Edge states and supersymmetric sigma models Roberto Bondesan To cite this version: Roberto Bondesan. Edge states and supersymmetric sigma models. Mesoscopic Systems and Quantum Hall Effect [cond-mat.mes-hall]. Université Pierre et Marie Curie - Paris VI, 2012. English. tel- 00808736 HAL Id: tel-00808736 https://tel.archives-ouvertes.fr/tel-00808736 Submitted on 6 Apr 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE´ PARIS 6 - PIERRE ET MARIE CURIE et INSTITUT DE PHYSIQUE THEORIQUE´ - CEA/SACLAY Th`ese de doctorat Sp´ecialit´e Physique Th´eorique Sujet de la th`ese: Edge states and supersymmetric sigma models pr´esent´eepar Roberto BONDESAN pour obtenir le grade de Docteur de l'Universit´eParis 6 Soutenue le 14 Septembre 2012 devant le jury compos´ede: M. Michel Bauer Membre invit´e M. Sergio Caracciolo Rapporteur M. Ilya Gruzberg Examinateur M. Jesper Jacobsen Membre invit´e(co-directeur de th`ese) M. Hubert Saleur Directeur de th`ese M. Kareljan Schoutens Rapporteur M. Jean-Bernard Zuber Pr´esident du jury Abstract A fundamental property of the quantum Hall effect is the presence of edge states at the boundary of the sample, robust against localization and responsible for the perfect quantization of the Hall conductance. The transition between integer quantum Hall plateaus is a delocalization transition, which can be identified as a strong-coupling fixed point of a 1 + 1-dimensional supersymmetric sigma model with topological θ-term. The conformal field theory describing this transition displays unusual features such as non- unitarity, and resisted any attempt of solution so far. In this thesis we investigate the role of edge states at quantum Hall transitions using lattice discretizations of super sigma models. Edge states correspond to twisted boundary conditions for the fields, which can be discretized as quantum spin chains or geometrical (loop) models. For the spin quantum Hall effect, a counterpart of the integer quantum Hall effect for spin transport (class C), our techniques allow the exact computation of critical exponents of the boundary conformal field theories describing higher plateaus transitions. Our predictions for the mean spin conductance are validated by extensive numerical simulations of the related localization problems. Envisaging applications to transport in network models of 2 + 1-dimensional disor- dered electrons, and to quenches in one-dimensional gapless quantum systems, in this thesis we have also developed a new formalism for dealing with partition functions of critical systems in rectangular geometries. As an application, we derive formulas for probabilities of self-avoiding walks. Key words: Loop models, quantum spin chains, boundary conformal field theory, edge states, supersymmetry, quantum Hall effect R´esum´e Une propri´et´efondamentale de l’effet Hall quantique est la pr´esencedes ´etatsde bord. Ils r´esistent `ala localisation et sont responsables de la quantification parfaite de la conductance de Hall. La transition entre les plateaux d'effet Hall quantique entier est une transition de d´elocalisation, qui peut ^etreidentifi´eecomme un point fixe de couplage fort d'un mod`elesigma supersym´etriqueen 1 + 1-dimensions avec terme topologique θ. La th´eorieconforme d´ecrivant cette transition pr´esente des caract´eristiquesinhabituelles telles que la non-unitarit´e,et a r´esist´e`atoute tentative de r´esolutionjusqu’`apr´esent. Dans cette th`ese,nous ´etudionsle r^oledes ´etatsde bord dans les transitions d’effet Hall, en utilisant des discr´etisationssur r´eseaude mod`elessigma. Les ´etatsde bord correspondent aux conditions aux bord pour les champs des mod`elessigma, et peuvent ^etrediscr´etis´esen terme de cha^ınesde spins quantiques ou de mod`elesg´eom´etriques(de boucles). Pour l’effet Hall de spin, un ´equivalent de l’effet Hall entier pour le transport de spin (classe C), nos techniques permettent le calcul exact des exposants critiques des th´eoriesconformes avec bord d´ecrivant les transitions entre plateaux ´elev´es. Nos pr´edictionspour la moyenne de la conductance de spin sont valid´eespar des simulations num´eriquesdes probl`emesde localisation correspondant. Dans cette th`ese,envisageant des applications au transport dans les mod`elessur r´eseaudes ´electronsd´esordonn´esen 2 + 1-dimensions, et aux trempes dans des syst`emes quantiques `aune dimension, nous avons ´egalement d´evelopp´eun nouveau formalisme pour calculer des fonctions de partition de syst`emescritiques sur un rectangle. Comme application, nous d´erivons des