Narrow Results

We continue the analysis started in [14] on a model describing a two-dimensional rotating Bose–Einstein condensate. This model consists in minimizing under the unit mass constraint, a Gross–Pitaevskii energy defined in ℝ². In this contribution, we estimate the critical rotational speeds Ω_d for having exactly d vortices in the bulk of the condensate and we determine their topological charge and their precise location. Our approach relies on asymptotic energy expansion techniques developed by Serfaty [20–22] for the Ginzburg–Landau energy of superconductivity in the high κ limit.

The dynamical (super)symmetries for various monopole systems are reviewed. For a Dirac monopole, non-smooth Runge–Lenz vector can exist; there is, however, a spectrum-generating conformal o(2,1) dynamical symmetry that extends into osp(1/1) or osp(1/2) for spin 1/2 particles. Self-dual 't Hooft–Polyakov-type monopoles admit an su(2/2) dynamical supersymmetry algebra, which allows us to reduce the fluctuation equation to the spin 0 case. For large r, the system reduces to a Dirac monopole plus a suitable inverse-square potential considered before by McIntosh and Cisneros, and by Zwanziger in the spin 0 case, and to the "dyon" of D'Hoker and Vinet for spin 1/2. The asymptotic system admits a Kepler-type dynamical symmetry as well as a "helicity-supersymmetry" analogous to the one Biedenharn found in the relativistic Kepler problem. Similar results hold for the Kaluza–Klein monopole of Gross–Perry–Sorkin. For the magnetic vortex, the N = 2 supersymmetry of the Pauli Hamiltonian in a static magnetic field in the plane combines with the o(2) × o(2,1) bosonic symmetry into an o(2) × osp(1/2) dynamical superalgebra.

We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantized twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit.

Three different forcing functions are used with the lattice Boltzmann method (LBM) to simulate the forced isotropic turbulence in periodic boxes at different resolutions ranging from $32^3$ to $256^3$ grid points using the D3Q19 model. The aims of this study are to examine the effect of using different forcing functions on the LBM stability; to track the development of the turbulent fields at several resolutions, to investigate the effect of the weak compressibility due to change of fluid density on the flow simulations, and to identify the effective force type. The injection is performed through adding the force randomly to the collision term. The three forcing methods depend on sine and cosine as functions of the wave numbers and space. The forcing amplitude values of $A=10^{-4}$ and the relaxation time $\tau =0.503$ are fixed in all cases. The single relaxation time model is found stable at such values of the forcing amplitude and the relaxation time. However, the development of the turbulent data at the different resolutions needs about 10000 time-steps to reach the required statistical state including clear visualizations of fine scale vortices. Many simulations have been tested using different values of the relaxation time $\tau$ and the development of the turbulent fields is found faster with fewer time-steps but the stability of the LBM is broken at some resolutions (not necessary the higher resolution). The statistical features of all fields, such as the Taylor and the Kolmogorov micro-scales, the Taylor Reynolds number, the flatness and the skewness, are calculated and compared with the previous efforts. The worm-like vortices are visualized at all cases and it is found that more fine vortices can be extracted as the resolution increases. The energy spectrum has a reasonable Kolmogorov power law at the resolutions of $128^3$ and $256^3$, respectively. Results show that the third forcing method that uses a cosine disturbance function has the best statistical features and the finest visualized vortical structures especially at higher resolutions. Extensive discussions about the density field and its evolution with time at different forcing functions, comparison to Navier–Stokes solutions and the time development of the energy spectra for all cases are also carried out.

We present a complete set of practical expressions for fast and accurate evaluation of Coulomb interactions, forces and energies, in any rectangular periodic medium. This analysis also includes explicit expressions for energy constants and self-energies of charged particles in periodic lattices. Energy constants and self-energies are further evaluated for logarithmic interactions in two-dimensional rectangular systems with periodic boundary conditions. The expressions are optimized for computational speed and tested by direct numerical implementation.

