Experimental observation of non-Abelian topological charges and edge states.

In the last few decades, topological phase1-11 has emerged as a new classification of matter states beyond the Ginzburg-Landau symmetry-breaking paradigm. The underlying global invariant is usually well characterized by integers, such as Chern numbers or winding numbers-the Abelian charges12-15. Very recently, researchers proposed the notion of non-Abelian topological charges16-19, which possess non-commutative and fruitful braiding structures with multiple (more than one) bandgaps tangled together. Here we experimentally observe the non-Abelian topological charges in a time-reversal and inversion-symmetric transmission line network. The quaternion-valued non-Abelian topological charges are clearly mapped onto an eigenstate-frame sphere. Moreover, we find a non-Abelian quotient relation that provides a global perspective on the distribution of edge/domain-wall states. Our work opens the door towards characterization and manipulation of non-Abelian topological charges, which may lead to interesting observables such as trajectory-dependent Dirac/Weyl node collisions in two-dimensional systems16,17,20, admissible nodal line configurations in three dimensions16,19,20, and may provide insight into certain strongly correlated phases of twisted bilayer graphene21.

Scalar topological photonic nested meta-crystals and skyrmion surface states in the light cone continuum.

Topological photonics is rapidly expanding. However, discovering three-dimensional topological electromagnetic systems can be more challenging than electronic systems for two reasons. First, the vectorial nature of electromagnetic waves results in complicated band dispersions, and simple tight-binding-type predictions usually fail. Second, topological electromagnetic surface modes inside the light cone have very low quality factors (Q factors). Here, we propose the concept of scalar topological photonics to address these challenges. Our approach is experimentally validated by employing a nested meta-crystal configuration using connected coaxial waveguides. They exhibit scalar-wave-like band dispersions, making the search for photonic topological phases an easier task. Their surface states have skyrmion-like electric field distributions, resulting in a whole, bright surface state band inside the light cone continuum. As such, the topological surface states in our three-dimensional nested crystals can be exposed to air, making such systems well-suited for practical applications.

Near-Field Spin Chern Number Quantized by Real-Space Topology of Optical Structures.

The Chern number has been widely used to describe the topological properties of periodic structures in momentum space. Here, we introduce a real-space spin Chern number for the optical near fields of finite-sized structures. This new spin Chern number is intrinsically quantized and equal to the structure's Euler characteristic. The relationship is robust against continuous deformation of the structure's geometry and is irrelevant to the specific material constituents or external excitation. Our Letter enriches topological physics by extending the Chern number to real space, opening exciting possibilities for exploring the real-space topological properties of light.

Topological Photonic Alloy.

We present the new concept of photonic alloy as a nonperiodic topological material. By mixing nonmagnetized and magnetized rods in a nonperiodic 2D photonic crystal configuration, we realized photonic alloys in the microwave regime. Our experimental findings reveal that the photonic alloy sustains nonreciprocal chiral edge states even at very low concentration of magnetized rods. The nontrivial topology and the associated edge states of these nonperiodic systems can be characterized by the winding of the reflection phase. Our results indicate that the threshold concentrations for the investigated system within the first nontrivial band gap to exhibit topological behavior approach zero in the thermodynamic limit for substitutional alloys, while the threshold remains nonzero for interstitial alloys. At low concentration, the system exhibits an inhomogeneous structure characterized by isolated patches of nonpercolating magnetic domains that are spaced far apart within a topologically trivial photonic crystal. Surprisingly, the system manifests chiral edge states despite a local breakdown of time-reversal symmetry rather than a global one. Photonic alloys represent a new category of disordered topological materials, offering exciting opportunities for exploring topological materials with adjustable gaps.

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.

Photonic Z_{2} Topological Anderson Insulators.

