Nonsingular Vortices Research Articles

The Abrikosov vortex in curved space

We study the self-gravitating Abrikosov vortex in curved space with and with-out a (negative) cosmological constant, considering both singular and non-singular solutions with an eye to hairy black holes. In the asymptotically flat case, we find that non-singular vortices round off the singularity of the point particle’s metric in 3 dimensions, whereas singular solutions consist of vortices holding a conical singularity at their core. There are no black hole vortex solutions. In the asymptotically AdS case, in addition to these solutions there exist singular solutions containing a BTZ black hole, but they are always hairless. So we find that in contrast with 4-dimensional ’t Hooft-Polyakov monopoles, which can be regarded as their higher-dimensional analogues, Abrikosov vortices cannot hold a black hole at their core. We also describe the implications of these results in the context of AdS/CFT and propose an interpretation for their CFT dual along the lines of the holographic superconductor.

Singular and regular vortices on top of a background pulled to the center

A recent analysis has revealed singular but physically relevant localized two-dimensional vortex states with density ∼r −4/3 at and a convergent total norm, which are maintained by the interplay of the potential of the attraction to the center, ∼−r −2, and a self-repulsive quartic nonlinearity, produced by the Lee-Huang-Yang correction to the mean-field dynamics of Bose–Einstein condensates. In optics, a similar setting, with the density singularity ∼r −1, is realized with the help of quintic self-defocusing. Here we present physically relevant antidark singular-vortex states in these systems, existing on top of a flat background. Numerical solutions for them are very accurately approximated by the Thomas–Fermi wave function. Their stability exactly obeys an analytical criterion derived from the consideration of small perturbations. The singular-vortex states exist as well in the case when the effective potential is weakly repulsive. It is demonstrated that the singular vortices can be excited by the input in the form of the ordinary nonsingular vortices, hence the singular modes can be created in the experiment. We also consider regular (dark) vortices maintained by the flat background, under the action of the repulsive central potential ∼+r −2. The dark modes with vorticities l = 0 and 1 are completely stable. In the case when the central potential is attractive, but the effective one, which includes the centrifugal term, is repulsive, and, in addition, a weak trapping potential ∼r 2 is applied, dark vortices with l = 1 feature an intricate pattern of alternating stability and instability regions. Under the action of the instability, states with l = 1 travel along tangled trajectories, which stay in a finite area defined by the trap. The analysis is also reported for dark vortices with l = 2, which feature a complex structure of alternating intervals of stability and instability against splitting. Lastly, simple but novel flat vortices are found at the border between the anidark and dark ones.

Controlled creation and decay of singly-quantized vortices in a polar magnetic phase

Quantized vortices appear in physical systems from superfluids and superconductors to liquid crystals and high energy physics. Unlike their scalar cousins, superfluids with complex internal structure can exhibit rich dynamics of decay and even fractional vorticity. Here, we experimentally and theoretically explore the creation and time evolution of vortex lines in the polar magnetic phase of a trapped spin-1 87Rb Bose–Einstein condensate. A process of phase-imprinting a nonsingular vortex, its decay into a pair of singular spinor vortices, and a rapid exchange of magnetic phases creates a pair of three-dimensional, singular singly-quantized vortex lines with core regions that are filled with atoms in the ferromagnetic phase. Atomic interactions guide the subsequent vortex dynamics, leading to core structures that suggest the decay of the singly-quantized vortices into half-quantum vortices.

Non-singular vortices with positive mass in 2+1-dimensional Einstein gravity with AdS3 and Minkowski background

In previous work, black hole vortex solutions in Einstein gravity with AdS3 background were found where the scalar matter profile had a singularity at the origin r = 0. In this paper, we find numerically static vortex solutions where the scalar and gauge fields have a non-singular profile under Einstein gravity in an AdS3 background. Vortices with different winding numbers n, VEV v and cosmological constant Λ are obtained. These vortices have positive mass and are not BTZ black holes as they have no event horizon. The mass is determined in two ways: by subtracting the numerical values of two separate asymptotic metrics and via an integral that is purely over the matter fields. The mass of the vortex increases as the cosmological constant becomes more negative and this coincides with the core of the vortex becoming smaller (compressed). We then consider the vortex with gravity in asymptotically flat spacetime for different values of the coupling α = 1/(16πG). At the origin, the spacetime has its highest curvature and there is no singularity. It transitions to an asymptotic conical spacetime with angular deficit that increases significantly as α decreases. For comparison, we also consider the vortex without gravity in flat spacetime. For this case, one cannot obtain the mass by the first method (subtracting two metrics) but remarkably, via a limiting procedure, one can obtain an integral mass formula. In the absence of gauge fields, there is a well-known logarithmic divergence in the energy of the vortex. With gravity, we present this divergence in a new light. We show that the metric acquires a logarithmic term which is the 2 + 1 dimensional realization of the Newtonian gravitational potential when General Relativity is supplemented with a scalar field. This opens up novel possibilities which we discuss in the conclusion.

