Dans le cadre de cette thèse nous utilisons les techniques du groupe de renormalisation pour étudier des systèmes vitreux. Plus précisément, nous étudions des modèles de verres de spins et de verres structuraux.Le modèles de verres de spin représentent des matériaux magnétiques désordonnés uniaxaux, comme une solution diluée de Mn en Cu, donnée par un réseau de spins situés sur le Mn et disposés aléatoirement dans le réseau des atomes de Cu. Ces spins interagissent entre eux avec un potentiel qui oscille en fonction de la séparation entre les spins. Quant aux modèles de verres structuraux, ils représentent des liquides qui ont été refroidis assez rapidement pour ne pas cristalliser, comme le o-Terphényle ou le Glycérol. Les verres se spin et les verres structuraux sont intéressants physiquement parce que leurs propriétés critiques ne sont connues que dans la limite où la dimension de l'espace tends ver l'infinie, c'est-à-dire dans l'approximation de champ moyen. Une question fondamentale est si les propriétés physiques qui caractérisent ces systèmes dans le cas du champ moyen restent ou pas valables pour des verres de spin et des verres structuraux réels, qui sont dans un espace avec un nombre finie de dimensions.Les modèles de verres de spin et de verres structuraux que nous étudions dans ce travail de thèse sont des des modèles construits sur des réseaux hiérarchiques, qui sont les systèmes non-champ moyen les plus simples où l'approche du groupe de renormalisation peut être implémentée de façon naturelle. Les propriétés qui émergent de l'implémentation de la transformation du groupe de renormalisation clarifient le comportement critique de ces systèmes. En ce qui concerne le modèle de verre de spin en dimension finie que nous avons étudié, nous avons développé une nouvelle technique pour implémenter la transformation du groupe de renormalisation pour les verres de spin en dimension finie. Cette technique montre que le système a une transition de phase, caractérisée par un point critique où la longueur de corrélation du système devient infinie. Quant au modèle de verre structural en dimension finie que nous avons étudié, ceci est le premier modèle de verre structural pour lequel on a démontré l'existence d'une transition de phase au delà du champ moyen. Les idées introduites dans ce travail peuvent être développées dans le but de comprendre la structure de la phase de basse température de ces systèmes, et dans le but comprendre si les propriétés de la phase de basse température du champ moyen continuent à être valables pour les systèmes vitreux en dimension finie. / In this thesis we investigate the employ of the renormalization group for glassy systems. More precisely, we focus on models of spin glasses and structural glasses. Spin-glass models represent disordered uniaxial magnetic materials, such as a dilute solution of Mn in Cu, modeled by an array of spins on the Mn arranged at random in the matrix of Cu, and interacting with a potential which oscillates as a function of the separation of the spins. Structural glasses are liquids that have been cooled fast enough to avoid crystallization, like o-Terphenyl or Glycerol. Spin and structural glasses are physically interesting because their critical properties are known only in the limit where the space dimensionality tends to infinity, i. e. in the mean-field approximation. A fundamental question is whether the physical properties characterizing these systems in the mean-field case still hold for real spin or structural glasses, which live in a space with a finite number of dimensions. The spin and structural glasses that we study in this thesis are models built up on hierarchical lattices, which are the simplest non-mean field systems where the renormalisation group approach can be implemented in a natural way. The features emerging from this implementation clarify the critical behavior of these systems. As far as the finite-dimensional spin glass studied in this thesis is concerned, we developed a new technique to implement the renormalization group transformation for finite-dimensional spin glasses. This technique shows that the system has a finite-temperature phase transition characterized by a critical point where the system's correlation length is infinite. As far as the structural glass studied in this thesis is concerned, this is the first structural glass model where we showed the existence of a phase transition beyond mean field. The ideas introduced in this work can be further developed in order to understand the structure of the low-temperature phase of these systems, and in order to establish whether the properties of the low-temperature phase holding in the mean-field case still hold for finite-dimensional glassy systems.

