Using three-dimensional (3D) sonic crystals as acoustic higher-order topological insulators (HOTIs), we discover two-dimensional (2D) surface states described by spin-1 Dirac equations at the interfaces between the two sonic crystals with distinct topology but the same crystalline symmetry. We find that the Dirac mass can be tuned by the geometry of the two sonic crystals. The sign reversal of the Dirac mass reveals a surface topological transition where the surface states exhibit zero refractive index behavior. When the surface states are gapped, one-dimensional (1D) hinge states emerge due to the topology of the gapped surface states. We confirm experimentally the zero refractive index behavior and the emergent topological hinge states. Our study reveals a multidimensional Wannier orbital control that leads to extraordinary properties of surface states and unveils an interesting topological mechanism for the control of surface waves.
Acoustic systems that are without limitations imposed by the Fermi level have been demonstrated as significant platform for the exploration of fruitful topological phases. By surrounding the nontrivial domain with trivial "environment", the domain-wall topological states have been theoretically and experimentally demonstrated. In this work, based on the topological crystalline insulator with a kagome lattice, we rigorously derive the corresponding Hamiltonian from the traditional acoustics perspective, and exactly reveal the correspondences of the hopping and onsite terms within acoustic syste…
The recent discovery of higher-order topological insulators (TIs) has opened new possibilities in the search for novel topological materials and metamaterials. Second-order TIs have been implemented in two-dimensional (2D) systems exhibiting topological 'corner states', as well as three-dimensional (3D) systems having one-dimensional (1D) topological 'hinge states'. 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 diamond-latt…
The interplay between real-space topological lattice defects and the reciprocal-space topology of energy bands can give rise to novel phenomena, such as one-dimensional topological modes bound to screw dislocations in three-dimensional topological insulators. We obtain direct experimental observations of dislocation-induced helical modes in an acoustic analog of a weak three-dimensional topological insulator. The spatial distribution of the helical modes is found through spin-resolved field mapping, and verified numerically by tight-binding and finite-element calculations. These one-dimensiona…
A quadrupole topological insulator, being one higher-order topological insulator with nontrivial quadrupole quantization, has been intensely investigated very recently. However, the tight-binding model proposed for such emergent topological insulators demands both positive and negative hopping coefficients, which imposes an obstacle in practical realizations. Here we introduce a feasible approach to design the sign of hopping in acoustics, and construct the first acoustic quadrupole topological insulator that stringently emulates the tight-binding model. The inherent hierarchy quadrupole topol…
We present a theoretical study and experimental realization of a system that is simultaneously a four-dimensional (4D) Chern insulator and a higher-order topological insulator (HOTI). The system sustains the coexistence of (4-1)-dimensional chiral topological hypersurface modes (THMs) and (4-2)-dimensional chiral topological surface modes (TSMs). Our study reveals that the THMs are protected by second Chern numbers, and the TSMs are protected by a topological invariant composed of two first Chern numbers, each belonging a Chern insulator existing in sub-dimensions. With the synthetic coordinat…
Dirac cones (DCs) play a pivotal role in various unique phenomena ranging from massless electrons in graphene to robust surface states in topological insulators (TIs). Recent studies have theoretically revealed a full Dirac hierarchy comprising an eightfold bulk DC, a fourfold surface DC, and a twofold hinge DC, associated with a hierarchy of topological phases including first-order to third-order three-dimensional (3D) topological insulators, using the same 3D base lattice. Here, we report the first experimental observation of the Dirac hierarchy in 3D acoustic TIs. Using acoustic measurement…
Topological insulators are new states of matter in which the topological phase originates from symmetry breaking. Recently, time-reversal invariant topological insulators were demonstrated for classical wave systems, such as acoustic systems, but limited by inter-pseudo-spin or inter-valley backscattering. This challenge can be effectively overcome via breaking the time-reversal symmetry. Here, we report the first experimental realization of acoustic topological insulators with nonzero Chern numbers, viz., acoustic Chern insulator (ACI), by introducing an angular-momentum-biased resonator arra…
