Higher-order Topological States Research Articles

Higher-order topological states in T-graphene and their realization in photonic crystals.

Higher-order topological states extend the power of nontrivial topological states beyond the bulk-edge correspondence. Here we study the higher-order topological states (corner states) in an open-boundary two-dimensional T-graphene lattice. Unlike the common zero-energy corner states, our findings reveal non-zero energy corner states in such lattice systems, and the energy could be controlled by modifying the hopping parameters. Moreover, the corner states could be transferred away from the lattice corners by designing the position-specific vacancy defects. The strong robustness of the corner states is also demonstrated against the uniaxial strain and vacancy defects, respectively. A plasmonic crystal is constructed to testify to the theory, in which the corner states are realized in optical modes and their higher-order topological properties are verified. Our results open the avenue of corner-states engineering, which holds significant physical implications of higher-order topological states for the design of photonic and electronic devices with specialized functionalities.

Creation and manipulation of higher-order topological states by altermagnets

Creation and manipulation of higher-order topological states by altermagnets

Hermitian and non-hermitian higher-order topological states in mechanical metamaterials

Higher-order topological insulators exhibit clear hierarchical boundary states, providing an amazing platform for robust wave manipulations in mechanical or acoustic systems. Recently, this theory has been extended to non-Hermiticity-induced higher-order topology, which enables actively controllable topological transport. Nevertheless, most related studies focus on the non-Hermitian quadrupole topology, associated with the coexistence of negative and positive couplings to constitute a π-flux lattice, which hindered the realizations in classical wave systems. Here, we propose a novel tight-binding model without negative couplings, which supports non-Hermiticity-induced higher-order topological states. In the Hermitian case, the middle band gaps host the vanishing bulk polarization and the associated topology is dependent only upon the associated edge polarization. By introducing the external loss as non-Hermitian components, we reveal that therein the trivial structures can also host topological corner states, featured by the quantized nonzero edge polarizations in the biorthogonal basis. Furthermore, such a topological phase transition induced by non-Hermiticity can be realized for almost all of our lattices, except for the isotropic case where the middle band gaps never open. Basing on the theoretical design, we map the lattice models into the mechanical metamaterials to access the analogous higher-order topology in the elastic wave systems for both Hermitian and non-Hermitian physics. The simulated band structures match well with theoretical solutions. And the robust higher-order topological states are also identified. Our work paves the way to implement the non-Hermitian higher-order topology and offers the possibility for wave manipulations in non-Hermitian systems.

Bogoliubov corner excitations in a conventional s-wave superfluid

Higher-order topological superconductors and superfluids have triggered a great deal of interest in recent years. While Majorana zero-energy corner or hinge states have been studied intensively, whether superconductors and superfluids host higher-order topological Bogoliubov excitations with finite energies remain elusive. In this work, we propose that Bogoliubov corner excitations with finite energies can be induced through only mirror-symmetric local potentials from a trivial conventional s-wave superfluid. The topological Bogoliubov excited modes originate from the nontrivial Bogoliubov excitation bands. These modes are protected by the mirror symmetry and are robust against mirror-symmetric perturbations as long as the Bogoliubov energy gap remains open. Our work provides a new insight into higher-order topological excitation states in superfluids and superconductors.

Acoustic Higher-Order Topological Insulators Induced by Orbital-Interactions.

The discovery of higher-order topological insulator metamaterials, in analogy with their condensed-matter counterparts, has enabled various breakthroughs in photonics, mechanics, and acoustics. A common way of inducing higher-order topological wave phenomena is through pseudo-spins, which mimic the electron spins as a symmetry-breaking degree of freedom. Here, this work exploits degenerate orbitals in acoustic resonant cavities to demonstrate versatile, orbital-selective, higher-order topological corner states. Type-II corner states are theoretically investigated and experimentally demonstrated based on tailored orbital interactions, without the need for long-range hoppings that has so far served as a key ingredient for Type-II corner states in single-orbital systems. Due to the orthogonal nature of the degenerate p orbitals, this work also introduces a universal strategy to realize orbital-dependent edge modes, featuring high-Q edge states identified in bulk bands. These findings provide an understanding of the interplay between acoustic orbitals and topology, shedding light on orbital-related topological wave physics, as well as its applications for acoustic sensing and trapping.