Vidros de spins são sistemas extremamente complexos caracterizados por um número enorme de estados estáveis e meta estáveis. Se identificarmos cada um desses estados com uma informação memorizada, esses sistemas podem ser utilizados como memórias associativas ou endereçáveis por conteúdo. O modelo de vidro de spins passa então a ser chamado de rede de neurônios. Neste trabalho estudamos a termodinâmica e alguns aspectos dinâmicos de uma rede de neurônios com processamento paralelo ou síncrono - o Modelo de Little de memória associativa - no regime em que o número de informações memorizadas p cresce como p = αN, onde N é o número de neurônios. Usando a teoria simétrica em relação às réplicas obtemos o diagrama de fases no espaço de parâmetros do modelo no qual incluímos um termo de autointeração dos neurônios.A riqueza do diagrama de fases que possui uma superfície de pontos tricríticos é devida à competição entre os dois regimes assintóticos da dinâmica síncrona: pontos fixos e ciclos de período dois. / Spin glasses are very complex systems characterized by a huge number of stable and metastable states. If we identify each state with a memorized information then spin glasses may be used as associative or content addressable memories. This spin glass model is then called a neural network. In this work we study the thermodynamics and some dynamical aspects of a neural network with parallel or synchronous processing - Little\'s model of associative memory -in the regime where the number of memorized informations p grows as p = αN, where N is the number of neurons. Using the replica symmetric theory we determine the phase diagram in the space of the model\'s parameters, in which we include a neural self interaction term. The richness of the phase diagram which possesses a surface of tricritical points is due to the competition between the two asymptotic dynamical behaviours of the synchronous dynamics: fixed points and cycles of lenght two.

Neste trabalho investigamos as propriedades estáticas de um modelo de vidro de spin esférico antiferromagnético com interação de multi-spins por meio do método das réplicas. O objetivo consiste em modelar materiais antiferromagnéticos diluídos como FexMg1-xCl2 e compostos antiferromagnéticos mistos como FexMn1-xTiO3 que apresentam evidências de comportamento característico da fase vidro de spin para um intervalo de valores de concentração x. Analisamos a solução réplica simétrica e a solução de Parisi com uma etapa de quebra de simetria entre réplicas, a qual espera-se que seja a solução mais geral para este modelo. Quatro fases são obtidas no diagrama de fases: paramagnética, vidro de spin, antiferromagnética e antiferromagnética com quebra de simetria ou fase mista. As linhas de transição podem ser contínuas ou descontínuas. / In this work we investigate the static properties of a multi-spin antiferromagnetic spherical spin-glass model using the replica method. The aim is to try to model diluted antiferromagnetic materials (e.g. FexMg1-xCl2) and mixed antiferromagnetic compounds (e.g. FexMn1-xTiO3) that present evidences of a spin-glass behavior for certain range of x values. The replica-symmetric and the one-step replica-symmetry-breaking solution given by Parisi are analised, the last one being expected to be the most general solution for this model. Four phases are found in the phase diagram: paramagnetic, spin glass, antiferromagnetic and mixed or glassy antiferromagnetic phase. The transition lines may be either continuous or discontinuous.

