Corner Modes Research Articles

The topological dynamics of continuum lattice grid structures

Continuum lattice grid structures which consist of joined elastic beams subject to flexural deformations are ubiquitous. In this work, we establish a theoretical framework of the topological dynamics of continuum lattice grid structures, and discover the topological edge and corner modes in these structures. We rigorously identify the infinitely many topological edge states within the bandgaps via a theorem, with a clear criterion for the infinite number of topological phase transitions. Then, we obtain analytical expressions for the topological phases of bulk bands, and propose a topological index related to the topological phases that determines the existence of the edge states. The theoretical approach is directly applicable to a broad range of continuum lattice grid structures including bridge-like frames, square frames, kagome frames, continuous beams on elastic springs. The frequencies of the topological modes are precisely obtained, applicable to all the bands from low to high frequencies. Continuum lattice grid structures serve as excellent platforms for exploring various kinds of topological phases and demonstrating the topological modes at multiple frequencies on demand. Their topological dynamics has significant implications in safety assessment, structural health monitoring, and energy harvesting.

Chiral Bulk Solitons in Photonic Graphene with Decorated Boundaries

AbstractA chiral bulk soliton in a nonlinear photonic lattice with decorated boundaries, presenting a novel approach to manipulate photonic transport without extensive bulk modifications is proposed. Unlike traditional methods that rely on topological edge and corner modes, this strategy leverages the robust chiral propagation of bulk modes. By introducing nonlinearity into the system, a stable bulk soliton, akin to the topological valley Hall effects is found. The chiral bulk soliton exhibits remarkable stability; the energy does not decay even after a long‐distance propagation; and the corresponding Fourier spectrum confirms the absence of inter‐valley scattering indicating a valley‐locking property. The findings not only contribute to the fundamental understanding of nonlinear photonic systems but also hold significant practical implications for the design and optimization of photonic devices.

Hierarchical bound states in heat transport

Higher-order topological phases in non-Hermitian photonics revolutionize the understanding of wave propagation and modulation, which lead to hierarchical states in open systems. However, intrinsic insulating properties endorsed by the lattice symmetry of photonic crystals fundamentally confine the robust transport only at explicit system boundaries, letting alone the flexible reconfiguration in hierarchical states at arbitrary positions. Here, we report a dynamic topological platform for creating the reconfigurable hierarchical bound states in heat transport systems and observe the robust and nonlocalized higher-order states in both the real- and imaginary-valued bands. Our experiments showcase that the hierarchical features of zero-dimension corner and nontrivial edge modes occur at tailored positions within the system bulk states instead of the explicit system boundaries. Our findings uncover the mechanism of non-localized hierarchical non-trivial topological states and offer distinct paradigms for diffusive transport field management.

Corner modes in non-Hermitian next-nearest-neighbor hopping model

Corner modes in non-Hermitian next-nearest-neighbor hopping model

Construction of topological quantum magnets from atomic spins on surfaces.

Artificial quantum systems have emerged as platforms to realize topological matter in a well-controlled manner. So far, experiments have mostly explored non-interacting topological states, and the realization of many-body topological phases in solid-state platforms with atomic resolution has remained challenging. Here we construct topological quantum Heisenberg spin lattices by assembling spin chains and two-dimensional spin arrays from spin-1/2 Ti atoms on an insulating MgO film in a scanning tunnelling microscope. We engineer both topological and trivial phases of the quantum spin model and thereby realize first- and second-order topological quantum magnets. We probe the many-body excitations of the quantum magnets by single-atom electron spin resonance with an energy resolution better than 100 neV. Making use of the atomically localized magnetic field of the scanning tunnelling microscope tip, we visualize various many-body topological bound modes including topological edge states, topological defects and higher-order corner modes. Our results provide a bottom-up approach for the simulation of exotic quantum many-body phases of interacting spins.

