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Vorticité dans des systèmes de spins à symétrie continue

Abstract : This Thesis concerns spin systems with continuous symmetry on a 2-D lattice. For XY model we consider second order phase transitions [Berezinskii, Kosterlitz and Thouless], related with vorticity of Gibbs states or order parameters (minimizers of free energy $\cal F$). Vortices are analogues with interfaces in Ising model; but internal continuous symmetry smears out phase transitions, excluding in 2-D spontaneous aimantation and allowing for a delay of correlation functions, even at low temperature. For Heisenberg model with Kac potential, vortices are replaced by instantons.
In Part I, we recall some properties of nearest neighbors interactions for the rotator, or its simplified version (Villain model). We also introduce the mean field model.
Kac's model, which relates various aspects of these models is presented in Part II. By homogenization, we essentially reduce properties of Gibbs measure in finite volume to these of $\cal F$, generalizing techniques used for the Ising model.
In Part III we analyze vorticity for Kac's model, and determine extrema for $\cal F$, with boundary conditions. We thus arrive at configurations similar to solutions of Ginzburg-Landau equations.
In Part IV, we turn to the quantum case, introducing "vorticity matrices" at inverse temperature $\beta$, and compute their "non commutative degree". Thus we obtain for the XY model with spin 1/2, configurations as those we met in the classical case.
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Contributor : Hicham El Bouanani <>
Submitted on : Saturday, May 16, 2009 - 9:41:48 AM
Last modification on : Thursday, March 15, 2018 - 4:56:04 PM
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  • HAL Id : tel-00381684, version 2


Hicham El-Bouanani. Vorticité dans des systèmes de spins à symétrie continue. Mathématiques [math]. Université du Sud Toulon Var, 2008. Français. ⟨tel-00381684v2⟩



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