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Machine Learning and Quantum Phases of Matter Hugo Théveniaut Machine learning and quantum phases of matter Hugo Théveniaut To cite this version: Hugo Théveniaut. Machine learning and quantum phases of matter. Quantum Physics [quant-ph]. Université Paul Sabatier - Toulouse III, 2020. English. NNT : 2020TOU30228. tel-03157002v2 HAL Id: tel-03157002 https://tel.archives-ouvertes.fr/tel-03157002v2 Submitted on 22 Apr 2021 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. THÈSE En vue de l'obtention du DOCTORAT DE L’UNIVERSITÉ DE TOULOUSE Délivré par l'Université Toulouse 3 - Paul Sabatier Présentée et soutenue le 24 novembre 2020 par : Hugo THÉVENIAUT Méthodes d'apprentissage automatique et phases quantiques de la matière Ecole doctorale : SDM - SCIENCES DE LA MATIERE - Toulouse Spécialité : Physique Unité de recherche : LPT-IRSAMC - Laboratoire de Physique Théorique Thèse dirigée par Fabien ALET Jury M. Nicolas REGNAULT, Rapporteur M. Simon TREBST, Rapporteur M. David GUéRY-ODELIN, Examinateur M. Evert VAN NIEUWENBURG, Examinateur Mme Eliska GREPLOVA, Examinatrice M. Fabien ALET, Directeur de thèse Remerciements On exprime sans doute trop rarement sa gratitude, c'est donc avec plaisir et une ´emotioncertaine que je me plie `al'exercice qui va suivre, celui des remerciements de th`ese.J'ai eu de nombreux bienfaiteur.e.s pendant cette th`eseet je leur dois enti`erement la r´eussitede ces trois derni`eres ann´ees. J'adresse d'abord ma reconnaissance `amon directeur de th`ese,Fabien Alet. Je le remercie d'avoir ´et´esi impliqu´e,autant pr^et`adebugger mon code qu’`arelire (tr`es)attentivement ce manuscrit, d'avoir ´et´ecompr´ehensiflorsque j'avais des moments de doute quand la th`esebloquait ou que j'avais des soucis personnels, de m'avoir encadr´edans des projets vari´eset int´eressants, et de m'avoir fait confiance lorsque je pr´esentais nos travaux ou au moment de la r´edaction d'articles. Il m'a aussi donn´ela chance de pouvoir assister `aplusieurs workshops en Allemagne et au Canada (je regrette seulement les 5 tonnes de CO2 que j'ai ´emisespour y participer..). Remerciement particulier pour cette ´ecoled'´et´ed'un mois aux Houches qui a ´et´eune occasion unique de rencontrer les experts mondiaux de notre discipline dans un cadre de r^eve, au pied du mont Blanc. Je suis tr`esheureux d'avoir pu rencontrer Evert van Nieuwenburg, d'abord `aun workshop `a Dresde puis au Perimeter Insitute `aWaterloo. Fabien a bien fait de me pousser `aaller travailler avec lui au Danemark. Ce fut un excellent s´ejourd'un mois et demi `aCopenhague dans son ´equipe. J'ai eu plaisir `acollaborer avec lui sur un projet original qui m'a plu au del`ade mes attentes. Je garderai un tr`esbon souvenir de l'institut Niels Bohr et de ses doctorants, moment que j'ai d^umalheureusement ´ecourter pour cause de pand´emiemondiale. Je remercie ensuite les rapporteurs Nicolas Regnault et Simon Trebst pour leur lecture attentive de ce manuscrit, j'ai trouv´eleurs commentaires justes et instructifs. Je remercie ´egalement les examinateurs Eliska Greplova, David Gu´ery-Odelinet Evert van Nieuwenburg pour leur participation `ama soutenance ainsi qu'`ala discussion que nous avons eu lors de mon oral. Je suis heureux d'avoir pu terminer ma th`esede cette fa¸con. Je remercie Malika et de Sandrine pour leur sympathie et leur disponibilit´e,ainsi que les membres du LPT, en particulier du groupe FFC, dont la pr´esenceun peu intimidante au d´ebut s'est r´ev´el´ee agr´eableet chaleureuse au fil du temps. Je remercie tr`esvivement la fondation CFM sans qui tout ceci n'aurait pas eu lieu. En particulier, le budget destin´e`ala mobilit´equi a ´et´eenglouti dans mon Airbnb `aCopenhague (!). Je remercie sinc`erement mes coll`egueset ami.e.s du "bureau des doctorants" : les deux Benjamin, Julie, Jordan, Th´eoet Yassir. Gr^ace`avous, ce bureau ´etaitun endroit o`uje me sentais bien. Mention sp´eciale`ames camarades de la promotion th`ese2017-2020, Julie et Jordan, qui ont ´et´eles meilleurs (!), j'ai aim´enos discussions parfois l´eg`eres,parfois profondes, toujours agr´eables.Je pense aux doctorants et post-doctorants de tous les ´etagesavec qui j'ai partag´edes repas, pauses caf´eset plus aussi : les trois Maxime, le docte Olivier, Julie du LCAR, Julien, Nathan, Faedi, Ulysse, Pranay et Juraj. Pens´ee ´egalement `ames ´etudiants plus ou moins turbulents et/ou int´eress´es,j'ai appr´eci´emes diff´erentes exp´eriencesd'enseignement qui me changeaient les id´eesquand la recherche n'avan¸caitpas. Je remercie enfin toutes les personnes qui m'ont accompagn´eesau quotidien `aToulouse, leur pr´esencea ´et´eimportante pour moi : Lisa ma premi`erecoloc, Vincent du labo puis plus du labo puis en coloc aussi, Gabriel meilleur compagnon d'escalade et de sorties techno, Mariane avec qui j'ai d´ecouvert plein de choses et fait de beaux voyages, Agn`esdont j'appr´eciebeaucoup la compagnie musicale ou sportive, Nicolas et Julia dont la pr´esencebienveillante et joyeuse a transform´eles difficult´esde la derni`ereann´ee(entre confinements et r´edaction)en souvenirs richtig agr´eables,et enfin Marion qui apporte ´enorm´ement de gentillesse et de douceur `ama vie. Je n'oublie pas mes vieux amis qui comptent pour moi : Ars`ene,Lamia, Jules et Augustin. Et enfin, mes tantes et oncles, cousins, cousines, mes grands-parents, en particulier `ama Mouli qui est partie paisiblement l'´et´edernier. Tout ceci ne serait pas possible sans l'amour et le soutien constants de mes adelphes Zo´eet Robin et de mes parents Herv´eet Agn`es,merci du fond du cœur. ii Contents 1 Introduction 1 1.1 Machine learning....................................1 1.2 Numerical methods for condensed matter physics..................3 1.3 Machine learning and condensed matter physics...................6 1.4 Organization of the manuscript............................9 2 Methods of machine learning 11 2.1 Machine learning basics................................ 12 2.1.1 Dataset..................................... 12 2.1.2 Model...................................... 13 2.1.3 Cost function.................................. 14 2.1.4 Optimization.................................. 14 2.1.5 Examples of machine learning problems................... 15 2.2 A paradigmatic example: Polynomial regression................... 16 2.2.1 Training of the model............................. 16 2.2.2 Evaluation of the model............................ 18 2.2.3 The bias-variance tradeoff........................... 20 2.3 Neural networks.................................... 21 2.3.1 Feed-forward neural networks (FFNN).................... 22 2.3.2 Convolutional neural networks (CNN).................... 24 2.3.3 Restricted Boltzmann machines (RBM)................... 25 2.3.4 Training of neural networks.......................... 26 2.3.5 Regularization................................. 28 2.3.6 Software..................................... 29 2.4 Dimensional reduction................................. 29 2.4.1 Principal component analysis......................... 29 2.4.2 Variational autoencoders........................... 30 2.5 Reinforcement learning................................. 31 2.5.1 A practical example: the game of chess................... 31 2.5.2 Mathematical formalization.......................... 33 3 Automatic classification of phases of matter 35 3.1 Quantum phase transitions.............................. 36 3.1.1 Critical points and universality........................ 36 3.1.2 The finite-size scaling method......................... 37 3.2 Automatic classification of phases of matter..................... 38 3.2.1 Data in condensed matter physics....................... 38 3.2.2 Supervised method............................... 39 3.2.3 Unsupervised method............................. 40 3.2.4 Semi-supervised method............................ 41 3.2.5 Advantages and drawbacks.......................... 42 Contents 3.3 Many-body localization................................ 43 3.3.1 Thermalization in closed systems....................... 44 3.3.2 The random-field Heisenberg model..................... 44 3.4 Application to the ETH-MBL transition in a 1D model.............. 46 3.4.1 Description of the dataset........................... 47 3.4.2 Dataset analysis with PCA.......................... 50 3.4.3 Neural networks as phase classifiers...................... 52 3.4.4 Results: Neural-network output analysis................... 54 3.4.5 Neural-network setups............................. 57 3.4.6 Results: Single system size training...................... 58 3.4.7 Results: Multiple system size training.................... 64 3.4.8 Results: System size adversarial training................... 67 3.4.9 Discussion.................................... 70 3.5 Application to the ETH-MBL transition in a 2D model.............. 71 3.5.1 The quantum dimer model with random disorder.............. 72 3.5.2 Results: Spectral statistics.......................... 72 3.5.3 Results: Machine learning this transition.................. 73 3.5.4 Discussion.................................... 76 4 Neural-network quantum states 77 4.1 Neural-network quantum states...........................
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