Corner Modes Research Articles

Phononic Weyl pair, phononic Weyl complex, phononic real Chern insulator state, and phononic corner modes in 2D Kekulé-order graphene

The conceptual framework of topological states has recently been extended to bosonic systems, particularly phononic systems. In this work, we chose the recently experimentally prepared two-dimensional (2D) Kekulé-order graphene as a target to propose the coexistence of gapless and gapped topological phonon states in its phonon curves. This is the first work to investigate rich gapped and gapless topological phonon states in experimentally feasible 2D materials. For the gapped topological phonons, 2D Kekulé-order graphene hosts phononic real Chern insulator states, i.e., second-order topological states, and corner vibrational modes inside frequency gaps at 27.96 and 37.065 THz. For the gapless topological phonons, 2D Kekulé-order graphene hosts a phononic Weyl pair [comprising two linear Weyl points (LWPs)] and a phononic Weyl complex [comprising one quadratic nodal point (QNP) and two LWPs] around 7.54 and 47.3 THz (39.2 THz), respectively. Moreover, the difference between the phononic Weyl pair and the phononic Weyl complex was investigated in detail. Our study not only promotes 2D Kekulé-order graphene as a concrete material platform for exploring the intriguing physics of phononic second-order topology but also proposes the coexistence of different categories of Weyl phonons, i.e., a Weyl complex that comprises two LWPs and one QNP, in two dimensions. Our work paves the way for new advancements in topological phononics comprising gapless and gapped topological phonons.

Identifying s-wave pairing symmetry in single-layer FeSe from topologically trivial edge states

Determining the pairing symmetry of single-layer FeSe on SrTiO3 is the key to understanding the enhanced pairing mechanism. It also guides the search for superconductors with high transition temperatures. Despite considerable efforts, it remains controversial whether the symmetry is the sign-preserving s- or the sign-changing s±-wave. Here, we investigate the pairing symmetry of single-layer FeSe from a topological point of view. Using low-temperature scanning tunneling microscopy/spectroscopy, we systematically characterize the superconducting states at edges and corners of single-layer FeSe. The tunneling spectra collected at edges and corners show a full energy gap and a substantial dip, respectively, suggesting the absence of topologically non-trivial edge and corner modes. According to our theoretical calculations, these spectroscopic features can be considered as strong evidence for the sign-preserving s-wave pairing in single-layer FeSe.

Majorana Corner Modes and Flat-Band Majorana Edge Modes in Superconductor/Topological-Insulator/Superconductor Junctions

Recently, superconductors with higher-order topology have stimulated extensive attention and research interest. Higher-order topological superconductors exhibit unconventional bulk-boundary correspondence, thus allow exotic lower-dimensional boundary modes, such as Majorana corner and hinge modes. However, higher-order topological superconductivity has yet to be found in naturally occurring materials. We investigate higher-order topology in a two-dimensional Josephson junction comprised of two s-wave superconductors separated by a topological insulator thin film. We find that zero-energy Majorana corner modes, a boundary fingerprint of higher-order topological superconductivity, can be achieved by applying magnetic field. When an in-plane Zeeman field is applied to the system, two corner modes appear in the superconducting junction. Furthermore, we also discover a two-dimensional nodal superconducting phase which supports flat-band Majorana edge modes connecting the bulk nodes. Importantly, we demonstrate that zero-energy Majorana corner modes are stable when increasing the thickness of topological insulator thin film.

Mirror symmetry protected higher-order topological zero-frequency boundary and corner modes in Maxwell lattices

Mirror symmetry protected higher-order topological zero-frequency boundary and corner modes in Maxwell lattices

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 topology in a Creutz ladder

A Creutz ladder, is a quasi one dimensional system hosting robust topological phases with localized edge modes protected by different symmetries such as inversion, chiral and particle-hole symmetries. Non-trivial topology is observed in a large region of the parameter space defined by the horizontal, diagonal and vertical hopping amplitudes and a transverse magnetic flux that threads through the ladder. In this work, we investigate higher order topology in a two dimensional extrapolated version of the Creutz ladder. To explore the topological phases, we consider two different configurations, namely a torus (periodic boundary) and a ribbon (open boundary) to look for hints of gap closing phase transitions. We also associate suitable topological invariants to characterize the non-trivial sectors. Further, we find that the resultant phase diagram hosts two different topological phases, one where the higher order topological excitations are realized in the form of robust corner modes, along with (usual) first order excitations demonstrated via the presence of edge modes in a finite lattice, for the other.

