From the Ginzburg-Landau Model to Vortex Lattice Problems

Abstract

We introduce a "Coulombian renormalized energy" W which is a logarithmic type of interaction between points in the plane, computed by a "renormalization." We prove various of its properties, such as the existence of minimizers, and show in particular, using results from number theory, that among lattice configurations the triangular lattice is the unique minimizer. Its minimization in general remains open.Our motivation is the study of minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields H c 1 and H c 2 . In that regime, minimizing configurations exhibit densely packed triangular vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, W as a -limit of the Ginzburg-Landau energy. More precisely we show that the vortices of minimizers of Ginzburg-Landau, blown-up at a suitable scale, converge to minimizers of W , thus providing a first rigorous hint at the Abrikosov lattice. This is a next order effect compared to the mean-field type results we previously established.The derivation of W uses energy methods: the framework of -convergence, and an abstract scheme for obtaining lower bounds for "2-scale energies" via the ergodic theorem, that we introduce.

Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma

“…This change of behaviour of the Coulomb gas is at the heart of the change of speed in the large deviations tails; b) Based on the analysis of the next-to-leading order corrections to (6), which we are also able to obtain (see (51)), we show that a genuine 1/N -expansion of the excess free energy for the 2D-OCP fails to exist if the fluid phase assumption is violated (i.e. when the plasma distribution becomes singular).…”

“…This statistical mechanics fluid model has appeared in several areas of physics and mathematics. Indeed, the logarithmic repulsion in (2) does occur as interaction between vortices and dislocations in real systems such as superconductors [51], superfluids, rotating BoseEinstein condensates [18,19,50] (we refer to [2] and [54,55] for detailed reviews). There is also a well-known analogy between the canonical measure (3) of the 2D-OCP and the Laughlin trial wave function [44] (in the symmetric gauge) in the theory of fractional quantum Hall effect [11,12].…”

“…For Hermitian matrix models, the large-N eigenvalue density with a constraint on the largest eigenvalue is absolutely continuous [9] and hence the term O(log N /N ) in the large deviation asymptotics is absent. The expansion (51) has been compared with the exact β = 2 formula. See Fig.…”

Low Density Phases in a Uniformly Charged Liquid

“…It is widely believed that the solution of this problem should be given by a hexagonal lattice, which in the context of type-II superconductors is called the "Abrikosov lattice" [52]. Proving this result rigorously is a formidable task, and to date such a result has been obtained only within a much reduced class of Bravais lattices [50,53].…”

Large Deviations for the Two-Dimensional Two-Component Plasma

“…This does not matter however, since the energy of the true electric field ∇V ′ N,r generated by the configuration C will be shown to be smaller than that of E by Helmholtz projection. In the case of positive charges with a neutralizing background, this procedure was introduced in [SS12] and further refined in [SS15, RS15, PS15, LS15], to which we refer for more detail. We implement this program, with the appropriate adaptations needed for controlling dipoles, in the rest of this section.…”