Holographic Condensed Matter Theories and Gravitational Instability

JIANYANG HE

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

Doctor of Philosophy

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 ω 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 ψ(r) at Teff = 0.036...... 66 4.8 The pole’s track of Green’s function on complex ω 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, ω) as a function of coupling...... 71 4.12 Pole position (k, ω) 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 {ψ(z),A√x(z)} at different parameters...... 79 5.5 The condensation √Ψ2/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, β|φ|) plane in the one charge case. . . . 123 6.3 Instability in the one charge case...... 124 6.4 The existence of bubbles in the (β, β|φ|) plane in the three equal charges case. 125 6.5 Instability in the three equal charge case...... 125 6.6 Vacua in different size of spatial circle β...... 128 6.7 tr(eiβα)tr(e−iβα)/N 2 as a function of θ...... 130 6.8 Wilson loop and string’s world sheet...... 133

A.1 Oscillations in the susceptibility with varying magnetic ﬁeld...... 153 ix

Acknowledgements

During my Ph.D. program, many people have helped me in various ways. Prof. Moshe Rozali, my supervisor, is a very nice person, and helped me a lot both in my research and in life. I would like to thank him for his patience and guidance in my program. I thank Prof. Mark Van Raamsdonk for his generous help whenever I dropped by his oﬃce for trivial questions. I would like to thank my collaborators P. Basu, A. Mukherjee, B. Shieh for their help and appreciate their cooperation in our research. I am grateful to C. Chen, L. Jin, J. Karczmarek, J. Ng, G. Semenoﬀ, K. Schleich, Z. Zhu and the string theory group at UBC. I should thank my mother for her support to my choice of life, no matter how unreal it is from her point of view. 1

Part I

Introduction 2 Chapter 1

Introduction

As one of the most promising candidates of high energy theory, string theory provides us with a new way to understand the world, a potential unification of quantum theory and gravity, and a connection between the largest scale of cosmology and the smallest scale of particle physics. In the last decade, a conjecture of a correspondence between field theory and gravity[6– 9] was studied. It is called the AdS/CFT correspondence, since the simplest example is the 5 duality between supergravity in AdS5 × S spacetime and N = 4 super-Yang-Mills theory. This strong-weak correspondence can help us to avoid the usual problem of obtaining results beyond perturbation theory, and is helpful in understanding quantum gravity in highly curved geometries, such as arise in the early universe and black hole physics. Although the original example of field theory was N = 4 super-Yang-Mills theory, more and more realistic models, for example QCD-like theories, were investigated since then. The AdS/CFT correspondence, assuming it is a true statement, is a new method to understand physics, i.e. one can define a field theory from its dual gravity or vice versa. This from a philosophical point of view results in a question: which theories are more fundamental? A similar question can also be asked of a recent work [10], which aims to interpret the origin of gravity as a thermal property, i.e. entropy. I will not try to answer these questions in this dissertation, and keep them in my future interest. Recently some holographic condensed matter theories (CMT) were constructed, a duality which is sometimes named AdS/CMT. Various phase transitions exist in condensed matter physics, and the phase diagrams are usually complicated, one of the examples of which is given in Fig. 3.1. Some of condensed matter phenomena still lack have very well developed theories, for example high Tc superconductivity. The AdS/CFT correspondence, and the holographic models, provide us with a new tool to shed light on these issues. For example, it is well known that the (near-)phase transition regimes typically involve some scaling symmetries, which is a property of CFT(conformal field theory). One of the advantages is that the dual gravity theory is that it can reproduce the same scaling properties, and is able to tell us a more complete story about quantum critical points in CMT. It is exciting to find well-known condensed matter phenomena, e.g. quantum Hall- effect[11, 12], superconductivity[13], and Fermi-surface[14] in a gravity theory context. Our recent work[1, 3, 4, 15, 16] focused on these phenomena. Conductivity can be investigated in the dual gravity theory, which is usually an Abelian scalar model, or a similar model, in a spacetime background with or without a black hole. The scalar can condense at low temperature, and one finds that the system turns to be a superconductor. We further studied a non-Abelian model, showed the existence of pion-like condensation, and recently we found a hard-gapped p-wave superconductivity, different from the Abelian case and more precisely mimicking real world physics. Chapter 1. Introduction 3

Holographic non-Fermi liquid was found by evaluation of Fermion operators. With a constant magnetic field turned on, we extended the calculation to a dyonic black hole background, found a discrete spectrum, and discussed the symmetry of the parameters, e.g. the magnetic field, the charge of the fermion, the chemical potential, etc. In future work we are interested in the effect of interactions between a condensed scalar field and the probe Fermion: does the condensation affect the existence of the Fermi surface? Another direction of study is the parameter space (T, µ), where we know in some limit the field theory is a non-Fermi liquid, with some critical scaling behavior different from Fermi liquid, and we believe at some other limit the exact Fermi liquid should appear. With the Landau level-like discrete spectrum provided by the magnetic field, one can vary the parameters to change the fermion occupation number of the top Landau level, and might be able to explore some integer or fractional quantum Hall effect. More dual gravity theory might be found for other condensed matter phenomena, e.g. supersolid, type 1.5 superconductivity etc. We will continue working on these directions. In addition, in the parameter space (T, µ), a neutron star can be discussed by looking at the low temperature and large chemical potential regime, it would be interesting to see whether and what AdS/CFT can tell us about such issues. Additionally, fermion star and fermion black hole are also some directions worthy of investigation. Phase transition is a very common feature in physics, from the triple point in water to the confinement(hadron)-deconfinement(free quark) phenomena in particle physics. Ac- cording to the AdS/CFT correspondence, which is a strong/weak duality, one is not able to simultaneously obtain quantitative answers on both sides. On the other hand, phase transition is one of the qualitative properties one can check at both sides. From investigation on existence and instability of bubble of nothing, we argue for and construct a connection to the dual field theory[5]. Basically, we consider a string worldsheet ending on a Wilson loop/line at the boundary, which is a gauge invariant and comparatively straightforward quantity to evaluate, in various background geometries, and seek for transitions and the Euclidean version—–decays. One particular interesting example is the gravity dual of Un- ruh effect which can be derived from an accelerating particle’s path, a Euclidean extension of the Wilson loop. The plan of this dissertation is as follows: In part I, I will first give a brief review of the history of the development of string theory, then I will give an introduction to the background knowledge, including conformal field theory and the AdS/CFT correspondence, and list some examples of gravitational instabilities. Part II is the main part, which is dedicated to holographic condensed matter physics. I describe the basic ideas and the general methods of holographic condensed matter theory in chapter 3. Then I discuss the holographic Fermi surface behavior in chapter 4. In Sec. 4.2 an external magnetic field is turned on and results in a Landau level structure, then a similar calculation in global coordinate AdS4 spacetime is discussed in Sec. 4.3, which hints of the existence of an exact Fermi liquid. The extension to finite temperature is given in Sec. 4.4. Chapter 5 explores the holographic superconductivity. The Abelian scalar model, the simplest example, is shown in Sec. 5.1, then a model in string context, holographic Pion superfluid, is discussed in Sec. 5.2. Sec. 5.3 describes the gapless theorem in the Abelian Higgs model and the hard-gap existence in non-Abelian theory. We also discuss Chapter 1. Introduction 4 some specific properties of the conductivity in Sec. 5.4. Finally, chapter 6 studies the gravitational instability of bubble of nothing and the dual field theory. 5 Chapter 2

Background and Brief Review

In this chapter I will make a brief review of the relevant background knowledge. Firstly in Sec. 2.1 I will review the history of the string theory’s development. Then the AdS/CFT correspondence is described in Sec. 2.2. The last section 2.3 is a preparation for part III, listing some examples of gravitational instabilities and corresponding phase transitions in the dual ﬁeld theory.

2.1 Aspects of String Theory

Before we go to the detailed discussion, it might be interesting to review the history of string theory brieﬂy, not going over the complete history and developments here. First of all, here is a very short list of string theory’s main developments:

• 1960s, string theory arose in the context of the strong interactions.

• 1970s, supersymmetry and supergravity.

• 1980s, the ﬁve superstring theories.

• 1984, the ﬁrst revolution of string theory: Green-Schwarz mechanism, compactiﬁca- tion on Calabi-Yau manifolds.

• 1990s, the second revolution of string theory: supersymmetric gauge theory, dualities, M-theory, D-branes, the AdS/CFT correspondence, brane cosmology, etc.

• 2000s, string landscape, moduli stabilization, developments of AdS/CFT, etc.

I will give little more details below, organized mainly chronologically. Although strings are very common objects in classical dynamics, the appearance of strings in the 1960s’ modern physics was unexpected. Modern string theory originally arose in the context of the strong interaction in the late 1960s, explaining the observed s − t channel duality. However, it did not match the observed the proporty of “asymptotic freedom” : the property of having weak coupling at short distance and strong coupling at long distance. Therefore, another theory, i.e. QCD(quantum chromodynamics) turned to be more successful as a description of the strong interactions. In the 1970s, a brand new idea was introduced in physics, supersymmetry. Supersymmetry is a symmetry between fermions and bosons, which requires that every fermionic degree of freedom has a corresponding bosonic partner, and vice versa. Although supersymmetry came out firstly in string theory, it has many applications in particle physics. Some of its advantages of supersymmetry are zero vacuum energy and unification of the running coupling constants Chapter 2. Background and Brief Review 6 of electromagnetism, weak and strong interactions. A supersymmetric theory has two main parameters: the number of copies of supersymmetry N and the spacetime dimensions d. The models with N ≥ 2 are called extended supersymmetries. It was shown that there are some relations between N and d, for instance a d = 4, N = 8 supersymmetry can be obtained from d = 11, N = 1 model by dimensional reduction or compactification. A priori, both N and d can be increased without bound, but any example with N > 8 in d = 4 requires high spin fields s > 2. Since the highest spin field we are familiar with is the graviton(s = 2), and we do not know how to construct consistent interacting theories for a finite number of fields with spin s > 2, the discussion is usually limited up to N = 8, d = 4 supergravity. The awkward fact is that we have not yet found any supersymmetry in our real world. The way out is typically to argue that supersymmetry breaks at some energy level higher than we probed, and the new physics phenomena (including supersymmetry) are expected to appear in the LHC(Large Hadron Collider) soon. On the other hand, the two most famous branches in physics in last century, quantum mechanics and general rela- tivity, are well-known to be difficult to unify, while a grand unification is one of an utmost goals in physics, therefore physicists were excited in the 1970s when the graviton appeared naturally in string theory, together with other fields (scalars, vectors and others, with spins s ≤ 2). It was around the year 1980 when a supersymmetric string (superstring) theory was studied “thoroughly”. Without supersymmetry, there are bosonic string theories. The ground state of a bosonic string theory is a tachyon, which has negative mass squared and usually taken to be unphysical. The mass of the ground and excited states of both bosonic and supersymmetric string theories depend on the number of dimensions. If Lorentz invariance is required, the theories have critical dimensions, where bosonic string has dc = 26 while superstring theories need dc = 10. The string theories with the critical dimensions are called critical string theories, while the others are non-critical theories. We are mostly interested in the critical theories. At that time some problems prevented more interest in string theory. One of those problems is that too many different string theories existed, while we believe that a good theory should be unique or with only finitely many different versions. Another problem is the existence of gauge and gravitational anomalies. Until 1984, when the first revolution of string theory happened, not many people believed it. At 1984, it was pointed out[17] that both gauge and gravitational anomalies are canceled in type I string theory with gauge group SO(32). Type I string theory is interesting since it can give chiral fermions. Later, heterotic strings [18, 19], with chiral fermions, were constructed. Another important work in that year might be [20], where the authors argued that the compactification of type I or heterotic string theories on a Calabi-Yau manifold leads to N = 1 d = 4 supersymmetric theory, and three generations of fields. This work initiated low dimensional string theory, and the compactification on Calabi-Yau manifold is the first example connecting string theory to algebraic geometry. Since then, physicists have found five different anomaly free superstring theories, and I should pause here to make them clear first. In the name of type I or II, the Roman number signifies the number of supersymmetries at d = 10. Type I superstring has the gauge group SO(32) and includes both open and closed strings. Closed strings appear naturally in a open string theory, since the ends of open strings can join together. For a Chapter 2. Background and Brief Review 7 closed string, besides the trivial movement of the center mass of string, there are two sets of modes moving along the string, which are usually named left- and right-movers. The chiralities of fermionic spectrum can be constructed independently for the two opposite sets of left and right movers. Type IIA and IIB superstring theories contain only closed string spectrum, while the name of IIA or IIB corresponds to the chiralities, opposite or identical in both sectors, respectively. The other superstring theories are both heterotic strings. The left-movers of a heterotic string is bosonic and live in 26 dimensions, while 16 of them are compactified. On the other hand, the right-movers are supersymmetric, with gauge group SO(32) or E8 × E8. String theory can be quantizd and a scattering amplitude can be expanded, using the string’s Feynman diagrams, order by order. The key parameter is the string coupling constant gs. Similar to its much more familiar cousin in quantum field theory, some of the properties can be extracted from the tree-level and the first orders of loop levels, if gs is small. This algorithm is called string perturbation theory. As was discussed earlier, there is ar obvious UV divergence from the vertices. In 1986, Witten discussed non-commutative geometry (NCG) from string field theory point of view[21], which was the first time when NCG appeared in string theory. Later it was proved that this theory can be used to derive perturbative open string theory, and the non-perturbative properties were explored more recently. In 1989, a now so called “old” matrix model was discovered[22–24]. A matrix model was earlier introduced in field theory by ’t Hooft[25] where he used an N-ranked matrix to discuss a non-Abelian gauge theory. In the large N limit, the model can be expanded in the parameter 1/N, and the Feynman diagrams are smoothed into 2d compact Riemann surfaces. In addition, the expansion is controlled by the genus of the Riemann surfaces. As we know, the worldsheet of strings are 2-dimensional surfaces, and the expansions of string’s scattering amplitude are very similar to the expansions of the matrix model in the large N limit. It was pointed out that these expansions are connected with random surface theory. The 2d surfaces themselves are naturally described by a 2d gravity, while if scalars are added into the model, they can be interpreted as the embedding of the string’s worldsheet into a spacetime. Following the discovery of the “old” matrix model, many followup work has been done. For example, Witten discovered 2d black hole in 1991, which is a solvable CFT from the string’s worldsheet viewpoint. The following decade, the 1990s, was a decade of dualities. The first was T-duality. For a closed string, if it is compactified on a circle with radius R, the spectrum is parameterized by two sets of modes: the winding number (energy) nw ∝ R and the momentum nq ∝ 1/R. The spectrum is invariant under the transformation that the circle is mapped into another with radius 1/R, with (nw ↔ nq). This is one of the examples of T-duality. Since particles do not have winding modes, there is no such T-duality in field theory. One example related to the T-duality in the superstring is the mirror symmetry, which maps a type IIA superstring compactified on a Calabi-Yau manifold to a type IIB superstring compactified on another Calabi-Yau manifold, and vice versa, i.e. it claims that Calabi-Yau manifolds appear in pairs with “mirror symmetry” between each other. Another important duality is S-duality, which for simplicity can be described as a strong- weak coupling duality. S-duality can be traced back to the Dirac’s magnetic monopole, where he predicted the existence of monopoles, and gave the so-called Dirac’s quantization Chapter 2. Background and Brief Review 8

condition qeqm ∈ integers, where qe and qm are electric and magnetic charges respectively. On the other hand, the coupling between electric charges ge is inversely proportional to the one between magnetic charges, ge ∝ 1/gm, i.e. there is a strong-weak coupling duality. It was Sen who first demonstrated S-duality in string theory[26, 27], by proving the existence of a bound state with two magnetic charges and one electric charge in string theory. One is usually told that the second revolution of string theory started with Seiberg and Witten’s work[28]. In that paper, the authors investigated N = 2 supersymmetric gauge in 4 dimensions with gauge group SU(2). They proved that exact formulas for some quantities of the theory can be obtained, and showed the existence of a version of strong-weak coupling duality. This strong-weak coupling duality provides us a method to explore a with a strong coupling, by using its weak coupled dual theory. Months later, Hull and Townsend found another duality[29], U-duality. Here “U” means “unity”. U-duality is the unity of T- and S-duality. They considered N = 8, 4 dimensional superstring where the other 6 dimensions are compactified on a torus T 6. All the BPS states1 can be obtained from one single BPS state by T-dualities. Under the transformation of U-duality, some other BPS states can be found, while not all of them are string states. The authors conjectured that these non-string states are solitonic extremal black holes. Furthermore, they pointed out 4d heterotic string compactified on T 6 is dual to 4d type II superstrings, and conjectured that the duality was probably derived from 11d supergravity. Shortly after that, Witten argued[30] 11d supergravity arose as the strong coupling limit of type IIA superstring, and discussed U-duality in variant dimensions d > 4. Half a year later, he also found[31] that 11d supergravity is again dual to the strong coupling limit of heterotic string with gauge group E8 × E8. Around that time, a crucial object in string theory was discovered. Polchinski pointed out the existence of Dirichlet-Branes or D-branes[32]. D-brane is a solitonic object with charge and tension, where the tension is inversely dependent on string’s coupling T ∝ 1/gs. A Dp-brane is the source of (p + 1)-form Ramond-Ramond(R-R) charge, extends along p spatial dimensions, and is a (p + 1)-d object. Embedded in a D-dimensional spacetime, a Dp-brane can have some excitations—-open strings with their ends attached on the branes, while the closed strings move in the bulk. The name “D” comes from the Dirichlet boundary conditions of the ends of the open strings, which means that the ends satisfy Dirichlet boundary conditions along the (D − p − 1) directions normal to the brane and Neumann boundary conditions along the p + 1 tangent directions. A D-brane is not a rigid object, and can have the dynamic behavior. For example, if a string is attached to it, the attached point should have some deformation of the position in the embedded spacetime. D-brane is a BPS solitonic object, conserving part of the supersymmetry. The fields living on D-branes include scalars, vectors and fermions. The scalars describe the positions of the D-brane in the spacetime, and the vectors interpret the vibrations along the brane, while the fermions are the Goldstones of the broken supersymmetry. One special property is that there is no interaction between parallel D-branes, which together have the same supersymmetry as single D-brane and so is a BPS state as well. The attraction force

1BPS(Bogomolnyi-Prasad-Sommerfield) state is a state invariant under a nontrivial subalgebra of the full supersymmetry algebra. It has a shorter representation than the non-BPS states, and sometimes can be thought as a limit of the non-BPS states, for instance extremal black hole is a BPS solution which is a limit of the non-extremal black hole solutions. Chapter 2. Background and Brief Review 9 provided from gravitons and dilaton cancels with the repelling force from the R-R charges. One interesting behavior is shown in Fig.2.1. Closed strings can move in the bulk, while open strings have their ends attached to the branes. For the cylinder connecting the two branes, there are two totally different versions of story. One can think it is an open string with its ends on two different branes, moving around a circle. Alternatively it can be taken as a closed string emitted from one brane, moving to the other and eventually absorbed by it. Heuristically the classical open string moving is dual to the quantum creation and annihilation of the closed string, which also hints of the soon to be discovered the AdS/CFT correspondence. To make a gravitational effect, one usually needs to set up large N copies of coincident D-branes, where for the fields living on the branes the gauge group is now U(N).

Figure 2.1: Branes and strings. The blue curves are strings, where one of them is closed string moving in the bulk, and another is open string with both of its ends attached on the left D-brane. The cylinder connecting the two branes can be realized in two diﬀerent ways: a closed string emitted by one brane and absorbed by the other, or an open string connecting the two branes moving along a circle. The closed string story is a quantum behavior, while the open string one is in a classical level.

Generally a Calabi-Yau manifold has one or more singular points (conifolds), where three of the six dimensions of the manifold shrinks to a point, or more precisely a submanifold shrinks to zero size[33–36]. Strominger and his collaborators gave a physical understanding of the transition from one Calabi-Yau manifold into another through a conifold, and showed the process by an example with a massless D3-brane [37, 38]. It was conjectured that any two Calabi-Yau manifolds can be connected with such conifolds, and then string theory is uniﬁed in a single moduli space. Chapter 2. Background and Brief Review 10

By then physicists have found many dualities among the five superstring theories. The 10 dimensional string theories also seem to be connected to 11d supergravity. All of these connections lead people to conjecture that there is a single underlying theory, which is named by Witten “M-theory”[39]. Here “M” might stand for magic, mystery or membrane, and later was used by others as matrix or mother2. Some of the dualities are shown in Fig.2.2. The 11 dimensional limit compactified on S1 is dual to type IIA superstring, while the compactification on a section of line (effectively S1/Z2) is dual to heterotic string theory with gauge group E8 × E8. The compactification of type IIA on a circle is a T-duality of type IIB string theory, and similarly the two heterotic string theories are T-dual to each other when compactified on a circle. Type I string theory is connected by a S-duality from heterotic string theory with gauge group SO(32) and a projection from type IIB superstring theory.

Figure 2.2: The dualities of M-theory and between string theories.

Since string theory is one of the candidates for a theory of quantum gravity, black holes might be the best objects to test the theory. Since there are many long distance interactions in string theory, it is not very difficult to construct “black” solutions, e.g. black branes. Branes are BPS state, which sometimes are called extremal black holes. These single charge “extremal black holes” have zero temperature and zero size horizon. However, spme extremal black hole solutions in 4d gravity involve more than one “charge”, such as mass and electric or magnetic charges for a Reissner-Nordströmblack hole, or mass and angular momenta for a Kerr black hole, and the entropy does not vanish although the temperature is zero. To get an extremal black hole with finite size horizon, the solution

2As a comparison to Vafa’s F-theory, which someone named as “Father-theory”. Chapter 2. Background and Brief Review 11 has to contain multi-charges. One of the first examples was given by Strominger and Vafa[40], and a simpler solution was found by Callan and Maldacena[41]. The main idea of the work is to compactify a type II superstring theory on a 5d torus T 5. In type IIB theory, bound states of D5 and D1 branes exist where D1-branes are parallel to D5 which wind around the torus. Besides the the number of the two kinds of branes N5 and N1, another parameter is the√ momentum q along the D1 brane. The authors found the formula of the entropy S ∝ qN5N1, and also calculated the Hawking temperature of the solution which is slightly different from the extremal state. More detailed calculation of the Hawking radiation and decay rate was done by Das and Mathur[42], where they show the probability of a left-handed and a right-handed open string states annihilating into a closed string and the decay rate was consistent with Hawking’s blackbody radiation. Similar black hole solutions were also given from M-theory’s point of view, for example the three sets of M5-branes model[43]. The now known “matrix theory”, different from the “old” matrix model, arose around 1996. One of the first works is [44], where Banks and his collaborators conjectured that the perturbative quantum theory of M-theory is a limit of multi D0-branes. Before that, Shenker and others showed a method[45] to investigate interactions between D0-branes with a low energy theory, i.e. matrix quantum mechanics. Some followup works, such as [46–48], appliued the matrix model to M-theory and string theory. Matrix theory is a holographic theory, since it explores the 11d M-theory, while only 10 dimensions manifestly appear in the matrix model. The 11-th dimension arises from the bound states of the D0-branes. It is difficult to find a Lorentz covariant version of matrix theory, for the reason that the space is described by matrices themselves, e.g. the positions are given when a matrix is diagonalized. Nevertheless it provides us a new method to approach and understand M- theory, and plays an indispensable role in the appearance of the AdS/CFT correspondence conjecture. The AdS/CFT correspondence is the main interest in string theory in the last decade. It was first conjectured by Maldacena[6] that the large N limit of superconformal field theory is dual to supergravity, and was very soon supported by Gubser, Klebanov and Polyakov[7], as well as Witten[8, 9]. The simplest example comes from large N copies 5 coincident D3-branes, the near horizon of which has effective metric of AdS5 × S . It 5 was argued that the low energy limits, supergravity in AdS5 × S and N = 4, d = 4 super-Yang-Mills (superconformal) theory living on the D3-branes, are dual to each other. This is the reason why the correspondence is named AdS/CFT. It is a holographic duality, since the bulk gravity corresponds to the field theory living on the boundary of the AdS space. The conjecture was generalized to various dualities between field theories and string theory or M-theory. Although we are not able to prove the correspondence completely, there is more and more evidence supporting it. Recently there is more interest in dualities between gravity and condensed matter theory, with the abbreviation of AdS/CMT, which will be discussed in more detail later. Our work mainly focus on this field. The AdS/CFT correspondence is a strong-weak duality, i.e. weakly coupled gravity is dual to strongly coupled field theory. Thus it is a very useful tool to understand the non-perturbative properties of the physics, which we are usually incapable to do. Since the development of AdS/CFT, there are many unrelated avenues of progress in string theory. For example, it was argued that with appropriate configuration, the mod- Chapter 2. Background and Brief Review 12

uli of Calabi-Yau manifolds can be stabilized[49, 50], which results in a string (vacuum) landscape[51]. The landscape is very complicated, while we want to find one point corresponding to our universe, which is de-Sitter space-like with a very small positive cosmological constant. As argued by Susskind, there is a very large number of vacua, with the magnitude of 10500, and it seems not feasible to find our universe by any calculation. The most possible theory we are able to rely on might be the anthropic principle. Some people also argued the existence of multi-verse(multi- or parallel universes) motivated by the string landscape. Another interesting field is string theory’s application on cosmology. An early work is brane world, for example [52], which showed a process of brane-anti-brane annihilation and argued it results in inflation and reheating. Some related models include string gas cosmology and recent arguments that string theory may resolve singularities, such as the big-bang singularity.

2.2 The AdS/CFT Correspondence

2.2.1 Conformal Field Theory Symmetry plays a crucial role in modern physics. The most familiar example in quantum field theory are the Lorentz group and the Poincare group. In addition, scaling symmetries appear frequently in physics, e.g. in statistical mechanics and renormalization group discussion. Combined with the Poincare group, scaling gives an extension: the conformal group. The conformal transformation symmetry applies to many fields in physics. The scaling symmetry on a quantum field theory requires the vanishing of masses, which in general is not true. However in the high energy regime, the masses of particles can be effectively taken to be zero, and the “massless” fields do have the scaling symmetry approximately. One other example happens in condensed matter physics, where critical phenomena play a very important role. For example in a lattice system, one can calculate the correlation between two separated points, and the system can be characterized by a scale called the correlation length. Correlation length usually depends on the parameters of the system, e.g. temperature, chemical potential, pressure, etc. At some critical value of temperature or other parameters, the correlation length grows very quickly, becoming larger than the size of the whole system, or effectively one can say it goes to infinity. Meanwhile, some physical quantities diverge, depending on the difference of parameters from the critical value by inverse power-law. For instance, in a two-dimensional Ising model, 7 χ ∝ (T − T )−γ, γ = , c 4 −ν ξ ∝ (Tc − T ) , ν = 1, (2.1) where χ and ξ are susceptibility and correlation length of the system. In addition, string theorists are familiar with CFT as well. The first application in string theory is the 2d CFT discussion of string’s worldsheet, which is one of the basics of string perturbation theory. In the last decade, with the development of the hypothesis of the AdS/CFT correspondence, interest is more focused on higher dimensional CFTs. In Chapter 2. Background and Brief Review 13

our work and in this dissertation, since only AdS4 and AdS5 are discussed, the dual ﬁeld theories (including CFTs) are restricted to 3− and 4− dimensions.

Conformal Group and Algebra From the name “conformal”, one can guess the conformal transformation is a mapping which preserves angles between any two lines. Formally it is a coordinate transformation mapping xµ → x0µ, with the metric invariant up to a scale,

0 0 2 gµν(x) → gµν(x ) = Ω (x)gµν(x). (2.2)

It can be veriﬁed that conformal group is the minimal group containing Poincare group and inversion symmetry, xµ → xµ/x2. As far as the generators to be concerned, as well as the Lorentzian boosts and rotations Mµν, translations Pµ, which together combined the Poincare group, conformal group contains also scale transformation D, and special conformal transformation Kµ. The ﬁnite coordinate transformations of the latter two can be described as

scale transformation (dilatation)D : xµ → λxµ, µ µ 2 µ x + a x special conformal transformation Kµ : x → ν 2 2 . (2.3) 1 + 2x aν + a x One can check that the special conformal transformation is also a speciﬁc translation

x0µ xµ = − aµ. (2.4) x02 x2 The algebra of the generators are

[Mµν,Pρ] = −i(ηµρPν − ηνρPµ); [Mµν,Kρ] = −i(ηµρKν − ηνρKµ);

[Mµν,Mρσ] = −iηµρMνσ ± permutations; [Mµν,D] = 0; [D,Kµ] = −iKµ;

[D,Pµ] = −iPµ;[Pµ,Kν] = 2iMµν − 2iηµνD, (2.5) with all the other commutations vanish. As discussed above, scaling invariance is part of conformal invariance, however there are few natural examples in physics which are scaling invariant but not conformally invariant, so a scaling invariance is usually taken to “be equivalent to” a conformal invariance. Although we are more interested in the CFT in Minkowski spacetime, sometimes it is simpler to discuss it in Euclidean spacetime, which is a Wick rotation of the former. The Euclidean conformal group is isometric to the algebra of SO(d + 1, 1), the Wick rotation of SO(d, 2) in Minkowski version. One can deﬁne the new operators as following, 1 1 J = M ,J = (P − K ),J = D,J = (P + K ). (2.6) µν µν −1,µ 2 µ µ −1,0 0,µ 2 µ µ Chapter 2. Background and Brief Review 14

The operators satisfy the commutation relations,

[Jab,Jcd] = i(ηadJbc + ηbcJad − ηacJbd − ηbdJac), (2.7)

where a, b ∈ {−1, 0, 1, . . . , d} and ηab is the Minkowski metric.

Primary Fields, Stress Tensor and Correlation Functions

When acting on local ﬁelds, one can introduce a matrix representation Sµν of Lorentz group on a ﬁeld φ(x) satisfying

Mµνφ(0) = Sµνφ(0), (2.8)

µ where Sµν is the spin operator. The action can be extended to x 6= 0 by translation. For example, Lµνφ(x) = i(xµ∂ν − xν∂µ)φ(x) + Sµνφ(x). (2.9) For simplicity, we only consider spinless(scalar) ﬁeld here. A scalar transforms

∂x0 −∆/d φ(x) → φ0(x0) = φ(x), (2.10) ∂x as xµ → x0µ. Under a conformal transformation 2.2,

φ0(x0) = Ω(x)∆φ(x), (2.11)

i.e. the scalar φ(x) undergoes a scaling transformation, while ∆ is usually called the scaling dimension of the quasi-primary ﬁeld φ(x). A two-point correlation function of φ(x) is deﬁned as

1 Z hφ (x )φ (x )i = [dφ]φ (x )φ (x )exp − S[φ] , (2.12) 1 1 2 2 Z 1 1 2 2

where Z and S[φ] are partition function and action respectively. Under a scaling transformation xµ → λxµ, ∆1+∆2 hφ1(x1)φ2(x2)i = λ hφ1(λx1)φ2(λx2)i. (2.13)

without losing generality, one can take (x1, x2) → (x, 0). The rotation invariance (in Eu- clidean spacetime) and the translation invariance require the 2-point function is a function of |x|, and the clean result is 1 1 hφ(0)φ(x)i ∝ ≡ . (2.14) |x|2∆ (x2)∆

The 3-point functions can be found similarly 1 hφi(x1)φj(x2)φk(x3)i ∝ ∆ +∆ −∆ ∆ +∆ −∆ ∆ +∆ −∆ . (2.15) |x1 − x2| 1 2 3 |x1 − x3| 1 3 2 |x2 − x3| 2 3 1 Chapter 2. Background and Brief Review 15

However any higher point correlation function, e.g. hφ4i, has an arbitrary function which cannot be ﬁxed by the conformal invariance. Another important operator is the stress tensor(energy-momentum tensor) Tµν. Under an inﬁnitesimal transformation xµ → xµ + µ, the stress tensor can be introduced, Z d µν δS = d xT ∂µnu. (2.16)

With the properties of a conformal transformation, the above equation can be written as 1 Z δS = ddxT µ ∂ ν. (2.17) d µ ν Obviously, the vanishing of the trace of stress tensor indicates the invariance of the action. A quantity related to stress tensor is current, which is usually deﬁned as

µ µν j = T ν. (2.18)

µ The conservation of current ∂µj = 0 results in

µν ∂µT = 0 (2.19)

as expected. One important tool of conformal ﬁeld theory is OPE (operator product expansion). Generally two operators, say O1 and O2 do not commute. If we bring them close to the same point, the product can be written in a sum of other local operators,

X n O1(x)O2(y) → C12(x − y)On(y), (2.20) n

n ∆1+∆2−∆n where C12(x − y) ∝ 1/(x − y) . The leading term of the expansion is determined by conformal algebra, for instance the stress tensor operator imposed on a scalar, 1 T (x)φ(0) ∝ ∆φ(0)∂ ∂ ( ) + ··· . (2.21) µν µ ν x2

Superconformal algebra and Field theories According to the AdS/CFT correspondence, which we will discuss later, the 10−dimensional string theory is dual to a ﬁeld theory. Since string theory is supersymmetric, the ﬁeld theory should include supersymmetry as well. For example, in the most familiar case, large 5 N copies of coincident D3−branes result in an AdS5 × S geometry, which has a bosonic symmetry group SO(4, 2) × SO(6). The symmetry group is not complete until the supersymmetry generators Q are added into the algebra. I am not going to give more details here since our work does not involve much of the supersymmetry. Chapter 2. Background and Brief Review 16

2.2.2 The Basics of the AdS/CFT Correspondence It was in the late 1990s when an impressive and prominent conjecture[7–9] came to be, about the correspondence between some specific theories of gravity and gauge theories, which is usually called the AdS/CFT correspondence.(For a review see [53]) A more precise 5 description is that type IIB string theory in geometry AdS5 × S is dual to N = 4 large N SU(N) super-Yang-Mills theory. The motivation of the conjecture comes from low energy states of string theory. Considering N parallel coincident D3-branes in 10 dimensional spacetime. In the low energy limit, we need to consider the massless string states only. The closed string massless states give a gravity supermultiplet in 10 dimensional space, with the low energy effective Lagrangian as type IIB supergravity. On the other hand, the open string massless states relate to N = 4 vector supermultiplet in (3+1) dimensions, and the low energy effective Lagrangian is N = 4 U(N) super-Yang-Mills theory. At the low energy limit, one can write the action in terms of the open string excitations on the branes, closed strings in the bulk, and the interactions between them,

S = Sbulk + Sbrane + Sinteraction. (2.22)

Sbulk is the action of 10 dimensional supergravity, plus some higher derivative corrections. The brane action Sbrane is defined in (3+1) dimensions, containing the N = 4 super- Yang-Mills Lagrangian plus some higher derivative corrections. Sinteraction describes the interactions between branes modes and bulk modes. For small fluctuations, one can expand the metric gµν = ηµν + κhµν, where κ is a small gravitational constant, and the bulk action can be expanded as following, 1 Z √ Z S ∼ d10x g R + · · · ∼ d10x[(∂h)2 + κh(∂h)2 + ... ], (2.23) bulk 2κ2 where in the last part the first term of is the free massless modes, followed by the interactions with positive powers of Newton’s constant κ. The interactions become weaker at low energies, and one can say it is an IR-free gravity. The similar expansion of Sinteraction gives also terms proportional to κ,

Z √ Z δ S ∼ d4x g T r[F 2] + ... ∼ κ d4x h T r F 2 − µν F 2 + ..., (2.24) int µν µν 4

where Fµν is the gauge ﬁeld strength. In the low energy limit, the DBI action of Dp-brane should take the form " # 1 Z X S = dp+1ξ e−φ Tr F 2 + 2(D XI )2 + [XI ,XJ ]2 + fermion terms N,p−brane 4g2 µν µ Y M,p I,J + (higher order corrections), (2.25)

where 2 p−4 gY M,p = (2πls) lsgs. (2.26) Chapter 2. Background and Brief Review 17

If we take a limit at which energy is small and fixed, also ls → 0 keeping all other 4 dimensionless parameters like N or gs finite. In this limit κ ∼ gsls → 0, thus in the action (2.22), Sbulk has only the free gravity term, Sinteraction goes to zero, and in the DBI action Sbrane, all higher derivative terms are proportional to positive powers of ls thus also vanish. Finally we have two parts, free bulk supergravity and N = 4, U(N) super Yang-Mills theory, decoupled from each other. For this reason, the limit is also called decoupling limit. This specific super Yang-Mills theory is a scale invariant theory, and actually a 4-dim conformal field theory(CFT). One can study the system from another point of view. The D3-brane metric is

2 −1/2 2 1/2 2 2 2 ds = H (−dt + d~x · d~x) + H (dr + r dΩ5), (2.27)

where the dilation is constant, the convenient fact of which is true only for D3-branes. The self-dual 4-form C0123 = 1 − 1/H, and

R4 H = 1 + ,R4 = 4πg l4N. (2.28) r4 s s

Since gtt is not constant, the energy measured is r dependent. For an observer at inﬁnity r → ∞, there is a red-shift eﬀect

−1/4 E∞ = H(r) Er. (2.29)

If the observer tries to measure the energy of a particle moving to r = 0, it gets lower and lower. From this observer’s view, there are two sets of excitations, one is the standard massless large-wavelength excitations propagating in the whole bulk, the other is the excitations approaching r = 0(near-horizon region). One can calculate that the coupling between these two sets of modes,

8 3 σabsorption ∼ R ω , (2.30) which is weak at low energy and ls → 0, or one can say these two sets decouple with each other in the low energy decoupling limit. Around the horizon r = 0, the metric is approximately

R2 r2 ds2 = dr2 + (−dt2 + d~x · d~x) + R2dΩ2 (2.31) r2 R2 5 R2 R2 = du2 − dt2 + d~x · d~x + R2dΩ2, u = , (2.32) u2 5 r

5 which is exactly the metric of AdS5 × S . These two non-interacting sets of excitations can be viewed as type IIB supergravity in the asymptotically ﬂat 10dim spacetime and IIB 5 supergravity in AdS5 × S . Comparing these two versions of theory, one may make a conjecture that N = 4 U(N) 5 super Yang-Mills theory in 4 dimensions is dual to type IIB supergravity in AdS5 × S , at least in the low energy decoupling limit. Chapter 2. Background and Brief Review 18

Facts of the correspondence In this part, let us check some facts of the correspondence. Firstly it is interesting to compare the global symmetry of both theories. N = 4 SU(N) SYM theory is conformal, and for N copies of D3-brane in the 10 dimensional spacetime, it has a conformal group SO(2,4), and R-symmetry group SO(6), which is obviously the same as the Killing symmetries of 5 AdS5 × S background. On the other side, this super Yang-Mills theory has fermionic symmetries. There are 16 supercharges in the usual N = 4 supersymmetry, and the CFT also contribute extra 5 16 supercharges, totally 32 ones. In the AdS5 × S background, string theory is invariant under 32 supercharges. Thus both sides have the same number of supercharges. Two dimensionless characteristic parameters in gauge theory are gYM ,N, which can 2 be combined as the ’t Hooft coupling constant λ = gYM N[25]. Schematically the SU(N) Yang-Mills action is

ijk 2 ijkl S = Tr[dXidXi + gYM c XiXjXk + gYM d XiXjXkXl + ... ]. (2.33)

After rescaling the ﬁelds Xi → Xi/gYM ,

1 ijk ijkl S = 2 Tr[dXidXi + c XiXjXk + d XiXjXkXl + ... ]. (2.34) gYM Thus for a Feynman diagram, each interaction vertex has a factor of N/λ, each propagator gives a λ/N, and each closed loop contributes a factor of N, thus for a vacuum diagram with V vertices, E propagators(edges), and F loops(faces), there is a total coeﬃcient

λ E N V N F = N χλE−V , (2.35) N λ

where χ = V −E +F is the Euler number. On the other hand, for compact closed surfaces, χ = 2 − 2g, with g the number of handles, therefore the perturbation of Feynman diagrams at large N is given by

∞ ∞ ∞ X 2−2g X i X 2−2g N ci,gλ ≡ N Zg(λ). (2.36) g=0 i=0 g=0

At large N limit, the result is dominated by the minimal g, number of handles, surface, which is usually the planer diagram. All the subleading ones are suppressed by 1/N 2 order by order. The form of perturbation (2.36) is similar to genus expansion of string theory, where the subleading diagrams are suppressed by gs = 1/N order by order, from which one might 2 guess gs ∼ gYM . Some other relations are

4 2 2 R 2 2κ10 G10 7 2 5 λ 4 = 4πgsN = gYM N = λ, 8 = 16π 8 = (2π) gs = 8π 2 , (2.37) ls ls ls N

where ls is the string length, R is the AdS radius, and G10 is the 10 dimensional Newton’s Chapter 2. Background and Brief Review 19

constant. 2 In the string theory, from equation (2.37), G10 ∼ 1/N , the quantum gravity eﬀect is suppressed in large N limit. And higher string corrections are small if the background satisﬁes R ls, which from (2.37) gives λ 1. Thus this discussion also support that the large N limit strong ’t Hooft coupling λ gauge theory is dual to classical limit of IIB 5 supergravity in AdS5 × S . A fact we need to emphasize here is that it is a strong/weak duality, which means the strongly coupled super Yang-Mills theory is dual to the weakly curved supergravity.

