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Baryons, branes, and (striped) black holes : applications of the gauge / gravity duality to quantum chromodynamics… Stang, Jared Brendan 2014

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Baryons, branes,and (striped) black holesApplications of the gauge / gravity duality to quantumchromodynamics and condensed matter physicsbyJared Brendan StangB.Sc. (Honours), The University of Toronto, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2014c© Jared Brendan Stang 2014AbstractThe gauge / gravity duality, or holographic correspondence, is a theoreticaltool that allows the description of strongly coupled field theory through adual classical gravity theory. In this thesis, we advance the use of numericalmethods in applications of the holographic correspondence to the study ofstrongly coupled field theories in three situations.Firstly, we study the relationship between chemical potential and chargedensity across myriad examples of Lorentz invariant 3 + 1 dimensional holo-graphic field theories with the minimal structure of a conserved charge.Solving for the classical gravitational configurations dual to the field theo-ries and extracting the charge density and chemical potential, we enumeratethe relationships that can exist in a wide range of holographic theories.Secondly, we study the spontaneous formation of inhomogeneous (striped)order, a phenomenon that has been observed in the cuprates, in a 2 + 1dimensional strongly coupled field theory. By numerically solving the equa-tions of motion using finite difference techniques, we construct the full non-linear striped black brane solutions that provide the gravity dual to this fieldtheory. We evaluate the thermodynamics and show that the system under-goes a second order phase transition to the striped phase as the temperatureis lowered.Finally, we apply the holographic correspondence to study particular as-pects of quantum chromodynamics (QCD). First, we develop a phenomeno-logical holographic model to describe the colour superconductivity phase ofQCD, which is believed to exist at large quark density. We construct thephase diagram for our model, which includes confined, deconfined, and su-perconducting phases. In a separate project, we revisit the construction ofthe baryon in the Sakai-Sugimoto model of holographic QCD. In this model,gauge field configurations on the probe D8 flavour branes with non-trivialtopological charge (instantons) correspond to baryons in the dual field the-ory. In order to extend previous studies, we relax an assumption of sphericalsymmetry and, utilizing pseudospectral methods, numerically construct thedeformed instanton in the bulk. Compared to previous studies, we findsignificantly more realistic values for the mass and size of the baryon.iiPrefaceA version of chapter 2 has been published: Fernando Nogueira and JaredB. Stang, Density versus chemical potential in holographic field theories,Physical Review D86, 026001 (2012) [1]. This work was an equal collabora-tion between the thesis author and a colleague. The thesis author performedand checked all analytical and numerical calculations and produced all fig-ures for the manuscript. Writing and editing the work was a collaborativeeffort.Versions of chapters 3 and 4, based on the same project, have beenpublished: i. Moshe Rozali, Darren Smyth, Evgeny Sorkin, and Jared B.Stang, Holographic Stripes, Physical Review Letters 110, 201603 (2013) [2],and; ii. Moshe Rozali, Darren Smyth, Evgeny Sorkin, and Jared B. Stang,Striped Order in AdS/CFT correspondence, Physical Review D87, 126007(2013) [3]. The thesis author was primarily responsible for numerical calcu-lations and provided the majority of the results used in the manuscripts. Formanuscript i, the thesis author was responsible for editing and presentation,and contributed figures 3.3 through 3.5. For manuscript ii, the thesis authorwas primarily responsible for the writing of the paper, with the exceptionof section 4.3, which was first drafted by Evgeny Sorkin.A version of chapter 5 has been published: Pallab Basu, FernandoNogueira, Moshe Rozali, Jared B. Stang, and Mark Van Raamsdonk, To-wards A Holographic Model of Color Superconductivity, New Journal ofPhysics 13, 055001 (2011) [4]. In addition to performing analytical and nu-merical computations throughout the project and editing the manuscript,the thesis author contributed results that appeared in section 5.5, includingfigures 5.9 through 5.12.A version of chapter 6 has been published: Moshe Rozali, Jared B. Stang,and Mark Van Raamsdonk, Holographic baryons from oblate instantons,Journal of High Energy Physics, 1402, 044 (2014) [5]. The thesis authoracted as principal investigator, being responsible for all analytical and nu-merical computations and for writing the majority of the manuscript. Theexceptions were sections 6.1 and 6.5, for which the thesis author’s draftswere significantly modified by Mark Van Raamsdonk.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Introduction to the gauge / gravity duality . . . . . . . . . . 31.2.1 The gauge / gravity duality . . . . . . . . . . . . . . 41.2.2 More justification for the correspondence . . . . . . . 71.2.3 The holographic dictionary . . . . . . . . . . . . . . . 131.2.4 Holography and numerics . . . . . . . . . . . . . . . . 211.3 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.1 Density versus chemical potential in holographic probetheories . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.2 Holographic stripes . . . . . . . . . . . . . . . . . . . 251.3.3 Towards a holographic model of colour superconduc-tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.3.4 Holographic baryons from oblate instantons . . . . . 292 Density versus chemical potential in holographic probe the-ories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 CFT thermodynamics . . . . . . . . . . . . . . . . . . . . . . 37ivTable of Contents2.3 General holographic field theories at finite density . . . . . . 392.3.1 Finite density . . . . . . . . . . . . . . . . . . . . . . 392.3.2 Gauge field actions . . . . . . . . . . . . . . . . . . . 402.4 Holographic probes . . . . . . . . . . . . . . . . . . . . . . . 422.4.1 Probe branes and the Born-Infeld action . . . . . . . 432.4.2 Bottom-up models and the Einstein-Maxwell action . 462.5 ρ− µ in backreacted systems . . . . . . . . . . . . . . . . . . 522.5.1 Charged black holes . . . . . . . . . . . . . . . . . . . 532.5.2 Hairy black holes . . . . . . . . . . . . . . . . . . . . 542.5.3 Backreacted soliton . . . . . . . . . . . . . . . . . . . 552.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Holographic stripes . . . . . . . . . . . . . . . . . . . . . . . . . 583.1 Introduction and summary . . . . . . . . . . . . . . . . . . . 583.2 The holographic setup . . . . . . . . . . . . . . . . . . . . . . 603.3 The solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 654 Striped order in the AdS/CFT correspondence . . . . . . . 684.1 Introduction and summary . . . . . . . . . . . . . . . . . . . 684.2 Numerical set-up: Einstein-Maxwell-axion model . . . . . . . 714.2.1 The model and ansatz . . . . . . . . . . . . . . . . . . 724.2.2 The constraints . . . . . . . . . . . . . . . . . . . . . 734.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . 744.2.4 Parameters and algorithm . . . . . . . . . . . . . . . 794.3 The solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3.1 Metric and fields . . . . . . . . . . . . . . . . . . . . . 804.3.2 The geometry . . . . . . . . . . . . . . . . . . . . . . 814.4 Thermodynamics at finite length . . . . . . . . . . . . . . . . 864.4.1 The first law . . . . . . . . . . . . . . . . . . . . . . . 864.4.2 Phase transitions . . . . . . . . . . . . . . . . . . . . 894.5 Thermodynamics for the infinite system . . . . . . . . . . . . 955 Towards a holographic model of colour superconductivity 975.