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The intensities and phase profiles of the elliptic optics vortex beam both are determined by the topological charge (TC) and ellipticity. This paper presents an efficient and simple method for measuring the TC and ellipticity of elliptic vortex. By observing the diffraction patterns, the TC modulus and ellipticity can be probed with the suitable rectangular aperture, and the TC sign can be detected with the suitable right triangular aperture. The scheme works well even for high-order elliptic vortex with topological charge value as high as ± 18, and ellipticity range from 0.5 to 2 at least. Obviously, the measurement of TC and ellipticity have important research value for promoting the application of elliptical vortex.

We introduce a minimal model to describe the charge degree of freedom in cuprates with the on-site Hilbert space reduced to only the three valence states CuO$_4^{7-,6-,5-}$ (nominally Cu$^{1+,2+,3+}$) and make use of the S=1 pseudospin formalism. The formalism constitutes a powerful method to study complex phenomena in interacting quantum systems characterized by the coexistence and competition of various ordered states. Overall, such a framework provides a simple and systematic methodology to predict and discover new kinds of orders. In particular, the pseudospin formalism provides the most effective way to describe different topological structures, in particular, due to a possibility of a geometrical two-vector description of the on-site states. We introduce and analyze effective pseudospin Hamiltonian with on-site and inter-site charge correlations, two types of a correlated one-particle transfer and two-particle, or the composite boson transfer. The 2D S=1 pseudospin system is prone to a creation of different topological structures, which form topologically protected inhomogeneous distributions of the eight local S=1 pseudospin order parameters. We present a short overview of localized topological structures, typical for S=1 (pseudo)spin systems, focusing on unexpected antiphase domain walls in parent cuprates and so-called quadrupole skyrmion, which are believed to be candidates for a topological charge excitation in parent or underdoped cuprates. Puzzlingly, these unconventional structures can be characterized by an uniform distribution of the mean on-site charge, that makes these invisible for X-rays. Quasiclassical approximation and computer simulation are applied to analyze localized topological defects and evolution of the domain structures in "negative-$U$" model under charge order-superfluid phase transition., 25 pages, 7 figures. arXiv admin note: text overlap with arXiv:1409.5064

In this chapter we introduce and discuss the following fundamental notions that will be used throughout the book: topological equivalence of dynamical systems and their classification, bifurcations and bifurcation diagrams, and topological normal forms for bifurcations. The last section is devoted to the more abstract notion of structural stability. In this chapter we will be dealing only with dynamical systems in the state space X = ℝ n .

An orbital angular momentum (OAM) antenna based on chiral harmonic diffraction of spoof surface plasmon polaritons (SSPP) is demonstrated in this paper. Due to the diffraction of the helical wire, SSPP is transformed into the high-purity OAM mode. The operation frequency is determined by SSPP, and the topological charge is manipulated by modulating the harmonic order. Additionally, this system achieves a frequency-controlled beam-scanning characteristic in a wide bandwidth from 7.8 GHz to 12.6 GHz, and the measurement data agrees well with the theoretical analysis and the simulation results. Furthermore, the OAM antenna validated in this work is compact, easy-fabricated, and robust. The underlying principle opens up a new train of thought for generating OAM mode and paves the versatile applications involving wireless communication

[EN]A hedgehog or Bloch point is a point-like 3D magnetization configuration in a ferromagnet. Regardless of widely spread treatment of a Bloch point as a topological defect, its 3D topological charge has never been calculated. Here, applying the concepts of the emergent magnetic field and Dirac string, we calculate the 3D topological charge (Hopf index) of a Bloch point and show that due to the magnetostatic energy contribution it has a finite, non-integer value. Thus, Bloch points form a new class of hopfions-3D topological magnetization configurations. The calculated Bloch point non-zero gyrovector leads to important dynamical consequences such as the appearance of topological Hall effect. F.T. acknowledges support by ANID-PFCHA/Postdoctorado Becas Chile 74200122. K.G. acknowledges support by IKERBASQUE (the Basque Foundation for Science). This work was supported by the Spanish Ministry of Science and Innovation under grants FIS2016-78591-C3-3-R, PID2019-108075RB-C31, PID2019-108075RB-C33 and PID2019-106802GB-I00 /AEI/1013039/501100011033.

Terahertz (THz) vortex beams carrying orbital angular momentum (OAM) with high purity and tunable topological charge (TC) will undoubtedly bring extraordinary capacities for advanced imaging or communication systems. We propose a convenient and efficient method to generate a tunable and broadband vortex beam that is excited by superradiant Smith Purcell radiation (SSPR) on a helical grating. This scheme fully explores the advantages of natural broadband evanescent wave carried by the electron beam and the Bloch’s theorem regulated helical periodic systems. An explicit relation is established between the index of the spatial harmonic wave on the grating and the topological charge of the vortex beam. The electron energy in the SSPR can be customized to ignite the specific spatial harmonics and manipulate the OAM beam, accordingly. The separated radiation region also promises the high purity of the OAM spectrum. The harmonic excitation reduces the communality between the wavelength and the device size and alleviates the difficulty of device fabrication in the THz band. The proposed vortex scheme can not only be handily scaled to microwave and mid‑infrared regions, but also bring possibilities to applications based on compact tunable vortex beam sources.

