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}
}
  • E. Sandier, S. Serfaty
  • Published 20 November 2010
  • Mathematics, Physics, Computer Science
  • Communications in Mathematical Physics
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… 
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References

SHOWING 1-10 OF 72 REFERENCES
Vortices in the Magnetic Ginzburg-Landau Model
With the discovery of type-II superconductivity by Abrikosov, the prediction of vortex lattices, and their experimental observation, quantized vortices have become a central object of study in
Ginzburg-Landau Vortices
The mathematics in this book apply directly to classical problems in superconductors, superfluids and liquid crystals. It should be of interest to mathematicians, physicists and engineers working on
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
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
The Decrease of Bulk-Superconductivity Close to the Second Critical Field in the Ginzburg-Landau Model
TLDR
It is shown how, for energy-minimizers, superconductivity decreases in average in the bulk of the sample when the applied field increases to Hc2, just below the "second critical field" H c2.
The Gamma-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement at the sharp interface level via the renormalized energy
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
On the Third Critical Field in Ginzburg-Landau Theory
AbstractUsing recent results by the authors on the spectral asymptotics of the Neumann Laplacian with magnetic field, we give precise estimates on the critical field, $$H_{C_3}$$, describing the
A Proof of Crystallization in Two Dimensions
TLDR
This work shows rigorously that under suitable assumptions on the potential V which are compatible with the growth behavior of the Lennard-Jones potential the ground state energy per particle converges to an explicit constant E*: where E* ∈ ℝ is the minimum of a simple function on [0,∞).
Droplet Phases in Non-local Ginzburg-Landau Models with Coulomb Repulsion in Two Dimensions
We establish the behavior of the energy of minimizers of non-local Ginzburg-Landau energies with Coulomb repulsion in two space dimensions near the onset of multi-droplet patterns. Under suitable
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