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3 Review: ψ = 0 solutions . . . . . . . . . . . . . . . . . . . . . 1095.3.1 AdS soliton solution . . . . . . . . . . . . . . . . . . . 1095.3.2 Reissner-Nordstrom black hole solution . . . . . . . . 1095.4 Neutral scalar field: Colour superconductivity . . . . . . . . 111vTable of Contents5.4.1 Numerical evaluation of solutions . . . . . . . . . . . 1135.4.2 Critical temperature . . . . . . . . . . . . . . . . . . . 1145.4.3 Properties of the superconducting phase . . . . . . . 1155.5 Charged scalar field: Flavour superconductivity . . . . . . . 1165.5.1 Low-temperature horizon free solutions with scalar . 1175.5.2 Hairy black hole solutions . . . . . . . . . . . . . . . 1205.5.3 Phase diagrams . . . . . . . . . . . . . . . . . . . . . 1225.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226 Holographic baryons from oblate instantons . . . . . . . . . 1276.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2 Baryons as solitons in the Sakai-Sugimoto model . . . . . . . 1306.3 Numerical setup and boundary conditions . . . . . . . . . . . 1336.3.1 Gauge fixing . . . . . . . . . . . . . . . . . . . . . . . 1336.3.2 Ansatz and boundary conditions . . . . . . . . . . . . 1346.3.3 Numerical procedure . . . . . . . . . . . . . . . . . . 1366.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.4.1 The mass-energy . . . . . . . . . . . . . . . . . . . . . 1386.4.2 The baryon charge . . . . . . . . . . . . . . . . . . . . 1396.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . 1477.2.1 Inhomogeneous holography and condensed matter . . 1477.2.2 Baryons in holographic QCD . . . . . . . . . . . . . . 1497.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 152Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154AppendicesA Review of numerical techniques used . . . . . . . . . . . . . 167A.1 Generic numerics for elliptic partial differential equations . . 168A.2 The numerical approach for holographic stripes . . . . . . . . 169A.2.1 Finite differencing on a rectangular grid . . . . . . . . 169A.2.2 Relaxation iteration . . . . . . . . . . . . . . . . . . . 170A.3 The numerical approach for holographic baryons . . . . . . . 172A.3.1 Pseudospectral differentiation and Chebyshev grids . 172viTable of ContentsA.3.2 Newton’s method for matrix equations . . . . . . . . 175B Striped order supplementary material . . . . . . . . . . . . . 178B.1 Asymptotic charges . . . . . . . . . . . . . . . . . . . . . . . 178B.1.1 Deriving the charges . . . . . . . . . . . . . . . . . . . 178B.1.2 Explicit expressions for the charges . . . . . . . . . . 180B.1.3 Consistency of the first laws . . . . . . . . . . . . . . 181B.2 Further details about the numerics . . . . . . . . . . . . . . . 183B.2.1 The linearized analysis . . . . . . . . . . . . . . . . . 183B.2.2 The equations of motion . . . . . . . . . . . . . . . . 184B.2.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . 187B.2.4 Generating the action density plot . . . . . . . . . . . 191B.2.5 Convergence and independence of numerical parame-ters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191C Colour superconductivity supplementary material . . . . . 198C.1 Large charge limit . . . . . . . . . . . . . . . . . . . . . . . . 198C.2 Critical µ for solutions with infinitesimal charged scalar . . . 202viiList of Tables2.1 The power α in the relationship ρ ∝ µα at large ρ for 3 + 1dimensional field theories dual to the given brane backgroundwith the indicated probe brane. . . . . . . . . . . . . . . . . . 342.2 The power α in the relationship ρ ∝ µα at large ρ for 3 +1 dimensional field theories dual to the given gravitationalbackground with the stated fields. . . . . . . . . . . . . . . . 36B.1 The maximum critical temperatures and corresponding criti-cal wavenumbers for varying c1. . . . . . . . . . . . . . . . . . 184B.2 Behaviour of physical quantities with the cutoff for c1 = 8and Lµ/4 = 0.75 and for fixed grid resolution dρ, dx ∼ 0.02. . 194B.3 Comparison of the constraint violation, measured by the schematicconstraint equation∑i hi, to the scale set by the individualterms,∑i |hi|, for grid size dρ, dx ∼ 0.01. . . . . . . . . . . . 196viiiList of Figures1.1 A visualization of the geometries involved in the holographiccorrespondence for the example of classical global anti-de Sit-ter space in 2 + 1 dimensions (AdS3, pictured at left, is con-formal to a solid cylinder) dual to strongly coupled conformalfield theory on S1 ×R (right, on the boundary of a cylinder). 71.2 Motivating the gauge / gravity duality via string theory. . . . 111.3 A caricature of AdS space, as described by equation (1.9). . . 151.4 The UV/IR relationship between the bulk and boundary. . . 161.5 Including an electric field in the bulk is dual to a field theoryat finite charge density. . . . . . . . . . . . . . . . . . . . . . 201.6 In chapter 2, we study holographic field theories with theminimal structure of a conserved charge, which correspondsto having a gravity bulk with an electric field. . . . . . . . . . 241.7 A schematic phase diagram for the cuprates. . . . . . . . . . 261.8 A schematic of the conjectured QCD phase diagram. . . . . . 281.9 A visualization of the duality between the instanton in thebulk (at the left, viewed side-on) and the baryon in the bound-ary field theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 302.1 Charge density versus chemical potential for the probe gaugeand scalar fields in the d = 4 black hole background, on alog-log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.2 Charge density versus chemical potential for the probe gaugeand scalar fields in the soliton background and the d = 5black hole background. . . . . . . . . . . . . . . . . . . . . . . 523.1 Metric functions for θ ' 0.11 and c1 = 4.5. . . . . . . . . . . . 623.2 Left panel: The variation along x of the size of the horizonin the y direction includes alternating ‘necks’ and ‘bulges’.Right panel: Ricci scalar relative to that of RN black hole,R/RRN − 1 for θ ' 0.003 over half the period. . . . . . . . . 64ixList of Figures3.3 Difference in the thermodynamic potentials between the in-homogeneous phase and the RN solution for c1 = 8, plottedagainst the temperature. . . . . . . . . . . . . . . . . . . . . . 653.4 The entropy of the inhomogeneous solution for c1 = 8 (pointswith dotted line) and of the RN solution (solid line). . . . . . 663.5 A contour plot of the free energy density, relative to the ho-mogenous solution. . . . . . . . . . . . . . . . . . . . . . . . . 674.1 A summary of the boundary conditions on our domain. . . . 754.2 Metric functions for T/Tc ' 0.11. . . . . . . . . . . . . . . . . 814.3 At relative to the corresponding RN solutions, Ay and ψ forT/Tc ' 0.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.4 Magnetic field lines for solution with T/Tc ' 0.07. . . . . . . 824.5 Ricci scalar relative to that of RN black hole, R/RRN − 1,RRN = −24, for T/Tc ' 0.054 over half the period. . . . . . . 834.6 The embedding diagram of constant x spatial slices, as a func-tion of x at given y for T/Tc ' 0.035. . . . . . . . . . . . . . . 844.7 Radial dependence of the normalized proper length along xfor T/Tc ' 0.054. . . . . . . . . . . . . . . . . . . . . . . . . . 854.8 Temperature dependence of the proper length of the horizonalong the stripe. . . . . . . . . . . . . . . . . . . . . . . . . . 864.9 The extent of the horizon in the transverse direction, ry, as afunction of x for T/Tc ' 0.054 in x ∈ [−L/2, L/2]. . . . . . . 874.10 The dependence of the size of the neck and the bulge ontemperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.11 The ratio of the transverse extents of the neck and the bulgeshrinks as rnecky /rbulgey ∼ (T/Tc)1/2 at small temperatures,indicating a pinch-off of the horizon in the limit T → 0. . . . 