Unit Topological Charge Research Articles

Baryons as vortexes on the η′ domain wall

We show that the recent construction of Nf = 1 baryons on the η′ domain wall can be understood as vortexes of the principal effective theory — the Chern-Simons-Higgs theory — on a 2+1-dimensional sheet. This theory has a series of vortex solutions, and the vortex with unit topological charge naturally spins Nc/2, which coincides with the spin of the one-flavor baryon in QCD. Since the Nc scaling of the vortexes is the same as that of baryons, the latter can be regarded as the former. By virtue of the particle-vortex symmetry, the dual Zhang-Hansson-Kivelson theory indicates that the quark carries topological charge 1/Nc and obeys fractional statistics. The generalization to arbitrary Nf is also discussed.

Vortex position detection using a scanning triangular aperture in a digital micromirror device

Optical vortices, as solutions to the wave equation, remain stable under ideal theoretical conditions. However, in experiments where beams produced with holograms and spatial light modulators introduce perturbations, their phase structures can be disrupted, leading to instability. Specifically, higher-order optical vortices (with topological charge |ℓ|>1 ) are inherently unstable in that they tend to separate in a series of optical vortices with unit topological charge (|ℓ|=1) even with small perturbations. In this work, we demonstrate a technique to detect the positions of optical vortices using a scanning triangular aperture in a digital micromirror device and a digital pinhole. We observe the diffraction patterns of a vortex through the triangular aperture, and we show that we can determine the center of an optical vortex (OV) by measuring the intensity at the center of a pattern using a digital pinhole. We investigate the changes in the measured signal for different triangular apertures and pinhole sizes. We observe that while smaller pinholes lead to a decrease in light intensity, the detected position of a single OV with unit topological charge is similar for all pinhole sizes. We also find that the triangular aperture size becomes important when locating vortex pairs. Using a triangular aperture with a radius R = 0.52 mm, we can resolve OV pairs at least 86.4μ m apart in our experiments. Our methods help pave the way to understanding the fundamental behavior of multiple vortex interactions in optical beams and also can be used when determining the position of an OV in metrology.

Solitons and vortices formation in deformed photonic graphene

We investigate the formation and evolution of two-dimensional solitons and vortices in deformed photonic graphene, encompassing fundamental solitons, multipole solitons, and vortex solitons. Using the plane-wave expansion method, we analyze the bandgap structure of photonic graphene, revealing Dirac cones whose characteristics are influenced by deformation. Soliton generation results from lattice potential deformation, linear energy gap modulation, and nonlinear effects. Specifically, fundamental and dipole solitons, exhibiting out-of-phase behavior, propagate robustly within photonic graphene, maintaining structural integrity over extended distances, while other soliton solutions are unstable. Furthermore, we explore the realization and propagation characteristics of vortex solitons with unit topological charge within photonic graphene.

Fundamental, dipole, and vortex solitons in fractional nonlinear Schrödinger equation with a parity-time-symmetric periodic potential

We investigate the existence and stability of two-dimensional (2D) solitons in the framework of the nonlinear fractional Schrödinger equation (NLFSE) with focusing nonlinearity and a parity-time (PT) symmetric periodic potential. Numerical studies reveal that the fundamental, dipole, and localized vortex solitons exist in a semi-infinite band gap. The Lévy index (α) cannot change the phase transition point of the PT-symmetric periodic potential, but it affects the existence and stability of these solitons significantly. With the growth of the Lévy index, the stability regions become markedly wider for the fundamental, dipole solitons and vortex solitons with the topological charge of 2 (S=2). But for a unit topological charge (S=1), the vortex soliton cannot maintain its stability under the usual diffraction effect (α=2). The result shows that the fractional-order diffraction helps to stabilize the vortex solitons with S=1.

