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Universality of the Random-Cluster Model and Applications to Quantum Systems

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Universality of the Random-Cluster Model and Applications to Quantum Systems UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES Section de Mathématiques Professeur Hugo Duminil-Copin Section de Mathématiques Professeur Stanislav Smirnov Universality of the Random-Cluster Model and Applications to Quantum Systems THÈSE Présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de Docteur ès Sciences, mention Mathématiques par Jhih-Huang LI de Taïpei (Taïwan) Thèse N°5155 GENÈVE Atelier d’impression ReproMail 2017 x而不思GT，思而不xG殆。 TP Abstract This thesis is divided into two parts: the rst part being dedicated to the universality of the random-cluster model and the second to its quantum counterpart and, in particular, to the quantum Ising model. The random-cluster model is a generalization of Bernoulli percolation, the Ising model and the q-color Potts model. It can be seen as a reweighted Bernoulli percolation with an additional (real) weight parameter q > 1, which is also the number of colors in the Potts model when it is an integer. These models have been widely studied on planar regular graphs, especially on the square lattice. Critical parameters are known and behaviors at the criticality and away from the criticality are also fairly well understood. Moreover, for the Ising model, by means of the parafermionic observable, we can prove the conformal invariance of interfaces separating dierent connected components. In the thesis, we study the random-cluster model on a wider family of (planar) graphs, called isoradial, by proving that the same properties also hold. This family of graphs is in- teresting due to the following reasons. A parafermionic observable can be dened on such graphs for our model of interest and nice combinatorial properties can be deduced. Along with the complex analysis on isoradial graphs, we may also get some exact relations at the discrete level. Moreover, under star-triangle transformations, the main tool that we introduce in the thesis, random-cluster measures are preserved. This allows us to transport properties, even only conjecturally known, from the square lattice to other isoradial graphs. In particu- lar, results using methods which are specic to the square lattice (such as the transfer matrix formulation) can be obtained on isoradial graphs in this way. A (d + 1)-dimensional quantum model consists of d dimensions in space and 1 dimen- sion in time, representing evolution of a quantum state under the action of a Hamiltonian. If we represent both the space and time dimensions graphically, a (1 + 1)-quantum model has a planar representation, and thus, we expect to nd the same properties as its planar counterpart. We dene the quantum version of the aforementioned random-cluster model, then com- pute its critical parameters and determine its behavior at the criticality and away from it. This is an application of the previous part: the quantum random-cluster model can be seen as the limit of its discrete counterpart dened on more and more attened isoradial graphs. Then, by proving uniform probability bounds on crossing events, we obtain the same properties for the quantum model. iii Abstract To conclude the thesis, we prove, for the (1 + 1)-dimensional quantum Ising model, a classical result of the 2D Ising model: the conformal invariance. We work directly in the quantum setting; in other words, on the semi-discrete lattice Z R. × iv Résumé Cette thèse comprend deux parties : la première portant sur l’universalité du modèle de ran- dom-cluster et la seconde sur sa version quantique, et plus précisément, sur le modèle d’Ising quantique. Le modèle de random-cluster est une généralisation de la percolation de Bernoulli, le mo- dèle d’Ising et le modèle de Potts à q couleurs. Ce modèle peut être vu comme une percolation de Bernoulli pondérée à l’aide d’un paramètre supplémentaire q > 1, qui est aussi le nombre de couleurs dans le modèle de Potts lorsque ce dernier est un entier. Ces modèles ont été beaucoup étudiés sur les graphes réguliers, dont le réseau carré en particulier. Les paramètres critiques sont connus et les comportements au point critique et en dehors du point critique sont plutôt bien compris. De plus, l’observable fermionique pour le modèle d’Ising nous permet de prouver l’invariance conforme des interfaces qui séparent des composantes connexes distinctes. Dans cette thèse, nous étudions le modèle de random-cluster sur une famille de graphes (planaires) plus large, appelés isoradiaux et nous démontrons que les mêmes propriétés sont aussi satisfaites. Cette famille de graphes ont un intérêt particulier pour les raisons suivantes. Une observable parafermionique peut être dénie sur de tels graphes pour les modèles qui nous intéressent et à partir de celle-ci, on peut déduire de bonnes propriétés combinatoires. De