Singular Vortex Research Articles

Symmetry breaking and singularity structure in Bose-Einstein condensates

We determine the trajectories of vortex singularities that arise after a single vortex is broken by a discretely symmetric impulse in the context of Bose-Einstein condensates in a harmonic trap. The dynamics of these singularities are analyzed to determine the form of the imprinted motion. We find that the symmetry-breaking process introduces two effective forces: a repulsive harmonic force that causes the daughter trajectories to be ejected from the parent singularity and a Magnus force that introduces a torque about the axis of symmetry. For the analytical noninteracting case we find that the parent singularity is reconstructed from the daughter singularities after one period of the trapping frequency. The interactions between singularities in the weakly interacting system do not allow the parent vortex to be reconstructed. Analytic trajectories were compared to the actual minima of the wave function, showing less than $0.5%$ error for an impulse strength of $v=0.00005$. We show that these solutions are valid within the impulse regime for various impulse strengths using numerical integration of the Gross-Pitaevskii equation. We also show that the actual duration of the symmetry-breaking potential does not significantly change the dynamics of the system as long as the strength is below $v=0.0005$.

Superconductive proximity in a topological insulator slab and excitations bound to an axial vortex

We consider the proximity effect in a topological insulator (TI) sandwiched between two conventional superconductors by comparing $s$-wave spin singlet superconducting pairing correlations and odd-parity triplet pairing correlations with a zero spin component orthogonal to the slab (``polar'' phase). A superconducting gap opens in the Dirac dispersion of the surface states existing at the interfaces. An axial vortex is included, piercing the slab along the normal to the interfaces with the superconductors. It is known that, when proximity is $s$-wave, quasiparticles in the gap are Majorana bound states, localized at opposite interfaces. We report the full expression for the quantum field associated to the midgap neutral fermions, as derived in the two-orbital band model for the TI. When proximity involves odd-parity pairing, midgap modes are charged surface Andreev bound states, and they originate from interfacial circular states of definite chirality, centered at the vortex singularity and decaying in the TI film with oscillations. When the chemical potential is moved away from midgap, extended states along the vortex axis are also allowed. Their orbital structure depends on the symmetry of the bulk band from where the quasiparticle level splits off.

A Spin Triplet Superconductor UPt3

Motivated by a recent angle-resolved thermal conductivity experiment that shows a twofold gap symmetry in the high-field and low-temperature C phase in the heavy-fermion superconductor UPt$_3$, we group-theoretically identify the pairing functions as $E_{1u}$ with the $f$-wave character for all the three phases. The pairing functions are consistent with the observation as well as with a variety of existing measurements. By using a microscopic quasi-classical Eilenberger equation with the identified triplet pairing function under applied fields, we performed detailed studies of the vortex structures for three phases, including the vortex lattice symmetry, the local density of states, and the internal field distribution. These quantities are directly measurable experimentally by SANS, STM/STS, and NMR, respectively. It is found that, in the B phase of low $H$ and low $T$, the double-core vortex is stabilized over a singular vortex. In the C phase, thermal conductivity data are analyzed to confirm the gap structure proposed. We also give detailed comparisons of various proposed pair functions, concluding that the present scenario of $E_{1u}$ with the $f$-wave, which is an analogue to the triplet planar state, is better than the $E_{2u}$ or $E_{1g}$ scenario. Finally, we discuss the surface topological aspects of Majorana modes associated with the $E_{1u}^f$ state of planar like features.

Energetically stable singular vortex cores in an atomic spin-1 Bose-Einstein condensate

We analyze the structure and stability of singular singly quantized vortices in a rotating spin-1 Bose-Einstein condensate. We show that the singular vortex can be energetically stable in both the ferromagnetic and polar phases despite the existence of a lower-energy nonsingular coreless vortex in the ferromagnetic phase. The spin-1 system exhibits an energetic hierarchy of length scales resulting from different interaction strengths and we find that the vortex cores deform to a larger size determined by the characteristic length scale of the spin-dependent interaction. We show that in the ferromagnetic phase the resulting stable core structure, despite apparent complexity, can be identified as a single polar core with axially symmetric density profile which is nonvanishing everywhere. In the polar phase, the energetically favored core deformation leads to a splitting of a singly quantized vortex into a pair of half-quantum vortices that preserves the topology of the vortex outside the extended core region, but breaks the axial symmetry of the core. The resulting half-quantum vortices exhibit nonvanishing ferromagnetic cores.

