Quadrupole Topological Insulator Research Articles

Tunable cornerlike modes in generalized quadrupole topological insulators

Tunable cornerlike modes in generalized quadrupole topological insulators

Realization of a quadrupole topological insulator phase in a gyromagnetic photonic crystal

ABSTRACT The field of topological photonics was initiated with the realization of a Chern insulator phase in a gyromagnetic photonic crystal (PhC) with broken time-reversal symmetry (T), hosting chiral edge states that are topologically protected propagating modes. Along a separate line of research, a quadrupole topological insulator was the first higher-order topological phase supporting localized corner states, but has been so far limited to T-invariant systems, as T is a key ingredient in early models. Here we report the realization of a quadrupole topological insulator phase in a gyromagnetic PhC, as a consequence of topological phase transition from the previously demonstrated Chern insulator phase. The phase transition has been demonstrated with microwave measurements, which characterize the evolution from propagating chiral edge states to localized corner states. We also demonstrate the migration of topological boundary states into the continuum, when the gyromagnetic PhC is magnetically tuned. These results extend the quadrupole topological insulator phase into T-broken systems, and integrate topologically protected propagating and localized modes in a magnetically tunable photonic crystal platform.

Dislocation defect states in acoustic quadrupole topological insulators

Quadrupole topological insulator (QTI) is the first proposed higher-order topological phase of matter with quantized quadrupole moment but zero dipole moment. The QTI has expanded widely the traditional bulk-boundary correspondence, thereby the lower-dimensional topological boundary state can be observed. The recent interest has turned to the bulk-dislocation correspondence, which dominates the topological states localized to disclinations, and links the reciprocal-space topology of lattices with the appearance of dislocation states. Recently, many research groups have turned the studies of dislocation defects to classical wave systems. In these researches, the method of inducing dislocation defects is to remove a portion of the lattices of topological insulator and then rearrange the remaining lattices of the topological insulator. Through such a method, the micro structure of the lattices is changed, but it is difficult to realize in the actual operation. In this work, we study the dislocation defect states in acoustic QTIs. The acoustic QTI is designed by reversing the magnitude of the intracellular and extracellular coupling in the system, and the bulk energy bands and topological corner states are studied. Subsequently, by introducing partial trivial lattices into acoustic QTI structure, the dislocation bound states are generated in the corner formed by two different topological phases, which can be characterized by a 1/2 quantized fractional charge. The robustness of the topological dislocation states is verified by introducing the imperfection inside the system. Further, it is demonstrated that the dislocation positions can be designed at will. Without changing the microstructure of the lattice, we successfully modulate the line dislocation states and bulk dislocation states. The topological dislocation states studied in this work broaden the types of higher-order topological states in artificial structures, and provide new insights into the acoustic applications of higher-order topological insulators, such as sensing and high-performance energy harvesting.

A robust and tunable Luttinger liquid in correlated edge of transition-metal second-order topological insulator Ta2Pd3Te5

The interplay between topology and interaction always plays an important role in condensed matter physics and induces many exotic quantum phases, while rare transition metal layered material (TMLM) has been proved to possess both. Here we report a TMLM Ta2Pd3Te5 has the two-dimensional second-order topology (also a quadrupole topological insulator) with correlated edge states - Luttinger liquid. It is ascribed to the unconventional nature of the mismatch between charge- and atomic- centers induced by a remarkable double-band inversion. This one-dimensional protected edge state preserves the Luttinger liquid behavior with robustness and universality in scale from micro- to macro- size, leading to a significant anisotropic electrical transport through two-dimensional sides of bulk materials. Moreover, the bulk gap can be modulated by the thickness, resulting in an extensive-range phase diagram for Luttinger liquid. These provide an attractive model to study the interaction and quantum phases in correlated topological systems.

Reconfigurable topological modes in acoustic non-Hermitian crystals

Recent explorations show that the non-Hermiticity, associated with loss and/or gain, can play a surprisingly significant role in topological insulators, where the topological states can only penetrate the bulk structure if the topological phase changes in the interior. In this work, by introducing non-Hermiticity into an acoustic quadrupole topological insulator, the authors show experimentally that the topological corner and edge states could be engineered at any desired positions without a topological phase transition, making the bulk sites useful.

