Topological Boundary Research Articles

3D 3,44T19 Type Carbon Allotrope (3,44T19‐CA): An Ideal Candidate with Phononic Stiefel–Whitney Topology and Multi–Dimensional Boundary Vibrational Modes

The Stiefel–Whitney (SW) insulator, which lacks spin‐orbit coupling and has space–time inversion () symmetry, has attracted significant interest due to its fragile topological state and unconventional bulk‐boundary correspondence. Previously, the identification of SW insulators has been widely proposed for 2D phononic systems, but it has received little attention for 3D phononic systems. Herein, it is recognized that the 3D 3,44T19 type carbon allotrope (3,44T19‐CA) possesses the possibility to attain the nontrivial phononic SW topology characterized by a nontrivial second SW number, specifically w2 = 1. In addition, the 3D 3,44T19‐CA has 2D surface vibrational modes and 1D hinge vibrational modes, providing an excellent platform for multi–dimensional topological phononic boundaries research.

Uniform Diffusion of Cooper Pairing Mediated by Hole Carriers in Topological Sb2Te3/Nb.

Spin-helical Dirac Fermions at a doped topological insulator's boundaries can support Majorana quasiparticles when coupled with s-wave superconductors, but in n-doped systems, the requisite induced Cooper pairing in topological states is often buried at heterointerfaces or complicated by degenerate coupling with bulk conduction carriers. Rarely probed are p-doped topological structures with nondegenerate Dirac and bulk valence bands at the Fermi level, which may foster long-range superconductivity without sacrificing Majorana physics. Using ultrahigh-resolution photoemission, we report proximity pairing with a large decay length in p-doped topological Sb2Te3 on superconducting Nb. Despite no momentum-space degeneracy, the topological and bulk states of Sb2Te3/Nb exhibit the same isotropic superconducting gaps at low temperatures. Our results unify principles for realizing accessible pairing in Dirac Fermions relevant to topological superconductivity.

Rectifiability, finite Hausdorff measure, and compactness for non‐minimizing Bernoulli free boundaries

AbstractWhile there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less is known about critical points of the corresponding energy. Saddle points of the energy (or of closely related energies) and solutions of the corresponding time‐dependent problem occur naturally in applied problems such as water waves and combustion theory. For such critical points —which can be obtained as limits of classical solutions or limits of a singular perturbation problem—it has been open since (Weiss, 2003) whether the singular set can be large and what equation the measure satisfies, except for the case of two dimensions. In the present result we use recent techniques such as a frequency formula for the Bernoulli problem as well as the celebrated Naber–Valtorta procedure to answer this more than 20 year old question in an affirmative way: For a closed class we call variational solutions of the Bernoulli problem, we show that the topological free boundary (including degenerate singular points , at which as ) is countably ‐rectifiable and has locally finite ‐measure, and we identify the measure completely. This gives a more precise characterization of the free boundary of in arbitrary dimension than was previously available even in dimension two. We also show that limits of (not necessarily minimizing) classical solutions as well as limits of critical points of a singularly perturbed energy are variational solutions, so that the result above applies directly to all of them.

Topology optimization based on the improved high-order RAMP interpolation model and bending properties research for curved square honeycomb sandwich structures

Curved honeycomb sandwich structures with larger surface areas effectively reduce the number of fasteners and connectors, resulting in weight reduction, cost savings, and reliability improvement. Square honeycombs exhibit higher in-plane tensile strength, and are more compatible with mechanical components. Topology optimization can realize the variable-density design of square honeycombs, enhancing the strength and stiffness of curved sandwich structures, but the high-order Rational Approximation of Material Properties (RAMP) interpolation model with a fast convergence rate has the relatively poor clarity and stability in topology boundaries. Hence, a variable-density topology optimization method based on the improved high-order RAMP model is developed for curved square honeycomb sandwich structures. The high-order RAMP interpolation model is improved by incorporating a minimum modulus term into the material interpolation function and employing the Bi-directional Evolutionary Structural Optimization (BESO) to refine the Optimality Criteria (OC) method. A functional relationship between the wall thickness of cross-shaped cells (i.e., simplified models of four adjacent square cells arranged in a cross) and the relative density of topology units is constructed using a density mapping method, and the topology optimization with the relative density of cross-shaped cells as the design variable is performed to minimize the compliance of the core. Three-point bending tests are conducted on 3D printed non-optimized and optimized curved honeycomb sandwich structures with different core material retention rates and panel thicknesses, and the experimental results are compared with those of simulations to explore the bending properties and failure behaviors. The results indicate that the modified high-order RAMP interpolation model enhances the clarity and stability of topology structures, the variable-density topology optimization significantly improves the bending properties of curved square honeycomb sandwich structures, and the experimental and numerical results are largely consistent.

