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Gauge/Gravity Duality applied to Condensed Matter Systems Martin Matthias Ammon M¨unchen 2010 Gauge/Gravity Duality applied to Condensed Matter Systems Martin Matthias Ammon Dissertation an der Fakult¨at f¨ur Physik der Ludwig–Maximilians–Universit¨at M¨unchen vorgelegt von Martin Matthias Ammon aus Roth, Mittelfranken. M¨unchen, den 31. Mai 2010 This thesis is based on the author’s work partly published in [1–9] conducted from November 2007 until May 2010 at the Max-Planck-Institut f¨ur Physik (Werner- Heisenberg-Institut), M¨unchen under supervision of Priv.-Doz. Dr. Johanna Karen Erdmenger. Erstgutachter: Priv.-Doz. Dr. Johanna Karen Erdmenger Zweitgutachter: Prof. Dr. Dieter L¨ust Tag der m¨undlichen Pr¨ufung: 07. Juli 2010 Zusammenfassung In der hier vorliegenden Arbeit werden mit Hilfe der AdS/CFT-Korrespondenz Ph¨anomene stark gekoppelter quantenkritischer Systeme untersucht, wie sie beispielsweise in Syste- men der kondensierten Materie auftreten. Die AdS/CFT-Korrespondenz ist dabei eine aus der Stringtheorie hervorgegangene Dualit¨at zwischen Eich- und Gravitationstheorien, die Rechnungen in starker Kopplung auf der Eichtheorieseite in perturbative Rechnun- gen auf der Gravitationsseite ubersetzt.¨ Dabei setzt die ursprungliche,¨ von Maldacena im Jahr 1997 entwickelte Vermutung vierdimensionale, = 4 supersymmetrische Yang- N Mills-Theorie und Typ IIB-Supergravitation im funfdimensionalen¨ Anti-de Sitter-Raum in Beziehung. Diese Vermutung kann auf verschiedene Weise verallgemeinert werden. So k¨onnen auf der Eichtheorieseite Zust¨ande mit endlicher Temperatur und Dichte betrachtet werden oder Freiheitsgrade hinzugefugt¨ werden, die sich in der fundamentalen Darstellung der Eichgruppe transformieren, die sogenannten Flavor-Freiheitsgrade Diese Deformationen der Korrespondenz werden in der hier vorliegenden Arbeit verwen- det, um stark gekoppelte Systeme in der N¨ahe von quantenkritischen Punkten besser zu verstehen. Wir approximieren hierbei die Feldtheorie am quantenkritischen Punkt durch = 4 supersymmetrische Yang-Mills-Theorie. Die Ladungstr¨ager des Systems werden N durch supersymmetrische Flavor-Felder eingefuhrt.¨ Dieses Modell ist der Ausgangspunkt von vielen Uberlegungen¨ und Untersuchungen, die im Rahmen der Dissertation durch- gefuhrt¨ wurden. Beispielsweise wird in der hier vorliegenden Arbeit im Fall von zwei Flavor Feldern, die die gleiche Masse haben, ein chemisches Potential fur¨ den Isospin betrachtet und das Phasen- diagramm untersucht. Das Isospin-chemische Potential bricht hierbei den nicht-Abelschen Anteil der Flavor-Symmetrie, SU(2), auf U(1). Wird ein kritischer Wert des Isospin- chemischen Potentials uberstiegen,¨ so zeigt unsere Rechnung, dass das stark-gekoppelte System gegenuber¨ Fluktuationen instabil wird. Es bildet sich ein neuer thermodynamisch favorisierter Zustand aus. Dieser Zustand bricht die restliche U(1) Flavorsymmetrie spon- tan und kann daher als eine Supraflussigkeit¨ angesehen werden. Ist nun die U(1) geeicht, erh¨alt man auf diese Weise einen Supraleiter. Die Wechselstrom-Leitf¨ahigkeit geht in der supraleitenden Phase fur¨ kleine Frequenzen gegen Null. Die Gleichstrom-Leitf¨ahigkeit je- doch ist unendlich, wie das auch von einem Supraleiter erwartet wird. Da der Supraleiter stark gekoppelt ist, kann dieser nicht durch die BCS Theorie beschrieben werden. Des Weiteren berechnen wir auch die Fermifl¨ache in der supraleitenden Phase und beobach- ten, dass diese auf vier ausgezeichnete Punkte kollabiert. Außerdem wird eine holographische Methode zur Berechnung der Leitf¨ahigkeit des Gleich- stroms auf beliebige konstante elektrische und magnetische Felder weiterentwickelt. Hierbei wird die Leitf¨ahigkeit nicht mit Hilfe von Linearer-Antwort-Theorie beschrieben, sondern ist selbst vom elektrischen Feld abh¨angig. Schließlich wird auch ein zweites Modell fur¨ die Feldtheorie am quantenkritischen Punkt, eine Chern-Simons-Materie Theorie in (2+1)-Dimensionen, genauer untersucht. Die Chern- Simons-Materie Theorie hat dynamische Freiheitsgrade und kann als Niederenergie-Limes vi Zusammenfassung von (2+1)-dimensionalen Membranen, sogenannter M2-Branen, aufgefasst werden. Auf verschiedene Weisen werden in der vorliegenden Arbeit auf der Eichtheorieseite Ladungs- tr¨ager in Form von fundamentalen supersymmetrischen Feldern hinzugefugt¨ und die effek- tive Kopplung an die Felder der Chern-Simons-Materie bestimmt. Auf der Gravitations- seite werden dabei h¨oherdimensionale Membranen und andere nicht-perturbative Objekte, sogenannte KK-Monopole, in M-Theorie bzw. dessen Typ IIA Limes eingebettet. Insgesamt erhoffen wir uns von diesen Untersuchungen an Modelltheorien, die sowohl Berechnungen bei starker als auch bei schwacher Kopplung zulassen, neue Impulse fur¨ die Theorie der kondensierten Materie. Dabei haben wir insbesondere quantenkritische Systeme, wie beispielsweise Hochtemperatur-Supraleiter, im Blick. Contents Zusammenfassung v 1 Introduction and Overview 1 1.1 String Theory and strongly coupled systems . ....... 