THE NONEQUILIBRIUM DYNAMICS OF QUANTUM INTEGRABLE MODELS BY DEEPAK IYER A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Physics and Astronomy Written under the direction of Professor Natan Andrei and approved by New Brunswick, New Jersey October, 2013 ABSTRACT OF THE DISSERTATION The nonequilibrium dynamics of quantum integrable models by Deepak Iyer Dissertation Director: Professor Natan Andrei The primary goals of this dissertaion are to describe and elucidate a new formalism to study the out of equilibrium dynamics of integrable models, and to apply this formal- ism to some specific problems. In particular, I describe the “quench dynamics” of a gas of interacting bosons in one dimension, and provide details of some preliminary work on the isotropic Heisenberg chain. The formalism, based on earlier work by Yudson, provides a way of getting around some of the difficulties involved in calculating the time evolution of arbitrary initial states evolved with integrable Hamiltonians. In addition to this, I aim to discuss some of the general aspects of quench dynam- ics in quantum systems. The description of the nonequilibrium dynamics of a given system depends significantly on initial states, and the time scales at which the system is probed compared with the various inherent time scales in the system. I present the experimental context and motivation for these studies, and survey the existing tech- niques and efforts at understanding the relaxation of systems that are far from their equilibrium states or ground states. ii Preface Parts of this thesis, in particular a large part of Chapter 5 have been published in Refs. [71] and [72]. The main content of Chapters 5 and 6 is work done in collaboration with my advisor, Natan Andrei. iii Acknowledgements The last several years of my life in graduate school have been influenced in ways be- yond my own comprehension, and possibly knowledge, by the large number of won- derful people I have had the opportunity to meet and get to know. It would therefore be unfair for me to acknowledge only a subset of these people. That said, it is obvi- ously not possible to produce such a list, even in principle, though I will try to be as complete as possible. A profound influence on my interests, my thought process, and my graduate ex- perience has been due to my mentor and advisor, Natan Andrei. Natan has been a most patient, considerate, and thoughtful advisor, and my success here derives in large parts from his support and confidence in me. Natan is also an excellent teacher and I have learned much about physics and science under his guidance. The multi- tudes of discussions we’ve had on topics ranging from high energy physics to politics to food made my graduate experience colorful. While Natan has been my most constant point of contact in the department, I have benefited a lot from the other faculty here. My brief and local swim in the vast seas of material science with David Vanderbilt and Karin Rabe opened my eyes to several interesting and exciting problems. The down and dirty first principles approach that I worked with gave me a new way of looking at physics, and broadened my mind. Discussions with Emil Yuzbashyan greatly benefited my approach in my own work, and provided vital insights. My couple of year collaboration with Matthew Foster has also been very fruitful in teaching me about the exciting field of disorder, and giving me the excuse to go back to relativistic quantum field theories. Matt has also been an advisor of sorts in being very patient with my slow pace of work, and filling many holes in my education. I also thank my dissertation committee for the feedback during iv committee meetings, and hard questions that made me think. To all of these wonderful people, I am very grateful. Outside of my research life here at Rutgers, I am lucky to have had a compas- sionate, helpful, and truly wonderful set of friends who I’ve lived with, cooked with, and complained to. I can say for certain that many phases during my graduate career would have been nearly impossible to deal with was it not for their constant sup- port, and cheerful distractions. A large part of this support came from Darakhshan Mir, who has probably been witness to the worst of these phases, and gave me much needed clarity of thought at these times. I am also grateful to Aatish Bhatia for his per- petual excitement about science, to Anindya Roy for arguments that twisted logic to new levels, to Chuck-Hou Yee for excellent discussions about physics and some great cooking, to Kshitij Wagh for being a buddy throughout and exposing me to some great music, to Vijay Ravikumar for the artsy side of things, and being a most comforting