Non-Hermitian non-Abelian lattice gauge fields exhibit noncommutative and nonunitary gauge structures, giving rise to novel geometric and topological phenomena. However, their experimental realization has remained elusive. Here, we implement a synthetic nonreciprocal SU(2) gauge field in a one-dimensional spinful chain by employing electric circuit networks with highly tunable asymmetric couplings. We observe a non-Hermitian non-Abelian Aharonov-Bohm effect in a single plaquette, where the final states exhibit an uncorrelated response under non-conjugated loop operations. Furthermore, we reveal the high-order nontrivial braiding and spin-hybridized unidirectional and bidirectional skin states, which are distinctive features of non-Hermitian non-Abelian lattice gauge fields. Our work paves the way for exploring rich non-Abelian phenomena in open systems and offers a versatile platform to implement exotic synthetic gauge fields.

The refraction of waves at the interface between two different media is one of the most common phenomena in nature, which is always accompanied by reflection. Topological phononic crystals (PCs) offer unprecedented opportunities to manipulate the refraction of acoustic waves without generating reflected waves. Here, we propose and demonstrate the topological birefraction of elastic waves in PC by simulation. Starting from the perforated elastic plate with a honeycomb-like structure, we designed a circular trajectory based on structural parameters and revealed that two points on the trajectory with a π phase difference correspond to distinct topological phases. By engineering a non-uniform PC along the circular trajectory, we demonstrate the one-way valley switches and topological birefraction of elastic edge states. Specifically, we design the topological heterostructure by expanding the parameter trajectory to a 2D circular region; the edge states extend to the bulk states; therefore, the topological birefraction of elastic bulk states is demonstrated. Our results may have significance in elastic beam splitters, switches, and filters and are also applicable to on-chip elastic surface waves with higher frequencies, which are of significance in information processing and sensing.

Topological phases have been extensively studied primarily in crystalline systems with translational symmetry. Recent theoretical studies, however, have demonstrated the existence of topological phases in quasicrystals that are absent in crystals. Despite numerous experimental observations of topological phases in various crystalline systems, observing these phases without crystalline counterparts remains challenging due to very complex models. Here, we design a practically realizable tight-binding model with nearest-neighbor hopping on the Ammann-Beenker quasicrystalline lattice. This model respects eight-fold rotational and chiral symmetries, resulting in a higher-order topological phase with eight zero-energy corner modes that have no crystalline counterparts. We experimentally explore the topological phase in an acoustic quasicrystal. Surprisingly, we also discover symmetry-protected zero-energy modes near the center of the quasicrystal in a topologically trivial phase, a phenomenon not seen in crystals. We further experimentally observe these modes in a topologically trivial acoustic quasicrystal. Our work represents the first experimental observation of topological phases in quasicrystals without crystalline counterparts, paving the way for the study of exotic topological physics in quasicrystals.

Landau levels play a crucial role in exploring fundamental phenomena of condensed-matter physics. Recently, a new paradigm for generating Landau levels by means of non-Hermiticity has been proposed. Notably, the zeroth Landau level states here are unpolarized, unlike the sublattice-polarized states in Hermitian Dirac systems. However, due to the challenges of implementing spatially varying dissipation and large gain-loss terms, the experimental demonstration of such Landau level has remained elusive. Here, we present an experimental observation in honeycomb circuit networks. Specifically, we construct a circuit lattice consisting of two types of non-Hermitian configurations to realize a large pseudomagnetic field. Through measuring the admittance spectrum and voltage responses, we successfully observe the unpolarized zeroth Landau level and Hall-like edge states in the non-Hermitian Dirac circuit lattice. Extending the system to a synthetic 3D non-Hermitian Weyl circuit, we further observe the non-Hermiticity-induced chiral LLs and surface modes. Our work paves the way for studying non-Hermitian quantum Hall physics and may be useful for designing novel topological devices.