Trailing-edge noise generation from a flat-plate aerofoil interacting with a prescribed vortex

Aerodynamic noise generated by aerofoil-vortex interaction has widely been studied in the past where the primary noise source mechanism is the scattering of the vortex at the leading edge (LE) of the aerofoil. In this paper the secondary source mechanism – the subsequent vortical scattering at the trailing edge (TE) – is investigated in detail, which shows that this mechanism is essentially due to nonlinear effects as predicted by some analytical studies in the past. The present study is performed by employing high-resolution numerical simulations based on a prescribed non-singular vortex impinging on a flat-plate aerofoil with zero mean loading. The present work investigates both inviscid and viscous flow conditions. The inviscid flow condition is intended to support and extend from the existing theoretical works, whereas the viscous one leads to more realistic findings. The current viscous study involves laminar boundary layers and their convective instability travelling with the impinged vortex. One of the most notable observations made in this work is that the vortical scattering at the TE (in the absence of turbulent boundary layers) may become a dictating source of noise at high frequencies (surpassing the primary source at the LE) across a wide range of observer angles for both inviscid and viscous flow conditions. It is found that the high-frequency dominant source is produced by secondary near-wall vortices that are induced as a result of nonlinear interactions between the aerofoil and the impinging vortex. This new discovery makes a contrast to the existing knowledge on aerofoil-vortex interaction noise in which the secondary source is normally assumed inferior to the primary at all frequencies.

Vortex lattice in rotating neutron spin-triplet superfluid

ABSTRACT In the Ginzburg–Landau limit, a possible texture without singular cores in a rotating neutron spin-triplet superfluid is studied. The lattice of non-singular vortices is constructed with a vorticity diffusely distributed over the entire unit cell. The upper limit of the free energy associated with this structure is estimated, and the result shows that non-singular vortices are more preferable in the core of rotating neutron stars than ordinary linear singular vortices. The order parameter of the studied neutron system belongs to the unitary class. This implies that the superfluid under consideration does not have spin polarization and the core-less 3P2 vortices do not have magnetization, which makes electron scattering by vortices ineffective. For the same reason, pinning with flux tubes is also suppressed. This can affect the hydrodynamics of superfluid neutron stars.

Numerical solutions of the singular vortex problem

This study develops a finite-difference numerical formulation to describe the motion of a singular monopole in a quasigeostrophic β-channel model with scale-selective frictional damping, using parameter values typical for the middle-latitude atmosphere and a wide range of viscosities. In this model, the “theoretical” singular vortex is replaced by the equivalent nonsingular vortex of a finite amplitude, consistent with the finite spatial resolution of the model. Numerical experiments demonstrate that at initial stages of the singular-vortex (SV) evolution, this model accurately reproduces the behavior expected from the theoretical considerations of the inviscid case. The numerical model also approximately conserves several invariants of motion derived from the continuous equations and accurately represents their modifications in the presence of friction. The evolution of a singular cyclone in the Northern Hemisphere starts with the development of the dipolar β gyres in the regular component of the flow; these gyres induce initial northward displacement and subsequent westward bending of the cyclone trajectory. At larger times, the β gyres gradually disintegrate, and the singular cyclone in the Northern Hemisphere continues to move northwestward by forming a dipolelike system with the localized secondary regular-field anticyclone northeast of the singular-cyclone center resulting eventually in a friction-assisted steady-state regime. The SV trajectories tend to become more zonally elongated for large vortices and small values of viscosity. These results lay a foundation for numerical consideration of systems of multiple singular vortices, which could provide further insights into our fundamental understanding of the processes underlying the multiscale atmospheric and oceanic variability.

Motion of vortices in ferromagnetic spin-1 BEC

The paper investigates dynamics of nonsingular vortices in a ferromagnetic spin-1 BEC, where spin and mass superfluidity coexist in the presence of uniaxial anisotropy (linear and quadratic Zeeman effect). The analysis is based on hydrodynamics following from the Gross-Pitaevskii theory. Cores of nonsingular vortices are skyrmions with charge, which is tuned by uniaxial anisotropy and can have any fractal value between 0 and 1. There are circulations of mass and spin currents around these vortices. The results are compared with the equation of vortex motion derived earlier in the Landau-Lifshitz-Gilbert theory for magnetic vortices in easy-plane ferromagnetic insulators. In the both cases the transverse gyrotropic force (analog of the Magnus force in superfluid and classical hydrodynamics) is proportional to the charge of skyrmions in vortex cores.