We derive a formula for the entropy of two-dimensional incompressible inviscid flow, by determining the volume of the space of vorticity distributions with fixed values for the moments Q_k = ∫ ω(x)^kd²x. This space is approximated by a sequence of spaces of finite volume, by using a regularization of the system that is geometrically natural and connected with the theory of random matrices. By taking the limit we get a simple formula for the entropy of a vortex field. We predict vorticity distributions of maximum entropy with given mean vorticity and enstrophy; we also predict the cylindrically symmetric vortex field with maximum entropy. This could be an approximate description of a hurricane.

The interaction of a magnetic flux vortex with weak external fields is considered in the framework of the Abelian Higgs model. The approach is based on the calculation of the zero-mode excitation probability in the external field. The excitation of the field configuration is found perturbatively. As an example we consider the effect of interaction with an external current. The linear in the scalar field perturbation is also considered.

We show that vortices of Yang–Mills–Higgs model in R² space can be regarded as instantons of Yang–Mills model in R² × Z₂ space. For this, we construct the noncommutative Z₂ space by explicitly fixing the Z₂ coordinates and then show, by using the Z₂ coordinates, that BPS equation for the vortices can be considered as a self-dual equation. We also propose the possibility to rewrite the BPS equations for vortices as ADHM equations through the use of self-dual equation.

We first review the spontaneous Lorentz symmetry breaking in the presence of massless gauge fields and infraparticles. This result was obtained long time ago in the context of rigorous quantum field theory (QFT) by Fröhlich, Morchio and Strocchi [Ann. Phys.119, 241 (1979); Phys. Lett. B89, 61 (1979)] and reformulated by Balachandran and Vaidya (arXiv:1302.3406) using the notion of superselection sectors and direction-dependent test functions at spatial infinity for gauge transformations. Inspired by these developments and under the assumption that the spectrum of the electric charge is quantized (in units of a fundamental charge e), we construct a family of vertex operators which create winding number k, electrically charged Abelian vortices from the vacuum (zero winding number sector) and/or shift the winding number by k units. Vortices created by this vertex operator may be viewed both as a source and as a probe for inducing and detecting the breaking of spontaneous Lorentz symmetry.

We find that for rotating vortices, the vertex operator at level k shifts the angular momentum of the vortex by , where is the electric charge of the quantum state of the vortex and q is the charge of the vortex scalar field under the U(1) gauge field. We also show that, for charged-particle-vortex composites, angular momentum eigenvalues shift by being the electric charge of the charged-particle-vortex composite. This leads to the result that for half-odd integral and for odd k, our vertex operators flip the statistics of charged-particle-vortex composites from bosons to fermions and vice versa. For fractional values of , application of vertex operator on charged-particle-vortex composite leads in general to composites with anyonic statistics.

This work introduces a procedure to obtain vortex configurations described by first-order equations in generalized Maxwell–Chern–Simons models without the inclusion of a neutral field. The results show that the novel methodology is capable of inducing important modification in the vortex core, leading to vortex configurations with unconventional features.

We discuss general properties and possible types of magnetic vortices in non-Abelian gauge theories (we consider here G = SU(N), SO(N), USp(2N)) in the Higgs phase. The sources of such vortices carry "fractional" quantum numbers such as Z_n charge (for SU(N)), but also full non-Abelian charges of the dual gauge group. If such a model emerges as an effective dual magnetic theory of the fundamental (electric) theory, the non-Abelian vortices can provide for the mechanism of quark confinement in the latter.

Topological structure of vortex line in quantum mechanics is studied and classified by Hopf index, linking number in geometry. A mechanism of generation or annihilation of vortex line is given by method of phase singularity theory. The dynamic behavior of vortex line at the critical points is discussed in detail, and three kinds of length approximation relations are given.