We study a simple lattice model with local symmetry, whose construction is based on a crossed module of finite groups. Its dynamical degrees of freedom are associated both to links and faces of a four-dimensional lattice. In special limits the discussed model reduces to certain known topological quantum field theories. In this work we focus on its dynamics, which we study both analytically and using Monte Carlo simulations. We prove a factorization theorem which reduces computation of correlation functions of local observables to known, simpler models. This, combined with standard Krammers-Wannier type dualities, allows us to propose a detailed phase diagram, which form is then confirmed in numerical simulations. We describe also topological charges present in the model, its symmetries and symmetry breaking patterns. The corresponding order parameters are the Polyakov loop and its generalization, which we call a Polyakov surface. The latter is particularly interesting, as it is beyond the scope of the factorization theorem. As shown by the numerical results, expectation value of Polyakov surface may serve to detects all phase transitions and is sensitive to a value of the topological charge., Small revision before journal submission

Energy flux density for higher-order cylindrical vector vortex (HCVV) beam focused through a dielectric interface is analyzed numerically based on vector diffraction theory. Numerical results shows variation in the direction of the energy flux or Poynting vector distribution both in transverse and longitudinal plane upon suitably tuning the topological charge and polarization order of HCVV beam. We also note that the strong reverse flow of energy occur at the centre of the focal plane for certain values of the topological charge and polarization order. These results may find potential applications in optical trapping and optical manipulations.

The statistical mechanics of an ideal polymer chain entangled with static topological constraints is studied using a superspace approach, in which the probability distribution of the polymer is obtained as solutions of the Fokker-Planck equation in a superspace with an inner structure characterized by the n-generator free group. The theory predicts that the force-extension curve of the polymer under the topological constraints has the generic form F=kl+Z/l, where l is an effective extension. Aside from the elastic term that is linear in l, the force-extension curve contains a universal term of the form Z/l. The magnitude of this topological term is determined by the topological charge number Z, which characterizes the topological nature of the static constraints. The theoretical results are further verified by a scaling analysis based on a blob model of the chain conformations.

We argue that the topological charge density wave phase in the quasi-2D Kagome superconductor AV$_3$Sb$_5$ is a chiral flux phase. Considering the symmetry of the Kagome lattice, we show that the chiral flux phase has the lowest energy among those states which exhibit $2\times2$ charge orders observed experimentally. This state breaks the time-reversal symmetry and displays anomalous Hall effect. The explicit pattern of the density of this state in real space is calculated. These results are supported by recent experiments and suggest that these materials are a new platform to investigate the interplay between topology, superconductivity and electron-electron correlations., Comment: 5 pages, 3 figures, to be published in Science Bulletin

Terahertz (THz) radiation has recently received a lot of interests in science and technology. The potential of orbital angular momentum (OAM) of light in the THz region is a novel idea for researchers with increasing applications in broadband communications. THz waveguides have rarely been reported to generate and guide OAM modes. In this paper, we propose for the first time hollow-core optical fibers consist of circular air-hole arrays for guiding THz OAM modes with the topological charge order up to l = 3. Using numerical simulations, two low-loss transmission windows over 0.47 – 0.65 THz (window1) and 0.95 – 1.20 THz (window2) are recognized to guide OAM modes. Results show that both windows of the THz fiber exhibit the flat near-zero dispersion as well as low confinement loss (∼ 10-3 dB/cm) with simultaneously high purity of the OAM modes (higher than 90%). Finally, two designs of THz optical fibers consist of one and two rings of air-hole arrays in their cladding are examined and it is shown that difference of the confinement loss and the modal purity between two designs are negligible. This implies that hollow-core optical fibers consist of one ring of air-hole array in their cladding are very promising for efficient transmission of OAM modes in the THz region.

Abstract The Heisenberg model for an isotropic magnet is considered in this work. A substitution that reduces the corresponding equations to equations with a simpler geometric interpretation is applied. One of the solutions of the latter describes a new magnetic structure comprised of two straight intersecting vortex filaments, which change the topological charge after intersection.

Terahertz (THz) vortex waves carrying orbital angular momentum (OAM) hold great potential in dealing with the capacity crunch in wireless high-speed communication systems. Nevertheless, it is quite a challenge for the widespread applications of OAM in the THz regime due to the beam divergence and stringent alignment requirement. To address this issue, an all-dielectric lens (ADL) is proposed for the arbitrary manipulation of quasi-nondiffractive THz OAM waves (QTOWs). On the basis of the concept of the optical conical lens and the multivorticity metasurface, the beam number, the topological charge (TC), and the deflection angle as well as the nondiffractive depth of the generated THz OAM waves are controllable. For proof-of-concept, two ADLs are 3D printed to create single and dual deflected QTOWs, respectively. Remarkably, measured by a THz imaging camera, the desired QTOWs with high mode purity are observed in predesigned directions with a nondiffractive depth predefined theoretically. The proposed designs and experiments, for the first time, verified that the QTOWs could be achieved with a nondiffractive range of 55.58λg (λg = wavelength at 140 GHz) and large deflection angles of 30° and 45°.