Spin and mass superfluidity in a ferromagnetic spin-1 Bose-Einstein condensate

The paper investigates the coexistence and interplay of spin and mass superfluidity in a ferromagnetic spin-1 Bose-Einstein condensate. Superfluidity is possible only in the presence of uniaxial anisotropy (linear and quadratic Zeeman effect). This follows from the topology of the order-parameter space (vacuum manifold). According to the Landau criterion, the critical phase gradients, both for mass and spin supercurrents, vanish at the phase transition from the easy-plane to the easy-axis anisotropy. However, mass superfluidity is still possible at the phase transition. This is because the Landau criterion signals instability only with respect to nonsingular vortices with special ratio between circulations of mass and spin currents. Phase slips produced by these vortices are not sufficient for complete decay of supercurrents. Full decay of supercurrents requires phase slips with vortices of another topological class and larger energy. These phase slips are suppressed by energetic barriers up to the upper critical velocity (gradient) exceeding the Landau critical velocity. The upper critical velocity does not vanish nor has any anomaly in the critical point at the phase transition from the easy-plane to the easy-axis anisotropy.

Internal structure and stability of vortices in a dipolar spinor Bose-Einstein condensate

We demonstrate how dipolar interactions can have pronounced effects on the structure of vortices in atomic spinor Bose-Einstein condensates and illustrate generic physical principles that apply across dipolar spinor systems. We then find and analyze the cores of singular vortices with non-Abelian charges in the point-group symmetry of a spin-3 $^{52}$Cr condensate. Using a simpler model system, we analyze the underlying dipolar physics and show how a characteristic length scale arising from the magnetic dipolar coupling interacts with the hierarchy of healing lengths of the s-wave scattering, and leads to simple criteria for the core structure: When the interactions both energetically favor the ground-state spin condition, such as in the spin-1 ferromagnetic phase, the size of singular vortices is restricted to the shorter spin-dependent healing length. Conversely, when the interactions compete (e.g., in the spin-1 polar phase), we find that the core of a singular vortex is enlarged by increasing dipolar coupling. We further demonstrate how the spin-alignment arising from the interaction anisotropy is manifest in the appearance of a ground-state spin-vortex line that is oriented perpendicularly to the condensate axis of rotation, as well as in potentially observable internal core spin textures. We also explain how it leads to interaction-dependent angular momentum in nonsingular vortices as a result of competition with rotation-induced spin ordering. When the anisotropy is modified by a strong magnetic field, we show how it gives rise to a symmetry-breaking deformation of a vortex core into a spin-domain wall.

Half-quantum vortices

Unlike to superfluid 4He the superfluid 3He-A support the existence of vortices with half-quantum of circulation as well as single quantum vortices. The singular single quanta vortices as well as nonsingular vortices with 2 quanta of circulation have been revealed in rotating 3He-A. However, the half-quantum vortices in open geometry always possess an extra energy due to spin-orbit coupling leading to formation of domain wall at distances larger than dipole length ∼10−3 cm from the vortex axis. Fortunately the same magnetic dipole-dipole interaction does not prevent the existence of half-quantum vortices in the polar phase of superfluid 3He recently discovered in peculiar porous media “nematically ordered” aerogel. Here we discuss this exotic possibility. The discoveries of half-quantum vortices in triplet pairing superconductor Sr2RuO4 as well in the exciton–polariton condensates are the other parts of the story about half-quantum vortices also described in the paper.

On the position of a vortex in a two-dimensional model of atmosphere

As a continuation of our work, Rozanova et al. (2010) [1] we study possible trajectories of a long time existing vortex in a model of the atmosphere dynamics, where the vortex can be interpreted as a tropical cyclone. The model can be obtained from the system of primitive equations governing the motion of air over the Earth’s surface after averaging over the height. We consider approximations of l-plane and β-plane used in geophysics for modeling of middle scale processes and equations on the whole sphere as well. We associate with a cyclone a special class of smooth solutions having a form of a localized steady non-singular vortex moving with a bearing field. We show that the solutions satisfy the equations of the model either exactly or with a discrepancy which is small in a neighborhood of the trajectory of the center of vortex. We show both analytically and numerically that the trajectory of a localized vortex keeps the features of trajectory of vortex with a linear profile of velocity, where the exact solution can be obtained.