plus, avec la théorie de l’analyse complexe sur ces graphes, nous obtenons aussi des rela- tions exactes au niveau discret. Nous introduisons aussi les transformations triangle-étoile, qui jouent le rôle central dans cette thèse. Ce sont des transformations qui préservent les mesures de random-cluster qui transportent des propriétés, même si elles sont seulement conjecturales, du réseau carré à n’importe quel autre graphe isoradial. Ce qui est particuliè- rement intéressant est que les résultats qui découlent des méthodes propres au réseau carré (les matrices de transfert par exemple) peuvent aussi être obtenus de cette manière. Un modèle quantique de dimension (d + 1) contient d dimensions en espace et une en temps, qui représente l’évolution d’un état quantique sous l’action d’un Hamiltonien. Si nous représentons l’espace et le temps graphiquement, un modèle quantique de dimension (1 + 1) admet une représentation planaire, et nous nous attendons à trouver sur ce dernier les mêmes propriétés que son homologue planaire. Nous dénissons le modèle de random-cluster quantique, calculons ses paramètres cri- tiques et déterminons ses comportements au point critique et en dehors du point critique. Ceci est une application de la partie précédente : le modèle quantique peut être vu comme la limite du modèle discret déni sur des graphes isoradiaux de plus en plus plats. Nous établis- v Résumé sons des bornes uniformes sur les probabilités de croisement an d’étudier les comportements mentionnés ci-dessus. Pour conclure cette thèse, nous démontrons un résultat classique du modèle d’Ising pla- naire pour le modèle d’Ising quantique de dimension (1 + 1) : l’invariance conforme. Nous travaillons directement dans le cadre quantique, en d’autres termes, sur le réseau semi-discret Z R. × vi Remerciements Partir loin de chez soi pour les études n’est pas une chose facile. Aller dans un pays où l’on comprend à peine la langue et la culture, y construire une vie et en être content, c’est en- core moins évident. Tout cela sans parler de l’apprentissage mathématique, une chose qui demande de la patience et beaucoup de réexion. Ces neuf ans et demi à l’étranger, durant lesquels se sont passés d’innombrables événements inoubliables, se sont envolés comme le vent. J’aimerais dédier ce paragraphe de la thèse à ces personnes avec qui j’ai partagé ces moments précieux et je demande aussi pardon à ceux que j’aurais oubliés par erreur. L’aboutissement de cette thèse n’aurait pas été possible sans l’aide de mes directeurs de thèse, Hugo Duminil-Copin et Stanislav Smirnov. Je les remercie pour leur disponibilité, leur patience, et leur connaissance mathématique, dont j’ai encore beaucoup à apprendre. Je suis également reconnaissant envers Vincent Beara, qui a dirigé mon mémoire de master et qui m’a initié au domaine de la physique statistique. J’aimerais aussi remercier Ioan Manolescu, avec qui j’ai fait une grande partie de cette thèse et grâce à qui j’ai appris la rigueur mathéma- tique dans la rédaction après de nombreuses corrections et réécritures. Mes remerciements vont également à Yvan Velenik, qui a accepté de faire partie du jury et qui m’a aidé à plu- sieurs reprises au cours de mon doctorat. Il ne faut pas non plus oublier d’autres membres du groupe de probabilités à Genève avec qui j’ai eu des discussions mathématiques enri- chissantes : Maxime, Sébastien, Roma, Matan, Eveliina, Hao, Sasha, Vincent, Marcelo, Aran, Misha, Dasha et Marianna. Les enseignants que j’ai eus en classe préparatoire m’ont donné la première motivation pour la recherche mathématique. Je tiens à remercier Monsieur Dupont et Monsieur Duval pour leur pédagogie ainsi que la culture générale mathématique qu’ils m’ont apportée. Je pense aussi à Henri Guenancia pour son apport et son encadrement au cours de mon projet TIPE ; Gabriel Scherer et Irène Walspurger pour les colles d’informatiques et mathématiques qui m’ont été très utiles. Mon tuteur Thierry Bodineau a joué un rôle important durant mes années d’ENS, qui m’a encouragé et m’a donné des informations sur les conférences de mé- canique statistique. Je tiens aussi à mentionner Jean-François Le Gall, Wendelin Werner et Raphaël Cerf pour leurs cours, leurs conseils professionnels et leurs disponibilités pendant mon année de M2. Je n’oublie pas non plus Sébastien Martineau, qui a partagé avec moi sa passion des mathématiques, plus précisement de la vulgarisation mathématique, et avec qui j’ai pu réaliser quelques petites expériences intéressantes. N’étant pas né francophone, je suis reconnaissant à ceux qui ne se sont jamais montrés impatients avec mon français et qui m’ont aidé à l’améliorer. Merci à mon professeur de fran- çais et amie épistolaire Angélique pour ses encouragements et ses petits mots ainsi qu’à deux vii Remerciements autres professeurs de l’Institut français de Taïpei, Sonia et Thierry, pour m’avoir beaucoup poussé durant les heures de cours intensifs.
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