Symmetry plays a critical role in classifying phases of matter. This is exemplified by how crystalline symmetries enrich the topological classification of materials and enable unconventional phenomena in topologically nontrivial ones. After an extensive study over the past decade, the list of topological crystalline insulators and semimetals seems to be exhaustive and concluded. However, in the presence of gauge symmetry, common but not limited to artificial crystals, the algebraic structure of crystalline symmetries needs to be projectively represented, giving rise to unprecedented topologica…
The unique conduction properties of condensed matter systems with topological order have recently inspired a quest for similar effects in classical wave phenomena. Acoustic topological insulators, in particular, hold the promise to revolutionize our ability to control sound, allowing for large isolation in the bulk and broadband one-way transport on their edges, with topological immunity against structural defects and disorder. So far these fascinating properties have been obtained relying on moving media, which inherently introduce noise and absorption losses, hindering the practical applicab…
Surface Acoustic Wave Propagation and Inhomogeneities in Low Density Two-Dimensional Electron Systems Near the Metal-Insulator Transition
我们测量了零磁场下低密度区（$<$ $10^{10}$ cm $^{-2}$）中含有二维电子系统的GaAs/AlGaAs异质结构（2DES）中的声表面波速度漂移。的表面声波与2DES的相互作用是不是很好地描述了一个简单的模型，使用低频电导率测量。我们推测，这种冲突是由于在2DES的不均匀性，这在低密度变得非常重要。这对二维中假定的金属-绝缘体转变有影响。
The quantum anomalous Hall effect has been theoretically predicted and experimentally verified in magnetic topological insulators. In addition, the surface states of these materials exhibit a hedgehog-like "spin" texture in momentum space. Here, we apply the previously formulated low-energy model for Bi$_2$Se$_3$, a parent compound for magnetic topological insulators, to a slab geometry in which an exchange field acts only within one of the surface layers. In this sample set up, the hedgehog transforms into a skyrmion texture beyond a critical exchange field. This critical field marks a transi…
Transient reflectivity traces measured for nanometer-sized films of the topological insulator Bi2Se3 revealed GHz-range oscillations driven within the relaxation of hot carriers photoexcited with ultrashort laser pulses of 1.51 eV photon energy. These oscillations have been suggested to result from acoustic phonon dynamics, including coherent longitudinal acoustic phonons in the form of standing acoustic waves. An increase of oscillation frequency from ~35 to ~70 GHz with decreasing film thickness from 40 to 15 nm was attributed to the interplay between two different regimes employing travelin…
Topological edge states are predicted to be responsible for the high efficient thermoelectric response of topological insulators, currently the best thermoelectric materials. However, to explain their figure of merit the coexistence of topological electrons, entropy and phonons can not be considered independently. In a background that puts together electrodynamics and topology, through an expression for the topological intrinsic field, we treat relativistic phonons within the topological surface showing their ability to modulate the Berry curvature of the bands and then playing a fundamental r…
The entanglement of the charge, spin and orbital degrees of freedom can give rise to emergent behavior especially in thin films, surfaces and interfaces. Often, materials that exhibit those properties require large spin orbit coupling. We hypothesize that the emergent behavior can also occur due to spin, electron and phonon interactions in widely studied simple materials such as Si. That is, large intrinsic spin-orbit coupling is not an essential requirement for emergent behavior. The central hypothesis is that when one of the specimen dimensions is of the same order (or smaller) as the spin d…
Square-root topological states are new topological phases, whose topological property is inherited from the square of the Hamiltonian. We realize the first-order and second-order square-root topological insulators in phononic crystals, by putting additional cavities on connecting tubes in the acoustic Su-Schrieffer-Heeger model and the honeycomb lattice, respectively. Because of the square-root procedure, the bulk gap of the squared Hamiltonian is doubled. In both two bulk gaps, the square-root topological insulators possess multiple localized modes, i.e., the end and corner states, which are …
