Total Topological Charge Research Articles

Splitting of an edge dislocation by a vortex emergent from a nonparaxial beam

The splitting of a linear or circular edge dislocation by a vortex nested in a nonparaxial Gaussian beam is studied. It is found that the edge dislocations are unstable and vanish, while new vortices may take place and the total topological charge may change with propagation. The location of the vortex, the intercept and slope of the edge dislocation, or the location and radius of the circular edge dislocation in the initial beam may affect the number and positions of vortices occurring in the field. The results are compared with previous works.

Phononic Weyl points and one-way topologically protected nontrivial phononic surface arc states in noncentrosymmetric WC-type materials

By means of first-principles calculations, we have predicted the existence of phononic Weyl points (WPs) along the high-symmetric K($\frac{1}{3},\frac{1}{3},0$)-H($\frac{1}{3},\frac{1}{3},\frac{1}{2}$) line in the Brillouin zone for a series of noncentrosymmetric hexagonal WC-type materials (HfS, HfSe, IrB, MoC, MoN, MoP, NbN, NbS, TaN, TaS, TiO, TiS, WN, ZrS, ZrSe, and ZrTe) and these WPs carry nonzero topological charges. In terms of symmetry analysis, we have further derived two-band $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ models to describe these WPs. For most WPs a first-order theory is sufficient to reproduce well the phonon spectra around these WPs. However, for some WPs a second-order theory has to be required. Particularly, when the second-order term plays a leading role, a topological charge $\ifmmode\pm\else\textpm\fi{}1\phantom{\rule{0.16em}{0ex}}\mathrm{WP}$ on the K-H line is accompanied by three nearby WPs with the opposite charges ($\ensuremath{\mp}1$) off the K-H line. These four spatially close (in the reciprocal space) WPs have a total topological charge of $\ensuremath{\mp}2$. On the crystal ($10\overline{1}0$) and ($01\overline{1}0$) surfaces, we have observed clear one-way topologically protected nontrivial phononic surface arc states connecting two WPs with opposite chirality. Our results pave the way for future experimental studies of the topological phonons for those WC-type materials.

Topological transport of vorticity in Heisenberg magnets

We study a robust topological transport carried by vortices in a thin film of an easy-plane ferromagnetic insulator between two metal contacts. A vortex, which is a nonlocal topological spin texture in two-dimensional magnets, exhibits some beneficial features as compared to skyrmions, which are local topological defects. In particular, the total topological charge carried by vorticity is robust against local fluctuations of the spin order-parameter magnitude. We show that an electric current in one of the magnetized metal contacts can pump vortices into the insulating bulk. Diffusion and nonlocal Coulomb-like interaction between these vortices will establish a steady-state vortex flow. Vortices leaving the bulk produce an electromotive force at another contact, which is related to the current-induced vorticity pumping by the Onsager reciprocity. The voltage signal decays algebraically with the separation between two contacts, similarly to a superfluid spin transport. Finally, the vorticity and closely related skyrmion type topological hydrodynamics are generalized to arbitrary dimensions, in terms of nonsingular order-parameter vector fields.

Near-field imaging of surface-plasmon vortex-modes around a single elliptical nanohole in a gold film

We present scanning near-field images of surface plasmon modes around a single elliptical nanohole in 88 nm thick Au film. We find that rotating surface plasmon vortex modes carrying extrinsic orbital angular momentum can be induced under linearly polarized illumination. The vortex modes are obtained only when the incident polarization direction differs from one of the ellipse axes. Such a direct observation of the vortex modes is possible thanks to the ability of the SNOM technique to obtain information on both the amplitude and the phase of the near-field. The presence of the vortex mode is determined by the rotational symmetry breaking of the system. Finite element method calculations show that such a vorticity originates from the presence of nodal points where the phase of the field is undefined, leading to a circulation of the energy flow. The configuration producing vortex modes corresponds to a nonzero total topological charge (+1).