ZONA6 Versus the Doublet-Lattice Method for Unsteady Aerodynamics on Lifting Surfaces

T HE purpose of this Note is to clarify the frequency-dependence issue of ZONA6 and the doublet-lattice method (DLM) commented on by Rodden [1]. The DLM [2] is an unsteady flow extension of the vortex-lattice method (VLM) that was proposed by Hedman [3] and Belotserkovskii [4] during the mid-1960s to compute the steady aerodynamics on lifting surfaces. The VLM is a low-order steady aerodynamic panel method, because the vortex singularity is concentrated along the quarter-chord line on each aerodynamic box. By the mid-1970s, Woodward’s method [5] further advanced the steady panel methods with a distributed singularity over the complete aerodynamic box, and they are therefore higher-order methods than VLM. ZONA6 [6] is an unsteady flow extension of Woodward’s method and is therefore a higher-order method than the DLM. In this Note, we first clarify the meaning of low-order and highorder methods from the steady aerodynamic sense. Then the comparison between the ZONA6 and DLM solutions on the BAH wing [7] is presented to demonstrate that ZONA6 requires fewer aerodynamic boxes than the DLM for a converged unsteady aerodynamic solution.

Rotation of quantum liquid without singular vortex lines

The operator equations for quantum hydrodynamics are discussed and solved in a simple cylindrical geometry. We find a solution with the velocity curl ‘frozen’ into a density of the liquid in the absence of singular vortex lines. The spectrum of small oscillations around this solution is found as a generalization of the standard phonon spectrum, and the stability of the solution is demonstrated.

Inviscid and viscous vortex models for Richtmyer–Meshkov instability

We present simple inviscid and viscous vortex models for the Richtmyer–Meshkov (RM) instability, in the limit of fluids of the same densities. The evolution of the interface is driven by a row of alternating point vortices. A regularization method for the singular vortices is applied to the inviscid model for stable calculations. The regularized vortex model provides fine resolutions for the interface, improving previous results on the evolution of the RM instability. The point vortex model, for the inviscid fluid, is validated by comparing with the results of the full vortex sheet model. The viscous vortex model for the unstable interface is based on the multi-Lamb–Oseen vortices. We show that the viscous vortex model gives the main qualitative features of the evolution of the viscous RM instability.

Zero Energy Modes and Statistics of Vortices in Spinful Chiralp-Wave Superfluids

The possible stable singular vortex (SV) and half-quantum vortex (HQV) of the superfluid $^3$He-A phase confined in restricted geometries are investigated. The associated low-energy excitations are calculated in connection with the possible existence of Majorana zero modes obeying non-Abelian statistics. The energetics between those vortices is carefully examined using the standard Ginzburg-Landau (GL) functional with a strong-coupling correction. The Fermi liquid effect, which is not included in the GL functional, is considered approximately within the London approach. This allows us to determine the stability regions in pressure, temperature, and applied field for SV and HQV. The existence of the Majorana zero mode and its statistics, either Abelian or non-Abelian under braiding of SVs, is studied by solving the Bogoliubov-de Gennes equation for spinful chiral p-wave superfluids at sufficiently low temperatures. We determined several conditions controllable external parameters for realizing the non-Abelian statistics of Majorana zero modes e.g., pressure, field direction, and strength.

The effects of chaotic advection in a three-layer ocean model

A three-layer model of an inviscid incompressible fluid is considered in a quasigeostrophic approximation within the concept of background currents. A singular topographic obstacle yields the formation of a vortical motion in the layers. It is always present in the lower layer, while in the upper and middle layers it can be observed only at certain velocities of the external flows. Three current functions describing a singular topographic flow in the lower layer and two regular flows in the upper layer are obtained. In case of a nonstationary harmonic perturbation of the external flow, chaotic particle transport is possible in these flows. This paper analyzes the chaotic properties of this model. Depending on the type of unperturbed frequency curves of the fluid particle revolution is determined by the model parameters (stratification) and the incident flow, the patterns of the chaotic transport will substantially differ. Two of the limiting dependences of the revolution frequency are determined, namely, the dependence for a singular vortex and that for a regular vortex with the smallest area. Other dependences are intermediate ones. Two limiting types of revolution frequencies determine the two different scenarios of chaotic advection.

Vortex–Antivortex Dynamics in a Mesoscopic Superconducting Prism with a Centered Antidot

In this work, we investigated theoretically the dynamics of the annihilation of a vortex–antivortex pair in a superconducting mesoscopic prism of square transversal section with a square antidot inserted at its center. The sample is immersed in a magnetic field applied perpendicular to the sample plane. It is assumed that the inner hole is made of a material whose properties are accounted on de Gennes boundary conditions via de Gennes extrapolation length (b parameter). We analyze the nucleation of vortices and antivortices by increasing the magnetic field from zero until the first vortex is created and then reversing the polarity of the applied magnetic field until an antivortex is also created. Depending on the b parameter, the vortex–antivortex encounter can take place at the hole or at the superconducting region around it. In the framework of the time dependent Ginzburg–Landau theory, we calculate the magnetization, order parameter topology, the position and the velocity of the vortex and antivortex singularities as a function of time.