That disorder can induce nontrivial topology is a surprising discovery in topological physics. As a typical example, Chern topological Anderson insulators (TAIs) have been realized in photonic systems, where the topological phases exist without symmetry protection. In this Letter, by taking transverse magnetic and transverse electric polarizations as pseudospin degrees of freedom, we theoretically propose a scheme to realize disorder-induced symmetry-protected topological phase transitions in two-dimensional photonic crystals with a combined time-reversal, mirror, and duality symmetry T_{f}=TM_{z}D. In particular, we demonstrate that the disorder-induced symmetry-protected topological phase persists even without pseudospin conservation, thereby realizing a photonic Z_{2} TAI, in contrast to a Z-classified quantum spin Hall (QSH) TAI with decoupled spins. By formulating a new scattering approach, we show that the topology of both the QSH and Z_{2} TAIs can be manifested by the accumulated spin rotations of the reflected waves from the photonic crystals. Using a transmission structure, we also illustrate the trivialization of a disordered QSH phase with an even integer topological index caused by spin coupling.

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.

Straight Photonic Nodal Lines with Quadrupole Berry Curvature Distribution and Superimaging "Fermi Arcs".

In periodic systems, nodal lines are loops in the three-dimensional momentum space with each point on them representing a band degeneracy. Nodal lines exhibit rich topological features, as they can take various configurations such as rings, links, chains, and knots. These line nodes are generally protected by mirror or PT symmetry and frequently accompanied by drumhead surface states. Here, we propose and demonstrate a novel type of photonic straight nodal lines in a D_{2D} metacrystal, which are protected by an unusual rotoinversion time (roto-PT) symmetry. These nodal lines are located at the central axis and hinges of the Brillouin zone. They appear as quadrupole sources of Berry curvature flux in contrast to the Weyl points, which are monopoles. Interestingly, topological surface states exist at all three cutting surfaces, as guaranteed by π-quantized Zak phases along all three directions. As frequency changes, the surface state equifrequency contours evolve from closed to open and become straight lines at a critical transition frequency, at which diffractionless surface wave propagations are experimentally demonstrated, paving the way toward development of superimaging topological devices.

Observation of Non-Abelian Charged Nodes Linking Nonadjacent Gaps.

Nodal links are special configurations of band degeneracies in the momentum space, where nodal line branches encircle each other. In PT symmetric systems, nodal lines can be topologically characterized using the eigenvector frame rotations along an encircling loop and the linking structure can be described with non-Abelian frame charges involving adjacent bands. While the commutation rules between the frame charges are well established, the underlying relationship between distant band gap closing nodes remains to be explored. In this Letter, we present a photonic multiple nodal links system, where the nodal lines of nonadjacent bands are investigated with symmetry constraints on frame charges. Through an orthogonal nodal chain, the nodal line from the lower two bands predicts the existence of nodal lines formed between the higher bands. We designed and fabricated a metamaterial, with which the multiple nodal links and the topological connection between nonadjacent nodal lines are experimentally demonstrated.

Optical forces on a cylinder induced by surface waves and the conservation of the canonical momentum of light.

Based on rigorous derivations using the electromagnetic energy-momentum tensor, we established a generic relationship between the longitudinal optical force (along the surface wave propagating direction) on a cylinder induced by surface waves and the energy flux of each surface mode supported on the interface between air and a lossless substrate possessing continuous translational symmetry along the longitudinal direction. The longitudinal optical force is completely attributed to the canonical momentum of light. Our theory is valid for generic types of surface waves and lays the theoretical foundation for the research and applications of optical manipulations by surface waves.

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.

Topological Corner Modes Induced by Dirac Vortices in Arbitrary Geometry.

Recently, higher-order topologies have been experimentally realized, featuring topological corner modes (TCMs) between adjacent topologically distinct domains. However, they have to comply with specific spatial symmetries of underlying lattices, hence their TCMs only emerge in very limited geometries, which significantly impedes generic applications. Here, we report a general scheme of inducing TCMs in arbitrary geometry based on Dirac vortices from aperiodic Kekulé modulations. The TCMs can now be constructed and experimentally observed in square and pentagonal domains incompatible with underlying triangular lattices. Such bound modes at arbitrary corners do not require their boundaries to run along particular lattice directions. Our scheme allows an arbitrary specification of numbers and positions of TCMs, which will be important for future on-chip topological circuits. Moreover, the general scheme developed here can be extended to other classical wave systems. Our findings reveal rich physics of aperiodic modulations, and advance applications of TCMs in realistic scenarios.

