Elasticity meets topology.

Harnessing the unique vectorial properties of elastic waves, Wu et al. find new degrees of freedom for realizing novel topological phases.

Optimizing multi-user indoor sound communications with acoustic reconfigurable metasurfaces.

Sound in indoor spaces forms a complex wavefield due to multiple scattering encountered by the sound. Indoor acoustic communication involving multiple sources and receivers thus inevitably suffers from cross-talks. Here, we demonstrate the isolation of acoustic communication channels in a room by wavefield shaping using acoustic reconfigurable metasurfaces (ARMs) controlled by optimization protocols based on communication theories. The ARMs have 200 electrically switchable units, each selectively offering 0 or π phase shifts in the reflected waves. The sound field is reshaped for maximal Shannon capacity and minimal cross-talk simultaneously. We demonstrate diverse acoustic functionalities over a spectrum much larger than the coherence bandwidth of the room, including multi-channel, multi-spectral channel isolations, and frequency-multiplexed acoustic communication. Our work shows that wavefield shaping in complex media can offer new strategies for future acoustic engineering.

Non-Abelian physics in light and sound.

Non-Abelian phenomena arise when the sequence of operations on physical systems influences their behaviors. By possessing internal degrees of freedom such as polarization, light and sound can be subjected to various manipulations, including constituent materials, structured environments, and tailored source conditions. These manipulations enable the creation of a great variety of Hamiltonians, through which rich non-Abelian phenomena can be explored and observed. Recent developments have constituted a versatile testbed for exploring non-Abelian physics at the intersection of atomic, molecular, and optical physics; condensed matter physics; and mathematical physics. These fundamental endeavors could enable photonic and acoustic devices with multiplexing functionalities. Our review aims to provide a timely and comprehensive account of this emerging topic. Starting from the foundation of matrix-valued geometric phases, we address non-Abelian topological charges, non-Abelian gauge fields, non-Abelian braiding, non-Hermitian non-Abelian phenomena, and their realizations with photonics and acoustics and conclude with future prospects.

Experimental Realization of Geometry-Dependent Skin Effect in a Reciprocal Two-Dimensional Lattice.

Recent studies of non-Hermitian periodic lattices unveiled the non-Hermitian skin effect (NHSE), in which the bulk modes under the periodic boundary conditions (PBC) become skin modes under open boundary conditions. The NHSE is a topological effect owing to the nontrivial spectral winding, and such spectral behaviors appear naturally in nonreciprocal systems. Hence prevailing approaches rely on nonreciprocity to achieve the NHSE. Here, we report the experimental realization of the geometry-dependent skin effect in a two-dimensional reciprocal system, in which the skin effect occurs only at boundaries whose macroscopic symmetry mismatches with the lattice symmetry. The role of spectral reciprocity and symmetry is revealed by connecting reflective channels at given boundaries with the spectral topology of the PBC spectrum. Our work highlights the vital role of reciprocity, symmetry, and macroscopic geometry on the NHSE in dimensionality larger than one and opens new routes for wave structuring using non-Hermitian effects.

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.

Realization of a Z-Classified Chiral-Symmetric Higher-Order Topological Insulator in a Coupling-Inverted Acoustic Crystal.

Higher-order topological band theory has transformed the landscape of topological phases in quantum and classical systems. Here, we experimentally demonstrate a two-dimensional higher-order topological phase, referred to as the multiple chiral topological phase, which is protected by a multipole chiral number (MCN). Our realization differs from previous higher-order topological phases in that it possesses a larger-than-unity MCN, which arises when the nearest-neighbor couplings are weaker than long-range couplings. Our phase has an MCN of 4, protecting the existence of 4 midgap topological corner modes at each corner. The multiple topological corner modes demonstrated here could lead to enhanced quantum-inspired devices for sensing and computing. Our study also highlights the rich and untapped potential of long-range coupling manipulation for future research in topological phases.

Realization and topological properties of third-order exceptional lines embedded in exceptional surfaces.

