Zero-dimensional Corner States Research Articles

Pseudospin-dependent acoustic topological edge and corner states in silica aerogel metamaterialsa).

Fueled by the concepts of topological insulators, analogous topological acoustics offer an alternative approach to manipulate sound. Theoretical proposals for subwavelength acoustic topological insulators are considered to be ideal effective parameters or utilizeing artificial coiling-space metamaterials. However, the corresponding realization using realistic soft metamaterials remains challenging. In this study, we present the design of an acoustic subwavelength second-order topological insulator using nanoscale porous solid material, silica aerogel, which supports pseudospin-dependent topological edge and corner states simultaneously. Through simulations and experiments, we demonstrate that silica aerogel can function as a soft acoustic metamaterial at the subwavelength scale. By embedding silica aerogel in an air matrix to construct a honeycomb lattice, a double Dirac cone is obtained. A topological phase transition is induced by expanding or contracting the supercell, resulting in band inversion. Additionally, we propose topologically robust acoustic transmission along the one-dimensional edge. Furthermore, we discover that the proposed sonic crystal sustains zero-dimensional corner states, which can efficiently confine energy at subwavelength corners. These findings offer potential for the realization of subwavelength topological acoustic devices using realistic soft metamaterials.

Corner states and particle trapping in waterborne acoustic crystals

Higher-order topological insulators exhibit intriguing capacity to confine energy on lower-dimensional boundaries owing to the unique bulk-boundary correspondence. Many potential practical applications of the higher-order states in classical wave systems have been proposed and achieved. In this work, we implement a second-order topological insulator in a waterborne acoustic crystal by drilling and grooving a copper plate based on a two-dimensional Su–Schrieffer–Heeger lattice with different intracell and intercell couplings. The far-field transmission spectrum and near-field pressure field distributions verify the existence of the one-dimensional edge states and zero-dimensional corner states in the bulk bandgap. Due to the highly localized edge and corner states, the polystyrene particles are trapped at the edges and corners by the acoustic gradient force. Our findings provide a good platform to manipulate underwater acoustic wave and may inspire topological acoustic applications.

Antiferromagnetic second-order topology in two-dimensional NiRuCl6

In recent years, 2D second-order topological insulators (SOTIs) have garnered considerable interest because of their unique properties. However, only the FeSe monolayer with four corner states (two occupied and two unoccupied states) near the Fermi level has been reported to be a candidate for 2D intrinsic antiferromagnetic SOTIs in theory. The limited amount of antiferromagnetic SOTIs has hindered future research, and corner states should be at the Fermi level in order to manifest interesting physics. Herein, we propose NiRuCl6 as a candidate for 2D antiferromagnetic SOTIs with corner states strictly at the Fermi level. Without spin–orbit coupling (SOC), NiRuCl6 is an antiferromagnetic half-metal with a compensating magnetic moment and decoupled spin bands. In the spin-up channel, NiRuCl6 hosts a nontrivial gap of 1.11 eV, where zero-dimensional corner states appear. In the spin-down channels, NiRuCl6 hosts metallically behaved bands, where a spin-polarized quadratic Weyl point emerges. With SOC, two spin bands are coupled, and NiRuCl6 becomes an antiferromagnetic SOTI with three degenerate corner states at the Fermi level inside the SOC-induced gap with a value of 0.11 eV. Remarkably, the corner states in NiRuCl6 are resistant to changes in SOC strength and magnetization orientation. We also reveal that the phononic second-order topology and corner vibrational modes appear in the phonon dispersion curves of NiRuCl6. The presented results improve the general understanding of antiferromagnetic SOTIs and contribute to the prediction of materials with ideal corner states at the Fermi level, thereby advancing the field of topological antiferromagnetic spintronics.

Magnetic second-order topological physics in two-dimensional NiZrI6

Two-dimensional (2D) nonmagnetic second-order topological insulators (SOTIs) have been exhaustively studied, whereas 2D magnetic SOTIs have received little attention from researchers. In this work, we have demonstrated using first-principles calculations that the 2D NiZrI6 monolayer is a 2D magnetic SOTI with a ferromagnetic ground state. Two spin channels in the 2D NiZrI6 monolayer exhibit nontrivial gaps. According to the higher-order bulk-boundary correspondence, the 2D NiZrI6 monolayer shows topologically protected corner states with quantized fractional charge (e/3) that are spin-polarized and pinned at the sample's corners in real space. In addition, the zero-dimensional corner states of the 2D NiZrI6 monolayer are resistant to spin–orbit coupling (SOC) effects. It is anticipated that 2D NiZrI6 monolayer with magnetic higher-order topology will contribute to topo-spintronics advancement.