884.12 The grand free energy relative to the RN solution for severalsolutions of different fixed lengths at c1 = 8. . . . . . . . . . . 904.13 The grand free energy relative to the RN solution for c1 = 4.5and fixed Lµ/4 = 2.08. . . . . . . . . . . . . . . . . . . . . . . 914.14 The observables in the grand canonical ensemble for c1 = 8and Lµ/4 = 1.21 (points with dotted line) plotted with thecorresponding quantities for the RN black hole (solid line). . 924.15 The difference in canonical free energy, at c1 = 8 and fixedlength LN/4 = 1.25, between the striped solution and theRN black hole. . . . . . . . . . . . . . . . . . . . . . . . . . . 934.16 The entropy of the inhomogeneous solution for c1 = 8 (pointswith dotted line) and of the RN solution (solid line). . . . . . 94xList of Figures4.17 Action density for c1 = 8 system relative to the RN solution. 965.1 Phase diagram of our model gauge theory with m2 = −6,R = 2/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.2 Phase diagram of our model gauge theory with m2 = −6,R = 2/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.3 Phase diagram of our model gauge theory with m2 = −6,R = 2/5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.4 Phase diagram without scalar field, in units where R = 2/5. . 1115.5 Critical T/µ vs m2 of neutral scalar (filled circles). . . . . . . 1155.6 Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 2. . . . . . . . . . . . . . . . . . . . 1195.7 Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 1.3. . . . . . . . . . . . . . . . . . . 1195.8 Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 1.2. . . . . . . . . . . . . . . . . . . 1205.9 Phase diagram for m2 = −6 and q = 2. . . . . . . . . . . . . . 1235.10 Phase diagram for m2 = −6 and q = 1.3. . . . . . . . . . . . . 1235.11 Small temperature region of phase diagram for m2 = −6 andq = 1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.12 Phase diagram for large q, m2 = −6. . . . . . . . . . . . . . 1246.1 The convergence of the value ∆u = |u(NR)−u(NR−2)|/NRNθ,where u(NR) denotes the solution for the five fields {ψ1, ψ2, ar, az, s}on the grid with NR points in the R direction and Nθ = NR/2points in the θ direction. . . . . . . . . . . . . . . . . . . . . . 1376.2 The energy density ρE(r, z) in the (r, z) plane. . . . . . . . . 1386.3 The logarithm of the energy density ρE(r, z) in the (r, z)plane, on the same domain as the corresponding plots in Fig-ure 6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.4 The total mass of the soliton as a function of γ, normalized bythe mass M0 = 8pi2κ of a D4 brane wrapping the sphere direc-tions (equivalently the mass of a point-like SO(4) instantonat γ = 0 in the effective theory). . . . . . . . . . . . . . . . . 1406.5 The instanton number density 18pi2 trF ∧ F in the (r, z) plane. 1416.6 The logarithm of the instanton number density 18pi2 trF ∧ Fin the (r, z) plane, on the same domain as the correspondingplots in Figure 6.5. . . . . . . . . . . . . . . . . . . . . . . . . 1416.7 Left: The charge density ρB(r) for γ = 4, 12, 20, 28, from topto bottom. Right: The same data on a log-log axis. . . . . . . 142xiList of Figures6.8 The charge density ρB(r) for varying γ. . . . . . . . . . . . . 1436.9 The baryon charge radius 〈r2〉 =∫r2(4pir2ρB(r))dr as afunction of γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 144B.1 The critical temperatures at which the Reissner Nordstromblack brane becomes unstable, for varying axion coupling c1. 185B.2 The data underlying Figure 4.17. . . . . . . . . . . . . . . . . 191B.3 The behaviour of the L2 norm of the residual during the re-laxation iterations for c1 = 8, T0 = 0.04 and Lµ/4 = 0.75. . . 192B.4 The value of the scalar field condensate for varying grid sizesfor c1 = 8 and Lµ/4 = 0.75. . . . . . . . . . . . . . . . . . . . 193B.5 The weighted constraints for c1 = 8 and Lµ/4 = 1.21. . . . . 195C.1 Critical values of µq vs m2 for scalar condensation in large qlimit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201C.2 Critical T/µ vs charge q for condensation of m2 = −6 scalarfield in Reissner-Nordstrom background. . . . . . . . . . . . . 203xiiAcknowledgementsThere are many people to thank for their contributions to my personal andprofessional development over the years of my graduate studies.Firstly, for significant contributions to my development as a researcherand a physicist, guidance in navigating both the subject of string theory andthe process of completing a doctoral dissertation, and support to pursue myinterests both in my physics research and elsewhere, I thank my supervisor,Mark Van Raamsdonk.In terms of my professional activities, I also extend my gratitude to:Moshe Rozali, for many hours of discussion and mentorship, even thoughhe had no formal obligation to do so; my supervisory committee (JoannaKarczmarek, Mark Halpern, and Jo¨rg Rottler), for providing useful feedbackand benchmarks for me as I progressed through the stages of my degree; mycollaborators (Pallab Basu, Fernando Nogueira, Darren Smyth, and EvgenySorkin), for shared discoveries and frustrations; Ido Roll, for taking me on topursue an interesting side project; members of PHASER-G (especially GeorgRieger and Joss Ives), for providing productive distractions from my physicsresearch; the Natural Science and Engineering Research Council of Canadaand the Killam Trusts, for financial support; Matt Choptuik, for enablingmy access to computational resources; and my fellow graduate students, forproviding a stimulating and enjoyable environment throughout my studies.In terms of my personal life, I thank my brothers (Dustin and ConnerStang) for their camaraderie. I thank Brenda Shkuratoff dearly for hercontinual love, support, and friendship. Finally, and most importantly, Ithank my parents (Bernadine and Bruce Stang) for making it all possible:although I walked through the doors, you held them open for me.xiiiTo my parents.xivChapter 1Introduction1.1 MotivationQuantum field theory is one of the most ubiquitous and useful theoreticalframeworks ever developed. It provides the backbone underlying two of thelargest and most prominent areas of modern physics: particle physics andcondensed matter physics. Within these contexts, the framework has beenextremely successful as a theoretical description of experimentally observedphenomena. In particular, quantum electrodynamics (quantum field theoryapplied to the theory of photons and electrons) has provided the most precisematch between theory and experiment ever.1A key feature of quantum electrodynamics, which allows such precisecalculations and which is shared by many of the situations successfully de-scribed by quantum field theory, is that the theory is weakly coupled. Phys-ically, this means that if the system is perturbed slightly (say we grab andshake an electron), the configuration will not change very much (other elec-trons will only be slightly bothered). Mathematically, weakly coupled the-ories admit a perturbative description, in which the effect of interactionscan be computed by expanding around the simpler non-interacting system,resulting in a series expression for the result. Depending on the precisionneeded for a calculation, this series may be truncated at some point andhigher order interactions may be neglected.However successful the perturbative approach is, it is fundamentally lim-ited to the description of weakly coupled theories. There are many systemsin nature which are properly described as strongly coupled quantum field the-ories. A prime example of this is quantum chromodynamics (QCD, quantumfield theory for quarks and gluons) at low energies. For these theories, if thesystem is perturbed slightly (if we grab and shake a quark), the entire con-figuration may change significantly. From a calculational perspective, theperturbative approach is no longer applicable, as the first few terms in the1This is for the magnetic moment of the electron, the most