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 evidently confirmed by our calculations and experimental observations. We further propose a second-order square-root topological semimetal by stacking the decorated honeycomb lattice to three dimensions.
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…
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…
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…
We have measured the surface acoustic wave velocity shift in a GaAs/AlGaAs heterostructure containing a two-dimensional electron system (2DES) in a low-density regime ($<$ $10^{10}$ cm$^{-2}$) at zero magnetic field. The interaction of the surface acoustic wave with the 2DES is not well described by a simple model using low-frequency conductivity measurements. We speculate that this conflict is a result of inhomogeneities in the 2DES which become very important at low density. This has implications for the putative metal-insulator transition in two dimensions.
Notion of square-root topological insulators have been recently generalized to higher-order topological insulators. In two-dimensional square-root higher-order topological insulators, emergence of in-gap corner states are inherited from the squared Hamiltonian which hosts higher-order topology. In this paper, we propose that the martini lattice model serves as a concrete example of higher-order topological insulators. Furthermore, we also propose a suquare-root higher-order topological insulator based on the martini model. Specifically, we propose that the honeycomb lattice model with two-site…
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…
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…
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…
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…
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 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…
Recently, we have introduced in [A. M. Marques et al., Phys. Rev. B 103, 235425 (2021)] the concept of $2^n$-root topology and applied it to one-dimensional systems. These models require $n$ squaring operations to their Hamiltonians, intercalated with different constant energy downshifts at each level, in order to arrive at a decoupled block corresponding to a known topological insulator (TI) that acts as the source of the topological features of the starting $2^n$-root TI ($\sqrt[2^n]{\text{TI}}$). In the process, $n$ non-topological residual models with degenerate spectra and in-gap impurity…
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…
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…
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…
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 present numerical and analytical results for the optical and acoustic plasmon collective modes of coupled massless-Dirac two-dimensional electron systems. Our results apply to topological insulator (TI) thin films and to two graphene sheets separated by a thin dielectric barrier layer. We find that because of strong bulk dielectric screening TI acoustic modes are locked to the top of the particle-hole continuum and therefore probably unobservable.
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…
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 …
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…
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…
Square-root topology is a recently emerged subfield describing a class of insulators and superconductors whose topological nature is only revealed upon squaring their Hamiltonians, i.e., the finite energy edge states of the starting square-root model inherit their topological features from the zero-energy edge states of a known topological insulator/superconductor present in the squared model. Focusing on one-dimensional models, we show how this concept can be generalized to $2^n$-root topological insulators and superconductors, with $n$ any positive integer, whose rules of construction are sy…
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 …
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…
We propose a general scheme to construct a Hamiltonian $H_{\text{root}}$ describing a square root of an original Hamiltonian $H_{\text{original}}$ based on the graph theory. The square-root Hamiltonian is defined on the subdivided graph of the original graph of $H_{\text{original}}$, where the subdivided graph is obtained by putting one vertex on each link in the original graph. When $H_{\text{original}}$ describes a topological system, there emerge in-gap edge states at non-zero energy in the spectrum of $H_{\text{root}}$, which are the inherence of the topological edge states at zero energy …
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…
Square-root topological insulators are recently-proposed intriguing topological insulators, where the topologically nontrivial nature of Bloch wave functions is inherited from the square of the Hamiltonian. In this paper, we propose that higher-order topological insulators can also have their square-root descendants, which we term square-root higher-order topological insulators. There, emergence of in-gap corner states is inherited from the squared Hamiltonian which hosts higher-order topology. As an example of such systems, we investigate the tight-binding model on a decorated honeycomb latti…
An acoustic topological insulator (TI) is synthesized using topology optimization, a free material inverse design method. The TI appears spontaneously from the optimization process without imposing requirements on the existence of pseudo spin-1/2 states at the TI interface edge, or the Chern number of the topological phases. The resulting TI is passive; consisting of acoustically hard members placed in an air background and has an operational bandwidth of $\approx$12.5\% showing high transmission. Further analysis demonstrates confinement of more than 99\% of the total field intensity in the T…
