Many-body Localization of Two-dimensional Disordered Bosons These` de doctorat de l’Universite´ Paris-Saclay prepar´ ee´ a` Universite´ Paris-Sud Ecole doctorale n◦564 Physique en ˆIle-de-France (EDPIF) Specialit´ e´ de doctorat : Physique NNT : 2019SACLS031 These` present´ ee´ et soutenue a` Orsay, le 5.2.2019, par GIULIO BERTOLI Composition du Jury : Nicolas Pavloff Professeur, Universite´ Paris-Sud (LPTMS, UMR 8626) President´ Anna Minguzzi Directrice de Recherche, Universite´ Grenoble Alpes (LPMMC, UMR 5493) Rapporteur Vladimir Yudson Professeur, Russian quantum center Rapporteur Vladimir Kravtsov Professeur em´ erite,´ International center for theoretical physics (CMSP) Examinateur Georgy Shlyapnikov Professeur em´ erite,´ Universite´ Paris-Sud (LPTMS, UMR 8626) Directeur de these` ` ese de doctorat Th To my parents iii UNIVERSITE´ PARIS SUD LPTMS { Laboratoire de Physique Th´eoriqueet Mod`elesStatistiques Abstract The study of the interplay between localization and interactions in disordered quan- tum systems led to the discovery of the interesting physics of many-body localization (MBL). This remarkable phenomenon provides a generic mechanism for the breaking of ergodicity in quantum isolated systems, and has stimulated several questions such as the possibility of a finite-temperature fluid-insulator transition. At the same time, the domain of ultracold interacting atoms is a rapidly growing field in the physics of disordered quantum systems. In this Thesis, we study many-body localization in the context of two-dimensional disor- dered ultracold bosons. After reviewing some importance concepts, we present a study of the phase diagram of a two-dimensional weakly interacting Bose gas in a random poten- tial at finite temperatures. The system undergoes two finite-temperature transitions: the MBL transition from normal fluid to insulator and the Berezinskii-Kosterlitz-Thouless transition from algebraic superfluid to normal fluid. At T = 0, we show the existence of a tricritical point where the three phases coexist. We also discuss the influence of the truncation of the energy distribution function at the trap barrier, a generic phenomenon for ultracold atoms. The truncation limits the growth of the localization length with energy and, in contrast to the thermodynamic limit, the insulator phase is present at any temperature. Finally, we conclude by discussing the stability of the insulating phase with respect to highly energetic particles in systems defined on a continuum. Acknowledgements I would like to thank first my thesis supervisor, prof. dr. Georgy Shlyapnikov. During the three years and five months of my PhD, I had the opportunity of learning from one of the greatest minds that I ever came across in my life. As the saying goes: . nos esse quasi nanos gigantium humeris insidentes. The second person that I want to thank is Boris Altshuler. As far as giants are concerned, he truly stands out as one of the greatest physicist that I ever met. Collaborating with him has been a formative experience that I will carry with me throughout my life. During my staying at the LPTMS, I came into contact with a great number of people. This is a general \Thank you", to all of them. I would like also to extend a personal acknowledgment to some of the lab members. Very much in the flavor of this Thesis, I will thank them in random order. The first one is Claudine. Always smiling, her efficiency is second only to her kindness. The lab would be lost without her. Special mention goes as well to my office mates Aleksey and Mehdi. Our conversations have been a great help in my everyday life at the lab. I also had the pleasure of sharing offices with Ada, Tirthankar and Thibaud. Vincent helped me during the first months of my PhD, when I was trying to make sense of the localization idea. Micha, thank you for the animated conversations on great many topics. I also acknowledge the many discussion with Giovanni. Thanks as well to my fellow PhD students for sharing this experience with me and reminding me the importance of the letter \C". Thanks also to Denis for the company and support in the final days of writing. Concerning this Thesis, I would like to thank Jean Dalibard for helping me with the references on the BKT transition, and the jury members for agreeing to be part of the committee. Thanks as well to Christophe Texier for helping me out with the preparation of the defence. I would like to mention as well the great many people who supported me and shared special moments with me in my life outside the lab. To all my friends in Paris, Am- sterdam, Florence and around the world: you are too many to write all your names!!! Nevertheless, I would like to mention Tommaso and Bernardo as the two people on whom I relied the most during my Parisian