Singular and Half-Quantum Vortices and Associated Majorana Particles in Superfluid3He-A between Parallel Plates

Motivated by a recent experiment on superfluid 3 He-A confined in narrow parallel plates using a rotating cryostat, we explore possible vortices that are stable under a magnetic field applied to an arbitrary angle relative to the plates in order to seek vortices that can accommodate the Majorana zero mode in the core. After proving that the singular vortex with the unit winding number provides the Majorana mode in a spinful situation, we establish the phase diagram in the plane; the rotation frequency Ω vs the system size R by finding possible order parameter textures within the Ginzburg–Landau framework. We also analyze the stability for a single and a pair of half-quantum vortices, which possess the Majorana mode in their core. It is concluded from the above-mentioned results that the Majorana zero mode can be found in the present on-going experiment performed at ISSP, University of Tokyo.

Stability of the Singular Vortex and Associated Majorana Zero Modes in Trapped p-Wave Resonant Superfluids of Neutral Cold Atoms

The stability conditions for the singular vortex which accompanies Majorana zero modes at the core are investigated for p-wave resonant superfluids of atomic Fermi gases. Within the Ginzburg-Landau framework we determine the stable conditions in the parameter space for the external rotation frequency and the harmonic trap frequency. There exists the narrow stable region in this parameter space for quasi-two-dimensional condensates. We also describe the detailed characterizations of the spatial structure of the order parameter in the chiral p-wave superfluids under rotation.

Homostrophic vortex interaction under external strain, in a coupled QG-SQG model

The interaction between two co-rotating vortices, embedded in a steady external strain field, is studied in a coupled Quasi-Geostrophic — Surface Quasi-Geostrophic (hereafter referred to as QG-SQG) model. One vortex is an anomaly of surface density, and the other is an anomaly of internal potential vorticity. The equilibria of singular point vortices and their stability are presented first. The number and form of the equilibria are determined as a function of two parameters: the external strain rate and the vertical separation between the vortices. A curve is determined analytically which separates the domain of existence of one saddle-point, and that of one neutral point and two saddle-points. Then, a Contour-Advective Semi-Lagrangian (hereafter referred to as CASL) numerical model of the coupled QG-SQG equations is used to simulate the time-evolution of a sphere of uniform potential vorticity, with radius R at depth −2H interacting with a disk of uniform density anomaly, with radius R, at the surface. In the absence of external strain, distant vortices co-rotate, while closer vortices align vertically, either completely or partially (depending on their initial distance). With strain, a fourth regime appears in which vortices are strongly elongated and drift away from their common center, irreversibly. An analysis of the vertical tilt and of the horizontal deformation of the internal vortex in the regimes of partial or complete alignment is used to quantify the three-dimensional deformation of the internal vortex in time. A similar analysis is performed to understand the deformation of the surface vortex.

Geometric stabilization of extendedS=2vortices in two-dimensional photonic lattices: Theoretical analysis, numerical computation, and experimental results

In this work, we focus our studies on the subject of nonlinear discrete self-trapping of S=2 (doubly-charged) vortices in two-dimensional photonic lattices, including theoretical analysis, numerical computation and experimental demonstration. We revisit earlier findings about S=2 vortices with a discrete model, and find that S=2 vortices extended over eight lattice sites can indeed be stable (or only weakly unstable) under certain conditions, not only for the cubic nonlinearity previously used, but also for a saturable nonlinearity more relevant to our experiment with a biased photorefractive nonlinear crystal. We then use the discrete analysis as a guide towards numerically identifying stable (and unstable) vortex solutions in a more realistic continuum model with a periodic potential. Finally, we present our experimental observation of such geometrically extended S=2 vortex solitons in optically induced lattices under both self-focusing and self-defocusing nonlinearities, and show clearly that the S=2 vortex singularities are preserved during nonlinear propagation.

Singular vortices in regular flows

A review of the theory of quasigeostrophic singular vortices embedded in regular flows is presented with emphasis on recent results. The equations governing the joint evolution of singular vortices and regular flow, and the conservation laws (integrals) yielded by these equations are presented. Using these integrals, we prove the nonlinear stability of a vortex pair on the f-plane with respect to any small regular perturbation with finite energy and enstrophy. On the β-plane, a new exact steady-state solution is presented, a hybrid regular-singular modon comprised of a singular vortex and a localized regular component. The unsteady drift of an individual singular β-plane vortex confined to one layer of a two-layer fluid is considered. Analysis of the β-gyres shows that the vortex trajectory is similar to that of a barotropic monopole on the β-plane. Non-stationary behavior of a dipole interacting with a radial flow produced by a point source in a 2D fluid is examined. The dipole always survives after collision with the source and accelerates (decelerates) in a convergent (divergent) radial flow.