As the counterpart of Hermitian nodal structures, the geometry formed by exceptional points (EPs), such as exceptional lines (ELs), entails intriguing spectral topology. We report the experimental realization of order-3 exceptional lines (EL3) that are entirely embedded in order-2 exceptional surfaces (ES2) in a three-dimensional periodic synthetic momentum space. The EL3 and the concomitant ES2, together with the topology of the underlying space, prohibit the evaluation of their topology in the eigenvalue manifold by prevailing topological characterization methods. We use a winding number associated with the resultants of the Hamiltonian. This resultant winding number can be chosen to detect only the EL3 but ignores the ES2, allowing the diagnosis of the topological currents carried by the EL3, which enables the prediction of their evolution under perturbations. We further reveal the connection between the intersection multiplicity of the resultants and the winding of the resultant field around the EPs and generalize the approach for detecting and topologically characterizing higher-order EPs. Our work exemplifies the unprecedented topology of higher-order exceptional geometries and may inspire new non-Hermitian topological applications.

Experimental realization of non-Abelian permutations in a three-state non-Hermitian system.

Eigenstates of a non-Hermitian system exist on complex Riemannian manifolds, with multiple sheets connecting at branch cuts and exceptional points (EPs). These eigenstates can evolve across different sheets-a process that naturally corresponds to state permutation. Here, we report the first experimental realization of non-Abelian permutations in a three-state non-Hermitian system. Our approach relies on the stroboscopic encircling of two different exceptional arcs (EAs), which are smooth trajectories of order-2 EPs appearing from the coalescence of two adjacent states. The non-Abelian characteristics are confirmed by encircling the EAs in opposite sequences. A total of five non-trivial permutations are experimentally realized, which together comprise a non-Abelian group. Our approach provides a reliable way of investigating non-Abelian state permutations and the related exotic winding effects in non-Hermitian systems.

Chirality-switchable acoustic vortex emission via non-Hermitian selective excitation at an exceptional point.

Artificial structures provide an efficient method to generate acoustic vortices carrying orbital angular momentum (OAM) essential for applications ranging from object manipulation to acoustic communication. However, their flexibility in terms of chirality control has thus far been limited by the lack of reconfigurability and degrees of freedom like spin-orbit coupling. Here we show that this restriction can be lifted by controlling the individual on-off states of two coherent monopolar sources inside a passive parity-time-symmetric ring cavity at an exceptional point where the counter-propagating waves coalesce into one chiral eigenmode. One of the sources satisfies the chirality-reversal condition, generating a travelling wave field fully decoupled from and opposite to the chiral eigenmode, while the other source is phase-shifted such that the wave generated by the first source can be canceled out, and the remaining sound field circulates in the same direction as the chiral eigenmode. Such non-Hermitian selective excitation enables our experimental realization of acoustic vortex emission with switchable OAM but free of system reconfiguration. Our work offers opportunities for chiral sound manipulation as well as integrated and tunable acoustic OAM devices.

Experimental Realization of Weyl Exceptional Rings in a Synthetic Three-Dimensional Non-Hermitian Phononic Crystal.

Weyl points-topological monopoles of quantized Berry flux-are predicted to spread to Weyl exceptional rings in the presence of non-Hermiticity. Here, we use a one-dimensional Aubry-Andre-Harper model to construct a Weyl semimetal in a three-dimensional parameter space comprising one reciprocal dimension and two synthetic dimensions. The inclusion of non-Hermiticity in the form of gain and loss produces a synthetic Weyl exceptional ring (SWER). The topology of the SWER is characterized by both its topological charge and non-Hermitian winding numbers. We experimentally observe the SWER and synthetic Fermi arc in a one-dimensional phononic crystal with the non-Hermiticity introduced by active acoustic components. Our findings pave the way for studying the high-dimensional non-Hermitian topological physics in acoustics.

Non-Hermitian morphing of topological modes.

Topological modes (TMs) are usually localized at defects or boundaries of a much larger topological lattice1,2. Recent studies of non-Hermitian band theories unveiled the non-Hermitian skin effect (NHSE), by which the bulk states collapse to the boundary as skin modes3-6. Here we explore the NHSE to reshape the wavefunctions of TMs by delocalizing them from the boundary. At a critical non-Hermitian parameter, the in-gap TMs even become completely extended in the entire bulk lattice, forming an 'extended mode outside of a continuum'. These extended modes are still protected by bulk-band topology, making them robust against local disorders. The morphing of TM wavefunction is experimentally realized in active mechanical lattices in both one-dimensional and two-dimensional topological lattices, as well as in a higher-order topological lattice. Furthermore, by the judicious engineering of the non-Hermiticity distribution, the TMs can deform into a diversity of shapes. Our findings not only broaden and deepen the current understanding of the TMs and the NHSE but also open new grounds for topological applications.