days. I am very grateful to be your friend. vi Thanks to my family as well, for always being with me. I have always felt supported by every single one of you, and I am extremely grateful for that. My greatest thanks goes to my parents, to whom I dedicate this Thesis. Special mention goes to my cat, Cleopatra, for sitting on my notes and on my lap while I was working. Saving the best for last, Jasmine. No words can express the amount of gratitude I have for you. I wouldn't have been able to write this Thesis without you, your support and your love. Thank you for being exactly as you are. I am grateful that life brought us together, every day. Our life awaits us. Contents Abstract v Acknowledgements vi Contents viii Introduction1 1 Localization: from single-particle to many-body physics5 1.1 Anderson Localization.............................5 1.1.1 Introduction..............................5 1.1.2 Perturbation theory in AL.......................8 1.1.3 Phenomenology of AL......................... 15 1.1.3.1 Spectral properties...................... 16 1.1.3.2 Absence of dc transport................... 17 1.1.3.3 Spectral statistics...................... 17 1.1.4 Scaling theory and dimensionality.................. 19 1.1.5 The Bethe lattice solution....................... 22 1.2 Thermalization................................. 24 1.3 From AL to MBL................................ 27 1.3.1 Localization in Fock space....................... 30 1.3.2 Properties of MBL........................... 31 1.3.2.1 Local integrals of motion.................. 32 1.3.3 Effects of rare regions......................... 34 1.3.3.1 Avalanche scenario...................... 34 1.3.3.2 Many-body mobility edges................. 36 1.3.4 Non-ergodic extended phase...................... 39 2 Disordered ultracold atoms 41 2.1 Introduction................................... 41 2.1.1 Reaching the ultracold limit...................... 42 2.1.2 A playground for disordered interacting systems.......... 43 2.1.3 Two dimensions and long-range order................ 44 ix Contents x 2.2 The Berezinskii-Kosterlitz-Thouless transition................ 45 2.2.1 Introduction.............................. 45 2.2.2 Bogoliubov method........................... 45 2.2.3 Decay of correlations.......................... 47 2.2.4 Vortices and the BKT transition................... 49 2.2.5 Topological nature of the BKT transition.............. 52 2.3 Disordered Bose fluids............................. 54 2.4 Experimental activity............................. 56 3 Finite-Temperature Disordered Bosons in Two Dimensions 59 3.1 Introduction................................... 59 3.1.1 Length and energy scales of the problem............... 59 3.2 Many-Body Localization-Delocalization criterion.............. 62 3.3 Temperature dependence of the MBLDT................... 65 3.3.1 Discussion................................ 67 3.4 Influence of the disorder on the BKT transition............... 69 3.5 Phase diagram................................. 72 3.5.1 Tricritcal point at T = 0........................ 73 3.5.2 Experimental outlook......................... 75 4 Stability of many-body localization in continuum systems 77 4.1 Introduction................................... 77 4.2 The criterion for the localization-delocalization transition......... 79 4.3 The MBDLT criterion revisited: the case of two-dimensional bosons........................... 82 4.4 Results on the stability of the MBL phase.................. 85 5 Concluding remarks 89 A.1 Temperature dependence of the MBLDT................... 91 A.2 MBLDT for the truncated distribution function............... 95 A.3 MBLDT in the thermodynamic limit..................... 97 A.4 Average of small occupation numbers..................... 99 A.5 Fourier transform and correlator....................... 102 R´esum´een fran¸cais 105 Bibliography 111 Introduction Disorder is ubiquitous in Nature. Its presence is often unavoidable in a wide variety of physical systems and, as a consequence, it plays a crucial role in our understanding of the laws of physics. In condensed matter systems especially, disorder is responsible for a rich phenomenology that has far-reaching consequences on the transport properties of a material. Sixty years ago, P. W. Anderson showed that randomness has powerful effects on isolated quantum systems [1]. Transport may be absent, as wavefunctions show an exponential decay in real space such that particle diffusion is suppressed. The eigenstates are said to be \localized" in real space, an effect that is particularly dramatic in low dimensionality. A new wave of interest to this problem was inspired by the observation of Anderson localization (AL) in dilute quasi-one-dimensional clouds of cold bosonic atoms with a negligible