Finite vortex numbers and symmetric vortex structures in a rotating trapped Fermi gas in the BCS-BEC crossover

The ground state of a three-dimensional (3D) rotating trapped superfluid Fermi gas in the BCS-BEC crossover is mapped to finite N _v -body vortex states by a simple ansatz. The total vortex energy is measured from the ground-state energy of the system in the absence of the vortices. The vortex state is stable since the vortex potential and rotation energies are attractive while the vortex kinetic energy and interaction between vortices are repulsive. By combining the analytical and numerical works for the minimal vortex energy, the 2D configurations of N _v vortices are studied by taking into account of the finite size effects both on xy-plane and on z-direction. The calculated vortex numbers as a function of the interaction strength are appropriate to the renew experimental results by Zwierlein in [High-temperature superfluidity in a ultracold Fermi gas, Ph.D. thesis, Massachusetts Institute of Technology, 2006]. The numerical results show that there exist two types of vortex structures: the trap center is occupied and unoccupied by a vortex, even in the case of N _v < 10 with regular polygon and in the case of N _v ≥ 10 with finite triangle lattice. The rotation frequency dependent vortex numbers with different interaction strengths are also discussed.     

On the vortex formation at the moving front of lightweight granular particles

The formation of vortices at a moving front of lightweight granular particles is investigated experimentally. The particles used in this study are made of polystyrene foam with three different diameters of nearly uniform size. Pairs of vortices are found to emerge at the moving front at regular intervals, thereby forming a wavy pattern. Once the vortices are produced, the flow velocity tends to increase. A simple analysis suggests the existence of a velocity boundary layer at the moving front, whose thickness increases with increasing particle diameter. The frontal radius of each vortex pair is about the size of this boundary layer; when the radius exceeds this size, the front tends to bifurcate into a train of vortices with the size of the boundary layer. The formation of twin vortices leads to a reduction in the air drag force exerted on the system, and thereby the system attains a higher flow velocity, i.e., a higher conversion rate of gravitational potential energy to the kinetic energy of the particle motion. The higher conversion rate of potential energy thus feeds back to the development of the vortex motion, resulting in the twin vortex formation.     

Spectrum of the non-abelian phase in Kitaev’s honeycomb lattice model

The spectral properties of Kitaev’s honeycomb lattice model are investigated both analytically and numerically with the focus on the non-abelian phase of the model. After summarizing the fermionization technique which maps spins into free Majorana fermions, we evaluate the spectrum of sparse vortex configurations and derive the interaction between two vortices as a function of their separation. We consider the effect vortices can have on the fermionic spectrum as well as on the phase transition between the abelian and non-abelian phases. We explicitly demonstrate the 2ⁿ-fold ground state degeneracy in the presence of 2n well separated vortices and the lifting of the degeneracy due to their short-range interactions. The calculations are performed on an infinite lattice. In addition to the analytic treatment, a numerical study of finite size systems is performed which is in exact agreement with the theoretical considerations. The general spectral properties of the non-abelian phase are considered for various finite toroidal systems.     

Matter-wave vortices and solitons in anisotropic optical lattices

Using numerical methods, we construct families of vortical, quadrupole, and fundamental solitons in a two-dimensional (2D) nonlinear-Schrödinger/Gross-Pitaevskii equation which models Bose-Einstein condensates (BECs) or photonic crystals. The equation includes the attractive or repulsive cubic nonlinearity and an anisotropic periodic potential. Two types of anisotropy are considered, accounted for by the difference in the strengths of the 1D sublattices, or by a difference in their periods. The limit case of the quasi-1D optical lattice (OL), when one sublattice is missing, is included too. By means of systematic simulations, we identify stability limits for two species of vortex solitons and quadrupoles, of the rhombus and square types. In the attraction model, rhombic vortices and quadrupoles remain stable up to the limit case of the quasi-1D lattice. In the same model, finite stability limits are found for vortices and quadrupoles of the square type, in terms of the anisotropy parameter. In the repulsion model, rhombic vortices and quadrupoles are stable in large parts of the first finite bandgap (FBG). Another species of partly stable anisotropic states is found in the second FBG, subfundamental dipoles, each squeezed into a single cell of the OL. Square-shaped quadrupoles are completely unstable in the repulsion model, while vortices of the same type are stable only in weakly anisotropic OL potentials.     

