Corner States Research Articles

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.

Metal-organic framework as high-order topological insulator with protected corner modes

The capacity to design and manipulate the chemical characteristics of metal-organic frameworks (MOFs) presents a promising approach for developing functional materials, particularly with regard to their topological properties. Here, based on first-principles calculations, we predict Ni3(C6S6)2 as a two-dimensional (2D) higher-order topological insulator (HOTI) with protected corner modes, which is the first material example of such topological phase in MOFs reported so far. The non-trivial topology of the material is identified by two different classes of bulk topological invariants: the real Chen number induced by spacetime inversion symmetry, and the fractional corner charge induced by rotational symmetry. The presence of corner states can serve as a diagnostic of its nontrivial topology. Moreover, the HOTI natural is demonstrated to be robust against symmetry perturbations and spin-orbit coupling. Inspired by the topological properties of Ni3(C6S6)2, we can explore regulating organic ligands or metal atoms in metal-organic frameworks to realize other 2D HOTIs, thereby expanding the family of 2D HOTIs. Our findings not only broaden the understanding of physical properties of metal-organic materials, but also extend the study of high-order topological insulators to organic platforms.

Higher-order topological heat conduction on a lattice for detection of corner states.

A heat conduction equationon a lattice composed of nodes and bonds is formulated assuming the Fourier law and the energy conservation law. Based on this equation, we propose a higher-order topological heat conduction model on the breathing kagome lattice. We show that the temperature measurement at a corner node can detect the corner state which causes rapid heat conduction toward the heat bath, and that several-nodes measurement can determine the precise energy of the corner states.

Customizable multiband second-order sonic topological insulators via inverse design

The second-order sonic topological insulators (SSTIs) with topologically protected corner states offer promising opportunities for developing novel acoustic devices. However, most of the current SSTIs are designed via trial-and-error and are only able to host the second-order topological phases within a single bandgap, leaving the topic of second-order topological phases within multiple bandgaps rarely studied. Here, we exploit a topology optimization method to customize and optimize multiband SSTIs. To begin with, we create multiple dual-band SSTIs with customizable dual bandgaps for hosting dual-band corner states. On that basis, a three-band SSTI with three bandgaps is constructed for hosting three-band corner states. Experimental validation is performed to prove the existence of the three-band corner states. This study ushers in a route for customizing high-performance multiband SSTIs, and the designed multiband SSTIs have potential for designing robust multiband acoustic devices.

Magnetic Second‐Order Topological Insulator: An Experimentally Feasible 2D CrSiTe3

Abstract2D second‐order topological insulators (SOTIs) have sparked significant interest, but currently, the proposed realistic 2D materials for SOTIs are limited to nonmagnetic systems. In this study, for the first time, a single layer of chalcogenide CrSiTe3—an experimentally realized transition metal trichalcogenide is proposed with a layer structure—as a 2D ferromagnetic (FM) SOTI. Based on first‐principles calculations, this study confirms that the CrSiTe3 monolayer exhibits a nontrivial gapped bulk state in the spin‐up channel and a trivial gapped bulk state in the spin‐down channel. Based on the higher‐order bulk–boundary correspondence, it demonstrates that the CrSiTe3 monolayer exhibits topologically protected corner states with a quantized fractional charge () in the spin‐up channel. Notably, unlike previous nonmagnetic examples, the topological corner states of the CrSiTe3 monolayer are spin‐polarized and pinned at the corners of the sample in real space. Furthermore, the CrSiTe3 monolayer retains SOTI features when the spin–orbit coupling (SOC) is considered, as evidenced by the corner charge and corner states distribution. Finally, by applying biaxial strain and hole doping, this study transforms the magnetic insulating bulk states into spin‐gapless semiconducting and half‐metallic bulk states, respectively. Importantly, the topological corner states persist in the spin‐up channel under these conditions.

Magnetic Second-Order Topological Insulators in 2H-Transition Metal Dichalcogenides.

The transition metal dichalcogenides, 2H-VX2 (X = S, Se, Te), are identified astwo-dimensional second-order topological insulator (SOTI) with a ferromagnetic ground state by first-principles calculations. The 2H-VX2 (X = S, Se, Te) materials have a nontrivial band gap in two spin channels is found and exhibit topologically protected corner states with spin-polarization. These corner states only accommodate the quantized fractional charge (e/3). And the charge is bound at the corners of the nanodisk geometry 2H-VX2 (X = S, Se, Te) in real space. The corner states are robust against symmetry-breaking perturbations, which makes them more easily detectable in experiments. Further, it is demonstrated that the SOTI properties of 2H-VX2 (X = S, Se, Te) materials can be maintained in the presence of spin-orbit coupling and are stable against magnetization. Overall, the results reveal 2H-VX2 (X = S, Se, Te) as an ideal platform for the exploration of magnetic SOTI and suggest its great potential in experimental detection.

