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Lifshitz Symmetries and Nonrelativistic Holography

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Lifshitz Symmetries and Nonrelativistic Holography Lifshitz symmetries and nonrelativistic holography About the cover: during the bachelor and master studies of the author of this dissertation, the main postal office of the Netherlands, a magnificent building built in the expressionistic architectural style of the Amsterdamse school and located in Utrecht, was overtaken by time and forced to close down. In its impressive main hall there are statues embodying the different continents, clarified by a wonderful expressionist font. Saddened by the thought that these elements could be hidden from the general public forever, a collective of people digitalized the font and as a result, it is now found on, among other places, the cover of this dissertation. The font was dubbed Neude, after the plaza at which this building is situated. Coinciding with the time of the author obtaining his doctoral title, it became clear that the building will reopen as a library. As such, the building will once again execute an honorable function and be accessible for the general public. ISBN: 978-94-028-0667-0 Lifshitz symmetries and nonrelativistic holography Lifshitz-symmetrie¨en en niet-relativistische holografie (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof. dr. G. J. van der Zwaan, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op woensdag 12 juni 2017 des middags te 2.30 uur door Zhao-He Watse Sybesma geboren op 22 februari 1989 te Leiderdorp Promotor: prof. dr. S. J. G. Vandoren Publication list The content of the following chapters is based on Chapter 5 • Lifshitz quasinormal modes and relaxation from holography Watse Sybesma and Stefan Vandoren Journal of High Energy Physics [1] Chapter 6 • Holographic equilibration of nonrelativistic plasmas Umut G¨ursoy, Aron Jansen, Watse Sybesma, and Stefan Vandoren Physical Review Letters [2] Chapter 7 • Correlation functions in theories with Lifshitz scaling Ville Ker¨anen,Watse Sybesma, Philip Szepietowski, and L´arusThor- lacius Journal of High Energy Physics [3] Chapter 3 • A general theory of fluids with(out) boost symmetries (in preparation) Jan de Boer, Jelle Hartong, Niels Obers, Watse Sybesma, and Stefan Vandoren Work which is not part of this dissertation and popular science work Hyperscaling violating chemistry (in preparation) • Diego Cohen, Juan Pedraza, Watse Sybesma, and Manus Visser Nieuwe vloeistof, nieuwe kansen • Watse Sybesma Nederlands Tijdschrift van de Natuurkunde 83-6 (2017) Femke en het zwarte gat dat haar sok opvrat (in preparation) • Diego Cohen and Watse Sybesma i Contents 1 Introduction and synopsis 1 1.1 Quantum criticality and Lifshitz symmetries . .2 1.2 The holographic duality conjecture . .5 1.3 Lifshitz and finite temperature holography . .8 1.4 Autocorrelation functions and the geodesic approximation . .9 1.5 Good vibrations and dissipations . 11 1.6 The curious case of quasinormal modes versus back-reaction . 13 1.7 Nonrelativistic hydrodynamics . 14 1.8 Outline . 15 2 Nonrelativistic symmetries and field theories 17 2.1 Introduction . 17 2.2 Symmetries and algebras . 18 2.2.1 Translation, rotations, and boosts . 19 2.2.2 Particle number conservation and Lifshitz . 21 2.2.3 Schr¨odinger and the conformal group . 22 2.3 Nonrelativistic field theoretical realizations . 23 2.3.1 Galilean Lagrangian from the nonrelativistic limit . 23 2.3.2 Quantum Lifshitz Lagrangian and its propagators . 26 2.3.3 Galilean propagators . 30 2.4 Energy-momentum tensors and a nod to Newton-Cartan . 31 2.4.1 The scale invariant relativistic scalar example . 31 2.4.2 Improved Galilean energy-momentum tensor . 33 2.4.3 Lifshitz energy-momentum tensor improved using Newton- Cartan . 34 3 Fluids without boosts 37 3.1 Introduction . 37 3.2 Ideal fluids without boost symmetries . 38 3.2.1 Thermodynamics . 38 iii Contents 3.2.2 Ideal fluid energy-momentum tensor . 39 3.2.3 Boost invariant equations of state for mass density ρ . 42 3.2.4 Formula for speed of sound . 44 3.3 Ideal gases of Lifshitz particles . 45 3.3.1 Boltzmann gas . 46 3.3.2 Quantum gases . 51 3.4 Discussion . 58 4 Lifshitz holography 61 4.1 Introduction . 61 4.2 Vacuum Lifshitz spacetime . 62 4.2.1 Anti-de Sitter geometry . 62 4.2.2 Lifshitz geometry . 63 4.2.3 Different matter, same Lifshitz geometry . 64 4.3 Lifshitz black branes and thermal field theories . 65 4.3.1 Lifshitz black branes . 65 4.3.2 holographic conjecture applied . 67 4.4 The geodesic approximation . 70 4.4.1 Geodesic approximation applied . 70 4.4.2 Comparing correlators . 73 5 Lifshitz quasinormal modes and relaxation form holography 75 5.1 Introduction . 75 5.2 Schr¨odingerproblems for Lifshitz geometries . 77 5.2.1 Lifshitz brane . 77 5.2.2 Quasinormal modes of a scalar probe . 78 5.3 Obtaining quasinormal modes . 81 5.3.1 Analytic solutions, z = d ................. 