Dynamics of two-sign point vortices in positive and negative temperature states

Dynamics of two-sign point vortices in two-dimensional circular boundary is examined by numerical simulations with MDGRAPE-2. The vortex system is characterized by the inverse temperature beta as determined from the density of states of the microcanonical ensemble of numerically generated 10(7) states. The massive simulation shows that different configurations appear in the time-asymptotic state depending on the sign of beta. Condensation of the same-sign vortices is observed when beta<0, while the both-sign vortices tend to be uniformly neutralized when beta>0. During the condensation, a part of the vortices gains energy to form clumps (patches), and the other part of the vortices loses energy to keep the total energy constant and mixes with vortices of the other sign. This observation demonstrates a characteristic feature of negative beta states that the system energy concentrates into the clumps of the same-sign vortices.     

Hyperbolic Bending of Vortex Lines with Finite Number and Length in Rotating Trapped Bose-Einstein Condensates

The minimal energy configurations of hyperbolic bending vortex lines in the rotating trapped Bose-Einstein condensates are investigated by using a variational ansatz and numerical simulation. The theoretical calculation of the energy of the vortex lines as a function of the rotation frequency gives self-consistently vortex number, curvature and configuration. The numerical results show that bending is more stable than straight vortex line along the z-axis, and the vortex configuration in the xy-plane has a little expansion by increasing z.     

Helical instability of a straight Abrikosov vortex in an anisotropic superconductor

Because of attraction of the parallel currents forming an Abrikosov vortex, the vortex energy per unit length decreases, under bending of the vortex, by a quantity proportional to the square of the curvature. Solving the London equation in an approximation allowing for this effect makes it possible to calculate the energy of an Abrikosov vortex in the form of a helix whose length and pitch are much larger than the correlation length, whose curvature is small compared to the reciprocal London length, and whose slope in relation to an axis coinciding with the direction in which the vortex energy is the highest is also small. When the anisotropy is large, which is characteristic of high-T _c superconductors, the energy of such an Abrikosov vortex is lower than that of a straight Abrikosov vortex. Certain consequences of the fact that the Abrikosov vortices in a high-T _c superconductor are helical are discussed. Among these is a phase transition that breaks the symmetry between Abrikosov vortices shaped like right-and left-hand helixes in relation to the magnetic field. Zh. éksp. Teor. Fiz. 111, 1869–1878 (May 1997)     

Structure and energy of a line vortex in a three-dimensional ordered Josephson medium

Two possible equilibrium configurations of line vortices in a three-dimensional ordered Josephson medium for any value of structural factor b are considered: the center of the vortex coincides with the center of one of the cells and the center of the vortex is on one of the contacts. Infinite sets of equations describing these configurations are derived. The infinite set can be made finite if currents away from the center are neglected. The assumption b = 0 is shown to be valid if pinning parameter I is less than 0.25. For I > 0.25, the structures and energies of both configurations of line isolated vortices are calculated throughout the range of structural factor b. As structural factor b increases, phase jumps at the contacts, currents in the central part of the vortex, and the total energies of the vortices decrease in both configurations. This leads to a decrease in critical field H _c1. For all values of I and b, the energy of the vortex centered on the contact is higher than that of the vortex centered in the middle of the cell.     

Lyapunov-Schmidt reduction algorithm for three-dimensional discrete vortices

We address the persistence and stability of three-dimensional vortex configurations in the discrete nonlinear Schrödinger equation and develop a symbolic package based on Wolfram’s MATHEMATICA for computations of the Lyapunov-Schmidt reduction method. The Lyapunov-Schmidt reduction method is a theoretical tool which enables us to study continuations and terminations of the discrete vortices for small coupling between lattice nodes as well as the spectral stability of the persistent configurations. The method was developed earlier in the context of the two-dimensional lattice and applied to the onsite and offsite configurations (called the vortex cross and the vortex cell) by using semianalytical computations [D.E. Pelinovsky, P.G. Kevrekidis, D. Frantzeskakis, Physica D 212 (2005) 20-53; P.G. Kevrekidis, D.E. Pelinovsky, Proc. R. Soc. A 462 (2006) 2671-2694]. The present treatment develops a full symbolic computational package which takes a desired waveform at the anticontinuum limit of uncoupled sites, performs a required number of Lyapunov-Schmidt reductions and outputs the predictions on whether the configuration persists, for finite coupling, in the three-dimensional lattice and whether it is stable or unstable. It also provides approximations for the eigenvalues of the linearized stability problem. We report a number of applications of the algorithm to important multisite three-dimensional configurations, such as the simple cube, the double cross and the diamond. For each configuration, we identify exactly one solution, which is stable for small coupling between lattice nodes.     