Entanglement spectroscopy of non-Abelian anyons: Reading off quantum dimensions of individual anyons

We study the entanglement spectrum of topological systems hosting non-Abelian anyons. Akin to energy levels of a Hamiltonian, the entanglement spectrum is composed of symmetry multiplets. We find that the ratio between different eigenvalues within one multiplet is universal and is determined by the anyonic quantum dimensions. This result is a consequence of the conservation of the total topological charge. For systems with non-Abelian topological order, this generalizes known degeneracies of the entanglement spectrum, which are hallmarks of topological states. Experimental detection of these entanglement spectrum signatures may become possible in Majorana wires using multicopy schemes, allowing the measurement of quantum entanglement and its symmetry resolution.

Defects in vertically vibrated monolayers of cylinders

We analyse liquid-crystalline ordering in vertically vibrated monolayers of cylinders confined in a circular cavity. Short cylinders form tetratic arrangements with C4 symmetry. This symmetry, which is incompatible with the geometry of the cavity, is restored by the presence of four point defects with total topological charge +4. Equilibrium Monte Carlo simulations predict the same structure. A new method to measure the elastic properties of the tetratic medium is developed which exploits the clear similarities between the vibrated dissipative system and the thermal equilibrium system. Our observations open up a new avenue to investigate the formation of defects in response to boundary conditions, an issue which is very difficult to realise in colloidal or molecular systems.

Optical angular momentum manipulations in a four-wave mixing process

We investigate the spatial and quantum intensity correlations between the probe and Stokes optical fields produced via four-wave mixing in a double-Λ configuration, when both incoming probe and control fields carry non-zero optical orbital angular momentum (OAM). We observed that the topological charge of the generated Stokes field obeyed the OAM conservation law. However, the maximum values and optimal conditions for the intensity squeezing between the probe and Stokes fields were largely independent of the angular momenta of the beams, even when these two fields had significantly different OAM charges. We also investigated the case of a composite-vortex pump field, containing two closely positioned optical vortices, and showed that the generated Stokes field carried the OAM corresponding to the total topological charge of the pump field, further expanding the range of possible OAM manipulation techniques.

Skyrmionium \u2013 high velocity without the skyrmion Hall effect

The lateral motion of a magnetic skyrmion, arising because of the skyrmion Hall effect, imposes a number of restrictions on the use of this spin state in the racetrack memory. A skyrmionium is a more promising spin texture for memory applications, since it has zero total topological charge and propagates strictly along a nanotrack. Here, the stability of the skyrmionium, as well as the dependence of its size on the magnetic parameters, such as the Dzyaloshinskii–Moriya interaction and perpendicular magnetic anisotropy, are studied by means of micromagnetic simulations. We propose an advanced method for the skyrmionium nucleation due to a local enhancement of the spin Hall effect. The stability of the skyrmionium being in motion under the action of the spin polarized current is analyzed.

Artificial graphenes: Dirac matter beyond condensed matter

After the discovery of graphene and its many fascinating properties, there has been a growing interest for the study of "artificial graphenes". These are totally different and novel systems which bear exciting similarities with graphene. Among them are lattices of ultracold atoms, microwave or photonic lattices, "molecular graphene" or new compounds like phosphorene. The advantage of these structures is that they serve as new playgrounds for measuring and testing physical phenomena which may not be reachable in graphene, in particular: the possibility of controlling the existence of Dirac points (or Dirac cones) existing in the electronic spectrum of graphene, of performing interference experiments in reciprocal space, of probing geometrical properties of the wave functions, of manipulating edge states, etc. These cones, which describe the band structure in the vicinity of the two connected energy bands, are characterized by a topological "charge". They can be moved in reciprocal space by appropriate modification of external parameters (pressure, twist, sliding, stress, etc.). They can be manipulated, created or suppressed under the condition that the total topological charge be conserved. In this short review, I discuss several aspects of the scenarios of merging or emergence of Dirac points as well as the experimental investigations of these scenarios in condensed matter and beyond.

Nematic topological defects positionally controlled by geometry and external fields.