The term Poincaré beam, which describes the space-variant polarization of a light beam carrying spin angular momentum (SAM) and orbital angular momentum (OAM), plays an important role in various optical applications. Since the radius of a Poincaré beam conventionally depends on the topological charge number, it is difficult to generate a stable and high-quality Poincaré beam by two optical vortices with different topological charge numbers, as the Poincaré beam formed in this way collapses upon propagation. Here, based on an all-dielectric metasurface platform, we experimentally demonstrate broadband generation of a generalized perfect Poincaré beam (PPB), whose radius is independent of the topological charge number. By utilizing a phase-only modulation approach, a single-layer spin-multiplexed metasurface is shown to achieve all the states of PPBs on the hybrid-order Poincaré Sphere for visible light. Furthermore, as a proof-of-concept demonstration, a metasurface encoding multidimensional SAM and OAM states in the parallel channels of elliptical and circular PPBs is implemented for optical information encryption. We envision that this work will provide a compact and efficient platform for generation of PPBs for visible light, and may promote their applications in optical communications, information encryption, optical data storage and quantum information sciences., Well-controlled Poincaré beams are potentially useful in optical communications applications. Here, the authors present phase-only metasurfaces that generate broadband, perfect Poincaré beams in the visible, with radius independent of the topological number.

Spiral waves in the cardiac tissue may cause life-threatening arrhythmia. Such waves can be anchored to a local heterogeneity and form stable pinned waves, which are difficult to be eliminated. Here, a numerical simulation using the FHN model provides the parameter area of the obstacle size and the perturbation period for successful unpinning and elimination by the method of a periodic perturbation near the boundary. The termination of pinned spirals may be successful when the obstacle size is small, and the optional range of the perturbation period is small. It is also illustrated that the pinned spiral wave can be coincidentally eliminated for individual parameter values outside the area. A new strategy of the periodic perturbation is proposed to liberate pinned spiral waves, where there is no limit to the size of the obstacle. In this approach, a free end is generated near the boundary, and is carried to the obstacle by the wave train, thereby changing the total topological charge around the obstacle and making the pinned spiral be eliminated. To better implement the control scheme, the dynamics of the carried defect is also discussed in detail.

Optical orbital angular momentum (OAM), carried by optical vortices, is a fundamental property of light that grabs broad interests. The detection of OAM modes, including fractional OAM is of great importance in diverse fields. Here, we firstly propose and develop the limited integer-component measurement method to determine the fractional topological charge (TC) of OAM states. This method indicates that a 2-integer-mode responsive device could be applied for the detection of fractional TC. An orthogonal long-period fiber grating (OLPFG) element is used as a fractional OAM TC probe to verify the Applicability. We experimentally demonstrate the fractional TC detection, ranging from −1 to 0, and recognize different fractional OAM modes with the minimum TC interval of 0.01. This scheme provides a concise, precise, continuous, and fiber-compatible fractional TC detection method that might have great potential in micro-detection, optical communication, quantum information, and chiral sensing applications.

A harmonic vortex beam is a typical vector beam with a helical wavefront at harmonic frequencies (e.g., second and third harmonics). It provides an additional degree of freedom beyond spin- and orbital-angular momentum, which may greatly increase the capacity for communicating and encoding information. However, conventional harmonic vortex beam generators suffer from complex designs and a low nonlinear conversion efficiency. Here, we propose and experimentally demonstrate the generation of a large second-harmonic (SH) vortex beam with quasi-nonlinear spin–orbit interaction (SOI). High-quality SH vortex beams with large topological charges up to 28 are realized experimentally. This indicated that the quasi-angular-momentum of a plasmonic spiral phase plate at the excitation wavelength (topological charge, q) could be imprinted on the harmonic signals from the attached WS2 monolayer. The generated harmonic vortex beam has a topological charge of ln=2nq (n is the harmonic order). The results may open new avenues for generating harmonic optical vortices for optical communications and enables novel multi-functional hybrid metasurface devices to manipulate harmonic beams.

Here we show theoretically that a superposition of two Bessel-Gaussian (BG) beams with different topological charges (TC) and different scaling factors (radial components of the wave vectors) has the TC equal to that of the BG beam with the larger scaling factor. If the scaling factors of the BG beams are equal, then TC of the whole superposition equals TC of the BG beam with the larger (in absolute value) weight coefficient in the superposition (i.e. with larger power). If the constituent BG beams are also same-power, TC of the superposition equals the average TC of the two BG beams. Therefore, if the sum of TCs of both beams is odd, TC of the superposition is a half-integer number. In practice, however, TC is calculated over a finite radius circle and, hence, the half-integer TC for the degenerated case cannot be obtained. Instead of the half-integer TC, the lower of the two integer TCs is obtained. Numerical simulation reveals that if the weight coefficients in the superposition are slightly different, TC of the superposition is not conserved on propagation. In the near field and in the Fresnel diffraction zone, TC is equal to the highest TC of the two BG beams, while in the far field it is equal to the lower TC. What is more, TC changes its value from high to low not instantly, but continuously at some propagation distance. In the intermediate zone TC is fractional.

