The aim of this thesis is to generalise Reid's recipe as first defined by Reid for $G-\Hilb(\mathbb{C}^3)$ ($G$ a finite abelian subgroup of $\SL(3, \mathbb{C})$) to the setting of consistent dimer models. We study the $\theta$-stable representations of a quiver $Q$ with relations $\mathcal{R}$ dual to a consistent dimer model $\Gamma$ in order to introduce a well-defined recipe that marks interior lattice points and interior line segments of a cross-section of the toric fan $\Sigma$ of the moduli space $\mathcal{M}_A(\theta)$ with vertices of $Q$, where $A=\mathbb{C}Q/\langle \mathcal{R}\rangle$. After analysing the behaviour of 'meandering walks' on a consistent dimer model $\Gamma$ and assuming two technical conjectures, we introduce an algorithm - the arrow contraction algorithm - that allows us to produce new consistent dimer models from old. This algorithm could be used in the future to show that in doing combinatorial Reid's recipe, every vertex of $Q$ appears 'once' and that combinatorial Reid's recipe encodes the relations of the tautological line bundles of $\mathcal{M}_A(\theta)$ in $\Pic(\mathcal{M}_A(\theta))$.

Resolvemos o modelo de d´meros em duas redes planas diferentes, a rede 4-8 e a rede hexagonal (favo de mel). Na rede 4-8 ocorre uma transição do tipo Ising (bidimensional); na rede hexagonal há uma transição conhecida como 3/2. Após a definição do modelo mostramos que o cálculo da função de partição pode ser formulado em termos do traço de uma matriz de transferência escrita numa representação de matrizes de Pauli. Usando a transformação de Jordan-Wigner, os operadores de Pauli são transformados em operadores de criação e aniquilação de férmions, e a matriz de transferência pode ser diagonalizada pela redução a um problema de férmions livres. Comparamos as soluções do modelo de dímeros na rede 4-8 e do modelo de Ising bidimensional; em particular, comparamos o comportamento do calor específico e analisamos o espectro da matriz de transferência. Verificamos que as nossas soluções concordam com resultados obtidos pelas técnicas combinatórias. Utilizamos a formulação da matriz de transferência para construir uma versão de tempo contínuo dos modelos de dímeros nas redes quadrada, 4-8 e hexagonal. Ao contrário do modelo de Ising, no caso dos dímeros essa aproximação de tempo contínuo altera a natureza do comportamento crítico. / We solve the dimer model on two different planar lattices, the 4-8 lattice and the honeycomb lattice. In the dimer model on the 4-8 lattice there is a phase transition of the (two-dimensional) Ising type; on the honeycomb lattice there is a phase transition known as 3/2. After defining the model we show that the calculation of the partition function can be formulated as the trace of a transfer matrix that is written in terms of Pauli matrices. Using the Jordan-Wigner transformation, the Pauli matrices give rise to fermion creation and annihilation operators, and the problem is reduced to the diagonalization of a system of free fermions. We compare the solutions of the dimer model on the 4-8 lattice and of the two-dimensional Ising model; in particular, we compare the behavior of the specific heat and we analyze the spectrum of the transfer matrix. These solutions agree with well-known results from combinatorial techniques. We then use the transfer matrix approach to obtain a continuum time formulation for the dimer models on the square, 4-8 an d honeycomb lattices. In contrast to the Ising case, for the dimer models this approximation changes the nature of critical behavior.

