit of vanishing bulk to shear modulus ratio, which provides perfect unidirectional and backscattering-free edge propagation that is immune to any edge roughness and has no limitation on its frequency (instead of residing in gaps between bulk bands). We further show that such modes are characterized by a topological winding number that protects the linear momentum of the wave along the edge. These passive and backscattering-free edge waves have the potential to enable phononic devices in the form of lattices or continua that work in previously inaccessible frequency ranges.

Nonvolatile Ferroic and Topological Phase Control under Nonresonant Light

Light-matter interaction is a long-standing promising topic that can be dated back to a few centuries ago and has witnessed the long-term debate between the particle and wave nature of light. In modern condensed matter physics and materials science, light usually serves as a detection tool to effectively characterize the physical and chemical features of samples. The light modulation on intrinsic properties of materials, such as atomic geometries, electronic bands, and magnetic behaviors, is more intriguing for information control and storage. This corresponds to a light-induced order parameter switch in the phase space. Most prior works focus on the situation when photon energy is larger than the material band gap, in which the photon is absorbed by the electron subsystem and then transfers its energy into other subsystems such as phonon and spin. This can be described by the imaginary part of the dielectric function. In contrast, recent theoretical predictions and experimental advances have suggested that the real part of dielectric function could also vary the energy landscape in phase space, so that it triggers phase transition in an athermic approach (without direct photon absorption). In this Perspective, we review some recent theoretical, computational, and experimental developments of such a low-frequency light-induced phase transition, focusing on ferroic and topological order parameters. We also elucidate its fundamental mechanisms by comparing it with the optical tweezers technique, and light irradiation could trigger impulsive stimulated Raman phonon excitation. Finally, we propose some further developments and challenges in such a nonresonant light-matter interaction.

Magnetophononics and the chiral phonon misnomer

The direct, ultrafast excitation of polar phonons with electromagnetic radiation is a potent strategy for controlling the properties of a wide range of materials, particularly in the context of influencing their magnetic behavior. Here, we show that, contrary to common perception, the origin of phonon-induced magnetic activity does not stem from the Maxwellian fields resulting from the motion of the ions themselves or the effect their motion exerts on the electron subsystem. Through the mechanism of electron-phonon coupling, a coherent state of circularly polarized phonons generates substantial non-Maxwellian fields that disrupt time-reversal symmetry, effectively emulating the behavior of authentic magnetic fields. Notably, the effective fields can reach magnitudes as high as 100 T, surpassing by a factor of α - 2 ≈ 2 × 10 4 the Maxwellian fields resulting from the inverse Faraday effect; α is the fine-structure constant. Because the light-induced nonreciprocal fields depend on the square of the phonon displacements, the chirality the photons transfer to the ions plays no role in magnetophononics.

Gigahertz Surface Acoustic Wave Topological Rainbow in Nanoscale Phononic Crystals

Precisely engineered gigahertz surface acoustic wave (SAW) trapping enables diverse and controllable interconnections with various quantum systems, which are crucial to unlocking the full potential of phonons. The topological rainbow based on synthetic dimension presents a promising avenue for facile and precise localization of SAWs. In this study, we successfully developed a monolithic gigahertz SAW topological rainbow by utilizing a nanoscale translational deformation as a synthetic dimension. We observed a gapless topological boundary state, which originates from a 2π phase winding in the Zak phase during translation. These boundary states enable on-chip single-mode rainbowlike filters with an extensive range of adjustable operating frequencies. Furthermore, we construct nanoscale wedge-shaped grooves, realizing the Born-von-Karman interface. The interface generates topological rainbow resonators with high quality and small mode volume, which can trap topological phononic states of different frequencies into different positions. This study underscores the immense potential of topological acoustics in synthetic dimensions for microwave acoustics, providing a robust design framework for the precise manipulation of SAWs.

Heavy-boundary mode patterning and dynamics of topological phonons in polymer chains and supramolecular lattices on surfaces

In topological band theory, phonon boundary modes consequence of a topologically non-trivial  band structure feature desirable properties for atomically-precise technologies, such as robustness against defects, waveguiding, and one-way transport. These topological phonon boundary modes remain to be studied both theoretically and experimentally in synthetic materials, such as polymers and supramolecular assemblies at the atomistic level under thermal fluctuations. Here we show by means of molecular simulations, that surface-confined Su-Schrieffer-Heeger (SSH) phonon analogue models express robust topological phonon boundary modes at heavy boundaries and under thermal fluctuations. The resulting bulk-heavy boundary correspondence enables patterning of boundary modes in polymer chains and weakly-interacting supramolecular lattices. Moreover, we show that upon excitation of a single molecule, propagation along heavy-boundary modes differs from free boundary modes. Our work is an entry to topological vibrations in supramolecular systems, and may find applications in the patterning of phonon circuits and realization of Hall effect phonon analogues at the molecular scale.