Spectrum of bound fermion states on vortices in 3He-B

We study subgap spectra of fermions localized within vortex cores in $^3$He-B. We develop an analytical treatment of the low-energy states and consider the characteristic properties of fermion spectra for different types of vortices. Due to the removed spin degeneracy the spectra of all singly quantized vortices consist of two different anomalous branches crossing the Fermi level. For singular $o$ and $u$ vortices the anomalous branches are similar to the standard Caroli-de Gennes -Matricon ones and intersect the Fermi level at zero angular momentum yet with different slopes corresponding to different spin states. On the contrary the spectral branches of nonsingular vortices intersect the Fermi level at finite angular momenta which leads to the appearance of a large number of zero modes, i.e. energy states at the Fermi level. Considering the $v$, $w$ and $uvw$ vortices with superfluid cores we show that the number of zero modes is proportional to the size of the vortex core.

Singular and non-singular vortices and their excitation spectra of neutral fermi superfluids in restricted geometries

We study stable vortices and their bound excitation spectra in superfluid 3He-A phase under restricted geometries. By choosing appropriate geometry we can control the l⃗-vector and d⃗-vector of the A phase order parameter (OP), making it possible to select a particular vortex among a variety of singular or non-singular vortices. This tunability through the boundary condition and external field is demonstrated to be useful. Here we take up two examples of the restricted geometries under rotation: A narrow cylinder whose radius is 50 μm under field parallel to the rotation axis and thin film configuration sandwiched by parallel plates whose gap is an order of μm or less. We examine the ongoing experiment by rotating cryostat at ISSP, University of Tokyo, reporting on new vortex state for a cylinder in a slow rotation speed region near Tc. For parallel plates in a field parallel to the film surface we analyze the conditions under which singular vortex is stable. This study allows us to examine experimental realization of so-called Majorana zero energy state bound in a core in 3He-A chiral phase.

Einstein–de Haas Effect in Dipolar Bose-Einstein Condensates

The general properties of the order parameter for a dipolar spinor Bose-Einstein condensate are discussed based on symmetries of interactions. An initially spin-polarized dipolar condensate is shown to dynamically generate a nonsingular vortex via spin-orbit interactions--a phenomenon reminiscent of the Einstein-de Haas effect in ferromagnets.

Singular and Nonsingular Vortex Structures in High-Temperature Superconductors

In this paper we focus on the comparison of the equilibrium and transport properties of singular and nonsingular vortex structures in high-temperature compounds. Using the time-dependent Ginzburg–Landau theory we study the dynamics of vortex structures in superconductors with (d+s)-wave pairing. We calculate the angular dependent correction to the viscosity tensor of a singular flux line, which appears due to the admixture of subdominant s-wave order parameter component in the vortex core. The second order phase transition between singular and nonsingular vortices in high-temperature superconductors was simulated within the time-dependent Ginzburg-Landau equations. The semiclassical analysis of the positions of gap nodes in the vortex core is performed both for singular and nonsingular flux lines.

Singular and nonsingular vortices in high-temperature superconductors

In this paper we focus on the comparison of equilibrium and transport properties of singular and nonsingular vortex structures in high-temperature compounds. Using the time-dependent Ginzburg–Landau (GL) theory we study the dynamics of vortex structures in superconductors with (d+s)-wave pairing. We calculate the angular dependent correction to the viscosity tensor of a singular flux line, which appears due to the admixture of subdominant s-wave order parameter component in the vortex core. The second order phase transition between singular and nonsingular vortices in high-temperature superconductors was simulated within time-dependent GL equations.

Kaluza-Klein vortices

We study static vortex type solutions of pure gravity for $D \geq 4+1 $. Non-singular vortex solutions can be obtained by considering periodic Kaluza-Klein monopoles. We also show that away from the center of the vortices the space is described by the gravitational instantons derived from minimal surfaces.

Nonsingular vortices in(d+s)-wave superconductors

The structure of a single flux line in (s+d)-wave superconductors has been analyzed within the Ginzburg-Landau (GL) model generalized for two order parameter components. The fourfold symmetric singular vortex solution is shown to be unstable in a certain range of the GL parameters with respect to the mutual shift of s- and d- wave unit vortices. The resulting nonsingular vortex structure is studied both analytically and numerically.

Global vortex and black cosmic string

We study global vortices coupled to (2+1) dimensional gravity with negative cosmological constant. We found nonsingular vortex solutions in $\phi^4$-theory with a broken U(1) symmetry, of which the spacetimes do not involve physical curvature singularity. When the magnitude of negative cosmological constant is larger than a critical value at a given symmetry breaking scale, the spacetime structure is a regular hyperbola, however it becomes a charged black hole when the magnitude of cosmological constant is less than the critical value. We explain through duality transformation the reason why static global vortex which is electrically neutral forms black hole with electric charge. Under the present experimental bound of the cosmological constant, implications on cosmology as a straight black cosmic string is also discussed in comparison with global U(1) cosmic string in the spacetime of the zero cosmological constant.