Investigamos analiticamente as propriedades estatísticas dos mínimos locais (estados metaestáveis) de vidros de spin de Ising com interações de p-spins na presença de um campo magnético h. O número médio de mínimos, assim como a sobreposição típica entre pares de mínimos idênticos são calculados para qualquer valor de p. Para p &#62 2 e h pequeno mostramos que a sobreposição típica qt é uma função descontínua da energia. O tamanho na descontinuidade em qt cresce com p e decresce com h, indo a zero para valores finitos do campo magnético [1]. Investigamos as correções ao alcance infinito para o caso em que h = 0 e encontramos que o número de estados metaestáveis aumenta quando o efeito de conectividade finita é considerado e esse aumento torna-se mais pronunciado à medida que p aumenta [2]. Ainda, estudamos a termodinâmica deste modelo utilizando o método das réplicas. Demos ênfase à análise da transição entre os regimes de simetria de réplicas e o primeiro passo de quebra de simetria de réplicas. Em particular, derivamos condições analíticas para o início da transição contínua, assim como para a localização do ponto tricrítico onde a transição entre os dois regimes torna-se descontínua [3]. Como aplicação de interações de ordem alta em sistemas de spins contínuos, estudamos analiticamente as propriedades estatísticas de um ecossistema composto de N espécies interagindo através de interações Gaussianas aleatórias de ordem p &#8805 2 e auto-interações determinísticas u &#8805 0. Para o caso u &#8800 0, o aumento na ordem das interações faz com que o sistema se torne mais cooperativo. Para p &#62 2 há um limite inferior para a concentração de espécies sobreviventes, prevenindo a existência de espécies raras e, conseqüentemente, aumentando a robustez do ecossistema contra perturbações externas [4]. / The statistical properties of the local optima (metastable states) of the infinite range Ising spin glass with p-spin interactions in the presence of an external magnetic field h are investigated analytically. The average number of optima as well as the typical overlap between pairs of identical optima are calculated for general p. For p &#62 2 and small h we show that the typical overlap qt is a discontinuous function of the energy. The size of the jump in qt increases with p and decreases with h, vanishing at finite values of the magnetic field [1]. We study the corrections to the infinite range model for h = 0 and find that the number of local optima increases as the effect of the finite connectivity is considered, and that this increase becomes more pronounced for large p [2]. Furthermore, we study analytically the thermodynamics of this model using the replica method, giving emphasis to the analysis of the transition between the replica symmetric and the one-step of replica symmetry breaking regimes. In particular, we derive analytical conditions for the onset of the continuous transition, as well as for the location of the tricritical point at which the transition between those two regimes becomes discontinuous [3]. As an application of high-order interactions in systems of continuous spins, we study the statistical properties of an ecosystem composed of N species interacting via random Gaussian interactions of order p &#8805 2, and deterministic self-interactions u &#8805 0. For nonzero u the increase of the order of the interactions makes the system more cooperative. We find that for p &#62 2 there is a threshold value which gives a lower bound to the concentration of the surviving species, preventing then the existence of rare species and, consequently, increasing the robustness of the ecosystem to external perturbations [4].

In this thesis we investigate the employ of the renormalization group for glassy systems. More precisely, we focus on models of spin glasses and structural glasses. Spin-glass models represent disordered uniaxial magnetic materials, such as a dilute solution of Mn in Cu, modeled by an array of spins on the Mn arranged at random in the matrix of Cu, and interacting with a potential which oscillates as a function of the separation of the spins. Structural glasses are liquids that have been cooled fast enough to avoid crystallization, like o-Terphenyl or Glycerol. Spin and structural glasses are physically interesting because their critical properties are known only in the limit where the space dimensionality tends to infinity, i. e. in the mean-field approximation. A fundamental question is whether the physical properties characterizing these systems in the mean-field case still hold for real spin or structural glasses, which live in a space with a finite number of dimensions. The spin and structural glasses that we study in this thesis are models built up on hierarchical lattices, which are the simplest non-mean field systems where the renormalisation group approach can be implemented in a natural way. The features emerging from this implementation clarify the critical behavior of these systems. As far as the finite-dimensional spin glass studied in this thesis is concerned, we developed a new technique to implement the renormalization group transformation for finite-dimensional spin glasses. This technique shows that the system has a finite-temperature phase transition characterized by a critical point where the system's correlation length is infinite. As far as the structural glass studied in this thesis is concerned, this is the first structural glass model where we showed the existence of a phase transition beyond mean field. The ideas introduced in this work can be further developed in order to understand the structure of the low-temperature phase of these systems, and in order to establish whether the properties of the low-temperature phase holding in the mean-field case still hold for finite-dimensional glassy systems.