FreeTO - Freeform 3D topology optimization using a structured mesh with smooth boundaries in Matlab

Topology optimization has revolutionized the design of structures for various applications, particularly with the advancement of additive manufacturing. However, existing open-source codes for topology optimization have limitations, such as restricted domain initialization and lack of a CAD output after optimization. A novel open-source Matlab code, FreeTO, is presented, and it addresses these limitations by enabling the initialization of 3D arbitrary geometries and providing an STL file post-optimization. FreeTO utilizes a structured mesh and a smooth-edge (boundary) algorithm to generate smooth topological boundaries. The code is demonstrated through six practical design cases, showcasing its effectiveness in compliance minimization, compliant mechanisms, and self-supporting problems. FreeTO offers a user-friendly, all-in-one topology optimization package, making it an invaluable tool for educators, researchers, and practitioners. Future developments will focus on eliminating a few geometrical deviations in the optimized topologies, incorporating speedups, and extending the code to apply to more applications.

Braids and higher-order exceptional points from the interplay between lossy defects and topological boundary states

We show that the perturbation of the Su-Schrieffer-Heeger chain by a localized lossy defect leads to higher-order exceptional points (HOEP). Depending on the location of the defect, third- and fourth-order exceptional points (EP3 and EP4) appear in the space of Hamiltonian parameters. On the one hand, they arise due to the non-Abelian braiding properties of exceptional lines (EL) in parameter space. Namely, the HOEPs lie at intersections of mutually noncommuting ELs. On the other hand, we show that such special intersections happen due to the fact that the delocalization of edge states, induced by the non-Hermitian defect, hybridizes them with defect states. These can then coalesce together into an EP3. When the defect lies at the midpoint of the chain, a special symmetry of the full spectrum can lead to an EP4. In this way, our model illustrates the emergence of interesting non-Abelian topological properties in the multiband structure of non-Hermitian perturbations of topological phases. Published by the American Physical Society 2024

Observation of multifunctional robust topological states based on asymmetric C4 photonic crystals

Recent advancements in high-order topological insulators have heralded new opportunities for the innovation and utilization of optical devices. This paper presents a composite asymmetric C4 photonic crystal to achieve multifunctional, robust topological states. Through detailed analysis of the fine changes in the topological bandgap induced by distortion parameters, we facilitate the realization of topological edge states in wavelength division multiplexing applications. We utilize both trivial and nontrivial properties of the topological bandgap to precisely manipulate zero-dimensional angular states, one-dimensional topological boundary states, and two-dimensional body states. Through simulations and experimental results, our advanced asymmetric C4 photonic crystal structure demonstrates superior robustness for the transmission of topological edge states. Our research paves the way for the deployment of more robust topological boundary state transmission systems and advances the application potential of higher-order topological states.

Topological modes, vibration attenuation, and energy harvesting in electromechanical metastructures

The dynamics of topological boundary modes in both periodic and quasi-periodic electromechanical metastructures is investigated, with a focus on their applications to energy harvesting and vibration reduction. The metastructure analyzed in this study is based on a shunted array of piezoelectric patches, with electrical parameters modulated according to the 1D Aubry–André–Harper model. As a result of this modulation, a fractal spectrum is generated near the central frequency of the resonators, a hallmark of nontrivial topology that enables the emergence of digitally controllable edge states and ensuing localization phenomena at subwavelength frequencies. In this framework, a detailed analysis of the metastructure spectral characteristics is conducted to investigate the influence of the modulation parameters on mode localization, both at the boundaries and within the interior of the beam. Such localization effects are then studied in relation to the energy harvesting, attenuation, and wave transport capabilities of the system. These functionalities point toward the realization of self-powered structures with low frequency and digitally controllable vibration attenuation capabilities, and are considered of significant technological interest in applications involving elastic waves and vibrations, where the ability to precisely control and harness these phenomena could lead to innovative solutions in energy-efficient and adaptive systems.

Design of a beam splitter based on an adjustable beam-splitting ratio of a hexagonal star-like topological photonic crystal

We propose a new, to the best of our knowledge, mechanism to realize topological phase transition, that is, in a hexagonal star-like honeycomb lattice photonic crystal (PC), the optical quantum spin Hall effect (QSHE) can be realized by changing the materials of the outer or inner ring dielectric rods in the cells. We calculated the energy band and analyzed the topological phase transition law of a hexagonal star-like honeycomb PC. By changing the permittivity of the PC, the disturbance is introduced to the edge state. It is found that with the decrease of the permittivity of the PC, the gap decreases, the lower boundary state gradually redshifts, and the maximum transmittance in the straight waveguide can reach 98.8%. On this basis, a topological beam splitter was designed and analyzed. Results show that the beam splitting ratio obtained by the system is in the wide range of 0.2–3.5. Our research enriches the implementation of topological photonics, provides potential applications for topological boundary states in terahertz technology, and offers a new avenue for the design of current optical integrated devices.