Variable-order topological insulators

Topological insulators, either the first-order or the higher-order, experience generally a transition to a trivial phase or a topological one of the same order through the gap closing and reopening procedure. Here, we report a topological insulator, which switches directly between the first and higher orders, with only varying the hoppings and without breaking the symmetry. The phase transition of the first and higher orders is originated from a competition mechanism between the nearest and second-nearest neighbor interactions. This variable-order topological insulator is implemented in a two-dimensional phononic crystal, and the one-dimensional helical edge states, which signal the first-order phase, and the zero-dimensional corner states, which signal the second-order one, are demonstrated in the simulations and experiments. Our study gives insight to the topological states of different orders.

Second-order topological corner states in zigzag graphene nanoflake with different types of edge magnetic configurations

We study the energy spectrum and energy levels of the extended Kane-Mele model with magnetic atoms on their zigzag edges. It is demonstrated that the edges of ferromagnetism or antiferromagnetism are enough to break the time-reversal symmetry and host one-dimensional gapped edge states. Thus, a second-order topological phase transition could happen, which leads to the emergence of topological in-gap zero-dimensional corner states. We also prove that the in-gap corner states are robust against corner defects and magnetic disorder. Our proposal based on the edge antiferromagnetism shows that the high-order topological states can be realized, although the net magnetization of the material is zero. In addition, we discuss the influence of spin magnetization orientation on the degeneracy and energy of in-gap corner states.

Corner states in second-order two-dimensional topological photonic crystals with reversed materials

Recently, topological corner states have been extensively investigated in second-order topological photonic crystals (STPCs). However, most square lattice STPCs are proposed based on the two-dimensional (2D) Su-Schrieffer--Heeger (SSH) model. In this work, we propose a photonic crystal (PC) that goes beyond the 2D SSH model. The unit cell (UC) of the PC is only composed of a dielectric cylinder in an air background (or an air hole in a dielectric slab). The topological trivial or nontrivial photonic band gap of the cylinder-in-air or hole-in-slab UC can be confirmed from the value of 2D polarization. Remarkably, photonic band gaps form directly without any intermediate transition. 2D bulk states, one-dimensional edge states, and zero-dimensional corner states are generated hierarchically in the box-type combination structures composed of trivial and nontrivial PCs. By comparing the perfect and defective structures, the corner states show strong robustness against the defects. The proposed configurations provide a simpler platform for exploring topological corner states in photonic systems, which have great potential for application in integrated nanophotonic devices.

Topological Corner States in Non-Unitary Coinless Discrete-Time Quantum Walks

The discrete-time quantum walk provides a versatile platform for exploring abundant topological phenomena due to its intrinsic spin-orbit coupling. In this work, we study the non-Hermitian second-order topology in a two-dimensional non-unitary coinless discrete-time quantum walk, which is realizable in the three-dimensional photonic waveguides. By adding the non-unitary gain-loss substep operators into the one-step operator of the coinless discrete-time quantum walk, we find the appearance of the four-degenerate zero-dimensional corner states at ReE = 0 when the gain-loss parameter of the system is larger than a critical value. This intriguing phenomenon originates from the nontrivial second-order topology of the system, which can be characterized by a second-order topological invariant of polarizations. Finally, we show that the exotic corner states can be observed experimentally through the probability distributions during the multistep non-unitary coinless discrete-time quantum walks. Our work potentially pave the way for exploring exotic non-Hermitian higher-order topological states of matter in coinless discrete-time quantum walks.

Higher-order valley vortices enabled by synchronized rotation in a photonic crystal

Synchronized rotation of unit cells in a periodic structure provides a novel design perspective for manipulation of band topology. We then design a two-dimensional version of higher-order topological insulator (HOTI) by such rotation in a triangular photonic lattice with C 3 symmetry. This HOTI supports the hallmark zero-dimensional corner states and, simultaneously, the one-dimensional edge states. We also find that our photonic corner states carry chiral orbital angular momenta locked by valleys, whose wave functions are featured by the phase vortex (singularity) positioned at the maximal Wyckoff points. Moreover, when excited by a fired source with various frequencies, the valley topological states of both one-dimensional edges and zero-dimensional corners emerge simultaneously. Extendable to higher or synthetic dimensions, our paper provides access to a chiral vortex platform for HOTI realizations in the terahertz photonic system.

