From the Ginzburg-Landau model to vortex lattice problems

We study minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields. In this regime, minimizing configurations exhibit densely packed hexagonal vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, a limiting interaction energy between points in the plane, $W$, which we prove has to be minimized by limits of energy-minimizing configurations, once blown-up at a suitable scale. This is a next order effect compared to the mean-field type results we previously established. The limiting "Coulombian renormalized energy" $W$ is a logarithmic type of interaction, computed by a "renormalization," and we believe it should be rather ubiquitous. We study various of its properties, and show in particular, using results from number theory, that among lattice configurations the hexagonal lattice is the unique minimizer, thus providing a first rigorous hint at the Abrikosov lattice. Its minimization in general remains open. The derivation of $W$ uses energy methods: the framework of $\Gamma$-convergence, and an abstract scheme for obtaining lower bounds for "2-scale energies" via the ergodic theorem.Comment: 107 page

Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement

We prove some improved estimates for the Ginzburg-Landau energy (with or without magnetic field) in two dimensions, relating the asymptotic energy of an arbitrary configuration to its vortices and their degrees, with possibly unbounded numbers of vortices. The method is based on a localisation of the ``ball construction method" combined with a mass displacement idea which allows to compensate for negative errors in the ball construction estimates by energy ``displaced" from close by. Under good conditions, our main estimate allows to get a lower bound on the energy which includes a finite order ``renormalized energy" of vortex interaction, up to the best possible precision i.e. with only a $o(1)$ error per vortex, and is complemented by local compactness results on the vortices. This is used crucially in a forthcoming paper relating minimizers of the Ginzburg-Landau energy with the Abrikosov lattice. It can also serve to provide lower bounds for weighted Ginzburg-Landau energies.Comment: 43 pages, to appear in "Analysis & PDE

The Regulation of Toothpaste

Humankind invented toothpaste for a variety of reasons, the most important of which is the prevention of this tooth decay. Usually, we, the buying public, do not see this product until it is in its familiar plastic tubing at the supermarket, complete with an easy-to-read (usually) label and brand name conveniently on the front. However, before their product reaches the shelves, manufacturers of toothpaste in this country are regulated by the Food and Drug Administration with various practices designed to protect the public. These practices include official FDA standards on toothpaste safety and effectiveness, as well as regulation of how much and which kind of ingredients are permitted in the formula

Energy and Vorticity in Fast Rotating Bose-Einstein Condensates

We study a rapidly rotating Bose-Einstein condensate confined to a finite trap in the framework of two-dimensional Gross-Pitaevskii theory in the strong coupling (Thomas-Fermi) limit. Denoting the coupling parameter by 1/\eps^2 and the rotational velocity by $\Omega$, we evaluate exactly the next to leading order contribution to the ground state energy in the parameter regime |\log\eps|\ll \Omega\ll 1/(\eps^2|\log\eps|) with \eps\to 0. While the TF energy includes only the contribution of the centrifugal forces the next order corresponds to a lattice of vortices whose density is proportional to the rotational velocity.Comment: 19 pages, LaTeX; typos corrected, clarifying remarks added, some rearrangements in the tex

Rotating superfluids in anharmonic traps: From vortex lattices to giant vortices

We study a superfluid in a rotating anharmonic trap and explicate a rigorous proof of a transition from a vortex lattice to a giant vortex state as the rotation is increased beyond a limiting speed determined by the interaction strength. The transition is characterized by the disappearance of the vortices from the annulus where the bulk of the superfluid is concentrated due to centrifugal forces while a macroscopic phase circulation remains. The analysis is carried out within two-dimensional Gross-Pitaevskii theory at large coupling constant and reveals significant differences between 'soft' anharmonic traps (like a quartic plus quadratic trapping potential) and traps with a fixed boundary: In the latter case the transition takes place in a parameter regime where the size of vortices is very small relative to the width of the annulus whereas in 'soft' traps the vortex lattice persists until the width of the annulus becomes comparable to the vortex cores. Moreover, the density profile in the annulus where the bulk is concentrated is, in the 'soft' case, approximately gaussian with long tails and not of the Thomas-Fermi type like in a trap with a fixed boundary.Comment: Published version. Typos corrected, references adde

Ginzburg-Landau vortex dynamics with pinning and strong applied currents

We study a mixed heat and Schr\"odinger Ginzburg-Landau evolution equation on a bounded two-dimensional domain with an electric current applied on the boundary and a pinning potential term. This is meant to model a superconductor subjected to an applied electric current and electromagnetic field and containing impurities. Such a current is expected to set the vortices in motion, while the pinning term drives them toward minima of the pinning potential and "pins" them there. We derive the limiting dynamics of a finite number of vortices in the limit of a large Ginzburg-Landau parameter, or \ep \to 0, when the intensity of the electric current and applied magnetic field on the boundary scale like \lep. We show that the limiting velocity of the vortices is the sum of a Lorentz force, due to the current, and a pinning force. We state an analogous result for a model Ginzburg-Landau equation without magnetic field but with forcing terms. Our proof provides a unified approach to various proofs of dynamics of Ginzburg-Landau vortices.Comment: 48 pages; v2: minor errors and typos correcte

The bifurcation diagrams for the Ginzburg-Landau system for superconductivity

In this paper, we provide the different types of bifurcation diagrams for a superconducting cylinder placed in a magnetic field along the direction of the axis of the cylinder. The computation is based on the numerical solutions of the Ginzburg-Landau model by the finite element method. The response of the material depends on the values of the exterior field, the Ginzburg-Landau parameter and the size of the domain. The solution branches in the different regions of the bifurcation diagrams are analyzed and open mathematical problems are mentioned.Comment: 16 page