recent measurement ofwhich was reported in [6].11.1. Motivationseries will not be a good approximation of the exact result, and we needdifferent methods with which to study the theory.One possible route to study strongly coupled field theory is throughlattice techniques. Schematically, one Euclideanizes the theory before dis-cretizing spacetime by putting it on a lattice. The problem of solving thetheory becomes the problem of minimizing the (Euclidean) action on thislattice. Established techniques, such as the Monte Carlo method, can thenbe applied to find the minimized action. Lattice techniques, however, haveseveral shortcomings. Firstly, the computational power needed to studythese theories becomes very large very quickly as one tries to increase theaccuracy of calculations. Secondly, and perhaps more importantly, thesetechniques are limited to field theory at zero charge density. Finite densitysystems are interesting in a variety of situations, including, for example,neutron stars and superconductors. Upon analytically continuing the ac-tion, a chemical potential term will become imaginary, resulting in highlyoscillatory behaviour in the path integral, which is not amenable to studywith Monte Carlo techniques. Thus, lattice techniques are limited in theirease of implementation and applicability.A more contemporary and flexible approach to the study of stronglycoupled field theory, the application of which is the topic of this thesis, isthe gauge / gravity duality.2 First proposed in 1997 [7–9], this remarkablecorrespondence states that a strongly coupled field theory describes identicalphysics as a classical theory of gravity (in one higher dimension). Under theduality, the problem of calculating observables in the field theory is mappedto the relatively simpler problem of solving classical equations of motion.The correspondence provides calculational access to many novel regimes ofstrongly coupled field theory, including field theory at finite density.In the years since the discovery of the holographic correspondence, it hasbeen applied to the study of strongly coupled field theories in many interest-ing ways. One example and success story is a study of viscosity in stronglycoupled field theories [10], which provides some theoretical explanation forthe small viscosities of the quark-gluon plasma seen at the Relativistic HeavyIon Collider. Further examples of applications of the correspondence include2The holographic correspondence is a general term to describe the correspondence be-tween a gravitational theory in d + 1 dimensions and a theory without gravity in d di-mensions. Specifying to the gauge / gravity duality identifies the d-dimensional theory asa gauge field theory. Finally, the AdS/CFT correspondence (anti-de Sitter / conformalfield theory) refers to a specific class of theories that exhibit this relationship, in which agravity theory with anti-de Sitter asymptotics is dual to a conformal field theory. In thisthesis, as is common, we use these three labels synonymously.21.2. Introduction to the gauge / gravity dualitymodels of superconductors [11], superfluids [12], and QCD (for example,[13]). While exact holographic descriptions of QCD or high-Tc supercon-ductivity, as seen in experiments, are not yet possible, models that elucidateuniversal features of similar classes of systems are available for study. Inthis way, the correspondence currently offers tremendous potential for gen-eral results from which qualitative and (in some cases) quantitative lessonsmay be drawn.In this thesis, the gauge / gravity duality is applied to the study ofvarious strongly coupled field theoretic phenomena, with the goal of con-tributing to and advancing the literature on holographic techniques andresults. To this end, we describe projects focused in three domains: generalholographic field theories, applications to condensed matter systems, andapplications to QCD. To facilitate these studies, we make extensive use ofnumerical techniques, from the application of standard ‘blackbox’ solvers forordinary differential equations to the use of finite difference and pseudospec-tral methods for the partial differential classical field equations that arisein inhomogeneous situations. The studies applying the latter techniquescontribute to the forefront of the emerging research direction combining nu-merical techniques and holographic methods.A brief outline of this introductory chapter is as follows. In section 1.2,we provide a more detailed background for the gauge / gravity duality,including a more precise statement of the correspondence, a sketch of theoriginal ‘derivation’, and examples of the explicit mapping between fieldtheory and gravitational observables. In section 1.3, we provide a summaryof each of the four projects that comprise the content of this thesis.1.2 Introduction to the gauge / gravity dualityHere we provide a brief review of certain salient features of the gauge / grav-ity duality, the main theoretical tool used in this thesis.3 In section 1.2.1,we describe the gauge / gravity duality in generality, defining it more pre-cisely than above and providing a conceptual argument as to why it maybe true. In section 1.2.2, we review the specific construction that moti-vated the original statement of the conjecture. This is the duality betweenstrongly coupled N = 4 SU(N) super-Yang-Mills theory on 3 + 1 dimen-sional Minkowski space and type IIB string theory on AdS5 × S5. Thissubsection contains technical details which depend on some prior knowledge3There are many existing reviews of the holographic correspondence, including [14–17].Reports the thesis author found particularly useful are [18–21].31.2. Introduction to the gauge / gravity dualityof string theory. In section 1.2.3, we provide some of the standard entries inthe holographic ‘dictionary’, which relates observables and constructions oneither side of the correspondence. Finally, section 1.2.4 briefly motivates theuse of and need for numerical techniques in the study of strongly coupledfield theory using the holographic correspondence.1.2.1 The gauge / gravity dualityIn this section, we discuss the gauge / gravity duality in generality, limitingourselves to features that are present across examples, and providing anargument as to why such a correspondence may exist. For a particularconstruction manifesting the correspondence, see section 1.2.2.The gauge / gravity duality, in general, is the conjectured equivalencebetween a quantum field theory in d spacetime dimensions and a theoryof quantum gravity in d + 1 spacetime dimensions.4 This equivalence is acomplete equivalence of the physical spectra at any value of the parametersin the theory, “including operator observables, states, correlation functionsand full dynamics” [16]. There is an established dictionary (see section 1.2.3)that describes the precise mapping between objects in the field theory andobjects on the quantum gravity side. If computational tools were availablefor both theories in every region of parameter space, one could in principleprecisely match up every result in each theory: every physical question andcorresponding result in one theory has a dual version in the theory on theother side of the correspondence. Put very shortly, the two theories describethe same physics.How could such a correspondence exist? At first blush, it sounds absurdto claim that a quantum field theory and a theory of quantum gravity couldbe alternate descriptions of the same physics. On one side of the duality is astandard field theory on a fixed spacetime background, which in particularcannot contain a massless spin-2 (graviton) field [22]. On the other sideis a theory of quantum gravity,5 which necessarily contains a spin-2 field,in one higher dimension. Further, the descriptions of these theories aremanifestly different. In particular, quantum gravity must admit phenomenaand characteristics