Bulk ﬁelds and boundary operators

At the boundary of AdSd+1, one typical basic object in the CFT is local operators O, with R the coupling boundary φ0O. One can calculate the correlation functions hO(x1)O(x2) ... O(xn)i, where (x1, x2, . . . , xn) are positions on the boundary spacetime, and naturally to define the generating function Z hexp( φ0O)iCFT . (2.38) boundary On the gravity side, with a fixed boundary, generally there are more than one form of bulk metric to make the system consistent, for instance, with a Euclidean AdS5 boundary 3 1 4 1 3 2 S × S , two explicit bulk solutions exist, B × S labeled as (X1) and S × B labeled as (X2). According to the approach of Euclidean quantum gravity[54], one should sum up their contributions. For each solution, the fields should be consistent with the boundary conditions, i.e. φ(~x,r)|r→∞ = φ0(~x), (2.39) where ~x are the boundary coordinates, r is the radial coordinate, and the boundary is at r → ∞. We know for a classical supergravity solution,

ZS(φ0) = exp(−IS(φ)), (2.40)

where IS is the classical supergravity action. The ansatz for the relation is Z exp φ0O = ZS(φ0), (2.41) boundary CFT

or more explicitly, one need to add all the bulk solutions (Xi) up, which have the same boundary as we have,

Z X exp φ0O = ZS(φ0,Xi), (2.42) boundary CFT saddle points(Xi)

The bulk with the minimal gravity action should dominate. Since the comparison is by exponential functions(partition functions), and a small diﬀerence between the bulk actions usually results in a large diﬀerence in the partition functions, most of the time the contributions from the other bulk solutions are negligible. A similar equation as (2.41) for gauge Chapter 2. Background and Brief Review 20

ﬁeld is Z a exp JaA0 = ZS(A0), (2.43) boundary CFT

It is helpful for understanding the correlation functions hO(x1)O(x2) ... O(xn)i to do a sample calculation. For simplicity, consider a massless scalar φ in AdS background with action 1 Z √ I(φ) = dd+1y g|dφ|2. (2.44) 2 Bd+1

The bulk ﬁeld φ should have the boundary value φ0. In the metric

d 1 X ds2 = (dx2), (2.45) x2 i 0 i=0 the Laplace equation for a Green’s function is

d −d+1 d x0 K(x0) = 0, (2.46) dx0 dx0 and the solution vanishing at x0 = 0 is

d K(x0) = cx0, (2.47) where c is a constant. One might want to do a SO(1,d+1) transformation to avoid some singularity problems, xi xi → , i = 0, . . . , d, (2.48) 2 Pd 2 x0 + j=1 xj and the Green’s function now is xd K(x) = c 0 . (2.49) 2 Pd 2 d (x0 + j=1 xj ) From this expression, we can smell some hint that

hO(x)O(x0)i ∼ |x − x0|−2d, (2.50) but let us do further to see more explicitly. Using the Green’s function, one can write a bulk ﬁeld as Z d 0 x0 0 φ(x0, xi) = c d~x 2 0 2 d φ0(xi). (2.51) (x0 + |~x − ~x | )

For x0 → 0, Z 0 ∂φ d−1 0 φ0(x ) d+1 ∼ dcx0 d~x 0 2d + O(x0 ). (2.52) ∂x0 |~x − ~x | Chapter 2. Background and Brief Review 21

The action I(φ) can be described as a surface integral Z √ ~ I(φ) = lim→0 d~x hφ(~n · ∇)φ, (2.53) T

where T is the surface with x0 = , h is the induced metric, and ~n is a unit normal vector to T. Using the boundary condition φ → φ0 for x0 → 0, cd Z φ (~x)φ (~x0) I(φ) = d~xd~x0 0 0 , (2.54) 2 |~x − ~x0|2d

and the two point function of operator O is

hO(~x)O(~x0)i ∼ |~x − ~x0|−2d, (2.55)

the same as for a ﬁeld O with conformal dimension d. The case for massive scalar ﬁeld is more complicated, and the conformal dimension is √ d + d2 + 4m2 ∆ = . (2.56) 2

Wilson loops A important object in Yang-Mills theory is the Wilson loop

h i H Ai W (C) = Tr Pe C , (2.57)

which is a path-ordered integral over a closed path C. The Wilson loop describes a process that a pair of quark-anti-quark created in vacuum and annihilate again at somewhere else. In the N = 4 theory, with the scalars added, the form of Wilson loop transforms into

√ h H [iA x˙ µ+n·φ x˙ 2]dτ i W (C) = Tr P e C µ , (2.58)

where the integral is in the Euclidean spacetime. The modiﬁed Wilson loop is called Maldacena-Wilson loop. On the other side, the expectation value of Wilson loop in string theory is given by the string partition function with a string worldsheet ending along the closed loop on boundary[55, 56]. At λ 1, it is well described by the classical supergravity approximation, while the expectation value of Maldacena-Wilson loop is amounted by the area of the worldsheet.

Conﬁnement/deconﬁnement transition From equation (2.42), one needs to count in all the saddle point solutions of the supergravity, so it might be possible that in some regimes of parameters, one solution dominates, and in some other regimes, one other solution dominates, thus a transition happens. A Chapter 2. Background and Brief Review 22

simple example is shown by Witten, with a transition between AdS black hole and thermal AdS[8, 9]. As discussed above, the expectation value of Wilson loop can be calculated from either super Yang-Mills theory or string theory, thus

hW (C)i ∼ e−βF (2.59) Z ∼ dµ e−α(D), (2.60) D where β is the inverse temperature, D is the space of string world sheet obeying the boundary conditions, dµ is the measure of the worldsheet path integral, and α(D) is the regularized area. If a system is confined, its free energy F at infinity should go to infinity, F → ∞, and the expectation value of Wilson loop vanishes, hW (C)i = 0. In the string world sheet point of view, if the closed loop C is not a boundary of some manifold, the expectation value of Wilson loop also vanishes, and the system is confined. In the Euclidean AdS at finite temperature, its topology is R4 ×S1, or B4 ×S1 including 4 3 1 the point at infinity of R , one may name it X1. The boundary of X1 is S × S , and it is 3 2 also the boundary of S × B , labeled as X2. One can also see that a Euclidean AdS black hole has the topology X2, which might remind us the Hawking-Page transition, where a thermal AdS is preferred at low temperature and the AdS black hole solution dominates at high temperature. If we do not consider any other bulk solution with the same boundary, the CFT on the boundary is dual to the gravity theory with contribution from both X1 1 3 1 and X2. If one set the closed path C temporal, that is along S in the boundary S × S , it behaves different in the two bulk manifolds. In X1, C is not a boundary, thus hW (C)i = 0, 2 the system is confined. And in X2, C is the boundary of B , hW (C)i 6= 0, the system is deconfined. The calculation of Euclidean action in IIB string theory shows that the Hawking-Page transition also happens here.

Generally, a correspondence could be valid between d dimensional conformal field theory and a bulk theory on AdSd+1 ×K where K is a compact manifold. The stress-energy tensor in CFT indicates the existence of spin-2 particle in the bulk theory, and the conservation of stress-energy tensor implies the diffeomorphism invariance associated with the spin-2 particle, i.e. the graviton. There are also some discussions of the non-AdS/non-CFT correspondence [57], therefore the general bulk-boundary correspondence means that a field theory in d dimensions is dual to string theory in the bulk with the d dimensional boundary.

2.3 Examples of Instability and Phase Transition

In this section, we list and review brieﬂy some of the examples of instability and phase transition in string theory and the relevant context. Chapter 2. Background and Brief Review 23

2.3.1 Coleman-De Luccia Decay A series of Coleman’s papers discussed decay of false vacuum, [58–60]. For simplicity, one can consider a potential U(φ) with two minima, (φ+, φ−), with U(φ+) > U(φ−). In the classical theory, the two minima are both stable, but in quantum theory, U(φ−) is a true vacuum, and U(φ+) is a false vacuum. Considering a Euclidean action

Z 1 S = d4x (∂ φ)2 + U(φ) , (2.61) E 2 µ there are two standard saddle point solutions with action SE(φ+) and SE(φ−), and a speciﬁc “bounce” solution as well, with Euclidean action SE(φ). The decay/tunnelling rate is given by −B/ Γ/V = Ae ~[1 + O(~)] (2.62) where B is calculated in [58] and A in [59]. They showed that

B = SE(φ) − SE(φ+), (2.63) which is consistent with Hawking-Page’s claim[61]. Consider an O(4) symmetric system, the Euclidean action can be simpliﬁed,

Z ∞ 2 3 1 0 2 SE = 2π ρ dρ (φ ) + U(φ) , (2.64) 0 2 One can do an explicit approximation in the limit of small energy-density diﬀerence between the two vacua. Deﬁne = U(φ+) − U(φ−), (2.65) and write U(φ) = U0(φ) + O(), (2.66) where U0(φ+) = U0(φ−). One can solve that µ 1 φ = √ tanh[ µ(ρ − ρ¯)], (2.67) λ 2 with 1 U = λ(φ2 − µ2/λ)2. (2.68) 0 8

Assume that ρµ 1, it looks like a thin wall ball, with true vacuum, φ = φ− inside, in a sea of false vacuum φ = φ+. In the thin-wall approximation, one can ﬁnd a stationary solution atρ ¯0 = 3S1/ with Z S1 = 2 dρ [U0(φ) − U0(φ+)] , (2.69) and 2 4 3 B0 = 27π S1 /2 . (2.70) Chapter 2. Background and Brief Review 24

Returning back to the Lorentzian spacetime,

ρ → (|~x|2 − t2)1/2, (2.71)

the stationary wall ρ =ρ ¯0 in Euclidean space means an expanding bubble in Lorentzian space. The decay rate mentioned above can be understood as the probability of forming a true vacuum bubble from the false vacuum sea. One point needs to be emphasized is that only the bounce solution with one and with only one negative eigenvalue can be described as a decay process. Now let us include the gravity in our discussion[60], if we still assume the spherical symmetry, the metric can be written

ds2 = dξ2 + ρ(ξ)2dΩ2. (2.72)

And the Euclidean action is changed into

Z 1 3 S = 2π2 dξ ρ3( φ02 + U) + (ρ2ρ00 + ρρ02 − ρ) E 2 κ Z 3ρ = 4π2 dξ ρ3U − . (2.73) κ

Without gravity, the only relevant aspect of the potential is the difference = U(φ+) − U(φ−), however, with gravity included, there are some differences, which we will show below. Let us consider two specific cases, one is

U(φ+) = , U(φ−) = 0. (2.74)

The stationary solution is now at

ρ¯0 ρ¯ = 2 , (2.75) 1 + (¯ρ0/2Λ) and B0 B = 2 2 . (2.76) [1 + (¯ρ0/2Λ) ] While in another case, we choose

U(φ+) = 0,U(φ−) = −. (2.77)

The new stationary solution is at

ρ¯0 ρ¯ = 2 , (2.78) 1 − (¯ρ0/2Λ) and B0 B = 2 2 . (2.79) [1 − (¯ρ0/2Λ) ] Chapter 2. Background and Brief Review 25

The solutions look not so different, with only slight change resulted from gravity, however one can see the picture is very different for some values of parameters. In the first case, when bothρ ¯ and B are reduced, the bubble is smaller than before and the decay rate is enhanced. In the second case, it is the opposite situation, i.e. the bubble is larger and it is more difficult to decay from the false vacuum to true vacuum. With the potential difference small enough, the size of bubble goes to infinity, and actually no decay can happen. One can understand that fact from another point of view. The energy is conserved in all these decay processes. Without gravity, the energy is sum of a negative volume term and a positive surface term, and totally the energy 4π 4π E = − ρ¯3 + 4πS ρ¯2 = ρ¯2(¯ρ − ρ¯) (2.80) 3 1 3 0 is zero at the stationary sizeρ ¯ =ρ ¯0. If the size increases more thanρ ¯0, the energy turns to be negative. Now with gravity turned on and with negative energy density in the true vacuum, it changes the geometry inside the bubble, and one can calculate the energy contribution from gravity 5 2 Egrav = πρ¯0/3Λ > 0, (2.81) Thus the stationary size of bubble is larger than the one without gravity, to make the total energy zero. Note thatρ ¯0 = 3S1/, if is small enough,ρ ¯0 goes to large, and it is possible that one cannot find a solution forρ ¯, even the bubble is very large, to make the total energy vanish. This explains the behavior of the second specific case.

2.3.2 Witten’s Bubble Witten showed that asymptotically ﬂat Klein-Kaluza space is unstable due to a process of semiclassical decay. The underlying reason is the argument that the positive energy conjecture does not hold for a Klein-Kaluza space. The decay can be described by a black hole solution, after double analytic continuations[62]. Let us have a brief review here. One can start from a 5 dimensional Schwarzschild black hole solution,

2! 2 2 R 2 dr 2 2 ds = − 1 − dt + + r dΩ3 (2.82) r R 2 1 − r 2! 2 R 2 dr 2 2 = 1 − dφ + + r dΩ3, (2.83) r R 2 1 − r where we introduce Euclidean time φ = it in the second line, and so φ ∼ φ + β is periodic, and here β is the inverse temperature of black hole. Since

2 2 2 2 dΩ3 = dθ + sin θdΩ2, (2.84) Chapter 2. Background and Brief Review 26

1 one can make an analytic continuation, θ → 2 π +iψ, and the metric looks Lorentzian again

2! 2 2 R 2 dr 2 2 2 2 2 ds = 1 − dφ + − r dψ + r cosh ψdΩ2. (2.85) r R 2 1 − r

If we drop the factor (1 − (R/r)2), the (r, ψ) coordinates are the 2d Minkowski space

ds2 = dr2 − r2dψ2 = dx2 − dt2, (2.86) with x = r cosh ψ, t = r sinh ψ. (2.87)

Now we have a constraint, since r > R for the original black hole solution, thus

x2 − t2 = r2 > R2, (2.88) which gives an accelerating expanding bubble, and at late time it expands at the speed of light.

2.3.3 Other Spacetime Transitions There are some other spacetime transitions which we are interested in.

Hawking-Page Transition and AdS Bubbles

In [61], the authors discussed some spacetime structures which are asymptotically AdS4 with finite temperature. One of them is the thermal AdS space. For a fixed temperature T ∼ 1/β > T0, they found two Schwarzschild black hole solutions, with different mass, which are not the same as in asymptotically flat space. However if the temperature is too small T < T0, such black hole solutions do not exist. The black hole with large mass is classically and locally stable with a positive specific heat, while the small mass solution is not, with a negative specific heat. The small mass solution tends to decay into thermal radiation or the large mass solution. And when T0 < T < T1, where at T1 the thermal AdS has the same free energy with large mass black hole, the large mass solution is unstable in quantum mechanical sense. It has a positive free energy compared with the thermal AdS, and thus a radiation to thermal states would reduce the free energy. The authors claimed that the tunneling probability is like

Γ = Ae−B, (2.89) where A is some determinant and B is the diﬀerence between the actions of the lower and higher mass solutions at the same temperature. Compare with Coleman-De Luccia decay, one might say the Euclidean small black hole solution is exactly the “bounce”. In the AdS5 space, there is a similar story[63]. At temperature large enough, two black hole solutions exist, and roughly speaking one is larger than the AdS radius and the other one is smaller. One can construct a “topological black hole” by identifying a global AdS5 space along a boost, which can be shown to be related to the thermal AdS space. As Chapter 2. Background and Brief Review 27

the Witten’s bubble in the asymptotically flat space, the AdS bubble of nothing, which is related to Euclidean Schwarzschild AdS black hole by analytic continuation, also mediates the decay of the topological black hole. One different point from the flat space is in the Lorentzian space, it is time-dependent, thus the energy is not conserved. However for the decay to happen, one finds that the nucleated bubble is accompanied by a bath of energy with the difference between the topological black hole and its decay product.

Closed String Tachyon Condensation In [64, 65], the authors argued that closed string tachyons could drive topology change. One can understand the argument in a simple way. Type II string theory can be compactiﬁed into 8 dimensions on a Riemann surface, with Euler character,

χ = 2 − 2h, (2.90)

where h is the number of handles. Each handle adds energy, and for a constant curvature Riemann surface, the 8 dimensional Einstein frame energy density is

2/3 1 gs U8E ∼ 8 2 (2h − 2), (2.91) l8 VΣ

with 8d Planck length l8, string coupling gs, and volume VΣ. If a Riemann surface decouples into two components, with number of handles h1 and h2 respectively, then h = h1 + h2. Then from the energy density formula above, one find the final geometries have less energy than the initial one, U8E > U1 + U2, thus this might indicate a gravitational decay. Let us consider small string coupling region, and to be controllable, we might need to restrict in surface with weak curvature everywhere. In Fig. 2.3, there is a long thin tube, with small curvature Rα0 1, that is the tube is near flat. Since it is of string scale, string theory may play an important role in the discussion. Let us consider a Scherk-Schwarz string winding around the thin tube, with antiperiodic condition around the circle, the world- 2 2 sheet vacuum energy in the nth twisted sector is −1/2 + n L , where Lls is the radius of Scherk-Schwarz circle. For sufficiently small L, the vacuum energy is negative, yielding tachyonic modes. The authors argued that the tachyons condense, and make a topology transition as discussed above.

Field Theory Transition In a series of work [66–68], weakly coupled, large NSU(N) Yang-Mills gauge theory was considered on S3×time. A confinement/deconfinement transition happens at some temperature. The low temperature states have free energy of order O(1), while the high temperature phase has O(N 2) free energy. A typical transition is first order transition, as shown in Fig. 2.4.

iβα Here we deﬁned the parameter u1 = tr(e ), with β = 1/kT the inverse temperature, Chapter 2. Background and Brief Review 28

Figure 2.3: A Riemann surface with a long thin tube.

Figure 2.4: In the plots, the vertical axis is free energy, and the horizontal axis is u1 = tr(eiβα). The temperature is increasing from left to right side ﬁgures. One can see at low temperature the u1 = 0 solution dominates, while at high temperature another solution is more important, with less free energy. and α is the average of A0 over the sphere S3, Z 1 0 α ≡ 3 A . (2.92) vol(S ) S3

inβα Similarly one can define the general un ≡ tr(e ), which together should give a complicated parameter space, and the related transition might be more difficult to analyze. It is interesting to see what is the dual gravity theory corresponding to this field theory transition. Naively, it might be Hawking-Page like transition, between AdS black hole and thermal AdS space. However one has to notice that the usual gauge theory in the AdS/CFT correspondence is N = 4 super Yang-Mills theory, not the pure YM theory, and there is more discussion in [66–68] and other further works. 29

Part II

Holographic Methods for Condensed Matter Physics 30 Chapter 3

Introduction to Holographic Condensed Matter Theory

After the appearance of the original the AdS/CFT correspondence conjecture[6–9], nu- merous extension works have been done. One interesting application is on holographic condensed matter theory. One of the advantages of the correspondence is that it is a strong-weak relation, i.e. the weakly-curved gravity can help calculate in the dual strongly coupled field theory, which usually has no good way to be analyzed. On the other hand, the correspondence provides a new view of nature: one can find gravity from condensed matter physics, or vice versa. The present situation is that physicists are more able to do experiments in condensed matter physics, rather than the highly curved geometries. In addition, various models attempt the explanation of complicated phenomena, while many types of behavior are still not well described theoretically. So we are at a stage of trying to understand the condensed matter physics holographically from the dual gravity theories. In this chapter, I give a brief review of the methodology of the holographical condensed matter theory 1. Typically critical points in condensed matter theory(CMT) have very special properties, such as vanishing energy gap, divergent coherence length, scaling invariance and critical exponents. The condensed matter theorists usually apply conformal field theory(CFT) to study these behavior. It hints that the AdS/CFT correspondence can be very useful in CMT. The figure 3.1 shows a typical quantum critical point in condensed matter theory. At the region where the energy gap is negligible compared to the temperature, the theory can be extended to finite temperature quantum critical points.

3.1 Operators and the Expectation Values

In a ﬁeld theory, the relevant operators can be introduced without destroying the UV ﬁxed points, while the irrelevant operators do. According to the AdS/CFT correspondence, the relevant operators keep the asymptotic AdS behavior, while the irrelevant ones take the theory out from the best understood AdS/CFT framework. For simplicity, let us restrict our discussion with only the relevant operators.

1Thanks to S. Sachdev[69], C. Herzog[70], S. Hartnoll[71], P. Kovtun[72], R. Myers and many others for their splendid works and good lectures. Chapter 3. Introduction to Holographic Condensed Matter Theory 31

Quantum critical point/region

Superfluid Insulator

Figure 3.1: Typical quantum critical points in condensed matter theory. x is a order parameter. The system is in superfluid or insulator state with smaller or larger values. The regime between the two curves has quantum critical properties, and it shrinks to a point at zero temperature. Similar complicated phase diagrams exist for other condensed matter systems, for example, the labels of the different states in the figure might be replaced by non-Fermi liquid, electrons mostly localized/delocalized, etc. Depending on the systems, the order of the phase transitions varies from first to second order.

One familiar operator is the energy-momentum tensor δS T µν = , (3.1) δg(0)µν where g(0)µν is the boundary value of the bulk metric. If the metric is perturbed, with 0 the perturbation of the boundary value g(0) → g(0) = g(0) + δg(0), the action is changed by R d √ µν δS = d x −g(0)δg(0)µνT , and the partition function Z 0 D d p µν E Zbulk[g(0)] = exp iS + i d x −g(0)δg(0)µνT . (3.2) F.T.

Another example is the bulk Maxwell ﬁeld. A current J µ can be introduced by a source of a background electromagnetic ﬁeld, δS J µ = . (3.3) δA(0)µ

Here A(0)µ is the perturbation of the boundary value of U(1) electromagnetic ﬁeld, with R d √ µ the change of the action δS = d x −g(0)δA(0)µJ . Now we have the correspondence between the operators {J µ,T µν} and the ﬁelds in the bulk spacetime, {Aµ, gµν}. In general, one can introduce such correspondence for any Chapter 3. Introduction to Holographic Condensed Matter Theory 32

relevant operator O,

operator O(in ﬁeld theory) ←→ dynamical ﬁeld φ(in bulk), (3.4) with the partition function, Z D d p E Zbulk[φ → δφ(0)] = exp iS + i d x −g(0)δφ(0)O . (3.5) F.T.

For simplicity, assume the operator O is a Lorentz scalar, then the spacetime is expected to be dr2 h(r)(−dt2 + d~x2) ds2 = L2 + , (3.6) r2 r2 which has a Lorentz boundary spacetime. Working with the minimal Einstein-scalar action, the general behavior of the scalar ﬁeld near the boundary is

r d−∆ r ∆ φ(r) = φ + φ + ··· , as r → 0, (3.7) L (0) L (1) with ∆ the solution of

(Lm)2 = ∆(∆ − d), (3.8)

where m is the mass of the scalar. With the partition function (3.5), the expectation value of the scalar operator O can be calculated,

δZ [φ ] δS[φ ] hOi = −i bulk (0) N → ∞ (0) , (3.9) δφ(0) −−−−−→ δφ(0)

iS where the last equality is taken in the semiclassical limit N → ∞, Zbulk = e . With the boundary term of the action included, the momentum is introduced, δS[φ(0)] δS[φ(0)] δSbdy[φ(0)] = lim − + ≡ lim Π[φ(0)]. (3.10) δφ(0) r→0 δ∂rφ(0) δφ(0) r→0

∆−d To be consistent with equation (3.7), here we deﬁne ∂rφ(0) = ∂r[(r/L) φ]. The boundary term is necessary to assure the action is on shell. Furthermore, counter-terms might be also required to cancel the possible divergences. If φ(0) is the slower falloﬀ, i.e. d − ∆ < ∆, the boundary action should be added as Z ∆ − d d √ 2 Sbdy. = d x γφ . (3.11) 2L r→0

Else if φ(0) is the faster falloﬀ, i.e. d/2 ≥ ∆ ≥ (d − 2)/2, where the other restriction comes Chapter 3. Introduction to Holographic Condensed Matter Theory 33

from CFT unitary bound, the boundary action should be Z d √ µ ∆ 2 Sbdy. = − d x γ φn ∇µφ + φ . (3.12) r→0 2L Substituting the φ(r) behavior (3.7) back into (3.9,3.10), one ﬁnds the relation

2∆ − d hOi = φ . (3.13) L (1)

And it is concluded the “non-normalizable” falloff φ(0) is the source and the “normalizable” falloff φ(1) gives the expectation value. Similar results exist for metric and Maxwell field. Let us write down explicitly here the Maxwell example. The chemical potential µ = At(0) is the source for the charge density t J . One can introduce an effective scalar φ = rAt. For the charge density, the conformal dimension is found ∆ = d − 1, and the expectation value is

µ(d − 2)Ld−3 hJ µi = . (3.14) 2 d−2 g r+

3.2 Near Equilibrium Dynamics

The analysis above is in equilibrium, and now let us consider some perturbation around it. One basic structure we can study is the retarded Green’s function, which relates expectation values to the corresponding sources. In the coupling between operators OA and OB, the retarded Green’s function is in the form,

δhO i(ω, k) = GR (ω, k) δφ (ω, k). (3.15) A OAOB B(0)

In the dual geometry, suppose the scalar φA(r) has the boundary value φA(0). And the perturbation can be turned on

−iωt+ik·x φA(r) → φA(r) + δφA(r)e . (3.16)

Asymptotically the perturbation has the form similar to 3.7. In addition, the perturbation needs to satisfy the infalling condition near horizon, which is diﬀerent at ﬁnite temperature 2 1 and at zero temperature. If the temperature vanishes, gtt ∼ (r − r+) instead of (r − r+) , and the near horizon metric appears to be AdS2. The emergence of near horizon AdS2 is crucial in the condensation since it provides a higher BF-bound than the ordinary AdSd+1. The infalling boundary condition breaks the time reversal symmetry, and makes us only able to calculate the retarded Green’s function, but not the advanced one. Chapter 3. Introduction to Holographic Condensed Matter Theory 34

Using the discussion above, one ﬁnds

δhO i δΠ 2∆ − d δφ GR = A = lim A = A A(1) . (3.17) OAOB δφB(0) r→0 δφB(0) L δφB(0) δφ=0 δφ=0

Then the retarded Green’s function is exactly how the “normalizable” falloﬀ δφA(1) depends on the “non-normalizable” falloﬀ δφB(0). In the above argument, the Green’s function is considered as the ratio of a one point function(the expectation value) and the source, while some early work studied it as a two point function[73–75].

Conductivities I will show the calculation of the electrical conductivity ﬁrst, and then give a little extension to thermal one. For simplicity I set k = 0 here. Suppose the temperature is the same everywhere, and one needs only consider the perturbation of gauge ﬁeld and thus the electrical conductivity. Ohm’s law is given as

hJ~i = σE.~ (3.18)

iωt From the perturbation’s form of gauge ﬁeld, δAj ∝ e , one has

Ej = iωδAj(0). (3.19)

The resulted change of action is, supposing only components along x direction are not vanishing, Z d−1 p x δS = d xdt −g(0)(J δAx(0)). (3.20)

The conductivity is then

hJ i hJ i −iG σ = x = x = JxJx . (3.21) Ex iωδAx ω

In the AdS/CFT methodology, one needs to solve the equations of perturbations δAx and δgtx in the bulk. Fortunately at zero momentum k = 0, the perturbations do not source any other ﬁelds. In the background of a Reissner-Nordstrom-AdS black hole,

L2 dr2 ds2 = −f(r)dt2 + + dxidxi , (3.22) r2 f(r) Chapter 3. Introduction to Holographic Condensed Matter Theory 35 where

2 2 d 2 2(d−1) r+µ r r+ r f(r) = 1 − 1 + 2 + 2 , (3.23) γ r+ γ r+ 2 2 " d−2)# 2 (d − 1)g L r γ = 2 ,At = µ 1 − , (3.24) (d − 2)κ r+ the linearized equations are

2 4L2 δg0 + δg + A0 δA = 0, tx r tx γ2 t x ω2 r2A0 2 (fδA0 )0 + δA + t δg0 + δg = 0. (3.25) x f x L2 tx r tx

The equation of Ax can be decoupled from the above equations,

2 2 2 0 0 ω 4µ r (fδAx) + δAx − 2 2 δAx = 0. (3.26) f γ r+ It has a near boundary behavior, r δA = δA + δA + ··· , as r → 0. (3.27) x x(0) L x(1)

It is clear from above equation that Ax(1) depends linearly on Ax(0). Compare equations (3.21) with equation (3.17), and using the form of δAx (3.27), one can write explicitly the conductivities,

−i δAx(1) σ(ω) = 2 (3.28) g Lω δAx(0)

iωt From the discussion above, assuming Ex ∼ e , the near boundary behavior of Ax can also be written as E A = x + g2J xr + ··· , (3.29) x iω with J x σxx = . (3.30) Ex The thermal conductivity can also be derived in similar way. Combined with thermal eﬀects, Ohm’s law can be extended into

hJ i σ αT E x = x . hQxi αT κT¯ −(∇xT )/T

The heat current is introduced as Qx = Ttx − µJx. Now there are three conductivities: Chapter 3. Introduction to Holographic Condensed Matter Theory 36

electrical (σ), thermal (¯κ), and thermoelectric (α). α andκ ¯ can be found similarly,

−iGR (ω) iρ α(ω)T = QxJx = − µσ(ω),, ω ω −iGR (ω) i( + P − 2µρ) κT¯ = QxQx = + µ2σ(ω), (3.31) ω ω

2 2 2 2 2 3 where the energy density is introduced, = −L (1 + r+µ /γ )/(κ r+), while ρ and P are density and pressure respectively.

The London Equation The London equation appears in these holographic models as ~ ~ J(ω, k) = −nsA(ω, k), (3.32)

which is expected to explain the infinite DC conductivity and the Meissner effect of the superconductors. In the limit ~k = 0 and ω → 0, the equation above can be interpreted in another form, in J(ω, 0) = s E(ω, 0), (3.33) ω −iωt with the assumption Ex ∼ e . It explains the appearance of the infinity of DC conductivity. In the other limit ω = 0 and ~k → 0, the curl of London equation yields

∇ × J(~x) = −nsB(~x). (3.34)

Together with the Maxwell’s equation ∇ × B = 4πJ, the limit of London equation implies the magnetic ﬁeld is excluded from superconductors,

2 − ∇ B = ∇ × (∇ × B) = 4π∇ × J = −4πnsB, (3.35) which has the exponentially damping solution. The penetration length square of magnetic 2 ﬁeld is λ = 1/4πns, proportional to the inverse of superﬂuid density ns.

