Applications of Holographic Duality: Black Hole Metals and Supergravity Superconductors by Oscar Karl Johannes Henriksson B.S., Uppsala University, 2011 M.S., University of Colorado Boulder, 2013 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics 2017 This thesis entitled: Applications of Holographic Duality: Black Hole Metals and Supergravity Superconductors written by Oscar Karl Johannes Henriksson has been approved for the Department of Physics Prof. Oliver DeWolfe Prof. Senarath de Alwis Prof. Thomas DeGrand Prof. Andrew Hamilton Prof. Paul Romatschke Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Henriksson, Oscar Karl Johannes (Ph.D., Physics) Applications of Holographic Duality: Black Hole Metals and Supergravity Superconductors Thesis directed by Prof. Oliver DeWolfe We apply holographic duality to the study of strongly interacting quantum matter. The cor- respondence between the four-dimensional = 8 gauged supergravity and the three-dimensional N superconformal ABJM quantum field theory allows us to study the latter theory by performing computations in the former. Asymptotically anti-de Sitter spacetimes satisfying the classical su- pergravity equations of motion are interpreted as states of strongly interacting ABJM theory. If such a spacetime sources an electric field, the dual state is at non-zero charge density. Interesting observables of such states include spectral functions of fermionic operators | we compute these by solving Dirac equations in a variety of spacetimes. In a family of extremal charged black holes, we find Fermi surface singularities with non-Fermi liquid characteristics. In a special \three-charge" black hole, an interval appears in the spectral functions within which the fermionic excitations are perfectly stable. We then study three different domain wall spacetimes dual to zero-temperature states with a broken U(1) symmetry. In these \holographic superconductors", we find features similar to conventional superconductors such as the development of a gap in the fermionic spectra. Finally, we investigate the question of how bosonic properties, for example susceptibilities, are affected by fermionic properties, such as Fermi surface singularities, in holographic states of matter. We do this by computing the static charge susceptibility in the three-charge black hole state. Our results reveal singularities at complex momenta, with a real part approximately equal to the largest Fermi momentum in the state. Dedication To my parents, Ingvar and Marita Henriksson, with gratitude. v Acknowledgements During my years in Boulder it has been a true pleasure to work with my advisor Oliver DeWolfe. I have benefited from his great knowledge in physics, his clear explanations and his kindness from the first class I took from him and throughout the writing of this dissertation. In similar ways, Christopher Rosen has been a fantastic collaborator and a source of help and motivation for most of the work herein. I have also gained much from fruitful collaborations with Steven Gubser, Paul Romatschke and Chaolun Wu. Various other physicists at CU Boulder, including Shanta de Alwis, Andrea Carosso, Anqi Cheng, Tom DeGrand, Dan Hackett, Anna Hasenfratz, Takaaki Ishii, Will Jay, Ethan Neil and Greg Petropoulos, taught me a great deal about physics, as well as other topics, during lectures, office chats and 10:30 coffee breaks. I was lucky to share the wonderful and strange experience of graduate school with a strangely wonderful group of fellow students. I hope to enjoy conversations and adventures with Andrew and Helen, Jack, Nico and Dani, Paige, Scott, Tyler, and many others, for many years to come. During my time in the U.S., the friendships of (and constant stream of messages from) my friends from the Aland˚ Islands | Christoffer, Fredrik, Jonas, Jonathan and Kim | have made the long distance home feel a lot shorter. My girlfriend Marion Boulet has been a wonderful, beautiful and reliable source of support, laughter and tasty cookies throughout our time together | she truly has a Heart of Gold. Finally, this dissertation is dedicated to my parents for the unrelenting support and encour- agement they have provided throughout my life, making all of this possible. Contents Chapter 1 Introduction and Overview 1 1.1 A Brief Tour of String Theory...............................2 1.2 An Introduction to AdS/CFT...............................5 1.2.1 The Dictionary...................................8 1.3 Applications of AdS/CFT.................................9 1.3.1 Condensed Matter Physics and AdS/CFT.................... 10 1.4 Summary of Dissertation.................................. 