Topological Phononic Fiber of Second Spin-Chern Number

The discovery of quantum spin Hall effect characterized by the first spin-Chern numbers in 2D systems has significantly advanced topological materials. To explore its 4D counterpart is of fundamental importance, but so far remains elusive in experiments. Here, we realize a topological phononic fiber protected by the second spin-Chern number in a 4D manifold, using a 3D geometric structure combined with a 1D rotational parameter space. We experimentally observe spin-momentum-locked core states traveling along a vortex line where the Dirac mass varies continuously. A novel higher-order face-centered bound state is further demonstrated. These findings underscore the interplay between higher-dimensional topological physics and defects, opening up a topological path for fibers.

Thermally tunable add-drop filter based on valley photonic crystals for optical communications

Valley photonic crystals (VPCs) provide an intriguing approach to suppress backscattering losses and enable robust transport of light against sharp bends, which could be utilized to realize low-loss and small-footprint devices for on-chip optical communications. However, there are few studies on how to achieve power-efficient tunable devices based on VPCs, which are essential for implementing basic functions such as optical switching and routing. Here, we propose and experimentally demonstrate a thermally tunable add-drop filter (ADF) based on VPCs operating at telecommunication wavelengths. By leveraging the topological protection of the edge state and the distinct property of negligible scattering at sharp bends, a small footprint of 17.4 × 28.2 μm2 and a low insertion loss of 2.7 dB can be achieved for the proposed device. A diamond-shaped microloop resonator is designed to confine the light and enhance its interaction with the thermal field generated by the microheater, leading to a relatively low power of 23.97 mW needed for switching the output signal from one port to the other. Based on the thermally tunable ADF under the protection of band topology, robust data transmission is implemented with an ultrahigh data rate of 132 Gb/s. Our work shows great potential for developing high-performance topological photonic devices with the thermally tunable silicon-based VPCs, which offers unprecedented opportunities for realizing topologically protected and reconfigurable high-speed datalinks on a chip.

High-Order Topological Pumping on a Superconducting Quantum Processor

High-order topological phases of matter refer to the systems of n-dimensional bulk with the topology of m-th order, exhibiting (n-m)-dimensional boundary modes and can be characterized by topological pumping. Here, we experimentally demonstrate two types of second-order topological pumps, forming four 0-dimensional corner localized states on a 4×4 square lattice array of 16 superconducting qubits. The initial ground state of the system at half-filling, as a product of four identical entangled 4-qubit states, is prepared using an adiabatic scheme. During the pumping procedure, we adiabatically modulate the superlattice Bose-Hubbard Hamiltonian by precisely controlling both the hopping strengths and on-site potentials. At the half pumping period, the system evolves to a corner-localized state in a quadrupole configuration. The robustness of the second-order topological pump is also investigated by introducing different on-site disorder. Our Letter studies the topological properties of high-order topological phases from the dynamical transport picture using superconducting qubits, which would inspire further research on high-order topological phases.

Fully Polarized Topological Isostatic Metamaterials in Three Dimensions

Topological surface states are unique to topological materials and are immune to disturbances. In isostatic lattices, mechanical topological floppy modes exhibit softness depending on the polarization relative to the terminating surface. However, in three dimensions, the polarization of topological floppy modes is disrupted by the ubiquitous mechanical Weyl lines. Here, we demonstrate, both theoretically and experimentally, the fully polarized topological mechanical phases free of Weyl lines. Floppy modes emerge exclusively on a particular surface of the three-dimensional isostatic structure, leading to the strongly asymmetric stiffness between opposing boundaries. Additionally, uniform soft strains can reversibly shift the lattice configuration to Weyl phases, switching the stiffness contrast to a trivially comparable level. Our work demonstrates the fully polarized topological mechanical phases in three dimensions, and paves the way towards engineering soft and adaptive metamaterials.

Chiral edge waves in a dance-based human topological insulator

Topological insulators are insulators in the bulk but feature chiral energy propagation along the boundary. This property is topological in nature and therefore robust to disorder. Originally discovered in electronic materials, topologically protected boundary transport has since been observed in many other physical systems. Thus, it is natural to ask whether this phenomenon finds relevance in a broader context. We choreograph a dance in which a group of humans, arranged on a square grid, behave as a topological insulator. The dance features unidirectional flow of movement through dancers on the lattice edge. This effect persists when people are removed from the dance floor. Our work extends the applicability of wave physics to dance.

Localized and delocalized topological modes of heat

The topological physics has sparked intensive investigations into topological lattices in photonic, acoustic, and mechanical systems, powering counterintuitive effects otherwise inaccessible with usual settings. Following the success of these endeavors in classical wave dynamics, there has been a growing interest in establishing their topological counterparts in diffusion. Here, we propose an additional real-space dimension in diffusion, and the system eigenvalues are transformed from "imaginary" to "real." By judiciously tailoring the effective Hamiltonian with coupling networks, localized and delocalized topological modes are realized in heat transfer. Simulations and experiments in active thermal lattices validate the effectiveness of the proposed theoretical strategy. This approach can be applied to establish various topological lattices in diffusion systems, offering insights into engineering topologically protected edge states in dynamic diffusive scenarios.