High-order optical Hall effect at the tight focus of laser radiation

In this work, by the Richards-Wolf method, which describes the behavior of electromagnetic radiation at the tight focus, it is shown that high-order spin and orbital Hall effects take place in the focal plane. It is shown that when focusing a linearly polarized optical vortex with unit topological charge, four local subwavelength regions are formed in the focal plane, in which directions of the longitudinal projection of the spin angular momentum are opposite in the neighboring regions. That is, photons falling into neighboring regions in the focus have the opposite spin. This is the spin Hall effect of the 2nd order. It is also shown that when tightly focusing of superposition of cylindrical vector beams of the m-th order and zero order, 2m subwavelength regions are formed in the plane of tight focus, in which directions of the longitudinal projection of the orbital angular momentum are opposite in the neighboring regions. That is, photons falling into the neighboring regions at the focus have the opposite-sign on-axis projections of the orbital angular momentum. This is the orbital Hall effect of the m-th order.

Vortex beam generator based on exceptional topological phase in terahertz all-dielectric metasurface

In this study, an all-dielectric metasurface with an F-shaped chiral structure in the terahertz region has been proposed. The coupling between the electric and magnetic dipole modes supported by the metasurface can be adjusted by varying the structural parameters. In the parameter space composed of l l and s, any closed path around exceptional point in the circular polarization conversion channel t rl generates a topologically protected exceptional topological phase. A vortex beam generator with a unit topological charge was proposed to provide a novel concept for designing terahertz metasurface wavefront modulation devices.

Bloch points and topological dipoles observed by X-ray vector magnetic tomography in a ferromagnetic microstructure

Bloch points are small 3D magnetic textures with unit topological charge that require the use of advanced vector imaging techniques for their direct experimental characterization. Here we show results from the reconstruction of the magnetization field {{{{{bf{m}}}}}}left({{{{{bf{r}}}}}}right) of an elongated permalloy microstructure by X-ray vector magnetic tomography. A central asymmetric Bloch domain wall is observed, decorated by Bloch points arranged in several dipoles and a triplet. The analysis of the {{{{{bf{m}}}}}}left({{{{{bf{r}}}}}}right) map in terms of topological concepts provides a quantitative description of the Bloch points as topological monopoles connected by bundles of emergent magnetic field lines carrying a fractional topological flux. It also reveals the topological constraints that determine chirality transitions of the central domain wall in the vicinity of Bloch points, independent of specific material properties. This approach could be readily extended to the study of magnetic microstructures in arbitrary remanent configurations.

High-Order Orbital and Spin Hall Effects at the Tight Focus of Laser Beams

In this paper, using a Richards–Wolf method, which describes the behavior of electromagnetic waves at the sharp focus, we show that high-order spin and orbital Hall effects take place at the focal plane of tightly focused laser beams. We reveal that four local subwavelength regions are formed at the focus of a linearly polarized optical vortex with unit topological charge, where the spin angular momentum behaves in a special way. Longitudinal projections of the spin angular momentum are oppositely directed in the adjacent regions. We conclude that this is because photons falling into the neighboring regions at the focus have the opposite spin. This newly observed phenomenon may be called a spin Hall effect of the 4-th order. We also show that tightly focusing the superposition of cylindrical vector beams of the m-th and zero-order produces 2m subwavelength regions in the focal plane, such that longitudinal projections of the orbital angular momentum are oppositely directed in the neighboring regions. This occurs because photons falling into the neighboring regions at the focus have the opposite signs of the on-axis projections of the orbital angular momentum. This phenomenon may be termed an orbital Hall effect of the 2m-th order.

Conformality and self-duality of Nf = 2 QED3

We study the IR phase of three dimensional quantum electrodynamics (QED3) coupled to Nf=2 flavors of two-component Dirac fermions, which has been controversial for decades. This theory has been proposed to be self-dual with symmetry enhancement (SU(2)f×U(1)t)/Z2→O(4) at the IR fixed point. We focus on the four-point correlator of monopole operators with unit topological charge of U(1)t. We illustrate the O(4)→SU(2)f×U(1)t branching rules based on an O(4) symmetric positive structure in the monopole four-point crossing equations. We use conformal bootstrap method to derive nonperturbative constraints on the CFT data and test the conformality and self-duality of Nf=2 QED3. In particular we find the CFT data obtained from previous lattice simulations can be ruled out by introducing irrelevant assumptions in the spectrum, indicating the IR phase of Nf=2 QED3 is not conformal.

