Universidad Consejo Superior Aut´onoma de Madrid de Investigaciones Cient´ıficas Facultad de Ciencias Instituto de F´ısica Te´orica Departamento de F´ısica Te´orica IFT–UAM/CSIC Applications of Holography to Strongly Coupled Hydrodynamics Irene Rodr´ıguez Amado, Madrid, Mayo 2010. Universidad Consejo Superior Aut´onoma de Madrid de Investigaciones Cient´ıficas Facultad de Ciencias Instituto de F´ısica Te´orica Departamento de F´ısica Te´orica IFT–UAM/CSIC Applications of Holography to Strongly Coupled Hydrodynamics Memoria de Tesis Doctoral realizada por D˜na. Irene Rodr´ıguez Amado, presentada ante el Departamento de F´ısica Te´orica de la Universidad Aut´onoma de Madrid para la obtenci´on del T´ıtulo de Doctor en Ciencias. Tesis Doctoral dirigida por Dr. D. Karl Landsteiner, Cient´ıfico Titular del Consejo Superior de Investigaciones Cient´ıficas. Madrid, Mayo 2010. Contents Preface v I Background: Strongly Coupled Systems and Holography 1 1 Strongly coupled systems 3 1.1 Quark-GluonPlasma............................... 4 QCDphasediagram ............................... 4 RHICandthesQGP............................... 8 1.2 Quantumphasetransitions . 10 Quantumcriticality................................ 11 Superfluids and Superconductors. ........................ 14 1.3 LinearresponseandHydrodynamics . 15 Hydrodynamics. ................................. 16 2 The AdS/CFT correspondence 21 2.1 TheConjecture.................................. 22 LargeNandstrings ............................... 22 Openstringsvs.closedstrings . 24 Thecanonicalexample.............................. 25 Thedictionary .................................. 29 2.2 FiniteTemperature................................ 31 Addingablackhole ............................... 31 Confinement-Deconfinement . ........................ 33 2.3 Finite chemical potential . 34 Addingaglobalcurrent ............................. 35 Spontaneoussymmetrybreaking . 36 ii Contents 3 Thermal correlators and quasinormal modes 39 3.1 The Lorentzian propagators . ........................ 39 Retarded two-point function prescription . 41 Schwinger-Keldish propagators . ........................ 43 3.2 Quasinormalmodes ............................... 45 3.3 Stability issues . ............................... 48 II Aplications of the AdS/CFT correspondence 53 4 AdS black holes as reflecting cavities 55 4.1 AdSwithamirror ................................ 56 4.2 The eikonal approximation in AdS black holes . 58 Matching of eikonal solutions and bouncing rays . 61 4.3 Black holes as reflecting cavities . 63 4.4 Conclusions .................................... 65 5 Beyond hydrodynamics in the sQGP 69 5.1 Hydrodynamic scales and linear response theory . 71 5.2 Includinghigherthermalresonances . 73 5.3 R-chargecurrentcorrelators . 75 5.4 Energy-momentum tensor correlators . 80 Residuesandhigherthermalresonances . 84 5.5 Conclusions .................................... 87 6 Holographic superconductors 89 6.1 TheModel .................................... 92 6.2 Quasinormal Frequencies in the Unbroken Phase . 95 GreenFunctions ................................. 96 QuasinormalModesfromDeterminats . 97 HydrodynamicandhigherQNMs . .100 Contents iii 6.3 Quasinormal Frequencies in the Broken Phase . 102 Application of the determinant method . 103 HydrodynamicandGoldstonemodes . .105 Higherquasinormalmodes. .109 6.4 Conclusions ....................................111 Summary and Outlook 113 Appendix 115 A.1 Computingquasinormalmodesandresidues . 115 Energy-momentumtensor . .115 Globalcurrent ..................................116 A.2 Zerosofhydrodynamicresidues . .117 A.3 Frontvelocity...................................119 III Secciones en castellano 121 Introducci´on 123 Conclusiones 127 Agradecimientos 129 Bibliography 131 Preface This thesis is about the application of the AdS/CFT correspondence to the dynamics of strongly coupled gauge theories, paying special attention to the hydrodynamic behavior of these systems. In particular, we study the response to small external perturbations of a strongly coupled quark-gluon plasma (sQGP) using holographic techniques, with an special interest in the exploration of the validity of the hydrodynamic approximation in such a system. We also study the hydrodynamic regime of high Tc superconductors within the frame of the gauge/gravity duality. The AdS/CFT correspondence establishes an equivalence between a quantum field theory and string theory in a curved background with the peculiarity of being a strong- weak coupling duality. When the gauge coupling is taken to be large, the field theory is in a non-perturbative regime, whereas the string theory can be approximated by its classical low energy limit, supergravity. It provides us with a theoretical tool to describe the