Linear and nonlinear edge and corner states in graphenelike moiré lattices

Linear and nonlinear edge and corner states in graphenelike moiré lattices

Realization of the square-root higher-order topology in decorated Su–Schrieffer–Heeger electric circuits

In this work, we experimentally implemented square-root higher-order topological insulators (HOTIs) in topological electric circuits with decorated Su–Schrieffer–Heeger (SSH) lattice. Due to the square-root operation, the corner states of the square-root HOTIs typically appear on the finite energy. Through theory and experiments, the corner states in both the parent and square-root HOTIs models, as well as their spectrum connection, were demonstrated. Our work not only experimentally substantiates the existence of square-root higher-order topology in decorated SSH lattice model, from the perspective of electric circuits, but also extends the possibilities for observing various topological phases in acoustic and mechanical systems.

Combining sensitivity and robustness: EIT-like characteristic in a 2D topological photonic crystal.

The study of topological photonics has gained significant attention due to its potential application for robust and efficient light manipulation. In this work, we theoretically investigate a two-dimensional photonics crystal that exhibits a topological edge state (TES) and a topological corner state (TCS). Furthermore, we also achieve a coupling between a topological corner state and a trivial cavity (TC), resulting in a phenomenon similar to the electromagnetically induced transparency (EIT) effect. To verify the stability of the EIT-like effect, disorders around TES and TCS are introduced, and the theoretical results show that this structure is immune to the disorders. The achievement of the coupling between topological states can have potential applications in the areas of waveguiding, sensing, and logic gates. It is hoped that this work will contribute to the ongoing efforts in the exploration and utilization of topological photonics.

Fractional mode charge in Cn-symmetric higher-order topological crystalline insulators

The measurement of topological insulators is usually achieved by measuring spectra, which is inadequate in higher-order topological insulators because the separation of bandgaps may be blurred, such as body boundaries and corner state hybridization, and the above measurement requires that the energy of the topological mode must be located in the bulk bandgap to be measured. Recent experiments have shown a strong relationship between the fractional mode charge and the location of the Wannier center in the crystal body. The fractional quantization of the corner charge originates from the filling anomaly, and the fractional mode charge can be calculated from experimental measurements of the local density of states to describe the topological state of the system, which has been demonstrated in optical systems. We have achieved a similar effect in acoustics by utilizing a coupled acoustic-cavity system featuring C3, C4, and C6 symmetries for simulation. Furthermore, we extended our study to 3D third-order topological crystalline insulators to validate the fractional-corner charge approach. The simulation and theoretical results demonstrate that the fractional mode charge can clearly identify the high-order topology, even without gap states.

Selective activation of topological valley corner states in C3-symmetric photonic crystals

Higher-order topological insulators have drawn great research attention in nanophotonics due to their ability to both support robust edge states and lower dimensional corner states. In this work, we present a theoretical proposal for achieving topologically switchable and valley-selective corner states based on two-dimensional C3-symmetric photonic crystals (PCs), with breaking of inversion symmetry. Through the concatenation of two valley PCs with contrasting topological indices, we demonstrate the emergence of two types of valley-locked chiral topological edge states resulting from the valley–valley interaction. More importantly, we find that the system exhibits two distinct types of corner states, characterized by strong robustness and high localization, when the PCs are spliced at a 60° angle. However, the corner states are absent when the splicing angle is set as 120°. According to the theoretical analysis, the selective activation of topological valley corner states is related to the sign flip of valley Chern number at the corner. Based on this feature, we further propose a topological photonic switching device, in which the corner can be lighted up or off selectively. By combining the benefits of higher-order topology and valley degree of freedom, our work provides an efficient and flexible method for light manipulation.

Competition of non-Hermitian skin effect and topological localization of corner states observed in circuits

Competition of non-Hermitian skin effect and topological localization of corner states observed in circuits

Corner states, energy spectrum and light absorption of [formula omitted]graphyne quantum dots

Recently, γ−graphyne has been recognized as a two-dimensional high-order topological insulator with typical corner states. In this work, we examine the condition for the existence of corner states in a hexagonal quantum dot made of γ−graphyne (GYQD). Based on a single-orbital tight-binding model, we find that corner states appear only when there are partially-bonding carbon atoms located at the corners. This indicates a potential instability of corner states. As the size of GYQDs increases, mid-gap states in the energy spectrum are either prohibited or exhibited as corner-only or edge-only states. Additionally, we find that the light absorption of GYQDs has a relatively weak polarization anisotropy. The presence of corner states leads to the emergence of sub-gap absorption and enhancement of above-gap absorption.