3.3 Holographic Hydrodynamics

This section will describe the very basic idea of holographic hydrodynamics, and some discussion of the ratio η/s. Consider a theory of pure gravity with negative cosmological constant in 5 dimensions, besides the well-known AdS5, there is another solution called “boosted black branes”,

2 µ 2 µ ν 2 µ ν ds = −2uµdx dr − r f(br)uµuνdx dx + r Pµνdx dx , (3.36) Chapter 3. Introduction to Holographic Condensed Matter Theory 37

with 1 1 β f(r) = 1 − , ut = , ui = i . (3.37) r4 p1 − β2 p1 − β2

1 2 j The temperature is T = πb and velocities βi are all constants with β = βjβ , and P µν = uµuν + ηµν (3.38)

is the projection onto spatial directions. Now if we replace the constants (b, βi) by slowly µ µ varying functions (b(x ), βix ) of the boundary conditions, the metric turns to be

2 α µ 2 α α α µ ν 2 α µ ν ds = −2uµ(x )dx dr − r f(b(x )r)uµ(x )uν(x )dx dx + r Pµν(x )dx dx . (3.39)

Although it is generally not the solution of Einstein equations, the slowly varying functions give us a good approximation of the solutions. The Einstein equation can be expanded order by order, and according to the AdS/CFT correspondence, one can ﬁnd the expansion of the energy-momentum tensor in the dual ﬁeld theory[76, 77],

T µν = (πT )4(ηµν + 4uµuν) − 2(πT )3σµν ! 2 µν µν 1 µν µν µν +(πT ) (ln2)T2a + 2T2b + (2 − ln2) 3 T2c + T2d + T2e , (3.40)

where 1 σµν = P µαP νβ∂ u − P µν∂ uα, (α β) 3 α µν αβγ(µ ν) T2a = σ γuαlβ, 1 T µν = σµασν − P µνσαβσ , 2b α 3 αβ µν α µν T2c = ∂αu σ , (3.41) 1 T µν = DuµDuν − P µνDuαDu , 2d 3 α 1 T µν = P µαP νβD(∂ u ) − P µνP αβD(∂ u ), 2e (α β) 3 α β α β γ lmu = αβγµu ∂ u .

Here we used the convention of

0123 α 0123 = − = 1, D = u ∂α, (3.42) and the brackets () of the indices mean symmetrization, i.e. a(αbβ) = (aαbβ + aβbα)/2. Apparently the first terms of (3.40) describes a perfect fluid with pressure π4T 4, and so the entropy density s = 4π4T 3. The viscosity η of this is read from the coefficient of σµν, Chapter 3. Introduction to Holographic Condensed Matter Theory 38

η = π3T 3, and thus one gets the ratio η 1 = , (3.43) s 4π which is consistent with [78], and we will discuss the ratio in more details below. It can be checked that the stress tensor (3.40) transforms homogeneously under Weyl transformations. For example, under a Weyl transform of the boundary metric,

2φ µ −φ µ −φ ˜ gµν = e g˜µν, u = e u˜ ,T = e T, (3.44) the stress tensor transforms as T µν = e−6φT˜µν. (3.45) Consider a static bath of homogeneous fluid at temperature T , one can solve for the spectrum of small oscillations of fluid dynamics. Let us introduce the fluctuations as the form

j j iωv+ikj x βi(v, x ) = δβie , j T (v, xj) = 1 + δT eiωv+ikj x , (3.46) two modes of spectrum can be found,

k ik2 3 − ln4 sound mode : ω(k) = ±√ + ± √ k3 + O(k4), 3 6 24 3 ik2 i shear mode : ω(k) = + (2 − ln2)k4 + O(k6), (3.47) 4 32 where (ω, k) are rescaled quantities,

ω = ω/πT, k = k/πT. (3.48)

The Shear Viscosity and Entropy Density Ratio η/s There are variants methods to calculate the shear viscosity and entropy density ration η/s, some of which are listed below: 1. Using the relation between graviton’s absorption cross section and the imaginary part of the retarded Green’s function for Txy[79]; 2. Via direct AdS/CFT calculation of the correlation function in Kubo’s formula. [80]; 3. From membrane paradigm calculation [81]. In this simple case, via membrane paradigm way, one ﬁnds that η 1 = . (3.49) s 4π Chapter 3. Introduction to Holographic Condensed Matter Theory 39

And in N = 4 SYM theory, to the next order in the inverse ’t Hooft coupling expansion,[82]

η 1 135ζ(3) = 1 + , (3.50) s 4π 8(g2N)3/2

in the limit of g2N → 0, η/s → ∞. From the result above, one might conjecture a universal bound that η 1 ≥ . (3.51) s 4π However it might be not right, as discussed in [83]. The authors consider a generalized gravity action, Z 1 5 √ 2 2 µν µνρσ I = d x −g R − 2Λ + L (α1R + α2RµνR + α3R Rµνρσ) , (3.52) 16πGN

2 0 2 where Λ = −6/L . Assuming αi ∼ α /L 1, one ﬁnds that η 1 = (1 − 8α ) + O(α2), (3.53) s 4π 3 i

which is independent on α1, α2. Taking a speciﬁc form as the Gauss-Bonnet gravity action Z 1 5 √ λGB 2 2 µν µνρσ I = d x −g R − 2Λ + L (R − 4RµνR + R Rµνρσ) , (3.54) 16πGN 2 it turns out η 1 = (1 − 4λ ), (3.55) s 4π GB

λGB is bounded above by 1/4, and η/s > 0. However the conjectured bound η/s ≥ 1/4π is 9 violated when λGB > 0. A numerical analysis shows that when λGB > 100 , the local speed of graviton is larger than expected speed of light, and the causality is violated. However 9 the regime 0 < λGB < 100 is not clear. 40 Chapter 4

The Appearance of Fermi Surface

There are various phenomena in condensed matter theory which have been explored holographically with the AdS/CFT correspondence. The most worked out example might be the holographic superfluid 1 and holographic superconductivity[1, 13, 84], which we will discuss in the next chapter. Besides, Fermi-surface is also found in the AdS/CFT context[14]. We will describe the existence of Fermi surface at zero temperature briefly in the first section, and show that it is a non-Fermi liquid, which is attractive since it’s believed that the normal states of high-Tc superconductors, and metals close to quantum critical points are non-Fermi liquids[85, 86]. Then we add external magnetic field into the system, and find a Landau level structure. The global spacetime calculation has an extra parameter and seems to provide the exact Fermi liquid. In the end of this chapter, finite temperature case is discussed, while the system has a thermal instability and the poles run into the low half of complex ω plane.

4.1 The Fermi Surface at Zero Temperature

Set-up

The action of a electric ﬁeld in AdS4 geometry is

Z 2 1 4 √ 6 R MN S = 2 d x −g R − 2 − 2 FMN F , (4.1) 2κ R gF

2 where gF is an eﬀective dimensionless gauge coupling, and R the radius of AdS4. The metric of the charged RN black hole is

r2 R2dr2 ds2 = (−fdt2 + dx2) + , (4.2) R2 i r2f with

2 Q M r0 gF Q f = 1 + 4 − 3 ,A0 = µ 1 − , µ ≡ 2 , (4.3) r r r R r0 where r0 is the horizon radius of the black hole, and µ is the chemical potential in boundary

1Some works are [72, 78, 79], and one of the reviews is [70]. Chapter 4. The Appearance of Fermi Surface 41

theory. With the rescaling,

2 R r0 3 2 r → r0r, (t, ~x) → (t, ~x),A0 → 2 A0,M → Mr0,Q → Qr0, (4.4) r0 R the metric becomes ds2 1 dr2 ≡ g dxM dxN = r2(−fdt2 + d~x2) + , (4.5) R2 MN r2 f with the new forms of the functions,

Q2 1 + Q2 1 f = 1 + − ,A = µ 1 − , µ = g Q, (4.6) r4 r3 0 r F

and the horizon is at r = 1. The dimensionless temperature is 1 T = (3 − Q2). (4.7) 4π √ 2 At zero temperature, Q = 3, and the near horizon limit is AdS2 × R with the radius of AdS2, R R2 = √ . (4.8) 6 We want to explore the spectral function of a fermion operator O, so turn on the perturbation of Dirac ﬁeld in bulk, Z √ d+1 ¯ M ¯ Sspinor = d x −gi(ψΓ DM ψ − mψψ), (4.9)

where 1 ψ¯ = ψ+Γt, D = ∂ + ω Γab − iqA , (4.10) M M 4 abM M

ab 1 a b with ωabM the spin connection, and Γ = 2 [Γ , Γ ]. One can choose appropriate Γ matrices, and here we use

1 0 0 γµ Γr = , Γµ = , 0 −1 γµ 0 where µ = {t, ~x}, and γµ are gamma matrices in (2 + 1) dimensional theory. The Dirac ﬁeld can be decomposed as ψ ψ = + , ψ− with each of ψ± a two-component spinor. Assuming the solution is translational invariant Chapter 4. The Appearance of Fermi Surface 42

along ~x, it can be written in the form below,

i rr −1/4 −iωt+ikix ψ± = (−g g ) e φ±, (4.11)

the Dirac equation becomes r gii √ µ ∓ (∂r ∓ m grr)φpm = ∓iKµγ φ , (4.12) grr with r gii 1 Kµ(r) = (−u(r), ki), u = (ω + µq(1 − )). (4.13) −gtt r

At r → ∞, u → ω + µq, which means ω is the diﬀerence of energy from the chemical potential, i.e. ω = 0 is the Fermi energy. The method is basically to impose the infalling wave condition near horizon, and solve for the asymptotic behavior, identifying the source and expectation value respectively. From the Dirac equation, the asymptotic form of ψ± can be found easily,

m −m−1 m−1 −m φ+ = Ar + Br , φ− = Cr + Dr , (4.14)

with iγµk iγµk C = µ A, B = µ D, k = (−(ω + µ ), k ). (4.15) 2m − 1 2m + 1 µ q i It can be proven that the coeﬃcients D corresponds to expectation value while A is the source. Suppose they have the relation D = SA, then the retarded Green’s function can be written as[87] 0 GR = −iSγ . (4.16) Choosing the basis of gamma matrices,

0 1 2 γ = iσ2, γ = σ1, γ = σ3, (4.17)

and set k2 = 0 for simplicity, the Dirac equation can transform to the equations, r gii √ (∂r ∓ m grr)y± = ∓i(k1 − u)z∓, grr r gii √ (∂r ± m grr)z∓ = ±i(k1 + u)y±, (4.18) grr where we wrote y± φ± = . z± Introducing the ratios, iy− iz− ξ+ = , ξ− = − , (4.19) z+ y+ Chapter 4. The Appearance of Fermi Surface 43 the retarded Green’s function becomes −2m ξ+ 0 GR = lim . →0 0 ξ− 1 r=

Then the equations (4.18) can be written in terms of ξ±, r gii √ 2 ∂rξ± = −2m giiξ± ∓ (k1 ∓ u) ± (k1 ± u)ξ±. (4.20) grr The infalling wave condition turns to be

ξ±|r=1 = i. (4.21)

The equations (4.20) are nonlinear, and cannot be treated as Schrodinger’s form directly. Before going to the numerical result, let us list some properties of the retarded Green’s function ﬁrst. It has some symmetries of (k, q, ω)

∗ G22(ω, k) = G11(ω, −k),G11(ω, k; −q) = −G22(−ω, k; q). (4.22)

At m = 0, 1 G22(ω, k) = − ,G11(ω, k = 0) = G22(ω, k = 0) = i. (4.23) G11(ω, k) For numerical calculation, we set m = 0, q = 1, and the imaginary parts of the retarded Green’s function’s components are plotted in Fig. 4.1. From the symmetries of GR discussed above, one only needs to show the momentum space k > 0. From the ﬁgure, ImG11 is pretty smooth everywhere, while ImG22 has a sharp peak around (ω = 0, k = kF ≈ 0.92). At k > kF , there is a hump with linear relation of (ω, k). Since the peak is at ω = 0, we say it is a Fermi surface. The Green’s function has a quasi-particle-hole peak at ω < 0 with k < kF , as in Fig. 4.1. Meanwhile the existence of hump is consistent with Landau’s Fermi liquid discussion as well. Denoting the position of the quasi-particle peak as ω∗(k⊥) with k⊥ = k − kF < 0, it − scales with k⊥ → 0 , z ω∗(k⊥) ∼ k⊥, z = 2.09 ± 0.01, (4.24) and the height of ImG22 at the maximum scales as

−α ImG22(ω∗(k⊥), k⊥) ∼ k⊥ , α = 1.00 ± 0.01. (4.25) These scaling strongly suggest that the Green’s function has the scaling form

z −α ImG22(λ ω, λk⊥) = λ ImG22(ω, k⊥), (4.26) with z = 2.09 ± 0.01, α = 1.00 ± 0.01. (4.27) Chapter 4. The Appearance of Fermi Surface 44

Figure 4.1: The imaginary parts of the retarded Green’s function’s components. ImG11 is pretty smooth, while ImG22 has a sharp peak around (ω = 0, k ≈ 0.92). The plots are taken from [14].

Re@G22D, Im@G22D 300

200

100

0 Ω

-100

-200 -0.002 -0.001 0.000 0.001 0.002

Figure 4.2: ReG22(ω)(blue) and ImG22(ω)(orange) at k = 0.9 < kF . There is a quasi- particle peak at ω < 0. The plots are taken from [14]. Chapter 4. The Appearance of Fermi Surface 45

Note that Landau Fermi liquid has the exponents z = α = 1, thus what we ﬁnd here is a non-Fermi liquid, one example of the more general discussed solutions[88, 89]. The particle(√ k⊥ > 0)-hole(k⊥ < 0) asymmetry indicates the non-Fermi liquid as well. Since µq = 3 > kF , the system is with a repulsive interaction.

Figure 4.3: ImG11(left) and ImG22(right) on the parameter space (k, q), at ω = −0.001 and temperature T = 2.76 × 10−6. The Green’s function has a periodic property along q. The bumps at low momentum k turns into poles when k is large. The plots are taken from [14].

Another interesting property of the Green’s function is that it is somehow periodic along q direction, which one might explain as multi-particles system. As shown in Fig. 4.1, at very small negative frequency ω = −0.001 and low temperature T = 2.76 × 10−6, the Green’s function components keep as bumps at small momentum k, and the bumps turn into poles at large k. The transition depends on the speed of light in the infra spacetime, discussed more generally in [4, 90].

4.2 Landau Levels with External Magnetic Field

It is interesting to explore more properties of the holographic (non-)Fermi liquid. In this section2 we extend such studies to the case of non-zero magnetic fields. The study of a fermionic system in a non-zero magnetic field is by itself a vast and important subject. In this preliminary study, we are only able to explore a small part of it. We concentrate on an extremal dyonic AdS black hole geometry. Dyonic black holes are both magnetically and electrically charged and correspond to a phase of 2 + 1 dimensional boundary CFT with a non-zero chemical potential and a magnetic field. Following [14, 91] we introduce a probe fermion in this geometry to study the properties of the resulting Fermi surface.

2This section is based on our work [2]. Chapter 4. The Appearance of Fermi Surface 46

We find that the introduction of magnetic field changes (decreases) the effective charge of the probe fermion. This observation enables us to address the question of the change in the nature of the Fermi surface in presence of a non-zero magnetic field. For a positive fermionic mass, the Fermi surface gets dissolved as we increase the magnetic field. We find a discrete spectrum analogous to Landau levels and discuss phenomena similar to the de Haas-van Alphen effect.

4.2.1 Dyonic Black Hole The Maxwell-Einstein action with a negative cosmological constant in four dimensions is written as Z 2 1 4 √ 6 L MN S = 2 d x −g R − 2 − 2 FMN F , (4.28) κ L gF Here we consider the case of a dyonic black hole (i.e. with both electric and magnetic charges) in AdS4 space. The metric is given by[92]

ds2 dz2 1 = −f(z)dt2 + + (dx2 + dy2), (4.29) L2 z4f(z) z2 1 2 2 4 3 1 2 2 f(z) = 2 1 + (h + Q )z − z 3 + z+(h + Q ) (4.30) z z+ and the gauge ﬁelds are given by

Ay = Az = 0,At(z) = µ − Qz, Ax(y) = −hy, (4.31)

with µ = Qz+. The Hawking temperature in this background is

1 4 2 2 TH = 3 − z+(Q + h ) . (4.32) 4πz+

We may rescale the co-ordinates to set z+ = 1 from now on. We wish to study the system at zero temperature i.e., in the extremal limit. This gives us the following condition

h2 + Q2 = 3. (4.33)

Note that all dimensionful quantities are now measured in units of 1/z+. The relevant dimensionless parameter related to the strength of the magnetic ﬁeld is H = h/Q = √ h 3−h2 which goes from zero to inﬁnity.

4.2.2 Probe Fermion We will consider a bulk Dirac fermion ﬁeld Ψ as a probe to the system. The action for Ψ is given by 1 Z √ S = d4x −gi Ψ¯ eM ΓaD Ψ − mΨΨ¯ . (4.34) bulk κ2 a M Chapter 4. The Appearance of Fermi Surface 47

with the covariant derivative 1 D Ψ = ∂ Ψ + ωab[Γ , Γ ]Ψ − iqA Ψ . (4.35) M M 8 M a b M

M a Here e a is the inverse vielbein and the non-zero components of spin-connection ωM b are 1 ω t = −ω z = z2f 0, ω z = −ω x = pf = ω z = −ω y . (4.36) t z t t 2 x x x z y y y z The ﬁeld Ψ corresponds to a boundary fermionic operator O. The fermion charge q determines the charge of the operator O, while its dimension ∆ is determined by the mass m of Ψ according to the formula 3 ∆ = m + . (4.37) 2 The Dirac equation (/D + m)Ψ = 0 can be written in the form

(U(z) + V (y))Ψ = 0, (4.38)

where 1 f 0 m ∂ − iqA U(z) = Γz · zpf ∂ − + + + Γt · t √ t , z z 4f z z f x y V (y) = Γ · (∂x − iqAx(y)) + Γ · ∂y. (4.39)

If we now perform a Fourier transform along (t, x), then ∂t → −iω, ∂x → ikx. For Simplicity, we choose the Gamma matrices as follows

11 γµ Γz = , Γµ = , with γt = iσ , γx = σ , γy = σ . −11 γµ 3 1 2

Then we have D iσ C 0 σ C + σ ∂ U = + 3 t ,V = 1 y 2 y , iσ3Ct −D− σ1Cy + σ2∂y 0

where f 0 1 m D = zpf(∂ + C ),C = − ± √ , (4.40) ± z ± ± 4f z z2 f −iω − iqA C = √ t ,C = i(k + qhy). t z f y x One can easily check that the matrices U and V do not commute. However, it is possible to ﬁnd a constant matrix M such that [MU, MV ] = 0. In order to separate the variables, one can left multiply Eqn. (4.38) by M (see [93]). It turns out that a convenient choice for M is 0 σ M = 3 . −σ3 0 Chapter 4. The Appearance of Fermi Surface 48

Here iC −σ D σ (σ C + σ ∂ ) 0 MU = t 3 − ,MV = 3 1 y 2 y . −σ3D+ −iCt 0 −σ3(σ1Cy + σ2∂y)

As MU and MV are two commuting hermitian matrices, we may look for solutions of the eigenvalue equation of the form

MUΨ = −MV Ψ = LΨ. (4.41) with real L. Suppose Ψ Ψ = + , Ψ− At ﬁrst we will try to solve the y dependent part of the equation. It turns out that this part of the equation is identical to that of a massless free fermion in 2 + 1 dimensions

σ (σ C + σ ∂ )Ψ Ψ MV Ψ = 3 1 y 2 y + = −L + . −σ3(σ1Cy + σ2∂y)Ψ− Ψ−

For the Ψ+ part, let us left multiply σ3(σ1Cy + σ2∂y), then

2 [σ3(σ1Cy + σ2∂y)] [σ3(σ1Cy + σ2∂y)] Ψ+ = L Ψ+. (4.42)

After simpliﬁcation we get

2 2 2 0 ∂y Ψ+ + (L + Cy − iσ3Cy)Ψ+ = 0. (4.43)

T Suppose Ψ+ = (R1(z)S1(y),R2(z)S2(y)). Since the equation above does not depend on z, one can set R1 = R2, leading to

2 − ∂y S1,2 + P±(y)S1,2 = 0, (4.44) where

2 2 0 2 2 P±(y) = −(L + Cy ∓ iCy) = (kx + qhy) − L ∓ qh, (4.45) which is exactly√ a simple harmonic oscillator potential. To see this more explicitly, let us kx deﬁne η = qh(y + qh ). Then

L2 − ∂2S + η2S = ± 1 S . (4.46) η 1,2 1,2 qh 1,2

The eigenvalues and eigenvectors are

1 L2 1 E = ± 1 = n + , n = 0, 1, 2,... n 2 qh 2 −η2/2 S1,2 = Nne Hn(η) ≡ In(η), (4.47) Chapter 4. The Appearance of Fermi Surface 49

where Nn are normalized factors. Substituting back into the ﬁrst order equations for Ψ+ with the same eigenvalues we get In(η)R1 Ψ+ = , −iIn−1(η)R1

2 corresponding to eigenvalues Ln = 2nqh, n = 0, 1, 2, ··· . For completeness, we deﬁne I−1(η) = 0. Combining with the expression for Ψ− which is obtained similarly, we have   In(η)R1  −iIn−1(η)R1  Ψ =   . (4.48)  In(η)R2  iIn−1(η)R2

Another independent solution comes from the values corresponding to −Ln, i.e.

0 0 0 MUΨ = −MV Ψ = −LnΨ , (4.49)

with  ¯  −iInR1 ¯ 0  In−1R1  Ψ =  ¯  . (4.50)  −iInR2  ¯ −In−1R2 ¯ The equations for Ri and Ri are respectively,

D+R1 = −(iCt + Ln)R2,D−R2 = (iCt − Ln)R1, ¯ ¯ ¯ ¯ D+R1 = −(iCt − Ln)R2,D−R2 = (iCt + Ln)R1, (4.51)

Note that for n = 0,L0 = 0, and the solution is   I0(η)R1 0  0  Ψ = −iΨ =   ,  I0(η)R2  0

¯ with Ri = Ri. Thus in this case there is only one independent solution. Chapter 4. The Appearance of Fermi Surface 50

Green’s Function Let’s ﬁrst look at the general form of the solutions3 ! ! X iR(n)e(n) −iR¯(n)e(n) Ψ = Ψ(n), Ψ(n) = Ψ(Ln) + Ψ0(−Ln) = 1 1 + 1 2 , (n) (n) ¯(n) (n) n R2 e2 −R2 e1 i.e. (n) (n) (n) ¯(n) (n) (n) ¯(n) (n) (n) (n) Ψ+ = iR1 e1 − iR1 e2 , Ψ− = −R2 e1 + R2 e2 . (4.52)

For simplicity, we have introduced a set of two-component basis spinors (n) iIn(η) (n) In(η) e1 = , e2 = , n ≥ 1. In−1(η) iIn−1(η)

The orthonormality of the solutions to the y dependent equations gives

Z ∗ Z ∗ ∗ ∗ (n) (n0) (n) (n0) R (n) (n0) R (n) (n0) dη e1 e2 = dη e2 e1 = 0, dη e1 e1 = dη e2 e2 = 2δnn0 ,

(n) (n) (n) (n) iσ3e1 = −e2 , iσ3e2 = e1 . (4.53)

(n) If we express the spinors ei as 1 0 e(n) ⇒ (n) = , e(n) ⇒ (n) = , 1 1 0 2 2 1

0 1 then in this basis iσ = . 3 −1 0 We now have everything we need to calculate the Green’s function. Given the orthonormality property introduced above, we can as usual substitute the solutions into the boundary term in the action

Z √ 3 ¯ Sb = d x −g3Ψ+Ψ−. (4.54)

(n) (n) In the basis of (1 , 2 )

 R¯(n)  (n) ! (n) ! 2 ¯ 0 (n) (n) iR (n) −R R¯ (n) (n) Ψ = 1 , Ψ = 2 = −i  1  Ψ ≡ S(n)Ψ + (n) − (n) R(n) + + −iR¯ R  2  1 2 (n) 0 R1

3Here we have used a somewhat nonconventional notation for the expansion. We have absorbed the (n) coefficients of the eigenfunctions into the definitions of R1,2 . Explicitly, these coefficients are specified by assigning appropriate asymptotic boundary conditions. This is to facilitate comparisons to the Green’s functions in [14, 94]. Chapter 4. The Appearance of Fermi Surface 51

the retarded Green’s function is given by [14],

 R¯(n)   R¯(n)   R(n)  0 2 0 2 − 2 0 (n) R¯(n) 0 1 R¯(n) R(n) t (n)  1   1   1  GR = −iσ S = −i·iσ3·(−i) (n) = − (n) = ¯(n) .  R2  −1 0  R2   R2  (n) 0 (n) 0 0 ¯(n) R1 R1 R1 The spectral function is ! R(n) R¯(n) A(n) ∼ −Im(Tr iσtS(n)) ∼ Im − 2 + 2 . (4.55) (n) ¯(n) R1 R1 The special case of n = 0 needs to be considered separately. It is easy to show that

0 R(0) Ψ(0) = iI R(0) = − 2 Ψ(0) ≡ S(0)Ψ(0), − 0 2 1 (0) + + R1 and the Green’s function is

R(0) 1 0 G(0) = −iσtS(0) = σ3S(0) = − 2 , R (0) 0 −1 R1 and thus the spectral function vanishes,

A(0) ∼ −Im(Tr iσtS(0)) = 0. (4.56)

R(n) R¯(n) (n) 2 ¯(n) 2 Let us introduce ξ = (n) , ξ = ¯(n) to discuss the general behavior of Green’s R1 R1 function. The equations for them are 2m zpf∂ ξ(n) = ξ(n) + (ξ(n))2(iC + L ) + (iC − L ), z z t n t n 2m zpf∂ ξ¯(n) = ξ¯(n) + (ξ¯(n))2(iC − L ) + (iC + L ), (4.57) z z t n t n

¯(n) (n) which are exactly the same equations appearing in (24) of [14] with ξ ∼ ξ−, ξ ∼ ξ+ and Ln ∼ k1. The functions ξ and ξ¯ satisfy ¯ ξ(Ln) ∼ ξ(−Ln), (4.58)

and for m = 0 1 ξ¯(n) = − −→ ξ(n = 0) = ξ¯(n = 0) = i. (4.59) ξ(n) Chapter 4. The Appearance of Fermi Surface 52

Boundary Conditions at the Horizon In order to solve for the Green’s function, one needs a boundary condition at the horizon, ˜ 1/4 z → 1. To simplify the equations in the near horizon limit, let us introduce Ri = f Ri, and note that for an extremal black hole

f = 1 + 3z4 − 4z3 → 6(1 − z)2. (4.60)

In the near horizon limit we get Z dz 1 ∂2R˜ = −ω2R˜ , with u ∼ ∼ − . (4.61) u i i f(z) 6(1 − z)

To compute the retarded Green’s function we will impose the in-falling wave condition at the horizon

1 iω R1 ∼ √ e 6(1−z) ,R2 = iR1. (4.62) 1 − z

For ω = 0, the above mentioned in-falling boundary conditions do not apply. Here (see [14]), q 2 m − k2 + m2 − (qQ) − i ξ(ξ¯) = 6 (4.63) √qQ + (−)k 6

4.2.3 Results In order to understand the Fermi level structure and the associated critical behavior we plot the spectral function (Eqn. (4.55)) as a function of Ln (or n) and ω. Due to quantization of the energy levels in presence of a magnetic field the quantity Ln is discrete. However, we will at first try to solve the equations Eqn. (4.57) as if Ln is a continuous parameter. The charge of the fermion q appears in Eqn. (4.57) only in the form qQ, where Q is the charge of the black hole. Hence as long as we keep qQ fixed the nature of the solution q h2 should not change. For a given magnetic field h we can define qeff = q 1 − 3 , such that a system with charge q and magnetic field h is equivalent to a system with charge qeff and zero magnetic field. This can be written schematically as in Eqn. (4.64). This decrease in the effective charge is not surprising because for a fixed horizon radius of the black hole, chemical potential decreases with increasing h ( Eqn. (4.33)). r h2 (q, h) → (q = q 1 − , 0) (4.64) eff 3 From the above argument we can conclude that the physical properties of our system are similar to what has been calculated in the zero magnetic field case [94]. However for a fixed probe fermion charge q if we change the magnetic field, qeff changes. This leads to a subtle relation between fermi level structure and the applied magnetic field. Here we will not try to calculate all the properties of the system. Instead we will just briefly summarize a few Chapter 4. The Appearance of Fermi Surface 53

pertinent aspects using the results of [94]. For a more detailed and complete discussion we refer to the original literature. At zero magnetic ﬁeld the role of Ln is played by the momentum k. As long as we use Ln as a continuous parameter the discreteness of the energy levels does not play any role in our discussion. Of course, after we have a proﬁle for the spectral function we have to put back the discrete energy levels for a physical interpretation. • The existence of Fermi surface is manifested by the existence of one or more ω = 0 bound states of Eqn. (4.57). A bound state corresponds to a sharp peak in 22 Im GR (0, k). At large q there exists one or more bound states of Eqn. (4.57). These bound states can be found numerically or by WKB approximation. In the region ω → 0, k → kF h G22(k, ω) = 1 (4.65) R 1 iγk 2νk k − kF − ω − h2e F ω F vF The pole of the Green’s function in the complex ω plane is located at

ωc(k) = ω∗(k) − iΓ(k), (4.66)

where we have

z ω∗(k) ∝ (k − kF ) (4.67) δ Γ(k) ∝ (k − kF )

The critical exponents are given by

1 1 2ν , νkF < 2 kF z = { 1 (4.68) 1, νkF > 2

and 1 1 2ν , νkF < 2 kF δ = { 1 (4.69) 2νkF , νkF > 2 . From such critical behavior it follows that the ratio Γ goes to zero if ν > 1 and to ω∗ kF 2 1 a non-zero constant otherwise. Hence for νkF < 2 the imaginary part of the pole is always comparable to the real part and thus the quasi-particle is never stable. Another important feature of the pole is that its residue goes to zero as the Fermi surface is

approached. In particular, the smaller νkF , the faster the residue approaches zero. 1 For νk = 2 , the Green’s function has logarithmic corrections. νkF can be calculated exactly from looking at the near horizon (IR) AdS2 region. In the IR geometry, the conformal dimension δk of the operator corresponding to the probe fermion with momentum k is given by r 1 k2 + m2 q2 δ = + ν , ν = − . (4.70) k 2 k k 6 12

• For a ﬁxed m there always exists an “oscillatory region” in the (q, k) parameter space Chapter 4. The Appearance of Fermi Surface 54

(Fig. 4.4) characterized by an imaginary νk. If m ≥ 0 and q is gradually decreased the Fermi levels dissolve in the oscillatory region (Fig. 4.4).At this point there will no longer be any bound state solutions to Eqn. (4.57). For m < 0 the Fermi level still exists for small q.

• If m 6= 0 then decreasing q leads to a phase where there is no oscillatory region (Fig. 4.4). For m = 0 however the oscillatory region persists down to q = 0.

• There√ are other notable features of the Green’s function like a ﬁnite peak at ω ≈ k − 3q etc. We refer the reader to [14] for a more elaborate discussion.

If we start with a value of q = qini, h = 0 and begin to increase h then qeff decreases. The evolution of the system is understood form Fig. 4.4. Generically kF decreases with 1 increasing magnetic ﬁeld, it is expected that νk will also decrease. For νkF < 2 the quasi particle becomes unstable. For m > 0, it seems that there exist a h = hc(m, qini) such that

νkF becomes zero at this point. The system enters the oscillatory region then and there is no Fermi surface for h > hc. As shown in [94] using a WKB approximation, the Fermi 2 2 surfaces only exist for m < qeff /3. Hence s m2 hc = 3(1 − 3 2 ) (4.71) qini

It should be noted that at the cross over phase, νK ≈ 0 and the system is far from being a fermi liquid. The oscillatory region itself goes away for a greater value of h.

6 6 6 4 4 4 2 2 2

q 0 q 0 q 0 -2 -2 -2 -4 -4 -4 -6 -6 -6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 k k k

Figure 4.4: Fig 5 of [94]. The values of kF as a function of q are shown by the solid lines for m = 0.4, m = −0.4, m = 0. The oscillatory region with imaginary νk is shaded. As we q h2 turn on a non-zero magnetic ﬁeld q changes to qeff = q 1 − 3 . From this diagram one can easily see what happens to the system.

For the case m ≤ 0, when the magnetic field increases, the pole seems to persists for arbitrary small qeff . Hence there will be no disappearance of fermi surface at strong magnetic field. Here we have plotted (Fig. 4.5) how the location of the pole (Fermi surface) moves with back ground magnetic field h. Chapter 4. The Appearance of Fermi Surface 55

ImHΞL ImHΞL

400 400

300 300

200 200

100 100

n 0 L 0.5 1.0 1.5 2.0 2.5 0.2 0.4 0.6 0.8 1.0

Figure 4.5: The poles of Im ξ¯ at ω = −10−9, m = 0 with diﬀerent magnetic ﬁelds h and qini = 1: from right to left, h = 0.205, 0.39, 0.5, 0.7, 1.0, 1.4 respectively.

Discrete Nature of Ln The above discussion is not complete. One important aspect of our solution is that n = 0, 1, 2, ··· is discrete (and hence so is Ln), rather than the continuous the plane wave number k in [14]. This is the analogue of Landau levels. However, discrete values of Ln do not necessarily imply a discrete spectrum in the dyonic black hole background. The holographic dual of a black hole is a strongly coupled large N gauge theory at finite energy density. Hence the spectrum is most likely continuous due to interactions [95]. Our situation might not be very different from the case of a global AdS black hole, where due to the compactness of the boundary the S3 harmonics are quantized, although the spectrum in a black hole background is not discrete. Due to the discrete nature of Ln √the entire (ω, Ln) space is not accessible. The only physically allowed values are Ln = 2nqh. The distance between consecutive levels is q proportional to qh . Hence for large n, the levels become densely packed. One recovers n √ a continuum at zero magnetic field by defining a limit keeping Ln = 2nqh fixed while h → 0, n → ∞. At finite h, existence of a pole of the spectral function at Ln = kF , ω = 0 does not imply that kF is a physical value of Ln. However, as the value√ of h is gradually changed the pole passes through allowed values of Ln (i.e. kF (qeff ) = 2ngh for some n and h = hn). In the (n) limit h → hn, the position of the pole of Im G22 (ω) on the complex ω plane approaches ω = 0. Just like Eqn. (4.66) and Eqn. (4.67) one may define two critical exponent,

α ω∗(h) ∼ (h − hn) (4.72) β Γ∗(h) ∼ (h − hn) .

q 2 hn It is not diﬃcult to see that α = z(qeff ) and β = δ(qeff ), where qeff = qini 1 − 3 . (see Eqn. (4.68), Eqn. (4.69)) Generically the pole moves to smaller values of n with increasing h (Fig. 4.5). Due to the gradual passing of poles through physical values of Ln, the spectral function diverges Chapter 4. The Appearance of Fermi Surface 56

periodically with changing magnetic field. It is expected that other quantities associated with the system will also show a similar periodic divergence. This is similar to the de Hass-van Alphen effect (see Appendix. A) observed in condensed matter systems. We can indeed derive a formula similar to Onsager’s relation (A.2) in the limit of a small magnetic field.

Small magnetic ﬁeld

The divergence√ in the imaginary part of the Green’s function occurs when the condition kF (geff ) = 2ngh is satisfied. For small h, q ≈ qeff at linear order in h. Hence if the condition is satisfied for two adjacent level with quantum number n and n − 1, where the magnetic field takes the values h and h + δh respectively then p p 2nqh = kF = 2(n − 1)q(h + δh) (4.73) 1 ⇒ (1 − )(h − δh) = h (4.74) n h ⇒ δh = − . (4.75) n

1 1 Here we have neglected the quadratic terms in both h and n , as n ∼ O(h). Now using 2 n = kF /2qh we get 2qh2 δh = − 2 (4.76) kF 1 2πq 2πq ⇒ δ( ) = 2 = , (4.77) h πkF AF

where AF is the area of the Fermi surface. This is our version of Onsager’s relation.