Mixed higher-order topology, and nodal and nodeless flat band topological phases in a superconducting multiorbital model

We investigate the topological phases that appear in an orbital version of the Benalcazar-Bernevig-Hughes (BBH) model in the presence of conventional spin-singlet s-wave superconductivity and with the possibility of tuning an in-plane magnetic field. We chart out the phase diagram by considering different boundary conditions, with the topology of the individual phases further examined by considering both the Wannier and entanglement spectra, as well as the Majorana polarization. For weak to moderate values of magnetic field and superconducting pairing amplitude, we find a second-order topological superconductor phase with eight zero-energy corner modes. Further increasing field or pairing, half of the corner states can be turned into zero-energy edge-localized modes, thus forming what we name hybrid-order phase. Then, we find two different putative first-order topological phases, a nodal and a nodeless phase, both with zero-energy flat bands localized along mirror-symmetric open edges. For the nodal phase, the flat bands are, as expected, localized between the nodes in reciprocal space, while in the nodeless phase, the zero-energy boundary flat band instead spans the whole Brillouin zone and appears disjoint from the fully gapped bulk spectrum. As a consequence, this model present several unexpected phases with unusual surface states that can be tuned via an external magnetic field. Published by the American Physical Society 2024

ExSitu Production and Storage of Exciton-Polariton Vortices in Higher-Order Topological Corner Modes.

We propose a scheme for producing and manipulating quantized exciton-polariton vortices in the higher-order topological corner modes of a two-dimensional array of micropillars. By nonresonantly exciting p-orbital condensates with different orientations at two input corners, polariton vortices carrying the required topological charges can be controllably created at output corners away from the pumping spots. Besides, polariton vortices formed at input corners can be copied to the output corners through the topological edge states. Our scheme provides topological double insurance for intrinsic binary information memory and holds potential applications in remote information processing.

Multiple phononic corner modes in 2D buckled arsenene

The theoretical framework of topological states has recently been expanded to include phononic systems. This study focuses on investigating the presence of multiple phononic corner modes in the phonon dispersion curves of 2D buckled arsenene. The phononic corner modes manifest within the frequency range of 5.0–5.4 THz, specifically in the frequency gap between the optical and acoustical phononic bands, for the 2D buckled arsenene. The topologically nontrivial behavior of the frequency gaps can be confirmed by the secondary topological index. In addition to the 0D phononic corner modes, 1D phononic edge modes also appear in the gap of the above frequency range. Therefore, this research establishes the foundation for future progress in the field of 2D topological phononics, specifically focusing on boundary states with multiple dimensions (1D and 0D).

Erratum: Braiding Majorana corner modes in a second-order topological superconductor [Phys. Rev. Research 2 , 032068(R) (2020)

Erratum: Braiding Majorana corner modes in a second-order topological superconductor [Phys. Rev. Research <b>2</b> , 032068(R) (2020)

Topological Anderson phases in heat transport

Topological Anderson phases (TAPs) offer intriguing transitions from ordered to disordered systems in photonics and acoustics. However, achieving these transitions often involves cumbersome structural modifications to introduce disorders in parameters, leading to limitations in flexible tuning of topological properties and real-space control of TAPs. Here, we exploit disordered convective perturbations in a fixed heat transport system. Continuously tunable disorder-topology interactions are enabled in thermal dissipation through irregular convective lattices. In the presence of a weak convective disorder, the trivial diffusive system undergos TAP transition, characterized by the emergence of topologically protected corner modes. Further increasing the strength of convective perturbations, a second phase transition occurs converting from TAP to Anderson phase. Our work elucidates the pivotal role of disorders in topological heat transport and provides a novel recipe for manipulating thermal behaviors in diverse topological platforms.

Edge-dependent Majorana corner modes in an s-wave superconductor

In contrast to Majorana edge modes in first-order topological superconductors, Majorana corner modes are usually related to spatial symmetries and further rely on specific corner of the sample. In this paper, we study Majorana corner modes in an extended Kane-Mele model with arbitrary phase. Majorana corner mode exists at both corner between zigzag edge and armchair edge and corner between zigzag edge and zigzag edge for the phase at π/2. However, while the Majorana corner mode at the corner between zigzag edge and armchair edge remains as the phase deviates from π/2, Majorana mode at the corner between zigzag edge and armchair edge disappears. These Majorana corner modes are edge dependent and originate from the sign change of Dirac mass or Dirac velocity for edge states at adjacent edges tuned by the phase.