We report a method for laser-assisted electrophoretic deposition using a Laguerre–Gaussian beam for maskless patterning of metal rings and tubes. These are structures of utmost importance, particularly in photonic devices. Metal nanoparticles, which are gathered in a colloidal solution by a laser trapping technique, are deposited on a substrate via electrophoresis. The deposition pattern is dependent on the focused spot shape during laser trapping. The intensity distribution of the Laguerre–Gaussian beam is ring-shaped. Rings with different inner diameters can be fabricated by varying the topological charge of the Laguerre–Gaussian beam. The equivalent inner diameters of the deposited rings with topological charges of 1, 3, and 5 were 0.30, 0.78, and 1.45 µm, respectively. The inner diameter of the deposited ring with a topological charge of 1 was smaller than the wavelength of the laser beam (532 nm). A tube was also fabricated with a topological charge of 3 by vertical displacement of the deposition cell. The deposition technology developed here using Laguerre–Gaussian beam will contribute to advancements in the fabrication of photonic and nano-fluidic devices.

Orientation of nematic liquid crystal (NLC) around a topological defect is expressed by the equation, ψ=sϕ+ϕ0, where ψ, s, ϕ and ϕ0 are the director spatial phase, topological charge, azimuthal ang...

Topological states of matter in two-dimensional (2D) materials have received increasing attention due to their potential applications in nanoscale spintronics. Here, we report the presence of unique topological electronic properties in a 2D planar octagon TiB4 compound. Particularly, without considering the spin–orbit coupling (SOC), we found that the material showed a coexistence of novel quadratic node (QN), and two different types of nodal rings (NRs), namely type-I and type-II. The protection mechanism of fermions has been fully clarified in this study. Furthermore, these fermions showed clear edge states. It is worth noting that QN had a topological charge of 2 since it is different from linear nodes and exhibit clear Fermi arc edge states. Under lattice strain, we found that the system could further exhibit rich topological phase transition. When SOC was included, we determined that these crossing points open very tiny energy gaps, which were smaller than previously reported 3D and 2D examples. These results show that monolayer TiB4 is an excellent nodal point and nodal ring semimetal, which also provides a feasible member for studying potential entanglements among multiple fermions.

Multi-channel structured light with orbital angular momentum (OAM) can be applied in different applications. For example, OAM modulation and OAM multiplexing in fiber optics communications, high-dimensional quantum cryptography-based OAM states for transmitting secure information across free-space, and independent data streams through OAM beams multiplexed free-space optical links. Using a simple and efficient system consisting of a spiral phase element (SPE) and a multi-channel vortex filter (MVF), we have converted input Gaussian beams into multi-channel OAM-based vortex beams for infrared wavelengths. An SPE has been designed, which generates optical vortices with wavelength-dependent topological charge (including fractional values). The resulting complex fields are optically relayed on a binary MVF designed by modulo- $2\pi $ phase addition of multiple fork gratings with topological charges 1, 2, and 3 and azimuthal orientations. In this way, the MVF generates beams with different OAM states for different carrier waves with different angles and maps them at desired locations in the detector plane. In this study, both $3\times 3$ , as well as a hexagonal configuration, were used. Furthermore, the experimentally obtained OAM spectrum qualitatively agrees with the results of numerical simulations, thus verifying our approach. The presented approach opens a new pathway for developing an efficient multi-channel OAM beams generator designed for a specific wavelength and illuminated by an input beam of different wavelengths over a broad spectral range.

A terahertz (THz) superimposed perfect vortex (SPV) beam is generated by using a designed superimposed optimal phase element (SOPE) at 0.3 THz. As the superimposed topological charge changes, the ring radius of THz SPV beams remains almost constant in the experiment, whose relative error is only 0.96%. The perfect vortex property is confirmed. The intensity profile of THz SPV beams is a ring with equally spaced breakpoints. The superimposed topological charge carried can be judged directly by identifying the number of breakpoints. The ring radius is linearly related to the radial wave vector, which can be adjusted. Such THz SPV beams have promising applications for THz fiber communication systems based on vortex beams.

