Corner States Research Articles

Higher-Order Topological Peierls Insulator in a Two-Dimensional Atom-Cavity System.

In this work, we investigate a two-dimensional system of ultracold bosonic atoms inside an optical cavity, and show how photon-mediated interactions give rise to a plaquette-ordered bond pattern in the atomic ground state. The latter corresponds to a 2D Peierls transition, generalizing the spontaneous bond dimerization driven by phonon-electron interactions in the 1D Su-Schrieffer-Heeger (SSH) model. Here the bosonic nature of the atoms plays a crucial role to generate the phase, as similar generalizations with fermionic matter do not lead to a plaquette structure. Similar to the SSH model, we show how this pattern opens a nontrivial topological gap in 2D, resulting in a higher-order topological phase hosting corner states, that we characterize by means of a many-body topological invariant and through its entanglement structure. Finally, we demonstrate how this higher-order topological Peierls insulator can be readily prepared in atomic experiments through adiabatic protocols. Our work thus shows how atomic quantum simulators can be harnessed to investigate novel strongly correlated topological phenomena beyond those observed in natural materials.

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.

Higher-order obstructed atomic insulator phase in pentagonal monolayer PdSe2

Abstract We investigate a pentagonal monolayer of palladium diselenide, a stable two-dimensional system, as a material realization of a crystalline phase with nontrivial topological electronic properties. We find that its electronic structure involves an atomic obstructed insulator related to higher-order topology, which is a consequence of the selenium-selenium bond dimerization along with inversion and time-reversal symmetry). By means of first-principles calculations and the analysis of symmetry indicators and topological invariants, we also characterize the electronic corner states associated with the atomic obstruction and compute the corresponding corner charge for a finite geometry, which is found to be not quantized but still inversion-protected. Applying tensile strain to the finite geometry we verify the robustness of the corner states and also achieve a strain-controlled variation of the corner charge magnitude.

Quadrupole photonic topological corner states in generalized non-square lattices with translation symmetry

Quadrupole topological insulators have recently attracted great attention in the field of topological physics, while they are limited to square and hexagonal lattices. In this work, we theoretically show that nontrivial quadrupole topology can be obtained in generalized non-square lattice photonic crystals with translation symmetry, which are composed of parallelogram-shaped unit cells. The translation symmetry is described by the fractional linear combination of primary lattice vectors, leading to the quantization of fractional quadrupole moment in conjunction with an additional symmorphic symmetry. For parallelogramatic lattice with inversion symmetry, in particular, the quantization of the quadrupole moment is independent of the choice of primary lattice vectors, enabling cavity structures with arbitrary angles. For the change of structural parameters, quadrupole bandgaps undergo second-order topological phase transitions, accompanying with double band inversions. Nontrivial quadrupole phases are manifested by the appearance of disorder-immune in-gap corner states localized at the topological interfaces. Furthermore, the proposed parallelogramatic lattice photonic crystal has multiple quadrupole bandgaps for proper structural parameters, exhibiting multiband second-order topological corner states. The presented results will further extend the class of quadrupole topological photonic crystals and pave a broad way towards their practical applications due to improved design flexibility.

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.

Multi-dimensional wave manipulation with 3D mechanical higher-order topological insulators

High-order topological insulators (HOTIs) have emerged as a promising platform for controlling wave propagation and trapping due to their unique topological properties. In this study, we investigate the multi-dimensional manipulation of elastic waves in two types of 3D HOTIs induced by the Dirac hierarchy. These two types of 3D HOTIs are characterized by molecule-zigzag hinge configurations and armchair hinge configurations, both derived from a 3D stack-layered structure featuring an eightfold-degenerate Dirac cone (DC). By gradually introducing alternating interlayer couplings in the vertical direction, varying the intercell and intracell couplings in the horizontal plane, and breaking mirror symmetry, we lift the eightfold-degenerate DC and construct the 3D mechanical HOTI with molecule-zigzag hinge configurations. Remarkably, the realization of 3D mechanical HOTIs with armchair hinge configurations does not require breaking mirror symmetry. Both of these 3D mechanical HOTIs exhibit intriguing hierarchical topological states, including surface states, hinge states, and corner states. Our findings provide new insights into the manipulation of elastic waves and offer potential applications in robust waveguiding, vibration signal processing and energy harvesting.