Using a Landau–de Gennes approach, we study the impact of confinement topology, geometry and external fields on the spatial positioning of nematic topological defects (TDs). In quasi two-dimensional systems we demonstrate that a confinement-enforced total topological charge of m > 1/2 decays into elementary TDs bearing a charge of m = 1/2. These assemble close to the bounding substrate to enable essentially bulk-like uniform nematic ordering in the central part of a system. This effect is reminiscent of the Faraday cavity phenomenon in electrostatics. We observe that in certain confinement geometries, varying the correlation length size of the order parameter could trigger a global rotation of an assembly of TDs. Finally, we show that an external electric field could be used to drag the boojum fingertip towards the interior of the confinement cell. Assemblies of TDs could be exploited as traps for appropriate nanoparticles, opening several opportunities for the development of functional nanodevices.

Points, skyrmions and torons in chiral nematic droplets.

Chiral nematic droplets with perpendicular surface alignment of liquid crystalline molecules frustrate the helical structure into convoluted 3D textures with complex topology. We observe the droplets with fluorescent confocal polarising microscopy (FCPM), and reconstruct and analyse for the first time the topology of the 3D director field using a novel method of director reconstruction from raw data. We always find an odd number of topological defects, which preserve the total topological charge of the droplet of +1 regardless of chirality. At higher chirality, we observe up to 5 point hedgehog defects, which are elastically stabilized with convoluted twisted structures, reminiscent of 2D skyrmions and toron-like structure, nested into a sphere.

Novel Z(2) Topological Metals and Semimetals.

We report two theoretical discoveries for Z(2) topological metals and semimetals. It is shown first that any dimensional Z(2) Fermi surface is topologically equivalent to a Fermi point. Then the famous conventional no-go theorem, which was merely proven before for Z Fermi points in a periodic system without any discrete symmetry, is generalized so that the total topological charge is zero for all cases. Most remarkably, we find and prove an unconventional strong no-go theorem: all Z(2) Fermi points have the same topological charge ν(Z(2))=1 or 0 for periodic systems. Moreover, we also establish all six topological types of Z(2) models for realistic physical dimensions.

Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium

Expressions for the electric field at a sum frequency generated by a collinear elliptically polarized Gaussian beam and circularly polarized Laguerre-Gaussian beam in an isotropic chiral nonlinear medium are obtained in quadratures. The amount and locations of $C$ points in the cross section of a signal beam at a sum frequency are shown to be dependent on frequency and diffraction lengths ratios of fundamental beams and on the ellipticity degree of the Gaussian beam's polarization ellipse. Possible values of total topological charges of the emergent $C$ points are determined by the topological charge of the Laguerre-Gaussian beam and remain constant while the radiation propagates in nonlinear media. In case of nonzero total topological charge $C$ lines form helical structures, the parameters of which depend on the wave-vector mismatch. Otherwise, $C$ lines form a loop. As the wave-vector mismatch grows the loop undergoes deformation and breaks up, creating new $C$ lines.

Defect unbinding on a toroidal nematic shell.

We study nematic liquid crystal textures exhibiting topological defects (TDs) on a two-dimensional (2D) toroidal shell. For the toroidal topology the total topological charge of TDs is equal to zero. We use a mesoscopic Landau-de Gennes approach which features a 2D nematic order tensor Q. We show that fat tori unbind TDs. If no extrinsic free energy couples Q with the Weingarten tensor of the torus, then defects and antidefects are assembled along the innermost and the outermost circles of the torus, respectively. In this case, we estimate the critical condition for the onset of TDs using an electrostatic analogy. If, on the other hand, an extrinsic free energy is present, then defects are repelled from these regions.

Extended topological defects as sources and outlets of dislocations in spherical hexagonal crystals

Extended topological defects (ETDs) arising in spherical hexagonal crystals due to their curvature are considered. These prevalent defects carry a unit total topological charge and are surrounded by scalene pentagonal boundaries. Topological peculiarities of reactions between ETDs and dislocations are considered. Similarly to boundaries of the usual planar crystalline order the ETDs emit and absorb the dislocations without preservation of their dislocational charge. Dislocations located inside the ETD area lose it and the enforced ETD decay can proceed in different ways without conservation of the total Burgers vector of the dislocations emitted.