Resolvemos o modelo de d´meros em duas redes planas diferentes, a rede 4-8 e a rede hexagonal (favo de mel). Na rede 4-8 ocorre uma transição do tipo Ising (bidimensional); na rede hexagonal há uma transição conhecida como 3/2. Após a definição do modelo mostramos que o cálculo da função de partição pode ser formulado em termos do traço de uma matriz de transferência escrita numa representação de matrizes de Pauli. Usando a transformação de Jordan-Wigner, os operadores de Pauli são transformados em operadores de criação e aniquilação de férmions, e a matriz de transferência pode ser diagonalizada pela redução a um problema de férmions livres. Comparamos as soluções do modelo de dímeros na rede 4-8 e do modelo de Ising bidimensional; em particular, comparamos o comportamento do calor específico e analisamos o espectro da matriz de transferência. Verificamos que as nossas soluções concordam com resultados obtidos pelas técnicas combinatórias. Utilizamos a formulação da matriz de transferência para construir uma versão de tempo contínuo dos modelos de dímeros nas redes quadrada, 4-8 e hexagonal. Ao contrário do modelo de Ising, no caso dos dímeros essa aproximação de tempo contínuo altera a natureza do comportamento crítico. / We solve the dimer model on two different planar lattices, the 4-8 lattice and the honeycomb lattice. In the dimer model on the 4-8 lattice there is a phase transition of the (two-dimensional) Ising type; on the honeycomb lattice there is a phase transition known as 3/2. After defining the model we show that the calculation of the partition function can be formulated as the trace of a transfer matrix that is written in terms of Pauli matrices. Using the Jordan-Wigner transformation, the Pauli matrices give rise to fermion creation and annihilation operators, and the problem is reduced to the diagonalization of a system of free fermions. We compare the solutions of the dimer model on the 4-8 lattice and of the two-dimensional Ising model; in particular, we compare the behavior of the specific heat and we analyze the spectrum of the transfer matrix. These solutions agree with well-known results from combinatorial techniques. We then use the transfer matrix approach to obtain a continuum time formulation for the dimer models on the square, 4-8 an d honeycomb lattices. In contrast to the Ising case, for the dimer models this approximation changes the nature of critical behavior.

This thesis is devoted to the study of orientifolds and dynamical supersymmetry breaking in configurations of D-branes on toric Calabi-Yau singularities, through the lens of dimer models. We first review the basic ingredients of string theory that led to the formulation of gauge/gravity dualities in terms of dimers. Then, we discuss the non-abelian anomaly cancellation conditions for the supersymmetric gauge theories arising on D-branes and provide necessary geometric criteria to determine whether an orientifold projection can be safely introduced. We also find a new realization of orientifold projection without fixed loci in dimer models and expand on its physical features. We argue that it exhausts the possibilities of orientifolding dimer models. In the subsequent part of the thesis, we investigate dynamical supersymmetry breaking vacua in the same class of models and their typical instability along N=2 Coulomb branches. This leads us to formulate a no-go theorem against their stability based on geometrical features of the singularity, and then to establish a precise way to circumvent it. We eventually find the first instance of stable dynamical supersymmetry breaking vacuum in string theory from D-branes on a toric Calabi-Yau singularity, the Octagon. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Les modèles de dimères quantiques (QDM's) ont une série de comportements intéressants, comme de l'ordre topologique et des phases de liquides de spin. Dans cette thèse, nous explorons ces modèles pour un réseaux hexagonal, ainsi que leur équivalence aux problèmes de partitions, un sujet qui fait partie du domaine de la combinatoire. Premièrement, nous étudions le modèle RK, pour lequel la question sur la présence d'une phase avec un gap non-nul restait encore ouverte. Nous décrivons un algorithme Monte-Carlo qui nous permet, entre autres résultats, d'accéder directement au gap du système. Deuxièmement, nous proposons une généralisation de ce modèle. Nous trouvons un diagramme de phase beaucoup plus complexe, avec des transitions de phase entre différents secteurs topologiques, et compatible avec le déconfinement de Cantor. Troisièmement, nous étudions l'application du modèle RK à des réseaux hexagonales associés à des problèmes de partitions planaires. Cela impose des nouvelles conditions de bord, et nous trouvons un nouveau comportement du modèle. Nous proposons aussi une méthode que utilise les propriétés de l'espace de configurations des problèmes de partitions pour réduire la complexité du QDM.Finalement, nous modélisons les problèmes de croissance et effondrement de coin de cristaux classiques dans le cadre des problèmes de partition, trouvant une transition souple entre des interfaces limites du type "amibe" et le cercle arctique. / The quantum dimer models (QDM's) have a series of interesting behaviors, such as topological order and spin liquid phases. In this thesis, we study these models for an honeycomb lattice, and also their equivalence with the partition problems, a subject of the domain of combinatorics. Firstly, we study the RK model, for which the question on whenever one of its phases is gapped or not was still open. We describe an Monte-Carlo algorithm that allows to, among other results, access this gap directly. Secondly, we propose a generalization of this model. We find a more complex phase diagram, with phase transitions between the different topological sectors, and compatible with the Cantor deconfinement. Thirdly, we study the application of the RK model to honeycomb lattices associated to the planar partition problems. This imposes new boundary conditions, and we find a new model behavior. We also propose a méthod that uses the properties of the partition problem's configuration space to reduce the complexity of the QDM. Finally, we modelize the problems of classical crystal corner growth and melting with the formalism of the partition problems, finding a smooth transition between the limit interfaces of type "amoebae" and the arctic circle.