Observation of Degenerate Zero-Energy Topological States at Disclinations in an Acoustic Lattice.

Building upon the bulk-boundary correspondence in topological phases of matter, disclinations have recently been harnessed to trap fractionally quantized density of states (DOS) in classical wave systems. While these fractional DOS have associated states localized to the disclination's core, such states are not protected from deconfinement due to the breaking of chiral symmetry, generally leading to resonances which, even in principle, have finite lifetimes and suboptimal confinement. Here, we devise and experimentally validate in acoustic lattices a paradigm by which topological states bind to disclinations without a fractional DOS but which preserve chiral symmetry. The preservation of chiral symmetry pins the states at the midgap, resulting in their protected maximal confinement. The integer DOS at the defect results in twofold degenerate states that, due to symmetry constraints, do not gap out. Our study provides a fresh perspective about the interplay between symmetry protection in topological phases and topological defects, with possible applications in classical and quantum systems alike.

Extended State in a Localized Continuum.

Bound state in a continuum (BIC) is a spatially confined resonance with its energy embedded in a continuous spectrum of propagative modes, yet their coupling is prohibited. In this Letter, we report the discovery of a generic non-Hermitian phenomenon that we call an "extended state in a localized continuum" (ELC). As the name suggests, the ELC is the inversion of the BIC-a single extended state embedded in a continuous spectrum entirely consisting of localized modes, and its emergence rests in the interplay between the BIC and the non-Hermitian skin effect (NHSE). Herein, the BIC is a zero-energy corner mode that spectrally overlaps with a bulk band in a Hermitian kagome lattice. The ELC emerges with the introduction of the NHSE in a particular way, such that it turns all the bulk states into corner skin modes and simultaneously delocalizes the corner mode. We experimentally realize the ELC using an active mechanical lattice. Our findings not only demonstrate the rich potential of the NHSE but may also spark new wave-based applications.

Synthetic Three-Dimensional Z×Z_{2} Topological Insulator in an Elastic Metacrystal.

We report a three-dimensional (3D) topological insulator (TI) formed by stacking identical layers of Chern insulators in a hybrid real-synthetic space. By introducing staggered interlayer hopping that respects mirror symmetry, the bulk bands possess an additional Z_{2} topological invariant along the stacking dimension, which, together with the nontrivial Chern numbers, endows the system with a Z×Z_{2} topology. A 4-tuple topological index characterizes the system's bulk bands. Consequently, two distinct types of topological surface modes (TSMs) are found localized on different surfaces. Type-I TSMs are gapless and are protected by Chern numbers, whereas type-II gapped TSMs are protected by Z_{2} bulk polarization in the stacking direction. Remarkably, each type-II TSM band is also topologically nontrivial, giving rise to second-order topological hinge modes (THMs). Both types of TSMs and the THMs are experimentally observed in an elastic metacrystal.

Spin-orbit interactions of transverse sound.

Spin-orbit interactions (SOIs) endow light with intriguing properties and applications such as photonic spin-Hall effects and spin-dependent vortex generations. However, it is counterintuitive that SOIs can exist for sound, which is a longitudinal wave that carries no intrinsic spin. Here, we theoretically and experimentally demonstrate that airborne sound can possess artificial transversality in an acoustic micropolar metamaterial and thus carry both spin and orbital angular momentum. This enables the realization of acoustic SOIs with rich phenomena beyond those in conventional acoustic systems. We demonstrate that acoustic activity of the metamaterial can induce coupling between the spin and linear crystal momentum k, which leads to negative refraction of the transverse sound. In addition, we show that the scattering of the transverse sound by a dipole particle can generate spin-dependent acoustic vortices via the geometric phase effect. The acoustic SOIs can provide new perspectives and functionalities for sound manipulations beyond the conventional scalar degree of freedom and may open an avenue to the development of spin-orbit acoustics.

