Observation of Higher-Order Nodal-Line Semimetal in Phononic Crystals.

Higher-order topological insulators and semimetals, which generalize the conventional bulk-boundary correspondence, have attracted extensive research interest. Among them, higher-order Weyl semimetals feature twofold linear crossing points in three-dimensional momentum space, 2D Fermi-arc surface states, and 1D hinge states. Higher-order nodal-point semimetals possessing Weyl points or Dirac points have been implemented. However, higher-order nodal-line or nodal-surface semimetals remain to be further explored in experiments in spite of many previous theoretical efforts. In this work, we realize a second-order nodal-line semimetal in 3D phononic crystals. The bulk nodal lines, 2D drumhead surface states guaranteed by Zak phases, and 1D flat hinge states attributed to k_{z}-dependent quadrupole moments are observed in simulations and experiments. Our findings of nondispersive surface and hinge states may promote applications in acoustic sensing and energy harvesting.

Topological negative refraction of surface acoustic waves in a Weyl phononic crystal.

Reflection and refraction of waves occur at the interface between two different media. These two fundamental interfacial wave phenomena form the basis of fabricating various wave components, such as optical lenses. Classical refraction-now referred to as positive refraction-causes the transmitted wave to appear on the opposite side of the interface normal compared to the incident wave. By contrast, negative refraction results in the transmitted wave emerging on the same side of the interface normal. It has been observed in artificial materials1-5, following its theoretical prediction6, and has stimulated many applications including super-resolution imaging7. In general, reflection is inevitable during the refraction process, but this is often undesirable in designing wave functional devices. Here we report negative refraction of topological surface waves hosted by a Weyl phononic crystal-an acoustic analogue of the recently discovered Weyl semimetals8-12. The interfaces at which this topological negative refraction occurs are one-dimensional edges separating different facets of the crystal. By tailoring the surface terminations of the Weyl phononic crystal, constant-frequency contours of surface acoustic waves can be designed to produce negative refraction at certain interfaces, while positive refraction is realized at different interfaces within the same sample. In contrast to the more familiar behaviour of waves at interfaces, unwanted reflection can be prevented in our crystal, owing to the open nature of the constant-frequency contours, which is a hallmark of the topologically protected  surface states in Weyl crystals8-12.

Topological phononic metamaterials.

The concept of topological energy bands and their manifestations have been demonstrated in condensed matter systems as a fantastic paradigm toward unprecedented physical phenomena and properties that are robust against disorders. Recent years, this paradigm was extended to phononic metamaterials (including mechanical and acoustic metamaterials), giving rise to the discovery of remarkable phenomena that were not observed elsewhere thanks to the extraordinary controllability and tunability of phononic metamaterials as well as versatile measuring techniques. These phenomena include, but not limited to, topological negative refraction, topological 'sasers' (i.e. the phononic analog of lasers), higher-order topological insulating states, non-Abelian topological phases, higher-order Weyl semimetal phases, Majorana-like modes in Dirac vortex structures and fragile topological phases with spectral flows. Here we review the developments in the field of topological phononic metamaterials from both theoretical and experimental perspectives with emphasis on the underlying physics principles. To give a broad view of topological phononics, we also discuss the synergy with non-Hermitian effects and cover topics including synthetic dimensions, artificial gauge fields, Floquet topological acoustics, bulk topological transport, topological pumping, and topological active matters as well as potential applications, materials fabrications and measurements of topological phononic metamaterials. Finally, we discuss the challenges, opportunities and future developments in this intriguing field and its potential impact on physics and materials science.

Antichirality Emergent in Type-II Weyl Phononic Crystals.

Chiral anomaly as the hallmark feature lies in the heart of the researches for Weyl semimetal. It is rooted in the zeroth Landau level of the system with an applied magnetic field. Chirality or antichirality characterizes the propagation property of the one-way zeroth Landau level mode, and antichirality means an opposite group velocity compared to the case of chirality. Chirality is commonly observed for Weyl semimetals. Interestingly, the type-II Weyl point, with the overtilted dispersion, may flip the chirality to the antichirality, which, however, is yet to be evidenced despite numerous previous experimental efforts. Here, we implement the type-II Weyl point in sonic crystals, and by creating the pseudomagnetic fields with geometric deformation, the chirality flip of zeroth Landau levels is unambiguously demonstrated. Our Letter unveils the novel antichiral transport in the presence of time-reversal symmetry, and paves the way toward the state-of-the-art manipulation of sound waves.