Biskyrmion states and their current-driven motion in a layered manganite

The magnetic skyrmion is a topologically stable spin texture in which the constituent spins point to all the directions wrapping a sphere. Generation and control of nanometric magnetic skyrmions have large potential, for example, reduced power consumption, in spintronics device applications. Here we show the real-space observation of a biskyrmion, as defined by a molecular form of two bound skyrmions with the total topological charge of 2, realized under magnetic field applied normal to a thin plate of a bilayered manganite with centrosymmetric structure. In terms of a Lorentz transmission electron microscopy (TEM), we have observed a distorted-triangle lattice of biskyrmion crystal, each composed of two bound skyrmions with oppositely swirling spins (magnetic helicities). Furthermore, we demonstrate that these biskyrmions can be electrically driven with orders of magnitude lower current density (<10(8) A m(-2)) than that for the conventional ferromagnetic domain walls.

Topological Defects in Flat Geometry: The Role of Density Inhomogeneity

Topological defects are found in particles confined to planar disks interacting via the 1/r Coulomb potential. The total interior topological charge is found to monotonically converge to a negative value as the energy decreases during the relaxation process regardless of initial configurations; it is more negative in a larger cluster. The comparison with a uniform hyperbolic tessellation reveals an underlying hyperbolic structure in a low-energy configuration where the particle density increases from the center of the disk to its boundary. An elliptic structure is identified in an opposite particle distribution where the particle density decreases from the center to the edge of the disk. The novel mechanism of density inhomogeneity driven topological defects as well as the underlying geometric structure may shed light in understanding a wide variety of relevant systems.

Propagation of Riemann–Silberstein vortices through an astigmatic lens

The propagation of Riemann-Silberstein (RS) vortices for Gaussian vortex beams with topological charges m=+1 through a lens is studied. It is shown that if there is an ideal lens, a RS vortex and a circular edge dislocation appear for Gaussian on-axis vortex beams, while only RS vortices take place for Gaussian off-axis vortex beams. In the presence of an astigmatic lens, there exist RS vortices but no edge dislocations for both Gaussian on-axis and off-axis beams. By varying the astigmatic coefficient, the off-axis parameter, and the propagation distance, the motion, creation, and annihilation of vortices may take place, and in the process, the total topological charge of RS vortices remains unchanged.

An ideal toy model for confining, walking and conformal gauge theories: the O(3) sigma model with ϑ-term

A toy model is proposed for four dimensional non-abelian gauge theories coupled to a large number of fermionic degrees of freedom. As the number of flavors is varied the gauge theory may be confining, walking or conformal. The toy model mimicking this feature is the two dimensional O(3) sigma model with a theta-term. For all theta the model is asymptotically free. For small theta the model is confining in the infra red, for theta = pi the model has a non-trivial infra red fixed point and consequently for theta slightly below pi the coupling walks. The first step in investigating the notoriously difficult systematic effects of the gauge theory in the toy model is to establish non-perturbatively that the theta parameter is actually a relevant coupling. This is done by showing that there exist quantities that are entirely given by the total topological charge and are well defined in the continuum limit and are non-zero, despite the fact that the topological susceptibility is divergent. More precisely it is established that the differences of connected correlation functions of the topological charge (the cumulants) are finite and non-zero and consequently there is only a single divergent parameter in Z(theta) but otherwise it is finite. This divergent constant can be removed by an appropriate counter term rendering the theory completely finite even at theta > 0.

Coupling of spin and angular momentum of light in plasmonic vortex

We present that two distinct optical properties of light, the spin angular momentum (SAM) and the orbital angular momentum (OAM), can be coupled in the plasmonic vortex. If a plasmonic vortex lens (PVL) is illuminated by the helical vector beam (HVB) with the SAM and OAM, then those distinct angular momenta contribute to the generation of the plasmonic vortex together. The analytical model reveals that the total topological charge of the generated plasmonic vortex is given by a linear summation of those of the SAM and OAM, as well as the geometric charge of the PVL. The generation of the plasmonic vortex and the manipulation of the fractional topological charge are also presented.