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Recently discovered1,2 valley photonic crystals (VPCs) mimic many of the unusual properties of two-dimensional (2D) gapped valleytronic materials [3][4][5][6][7][8][9] . Of the utmost interest to optical communications is their ability to support topologically protected chiral edge (kink) states [3][4][5][6][7][8][9] at the internal domain wall between two VPCs with opposite valley-Chern indices. Here we experimentally demonstrate valley-polarized kink states with polarization multiplexing in VPCs, designed from a spin-compatible four-band model. When the valley pseudospin is conserved, we show that the kink states exhibit nearly perfect out-coupling e ciency into directional beams, through the intersection between the internal domain wall and the external edge separating the VPCs from ambient space. The out-coupling behaviour remains topologically protected even when we break the spin-like polarization degree of freedom (DOF), by introducing an e ective spin-orbit coupling in one of the VPC domains. This also constitutes the first realization of spin-valley locking for topological valley transport.

Confining photons in a finite volume is in high demand in modern photonic devices. This motivated decades ago the invention of photonic crystals, featured with a photonic bandgap forbidding light propagation in all directions 1-3 . Recently, inspired by the discoveries of topological insulators (TIs) 4,5 , the confinement of photons with topological protection has been demonstrated in two-dimensional (2D) photonic structures known as photonic TIs 6-8 , with promising applications in topological lasers 9,10 and robust optical delay lines 11 . However, a fully three-dimensional (3D) topological photonic bandgap has never before been achieved. Here, we experimentally demonstrate a 3D photonic TI with an extremely wide (> 25% bandwidth) 3D topological bandgap. The sample consists of split-ring resonators (SRRs) with strong magneto-electric coupling and behaves as a "weak TI", or a stack of 2D quantum spin Hall insulators. Using direct field measurements, we map out both the gapped bulk bandstructure and the Dirac-like dispersion of the photonic surface states, and demonstrate robust photonic propagation along a non-planar surface. Our work extends the family of 3D TIs from fermions to bosons and paves the way for applications in topological photonic cavities, circuits, and lasers in 3D geometries.

Recent theories have proposed a concept of valley photonic crystals as an analog of gapped valleytronic materials such as MoS 2 and bilayer graphene. Here, we further extend the applicability of valley photonic crystals to surface electromagnetic waves and experimentally demonstrate a valley surface-wave photonic crystal on a single metal surface as a photonic duplicate of MoS 2 . Both bulk transport and edge transport are directly mapped with a near-field microwave imaging system. The photonic valley pseudospins are demonstrated, together with the photonic valley Hall effect that splits the opposite photonic valley pseudospins into two opposite directions. The valley edge transport in MoS 2 or other transition-metal dichalcogenide monolayers, which is different from bilayer graphene but still stays unrealized in condensed-matter systems, is demonstrated on this MoS 2 -like photonic platform. Our study not only offers a tabletop platform to study the valleytronic physics, but also opens a venue for on-chip integrated photonic device applications using valley-polarized information.