In this paper, we construct the first analytic examples of (3+1)-dimensional self-gravitating regular cosmic tube solutions which are superconducting, free of curvature singularities and with non-trivial topological charge in the Einstein-SU(2) non-linear sigma-model. These gravitating topological solitons at a large distance from the axis look like a (boosted) cosmic string with an angular defect given by the parameters of the theory, and near the axis, the parameters of the solutions can be chosen so that the metric is singularity free and without angular defect. The curvature is concentrated on a tube around the axis. These solutions are similar to the Cohen-Kaplan global string but regular everywhere, and the non-linear sigma-model regularizes the gravitating global string in a similar way as a non-Abelian field regularizes the Dirac monopole. Also, these solutions can be promoted to those of the fully coupled Einstein-Maxwell non-linear sigma-model in which the non-linear sigma-model is minimally coupled both to the U(1) gauge field and to General Relativity. The analysis shows that these solutions behave as superconductors as they carry a persistent current even when the U(1) field vanishes. Such persistent current cannot be continuously deformed to zero as it is tied to the topological charge of the solutions themselves. The peculiar features of the gravitational lensing of these gravitating solitons are shortly discussed., Comment: 30 pages and 9 figures included. In the new version the title has been slightly changed. The details on why the ansatz does work have been included in the appendix. An extra section on the flat limit has been included. Various clarifying comments on the existing literature have also been inserted. Final version was accepted for publication on European Physical Journal C

Searching for new topological phases of matter has long been a hot topic in condensed matter physics and materials science based on its fundamental physics and promising device applications. Here we report a systematic ab initio study on the topological electronic properties of CoAs3, RhAs3, and RhSb3 binary compounds. Without spin–orbit coupling, there is a six-fold band crossing node at the high-symmetric Γ point with topological charge , which is denoted as “six-fold excitation”. This nodal point is formed by the highest occupied band and two of the lowest unoccupied bands, and protected by time-reversal symmetry, and spatial-inversion symmetry and stabilized by the two-fold rotational symmetry. Detailed band structure and elementary band representation analysis shows that the six-fold band degeneracy at the Γ point near the Fermi level is formed by the bands of Ag@8c originating from the d-orbital of metal atoms. Meanwhile, with spin–orbit coupling, the six-fold nodal point becomes a four-fold degenerate quadratic Dirac point with the topological charge conserved as . These results provide a systematic understanding of the electronic properties of the skutterudite compounds and enrich the families of topological fermions in condensed matter systems.

We introduce a consistent ansatz for the baby Skyrme model in (2+1)-dimensions which is able to reduce the complete set of field equations to just one equation for the profile function in situations in which the baby baryon charge can be arbitrary. Many analytic solutions both with and without the inclusion of the effects of the minimal coupling with the Maxwell field are constructed. Linear stability and other physical properties are discussed. These analytic gauged baby Skyrmions generate a persistent $U(1)$ current which cannot be turned off continuously as it is tied to the topological charge of the baby Skyrmions themselves. In the simplest non-trivial case of a gauged baby Skyrmion, a very important role is played by the Mathieu equation with an effective coupling constant which can be computed explicitly. These configurations are a very suitable arena to test resurgence in a non-integrable context., Version accepted for publication in the European Physical Journal C

Integrating multiple functionalities into one single device is of great importance for ever-growing demand of photonic device. Optical vortex beam contains orbital angular momentum (OAM) and spin angular momentum (SAM) associated with phase and polarization singularities respectively, which has contributed to various application including optical manipulation, optical trapping and high-speed optical communication. Traditional methods to detect its phase and polarization mostly focus on the functionality of single dimensional optical field. However, quite different from the previous reports, we here propose a novel approach to measure the phase and polarization state based on the full use of near and far-field light functionalities in one single metadevice, which supports spin-controlled surface plasmon polaritons (SPPs) and topological charge-dependent focus spot position. On the one hand, this design makes the most of multidimensional field optical scattering characteristics to rich the functions of metadevice. On the other hand, it provides a novel concept to recognize various angular momentum information of vortex beam. We believe that such design will have great potential for the integration of different sub-wavelength optical components.

While active systems possess notable potential to form the foundation of new classes of autonomous materials, designing systems that can extract functional work from active surroundings has proven challenging. In this work, we extend these efforts to the realm of designed active liquid crystal/colloidal composites. We propose suspending colloidal particles with Janus anchoring conditions in an active nematic medium. These passive Janus particles become effectively self-propelled once immersed into an active nematic bath. The self-propulsion of passive Janus particles arises from the effective $+1/2$ topological charge their surface enforces on the surrounding active fluid. We analytically study their dynamics and the orientational dependence on the position of a companion $-1/2$ defect. We predict that at sufficiently small activity, the colloid and companion defect remain bound to each other, with the defect strongly orienting the colloid to propel either parallel or perpendicular to the nematic. At sufficiently high activity, we predict an unbinding of the colloid/defect pair. This work demonstrates how suspending engineered colloids in active liquid crystals may present a path to extracting activity to drive functionality., Comment: 14 pages, 9 figures

Differentiation has widespread applications, particularly in image processing for edge detection. Significant advances have been made in using nanophotonic structures and metamaterials to perform such operations. In particular, a recent work demonstrated a topological differentiator in which the transfer function exhibits a topological charge, making the differentiation operation robust to variation of operating conditions. The demonstrated topological differentiator, however, operates in the reflection mode at off-normal incidence and is difficult to integrate into compact imaging systems. In this work, we design a topological differentiator that operates isotropically in transmission mode at normal incidence. The device exhibits an optical transfer function with a symmetry-protected, topological charge of $\pm 2$ and performs second-order differentiation. Our work points to the potential of harnessing topological concepts for optical computing applications., Comment: 5 pages, 5 figures

