From the Ginzburg-Landau Model to Vortex Lattice Problems

By Etienne Sandier, Sylvia Serfaty. 2010

164 citations recorded by the source corpus.

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 Hc1 and Hc2. In that regime, minimizing configurations exhibit densely packed triangular vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, W as a Γ-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 Γ-convergence, and an abstract scheme for obtaining lower bounds for “2-scale energies” via the ergodic theorem, that we introduce.

Subfields: Computational Theory and Mathematics, Condensed Matter Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/4235eff7086dc4fc5adf77423fe9d5c4ba136496

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Citations in the corpus

2D Coulomb Gases and the Renormalized Energy (2012, methodology): “ect triangular lattices (i.e. with 60◦ angles), named Abrikosov lattices after the physicist who predicted them [Ab]. The Abrikosov lattices are indeed observed in experiments on superconductors1. In [SS1] we made this partly rigorous by showing that minimizers of the Ginzburg-Landau energy have vor”
Coulomb Gases and Ginzburg - Landau Vortices (2014, methodology): “a function of one entry. 2the constants involved were dened in (4.15) 70 5 Denition(s) and properties of the renormalized energy We wrote down here an equivalent for dimension d3 of the Wdened in [SS6] in dimension 2. It suced to replace 2ˇby c d and logby g(). However, the good properties we need f”
Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals (2013, methodology): “In [67] it was shown, however, that in dimension 2, if the minimizer is a lattice, then it has to be the triangular one, called the”
Bogoliubov Spectrum of Interacting Bose Gases (2012, result): “Our study in this section was motivated by a recent work of Sandier and Serfaty [52, 51].”
Large deviation principle for empirical fields of Log and Riesz gases (2015, background): “ying min A 1 Wis open, and we expect some LARGE DEVIATION PRINCIPLE FOR EMPIRICAL FIELDS OF LOG AND RIESZ GASES 19 (Bravais) lattice conguration to achieve the minimum, at least in low dimension. In [SS12b] it is proven that in the case (1.3), Wachieves its minimum over lattice congurations of volum”