A general phenomenon of the Cherenkov radiation known in optics or acoustics of conventional materials is a formation of a forward cone of, respectively, photons or phonons emitted by a particle accelerated above the speed of light or sound in those materials. Here we suggest three-dimensional topological insulators as a unique platform to fundamentally explore and practically exploit the acoustic aspect of the Cherenkov effect. We demonstrate that applying an in-plane magnetic field to a surface of a three-dimensional topological insulator one may suppress the forward Cherenkov sound up to ze…
The concept of topological insulator (TI) has introduced a new point of view to condensed-matter physics, relating a priori unrelated subfields such as quantum (spin, anomalous) Hall effects, spin-orbit coupled materials, some classes of nodal superconductors and superfluid $^3$He, etc. From a technological point of view, topological insulator is expected to serve as a platform for realizing dissipationless transport in a non-superconducting context. The topological insulator exhibits a gapless surface state with a characteristic conic dispersion (a surface Dirac cone). Here, we review peculia…
Low-energy collective electronic excitations exhibiting sound-like linear dispersion have been intensively studied both experimentally and theoretically for a long time. However, coherent acoustic plasmon modes appearing in time-domain measurements are rarely observed due to Landau damping by the single-particle continua. Here we report on the observation of coherent acoustic Dirac plasmon (CADP) modes excited in indirectly (electrostatically) opposite-surface coupled films of the topological insulator Bi2Se3. Using transient second harmonic generation, a technique capable of independently mon…
We numerically study silicon waveguides on silica showing that it is possible to simultaneously guide optical and acoustic waves in the technologically important silicon on insulator (SOI) material system. Thin waveguides, or fins, exhibit geometrically softened mechanical modes at gigahertz frequencies with phase velocities below the Rayleigh velocity in glass, eliminating acoustic radiation losses. We propose slot waveguides on glass with telecom optical frequencies and strong radiation pressure forces resulting in Brillouin gains on the order of 500 and 50,000 1/(Wm) for backward and forwar…
The realization of the quantum spin Hall effect in HgTe quantum wells has led to the development of topological materials which, in combination with magnetism and superconductivity, are predicted to host chiral Majorana fermions. However, the large magnetization ($\sim$ a few tesla) in conventional quantum anomalous Hall system, makes it challenging to induce superconductivity. Here, we report two different emergent quantum Hall effects in HgTe quantum wells dilutely alloyed with Mn. Firstly, a novel quantum Hall state emerges from the quantum spin Hall state at an exceptionally low magnetic f…
Recent experiments in graphene heterostructures have observed Chern insulators - integer and fractional Quantum Hall states made possible by a periodic substrate potential. Here we study theoretically the competition between different Chern insulators, which can be tuned by the amplitude of the periodic potential, leads to a new family of quantum critical points described by QED$_3$-Chern-Simons theory. At these critical points, $N_f$ flavors of Dirac fermions interact through an emergent U$(1)$ gauge theory at Chern-Simons level $K$, and remarkably, the entire family (with any $N_f$ or $K$) c…
We explore the phonon dynamics of thin films of the topological insulator material Bi2Se2Te using ultrafast pump-probe spectroscopy. The time resolved differential reflectivity of the films exhibit fast and slow oscillations. We have given a careful analysis of variation of phonon frequency as a function of film thickness attributing this to existence of standing acoustic modes. However, no variation in the frequency of the optical phonon modes was found with film thickness. This indicates that the optical phonons intrinsically belong to the surface of the topological insulators. The fact that…
Recently extended from the modern theory of electric polarization, quantized multipole topological insulators (QMTIs) describe higher-order multipole moments, lying in nested Wilson loops, which are inherently quantized by lattice symmetries. Overlooked in the past, QMTIs reveal new types of gapped boundaries, which themselves represent lower-dimensional topological phases and host topologically protected zero-dimensional (0D) corner states. Inspired by these pioneering theoretical predictions, tremendous efforts have been devoted to the experimental observation of topological quantized quadru…
