Dobrushin Interfaces via Reflection Positivity

We study the interfaces separating different phases of 3D systems by means of the Reflection Positivity method. We treat discrete non-linear sigma-models, which exhibit power-law decay of correlations at low temperatures, and we prove the rigidity property of the interface. Our method is applicable to the Ising and Potts models, where it simplifies the derivation of some known results. The method also works for large-entropy systems of continuous spins.Comment: 48 pages, 4 figures; updated for publication (to appear in CMP

Potts models in the continuum. Uniqueness and exponential decay in the restricted ensembles

In this paper we study a continuum version of the Potts model. Particles are points in R^d, with a spin which may take S possible values, S being at least 3. Particles with different spins repel each other via a Kac pair potential. In mean field, for any inverse temperature there is a value of the chemical potential at which S+1 distinct phases coexist. For each mean field pure phase, we introduce a restricted ensemble which is defined so that the empirical particles densities are close to the mean field values. Then, in the spirit of the Dobrushin Shlosman theory, we get uniqueness and exponential decay of correlations when the range of the interaction is large enough. In a second paper, we will use such a result to implement the Pirogov-Sinai scheme proving coexistence of S+1 extremal DLR measures.Comment: 72 pages, 1 figur

Coexistence of ordered and disordered phases in Potts models in the continuum

This is the second of two papers on a continuum version of the Potts model, where particles are points in $\mathbb R^d$, $d\ge 2$, with a spin which may take $S\ge 3$ possible values. Particles with different spins repel each other via a Kac pair potential of range \ga^{-1}, \ga>0. In this paper we prove phase transition, namely we prove that if the scaling parameter of the Kac potential is suitably small, given any temperature there is a value of the chemical potential such that at the given temperature and chemical potential there exist $S+1$ mutually distinct DLR measures.Comment: 57 pages, 1 figur

Interfaces rigides des modèles sur réseau : une application de la positivité par réflexion

We study interfaces for four spin models on a lattice: the Ising model at low temperature, the Potts model at critical point, a continuous symmetry model and its associated clock approximation. For each model, we use specific boundary conditions that force the existence of an interface; related Gibbs measures then satisfy the so-called chessboard estimates, which are powerful correlation inequalities. These estimates are a simple way to show that such interfaces are rigid, in the sense that they deviate only locally from a perfect hyperplane. This method is a restricted version of the so-called reflection positivity method, since reflections in certain directions are forbidden by the chosen boundary conditions.In the case of Ising and Potts models, our method drastically simplifies historical proofs respectively since its requires neither Pirogov-Sinai theory nor cluster expansions. Besides, PS-theory is not directly available for the continuous model and its clock approximation, their ground states being infinitely degenerated; our method is thus a real alternative to such techniques.Nous étudions les interfaces de quatre modèles de spins sur réseau: le modèle d'Ising à basse température, le modèle de Potts à la température critique, un modèle à symétrie continue et son approximation d'horloge.Pour chacun de ces modèles, nous imposons des conditions au bord spécifiques qui assurent l'existence d'une interface ; les mesures de Gibbs associées à de telles conditions au bord satisfont alors de puissantes inégalités de corrélation. Ces inégalités nous permettent de montrer que les interfaces considérées sont rigides, au sens où ce sont des hyperplans légèrement déformés par des aspérités locales. Cette méthode est une version restreinte de la méthode de positivité par réflexion, l'une des directions de réflexion étant prohibée par les conditions au bord choisies.Pour Ising et Potts, notre méthode simplifie considérablement les démonstrations historiques, puisque ni la théorie de Pirogov-Sinai, ni les développements en amas ne sont nécessaires à son application. Par ailleurs, la théorie-PS n'est directement envisageable ni pour le modèle continu ni son approximation car leurs états fondamentaux sont infiniment dégénérés; notre méthode est donc une réelle alternative à ces techniques

Interfaces rigides des modèles sur réseau (une application de la positivité par réflexion)