Photonic topological phase transition induced by material phase transition

Photonic topological insulators (PTIs) have been proposed as an analogy to topological insulators in electronic systems. In particular, two-dimensional PTIs have gained attention for the integrated circuit applications. However, controlling the topological phase after fabrication is difficult because the photonic topology requires the built-in specific structures. This study experimentally demonstrates the band inversion in two-dimensional PTI induced by the phase transition of deliberately designed nanopatterns of a phase change material, Ge2Sb2Te5 (GST), which indicates the first observation of the photonic topological phase transition in two-dimensional PTI with changes in the Chern number. This approach allows us to directly alter the topological invariants, which is achieved by symmetry-breaking perturbation through GST nanopatterns with different symmetry from original PTI. The success of our scheme is attributed to the ultrafine lithographic alignment technologies of GST nanopatterns. These results demonstrate how to control photonic topological properties in a reconfigurable manner, providing insight into the possibilities for reconfigurable photonic processing circuits.

A blueprint for truncation resonance placement in elastic diatomic lattices with unit cell asymmetrya)

Elastic periodic lattices act as mechanical filters of incident vibrations. By and large, they forbid wave propagation within bandgaps and resonate outside them. However, they often encounter "truncation resonances" (TRs) inside bandgaps when certain conditions are met. In this study, we show that the extent of unit cell asymmetry, its mass and stiffness contrasts, and the boundary conditions all play a role in the TR location and wave profile. The work is experimentally supported via two examples that validate the methodology, and a set of design charts is provided as a blueprint for selective TR placement in diatomic lattices.

Topological Network Modes in a Twisted Moiré Phononic Crystal

Twisted moiré materials, a new class of layered structures with different twist angles for neighboring layers, are attracting great attention because of the rich intriguing physical phenomena associated with them. Of particular interest are the topological network modes, first proposed in the small angle twisted bilayer graphene under interlayer bias. Here we report the observations of such topological network modes in twisted moiré phononic crystals without requiring the external bias fields. Acoustic topological network modes that can be constructed in a wide range of twist angles are both observed in the domain walls with and without reconstructions, which serve as the analogy of the lattice relaxations in electronic moiré materials. Topological robustness of the topological network modes is observed by introducing valley-preserved defects to the network channel. Furthermore, the network can be reconfigured into two-dimensional patterns with any desired connectivity, offering a unique prototype platform for acoustic applications.

Topological Hong-Ou-Mandel interference

The interplay of topology and optics provides a route to pursue robust photonic devices, with the application to photonic quantum computation in its infancy. However, the possibilities of harnessing topological structures to process quantum information with linear optics, through the quantum interference of photons, remain largely uncharted. Here, we present a Hong-Ou-Mandel interference effect of topological origin. We show that this interference of photon pairs-ranging from constructive to destructive-is solely determined by a synthetic magnetic flux, rendering it resilient to errors on a fundamental level. Our implementation establishes a quantized flux that facilitates exclusively destructive quantum interference. Our findings pave the way toward the development of next-generation photonic quantum circuitry and scalable quantum computing protected by virtue of topologically robust quantum gates.

Observation of non-Hermitian skin effect in thermal diffusion

The paradigm shift of Hermitian systems into the non-Hermitian regime profoundly modifies inherent property of the topological systems, leading to various unprecedented effects such as the non-Hermitian skin effect (NHSE). In the past decade, the NHSE has been demonstrated in quantum, optical and acoustic systems. Beside those wave systems, the NHSE in diffusive systems has not yet been observed, despite recent abundant advances in the study of topological thermal diffusion. In this work, we design a thermal diffusion lattice based on a modified Su-Schrieffer-Heeger model and demonstrate the diffusive NHSE. In the proposed model, the asymmetric temperature field coupling inside each unit cell can be judiciously realized by appropriate configurations of structural parameters. We find that the temperature fields trend to concentrate toward the target boundary which is robust against initial excitation conditions. We thus experimentally demonstrated the NHSE in thermal diffusion and verified its robustness against various defects. Our work provides a platform for exploration of non-Hermitian physics in the diffusive systems, which has important applications in efficient heat collection, highly sensitive thermal sensing and others.

Observation of D-class topology in an acoustic metamaterial

Topological materials and metamaterials opened new paradigms to create and manipulate phases of matter with unconventional properties. Topological D-class phases (TDPs) are archetypes of the ten-fold classification of topological phases with particle-hole symmetry. In two dimensions, TDPs support propagating topological edge modes that simulate the elusive Majorana elementary particles. Furthermore, a piercing of π-flux Dirac-solenoids in TDPs stabilizes localized Majorana excitations that can be braided for the purpose of topological quantum computation. Such two-dimensional (2D) TDPs have been a focus in the research frontier, but their experimental realizations are still under debate. Here, with a novel design scheme, we realize 2D TDPs in an acoustic crystal by synthesizing both the particle-hole and fermion-like time reversal symmetries for a wide range of frequencies. The design scheme leverages an enriched unit cell structure with real-valued couplings that emulate the targeted Hamiltonian of TDPs with complex hoppings: A technique that could unlock the realization of all topological classes with passive metamaterials. In our experiments, we realize a pair of TDPs with opposite Chern numbers in two independent sectors that are connected by an intrinsic fermion-like time-reversal symmetry built in the system. We measure the acoustic Majorana-like helical edge modes and visualize their robust topological transport, thus revealing the unprecedented D and DIII class topologies with direct evidence. Our study opens up a new pathway for the experimental realization of two fundamental classes of topological phases and may offer new insights in fundamental physics, materials science, and phononic information processing.