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 haveonly been observed along a single hinge, and a natural question arises: whether the hinge states can exist simultaneou…
Discovery of novel topological orders of condensed matters is of a significant interest in both fundamental and applied physics due to the associated quantum conductance behaviors and unique symmetry-protected backscattering-immune propagation against defects, which inspired similar fantastic effects in classical waves system, leading to the revolution of the manipulation of wave propagation. To date, however, only few theoretical models were proposed to realize acoustic topological states. Here, we theoretically and experimentally demonstrate a two dimensional acoustic topological insulators …
Recent breakthrough on topological Anderson insulators revealed the breakdown of the traditional perception that sufficiently strong disorder may induce the appearance of topological protected transport states instead of destruction. Although topological Anderson insulators have been observed in various time-reversal symmetry breaking systems, the observation of topological Anderson insulators protected by time-reversal symmetry remains scarce, which are considered to be more promising in applications such as the integrated devices. Here, we report the experimental observation of topological A…
The electronic band structure of iron pnictides exhibits four Dirac cones, which are due to crystal symmetry and orbital bonding orientation. This hallmark signature presents the pnictide family as an ideal candidate in the search for quasi-two-dimensional topological crystalline insulators. In this report, we explore interaction-induced topological phases which cannot be described by conventional local order parameters. Based on a model Hamiltonian our symmetry analysis shows that sponta- neous novel topological phases may be realized in compounds with tetragonal crystal field symmetry, where…
The higher-order topological insulators (HOTIs), with such as the topological corner states, emerge as a thriving topic in the field of topological physics. But few connections have been found for the HOTIs with the well explored first-order topological insulators described by the Z_2 index. However, most recently, a proposal asserts that a significant bridge can be established between the HOTIs and the Z_2 topological insulators. When subject to an in-plane Zeeman field, the corner states, the signature of the HOTIs, can be induced in a Z_2 topological insulator. Such Zeeman field can be prod…
Higher-order topological insulators, which support lower-dimensional topological boundary states than the first-order topological insulators, have been intensely investigated in the integer dimensional systems. Here, we provide a new paradigm by presenting experimentally a higher-order topological phase in a fractal-dimensional system. Through applying the Benalcazar, Bernevig, and Hughes model into a Sierpinski carpet fractal lattice, we uncover a squeezed higher-order phase diagram featuring the abundant corner states, which consist of zero-dimensional outer corner states and 1.89-dimensiona…
Conversion of thermal energy into magnonic spin currents and/or effective electric polarization promises new device functionalities. A versatile approach is presented here for generating and controlling open circuit magnonic spin currents and an effective multiferroicity at a uniform temperature with the aid of spatially inhomogeneous, external, static electric fields. This field applied to a ferromagnetic insulator with a Dzyaloshinskii-Moriya type coupling changes locally the magnon dispersion and modifies the density of thermally excited magnons in a region of the scale of the field inhomog…
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 posse…
The Berry phase associated with energy bands in crystals can lead to quantized quantities, such as the quantization of electric dipole polarization in an insulator, known as a one-dimensional (1D) topological insulator (TI) phase. Recent theories have generalized such quantization from dipole to higher multipole moments, giving rise to the discovery of multipole TIs, which exhibit a cascade hierarchy of multipole topology at boundaries of boundaries: A quantized octupole moment in the three-dimensional (3D) bulk can induce quantized quadrupole moments on its two-dimensional (2D) surfaces, whic…