Analogous quadrupole topology induced by non-Hermiticity

Non-Hermiticity can alter the topological phases of Hermitian lattices and induce gapped edge states and in-gap corner states for the topologically trivial structures in Hermitian cases. To date, however, the realizations of non-Hermiticity-induced higher-order topological phases have often needed intricate designs, where the non-Hermitian systems on a square lattice are comprised of 16 atoms per unit cell with staggered on-site non-Hermitian components at least. It is still necessary to further investigate their topological characteristics in both Hermitian and non-Hermitian systems. In this paper, we suggest an extended quadrupole topological insulator that contains eight atoms in each unit cell and whose topological phases in Hermitian cases are governed by the couplings between adjacent atoms. By introducing external losses into such a higher-order topological insulator, we show that therein the trivial lattices can also give rise to nontrivial quadrupole topology with higher-order corner states. Calculating the biorthogonal nested Wilson loop characterizes the topology for such a non-Hermitian system. Interestingly, the non-Hermitian system has four Wannier bands, with two near zero and two deviating from zero and distributing symmetrically in the negative and positive sections, implying a topology similar to the nontrivial phase in the Hermitian case. Furthermore, both the positive and negative Wannier bands exhibit a nontrivial polarization of 0.5 whereas the remain Wannier bands near zero carry a trivial polarization, indicating that the non-Hermitian system hosts a nontrivial quadrupole topology. We also propose a complete design with acoustic resonators to implement the homologous non-Hermitian topology in acoustic systems to validate our theory. The numerical results match the analytical solution perfectly. Our work introduces an alternative method for fabricating non-Hermitian-induced quadrupole topological insulators and paves the way for future research into the non-Hermitian higher-order topology.

Photonic quadrupole topological insulator using orbital-induced synthetic flux

The rich physical properties of multiatomic crystals are determined, to a significant extent, by the underlying geometry and connectivity of atomic orbitals. The mixing of orbitals with distinct parity representations, such as s and p orbitals, has been shown to be useful for generating systems that require alternating phase patterns, as with the sign of couplings within a lattice. Here we show that by breaking the symmetries of such mixed-orbital lattices, it is possible to generate synthetic magnetic flux threading the lattice. We use this insight to experimentally demonstrate quadrupole topological insulators in two-dimensional photonic lattices, leveraging both s and p orbital-type modes. We confirm the nontrivial quadrupole topology by observing the presence of protected zero-dimensional states, which are spatially confined to the corners, and by confirming that these states sit at mid-gap. Our approach is also applicable to a broader range of time-reversal-invariant synthetic materials that do not allow for tailored connectivity, and in which synthetic fluxes are essential.

Quadrupole topological insulators in Ta2M3Te5 (M = Ni, Pd) monolayers

Higher-order topological insulators have been introduced in the precursory Benalcazar-Bernevig-Hughes quadrupole model, but no electronic compound has been proposed to be a quadrupole topological insulator (QTI) yet. In this work, we predict that Ta2M3Te5 (M = Pd, Ni) monolayers can be 2D QTIs with second-order topology due to the double-band inversion. A time-reversal-invariant system with two mirror reflections (Mx and My) can be classified by Stiefel-Whitney numbers (w1, w2) due to the combined symmetry TC2z. Using the Wilson loop method, we compute w1 = 0 and w2 = 1 for Ta2Ni3Te5, indicating a QTI with qxy = e/2. Thus, gapped edge states and localized corner states are obtained. By analyzing atomic band representations, we demonstrate that its unconventional nature with an essential band representation at an empty site, i.e., Ag@4e, is due to the remarkable double-band inversion on Y–Γ. Then, we construct an eight-band quadrupole model with Mx and My successfully for electronic materials. These transition-metal compounds of A2M1,3X5 (A = Ta, Nb; M = Pd, Ni; X = Se, Te) family provide a good platform for realizing the QTI and exploring the interplay between topology and interactions.

Tailoring quadrupole topological insulators with periodic driving and disorder

The quadrupole topological insulator (QTI) has attracted intense studies as a prototype of symmetry-protected higher-order topological phases of matter with a quantized quadrupole moment. The realization of QTIs has been reported in various static settings with periodic structures. Here, we theoretically investigate topological phase transitions and establish the QTI phase in a periodically driven system with disorder. In the clean limit, the Floquet QTI phase emerges from a topologically trivial band structure driven by elliptically polarized irradiation. More strikingly, starting from a pure and static system with trivial topology, we unveil an intriguing QTI phase which necessitates the simultaneous presence of disorder and periodic driving. Furthermore, we reveal that particle-hole symmetry is sufficient to protect the QTI. Our work not only establishes a new strategy to design QTIs but also enriches the symmetry-protected mechanism of higher-order topology.