Topological corner states in acoustic honeycomb structure

In recent years, a new type of topological insulator, termed higher-order topological insulator, has attracted tremendous research interest. Such exotic lower-dimensional topological boundary states have been extended and reproduced in classical systems, such as optics and acoustics. In this paper, a two-dimensional acoustic honeycomb structure with a triangle resonant cavity is numerically studied. Topological phase transition is induced by gradually adjusting the intracell and intercell coupling, and then the topological phase is used to construct a second-order topological insulator. The topological properties of second-order topological insulators can be characterized by using the quantized quadrupole moments. When quantized quadrupole <inline-formula><tex-math id="M4">\begin{document}$ {Q_{ij}} = 0 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20211848_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20211848_M4.png"/></alternatives></inline-formula>, the system is trivial, while <inline-formula><tex-math id="M5">\begin{document}$ {Q_{ij}} = 1/2 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20211848_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20211848_M5.png"/></alternatives></inline-formula>, the system is topologically nontrivial. We investigate the acoustical higher-order states of triangular and hexagonal structures, respectively. The gapped zero-dimensional corner states are observed in both structures, but the robustness properties of the corner states emerge only in the hexagonal structures but not in the triangular-shaped ones. The topological corner modes will offer a new way to robustly confine the sound in a compact acoustic system.

Hybrid-Order Topological Insulators in a Phononic Crystal.

Topological phases, including the conventional first-order and higher-order topological insulators and semimetals, have emerged as a thriving topic in the fields of condensed-matter physics and materials science. Usually, a topological insulator is characterized by a fixed order topological invariant and exhibits associated bulk-boundary correspondence. Here, we realize a new type of topological insulator in a bilayer phononic crystal, which hosts simultaneously the first-order and second-order topologies, referred to here as the hybrid-order topological insulator. The one-dimensional gapless helical edge states, and zero-dimensional corner states coexist in the same system. The new hybrid-order topological phase may produce novel applications in topological acoustic devices.

Octupole corner state in a three-dimensional topological circuit

Higher-order topological insulators (HOTIs) represent a new family of topological materials featuring quantized bulk polarizations and zero-dimensional corner states. In recent years, zero-dimensional corner states have been demonstrated in two-dimensional systems in the form of quadrupole modes or dipole modes. Due to the challenges in designing and constructing three-dimensional systems, octupole corner modes in 3D have not been observed. In this work, we experimentally investigate octupole topological phases in a three-dimensional electrical circuit, which can be viewed as a cubic lattice version of the Hofstadter model with a π-flux threading each plaquette. We experimentally observe in our higher-order topological circuit a 0D corner state manifested as a localized impedance peak. The observed corner state in the electrical circuit is induced by the octupole moment of the bulk circuit and is topologically protected by anticommuting spatial symmetries of the circuit lattice. Our work provides a platform for investigating higher-order topological effects in three-dimensional electrical circuits.

Realization of hierarchical topological transitions and high-Q-response corner states in second-order topological photonic crystals

Topological photonic crystals (PCs) provide a convenient method of electromagnetic wave processing. Recently, higher-order photonic topological insulators have been investigated as an intriguing topological phase beyond the usual bulk–edge correspondence. Previous studies of corner states in higher-order topological insulators mainly focus on topological multipole systems realized by introducing a negative coupling between constituent units or based on the two-dimensional Su–Schrieffer–Heeger model. In this work, we study a second-order topological insulator of square lattice that can host one-dimensional edge states and zero-dimensional corner states by means of breaking the symmetry of the structure. A corner mode with a high quality factor over 22 000 is formed by combining two types of such PCs with different topological phases. The significant topological phase transitions in the bulk and at the edges can be realized by simply changing the diameter of the constituent cylinders. Our results provide a topological way to guide and trap electromagnetic waves.

In-Plane Second-Order Topologically Protected States in Elastic Kagome Lattices

Second-order topological insulators, which exhibit capability of hosting topologically protected zero-dimensional corner states distinct from the well-studied topological edge states, unveil a horizon beyond the conventional bulk-edge correspondence. Motivated by recent experimental observation of Wannier-type second-order corner states in acoustic structures, we investigate numerically and demonstrate experimentally the in-plane edge and corner states in a mechanical kagome lattice. By manipulating simply lattice geometry and quantized characterization, we exploit that the emerging corner states are topologically robust against disorders. We further present a second-order topological insulator with multi-interfaces such that the mechanical energy can be localized in multiple locations, which provides the possibility of practical application in energy harvesting devices. The present study is the physical observation to extend the second-order topological insulator to in-plane elastic dynamics, and modal coupling of in-plane elasticity makes it more challenging to be measured experimentally.

Second-order topological insulator in a coinless discrete-time quantum walk

Higher-order topological insulators not only exhibit exotic bulk-boundary correspondence principle, but also have an important application in quantum computing. However, they have never been achieved in quantum walk. In this paper, we construct a two-dimensional coinless discrete-time quantum walk to simulate second-order topological insulator with zero-dimensional corner states. We show that both of the corner and edge states can be observed through the probability distribution of the walker after multi-step discrete-time quantum walks. Furthermore, we demonstrate the robustness of the topological corner states by introducing the static disorder. Finally, we propose a possible experimental implementation to realize this discrete-time quantum walk in a three-dimensional integrated photonic circuits. Our work offers a new route to explore exotic higher-order topological matters using discrete-time quantum walks.