Acoustic Higher-Order Weyl Semimetal with Bound Hinge States in the Continuum.

Higher-order topological phases have raised widespread interest in recent years with the occurrence of the topological boundary states of dimension two or more less than that of the system bulk. The higher-order topological states have been verified in gapped phases, in a wide variety of systems, such as photonic and acoustic systems, and recently also observed in gapless semimetal phase, such as Weyl and Dirac phases, in systems alike. The higher-order topology is signaled by the hinge states emerging in the common band gaps of the bulk states and the surface states. In this Letter, we report our first prediction and observation of a new type of hinge states, the bound hinge states in the continuum (BHICs) bulk band, in a higher-order Weyl semimetal implemented in phononic crystal. In contrast to the hinge state in gap, which is characterized by the bulk polarization, the BHIC is identified by the nontrivial surface polarization. The finding of the topological BHICs broadens our insight to the topological states, and may stimulate similar researches in other systems such as electronic, photonic, and cold atoms systems. Our Letter may pave the way toward high-Q acoustic devices in application.

Higher-order topological semimetal in acoustic crystals.

The notion of higher-order topological insulators has endowed materials with topological states beyond the first order. Particularly, a three-dimensional (3D) higher-order topological insulator can host topologically protected 1D hinge states, referred to as the second-order topological insulator, or 0D corner states, referred to as the third-order topological insulator. Similarly, a 3D higher-order topological semimetal can be envisaged if it hosts states on the 1D hinges. Here we report the realization of a second-order topological Weyl semimetal in a 3D-printed acoustic crystal, which possesses Weyl points in 3D momentum space, 2D Fermi arc states on surfaces and 1D gapless states on hinges. Like the arc surface states, the hinge states also connect the projections of the Weyl points. Our experimental results evidence the existence of the higher-order topological semimetal, which may pave the way towards innovative acoustic devices.

Experimental Observation of Non-Abelian Earring Nodal Links in Phononic Crystals.

Nodal lines are symmetry-protected one-dimensional band degeneracies in momentum space, which can appear in numerous topological configurations such as nodal rings, chains, links, and knots. Very recently, non-Abelian topological physics have been proposed in space-time inversion (PT) symmetric systems. One of the most special configurations in such systems is the earring nodal link, composing of a nodal chain linking with an isolated nodal line. Such earring nodal links have not been observed in real systems. We designed phononic crystals with earring nodal links, and experimentally observed two different kinds of earring nodal links by measuring the band structures. We found that the order of the nodal chain and line can be switched after band inversion but their link cannot be severed. Our Letter provides experimental evidence for phenomena unique to non-Abelian band topology and our acoustic system provides a convenient platform for studying the new materials carrying non-Abelian charges.

Hybrid-Order Topological Insulators in a Phononic Crystal.

Topological phases, including the conventional first-order and higher-order topological insulators and semimetals, have emerged as a thriving topic in the fields of condensed-matter physics and materials science. Usually, a topological insulator is characterized by a fixed order topological invariant and exhibits associated bulk-boundary correspondence. Here, we realize a new type of topological insulator in a bilayer phononic crystal, which hosts simultaneously the first-order and second-order topologies, referred to here as the hybrid-order topological insulator. The one-dimensional gapless helical edge states, and zero-dimensional corner states coexist in the same system. The new hybrid-order topological phase may produce novel applications in topological acoustic devices.

Pseudomagnetic Fields Enabled Manipulation of On-Chip Elastic Waves.

The physical realization of pseudomagnetic fields (PMFs) is an engaging frontier of research. As in graphene, elastic PMF can be realized by the structural modulations of Dirac materials. We show that, in the presence of PMFs, the conical dispersions split into elastic Landau levels, and the elastic modes robustly propagate along the edges, similar to the quantum Hall edge transports. In particular, we reveal unique elastic snake states in an on-chip heterostructure with two opposite PMFs. The flexibility in the micromanufacture of silicon chips and the low loss of elastic waves provide an unprecedented opportunity to demonstrate various fascinating topological transports of the edge states under PMFs. These properties open new possibilities for designing functional elastic wave devices in miniature and compact scales.

3D Hinge Transport in Acoustic Higher-Order Topological Insulators.