Generalized bulk-boundary correspondence in periodically driven non-Hermitian systems

We present a pedagogical review of the periodically driven non-Hermitian systems, particularly on the rich interplay between the non-Hermitian skin effect and the topology. We start by reviewing the non-Bloch band theory of the static non-Hermitian systems and discuss the establishment of its generalized bulk-boundary correspondence (BBC). Ultimately, we focus on the non-Bloch band theory of two typical periodically driven non-Hermitian systems: harmonically driven non-Hermitian system and periodically quenched non-Hermitian system. The non-Bloch topological invariants were defined on the generalized Brillouin zone and the real space wave functions to characterize the Floquet non-Hermtian topological phases. Then, the generalized BBC was established for the two typical periodically driven non-Hermitian systems. Additionally, we review novel phenomena in the higher-dimensional periodically driven non-Hermitian systems, including Floquet non-Hermitian higher-order topological phases and Floquet hybrid skin-topological modes. The experimental realizations and recent advances have also been surveyed. Finally, we end with a summarization and hope this pedagogical review can motivate further research on Floquet non-Hermtian topological physics.

Non-Hermitian non-equipartition theory for trapped particles

The equipartition theorem is an elegant cornerstone theory of thermal and statistical physics. However, it fails to address some contemporary problems, such as those associated with optical and acoustic trapping, due to the non-Hermitian nature of the external wave-induced force. We use stochastic calculus to solve the Langevin equation and thereby analytically generalize the equipartition theorem to a theory that we denote the non-Hermitian non-equipartition theory. We use the non-Hermitian non-equipartition theory to calculate the relevant statistics, which reveal that the averaged kinetic and potential energies are no longer equal to kBT/2 and are not equipartitioned. As examples, we apply non-Hermitian non-equipartition theory to derive the connection between the non-Hermitian trapping force and particle statistics, whereby measurement of the latter can determine the former. Furthermore, we apply a non-Hermitian force to convert a saddle potential into a stable potential, leading to a different type of stable state.

Parity-time-symmetric photonic topological insulator

Topological insulators are a concept that originally stems from condensed matter physics. As a corollary to their hallmark protected edge transport, the conventional understanding of such systems holds that they are intrinsically closed, that is, that they are assumed to be entirely isolated from the surrounding world. Here, by demonstrating a parity-time-symmetric topological insulator, we show that topological transport exists beyond these constraints. Implemented on a photonic platform, our non-Hermitian topological system harnesses the complex interplay between a discrete coupling protocol and judiciously placed losses and, as such, inherently constitutes an open system. Nevertheless, even though energy conservation is violated, our system exhibits an entirely real eigenvalue spectrum as well as chiral edge transport. Along these lines, this work enables the study of the dynamical properties of topological matter in open systems without the instability arising from complex spectra. Thus, it may inspire the development of compact active devices that harness topological features on-demand.

Bloch oscillations, Landau-Zener transition, and topological phase evolution in an array of coupled pendula

We experimentally and theoretically study the dynamics of a one-dimensional array of pendula with a mild spatial gradient in their self-frequency and where neighboring pendula are connected with weak and alternating coupling. We map their dynamics to the topological Su-Schrieffer-Heeger model of charged quantum particles on a lattice with alternating hopping rates in an external electric field. By directly tracking the dynamics of a wave-packet in the bulk of the lattice, we observe Bloch oscillations, Landau-Zener transitions, and coupling between the isospin (i.e., the inner wave function distribution within the unit cell) and the spatial degrees of freedom (the distribution between unit cells). We then use Bloch oscillations in the bulk to directly measure the nontrivial global topological phase winding and local geometric phase of the band. We measure an overall evolution of 3.1 [Formula: see text] 0.2 radians for the geometrical phase during the Bloch period, consistent with the expected Zak phase of [Formula: see text]. Our results demonstrate the power of classical analogs of quantum models to directly observe the topological properties of the band structure and shed light on the similarities and the differences between quantum and classical topological effects.

Three-dimensional [Formula: see text] topological insulators without reflection symmetry

In recent decades, the Altland-Zirnabuer (AZ) table has proven incredibly powerful in delineating constraints for topological classification of a given band-insulator based on dimension and (nonspatial) symmetry class, and has also been expanded by considering additional crystalline symmetries. Nevertheless, realizing a three-dimensional (3D), time-reversal symmetric (class AII) topological insulator (TI) in the absence of reflection symmetries, with a classification beyond the [Formula: see text] paradigm remains an open problem. In this work we present a general procedure for constructing such systems within the framework of projected topological branes (PTBs). In particular, a 3D projected brane from a "parent" four-dimensional topological insulator exhibits a [Formula: see text] topological classification, corroborated through its response to the inserted bulk monopole loop. More generally, PTBs have been demonstrated to be an effective route to performing dimensional reduction and embedding the topology of a [Formula: see text]-dimensional "parent" Hamiltonian in d dimensions, yielding lower-dimensional topological phases beyond the AZ classification without additional symmetries. Our findings should be relevant for the metamaterial platforms, such as photonic and phononic crystals, topolectric circuits, and designer systems.