The quasicrystal(QC)s have unique lattice structure with the rotational symmetry forbidden in the periodic crystals. The electric properties are far from complete understanding. It has been unresolved whether the magnetic long-range orders are realized in the QC. Here we report our theoretical discovery of the ferromagnetic long-range order in the Tb-based QC. The difficulty in past theoretical studies on the QC was lack of the microscopic theory of the crystalline electric field (CEF), which is crucially important in the rare-earth systems. By analyzing the CEF in the Tb-based QC, we clarify that magnetic anisotropy plays a key role in realizing unique magnetic textures in the Tb-based QC and approximant crystal (AC). By constructing the minimal model, we show that various magnetic textures on the icosahedron at whose vertices Tb atoms are located. We find that the hedgehog state is characterized by the topological charge of one and the whirling-moment state is characterized by unusually large topological charge of three. The hedgehog and whirling-moment states are shown to be realized as antiferromagnetic orders transcribed as the emergent monopole and antimonopole in the 1/1 AC. We find that these states exhibit the topological Hall effect under applied magnetic field accompanied by the topological as well as metamagnetic transition. Our model and the determined phase diagram are expected to be relevant to the broad range of the rare-earth based QCs and ACs with strong magnetic anisotropy, which are useful not only to understand magnetism but also to explore novel topological properties., Comment: 15 pages, 5 figures

This works deals with the presence of localized planar structures in magnetic materials that admit integer or half-integer topological charge. We study models in which the internal disposition of magnetization is driven by a single parameter, that controls the topological charge density of the magnetic structure. In particular, we focus mainly on the configurational entropy of these skyrmions and half-skyrmions and show how to increase or diminish their informational capabilities., Comment: 9 pages, 8 figures

Summary For electrochemical hydrogen evolution reaction (HER), developing high-performance catalysts without containing precious metals have been a major research focus in the present. Herein, we show the feasibility of HER catalytic enhancement in Ni-based materials based on topological engineering from hybrid Weyl states. Via a high-throughput computational screening from ∼140,000 materials, we identify that a chiral compound NiSi is a hybrid Weyl semimetal (WSM) showing bulk type-I and type-II Weyl nodes and long surface Fermi arcs near the Fermi level. Sufficient evidences verify that topological charge carriers participate in the HER process, and make the certain surface of NiSi highly active with the Gibbs free energy nearly zero (0.07 eV), which is even lower than Pt and locates on the top of the volcano plots. This work opens up a new routine to develop no-precious-metal-containing HER catalysts via topological engineering, rather than traditional defect engineering, doping engineering, or strain engineering., Graphical abstract, Highlights • A new designation to catalytic enhancement via topological engineering is constructed • An ideal hybrid Weyl catalyst NiSi is identified by a high-throughput material screening • Sufficient evidences verify topological charge participates in catalytic enhancement • NiSi has a higher theoretical hydrogen evolution activity than precious-metal Pt, Chemistry; Catalysis; Computational chemistry; Computational method in materials science

Skyrmions are considered promising candidates to be the information carriers in the next generation of data storage and logic devices, due to its stability and easy control under the application of an electric current. For future technological applications in spintronic devices, it is important to study the properties behavior of these topological excitations during its movement on magnetic nanotracks, specially because in ferromagnetic materials they suffer a kind of magnus effect which tends to spell the skyrmion through the borders, preventing its transport throughout the nanotrack. We used micromagnetic simulations to study the dynamics of a skyrmion on a magnetic nanotrack induced by a spin polarized electric current. We considered thin magnetic nanotrack made of cobalt and platinum multilayers, whose magnetic state is perpendicular to the track plane and contain a single Néel-type Skyrmion. To describe this magnetic system, we used a Hamiltonian containing exchange, Dzyaloshinskii-Moriya, perpendicular magnetic anisotropy and dipole-dipole interactions. In our study we observed the well-known Skyrmion Hall effect and changes in the structure of the skyrmion when it approaches of the border. This alteration can be measured by determining the radius and the topological charge of the Skyrmion. Our simulation results show that both the radius and the topological charge decrease when it approaches of the border. Our study also demonstrates that the skyrmion-border interaction is repulsive, but there is a minimum distance from the border at which the interaction becomes attractive. If the skyrmion exceeds this critical position yc, it will be attracted and annihilated at the border of the nanotrack. We also performed simulations to obtain the limit value jc of the applied current density that the skyrmion can be transported along of the nanotrack without escaping from the side edge. From a technological point of view for possible applications in spintronic devices, the estimate of jc is of crucial importance.