Non-Hermitian skin localization, as a hallmark feature unique to non-Hermitian systems, has attracted enormous interests. Recent theoretical work reveals that the coupling between a nonreciprocal chain with skin-localized states and a Hermitian chain with extended states can induce a phase with a pseudo-mobility-edge (PME) under the weak interchain couplings. However, such a phenomenon remains to be confirmed in experiments. Here, we experimentally realize the non-Hermitian PME by designing a hybridized non-Hermitian circuit ladder. Through measuring the admittance spectrum and eigenstates, the transition from the PME phase to the skin-localized phase for the open boundary condition and to the extended phase for the mixed boundary condition (MBC) are, respectively, successfully observed with increasing the ac current feed frequency. Moreover, these localization behaviors are also manifested by measuring the voltage response to a local current feed. In addition, we observe the bulk-boundary correspondence between the spectral winding topology and the PME transition under the MBC. Our paper provides a fertile platform to study unconventional non-Hermitian localization.

A two-dimensional topological metal with anti-helical-like edge states has been predicted recently but has not been confirmed experimentally. In this paper, we report an experimental realization of this topological metal in acoustic metamaterial by introducing a time-reversal symmetry protected square lattice. The edge states appearing in gapless bulk bands are observed by measuring the projected dispersions and acoustic pressure field distributions. Moreover, these edge states propagate in the same direction when simultaneously exciting two sources with a fixed phase difference. Interestingly, by simply changing the coupling tubes, we realized the transformation of an acoustic topological metal to a topological insulator. Our work not only pushes forward the studies of topological metals but also inspires the design of multifunctional acoustic devices.

For three-dimensional systems, discussing the possibility of implementing quantum Hall physics has been an enduring topic. Based on Weyl semimetals, pioneering works have proposed chiral Landau levels and three-dimensional quantum Hall effect. When the degenerate band extends from nodal points to nodal lines, the magnetic field can lead to various quantum Hall phenomena, such as the three-dimensional flat Landau levels. Usually, the above exotic phenomena are explored under a uniform magnetic field with a fixed direction and magnitude. Here, we obtain a momentum-dependent pseudomagnetic field in a nodal line semimetal by designing a gradient coupling. The resulting quantum Hall surface states exhibit momentum-dependent chiral and helical transport features. These findings are experimentally confirmed in the phononic crystal platform. The novel sound transport may pave the way for acoustic devices with unconventional functions.

Landau levels (LLs) are of great importance for understanding the quantum Hall effect and associated many-body physics. Recently, their three-dimensional (3D) counterparts, i.e., dispersionless 3D LLs with well-defined quantum numbers, have attracted significant attention but have not yet been reported. Here we theoretically propose and experimentally observe 3D LLs with a sharply quantized spectrum in a diamond acoustic lattice, where the eigenstates are characterized by SU(3) quantum numbers. The engineered inhomogeneous hopping strengths not only introduce pseudomagnetic fields that quantize the nodal lines into LLs but also provide three bosonic degrees of freedom, embedding a generic SU(3) symmetry into the LLs. Using a phased array of acoustic sources, we selectively excite distinct eigenstates within the degenerate LL multiplets and visualize their 3D eigenmodes. Importantly, our approach enables the precise reconstruction of SU(3) quantum numbers directly from eigenmode correlations. Our results establish SU(3) LLs as a tractable model in artificial platforms, and pave the way for synthesizing LLs with zero dispersion and countable quantum numbers in arbitrary dimensions.

Non-Hermitian Dirac point plays an important role in topological transition as their Hermitian counterpart and connect non-Hermitian physics with band topology. Instead of being exceptional point or exceptional ring, we here reveal that the Dirac points can be survived in the presence of gain and loss obeying anti-parity-time symmetry based on the two-dimensional inclined Su-Schrieffer-Heeger model. Particularly, such non-Hermitian parameters enable the engineering of non-Hermitian Dirac states, including shift of the Dirac points and topological transition from Dirac semimetal to weak topological insulator. We experimentally demonstrate these non-Hermitian Dirac states in acoustic crystal, where the gain and loss are, respectively, controlled by the active acoustic components and absorbing materials. Through varying the strength of gain and loss, the shifting and opening of the Dirac points, together with topological edge states, are observed. Our system serves as an ideal and highly tunable platform for exploring the non-Hermitian topological physics and has potential applications in designing acoustic devices.