Nous étudions les interfaces de quatre modèles de spins sur le réseau Zd, d>= 3 : le modèle d Ising à basse température, le modèle de Potts à la température critique, un modèle à symétrie continue et son approximation d horloge ; les interfaces de ces deux derniers modèles n ont jamais été étudiées auparavant. Pour chacun de ces modèles, nous imposons des conditions au bord spécifiques qui assurent l existence d une interface ; les mesures de Gibbs associées à de telles conditions au bord satisfont alors de puissantes inégalités de corrélation, dites de l échiquier. Ces inégalités nous permettent de montrer que les interfaces conside rées sont rigides, au sens où ce sont des hyperplans légèrement déformés par des aspérités locales. Cette méthode est une version restreinte de la méthode de positivité par réflexion, l une des directions de réflexion étant prohibée par les conditions au bord choisies. Pour Ising et Potts, notre méthode simplifie considérablement les démonstrations historiques respectivement dues à Dobrushin et Messager et al., puisque ni la théorie de Pirogov-Sina ı, ni les développements en amas ne sont nécessaires à son application. Par ailleurs, la théorie-PS n est directement envisageable ni pour le modèle continu ni pour son approximation car leurs états fondamentaux sont infiniment dégénérés ; notre méthode est donc une réelle alternative à ces techniques.We study interfaces for four spin models on the lattice Zd, d >= 3: the Ising model at low temperature, the Potts model at critical point, a continuous symmetry model and its associated clock approximation. Interfaces of the last two models were not previously dealt with. For each model, we use specific boundary conditions that force the existence of an interface; related Gibbs measures then satisfy the so-called chessboard estimates, which are powerful correlation inequalities. These estimates are a simple way of showing that such interfaces are rigid, in the sense that they deviate only locally from a perfect hyperplane. This method is a restricted version of the so-called reflection positivity method, since reflections in certain directions are forbidden by the chosen boundary conditions. In the case of Ising and Potts models, our method drastically simplifies historical proofs respectively devised by Dobrushin and Messager et al., since its requires neither Pirogov-Sina ı theory nor cluster expansions. Besides, PS-theory is not directly available for the continuous model and its clock approximation, their ground states being infinitely degenerated; our method is thus a true alternative to such techniques.AIX-MARSEILLE2-BU Sci.Luminy (130552106) / SudocSudocFranceF

Influence of growth conditions on the structural, optical and electrical quality of MBE grown InAlAs/InGaAs metamorphic HEMTs on GaAs

InAlAs/InGaAs metamorphic HEMTs on GaAs have demonstrated low noise figures and high output powers with obvious advantages over structures grown on InP substrates. Indeed, from a processing viewpoint, the GaAs substrate is less brittle, less expensive, available in size up to 6 inches in diameter and then it is preferred for the production of high performance monolithic integrated circuits. Furthermore, the metamorphic scheme allows one to arbitrary choose the indium content in the InAlAs/InGaAs layers, which is a supplementary degree of freedom for the optimization of the active layers. Various buffer layers have been developed to accommodate the lattice mismatch between the active layers and the substrate. Production tools allows the growth of ternary as well as quaternary graded buffer layers; although the growth of a ternary alloy is more simple, it still requires the optimization of growth parameters like temperatures and arsenic fluxes. The layers presented here are based on thick InAlAs with a graded indium content from 1%-10% to 49% and terminated with an inverse step to obtain a highly relaxed In/sub 0.42/Al/sub 0.58/As/In/sub 0.43/Ga/sub 0.57/As structure. In the present work, the quality of the metamorphic HEMT structures is investigated by varying the growth parameters for the graded buffer layer as well as for the active layers. The structural quality is studied with high resolution X-ray diffraction, transmission electron microscopy (TEM) and atomic force microscopy, while the optical quality and the electrical quality of the HEMTs are studied with photoluminescence and Hall effect measurements respectively