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Th`Ese De Doctorat Indecomposability in Field Theory UNIVERSITE´ PIERRE ET MARIE CURIE - PARIS VI et INSTITUT DE PHYSIQUE THEORIQUE´ - CEA/SACLAY Ecole´ Doctorale de Physique de la R´egion Parisienne - ED 107 Th`ese de doctorat Sp´ecialit´e: Physique Th´eorique Indecomposability in field theory and applications to disordered systems and geometrical problems pr´esent´ee par Romain Vasseur pour obtenir le grade de Docteur de l’Universit´ePierre et Marie Curie Th`ese pr´epar´ee sous la direction de Hubert Saleur et de Jesper Jacobsen Soutenue le 27 Septembre 2013 devant le jury compos´ede : Denis Bernard Examinateur Ecole´ Normale Sup´erieure John Cardy Rapporteur Oxford University Matthias Gaberdiel Rapporteur ETH Z¨urich Jesper Jacobsen Membre invit´e(co-directeur) Ecole´ Normale Sup´erieure Nicholas Read Examinateur Yale University Hubert Saleur Directeur de th`ese CEA Saclay Jean-Bernard Zuber Pr´esident du jury Universit´ePierre et Marie Curie ii Indecomposabilit´edans les th´eories des champs et applications aux syst`emes d´esordonn´es et aux probl`emes g´eom´etriques R´esum´e: Les th´eories des champs conformes logarithmiques (LCFTs) sont cruciales pour d´ecrire le comportement critique de syst`emes physiques vari´es : les transitions de phase dans les syst`emes ´electroniques d´esordonn´es sans interaction (comme par exemple la transition entre plateaux dans l’effet Hall quantique entier), les points critiques d´esordonn´es dans les syst`emes statistiques classiques (comme le mod`ele d’Ising avec liens al´eatoires), ou encore les mod`eles g´eom´etriques critiques (comme la percolation ou les marches al´eatoires auto-´evitantes). Les LCFTs d´ecrivent des th´eories non unitaires, qui ne seraient probablement pas pertinentes dans le contexte de la physique des parti- cules, mais qui apparaissent naturellement en mati`ere condens´ee et en physique statis- tique. Sans cette condition d’unitarit´e, toute la puissance alg´ebrique qui a fait le succ`es des th´eories conformes est fortement compromise `acause de “l’ind´ecomposabilit´e” de la th´eorie des repr´esentations sous-jacente. Ceci a pour cons´equence de modifier les fonc- tions de corr´elation alg´ebriques par des corrections logarithmiques, et r´eduit s´ev`erement l’espoir d’une classification g´en´erale. Le but de cette th`ese est d’analyser ces th´eories logarithmiques en ´etudiant leur r´egularisation sur r´eseau, l’id´ee principale ´etant que la plupart des difficult´es alg´ebriques caus´ees par l’ind´ecomposabilit´esont d´ej`apr´esentes dans des syst`emes de taille finie. Notre approche consiste `aconsid´erer des mod`eles statistiques critiques avec matrice de transfert non diagonalisable (ou des chaˆınes de spins critiques avec Hamiltonien non diagonalisable) et d’analyser leur limite thermodynamique `al’aide de diff´erentes m´ethodes num´eriques, alg´ebriques et analytiques. On explique en particulier comment mesurer num´eriquement les param`etres universels qui caract´erisent les repr´esentations ind´ecomposables qui apparaissent `ala limite continue. L’analyse d´etaill´ee d’une vaste classe de mod`eles sur r´eseau nous permet ´egalement de conjecturer une classification de toutes les LCFTs chirales pertinentes physiquement, pour lesquelles la seule sym´etrie est donn´ee par l’alg`ebre de Virasoro. Cette approche est aussi partiellement ´etendue aux th´eories non chirales, avec une attention particuli`ere port´ee au probl`eme bien connu de la formulation d’une th´eorie des champs coh´erente qui d´ecrirait la percolation en deux dimensions. On montre que les mod`eles sur r´eseaux p´eriodiques ou avec bords peuvent ˆetre reli´es alg´ebriquement seulement dans le cas des mod`eles minimaux, impliquant des cons´equences int´eressantes pour les th´eories des champs sous-jacentes. Un certain nombre d’applications aux syst`emes d´esordonn´es et aux mod`eles g´eom´etriques sont ´egalement abord´ees, avec en particulier une discussion d´etaill´ee des observables avec comportement logarithmique au point critique dans le mod`ele de Potts en dimension arbitraire. Mots clefs : Th´eories des Champs Conformes Logarithmiques, Repr´esentations inde- composables, Mod`eles sur r´eseau, Syst`emes d´esordonn´es, Probl`emes g´eom´etriques. iii Indecomposability in field theory and applications to disordered systems and geometrical problems Abstract: Logarithmic Conformal Field Theories (LCFTs) are crucial for describing the crit- ical behavior of a variety of physical systems. These include phase transitions in dis- ordered non-interacting electronic systems (such as the transition between plateaus in the integer quantum Hall effect), disordered critical points in classical statistical mod- els (such as the random bond Ising model), or critical geometrical models (such as polymers and percolation). LCFTs appear when one has to give up the unitarity con- dition, which is natural in particle physics applications, but not in statistical mechanics and condensed matter physics. Without unitarity, the powerful algebraic