that do not seem to have any obvious description in a4Although the focus of this thesis is on strongly coupled quantum field theory, we beginby discussing the holographic correspondence as applied to a general field theory, beforespecializing to the limit of strong coupling (and the corresponding region of validity forclassical gravity) below.5In the earliest examples of the duality the theory of quantum gravity was a stringtheory.41.2. Introduction to the gauge / gravity dualityfield theory, including black holes, wormholes, and diffeomorphism invari-ance. Thus, we have two theories with two very different descriptions thatpurportedly describe the same physics.A key realization lies in the fact that quantum gravity is holographicin that the number of degrees of freedom in a region is proportional to thesurface area surrounding the region, and not the volume as is the case fora local quantum field theory.6 To see why this must be the case, considerfor a moment the following gedanken experiment in a theory of gravity.Assume you have some volume V of space (bounded by the area A) whichcontains more information, or entropy, than a corresponding black hole ofthe same size: S > SBH . For a sufficiently large volume, general relativitywill provide a good description irrespective of the underlying theory, sothat the entropy of the black hole will be given by the Bekenstein-Hawkingformula SBH = A/4GN . Since our configuration is not a black hole, its massis less than the critical mass for the volume V . Now, add matter to theregion V such that its mass exceeds the critical mass; the configuration willgravitationally collapse, forming a black hole. Before adding the matter, thetotal entropy of the system was Sbefore = S+Sout, where Sout is the entropyof the matter that we threw into the region. After the region collapses, thetotal entropy of the system is described by the Bekenstein-Hawking formula:Safter = SBH . By our initial assumption, we have Sbefore > Safter, showingthat this process violates the second law of thermodynamics. Therefore, theassumption that we could have a region with larger entropy than a blackhole of the same size must be false. We arrive at the conclusion that, in atheory of quantum gravity, the information in the region V is bounded bySBH = A/4GN .Given the above discussion, we see that the number of degrees of free-dom in a theory of quantum gravity in d+ 1 dimensions scales as a volumein d dimensions. This is the same behaviour one expects from a d dimen-sional quantum field theory.7 Thus, it becomes at least plausible that theholographic correspondence could connect two such theories.A particular salient feature of the duality is the relationship between theparameters on either side of the correspondence. The dimensionless couplingλ of the field theory is directly related to the typical length scale of the cur-vature in the quantum gravity theory: at large coupling λ the geometry of6See [23, 24] for discussions of the holographic principle. The entropy bound describedhere is called the Bekenstein bound [25]. See, for example, [21] for a version of theargument given here.7In section 1.2.2, we will match the number of degrees of freedom more precisely in aparticular example of the duality.51.2. Introduction to the gauge / gravity dualitythe dual theory is weakly curved and classical gravity is a good approxima-tion, while for small λ the curvature of the gravity side is large in units ofthe string length and the full quantum gravity theory is needed. Thus, forlarge λ, the gravity side is accessible with current tools for classical generalrelativity, while for small λ, the field theory side is accessible via perturba-tive quantum field theory. This mutual exclusivity of reliable calculationaldomains makes the holographic correspondence difficult to prove, as explicitmatching of the sides is only possible in certain symmetrical situations.However, it makes it extremely useful as a tool for the study of stronglycoupled field theories. In order to describe the physics of the strongly cou-pled quantum field theory, one simply has to find classical saddle-point ofthe gravity action. In cases with gravitational backreaction, this reduces tosolving the Einstein equations of general relativity. While this may be tech-nically difficult, established techniques exist for this integration, in contrastto the dual problem of studying a strongly coupled quantum theory.In this way, using the holographic correspondence, many questions aboutstrongly coupled quantum field theories may be phrased in the context ofa classical gravity theory. By considering different theories on the gravita-tional side (for example, different geometries or different matter content),one may study a variety of strongly coupled field theories which display arich array of behaviours. Relatively straightforward classical gravity com-putations thus provide access to certain previously intractable field theorycalculations. This thesis utilizes the gauge / gravity duality in this way tostudy strongly coupled field theory in various contexts.It is useful to have a mental image of the correspondence. Let us spe-cialize to large λ and consider the particular example of the classical globalanti-de Sitter (AdS) space in 2 + 1 dimensions (AdS3, a solution to Einsteingravity with a negative cosmological constant), dual to strongly coupledconformal field theory on S1 × R. In Figure 1.1, we provide a visualiza-tion of the geometries involved in this correspondence. The gravity side,AdS3, is conformal to the bulk of a cylinder. On the field theory side, atthe right of the figure, we have S1 ×R, which maps out the boundary of acylinder, and which is conformal to the boundary of AdS3. Then, the de-grees of freedom of the field theory live on a space which can be conformallymapped to the boundary of the gravity side; this motivates the label of theholographic correspondence. The extra direction on the gravity side in thiscase is parametrized by the radial coordinate from the axis of the cylinder.Often, these two images are amalgamated into one, with the field theoryconsidered to be defined on the boundary of the bulk spacetime.61.2. Introduction to the gauge / gravity dualityx txtgauge /gravityAdS3 CFT onS1 ×RFigure 1.1: A visualization of the geometries involved in the holographiccorrespondence for the example of classical global anti-de Sitter space in 2+1dimensions (AdS3, pictured at left, is conformal to a solid cylinder) dual tostrongly coupled conformal field theory on S1 ×R (right, on the boundaryof a cylinder). The physics of the gravity theory in the solid cylinder iscompletely encoded in the field theory on the cylindrical surface, and viceversa. In these images, the time direction is vertical; the shaded disc on AdS3at left represents a spatial slice of the gravity theory, at constant time, whilethe drawn-in ring on S1 × R at right is the corresponding spatial slice inthe field theory. The geometry of the field theory, S1 × R, is conformallyequivalent to the boundary of the gravity spacetime. Due to this, thesepictures are often amalgamated into one image, with the field theory mappedonto the boundary of the bulk spacetime. (In Figure 1.4 and beyond weadopt this representation.)1.2.2 More justification for the correspondenceThe original and perhaps most concrete example of the gauge / gravity dual-ity is the equivalence of type IIB string theory on AdS5×S5 and strongly cou-pled N = 4 SU(N) super-Yang-Mills theory on 3+1 dimensional Minkowskispace.8 It was through this example that this remarkable correspondencewas first proposed by Maldacena [7].9 In this technical section, using results8N = 4 SU(N) super-Yang-Mills theory is a supersymmetric field theory with gaugefields, fermions, and scalars connected by the N = 4 supersymmetry generators. Thegauge fields in this theory transform in the adjoint of SU(N) and are