The discovery of topologically protected boundary states in topological insulators opens a new avenue toward exploring novel transport phenomena. The one-way feature of boundary states against disorders and impurities prospects great potential in applications of electronic and classical wave devices. Particularly, for the 3D higher-order topological insulators, it can host hinge states, which allow the energy to transport along the hinge channels. However, the hinge states have only been observed along a single hinge, and a natural question arises: whether the hinge states can exist simultaneously on all the three independent directions of one sample? Here we theoretically predict, numerically simulate, and experimentally observe the hinge states on three different directions of a higher-order topological phononic crystal, and demonstrate their robust one-way transport from hinge to hinge. Therefore, 3D topological hinge transport is successfully achieved. The novel sound transport may serve as the basis for acoustic devices of unconventional functions.

Metafluids beyond the Bulk Modulus.

It is well known that the acoustic properties of fluid are characterized by mass density and bulk modulus. Metafluids, the fluid metamaterials, generalize the natural fluid, which can accommodate extreme and/or negative values of these two parameters. Here, we further show that the metafluids, composed of periodic thin-walled hollow cylinders immersed in fluid, can provide not only the designable effective mass density and bulk modulus, but also a completely new effective parameter, which appears in the wave velocities as a role similar to the shear modulus of solid. The new effective parameter, describing the response of the fluid to the quadrupolar component of waves, is obtained by generalizing the effective medium theory (EMT) to include the second-order effects, which is vanishing and neglected in the conventional EMT, but giant here in the metafluids with built-in quadrupolar resonances. With the discovery of the metafluids of shearlike moduli, our Letter extends the concept of metafluids and will have a great significance in the field of metamaterials.

Acoustic Realization of Quadrupole Topological Insulators.

A quadrupole topological insulator, being one higher-order topological insulator with nontrivial quadrupole quantization, has been intensely investigated very recently. However, the tight-binding model proposed for such emergent topological insulators demands both positive and negative hopping coefficients, which imposes an obstacle in practical realizations. Here, we introduce a feasible approach to design the sign of hopping in acoustics, and construct the first acoustic quadrupole topological insulator that stringently emulates the tight-binding model. The inherent hierarchy quadrupole topology has been experimentally confirmed by detecting the acoustic responses at the bulk, edge, and corner of the sample. Potential applications can be anticipated for the topologically robust in-gap states, such as acoustic sensing and energy trapping.

Ideal Type-II Weyl Phase and Topological Transition in Phononic Crystals.

Ideal Weyl points, which are related by symmetry and thus reside at the same frequency, could offer further insight into the Weyl physics. The ideal type-I Weyl points have been observed in photonic crystals, but the ideal type-II Weyl points with tilted conelike band dispersions are still not realized. Here we present the observation of the ideal type-II Weyl points of the minimal number in three-dimensional phononic crystals and, in the meantime, the topological phase transition from the Weyl semimetal to the valley insulators of two distinct types. The Fermi-arc surface states are shown to exist on the surfaces of the Weyl phase, and the Fermi-circle surface states are also observed, but on the interface of the two distinct valley phases. Intriguing wave partition of the Fermi-circle surface states is demonstrated.

On-chip valley topological materials for elastic wave manipulation.

Valley topological materials, in which electrons possess valley pseudospin, have attracted a growing interest recently. The additional valley degree of freedom offers a great potential for its use in information encoding and processing. The valley pseudospin and valley edge transport have been investigated in photonic and phononic crystals for electromagnetic and acoustic waves, respectively. In this work, by using a micromanufacturing technology, valley topological materials are fabricated on silicon chips, which allows the observation of gyral valley states and valley edge transport for elastic waves. The edge states protected by the valley topology are robust against the bending and weak randomness of the channel between distinct valley Hall phases. At the channel intersection, a counterintuitive partition of the valley edge states manifests for elastic waves, in which the partition ratio can be freely adjusted. These results may enable the creation of on-chip high-performance micro-ultrasonic materials and devices.

Generating arbitrary photoacoustic fields with a spatial light modulator.

