University of cambridge Spectral and dynamical properties of disordered and noisy quantum spin models Daniel Alexander Rowlands Corpus Christi College This dissertation is submitted for the degree of Doctor of Philosophy Cavendish Laboratory August 2018 1 Spectral and dynamical properties of disordered and noisy quantum spin models by Daniel Alexander Rowlands Abstract This thesis, divided into two parts, is concerned with the analysis of spectral and dynamical characteristics of certain quantum spin systems in the presence of either I) quenched disorder, or II) dynamical noise. In the first part, the quantum random energy model (QREM), a mean-field spin glass model with a many-body localisation transition, is studied. In Chapter 2, we attempt a diagrammatic perturbative analysis of the QREM from the ergodic side, proceeding by analogy to the single- particle theory of weak localisation. Whilst we are able to describe diffusion, the analogy breaks down and a description of the onset of localisation in terms of quantum corrections quickly becomes intractable. Some progress is possible by deriving a quantum kinetic equation, namely the relaxation of the one-spin reduced density matrix is determined, but this affords little insight and extension to two-spin quantities is difficult. We change our approach in Chapter 3, studying instead a stroboscopic version of the model using the formalism of quantum graphs. Here, an analytic evaluation of the form factor in the diagonal approximation is possible, which we find to be consistent with the universal random matrix theory (RMT) result in the ergodic regime. In Chapter 4, we replace the QREM’s transverse field with a random kinetic term and present a diagrammatic calculation of the average density of states, exact in the large-N limit, and interpret the result in terms of the addition of freely independent random variables. In the second part, we turn our attention to noisy quantum spins. Chapter 5 is concerned with noninteracting spins coupled to a common stochastic field; correlations arising from the common noise relax only due to the spins’ differing precession frequencies. Our key result is a mapping of the equation of motion of n-spin correlators onto the (integrable) non-Hermitian Richardson-Gaudin model, enabling exact calculation of the relaxation rate of correlations. The second problem, addressed in Chapter 6, is that of the dynamics of operator moments in a noisy Heisenberg model; qualitatively different behaviour is found depending on whether or not the noise conserves a component of spin. In the case of nonconserving noise, we report that the evolution of the second moment maps onto the Fredrickson-Andersen model – a kinetically constrained model originally introduced to describe the glass transition. This facilitates a rigorous study of operator spreading in a continuous-time model, providing a complementary viewpoint to recent investigations of random unitary circuits. 3 Preface The first chapter of this dissertation provides an introduction to the main concepts and methods we shall encounter in later chapters, whilst concluding remarks are contained in the final chapter. The intermediate chapters contain original material that is either published or in preparation for submission to peer review as follows: Chapters 2 & 4: Daniel A. Rowlands and Austen Lamacraft. Diagrammatic approaches to quantum random energy models. Chapter 3: Stroboscopic quantum random energy model. Whilst in preparation, the same result for the form factor appeared in (a preprint of) Ref. [1]. Chapter 5: Daniel A. Rowlands and Austen Lamacraft. Noisy spins and the Richardson-Gaudin model, Physical Review Letters 120 (9), 090401. Chapter 6: Daniel A. Rowlands and Austen Lamacraft. Noisy coupled qubits: operator spreading and the Fredrickson-Andersen model. To appear in Physical Review B. This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the Preface and specified in the text. It is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. I further state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. This dissertation does not exceed 60,000 words, including summary/abstract, tables, foot- notes and appendices, but excluding table of contents, photographs, diagrams, figure captions, list of figures/diagrams, list of abbreviations/acronyms, bibliography and acknowledgements. Daniel Alexander Rowlands On the 6th of August, the Feast of the Transfiguration of the Lord, 2018 5 Acknowledgements In the first place I express sincere gratitude to my supervisor Austen Lamacraft for his outstanding guidance over the last three years that has made this