Higher-order topological states in dual-band valley sonic crystals

As a quantum state of frequency extrema in the momentum space of acoustic systems, sonic valley pseudospin provides a new degree of freedom for controlling acoustic waves. Higher-order topological insulators (HOTIs) have extended the traditional bulk-edge correspondence principle and are a crucial concept for classic wave regulation. However, HOTIs in valley sonic crystals (VSCs) only appear in a single bandgap, which limits the multi-frequency selectivity of the corner state and is not conducive to the design of multi-frequency acoustic communication devices. Here, we demonstrate “Y-shaped” acoustic crystals with C3 symmetry that form a double-band VSC, and the topological phase transitions in both low- and high-frequency band gaps coincide. We realize theoretically and experimentally higher-order states in dual-band valley sonic crystals. Our work enriches the application of HOTIs in acoustic multi-frequency regulatory systems and provides different avenues for designing of multi-band acoustic devices.

Three-dimensional non-Abelian Bloch oscillations and higher-order topological states

Recently, higher-order topological insulators (HOTIs) have been introduced, and were shown to host topological corner states under the theoretical framework of Benalcazar-Bernevig-Hughes. Here we unveil some topological effects in HOTIs by studying the three-dimensional (3D) non-Abelian Bloch oscillations (BOs). In HOTIs, BOs with a multiplied period occur when a force with a special direction is applied due to the effect of the non-Abelian Berry curvature. Along the direction of the oscillations we find a higher-order topological state that goes beyond the theoretical framework of multipole moments. The emergence of such a higher-order topological state coincides with the appearance of the 3D non-Abelian BOs. That is, the 3D non-Abelian BOs can be used as a tool to probe higher-order topological states. These phenomena are observed experimentally with designed electric circuit networks. Our work opens up a way to detect topological phases theoretically and experimentally.

Acoustic higher-order topological states in kagome lattice with split-ring resonators

The next-nearest-neighbour (NNN) hoppings can induce many unique topological phenomena but are rarely considered in acoustics. This work investigates the higher-order topological states induced by the NNN hoppings in acoustic kagome lattice with split-ring resonators (SRRs). We build an extended tight-binding model (TBM) considering NNN hoppings to predict the topological corner modes (CMs) of the acoustic system and obtain the coupling strengths between the SRRs in different rotational angles. During rotating the SRRs, two nontrivial and one trivial systems characterized by bulk polarization can be obtained. In these two nontrivial systems, four novel CMs induced by the NNN hoppings can be observed in the first bandgap. Furthermore, the interactions between resonant and Bragg scattering effect of the SRRs are researched and the defect modes (DMs) are realized in the second bandgap. Finally, the CMs predicted by the extended TBM and DMs match well with the numerical and experimental ones. These discoveries unveil the intricate physics driven by the NNN hoppings, paving the way for innovative approaches to the design of tunable sound energy localization and multi-band metamaterials.

Observation of higher-order topological corner states in photonic two-dimensional trimer lattices.

We demonstrate the first, to the best of our knowledge, experimental observation of higher-order topological corner states in the photonic two-dimensional (2D) trimer lattices. Using a femtosecond laser direct writing technology, we experimentally fabricate a series of 2D trimer lattices with different open boundary conditions and thereby observe two kinds of 0D topological corner states, i.e., topological corner states and topological defect corner states. Interestingly, these corner states and defect corner states can not only exist in the bandgap but also coexist with the bulk states and show obvious localization properties. This work provides fresh perspectives on higher-order topology in artificial microstructures.