Ultrasound fields have broad applications in imaging, sensing, medical therapy, etc. In these applications, it is of great importance to generate desired ultrasound fields. The generation of arbitrary ultrasound fields is challenging using phased array transducers or a monolithic acoustic hologram. In this work, by taking advantage of the photoacoustic effect and spatial light modulating technique, we demonstrate that dynamic and high-resolution arbitrary acoustic fields in liquid can be realized. We clearly show ultrasonic vortex and arbitrary 2D/3D ultrasonic fields in our photoacoustic system. All the measured pressure fields agree well with the desired ones. We anticipate these rapidly tunable photoacoustic fields will find applications in dynamic acoustic tweezers and ultrasonic imaging.

Experimental Realization of Type-II Weyl Points and Fermi Arcs in Phononic Crystal.

Weyl points (WPs), as the doubly degenerate points in three-dimensional momentum band structures, carry quantized topological charges and give rise to a variety of extraordinary properties, such as robust surface wave and chiral anomaly. Type-II Weyl semimetals, which have conical dispersions in Fermi surfaces and a strongly tilted dispersion with respect to type I, have recently been proposed in condensed-matter systems and photonics. Although the type-II WPs have been theoretically predicted in acoustics, the experimental realization in phononic crystals has not been reported so far. Here, we experimentally realize a type-II Weyl phononic crystal. We demonstrate the topological transitions observed at the WP frequencies and the topological surface acoustic waves between the Weyl frequencies. The experiment results are in good accordance with our theoretical analyses. Due to the violation of the Lorentz symmetry, the type-II WPs only exist in low energy systems. As the analog counterpart in classical waves, the phononic crystal brings a platform for the research of type-II WPs in macroscopic systems.

Probing Weyl Physics with One-Dimensional Sonic Crystals.

Recently, intense efforts have been devoted to realizing classical analogues of various topological phases of matter. In this Letter, we explore the intriguing Weyl physics by a simple one-dimensional sonic crystal, in which two extra structural parameters are combined to construct a synthetic three-dimensional space. Based on our ultrasonic experiments, we have not only observed the synthetic Weyl points, but also probed the novel reflection phase singularity that connects inherently with the topological robustness of Weyl points. The presence of topologically nontrivial interface modes has been demonstrated further. As the first realization of topological acoustics in synthetic space, our study exhibits great potential of probing high-dimensional topological phenomena by such easily fabricated and detected low-dimension acoustic systems.

Valley Physics in Non-Hermitian Artificial Acoustic Boron Nitride.

The valley can serve as a new degree of freedom in the manipulation of particles or waves in condensed matter physics, whereas systems containing combinations of gain and loss elements constitute rich building units that can mimic non-Hermitian properties. By introducing gain and loss in artificial acoustic boron nitride, we show that the acoustic valley states and the valley-projected edge states display exotic behaviors in that they sustain either attenuated or amplified wave propagation. Our findings show how non-Hermiticity introduces a mechanism in tuning topological protected valley transports, which may have significance in advanced wave control for sensing and communication applications.

Valley Topological Phases in Bilayer Sonic Crystals.

Recently, the topological physics in artificial crystals for classical waves has become an emerging research area. In this Letter, we propose a unique bilayer design of sonic crystals that are constructed by two layers of coupled hexagonal array of triangular scatterers. Assisted by the additional layer degree of freedom, a rich topological phase diagram is achieved by simply rotating scatterers in both layers. Under a unified theoretical framework, two kinds of valley-projected topological acoustic insulators are distinguished analytically, i.e., the layer-mixed and layer-polarized topological valley Hall phases, respectively. The theory is evidently confirmed by our numerical and experimental observations of the nontrivial edge states that propagate along the interfaces separating different topological phases. Various applications such as sound communications in integrated devices can be anticipated by the intriguing acoustic edge states enriched by the layer information.

Valley Vortex States in Sonic Crystals.

Valleytronics is quickly emerging as an exciting field in fundamental and applied research. In this Letter, we study the acoustic version of valley states in sonic crystals and reveal a vortex nature of such states. In addition to the selection rules established for exciting valley polarized states, a mimicked valley Hall effect of sound is proposed further. The extraordinary chirality of valley vortex states, detectable in experiments, may open a new possibility in sound manipulations. This is appealing to scalar acoustics that lacks a spin degree of freedom inherently. In addition, the valley selection enables a handy way to create vortex matter in acoustics, in which the vortex chirality can be controlled flexibly. Potential applications can be anticipated with the exotic interaction of acoustic vortices with matter, such as to trigger the rotation of the trapped microparticles without contact.