described by theYang-Mills Lagrangian.9See also [8, 9] for important early developments.71.2. Introduction to the gauge / gravity dualityfrom string theory, we will briefly review the argument for the equivalenceof these theories, focussing on the limit in which the gravity side becomesweakly curved and classical gravity is a good approximation. Given thisspecific example, we go on to make explicit the relationship between param-eters on either side of the duality and more precisely compare the numbersof degrees of freedom on either side of the correspondence. The contentin this section is intended to provide a more detailed motivation for thecorrespondence.The basic argument for the correspondence, in this case, relies on thecommutativity of certain limits or scalings. We will begin with a well-definedstring theory construction before taking two limits: the large coupling λlimit10 and the low-energy limit. If we perform these operations in differ-ent orders, we arrive at the disparate theories. Assuming that the limitscommute then gives the correspondence. The string theory construction weconsider is a stack of N coincident D3 branes (in type IIB string theory),where N  1. The parameters in the string theory will be the number ofbranes N and the string coupling g, which controls the strength of stringinteractions. Recall that D-branes are surfaces on which strings can end andare dynamical objects themselves in string theory. Two aspects of D-branesthat will be important here are that the string endpoints generate a fieldtheory on the world-volume of the branes (whose massless states includegauge fields described by a Yang-Mills theory) and that the branes carryenergy, which causes gravitational effects around the branes. First, we willtake the low-energy limit before going to strong coupling, giving the fieldtheory part of the correspondence, before performing the operations in thereverse order to arrive at the gravity side.To find the field theory part of the correspondence, consider the physicson the world-volume of the branes. The string endpoints give a field theoryon the branes, the effective coupling (gYM) of which is related to the stringcoupling as g2YM = g. However, for a large number N of coincident branes,the gauge group of the field theory has large rank (is a ‘large-N ’ theory),so that the relevant coupling is the ‘t Hooft coupling λ = g2YMN = gN[26]. If we consider these branes at small coupling λ, the physics can bedescribed perturbatively using field theory techniques on the branes. Ifwe now take the low-energy limit, the massive open string states decouple,leaving massless open string states on the branes, which, in this case, givesprecisely the N = 4 SU(N) super-Yang-Mills theory, with coupling λ, on10λ refers to coupling on the field theory side. Below, we describe the interpretation ofλ in terms of parameters on the gravity side.81.2. Introduction to the gauge / gravity duality3 + 1 dimensional Minkowski space (on the world-volume of the branes).11Taking the coupling λ large, we get the super-Yang-Mills theory at strongcoupling.Now, by taking the limits in the opposite order, we may find the gravityside of our correspondence. We begin again at our stack of N D3 branes,where N  1. We would like to replace this configuration by a classicalsupergravity geometry, which will provide a good description of the systemwhen the typical curvatures in the geometry are small. Given that D-branescarry energy (and charge), we solve the classical supergravity equations tofind the spacetime that describes the brane configuration. The resultingspacetime is the p-brane supergravity solution, which is somewhat similarto a standard black hole solution (but with different geometry and in adifferent number of dimensions). The characteristic length of curvature inthis spacetime is proportional to λ; if we take λ large, the branes sourcea spacetime with small curvature, and we can effectively replace the stackof D3 branes with this supergravity geometry. Then, after taking the largeλ limit, we are left with type IIB string theory on this supergravity back-ground. At this point, we now take the low-energy limit. From the pointof view of an observer infinitely far from the branes, all string states suffi-ciently close to the horizon of the branes have vanishing energy. Therefore,the low-energy limit is synonymous with the ‘near-horizon’ limit, where wefocus in on the geometry very close to the branes and keep all the stringstates. This near-horizon geometry is AdS5 × S5, which may be written asds2 = r2L2 (ηµνdxµdxν) + L2r2 dr2 + L2dΩ2S5 , (1.1)where L is the characteristic length scale of the geometry and ηµν is the 3+1dimensional Minkowski metric. The field theory directions, or those parallelto the D3 branes, are described by the ηµνdxµdxν part of the metric, whiler labels the radial distance from the branes (the horizon being at r = 0).The result of taking the limits in this way is that we are left with type IIBstring theory on the near-horizon geometry (1.1).1211In addition to the theory on the branes, the physical degrees of freedom in this situa-tion include closed strings away from the branes. In the low-energy limit, massive closedstring states decouple, leaving massless closed string states (supergravity) in the bulk,and interactions between the closed strings in the bulk and the open strings on the braneare suppressed. Thus, we have also bulk free supergravity in the space away from thebranes, which is decoupled from the field theory on the branes. Below, we will see thatwe have an identical decoupled supergravity sector in additional to the gravity side of thecorrespondence, so that this sector does not play a role in the duality.12In the low-energy limit, we also keep massless string states away from the near-horizon91.2. Introduction to the gauge / gravity dualityIn summary, taking the low-energy and strong coupling λ limits in dif-ferent orders (and with N  1), we have arrived at either strongly coupledN = 4 SU(N) super-Yang-Mills theory on 3 + 1 dimensional Minkowskispace or type IIB string theory on AdS5 × S5. Assuming that the limitscommute gives us the conjectured equivalence of these two theories. Thisdiscussion is summarized in Figure 1.2.Given this explicit construction, we can identify relationships betweenthe parameters of the string theory (or gravity side) and the field theory (orgauge theory side). The parameters of the gravity side can be taken to bethe AdS radius L/ls (in units of the string length ls) and the string couplingg. On the field theory side, we have the rank of the gauge group N andthe ‘t Hooft coupling λ. In the duality described here, the relations betweenthese are given byλ = gN, L4 = 4piα′2gN, (1.2)where α′ = l2s . We see here the precise dependence of the characteristic sizeL on the coupling λ = gN .For classical supergravity to be a good approximation on the gravityside, the characteristic length scale L of the curvature must be large sothat the space is weakly curved. The relevant length scales for stringy andquantum effects are the string length ls and the Planck length lP ; L mustbe large compared to these for the classical supergravity description. UsingGN = l8P = g2l8s for this ten dimensional space, we haveLlP∼ N1/4  1 (1.3)as the condition which suppresses quantum effects. Using the string length,we find thatLls∼ λ1/4  1 (1.4)will control corrections from the tower of massive string states.13Now, using the specific duality developed above, we can again comparethe degrees of freedom in each theory, as in section 1.2.1. This time, givenmore information about the theories involved and the relations (1.2) we willregion, which gives us free supergravity away from the branes and which decouples fromthe brane physics. This is precisely the same as the supergravity sector we arrived at awayfrom the branes in the alternative ordering above, so that