Lasing at Multidimensional Topological States in a Two-Dimensional Photonic Crystal Structure

Recently, topologically engineered photonic structures have garnered significant attention as their eigenstates may offer a new insight on photon manipulation and an unconventional route for nanophotonic devices with unprecedented functionalities and robustness. Herein, we present lasing actions at all hierarchical eigenstates that can exist in a topologically designed single two-dimensional (2D) photonic crystal (PhC) platform: 2D bulk, one-dimensional edge, and zero-dimensional corner states. In particular, multiple topological eigenstates are generated in a hierarchical manner with no bulk multipole moment. The unit cell of the topological PhC structure is a tetramer composed of four identical air holes perforated into an InGaAsP multiple-quantum-well epilayer slab. A square area of a topologically nontrivial PhC structure is surrounded by a topologically trivial counterpart, resulting in multidimensional eigenstates of one bulk, four side edges, and four corners within and at the boundaries. Spatially resolved optical excitation spontaneously results in lasing actions at all nine hierarchical topological states. Our experimental findings may provide insight into the development of sophisticated next-generation nanophotonic devices and robust integration platforms.

Demonstration of a quantized acoustic octupole topological insulator

Recently introduced quantized multipole topological insulators (QMTIs) reveal new types of gapped boundary states, which themselves represent lower-dimensional topological phases and host symmetry protected zero-dimensional corner states. Inspired by these predictions, tremendous efforts have been devoted to the experimental observation of quantized quadrupole topological phase. However, due to stringent requirements of anti-commuting reflection symmetries, it is challenging to achieve higher-order quantized multipole moments, such as octupole moments, in a three-dimensional structure. Here, we overcome this challenge, and experimentally realize the acoustic analogue of a quantized octupole topological insulator using negatively coupled resonators. We confirm by first-principle studies that our design possesses a quantized octupole topological phase, and experimentally demonstrate spectroscopic evidence of a hierarchy of boundary modes, observing 3rd order topological corner states. Furthermore, we reveal topological phase transitions from higher- to lower-order multipole moments. Our work offers a pathway to explore higher-order topological states in 3D classical platforms.

Higher-Order Topological States in Surface-Wave Photonic Crystals.

Photonic topological states have revolutionized the understanding of the propagation and scattering of light. The recent discovery of higher‐order photonic topological insulators opens an emergent horizon for 0D topological corner states. However, the previous realizations of higher‐order topological insulators in electromagnetic‐wave systems suffer from either a limited operational frequency range due to the lumped components involved or a bulky structure with a large footprint, which are unfavorable for achieving compact photonic devices. To overcome these limitations, a planar surface‐wave photonic crystal realization of 2D higher‐order topological insulators is hereby demonstrated experimentally. The surface‐wave photonic crystals exhibit a very large bulk bandgap (a bandwidth of 28%) due to multiple Bragg scatterings and host 1D gapped edge states described by massive Dirac equations. The topology of those higher‐dimensional photonic bands leads to the emergence of in‐gap 0D corner states, which provide a route toward robust cavity modes for scalable compact photonic devices.

Second-order topological corner states with ultracold atoms carrying orbital angular momentum in optical lattices

We propose a realization of a two-dimensional higher-order topological insulator with ultracold atoms loaded into orbital angular momentum (OAM) states of an optical lattice. The symmetries of the OAM states induce relative phases in the tunneling amplitudes that allow to describe the system in terms of two decoupled lattice models. Each of these models displays one-dimensional edge states and zero-dimensional corner states that are correlated with the topological properties of the bulk. We show that the topologically non-trivial regime can be explored in a wide range of experimentally feasible values of the parameters of the physical system. Furthermore, we propose an alternative way to characterize the second-order topological corner states based on the computation of the Zak's phases of the bands of first-order edge states.

Nonlinear Second-Order Topological Insulators.

We demonstrate, both theoretically and experimentally, the concept of nonlinear second-order topological insulators, a class of bulk insulators with quantized Wannier centers and a bulk polarization directly controlled by the level of nonlinearity. We show that one-dimensional edge states and zero-dimensional corner states can be induced in a trivial crystal insulator made of evanescently coupled resonators with linear and nonlinear coupling coefficients, simply by tuning the intensity. This allows global external control over topological phase transitions and switching to a phase with nonzero bulk polarization, without requiring any structural or geometrical changes. We further show how these nonlinear effects enable dynamic tuning of the spectral properties and localization of the topological edge and corner states. Such self-induced second-order topological insulators, which can be found and implemented in a wide variety of physical platforms ranging from electronics to microwaves, acoustics, and optics, hold exciting promises for reconfigurable topological energy confinement, power harvesting, data storage, and spatial management of high-intensity fields.