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…
Higher-order topological insulator (HOTI) represents a new phase of matter, the characterization of which goes beyond the conventional bulk-boundary correspondence and is attracting significant attention by the broad community. Using a square-root operation, it has been suggested that a square-root HOTI may emerge in a hybrid honeycomb-kagome lattice. Here, we report the first experimental realization of the square-root HOTI in topological LC circuits. We show theoretically and experimentally that the square-root HOTI inherits the feature of wave function from its parent, with corner states pi…
In condensed matter physics, symmetry profoundly governs the fundamentals of topological matter. The emergence of new topological phase is typically linked to the enrichment of symmetries. Different parity-time symmetry relations distinguish between spinless and spinful physical systems. In spinless systems, creating pseudo-spins can realize fragile topological phase but not break the time-reversal symmetry. Therefore, growing attentions were recently focused on the strong topological phase in spinless systems. Here we break the framework of crystallographic symmetry groups by utilizing the pr…
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 t…
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…
Square-root higher-order topological insulators (HOTIs) are recently discovered new topological phases, with intriguing topological properties inherited from a parent lattice Hamiltonian. Different from conventional HOTIs, the square-root HOTIs typically manifest two paired non-zero energy corner states. In this work, we experimentally demonstrate the second-order square-root HOTIs in photonics for the first time to our knowledge, thereby unveiling such distinct corner states. The specific platform is a laser-written decorated honeycomb lattice (HCL), for which the squared Hamiltonian represen…
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…
Floquet phases of matter have attracted great attention due to their dynamical and topological nature that are unique to nonequilibrium settings. In this work, we introduce a generic way of taking any integer $q$th-root of the evolution operator $U$ that describes Floquet topological matter. We further apply our $q$th-rooting procedure to obtain $2^n$th- and $3^n$th-root first- and second-order non-Hermitian Floquet topological insulators (FTIs). There, we explicitly demonstrate the presence of multiple edge and corner modes at fractional quasienergies $\pm(0,1,...2^{n})π/2^{n}$ and $\pm(0,1,.…
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…
This letter investigates a flow-free, pseudospin-based acoustic topological insulator. Zone folding, a strategy originated from photonic crystal, is used to form double Dirac cones in phononic crystal. The lattice symmetry of the phononic crystal is broken by tuning the size of the center "atom" of the unit cell in order to open the nontrivial topological gap. Robust sound one-way propagation is demonstrated both numerically and experimentally. This study provides a flexible approach for realizing acoustic topological insulators, which are promising for applications such as noise control and w…
We provide a parton construction of wavefunctions and effective field theories for fractional Chern insulators. We also analyze a strong coupling expansion in lattice gauge theory that enables us to reliably map the parton gauge theory onto the microsopic Hamiltonian. We show that this strong coupling expansion is useful because of a special hierarchy of energy scales in fractional quantum Hall physics. Our procedure is illustrated using the Hofstadter model and then applied to bosons at 1/2 filling and fermions at 1/3 filling in a checkerboard lattice model recently studied numerically. Becau…
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…
We investigate the transient photoexcited lattice dynamics in a layered perovskite Mott insulator Sr2IrO4 by femtosecond X-ray diffraction using a laser plasma-based X-ray source. Ultrafast structural dynamics of Sr2IrO4 thin films are determined by observing the shift and broadening of the (0012) Bragg diffraction after excitation by 1.5 eV and 3.0 eV pump photons for films with different thicknesses. The observed transient lattice response can be well interpreted as a distinct three-step dynamics due to the propagation of coherent acoustic phonons generated by the photoinduced quasiparticles…
We explore the temperature-dependent plasmonic modes of an n-doped double-layer silicene system which is composed of two spatially separated single layers of silicene with a distance large enough to prevent the interlayer electron tunneling. By applying an externally applied electric field, we numerically obtain the poles of the loss function within the so-called random phase approximation, to investigate the effects of temperature and geometry on the plasmon branches in three different regimes: topological insulator, valley-spin polarized metal, and band insulator. Also, we present the finite…
Topological Insulators are a novel state of matter where spectral bands are characterized by quantized topological invariants. This unique quantized non-local property commonly manifests through exotic bulk phenomena and corresponding robust boundary effects. In our work, we report a new type of topological insulator exhibiting spectral bands with non-quantized topological properties, but with a quantization that arises in a corresponding system where the square of the Hamiltonian is taken. We provide a thorough theoretical analysis as well as an experimental demonstration based on photonic Ah…