Spin resolved topological bulk state in acoustics

Extremely rare topologically protected acoustic energy sink is presented in this article. Acoustic topological phenomena are generally described using quantum anomalous hall effects (QAHE), quantum valley hall effects (QVHE), and quantum spin hall effects (QSHE) where spin orbit coupling is predominant. Topological edge states are demonstrated by bulk-boundary distinction when the bulk is insulated. In this article topological acoustic conductor and its phenomena are theoretically demonstrated where the boundaries are insulated. This is exactly opposite to the behavior of a topological acoustic insulator. Phenomena presented in this article could not be explained by any of the trio Quantum Hall effects. To explain the phenomenon phononic crystals or PnCs were designed to obtain accidental triple degeneracies, resulting a Dirac-like cone at the Γ point ([Formula: see text]). The phenomenon is microarchitecture and microrotation field independent. Here time reversal symmetry or the space inversion symmetry is not broken, and the degenerated 'Deaf band' dominates the local dispersion with a syncline top band. This scenario results in continuously changing 'up spin' and 'down spin' of the wave energy in the media and remain trapped without specific preferential direction of wave transport. The spin was found to generate the spin angular momentum, causing the switching in geometric phase from [Formula: see text] in cyclic pattern, keeping the energy trapped inside the bulk media.

Odd elasticity and topological waves in active surfaces

Odd elasticity describes active elastic systems whose stress-strain relationship is not compatible with a potential energy. As the requirement of energy conservation is lifted from linear elasticity, new antisymmetric (odd) components appear in the elastic tensor. In this work we study the odd elasticity and non-Hermitian wave dynamics of active surfaces, specifically plates of moderate thickness. These odd moduli can endow the vibrational modes of the plate with a nonzero topological invariant known as the first Chern number. Within continuum elastic theory, we show that the Chern number is related to the presence of unidirectional shearing waves that are hosted at the plate's boundary. We show that the existence of these chiral edge waves hinges on a distinctive two-step mechanism. Unlike electronic Chern insulators where the magnetic field at the same time gaps the spectrum and imparts chirality, here the finite thickness of the sample gaps the shear modes, and the odd elasticity makes them chiral.

Model-free characterization of topological edge and corner states in mechanical networks

Topological materials can host edge and corner states that are protected from disorder and material imperfections. In particular, the topological edge states of mechanical structures present unmatched opportunities for achieving robust responses in wave guiding, sensing, computation, and filtering. However, determining whether a mechanical structure is topologically nontrivial and features topologically protected modes has hitherto relied on theoretical models. This strong requirement has limited the experimental and practical significance of topological mechanics to laboratory demonstrations. Here, we introduce and validate an experimental method to detect the topologically protected zero modes of mechanical structures without resorting to any modeling step. Our practical method is based on a simple electrostatic analogy: Topological zero modes are akin to electric charges. To detect them, we identify elementary mechanical molecules and measure their chiral polarization, a recently introduced marker of topology in chiral phases. Topological zero modes are then identified as singularities of the polarization field. Our method readily applies to any mechanical structure and effectively detects the edge and corner states of regular and higher-order topological insulators. Our findings extend the reach of chiral topological phases beyond designer materials and allow their direct experimental investigation.

Experimental Realization of Stable Exceptional Chains Protected by Non-Hermitian Latent Symmetries Unique to Mechanical Systems

Lines of exceptional points are robust in the three-dimensional non-Hermitian parameter space without requiring any symmetry. However, when more elaborate exceptional structures are considered, the role of symmetry becomes critical. One such case is the exceptional chain (EC), which is formed by the intersection or osculation of multiple exceptional lines (ELs). In this Letter, we investigate a non-Hermitian classical mechanical system and reveal that a symmetry intrinsic to second-order dynamical equations, in combination with the source-free principle of ELs, guarantees the emergence of ECs. This symmetry can be understood as a non-Hermitian generalized latent symmetry, which is absent in prevailing formalisms rooted in first-order Schrödinger-like equations and has largely been overlooked so far. We experimentally confirm and characterize the ECs using an active mechanical oscillator system. Moreover, by measuring eigenvalue braiding around the ELs meeting at a chain point, we demonstrate the source-free principle of directed ELs that underlies the mechanism for EC formation. Our Letter not only enriches the diversity of non-Hermitian exceptional point configurations, but also highlights the new potential for non-Hermitian physics in second-order dynamical systems.

