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 r...

We investigate the effects of material anisotropy on the superheating field of layered superconductors. We provide an intuitive argument both for the existence of a superheating field, and its dependence on anisotropy, for κ = λ/ξ (the ratio of magnetic to superconducting healing lengths) both large and small. On the one hand, the combination of our estimates with published results using a two-...

We define and investigate deformed n-Calabi-Yau completions of homologically smooth differential graded (=dg) categories. Important examples are: deformed preprojective algebras of connected non Dynkin quivers, Ginzburg dg algebras associated to quivers with potentials and dg categories associated to the category of coherent sheaves on the canonical bundle of a smooth variety. We show that defo...

This paper presents an introduction to phase transitions and critical phenomena on the one hand, and nonequilibrium patterns on the other, using the Ginzburg-Landau theory as a unified language. In the first part, meanfield theory is presented, for both statics and dynamics, and its validity tested self-consistently. As is well known, the mean-field approximation breaks down below four spatial ...

The purpose of this paper is to prove connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg–Landau polynomials. The model under study is a mean-field version of a lattice spin model due to Blume and Capel. It is defined by a probability distribution that depends on the parameters β and K , whi...

It is an open challenge to analyze the crystalline color superconducting phases that may arise in cold dense, but not asymptotically dense, three-flavor quark matter. At present the only approximation within which it seems possible to compare the free energies of the myriad possible crystal structures is the Ginzburg-Landau approximation. Here, we test this approximation on a particularly simpl...

We consider the Taubes correspondencebetween solutions of the Seiberg– Witten equations on a compact 4-dimensional symplectic manifold and pseudoholomorphic curves on this manifold. We compare it with its 3-dimensional analogue in which the roles of Seiberg–Witten equations and pseudoholomorphic curves are played respectively by the Ginzburg–Landau equations and adiabatic paths in the moduli sp...