Non-Hermitian higher-order topological corner states on the extended kagome lattice

Exploring the interaction between topological phases and non-Hermitian potentials such as gain and loss can benefit designing robust optical devices. Recent studies have revealed topological phases can be simply from gain and loss in non-Hermitian systems. Here, we propose an extended kagome lattice model, where the non-Hermitian potentials drive the system from a trivial phase to a higher-order topological phase. Higher-order topological insulators exhibit lower-dimensional boundary states on corners or hinges. We construct two-dimensional higher-order topological insulators on different arrays of the extended kagome lattice model. Topologically protected states emerge at the corner with a 1/3 fractional charge at each corner as the strength of the gain and loss increases. The topologically protected corner states are characterized by the quantized polarization as the topological index. We find that non-Hermitian potentials provide an extra degree of freedom to switch on and off the higher-order topological corner states. The proposed system can be verified through many experimental platforms, including coupled optical resonating cavities and waveguides. Our work indicates the great prospects for constructing integrated photonics platforms and designing actively reconfigurable photonic devices.

Broad and colossal edge supercurrent in Dirac semimetal Cd3As2 Josephson junctions

Edge supercurrent has attracted great interest recently due to its crucial role in achieving and manipulating topological superconducting states. Proximity-induced superconductivity has been realized in quantum Hall and quantum spin Hall edge states, as well as in higher-order topological hinge states. Non-Hermitian skin effect, the aggregation of non-Bloch eigenstates at open boundaries, promises an abnormal edge channel. Here we report the observation of broad edge supercurrent in Dirac semimetal Cd3As2-based Josephson junctions. The as-grown Cd3As2 nanoplates are electron-doped by intrinsic defects, which enhance the non-Hermitian perturbations. The superconducting quantum interference indicates edge supercurrent with a width of ~1.6 μm and a magnitude of ~1 μA at 10 mK. The wide and large edge supercurrent is inaccessible for a conventional edge system and suggests the presence of non-Hermitian skin effect. A supercurrent nonlocality is also observed. The interplay between band topology and non-Hermiticity is beneficial for exploiting exotic topological matter.

Dual-band higher-order topological states and four-wave mixing in plasmonic valley-Hall metasurfaces

Higher-order topological states in plasmonic metasurfaces are of great importance in practice for their tight confinements and high field enhancements, providing many promising nonlinear applications such as robust quantum light source through four-wave mixing (FWM). However, previous works are limited to single-band modes and FWM processes based on second-order topological corner states have still not been revealed. In this work, we propose a plasmonic valley-Hall topological metasurface, which is modelled based on coupled dipole approximation with dyadic Green's functions containing retarded long-range interaction terms. Simultaneous lifting of Dirac degeneracies in two bandgaps due to inversion-symmetry breaking results in dual-band second-order topological states, which are robust against structural disorder. We present a theoretical approach for considering the nonlinear interaction in dipole arrays and numerically show nonlinear generation of corner mode at signal frequency via the FWM process by pumping with edge modes. The results presented in this work will pave a broad way towards the wide applications of topological plasmonic metasurfaces.

Quadrupole higher-order topological phases in static mechanical metamaterials

Higher-order topological insulators (HOTIs), an extended concept of traditional first-order topological insulators, are capable of hosting multidimensional topological states and provide an unprecedented platform for robust wave manipulation and energy enhancement. HOTIs have recently been extended to mechanical systems. However, most studies focus primarily on dynamic wave motion and signal transmission. In this paper, we show a realization of quadrupole HOTIs using static Rayleigh deformation modes in three-dimensional layered mechanical metamaterials. The quadrupole topological phase, usually shown in dynamic systems as charge accumulation or energy concentration at corners, is present in the mechanical metamaterial through its patterned static deformation. It is demonstrated analytically that for the topologically protected corner modes, an externally applied corner point load can be fully confined within a localized region for any layer, thereby constructing a deformation isolation device capable of facilitating deformation shielding for a macroscopic bulk region. The obtained results shed light on the exploration of higher-order topological states with a static system and the control of bulk and boundary static deformations in various dimensions.