P‐Orbital Higher‐Order Topological Corner States in 2D Photonic Su–Schrieffer–Heeger Lattices

AbstractHigher‐order topological insulators (HOTIs) have attracted much attention in photonics partly because of the robust and highly confined corner modes they support. The growing availability of synthetic multi‐orbital platforms has recently stimulated research focus on the interplay between HOTIs and orbital degree of freedom. In this work, the topological properties of the two‐dimensional (2D) Su–Schrieffer–Heeger (SSH) model with ‐orbital degree of freedom are explored and ‐orbital topological corner states in a laser‐written photonic 2D SSH lattice are experimentally observed. It is shown that the ‐band 2D SSH model is a HOTI, where zero‐energy orbital corner states appear either as in‐gap states or as bound states in the continuum (BICs). Chiral symmetry is sufficient for topological protection of the in‐gap ‐orbital corner states, but not for their BIC counterparts because of the hybridization effect within the bulk continuum. The multipole chiral number is employed as an optimal topological invariant to characterize the ‐orbital HOTIs. The flexibility of tuning the orbital couplings offers an extra degree of manipulation for the BICs, which may be useful for the development of novel photonic devices.

Machine learning guided discovery of stable, spin-resolved topological insulators

Identification of a nontrivial Z2 index in a spinful two-dimensional insulator indicates the presence of an odd, quantized (pseudo)spin-resolved Chern number, Cs=(C↑−C↓)/2. However, the statement is not biconditional. An odd spin-Chern number can survive when the familiar Z2 index vanishes. Identification of solid-state systems hosting an odd, quantized Cs and trivial Z2 index is a pressing issue due to the potential for such insulators to admit band gaps optimal for experiments and quantum devices. Nevertheless, they have proven elusive due to the computational expense associated with their discovery. In this work, a neural network capable of identifying the spin-Chern number is developed and used to identify the first solid-state systems hosting a trivial Z2 index and odd Cs. We demonstrate the potential of one such system, Ti2CO2, to support Majorana corner modes via the superconducting proximity effect. Published by the American Physical Society 2024

Solitons in higher-order topological insulator created by unit cell twisting

We show that higher-order topological insulators can be created from usual square structure by twisting waveguides in each unit cell around the axis passing through the center of the unit cell, even without changing intracell distance between waveguides. When applied to usual square array, this approach produces two-dimensional generalization of Su-Schrieffer-Heeger (SSH) structure supporting topological corner modes with propagation constants belonging to two forbidden spectral gaps opening only for twist angles from certain interval. In contrast to usual SSH arrays, where higher-order topology is typically introduced by diagonal waveguide shifts and only one type of corner states exists, our SSH-like structure in topological phase supports two co-existing types of in-phase and out-of-phase corner modes appearing in two different topological gaps that open in the spectrum. Therefore, twisting of the unit cell qualitatively changes topological properties of the system, offering a new degree of freedom in creation of higher-order topological phases. In material with focusing cubic nonlinearity two coexisting types of topological corner solitons emerge from these modes, whose existence and stability properties are studied here. Despite different internal structure, both such modes can be simultaneously dynamically stable in the appreciable part of the topological gap.

Topological Fano-resonance with type-II and type-III corner states.

Topological corner states have been used to develop topologically robust Fano-resonant systems immune to structural perturbations while preserving the ultra-sensitive profiles under external factors. In this work, we have extended the possibility of obtaining Fano-resonant systems by introducing type-II and type-III corner states with a large modal surface to this class of resonance. Through photonic lattices with low symmetry, such as C2, it is easy to obtain type-II and type-III corner states due to the tailoring of long-range interactions. Subsequently, one can combine topological cavities of type-II and type-III corner modes with topological waveguides obtained from a first-order topological insulating phase. Our results may pave the way to generate devices suitable for creating non-classical light applicable in quantum computing and ultra-sensitive sensors employing large-area topological states.

Interaction-driven first-order and higher-order topological superconductivity

We investigate topological superconductivity in the Rashba-Hubbard model, describing heavy-atom superlattice and van der Waals materials with broken inversion. We focus in particular on fillings close to the van Hove singularities, where a large density of states enhances the superconducting transition temperature. To determine the topology of the superconducting gaps and to analyze the stability of their surface states in the presence of disorder and residual interactions, we employ an fRG+MFT approach, which combines the unbiased functional renormalization group (fRG) with a real-space mean-field theory (MFT). Our approach uncovers a cascade of topological superconducting states, including A1 and B1 pairings, whose wave functions are of dominant p- and d-wave character, respectively, as well as a time-reversal breaking A1+iB1 pairing. While the A1 and B1 states have first-order topology with helical and flat-band Majorana edge states, respectively, the A1+iB1 pairing exhibits second-order topology with Majorana corner modes. We investigate the disorder stability of the bulk superconducting states, analyze interaction-induced instabilities of the edge states, and discuss implications for experimental systems. Published by the American Physical Society 2024

Hidden Real Topology and Unusual Magnetoelectric Responses in Two-Dimensional Antiferromagnets.