Moiré flatbands, occurring, e.g., in twisted bilayer graphene at magic angles, have attracted ample interest due to their high degree of experimental tunability and the intriguing possibility of generating novel strongly interacting phases. Here we consider the core problem of Coulomb interactions within fractionally filled spin and valley polarized Moiré flatbands and demonstrate that the dual description in terms of holes, which acquire a nontrivial hole dispersion, provides key physical intuition and enables the use of standard perturbative techniques for this strongly correlated problem. I…
Time reversal (T) invariant topological insulator is widely recognized as one of the fundamental discoveries in condensed matter physics, for which the most fascinating hallmark is perhaps a spin based topological protection, the total cancellation of scattering of conduction electrons with certain spins on matter surface. Recently, it has created a paradigm shift for topological insulators, from electronics to photonics, phononics as well as mechanics, bringing about not only involved new physics but also potential applications in robust wave transport. Despite the growing interests in realiz…
High-order topological insulators (TIs) are a family of recently-predicted topological phases of matter obeying an extended topological bulk-boundary correspondence principle. For example, a two-dimensional (2D) second-order TI does not exhibit gapless one-dimensional (1D) topological edge states, like a standard 2D TI, but instead has topologically-protected zero-dimensional (0D) corner states. So far, higher-order TIs have been demonstrated only in classical mechanical and electromagnetic metamaterials exhibiting quantized quadrupole polarization. Here, we experimentally realize a second-ord…
We investigate the ground state and quantum dynamics of an interacting bosonic chain with the nonreciprocal hopping. In sharp contrast to its Hermitian counterpart, the ground state can support Mott insulators in systems with noninteger filling due to the competition between nonreciprocal hopping and the on-site interaction. For the quantum dynamics, conservation laws for non-Hermitian systems manifest a stark difference from their Hermitian counterpart. In particular, for any Hermitian operator that commutes with the Hamiltonian operator, its expectation value is guaranteed to be nonconserved…
Motivated by the recent work of QED$_3$-Chern-Simons quantum critical points of fractional Chern insulators (Phys. Rev. X \textbf{8}, 031015, (2018)), we study its non-Abelian generalizations, namely QCD$_3$-Chern-Simons quantum phase transitions of fractional Chern insulators. These phase transitions are described by Dirac fermions interacting with non-Abelian Chern-Simons gauge fields ($U(N)$, $SU(N)$, $USp(N)$, etc.). Utilizing the level-rank duality of Chern-Simons gauge theory and non-Abelian parton constructions, we discuss two types of QCD$_3$ quantum phase transitions. The first type h…
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. The integer quantum Hall effect has delivered on that promise in the electronic realm through high-precision metrology devices. 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 use a two-dimensi…
Three-dimensional topological insulators (TIs) are a perfectly tuned quantum-mechanical machinery in which counter-propagating and oppositely spin-polarized conduction channels balance each other on the surface of the material. This topological surface state crosses the bandgap of the TI, and lives at the interface between the topological and a trivial material, such as vacuum. Despite its balanced perfection, it is rather useless for any practical applications. Instead, it takes the breaking of time-reversal symmetry (TRS), and the appearance of an exchange gap to unlock hidden quantum states…
Topological insulators with unique gapless edge states have revolutionized the understanding of electronic properties in solid materials. These gapless edge states are dictated by the topological invariants associated with the quantization of generalized Berry phases of the bulk energy bands through the bulk-edge correspondence, a paradigm that can also be extended to acoustic and photonic systems. Recently, high-order topological insulators (HOTIs) are proposed and observed, where the bulk topological invariants result in gapped edge states and in-gap corner or hinge states, going beyond the …
Dislocations are ubiquitous in three-dimensional solid-state materials. The interplay of such real space topology with the emergent band topology defined in reciprocal space gives rise to gapless helical modes bound to the line defects. This is known as bulk-dislocation correspondence, in contrast to the conventional bulk-boundary correspondence featuring topological states at boundaries. However, to date rare compelling experimental evidences are presented for this intriguing topological observable, owing to the presence of various challenges in solid-state systems. Here, using a three-dimens…
