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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.

We have investigated superconductivity in thin mesoscopic regular polygons in a perpendicular magnetic field. For that, we have used an analytical gauge transformation for the vector potential which takes into account the superconductor/vacuum boundary condition. As reported previously, this has opened the way to investigate the nucleation of superconductivity in nanosu-perconductors at the normal-superconducting phase boundary in the framework of the linearized Ginzburg-Landau (GL) equations. We report now on the extension of this work to the superconducting phase beyond the phase boundary, by using the full non-linear GL equations. The stability in the superconducting phase of the symmetry induced vortex-antivortex molecules has been examined in detail for squared and equilateral triangle geometries.

New aspects of collective nonlinear dust plasma interactions are presented. These are the formation of dust ion-acoustic shocks and Langmuir envelope solitons in a uniform dusty plasma, as well as self-organization of incompressible dust fluid in the form of vortical structures in a nonuniform dusty plasma. We present relevant nonlinear models and their numerical simulations for those nonlinear waves and structures. The relevance of our investigation to laboratory and space dusty plasmas is discussed.

We report on an analytical approach to deduce the critical angular velocities for the appearance of a finite number of vortices and their degrees in rotating 2D Bose-Einstein condensates in the Thomas-Fermi regime of large coupling strength between the particles. Moreover, we derive the distribution of vortices in the condensate i.e. the vortex pattern.

We present a scenario for studying the dynamics of vortices in Bose–Einstein condensates. We construct a trapping potential that can sustain metastable off-axis vortices and propose a simple phase-imprinting method to numerically generate them. In contrast to previous works, we do not use a rotating frame or a centrifugal potential for their creation. Our method offers a good control in the number and location of the generated vortices. We include an analysis of the vortex energy as a function of its position.

We investigate the nucleation of vortices in a small superconducting disk. We formulate the Gibbs free energy of the disk with an arbitrary number of vortices (arranged in rings concentric with the disk) as a function of temperature and applied magnetic field and minimize the energy to obtain the optimal position of vortices for different applied fields and temperatures. We also analyze the stability of the different vortex states inside the disk and compare our results with those of other theoretical studies and with available experimental observations. Our results are in very good agreement with experiments.

We present results of a study of vortex proliferation in the Berezinskii-Kosterlitz-Thouless (BKT) regime on a two-dimensional (2D) array of Josephson-coupled Bose-Einstein condensates. In our lattice system, tunneling between nearest-neighbor condensates provides a Josephson coupling J which acts to keep the condensates' relative phases locked. A cloud of uncondensed atoms, on the other hand, interacts with the condensates and induces thermal phase fluctuations, which we observe as vortices. As long as the Josephson energy J exceeds the thermal energy T, the array is vortex-free, while with decreasing J/T, thermally activated vortices appear. We give an extended description of a time-to-length mapping technique that allows us to obtain information on the size of vortex pairs as J/T is varied.

The statistical mechanics of some electric models predicts exponential decay of space correlations (screening). This suggests that one look also for screening in 2- and 3-dimensional hydrodynamic turbulence.

Changes of field variables may lead to multivalued fields which do not satisfy the Schwarz integrability conditions. Their quantum field theory needs special care as is illustrated here in applications to superfluid and superconducting phase transitions. Extending the notions that first qantization governs fluctuating orbits while second quantization deals with fluctuating field, the theory of multivalued fields may be considered as a theory of third quantization. The lecture is an introduction to my new book on this subject.

We discuss the properties of classical string-like topological solitons in the A3M model. This model describes gauge-invariant interaction of Maxwell and "easy-axis" 3-component spin field.

This is a theoretical study of Newtonian trajectories governed by curl forces, i.e. position-dependent but not derivable from a potential, investigating in particular the possible existence of conserved quantities. Although nonconservative and nonhamiltonian, curl forces are not dissipative because volume in the position–velocity state space is preserved. A physical example is the effective forces exerted on small particles by light. When the force has rotational symmetry, for example when generated by an isolated optical vortex, particles spiral outwards and escape, even with an attractive gradient force, however strong. Without rotational symmetry, and for dynamics in the plane, the state space is four-dimensional, and to search for possible constants of motion we introduce the Volume of section: a numerical procedure, in which orbits are plotted as dots in a three-dimensional subspace. For some curl forces, e.g. optical fields with two opposite-strength vortices, the dots lie on a surface, indicating a hidden constant of motion. For other curl forces, e.g. those from four vortices, the dots explore clouds, in an unfamiliar kind of chaos, suggesting that no constant of motion exists. The curl force dynamics generated by optical vortices could be studied experimentally.