Recently, the real topology has been attracting widespread interest in two dimensions (2D). Here, based on first-principles calculations and theoretical analysis, the monolayer Cr2Se2O (ML-CrSeO) is revealed as the first material example of a 2D antiferromagnetic (AFM) real Chern insulator (RCI) with topologically protected corner states. Unlike previous RCIs, it is found that the real topology of the ML-CrSeO is rooted in one certain mirror subsystem of the two spin channels, and cannot be directly obtained from all the valence bands in each spin channel as commonly believed. In particular, due to antiferromagnetism, the corner modes in ML-CrSeO exhibit strong corner-contrasted spin polarization, leading to spin-corner coupling (SCC). This SCC enables a direct connection between spin space and real space. Consequently, large and switchable net magnetization can be induced in the ML-CrSeO nanodisk by electrostatic means, such as potential step and in-plane electric field, and the corresponding magnetoelectric responses behave like a sign function, distinguished from that of the conventional multiferroic materials. This work considerably broadens the candidate range of RCI materials, and opens up a new direction for topo-spintronics and 2D AFM materialsresearch.

Topological Corner Modes by Composite Wannier States in Glide‐Symmetric Photonic Crystal

AbstractSecond‐order topological insulators can be characterized by their bulk polarization, which is believed to be intrinsically connected to the center of the Wannier function. In this study, the existence of second‐order topological insulators is demonstrated that feature a pair of partially degenerate photonic bands. These arise from the nonsymmorphic glide symmetry in an all‐dielectric photonic crystal. The center of the maximally localized Wannier function (MLWF) is consistently located at the origin but is not equivalent with respect to the sum of constituent polarizations. As a result, topological corner modes can be identified by the distinctly hybridized MLWFs that truncate at the sample boundary. Through full‐wave numerical simulations paired with microwave experiments, the second‐order topology is clearly confirmed and characterized. These topological corner states exhibit notably unique modal symmetries, which are made possible by the inversion of the Wannier bands. These results provide an alternative approach to explore higher‐order topological physics with significant potential for applications in integrated and quantum photonics.

Corner- and edge-mode enhancement of near-field radiative heat transfer.

It is well established that near-field radiative heat transfer (NFRHT) can exceed Planck's blackbody limit1 by orders of magnitude owing to the tunnelling of evanescent electromagnetic frustrated and surface modes2-4, as has been demonstrated experimentally for NFRHT between two large parallel surfaces5-7 and between two subwavelength membranes8,9. However, although nanostructures can also sustain a much richer variety of localized electromagnetic modes at their corners and edges10,11, the contributions of such additional modes to further enhancing NFRHT remain unexplored. Here we demonstrate both theoretically and experimentally a physical mechanism of NFRHT mediated by the corner and edge modes, and show that it can dominate the NFRHT in the 'dual nanoscale regime' in which both the thickness of the emitter and receiver, and their gap spacing, are much smaller than the thermal photon wavelengths. For two coplanar 20-nm-thick silicon carbide membranes separated by a 100-nm vacuum gap, the NFRHT coefficient at room temperature is both predicted and measured to be 830W m-2 K-1, which is 5.5 times larger than that for two infinite silicon carbide surfaces separated by the same gap, and 1,400 times larger than the corresponding blackbody limit accounting for the geometric view factor between two coplanar membranes. This enhancement is dominated by the electromagnetic corner and edge modes, which account for 81% of the NFRHT between the silicon carbide membranes. These findings are important for future NFRHT applications in thermal management and energy conversion.

Sum and difference frequency generation in a valley-photonic-crystal-like topological system

Nonlinear sum frequency generation (SFG) and difference frequency generation (DFG) are fundamental methods to obtain new light sources for various applications. However, most of the on-chip SFG and DFG are based on conventional resonators, lacking robustness against fabrication defects. Here, we demonstrate topologically protected SFG and DFG in a second-order topological photonic system. The mechanism is based on the nonlinear interaction between three high-Q corner modes inside dual topological band gaps. The frequency matching condition for SFG and DFG is precisely satisfied by designing a valley-photonic-crystal-like topological system, which provides more freedoms to tune the corner modes. The topological SFG and DFG are achieved with high conversion efficiency, and the underlying topological physics is revealed. This work opens up avenues toward topologically protected nonlinear frequency conversion, and can find applications in the fields of on-chip single-photon detections and optical quantum memories with robustness against defects.