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Bifurcation vers l'etat d'Abrikosov et diagramme des phases

Mathieu Dutour
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Abstract

Nous étudions dans cette thèse la fonctionnelle de Ginzburg-Landau dans $\R^3$ sur des couples de fonctions $(ϕ, \overrightarrow{A})$ qui vérifient des conditions de périodicité de jauge en $x_3$ et selon un réseau discret de $(x_1,x_2)$. Nous montrons que le problème variationnel est équivalent au problème de la minimisation d'une autre fonctionnelle sur un tore. Dans le cadre de la démonstration, un fibré vectoriel non trivial apparaît. On se limite alors pour la suite à une quantification de 1. On montre ensuite que la fonctionnelle admet un minimum sur l'espace fonctionnel $H^{1}$ qui vérifie un système d'équations aux dérivées partielles appelé système de Ginzburg-Landau. Le minimum est $C^{\infty}$ par l'ellipticité du système d'équations de Ginzburg-Landau. On montre qu'il y a une bifurcation du couple $(0,0)$ pour le champ critique $H_{ext}=k$ où $k$ est un paramètre caractéristique du système. On étudie alors la stabilité de la solution bifurquée. On étudie la dépendance de l'énergie minimale à l'égard de la géométrie du tore. Enfin nous décrivons toutes les solutions du système d'équations de Ginzburg-Landau dans la limite $k$ tend vers l'infini. Dans le dernier chapitre, nous donnons pour notre modèle la structure du diagramme des phases en précisant quelles régions sont normales, supraconductrices pure, mixte.

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Mathieu Dutour. 2000-09-27. Bifurcation vers l'etat d'Abrikosov et diagramme des phases. https://arxiv.org/abs/math-ph/9912011

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