On a compris ces dernières années que certaines mesures d'intrications sont un outil efficace pour la compréhension et la caractérisation de phases nouvelles et exotiques de la matière, en particulier lorsque les méthodes traditionnelles basées sur l'identification d'un paramètre d'ordre sont insuffisantes. Cette thèse porte sur l'étude de quelques systèmes quantiques à basse dimension où un telle approche s'avère fructueuse. Parmi ces mesures, l'entropie d'intrication, définie via une bipartition du système quantique, est probablement la plus populaire, surtout à une dimension. Celle-ci est habituellement très difficile à calculer en dimension supérieure, mais nous montrons ici que le calcul se simplifie drastiquement pour une classe particulière de fonctions d'ondes, nommées d'après Rokhsar et Kivelson. L'entropie d'intrication peut en effet s'exprimer comme une entropie de Shannon relative à la distribution de probabilité générée par les composantes de la fonction d'onde du fondamental d'un autre système quantique, cette fois-ci unidimensionnel. Cette réduction dimensionnelle nous permet d'étudier l'entropie aussi bien par des méthodes numériques (fermions libres, diagonalisations exactes, ...) qu'analytiques (théories conformes). Nous argumentons aussi que cette approche permet d'accéder facilement à certaines caractéristiques subtiles et universelles d'une fonction d'onde donnée en général.Une autre partie de cette thèse est consacrée aux trempes quantiques locales dans des systèmes critiques unidimensionnels. Nous insisterons particulièrement sur une quantité appelée écho de Loschmidt, qui est le recouvrement entre la fonction d'onde avant la trempe et la fonction d'onde à temps t après la trempe. En exploitant la commensurabilité du spectre de la théorie conforme, nous montrons que l'évolution temporelle doit être périodique, et peut même être souvent obtenue analytiquement. Inspiré par ces résultats, nous étudions aussi la contribution de fréquence nulle à l'écho de Loschmidt après la trempe. Celle-ci s'exprime comme un simple produit scalaire -- que nous nommons fidélité bipartie -- et est une quantité intéressante en elle-même. Malgré sa simplicité, son comportement se trouve être très similaire à celui de l'entropie d'intrication. Pour un système critique unidimensionnel en particulier, notre fidélité décroît algébriquement avec la taille du système, un comportement rappelant la célèbre catastrophe d'Anderson. L'exposant est universel et relié à la charge centrale de la théorie conforme sous-jacente. / In recent years, it has been understood that entanglement measures can be useful tools for the understanding and characterization of new and exotic phases of matter, especially when the study of order parameters alone proves insufficient. This thesis is devoted to the study of a few low-dimensional quantum systems where this is the case. Among these measures, the entanglement entropy, defined through a bipartition of the quantum system, has been perhaps one of the most heavily studied, especially in one dimension. Such a quantity is usually very difficult to compute in dimension larger than one, but we show that for a particular class of wave functions, named after Rokhsar and Kivelson, the entanglement entropy of an infinite cylinder cut into two parts simplifies considerably. It can be expressed as the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum system. This dimensional reduction allows for a detailed numerical study (free fermion, exact diagonalizations, \ldots) as well as an analytic treatment, using conformal field theory (CFT) techniques. We also argue that this approach can give an easy access to some refined universal features of a given wave function in general.Another part of this thesis deals with the study of local quantum quenches in one-dimensional critical systems. The emphasis is put on the Loschmidt echo, the overlap between the wave function before the quench and the wave function at time t after the quench. Because