Waveguide–Cavity Coupling System Based on Topological Edge States and Corner States in Kagome Photonic Crystals

AbstractIn recent years, the topological edge states (TESs) and topological corner states (TCSs) based on high‐order topological photonic insulators have been researched and applied in various fields. Among them, the combination of topological photonics and waveguide–cavity coupling system has attracted much attention of researchers. In this paper, TESs and TCSs based on Kagome lattice photonic crystals (PCs) with rectangular dielectric columns are obtained. In addition, a topological waveguide–cavity coupling system is proposed, realizing the energy coupling between the TES waveguide and the TCS cavity. To further improve the performance of the system, a sandwich‐like waveguide and an extended waveguide are used to replace the original TES waveguide. Interestingly, the proposed extended waveguide–cavity coupling system cannot only achieve the excitation of two TCSs at different frequencies, but also has stronger electric field intensity, narrower resonance bandwidth, and higher quality factor than the other structures. This work provides a new path to design high‐performance micro‐nanointegrated optical devices such as filters, lasers, and resonators.

Reconfigurable higher-order topological electromechanical metamaterial

Higher-order topological insulators (HOTIs) exhibit hierarchical and multidimensional features of topological states, expanding the applications of topological insulators (TIs). Reconfigurable mechanical metamaterial-based HOTIs for the manipulation of elastic waves are highly desirable. Here, we propose a reconfigurable electromechanical metamaterial with first-order topological edge states and second-order topological corner states for the control of Lamb waves. The metamaterial consists of an aluminum layer sandwiched between two piezoelectric layers shunted through negative capacitance circuits. The topology of the metamaterial can be reconfigured by switching on or off the connected circuits, and the metamaterial is also tunable by changing the electrical parameters. Hierarchical and multidimensional topological waveguides are constructed and their reconfigurability is demonstrated. The metamaterial is continuous and thus facilitates waveguide miniaturization, such as on-chip device design.

Higher‐Order States in a Non‐Hermitian 3D Topological Photonic Crystal

AbstractHigher‐order topological insulators hosting topological modes at hinges and corners provide a new avenue for disorder‐immune light transport, which prospects great potential in applications of integrated photonics devices. However, the active control of topological modes in 3D higher‐order topological insulators is not realized yet, and its construction is too complicated and huge to analyze. Here, a method is proposed to construct effective Hamiltonians for higher‐order topological insulators, which provides a correct physical picture of the states and the corresponding energies at the domain walls. A non‐Hermitian 3D honeycomb lattice is constructed, which can generate arbitrary‐located, arbitrary‐shaped, and robust topological hinge states propagating at gain‐loss domain walls. The 3D honeycomb lattice can also appear as non‐Hermitian third‐order TIs exhibiting corner states, which can be dynamically controlled and show new topologically protected confinement rules. This work expands the understanding of the topological properties in non‐Hermitian systems and enables the dynamic control of topological states of different dimensions.

Adjustable topological corner states in terahertz valley photonic crystals

Adjustable topological corner states in terahertz valley photonic crystals

Nonlinear topological laser on the non-Hermitian Haldane model with higher-order corner states.

The non-Hermitian skin effect (NHSE) on the non-Hermitian Haldane model with gain and loss on the honeycomb lattice with the outline of a triangle is discussed. The NHSE only occurs on the edge of the lattice, transforming the edge modes into the higher-order corner modes. The NHSE can also occur on a lattice with only loss, which can be treated as a lattice with gain and loss as well as a global loss added to it. When the saturated gain is added to the three corner sites of the dissipative lattice, a single-mode laser system is obtained. When any one site is stimulated initially, the system will reach a saturated state depending on the distribution of the corner modes, and the stable laser light is emitted by sites at the corners.

Identifying topological corner states in two-dimensional metal-organic frameworks

Due to the diversity of molecular building blocks, the two-dimensional (2D) metal-organic frameworks (MOFs) are ideal platforms to realize exotic lattice models in condensed matter theory. In this work, we demonstrate the universal existence of topological corner states in 2D MOFs with a star lattice configuration, and confirm the intriguing higher-order nontrivial topology in the energy window between two Kagome-bands, or between Dirac-band and four-band. Furthermore, combining first-principles calculations and scanning tunneling microscopy measurements, the unique topological corner state is directly identified in monolayer Ni3(HITP)2 (HITP = 2,3,6,7,10,11-hexaiminotriphenylene) grown on the Au(111) substrate. Our results not only illustrate the first organic topological state in the experiments, but also offer an exciting opportunity to study higher-order topology in 2D MOFs with the large insulating band gap.

Growth and characterization of Bi(110)/CrTe2 heterostructures: Exploring interplay between magnetism and topology

The interplay between topology and magnetism is vital for realizing exotic quantum phenomena, significant examples including quantum anomalous Hall effect, axion insulators, and high-order topological states. These states host great potential for future applications in high-speed and low-consumption electronic devices. Despite being extensively investigated, practical platforms are still scarce. In this work, with molecular beam epitaxy (MBE), we provide the first experimental report on high-quality Bi(110)/CrTe2 magnetic heterostructure. By employing in-situ high-resolution scanning tunneling microscopy, we are able to examine the interaction between magnetism and topology. There is a potential edge state at an energy level above the Fermi level, but no edge states observed near the Fermi level The absence of high-order topological corner states near E F highlights the importance of lattice matching and interface engineering in designing high-order topological states. Our study provides key insights into the interplay between two-dimensional magnetic and topological materials and offers an important dimension for engineering magnetic topological states.