We propose as key of superconductivity in (hole-doped) cuprates a novel excitation of magnetic origin, characteristic of two-dimensions and of purely quantum nature: the antiferromagnetic spin vortices. In this formalism the charge pairing arises from a Kosterlitz-Thouless-like attraction between such vortices centered on opposite Néel sub-lattices. This charge pairing induces also the spin pairing through the action of a gauge force generated by the no-double occupation constraint imposed in the t-J model of the CuO planes of the cuprates. Superconductivity arises from coherence of pairs of exci tations describing Zhang-Rice singlets and it is not of standard BCS type. We show that many experimental features of the cuprates can find a natural explanation in this formalism.

Destruction of superconductivity in thin films was thought to be a simple instance of Berezinskii-Kosterlitz-Thouless physics in which only two phases exist: a superconductor with algebraic long range order in which the vortices condense and an insulator where the vortex-antivortex pairs proliferate. However, since 1989 this view has been challenged as now a preponderance of experiments indicate that an intervening bosonic metallic state obtains upon the destruction of superconductivity. Two key features of the intervening metallic state are that the resistivity turns on continuously from the zero resistance state as a power law, namely ρBM ∝ (g – gc)^α and the Hall conductance appears to vanish. We review here a glassy model which is capable of capturing both of these features. The finite resistance arises from three features. First, the disordered insulator-superconductor transition in the absence of fermionic degrees of freedom (Cooper pairs only), is controlled by a diffusive fixed point rather than the critical point of the clean system. Hence, the relevant physics that generates the Bose metal should arise from a term in the action in which different replicas are mixed. We show explicitly how such physics arises in the phase glass. Second, in 2D (not in 3D) the phase stiffness of the glass phase vanishes explicitly as has been shown in extensive numerical simulations. Third, bosons moving in such a glassy environment fail to localize as a result of the false minima in the landscape. We calculate the conductivity explicitly using Kubo response and show that it turns on as a power law and has a vanishing Hall response as a result of underlying particle-hole symmetry. We show that when particle-hole symmetry is broken, the Hall conductance turns on with the same power law as does the longitudinal conductance. This prediction can be verified experimentally by applying a ground plane to the 2D samples.

We study the nucleation of vortices in a thin mesoscopic superconducting disk and stable configurations of vortices as a function of the disk size, the applied magnetic field H and finite temperature T. We also investigate the stability of different vortex states inside the disk. Further, we compare the predictions from Ginzburg-Landau (GL) theory and London theory - the GL equations take the superconducting density into account, but the London equations do not. Our simulations from both theories show similar vortex states. As more vortices are generated, more superconducting regions will be destoryed. The GL Equations consider this effect and provide a more accurate estimate.

Maxwell equations are not logical consistent. This problem is caused by the implication that the divergence and the curl of a vector are not related. Based on Chen's S-R decomposition of a rank-two tensor, this logical un-consistency is discarded and, as a consequence, the classical Maxwell equations are reformulated to deduce London equations. From boundary field point, the relations between Josephson current and outside magnetic field are established, which shows that the Josephson current is produced by vortices of boundary magnetic field. From local field point, the first London equation corresponds to the local average rotation of electric field and the second London equation corresponds to the local average rotation of magnetic field. The relation between the Josephson effects and vortices of electromagnetic fields is discussed.

We present a survey of results obtained with Etienne Sandier on vortices in the minimizers of the 2D Ginzburg-Landau energy of superconductivity with an applied magnetic field, in the asymptotic regime of large kappa where vortices become point-like. We describe results which characterize the critical values of the applied field for which vortices appear, their numbers and locations. If the applied field is large enough, it is observed in experiments that vortices are densely packed and form triangular (hexagonal) lattices named Abrikosov lattices. Part of our results is the rigorous derivation of a mean field model describing the optimal density of vortices at leading order in the energy, and then the derivation of a next order limiting energy which governs the positions of the vortices after blow-up at their inter-distance scale. This limiting energy is a logarithmic-type interaction between points in the plane. Among lattice configurations it is uniquely minimized by the hexagonal lattice, thus providing a first justification of the Abrikosov lattice in this regime.

In joint work with Etienne Sandier, we studied the statistical mechanics of a classical two-dimensional Coulomb gas, particular cases of which also correspond to random matrix ensembles. We connect the problem to the “renormalized energy” W, a Coulombian interaction for an infinite set of points in the plane that we introduced in connection to the Ginzburg-Landau model, and whose minimum is expected to be achieved by the “Abrikosov” triangular lattice. Results include a next order asymptotic expansion of the partition function, and various characterizations of the behavior of the system at the microscopic scale. When the temperature tends to zero we show that the system tends to “crystallize” to a minimizer of W.