Observation of 3D acoustic quantum Hall states.

Quantum Hall effect, the quantized transport phenomenon of electrons under strong magnetic fields, remains one of the hottest research topics in condensed matter physics since its discovery in 2D electronic systems. Recently, as a great advance in the research of quantum Hall effects, the quantum Hall effect in 3D systems, despite its big challenge, has been achieved in the bulk ZrTe5 and Cd3As2 materials. Interestingly, Cd3As2 is a Weyl semimetal, and quantum Hall effect is hosted by the Fermi arc states on opposite surfaces via the Weyl nodes of the bulk, and induced by the unique edge states on the boundaries of the opposite surfaces. However, such intriguing edge state distribution has not yet been experimentally observed. Here, we aim to reveal experimentally the unusual edge states of Fermi arcs in acoustic Weyl system with the aid of pseudo-magnetic field. Benefiting from the macroscopic nature of acoustic crystals, the pseudo-magnetic field is introduced by elaborately designed the gradient on-site energy, and the edge states of Fermi arcs on the boundaries of the opposite surfaces are unambiguously demonstrated in experiments. Our system serves as an ideal and highly tunable platform to explore the Hall physics in 3D system, and has the potential in the application of new acoustic devices.

Brillouin Klein space and half-turn space in three-dimensional acoustic crystals.

The Bloch band theory and Brillouin zone (BZ) that characterize wave-like behaviors in periodic mediums are two cornerstones of contemporary physics, ranging from condensed matter to topological physics. Recent theoretical breakthrough revealed that, under the projective symmetry algebra enforced by artificial gauge fields, the usual two-dimensional (2D) BZ (orientable Brillouin two-torus) can be fundamentally modified to a non-orientable Brillouin Klein bottle with radically distinct manifold topology. However, the physical consequence of artificial gauge fields on the more general three-dimensional (3D) BZ (orientable Brillouin three-torus) was so far missing. Here, we theoretically discovered and experimentally observed that the fundamental domain and topology of the usual 3D BZ can be reduced to a non-orientable Brillouin Klein space or an orientable Brillouin half-turn space in a 3D acoustic crystal with artificial gauge fields. We experimentally identify peculiar 3D momentum-space non-symmorphic screw rotation and glide reflection symmetries in the measured band structures. Moreover, we experimentally demonstrate a novel stacked weak Klein bottle insulator featuring a nonzero Z2 topological invariant and self-collimated topological surface states at two opposite surfaces related by a nonlocal twist, radically distinct from all previous 3D topological insulators. Our discovery not only fundamentally modifies the fundamental domain and topology of 3D BZ, but also opens the door towards a wealth of previously overlooked momentum-space multidimensional manifold topologies and novel gauge-symmetry-enriched topological physics and robust acoustic wave manipulations beyond the existing paradigms.

Valley edge states as bound states in the continuum.

Bound states in the continuum (BICs) are spatially localized states with energy embedded in the continuum spectrum of extended states. The combination of BICs physics and nontrivial band topology theory givs rise to topological BICs, which are robust against disorders and meanwhile, the merit of conventional BICs is attracting wide attention recently. Here, we report valley edge states as topological BICs, which appear at the domain wall between two distinct valley topological phases. The robustness of such BICs is demonstrated. The simulations and experiments show great agreement. Our findings of valley related topological BICs shed light on both BICs and valley physics, and may foster innovative applications of topological acoustic devices.

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.

Non-Hermitian Topological Phononic Metamaterials.

Non-Hermitian (NH) physics describes novel phenomena in open systems that allow generally complex spectra. Introducing NH physics into topological metamaterials, which permits explorations of topological wave phenomena in artificially designed structures, not only enables the experimental verification of exotic NH phenomena in these flexible platforms, but also enriches the manipulation of wave propagation beyond the Hermitian cases. Here, a perspective on the advances in the research of NH topological phononic metamaterials is presented, which covers the exceptional points and their topological geometries, the skin effect related to the topology of complex spectra, the interplay of NH effects and topological states in phononic metamaterials, etc.