of the commensurability of the CFT spectrum, the time evolution turns out to be periodic, and can be obtained analytically in various cases. Inspired by these results, we also study the zero-frequency contribution to the Loschmidt echo after such a quench. It can be expressed as a simple overlap -- which we name bipartite fidelity -- and can be studied in its own right. We show that despite its simple definition, it mimics the behavior of the entanglement entropy very well. In particular when the one-dimensional system is critical, this fidelity decays algebraically with the system size, reminiscent of Anderson's celebrated orthogonality catastrophe. The exponent is universal and related to the central charge of the underlying CFT.

L’étude réalisée dans cette thèse porte sur les phénomènes critiques classiques et quantiques. En effet, les phénomènes critiques et les transitions de phases sont devenus des sujets fondamentaux en physique statistique moderne et en théorie des champs et nous proposons dans cette thèse d’étudier certains modèles qui présentent un comportement critique, à la fois à l’équilibre et hors de l’équilibre. Dans la première partie de la thèse, certaines propriétés du modèle de dimères à deux dimensions sont étudiées. Ce modèle a été largement étudié dans les communautés de physique statistique et de mathématiques et un grand nombre d’applications en physique de la matière condensée existent. Ici, nous proposons de mettre l’accent sur des solutions exactes du modèle et d’utiliser l’invariance conforme afin d’avoir une compréhension profonde de ce modèle en présence de monomères et/ou en présence de bords. Les mêmes types d’outils sont ensuite utilisés pour explorer un autre phénomène important apparaissant dans les modèles de dimères et de chaînes de spin : le cercle arctique. Le but étant de trouver une description adéquate en termes de théorie des champs de ce phénomène, en utilisant des calculs exacts ainsi que de l’analyse asymptotique. La deuxième partie de la thèse concerne les phénomènes critiques hors de l’équilibre dans le contexte des modèles de croissance d’interfaces. Ce domaine de recherche est très important de nos jours, principalement en raison de la découverte de l’équation Kardar-Parisi-Zhang et de ses relations avec les ensembles de matrices aléatoires. La phénoménologie de ces modèles en présence des bords est analysée via des solutions exactes et des simulations numériques, on montre alors que des comportements surprenants apparaissent proches des bords / The study carried in this thesis concerns classical and quantum critical phenomena. Indeed, critical behaviors and phase transitions are fundamental topics in modern statistical physics and field theory and we propose in this thesis to study some models which exhibit such behaviors both at equilibrium and out of equilibrium. In the first part of the thesis, some properties of the two-dimensional dimer model are studied. This model has been studied extensively in the statistical physics and mathematical communities and a lot of applications in condensed matter physics exist. Here we propose to focus on exact solutions of the model and conformal invariance in order to have a deep understanding of this model in presence of monomers, and/or boundaries. The same kind of tools are then used to explore another important phenomenon appearing in dimer models and spin chains: the arctic circle. The goal was to find a proper field theoretical description of this phenomenon using exact solutions and asymptotic analysis. The second part of the thesis concerns out of equilibrium critical phenomena in the context of interface growth models. This field of research is very important nowadays, mainly because of the Kardar-Parisi-Zhang equation and its relations with random matrix ensembles. The phenomenology of these models in presence of boundaries is studied via exact solutions and numerical simulations, we show that surprising behaviors appear close to the boundaries