Continuum Landau modes — predicted recently in a non-Hermitian Dirac Hamiltonian under a uniform magnetic field — are continuous bound states with no counterparts in Hermitian systems. However, they have still not been confirmed in experiments. Here, we report an experimental observation of continuum Landau modes in non-Hermitian electric circuits, in which the non-Hermitian Dirac Hamiltonian is simulated by non-reciprocal hoppings and the pseudomagnetic field is introduced by inhomogeneous complex on-site potentials. Through measuring the admittance spectrum and the eigenstates, we successfully verify key features of continuum Landau modes. Particularly, we observe the exotic voltage response acting as a rainbow trap or wave funnel through full-field excitation. This response originates from the linear relationship between the modes’ center position and complex eigenvalues. Our work builds a bridge between non-Hermiticity and magnetic fields, and thus opens an avenue to explore exotic non-Hermitian physics.

Topological phases of matter are classified based on symmetries, with nonsymmorphic symmetries like glide reflections and screw rotations being of particular importance in the classification. In contrast to extensively studied glide reflections in real space, introducing space-dependent gauge transformations can lead to momentum-space glide reflection symmetries, which may even change the fundamental domain for topological classifications, e.g., from a torus to a Klein bottle. Here we discover a new class of three-dimensional (3D) higher-order topological insulators, protected by a pair of momentum-space glide reflections. It supports gapless hinge modes, as dictated by the quadrupole moment and Wannier Hamiltonians defined on a Klein bottle manifold, and we introduce two topological invariants to characterize this phase. Our predicted topological hinge modes are experimentally verified in a 3D-printed acoustic crystal, providing direct evidence for 3D higher-order Klein bottle topological insulators. Our results not only showcase the remarkable role of momentum-space glide reflections in topological classifications, but also pave the way for experimentally exploring physical effects arising from momentum-space nonsymmorphic symmetries.

Manipulating elastic waves in lower-dimensional mechanical metamaterials has attracted much attention since it lays the foundation for the design of various elastic functional devices, especially for on-chip size. However, due to the experimental challenges, it is very difficult to control elastic waves in higher dimensions. In this Letter, we introduce an extra structural parameter to synthesize and investigate the on- chip Weyl physics in silicon-on-insulator system. Interestingly, we engineer an in-plane pseudomagnetic field to realize chiral Landau levels, which provides a bulk channel supporting robust energy transport. We also observe the pseudomagnetic field-induced boundary states near the diagonal corners, which are quite different from conventional higher order topological corner states. With the aid of the synthetic dimension, we can not only realize the multidimensional elastic wave manipulations, but more importantly, devise a novel strategy to explore the higher dimensional physics on an integrated platform.

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.

Hybrid-order topological insulators combine first- and higher-order topological properties and host topological boundary states with codimension one and more than one in different bandgaps. A Weyl semimetal (WSM) can possess two types of Weyl points: one class of Weyl points terminates the Fermi arc surface states, while another class of Weyl points not only launch Fermi arc surface states but also hinge arc states, exhibiting the hybrid-order topology. Here, we propose a hybrid-order WSM by stacking two-dimensional rhomboid lattices based on chiral nearest-neighbor and double-helix next-nearest interlayer couplings. The first type of Weyl point that only truncates the Fermi arc surface states exists at the crossing of any two-fold degeneracy of two adjacent bands, and the second type of Weyl point that connects the hinge arc states only appears at the crossing of the two middle bands. Our findings enrich the classification of topological semimetals in condensed matter physics.