Photonic non-Bloch quadrupole topological insulators in coupled ring resonators

We investigate the second-order topological phases in a two-dimensional ring resonator array with each plaquette occupied by \ensuremath{\pi} gauge flux and imaginary gauge field. The real and imaginary gauge fields are induced by shifting the displacement and integrating gain or loss into the two half perimeters of the auxiliary rings. The system supports topological corner modes with their emergence being determined by the non-Bloch topological invariant due to skin effects. The bulk modes, exhibiting second-order skin effects in both trivial and nontrivial phases, are accumulated at opposite corners depending on whether clockwise or counterclockwise modes are excited. By introducing an interface with different imaginary gauge fields, we show the bulk modes exist at the interface while the topological corner modes are localized at the physical corners. Furthermore, the skin effects are also presented in the passive ring resonators. The study may find applications in lasers and broadband light trapping.

Realization of quasicrystalline quadrupole topological insulators in electrical circuits

Quadrupole topological insulators are a new class of topological insulators with quantized quadrupole moments, which support protected gapless corner states. The experimental demonstrations of quadrupole-topological insulators were reported in a series of artificial materials, such as photonic crystals, acoustic crystals, and electrical circuits. In all these cases, the underlying structures have discrete translational symmetry and thus are periodic. Here we experimentally realize two-dimensional aperiodic-quasicrystalline quadrupole-topological insulators by constructing them in electrical circuits, and observe the spectrally and spatially localized corner modes. In measurement, the modes appear as topological boundary resonances in the corner impedance spectra. Additionally, we demonstrate the robustness of corner modes on the circuit. Our circuit design may be extended to study topological phases in higher-dimensional aperiodic structures.

Magnonic Quadrupole Topological Insulator in Antiskyrmion Crystals.

We uncover that antiskyrmion crystals provide an experimentally accessible platform to realize a magnonic quadrupole topological insulator, whose hallmark signatures are robust magnonic corner states. Furthermore, we show that tuning an applied magnetic field can trigger the self-assembly of antiskyrmions carrying a fractional topological charge along the sample edges. Crucially, these fractional antiskyrmions restore the symmetries needed to enforce the emergence of the magnonic corner states. Using the machinery of nested Wilson loops, adapted to magnonic systems supported by noncollinear magnetic textures, we demonstrate the quantization of the bulk quadrupole moment, edge dipole moments, and corner charges.

Buckled honeycomb antimony: Higher order topological insulator and its relation to the Kekulé lattice

Higher Order Topological Insulators (HOTI) are $d$-spatial dimensional systems featuring topologically protected gap-less states at their $(d-n)$-dimensional boundaries. With the help of \textit{ab-initio} calculations and tight binding models along with symmetry considerations we show that monolayer buckled honeycomb structures of group-V elements (Sb,As), which have already been synthesized, belong in this category and have a charge fractionalization of $\frac{e}{2}$ at the corner states as well as weak topological edge states, all protected by their properties under the inversion operation which classify this system as a quadrupole topological insulator.

Quadrupole topological phase and robust corner resonance in Kekulé hexagonal electric circuit

Two-dimensional (2D) quadrupole topological insulators, featured by topologically protected 0D corner modes, have recently attracted tremendous interest in condensed matter and materials physics. Herein, we construct a specific electric circuit made of capacitors and inductors forming a 2D Kekulé hexagonal lattice for quadrupole topological phase and corner modes. Trivial–nontrivial topological phase transition can be controlled by varying capacitance in the circuit, so that distinct topological edge states appear in 1D ribbons and corner states emerge in 0D flakes. We explore the field strength distribution and two-point impedance with respect to excitation frequency, and reveal that the topological corner resonance is robust against size of the LC network and randomness of the capacitors/inductors, a great benefit for experimental detection. Our results enrich the family of designer topoelectrical circuit as a flexible and tunable platform to achieve exotic quantum phases, which may have potential for future telecommunications, signal processing and quantum computing.