Perfectly localized Majorana corner modes in fermionic lattices

Perfectly localized Majorana corner modes in fermionic lattices

Effective impurity behavior emergent from non-Hermitian proximity effect

Non-Hermitian boundaries commonly take place in many open quantum systems locally coupled to a surrounding environment. Here, we reveal a type of non-Hermitian effect induced by non-Hermitian boundaries, the non-Hermitian proximity effect (NHPE), which describes the penetration of non-Hermiticity from the boundary into the bulk. For gapped quantum systems, the NHPE generates in-gap states with imaginary eigenenergies, termed “imaginary in-gap states". The imaginary in-gap states are localized at the system boundary and decay into the bulk, analogous to the behaviors of the conventional impurity states. However, in contrast to impurity states, the imaginary in-gap states exhibit distinct dynamical behaviors under time-evolution. Moreover, they are physically manifested as corner modes under open boundaries, as a combined result of the non-Hermitian skin effect (NHSE) and NHPE. These results not only uncover implicit similarities between quantum systems with non-Hermitian boundaries and impurity physics, but also point to intriguing non-Hermitian phenomena broadly relevant to open quantum systems.

Boundary-obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field

In this work, we show that a buckled honeycomb lattice can host a boundary-obstructed topological superconductor (BOTS) in the presence of f-wave spin-triplet pairing (fSTP). The underlying buckled structure allows for the manipulation of both chemical potential and sublattice potential using a double gate setup. Although a finite sublattice potential can stabilize the fSTP with a possible higher-order band topology, because it also breaks the relevant symmetry, the stability of the corner modes is not guaranteed. Here we show that the fSTP on the honeycomb lattice gives BOTS under nonzero sublattice potential, thus the corner modes can survive as long as the boundary is gapped. Also, by examining the large sublattice potential limit where the honeycomb lattice can be decomposed into two triangular lattices, we show that the boundary modes in the normal state are the quintessential ingredient leading to the BOTS. Thus the effective boundary Hamiltonian becomes nothing but the Hamiltonian for Kitaev chains, which eventually gives the corner modes of the BOTS.

High-temperature Majorana corner modes in a d+id′ superconductor heterostructure: Application to twisted bilayer cuprate superconductors

The realization of Majorana corner modes generally requires unconventional superconducting pairing or $s$-wave pairing. However, the bulk nodes in unconventional superconductors and the low $T_c$ of $s$-wave superconductors are not conducive to the experimental observation of Majorana corner modes. Here we show the emergence of a Majorana corner mode at each corner of a two-dimensional topological insulator in proximity to a $d+id'$ pairing superconductor, such as heavily doped graphene or especially a twisted bilayer of a cuprate superconductor, e.g., Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, which has recently been proposed as a fully gapped chiral $d_{x^2-y^2}+id_{xy}$ superconductor with $T_c$ close to its native 90 K, and an in-plane magnetic field. By numerical calculation and intuitive edge theory, we find that the interplay of the proximity-induced pairing and Zeeman field can introduce opposite Dirac masses on adjacent edges of the topological insulator, which creates one zero-energy Majorana mode at each corner. Our scheme offers a feasible route to achieve and explore Majorana corner modes in a high-temperature platform without bulk superconductor nodes.

Spin-dependent high-order topological states and corner modes in a monolayer FeSe/GdClO heterostructure

We propose that a spin-dependent second-order topological insulator can be realized in monolayer FeSe/GdClO heterostructure, in which substrate GdClO helps to stabilize and enhance the antiferromagnetic order in FeSe. The second-order topological insulator is free from spin-orbit coupling and in-plane magnetic field. We also find that there exist two types of distinct corner modes residing in intersections of two ferromagnetic edges and two antiferromagnetic edges, respectively. The underlying physics for ferromagnetic corner mode follows a sublattice-chirality-kink picture. More interestingly, ferromagnetic corner mode shows spin-dependent property, which is also robust against spin-orbit coupling. Unexpectedly, antiferromagnetic corner mode can be taken as a typical emergent and hierarchical phenomenon from an array of ferromagnetic corner modes. Remarkably, antiferromagnetic corner modes violate general kink picture and can be understood as bound states of a one-dimensional Schrodinger equation under a connected potential well. Our findings not only provide a promising second-order topological insulator in electronic materials, but uncover some new properties of corner modes in high-order topological insulator.

