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Holographic Condensed Matter Theories and Gravitational Instability

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Holographic Condensed Matter Theories and Gravitational Instability Holographic Condensed Matter Theories and Gravitational Instability by JIANYANG HE A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Physics) The University of British Columbia (Vancouver) October, 2010 c JIANYANG HE, 2010 ii Abstract The AdS/CFT correspondence, which connects a d-dimensional field theory to a (d + 1)-dimensional gravity, provides us with a new method to understand and explore physics. One of its recent interesting applications is holographic condensed matter theory. We investigate some holographic superconductivity models and discuss their properties. Both Abelian and non-Abelian models are studied, and we argue the p-wave solution is a hard-gapped superconductor. In a holographic system containing Fermions, the properties of a non-Fermi liquid with a Fermi surface are found. We show that a Landau level structure exists when external magnetic field is turned on, and argue for the existence of Fermi liquid when using the global coordinate system of AdS. Finite temperature results of the Fermion system are also given. In addition, a gravitational instability interpreted as a bubble of nothing is described, together with its field theory dual. THE OLD PREFACE PART, beginning here THE OLD PREFACE PART, ends here iii Preface This dissertation, as required by the university's policy, is a summary of my work in the Ph.D. program. I will review string theory and other relevant background knowledge briefly, and then describe my work. The main body of part II and part III comes from our published papers. I thank my collaborators for their permission to put the contents in my dissertation. For the holographic condensed matter papers[1{4], we divided the work, for example Anindya wrote the original Mathematica files, Brian and I focused on the analytic derivations, and Pallab wrote the papers, while we discussed the projects together for hours almost every day. If the work is to be separated into contributions from each person, it would be fair to say we had equal share of the work. For the work[5] with my supervisor Moshe, it was my first research project in the Ph.D. program. The idea was suggested by Moshe, and we did the work together. A list of publications of our work involved in this dissertation: 1. Pallab Basu, Jianyang He, Anindya Mukherjee, Moshe Rozali, Hsien-Hang Shieh (2010), \Comments on Non-Fermi Liquids in the Presence of a Condensate", [arXiv:1002.4929], chapter 4. 2. Pallab Basu, Jianyang He, Anindya Mukherjee, Hsien-Hang Shieh (2009), \Hard- gapped Holographic Superconductors", Phys.Lett.B689:45-50,2010, [arXiv: 0911.4999], chapter 5. 3. Pallab Basu, Jianyang He, Anindya Mukherjee, Hsien-Hang Shieh (2009),\Holo- graphic Non-Fermi Liquid in a Background Magnetic Field", Phys.Rev.D82:044036,2010, [arXiv:0908.1436], chapter 4. 4. Pallab Basu, Jianyang He, Anindya Mukherjee, Hsien-Hang Shieh, \Superconductivity from D3/D7: Holographic Pion Superfluid", JHEP 0911, 070 (2009), [arXiv:0810.3970], chapter 5. 5. Jianyang He, Moshe Rozali, \On Bubbles of Nothing in AdS/CFT ", JHEP 0709, 089 (2007), [arXiv:hep-th/0703220], chapter 6. iv Table of Contents Abstract ......................................... ii Preface .......................................... iii Table of Contents ................................... iv List of Figures ..................................... vii Acknowledgements .................................. ix I Introduction . 1 1 Introduction .....................................2 2 Background and Brief Review .........................5 2.1 Aspects of String Theory . .5 2.2 The AdS/CFT Correspondence . 12 2.2.1 Conformal Field Theory . 12 2.2.2 The Basics of the AdS/CFT Correspondence . 16 2.3 Examples of Instability and Phase Transition . 22 2.3.1 Coleman-De Luccia Decay . 23 2.3.2 Witten's Bubble . 25 2.3.3 Other Spacetime Transitions . 26 II Holographic Methods for Condensed Matter Physics . 29 3 Introduction to Holographic Condensed Matter Theory ......... 30 3.1 Operators and the Expectation Values . 30 3.2 Near Equilibrium Dynamics . 33 3.3 Holographic Hydrodynamics . 36 4 The Appearance of Fermi Surface ....................... 40 4.1 The Fermi Surface at Zero Temperature . 40 4.2 Landau Levels with External Magnetic Field . 45 4.2.1 Dyonic Black Hole . 46 4.2.2 Probe Fermion . 46 Table of Contents v 4.2.3 Results . 52 4.3 Global AdS4 Blackhole . 57 4.3.1 The General Setup . 57 4.3.2 Solution Along the Transverse Coordinates . 58 4.3.3 The Complete Solution . 61 4.3.4 The Scaling Symmetry and Two Limits . 63 4.3.5 Fermi Liquid? . 63 4.4 At Finite Temperature . 