Theory of nonlinear corner states in photonic fractal lattices

AbstractWe study linear and nonlinear higher-order topological insulators (HOTIs) based on waveguide arrays arranged into Sierpiński gasket and Sierpiński carpet structures, both of which have non-integer effective Hausdorff dimensionality. Such fractal structures possess different discrete rotational symmetries, but both lack transverse periodicity. Their characteristic feature is the existence of multiple internal edges and corners in their optical potential landscape, and the formal absence of an insulating bulk. Nevertheless, we show that a systematic geometric shift of the waveguides in the first generation of such fractal arrays, which affects the coupling strengths between sites of this building block as well as in subsequent structure generations, enables the formation of corner states of topological origin at the outer corners of the array. We find that, in contrast to HOTIs based on periodic arrays, Sierpiński gasket arrays always support topological corner states, irrespective of the direction of the shift of the waveguides, while in Sierpiński carpet structures, corner states emerge only for one direction of the waveguide shift. We also find families of corner solitons bifurcating from linear corner states of fractal structures that remain stable practically in the entire gap in which they form. These corner states can be efficiently excited by injecting Gaussian beams into the outer corner sites of the fractal arrays. Our results pave the way toward the investigation of nonlinear effects in topological insulators with non-integer dimensionality and enrich the variety of higher-order topological states.

Ferroelectric heterobilayer with tunable first- and higher-order topological states

Ferroelectric heterobilayer with tunable first- and higher-order topological states

Bonding-Driven Crystallization of Dimensionally Resolved Nanostructures Based on Quasi-1D Van Der Waals Building Blocks

The discovery of complex phenomena and strongly correlated behavior in the solid state has incessantly relied upon the creation of stable low-dimensional lattices approaching the atomic limit. Whereas 2D van der Waals (vdW) materials with atomic scale thicknesses have gained widespread interest, very little has been known about the chemistry and physics of their more confined 1D counterparts in the solid state. In these length scales and dimensionalities, unique physical properties arising from finite size effects such as ballistic electrical transport, size-dependent optical resonance modes, long carrier lifetimes, and exotic higher-order topological states become realizable. To this end, I will present the recent efforts in my group towards elucidating the chemical bonding interactions which define the structure and nanoscale dimensionality of structures derived from lattices which are comprised of 1D chains with sub-nm diameters that are held together by weak vdW interactions. By gaining anisotropy-driven control over these interactions via bottom-up chemical vapor growth, we demonstrate the creation of dimensionally-resolved nanostructures with various thicknesses ranging from 1D nanowire bundles to quasi-2D nanoribbons and nanosheets, all based on a the same 1D vdW lattice motif. Extensive electron diffraction and atomic-resolution microscopy of these resulting nanostructures reveal the strong dependence of the size and dimensionality of the 1D and quasi-2D nanostructures on anisotropic inter-chain vdW bonding interactions, precursor ratios, and growth temperatures. Using these results, we propose a crystallization mechanism defined by the competition between covalent and vdW bonding interactions and epitaxy. As physical properties directly correlate to the structure, dimensionality, and size in the solid state, I will also talk about how the unique topographies of resulting freestanding nanostructures dictate their electronic properties. I will describe how the synthetic control over the structural attributes led to our discovery of emergent properties such as the elusive indirect-to-direct band gap transition and higher-order topological states in 1D vdW nanostructures. Through these efforts, we define for the first time the anisotropy-driven growth of these emergent 1D vdW-derived nanostructures and how nanoscale morphologies in these nanostructures dictate their electronic structures. Altogether, this precise control over crystalline and electronic structure poises these emergent low-dimensional solids as building blocks towards programmable electrocatalysts, densified electronic and quantum devices in sensing, photonic, energy storage and conversion, and spintronic technologies.