Coexistence of Fano and electromagnetically induced transparency resonance line shapes in photonic topological insulators

Due to the bending resistance and defect immunity of the topological optical structure, topological photonics is extended to researching Fano line shapes or electromagnetically induced transparency-like (EIT-like) line shapes. In this paper, two-dimensional photonic crystal with Kekule´ distortion is designed to realize quantum spin Hall effect and quantum valley Hall effect simultaneously. The stable topological corner states and edge states are obtained. Corner states as dark modes interfere with edge states as bright mode to induce Fano or EIT-like line shapes. As far as we know, we first realize the coexistence of Fano and EIT-like line shapes based on the coexistence of quantum spin Hall effect and quantum valley Hall effect. Our results provide a new way to realize multi-channel sensing technology.

Square-root higher-order topological insulators in a photonic decorated SSH lattice.

Recently, there has been a surge of interest in square-root higher-order topological insulators (HOTIs) due to their unique topological properties inherited from their squared Hamiltonian. Different from conventional HOTIs, square-root HOTIs support paired corner states that exist in different bandgaps. In this work, we experimentally establish a series of two-dimensional photonic decorated Su-Schrieffer-Heeger (SSH) lattices by using the femtosecond-laser writing technique and thereby directly observe paired topological corner states. Interestingly, the higher-order topological properties of such square-root HOTIs are inherited from the parent Hamiltonian, which contains the celebrated 2D SSH lattice. The dynamic evolution of square-root corner states indicates that they exist in different bandgaps. This work not only provides a new platform to study higher-order topology in optics, it also brings about new possibilities for future studies of other novel HOTIs.

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.

Edge and corner states in two-dimensional finite phononic crystals: Simulation and experimental study

We investigate the mechanical vibration transmission in two-dimensional infinite and finite phononic crystals (PCs). The infinite PC consists of a periodic structure formed by square plates connected to the center of each of their nearest neighbors through a tiny beam. Numerical simulations using finite elements show a wide full bandgap for frequencies between 27 kHz and 32 kHz, approximately. Acoustic resonant spectroscopy was used to measure the PC frequency spectra for the different vibrations, using a finite PC consisting of 8 by 8 cells, which was designed with the same configuration as the infinite one. Experimental results corroborate the existence of a full complete bandgap predicted by the numerical method. However, the width was significantly reduced due to the appearance of edge and corner states. The border states were obtained numerically using a supercell. The measured wave amplitudes and the simulated ones present a great similarity. Some states appear located at the corners of the finite PC demonstrating that zero-dimensional states can also appear in two-dimensional phononic crystals.

Coexistence of magnetic and phononic second-order topological phases in two-dimensional NiZrCl6

Second-order topological phases (SOTPs) in two-dimensional (2D) magnetic and phononic systems are rarely reported. In this Letter, using first-principles calculations, we propose that the NiZrCl6 monolayer with space group P312 (No. 149) is a 2D ferromagnetic material with rich SOTPs: (i) magnetic SOTPs can be found in the band structures of both spin channels in NiZrCl6. NiZrCl6 hosts topologically protected corner states that have a quantized fractional charge (e/3) and are spin-polarized and pinned at the corners of the sample in real space. The SOTP nature in the NiZrCl6 monolayer is resistant to the spin–orbit coupling effect. (ii) Phononic SOTPs can be found in the phonon curves of NiZrCl6. The corner vibrational modes appear inside the frequency gap around 7.98 THz of the NiZrCl6 monolayer, and the secondary topological index can verify the nontrivial phase. The proposed 2D NiZrCl6 material can be a starting point for exploring higher-order topological phases in 2D magnetic and phononic systems.

Multiband, valley-selective corner states in phononic crystals

Multiband, valley-selective corner states in phononic crystals

Magnetic high-order topological insulator in 2D layered CrOCl

High-order topological insulators (HOTIs) have garnered significant interest in recent years due to their unique properties, but their material realization has thus far been limited to those protected by Cn (n = 3,4,6) rotation symmetry and to nonmagnetic crystals. In this study, we propose a HOTI in a recently synthesized two-dimensional (2D) magnetic material, CrOCl. Our findings reveal that the material's ground state, which has ferromagnetic ordering, features corner states carrying a quantized fractional charge. These states are enabled by a twofold rotation symmetry (C2z) and are robust against spin-orbital coupling (SOC). The corner states in CrOCl are distinct from those in nonmagnetic examples and even in magnetic examples, as they can achieve 100% spin polarization in the presence or absence of SOC. Remarkably, the corner states in CrOCl are also robust against layer stacking, symmetry breaking, and even magnetization. Furthermore, under a vertical electric field, the material undergoes a phase transition from HOTI to “hidden high-order topological insulator” phase, with the corner states submerged into edge or bulk states. Our study not only highlights the HOTI properties of 2D CrOCl, but also promotes it as a concrete material platform for exploring the intriguing physics of magnetic HOTIs. We also discuss possible experimental detections and applications of this material, which may pave the way for new advancements in materials sciences and condensed-matter physics.