Subsymmetry protected topology in topological insulators and superconductors

Exploration of topology protected by a certain symmetry is central in condensed matter physics. A recent idea of subsymmetry protected (SSP) topology—remains of a broken symmetry can still protect specific topological boundary states—has been developed and demonstrated in a photonic system [], but not in topological superconductors. Here, we extend this idea further by applying subsymmetry protecting perturbation (SSPP) to one-dimensional topological insulating and superconducting systems using the Su-Schrieffer-Hegger (SSH) and Kitaev models. Using the tight-binding and low-energy effective theory, we show that the SSP boundary states retain topological properties while the SSPP results in the asymmetry of boundary states. For the SSH model, an SSP zero-energy edge state localized on one edge possesses quantized polarization. In contrast, the other edge state is perturbed to have nonzero energy, and its polarization is not quantized. For topological superconductors, SSP zero-energy Majorana boundary states for spinful Kitaev models emerge on only one edge, contrary to the conventional belief that Majorana fermions emerge at opposite edges. Our findings can be used as a platform to expand our understanding of topological materials as they broaden our understanding of the symmetry in a topological system and a method to engineer Majorana fermions. Published by the American Physical Society 2024

Spin-Dependent Localization of Helical Edge States in a Non-Hermitian Phononic Crystal.

As a distinctive feature unique to non-Hermitian systems, non-Hermitian skin effect displays fruitful exotic phenomena in one or higher dimensions, especially when conventional topological phases are involved. Among them, hybrid skin-topological effect is theoretically proposed recently, which exhibits anomalous localization of topological boundary states at lower-dimensional boundaries accompanied by extended bulk states. Here, we experimentally realize the hybrid skin-topological effect in a non-Hermitian phononic crystal. The phononic crystal, before tuning to be non-Hermitian, is an ideal acoustic realization of the Kane-Mele model, which hosts gapless helical edge states at the boundaries. By introducing a staggered distribution of loss, the spin-dependent edge modes pile up to opposite corners, leading to a direct observation of the spin-dependent hybrid skin-topological effect. Our Letter highlights the interplay between topology and non-Hermiticity and opens new routes to non-Hermitian wave manipulations.

Design of nonlinear gradient sheet-based TPMS-lattice using artificial neural networks

Gradient triply periodic minimal surface (TPMS) structures are renowned for lightweight design and enhanced performance, but their complex and nonlinear configurations pose challenges in achieving targeted design goals. A new design methodology for the nonlinear gradient structure was proposed in this study, with the aim of achieving efficient and accurate modeling of complex and gradient sheet-based TPMS structures under specific performance objectives. This method utilized automated finite element (FE) simulations to obtain structure topology element densities under various boundary conditions. An artificial neural network (ANN) was then employed to efficiently predict the correspondence between these boundary conditions and topology element densities. A mapping was established between topology element densities and TPMS structural parameters, and the gradient structure was accurately constructed by using the voxel modeling technique. Taking a typical cantilever beam TPMS structure as an example of nonlinear gradient design, the results indicate that the error between the ANN-predicted and FE-simulated structure topology element densities is only 2.73 %, with prediction time being only 0.15 % of the simulation time. The thin regions of the gradient structure align with those geometrically removed in regular topology optimization scheme, achieving up to 65.45 % weight reduction, a 28.72 % improvement over the regular scheme, along with uniform structural stress transition and maximum stress reduction. TC4 alloy nonlinear gradient TPMS structures, printed by metal selective laser melting (SLM) technique, confirm the practical application value of this design method.

Topological boundary states in engineered quantum-dot molecules on the InAs(111)A surface: Odd numbers of quantum dots

Atom manipulation by scanning tunneling microscopy was used to construct quantum dots on the InAs(111)A surface. Each dot comprised six ionized indium adatoms. The positively charged adatoms create a confining potential acting on surface-state electrons, leading to the emergence of a bound state associated with the dot. By lining up the dots into N-dot chains with alternating tunnel coupling between them, quantum-dot molecules were constructed that revealed electronic boundary states as predicted by the Su-Schrieffer-Heeger (SSH) model of one-dimensional topological phases. Dot chains with odd N were constructed such that they host a single end or domain-wall state, allowing one to probe the localization of the boundary state on a given sublattice by scanning tunneling spectroscopy. We found probability density also on the forbidden sublattice together with an asymmetric energy spectrum of the chain-confined states. This deviation from the SSH model arises because the dots are charged and create a variation in on-site potential along the chain—which does not remove the boundary states but shifts their energy away from the midgap position. Our results demonstrate that topological boundary states can be created in quantum-dot arrays engineered with atomic-scale precision. Published by the American Physical Society 2024