The recent discovery of higher-order topological insulators (TIs) [1-5] has opened new possibilities in the search for novel topological materials and metamaterials. Secondorder TIs have been implemented in two-dimensional (2D) systems [6-19] exhibiting topological 'corner states', as well as three-dimensional (3D) systems having one-dimensional (1D) topological 'hinge states' [20]. Third-order TIs, which have topological states three dimensions lower than the bulk (which must thus be 3D or higher), have not yet been reported. Here, we describe the realization of a third-order TI in an anisotropic diamondlattice acoustic metamaterial. The bulk acoustic bandstructure has nontrivial topology characterized by quantized Wannier centers. By direct acoustic measurement, we observe corner states at two corners of a rhombohedron-like structure, as predicted by the quantized Wannier centers. This work extends topological corner states from 2D to 3D, and may find applications in novel acoustic devices.Higher-order TIs are a new class of topological materials supporting a generalization of the bulk-boundary correspondence principle, in which topological states are guaranteed to exist along boundaries two or more dimensions smaller than that of the bulk [1][2][3][4][5]. In standard TIs, topological edge states occur at one lower dimension than the bulk [21,22]; for instance, a quantum Hall insulator has a 2D bulk and topological states on 1D edges. By contrast, a 2D second-order TI supports zero-dimensional (0D) topological 'corner states'. Such a lattice was first devised based on quantized quadrupole moments [1,2] and quickly realized in mechanical [6], electromagnetic [7], and electrical [9] metamaterials. Later, another type of 2D second-order TIs based on quantized Wannier centers, was proposed [23][24][25] and demonstrated in acoustic metamaterials [10,11]. In 3D materials, second-order TI behavior has also been observed in the form of 1D topological 'hinge states' in bismuth [20].According to theoretical predictions, TIs of arbitrarily high order are possible. However, in real materials the bulk is at most 3D. Thus, barring the use of 'synthetic' dimensions [26,27], the only remaining class of high-order TI is a third-order TI with 3D bulk and 0D corner states.As of this writing, no such material has been reported in the literature, although there exists a theoretical proposal based on quantized octupole moments [1,2].Here, we realize a third-order TI in a 3D acoustic metamaterial, observing topological states at the corners of a rhombohedron-like sample. This third-order TI is based on the extension of Wannier-type second-order TIs to 3D [23,25], and can be regarded as a 3D generalization of the classic 1D Su-Schrieffer-Heeger (SSH) model [28]. Just as in the SSH case, the eigenmode polarizations are quantized by lattice symmetries, and the Wannier centers are pinned to highsymmetry points; the mismatch between the Wannier centers and lattice truncations gives rise to charge fractionalization and hence lower-dimensio...

Unconventional chiral particles have recently been predicted to appear in certain three dimensional (3D) crystal structures containing three-or more-fold linear band degeneracy points (BDPs). These BDPs carry topological charges, but are distinct from the standard twofold Weyl points or fourfold Dirac points, and cannot be described in terms of an emergent relativistic field theory. Here, we report on the experimental observation of a topological threefold BDP in a 3D phononic crystal. Using direct acoustic field mapping, we demonstrate the existence of the threefold BDP in the bulk

Berry phase associated with energy bands in crystals can lead to quantised observables like quantised dipole polarizations in one-dimensional topological insulators. Recent theories have generalised the concept of quantised dipoles to multipoles, resulting in the discovery of multipole topological insulators which exhibit a hierarchy of multipole topology: a quantised octupole moment in a three-dimensional bulk induces quantised quadrupole moments on its two-dimensional surfaces, which in turn induce quantised dipole moments on onedimensional hinges. Here, we report on the realisation of an octupole topological insulator in a three-dimensional acoustic metamaterial. We observe zero-dimensional topological corner states, one-dimensional gapped hinge states, two-dimensional gapped surface states, and three-dimensional gapped bulk states, representing the hierarchy of octupole, quadrupole and dipole moments. Conditions for forming a nontrivial octupole moment are demonstrated by comparisons with two different lattice configurations having trivial octupole moments. Our work establishes the multipole topology and its full hierarchy in three-dimensional geometries.

Topological phases of matter are classified based on their Hermitian Hamiltonians, whose real-valued dispersions together with orthogonal eigenstates form nontrivial topology. In the recently discovered higher-order topological insulators (TIs), the bulk topology can even exhibit hierarchical features, leading to topological corner states, as demonstrated in many photonic and acoustic artificial materials. Naturally, the intrinsic loss in these artificial materials has been omitted in the topology definition, due to its non-Hermitian nature; in practice, the presence of loss is generally considered harmful to the topological corner states. Here, we report the experimental realization of a higher-order TI in an acoustic crystal, whose nontrivial topology is induced by deliberately introduced losses. With local acoustic measurements, we identify a topological bulk bandgap that is populated with gapped edge states and in-gap corner states, as the hallmark signatures of hierarchical higher-order topology. Our work establishes the non-Hermitian route to higher-order topology, and paves the way to exploring various exotic non-Hermiticity-induced topological phases.