82 5.3.2 Numerical solutions, z = d ................ 84 6 5.3.3 Analysis of numerical output . 86 5.4 Relaxation times . 88 5.5 Discussion . 89 6 Holographic equilibration of nonrelativistic plasmas 93 6.1 Introduction . 93 6.2 Gravitational model . 94 6.3 Numerical methods . 97 iv Contents 6.4 Results . 99 6.5 Discussion . 102 7 Correlation functions in theories with Lifshitz scaling 105 7.1 Introduction . 105 7.2 The generalized quantum Lifshitz model . 107 7.2.1 Equal time correlation functions . 108 7.2.2 Operators inserted at generic points . 112 7.2.3 Vacuum autocorrelators . 114 7.2.4 Autocorrelators in a thermal state . 117 7.3 Holographic models with Lifshitz scaling . 118 7.3.1 Scalar two-point functions in the geodesic approximation120 7.3.2 Two-point function at equal time and the two-point autocorrelator . 122 7.3.3 Three-point vacuum autocorrelators . 123 7.4 Holographic thermal autocorrelators . 125 7.4.1 Holographic thermal two-point autocorrelators . 126 7.4.2 Holographic thermal three-point autocorrelators . 128 7.5 Excursions outside the time domain . 129 7.5.1 Correlation functions at ~q =0.............. 130 7.5.2 Quasinormal mode spectrum . 132 7.6 Non-equilibrium states . 136 7.6.1 States with translational and rotational symmetry . 136 7.6.2 A mass quench in the generalized quantum Lifshitz model . 138 7.6.3 Vaidya collapse spacetime in the holographic model . 139 7.7 Discussion . 140 Appendices 143 A Appendix to Chapter 5 . 143 B Appendix to Chapter 7 . 145 Outlook 159 Samenvatting 161 Acknowledgements 167 v Contents About the author 169 Bibliography 171 vi Preface We choose to merge the introduction with a synopsis in Chapter 1. In this way Chapter 1 is tailored towards giving a nontechnical overview of the topics of this dissertation, while also giving away parts of the conclusions and results of the research presented in this dissertation. Chapters 2 and 4 present technical background support and details. Ob- tained results, based on papers, are included in Chapter 3 and Chapters 5 through 7. Finally, apart from an appendix, there is an outlook, a summary in Dutch, and an acknowledgement at the end. We use ~ = c = GN = kB = ` = 1 (which stand for the reduced Planck constant, speed of light, Newton's constant, Boltzmann constant, Anti-de Sitter or Lifshitz radius, respectively), unless emphasize is needed. We re- serve d for the amount of spatial dimensions in a field theory and use the mostly plus metric sign convention ( + + ::: +). Finally, by z we denote − the dynamical critical exponent. I wish the reader an educational, but above all, joyful read, Watse Sybesma May 16, 2017, Utrecht vii 1 Introduction and synopsis \Ever since the dawn of civilization, people have not been content to see events as unconnected and inexplicable. They have craved an understanding of the underlying order in the world. Today we still yearn to know why we are here and where we came from. Hu- manity's deepest desire for knowledge is justification enough for our continuing quest. And our goal is nothing less than a complete de- scription of the universe we live in." { Steven Hawking, A brief history of time [4] About twenty years ago, a striking result from string theory [5] initiated the idea that seemingly unrelated fields of physics were, in fact, related. This result from string theory is called holography and, once extrapolated [6, 7], it has the potential to relate quantities in general relativity to quantities in, for instance, condensed matter, hydrodynamics, and collider physics. A remarkable fact is that, using holography, it can be possible to access the strongly coupled regime, something which is in general not possible using more conventional techniques. However, outside of this direct scope, holog- raphy has functioned as a catalyst for new developments in nonrelativistic theories [8], notions about the emergence of gravity [9], and hydrodynamics [10, 11], to name a few examples. The main topic of this dissertation revolves around a certain type of non- relativistic symmetry and its manifestation in holography and hydrodynam- ics. Hydrodynamics describes fluids, in the analogous way that the laws of Newton describe classical mechanics [12]. Hydrodynamics circumvents the particle picture of a fluid, which is convenient when the particle pic- ture breaks down, e.g. due to strong coupling. Examples of systems where the usual particle picture breaks down are quark-gluon plasma and certain quantum critical systems. Some of these systems have the characteristics of a Lifshitz fluid. Here, Lifshitz refers to a system with at least an anisotropic scale invariance between space and time and the absence of boost symme- 1 1 Introduction and synopsis tries. When hydrodynamics is applied to such Lifshitz systems, one can for example derive expressions for the speed of sound, transport coefficients, and study the relaxation behavior of pressure anisotropies.
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