Sinusoidal Gaussian optical vortex as a superposition of two hypergeometric beams

We analyze the propagation of hypergeometric beams with a parabolic initial wavefront in a homogeneous medium. While hypergeomentric beams have a central amplitude singularity in the initial plane and are of infinite energy, superposition of two such beams has no singularity and is of finite energy. A particular case of such a superposition we study in detail is a sinusoidal Gaussian beam with a unit topological charge. This beam belongs to the class of elegant laser beams since it is described by the same complex-argument function both in the initial plane and in the Fresnel diffraction zone. The diameter of the first light ring of the sinusoidal Gaussian beam is almost independent of the Gaussian beam waist radius.

A minimal subwavelength focal spot for the energy flux

It is shown theoretically and numerically that circularly and linearly polarized incident beams produce at the tight focus identical circularly symmetric distributions of an on-axis energy flux. It is also shown that the on-axis energy fluxes from radially and azimuthally polarized optical vortices with unit topological charge are equal to each other. An optical vortex with azimuthal polarization is found to generate the minimum focal spot measured for the intensity (all other parameters being equal). Slightly larger (by a fraction of a percent) is the spot size calculated for the energy flux for the circularly and linearly polarized light. The spot size in terms of intensity is of importance in light-matter interaction, whereas the spot size in terms of energy flux affects the resolution in optical microscopy.

Minimal Focal Spot Size Measured Based on Intensity and Power Flow.

It is shown, theoretically and numerically, that the distributions of the longitudinal energy flow for tightly focused light with circular and linear polarization are the same, and that the spot has circular symmetry. It is also shown that the longitudinal energy flows are equal for optical vortices with unit topological charge and with radial or azimuthal polarization. The focal spot has a minimum diameter (all other characteristics being equal), which is measured based on the intensity of an optical vortex with azimuthal polarization. The diameter of the focal spot calculated from the energy flow for light with circular or linear polarization is slightly larger (by a fraction of a percentage). The magnitude of the diameter based on the intensity plays a role in the interaction of light with matter, and the magnitude of the diameter based on the energy flux affects the resolution in optical microscopy which is crucial in sensorial applications.

Optical beams with an infinite number of vortices

In optical data transmission with using vortex laser beams, data can be encoded by the topo-logical charge, which is theoretically unlimited. However, the topological charge of a single sepa-rate vortex is limited by possibilities of its generating. Therefore, in this work, we analyze light beams with an unbounded (countable) set of optical vortices. The summary topological charge of such beams is infinite. Phase singularities (isolated intensity nulls) in such beams typically have a unit topological charge and reside equidistantly (or not equidistantly) on a straight line in the beam cross section. Such beams are form-invariant and, on propagation in space, change only in scale and rotate. Orbital angular momentum of such multivortex beams is finite, since only a finite number of optical vortices fall into the area, where the Gaussian beam has a notable intensity. Other phase singularities are located in the periphery (and at the infinity), where the intensity is almost zero.

Converting an nth-order edge dislocation to a set of optical vortices

We show theoretically that an astigmatic transformation of an edge dislocation (a straight line of zero intensity) of the n-th order forms n elliptical optical vortices (screw dislocations) with a unit topological charge at twice the focal distance from a cylindrical lens. The vortices are located on a straight line perpendicular to the edge dislocation at points with coordinates equal to the roots of an n-th order Hermite polynomial. The orbital angular momentum of an edge dislocation with an astigmatic phase is found to be proportional to n.

Transformation of a high-order edge dislocation to optical vortices (spiral dislocations)

We theoretically show that an astigmatic transformation of an nth-order edge dislocation (a zero-intensity straight line) produces n optical elliptical vortices (spiral dislocations) with unit topological charge at the double focal distance from the cylindrical lens, located on a straight line perpendicular to the edge dislocation, at points whose coordinates are the roots of an nth-order Hermite polynomial. The orbital angular momentum of the edge dislocation is proportional to the order n.

Skyrmions at vanishingly small Dzyaloshinskii–Moriya interaction or zero magnetic field

By introducing biquadratic together with usual bilinear ferromagnetic nearest neighbor exchange interaction in a square lattice, we find that the energy of the spin-wave mode is minimized at a finite wavevector for a vanishingly small Dzyaloshinskii–Moriya interaction (DMI), supporting a ground state with spin-spiral structure whose pitch length is unusually short as found in some of the experiments. Apart from reproducing the magnetic structures that can be obtained in a canonical model with nearest neighbor exchange interaction only, a numerical simulation of this model with further introduction of magnetic anisotropy and magnetic field predicts many other magnetic structures some of which are already observed in the experiments. Among many observed structures, nanoscale skyrmion even at vanishingly small DMI is found for the first time in a model. The model provides the nanoscale skyrmions of unit topological charge at zero magnetic field as well. We obtain phase diagrams for all the magnetic structures predicted in the model.