presence, to Sushmita Venugopalan for permitting me to ramble on about beautiful mathematics that I did not understand. I am also thankful to John Barton, Patrick O’Malley, Roberto Sepulveda, Rebecca Flint, Vandana Bajaj, Purba Rudra, for their fantastic and ever interesting company. I am also indebted to the white water kayaking community that showed me the joy of navigating tricky rapids and have taken me to some of the most scenic places in the Northeast. Dan, Ken, Jack, Ellen, Steve, Jessica, thanks! Thanks to Shannon Chase and the University Choir for a delightful two semesters of song that punctuated my research life. This would be woefully incomplete without a big thanks to my parents and my sister for providing me with a ton of opportunities, support, and trusting my decisions in life. And another final thanks to everyone. It has been a most exhilarating ride! v Dedication To my parents, for all the opportunities. vi Table of Contents Abstract .......................................... ii Preface ........................................... iii Acknowledgements ................................... iv Dedication ......................................... vi List of Figures ....................................... xi 1. Introduction ...................................... 1 1.1. “A Quantum Newton’s Cradle” ......................... 1 1.2. Quantumnonequilibriumphysics . 4 1.2.1. Whatisaquench? ........................... 4 1.2.2. Anoteon“nonequilibrium” . 6 1.2.3. Integrablemodels............................ 6 1.3. Anoverviewofnonequilibriumphenomena . 7 1.4. Outlineofthethesis.............................. 9 2. Quantum nonequilibrium dynamics ....................... 11 2.1. Experiments to probe out of equilibrium phenomena . ...... 11 2.1.1. Experimentswithcoldatoms . 11 2.1.2. Opticallattices ............................. 13 2.1.3. Feshbachresonances . 15 2.1.4. Recentdevelopments. 17 2.2. Theoreticalmethodsandtools. 19 2.2.1. Models.................................. 19 2.2.2. Fieldtheoreticmethods . 21 vii 2.2.3. Numericalmethods .......................... 22 2.2.4. Exactmethods.............................. 24 2.3. Dynamics of 1d isolatedmany-bodysystems . 25 2.4. Establishedresults .............................. 30 2.4.1. Eigenstatethermalization . 30 2.4.2. GeneralizedGibbsEnsemble . 31 2.5. Openquestions ................................. 33 2.5.1. Roleofinitialstates. 33 2.5.2. Roleofintegrabilityanditsbreaking. 34 3. The Bethe Ansatz ................................... 35 3.1. Integrablemodels................................ 36 3.1.1. Conservationlaws ........................... 37 3.2. AnoverviewoftheBetheAnsatz . 38 3.3. Examples..................................... 42 3.3.1. ResonantLevelModel . 42 3.3.2. Lieb-Linigermodel. 46 3.4. BetheAnsatzandquenches . 49 4. The Yudson representation ............................. 52 4.1. Acompletebasisininfinitevolume. 52 4.2. TheYudsonintegralrepresentation . 54 4.3. TheResonantLevelmodel-anexample . 56 4.3.1. YudsonrepresentationfortheRLM . 57 4.3.2. Timeevolutionofthedotoccupation . 58 4.3.3. Noteonfinitesystems . 64 4.4. Applicationtootherintegrablemodels. ..... 65 4.4.1. Andersonimpuritymodel. 66 5. Quench dynamics of the Lieb-Liniger model ................... 68 viii 5.1. TheLieb-Linigermodel ............................ 68 5.2. Yudsonrepresentation . 69 5.2.1. Repulsivecase.............................. 70 5.2.2. Attractivecase ............................. 71 5.3. Twoparticledynamics . 73 5.4. Multiparticledynamicsatlongtimes . ..... 79 5.4.1. Repulsiveinteractions-Asymptotics . 80 5.4.2. Attractiveinteractions . 88 5.4.3. Starting with a condensate - attractive and repulsive interactions 90 5.5. Conclusions and the dynamic RG hypothesis . 93 6. Quench dynamics of the Heisenberg chain .................... 97 6.1. XX model .................................... 97 6.1.1. The Yudson representation in the natural parametrization .... 98 6.1.2. The λ parametrization ......................... 99 6.1.3. Longtimeasymptotics. 101 6.2. XXX model ...................................101 6.2.1. Eigenstates................................102 6.2.2. Yudsonrepresentation . 105 6.3. Quench from a domain wall - XX model...................109 6.3.1. Spin current in the XX model - directly obtaining the steady state result...................................111 6.4. Quench from a domain wall - XXX model..................112 6.4.1. Conservationofthenormofthewavefunction. 113 6.4.2. Spin current in the XXX model....................117 7. Directions for future work .............................120 7.1. Extensionstoothermodelsandsituations . 120 7.2. Finitevolume ..................................121 7.3. Thermodynamic