thesis possible; his remarkable insight and ingenuity have been an inspiration. The research contained here has also been facilitated by the financial support of the Engineering and Physical Sciences Research Council and the accommodation provided by Corpus Christi College. The TCM group has provided an intellectually stimulating research environment. I thank in particular Gunnar Möller, Antonio García-García, David Khmelnitskii, and Michal Kwasigroch for useful discussions. I would also like to extend thanks to my fellow PhD students who could always be relied upon for engaging and diverse conversation, especially: Victor Jouf- frey, Corentin Morice, Sam Smith, Lars Schonenberg, Bruno Loureiro, Tom Whitehead, Beñat Mencia, Philipp Verpoort, and Bart Andrews. Discussions with members of groups outside Cambridge have also been much appreciated: Sarang Gopalakrishnan, Alex Altland, Marko Žnidarič, and Roland Speicher come to mind in particular. My thanks extends also to my examiners, John Chalker and Andreas Nunnnenkamp, for their careful reading of my work and perceptive comments. In addition to research, it has been a pleasure to supervise numerous undergraduates – I am grateful to the students for challenging me with good questions. Lastly, I thank all friends and family who have been of invaluable help through their willing- ness to listen, encourage, and advise; in particular, I am deeply thankful to my parents and sister for their constant support and love over the course of my life, and their particular generosity and patience during these years of doctoral study. 7 Contents Preface 3 Acknowledgements 5 1 Introduction 11 1.1 Disorder and localisation . 12 1.1.1 Many-body localisation: a brief history . 13 1.1.2 Weak localisation: semiclassical picture . 15 1.1.3 Weak localisation: diagrammatics . 17 Disorder-averaged Green’s function . 17 Born approximation . 18 Drude conductivity . 20 Diffusion . 21 Weak-localisation correction . 23 1.2 Random matrix theory . 25 1.3 Quantum graphs . 28 1.4 Operator spreading . 30 I Disordered spins 33 2 Quantum random energy model: a diagrammatic approach 35 2.1 Model . 35 2.1.1 Hamiltonian and Hilbert space . 35 2.1.2 Formal structure of the perturbation theory . 37 2.2 Disorder-averaged perturbation theory . 38 2.2.1 Model of disorder . 38 2.2.2 Born approximation . 39 2.2.3 Self-consistent Born approximation . 42 2.2.4 Diffusion ladder . 43 2.2.5 Classical diffusion on the hypercube . 45 2.3 Transverse spin-spin correlations . 48 2.3.1 Drude peak . 49 2.3.2 Schwinger time . 49 8 2.3.3 Cooperon ladder in the diffusive regime . 53 2.4 Dynamics of the average density matrix . 54 2.4.1 Derivation of the quantum master equation . 55 2.4.2 Spectrum . 57 2.4.3 Permutation symmetry and dimensional reduction . 57 2.4.4 Reduced density matrix . 60 2.5 Conclusions . 61 3 Stroboscopic quantum random energy model as a quantum graph 63 3.1 Introduction . 63 3.2 Spectral statistics . 64 3.2.1 Numerics . 64 3.3 Spectral form factor in the diagonal approximation . 66 3.4 Conclusions . 71 4 Quantum random energy model with a random kinetic term 73 4.1 Model . 73 4.2 Spectral statistics . 74 4.3 Perturbation theory . 76 4.3.1 Noncrossing approximation . 76 4.4 Density of states . 78 4.5 Binary disorder . 78 4.6 Free probability . 80 4.6.1 Free independence . 80 4.6.2 Connection to random matrix theory . 82 4.7 Single-particle analogue . 84 4.7.1 Density of states . 84 4.8 Conclusions . 85 II Noisy spins 87 5 Noisy spins and the Richardson–Gaudin model 89 5.1 Background and model . 89 5.2 Density matrix and correlation functions . 91 5.3 Mapping to the Richardson–Gaudin model . 92 5.4 Equivalence to stochastic evolution . 93 5.5 Exact solution . 95 5.6 Conclusions and outlook . 98 Contents 9 6 Noisy coupled qubits: operator spreading and the Fredrickson–Andersen model 99 6.1 Motivation . 100 6.2 Models and mapping . 101 6.2.1 Models . 101 6.2.2 Model C at strong noise: symmetric exclusion process for ........ 104 O 6.2.3 Model NC at strong noise: Fredrickson–Andersen model for .... 105 O ⊗ O 6.3 Phenomenology of fronts . 107 6.3.1 Fronts in the FA model . 107 6.3.2 Fronts in two dimensions . 107 6.4 Details of the numerical simulation . 109 6.5 Applied Phenomenology . 110 6.5.1 Out-of-time-order correlation functions . 110 6.5.2 Purity decay . 111 6.6 Conclusions . 113 7 Concluding remarks 115 N A Harmonic analysis on Z2 119 B Hamming scheme combinatorics 121 C Two-point correlators in the random kinetic quantum random energy model 123 D Effective Liouvillians for second-moment dynamics 129 D.1 Second-moment dynamics of Model C in the strong-noise limit .