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

2D Coulomb Gases and the Renormalized Energy

We study the statistical mechanics of classical two-dimensional "Coulomb gases" with general potential and arbitrary \beta, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case \beta=\infty corresponds to "weighted Fekete sets" and also falls within our analysis. It is known that in such a system points should be asymptotically distributed according to a macroscopic "equilibrium measure," and that a large deviations principle holds for this, as proven by Ben Arous and Zeitouni. By a suitable splitting of the Hamiltonian, we connect the problem to the "renormalized energy" W, a Coulombian interaction for points in the plane introduced in our prior work, which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of W have exponentially small probability. When \beta\to \infty, the estimate becomes sharp, showing that the system has to "crystallize" to a minimizer of W. In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of W, which are conjectured to be "Abrikosov" triangular lattices.Comment: 55 page

1D Log Gases and the Renormalized Energy: Crystallization at Vanishing Temperature

We study the statistical mechanics of a one-dimensional log gas with general potential and arbitrary beta, the inverse of temperature, according to the method we introduced for two-dimensional Coulomb gases in [SS2]. Such ensembles correspond to random matrix models in some particular cases. The formal limit beta infinite corresponds to "weighted Fekete sets" and is also treated. We introduce a one-dimensional version of the "renormalized energy" of [SS1], measuring the total logarithmic interaction of an infinite set of points on the real line in a uniform neutralizing background. We show that this energy is minimized when the points are on a lattice. By a suitable splitting of the Hamiltonian we connect the full statistical mechanics problem to this renormalized energy W, and this allows us to obtain new results on the distribution of the points at the microscopic scale: in particular we show that configurations whose W is above a certain threshhold (which tends to min W as beta tends to infinity) have exponentially small probability. This shows that the configurations have increasing order and crystallize as the temperature goes to zero.Comment: 44 pages, one figure, second version with improved statement

Vortex patterns and sheets in segregated two component Bose-Einstein condensates

We study minimizers of a Gross-Pitaevskii energy describing a two-component Bose-Einstein condensate set into rotation. We consider the case of segregation of the components in the Thomas-Fermi regime, where a small parameter $\epsilon$ conveys a singular perturbation. We estimate the energy as a term due to a perimeter minimization and a term due to rotation. In particular, we prove a new estimate concerning the error of a Modica Mortola type energy away from the interface. For large rotations, we show that the interface between the components gets long, which is a first indication towards vortex sheets

Solitons and solitonic vortices in a strip

We study the ground state of the Gross Pitaveskii energy in a strip, with a phase imprinting condition, motivated by recent experiments on matter waves solitons. We prove that when the width of the strip is small, the ground state is a one dimensional soliton. On the other hand, when the width is large, the ground state is a solitonic vortex. We provide an explicit expression for the limiting phase of the solitonic vortex as the size of the strip is large: it has the same behaviour as the soliton in the infinite direction and decays exponentially due to the geometry of the strip, instead of algebraically as vortices in the whole space

Bounded vorticity for the 3D Ginzburg-Landau model and an isoflux problem

We consider the full three-dimensional Ginzburg-Landau model of superconductivity with applied magnetic field, in the regime where the intensity of the applied field is close to the "first critical field" $H_{c_1}$ at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg-Landau parameter $\varepsilon$. This onset of vorticity is directly related to an "isoflux problem" on curves (finding a curve that maximizes the ratio of a magnetic flux by its length), whose study was initiated in [Rom\'an, C. On the First Critical Field in the Three Dimensional Ginzburg-Landau Model of Superconductivity. Commun. Math. Phys. 367, 317-349 (2019). https://doi.org/10.1007/s00220-019-03306-w] and which we continue here. By assuming a nondegeneracy condition for this isoflux problem, which we show holds at least for instance in the case of a ball, we prove that if the intensity of the applied field remains below $H_{c_1}+ C \log |\log \varepsilon|$, the total vorticity remains bounded independently of $\varepsilon$, with vortex lines concentrating near the maximizer of the isoflux problem, thus extending to the three-dimensional setting a two-dimensional result of [Sandier, E., Serfaty, S. Ginzburg-Landau minimizers near the first critical field have bounded vorticity. Cal Var 17, 17-28 (2003). https://doi.org/10.1007/s00526-002-0158-9]. We finish by showing an improved estimate on the value of $H_{c_1}$ in some specific simple geometries.Comment: 50 pages, 4 figure