Multiply charged vortex states of polariton condensates

The existence of quantized vortices is a key feature of Bose–Einstein condensates. In equilibrium condensates, only quantum vortices of unit topological charge are stable, due to the dynamical instabilities of multiply charged defects, unless supported by strong external rotation. Due to immense interest in the physics of these topological excitations, a great deal of work has been done to understand how to force their stability. Here we show that in photonic Bose–Einstein condensates of exciton–polariton quasiparticles pumped in an annular geometry, not only do the constant particle fluxes intrinsic to the system naturally stabilize multiply charged vortex states, but that such states can indeed form spontaneously during the condensate formation through a dynamic symmetry breaking mechanism. We elucidate the properties of these states, notably finding that they radiate acoustically at topologically quantized frequencies. Finally, we show that the vorticity of these photonic fluids is limited by a quantum Kelvin–Helmholtz instability, and therefore by the condensate radius and pumping intensity. This reported instability in a quantum photonic fluid represents a fundamental result in fluid dynamics.

Zeroth- and first-order long range non-diffracting Gauss\u2013Bessel beams generated by annihilating multiple-charged optical vortices

We demonstrate an alternative approach for generating zeroth- and first-order long range non-diffracting Gauss–Bessel beams (GBBs). Starting from a Gaussian beam, the key point is the creation of a bright ring-shaped beam with a large radius-to-width ratio, which is subsequently Fourier-transformed by a thin lens. The phase profile required for creating zeroth-order GBBs is flat and helical for first-order GBBs with unit topological charge (TC). Both the ring-shaped beam and the required phase profile can be realized by creating highly charged optical vortices by a spatial light modulator and annihilating them by using a second modulator of the same type. The generated long-range GBBs are proven to have negligible transverse evolution up to 2 m and can be regarded as non-diffracting. The influences of the charge state of the TCs, the propagation distance behind the focusing lens, and the GBB profiles on the relative intensities of the peak/rings are discussed. The method is much more efficient as compared to this using annular slits in the back focal plane of lenses. Moreover, at large propagation distances the quality of the generated GBBs significantly surpasses this of GBBs created by low angle axicons. The developed analytical model reproduces the experimental data. The presented method is flexible, easily realizable by using a spatial light modulator, does not require any special optical elements and, thus, is accessible in many laboratories.

Near-BPS baby Skyrmions

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 ϵ. Classical solutions of this model, to leading order in ϵ, are called restricted harmonic maps. In the BPS limit (ϵ → 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 ϵ with which we can calculate the corrections to the BPS mass. The leading order ( mathcal{O}left({upepsilon}^1right) ) 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 ϵ to capture the relevant terms in perturbation theory, however, the binding energy contributes to the total energy at order ϵ2. 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.

Vortical configurations in mesoscopic superconducting square loop with mixed pairing orders

Based on the microscopic Bogoliubov-de Gennes theory, we study vortical configurations in a mesoscopic superconducting square loop with mixed pairing orders. By choosing appropriate interaction strengths and chemical potential, the spin-triplet p-wave and subdominant spin-singlet d-wave pairing symmetries can be stabilized. In a small external magnetic field, unusual elliptical-like vortices in chiral p-wave orders as well as elongated vortices in the d-wave order tend to form in sample diagonals. For an enlarged flux, we find a two-quanta skyrmionic mode for the p-wave components, while the closed domain wall does not appear compared to the previous studies in superconducting systems with a pure p-wave state. Particularly, a novel multi-skyrmionic pattern containing four single-quanta skyrmions takes place with increasing flux. Such a single-quanta vortex structure consisting of two spatially separated half-quantum vortices is analogous to the nematic skyrmion with unit topological charge. In addition, several complex hybrid or multiple skyrmionic configurations are revealed in the strong flux range.