Studying magnetic properties of disordered alloys is important both for the understanding of phase transformations in alloys and from the point of view of fundamental issues of magnetism in solids. Disorder in a magnetic system can result in unconventional magnetic structures, such as spin glass, which have rather peculiar features. In this Thesis, a rather general approach to studying disordered magnetic alloys from first principles is presented. Phase transformations and magnetic behavior of crystalline substitutional alloys are considered. This approach is exemplified by calculations of an archetypical spin-glass material: the CuMn alloy. First, a general theoretical framework for the description of the thermodynamics of disordered magnetic alloys is given. It is shown that under certain conditions, a complex magnetic system can be reduced to an effective system containing no magnetic degrees of freedom. This substantially simplifies the investigation of phase transformations in magnetic alloys. The effective model is described in terms of material-specific interaction parameters. It is shown that interaction parameters can be obtained from the ground-state property of a disordered alloy which are in turn calculated from first principles by means of highly accurate up-to-date numerical techniques based on the Green's function method. The interaction parameters can subsequently be used in thermodynamic Monte-Carlo simulations to produce the atomic and magnetic structures of an alloy. An example of calculations for the Cu-rich CuMn alloy is given. It is demonstrated that the atomic and magnetic structure of the alloy obtained by the presented approach agrees very well with the results of neutron-scattering experiments for this system. Moreover, numerical simulations enable one to predict the ground state structure of the alloy, which is difficult to observe in experiment due to large atomic diffusion barriers at temperatures close to the temperature of the phase transformation. A general description of a spin glass is given, and difficulties of modeling this type of magnetic systems are discussed. To overcome the difficulties, improved Monte-Carlo methods, such as parallel tempering, overrelaxation technique, and finite-size scaling method of analysis, are introduced. The results for the CuMn alloy are presented.

Thesis (Ph.D.)--McMaster University (Canada), 1995. / Source: Dissertation Abstracts International, Volume: 57-03, Section: B, page: 1867. Adviser: T. Timusk.

Neste trabalho investigamos as propriedades estáticas de um modelo de vidro de spin esférico antiferromagnético com interação de multi-spins por meio do método das réplicas. O objetivo consiste em modelar materiais antiferromagnéticos diluídos como FexMg1-xCl2 e compostos antiferromagnéticos mistos como FexMn1-xTiO3 que apresentam evidências de comportamento característico da fase vidro de spin para um intervalo de valores de concentração x. Analisamos a solução réplica simétrica e a solução de Parisi com uma etapa de quebra de simetria entre réplicas, a qual espera-se que seja a solução mais geral para este modelo. Quatro fases são obtidas no diagrama de fases: paramagnética, vidro de spin, antiferromagnética e antiferromagnética com quebra de simetria ou fase mista. As linhas de transição podem ser contínuas ou descontínuas. / In this work we investigate the static properties of a multi-spin antiferromagnetic spherical spin-glass model using the replica method. The aim is to try to model diluted antiferromagnetic materials (e.g. FexMg1-xCl2) and mixed antiferromagnetic compounds (e.g. FexMn1-xTiO3) that present evidences of a spin-glass behavior for certain range of x values. The replica-symmetric and the one-step replica-symmetry-breaking solution given by Parisi are analised, the last one being expected to be the most general solution for this model. Four phases are found in the phase diagram: paramagnetic, spin glass, antiferromagnetic and mixed or glassy antiferromagnetic phase. The transition lines may be either continuous or discontinuous.