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 topological insulators that host the topological boundary states, which is insensitive to defects and disorder, have become an important research topic and provide completely new ways to manipulate acoustic waves. However, most of the acoustic topological boundary states generally appear in a single or two band gaps, hindering the applications in multiband acoustic devices. Compared to previous work, in this paper, we experimentally observe the acoustic multiple topological boundary states in four band gaps, including the end (corner) states in one-dimensional (two-dimensional) phononic crystals, based on the acoustic quartic-root topological insulators. These topological boundary states originate from two consecutive square-root procedures, which is similar to the square-root topological insulators. Our paper provides a brand-new approach to achieve the multiple boundary states by simply inserting additional cavities without elaborate designing the structure, which makes the manipulation of the acoustic more flexible.

Longer-range interactions, or couplings beyond the second nearest neighbors, have long been overlooked in topological models for either topological insulators or semimetals. For natural (electronic) materials, such ignorance is reasonable as near-range couplings are always dominant. However, artificially constructed metamaterials can break such a limitation, allowing the longer-range couplings comparable with the nearer ones, achieving alternative properties unattainable in natural materials. Here, we report a one-dimensional acoustic metamaterial designed in terms of an extended Su-Schrieffer-Heeger (SSH) model, containing the nearest, and in particular the third-nearest couplings. In contrast to the conventional SSH model with the topological phases of the winding number one, the extended SSH model can host topological phases of winding number two, besides one, exhibiting amazing rotonlike dispersions. The larger winding number implies more end states emerging. All these predictions are confirmed by the experiments with the fabricated acoustic metamaterials. Such longer-range couplings can also be extended to two- or three-dimensional lattices, to further achieve the topological states exceptionally in metamaterials.

Rapid developments for topological materials have promoted the search for topological transport in the fields of condensed-matter physics and materials science. However, topological network transport, proposed in twisted bilayer graphene and soon after being explored in other two-dimensional materials, is still elusive despite extensive experimental efforts. Here, we implemented on-chip phononic crystal networks based on a network unit cell with six channels consisting of triangular prisms on a silicon substrate arranged in a honeycomb array. The topological network bulk transport in the gap of the bulk states of the original phononic crystal and the network edge transport in the gap of the network bulk states were visualized directly. These results offer a controllable platform for exploring topological transport in networks and may enable the realization of on-chip microultrasonic devices, such as splitters, filters, and signal processing in a monolithic elastic network.

Recently, high-order topological insulators (HOTIs), accompanied by topologically nontrivial boundary states with codimension larger than one, have been extensively explored because of unconventional bulk-boundary correspondences. As a novel type of HOTIs, very recent works have explored the square-root HOTIs, where the topological nontrivial nature of bulk bands stems from the square of the Hamiltonian. In this paper, we experimentally demonstrate 2D square-root HOTIs in photonic waveguide arrays written in glass using femtosecond laser direct-write techniques. Edge and corner states are clearly observed through visible light spectra. The dynamical evolutions of topological boundary states are experimentally demonstrated, which further verify the existence of in-gap edge and corner states. The robustness of these edge and corner states is revealed by introducing defects and disorders into the bulk structures. Our studies provide an extended platform for realizing light manipulation and stable photonic devices.

The discovery of Weyl semimetals opens the door for searching topological semimetals in physical science. The Weyl points are generally recognized as conventional, quadratic, spin-1, and those of high topological charges. Here we report the observation of the quadruple Weyl point of charge 4, the highest topological charge a twofold degenerate node can carry. Besides the quadruple Weyl point, the phononic semimetal also hosts conventional, quadratic, and spin-1 Weyl points, which stands as a system with yet the richest types of Weyl points. The quadruple-helicoid surface states, specific to the quadruple Weyl point, are demonstrated. The finding of the high-charge Weyl point enriches the knowledge of Weyl semimetals and may stimulate related researches in other systems, such as photonic, mechanical and cold atom systems.