Observation of bulk quadrupole in topological heat transport

The quantized bulk quadrupole moment has so far revealed a non-trivial boundary state with lower-dimensional topological edge states and in-gap zero-dimensional corner modes. In contrast to photonic implementations, state-of-the-art strategies for topological thermal metamaterials struggle to achieve such higher-order hierarchical features. This is due to the absence of quantized bulk quadrupole moments in thermal diffusion fundamentally prohibiting possible band topology expansions. Here, we report a recipe for generating quantized bulk quadrupole moments in fluid heat transport and observe the quadrupole topological phases in non-Hermitian thermal systems. Our experiments show that both the real- and imaginary-valued bands exhibit the hierarchical features of bulk, gapped edge and in-gap corner states—in stark contrast to the higher-order states observed only on real-valued bands in classical wave fields. Our findings open up unique possibilities for diffusive metamaterial engineering and establish a playground for multipolar topological physics.

Higher order topological superconductivity in magnet-superconductor hybrid systems

Quantum engineering of topological superconductors and of the ensuing Majorana zero modes might hold the key for realizing topological quantum computing and topology-based devices. Magnet-superconductor hybrid (MSH) systems have proven to be experimentally versatile platforms for the creation of topological superconductivity by custom-designing the complex structure of their magnetic layer. Here, we demonstrate that higher order topological superconductivity (HOTSC) can be realized in two-dimensional MSH systems by using stacked magnetic structures. We show that the sensitivity of the HOTSC to the particular magnetic stacking opens an intriguing ability to tune the system between trivial and topological phases using atomic manipulation techniques. We propose that the realization of HOTSC in MSH systems, and in particular the existence of the characteristic Majorana corner modes, allows for the implementation of a measurement-based protocols for topological quantum computing.

Multiple higher-order topological phases with even and odd pairs of zero-energy corner modes in a C3symmetry broken model

Here we study emergent higher-order topological (HOTI) phases in the extended Haldane model without C 3 symmetry. For the inversion symmetric case, the QSHI and QAHI phases can embed the HOTI phases while the remaining QASHI phase does not yield any HOTI phases. Remarkably, four-fold degeneracy of zero-energy corner states can be reduced to two-fold under the application (withdrawal) of sub-lattice mass (Zeeman field) term. The sub-lattice mass and Zeeman field terms compete with each other to pin down the two mid-gap states at zero-energy. Interestingly, the bulk polarization can topologically characterize the second-order topological insulator phase with the mid-gap corner modes irrespective of their energies as long as inversion symmetry is preserved. Our study indicates that a hybrid symmetry can in principle protect the second-order topological insulator phases, however, spin-spectrum gap has to be essentially finite there.

Observation of Rotation-Induced Light Localization in Waveguide Arrays

We study both experimentally and theoretically the propagation of light in the fs-laser written rotating square waveguide arrays and present the first experimental evidence of light localization induced by the rotation of periodic structure in the direction of light propagation. Such linear light localization occurs either in the corners of truncated square array, where it results from the interplay between the centrifugal effect and total internal reflection at the borders of truncated array, or in the center of array, where rotation creates effective attractive optical potential. The degree of localization of linear bulk and corner modes emerging due to the rotation increases with the increase of rotation frequency. Consequently, corner and bulk solitons in rotating waveguide arrays become thresholdless for sufficiently large rotation frequencies, in contrast to solitons in nonrotating arrays that exist only above the power threshold. Focusing nonlinearity enhances the localization degree of corner modes, but surprisingly initially it leads to broadening of bulk nonlinear states, followed by their relocalization at high input powers. Our results open new prospects for control of evolution of nonlinear multidimensional excitations by dynamically varying potentials.

Observation of robust edge mode and in-gap corner mode in Kagome surface-wave photonic crystals.