Strong magnetic field √ p √ At strong magnetic field (h → 3), the first non-trivial level is at L1 = 2 3q. Interesting physical regions seem to lie inside this level. One expects to see something like quantum hall effect etc in this regime. A natural question is whether it is possible to relate the movement of the Fermi level to a change in the filling fractions of the degenerate Landau levels and investigate QHE in this setup. We leave this issue for scrutiny in a future work. Chapter 4. The Appearance of Fermi Surface 57

4.3 Global AdS4 Blackhole 4.3.1 The General Setup The global metric in AdS4 is

dz2 1 ds2 = −f(z)dt2 + + dθ2 + sin2(θ) dφ2 , (4.78) z4f(z) z2

where the factor f(z) takes the form,

2 2 1 z 2 2 1 2 2 1 f(z) = 1 − mz + q z + 2 2 = 1 − 1 + q z+ + 2 2 + q z + 2 2 , z = 1/r. l z z+ l z+ l z (4.79)

4 Here l is the AdS radius, and we set l = 1 for simplicity from now on , and z+ = 1/r+ is the inverse horizon radius. In dyonic black hole, the only difference is to make q2 → Q2 + h2. The gauge field is, if in dyonic black hole background, Q A = ,A = −h cos θ. (4.80) t r φ The temperature is easily to find,

1 2 2 4 T = z+ − q z+ + 3 . (4.81) 4πz+

a The nonzero components of spin connection ωµ b are

t z 1 2 0 θ z p ωt z = ωt t = 2 z f , ωθ z = −ωθ θ = f, φ z √ φ θ ωφ z = −ωφ φ = f sin θ, ωφ θ = −ωφ φ = − cos θ. (4.82) Similar to the Poincare coordinates, the Dirac equation (/D + m)Ψ = 0 can be reduced to

(U(z) + V (θ, φ))Ψ = 0, (4.83) where 1 zf 0 m U(z) = √ Γ0(∂ − iqA ) + Γ1 zpf∂ + √ − pf + , z f t t z 4 f z 1 ∂ − iqA V (θ, φ) = Γ2 ∂ + cot θ + Γ3 φ φ . (4.84) θ 2 sin θ

Notice here U(z) is the same form as before. If we use the same Γ matrices as in last

4We will restore the AdS radius l in the end of this section, and ﬁnd that it is the key factor to produce a Fermi liquid. Chapter 4. The Appearance of Fermi Surface 58

section, one still needs to left multiply the same constant matrix M, that is

11 γ0,2,3 Γ1 = , Γ0,2,3 = , with γ0 = iσ , γ2 = σ , γ3 = σ , −11 γ0,2,3 3 1 2 and

0 σ M = 3 . −σ3 0 It is easy to check that [MU, MV ] = 0, and then they can have eigenvalues simultaneously. One needs to ﬁnd the solution to

MUΨ = MV Ψ = LΨ. (4.85)

4.3.2 Solution Along the Transverse Coordinates Let us consider the V part ﬁrst,

σ (σ C + σ C ) 0 Ψ Ψ MV Ψ = 3 1 2 2 3 + = −L + , 0 −σ3(σ1C2 + σ2C3) Ψ− Ψ− where 1 ∂ − iqA C = ∂ + cot θ, C = φ φ . (4.86) 2 θ 2 3 sin θ Thus

σ3(σ1C2 + σ2C3)Ψ+ = −LΨ+, σ3(σ1C2 + σ2C3)Ψ− = LΨ−. (4.87)

For simplicity, one can turn off the magnetic field first, thus Aφ = 0, and write the two-component Ψ+ as S1 Ψ+ ∼ . S2 inφ Suppose one can write the solution in some mode expansion e , and thus i∂φ → −n, the Dirac equation turns out to be

2 00 0 h 2 x2+4nx−4n2−2 i (x − 1)S1 + 2xS1 + S1 −L + 4(x2−1) = 0, (x2−1)S0 +( x −n)S S = √1 2 1 , (4.88) 2 L 1−x2 where we introduced x = cos θ. Compared to the ordinary associate Legendre’s equation,

n2 (x2 − 1)y00 + 2xy0 + −λ + y = 0, (4.89) 1 − x2

one can find they are very similar. Actually we would like to guess that the equation (4.88) Chapter 4. The Appearance of Fermi Surface 59 is some form of Dirac equation with some S~ · J~ coupling, probably with some coordinate transformation or some angular rotation, however I have not succeeded in finding the right form yet. But we can do the numerical calculation as following. Mathematica finds the solution,

S1 = c1H1 + c2H2, (4.90) where c1,2 are some constants, and

1 n − 1 + n − − 1 1 − x H = (x − 1) 4 2 (x + 1) 4 2 F [−L, L, + n, ], 1 2 1 2 2 1 − n − 1 − n n 1 1 3 1 − x H = (x − 1) 4 2 (x + 1) 4 2 2 F [ − L − n, + L − n, − n, ]. (4.91) 2 2 1 2 2 2 2 Since the solution is on a sphere S2, it has to be regular on it, i.e. at −1 ≤ x = cos θ ≤ 1. One can check numerically that it can only be regular when n takes half-integer values, and

when n > 0, |L| > |n| and L ∈ integers,H1 is regular, but H2 is not.

when n < 0, |L| > |n| and L ∈ integers,H2 is regular, but H1 is not.

Also from the property of 2F1 function,

2F1(a, b, c, z) = 2 F1(b, a, c, z), (4.92) it is easy to see

S1(L) = S1(−L). (4.93)

Recalling that in the familiar quantum story of electrons, the good quantum numbers 2 ~ ~ ~ ~ are (~j , jz), where ~j = l + S, and l, S are angular momentum and spin respectively. Also jz takes values of half-integers, and l is with some integer value. It is natural to guess (L, n) ∼ (l, jz), which conﬁrm our hypothesis above in some sense.

An Alternative Approach to the Transverse Solution

From the equation (4.87) and the form of Ψ+, one can write, keeping the magnetic ﬁeld on,

(C2 − iC3)S2 = −LS1,

(C2 + iC3)S1 = +LS2, (4.94) and then

h i 2 L(C2 − iC3)S2 = (C2 − iC3) (C2 + iC3)S1 = −L S1, (4.95) Chapter 4. The Appearance of Fermi Surface 60

or more explicitly, introducing x = cos θ, and assume Ψ ∼ einφ,

h i 1 1 (x2 − 1)∂ (x2 − 1)∂ S + S x2( − q2h2 − L2) + x(n − 2nqh) − ( − qh + n2 − L2) = 0. x x 1 1 4 2 (4.96)

2 dx 1 1−x If we deﬁne (x − 1)∂x = ∂u, i.e. x2−1 = du or u ∼ 2 log 1+x , then u ∈ (−∞, ∞) as x ∈ (1, −1), θ ∈ (0, π), the equation above can be simpliﬁed,

2 − ∂uS1 + V (x)S1 = 0, (4.97)

where 1 1 V (x) = x2(L2 + q2h2 − ) − x(n − 2nqh) + ( − qh + n2 − L2). (4.98) 4 2 It is WKB equation of a half unit mass particle moving in potential V (x) with zero energy. One can show some characteristic values of the potential. First of all, it is a well-like potential, with the two maximum at the left and right ends,

1 2 1 2 V (x = 1) = n + (qh − ) ,V (x = −1) = n − (qh − ) , (4.99) max,1 2 max,2 2

and the minimum is 1 n2 V = −(L2 + qh − ) 1 − . (4.100) min 2 2 2 1 2 L + q h − 4 The zero point of the potential V (x) = 0 are at q 1 2 1 2 2 2 2 1 n(−qh + 2 ) ± (L + qh − 2 )(L + q h − n − 4 ) x = , (4.101) 1,2 2 2 2 1 L + q h − 4 which requires 1 1 L2 > −qh + ,L2 > n2 + − q2h2. (4.102) 2 4 In WKB approximation method, the wave needs to satisfy the bound state condition,

u x x p Z 2 Z 2 p du Z 2 −V (x) 1 p du = −V (x) dx = 2 dx = (l + )π, l = 0, 1, 2, ···(4.103) u1 x1 dx x1 x − 1 2 From the other approach, we have the tentative solution (at h = 0) with

1 3 1 L ∈ integers, n = ± , ± , ··· , ± |L| − , (4.104) 2 2 2 Chapter 4. The Appearance of Fermi Surface 61

we can check numerically of the WKB bound state condition, and ﬁnd at L larger (L2 1, |qh|, q2h2), the integrate goes closer to bound state condition, i.e.

Z u2 1 p du → (l + )π, l = 0, 1, 2, ··· (4.105) u1 2 It is reasonable since WKB works better at high excited states, recalling the minimum 2 2 2 potential in this limit Vmin → −(L − n ) and Vmax − Vmin ∼ L .

4.3.3 The Complete Solution Our solution is very similar to the case in Poincare coordinates, and the general forms are    ¯  S1(θ)R1(z) S1(θ)R1(z) ¯ (L,n) (L,n) (−L,n) (L,n)  S2(θ)R1(z)  (−L,n)  −S2(θ)R1(z)  Ψ = Ψ1 +Ψ2 , Ψ1 =   , Ψ2 =  ¯  ,  S1(θ)R2(z)   S1(θ)R2(z)  ¯ −S2(θ)R2(z) S2(θ)R2(z)

where Ψ1, Ψ2 correspond to eigenvalue L and −L respectively. Substitute the solution form back to the separated Dirac equations, i.e.

MUΨ1 = −MV Ψ1 = LΨ1,MUΨ2 = −MV Ψ2 = −LΨ2, (4.106)

one ﬁnds the radius part of equations are the same as before,

D+R1 = −(iCt + L)R2,D−R2 = (iCt − L)R1, ¯ ¯ ¯ ¯ D+R1 = −(iCt − L)R2,D−R2 = (iCt + L)R1, (4.107)

where if we assume Ψ ∼ e−iωt,

zf 0 m ω + qA D = zpf∂ + √ − pf ± , iC = √ t . (4.108) ± z 4 f z t z f

Orthogonality Let us introduce the bases in the transverse coordinates,

(L,n) ! (L,n) ! (L,n) S1 (L,n) S1 e1 = (L,n) , e2 = (L,n) , S2 −S2 Chapter 4. The Appearance of Fermi Surface 62

one can check the orthonormality numerically, and ﬁnd

Z 1 Z (L1,n) ∗ (L2,n) (L1,n) ∗ (L2,n) (e1 ) e1 dx = (e2 ) e2 dx −1 Z (L1,n) ∗ (L2,n) (L1,n) ∗ (L2,n) = ((S1 ) S1 + (S2 ) S2 )dx = const.(L1, n) δL1,L2 , Z 1 Z (L1,n) ∗ (L2,n) (L1,n) ∗ (L2,n) (e1 ) e2 dx = (e2 ) e1 dx −1 Z (L1,n) ∗ (L2,n) (L1,n) ∗ (L2,n) = ((S1 ) S1 − (S2 ) S2 )dx = 0, (4.109)

where the coeﬃcient const. are some constant function of (L1, n), which can be absorbed into the deﬁnition of S1,S2 to get the normalized forms. Also one can easily check some transformation

σ3e1 = e2, σ3e2 = e1. (4.110)

Write all the functions in the bases of 1 0 e ∼ , e ∼ , 1 0 2 1 then 0 1 σ ∼ , 3 1 0 and ! R(L,n) Ψ(L,n) = R(L,n)e(L,n) + R¯(L,n)e(L,n) ∼ 1 , + 1 1 1 2 ¯(L,n) R1

 R¯(L,n)  (L,n) ! 2 ¯ (L,n) (L,n) R R¯ (L,n) (L,n) Ψ ∼ 2 =  1  Ψ ≡ S(L,n)Ψ . − (L,n) R(L,n) + + R  2  2 (L,n) R1

Green’s Function One can calculate the Green’s function,

 R¯(L,n)   R(L,n)  2 2 (L,n) 1 R¯(L,n) R(L,n) t (L,n) (L,n)  1   1  GR = −iσ S = σ3S ∼ (L,n) = ¯(L,n) 1  R2   R2  (L,n) ¯(L,n) R1 R1 and then the spectral function is ! R(L,n) R¯(L,n) A(L,n) ∼ −Im(TriσtS(L,n)) ∼ Im 2 + 2 , (4.111) (L,n) ¯(L,n) R1 R1 Chapter 4. The Appearance of Fermi Surface 63

2m zpf∂ ξ(L,n) = ξ(L,n) + (ξ(L,n))2(iC + L) + (iC − L), z z t t 2m zpf∂ ξ¯(L,n) = ξ¯(L,n) + (ξ¯(L,n))2(iC − L) + (iC + L), (4.112) z z t t

R(L,n) R¯(L,n) (L,n) 2 ¯(L,n) 2 where we have deﬁned ξ = (L,n) , ξ = ¯(L,n) . R1 R1

4.3.4 The Scaling Symmetry and Two Limits To ﬁnd the pole behavior numerically, one needs to do some manipulation ﬁrst. Firstly, let z ˜ ˜ us do some rescaling, and introducez ˜ = , t˜ = z+t, Q = Qz+, l = lz+, the metric turns z+ out to be dz˜2 1 z2 ds2 = −f(˜z)dt˜2 + + dθ2 + (sin θ)2dφ2 , (4.113) + z˜4f(˜z) z˜2

To make the equations (4.112) invariant, it also requiresω ˜ = ω/z+, m˜ = m/z+, and from ˜ the gauge ﬁeld, At = At/z+, µ˜ = µ/z+. And the temperature is

z 3 T = + 1 − Q˜2 + . (4.114) 4π ˜l2 Thus if we are restricted in extremal black hole, i.e. T = 0, then 3 Q˜2 = 1 + . (4.115) ˜l2 4.3.5 Fermi Liquid? The discussion of section 4.1 is in Poincare coordinate, showing that the holographic ﬁeld theory system has a Fermi surface, and some non-Fermi liquid properties, some of which are described by equations (4.24) and (4.25),

z ω∗(k⊥) ∼ k⊥, z = 2.09 ± 0.01, −α ImG22(ω∗(k⊥), k⊥) ∼ k⊥ , α = 1.00 ± 0.01. (4.116) and the discussion around them. The global coordinates have an extra free parameter, eﬀectively the scaled AdS radius ˜l, and it turns out to be able to change the power z while α keeps as a constant. In Fig. 4.6, Poincare limit is at ˜l = 0, where z ≈ 2 consistent with the result in Sec. 4.1. In the other limit, at ˜l → ∞, z → 1, together with α ≈ 1, remind us the Landau’s Fermi liquid behavior. To make a deﬁnite statement of the Fermi liquid’s existence, we need more check of the other properties. It would be a very promising and interesting project for our future work. Chapter 4. The Appearance of Fermi Surface 64

z 2.0

1.8

1.6

1.4

1.2

L 5 10 15 20

Figure 4.6: The power z as function of ˜l.

4.4 At Finite Temperature

The main motivation of this section is to study of the effects of a scalar condensate on the fermionic system5. For this purpose, we extend the black hole background by turning on a scalar condensate; In this section[4] we will mostly be concerned with the non-extremal geometries. Since the condensate strength can become large, it becomes necessary to consider its effects on the gravitational background. We next introduce probe fermions in this geometry with a suitable coupling to the scalar. The coupling to the scalar, and other parameters we consider here (such as the fermion charge and the temperature) extend and generalize previous work [94]. We find that the system always contains sharp fermionic excitations, which are gapless and become stable in the zero temperature limit. The detailed properties of those fermionic excitations, depend strongly on the parameters chosen, in a way we discuss in detail. The robustness of the Fermi surface is perhaps surprising, and might be related to the large N limit. In this note we concentrated on the physics of the holographic superconductors obtained from the simplest Abelian Higgs model in the bulk [13, 101–103]. There is another set of models utilizing non-Abelian gauge fields in the bulk [104], which are more interesting in some ways. For instance, the bulk couplings are more constrained, and the model allows for both s-wave and p-wave [105] superconductors6.

4.4.1 Asymptotically AdS4 Black Holes

The set of backgrounds we are interested in asymptote to an AdS4 geometry, with possibly some profiles for the gauge and scalar fields. This corresponds to a 2+1 dimensional dual field theory at a finite temperature, with a finite chemical potential, and possibly resulting in a condensation of a scalar operator. We are interested in various such geometries, which dominate the thermodynamics of the dual field theory at different temperature ranges. The system is described by gravity coupled to a Maxwell field and a charged scalar field

5For previous work on such eﬀects see [94, 96–100]. 6For recent work on this set of models see [3, 106]. Chapter 4. The Appearance of Fermi Surface 65

(an Abelian Higgs model), with the Lagrangian: 6 1 L = R + − F µνF − |∇ψ − iq Aψ|2 − V (|ψ|), (4.117) L2 4 µν b

where F is the electromagnetic ﬁeld strength, and qb is the charge of the scalar ﬁeld ψ. We follow the conventions in [103]. Assuming solutions with spherical symmetry, the metric has the general form:

dr2 ds2 = −g(r)e−χ(r)dt2 + + r2(dx2 + dy2) (4.118) g(r)

with A = φ(r)dt and ψ = ψ(r). By a suitable gauge choice we can assume ψ to be real, and the scaling symmetries of the metric allow us to set L = 1. The full non-linear equations of motion are given by:

0 0 2 2 χ 0 00 g χ 2 0 qb φ e V (ψ) ψ + g − 2 + r ψ + g2 ψ − 2g = 0

0 2 2 00 χ 2 0 2qb ψ φ + 2 + r φ − g φ = 0 2 2 2 χ 0 02 rqb φ ψ e χ + rψ + g2 = 0 0 1 χ0 rφ02eχ rV (ψ) g + r − 2 g + 4 − 3r + 2 = 0, (4.119)

1 2 2 with the simple potential V (ψ) = 2 mψψ where mψ is the mass of the scalar ﬁeld. Fur- thermore, we will mostly take the scalar ψ to be massless (mψ = 0), unless otherwise indicated. The above ﬁeld equations have two scaling symmetries that will turn out to be useful. They are:

r → ar, (t, x, y) → (t, x, y)/a, g → a2g, φ → aφ, eχ → a2eχ, t → at, φ → φ/a. (4.120)

We are interested in non-extremal black hole solutions. We use the ﬁrst symmetry to set the black hole horizon radius r+ to 1, and the second symmetry to set χ to zero at the boundary. With these choices, the following boundary conditions can be used to ﬁx the remaining free parameters in Eq. (4.119):

0 φ(r+) = 0, φ (r+) = E, g(r+) = 0, 0 0 V (ψ+) ψ(r+) = ψ+, ψ (r+) = (4.121) 1 1 2 χ+ 2 3 − 2 V (ψ+) − 4 E e With the above choice of the boundary conditions, the solutions are determined by two parameters: ψ+ and E. The latter quantity can be interpreted as the electric ﬁeld at the horizon. From the equations of motion Eqs. (4.119), the general asymptotic form of the Chapter 4. The Appearance of Fermi Surface 66

ΨHrL gHrL 2.0 1.0

0.8 1.5

0.6 1.0 0.4

0.5 0.2

r 0.0 r 4 6 8 10 0 2 4 6 8 10 (a) Scalar ψ (b) Metric function g

Figure 4.7: (Colorful) The proﬁles for g(r) and ψ(r) at Teff = 0.036. For reference, the grey monotonically increasing curve is for the pure RN black hole.

scalar ﬁeld ψ near the boundary is of the form: ψ ψ ψ = 1 + 2 + ··· (4.122) r r2

The parameters ψ+ and E in Eqs. (4.121) can be traded for ψ1, ψ2. In what follows we 7 choose to set ψ1 to zero . This fixes E, and the remaining free parameter ψ+ can be used to vary the temperature of the solution. In Fig 4.7 we plot typical profile for the scalar field ψ(r) and the function g(r). The temperature is T ≡ T = 0.036. The expected asymptotic forms for this solution are eff Tc ψ ∼ 1/r2 and g ∼ r2 for large r. As the temperature is reduced, the thermodynamics is dominated by one of two different backgrounds. Above the critical temperature, there is no scalar hair, and the relevant background is the Reissner-Nordström(RN) black hole. Below the critical temperature, the scalar begins to condense; though around the critical temperature this can be treated as a small perturbation, as we lower the temperature further backreaction ultimately becomes important. Thus we have to use a non-extremal backreacted black hole with scalar hair, constructed in [103]. As the temperature approaches zero the system resembles the zero temperature backreacted black hole with a zero radius horizon, constructed in [84]. We find that in the zero temperature limit our results approach those of [90], as they should.

7 We set either ψ1 or ψ2 has to be zero, as we are interested in the ﬁeld theory in the absence of external sources. Massless scalars allow for two inequivalent quantization, where either ψ1 or ψ2 is interpreted as a normalizable mode. We choose the quantization for which ψ2 is normalizable; the alternative quantization gives results which are qualitatively similar to the ones discussed below. Chapter 4. The Appearance of Fermi Surface 67

4.4.2 Fermions We now concentrate on the eﬀect of a scalar condensate on non-Fermi liquid behavior. The basic idea, introduced in [14], is to introduce fermions in our geometries, whose Green’s functions probe the existence and properties of a Fermi surface in the boundary theory. In our case there can be also a scalar ﬁeld present in the bulk, and therefore we have to specify the coupling between the fermions and the scalar condensate. We take the action for the bulk fermion Ψ to be: Z √ d+1 ¯ M ¯ 2 ¯ SΨ = d x −g i(ΨΓ DM Ψ − mΨΨ − λ|ψ| ΨΨ), (4.123)

where ΓM are the curved space Gamma matrices and: 1 Ψ¯ = Ψ†Γt, D = ∂ + ω Γab − iq A . (4.124) M M 4 abM f M Here M denotes bulk spacetime indices while a, b denote tangent space indices. Greek letters denote indices along the boundary directions. Thus Γab are the tangent space Gamma matrices. We also choose units such that R = 1 in the AdS geometry. The charge of the fermionic ﬁeld is denoted by qf , and the fermion mass is m. We are mostly working with massless fermions, m = 0. The last term in Eq.(4.123) is a quartic coupling between the scalar ψ and the Dirac fermion Ψ, and λ is a tunable parameter controlling the coupling to the scalar, assumed to be positive. This is the most general coupling for general choices of scalar and fermion charges, though cubic couplings can exist when qb = 2qf [90]. The quartic coupling can be absorbed into an eﬀective (radial dependent) mass term for the fermions:

M(r) ≡ m + λ|ψ|2 (4.125) Since the quartic coupling does not necessitate a deﬁnite ratio of fermion and boson charge, we are free to vary their ratio. In the following we keep the charge of the complex scalar ﬁeld to be qb = 1, which is in the range where a phase transition occurs, but keep the freedom to vary qf . With the quartic coupling only, the mechanics of the fermionic equations remains more or less the same [14]. We choose the following basis for the Gamma matrices and the (4 component) spinors:

µ r 1 0 µ 0 γ Ψ+ Γ = , Γ = µ , Ψ = , (4.126) 0 −1 γ 0 Ψ−

µ where Ψ± are two-component spinors and γ are (2+1)-dimensional gamma matrices. We can now separate the radial and boundary coordinate dependencies in Ψ as follows: 1 i rr − −iωt+ikix y± Ψ± = (−gg ) 4 e Φ±, Φ± = (4.127) z±

µ 0 1 2 For the γ , we choose the basis γ = iσ2, γ = σ1, γ = σ3. We also use the rotational Chapter 4. The Appearance of Fermi Surface 68

symmetry of the system to set k2 = 0. The Dirac equations then reduce to two sets of decoupled equations: r gii √ (∂r ∓ M grr)y± = ∓i(k1 − u)z∓, grr r gii √ (∂r ± M grr)z∓ = ±i(k1 + u)y±, (4.128) grr with r gii 2 u = (ω + qf φ(r)) M = λ|ψ| (4.129) −gtt

We deﬁne the ratios ξ+ = iy−/z+, ξ− = −iz−/y+, in terms of which eqs. (4.128) can be written as: r gii √ 2 ∂rξ± = −2M giiξ± ∓ (k1 ∓ u) ± (k1 ± u)ξ±. (4.130) grr

The retarded Green’s function GR is given in terms of the quantities ξ± by: −2M ξ+ 0 G11 0 GR = lim ≡ (4.131) →0 0 ξ− 1 0 G22 r=

The spinors ξ± satisfy infalling boundary conditions at the black hole horizon, which is located at r+ = 1. This implies: lim ξ±(r) = i (4.132) r→1 This completes our review of the setup. We will be interested in the properties of the Fermi surface as we vary the temperature, or the quartic coupling λ8 In order to do that, we solve numerically the system of equations (4.130) in the various backgrounds obtained by solving the bosonic equations, Eqs. (4.119). From this, we can calculate the retarded Green’s function (Eq. (4.131)) and the fermionic spectral densities (related to the imaginary part of the retarded Green’s function), and investigate the eﬀects of turning on λ and varying the temperature.

4.4.3 Reissner-NordströmBlack Hole (T > Tc) The Reissner-Nordströmblack hole exists for any temperature, and is the dominant phase for T ≥ Tc. This background was originally discussed in [14], and we concentrate here on the behavior at non-zero temperature, to set up our notation and provide a comparison to other backgrounds. The Reissner-Nordströmmetric is given by:

dr2 ds2 = −f(r) dt2 + + r2(dx2 + dy2) (4.133) f(r)

8Note that the geometry includes back-reaction from the scalar condensate, therefore the condensate inﬂuences the fermions even when λ = 0. Chapter 4. The Appearance of Fermi Surface 69

2 Q2 1+Q2 1 with f(r) = r + r2 − r , φ(r) = µ 1 − r and µ = Q. Here we have used the scaling symmetries in Eqs. (4.120) to set L = 1 and horizon r+ = 1, then the temperature is 1 T = (3 − Q2), (4.134) 4π and the dimensionless temperature is

T 1 (3 − Q2) T ≡ = . (4.135) eff µ 4π Q Since the scalar ψ = 0, the black hole charge Q is the only tunable parameter. The discussion in [14] concentrated on the spectral density of the fermions at zero temperature, finding a signal of a Fermi surface by the presence of a delta function peak in the spectral function (or equivalently, a pole in the Green’s function), at critical value of the momentum (k = kF ) and zero frequency. As expected, we find that at finite temperature, the pole in the Green function moves off the real axis, to complex values of ω. In the following, we investigate how the pole evolves in the complex ω plane as the momentum k changes.

G G22 ImHΩL 120 ReHΩL -0.02 -0.01 0.01 0.02 0.03 100

- 0.005 80

60 -0.010 40

-0.015 20

Ω -0.020 -0.10 -0.05 0.05 0.10

Figure 4.8: (Colorful) The left figure shows the G22 pole location in the complex frequency plane as function of momentum, at different temperatures, Teff ≈ (0.00016, 0.0016, 0.008) respectively from top to bottom. The right figure shows a plot of Im G22 vs. Re ω at Teff ≈ 0.00016 for a range of values of k: from 0.8 to 1 for the curves from yellow to red(from the leftmost to the rightmost). The curve with highest Im G22 has k = 0.9.

In Fig 4.8 we plot the typical trajectory of the pole of the spectral function (Im(G22)) on the complex ω plane for a range of temperatures, in the left panel. The position of the pole changes as the momentum k is varied. The Fermi momentum corresponds to the point of closest approach to the real axis. The right panel shows the frequency dependence of the spectral function for one choice of the temperature. From this data we can determine how the minimum distance of the pole Γ from the real axis changes with temperature9. We ﬁnd that Γ grows linearly with temperature

9In all the plots of the poles in the complex frequency plane, we concentrate on the vicinity of primary Chapter 4. The Appearance of Fermi Surface 70

(Fig 4.9(a)). This is consistent with the results of [14], and extends them beyond the low temperature regime. We can also measure how the imaginary part of the pole position changes as a function of k − kF , near the Fermi momentum. For the RN black hole at ﬁnite temperature, Γ ∼ z (k−kF ) , with z ∼ 2. The critical exponent z does not depend strongly on the temperature, and is close to the value of the exponent found at [14] for T = 0. The dependence of Γ on the temperature and on k − kF is summarized in Fig 4.9(b).

G22 G G22 Im HΩ-Ω*L

0.10 k -k* -0.02 -0.01 0.01 0.02 0.03 0.04

0.08 -0.0002

0.06 -0.0004

0.04 -0.0006

0.02 -0.0008

Teff 0.01 0.02 0.03 0.04 0.05 (a) Γ vs. temperature (b) Pole position vs. momentum

Figure 4.9: Variation of the pole position with temperature and momentum. The plots are for G22.

4.4.4 Non Extremal Hairy Black Hole (T < Tc) We now consider the backreaction of the scalar on the black hole geometry, at temperatures below the critical one. The fermions are still treated as probes. At first we do not include any coupling between the scalar and fermions, concentrating on the behavior as a function of temperature and the fermion charge. The influence of the scalar on the behavior of the fermions comes through the backreaction on the geometry. We then turn on the quartic coupling λ discussed above, and demonstrate the influence of changing the quartic coupling, the fermion charge and the temperature on features of the spectral functions of the fermions. Fig. 4.10 shows a plot of Im(G22) vs. frequency ω for zero coupling. We see the typical “peak-dip-hump” found in [96]. In Fig. 4.11 we plot the location of the pole in the spectral density ImG22 as function of momentum and frequency, for various values of the coupling, including λ = 0. Note that unlike the high temperature case, discussed in the previous section, the spectral density ImG11 exhibits some non-analytic behavior, though for any momentum and frequency the peak of ImG11 is much broader than the one of ImG22. For that reason we concentrate exclusively on the spectral density ImG22 below.

Fermi surface, which is the range of momenta where the poles of the spectral function are closest to the real axis. Other branches of the curves shown exist in some cases. Chapter 4. The Appearance of Fermi Surface 71

ImG22 3.0

2.5

2.0

1.5

1.0

0.5

Ω 0 20 40 60 80 100

Figure 4.10: Im(G22) vs. frequency ω at the Fermi momentum. This is the so-called “peak- dip-hump” behavior, with some ripplesin the large ω region. (We are still not sure about the meaning of ripples yet.) The data is for Teff = 0.004, qF = 1, λ = 0.

Ω 20

k -4 -2 2 4 6 8 10

-5

-10

Figure 4.11: (Colorful) Peak position (k, ω) as a function of coupling λ. λ = 0, 0.5, 1, 2 from bottom to top (green to red).

The qualitative features of Fig. 4.11 correspond to the existence of a fairly sharp Fermi surface (smoothed out by finite temperature), with gapless fermionic excitations. Unlike the high temperature case, and similar to the discussion in [90] (for zero temperature), we find a line of sharp excitations in the frequency-momentum plane, rather than an isolated point. Note that we find sharp excitations for both signs of the momentum, or in other words both for particle and hole excitations. Fig. 4.12 is similar to Fig. 4.11, except that here the temperature is varied for a fixed coupling. We can also determine the behavior of the imaginary part of the pole position in the complex ω plane as a function of temperature, similar to what we did in Sec. 4.4.3. We find that in this case the temperature dependence is no longer linear, rather we find η that for λ = 0, Im ωp ∼ T , with η ∼ 5. Chapter 4. The Appearance of Fermi Surface 72

Ω 30

k -10 10 20 30

-10

-20

-30

Figure 4.12: (Colorful) Pole position (k, ω) as a function of temperature. Here both k and ω are real, and the tracks are with the similar meaning of the familiar dispersion relations. The temperature Teff = 0.023, 0.014, 0.004, 0.002, 0.0013, 0.0009 from yellow to red (left to right). The other parameters are qF = 1, λ = 0.

Turning on and varying the scalar-fermion coupling (in addition to the fermion charge) corresponds to scanning different boundary theories, as opposed to variation of physical parameters such as the temperature. In all those boundary theories, we find that the qualitative properties of the system do not change much: in the condensed phase we still have stable fermionic excitations (as zero temperature) which are gapless10. It is surprising to find such stable excitations for a large set of theories, particularly for both signs of the coupling λ. Perhaps this is somehow related to another feature that sets these theories apart, namely having a dual description in terms of classical gravity. In Fig 4.13 we plot the contour of poles for changing λ, at a fixed temperature. The coupling λ here determines the range of values of k which give rise to poles, and there are no gapped excitations in this picture. In the zero temperature limit , the gapless excitations become stable. We find that turning on λ (for a fixed temperature) seems to stabilize the fermionic excitations, for either sign of λ. For example, the critical exponent z defined above and in [14] increases monotonically with |λ|. We also checked the dependence of these feature on the fermion charge qF . These results are summarized in figure 4.14. In this case as well, we find that while qualitative features do not change as the fermion charge is varied, the detailed features do.

10These observations are consistent with the expectations in [90]. Chapter 4. The Appearance of Fermi Surface 73

0 l=-0.8 l=-0.6 l=-0.4 l=-0.2 -0.0002 l=0 l=0.2 l=0.4 l=0.6 -0.0004 l=0.8

-0.0006 ) ω Im( -0.0008

-0.001

-0.0012

-0.0014 -3 -2 -1 0 1 2 3 Re(ω)

Figure 4.13: Contour plots for varying λ for temperature Teff ∼ 0.004.

ImΩ kF

qF 0.6 0.8 1.0 1.2 1.4 12

10 -0.1

- 0.2 6

4 -0.3 2

-0.4 qF 0.5 1.0 1.5 2.0

Figure 4.14: The pole position as function of fermion charge qF . The plots are for Teff = 0.004. As qF increases, the curve in the complex ω plane moves closer to the real axis, and the fermion excitations become more stable. In the left panel, Im(ω) is plotted as function as qF , while the right panel shows the corresponding Fermi momenta kF . 74 Chapter 5

Holographic Superconductivity

Various phenomena in condensed matter systems have been discussed holographically with the AdS/CFT correspondence. Although many of them are not able to show any further information than the ordinary condensed matter theory by now, they do provide us with a new method to understand the physics. And the strong-weak duality enables us do some reliable calculation in the strong coupled regime. One interesting example is the high-Tc superconductivity, which has no well-developed theory in CMT yet.

5.1 Abelian Scalar Model

Firstly one should work in the simplest case, an Abelian scalar model[13]. From the dual gravity, it is found that the ﬁeld theory has a condensation phase transition, with a critical temperature, and the (DC) conductivity is calculated. Starting with a (3 + 1) dimensional Schwarzschild-anti-de Sitter black hole,

dr2 ds2 = −f(r)dt2 + + r2(dx2 + dy2), (5.1) f(r)

where f(r) = r2/L2 − M/r. Here L is the AdS radius, and M is the mass factor of the 3M 1/3 black hole. The temperature can be found, T = 4πL4/3 . In this background, one can introduce a Maxwell field and a charged complex scalar. The Lagrangian density is 1 L = − F F ab − V (|Ψ|) − |Ψ − iAΨ|2. (5.2) 4 ab The most common potential is taken as V (|Ψ|) = m2Ψ∗Ψ, and for simplicity, one can 2 2 2 set m = −2/L . For reference, the Breitenlohner-Freedman bound of AdS4 is mBF = 9 2 2 − 4L2 . We will see later that the value of m slightly higher than mBF is easy to make the scalar condensed1. The scalar model is a simplified model from M-theory, for instance, 7 the truncation of M-theory on AdS4 × S to N = 8 gauged supergravity has scalars and pseudoscalars with the mass, due to the bilinear operators trΦ(I ΦJ) and trΨ(I ΨJ) in the dual N = 8 super-Yang-Mills theory respectively. The simplification does not change the qualitative behavior. Suppose the temperature is not very low, and the model can be taken in the probe limit, i.e. backreaction on the metric is negligible. Let us work in a symmetric ansatz,

1 2 2 The radial dependent eﬀective mass squared meff is typically less than the set-up value m , and is 2 2 2 easier to approach the values below mBF for m ' mBF . Chapter 5. Holographic Superconductivity 75

Ψ = Ψ(r),At = φ(r), and take Ψ to be real. The equations of the ﬁelds are

f 0 2 φ2 2 Ψ00 + + Ψ0 + Ψ + Ψ = 0, f r f 2 L2f 2 2Ψ2 φ00 + φ0 − φ = 0, (5.3) r f

where we have set Ar = Ax = Ay = 0. From the equation above, the vector component φ has an eﬀective radius dependent mass 2Ψ2, while the eﬀective mass of scalar turns to be

φ2 m2 = −2 − . (5.4) eff f

2 The effective mass might be less than the BF bound mBF = −9/4 at some radius, and leads to the instability, i.e. the superconductivity. The fields take value at radius between r0 < r < ∞, where r0 is the radius of the horizon. At horizon, the gauge component has to 0 vanish, φ = 0, and then Ψ = −3r0Ψ /2. While on the other hand, the asymptotic behaviors of the fields are Ψ(1) Ψ(2) ρ Ψ = + + ··· , φ = µ − + ··· . (5.5) r r2 r Since both Ψ(1) and Ψ(2) are normalizable, one can set either of them vanish, and lead to two families of solutions. As shown in the general discussion section, the condensation of the scalar can be written as √ (i) hOii = 2Ψ , i = 1, 2, (5.6) √ where the coefficient 2 is introduced for future simplification, and the boundary condition (j) is set as ijΨ = 0. There is a scaling symmetry, which allows us freely to set AdS radius L = 1. Further- 2 more, the asymptotic behavior means that O1/T and O2/T are dimensionless quantities(on mass dimension). The numerical result is shown in Fig. 5.1. The right plot of the figure, 2 condensation of O2/T has a very similar property of BCS superconductivity, while the left seems more like type II superconductivity. The numerical fitting shows,

1/2 O1 ≈ 9.3Tc(1 − T/Tc) , as T → Tc, (5.7)

1/2 with Tc ≈ 0.226ρ , and

2 1/2 O2 ≈ 144Tc (1 − T/Tc) , as T → Tc, (5.8)

1/2 with Tc ≈ 0.118ρ . One needs to keep in mind that, at low temperature, the condensation is relevantly large, and the ratio µ/T diverges in the limit T → 0. In this limit, the probe limit is no Chapter 5. Holographic Superconductivity 76

10 8 8 6 1 6 Tc 4 4 Tc

2 2

0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 T T

Tc Tc

Figure 5.1: The two diﬀerent kinds of condensation modes, of hO1i and hO2i respectively. The ﬁgures are taken from [13]. longer valid, and the backreaction is not negligible anymore. However the full backreacted solution is qualitatively the same. And if we are only interested in the transition’s existence itself, the condensation is always very small.