Third-order square-root topological insulators on decorated diamond sonic crystals

The square-root operation can generate novel topological phases, whose nontrivial topological properties are inherited from the parent Hamiltonian. Here we report the acoustic realization of third-order square-root topological insulators by adding additional resonators between the site resonators of original diamond lattice. Due to the square-root operation, multiple acoustic localized modes appear in doubled bulk gaps. The bulk polarizations of the tight-binding models are employed to reveal the topological feature of the higher-order topological states. By tuning the coupling strength, we find the emergence of third-order topological corner states in doubled bulk gaps on tetrahedron-like and rhombohedron-like sonic crystals, respectively. The shape dependence of square-root corner states provides an extra degree of freedom for flexible manipulation on the sound localization. Furthermore, the robustness of the corner states in three-dimensional (3D) square-root topological insulator is well elucidated by introducing random disorders into the irrelevant bulk region of the proposed 3D lattices. This work extends square-root higher-order topological states into 3D system, and may find possible applications in selective acoustic sensors.

Higher-order hybrid topological bound states in a non-Hermitian system.

Higher-order topological states, such as the corner and pseudo-hinge states, have been discovered in both Hermitian and non-Hermitian systems. These states have inherent high-quality factors that make them useful in the application of photonic devices. In this work, we design a non-Hermiticity solely induced Su-Schrieffer-Heeger (SSH) lattice and demonstrate the existence of diverse higher-order topological bound states in the continuum (BICs). In particular, we first uncover some hybrid topological states that occur in the form of BICs in the non-Hermitian system. Furthermore, these hybrid states with an amplified and localized field have been demonstrated to excite nonlinear harmonic generation with high efficiency. The appearance of these topological bound states will advance the study of the interplay of topology, BICs, and non-Hermitian optics.

Higher-order topological corner state in a reconfigurable breathing kagome lattice consisting of magnetically coupled LC resonators

Higher-order topological insulators are attracting attention from fundamental interest to fascinating applications, owing to the topological properties with higher-order topological corner states. Breathing kagome lattice is a prospective platform which can support higher-order topological corner states. Here, we experimentally demonstrate that higher-order topological corner states are supported in a breathing kagome lattice consisting of magnetically coupled resonant coils. The winding direction of each coil is determined to hold C3 symmetry for each triangle unit cell, enabling to emerge higher-order topological corner states. In addition, topological and trivial phases can be switched by changing the distances between the coils. The emergence of corner states in the topological phase is experimentally observed through admittance measurements. As an illustration, wireless power transfer is performed between the corner states, and between the bulk and corner states. The proposed configuration is a promising platform for not only investigating topological properties of the breathing kagome lattice but also an alternative mechanism of selective wireless power transfer.

Tuning of Higher-Order Topological Corner States in a Honeycomb Elastic Plate

Different from the traditional bulk-edge correspondence principle, the discovery of higher-order topological states has generated widespread interest. In a second-order, two-dimensional elastic wave topological insulator, the fluctuation information can be confined to the corners, with the state being topologically protected. To better apply the topological corner states, a cleverly designed mechanical structure is used to achieve the adjustment of the topological corner states on a two-dimensional elastic plate. The energy-band inversion can be achieved by switching the material of the inner and outer brackets of the unit cell of the honeycomb elastic plate. The topological corner states can be observed at the boundary of two different topological phase structures in the finite lattice using finite-element software. The robustness of the topological corner state is verified by setting up defective control groups at the corner. In addition, the topological corner states of this honeycomb elastic plate are discussed in terms of positional tunability. The honeycomb elastic plate can provide a reference for the design of devices for the local control of elastic waves and energy harvesting due to the possibility of modulation of topological corner states, which is beneficial for the application of topological corner states in practice.