Vortex-antivortex annihilation dynamics in a square mesoscopic superconducting cylinder

The dynamics of the annihilation of a vortex-antivortex pair is investigated. The pair is activated magnetically during the run of a simulated hysteresis loop on a square mesoscopic superconducting cylinder with an antidot inserted at its center. We study the nucleation of vortices and antivortices by first increasing the magnetic field, applied parallel to the axis of the sample, from zero until the first vortex is created. A further increase in the field pulls the vortex in, until it reaches the antidot. As the polarity of the field is reversed, an antivortex enters the scene, travels toward the center of the sample, and eventually the pair is annihilated. Depending on the sample size, its temperature, and Ginzburg-Landau parameter, the vortex-antivortex encounter takes place at the antidot or at the superconducting sea around it. The position and velocity of the vortex and antivortex singularities were evaluated as a function of time. The current density, magnetization, and order-parameter topology were also calculated.

Unsteady Aerodynamic Coefficients Obtained by a Compressible Vortex Lattice Method

Unsteady solutions for the aerodynamic coefficients of a thin airfoil in compressible subsonic or supersonic flows were studied. The lift, the pitch moment, and pressure coefficients were obtained numerically for the following motions: the indicial response (unit step function) of the profile, i.e., a sudden change in the angle of attack; a thin profile penetrating into a sharp edge gust (for several gust velocity ratios); a thin profile penetrating into a one-minus-cosine gust and sinusoidal gust, i.e., a typical gust used in commercial aircraft design; oscillating airfoil; and also the interaction of the profile with a convected (from convection phenomenon) vortex passing under the profile, a phenomenon known in literature as BVI (Blade Vortex Interaction) or for a profile case AVI (Airfoil Vortex Interaction). The present work uses a numerical approach based on vortex singularity. The numerical model was created through the profile discretization in uniform segments and the compressible flow vortex singularity was used. The results available in the literature are based on approximated exponential equations, or computed via Computational Fluid Dynamics (CFD). Thus, the purpose of this method is to obtain a more accurate computation compared to those of approximated equations, and quite faster than those done via CFD. The results yielded by the present methodology were also compared with solutions available in the literature. The results were obtained for subsonic and supersonic flow in compressible environment.

Analysis of an immersed boundary method for three-dimensional flows in vorticity formulation

This article presents numerical analysis and practical considerations for three-dimensional flow computation using an implicit immersed boundary method. The Euler equations, or half a step of the Navier–Stokes equations when using fractional step algorithms, are investigated in their vorticity formulation. The context of flow computation around an arbitrarily shaped body is especially investigated. In conventional immersed boundary methods using vorticity, singular vortex are dispatched over the body surface. In the present study, one prefers using sources of potential velocity field, dispatched on the body, whose nature is not vorticity. Such a formulation is compatible to the Euler equations. In practice, these sources of potential flow produce a velocity through this surface, aiming in practice at cancelling a flow-through velocity. This article focuses on the use of the source-to-flow-through linear application, its properties being the key points for fast convergence. Its self-adjointness, or lack thereof, conditioning and preconditioning aspects are investigated. It follows that computing a velocity field with no-flow-through conditions in complex geometry, when using the source-to-flow-through linear application, can be achieved for 4/3 of the computational cost of standard Poisson equation in a Cartesian box. The robustness of immersed boundaries is especially interesting when used together with vortex-in-cell methods, well known for their robustness in time and their ability to compute accurately convective effects. A few examples, based on real-world geometries, illustrate the method capabilities.

Vortex singularities of optimal strategies at the beginning of motion in search problems on n-dimensional Riemannian manifolds

This work studies optimal strategies having unexpected singularities. So, in particular, in the problem of searching for an object E on a segment that has the probability distribution function of location going to infinity to both side of a pursuer P, the modulus of whose speed does not exceed some constant, the optimal search strategy has no derivative at the initial instant of time. During an arbitrary small interval of time, the pursuer P changes the direction of his motion infinitely many times in order to be at both sides where the probability of location of E is maximal. In the case of a two-dimensional manifold, with analogous singularities of the probability distribution function of location of E, the velocity of P executes countably many turns during an arbitrary small interval of time for the optimal search (this is the reason for which such singularities are said to be vortex singularities). It is more difficult to imagine optimal strategies arising in the search on manifolds of dimension more than 2. Here, during an arbitrary small initial interval of time, the player P tends to completely inspect a neighborhood of the boundary of the visibility domain at the initial instant of time.

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.