dynamics of strongly interacting gauge theories. Of particular interest is the strongly coupled quark-gluon plasma (sQGP) discovered in the Relativistic Heavy Ion Collider (RHIC) at Brookhaven. The sQGP behaves as a nearly perfect fluid and can be described in terms of hydrodynamics in the low energy long-wavelength limit. The RHIC accelerator creates very dense and hot matter, mostly made of quarks and gluons, by colliding heavy nuclei. The created fireball rapidly thermalizes, expands and cools down coming back to the hadronic gas phase, but in the meanwhile, when it is a quark-gluon plasma, it reproduces the conditions of the primordial plasma in the first microseconds after the 36 Big Bang. At the end of the Grand Unification era (10− seconds) the universe was 11 filled with a soup of free particles that was expanding and cooling down. Around 10− seconds after the explosion, the electroweak force decoupled from the strong force and particle interactions were energetic enough to create particles such as the Z and the W bosons. The Early Universe was then filled by the quark-gluon plasma, still cooling and expanding, until it reached a temperature of about 200 MeV that is 2 1012 K, when the × confinement/deconfinement phase transition took place. The QGP hadronized forming protons and neutrons. This happened a couple of microseconds after the Big Bang. Few minutes after that, the temperature was low enough to allow the binding of hadrons, thus the formation of light nuclei started. The state of the Early Universe just before the quark/hadron phase transition is roughly reproduced at RHIC, and soon it will also be tested at the ALICE experiment in the LHC at CERN. These experiments give valuable information about QCD in the deconfined phase and the physics of the primordial plasma. vi Preface Having a theoretical understanding of them is a hard, challenging and interesting task, and it seems that holography can help us in this venture. Recently, the AdS/CFT correspondence has emerged as a very useful tool in the frame of condensed matter physics as well, since there is a wide number of strongly correlated condensed matter systems that cannot be treated using the conventional paradigms, like for instance quantum critical systems. There are many strongly coupled materials that can be engineered and studied in laboratories that are challenging for condensed matter theory and for which seems possible that the gauge/gravity duality can be helpful to gain some insight into them. The high Tc superfluids and superconductors are expected to belong to this family of condensed matter theories to which holographic techniques might be applied. On the other hand, this large number of condensed matter systems also provides us with an enormous and rich variety of effective Lagrangians. Therefore it might as well be that using experimental techniques a material with a known gravity dual can be engineered, leading to experimental AdS/CFT and allowing for a better understanding of quantum gravity through atomic physics, thus reversing the usual direction of the correspondence. Consequently, the study of AdS/CM correspondence appears as a very exciting and rich topic. In this thesis we will apply the correspondence to model simplified condensed matter systems and see what kind of information we can get from it. This thesis is organized in two parts. The first part is devoted to present the back- ground ingredients and tools that will be used later on, namely, the properties of strongly coupled systems and the holographic techniques: Chapter 1: We present the main features and issues of strongly coupled systems, centered on the sQGP phase of QCD, exposing some of the most relevant results at RHIC experiment that point to the strong coupling behavior of the formed plasma, and on the phenomena of quantum criticality, specially for the case of high temperature supercon- ductivity and superfluidity. We also present some basics on linear response theory and hydrodynamics, the main tools to study perturbations in those setups. Chapter 2: This chapter is devoted to explain the basics of the AdS/CFT corre- spondence, avoiding as far as possible to enter in technical details that will not be useful in what follows. We center on those aspects of the correspondence and its extensions more relevant for applications to the systems already mentioned: how to add temperature and how to add finite chemical potentials to the gauge theory. Chapter 3: The power of the gauge/gravity duality relies on the ability to perform real-time computations. In this chapter we present the main computational tool for the Preface vii rest of the thesis: a prescription to holographically compute thermal correlators. We also present some results of the correspondence concerning