Classical analogue to driven quantum bits based on macroscopic pendula

Quantum mechanics increasingly penetrates modern technologies but, due to its non-deterministic nature seemingly contradicting our classical everyday world, our comprehension often stays elusive. Arguing along the correspondence principle, classical mechanics is often seen as a theory for large systems where quantum coherence is completely averaged out. Surprisingly, it is still possible to reconstruct the coherent dynamics of a quantum bit (qubit) by using a classical model system. This classical-to-quantum analogue is based on wave mechanics, which applies to both, the classical and the quantum world. In this spirit we investigate the dynamics of macroscopic physical pendula with a modulated coupling. As a proof of principle, we demonstrate full control of our one-to-one analogue to a qubit by realizing Rabi oscillations, Landau-Zener transitions and Landau-Zener-Stückelberg-Majorana interferometry. Our classical qubit demonstrator can help comprehending and developing useful quantum technologies.

Uncovering and Experimental Realization of Multimodal 3D Topological Metamaterials for Low-Frequency and Multiband Elastic Wave Control

Topological mechanical metamaterials unlock confined and robust elastic wave control. Recent breakthroughs have precipitated the development of 3D topological metamaterials, which facilitate extraordinary wave manipulation along 2D planar and layer-dependent waveguides. The 3D topological metamaterials studied thus far are constrained to function in single-frequency bandwidths that are typically in a high-frequency regime, and a comprehensive experimental investigation remains elusive. In this paper, these research gaps are addressed and the state of the art is advanced through the synthesis and experimental realization of a 3D topological metamaterial that exploits multimodal local resonance to enable low-frequency elastic wave control over multiple distinct frequency bands. The proposed metamaterial is geometrically configured to create multimodal local resonators whose frequency characteristics govern the emergence of four unique low-frequency topological states. Numerical simulations uncover how these topological states can be employed to achieve polarization-, frequency-, and layer-dependent wave manipulation in 3D structures. An experimental study results in the attainment of complete wave fields that illustrate 2D topological waveguides and multi-polarized wave control in a physical testbed. The outcomes from this work provide insight that will aid future research on 3D topological mechanical metamaterials and reveal the applicability of the proposed metamaterial for wave control applications.

Pseudospin Polarized Dual-Higher-Order Topology in Hydrogen-Substituted Graphdiyne

Although the topological band theory is applicable to both Fermionic and bosonic systems, the same electronic and phononic topological phases are seldom reported in one natural material. In this work, we show the presence of a dual-higher-order topology in hydrogen-substituted graphdiyne (H-GDY) by first-principles calculations. The intriguing enantiomorphic flat-bands are realized in both electronic and phononic bands of H-GDY, which is confirmed to be an organic 2D second-order topological insulator (SOTI). Most importantly, we found that the topological corner states are pseudospin polarized in H-GDY, exhibiting a clockwise or counterclockwise texture perpendicular to the radial direction. Our results not only identify the existence of the dual-higher-order topology in covalent organic frameworks but also uncover a unique pseudospin polarization-coordinate locking relation, further extending the well-known spin-momentum locking relation in conventional topological insulators.

Theory of nonlinear corner states in photonic fractal lattices

We study linear and nonlinear higher-order topological insulators (HOTIs) based on waveguide arrays arranged into Sierpiński gasket and Sierpiński carpet structures, both of which have non-integer effective Hausdorff dimensionality. Such fractal structures possess different discrete rotational symmetries, but both lack transverse periodicity. Their characteristic feature is the existence of multiple internal edges and corners in their optical potential landscape, and the formal absence of an insulating bulk. Nevertheless, we show that a systematic geometric shift of the waveguides in the first generation of such fractal arrays, which affects the coupling strengths between sites of this building block as well as in subsequent structure generations, enables the formation of corner states of topological origin at the outer corners of the array. We find that, in contrast to HOTIs based on periodic arrays, Sierpiński gasket arrays always support topological corner states, irrespective of the direction of the shift of the waveguides, while in Sierpiński carpet structures, corner states emerge only for one direction of the waveguide shift. We also find families of corner solitons bifurcating from linear corner states of fractal structures that remain stable practically in the entire gap in which they form. These corner states can be efficiently excited by injecting Gaussian beams into the outer corner sites of the fractal arrays. Our results pave the way toward the investigation of nonlinear effects in topological insulators with non-integer dimensionality and enrich the variety of higher-order topological states.

Higher-order topological heat conduction on a lattice for detection of corner states

A heat conduction equation on a lattice composed of nodes and bonds is formulated assuming the Fourier law and the energy conservation law. Based on this equation, we propose a higher-order topological heat conduction model on the breathing kagome lattice. We show that the temperature measurement at a corner node can detect the corner state which causes rapid heat conduction toward the heat bath, and that several-nodes measurement can determine the precise energy of the corner states.

Nonlinear Topological Mechanics in Elliptically Geared Isostatic Metamaterials

Despite the extensive studies of topological systems, the experimental characterizations of strongly nonlinear topological phases have been lagging. To address this shortcoming, we design and build elliptically geared isostatic metamaterials. Their nonlinear topological transitions can be realized by collective soliton motions, which stem from the transition of nonlinear Berry phase. Endowed by the intrinsic nonlinear topological mechanics, surface polar elasticity and dislocation-bound zero modes can be created or annihilated as the topological polarization reverses orientation. Our approach integrates topological physics with strongly nonlinear mechanics and promises multiphase structures at the micro- and macroscales.