Conductivities

For T < Tc, the scalar operators condense, and compared with the condensed matter theory, it is expected that superconductivity appears in this regime. −iωt Let us turn on a perturbation of Ax, with the form of Ax(r)e . It is easy to write the equation of motion,

f 0 2 2Ψ2 A00 + A0 + − A = 0. (5.9) x f x f 2 f x

It has the near horizon behavior,

−iω/3r0 Ax ∝ f , (5.10) and asymptotically

A(1) A = A(0) + x + ··· , as r → ∞. (5.11) x x r According to the AdS/CFT correspondence, the dual source and the expectation value of current are

(0) (1) Ax = Ax , hJxi = Ax . (5.12)

From Ohm’s law, one ﬁnds

hJ i hJ i ihJ i iA(1) σ(ω) = x = − x = − x = − x . (5.13) ˙ (0) Ex Ax Ax Ax Chapter 5. Holographic Superconductivity 77

1.2 1.2 1 1 0.8 0.8 Re@ΣD Re@ΣD 0.6 0.6 0.4 0.4 0.2 0.2

0 20 40 60 80 0 50 100 150 200 Ω Ω T T

Figure 5.2: The conductivities of the two kinds of condensation modes, hO1i and hO2i respectively from left to right. In each plot, a righter curve has lower temperature, and the rightmost curve has the temperature T/Tc = 0.0066(left) and T/Tc = 0.0026(right). A delta function exists at ω = 0 for each case. The ﬁgures are taken from [13].

In Fig. 5.2 we draw the real part of the conductivity of the two kinds of condensation. When T < Tc, a gap appears, and expands and gets deeper at lower temperature. The high frequency conductivity approaches a constant, which is expected to be consistent with normal phase behavior. There is a delta function at ω = 0, which comes from the translation invariance of the theory. Although it is diﬃcult to check numerically of the real part of conductivity, it is doable on the imaginary part. The Kramers-Kronig relation relates the real and imaginary parts, as

1 Z ∞ Re[σ(ω0)]dω0 Im[σ(ω)] = − P 0 . (5.14) π −∞ ω − ω For a delta function of real part, Re[σ(ω)] = πδ(ω), the imaginary part has a pole at ω = 0, Im[σ(ω)] = 1/ω, as in Fig. 5.3.

15 14 12.5 12 10 10 7.5 8 Im Σ Im@ΣD @ D 6 5 4 2.5 2 0

0 0.5 1 1.5 2 0 0.5 1 1.5 2 Ω Ω

< O1 > < O2 > Figure 5.3: The pole at ω = 0 of imaginary part of conductivity σ(ω) shows the delta function existence of the real part, from Kramers-Kronig relation. The ﬁgures are taken from [13].

The superﬂuid density is the coeﬃcient of the real part

Re[σ(ω)] ∼ πns(T )δ(ω), (5.15) Chapter 5. Holographic Superconductivity 78

which gives the imaginary part, Im[σ(ω)] ∼ ns(T )/ω as ω → 0. It can be shown that the superﬂuid density vanishes linearly near critical temperature,

ns(T ) ≈ Ci(Tc − T ), as T → Tc. (5.16)

The Ferrell-Glover sum rules states that the integrate R Re[σ]dω is independent of temperature. Thus the gap appeared at T < Tc is compensated by the delta function which also exists only when T < Tc, and numerical calculation of ns(T ) veriﬁes the sum rule.

Vector Hair for AdS Blackhole

In the simplest model, with an electric ﬁeld At and a complex scalar Ψ, it is shown superconductivity appears at low temperature. In that model, one turns on only the spatial component of the gauge ﬁeld Ax to be perturbation in calculating the conductivity. How- ever if this component is not so small, the black hole might have vector hair with Ax condensed[15]. The background is the same as in the simplest Abelian scalar model, with the La- grangian density (5.2). For convenience, we introduce z = 1/r, and the metric becomes

dz2 1 ds2 = −f(z)dt2 + + (dx2 + dy2), (5.17) z4f(z) z2

with f(z) = 1/z2 − z, where we have set L = M = 1. The horizon is at z = 1, and the asymptotic boundary is at z = 0. In the assumption that ψ = ψ(z),At = At(z),Ax = Ax(z),Ay = 0, and the radial component is gauged out Ar = 0, the equations of motion of the ﬁelds can be derived, f 0 1 A2 z2A2 2 ψ00 + ψ0 + t − x + ψ = 0 (5.18) f z4 f 2 f L2f ψ2 A00 − 2 A = 0, (5.19) t fz4 t 2 f 0 A A00 + + A0 − 2ψ2 x = 0. (5.20) x z f x z4f

As usual, regularity condition of At has to be imposed at horizon, At = 0 at z = 1, which leads to some constraints(at z = 1), 2 1 zψ0 = ψ − z2A2ψ2 (5.21) 3 3 x 2 ψ A0 = − ( )2A (5.22) x 3 z x At = 0. (5.23) Chapter 5. Holographic Superconductivity 79

The equations of motion also give the asymptotic behavior of the ﬁelds(at z = 0),

2 ψ ∼ Ψ1z + Ψ2z + ... (5.24)

At ∼ µ − ρz + ... (5.25)

Ax ∼ Sx + Jxz + ... (5.26)

According to the gravity/gauge field duality, µ, ρ are the chemical potential and the density of charge carrier in the dual field theory, and Jx corresponds to the current, while Sx is the dual current source. Either of Ψ1 or Ψ2 can be the expectation value, with the other turned off,

Ψi ∼ hOii, with ijΨj = 0. (5.27)

In this section and below, we mainly consider the case Ψ1 = 0, with the condensation of hO2i. √ hO i T Sx Jx 2 hO1i With the analysis of mass dimensions, the dimensionless parameters are ( µ , µ , µ2 , µ , µ ). In practice, we can fix the temperature and evaluate everything in unit of 1/µ, and effectively smaller 1/µ corresponds to lower temperature. 1 Sx Now one has two parameters { µ , µ }, and some typical values of fields are drawn in Fig. 5.4.

ΨHzL Ax 0.8 2.0

0.6 1.8

1.6 0.4

1.4 0.2 1.2

z z 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

ΨHzL Ax

0.030 7.015

0.025

0.020 7.010

0.015

0.010 7.005

0.005

z z 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

1 Sx Figure 5.4: The fields {ψ(z),Ax(z)} at different parameters, { µ , µ } = {0.174, 0.369} for top two plots, and {0.087, 0.609} for the bottom plots. The one with low temperature, 1 effectively with smaller µ , has vanishing ψ at horizon z = 1. Chapter 5. Holographic Superconductivity 80

√ It is natural to evaluate the fields and the condensation Ψ2/µ with Ψ1 = 0 with the parameters 1/µ or Sx/µ fixed respectively. Fig.5.5 shows the condensation with 1/µ fixed, while Fig.5.6 is for the case with Sx/µ fixed. An order changing of the phase transition is found in each case, and a clearer whole picture is drawn in Fig.5.7. The phase space and the transition order are described in the same figure as well. Basically, superconducting phase appears at small value of (1/µ, Sx/µ), while the normal phase is the one at higher value of these two parameters. After all, there is another condensation mode, with {Ψ2 = 0, Ψ1 6= 0}, and the appropriate dimensionless quantity to be evaluated is Ψ1/µ. It has a similar behavior of the parameters (1/µ, Sx/µ), and we draw the 3D plot in Fig.5.8.

5.2 Superconductivity from D3/D7: Holographic Pion Superﬂuid

The phase structure of QCD (and similar theories) at finite isospin and baryon chemical potential is an interesting issue. At high enough isospin density the color singlet mesonic flavor degrees of freedoms (e.g. pions) may go through a Bose-Einstein condensation. The physical motivation to study such pion superfluid formed at high isospin density is related to the investigation of neutron stars, isospin asymmetric nuclear matter and heavy ion collisions at intermediate energy. Unfortunately this set of problems are difficult to tackle numerically due to the complex nature of the action. Various approaches including lattice simulations are used to investigate the nature of the QCD phase diagram at finite isospin chemical potential and existence of a superfluid like state is argued [107–112]. One way to address various facts about gauge theory is to use the gauge gravity duality [6] and study a supergravity/tree level string theory to learn about large-N gauge theories. Although such examples do not include QCD or even pure YM theory yet, many qualitatively similar models has been constructed. At the idealized limit where the ratio of flavor and color degrees of freedom is small, one can introduce probe branes in the gravity background to study flavor physics [113]. In this scenario, the baryon/isospin chemical potential maps to chemical potential for various gauge fields living in the brane. The issue of baryon and isospin chemical potential has been addressed in various type of brane systems [114–120]. One issue which is relatively less discussed in string theory literature is the issue of flavor superconductivity. There are isospin charged bosonic states with mass of O(1) (e.g pions in QCD), which may be thought as strings which has endpoints on different flavor branes. Such a state may naturally condense as we turn on the isospin chemical potential. Here2 we discuss such a scenario in a D3/D7 system in the zero quark mass limit. We introduce a couple of branes with no separation between the branes and the branes extend through the horizon of an AdS5 black hole. We turn on a chemical potential corresponding to the SU(2) isospin gauge field living in the world volume of the branes and study the resulting superconducting phase transition and various properties associated with it.

2This section is basically from our work[1]. Chapter 5. Holographic Superconductivity 81

0.7

0.6 0.6

0.5 0.5

0.4 0.4 2 Y Μ "######### 2 Y 0.3 Μ 0.3 "#########

0.2 0.2

0.1 0.1

0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 S S Μ Μ (a) 1/µ = 0.146 (b) 1/µ = 0.217

2.5 0.10

2.0 0.08

1.5

II II 0.06 F F 1.0

0.04

0.5 I

0.0 0.02 I

-0.5 III 0.00 0.35 0.40 0.45 0.0 0.1 0.2 0.3 0.4 0.5 0.6 S S Μ Μ (c) 1/µ = 0.146 (d) 1/µ = 0.217 √ Figure 5.5: (Colorful) The condensation Ψ2/µ and the free energy F as function as Sx/µ with 1/µ ﬁxed. At smaller 1/µ, i.e. lower eﬀective temperature, the transition turns into 1st order from 2nd order. The one with smaller free energy dominates. Chapter 5. Holographic Superconductivity 82

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4 2 2 Y Y Μ Μ "######### "######### 0.3 0.3

0.2 0.2

0.1 0.1

0.0 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25 1 1 Μ Μ (a) 1/µ = 0.5 (b) 1/µ = 0.1

4 100

3 80

60 II II

F 1 F I 40

III 20 -1 I

0 -2 0.120 0.125 0.130 0.135 0.140 0.145 0.150 0.05 0.10 0.15 0.20 0.25 0.30 1 1 Μ Μ (c) 1/µ = 0.5 (d) 1/µ = 0.5 √ Figure 5.6: (Colorful) The condensation Ψ2/µ and the free energy F as function as 1/µ with Sx/µ ﬁxed. At higher value of Sx/µ, the transition turns into 1st order from 2nd order. The one with smaller free energy dominates. Chapter 5. Holographic Superconductivity 83

1.0

0.8

0.6 Normal 0.0 0.8 Μ S

0.6 0.2 0.4 !!!!!!!!!Y 2 S 0.4 x Superconducting Μ Μ 0.2 0.4 0.2

0.0 0.1 0.0 0.2 0.00 0.05 0.10 0.15 0.20 0.25 0.6 1 0.3 1 Μ Μ √ Figure 5.7: (Colorful) The condensation Ψ2/µ as function of parameters (1/µ, Sx/µ) is shown in the left plot. In the right plot, the phase space is divided into normal and superconducting in the parameter space 1/µ, Sx/µ. The nature of the phase transition changes from second order (blue curve) to ﬁrst order (red curve) at the special point (green dot).

5.2.1 General Setup

5 Let us consider AdS5 × S in Poincare co-ordinates, dr2 r2 ds2 = −f(r)dt2 + + (dx2 + dx2 + dx2) + L2dΩ2 (5.28) f(r) L2 1 2 3 5

with r2 M f(r) = − , (5.29) L2 r2 where L is the radius of the anti-de Sitter space and M is related to the mass of the black hole. In this section we will adopt the convention M = L = 1. The temperature of the black hole (and also of the boundary ﬁeld theory) is given by 1 T = . (5.30) π It is more convenient to analyze the system by making a coordinate transformation z = 1/r. The metric becomes: dz2 1 ds2 = −f(z)dt2 + + (dx2 + dx2 + dx2) + dΩ2 (5.31) z4f(z) z2 1 2 3 5 Chapter 5. Holographic Superconductivity 84

3 0.0

2 Y1 0.2 Μ Sx 1 Μ 0.4 0 0.4 0.6 0.8 0.6 1 1.0 Μ

Figure 5.8: (Colorful) With diﬀerent condensation mode Ψ2 = 0, the condensation Ψ1/µ as function of parameters (1/µ, Sx/µ). Chapter 5. Holographic Superconductivity 85

with 1 f(z) = − z2. (5.32) z2 The horizon is now at z = 1, while the conformal boundary lives at z = 0. In this background we will introduce two (or more) co-incident D7-branes. For simplicity we will consider the zero quark mass embedding, where the brane ﬁlls the whole AdS5 and wraps the maximal S3 of S5. In this limit the eﬀective induced metric on the brane will just be 3 the AdS5 × S metric. dz2 1 ds2 = −f(z)dt2 + + (dx2 + dx2 + dx2) + dΩ2 (5.33) brane z4f(z) z2 1 2 3 3

The eﬀective action for the brane ﬁelds is the Born-Infeld action, Z √ 8 0 SDBI = −T7 d x G + 2πα F. (5.34)

In order to consider the system at finite isospin chemical potential, we will add a pair of D7 brane probes. In this case F is a U(2) field strength on the world volume of the probes. We will not focus on baryonic U(1)B and only investigate terms containing SU(2) isospin gauge fields. The string states which have their endpoints on different branes are charged under the isospin SU(2). The exact form of the non-Abelian DBI action is unknown [121]. To proceed further we will expand the action to leading order in λYM4 keeping only Yang Mills terms.3 Such a simplification has also been employed in studying other aspects of holographic QCD such as the meson spectra and baryon masses[123–125]. Such an approximation will be more accurate in the limit where the non-Abelian field strengths are small. The effective action now takes the form 0 2 Z √ Z √ (2πα ) 8 2 8 2 SDBI = −T7 d x −GT rF ∝ Nc d x −GT rF (5.35) gs

where we have scaled out 7 + 1 dimensional Yang Mills coupling g7 and

F a = ∂Aa + abcAbAc (5.36)

On a phenomenological level (5.35) may be thought as an effective holographic model of flavor superconductivity. The setup is very similar to that of [104, 105, 126], where the non-Abelian gauge field

3The square root form of the DBI action was considered in [122]. Their main result that the existence of condensate and superconductivity (i.e. a pole of imaginary part of the conductivity at zero frequency) is similar. However they find out finer details of the frequency response like other poles. It should be noted that our approximation of neglecting all the higher order terms is not a controlled approximation, as the chemical potential µ (∝ F 2 in the unit of α0) is large (µ = 4) near the phase transition. Keeping the whole DBI action changes the exact quantitative results but order of the magnitudes of quantities do not seem to change much. For example we have checked that in our convention phase transition occurs at µ ≈ 7 for the DBI corrected action. The scalar condensates (see Sec. 5.2.5) forms at µ & 15, although at such a low temperature numerics is less reliable. Interestingly, near integral value µ ≈ 7 implies a possibility of finding an exact solution as in Sec. 5.2.2. Chapter 5. Holographic Superconductivity 86

is shown to condense at low temperatures in an AdS black hole background. Due to the non-Abelian nature of the SU(2) symmetry, the τ 1 and τ 2 components of the gauge field are charged under the τ 3 component. Hence turning on a chemical potential for the τ 3 component of the gauge field may lead to a condensation of the other two. This mechanism is very similar to condensation of a U(1) charged scalar discussed in [13]. Turning on a 3 chemical potential for the τ component of the gauge field breaks SU(2) to U(1)I . A con- 1 2 densation of the τ or τ component of the gauge field further breaks the U(1)I symmetry. It should be mentioned that unlike U(1)B where all charged states are baryonic and have masses of order O(N), this U(1)I theory has charged states (mesons) with O(1) masses and their condensation can naturally be studied in terms of the probe brane picture. We start with the ansatz4

3 1 A = Atτ dt + Bx1 τ dx1 (5.37)

We will assume spatial homogeneity in the field theory directions and our fields will only have dependence on the radial coordinate. The equations of motion for the fields in this coordinate system are:

A0 B2 A00 − t − 2 x1 A = 0 (5.38) t z z2f t f 0 1 1 B B00 + + B0 + A2 x1 = 0 (5.39) x1 f z x1 z2f t z2f

For regularity at the horizon we will have to set At = 0 at z = 1. Since we have a set of coupled equations, this will in turn give the following conditions at the horizon (z = 1).

0 Bx1 = 0 (5.40) At = 0 (5.41)

Examining the behavior of the ﬁelds near the boundary, we ﬁnd

2 At ∼ µ − ρz + ... (5.42) 2 Bx1 ∼ Mx + Wxz + ... (5.43)

Using gauge/gravity duality, µ, ρ are mapped to the isospin chemical potential5 and charge density in the dual ﬁeld theory, respectively. Wx is mapped to the expectation value of a meson operator which condenses at low temperatures. We will set the non-normalizable mode Mx to zero. In what follows we ﬁrst establish that the mesonic condensate forms below a critical temperature. We compute the time dependent conductivity by turning on

4Some other ansatzes with the spatial gauge field component along S3 are studied in appendix C 5It is known that such a system at finite isospin chemical potential and zero temperature is unstable due to runaway Higgs VEV as the zero VEV configuration becomes a local maximum of the effective potential [127]. What happens at finite temperature is not completely clear. However one may imagine a system where Higgs VEV is artificially fixed to zero. We would like to thank Nick Evans for pointing this out. Alternatively one may consider a supersymmetric D3/D5 system which does not have such instabilities. The resulting equations of motion are almost the same as in the case of a D3/D7 system. Chapter 5. Holographic Superconductivity 87

a spatial component for the isospin current as a ﬂuctuation.

iωt 3 Ax3 = X(z)e τ dx3 (5.44)

The equation of motion for Ax3 is 1 f 0 X ω2X X00 + ( + )X0 − B2 + = 0 (5.45) z f x1 z2f z4f 2

−iω/4 We will choose infalling boundary condition at the horizon Ax3 ∝ (z − 1) . Asymptot- ically, 2 X ∼ Sx + Jxz + ... (5.46)

Jx corresponds to the isospin current, while Sx gives the dual current source (superﬂuid velocity). The conductivity 6 is given by

J σ = Re x (5.47) iωSx X can be normalized to one at the horizon. As we will see, the conductivity has a pole at ω = 0. This suggests that there is a DC supercurrent solution. To ﬁnd such a solution we solve the following set of coupled time independent equations [15, 129]. Note here the

effects of Ax3 = X(z) on the other components of the gauge fields are taken into account, so this is not a fluctuation around the condensate formed by Bx1 . In this case the field Ax3 3 has the form Ax3 = X(z)τ dx3.

A0 B2 A00 − t − x1 A = 0 (5.48) t z z2f t f 1 1 B B00 + + B0 + A2 x1 − B3 − X2B = 0 (5.49) x1 f 0 z x1 z2f t z2f x1 x1 1 f 0 X X00 + ( + )X0 − B2 = 0 (5.50) z f x1 z2f

For the regularity of Ax3 at the horizon,

0 2 X X = − Bx1 (5.51) 4 z=1 As we will see shortly, the system (5.48) reveals a rich phase structure as the boundary

value of Ax3 is tuned. √3 √3 √3 T ω Sx Wx T ω Jx Wx The convenient physical parameters for us are ( µ , µ , µ , µ ) or ( µ , µ , µ , µ ). We √3 Wx T Sx will use µ is an order parameter and plot it as a function of ( µ , µ ). In practice, we choose to keep the temperature ﬁxed and vary µ in this section. The components of gauge ﬁelds on

6 iω There is a logarithmic correction to the conductivity in ﬁve dimensions [128], under which σ → σ + 2 . However such a term depends on the choice of renormalization and physical quantities like mass gap etc do not depend on it. Chapter 5. Holographic Superconductivity 88

Bx 0.25

0.20

0.15

0.10

0.05

z 0.2 0.4 0.6 0.8 1.0

Figure 5.9: Plot of the zero mode at µ = 4 the three-sphere may also condense through a similar mechanism. We will examine these cases in the appendix. An important question is whether the isospin chemical potential would modify the embedding of the ﬂavor branes. We will leave a detailed analysis of this for a future project. For now we will just check whether the transverse scalars also condense at low temperatures (Section 5.2.5).

5.2.2 Phase Diagram At high temperatures (or, equivalently, small values of µ) there is only one set of solutions to (5.39) given by

2 At = µ(1 − z ) (5.52)

Bx1 = 0 (5.53)

This should be interpreted as an isospin-charged black hole, where gauge fields are confined to the D7 brane. The dual gauge theory interpretation is a deconfined plasma with non-

zero isospin charge. As we increase µ the effective mass of Bx1 in (5.39) decreases and Bx1 develops a zero mode at µ = µc = 4. The existence of this zero mode can be analytically demonstrated. Substituting At from (5.52) into the second of (5.39) we get (this small fluctuation analysis does not depend on any possible cubic terms and is therefore true for other scalar field ansatzes considered later)

f 1 µ2(1 − z2)2 B00 + + B0 + B = 0 (5.54) x1 f 0 z x1 z4f 2 x1

The above equation has an analytic solution for µ = 4 given by

z2 B (z) = (5.55) x1 (1 + z2)2

The plot of this zero mode is shown in 5.2.2. Any further increment of µ leads to a

condensation of Bx1 . Hence for 1/µ < 0.25 the solution develops a new branch with a

non-zero value of Bx1 . Such a solution can be numerically constructed. The associated transition seems to be of second order from our numerics. In Fig. 5.2.2 we show a plot 1 of the condensate with µ (a plot of the corresponding free energy is provided in Section Chapter 5. Holographic Superconductivity 89

Wx Μ3

0.25

0.20

0.15

0.10

0.05

1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Μ

Figure 5.10: Plot of the condensate with 1/µ

5.2.5). 3 At low temperatures we ﬁnd that the condensate levels oﬀ at Wx/µ ≈ 0.26. In terms of 3 3 3 the critical temperature Tc the condensate strength can be expressed as Wx ≈ 0.264 π Tc ≈ 1 3 3 515.94Tc or Wx ≈ 8.01Tc.

Speed of Second Sound

5 The boundary field theory which is dual to the D3/D7 system in AdS5 × S behaves like a superfluid below the critical temperature. Superfluids are known to exhibit modes known as second sound which are basically temperature waves propagating through the fluid. For a hydrodynamic discussion see [129], where this was computed at zero superfluid velocity for the Abelian Higgs model on AdS4. We also compute the speed of second sound in our case. The superfluid velocity now corresponds to Sx/µ, which we set to zero for this computation. The main relation is (see Eq. (18) of [129]): ρ v2 = s , (5.56) 2 ∂2P µ ∂µ2

where ρs is the density of the superfluid component and P is the pressure. The pressure can be expressed in terms of the total fluid density ρ by using the equation of state of a perfect fluid P = µρ/(d − 1) where d = 4 is the dimension in the fluid (boundary) theory7. 2 Using this it is fairly simple to compute v2. We present the result in Fig. 5.11, where we 2 2 2 plot v2 as a function of 1/µ. At high values of µ, v2 approaches a limiting value v2 ≈ 0.32.

5.2.3 Frequency Response In this section we will study the frequency dependent conductivity of the spatial component

of the isospin current (5.44). In the absence of a condensate of Bx1 the frequency response can be exactly solved for [128], and is expressed in terms of digamma functions. In the

7The perfect ﬂuid approximation is valid here since we are not considering any backreaction due to the metric. Hence there is no viscosity correction which originates from ﬂuctuations of the metric. Chapter 5. Holographic Superconductivity 90

2 v2 0.35

0.30

0.25

0.20

0.15

0.10

0.05 1

0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 Μ

Figure 5.11: Speed of second sound as a function of 1/µ

presence of a condensate of the Bx1 field an analytic solution is difficult, but we can still numerically calculate the conductivity using (5.47). First, we solve (5.39) to determine the condensate. This solution is then taken as a fixed background on which we solve (5.45) for the conductivity σ. In Fig.5.12 we plot σ as a function of the frequency ω for ns various values of the parameter µ. We find that Im[σ(ω)] ∼ ω as ω → 0, where ns is the superfluid density. Fig.5.13 shows a plot of ns with 1/µ. Near µ = µc, ns becomes proportional to µ−µc, vanishing at µ = µc. Fitting a linear function near the critical point 2 we get ns ∝ 0.1µc (µ − µc). The pole at ω = 0 for Im[σ(ω)] implies Re[σ(0)] ∼ πnsδ(ω) + terms regular in ω. This delta function singularity of the real part of sigma is not captured in the numerics directly. However this corresponds to superconductivity/superfluidity and consequently we can find a supercurrent/superfluid solution (see Sec. 5.2.4). Unlike [122] we do not get any low temperature resonances in the conductivity. Our result is more similar to the zero mass Abelian-Higgs system presented in [128].8 As R Re[σ]dσ is a temperature invariant quantity, the delta function at ω = 0 is compensated by a dip in Re[σ] at low frequencies. The dip becomes more prominent as we lower the temperature (i.e., increase µ). It is clear from the plot (Fig.5.12) that at low temperatures ∆g (large µ) Re[σ] → 0. In fact it is expected that at low temperatures Re[σ] ∼ exp(− T ), where ∆g is the energy gap of the system. Also looking at the zero temperature limit of the real part of the conductivity we see that Re[σ] = 0 for ω ≤ ∆p. ∆p is similar to the energy of a “Cooper pair”. The ratio n = ∆p gives important information about the nature of g ∆g the condensate. From our numerics we calculate

ng ≈ 1.2 (5.57)

At high frequencies Re[σ] computed at diﬀerent temperatures (below Tc) approaches the zero condensate value.

8 It seems that in an Abelian-Higgs system in AdS5 resonances occur near the conformal mass. [130] Chapter 5. Holographic Superconductivity 91

Σ Re@ D Σ Μ Im@ D Μ

1.5

1.0

Ω 0.5 1.0 1.5 2.0 Μ

0.5 -2

Ω -4 0.5 1.0 1.5 2.0 Μ σ σ (a) Plot of real part of µ (b) Plot of imaginary part of µ

σ Figure 5.12: (Colorful) Plots of the real and imaginary part of µ with µ = 7.57µc, 2.52µc, 1.71µc, 1.37µc, 1.13µc (gradually from red to green curves). The blue curve is for the exact frequency response at µ = µc = 4.

ns Μ2

0.07

0.06

0.05

0.04

0.03

0.02

0.01 1 0.15 0.20 0.25 Μ

Figure 5.13: Plot of superﬂuid density with 1/µ. Chapter 5. Holographic Superconductivity 92

0.12

0.10

0.08 x 3 Μ W 0.06

0.04

0.02

0.00 0.2 0.3 0.4 0.5 0.6

Sx Μ

3 Figure 5.14: Plot of Wx/µ as a function of Sx/µ at diﬀerent values of µ. The values of µ range from 1.24µc to 1.97µc from left to right.

5.2.4 Effect of Stationary Isospin Current In this section we investigate the effects of turning on a finite time-independent isospin X a 3 1 3 field (recall Aµ(r) = At(r)τ dt + Bx1 (r)τ dx1 + X(r)τ dx3). We tune the boundary value of X, Sx/µ at a fixed isospin chemical potential µ to investigate the phase structure. At

high enough µ and in absence of X the gauge component Bx1 condenses. As we can see

from (5.48), the effect of X is to increase the effective mass of the Bx1 field. This will

cause the Bx1 condensate to weaken with increasing Sx/µ. Indeed, this happens, as we can see from Fig. 5.2.4. Here we plot the condensate strength as a function of the isospin current source Sx/µ for different chemical potentials µ. µ increases from left to right. For strong enough Sx/µ (above a critical value) there is a phase transition to the normal (non- superfluid) state. The order of this phase transition seems to be µ dependent. For high µ (compared to µc) the phase transition is first order, i.e. the system discontinuously jumps to the normal state above the critical current velocity Sx/µ. For µ close to µc the transition becomes second order. The order of the transition changes near µsp = 1.4µc. Note that in each case the condensate approaches a limiting value at low values of Sx/µ, and this limiting value decreases with decreasing µ. In order to properly see the transition from first to second order, we also plot the difference in free energies 9 of the normal and superconducting branches. In Fig. 5.15 the left hand plot is for µ = 1.97µc while the right hand plot is for µ = 1.24µc. We see the typical swallow tail shape indicating the abrupt change in dominance from the normal to the superconducting branch at low values of Sx/µ. We can understand this curve Fig. 5.15(a) in the following manner. As Sx/µ is lowered from above the critical value Sx,c/µ, at first there is only a normal branch (I). At some value Sx,N /µ = 0.571 two new branches

9Free energy is calculated from the action (5.34). The value of the Lorentzian action itself is the free energy. It should be noted that our evaluation of free energy is primarily numerical and there may be subtlety in the discussion of the phase transition. A better analytic understanding with possibly the full DBI action will be interesting. There may also be subtlety associated with boundary terms and gravity back reaction may be important in some situation. Chapter 5. Holographic Superconductivity 93

1.0

0.5 III -1 I

0.0 -2 F -3 II F -0.5

II -4 -1.0 -5 -1.5 -6 -2.0 0.550 0.555 0.560 0.565 0.570 0.575 0.580 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Sx Sx

Μ Μ

(a) µ = 1.97 (b) µ = 1.24

Figure 5.15: Plot of the diﬀerence in free energies (F ) of the normal and superconducting branches as a function of Sx/µ.

are nucleated: one of these is stable (II), while the other one is unstable (III). The stable branch starts out with a higher free energy than the normal branch, but as Sx/µ is lowered further these two branches intersect at Sx,c/µ = 0.568, where the branch (II) has the same free energy as branch (I). This is the ﬁrst order phase transition point below which the system jumps to the superconducting branch (II) which now has lower free energy. At lower values of the chemical potential µ shown in Fig. 5.15(b) the transition is continuous between branches (I) and (II) at Sx,c/µ = 0.038.

5.2.5 What Condenses and What Doesn’t

3 There are various type of bosonic fields living on the branes which are charged under Aµ and 3 may condense as we turn on a chemical potential for Aµ. These states come from strings ending on different branes. This includes gauge fields proportional to τ 1 and τ 2. Together with A3, the trio corresponds to a vector/isovector “meson”10 in the boundary theory. In this section we consider an ansatz like (5.37), but more generally one may consider 3 1 2 A = Atτ dt + Bx1 τ dx1 + Cxτ dx2. In this case the action contains a term proportional to 2 2 Bx1 Cx which describes a repulsive interaction between these two fields, so one may expect 1 2 that both do not condense together. Also, an ansatz of the form Bx1 (τ dx1 + τ dx2) [104] will generally have a higher free energy than what we have chosen. Another possibility is the condensation of transverse scalar fields which exist on the brane. Brane fields are invariant under only a SO(4) × SO(2) subgroup of the full R- symmetry group SO(6) of S5. A pair of transverse scalar fields correspond to one SO(2)-

10 As mq = 0 to begin with in our case, there is no stable meson. We only have quasinormal modes corresponding to various brane ﬁelds [131]. The dual interpretation of such modes are mesons decaying in the gauge theory plasma. Chapter 5. Holographic Superconductivity 94

S Μ4

0.07 0.06 0.05 0.04 0.03 0.02 0.01 1 0.05 0.10 0.15 0.20 0.25 Μ

Figure 5.16: Plot of action for phases Bx1 6= 0 (upper curve) and φ 6= 0 (lower curve). charged iso-vector scalar in the boundary theory. The Born-Infeld type eﬀective action for such scalars is given in the appendix. We choose a general ansatz suitable for our case,

1 2 Φ1 = φ1τ , Φ2 = φ2τ , (5.58)

The EOM’s are, 1 f 0 1 A2 φ00 + (− + )φ0 + t − φ2 φ = 0, (5.59) i z f i z4f f j i where i, j = 1, 2 and i 6= j. From the discussion in the previous paragraph it is clear that due to the repulsive interaction term, a preferred configuration will have one of the two φi’s turned off. One may try to find when such a field becomes unstable in a fixed At background. From our numerics we find out that for the solution (5.53) such a thing s happens at µc ≈ 6.57 = 1.64µc and greater than our µc = 4 value for Bx1 field. Hence when we gradually decrease the temperature of our system, Bx1 condenses before any scalar degree of freedom. One may further ask whether the resulting superconducting phase with a

Bx1 condensate has an instability towards φ fluctuations. Our numerics answer this question negatively. It seems that some type of ”blocking” mechanism stops further condensation of more fields. However that does not rule out the possibility of a first order transition between Bx1 condensed phase and φ condensed phase. We plot the free energy of both the phases to investigate a possible first order transition in Fig.5.16.

It seems that the Bx1 6= 0 phase always dominates. However our numerics is not very reliable for the parameter range µ > 10µc. Also, we did not exhaustively search for the possibility of various mixed phases. The case of gauge fields with S5 indices is similar to that of the scalar fields (see appendix), at least for small fluctuations. Hence the blocking mechanism discussed above will work for them too and these fields will not also condense. However we have not investigated the possible phases and possibility of a first order transition in great detail for these fields.

5.2.6 Conclusion Let us conclude this subsection here. We explored a model which provides a realization of holographic superconductivity or superﬂuidity in a string theory setup. We have studied a couple of probe D7 branes in Chapter 5. Holographic Superconductivity 95

5 an AdS5 × S background. We have introduced a finite isospin chemical potential (i.e. potential for some of the world volume gauge fields) and have found the existence of a flavored superconducting state at high enough values of this chemical potential. We have studied the frequency dependent conductivity and have found a delta function pole in the zero frequency limit. This indicates a superconductor-like phase. Consequently we have found a superfluid/supercurrent type solution and have studied the resulting phase diagram. The superconducting transition changes from second order to first order at a critical a superfluid velocity. The holographic dual of such a string theory system is large Nc, N = 4 supersymmetric gauge theory with Nf Nc. In a dual gauge theory such a superconducting state is characterized by mesonic condensates. We have studied various properties of the superconducting system like energy gap, second sound etc. We have discussed the possibility of a first order transition between various possible condensate. It is also important to check whether the isospin chemical potential modifies the embedding of the flavor branes and whether the transverse scalars actually condense. In our case they do not, but for some other setup they might. It would be interesting to address such questions in more detail. In QCD, mesonic condensates (pion superfluids) have been argued to exist at finite isospin chemical potential. A natural extension of our work will be to study more realistic holographic models like [119, 123] or AdS/QCD like theories. It would be interesting to study such phenomena in cases where the fundamental degrees of freedom are fermionic. Here we focused on the zero quark mass case, which may be thought of as a high temperature limit. It would be natural to extend our work to the case of finite quark mass. Another interesting issue to investigate is flavor backreaction [16, 132, 133]. The mesonic operators which condense in our case are color singlets and do not lead to color superconductivity. It would be interesting to study some models with color superconductivity.