Monopoles and instantons are sheets (membranes) and strings in d = 5+1 dimension, respectively, and instanton strings can terminate on monopole sheets. We consider a pair of monopole and antimonopole sheets which is unstable to decay and results in a creation of closed instanton strings. We show that when an instanton string is stretched between the monopole sheets, there remains a new topological soliton of codimension five after the pair annihilation, i.e. a twisted closed instanton string or a knotted instanton.

We study a $U(1)\times U(1)$ gauge theory discussing its vortex solutions and supersymmetric extension. In our set-up, the dynamics of one of two Abelian gauge fields is governed by a Maxwell term, the other by a Chern–Simons term. The two sectors interact via a BF gauge field mixing and a Higgs portal term that connects the two complex scalars. We also consider the supersymmetric version of this system which allows to find for the bosonic sector BPS equations in which an additional real scalar field enters into play. We study numerically the field equations finding vortex solutions with both magnetic flux and electric charge.

We solve the equations of motion of a complex ${\varphi }^4$ theory coupled to some given gauge field background. The solutions are given in both cylindrical and spherical coordinates and have finite energy.

Nonlinear field theories often possess the so-called soliton solutions that have localized energy densities and that are characterized by topological charges n. To explore the stability of solitons with $n>1$, we consider vortices in scalar electrodynamics which is one of the rare renormalizable models that contain soliton type solutions with different topological charges. We focus on the BPS case which is particularly interesting because the quantum corrections decide on the stability of vortices with $n>1$. We also explain how to overcome technical obstacles that are encountered when computing these corrections and that arise from the singular structure of the vortex profiles.

Angle and temperature dependent torque magnetization measurements are reported for the organic superconductor κ-(ET)₂Cu(NCS)₂, at extremely low temperatures (~ T_c/10³). Magneto-thermal instabilities are observed in the form of abrupt magnetization (flux) jumps. We carry out an analysis of the temperature and field orientation dependence of these flux jumps based on accepted models for layered type-II superconductors. Using a simple Bean model, we also find a critical current density of 4 × 10⁸A/m² from the remnant magnetization, in agreement with previous measurements.

Using a high-sensitivity cavity perturbation technique (40 to 180 GHz), we have probed the angle dependent interlayer magneto-electrodynamic response within the vortex state of the extreme two-dimensional organic superconductor κ-(BEDT-TTF)₂Cu(NCS)₂. A previously reported Josephson plasma resonance [M. Mola et al., Phys. Rev.B62, 5965 (2000)] exhibits a dramatic re-entrant behavior for fields very close (< 1°) to alignment with the layers. In this same narrow angle range, a new resonant mode develops which appears to be associated with the non-equilibrium critical state. Fits to the angle dependence of the Josephson plasma resonance provide microscopic information concerning the superconducting phase correlation function within the vortex state. We also show that the effect of an in-plane magnetic field on the temperature dependence of the interlayer phase coherence is quite different from what we have previously observed for perpendicular magnetic fields.

Small numbers N<5 of two-dimensional Coulomb-interacting electrons trapped in a parabolic potential placed in a perpendicular magnetic field are investigated. The reduced wave function of this system, which is obtained by fixing the positions of N-1 electrons, exhibits strong correlations between the electrons and the zeros. These zeros are often called vortices. An exact-diagonalization scheme is used to obtain the wave functions and the results are compared with results obtained from the recently proposed rotating electron molecule (REM) theory. We find that the vortices gather around the fixed electrons and repel each other, which is to a much lesser extend so for the REM results.

We study the nucleation of vortices in a thin mesoscopic superconducting disk in an applied magnetic field perpendicular to the disc. We write down an expression for the free energy of the system with an arbitrary number of vortices and anti-vortices. For a given applied field, we minimize the free energy to find the optimal position of the vortices and anti-vortices. We also calculate the magnetization of the disk as a function of the applied field and hence the determine the different configurations possible in which a fixed number of fluxoids can penetrate the disk.