Sylvia Serfaty

LOCAL MINIMIZERS FOR THE GINZBURG–LANDAU ENERGY NEAR CRITICAL MAGNETIC FIELD: PART I

We find local minimizers of the two-dimensional Ginzburg–Landau functionals depending on a large parameter κ, which describe the behavior of a superconductor in a prescribed exterior magnetic field hex. We prove an estimate on the critical value Hc1 of hex(κ), corresponding to the first phase-transition in which vortices appear in the superconductor; and we locate these vortices.

Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field

Abstract We prove that the global minimizer of the Ginzburg–Landau functional of superconductors in an external magnetic field is, below the first critical field, the vortex-less solution found in (S. Serfaty, to appear).

ON THE ENERGY OF TYPE-II SUPERCONDUCTORS IN THE MIXED PHASE

We study the Ginzburg–Landau energy of superconductors with high κ, put in a prescribed external field hex, for hex varying between the two critical fields Hc1 and Hc3. As κ → +∞, we give the leading term in the asymptotic expansion of the minimal energy and show that energy minimizers have vortices whose density tends to be uniform and equal to hex.

A rigorous derivation of a free-boundary problem arising in superconductivity

Abstract We study the Ginzburg–Landau energy of superconductors submitted to a possibly non-uniform magnetic field, in the limit of a large Ginzburg–Landau parameter κ . We prove that the induced magnetic fields associated to minimizers of the energy-functional converge as κ →+∞ to the solution of a free-boundary problem. This free-boundary problem has a nontrivial solution only when the applied magnetic field is of the order of the “first critical field”, i.e. of the order of log κ . In other cases, our results are contained in those we had previously obtained [15, 16, 14]. We also derive a convergence result for the density of vortices.

2D Coulomb gases and the renormalized energy

We study the statistical mechanics of classical two-dimensional “Coulomb gases” with general potential and arbitrary β, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case β=∞ 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 Petz and Hiai [In Advances in Differential Equations and Mathematical Physics (Atlanta, GA, 1997) (1998) Amer. Math. Soc.] and Ben Arous and Zeitouni [ ESAIM Probab. Statist. 2 (1998) 123–134]. 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 [ Comm. Math. Phys. 313 (2012) 635–743], 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 β→∞, 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.

A product-estimate for Ginzburg–Landau and corollaries

Abstract We prove a new inequality for the Jacobian (or vorticity) associated to the Ginzburg–Landau energy in any dimension. It allows to retrieve existing lower bounds on the energy, to extend them to the case of unbounded vorticity, and to get a few other corollaries. It also provides a new estimate on the time-variation for time-dependent families, which has applications for the study of Ginzburg–Landau dynamics.

Compactness, Kinetic Formulation, and Entropies for a Problem Related to Micromagnetics

Abstract We carry on the study of (Rivière T, Serfaty S. Limiting domain wall energy for a problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294–338.) on the asymptotics of a family of energy-functionals related to micromagnetics. We prove compactness for families of uniformly bounded energies releasing the LBP condition we had previously set. Such families converge to unit-valued divergence-free vector-fields that are tangent to the boundary of the domain, and we found in (Rivière T, Serfaty S. Limiting domain wall energy for a problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294–338.) that the energy-functionals Γ-converge to a limiting jump-energy of such configurations. We examine the behavior of certain truncated fields which serve to construct “entropies,” and to provide an improved lower bound. We give a kinetic formulation of the problem, and show that the limiting divergence-free problem is supplemented, in the case of minimizers, with a sign condition which can in turn, using the kinetic formulation, be interpreted as an entropy condition that plays a role in uniqueness questions.

Limiting vorticities for the Ginzburg-Landau equations

We study the asymptotic limit of solutions of the Ginzburg-Landau equations in two dimensions with or without magnetic field. We first study the Ginzburg-Landau system with magnetic field describing a superconductor in an applied magnetic field, in the “London limit” of a Ginzburg-Landau parameter κ tending to∞. We examine the asymptotic behavior of the “vorticity measures” associated to the vortices of the solution, and we prove that passing to the limit in the equations (via the “stress-energy tensor”) yields a criticality condition on the limiting measures. This condition allows us to describe the possible locations and densities of the vortices. We establish analogous results for the Ginzburg-Landau equation without magnetic field.

The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. I. Droplet Density

This is the first in a series of two papers in which we derive a Γ-expansion for a two-dimensional non-local Ginzburg–Landau energy with Coulomb repulsion, also known as the Ohta–Kawasaki model, in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating small “droplets” of the minority phase in a “sea” of the majority phase. In this paper we show that an appropriate setting for Γ-convergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the Ohta–Kawasaki energy functional Γ-converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain an asymptotic characterization of the energy minimizers in the diffuse interface model.

Large deviation principle for empirical fields of Log and Riesz gases

We study a system of N particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. We examine a microscopic quantity, the tagged empirical field, for which we prove a large deviation principle at speed N. The rate function is the sum of an entropy term, the specific relative entropy, and an energy term, the renormalized energy introduced in previous works, coupled by the temperature. We deduce a variational property of the sine-beta processes which arise in random matrix theory. We also give a next-to-leading order expansion of the free energy of the system, proving the existence of the thermodynamic limit.