Dynamic Phase Enabled Topological Mode Steering in Composite Su‐Schrieffer–Heeger Waveguide Arrays

AbstractTopological boundary states localize at interfaces whenever the interface implies a change of the associated topological invariant encoded in the geometric phase. The generically present dynamic phase, however, which is energy and time‐dependent, is known to be non‐universal, and hence not to intertwine with any topological geometric phase. Using the example of topological zero modes in composite Su‐Schrieffer‐Heeger (c‐SSH) waveguide arrays with a central defect is reported on the selective excitation and transition of topological boundary mode based on dynamic phase‐steered interferences. This work thus provides a new knob for the control and manipulation of topological states in composite photonic devices, indicating promising applications where topological modes and their bandwidth can be jointly controlled by the dynamic phase, geometric phase, and wavelength in on‐chip topological devices.

Experimental probe of point gap topology from non-Hermitian Fermi-arcs

The gap in spectra of a physical system is fundamental in physics, while gap topology further restricts possible occurrent gaps of topological boundary states. The emergence of non-Hermiticity unveils a unique gap type known as the point gap, which forecasts the wavefunction localization, known as the non-Hermitian skin effect. Therefore, experimentally identifying the point gap in the complex frequency plane through a real operating frequency can become a tool for the systematic investigation of skin effects. Here, we utilize a Weyl phononic crystal to demonstrate that the point gap constituted by bulk and Fermi-arc surface states can be observed experimentally by a real-space field mapping technique. The identified point gaps forecast various skin effects and their evolutions. We further experimentally demonstrate the hinge skin effect in a parallelogram structure. Our work provides a feasible recipe to explore point gap topology experimentally in a variety of systems and certainly stimulates the research on skin effects in three-dimensional systems.

Effects of topological boundary conditions on Bell nonlocality

Effects of topological boundary conditions on Bell nonlocality

Hybrid-order Weyl semimetal and its acoustic realizations

Abstract Hybrid-order topological insulators combine first- and higher-order topological properties and host topological boundary states with codimension one and more than one in different bandgaps. A Weyl semimetal (WSM) can possess two types of Weyl points: one class of Weyl points terminates the Fermi arc surface states, while another class of Weyl points not only launch Fermi arc surface states but also hinge arc states, exhibiting the hybrid-order topology. Here, we propose a hybrid-order WSM by stacking two-dimensional rhomboid lattices based on chiral nearest-neighbor and double-helix next-nearest interlayer couplings. The first type of Weyl point that only truncates the Fermi arc surface states exists at the crossing of any two-fold degeneracy of two adjacent bands, and the second type of Weyl point that connects the hinge arc states only appears at the crossing of the two middle bands. Our findings enrich the classification of topological semimetals in condensed matter physics.

Exceptional points of acoustic topological boundary states

Exceptional points of acoustic topological boundary states

Construction of Topological Bound States in the Continuum Via Subsymmetry.

Topological bound states in the continuum (BICs) are localized topological boundary modes coexisting with a continuous spectrum of extended modes. They have been realized in systems with symmetry-protected topological phases, where their immunity to defects and perturbations depends on the presence of symmetries. Here we propose a method that transforms an in-gap topological boundary state into a BIC by using the concept of subsymmetry. We design the coupling between a system possessing in-gap topological modes and a system possessing a continuum of states that results in topological BICs. We define the criteria for the coupling that yields the desired results. To implement this scheme, we construct representative topological BICs based on one-dimensional Su-Schrieffer-Heeger models and implement them in photonic lattices. Our results not only reveal novel physical phenomena but may also provide methods for designing a new generation of topological devices.

On the symmetry TFT of Yang-Mills-Chern-Simons theory

Three-dimensional Yang-Mills-Chern-Simons theory has the peculiar property that its one-form symmetry defects have nontrivial braiding, namely they are charged under the same symmetry they generate, which is then anomalous. This poses a few puzzles in describing the corresponding Symmetry TFT in a four-dimensional bulk. First, the braiding between lines at the boundary seems to be ill-defined when such lines are pulled into the bulk. Second, the Symmetry TFT appears to be too trivial to allow for topological boundary conditions encoding all the different global variants. We show that both of these puzzles can be solved by including endable (tubular) surfaces in the class of bulk topological operators one has to consider. In this way, we are able to reproduce all global variants of the theory, with their symmetries and their anomalies. We check the validity of our proposal also against a top-down holographic realization of the same class of theories.