Vortex states of a superconducting film from a magnetic dot array

Using Ginzburg-Landau theory, we find novel configurations of vortices in superconducting thin films subject to the magnetic field of a magnetic dot array, with dipole moments oriented perpendicular to the film. Sufficiently strong magnets cause the formation of vortex-antivortex pairs. In most cases, the vortices are confined to dot regions, while the antivortices can form a rich variety of lattice states. We propose an experiment in which the perpendicular component of the dot dipole moments can be tuned using an in-plane magnetic field. We show that in such an experiment the vortex-antivortex pair density shows broad plateaus as a function of the dipole strength. Many of the plateaus correspond to vortex configurations that break dot lattice symmetries. In some of these states, the vortex cores are strongly distorted. Possible experimental consequences are mentioned.     

Vortices in Bose-Einstein condensates: A review of the experimental results

Rotating dilute Bose-Einstein condensates (BEC) of alkali atoms offer a testing ground for theories of vortices in weakly interacting superfluids. In a rotating super-fluid, quantised vortices, with a vorticity h/m, form above a critical velocity. Such vortices have been generated in BEC of alkali atoms by different techniques such as (a) wave function engineering of a two-component BEC, (b) decay of solitons, (c) rotation of a thermal cloud before cooling it below the condensation temperature, (d) stirring with an ‘optical’ spoon, (e) rotating a deformation in the anisotropic trap in which the condensate is trapped and (f) by creating Berry phase by adiabatically reversing the axial magnetic field. Since the core of a vortex is a fraction of a micrometer in diameter, it cannot be directly imaged optically. The condensate with vortices is allowed to ballistically expand till the size increases by one order before the vortices are imaged. Surface wave spectroscopy and the change in aspect ratio of a rotating cloud are the other techniques used. Studies have been made on the creation and dynamics of single vortex and on systems with more than a hundred vortices. Results have been obtained on vortex nucleation, stability of vortex structures, nature of the vortex lattice and defects in such a lattice. Important results are: (a) evidence exists that vortex nucleation takes place by a surface mode instability; but this is not the only mechanism; (b) the vortex lattice is perfectly triangular right up to the edge; (c) in the initial stages of rotation of the cloud a tangled web of vortices is seen; it takes a few hundred milliseconds before the vortices arrange themselves in a lattice; this time appears to be independent of temperature; (d) the decay of vortices appears to arise from the transfer of energy to the rotating thermal component and is dependent on temperature; (e) defects in the lattices such as dislocations and grain boundaries are seen; (f) transverse oscillations (Tkachenko modes) of the vortex lattice have been observed; and (g) giant vortices have been produced. These will be discussed.     

Decay of MHD-scale Kelvin-Helmholtz vortices mediated by parasitic electron dynamics

We have simulated nonlinear development of MHD-scale Kelvin-Helmholtz (KH) vortices by a two-dimensional two-fluid system including finite electron inertial effects. In the presence of moderate density jump across a shear layer, in striking contrast to MHD results, MHD KH vortices are found to decay by the time one eddy turnover is completed. The decay is mediated by smaller vortices that appear within the parent vortex and stays effective even when the shear layer width is made larger. It is shown that the smaller vortices are basically of MHD nature while the seeding for these is achieved by the electron inertial effect. Application of the results to the magnetotail boundary layer is discussed.     

Pinning of planar vortices and magnetic field penetration in a three-dimensional Josephson medium

The behavior of planar (laminar) vortices in a three-dimensional, ordered Josephson medium as a function of the parameter I, which is proportional to the critical junction current and the cell size, is investigated with allowance for pinning due to the cellular structure of the medium. The minimum possible distances between two isolated vortices are calculated. A system of vortices formed in a sample in a monotonically increasing external magnetic field is analyzed. The minimum distance from the outermost vortex to the nearest neighbor is proportional to I ^−1.1. For I⩽1.3 each vortex contains a single flux quantum Φ₀, and the distance between them does not decrease in closer proximity to the boundary but remains approximately constant, implying that the magnetic field does not depend on the coordinate in the region penetrated by vortices. These facts contradict the generally accepted Bean model. The sample magnetization curve has a form typical of type II superconductors. Allowance for pinning raises the critical field H _c and induces a sudden jump in the curve at H=H _c. Zh. Tekh. Fiz. 67, 38–46 (September 1997)     

Two dimensional vortex structures in a superconductor slab at low temperatures

The dynamics of a two dimensional chain like structure of vortices is studied in the model of nonlinear time dependent Ginzburg–Landau equations (TDGL). The transition between different linear chains of vortices in a superconducting homogeneous slab with both surfaces in contact with a thin layer of metallic material is analyzed. The magnetization curve, vortex number, vortex configurations and modulus of the order parameter are studied as a function of the external magnetic field. We show how these vortex configurations are affected by the extrapolation length b (de Gennes boundary conditions), Ψ due to the proximity effects in a mesoscopic sample of area d_x × d_y, where d_y = 60ξ(0) and d_x varies discretely from 30ξ(0) to 12ξ(0). Possible connection with recent theoretical results in a two dimensional system of charged particles is discussed.     