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.

Observation of geometry-dependent skin effect in non-Hermitian phononic crystals with exceptional points.

Exceptional points and skin effect, as the two distinct hallmark features unique to the non-Hermitian physics, have each attracted enormous interests. Recent theoretical works reveal that the topologically nontrivial exceptional points can guarantee the non-Hermitian skin effect, which is geometry-dependent, relating these two unique phenomena. However, such novel relation remains to be confirmed by experiments. Here, we realize a non-Hermitian phononic crystal with exceptional points, which exhibits the geometry-dependent skin effect. The exceptional points connected by the bulk Fermi arcs, and the skin effects with the geometry dependence, are evidenced in simulations and experiments. Our work, building an experimental bridge between the exceptional points and skin effect and uncovering the unconventional geometry-dependent skin effect, expands a horizon in non-Hermitian physics.

Topological materials for full-vector elastic waves.

Elastic wave manipulation is important in a wide variety of applications, including information processing in small elastic devices and noise control in large solid structures. The recent emergence of topological materials has opened new avenues for modulating elastic waves in solids. However, because of the full-vector feature and the complicated couplings of the longitudinal and transverse components of elastic waves, manipulating elastic waves is generally difficult compared with manipulating acoustic waves (scalar waves) and electromagnetic waves (vectorial waves but transverse only). To date, topological materials, including insulators and semimetals, have been used for acoustic and electromagnetic waves. Although topological materials with elastic waves have also been reported, the observed topological edge modes lie on the domain wall. A natural question arises: Is there an elastic metamaterial with topological edge modes on its own boundary? Here, we report a 3D metal-printed bilayer metamaterial that topologically insulates elastic waves. By introducing chiral interlayer couplings, the spin-orbit couplings for elastic waves are induced, which give rise to nontrivial topological properties. Helical edge states with vortex features were demonstrated on the boundary of the single topological phase. We further show a heterostructure of the metamaterial that exhibits tunable edge transport. Our findings could be used in devices based on elastic waves in solids.

Acoustic higher-order topology derived from first-order with built-in Zeeman-like fields.

Higher-order topological insulators (HOTIs), with topological corner or hinge states, have emerged as a thriving topic in the field of topological physics. However, few connections have been found for HOTIs with well-explored first-order topological insulators. Recently a proposal asserted that a significant bridge can be established between the HOTIs and Z2 topological insulators. When subjected to an in-plane Zeeman field, corner states, the signature of the HOTIs, can be induced in a Z2 topological insulator. Such Zeeman fields can be produced, for example, by the ferromagnetic proximity effect or magnetic atom doping, which drastically increases the experimental complexity. Here, we show that a phononic crystal, designed as a bilayer of coupled acoustic cavities, exactly hosts the Kane-Mele model with built-in in-plane Zeeman fields. The helical edge states along the zigzag edges are gapped, and the corner states, localized spatially at the corners of the samples, appear in the gap. This verifies the Zeeman field induced higher-order topology. We further demonstrate the intriguing contrast properties of the corner states at the outer and inner corners in a hexagonal ring-shaped sample.

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.

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.

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.

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.

Acoustic spin-Chern insulator induced by synthetic spin-orbit coupling with spin conservation breaking.

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.

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.

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.

Nodal rings and drumhead surface states in phononic crystals.

Three-dimensional topological nodal lines, the touching curves of two bands in momentum space, which give rise to drumhead surface states, provide an opportunity to explore a variety of exotic phenomena. However, solid evidence for a flat drumhead surface state remains elusive. In this paper, we report a realization of three-dimensional nodal line dispersions and drumhead surface states in phononic crystal. Profiting from its macroscopic nature, the phononic crystal permits a flexible and accurate fabrication for materials with ring-like nodal lines and drumhead surface states. Phononic nodal rings of the lowest two bands and, more importantly, topological drumhead surface states are unambiguously demonstrated. Our system provides an ideal platform to explore the intriguing properties of acoustic waves endowed with extraordinary dispersions.

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.