these sectors can be triviallyidentified.13Schematically, the states of the string have masses m2 ∼ n/α′, where n is the level ofexcitation. For large λ, these masses become large and the massive string states are notaccessible.101.2. Introduction to the gauge / gravity dualityN D3 braneslow-energyN = 4 SYMλlow-energyλ 1λ 1strongly coupledSYM(string theory on)supergravityp-brane×S5⇐⇒Figure 1.2: Motivating the AdS/CFT correspondence via string theory. Be-ginning with a stack of N D3 branes (top left, shown with strings) andtaking the large coupling λ = gN and low-energy limits in different ways,one arrives at the correspondence. Taking the low-energy limit of the D3brane system gives the N = 4 super-Yang-Mills (SYM) theory on 3 + 1 di-mensional Minkowski space with coupling λ (top right). Then, we may takethe coupling λ large to get the strongly coupled theory. Starting again withthe stack of branes, and taking λ to be large first, one can replace the braneswith a corresponding classical supergravity p-brane geometry (bottom left).Taking the low-energy limit in this geometry leaves type IIB string theoryin the near-horizon region, AdS5 × S5. (See Figure 1.3 for a description ofAdS space, depicted here in green.) Finally, we identify the two theories inthe bottom right corner to arrive at the correspondence.111.2. Introduction to the gauge / gravity dualitybe able to observe the dependence on more parameters of the theories (inaddition to seeing the scaling with the field theory volume).We begin with the field theory side, which we will consider genericallyas an SU(N) field theory with matter content in the adjoint representa-tion. We will consider the theory in a finite volume V3, to regulate infrareddivergences, and introduce a short-distance cutoff δ, to control ultravioletdivergences. The number of cells in the volume will then be V3/δ3. For thistype of field theory, the field degrees of freedom in each cell will be N ×Nmatrices, with N2 degrees of freedom each. Therefore, the total number ofdegrees of freedom will be given byd.o.f.field theory ∼N2V3δ3 . (1.5)Now, consider the gravity side. As discussed above, the information ingravity is holographic, so that the degrees of freedom of the system shouldbe proportional to the area surrounding the system. In the metric (1.1),the boundary which surrounds the system is at r = ∞; it is the area ofthis boundary which we wish to compute. Before we calculate this area,however, we should also consider that we must regulate the gravity side inthe same way as we did our field theory. To impose an infrared regulator,we will simply consider a finite volume V3 of the directions parallel to theoriginal D3 branes (this is the ηµνdxµdxν term of the metric). To impose anultraviolet cutoff in the gravity theory is not so straight-forward. It turnsout (as discussed further below, section 1.2.3) that the near-boundary regionof the spacetime corresponds to the ultraviolet of the field theory. Thus, toimpose a similar cutoff, we should compute the area of the surface just insidethe boundary.For convenience in this calculation, we will change coordinates in (1.1)as r = L2/z, to get the metricds2 = L2z2(ηµνdxµdxν + dz2)+ L2dΩ2S5 . (1.6)The boundary, previously at r = ∞, is now at z = 0. By computing thearea of the surface near the boundary, at z = δ, we will be imposing anultraviolet cutoff as desired. In the background described by equation (1.6),the area of the surface at constant z = δ, constant time, and with volumeV3 in the field theory directions is given byA = V3 ·√L6δ6 · L5, (1.7)121.2. Introduction to the gauge / gravity dualitywhere the factor L5 gives the volume of the S5 part of the geometry. Thenumber of degrees of freedom, or information, is given by the Bekensteinformula S = A/4GN . Using also equation (1.2) and the string theory resultthat GN ∼ g2l8s = g2α′4, we find the information of the gravity theory to bed.o.f.gravity theory =A4GN∼ V3L8δ3g2α′4 ∼N2V3δ3 , (1.8)in agreement with the field theory result (1.5).In addition to finding the expected scaling with the field theory volumeV3, we have added the information of how the two theories behave withrespect to an ultraviolet cutoff δ and how the information scales with therank N of the field theory gauge group. We find that the results on bothsides of the correspondence match up as needed.1.2.3 The holographic dictionaryAbove, in section 1.2.2, we motivated the equivalence of strongly coupledN = 4 SU(N) super-Yang-Mills theory on 3 + 1 dimensional Minkowskispace and type IIB string theory on AdS5×S5. The construction discussedthere represents a precise example of the more general class of holographicrelationships known as the AdS/CFT correspondence.14 In this section, wezoom out from this specific situation and examine the holographic dualitywith the minimal ingredient of asymptotically AdS gravity, which, as we willdiscuss further below, is the minimal structure for the dual of a conformalfield theory. We will build some physical intuition for the correspondencewhile enumerating some of the standard entries in the holographic dictio-nary, which relates quantities on both sides of the duality. Through this,we will develop some of the practical computational tools one can use toaddress questions in holographic field theories.First, we elaborate on the structure of AdS space and connect the isome-tries of the spacetime to the symmetries of the field theory. Next, we discussthe direct correspondence between the path integrals on either side of theduality and detail how the classical gravitational action can be used to getinformation about the field theory partition function. Finally, from a phe-nomenological perspective, we discuss how one may add structure to thedual field theory by including different matter fields in the gravity action.14N = 4 SU(N) super-Yang-Mills theory is a conformally invariant field theory.131.2. Introduction to the gauge / gravity dualitySymmetries of AdS space and the UV/IR relationIn this subsection, we discuss AdS space in more detail, connecting thesymmetries of this spacetime to those in the field theory, and building anintuition about how dynamics at different coordinate positions in the gravityside correspond to dynamics at different energy scales in the field theory.We will consider AdSd+1, with metric given byds2 = r2L2 (ηµνdxµdxν) + L2r2 dr2, (1.9)where ηµν is the d-dimensional Minkowski metric. We will refer to xµ asthe field theory directions and r as the radial direction. In Figure 1.3, weprovide a caricature of this space. r varies from 0 to∞; r = 0 is the Poincare´horizon while r =∞ is the asymptotic boundary of the space. The geometryof the boundary at r = ∞ is conformal to d-dimensional Minkowski space,which is the spacetime on which the dual field theory is defined. AdSd+1 is aspace with constant negative curvature, which implies that radial geodesicsdiverge as they approach the asymptotic boundary. If we have a box witha certain area in the field theory directions, at a fixed r1, and we move thebox to a larger radial coordinate r2, the area of the box will increase by afactor rd−12 /rd−11 . The typical length scale of this curvature is L; if L is largein units of the Planck length, the space is weakly curved. Finally, we notethat AdSd+1 is a solution of the Einstein-Hilbert action with cosmologicalconstant:S =∫dd+1x√−g(R+ d(d− 1)L2). (1.10)It is this action which will define the partition function for the gravity side.AdS space is highly symmetric; that it is dual to a conformal field theory(which is also highly symmetric) is no coincidence. The isometries of theAdS space include Poincare´ transformations in the field theory directions,dilatations (scalings of the coordinates), and special coordinate transfor-mations. Taken together, the isometry group of AdSd+1 is isomorphic toSO(d, 2). Now, the conformal group in d spacetime dimensions is preciselySO(d, 