Optically Reconfigurable Spin-Valley Hall Effect of Light in Coupled Nonlinear Ring Resonator Lattice.

Scattering immune propagation of light in topological photonic systems may revolutionize the design of integrated photonic circuits for information processing and communications. In optics, various photonic topological circuits have been developed, which were based on classical emulation of either quantum spin Hall effect or quantum valley Hall effect. On the other hand, the combination of both the valley and spin degrees of freedom can lead to a new kind of topological transport phenomenon, dubbed spin-valley Hall effect (SVHE), which can further expand the number of topologically protected edge channels and would be useful for information multiplexing. However, it is challenging to realize SVHE in most known material platforms, due to the requirement of breaking both the (pseudo)fermionic time-reversal (T) and parity symmetries (P) individually, but leaving the combined symmetry S≡TP intact. Here, we propose an experimentally feasible platform to realize SVHE for light, based on coupled ring resonators mediated by optical Kerr nonlinearity. Thanks to the inherent flexibility of cross-mode modulation, the coupling between the probe light can be engineered in a controllable way such that spin-dependent staggered sublattice potential emerges in the effective Hamiltonian. With delicate yet experimentally feasible pump conditions, we show the existence of spin-valley Hall-induced topological edge states. We further demonstrate that both degrees of freedom, i.e., spin and valley, can be manipulated simultaneously in a reconfigurable manner to realize spin-valley photonics, doubling the degrees of freedom for enhancing the information capacity in optical communication systems.

Topological One-Way Large-Area Waveguide States in Magnetic Photonic Crystals.

We have theoretically and experimentally achieved large-area one-way transport by using heterostructures consisting of a domain of an ordinary photonic crystal sandwiched between two domains of magnetic photonic crystals. The nonmagnetized domain carries two orthogonal one-way waveguide states which have amplitude uniformly distributed over a large area. We show that such one-way waveguide states can be used to abruptly narrow the beam width of an extended state to concentrate energy, and the transport is robust against different kinds of defects and imperfections. They are also immune to the Anderson-type localization when large randomness is introduced.

Deterministic Scheme for Two-Dimensional Type-II Dirac Points and Experimental Realization in Acoustics.

Low-energy electrons near Dirac/Weyl nodal points mimic massless relativistic fermions. However, as they are not constrained by Lorentz invariance, they can exhibit tipped-over type-II Dirac/Weyl cones that provide highly anisotropic physical properties and responses, creating unique possibilities. Recently, they have been observed in several quantum and classical systems. Yet, there is still no simple and deterministic strategy to realize them since their nodal points are accidental degeneracies, unlike symmetry-guaranteed type-I counterparts. Here, we propose a band-folding scheme for constructing type-II Dirac points, and we use a tight-binding analysis to unveil its generality and deterministic nature. Through realizations in acoustics, type-II Dirac points are experimentally visualized and investigated using near-field mappings. As a direct effect of tipped-over Dirac cones, strongly tilted kink states originating from their valley-Hall properties are also observed. This deterministic scheme could serve as a platform for further investigations of intriguing physics associated with various strongly Lorentz-violating nodal points.

Subwavelength topological edge states based on localized spoof surface plasmonic metaparticle arrays.