Observation of Topologically Distinct Corner States in “Bearded” Photonic Kagome Lattices

AbstractKagome lattices represent an archetype of intriguing physics, attracting a great deal of interest in different branches of natural sciences, recently in the context of topological crystalline insulators. Here, two distinct classes of corner states in breathing Kagome lattices (BKLs) with “bearded” edge truncation are demonstrated, unveiling their topological origin. The in‐phase corner states are found to exist only in the topologically nontrivial regime, characterized by a nonzero bulk polarization. In contrast, the out‐of‐phase corner states appear in both topologically trivial and nontrivial regimes, either as bound states in the continuum or as in‐gap states depending on the lattice dimerization conditions. Furthermore, the out‐of‐phase corner states are highly localized, akin to flat‐band compact localized states, and they manifest both real‐ and momentum‐space topology. Experimentally, both types of corner states are observed in laser‐written photonic bearded‐edge BKLs, corroborated by numerical simulations. The results not only deepen the current understanding of topological corner modes in BKLs but also provide new insight into their physical origins, which may be applied to other topological BKL platforms beyond optics.

Quadrupole topological phases and filling anomaly in all-dielectric Lieb lattice photonic crystals.

While higher-order photonic topological corner states typically are created in systems with nontrivial bulk dipole polarization, they could also be created in systems with vanishing dipole polarization but with nontrivial quadrupole topology, which though is less explored. In this work, we show that simple all-dielectric photonic crystals in the Lieb lattice can host a topologically nontrivial quadrupole bandgap. Through a combination of symmetry analysis of the eigenmodes and explicit calculations of the Wannier bands and their polarization using the Wilson loop method, we demonstrate that the Lieb photonic crystals can have a bandgap with vanishing dipole polarization but with nontrivial quadrupole topology. The nontrivial bulk quadrupole moment could result in edge-localized polarization and topological corner states in systems with open edges. Interestingly, the indices of the corner states show an unusual "3+1" pattern compared to previously known "2+2" pattern, and this new pattern leads to unusual filling anomaly when the corner states are filled. Our work could not only deepen our understanding about quadrupole topology in simple all-dielectric photonic crystals but could also offer new opportunities for practical applications in integrated photonic devices.

Tuning corner states in proximitized second-order topological insulators with bulk-boundary obstruction

Tuning corner states in proximitized second-order topological insulators with bulk-boundary obstruction

Multi-topological state via the Brillouin zone overlap for nonlinear frequency conversion.

Multiband topological edge states (TESs) or topological corner states (TCSs) in photonic crystals provide effective ways to manipulate the nonlinear frequency conversions. However, the deliberate design and the limited number of multibands lead to the difficulty of experimental realization of the topological nonlinear frequency conversion or higher harmonic generation. Here, we propose an effective method to achieve multiple TESs and TCSs by combining the Brillouin zones of multiple different systems. It is shown that the spectra of the subsystems disperse into different energy levels due to the inter-system hopping. Based on this approach, we construct a topological photonic crystal based on the Brillouin zone overlapped SSH model, which enables the overlapped TCSs to participate in nonlinear frequency conversion. Our scheme can provide a significant way to realize the topological nonlinear frequency conversion with double resonances or multiple resonances.

Topological phase transitions and flat bands on an islamic lattice

We construct an islamic lattice by considering the nearest-neighbor (NN) hoppings with staggered magnetic fluxes and the next-NN hoppings. This model supports abundant quantum phases for various values of filling fractions. At filling, Chern insulator (CI) phases with Chern numbers and a zero-Chern-number topological insulator (ZCNTI) phase exist. At filling, several CI phases with Chern numbers and the ZCNTI phase are obtained. For the filling fraction 3/4, CI phases with Chern numbers and two ZCNTI phase areas appear. Interestingly, these ZCNTI phases host both robust corner states and gapless edge states which can be characterized by the quantized polarization and quadrupole moment. We further find that staggered magnetic fluxes can give rise to the ZCNTI state at and fillings. Phase diagrams for filling fractions , , and are presented as well. In addition, flat bands are obtained for various filling fractions by tuning the hopping parameters. At 1/8 filling, a best topological flat band (TFB) with flatness ratio about 12 appears. Several trivial flat bands but with total Chern number emerge in this model and exactly flat band is found at 3/8 filling. We further investigate fractional Chern insulate state when hard-core bosons fill into this TFB model.