Non-Hermitian topology in static mechanical metamaterials

The combination of broken Hermiticity and band topology in physical systems unveils a novel bound state dubbed as the non-Hermitian skin effect (NHSE). Active control that breaks reciprocity is usually used to achieve NHSE, and gain and loss in energy are inevitably involved. Here, we demonstrate non-Hermitian topology in a mechanical metamaterial system by exploring its static deformation. Nonreciprocity is introduced via passive modulation of the lattice configuration without resorting to active control and energy gain/loss. Intriguing physics such as the reciprocal and higher-order skin effects can be tailored in the passive system. Our study provides an easy-to-implement platform for the exploration of non-Hermitian and nonreciprocal phenomena beyond conventional wave dynamics.

Vector valley Hall edge solitons in distorted type-II Dirac photonic lattices

Topological edge states have recently garnered a lot of attention across various fields of physics. The topological edge soliton is a hybrid edge state that is both topologically protected and immune to defects or disorders, and a localized bound state that is diffraction-free, owing to the self-balance of diffraction by nonlinearity. Topological edge solitons hold great potential for on-chip optical functional device fabrication. In this report, we present the discovery of vector valley Hall edge (VHE) solitons in type-II Dirac photonic lattices, formed by breaking lattice inversion symmetry with distortion operations. The distorted lattice features a two-layer domain wall that supports both in-phase and out-of-phase VHE states, appearing in two different band gaps. Superposing soliton envelopes onto VHE states generates bright-bright and bright-dipole vector VHE solitons. The propagation dynamics of such vector solitons reveal a periodic change in their profiles, accompanied by the energy periodically transferring between the layers of the domain wall. The reported vector VHE solitons are found to be metastable.

Chiral phonons in quartz probed by X-rays

The concept of chirality is of great relevance in nature, from chiral molecules such as sugar to parity transformations in particle physics. In condensed matter physics, recent studies have demonstrated chiral fermions and their relevance in emergent phenomena closely related to topology1-3. The experimental verification of chiral phonons (bosons) remains challenging, however, despite their expected strong impact on fundamental physical properties4-6. Here we show experimental proof of chiral phonons using resonant inelastic X-ray scattering with circularly polarized X-rays. Using the prototypical chiral material quartz, we demonstrate that circularly polarized X-rays, which are intrinsically chiral, couple to chiral phonons at specific positions in reciprocal space, allowing us to determine the chiral dispersion of the lattice modes. Our experimental proof of chiral phonons demonstrates a new degree of freedom in condensed matter that is both of fundamental importance and opens the door to exploration of new emergent phenomena based on chiral bosons.

Topological Metamaterials

The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.

Higher-order topological polariton corner state lasing

Unlike conventional laser, the topological laser is able to emit coherent light robustly against disorders and defects because of its nontrivial band topology. As a promising platform for low-power consumption, exciton polariton topological lasers require no population inversion, a unique property that can be attributed to the part-light-part-matter bosonic nature and strong nonlinearity of exciton polaritons. Recently, the discovery of higher-order topology has shifted the paradigm of topological physics to topological states at boundaries of boundaries, such as corners. However, such topological corner states have never been realized in the exciton polariton system yet. Here, on the basis of an extended two-dimensional Su-Schrieffer-Heeger lattice model, we experimentally demonstrate the topological corner states of perovskite polaritons and achieved polariton corner state lasing with a low threshold (approximately microjoule per square centimeter) at room temperature. The realization of such polariton corner states also provides a mechanism of polariton localization under topological protection, paving the way toward on-chip active polaritonics using higher-order topology.

Higher-Order Topological States in Thermal Diffusion

Unlike conventional topological materials that carry topological states at their boundaries, higher-order topological materials are able to support topological states at boundaries of boundaries, such as corners and hinges. While band topology has been recently extended into thermal diffusion for thermal metamaterials, its realization is limited to a 1D thermal lattice, lacking access to the higher-order topology. In this work, the experimental realization is reported of a higher-order thermal topological insulator in a generalized 2D diffusion lattice. The topological corner states for thermal diffusion are observed in the bandgap of diffusion rate of the bulk, as a consequence of the anti-Hermitian nature of the diffusion Hamiltonian. The topological protection of these thermal corner states is demonstrated with the stability of their diffusion profile in the presence of amorphous deformation. This work constitutes the first realization of higher-order topology in purely diffusive systems and opens the door for future thermal management with topological protection beyond 1D geometries.