formules de probabilit´espour les marches auto-´evitantes. Mots cl´es: Mod`elesde boucles, cha^ınesde spins quantiques, th´eoriesconformes avec bord, ´etatsde bord, supersym´etrie, effet Hall quantique Acknowledgments I would like to express my gratitude to my supervisors Hubert and Jesper, for their constant support, teaching, encouragement and guidance during these three years of PhD. I have learned a lot from them. I am also very grateful to the institutes where I have been equally shared during my graduate studies, the IPhT Saclay and the LPTENS, for the warm and scientifically rich atmosphere. I thank Prof. Caracciolo and Prof. Schoutens for having accepted promptly to be referees on my manuscript and for their comments, and the other members of the com- mittee, Prof. M. Bauer, Prof. I. Gruzberg, and Prof. J.-B. Zuber for their interest in my work. I am also particularly thankful to Prof. Caracciolo for his guidance and encour- agement prior to the start of the PhD, to Prof. Gruzberg for his kind teaching and his support during the search of a postdoc, and to Dr. H. Obuse for our collaboration. Many thanks to the students and postdocs in Saclay and at the LPTENS, among whom I found very good friends. I benefit from discussions (not necessarily about physics) with Alexandre, Axel, Azat, Bruno, Cl´ement, Emanuele, Enrico, Eric, Francesco, Guillaume, Jean-Marie, J´er^ome,Hirohiko, Piotr, Romain, Tristan and Vittore. Special thanks to Cl´ement and Jean-Marie, and in particular to J´er^omefor our collaboration and extremely useful discussions. I am grateful to many people outside the scientific community for their support. First of all my family, and in particular my parents, who gave me the all the means to pursue my interests in life. Then, my tenderest thought goes to Laura, whose love completed my life and helped me to surmount the difficulties that doing a PhD abroad implies. I thank Axel, Adrien, B´eaand Charles for all the good time we had playing together, and Alessandro, Daniele and Luca for our everlasting friendship. Contents Introduction1 1 Boundary conformal field theory and loop models5 1.1 The Potts model, loops and super sigma-models...............5 1.1.1 Transfer matrices and the Temperley-Lieb algebra.........7 1.1.2 Replicas and supersymmetry..................... 12 1.1.3 Algebraic analysis of the spectrum.................. 20 1.1.4 From spin chains to non linear sigma models............ 28 1.2 Boundary conformal field theory....................... 35 1.2.1 The stress tensor and conformal data................ 35 1.2.2 Logarithmic conformal field theories................. 40 1.2.3 The annulus partition function.................... 44 1.2.4 Continuum limit of loop models.................... 52 1.2.5 Boundary loop models......................... 56 2 Fully open boundaries in conformal field theory 60 2.1 Conformal boundary state for the rectangular geometry.......... 60 2.1.1 Gluing condition............................ 60 2.1.2 Computation of the boundary states................. 62 2.1.3 Free theories.............................. 68 2.2 Rectangular amplitudes............................ 71 2.2.1 Homogeneous boundary conditions.................. 72 2.2.2 Conformal blocks and modular properties.............. 74 2.3 Geometrical description of conformal blocks................. 82 2.3.1 Lattice boundary states........................ 82 2.3.2 Probabilistic interpretation...................... 85 3 Quantum network models and classical geometrical models 90 3.1 The Chalker-Coddington model........................ 90 3.1.1 Integer quantum Hall effect...................... 90 3.1.2 Quantum transport in the network model.............. 96 3.1.3 Bosonic representation of observables................. 99 3.1.4 Supersymmetry method........................ 104 i 3.1.5 Geometrical models.......................... 109 3.1.6 Symmetry classes............................ 112 3.2 The spin quantum Hall transition....................... 116 3.2.1 Physics of the spin quantum Hall effect............... 116 3.2.2 The network model........................... 117 3.2.3 Exact solutions............................. 119 4 Boundary criticality at quantum Hall transitions 129 4.1 Edge states in network models........................ 129 4.1.1 Chalker-Coddington model with extra edge channels........ 129 4.1.2 Edge states in loop models and spin chains............. 132 4.1.3 Conformal boundary conditions in sigma models.......... 135 4.2 Exact exponents for the spin quantum Hall transition in