Vidros de spins são sistemas extremamente complexos caracterizados por um número enorme de estados estáveis e meta estáveis. Se identificarmos cada um desses estados com uma informação memorizada, esses sistemas podem ser utilizados como memórias associativas ou endereçáveis por conteúdo. O modelo de vidro de spins passa então a ser chamado de rede de neurônios. Neste trabalho estudamos a termodinâmica e alguns aspectos dinâmicos de uma rede de neurônios com processamento paralelo ou síncrono - o Modelo de Little de memória associativa - no regime em que o número de informações memorizadas p cresce como p = αN, onde N é o número de neurônios. Usando a teoria simétrica em relação às réplicas obtemos o diagrama de fases no espaço de parâmetros do modelo no qual incluímos um termo de autointeração dos neurônios.A riqueza do diagrama de fases que possui uma superfície de pontos tricríticos é devida à competição entre os dois regimes assintóticos da dinâmica síncrona: pontos fixos e ciclos de período dois. / Spin glasses are very complex systems characterized by a huge number of stable and metastable states. If we identify each state with a memorized information then spin glasses may be used as associative or content addressable memories. This spin glass model is then called a neural network. In this work we study the thermodynamics and some dynamical aspects of a neural network with parallel or synchronous processing - Little\'s model of associative memory -in the regime where the number of memorized informations p grows as p = αN, where N is the number of neurons. Using the replica symmetric theory we determine the phase diagram in the space of the model\'s parameters, in which we include a neural self interaction term. The richness of the phase diagram which possesses a surface of tricritical points is due to the competition between the two asymptotic dynamical behaviours of the synchronous dynamics: fixed points and cycles of lenght two.

Investigamos analiticamente as propriedades estatísticas dos mínimos locais (estados metaestáveis) de vidros de spin de Ising com interações de p-spins na presença de um campo magnético h. O número médio de mínimos, assim como a sobreposição típica entre pares de mínimos idênticos são calculados para qualquer valor de p. Para p &#62 2 e h pequeno mostramos que a sobreposição típica qt é uma função descontínua da energia. O tamanho na descontinuidade em qt cresce com p e decresce com h, indo a zero para valores finitos do campo magnético [1]. Investigamos as correções ao alcance infinito para o caso em que h = 0 e encontramos que o número de estados metaestáveis aumenta quando o efeito de conectividade finita é considerado e esse aumento torna-se mais pronunciado à medida que p aumenta [2]. Ainda, estudamos a termodinâmica deste modelo utilizando o método das réplicas. Demos ênfase à análise da transição entre os regimes de simetria de réplicas e o primeiro passo de quebra de simetria de réplicas. Em particular, derivamos condições analíticas para o início da transição contínua, assim como para a localização do ponto tricrítico onde a transição entre os dois regimes torna-se descontínua [3]. Como aplicação de interações de ordem alta em sistemas de spins contínuos, estudamos analiticamente as propriedades estatísticas de um ecossistema composto de N espécies interagindo através de interações Gaussianas aleatórias de ordem p &#8805 2 e auto-interações determinísticas u &#8805 0. Para o caso u &#8800 0, o aumento na ordem das interações faz com que o sistema se torne mais cooperativo. Para p &#62 2 há um limite inferior para a concentração de espécies sobreviventes, prevenindo a existência de espécies raras e, conseqüentemente, aumentando a robustez do ecossistema contra perturbações externas [4]. / The statistical properties of the local optima (metastable states) of the infinite range Ising spin glass with p-spin interactions in the presence of an external magnetic field h are investigated analytically. The average number of optima as well as the typical overlap between pairs of identical optima are calculated for general p. For p &#62 2 and small h we show that the typical overlap qt is a discontinuous function of the energy. The size of the jump in qt increases with p and decreases with h, vanishing at finite values of the magnetic field [1]. We study the corrections to the infinite range model for h = 0 and find that the number of local optima increases as the effect of the finite connectivity is considered, and that this increase becomes more pronounced for large p [2]. Furthermore, we study analytically the thermodynamics of this model using the replica method, giving emphasis to the analysis of the transition between the replica symmetric and the one-step of replica symmetry breaking regimes. In particular, we derive analytical conditions for the onset of the continuous transition, as well as for the location of the tricritical point at which the transition between those two regimes becomes discontinuous [3]. As an application of high-order interactions in systems of continuous spins, we study the statistical properties of an ecosystem composed of N species interacting via random Gaussian interactions of order p &#8805 2, and deterministic self-interactions u &#8805 0. For nonzero u the increase of the order of the interactions makes the system more cooperative. We find that for p &#62 2 there is a threshold value which gives a lower bound to the concentration of the surviving species, preventing then the existence of rare species and, consequently, increasing the robustness of the ecosystem to external perturbations [4].