Plasmonic cluster arrays have demonstrated rich physics in topological photonics, but they are seriously affected by the material loss and limited by the requirement of high-precision machining. Here, we propose a kind of ultra-thin metaparticle arrays which can mimic the coupled localized plasmonic resonances at lower frequency ranges and so that can overcome the loss and fabrication problems in real metal plasmonic systems. The metaparticle is a metallic disk with circuitous grooves that can support both spoof electric and magnetic localized resonances, and these resonances can be pushed to a subwavelength region through tuning the geometric parameters. In virtue of the highly field confinement of these localized resonances, it is thought to be an ideal experimental platform to be an analogy with various near-field interactions in topological materials. As a first proof-of-concept study to show this feasibility, the subwavelength topological edge states at the zigzag metaparticle chain boundaries are numerically and experimentally demonstrated at microwave ranges. Moreover, the subwavelength topological edge states in this zigzag chain can be excited simply by the plane wave incidence, and the edge modes at two ends can be selectively excited by controlling the polarization direction. Therefore, this kind of metaparticle array not only provides an ideal platform to experimentally study various near-filed interaction dominated topological systems but may also find massive potential applications.

Omnidirectional polarization beam splitter for white light.

As a key element in optical systems, a broadband and omnidirectional polarization beam splitter has been long desired. Here, based on anisotropic metamaterials, a perfect polarizing beam splitter is theoretically designed for the extremely broad frequency and angle bands without energy loss. When an electromagnetic wave is incident on the beam splitter, the transverse magnetic-polarized component suffers total reflection, while the transverse electric-polarized component is completely transmitted within the incident angle range [-90°, 90°] for the white light. This study provides a new approach to design an efficient polarizing beam splitter and may promote the development and applications of anisotropic metamaterials.

Metric-Torsion Duality of Optically Chiral Structures.

We develop a metric-torsion theory for chiral structures by using a generalized framework of transformation optics. We show that the chirality is uniquely determined by a metric with the local rotational degree of freedom. In analogy to the dislocation continuum, the chirality can be alternatively interpreted as the torsion tensor of a Riemann-Cartan space, which is mimicked by the anholonomy of the orthonormal basis. As a demonstration, we reveal the equivalence of typical three-dimensional chiral metamaterials in the continuum limit. Our theory provides an analytical recipe to design optical chirality.

Topological interface states in multiscale spoof-insulator-spoof waveguides.

The spoof-insulator-spoof (SIS) structure can serve as a waveguide for spoof surface plasmon polaritons (spoof SPPs). If a periodic geometry modulation in the wavelength scale is introduced to the SIS waveguide, this multiscale SIS (MSIS) waveguide possesses band gaps for spoof SPPs analogous to the band gaps in a photonic crystal. Inspired by the topological interface states found in photonic crystals, we construct an interface by connecting two MSIS waveguides with different topological properties (inverted Zak phases of bulk bands). The topological interface states in the MSIS waveguides are observed experimentally. The measured decay lengths of the interface states agree excellently with the numerical results. These localized interface states may find potential applications in miniaturized microwave devices.

Experimental realization of chiral Landau levels in two-dimensional Dirac cone systems with inhomogeneous effective mass.

Chiral zeroth Landau levels are topologically protected bulk states. In particle physics and condensed matter physics, the chiral zeroth Landau level plays a significant role in breaking chiral symmetry and gives rise to the chiral anomaly. Previous experimental works on such chiral Landau levels are mainly based on three-dimensional Weyl degeneracies coupled with axial magnetic fields. Their realizations using two-dimensional Dirac point systems, being more promising for future applications, were never experimentally realized before. Here we propose an experimental scheme for realizing chiral Landau levels in a two-dimensional photonic system. By introducing an inhomogeneous effective mass through breaking local parity-inversion symmetries, a synthetic in-plane magnetic field is generated and coupled with the Dirac quasi-particles. Consequently, the zeroth-order chiral Landau levels can be induced, and the one-way propagation characteristics are experimentally observed. In addition, the robust transport of the chiral zeroth mode against defects in the system is also experimentally tested. Our system provides a new pathway for the realization of chiral Landau levels in two-dimensional Dirac cone systems, and may potentially be applied in device designs utilizing the chiral response and transport robustness.