64 4.4.1 Asymptotically AdS4 Black Holes . 64 4.4.2 Fermions . 67 4.4.3 Reissner-Nordstr¨omBlack Hole (T > Tc)............... 68 4.4.4 Non Extremal Hairy Black Hole (T < Tc)............... 70 5 Holographic Superconductivity ......................... 74 5.1 Abelian Scalar Model . 74 5.2 Superconductivity from D3/D7: Holographic Pion Superfluid . 80 5.2.1 General Setup . 83 5.2.2 Phase Diagram . 88 5.2.3 Frequency Response . 89 5.2.4 Effect of Stationary Isospin Current . 92 5.2.5 What Condenses and What Doesn't . 93 5.2.6 Conclusion . 94 5.3 Gapless and Hard-gapped Holographic Superconductors . 95 5.3.1 Zero Temperature Holographic Superconductor . 96 5.3.2 Conductivity and Hard-gapless Theorem . 100 5.3.3 P-wave Holographic Superconductor and Hard-gapped Solution . 102 5.4 An Analytic Understanding of the Spikes of Conductivity . 111 III Instability of Gravitational Bubbles . 116 6 On Bubbles of Nothing in AdS/CFT ..................... 117 6.1 Introduction and Motivation . 117 6.2 Bubbles of Nothing in AdS/CFT . 118 6.2.1 R-Charged Bubbles . 118 6.2.2 Uncharged Case . 121 6.2.3 One Charge Case . 122 6.2.4 Three Equal Charges . 124 6.2.5 Features of the Phase Diagrams . 126 6.3 The Dual Field Theory . 128 6.3.1 S1 × R × S2 ............................... 128 6.4 Conclusions and Discussions . 131 6.4.1 Multi-wrapped Wilson Loop . 132 6.4.2 General D-brane Metric . 132 6.4.3 Polyakov and Polyakov-Maldacena Loop . 133 Table of Contents vi IV Conclusion . 135 7 Conclusion and Outlook ............................. 136 Bibliography . 140 Appendices . 152 A Condensed Matter Physics ........................... 152 B Kubo Formula for Electrical Conductivity .................. 156 C Transverse Scalars and Gauge Fields on S3 ................. 159 D Searching for Bound States on Complex ! Plane: WKB Method ... 162 vii List of Figures 2.1 Branes and strings. .9 2.2 The dualities of M-theory and between string theories. 10 2.3 A Riemann surface with a long thin tube. 28 2.4 A typical first order transition in field theory. 28 3.1 Typical quantum critical points in condensed matter theory. 31 4.1 The imaginary parts of the retarded Green's function's components. 44 4.2 Green's function at k < kF ........................... 44 4.3 Periodic behavior of Green's function. 45 4.4 Fermi momentum kF as a function as charge q. 54 4.5 Peaks moving as magnetic field h changes. 55 4.6 The power z as function of ~l........................... 64 4.7 The profiles for g(r) and (r) at Teff = 0:036. 66 4.8 The pole's track of Green's function on complex ! plane. 69 4.9 Variation of the pole position with temperature and momentum. 70 4.10 \Peak-dip-hump" behavior of Im(G22)..................... 71 4.11 Peak position (k, !) as a function of coupling. 71 4.12 Pole position (k, !) as a function of temperature. 72 4.13 Contour plots for varying λ for temperature Teff ∼ 0:004. 73 4.14 The pole position as function of fermion charge qF .............. 73 5.1 The two different kinds of condensation modes. 76 5.2 The conductivities of the two kinds of condensation modes. 77 5.3 The infinite DC conductivity'e effect on Im(σ). 77 5.4 The fields f (z);Apx(z)g at different parameters. 79 5.5 The condensation pΨ2/µ and the free energy F (µ fixed). 81 5.6 The condensation Ψ2/µ and the free energy F (Sx/µ fixed). 82 5.7 The phase space of normal and superconducting on (1/µ, Sx/µ) plane. 83 5.8 The phase space with condensation mode Ψ2 = 0. 84 5.9 Plot of the zero mode at µ =4......................... 88 5.10 Plot of the condensate with 1/µ ........................ 89 5.11 Speed of second sound as a function of 1/µ .................. 90 σ 5.12 µ at different value of µ............................. 91 5.13 Plot of superfluid density with 1/µ....................... 91 3 5.14 Plot of Wx/µ as a function of Sx/µ at different values of µ......... 92 5.15 First and second order phase transitions. 93 List of Figures viii 5.16 Plot of action for phases Bx1 6= 0 (upper curve) and φ 6= 0 (lower curve). 94 5.17 Zero and low temperature hairy black hole solutions. 98 5.18 α as function of q, giving a family of solution. 99 5.19 The potential at different temperature. 101 5.20 α as a function of q in non-Abelian model. 105 g 5.21 r2 and A as functions of rescaled rn...................... 106 5.22 Plot of the rescaled potential V (rn) at various values of q.......... 108 5.23 Schematic plot of the potential V (~r) for different T .............. 110 5.24 The potential and spikes of conductivity. 112 5.25 The number of spikes in AdS4 from WKB method. 113 5.26 The number of spikes in AdS5 from WKB method. 114 6.1 The actions of small and large bubbles. 122 6.2 The existence of bubbles in the (β; βjφj) plane in the one charge case.
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