When all routes are exceptional The exploration of non-Hermitian physics and parity-time symmetry have provided a route to develop a wealth of exotic physical effects. In such dissipative systems, the balance of gain and loss of the system lead to what are called exceptional points, or “sweet spots,” which relate to optimal device operation or material properties. The ability to tune the gain and loss over a range of system properties leads to exceptional arcs. Tang et al. show that systems can be designed in which the tuning of multiple parameters leads to a crossroads, or nexus, of exceptional arcs. Illustrating the effect in an acoustic system, the same properties should be attainable over various types of dissipative systems and thus provide a versatile route to fine-tune optimal performance of materials and devices. Science , this issue p. 1077

A plasmonic in-plane zone-plate (IPZP) is analytically proposed and numerically analyzed. The proposed scheme consists of several discretely metallic nano-slits, which couple incident vortices into SPP and form focal spots. Wave front information of incident beam is reflected in the field distribution along “focal plane” of IPZP, and therefore, the proposed structure can be used for topological charge detection. Besides, an orthogonal slit array IPZP achieving simultaneous spin and orbital angular momentum detection is elaborated. IPZP structure has relatively high fabrication tolerance and the parameters can be adjusted to fit various operating wavelength. The proposed structure is expected to find potential applications in orbital angular momentum communication and high-dimensional quantum entanglement.

We consider the baby-Skyrme model in the regime close to the so-called restricted baby-Skyrme model, which is a BPS model with area-preserving diffeomorphism invariance. The perturbation takes the form of the standard kinetic Dirichlet term with a small coefficient $\epsilon$. Classical solutions of this model, to leading order in $\epsilon$, are called restricted harmonic maps. In the BPS limit ($\epsilon\to 0$) of the model with the potential being the standard pion-mass term, the solution with unit topological charge is a compacton. Using analytical and numerical arguments we obtain solutions to the problem for topological sectors greater than one. We develop a perturbative scheme in $\epsilon$ with which we can calculate the corrections to the BPS mass. The leading order ($\mathcal{O}(\epsilon^1)$) corrections show that the baby Skyrmion with topological charge two is energetically preferred. The binding energy requires us to go to the third order in $\epsilon$ to capture the relevant terms in perturbation theory, however, the binding energy contributes to the total energy at order $\epsilon^2$. We find that the baby Skyrmions - in the near-BPS regime - are compactons of topological charge two, that touch each other on their periphery at a single point and with orientations in the attractive channel., Comment: LaTeX: 55 pages, 23 figures; V2: comments and references added, typos corrected

Nematic liquid crystals (LCs) normally contain both integer and half-integer topological charges. In this work, we investigate topological charge evolution dynamics in nematic LCs by the phase-fiel...

A (3 + 1)-dimensional Gross–Pitaevskii equation with partially nonlocal distributed coefficients under a linear potential is followed with interest. The mapping procedure from the distributed-coefficient to the constant-coefficient equations is provided. Via the procedure with solution of constant-coefficient NLSE from the variational principle, an approximate vortex soliton solution is reported. With the increase of value for topological charge m, torus-shaped vortex soliton turns into ring vortex soliton, and central region of ring gradually enlarges. The layer of vortex solitons and their phase along the z-direction is related to the number of $$q+1$$ with the Hermite parameter q. The branch number of spiral phase structures for vortex solitons is related to the value of topological charge m. The stability test from the direct numerical simulation indicates that vortex solitons with $$m=1,2,3,\, q=0$$ are stable, otherwise, vortex solitons with other values of m, q are unstable.

The use of spin waves (SWs) as data carriers in spintronic and magnonic logic devices offers operation at low power consumption, free of Joule heating. Nevertheless, the controlled emission and propagation of SWs in magnetic materials remains a significant challenge. Here, we propose that skyrmion-antiskyrmion bilayers form topological charge dipoles and act as efficient sub-100 nm SW emitters when excited by in-plane ac magnetic fields. The propagating SWs have a preferred radiation direction, with clear dipole signatures in their radiation pattern, suggesting that the bilayer forms a SW antenna. Bilayers with the same topological charge radiate SWs with spiral and antispiral spatial profiles, enlarging the class of SW patterns. We demonstrate that the characteristics of the emitted SWs are linked to the topology of the source, allowing for full control of the SW features, including their amplitude, preferred direction of propagation, and wavelength., 15 pages, 12 figures, 1 table

We developed and implemented an intensity moments technique for measuring amplitude and initial phase spectra, the topological charge (TC) and orbital angular momentum (OAM) of the Laguerre-Gauss (LG) beams decomposed into the basis of Hermite-Gaussian (HG) modes. A rigorous theoretical justification is given for measuring the TC of unperturbed LG beams with different values of radial and azimuthal numbers by means of an astigmatic transformation on a cylindrical lens. We have shown that the measured amplitude and phase spectra of the HG modes make it possible to find the orbital OAM and TC, as well as digitally sorting the HG modes and then restoring the initial singular beam.

Topological charges are calculated for a number of exact three-dimensional analytical solutions to the Landau–Lifshitz equation, which describe the distributions of the vector fields for the vectors of antiferromagnetism and antiferromagnet magnetization. It is shown that in the case of samples with dimensions that are comparable to the characteristic scales of topological objects of the antiferromagnetism and magnetization vector fields, there are modified characteristics that depend not only on the topological properties of these objects, but also on the geometry of the sample. These modified characteristics in samples with finite dimensions may assume non-integer values.