5.3 Gapless and Hard-gapped Holographic Superconductors

We have shown above at low temperature, generally a superconducting phase exists, and the normal phase is suppressed. The transition order can be either first order or second order at different order parameter(s). Let us keep in mind, all the discussion above are at low but non-zero temperature. As a matter of fact, nontrivial properties appear at zero temperature. As proved in [134], there is a no-hair theorem for extremal black branes. In the most familiar background of holographic superconductor, if the scalar condenses, the horizon of the black hole has to shrink to zero size. And the solution does not have the same scaling invariance as the finite superconductor solution, which is usually able to scale the horizon radius of some factor, and can be rescaled to be unit length. The zero temperature solution is firstly discussed in [84]. In the work [84], the authors discussed two families of solutions, one of which is with m = 0, and the other has m2 ≤ 0, q2 > |m2|/6. In both cases, the horizon shrinks to zero size in the extremal limit. The first one goes to AdS4 space limit, while the second has a null singularity at the vanishing horizon, r = 0. It is argued that since they are the smooth Chapter 5. Holographic Superconductivity 96

limit of black hole with smooth horizons, the null singularity is physical. When m2 = 0, the curvature is smooth everywhere, however the derivatives might diverge. There is a special value of the parameter q, with the solution completely smooth though the horizon. The ordinary superconductivity in AdS4, for instance in Fig.5.2, has a gap of Re[σ] at small value of frequency ω. It is usually thought that Re[σ] exponentially vanishes at zero temperature limit, and thus it is a hard-gap. However as shown in the work, the conductivity does not vanish so quickly which hints of a soft-gap, which is consistent with [135, 136]. Another interesting solution is with m2 > 0, some of value of parameter sets (m2, q) result in a Lifshitz-like infra spacetime, which is also found in [137]. We are interested in this solution and would postpone it as our future research project.

5.3.1 Zero Temperature Holographic Superconductor the Setup We are interested in the Lagrangian with Maxwell ﬁeld and a charged complex scalar in asymptotic AdS4 geometry, 6 1 L = R + − F µνF − |∇ψ − iqAψ|2 − V (|ψ|), (5.60) L2 4 µν and the metric ansatz,

dr2 ds2 = −g(r)e−χ(r)dt2 + + r2(dx2 + dy2), (5.61) g(r)

with the ﬁelds

A = φ(r)dt, ψ = ψ(r). (5.62)

The equations of motion can be found, working in the length unit of L = 1,

g0 χ0 2 q2φ2eχ V 0(ψ) ψ00 + − + ψ0 + ψ − = 0, (5.63) g 2 r g2 2g χ0 2 2q2ψ2 φ00 + + φ0 − φ = 0, (5.64) 2 r g rq2φ2ψ2eχ χ0 + rψ02 + = 0, (5.65) g2 1 χ0 rφ02eχ rV (ψ) g0 + − g + − 3r + = 0. (5.66) r 2 4 2

The above equations have two sets of scaling symmetries:

r → ar, (t, x, y) → (t, x, y)/a, g → a2g, φ → aφ, (5.67) Chapter 5. Holographic Superconductivity 97

and

eχ → a2eχ, t → at, φ → φ/a. (5.68)

It does not lose the generality to set the horizon r+ = 1, and χ = 0 at r → ∞. And the ﬁelds have asymptotic behavior

ρ ψ(λ) ψ(3−λ) φ = µ − , ψ = + , (5.69) r rλ r3−λ √ where λ = (3 + 9 + 4m2)/2, with m the mass of scalar ψ. As usual, we would set ψ(3−λ) = 0, and numerical calculation shows that it is much more stable and reliable than the other mode ψ(λ) = 0. At zero temperature, a natural comparison of the solution is to extremal Reisner- 2 Nordstrom AdS(RN-AdS) black hole, which has a peculiar near horizon limit, AdS2 × R , dr2 ds2 = −6(r − 1)2dt2 + + dx2 + dy2, (5.70) 6(r − 1)2 √ with the electric ﬁeld φ = 2 3(r − 1), and temperature T = (12 − ρ2)/16π → 0. The equation of scalar in the near horizon limit turns to be

2 m2 m2 − 2q2 ψ + ψ − eff ψ = 0, m2 = , (5.71) ,r˜r˜ r˜ ,r˜ r˜2 eff 6 where we introducedr ˜ = r − 1. One natural conjecture is the eﬀective mass of the scalar 2 should satisfy the Breitenlohner-Freedman(BF) bound of AdS2, mBF (AdS2) = −1/4, i.e. 3 m2 − 2q2 > − . (5.72) 2

2 2 For reference, we should remind ourselves that the BF-bound of AdS4 is m > mBF (AdS4) = 9 − 4 . The deviation of the diﬀerent BF-bound opens a window for the scalar’s instability.

Extremal Limit of Hairy Black Holes The potential is restricted in the simplest form, V (ψ) = m2ψ∗ψ, and the scalar can be taken at real value later. There are basically three families of solutions, with m = 0, (m2 < 0, q2 > |m2|/6), and m2 > 0 respectively. m2 = 0

The ansatz of the solutions are tried as

2+α 2(1+α) 2(1+α) 2 2(1+α) φ = r , ψ = ψ0 − ψ1r , χ = χ0 − χ1r , g = r (1 − g1r ). (5.73) Chapter 5. Holographic Superconductivity 98

The boundary conditions near horizon can be derived from the equations of motion,

α2 + 5α + 61/2 α2 + 5α + 6 qψ = , χ = eχ0 , (5.74) 0 2 1 4(α + 1) 1/2 α + 2 qeχ0 α2 + 5α + 6 g = eχ0 , ψ = . (5.75) 1 4 1 2(2α2 + 7α + 5) 2

One typical numerical solution is shown in Fig.5.17.

0.1

0.08 1.5 0.06 Ψ 1 g 0.04

0.5 0.02

0 0 0 0.1 0.2 1 0.4 0.5 0 0.1 0.2 1 0.4 0.5 2 3 2 3 rΜ rΜ

Figure 5.17: The dashed curve is at zero temperature, with the parameters λ = 3, q = 1, compared with successive low temperature hairy black hole solutions. Most of the scalar ψ seems to be constrained behind a sphere, and the metric factor g(r) appears to have a double zero at this radius. These plots are from [84].

Compared with some low temperature hairy black hole solution, it is clear the extremal solution is a smooth√ limit of them. In this state, most of the scalar sits behind a “horizon”, the value 1/2 3 in the ﬁgure, and the metric factor g(r) seems to drop to small value at this radius. It is a solitonic solution. The vanishing of g gives a AdS4 spacetime near origin. On the other hand, one can numerically check in a special limit of the parameters, the solution approaches extremal RN-AdS black hole. Varying scalar’s charge q can produce a family of solutions, and the variable α appearing in the powers can be drawn as function of q, Several properties√ of α(q) should be emphasized here. Firstly α seems to diverge in the limit of q → 3/2, which results in the extremal RN-AdS black hole solution. Another is that at some speciﬁc value of charge q ≈ 1.018, α = 0, the solution is completely smooth through the horizon, while the general has a diverging derivatives of the curvature. Furthermore, α approaches a constant at large q. m2 < 0, q2 > |m2|/6

The ansatz has to be diﬀerent from before, (near r = 0),

1/2 2 β 1/2 ψ = A(− log r) , g = g0r (− log r), φ = φ0r (− log r) , (5.76) Chapter 5. Holographic Superconductivity 99

0.3

0.2

Α 0.1

-0.1 0 3 2 4 6 8 10 2 q

Figure 5.18: α as function of q, giving a family of solution. These plots is from [84]. and with a speciﬁc choice

eχ = K(− log r)A2/4. (5.77)

To match the near horizon behavior, some restrictions are found, and the boundary conditions are determined,

ψ = 2(− log r)1/2, g = (2m2/3)r2 log r, (5.78) χ β 1/2 e = −K log r, φ = φ0r (− log r) , (5.79) with

m2 1 1 48q2 1/2 q2 > − , β = − + 1 − . (5.80) 6 2 2 m2

The horizon at r = 0 has a mild singularity, and the scalar is logarithmically diverging, with the near horizon metric

2 2 2 2 i dr ds = r (−dt + dxidx ) + 2 . (5.81) g0r (− log r)

1/2 1/2 If one introduce a new radius coordinate,r ˜ = −2(− log r) /g0 , the metric becomes

2 2 −g0r˜ /2 2 i 2 ds = e (−dt + dxidx ) + dr˜ , (5.82) which takes the horizon tor ˜ → −∞. Poincare invariance is restored near the horizon, which is also discussed in [138]. Similar behavior with diﬀerent form of potential are studies in bunch of papers, for example [136, 139]. Chapter 5. Holographic Superconductivity 100 m2 > 0

When m2 > 0, a Lifshitz-like near horizon solution arises. We are not going to the details here, but point out one fact: generally it is diﬃcult to match the special near horizon geometry to asymptotic AdS4 spacetime. However if a w-shaped potential is turned on,

V (ψ∗, ψ) = m2ψ∗ψ + u(ψ∗ψ)2, (5.83) an appropriate set of parameters (m, u, q) can match the interior and the asymptotic geometries well[137]. It would be part of our research project.

5.3.2 Conductivity and Hard-gapless Theorem In this section, we would show that the conductivity calculation can be transformed into a 1-dimensional Schrodinger’s problem, and the gap of the Re[σ] at low frequency does not suppressed as an exponential function, i.e. there is no hard-gap. −iωt Turning on a perturbation of spatial component of U(1) gauge ﬁeld, Ax(r)e , the equation of motion is

g0 χ0 ω2 φ02 2q2φ2 A00 + − A0 + − eχ − A = 0. (5.84) x g 2 x g2 g g x

One can introduce a new radial coordinate to simplify the equation,

eχ/2 dz = dr, (5.85) g the horizon is moved to z → −∞, and the asymptotic boundary is at z = 0. Now (5.84) is simply a Schrodinger’s form,

2 − Ax,zz + V (z)Ax = ω Ax, (5.86) with

2 2 2 −χ V (z) = g φ,r + 2q ψ e . (5.87)

First of all, the potential is always positive at −∞ < z < 0. At the asymptotic boundary, z = 0, V (z) = ρ2z2 + 2(qψ(λ))2z2(λ−1). Thus the potential vanishes if λ > 1, keeps as a constant if λ = 1, or diverges if 1/2 < λ < 1. On the other hand, the potential always vanishes at horizon z = −∞. If the temperature is ﬁnite, at the horizon g(r+) = 0, the potential vanishes, while at zero temperature, the potential approaches a ﬁnite limiting form which still goes to zero at z = −∞. To solve the Schrodinger’s equation above, one can extend the potential to z > 0,V (z) = 0. And clearly for z > 0, Ax has the solution

−iωz iωz Ax = e + Re , (5.88) Chapter 5. Holographic Superconductivity 101

and it takes the boundary value at z = 0,

Ax(0) = 1 + R, and Ax,z(0) = −iω(1 − R). (5.89)

(0) (1) As we shown in Sec. 5.1, if Ax = Ax + Ax /r, the conductivity is

(0) i Ax σ(ω) = − (1) . (5.90) ω Ax

(1) In terms of z, Ax = −Ax,z(0), and thus 1 − R σ(ω) = . (5.91) 1 + R The conductivity is simply connected to the reﬂection coeﬃcient, and the incident energy is the frequency ω2. Some typical potential with λ > 1 is drawn in Fig. 5.19. If the incident energy is higher 2 than the maximal potential, the conductivity is simply a constant. When ω < Vmax, there is a tunneling rate, which decreases as ω → 0. However the tunneling rate, as well as the conductivity, never vanishes since potential vanishes at horizon. Meanwhile the conductivity has a delta function density at ω = 0 due to the translational invariance, which together with the non-vanishing property indicates that the gap is never a hard-gap.

15 VHzL

Μ2 10

0 -10 -8 -6 -4 -2 0 2 z

Figure 5.19: In the ﬁgure, λ = 2, q = 10, and the higher curve corresponds to lower temperature. The potential increases as the temperature lower down. This plot is from [84]. Chapter 5. Holographic Superconductivity 102

5.3.3 P-wave Holographic Superconductor and Hard-gapped Solution In this section11 we consider non-Abelian gauge fields in the background of an extremal AdS4 black hole. The setup is quite similar to so-called “p-wave” holographic superconductors [105] and known to have a superconducting transition.12. We work at the zero temperature and found a fully gravity back-reacted solitonic solution to this system similar to [84]. Then we consider the frequency dependent conductivity by considering certain vector fluctuations on this background. This is an anisotropic system which shows different conductivity at different directions. We find the expected superconducting behavior for relevant currents. Interestingly we find that the corresponding effective potential does not vanish at the horizon (it goes to zero near the boundary). Following the argument in [84], we thus arrive at the conclusion that the holographic non-Abelian superconductor does have a finite gap for the relevant gauge field fluctuations.

Setup The Einstein-YM action for a non-Abelian gauge ﬁeld with a negative cosmological constant is given by [104],

Z √ 6 1 L = d4x −g R + − F µνF a , (5.92) l2 4 a µν where Fµν is the ﬁeld strength of an SU(2) gauge ﬁeld. To tally with our plan of considering anisotropy in the spatial direction, we choose the following ansatz for our metric

dr2 ds2 = −g(r)e−χ(r)dt2 + + r2 c(r)2dx2 + dy2 , (5.93) g(r)

and the gauge ﬁelds 13,

A = A(r)τ 3dt + B(r)τ 1dx. (5.94)

It is straightforward to ﬁnd the non-zero components of the ﬁeld strength

a a a a Fµν = ∂µAν − ∂νAµ + iq[Aµ,Aν] (5.95)

are

1 1 0 3 3 0 2 2 Frx = −Fxr = B (r),Frt = −Ftr = A (r),Fxt = −Ftx = qAB. (5.96) 11This subsection is basically from our work [3]. 12For other works in non-Abelian holographic superconductor look at [1, 104, 122, 126, 140–143]. 13Due to a repulsive term coming from the non-Abelian interactions, it is expected that a isotropic ansatz will have a quartic instability and would possibly have more free energy than the anisotropic ones [1, 105]. We left the detailed discussion of these issues for a future study. Chapter 5. Holographic Superconductivity 103

The energy momentum tensor is

1 δS 1 1 1 T = −√ matter = − g F 2 + gαβF a F a , (5.97) µν −g δgµν 2 µν 4 2 µα νβ and the non-zero components of the energy momentum tensor are,

1 g2 1 T = gA02 + e−χB02 + (qAB)2, tt 4 4r2c2 4r2c2 eχ 1 eχ T = − A02 + B02 + (qAB)2, rr 4gc2 4r2c2 4g2r2c2 1 g (qAB)2 T = eχr2c2A02 + B02 − eχ , xx 4 4 4g 1 g (qAB)2 T = eχr2A02 − B02 + eχ . (5.98) yy 4 4c2 4gc2

It is to be noted that here Txx 6= Tyy due our anisotropic ansatz. The Maxwell’s equations of A(r),B(r) are

3 00 0 2 χ0 c0 q2B2 At −→ A + A r + 2 + c − r2gc A = 0,

1 00 0 g0 χ0 c0 eχq2A2 Ax −→ B + B g − 2 − c + g2 B = 0. (5.99)

The potential term for A(r) is diﬀerent from the similar term in EYMH case due to the 1 existence of a factor r2 . The diagonal Einstein equations give, 1 c0 1 3c0 c00 eχ g q2A2B2 − g0 + − g + + + 3 = A02 + B02 + eχ , r 2c r2 rc c 4 4r2c 4r2gc χ0 c0 g0 eχq2A2B2 − + −χ0 + = , r c g g2r2c2 g0 2 χ0 B02 q2A2B2 cc00 + cc0 + − = − + eχ . (5.100) g r 2 2r2 2g2r2

The above equations are invariant under the following scaling symmetries:

2 r → a1r, (t, x, y) → (t, x, y)/a1, g → a1g, A → a1A, B → a1B, (5.101) χ 2 χ e → a2e , t → a2t, A → A/a2.

x → x/a3,B → a3B. c → a3c.

The second scaling symmetry may be used to set χ = 0 at infinity and the third scaling symmetry may be used to set c = 1 at infinity, so that the asymptotic metric is that of AdS4. The fields have the following asymptotic behavior:

ρ Bb A = µ − ,B = Bb + 1 , (5.102) r 0 r Chapter 5. Holographic Superconductivity 104

where µ is the chemical potential and ρ is the charge density in the boundary theory. In what follows we will only consider the solutions for the ﬁeld B which vanishes near the boundary, i.e. B0 = 0.

Reissner-Nordstr¨omBlack Hole Solution and Instability To see when one expects hairy black holes at low temperature, one can study linearized perturbations of the extremal Reissner-Nordstr¨omAdS (RN-AdS) black hole [84]. The general RN-AdS solution is given by, 1 c = 1, χ = B = 0, g = r2 − 1 + ρ2/4 + ρ2/4r2,A = ρ (1 − 1/r) (5.103) r The temperature of the black hole (5.103) is

[g0(g e−χ)0]1/2 T = | (5.104) 4π r=r+

2 For AdS-RN, T = (12 − ρ )/16π. It is known that ﬂuctuations of Bx in this background develops a tachyonic mode [105] at low temperature and for suﬃciently large q. The resulting solution is a superconducting black hole with vector hairs. Following [84], we argue that such√ an instability persists at the extremal limit. At the extremal limit (T = 0), 2 we have ρ = 2 3. The near-horizon limit of the extremal solution is AdS2 × R with a metric,

dr2 √ ds2 = −6(r − 1)2dt2 + + dx2 + dy2,A = 2 3(r − 1) (5.105) 6(r − 1)2

Plugging this into the wave equation of B(r), and changing variablesr ˜ = r − 1, we recover a wave equation for AdS2 with a new eﬀective mass,

2 m2 q2 B + B − eff B = 0, m2 = − . (5.106) ,r˜r˜ r˜ ,r˜ r˜2 eff 3 The instability to form SU(2) vector hair at low temperature then is just the instability of 2 scalar ﬁelds below the Breitenlohner-Freedman (BF) bound for AdS2: mBF = −1/4. Thus the condition for instability is 3 q2 > . (5.107) 4 It is reasonable to expect that such an instability takes the system to its superconducting ground state.

Zero Temperature Solution Being a single state without any degeneracy, a superconducting ground state does not have any entropy associated with it. This fact concur with the observations in [84],[136, 137] Chapter 5. Holographic Superconductivity 105

3 Α

0 1 2 3 4 5 6 7 8 q

Figure 5.20: α vs q plot shows a almost linear relation with a slight dip for α for a value of q < 2.5.

that the zero temperature limit of the superconducting black holes in EYMH theory have zero horizon size. Motivated by these facts we assume that same is true for a non-Abelian hairy black hole also and the ground state is a geometry with zero horizon size. We choose the following ansatz, near r → 0+, (similar to that of EYMH case [84])

2 A ∼ A0(r),B ∼ B0 − B1(r), χ ∼ χ0 − χ1(r), g ∼ r + g1(r), c ∼ c0 + c1(r).(5.108)

All the terms with subscript 1 are sub-leading and go to zero, faster than the leading part where it is applicable, as r → 0. Putting Eqn. (5.108) in Eqn. (5.99), we get ﬁnd out the various terms in Eqn. (5.108). For example from the equation of motion for A,

2 2 q B α 2 2 0 0 0 (− r ) r (r A0) = ⇒ A0 ∼ e , α = qB0/c0, (5.109) A0

where we have used the observation A0 → 0 at the horizon. Following similar procedures, we get

eχ0 q2A2 eχ0 A2 A ∼ A e−α/r,B ∼ B 1 − 0 e−2α/r , c ∼ c 1 + 0 e−2α/r , (5.110) 0 0 4α2 0 8r2 eχ0 A2α eχ0 A2α χ ∼ χ − 0 e−2α/r, g ∼ r2 − 0 e−2α/r, 0 2r 4r

where by rescaling (5.101) we set the coeﬃcients A0 = 1 and χ0 = 0, c0 = 1. After we solve the equations we again use the rescalings of g, c, χ to make the Asymptotic geometry the same as that of AdS4. For a given q, we choose α in such a fashion that B vanishes near the boundary [84]. The numerical evaluation shows us an almost linear relation between q and α (Fig 5.20),with a slight dip for a lower value of q. Fitting a linear relation between q and α with a range of 3 < q < 8, we get α ≈ 0.088 + 0.66q.

Our numerics is less reliable√ in the region q < 0.95. It is not clear what exactly happens 3 in this region. As q → 2 ≈ 0.866, the geometry of our solution seems to approaches extremal RN geometry (Fig 5.20). However the questions of phase transitions remain Chapter 5. Holographic Superconductivity 106

1.0 3.45

3.40 0.8 3.35 0.6

L 3.30 L n 2 n n r r r H H g 0.4 A 3.25

3.20 0.2 3.15 0.0 3.10 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 50 100 150 200

rn rn

g Figure 5.21: (Colorful) The left panel shows a plot of the rescaled r2 with the rescaled variable rn. From top to bottom q = 1.5, 1.1, 1, 0.9. The dashed line is for the RN black hole. The right panel shows a plot of A with rn for the same set of values of q, from bottom to top.

√ 3 unclear. It is expected that there is a second order phase transition at q = 2 from RN black hole to our solution. However one may not rule out the possibility of a first or any other type of phase transition(s). At least in the range of our numerics it seems that our solutions have lower free energy than the extremal RN black hole in a grand canonical ensemble. As expected the solitonic solution is more massive than the RN black hole, but they carry more charge for the same boundary chemical potential and consequently have a lower free energy. General properties of our solution is similar to EYMH case [84]. B, χ approaches a constant near r = 0, g = r2 and the gauge field A and its derivative (electric field) vanishes. The metric approaches AdS4 with the same value of the cosmological constant at infinity. The extremal horizon is just the Poincare horizon of near horizon geometry AdS4. In terms of the dual field theory, this means that the full conformal symmetry is restored in the infrared. Due to the asymmetry of our solution in x and y direction the ratio of speed of light in IR and UV is not only non-unity but also different in different directions. We have,

IR IR vx 1 χ0 vy χ0 UV = exp( ), UV = exp( ) (5.111) vx c0 2 vy 2 Our solution may be viewed as a static, charged soliton, where the electrostatic repulsion is balancing the gravitational and non-Abelian interactions. As can be checked easily from the asymptotic behaviors the energy of the soliton is ﬁnite. 14 One interesting feature is

14However it should be noted that B(r) is not square integrable. This is not a requirement of a valid solution to the Einstein’s equations and assuming that our numerics are reliable, such solutions exists. The question is that whether they play a role in the context of AdS/CFT. In a sense whether it belongs to same quantum ensemble as the extremal black hole. As our zero temperature solution is of finite energy this seems to be the case. There are well known solutions such as the BPST instanton [144] which also have a non square integrable gauge field configuration with a bounded action. In our case, just as in case of BPST instanton, the solution may be approached by configurations which are of finite norm with a β bounded energy (just replace B → exp(− r )B(r), β → 0). At finite temperature the gauge field B is likely Chapter 5. Holographic Superconductivity 107

− # the existence of an “essential singularity”(i.e. an e r behavior) of A(r) at r = 0. Similar singularities appear for other quantities in Eqn. (5.111). Hence all the sufficiently high derivative of the metric and gauge fields vanishes at r = 0. This is in contrast with the EYMH case [84], where sufficiently large curvature invariants diverge at r = 0 for a generic α and behavior of the fields near r = 0 is power law. In terms of the “tortoise” co-ordinates (5.118), the essential singularity is just the reflection of the fact that A is massive near the horizon and decays exponentially.

Conductivity

In order to calculate the conductivity of this system, we need to turn on a small perturbation in the vector potential. The gauge ﬁeld and the associated metric perturbations are of the form: 3 −iωt 3 −iωt Ay = a(r)e τ dy, gty = h(r)e , (5.112) where we assume that both the gauge and the metric perturbations are of the same order O(). We won’t be looking in the other component of the gauge ﬁeld perturbations which decouples from the above. From Maxwell’s equation we get,

3 00 0 g0 χ0 c0 eχω2 q2B2 Ay −→ a + a g − 2 + c + a g2 − gr2c2

eχh 00 0 h0 χ0 c0 q2B2 + g A + A h + 2 + c − r2gc2 A = 0. (5.113)

To linear order, Einstein’s equations give, 2 T −→ − h + h0 = −aA0, (5.114) ry r which combined with the equation of A(r) to give:

g0 χ0 c0 eχω2 q2B2 A02 a00 + a0 − + + a − − eχ = 0. (5.115) g 2 c g2 gr2c2 g

This can be written as a Schr¨odingerequation:

− a00 + V (˜r)a = c2ω2a, (5.116)

where q2B2 V (r) = g c2A02 + e−χ . (5.117) r2 and all the derivatives in Eqn. (5.116) are in terms of the new variable new variabler ˜

to be square integrable due to the cutoﬀ provided by the horizon. Our zero temperature solution may also be thought as a zero temperature limiting case of such a solution (see Sec. 5.3.3). Chapter 5. Holographic Superconductivity 108

2.0

1.5

1.0

0.5

2 4 6 8 10

Figure 5.22: (Colorful) Plot of the rescaled potential V (rn) for q = 1.5, 1.1, 1 (from the right) at T = 0 with rescaled co-ordinate rn. The Poincare horizon is at rn = 0 for the zero temperature solutions. V (rn) is non-zero at the horizon. The dashed curve is for the extremal black hole with a horizon at r = 1.

(“tortoise coordinate”) given by:

d d ≡ e−χ/2gc . (5.118) dr˜ dr

Here, c approaches to unity as r → ∞, so that the spacetime is asymptotically AdS4. It follows then from Eqn. (5.102) that the potential V (r) vanishes near the boundary. If we require g ∼ r2 near the horizon at r = 0 then the first term vanishes, while the second term is finite as B(r = 0) ≡ B0 6= 0 and χ√is also finite at the horizon. The value of the 3 potential at the horizon decreases as q → 2 (Fig 5.22). Note that since c → 1 near the boundary, the quantity ω can be interpreted as the frequency of the incoming wave. The superconducting nature of the system is argued from the existence of a supercurrent 3 solution. If we choose ω = 0 and integrate Ax from the horizon (with a regularity condition at the horizon), we are expected to get a non-trivial Ax. Existence of such a solution implies a δ function for the real part of conductivity at ω = 0 [15, 84] 15

Re(σ(ω)) ∼ δ(ω) + ﬁnite. (5.119)

The nature of the ﬁnite part of the conductivity can be inferred from the potential V (r). As argued before the potential vanishes near the boundary and is ﬁnite near the horizon. The fact that the potential is nonzero at the horizon makes it possible for this system to

15One should be more careful about the existence of a delta function for the real part of conductivity at ω = 0 and relation between superconductivity. Due to the coupling of gravity modes the potential V (r) is non-trivial even in the normal phase (Fig 5.22) and consequently a delta function is present in the real part of DC conductivity. Physical interpretation of such a delta function comes from the “translational invariance” of the theory. See [103] for further discussions. Chapter 5. Holographic Superconductivity 109

exhibit a hard gap. From Eqn. (5.116), the ﬁeld a has the following asymptotic behaviors near the horizon (˜r → −∞) and the boundary (˜r → 0):

b b a(˜r → 0) ∼ a0 + a1r˜ (5.120) iω˜r˜ a(˜r → ∞) = a0e , (5.121)

p 2 2 whereω ˜ = c0ω − V0, with ch, ah being the near-horizon values of c and a respectively. Here, we have chosen the incoming boundary condition near the horizon. The conductivity of the system is given by: b ia1 σ = − b (5.122) ωa0 It follows from Eqn. (5.116) that :

a∗a00 − aa∗00 = 0, (5.123) which implies that the quantity Λ = a∗a0 − aa∗0 = 2iIm(aa∗0 ) is a constant. Equating the values of Λ near the horizon and the boundary we get:

2 ( ω˜ |a0| 2 ω b 2 , ω˜ > 0 Re(σ) = |a0| (5.124) 0, ω˜2 < 0

Therefore,√ the real part of the conductivity will vanish wheneverω ˜ is imaginary, i.e. when ω < V0/c0, which deﬁnes the gap. As our solution is an anisotropic superconductor (only superconducting in Ay direction), 3 the excitations in Ax channel may not be gapped [105]. The gap seems to be a property of the anisotropic non-Abelian system, as the scalar-Abelian gauge ﬁeld system does not exhibit this gapped behavior in the extremal limit [84].

Finite Temperature Analysis Although the Eqn. (5.99) has been solved in a probe limit, the full set of equations with gravity backreaction (Eqn. (5.100)) is not yet solved at a non-zero temperature. Hence it is an issue whether our zero temperature solution may be realized as a zero temperature limit of the finite temperature hairy black hole (and whether such a low temperature hairy black hole solution at all exists.). One affirmative clue comes from the emergent conformal symmetry in the IR. In the near horizon region, i.e. α r, our solution approaches a AdS4 geometry with a background gauge field Ax = B0 and At = 0. One may consider a black hole situated deep inside this AdS space, such that the radius of the black hole r0 α. 2 r0 Here, g(r) near the horizon is given by, g(r) = r (1 − r ). However this Schwarzschild like solution will get correction due to a small non-zero A(r) near the horizon. The solution of A(r) in the r α is given by, α r A(r) ≈ exp(− )(1 − 0 ). (5.125) r r Chapter 5. Holographic Superconductivity 110

VHrL

1.5

1.0

0.5

r 5 10 15

Figure 5.23: Schematic plot of the potential V (˜r) for diﬀerent T . T decreases from the left. V (r) is gradually widened as T decreased. Horizon is atr ˜ = ∞.

A(r) is small in the region r α. In this region we can do a perturbation in A(r) for other quantities. Realizing that our scaling relations Eqn. (5.111) is just a perturbation in A(r), we can match a near horizon solution with the scaling solution (i.e. Eqn. (5.111)) in a region r0 r α. Hence we use the method of matched asymptotic expansion to argue that, r A (r) = A(r)(1 − 0 ) (5.126) T r r B (r) = B − (B(r) − B )(1 − 0 )2 T 0 0 r r c (r) = c + (c(r) − c )(1 − 0 )2 T 0 0 r r χ (r) = χ(r) + (χ(r) − χ )(1 − 0 ) T 0 r r g (r) = g(r)(1 − 0 ) T r is a solution to Einstein’s equations if r0 α. The quantities (A, B.c, χ) appear in the right hand side of the above equations are our numerical solution and the quantities with a subscript T are the finite temperature versions. Whether correction to such a solution is finite or not is an open question, which may be partially settled by a convincing finite temperature numerics. However due to the possible “smooth” nature of the equations Eqn. (5.99), i.e. a small change in the near horizon boundary condition leads to a small change near the boundary, it is reasonable to believe that corrections to Eqn. (5.127) are well controlled. What happens to the potential V (r) (see Eqn. (5.117)) in this limit is interesting. At any non-zero r0, the potential V (r) goes to zero at the horizon (r = r0) and the height of the potential is always finite. Hence the transmission through this potential is non-zero. In our language the (i.e. Eqn. (5.124))ω ˜ = ω is a real number. Hence the Chapter 5. Holographic Superconductivity 111

real part of the conductivity is non-zero for any value of ω. However, in the “tortoise co-ordinates” (5.118) the potential is gradually widened as r0 → 0 (Fig 5.3.3). Hence the transmission through the potential gradually decreases as the horizon size is decreased. We ∆ expect a exp(− T ) fall oﬀ for the real part of conductivity. A detailed discussion of the various gaps is left for a future work.

5.4 An Analytic Understanding of the Spikes of Conductivity

In the calculation of the conductivity, one generally finds some spikes, particularly at low enough temperature multi-spikes appear. As the parameters change, the spikes can go very sharp and high, seeming like poles. We will discuss that the spikes are actually quasi-normal modes, and show how approximately to calculate them. As argued in Sec. 5.3.2, the calculation of conductivity, in field Ax, can transform into a one dimensional Schrodinger’s problem. The potential of the Schrodinger field is finite, and both ends(near horizon and at asymptotic boundary) vanish16. Since one has to impose an infalling wave condition near horizon, an incident wave at boundary is required, as well as the reflected wave. With the equation (5.91,)

1 − R σ(ω) = , (5.127) 1 + R σ → ∞ as R = −1. The fact is since the potential is with finite height and width, the tunneling rate never vanishes, and thus the reflection coefficient R can never be −1. And the spikes are the quasi-normal modes on the complex ω plane. However at very low temperature, the potential is very high compared with the small incident energy ω2, and one can approximately treat the spikes as poles, i.e. bound states. The system can be approximately analyzed by WKB method. One needs to keep in mind WKB approximation works well when energy is much less than the potential’s maximum and the level of excited states should be much higher than the ground state. Although these two conditions are neither applicable here, the qualitative behavior is still reliable. Suppose we work in the Schwarzschild black hole limit, the electric field’s contribution in 17 the potential of Ax can be neglected , and the EOM of Ax gauge field can be transformed as d2A Z dr x + (ω2 − 2fψ2)A = 0, where y = , (5.128) dy2 x f(r) thus it is a quantum mechanics problem, to solve the eigenvalue of a particle in potential V (y) = 2fψ2. The basic properties of the conductivity have three components, (1) for ω > ωc, there is a plateau; (2) when ω < ωc, σ = 0 for most values of frequency; (3) some discrete spikes

16As discussed in Sec. 5.3.2, there are three possibilities of the boundary value, with parameters varying. However the spikes appear in only one of them, with Vbdy = 0, so we restrict our discussion in this case. 17 As a matter of fact, with the rescaling L = r+ = 1, the electric ﬁeld is not negligible, however the qualitative property is the same. Chapter 5. Holographic Superconductivity 112

0.7 2.5

0.6 2.0 0.5

0.4 1.5

0.3 1.0 0.2 0.5 0.1

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 1.2

2 2 Figure 5.24: The left ﬁgure is the regularized potential V (z) = 2fψ /µ for Ax, the 2 2 maximal of which is around Vmax = ωc ∼ 0.71 ∼ (0.84) . The right ﬁgure is the real part of conductivity, where the trivial step function jumps to a plateau at frequency around 2 ωc ∼ 0.8 − 0.9. The mass of scalar of these data is set to be m = −2.2.

2 2 may appear when m is close to mBF . We do not list the pole at zero frequency here, since A(1) (0) A(1) it is directly from the definition σ = + ··· , with Ax ∼ A + + ··· . iωA(0) r 2 In the QM argument, the plateau is straightforward with eigenvalue ω > Vmax, and we find it is consistent as shown in figure 5.24. We also checked for some other different mass, the magnitude of Vmax is consistent with the critical frequency in σ(ω). To explain the poles, the most natural way is the bound states trapped in the left side wedge of the potential. According with WKB approximation method, bound states should satisfy Z 0 3 dyp2(ω2 − V (y)) = (n + )π, n = 0, 1, 2, ··· (5.129) y1 4 2 where ω = V (y1). The constant 3/4 in above condition assures the amplitude vanishes at boundary z = 0, i.e. the reflection coefficient R = −1. If one integrates the real part of the conductivity, the Ferrell-Glover sum rules is compensated by the spikes.

