# From the Ginzburg-Landau Model to Vortex Lattice Problems

@article{Sandier2010FromTG, title={From the Ginzburg-Landau Model to Vortex Lattice Problems}, author={Etienne Sandier and Sylvia Serfaty}, journal={Communications in Mathematical Physics}, year={2010}, volume={313}, pages={635-743} }

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…

## 124 Citations

Vortex Patterns in Ginzburg-Landau Minimizers

- Physics, Mathematics
- 2010

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…

Local variational study of 2d lattice energies and application to Lennard–Jones type interactions

- Mathematics
- 2016

In this paper, we focus on finite Bravais lattice energies per point in two dimensions. We compute the first and second derivatives of these energies. We prove that the Hessian at the square and the…

Large vorticity stable solutions to the Ginzburg-Landau equations

- Mathematics, Physics
- 2011

We construct local minimizers to the Ginzburg-Landau functional of superconductivity whose number of vortices N is prescribed and blows up as the parameter epsilon, inverse of the Ginzburg-Landau…

Lattices energies and variational calculus

- Mathematics
- 2015

In this thesis, we study minimization problems for discrete energies and we search to understand why a periodic structure can be a minimizer for an interaction energy, that is called a…

A P ] 8 S ep 2 01 1 LARGE VORTICITY STABLE SOLUTIONS TO THE GINZBURG-LANDAU EQUATIONS

- Mathematics, Physics
- 2021

We construct local minimizers to the Ginzburg-Landau functional of superconductivity whose number of vortices N is prescribed and blows up as the parameter ε, inverse of the Ginzburg-Landau parameter…

Coulomb Gases and the Renormalized Energy

- Mathematics
- 2012

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…

Large systems with Coulomb interactions: Variational study and statistical mechanics

- Physics
- 2016

Systems with Coulomb and logarithmic interactions arise in various settings: an instance is the classical Coulomb gas which in some cases happens to be a random matrix ensemble, another is vortices…

The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. Droplet Arrangement via the Renormalized Energy

- Physics, Mathematics
- 2014

This is the second in a series of papers in which we derive a Γ-expansion for the two-dimensional non-local Ginzburg–Landau energy with Coulomb repulsion known as the Ohta–Kawasaki model in…

Renormalized Energy Concentration in Random Matrices

- Mathematics
- 2013

We define a “renormalized energy” as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line. The definition is inspired by ideas of…

Elliptical systems related to superconductor model

- Physics
- 2014

Our work focus on the elliptic partial differential Equations arising from the mathematical physics, especially from the superconductivity. Therefore most of our work is on the Ginzburg-Landau model,…

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This is the second in a series of papers in which we derive a $\Gamma$-expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in…

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