approach of conformal invariance encounters formidable technical difficulties due to ‘indecompos- ability’. This in turn yields logarithmic corrections to the power-law correlations at the critical point, and prevents the use of general classification techniques that have proven so powerful in the unitary case. The goal of this thesis is to understand LCFTs by studying their lattice regular- izations, the crucial point being that most algebraic complications due to indecompos- ability occur in finite size systems as well. Our approach is thus to consider critical statistical models with non-diagonalizable transfer matrices, or gapless quantum spin chains with non-diagonalizable hamiltonians, and to study their scaling limit by uti- lizing a variety of algebraic, numerical and integrable techniques. We show how to measure numerically universal parameters that characterize the indecomposable rep- resentations of the Virasoro algebra which emerge in the thermodynamic limit. An extensive understanding of a wide class of lattice models allows us to conjecture a ten- tative classification of all possible (chiral) LCFTs with Virasoro symmetry only. This approach is partially extended to the bulk case, for which we discuss how the long- standing bulk CFT formulation of percolation can be tackled along these lines. We also argue that boundary and periodic lattice models can be related algebraically only in the case of minimal models, and we work out the consequences for the underlying boundary and bulk field theories. Several concrete applications to disordered systems and geometrical problems are discussed, and we uncover a large class of geometrical observables in the Potts model that behave logarithmically at the critical point. Key words: Logarithmic Conformal Field Theory, Indecomposable representations, Temperley-Lieb algebra, Lattice models, Disordered systems, Geometrical problems. iv Remerciements Je voudrais commencer par remercier John Cardy et Matthias Gaberdiel, qui ont accept´ela p´enible tache de rapporteur, ainsi que les examinateurs, Denis Bernard, Nicholas Read et Jean-Bernard Zuber. Merci pour votre aide et vos conseils pour am´eliorer mon manuscrit. Je suis tr`es conscient du travail qu’ˆetre membre d’un comit´e de th`ese repr´esente, et je vous suis extrˆemement reconnaissant d’avoir accept´ede braver des heures de transport pour venir assister `ama soutenance. Je remercie aussi Hubert et Jesper, qui en plus de leur qualit´es scientifiques indis- cutables, m’ont beaucoup appris et aid´edans bien des domaines, en particulier dans mes moments de doute, ou pendant la p´eriode difficile des recherches de postdocs. Merci ´egalement de m’avoir soutenu durant mon changement de th´ematique apr`es ma premi`ere moiti´ede th`ese, mˆeme si cela repr´esentait un certain nombre de risques et de difficult´es pour vous aussi. Je sais que j’ai pu rˆaler `acertains moments, mais je suis vraiment ravi d’avoir fait ma th`ese avec vous deux, vous avez toujours ´et´edisponibles et `al’´ecoute en cas de probl`eme : travailler avec vous durant ces trois ann´ees a toujours ´et´eun plaisir. Je tiens ´egalement `aremercier tout particuli`erement Azat Gainutdinov qui a ´et´epostdoc `aSaclay durant toute ma th`ese : j’ai beaucoup appr´eci´etravailler avec toi sur tous ces projets, ¸ca a ´et´etr`es stimulant et b´en´efique pour moi. J’ai eu la chance d’avoir des bureaux dans deux laboratoires, au LPTENS et `a l’IPhT au CEA, ce qui me semble ˆetre un luxe `al’heure ou l’espace disponible pour les th´esards est chaque ann´ee plus restreint. Merci `atous les gens qui font la vie de ces laboratoires, je pense notamment `atous les membres du staff administratif, mais aussi au personnel informatique et `atous les chercheurs avec qui j’ai eu l’opportunit´ede discuter. Merci `aOlivier Golinelli et `aSt´ephane Nonnenmacher pour leurs conseils. Je tiens ´egalement `aremercier tous les th´esards et les postdocs qui ont crois´emon chemin `aSaclay (dans le d´esordre Piotr – sans toi Alexandre m’aurait encore plus empˆech´e de travailler – et Julien – sans toi, j’aurais sans doute oubli´edes noms dans cette liste – les anciens du M2, et aussi Thiago, Jean-Marie, Benoit, R´emi, Bruno, Richard, Guillaume, H´el`ene, Alexander, Hanna, Antoine, J´erˆome et tous les th´esards/postdocs qui continuent de manger ensemble le midi, ainsi que Katya, Nicolas et Thomas qui ont partag´emon bureau), `al’ENS ou `aJussieu (Michele, Tristan, Fabien, S´ebastien, Alexander, Julius et `atous ceux qui ont particip´ede pr`es ou de loin au s´eminaire jeunes cond-mat, qui je l’esp`ere continuera pendant de nombreuses ann´ees), ou au hasard v d’une conf´erence (je pense surtout `aJacopo et Loic r´ecemment). Il m’est impossible de citer tous les noms ici, mais je souhaite remercier tout particuli`erement Alexandre bien sˆur, pour toutes nos discussions scientifiques ou non, et Swann pour les sandwiches Libanais du mercredi et sa compagnie durant le M2 (je me rappellerai toujours du “that’s my spot” avec Pilou).
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