Type-II quadrupole topological insulators

Modern theory of electric polarization is formulated by the Berry phase, which, when quantized, leads to topological phases of matter. Such a formulation has recently been extended to higher electric multipole moments, through the discovery of the so-called quadupole topological insulator. It has been established by a classical electromagnetic theory that in a two-dimensional material the quantized properties for the quadupole topological insulator should satisfy a basic relation. Here we discover a new type of quadrupole topological insulator (dubbed type-II) that violates this relation due to the breakdown of the correspondence that a Wannier band and an edge energy spectrum close their gaps simultaneously. We find that, similar to the previously discovered (referred to as type-I) quadrupole topological insulator, the type-II hosts topologically protected corner states carrying fractional corner charges. However, the edge polarizations only occur at a pair of boundaries in the type-II insulating phase, leading to the violation of the classical constraint. We demonstrate that such new topological phenomena can appear from quench dynamics in non-equilibrium systems, which can be experimentally observed in ultracold atomic gases. We also propose an experimental scheme with electric circuits to realize such a new topological phase of matter. The existence of the new topological insulating phase means that new multipole topological insulators with distinct properties can exist in broader contexts beyond classical constraints.

Anomalous quadrupole topological insulators in two-dimensional nonsymmorphic sonic crystals

The discovery of quadrupole topology opens a new horizon in the study of topological phenomena. However, the existing experimental realizations of quadrupole topological insulators in symmorphic lattices with $\pi$-fluxes often break the protective mirror symmetry. Here, we present a theory for anomalous quadrupole topological insulators in nonsymmorphic crystals without flux, using 2D sonic crystals with $p4gm$ and $p2gg$ symmetry groups as concrete examples. We reveal that the anomalous quadrupole topology is protected by two orthogonal glide symmetries in square or rectangular lattices. The distinctive features of the anomalous quadrupole topological insulators include: (i) minimal four bands below the topological band gap, (ii) nondegenerate, gapped Wannier bands and special Wannier sectors with gapped composite Wannier bands, (iii) quantized Wannier band polarizations in these Wannier sectors. Remarkably, the protective glide symmetries are well-preserved in the sonic-crystal realizations where higher-order topological transitions can be triggered by symmetry or geometry engineering.

Quadrupole topological phases in the zero-dimensional optical cavity

The quadrupole topological insulator, which supports robust corner states, has been recently demonstrated in two-dimensional (2D) spatial lattices. Here, we design the first photonic quadrupole topological insulator in fully synthetic spaces with the utilization of 0D optical cavity. The frequency and orbital angular momentum (OAM) of light are used to form the 2D synthetic spaces. Four degenerate polarization states are mapped to the internal lattice sites within the unit cell. By suitably engineering the coupling between cavity modes with different frequencies, OAMs and polarizations, the ideal synthetic quadrupole topological insulator is obtained. By using the robust synthetic corner states, we present the possibility for the realization of topological transportation of multi-photon entangled states. Our designed synthetic photonic quadrupole insulator proposes a unique platform to investigate higher-order topological phases in lower-dimensional systems and possesses potential applications in quantum information and optical communication.

Twisted Quadrupole Topological Photonic Crystals

AbstractTopological manipulation of waves is at the heart of cutting‐edge metamaterial research. Quadrupole topological insulators were recently discovered in 2D flux‐threading lattices that exhibit higher‐order topological wave trapping at both the edges and corners. Photonic crystals (PhCs), lying at the boundary between continuous media and discrete lattices, however, are incompatible with the present quadrupole topological theory. Here, quadrupole topological PhCs triggered by a twisting degree‐of‐freedom are unveiled. Using a topologically trivial PhC as the motherboard, it is shown that twisting induces quadrupole topological PhCs without flux‐threading. The twisting‐induced crystalline symmetry enriches the Wannier polarizations and leads to the anomalous quadrupole topology. Versatile edge and corner phenomena are observed by controlling the twisting angles in a lateral heterostructure of 2D PhCs. This study paves the way toward topological twist photonics as well as the quadrupole topology in the quasi‐continuum regime for phonons and polaritons.

Acoustic Realization of Quadrupole Topological Insulators.

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 topology has been experimentally confirmed by detecting the acoustic responses at the bulk, edge, and corner of the sample. Potential applications can be anticipated for the topologically robust in-gap states, such as acoustic sensing and energy trapping.

Plasmon-polaritonic quadrupole topological insulators

Quadrupole topological insulator is a symmetry-protected higher-order topological phase with intriguing topology of Wannier bands, which, however, has not yet been realized in plasmonic metamaterials. Here, we propose a lattice of plasmon-polaritonic nanocavities which can realize quadrupole topological insulators by exploiting the geometry-dependent sign-reversal of the couplings between the daisy-like nanocavities. The designed system exhibits various topological and trivial phases as characterized by the nested Wannier bands and the topological quadrupole moment which can be controlled by the distances between the nanocavities. Our study opens a pathway toward plasmonic topological metamaterials with quadrupole topology.