Parallel numerical simulations for quantized vortices in Bose--Einstein condensates

We employ the parallel computing technology to study numerically the three-dimensional structure of quantized vortices of Bose--Einstein condensates. For anisotropic cases, the bending process of vortices is described in detail by the decrease of Gross--Pitaevskii energy. A completely straight vortex and the steady and symmetrical multiple-vortex configurations are obtained. We analyse the effect of initial conditions and angular velocity on the number and shape of vortices.     

Non-BCS superconductivity for underdoped cuprates by spin-vortex attraction

Within a gauge approach to the t-J model, we propose a new, non-BCS mechanism of superconductivity for underdoped cuprates. The gluing force of the superconducting mechanism is an attraction between spin vortices on two different Néel sublattices, centered around the empty sites described in terms of fermionic holons. The spin fluctuations are described by bosonic spinons with a gap generated by the spin vortices. Due to the no-double occupation constraint, there is a gauge attraction between holon and spinon binding them into a physical hole. Through gauge interaction the spin vortex attraction induces the formation of spin-singlet (RVB) spinon pairs with a lowering of the spinon gap. Lowering the temperature, the approach exhibits two crossover temperatures: at the higher crossover a finite density of incoherent holon pairs are formed leading to a reduction of the hole spectral weight, while at the lower crossover a finite density of incoherent spinon RVB pairs are also formed, giving rise to a gas of incoherent preformed hole pairs, and magnetic vortices appear in the plasma phase. Finally, at a even lower temperature the hole pairs become coherent, the magnetic vortices becoming dilute and superconductivity appears. The superconducting mechanism is not of BCS-type since it involves a gain in kinetic energy (for spinons) coming from the spin interactions.     

Switching of vortex polarization in 2D easy-plane magnets by magnetic fields

We investigate the dynamics of out-of-plane (OP) vortices, in a 2-dimensional (2D) classical Heisenberg magnet with a weak anisotropy in the coupling of z-components of spins (easy plane anisotropy), on square lattices, under the influence of a rotating in-plane (IP) magnetic field. Switching of the z-component of magnetization of the vortex is studied in computer simulations as a function of the magnetic field's amplitude and frequency. The effects of the size and the anisotropy of the system on the switching process are shown. An approximate dynamical equivalence of the system, in the bulk limit, to another system with both IP and OP static fields in the rotating reference frame is demonstrated, and qualitatively the same switching and critical behavior is obtained in computer simulations for both systems. We briefly discuss the interplay between finite size effects (image vortices) and the applied field in the dynamics of OP vortices. In the framework of a discrete reduced model of the vortex core we propose a mechanism for switching the vortex polarization, which can account qualitatively for all our results. A coupling between the IP movement (trajectories) of the vortex center and the OP core structure oscillations, due to the discreteness of the underlying lattice, is shown. A connection between this coupling and our reduced model is made clear, through an analogy with a generalized Thiele equation. Received 6 June 2002 / Received in final form 4 November 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: juan.zagorodny@uni-bayreuth.de     

Characteristic parameters of drift vortices coupled to Alfvén waves in an inhomogeneous space plasma

We present detailed measurements of ion scale vortices of drift type coupled to Alfvén waves in an inhomogeneous and collisionless space magnetoplasma. The two free parameters of a dipolar vortex, intensity and spatial radius, are measured. The vortices are driven by a strong density gradient on a boundary layer with scale size of the same order as the vortex diameter. Observations of vortices off the gradient show that symmetry-breaking conditions in a real inhomogeneous plasma can lead not only to cross-field but also to cross-boundary anomalous transport of particles and energy.     

Topological defects in incommensurate magnetic and crystal structures and quasi-periodic solutions of the elliptic sine-Gordon equation

Vortex-type singular solutions with a topological charge of the elliptic sine-Gordon equation have been studied. One- and two-dimensional vortex lattices on a homogeneous and periodic background are constructed in the explicit form using the Bäcklund transformation. The interaction of vortices is investigated and finite energy configurations are found. On the basis of the obtained results new topological defects in incommensurate magnetic and crystal structures are predicted and described. The interaction of vortex magnetic structures with nonlinear spin waves is considered.