In contrast to the orbital angular momentum (OAM), which is conserved on free space propagation, the topological charge (TC) of a paraxial optical vortex (OV) is not conserved in the general case. Here, we investigate a Gaussian beam with a fractional TC in the original plane and demonstrate both theoretically and numerically how the TC changes in the course of propagation. Depending on the proximity of the topological charge to an even or odd integer number, an optical vortex with the original fractional TC is shown to behave in a number of different ways. For simple OVs (Laguerre-Gaussian or Bessel-Gaussian modes), TC is conserved both in propagation and after weak phase distortions. An experiment shows that when scattered by a random phase screen, the integer TC of an OV is conserved right up to a random phase variation of π. Therefore, in the case of weak turbulences, it is expedient to measure a discretely varying TC instead of a continuously varying OAM.

The relation between center vortices and fractional topological charge has been studied in this work. Specifically, we investigate a vortex configuration with fractional topological charge analyzing the lowest modes of the adjoint Dirac operator in both overlap and staggered formulations. We discuss the effects leading to chiral symmetry breaking.

Magnetic skyrmions are topological magnetic spin structures exhibiting particle-like behaviour. They are of strong interest from a fundamental viewpoint and for application, where they have potential to act as information carriers in future low-power computing technologies. Importantly, skyrmions have high physical stability because of topological protection. However, they have potential to deform according to their local energy environment. Here we demonstrate that, in regions of high exchange energy density, skyrmions may exhibit such extreme deformation that spontaneous merging with nearest neighbours or spawning new skyrmions is favoured to attain a lower energy state. Using transmission electron microscopy and a high-speed imaging detector, we observe dynamics involving distinct configurational states, in which transitions are accompanied by spontaneous creation or annihilation of skyrmions. These observations raise important questions regarding the limits of skyrmion stability and topological charge conservation, while also suggesting a means of control of skyrmion creation and annihilation., 19 pages total. 4 figures at the end of the main text, supplementary material follows

We derive the analytical expression of the probability density of orbital angular momentum modes for elegant and standard Laguerre–Gaussian beams in turbulence. The results show that we can increase the initial topological charge of the beams to reduce the detrimental mode crosstalk induced by the turbulence. We also discuss the influence of the beam parameters on the probability of signal and the variances of crosstalk probability. The results obtained show that the beams with smaller beam width and longer wavelength have higher detection probabilities of signal and are less affected by turbulence. Comparing with the results of elegant and standard Laguerre–Gaussian beams, we find that elegant Laguerre–Gaussian beams display stronger turbulence resistance. Our results will be useful in free-space optical communications.

Measuring the topological charge(TC) of optical vortices is of great importance in applications of vortex beams. A flexible and convenient method to determine the TC of an optical vortex is proposed by using an elliptical airy phase mask (EAPM). The mask can be embedded into a typical vortex generation setup, in which no additional optical elements are required. The number and the sign of TC are respectively obtained by analyzing the number and the direction of dark stripes in the far-field diffraction patterns of the vortex beam companied with EAPM. It is experimentally demonstrated that this method is able to measure TCs of both conventional and perfect vortex up to ±120.

Based on the theory of transformation optics (TO), a new and simple method for generating multimodal orbital angular momentum (OAM) carrying beams in 3-D space is proposed in this letter. The proposed 3-D arbitrary-way multimodal OAM generator only requires an electric dipole as the source. It consists of a four-way plane waves converter and four multimodal OAM generators. Since this method only requires the operation of radial scaling map, different topological charge modes can be easily obtained by appropriately setting the sizes of the physical geometries.

A topological meron features a non-coplanar structure, whose order parameters in the core region are perpendicular to those near the perimeter. A meron is half of a skyrmion, and both have potential applications for information carrying and storage. Although merons and skyrmions in ferromagnetic materials can be readily obtained via inter-spin interactions, their behaviour and even existence in ferroelectric materials are still elusive. Here we observe using electron microscopy not only the atomic morphology of merons with a topological charge of 1/2, but also a periodic meron lattice in ultrathin PbTiO3 films under tensile epitaxial strain on a SmScO3 substrate. Phase-field simulations rationalize the formation of merons for which an epitaxial strain, as a single alterable parameter, plays a critical role in the coupling of lattice and charge. This study suggests that by engineering strain at the nanoscale it should be possible to fabricate topological polar textures, which in turn could facilitate the development of nanoscale ferroelectric devices. Merons are topological structures, but these have yet to be directly observed in ferroelectrics. Here, by epitaxially straining PbTiO3 on a SmScO3 substrate, electron microscopy and phase-field modelling allow the morphology and distribution of merons to be observed.

Perfect optical vortex beams (POV) have attracted considerable attention in many novel applications because they have the advantage of a radial profile that is independent of the topological charge (TC). Nowadays, there are few effective methods to measure both the integer and the fractional TCs of the POV. In this paper, we achieve the precise measurement of arbitrary TCs through the approach of dynamic angular double slits (ADS), which performs the transformation from the POV to the interference intensity patterns at the angular bisector direction of the ADS. The information of the TC can be obtained from the periodically changing interference pattern. The deviation is less than 2% by comparing the theoretical values with the fitting results, therefore the detection method is effective and reasonable.