AdS5 The conductivity in AdS5 is slightly diﬀerent. The real part of the conductivity at large frequency linearly depends on the frequency, which is consistent in a 5-dimensional theory. 2 And the BF-bound in AdS5 is smaller, mBF = −4, which we will show more spikes might appear in the conductivity. The equation of motion of Ax in AdS5 is little more complicated, 1 f 0 1 2ψ2 ω2 A00 + + A0 + − + A = 0, (5.130) x z f x z4 f f 2 x

(0) A(2) with z = 1/r. The asymptotic behavior of Ax is diﬀerent, with Ax ∼ A + r2 + ··· , and Chapter 5. Holographic Superconductivity 113

the conductivity is now 2A(2) iω σ = + . (5.131) iωA(0) 2 Being aware of Re(σ) ∝ ω at large ω, the right asymptotic form at large ω is

iω2/r2 −iω2/r2 Ax = c1e + c2e , (5.132) r→∞

±iω/r 2 instead of e as in AdS4. One needs to make a coordinate transformation, u = z , and thus 1 uf∂ (uf∂ A ) + (ω2 − 2fψ2)A = 0, (5.133) u u x 4u x or writing it in Schrodinger’s form, 1 du ∂2A + (ω2 − 2fψ2)A = 0, with dy = . (5.134) y x 4u x uf This is the Schrodinger equation of a unit mass particle with zero energy moving in a 1 2 2 potential V (y) = − 4u (ω − 2fψ ). Now the story is similar to AdS4 case.

Comments on WKB method

12 1.65

10 1.60

8 1.55 6

1.50 4

100 150 200 50 100 150

Figure 5.25: In AdS4 case, when energy approaches the maximum of potential, the inte- R yM p 2 gral 0 dy 2(ω − V ) gives the maximal number of bound states, according to equation (5.129). In the above figures, the maximum of integral is drawn as function of µ, the chemical potential and inverse temperature. In the left figure, the left steep straight line is for m2 = −2.25, and the gently increasing line is for m2 = −2.05, both of which seem linear. The dashed parallel lines indicate the number of bound states of 1,2,3,4, from bottom to top. The right figure is for m2 = −1.95, where the curve seems approaching an asymptotic constant at large µ. Since the integral never exceeds 2, which is the existence requirement of one bound state, there is no bound state for the right figure.

With the WKB method, we ﬁnd the number of bound states Nb.s. depends on a set 2 of parameters (m , µ). When the mass squared is close to BF mass in pure AdS, Nb.s. increases linearly with chemical potential µ. On the other hand, when m2 is much higher, Chapter 5. Holographic Superconductivity 114

6 12

5 10

4 8

50 100 150

14 16 18 20

2 Figure 5.26: In AdS5, maximum of integral as function of µ. The left ﬁgure is for m = −3.8 > m2 = −4, while for the dashed lines, N = 2, 3, 4. And in the BF (AdS5) bound states right ﬁgure, from top-left to bottom-right, the four curves with points correspond to m2 = −3.1, −3.0, −2.5, −2.0 respectively, the last of which has no bound states.

the number appears as a constant. In AdS4, one needs to satisfy the bound state condition,

Z 0 3 dyp2(ω2 − V (y)) = (n + )π, n = 0, 1, 2, ··· (5.135) y1 4

For given parameters (m2, µ), the potential V = 2fψ2 is fixed, thus the number of bound states only depends on the frequency ωm corresponding to the potential maximum Vmax = ω2 . We draw the integral R 0 dyp2(ω2 − V (y)) as function of µ in figure 5.25 with different m ym m m2 value. 2 2 When m is close to BF bound, mBF = −9/4, Nb.s. starts from 1 at low chemical 1 2 potential µ ∼ T , and gets 4 at high µ within our data region. For higher m = −2.05 the slope is much smaller, and for even higher m2 = −1.95, the curve seems to get an asymptotic constant at large µ, and there is no bound state, i.e. no spikes in conductivity. In AdS5, the condition is now

Z 0 3 dyp2(0 − V (y)) = (n + )π, n = 0, 1, 2, ··· (5.136) y1 4

1 2 2 with V (y) = − 4u (ω −2fψ ). The difference from AdS4 is the energy of the particle is fixed as zero, and one needs to tune the parameter ω to satisfy the condition (5.136) with integer n. When V = − 1 (ω2 − 2fψ2) = 0, the integral, similar to AdS case, determines max 4um m 4 2 2 the number of bound states(figure 5.26). When m = −3.8 & mBF = −4, the number of bound states varies from 1 to 4 “linearly” with µ. The right graph of figure 5.26 shows the 2 evolvement of Nb.s.’s of different m . The reason why we use WKB method here is to understand the pole behavior in Re(σ), while each bound state corresponds to a pole. It is helpful since it is difficult to see all the pole appearance in our Re(σ) figure, possibly due to some numeric technical difficulty. The existence of extra poles sometimes might be shown from little spikes on the Re(σ) = 0 Chapter 5. Holographic Superconductivity 115

basin. A good evidence of the poles’ existence is the zigzags of Imσ, which is well consistent with our WKB prediction in AdS4. However in AdS5 case, there seems a one pole/bound state discrepancy in WKB prediction and σ(ω) figures. For example, in Re(σ) and Im(σ) figures the existence of first pole comes about between m2 ∈ (−3.1, −3.0), while WKB method tells that the second pole starts to appear in this region. One may argue that if the pole is very narrow, the pole’s existence is difficult to be shown numerically, which is a general behavior in Re(σ) figure, and might happen in Im(σ) as well. In addition, one should be aware that WKB method works well only far away from the maximum of potential, while in our examples, approaching the maximum is necessary in counting the number of bound states. Thus the poles’ number and the corresponding frequencies might be not so accurate. Here are two more comments: 1. Since in the figures of σ(ω), only at zero temperature, the poles appear on real axis, the bound state frequencies predicted from WKB are not accurate since it assumes real poles. 2. since the potential at horizon is always less than the energy, the “bound states” can tunnel to the horizon and decay, thus they are not real bound states. Especially when the part of potential above the state’s energy is thin and not high, the WKB approximation is not valid anymore, which is much explicit in AdS5 case. 116

Part III

Instability of Gravitational Bubbles 117 Chapter 6

On Bubbles of Nothing in AdS/CFT

6.1 Introduction and Motivation

The AdS/CFT correspondence (for a review see [53]) provides a non-perturbative background independent definition of quantum gravity in asymptotically AdS spaces. It is useful then to investigate quantum gravity issues in this context. In this chapter1 we examine non-perturbative instabilities mediated by bubbles of nothing [62, 63, 145–147]. Semiclas- sical analysis suggests these correspond to vacuum decay, and indeed the false vacuum was identified in [63] as the topological black hole [148]. It is interesting then to discuss the process, and its place in the full non-perturbative definition of the theory. One goal is to clarify the role of the semi-classical analysis in the full theory, as such non-perturbative instabilities may play a crucial role in flux compactifications or in the context of eternal inflation. In our context, the dual field theory is a conventional (supersymmetric, conformal) field 1 theory formulated in curved space, namely dS3 × S [63]. This suggests that long distance features of the theory are captured in the matrix quantum mechanics of the lowest lying mode, which is the Wilson line wrapping the circle, which we call U. The time in that quantum mechanics is the non-compact time direction of the de-Sitter space 2 A priori that quantum mechanics has a time dependent Hamiltonian, but we will find that for our purposes that time dependence plays only a relatively minor role. We will calculate the effective action of that matrix quantum mechanics in the weak ’t Hooft coupling, small volume limit, following the work in [66]. Our purpose in this chapter is to collect evidence that the reduction to a quantum mechanics of a single degree of freedom u (defined as the condensate of the T r(U)) captures the essential features of the process, even in the supergravity (large ’t Hooft coupling) limit. For this purpose, we examine the phase diagram of the theory as function of the circle’s radius and three R-charge chemical potentials3. We find that generically the bulk analysis indicates a first order transition. The transition becomes enhanced (the bounce action becomes small) at specific loci in the phase diagram; we find that in precisely those loci the bulk developed a winding tachyon, which is precisely the bulk mode dual to u. These loci then correspond to the disappearance of the barrier in the effective potential, causing the instability to become perturbative and driven by the process of (winding) tachyon

1This part is based on our work [5]. 2In particular we do not expect the features of the decay to be captured by matrix integrals, such as the ones in [66], appropriately analytically continued, as suggested for example in [63, 149]. The process of bubble nucleation is time dependent, therefore degrees of freedom along the time direction of dS3 should not be integrated out. 3For previous analysis of the phase diagram see [150], for discussion of charged bubbles see also [151]. Chapter 6. On Bubbles of Nothing in AdS/CFT 118 condensation. This is precisely what one would expect in the quantum mechanical model of the winding condensate u. This analysis can be taken as an evidence for the conjecture in [152, 153] that the end point of the tachyon condensation is indeed the bubble of nothing. Inside that bubble is the true vacuum, the “nothing state”, which is then a winding mode condensate. That state is inherently stringy, and does not possess a conventional spacetime interpretation. This is despite the fact that no strong curvatures exist in the bubble spacetime. It would be interesting to further discuss this non-geometrical phase in terms of its field theory dual, understanding for example the disappearance of spacetime in this state. The outline of this note is as follows. In the next section we discuss the general bubble with up to three chemical potentials turned on. We demonstrate the features of the phase diagram for a few specific cases, and discuss the general qualitative features. Finally, we discuss the explanation of all these features in terms of the quantum mechanics of the variable u and its purported effective potential. We find that all those qualitative features, such as the phase boundaries, can be explained by a conventional quantum mechanical model, where the process is simply the familiar decay of the false vacuum. We present our conclusions and directions for future research in the final section.

6.2 Bubbles of Nothing in AdS/CFT

In this section we review and extend the bulk analysis of the decays mediated by the various bubble of nothing (BON) solutions. We focus on the qualitative features of the relevant gauge theory - in all cases the maximally supersymmetric Yang-Mills theory, formulated in curved space, in the large N and strong ’t Hooft coupling limits. Those qualitative features will be compared to those in the weakly coupled gauge theory in our future work. We will concentrate on exploring the phase diagram, looking for the regions in parameter space where the instability exists, and interpret these qualitative features in terms of a candidate eﬀective potential, such as the one existing in the dual gauge theory. The non-perturbative instability entails an existence of an appropriate bounce solution, a Eu- clidean solution of the equations of motion with a single (non-conformal) negative mode. The negative mode signals the existence of a local maximum in the eﬀective potential. Ad- ditionally we compare the Euclidean action of the bounce with that of the false vacuum: in order for the decay rate is small, corresponding to a meta-stable false vacuum, the action of the bounce has to be higher than that of the false vacuum. When the actions become comparable the false vacuum is no longer meta-stable.

6.2.1 R-Charged Bubbles Let us start with the most general black hole metric carrying up to 3 unequal R-charges. The bubble solutions we are interested in are formed by double analytic continuation as described below. The Lorentzian black hole metric is [154]

2 −2/3 2 1/3 −1 2 2 ds = −(H1H2H3) fdt + (H1H2H3) f dr + r dΩ3,k , (6.1) Chapter 6. On Bubbles of Nothing in AdS/CFT 119

where µ r2 q f = k − + H H H ,H = 1 + i , i = (1, 2, 3), (6.2) r2 l2 1 2 3 i r2

Here qi are 3 charge parameters, related to the physical charges in a manner specified below. To obtain a bubble of nothing solution we perform double analytic continuation. The reality conditions on all fields distinguish between the bubble interpretation and the more familiar black hole (thermal) one, which was analyzed in [154], thus resulting in a different phase diagram. In our case the circle parameterized by χ = it is interpreted as a spatial direction of the geometry, rather than Euclidean time, therefore the gauge field component in that direction Aχ has to be real. To ensure that we take the charge parameters qi to be negative, as opposed to having them take positive values for the black holes analyzed in [154]. 1 1 The Euclidean metric approaches asymptotically Mk ×S , where S is a circle of asymptotic radius β (which is a parameter of the solution), and Mk is an homogeneous space of 3 3 constant curvature k and metric dΩ3,k. For the case k = 1 it is S , if k = 0 it is R , and if k = −1 it is a hyperbolic space H3. Upon analytic continuation, one of the coordinates of 1 Mk becomes timelike, and the solution resemble a bubble, with the S being interpreted as a spatial circle. In the following we shall concentrate on the solutions with k = 1, in which case the analytically continued metric on Mk is that of dS3, the 3 -dimensional de-Sitter space. i i In addition to the metric the solution has three scalar fields X and gauge fields Aµ, which are of the form

i −1 1/3 i q˜i X = Hi (H1H2H3) ,At = 2 + φi, i = 1, 2, 3. (6.3) r + qi

The constants φi are adjusted such that the gauge potential at the Euclidean origin is zero. In that case they equal the gauge potentials at infinity, and thus are parameters of the dual gauge theory4. The physical chargesq ˜i are given in terms of the parameters qi as " # 1 Y q˜2 = q (r2 + q ) 1 + (r2 + q ) , (6.4) i i + i r2 + j + j6=i where r+ is the location of the outer horizon, namely the largest root of f(r) defined above. We will mostly work in “grand-canonical ensemble” where the fixed quantities are the potentials at infinity, given by

i q˜i φi ≡ −At(r+) = − 2 . (6.5) r+ + qi

Using the notations of [154] we take φi to be purely imaginary, which ensures the correct reality conditions upon analytic continuation to the Lorentzian bubble spacetime. To identify the false vacuum, decaying via the Euclidean solution, interpreted as a

4 No such parameters, or charges, are associated with the scalar ﬁelds Xi. Chapter 6. On Bubbles of Nothing in AdS/CFT 120

bounce, we look at the asymptotic behavior of the fields. The metric behaves asymptotically the same as in the solutions in [63], therefore in all cases the spacetime decaying is the topological black hole which has a non-contractible circle in the geometry (it is obtained from AdS by appropriate identifications, as discussed in [63]). The scalars fall off rapidly at infinity, and the gauge fields approach a constant. We conclude therefore that the false vacuum is the topological black hole with constant gauge potentials around the non- contractible circle5. To map out the general phase diagram we are first interested in the region of parameter space for which the Euclidean solution exists and has a single non-conformal negative mode. In all cases we fix the asymptotic radius β and the value of the potentials at infinity φi. In our coordinates there is a possible conical singularity at r = r+. Demanding regularity at the Euclidean origin determines as usual the asymptotic radius β. In this case we find 2 pQ 2 r+ i(r+ + qi) β = 2π 6 P 4 Q , (6.6) 2r+ + (1 + i qi)r+ − i qi where we set for convenience the AdS radius to unity, l = 1. Together with the formulas (6.4, 6.5) this determines the region of parameter space (spanned by β and φi for which a candidate Euclidean solution exists). To confirm that a candidate Euclidean solution is indeed a bounce one needs to check the existence of a non-conformal negative mode. In general this is a difficult problem, solved for the uncharged case in the classic reference [155]. However, it is known that “thermodynamic instability” means that the solution is a local maximum of the free energy, and therefore it is a sufficient condition for the existence of a negative mode [156]. Therefore, in order to check for the possibility of a negative mode we check the Hessian of the Euclidean action (with respect to the field theory parameters) in the region of parameter space relevant for the bubble interpretation. Finally, to check for the existence of a barrier in field space we calculate the Euclidean action (relative to the false vacuum which is locally simply AdS space). The Euclidean action then can be calculated directly from the above solution, it is found to be [154]

3 X I = β(M − q˜iφi) − S, (6.7) i=1 where the mass M is given by

3 3 3 X 3 Y X M = µ + q = (r2 + r4 H ) + q , (6.8) 2 i 2 + + i i i=1 i=1

and the “entropy”6 S is q A Y 2 S = = 2π (r+ + qi). (6.9) 4GN

5 −1 Furthermore the variables |φi| are periodic in units of β . 6This quantity has no interpretation as entropy in the analytic continuation leading to the bubble solution. Chapter 6. On Bubbles of Nothing in AdS/CFT 121

6.2.2 Uncharged Case The simplest case of the uncharged bubble was discussed extensively in [63, 145–147], and we discuss it here brieﬂy. In that case the Euclidean metric is

2 2 −1 2 2 2 2 ds = fdχ + f dr + r (dθ + sin θdΩ2), (6.10)

where µ f = 1 − + r2. (6.11) r2 The phase diagram is one dimensional, characterized by the size of the asymptotic circle β, which is given by 2πr+ β = 2 , (6.12) 1 + 2r+ where the horizon location r+ can take√ any positive value. We see then that since this function attains a maximum at βcrit = π/ 2, there cannot be a non-perturbative instability for β > βcrit. For β < βcrit one has two possible solutions (two values of r+), and one has to check for the existence of negative mode. The Euclidean action as a function of the parameter r+ is given by [63, 145–147]

3 2 πr+(r+ − 1) I = − 2 , (6.13) 1 + 2r+ where we have omitted an irrelevant reference constant. As explained above, to check for the existence of a negative mode we calculate the Hessian of the action, in this case this is simply the second derivative at ﬁxed β value7

2 2 ∂ IEuc 3π(2r+ − 1) 2 2 β = 2 (6.14) ∂(r+) 2r+(1 + 2r+)

We see that for β < βcrit there are two possible solutions, and one of them has a negative mode, corresponding to an instability. Therefore we conclude that for every β < βcrit a bounce exists. Additionally, as shown in Fig. 6.1, the Euclidean bubble has positive action (relative to the false vacuum, that is the topological black hole) for r+ < 1, which includes all the relevant region of instability. The action becomes small, corresponding to the disappearance of the barrier, for small radii β, signalling the onset of tachyon condensation as an alternative mode of instability, as discussed below. To make connection with the conjecture in [152], let us discuss the range of parameters for which a winding tachyon appears in the geometry. Since the analysis depends only on the geometry, and not on the background ﬁelds, it applies to all cases below as well. We start with the false vacuum, namely the topological black hole [63]

r2 r2 ds2 = (1 + )−1dr2 + β2(1 + )dχ2 + r2dΩ , (6.15) l2 l2 3 7This quantity would be the ﬁxed temperature speciﬁc heat in the thermal interpretation. Chapter 6. On Bubbles of Nothing in AdS/CFT 122

IBH IBH 0.4 0.2

0.2 small bubble 0.2 0.4 0.6 0.8 1 1.2 1.4 0 2 0.5 1 1.5 2 2.5 3 0 r+ β1 βmax β

±0.2 ±0.2

±0.4

±0.4 ±0.6 large bubble

±0.6 ±0.8

±1

Figure 6.1: In the left plot, we draw the Euclidean action of the bubbles as function of the bubble’s size r+. When the action is positive the false vacuum is metastable. The 2 1 conditions for negative Euclidean mode are satisﬁed for r+ < 2 . The action as function of inverse temperature β is shown in the right plot. The small and large bubble solutions are connected at β = βmax.

where l is the AdS radius, β is the periodicity of the circle, so that χ ∼ χ + 2π. The proper size of the circle is r r2 L(r) = β 1 + . (6.16) l2 It goes to a constant at r → 0, and to infinity at the boundary. For a winding tachyon to condense, two conditions are needed. The size of the circle has to be of the string scale ls, and it has to vary slowly (compared to the string scale). These conditions (for l ls ) are satisfied in the regime β < ls. This is precisely the locus of the phase diagram for which the Euclidean action becomes small, signalling the disappearance of the barrier in field space8.

6.2.3 One Charge Case

Now let turn on a single charge, say q1 = q, and q2 = q3 = 0, the ﬁeld theory parameters are given by

2 2 2 q˜ = q(r+ + q)(1 + r+), q˜ φ = − 2 , r+ + q p 2 2π r+ + q β = 2 (6.17) 2r+ + 1 + q

8The analysis is consistent with the winding tachyon found to exist in [153], since the AdS soliton is the limit for which the dS3 radius of curvature diverges, or by conformal transformation, the limit where β → 0. For additional support for the conjecture in [152] see also [157]. Chapter 6. On Bubbles of Nothing in AdS/CFT 123

The bubble solution, obtained by double analytic continuation from the black hole π t → iχ, θ → 2 + iτ, is given by [150]

2 −2/3 2 1/3 −1 2 2 2 2 2 ds = H fdχ + H (f dr − r dτ + r cosh τdΩ2), (6.18) where µ H = 1 + q/r2, f = 1 − + r2H. (6.19) r2 and q < 0 in the bubble interpretation. And it is easy to get the action and Hessian √ πR R + q(R − 1 + q) I = β(M − qφ˜ ) − S = − , 2R + 1 + q π2(3R2 + 3(1 + q)R + 4q)(2R2 + (1 + q)R − (1 − q)2) Hessian = . (6.20) 4q(2R + 1 + q)2(R + q)(1 + R)

2 where we deﬁned R = r+. First we look at the existence of bubbles in the parameter space spanned by β and β|φ|. This is the shaded region in Fig. 6.2. We see that, like in the neutral case, the bubbles exist for a ﬁnite region in parameters space, and in particular there is always a maximum radius βmax for which the instability disappears.

β|φ|

0 0.5 1 1.5 2 β

Figure 6.2: The existence of bubbles in the (β, β|φ|) plane in the one charge case.

Existence of the instability requires a negative mode9, and in addition we need to satisfy the condition Ibub > 0 for metastability. We see in Fig. 6.3 that the region of instability always satisfies the condition for metastability, the transition is always first order. Figure 3 shows the existence of instability, and the region of metastability, in parameter 2 diagrams (R = r+, q) and (β, β|φ|). Interestingly we find both β and β|φ| are bounded. As a check we note that all the features of the neutral instability are reproduces when setting

9We note that due to the analytic continuation compared to the thermal interpretation, the existence of negative mode corresponds to a positive Hessian, instead of negative value. Chapter 6. On Bubbles of Nothing in AdS/CFT 124

R 0.5 1 1.5 2 2.5 3 2 0 r+ β|φ|

5 C ±0.5 E 4

3 q ±1 A

±1.5 D 1 B ±2 q 0 0.5 1 1.5 2 β

Figure 6.3: In the left diagram, instability can exist only in region C. No bubbles exist in region A, and region E corresponds to disappearance of the barrier Ibub < 0. In the right (β, β|φ|) diagram, the shaded region describes the region of possible instability.

φ = 0.

6.2.4 Three Equal Charges

As an additional example we now consider the case of three equal charges, q1 = q2 = q3 = q, so that µ q f = 1 − + r2H,H = (1 + )3 (6.21) r2 r2

In this case all the scalars Xi are constants, therefore the black hole is simply the AdS Reissner-Nordstrom solution, investigated for example in [158]. In this case we ﬁnd that the physical parameters are

2πR(R + q)3/2 β = , 2R3 + R2(1 + 3q) − q3 1 q˜2 = q(R + q) 1 + (R + q)2 , R q˜ φ = − . (6.22) R + q where once again q < 0, corresponding to purely imaginary φ, for the bubble interpretation. Similar to the discussion above, we first look at the region of parameters corresponding to the existence of a bubble spacetime. This is depicted in Fig. 6.4. Finally, in Fig. 6.5 we show the regions for which the negative mode exists, and the region for which the topological black hole is meta-stable. As before we see that instability always occurs when the condition for metastability is satisfied, in other words the transition is always first order, and proceeds by tunneling over a potential barrier. Chapter 6. On Bubbles of Nothing in AdS/CFT 125

β|φ|

1.5

0.5

0 0.5 1 1.5 2 β

Figure 6.4: The existence of bubbles in the (β, β|φ|) plane in the three equal charges case.

R 2 4 6 8 10 2 0 r+ β|φ| 2

±2 D 1.5

±4

q C B 1 ±6

0.5 ±8 A

±10 0 0.5 1 1.5 2 q β

Figure 6.5: In the left diagram, instability can exist only in region B. No bubbles exist in region A, and region D corresponds to disappearance of the barrier Ibub < 0. In the right (β, β|φ|) diagram, the shaded region describes the region of possible instability. Chapter 6. On Bubbles of Nothing in AdS/CFT 126

6.2.5 Features of the Phase Diagrams Let us discuss qualitative features of the phase diagram, and how those are interpreted in terms of the quantum mechanics of a single variable (the winding condensate) and its effective potential. The winding mode (Wilson line) U is the lowest lying mode when compactifying on the circle, thus one can describe the long distance physics in terms of three dimensional scalar field theory living on dS3 space. We suggest that the physics of bubble nucleation can be effectively described in terms of the quantum mechanics of the spatially homogenous mode of U, integrating out the massive mode on the sphere S2 in global coordinates. This approximation will manifestly break down at late times where the S2 decompactifies, but may be sufficient for describing the bubble nucleation and early evolution. Furthermore, since the bubble is nucleated at t = 0 when dS3 is momentarily stationary, we expect that for the process of the bubble nucleation the time dependence is unimpor- tant10, though it may be important for a complete description of its subsequent evolution. 2 1 As further evidence we note that the analysis in [161] of the asymptotic Rt × S × S boundary suggests that qualitative features to do with bubble nucleation are insensitive to the time dependence. Therefore we suggest to describe the process with the aid of quantum mechanics with an approximately time-independent Hamiltonian. That Hamiltonian can be calculated in the weak coupling limit, here we detail its features that can be read off from the bulk analysis for strong ’t Hooft coupling. One distinctive qualitative feature in the phase diagrams is the point where the Eu- clidean action becomes small, which we would like to interpret as the disappearance in the barrier in the potential11. Indeed, this happens for a small enough circle radius, and we find in all cases that near that point (where the circle parameterized by χ becomes string scale) a tachyon developed, which is described in the quantum mechanics of u, the winding condensate, as maximum in the effective potential. Note that the process of the winding tachyon condensation corresponds precisely to moving along the u axis. This indicates also that the true vacuum, even in the regimes where the decay is non-perturbative, is simply a winding mode condensate. The picture of the decay by the expanding bubble of nothing is therefore the conventional one, where a bubble of a true vacuum nucleates inside the false vacuum, and then exponentially expands outwards. The only novelty is that the true vacuum is a stringy non-geometrical phase. The other striking feature of the phase diagrams is the disappearance of the instability. Indeed, in all cases there is a critical value of β, the circle’s radius, for which the instability stops. Equivalently, since the boundary theory is conformally invariant, there is a minimal radius of curvature of the space dS3 for which the instability stops. In particular, since the decay happens at the turning point t = 0 of dS3, there is a minimal size of the spatial sphere S2, and below that size the instability stops. In conventional quantum mechanics of the homogeneous mode u, the natural explanation of that point is that the false and true vacua become degenerate in energy, and the instability stops. The bounce in that case disappears, while its limiting action stays finite. This is exactly the behavior we find when discussing the bounce near the critical value of

10For discussions of issues to do with time dependence in the bubble spacetime see for example [159, 160]. 11For a similar statement using a spacetime analysis [162] see [163]. Chapter 6. On Bubbles of Nothing in AdS/CFT 127

β. The quantum mechanics of the homogeneous mode is not sensitive to spatial dependence of the process. However, using the Euclidean black hole as an instanton leads to a puzzle related to its de-Sitter symmetry. This symmetry means that the decay is spatially homogeneous from the boundary viewpoint, for example the expectation value of the Wilson loop, is homogeneous. This is contrary to the expectation one has from conventional bubble nucleation in first order transitions, where the typical situation involves an inhomogeneous pattern in space. This pattern of decay is likely to be an artifact of the gravity limit. For bubble nucleation in finite space one has to compare the size of the space to the radius of a typical bubble. The latter is determined in terms of parameters of the potential, which are a function of the ’t Hooft coupling λ. The suppression of the inhomogeneous decay is then an indication that typical bubble size is much larger than the size of the space, at least for large enough λ. In this case the decay will proceed at once throughout space. In order to discover inhomogeneous decays in the gravity dual, we would have to construct solutions which break the de-Sitter symmetry (or the spherical symmetry in Eu- clidean space), and have lower action than the spherically symmetric case. However, from bulk considerations, such instantons are unlikely to dominate at the gravity limit. The dual field theory interpretation leads us to conjecture then that at small enough λ there ought to be a transition to non-spherically symmetric bubbles, which will then dominate the decay process. Such transition is reminiscent of the Gregory-Laflamme instability, though in this case the localization is on a sphere.

βmax Problem From Fig. 6.1, 6.2, 6.4 and the discussion in this chapter, we see that β, radius of the spatial circle, can only take value below some critical value βmax. Solution for β beyond βmax does not exist, and we can safely say the decay does not happen with large enough size of the spatial circle. One can√ also analyze the βmax problem formally, and it always gives us a value β ≤ 2 βmax = π/ 2 at (r+, q) = (0.5, 0) for all the three particular cases above. 2 The discussion of a general parameter set (r+, qi) is diﬃcult, but we suppose they have a similar behavior.

String Tachyon Condensation and Phase Spaces We can draw the vacua behavior with the size β of spatial circle changing. In Fig. 6.6, the leftmost one is at β = ∞, where only one vacuum exist. When β decreases, another vacuum starts to appear. The new vacuum has larger Euclidean action, and is suppressed in the partition function. When β = βmax, the actions of the two vacua are the same. While β gets a little smaller than βmax, the left vacuum becomes a false vacuum, and the decay is mediated by a bubble of nothing. Such behavior holds until the circle’s size gets to string scale. When a slowly varying β is of such small scale, one has to consider closed string tachyon condensation, and the topology structure changes. The behavior looks similar as the bubble of decay, however one needs to note that the usual bubble of decay is like Chapter 6. On Bubbles of Nothing in AdS/CFT 128

a quantum tunnelling, while the decay driven by closed string tachyon condensation is perturbative.

V1 V2 β → ∞ β = β1 β → 0

Figure 6.6: Vacua in different size of spatial circle β. From left to right, β decreases from infinity to zero. In the left first diagram, β → ∞, only one vacuum exists. In the middle diagram, β = β1, two vacua have the same action, and after that, the left one becomes a false vacuum, and is possible to decay via a bubble. In the most right one, β ∼ ls, and with slow varying condition, closed string tachyons condense, and give a perturbative decay.

6.3 The Dual Field Theory

In the work [66–68], they discussed N = 4 YM theory on S1 × S3. On one-loop level calculation, both SYM and YM show that a transition exists at some temperature. And on the calculation up to 3-loop level, only S1 × S2 has a ﬁrst order transition, while S1 × S3 has a second order transition. For the Euclidean spacetime S1 × S3, if we interpret time as one of the dimension of 3 1 3 S , the Lorentzian space should be S × dS3, instead of R × S as in their case. If we are only interested in small time region, it is approximately S1 × R × S2.

6.3.1 S1 × R × S2 The Lagrangian of SU(N) YM theory is

2 SYM ∼ −tr [Dµ,Dν] , (6.23) where Dµ = ∂µ − igYM [Aµ, ∗]. (6.24) For one-loop level calculation, the only terms contributing are the quadratic terms of 2 1 2 Aµ, which in the metric R × S × S ,

2 2 2 2 ds = −dt + dχ + dΩ2, (6.25)

are proportional to

~2 ˜ 2 ˜ 2 i ~2 ˜ 2 0 ~2 ˜ 2 χ Ai(∂ + D0 + Dχ)A + A0(∂ + Dχ)A + Aχ(∂ + D0)A , (6.26) Chapter 6. On Bubbles of Nothing in AdS/CFT 129

where i = 1, 2, and the operators are covariant derivatives. Let us introduce two gauge ﬁeld relevant quantities 1 Z 1 Z α(t) ≡ Aχ, α¯(t) ≡ A0, (6.27) ω2 S2 βω2 S1×S2

where β is the period of S1. Also we ﬁx the gauge as

i ∂iA = 0, ∂χα = 0, ∂0α¯ = 0, (6.28)

thus we might callα ¯ and α(t) chemical potential and pseudo chemical potential respectively. Here we note thatα ¯ is a constant. The Fadeev Popov determinants conjugating to (6.28) are

Z i i −c∂¯ iD c 0 0 ∆1 = det∂iD = DcDce¯ , ∆2 = det (∂χ − i[α, ∗]), ∆3 = det (∂0 − i[¯α, ∗]). (6.29) Following the construction of eﬀective action in [66], we have

Z Z Z 0 Z −βF 0 −SYM (A ,α,α¯) −Seff (U) e = dα∆2 d¯α∆3 DA ∆1e = [dU]e , (6.30)

where Z e−Seff (U) = [dV ]e−Seff (U,V ), (6.31) and dα∆2 = [dU], d¯α∆3 = [dV ]. (6.32) Including the scalars and fermions, the eﬀective action can be written as Z Seff ∼ − dt Veff

∞ X Fk(x) ∼ const. + c T tr(U k)tr(U −k), (6.33) 1 k k=1 where k k+1 k Fk(x) = 1 − zB(x ) − (−1) zF (x ). (6.34)

And the Hagedorn radius βH is determined by

−βH F1(xH ) = 1 − zB(xH ) − zF (xH ) = 0, where xH = e . (6.35)

Thus in one loop calculation, a transition happens at the Hagedorn radius βH . 2 From the Yang-Mills action (6.23), there is a kinetic term of α(t), say from −tr ([D0,Dχ] ). Combined with the eﬀective action, we can in principle study the mechanical behavior. 1 In the case S × dS3, if we restrict t to be small, the eﬀective action has a similar expression. Chapter 6. On Bubbles of Nothing in AdS/CFT 130

To be clearer, let us have a close look at tr(eikβα)tr(e−ikβα). In Euclidean spacetime S1 × 3 S , α is only a constant matrix. If we write it in a diagonal way, α = diag(α1, α2, . . . , αN ). For simplicity, let us set the values,

θ θ θ n − 1 α = − , α = , α = −1 + 2 , n = 1...N (6.36) 1 β N β n β N − 1

tr(eiβα)tr(e−iβα)/N 2 1

0.8

0.6

0.4

0.2

0 0.5 1 1.5 2 2.5 3 π θ theta

iβα −iβα 2 Figure 6.7: tr(e )tr(e )/N as a function of θ, with βαn uniformly distributed between θ n−1 3 (−θ, θ) for the solid line, and αn = β −1 + 2 N−1 for the point line.

We ﬁnd that tr(eiβα)tr(e−iβα) is a monotonic function of θ, where θ corresponds to uniform distribution of chemical potential α on S1, and θ = 0 makes the distribution function as Delta function. We should remind us that in the S1 × S3 example with S1 as the Euclidean time, the uniform distribution is dual to thermal vacuum, and the Delta distribution gives black hole in gravity theory. θ n−1 3 However if we make the eigenvalues αn a little bit diﬀerent, say αn = β −1 + 2 N−1 , the tr(eiβα)tr(e−iβα) is not a monotonic function anymore. But in any event, the possible maximum and minimum are invariant, i.e. it is at uniform distribution,

iβα −iβα iβα −iβα min(tr(e )tr(e )) = tr(e )tr(e ) |uniform dist. = 0, (6.37) and at the Delta distribution,

1 1 max tr(eiβα)tr(e−iβα) = tr(eiβα)tr(e−iβα) | = 1. (6.38) N 2 N 2 delta dist.

The behavior between them might be complicated. Since

iβα −iβα 2 iβα −iβα 2 Seff ∼ F (x)(tr(e )tr(e ))/N , with (tr(e )tr(e )/N ) ∈ [0, 1], (6.39) Chapter 6. On Bubbles of Nothing in AdS/CFT 131

when β < βH , F (x) > 0, the uniform distribution dominates the effective action Seff , while at β > βH , F (x) < 0, the delta function distribution is most important. This shows that on 1-loop calculation, there is no first order transition from uniform distribution into Delta distribution. Since the gravity theory shows that the transition is first order, it is supposed to be the same in the dual gauge theory. We have not succeeded in it, and might pin our hope on the 2- and 3-loop calculation, and including the complete super Yang-Mills theory instead of pure Yang-Mills only. One can also understand the eigenvalue distribution from the QCD example. At low temperature, one observes baryons, which are color charge singlets, and at high temperature, quarks are supposed to be observable, each carrying a color index. It is similar here, the uniform distribution does not give us a color index preference, while the density function certainly makes a specific choice of the color index.