Recent theory has demonstrated that Kagome photonic crystals (PCs) support first-order and second-order topological phenomena. Here, we extend the topological physics of the Kagome lattice to surface electromagnetic waves and experimentally show a Kagome surface-wave PC. Under the protection of first-order and second-order topologies, both robust edge modes and in-gap corner modes are observed. The robust transport of edge modes is demonstrated by high transmission through the waveguide with a sharp bend. The localized corner mode is found at the corner with one isolated rod when a triangle-shaped sample is constructed. Our work not only shows a platform to mimic the topological physics in classical wave systems, but also offers a potential application in designing high-performance photonic devices.

Second-order topological corner states in two-dimensional elastic wave metamaterials with nonsymmorphic symmetries

This paper investigates the higher-order topological behavior of elastic wave metamaterials composed of L-shaped pillars attached to a plate. In this system, the multi-dimensional topological transitions can be developed for physical transfer among the corner, edge, and bulk elastic wave modes by tuning the rotation angle of the L-shaped pillars. Different from the four degenerate corner states in the symmorphic system, these corner states in this metamaterial evolve into two doublets because of the lack of mirror symmetries. With the energy highly trapped at the corners, it thus can immediately realize elastic resonators/sensors with high-quality factors. Besides, the coupling effect between corner and edge states is analyzed in the multi-dimensional topological system. The edge–corner coupling effect could be applied to relevant fields, such as high-sensitive detection, elastic energy harvesting, and optomechanical sensing.

Higher-order topological hyperbolic lattices

A hyperbolic lattice allows for any $p$-fold rotational symmetry, in stark contrast to a two-dimensional crystalline material, where only 2-fold, 3-fold, 4-fold, or 6-fold rotational symmetry is permitted. This unique feature motivates us to ask whether the enriched rotational symmetry in a hyperbolic lattice can lead to any new topological phases beyond a crystalline material. Here, by constructing and exploring tight-binding models in hyperbolic lattices, we theoretically demonstrate the existence of higher-order topological phases in hyperbolic lattices with 8-fold, 12-fold, 16-fold or 20-fold rotational symmetry, which is not allowed in a crystalline lattice. Since such models respect the combination of time-reversal symmetry and $p$-fold ($p=8$, 12, 16, or 20) rotational symmetry, $p$ zero-energy corner modes are protected. For the hyperbolic {8,3} lattice, we find a higher-order topological phase with a finite edge energy gap and a gapless phase. Our results thus open the door to studying higher-order topological phases in hyperbolic lattices.

Visualization of rainbow trapping effect in higher-order topological insulators

The rainbow trapping effect, which enables field amplification of multiple wavelengths in topological insulators, offers unprecedented opportunities for integrated ultrasonic sensors and energy harvesting. However, until now, the realization of the rainbow trapping effect has been limited to a one-dimensional (1D) or 2D interface in the electromagnetic and acoustic systems. Here, we propose a method to realize rainbow trapping based on higher-order topological corner modes (HOTCMs), where the elastic wave is localized at 0D corners rather than at the interface. We strategically design the two-dimensional elastic locally resonant metamaterial plate and construct multiple HOTCMs at the interfacial corners formed by trivial and nontrivial domains. We further investigate the interplay between nearest-neighbor couplings and next-nearest-neighbor couplings. Remarkably, the HOTCMs localized at corners with different geometric configurations are shown to be frequency dependent, providing a platform for trapping elastic waves at different corners. Moreover, we demonstrate rainbow trapping based on the HOTCMs at the corners and show the agreement between the experimental results and theoretical modeling. Our results provide a mechanism to realize the trapping of multiple frequencies and extend the use of elastic integrated devices toward higher order topological insulators.

Higher-order topological insulator in a modified Haldane-Hubbard model

We investigate the ground-state phase diagram of a modified spinless Haldane-Hubbard model, with broken threefold rotational symmetry, employing exact diagonalization calculations. The interplay of asymmetry, interactions, and topology gives rise to a rich phase diagram. The noninteracting limit of the Hamiltonian exhibits a higher-order topological insulator characterized by the existence of corner modes, in contrast to known chiral edge metallic states of the standard Haldane model. Our investigation demonstrates that these symmetry-protected states are robust to the presence of finite interactions. Furthermore, in certain regimes of parameters, we show that a topological Mott insulator exists in this model, where a nontrivial topological bulk coexists with an interaction-driven charge-density wave, whose emergence is characterized by a ${Z}_{2}$-symmetry breaking within the $3d\text{-Ising}$ universality class.