13 1.4.1 Top-down Non-Fermi Liquids and a Special Black Hole............ 13 1.4.2 Fermions in Supergravity Superconductors.................... 14 1.4.3 Correlations between Correlators: Charge Oscillations and Fermi Surfaces.. 16 2 The Two Theories 18 2.1 4D = 8 Gauged Supergravity.............................. 18 N 2.1.1 Bosonic Sector................................... 19 2.1.2 Quadratic Fermion Action............................. 23 2.2 The M2-brane ABJM Theory............................... 26 2.2.1 A Simplified Description.............................. 27 3 Fermi Surface Behavior in the ABJM M2-brane Theory 29 3.1 Introduction and Summary................................ 29 vii 3.1.1 Holographic Realizations of Non-Fermi Liquids................. 29 3.1.2 Fermionic Response in the M2-brane Theory.................. 32 3.2 Black Branes and Dirac Equations in Maximal Gauged Supergravity......... 35 3.2.1 Black Brane Solutions............................... 35 3.2.2 Linearized Dirac Equations............................ 38 3.3 Fermionic Green's Functions................................ 41 3.3.1 Solving the Dirac Equation............................ 41 3.3.2 Quantization of Fermi Fields and Green's Functions.............. 43 3.4 Regular Black Holes and Non-Fermi Liquids....................... 48 3.4.1 Regular Black Holes and Non-Fermi Liquids................... 50 3.4.2 The 3+1-Charge Black Hole............................ 53 3.4.3 The 2+2-Charge Black Hole............................ 62 3.5 The Extremal Three-Charge Black Hole and the Gap.................. 69 3.5.1 Near Horizon Analysis of the 3QBH....................... 70 3.5.2 Connection with Extremal (3+1)-Charge Black Holes............. 72 3.5.3 Fermion Fluctuations and Fermi Surfaces.................... 75 3.6 RG Flow Backgrounds: 2QBH and 1QBH........................ 78 3.6.1 The 1-Charge Black Hole............................. 80 3.6.2 The 2-Charge Black Hole............................. 82 4 Fermionic Response in Finite-Density ABJM Theory with Broken Symmetry 86 4.1 Introduction......................................... 86 4.2 The Bosonic Background Geometries........................... 89 4.2.1 The SO(3) SO(3) truncation.......................... 89 × 4.2.2 The Domain Wall Backgrounds.......................... 90 4.2.3 Holographic Interpretation............................ 95 4.2.4 Conductivities................................... 100 viii 4.3 Fermion Response in the Domain Wall Solutions.................... 102 4.3.1 Coupled Dirac equations and holographic operator map............ 102 4.3.2 Solving the Dirac equations and spinor Green's functions........... 106 4.3.3 Fermion Normal Modes.............................. 110 4.3.4 Spectral Functions................................. 113 4.3.5 Modifying Couplings................................ 118 4.4 Discussion.......................................... 122 5 Gapped Fermions in Top-down Holographic Superconductors 126 5.1 Overview.......................................... 126 5.2 The SU(4)− Flow..................................... 129 5.2.1 The SU(4)− Truncation.............................. 129 5.2.2 SU(4)−-invariant Domain Wall Solutions.................... 130 5.2.3 The Fermionic Sector............................... 132 5.2.4 Fermion Response................................. 135 5.2.5 Field Theory Operator Matching......................... 141 5.3 The H = SO(3) SO(3) Flows.............................. 142 × 5.3.1 The Fermionic Sector............................... 143 5.3.2 Fermion Response................................. 147 5.3.3 Field Theory Operator Matching......................... 151 5.4 Lessons for Strongly Coupled Systems.......................... 152 5.4.1 Top-down vs. Bottom-up Fermion Response................... 152 5.4.2 Extremal AdSRN and Effects of Broken Symmetry............... 153 5.4.3 Stability in Supergravity and Zero Temperature Response........... 157 6 \1kF " Singularities and Finite Density ABJM Theory at Strong Coupling 160 6.1 Overview.......................................... 160 6.2 The Three-Charge Black Hole............................... 166 ix 6.2.1 Thermodynamics and Field Theory Interpretation............... 167 6.3 Fluctuations and Linear Response............................ 171 6.3.1 Fermion Spectral Functions............................ 171 6.3.2 The Holographic Static Susceptibility Setup................... 173 6.4 Results............................................ 177 6.5 Discussion.......................................... 180 6.5.1 Singularities and Scaling Exponents....................... 180