Of the duality between gravity and gauge theory, one needs to note that the ﬁeld theory calculation here and in [66–68] is at weak coupling, then one can do the perturbation with 2 λ = gYM N, as X λβ X S ∼ c Tr(U m)Tr(U −m) + c Tr(U m)Tr(U n)Tr(U −m−n) eff m,−m N m,n,−m−n λ2β X + c Tr(U m)Tr(U n)Tr(U p)Tr(U −m−n−p) + ... (6.40) N 2 m,n,p,−m−n−p 2 2 4 4 ∼ f(T, λ)|u1| + λ b|u1| + O(λ ), (6.41)

where in the first line, the three sums are related with 1-, 2- and 3-loop calculations respectively, while in the last line, the coefficient b is determined by 2- and 3-loop calculations. Since λ = g2 ∼ g ∼ l−4, (6.42) N YM s s the first line approximation is valid in any strong or weak coupling λ in large N limit, while the second line perturbation is only good at weak coupling. But the gravity one can do the calculation easily is at small gs regime, or equivalently R ls. Thus the two dual theories are in two opposite limits, which might give some superficial discrepancies, or something more subtle. Such as in the gravity, it shows an existence of first order transition, while in the field theory, one has to calculate up to 3-loops to get such behavior.

6.4 Conclusions and Discussions

We have studied the phase diagram of the maximally supersymmetric gauge theory on 1 dS3 × S with various configurations of gauge fields on the non-contractible circle. The results were interpreted in terms of quantum mechanics for a single variable (expectation value of the Wilson line around the circle). We have seen that various boundaries in those phase diagrams can be interpreted as features of the effective potential of that quantum mechanical system. This supports an interpretation of the decay mediated by the bubble Chapter 6. On Bubbles of Nothing in AdS/CFT 132 of nothing as conventional vacuum decay, and the interpretation of the core of the bubble as the true vacuum, namely the tachyon condensate. This agrees with the interpretation of the tachyon condensation in the case of the AdS soliton, which is a limit of the bubble of nothing solutions we discuss. Even though it is plausible that the dynamics effectively reduces to that of a single degree of freedom, we are not able to calculate the effective potential in the strong ’t Hooft coupling regime. It is natural to attempt such calculation in other regimes of the gauge theory, and compare the qualitative features to those obtained here. The study of the same gauge theory at weak coupling is part of our interests of future work. In addition to providing a definition of the effective potential, this study can shed some light on the mysterious “nothing” state, by viewing it from a conventional quantum mechanical perspective. Additionally, it would be fascinating to probe the nothing state that apparently exists in the bulk bubble spacetime. It is likely that the picture in [164] applies here as well, and at the core of the bubble we have, in addition to the geometrical description, a winding mode condensate. String scattering in the background will then probe the winding condensate and will give indication that the geometrical description of the spacetime is incomplete. We list below some of future possible interests.

6.4.1 Multi-wrapped Wilson Loop Wilson/Polyakov loop plays an important role in gauge theory, and the expectation value of it, in AdS/CFT background, corresponds to a minimal area of string world sheet surface,

H i A0dτ −S −Area u1 = hW i = hP e i ∼ e ∼ e . (6.43)

However in the field theory beyond 1-loop calculation or when the spatial circle size is iβα small enough β 1, one also needs uk ∼ htr e i, which might connect to k-wrapped world sheet surfaces in string theory. The surfaces’ structures are subtle, usually combined with cone singularities and branch points. One also needs to note the difference and connection between k-wrapped surface and k times 1-wrapped surfaces.[165–167] It is interesting to see whether the k-wrapped world sheet surface or uk in gauge theory will change our picture much. Another interesting idea is embedding a bubble in the bulk, similar to a black hole in AdS, the world sheet might be nontrivially connected to it. One can calculate the new minimal surface area, and the effects on correlation functions.

6.4.2 General D-brane Metric The DBI action of a Dp-brane is Z p+1 p 0 S ∼ d σ det(gab + 2πα Fab). (6.44) Chapter 6. On Bubbles of Nothing in AdS/CFT 133

Figure 6.8: Gauge theory lives on the plane, and the circle on it displays a Wilson loop. The curved cone surface spanned by the loop and the red curve out the plane indicate a string world sheet ending on the loop.

With the flux F turned off, the metric for D-brane(2.27) is simple, and in the near-horizon 5 limit, the metric goes to AdS5 × S (2.31). It is natural to extend the discussion with non-vanishing F . In principle, following [168], one is able to calculate the new metric for the D-brane, and furthermore might be able to 5 find a “near-horizon” limit, some spacetime similar to AdS5 × S . On the gauge theory side, we are interested in the eigenvalue distribution of α(6.27). In the example S3 × S1, with S1 Euclidean time circle, one usually think uniform distribution corresponds to thermal AdS, and probably the delta function distribution to AdS black hole. If the message from the calculation mentioned above is positive, one can check and verify this hypotheses, while a further work would be the geometry dual to general eigenvalue distributions.

6.4.3 Polyakov and Polyakov-Maldacena Loop A Wilson loop wrapped around a Euclidean time circle is called Polyakov loop,

1 h i H x˙ µA i W (C) = Tr P e C µ . (6.45) N In the AdS/CFT correspondence, since the gauge theory is super-Yang-Mills instead of pure Yang-Mills, the more precise one is actually a modified function, called Polyakov-Maldacena loop, 1 h H ix˙ µA +|x˙|θiφi i W (C) = Tr P e C µ , (6.46) N Chapter 6. On Bubbles of Nothing in AdS/CFT 134 where φi are the six scalar fields of N = 4 SYM and θi is a unit vector in R6. We know in the strong coupling, the P-M loop, or generally a Wilson-Maldacena loop, is described by a string world sheet ending on the boundary of AdS5, and fixed by Dirichlet boundary conditions on the sphere, at point θi. The authors made a conjecture about the string dual of ordinary Wilson loop: the loop (6.45) might connect to a string worldsheet ending on the boundary of AdS, and with Neumann boundary conditions on the sphere S5. Then we would have something interesting to do. One is following either Neumann or Dirichlet boundary conditions, to calculate the minimal string world sheet area of 1- and k-wrapped surfaces ending on the loop. Firstly one can see the difference between the two cases, and what it gives. Since the ordinary Wilson loop has the gauge field only, it is natural to guess that the Neumann conditions break the supersymmetry, and the worldsheet surface with Neumann conditions relates with a pure Yang-Mills theory. One can check for the similarity between them. It is also interesting to see if we can get something from the result about eigenvalue distribution and transitions. 135

Part IV

Conclusion 136 Chapter 7

Conclusion and Outlook

This dissertation focuses on two fields of the AdS/CFT correspondence. In part II I discussed our work in holographic condensed matter theory, and then part III is dedicated to gravitational instability interpreted by bubbles and the dual field theory. The first chapters are introduction and reviews on string theory and the other relevant background knowledge. The organization is not according to the works done chronologically. Holographic condensed matter theory is a new research field of string theory. The AdS/CFT correspondence conjectures a duality between string theory and field theory. It is often called “holographic” because the most well known example is gravity in spacetime 5 AdS5 × S is dual to super-Yang-Mills theory living on the boundary of AdS5 space. The recent applications on condensed matter theory helps to understand the strongly coupled behavior and the quantum critical behavior, some of which have not been described well by the traditional condensed matter theory. Particularly this application is named AdS/CMT or holographic condensed matter theory. Since AdS/CFT is a strong-weak coupling duality, one is able to investigate a theory in weakly curved geometry to extract information of its dual strongly coupled CMT. On the other hand, phase transition is a general property in various fields of physics, and at low temperature condensed matter systems usually have complicated phase diagrams. Some works on AdS/CMT have already reproduced some phase diagrams successfully, and it is a promising direction for us. In chapter 4, I first described the appearance of Fermi surface in the boundary field theory, by the analysis of the dual bulk gravity. The calculation is done in an extremal charged black hole background at zero temperature. Then with the magnetic charges added in the model, the black hole turns into a dyonic black hole and the previous continuous spectrum discretizes, resulting in a Landau level like structure. Starting from the Dirac equations in AdS4 space, the fields can take the form of decoupled functions of radial coordinate and the transverse directions. To satisfy all the boundary conditions, the spectrum has to be discrete. We also numerically investigate the relation between the spectrum and the magnitude of the magnetic field. The analysis in Sec. 4.2 is in metric of planar coordinate system, and the discussion is converted to global coordinate in Sec. 4.3. Although the main behavior is expected similar since it is only a coordinate transformation, an interesting new phenomenon is hinted of. The global coordinate system provides one more parameter than the planar one, and covers a larger region. All together the new coordinate system enables us to approach one extra dimension in the phase diagram (T, µ), or effectively (T, ˜l), where ˜l is the effective AdS radius. In the planar coordinate calculation, although a Fermi surface exists, some other properties are not as a Fermi liquid, and the material is named non-Fermi liquid. On the other hand, the analysis in global coordinates shows us in an appropriate limit, it behaves as Fermi liquid. To justify this conjecture, future work need to be done to verify other expected properties. In Sec. 4.4 the work is extended to finite temperature Chapter 7. Conclusion and Outlook 137

regime. The system is now unstable due to the finite temperature. Since at some value of low temperature, there is a transition from the ordinary Reissner-Nordströmblack hole to a solitonic hairy black hole solution, the “Fermi surface” behavior is slightly different. The behavior of “Fermi surface” is described by the Green’s functions of the probe fermion. The trajectory of Green’s functions’ poles are numerically drawn on the complex ω plane and with temperature. Next, I showed some work on holographic superconductivity in chapter 5. In the background of planar AdS4 black hole, an Abelian scalar is found to condense below some low temperature, and the resulting material in the dual field theory behaves as a superconductor. The original model is set up by hand, without embedding in string theory. In Sec. 5.2 we study a string theory based configuration: two probe D7-branes in the background of large N copies of coincident D3-branes. Different from the previous example, the gauge group is now SU(2) and the spacetime background is AdS5. Besides the familiar compo- 3 1 nent of gauge field Atτ , we work in the ansatz with an extra spatial component Bxτ . It turns out that Bx plays a similar role of scalar in the previous model, and condenses below a specific low temperature. We plot the frequency dependent conductivity below the critical temperature, and calculate the speed of the second sound. The group SU(2) comes from the double D7-branes so actually it is an isospin symmetry, and the condensation is of a pion-like field. Then an argument of existence of gapped holographic superconductor is given in Sec. 5.3. Zero temperature is very special for holographic superconductivity, since the horizon disappears when the Abelian scalar condenses, and the final state is a solitonic object instead the ordinary condensed sphere around the horizon. By the discussion of the gauge field’s equation in a Schrodinger-like form, the conductivity is argued never vanishing in the near-zero frequency regime(the infinite DC conductivity has already been removed), and thus the holographic superconductor is gapless. The story is not the same in a p-wave superconductor, where the gauge group is SU(2) and we find the appropriate metric. Although the condensed state is again solitonic, it is anisotropic. The potential in the gauge field’s Schrodinger-like equation behaves differently, and the conductivity can vanish when ω → 0+, i.e. this superconductor is hard-gapped. One intriguing property of holographic superconductivity is that spikes appear in Re(σ(ω)) at low enough temperature and the mass of scalar is close to its BF-bound. The number of the spikes increases as either temperature or mass decreases. I use WKB method to understand this spike behavior analytically in Sec. 5.4. The result does not quantitatively agree with the numerical data in the above examples, nevertheless the qualitative behavior is consistent. Chapter 6 is our early work, describing a gravitational instability due to bubble of nothing and its dual field theory. Euclidean instantons often indicate decay processes in gravity. An AdS black hole solution, a Euclidean instanton and an AdS bubble of nothing are connected by a series of Wick rotations. The inverse temperature of black hole is changed to the radius of a spatial circle, i.e. the bubble’s metric is compactified around this spatial circle. From the familiar properties of AdS black hole, we are able to find the behavior of AdS bubble. In this work, we consider a R-charged AdS bubble, find the formulas of its temperature, Euclidean action, mass and entropy. With the analysis of the stability, we depict the phase diagrams for different cases. The non-perturbative phase transition depends on the size of bubble, or effectively the size of the spatial circle β. When the circle is large enough, there is no decay described by this solution. The dual field theory Chapter 7. Conclusion and Outlook 138

1 lives on S × dS3, however due to the difficulty of the analysis on de-Sitter space, we have a discussion on its approximated spacetime S1 × R × S2, where R is the time dimension. We argue that the phase transition in field theory is described by the difference of the actions between homogeneous and inhomogeneous distributions of the eigenvalues of the gauge field. The properties of an AdS bubble can be investigated by a Wilson loop and a closed string whose worldsheet ends on it. One of the possible extensions which might be helpful to understand is the inhomogeneous distributed eigenvalues is multi-wrapped Wilson loops. I will keep my interest on this project in the future. Besides superconductivity, superfluidity, Hall’s effect and Fermi surface which have been explored holographically, condensed matter has many more interesting phenomena. For example, it is generally true that the phase diagrams are complicated, which is mainly because of the sophisticated lattice structures of real materials. It would be interesting if one is able to reproduce these phase diagrams theoretically using the holographic method. This would be helpful for us to understand condensed matter physics and might tell us the underlying universality. Our latest work[169] has made a step in that direction. With so much evidence supporting the AdS/CFT correspondence, we believe that the conjecture is correct, and it should have many applications in condensed matters. In addition, there are recently works on other fields, such as holographic optics[170]. We will see more and more holographic applications appear in various fields. A duality in physics often hints of the existence of an underlying unity. For instance the duality between electric and magnetic fields comes from Maxwell’s electromagnetism, and the many dualities among the superstring theories are argued to be unified as M- theory. It would be interesting to ask similar question on the holographic theories: can we find more except for the apparent holographic dualities? Two familiar examples are black hole’s horizon and the AdS/CFT correspondence. In the former example, it seems that any information and theory can be encoded into gravity1, while in the second gravity also plays a crucial role. So what should be our goal: a unification, or gravity “is” everything? Most of our evidence of AdS/CFT considesr field theory without gravity. If more examples of gravity duality of field theories with complicated gravity background are found, it should be very helpful to answer the above question. Besides, the recent entropy force argument[10] seems to support the gravity=everything conclusion.

1It might be better to name it “gravity plus” since black holes can have charges other than mass. 139

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Appendices 152 Appendix A

Condensed Matter Physics

In this section we shall briefly review some aspects of the relevant condensed matter physics. We are mainly interested in the transport properties of strongly coupled electron systems. For this the most relevant physics comes from low energy excitations near the Fermi surface. We are also interested in the behavior of the Fermi surface in presence of a uniform magnetic field. For this purpose, we will summarize some well known experimental effects in condensed matter physics [171], the de Haas-van Alphen effect and quantum Hall effect. After that, superfluidity and superconductivity are very concisely reviewed.

de Haas-van Alphen Eﬀect

This effect was observed originally by de Haas and van Alphen in 1930 for Bismuth and subsequently by others for many different metals. When a metallic sample is placed in a high magnetic field H at low temperatures, many physical parameters oscillate as the magnetic field is varied. For example, the magnetization M to magnetic field ratio M/H shows characteristic oscillations with varying H. More precisely, the quantity that is measured is the magnetic susceptibility χ = dM/dH. Fig. A.1 shows a schematic plot of the susceptibility illustrating this effect. In presence of a uniform transverse magnetic field the electron orbits in a metallic sample form quantized Landau levels. For a 2+1 dimensional system of electrons the levels are characterized by a single integer ν. The energy of the level ν is given by 1 E = ν + ω (A.1) ν 2 ~ c

where ωc = eH/mc is the cyclotron frequency, e and m being the electronic charge and mass respectively. The corresponding wavefunctions are Hermite polynomials, similar to a harmonic oscillator. Each Landau level has a very high degeneracy proportional to the magnetic field H. The de Haas-van Alphen effect turns out to be a direct consequence of the above mentioned quantization of closed electronic orbits in an external magnetic field. The oscillations were first explained by Landau using this picture. It turns out that the susceptibility is actually a periodic function of the inverse magnetic field 1/H. A remarkable observation due to Onsager which connects the oscillations with the properties of the Fermi surface is expressed through the formula 1 2πe 1 ∆ = (A.2) H ~c A where A is an extremal cross-sectional area of the Fermi surface in a plane normal to the magnetic field. To derive this formula it is assumed that relevant Landau levels have very Appendix A. Condensed Matter Physics 153 H M

Figure A.1: Oscillations in the susceptibility with varying magnetic ﬁeld.

high level number (n). Magnitude of n is given by EF /[(e~/mc)H], where EF is the Fermi 8 4 energy. Typically EF /[e~/mc] ∼ 10 G. Hence n ∼ 10 even at a field strength of the order 104G. Let us now briefly describe the mechanism behind the above oscillatory behavior. In metals, it is expected that the dynamics near the Fermi surface governs most of the electronic properties. The density of electronic states near the Fermi level is thus an important factor. It can be shown that whenever the area of a quantized electronic orbit coincides with an extremal cross sectional area of Fermi surface in a plane normal to the magnetic field there is a singularity of the density of states at the Fermi level. As the magnetic field is varied the successive quantized orbits cross this extremal area. This is responsible for the observed oscillatory behavior. The de Haas-van Alphen effect is thus a very useful tool for probing the Fermi surface. In addition to the de Haas-van Alphen effect, some related phenomena exist for metals which can also provide some information about the Fermi surface. Some examples are the magneto-acoustic effect, ultrasonic attenuation, the anomalous skin effect, cyclotron resonance, size effects, etc.

Quantum Hall Eﬀect

The conventional Hall effect is the phenomenon where a conductor carrying an electrical current when placed in a transverse magnetic field develops a voltage across its edges transverse to both the magnetic field and the current. Classically, the effect is easily explained as a result from the deflection of the electrons carrying the current due to the Lorentz force in presence of the magnetic field. This picture changes as impurities are introduced into the system. This leads to localized bound states with energies between the Landau levels, at the expense of some of the degenerate Landau level states. Now as the electron density is increased beyond a filled Landau level, the extra electrons fill the localized states. The Hall conductivity is unchanged during this process as the localized states carry no current. Only after enough Appendix A. Condensed Matter Physics 154 electrons have been added to raise the Fermi energy to lie in the next highest Landau level does the conductivity begin to increase again. The process repeats itself, which leads to plateaus in the Hall conductivity. This phenomenon is known as the integer quantum Hall effect, as the plateaus occur after each Landau level is completely filled. The integer quantum Hall effect dominates when the electrons are weakly interacting; in presence of strong electron-electron interactions Hall plateaus can occur when only a fraction of the states at a particular Landau level is filled. This is known as the fractional quantum Hall effect. These effects are prominent at low temperature and high magnetic field. It would be interesting to investigate such effects in our system.

Superﬂuidity

At very low temperature, some liquids, typically Helium-4 or Helium-3 can overcome the friction by surface interaction, and the viscosity becomes zero. This quantum hydrodynamics behavior was discovered by P. Kapitsa, J. Allen and D. Misener in 1937. The phenomenological explanation(two-fluid model) was contributed by Landau. As- suming that sound waves are the most important excitations in 4He at low temperatures, he showed that 4He flowing past a wall would not spontaneously create excitations if the flow velocity was less than the sound velocity. In this model, the sound velocity is the ”critical velocity” above which superfluidity is destroyed. Following this idea, he can construct a so-called normal fluid state, which is zero at zero temperature. The ratio of normal fluid state increase with temperature, and all the superfluid turns into normal fluid above a critical temperature, which is named Lambda temperature TL. The superfluid component has very special properties: zero viscosity, zero entropy, and infinite thermal conductivity. One application of the properties results in a wave of heat conduction at relatively high velocity of 20 m/s, called second sound. In this phenomenon, heat transfers as a wave instead of the conventional mechanism of diffusion, and heat takes the place of the pressure in normal sound waves. Although both 4He and 3He have similar superfluidity behavior, they have different mechanisms. 4He atoms are bosons, and condense as Bose-Einstein condensation at low temperature. On the other hand, 3He atoms are fermions, and are expected to condense in Cooper-pairs.

Superconductivity

Superconductivity is a peculiar feature at low temperature. In a superconducting state, the DC electric resistance of the material drops to exact zero. It was discovered by H. K. Onnes in 1911. A famous property of this is Meissner eﬀect, which describes the phenomenon that magnetic ﬁeld is repelled from a superconductor. In a superconducting material, superconductivity behavior appears when temperature is low than a critical temperature Tc, and the DC electric resistance drops to exact zero. The typical critical temperatures of conventional superconductors range from 20K to below 1K, for example, for solid mercury Tc ≈ 4.2K. Appendix A. Condensed Matter Physics 155

The superconductors found in the early years are called conventional superconductors. They are nicely described by BCS(Bardeen-Cooper-Schrieffer) theory, the basic idea of which is the assumption of a condensation of electron-pair(Cooper pair) into a boson-like state. The highest critical temperature in conventional superconductors we have gotten is 39K in magnesium diboride. The conventional superconductors are classified into two types by their phase transition order: type I if the transition is of first order, and type II for second order phase transition. It was in 1986 when a revolution of superconductivity happened. Cuprate-based superconductors was found in laboratories, with typical much higher critical temperatures than the conventional types. As to now, the highest Tc in cuprate-based superconductors is 135K, and it might go as high as 164K at high pressure. Unfortunately, BCS theory is not able to explain the high-Tc superconductivity well, and the mechanism of which is still controversial. Furthermore, a new type of superconductors, iron-based ones, was found in 2008. Al- though they do not have the critical temperature as high as the cuprate-based partners, the new ones are helpful in understanding the mechanism of high-Tc superconductivity. Unlike the s-wave symmetric conventional superconductivity which is explained in electron- phonon attraction, high-Tc superconductivity is usually dealt with genuine electronic mechanism(i.e. by anti-ferromagnetic correlations), and d-wave symmetry instead of s-wave is substantial in cuprate-based superconductors. In addition, p-wave superconductors e.g. Sr2RuO4, are also found[172]. Besides the temperature, some other quantities can also play the role of order parameters of phase transition into superconductivity. The most familiar ones are pressure P and chemical potential µ. 156 Appendix B

Kubo Formula for Electrical Conductivity

A simple derivation of Kubo formula for electric conductivity is described in this section. The so-called Kubo formula is of the correlation functions describing the linear response. In a external electric ﬁeld background,

ext ext i(~q·~r−ωt) Eα (~r, t) = Ξα e , α = ~x, (B.1) and the linear response to external electric ﬁeld, i.e. the induced current is

X 0 ext Jα(~r, t) = σαβ(~q, ω)Eβ (~r, t). (B.2) β

The desired conductivity σ is the response to the total internal electric ﬁeld,

ext X Eα (~r, t) = (external and induced ﬁelds). (B.3)

Let us work in the assumptions: 1. Only long-wavelength excitations are studied(small ~q). And the most interested is the DC conductivity, in the limit ~q → 0, ω → 0. 2. Assume the system is linear and the perturbations at different ω’s act separately. The Hamiltonian can be written as Hˆ + Hˆ 0, (B.4) where Hˆ is the Hamiltonian without electric field, and Hˆ 0 is the interaction between electric field and particles, 2 X 1 h ei i X 1 Hˆ 0 = ~p − A~(~r , t) − ~p2. (B.5) 2m i c i 2m i i i For convenience, we work in Coulomb gauge from now on, ∇ · A~ = 0. From the Maxwell’s equations, one can write out the electric fields, ∂A i i E = − α −→ A = − E = − Ξ ei(~q·~r−ωt). (B.6) α ∂t α ω α ω α Since we are interested only with the linear response, the (A)2 terms can be dropped, and

e 1 X Hˆ 0 = − [p · A(r , t) + A(r , t) · p ]. (B.7) c 2m i i i i i Appendix B. Kubo Formula for Electrical Conductivity 157

The current operator can be introduced as

1 X ˆj(r) = [δ(r − r )p + p δ(r − r )], (B.8) 2m i i i i i so the perturbation Hamiltonian is 1 Z Hˆ 0 = − d3rˆj(r) · A(r, t). (B.9) c After the Fourier’s transformation, i Hˆ 0 = Ξ e−iωtˆj(q). (B.10) ω α The observable current is the average of particle velocities, e X J(r, t) = hvˆ i, (B.11) V i i where V is the volume. Notice h i ˆ ˆ 0 dr ri, H + H 1 h e i vˆ = i = = pˆ − A(r ) , (B.12) i dt i m i c i the current is separated into two parts,

e X e2 X J(r, t) = hpˆ i − A(r , t) mV i mcV i i i e2 = hˆj(r, t)i + i n E(r, t), (B.13) mω 0 P i ˆ where n0 = V =particle density. Assuming linear response, hj(r, t)i is proportional to E, which is called Kubo formula. In the linear response limit, only the linear terms of Hˆ 0 are kept,

Z t h i hˆj(r, t)i = hψ|ˆj(r, t)|ψi − i dt0hψ| ˆj(r, t), Hˆ 0(t0) |ψi, (B.14) −∞

where hψ|ˆj(r, t)|ψi = 0, since no current if no ﬁeld. Substituting equation (B.10,)

h i i 0 h i i 0 h i ˆj (r, t), Hˆ 0(t0) = Ξ e−iωt ˆj (r, t), ˆj (q, t0) = E (r, t)e−iq·reiω(t−t ) ˆj (r, t), ˆj (q, t0) . α ω β α β ω β α β (B.15) Appendix B. Kubo Formula for Electrical Conductivity 158

Thus the conductivity is

Z t 2 1 −iq·r 0 iω(t−t0) h 0 i n0e σαβ(q, ω) = e dt e hψ| ˆjα(r, t), ˆjβ(q, t ) |ψi + iδαβ. (B.16) ω −∞ mω After averaging r to get rid of atomic ﬂuctuations using, Z 3 −iq·r † d re jα(r, t) = jα(−q, t) = jα(q, t), (B.17) the Kubo formula is given

Z ∞ h i 2 1 iωt ˆ† ˆ n0e σαβ(q, ω) = e hψ| jα(q, t), jβ(q, 0) |ψi + iδαβ. (B.18) ωV 0 mω In terms of the retarded Green’s function iθ(t − t0) h i G (q, t − t0) = − hψ| ˆj† (q, t), ˆj (q, t0) |ψi, (B.19) αβ V α β with Fourier transform ∞ Z 0 h i i 0 iω(t−t ) ˆ† ˆ 0 Gαβ(q, ω) = − dtθ(t − t )e hψ| jα(q, t), jβ(q, t ) |ψi, (B.20) V −∞ the Kubo formula is " # i n e2 σ (q, ω) = G (q, ω) + 0 δ . (B.21) αβ ω αβ m αβ Taking the limit q → 0 and then ω → 0 gives us the DC conductivity, 1 Re[σαβ] = − lim Im[Gαβ(ω)]. (B.22) ω→0 ω

One relevant quantity often investigated is the spectral function Aαβ(ω) of Gαβ(ω) which can be introduced as 1 − Im[G (ω)] = A (ω) αβ 2 αβ π X = (1 − e−βω)eβΩ e−βEn hn|ˆj† |mihm|ˆj |niδ(ω + E − E ). V α β n m n,m (B.23) 159 Appendix C

Transverse Scalars and Gauge Fields on S3

5 Consider D7-branes embedded in an AdS5 × S background, dz2 1 ds2 = −f(z)dt2 + + (dx2 + dx2 + dx2) + dθ2 + sin2 θ dϕ2 + sin2 ϕ dΩ2 , (C.1) z4f(z) z2 1 2 3 3

5 where {t, z, ~x} are the AdS5 coordinates, while θ, φ, Ω3 are along S . Suppose the coincident 3 5 D7-branes ﬁll the AdS5 and wrap an S of S , i.e. they are in the coordinates {t, z, ~x, Ω3}, then the induced metric is dz2 1 ds2 = −f(z)dt2 + + (dx2 + dx2 + dx2) + sin2 θ sin2 ϕ dΩ2. (C.2) 7 z4f(z) z2 1 2 3 3

According to [121], the leading order DBI action of a Dp-brane is

0 2 Z Tp(2πα ) p+1 p ab a i i j Sp = − d ξ −GindTr FabF + 2DaΦiD Φ + [Φi, Φj][Φ , Φ ] , (C.3) 4gs where the scalars Φi ≡ Xi/(2πα0) are the transverse coordinates, and the covariant deriva- i tive is deﬁned as DaΦi = ∂aΦi + [Aa, Φi]. For D7-branes, one has two scalars Φ , i = θ, ϕ. 3 a a We turn on a gauge ﬁeld according to the ansatz A = Atτ dt, where τ = τ /2i with a b abc c i i a the commutation relations [τ , τ ] = τ . The scalars take the general form Φ = Φaτ . The non-zero components of DaΦi are

3 1 2 a DtΦi = [Atτ , Φi] = At(Φi,2τ − Φi,1τ ), DzΦi = ∂zΦi = (∂zΦi,a)τ . (C.4)

The eﬀective action can be written as

0 2 Z 3 3 ( T7(2πα ) 8 sin θ sin ϕ ab S7 = − d ξ 5 Tr FabF 4gs z " 3 # "1,2,3 #) X 1 X −g z4f(z) (∂ Φi)2 − f(z)A2 (Φi )2 + (Φi )2 − sin2 θ (ΦθΦϕ − Φθ Φϕ)2 , ii z l t 1 2 2 l m m l l=1 l6=m (C.5)

where gii are 10-dimensional metric components with i = θ or ϕ. Appendix C. Transverse Scalars and Gauge Fields on S3 160

i i i i i Introducing Φ± ≡ (Φ1 ± iΦ2)/2, we ﬁnd the equations of motion for (Φ3, Φ±), " # " # f(z) f f 0(z) sin2 θ g ∂2Φi + − + ∂ Φi = 2 Φi ((Φj )2 + Φj Φj ) − Φj (Φi Φj + Φi Φj ) , ii z z 3 z2 z z 3 z5 3 − + − 3 + − − + " # f(z) f f 0(z) g ∂2Φi + − + ∂ Φi ii z z + z2 z z + ( ) 1 g h i = − ii A2Φi + sin2 θ Φi Φj Φj + Φi (−(Φj )2 + (Φj )2) − Φi Φj Φj z5 f t + + + − − + 3 3 − 3 " # f(z) f f 0(z) g ∂2Φi + − + ∂ Φi ii z z − z2 z z − ( ) 1 g h i = − ii A2Φi + sin2 θ Φi Φj Φj + Φi (−(Φj )2 + (Φj )2) − Φi Φj Φj (C.6), z5 f t − − + − + − 3 3 + 3 where i, j ∈ (θ, ϕ) and i 6= j. Now let us turn on the ﬂuctuations. It does not lose the generality to set sin θ = 1, and thus gii : gθθ = gϕϕ = 1 in the above equations. If we restrict our discussion to the ansatz

1 2 Φ1 = φ1τ , Φ2 = φ2τ , (C.7) the EOM’s can be simpliﬁed

1 f 0 1 A2 φ00 + (− + )φ0 + t − φ2 φ = 0, (C.8) i z f i z4f f j i where i, j = 1, 2 and i 6= j. For the simplest case, with Φ = φ(z)(τ 1dx1 + τ 2dx2), the equation of motion is

1 f 0 1 A2 φ00 + (− + )φ0 + ( t φ − φ3) = 0. (C.9) z f z4f f If the gauge ﬁelds on the 3-sphere are turned on instead of the transverse scalars on the D7-branes, one would ﬁnd a similar condensation phenomenon. In an ansatz

3 1 A ∼ At(z)τ dt + Aθ(z)τ dθ, (C.10)

the equations of motion are

1 A2 A00 − A0 − θ A = 0, t z t z4f t 1 f 0 A2 A00 + − + A0 + t A = 0. (C.11) θ z f θ z4f 2 θ Appendix C. Transverse Scalars and Gauge Fields on S3 161

Or in another ansatz, 3 1 2 A ∼ Atτ dt + Aθ(τ dθ + τ sin θdϕ), (C.12) 3 where (θ, ϕ) are angle coordinates on 3-sphere of D7-brane worldvolume AdS5 × S . The EOM’s turn out to be 1 2A2 A00 − A0 − θ A = 0, t z t z4f t 1 f 0 1 A2 A00 + − + A0 + t A − A3 = 0. (C.13) θ z f θ z4f f θ θ 162 Appendix D

Searching for Bound States on Complex ω Plane: WKB Method

The discussion is restricted in AdS4 for simplicity. We have a unit mass particle moving in a 1-dimensional potential, V (y) = 2fψ2. To get the infalling wave function at horizon y → −∞, it is an ordinary tunneling problem of WKB approximation in textbooks. To be more familiar, let us deﬁne x = −y, thus the WKB wave functions in the three regions are  1 h i R x1 p(x0)dx0 −i R x1 p(x0)dx0 i √ A e x + B e x , 0 < x < x1, region I  p(x)  h R x 0 0 R x 0 0 i 1 x |p(x )|dx − x |p(x )|dx ψ(x) = √ C e 1 + D e 1 , x1 < x < x2, region II (D.1) p(x)  R x 0 0  1 i x p(x )dx √ F e 2 , x > x2, region III  p(x)

p 2 where 0 < x1 < x2 are the turning points, and p(x) = 2(ω − V (x)). Introduce γ = R x2 |p(x0)|dx0, one can ﬁnd the relation between the coeﬃcients, x1

c i π c −i π A = − i D e 4 ,B = + i D e 4 , 2 2 i −γ −i π γ −i π C = e e 4 F,D = e e 4 F, (D.2) 2

e−γ γ and thus A = i 4 − e F . Since the poles of σ(ω) require the boundary condition Ax|r→∞ ∼ 0, i.e. ψ(x = 0) = 0, thus i R x1 p dx0 −i R x1 p dx0 ψ(x)|x=0 = A e 0 + B e 0 = 0, (D.3) or write in the coeﬃcient F and function γ,

−γ −i π e i π i R x p dx0 −i π −i R x p dx0 γ i π i R x p dx0 −i π −i R x p dx0 i e 4 F e 4 e 0 + e 4 e 0 − e e 4 e 0 − e 4 e 0 = 0. 4 (D.4) Now if we analytically continue the real value of ω to complex plane, say ω = ω1 + iω2, then “momentum” is also complex, q p 2 2 2 p(x) = 2(ω − V (x)) = 2 ω1 − ω2 − V (x) + 2iω1ω2 . (D.5)

The WKB wave function can also be analytically continued, and the only diﬀerence Appendix D. Searching for Bound States on Complex ω Plane: WKB Method 163

from (D.1) is the part in region II,

−i π i π e 4 R x 0 e 4 R x 0 −i x p dx i x p dx ψII (x) = C e 1 + D e 1 , x1 < x < x2, (D.6) pp(x) pp(x)

from which one can determine γ = −i R x2 p(x0)dx0. One needs to be aware that the turning x1 points x1 and x2 are shifted with the existence of non-zero ω2. Let us ﬁrst make some simpliﬁcation for the integrals,

Z x1 0 In region I, x < x1, p(x) = p1(x) + ip2(x), i (p1 + ip2)dx ≡ iα1 − β1, 0 Z x2 0 In region II, x1 < x < x2, p(x) = p3(x) + ip4(x), −γ = i (p3 + ip4)dx ≡ iα2 − β2, x1 (D.7)

where all pi, αi, βi are real, and (α1, β1), (α2, β2) corresponds to the oscillation and decay in region I and II respectively. The boundary condition (D.4) turns to be

( −i π 1 −β −β β +β π 1 β −β −β +β π ie 4 F e 1 2 + e 1 2 cos(α + α + ) + e 1 2 − e 1 2 cos(α − α + ) 4 1 2 4 4 1 2 4 ) 1 π 1 π +i e−β1−β2 − eβ1+β2 sin(α + α + ) − eβ1−β2 + e−β1+β2 sin(α − α + ) = 0, 4 1 2 4 4 1 2 4 (D.8) thus both real and imaginary parts in {· · · } should vanish. Since in our numerical calibration, the poles are close to real axis of ω, i.e. |ω2| ω1, we can restrict our discussion in

|p1| |p2| −→ |α1| |β1|, and |p3| |p4| −→ |α2| |β2|. (D.9)

In the limit of real frequency, ω2 = 0, and then α2 = β1 = 0, the boundary condition (D.8) is now 1 π π F · e−β2 cos(α + ) − 2i eβ2 sin(α + ) = 0. (D.10) 2 1 4 1 4

Since β2 is large enough, the first term vanishes exponentially, and the second term requires π α + = nπ, n = 1, 2, ··· , (D.11) 1 4 which is exactly the bound state condition of real frequency as we discussed. In this limit, −γ with also β2 large, one can check that C/F ∼ e → 0, there is no backward wave in region II, and the potential is effectively infinite wide.