Direct Measurement of Topological Properties of an Exceptional Parabola.

Non-Hermitian systems can produce branch singularities known as exceptional points (EPs). Different from singularities in Hermitian systems, the topological properties of an EP can involve either the winding of eigenvalues that produces a discriminant number (DN) or the eigenvector holonomy that generates a Berry phase. The multiplicity of topological invariants also makes non-Hermitian topology richer than its Hermitian counterpart. Here, we study a parabola-shaped trajectory formed by EPs with both theory and acoustic experiments. By obtaining both the DNs and Berry phases through the measurement of eigenvalues and eigenfunctions, we show that the EP trajectory endows the parameter space with a nontrivial fundamental group. Our findings not only shed light on exotic non-Hermitian topology but also provide a route for the experimental characterization of non-Hermitian topological invariants.

Landau-Zener Transition in the Dynamic Transfer of Acoustic Topological States.

Topological notions in physics often emerge from adiabatic evolution of states. It not only leads to fundamental insight of topological protection but also provides an important approach for the study of higher-dimensional topological phases. In this work, we first demonstrate the transfer of topological boundary states (TBSs) across the bulk to the opposite boundary in an acoustic waveguide system. By exploring the finite-size induced minigap between two TBS bands, we unveil the quantitative condition for the breakdown of adiabaticity in the system by demonstrating the Landau-Zener transition with both theory and experiments. Our results not only serve as a foundation of future studies of dynamic state transfer but also inspire applications leveraging nonadiabatic transitions as a new degree of freedom.

Multi-dimensional wave steering with higher-order topological phononic crystal.

The recent discovery and realizations of higher-order topological insulators enrich the fundamental studies on topological phases. Here, we report three-dimensional (3D) wave-steering capabilities enabled by topological boundary states at three different orders in a 3D phononic crystal with nontrivial bulk topology originated from the synergy of mirror symmetry of the unit cell and a non-symmorphic glide symmetry of the lattice. The multitude of topological states brings diverse possibilities of wave manipulations. Through judicious engineering of the boundary modes, we experimentally demonstrate two functionalities at different dimensions: 2D negative refraction of sound wave enabled by a first-order topological surface state with negative dispersion, and a 3D acoustic interferometer leveraging on second-order topological hinge states. Our work showcases that topological modes at different orders promise diverse wave steering applications across different dimensions.

Exceptional nexus with a hybrid topological invariant.

Branch-point singularities known as exceptional points (EPs), which carry a nonzero topological charge, can emerge in non-Hermitian systems. We demonstrate with both theory and acoustic experiments an "exceptional nexus" (EX), which is not only a higher-order EP but also the cusp singularity of multiple exceptional arcs (EAs). Because the parameter space is segmented by the EAs, the EX possesses a hybrid topological invariant (HTI), which consists of distinct winding numbers associated with Berry phases accumulated by cyclic paths on different complex planes. The HTI is experimentally characterized by measuring the critical behaviors of the wave functions. Our findings constitute a major advance in the fundamental understanding of non-Hermitian systems and their topology, possibly opening new avenues for applications.

Three-Dimensional Acoustic Double-Zero-Index Medium with a Fourfold Degenerate Dirac-like Point.

We report the first realization of a three-dimensional (3D) acoustic double-zero-index medium (DZIM) made of a cubic lattice of metal rods. While the past decade has seen several realizations of 2D DZIM, achieving such a medium in 3D has remained an elusive challenge. Here, we show how a fourfold degenerate point with conical dispersion can be induced at the Brillouin zone center, such that the material becomes a 3D DZIM with the effective mass density and compressibility simultaneously acquiring near-zero values. To demonstrate the functionalities of this new medium, we have fabricated an acoustic waveguide of 3D DZIM in form of a "periscope" with two 90° turns and observed tunneling of a normally incident planar wave through the waveguide yielding undistorted planar wave front at the waveguide exit. Our findings establish a practical route to realize 3D DZIM as an effective acoustic "void space" that offers unprecedented control over acoustic wave propagation.