Topological skyrmion phases of matter

We introduce topological phases of matter defined by skyrmions in the ground state spin-or pseudospin-expectation value textures in the Brillouin zone, the chiral and helical topological skyrmion phases of matter. These phases are protected by a symmetry present in centrosymmetric superconductors. We consider a tight-binding model for spin-triplet superconductivity in transition metal oxides and find it realizes each of these topological skyrmion phases. The chiral phase is furthermore realized for a parameter set characterizing Sr2RuO4with spin-triplet superconductivity. We also find two types of topological phase transitions by which the skyrmion number can change. The second type occurs without the closing of energy gaps in a system described by a quadratic Hamiltonian without breaking the protecting symmetries when atomic spin-orbit coupling is non-negligible and there is a suitable additional degree of freedom. This contradicts the 'flat band' limit assumption important in use of entanglement spectrum and Wilson loops, and in construction of the ten-fold way classification scheme of topological phases of matter. We furthermore predict two kinds of bulk-boundary correspondence signatures-one for measurements which execute a partial trace over degrees of freedom other than spin, which yields quantized transport signatures-and a second resulting from skyrmions trapping defects with their own non-trivial topology that is discussed in a second work, which yields generalizations of unpaired Majorana zero-modes.

Ideal acoustic quantum spin Hall phase in a multi-topology platform

Fermionic time-reversal symmetry ([Formula: see text])-protected quantum spin Hall (QSH) materials feature gapless helical edge states when adjacent to arbitrary trivial cladding materials. However, due to symmetry reduction at the boundary, bosonic counterparts usually exhibit gaps and thus require additional cladding crystals to maintain robustness, limiting their applications. In this study, we demonstrate an ideal acoustic QSH with gapless behaviour by constructing a global T_f on both the bulk and the boundary based on bilayer structures. Consequently, a pair of helical edge states robustly winds several times in the first Brillouin zone when coupled to resonators, promising broadband topological slow waves. We further reveal that this ideal QSH phase behaves as a topological phase transition plane that bridges trivial and higher-order phases. Our versatile multi-topology platform sheds light on compact topological slow-wave and lasing devices.

Architected material with independently tunable mass, damping, and stiffness via multi-stability and kinematic amplification

We report on a class of architected material lattices that exploit multi-stability and kinematic amplification to independently adjust the local effective mass, damping, and stiffness properties, thereby realizing congruent alterations to the acoustic dispersion response post-fabrication. The fundamental structural tuning element permits a broad range in the effective property space; moreover, its particular design carries the benefit of tuning without altering the original size/shape of the emerging structure. The relation between the tuning element geometry and the achieved variability in effective properties is explored. Bloch's theorem facilitates the dynamic analysis of representative one- and two-dimensional (1D/2D) systems, revealing, e.g., bandgap formation, migration, and closure and positive/negative metadamping in accordance with the tuning element configuration. To demonstrate a utility, we improvise a waveguide by appropriately patterning the tuning element configuration within a 2D system. We believe that the proposed strategy offers a new way to expand the range of performance and functionality of architected materials for elastodynamics.

Programmable Polariton Topological Insulators All-Optically Controlled by the Stark Effect

Efficiently and flexibly manipulating unidirectional edge states is key to developing topological insulators as functional devices. In this work, we propose an all-optical method that utilizes the valley-selective optical Stark effect to realize programmable topological insulators. We pattern a two-dimensional honeycomb structure in an exciton-polariton platform resulting from a strong light-matter coupling in a monolayer transition metal dichalcogenide. The optical Stark effect is induced to generate a pseudo magnetic field, combined with spin-orbit coupling to form the topological one-way edge states of the polariton. On account of the ultrafast switching speed and precisely spatial controllability of the optical Stark effect, two applications, i.e., ports ratio tunable polariton splitter and programmable polariton router, were demonstrated, showing designable and rewritable functionality of all-optically controllable polariton topological insulators. This study paves the way to robustly and intelligently control/form polaritonic and spintronic devices for future classical and quantum information processing and application.

Spinful Topological Phases in Acoustic Crystals with Projective PT Symmetry

For the classification of topological phases of matter, an important consideration is whether a system is spinless or spinful, as these two classes have distinct symmetry algebra that gives rise to fundamentally different topological phases. However, only recently has it been realized theoretically that in the presence of gauge symmetry, the algebraic structure of symmetries can be projectively represented, which possibly enables the switch between spinless and spinful topological phases. Here, we report the experimental demonstration of this idea by realizing spinful topological phases in "spinless" acoustic crystals with projective space-time inversion symmetry. In particular, we realize a one-dimensional topologically gapped phase characterized by a 2Z winding number, which features double-degenerate bands in the entire Brillouin zone and two pairs of degenerate topological boundary modes. Our Letter thus overcomes a fundamental constraint on topological phases by spin classes.

Direction-selective non-reciprocal mechanical energy splitter

A direction-selective elastic micro-structured medium is proposed. The lattice model combines constitutive nonlinearity, a threshold activation displacement and the gyroscopic effect of an isolated spinner to induce a tunable wave deviation on a selected direction in a full non-reciprocal way. The direction-selective effect is quantified in terms of energy flux dependence on gyricity and propagation velocity of the incident wave. Three different regimes are identified, addressed as passing, highly directive and barrier. The nonlinear system sustains solitary waves, described both numerically and through an analytical approximation, which show that the wave is univocally determined by the propagation velocity. A final analysis demonstrates the non-reciprocal mechanism originated by the proposed micro-structured medium. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.