3 1.2 Strongly coupled condensed matter systems . ....... 4 1.3 AdS/CFTandcondensedmattersystems . ... 8 1.4 Planofthethesis................................. 13 I Foundations of AdS/CFT 15 2 The Anti-de Sitter/Conformal Field Theory Correspondence 17 2.1 Review: TheoriginofAdS/CFT . 18 2.2 Review: TheAdS/CFTDictionary . 28 2.3 Holographic renormalization for fermions . ........ 30 2.4 Review: Generalization to finite temperature . ........ 41 2.5 Review: Addingfundamentalflavor. 43 2.6 Other examples of gauge/gravity dualities . ........ 58 II Applications to Condensed Matter Systems 61 3 Response to external fields 63 3.1 Review: Linear response & retarded Green’s functions . .......... 63 3.2 Holographic Calculation: Linear response . ........ 65 3.3 Holographic calculation: Beyond linear response . .......... 75 4 Holographic Superconductors 87 4.1 Introductionandsummary . 88 4.2 Fieldtheorydescription . 90 4.3 DualGravitypicture.............................. 91 4.4 StringTheoryPicture ............................. 99 5 Fermions in AdS/CFT 103 5.1 IntroductionandSummary . 103 5.2 Worldvolume Fermions of Dp-branes . 106 5.3 EmergenceoftheFermisurface . 114 6 Non-relativistic systems: AC & DC conductivities 129 6.1 IntroductionandSummary . 129 6.2 Adding Flavor to Schr¨odinger Spacetime . .......132 viii CONTENTS 6.3 DCConductivityofProbeFlavor . 139 6.4 ACConductivityofProbeFlavor . 145 III Flavor in Chern-Simons-Matter Theories 151 7 Adding Flavor to AdS4/CF T3 153 7.1 Introductionandsummary . 154 7.2 AM-theoryPrimer................................156 7.3 ReviewofABJMTheory ............................159 7.4 General Analysis of Probe Flavor . 163 7.5 Codimension Zero =3SupersymmetricFlavor . 169 N 7.6 Codimension One = (0, 6) Supersymmetric Flavor . 173 N 7.7 SU(4)EquivalenceofProbeFlavor . .176 7.8 Other Flavor branes in AdS4/CF T3 ......................179 8 Conclusions and Outlook 181 A Glossary 189 A.1 BosonicStringTheory . .. .. .. .. .. .. .. .. 189 A.2 SuperstringTheory............................... 197 A.3 D-branes......................................205 B Appendix to Chapter 7 211 B.1 SupersymmetryofTypeIIBProbes . 211 B.2 TypeIIBtoM-theory ..............................214 B.3 SupersymmetryofM-theoryObjects . 219 B.4 = (0, 6)SupersymmetryTransformations . 223 N Acknowledgements 240 1 Introduction and Overview In the twentieth century, physicists made much progress in the theoretical understanding of many phenomena in nature, related to quantum theory and gravity. The basic build- ing blocks of the physical world are particles and their interactions. Today, four basic interactions are known: the electromagnetic force, gravity, the weak and strong nuclear forces. Except gravity, the remaining three interactions, as well as the matter fields, can be formulated in terms of a quantum field theory, known as the standard model of particle physics. Up to now, the standard model of particle physics has been very well tested at least up to energies of 100 GeV. Every high energy physics experiment has yielded results consistent with the Standard Model. All quantized forces, which are incorporated in the standard model, can be described in terms of exchange particles. For example, the electromagnetic force is described by the exchange of massless spin-1 particles, called photons, which we denote as γ. The quan- tized theory of electrodynamics, Quantum Electrodynamics (QED), is very precise. The theoretical prediction for the anomalous magnetic moment of the electron, for example, agrees with the measured value up to one part in 1010. 0 The weak nuclear force is described by three exchange particles, W ± and Z . In contrast to the photon, the W ± and Z are massive spin-1 gauge bosons, and therefore the force is short-ranged. The weak force is responsible for nuclear beta decay, for example. Although the electromagnetic and weak forces seem to be very different, they can be formulated in a unified theory, called the electro-weak theory, in terms of a SU(2) U(1) gauge × Y theory. The electro-weak symmetry breaking mechanism breaks the SU(2) U(1) gauge × Y symmetry to the electromagnetic U(1)em group, leaving only the photon massless and giving mass to the W ± and Z boson. This symmetry breaking mechanism also predicts the existence of a massive scalar particle, called the Higgs boson. At colliders, such as the Large Hadron Collider (LHC) at CERN, experimentalists aim at discovering this Higgs boson. Finally, the strong nuclear force can be described by Quantum Chromodynamics (QCD),
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