2) [27]. Thus, there is a direct relationship between isometries of thegravity side and the conformal symmetry of the field theory. The gravityside allows a geometrical realization of the conformal symmetry throughisometries of the spacetime.As an example of this relation, consider in particular scale transfor-mations in the field theory, which, through the duality, are directly ‘ge-ometrized’ in the extra gravity coordinate. In the field theory, these are141.2. Introduction to the gauge / gravity duality← r → ηµνdxµdxνFigure 1.3: A caricature of AdS space, as described by equation (1.9). Theradial direction, r, increases from left to right; at fixed r, the metric isproportional to d-dimensional Minkowski space. The space has a horizonat r = 0, past the left of the diagram, where the square will pinch off(the factor multiplying ηµνdxµdxν in the metric goes to zero). There is anasymptotic boundary as r →∞, on the right of the diagram. The geometryof the boundary of AdSd+1 is conformal to d-dimensional Minkowski space.AdSd+1 is a negatively curved space, which implies that radial geodesicsdiverge as they approach the asymptotic boundary, and which inspires theshape of this schematic.given by xµ → axµ. The energy in the field theory, conjugate to time, scalesas E → a−1E. The corresponding symmetry in the AdS space is the di-latation xµ → axµ, r → a−1r. Thus, scaling to high-energy processes (ordynamics at small distances) corresponds to moving the bulk process to-wards the boundary at r =∞. This behaviour is captured in the statementthat the radial coordinate on the gravity side behaves like an energy in thefield theory. Fields deep in the interior, at small r, represent processes in theinfrared of the field theory, while excitations near the boundary at r = ∞correspond to the ultraviolet of the field theory. On the gravity side, aninfrared cutoff would correspond to placing a cutoff before the asymptoticboundary, to regulate the long-range excitations. Thus, through the corre-spondence, the infrared of the gravity side is mapped to the ultraviolet of thefield theory, so that this is often called the UV/IR relation. See Figure 1.4for a schematic image of this relationship.1515A particular interesting implication of the UV/IR relation is that a cutoff of the gravity151.2. Introduction to the gauge / gravity dualityFigure 1.4: The UV/IR relationship between the bulk and boundary. Dy-namics at short distances, or high energy scales, in the boundary theorycorrespond to excitations near the boundary in the gravity theory (excita-tions at long distances, or in the infrared of the gravity theory). Infrareddynamics, or low-energy excitations of the field theory, correspond to dy-namics deep in the gravity bulk. (This image was inspired by a similar figurein [14].)Equivalence of path integrals and the computation of expectationvaluesIn this subsection, we make explicit the correspondence between the parti-tion functions on either side of the duality and discuss how, in the limit ofstrong coupling, we may approximate the partition function of the gravitytheory by using its classical action.In order to use the gauge / gravity duality to perform calculations, weneed a mathematical relationship between the two theories. The main prac-tical statement of the correspondence is the equivalence of the gravitationalpartition function with the generating functional of the field theory:ZAdS [certain b.c.s] = ZCFT [JA] =〈ei∫JAOA〉. (1.11)JA are sources with OA the corresponding operators in the field theory,where the label A includes all information about the operators, includingside at a minimum radius rmin (restricting r to (rmin,∞)) will introduce a minimumenergy for excitations, or a mass gap, into the field theory. This understanding gives someintuition about how we might use the correspondence to model theories with an energygap.161.2. Introduction to the gauge / gravity dualityLorentz indices. As we will detail shortly, the presence of sources for oper-ators in the field theory is dual to the existence of fields in the bulk gravityside. ZAdS must also be supplemented with particular boundary condi-tions (at the asymptotic boundary), as indicated in equation (1.11). Theseboundary conditions typically enforce the value of the source JA.In this work, we are interested in using the gravity side to answer ques-tions about the field theory. Through the relation (1.11), knowing the be-haviour of the gravity theory allows one to perform computations in the fieldtheory, using the standard expression〈OA〉 = −iδZCFT [JA]δJA. (1.12)Our main focus is on applying the correspondence to strongly coupled fieldtheories. In this limit, classical gravity is a good description of the bulk andwe can approximate the gravity partition function asZAdS ≈ eiS0 , (1.13)where S0 is the gravity action evaluated on the classical solution. Combiningequations (1.11), (1.12), and (1.13) gives〈OA〉 =δS0δJA. (1.14)Thus, to compute field theory expectation values, one may solve the classicalequations of motion on the gravity side to find the on-shell action S0 beforeusing (1.14) to arrive at the field theory result. The holographic correspon-dence translates the problem of computing correlation functions to findingthe classical gravity action.Mapping (gravity) fields to (field theory) operatorsGiven the equivalence of the partition functions of the theories on either sideof the correspondence, equation (1.11), and the prescription for computingcorrelation functions, equation (1.14), it is important to describe how wemay add structure to the field theory, in the form of sources JA and op-erators OA, through manipulations of the gravity theory (which we havefull control over). In this subsection, we precisely state how the content ofthe two theories connects and present three examples of this mapping. Theinformation reviewed here provides some of the necessary basis for buildingup holographic field theories in a phenomenological manner.171.2. Introduction to the gauge / gravity dualityAs touched on briefly above, the content of the field theory is determinedby sources for the operators of the theory. Generically, there are many typesof operators that could appear in a field theory (for example, with differentLorentz indices). What might be the corresponding content on the gravityside? Components that we can add to the gravity side, which also displaya rich array of possibilities, are fields of various types. Included in thestatement of the holographic correspondence is that (in the strong couplinglimit) classical fields in the gravity bulk are in correspondence with operatorsof the field theory. As we will see in the examples below, the source andexpectation value for the dual operator are encoded in the asymptotic, near-boundary behaviour of the classical field. Three important components ofthis mapping are:1. The Lorentz structure of the field in the bulk carries over to the oper-ator in the field theory.2. The mass of the field on the gravity side is in correspondence with thescaling dimension of the operator in the conformal field theory.3. Gauge symmetries in the bulk map to global symmetries on the bound-ary.These details begin to illuminate how we might use the holographic corre-spondence to build up conformal field theories with certain operator content.As a first example of how the duality between fields and operators worksin practice, let us consider the simplest case of a massive real scalar field ψin AdS space. From an effective-theory or naturalness perspective, we willuse the actionSψ =∫dd+1x√−g{12(∂ψ)2 + 12m2ψ2}, (1.15)where m is the mass of the scalar field. Since ψ is a scalar field, it willbe dual to some scalar operator in the field theory, which we will call Oψ.Including this scalar field on the gravity side corresponds to considering asource ψ(0) for Oψ, so that we are studying the generating functionalZCFT [ψ(0)] =〈ei∫ψ(0)Oψ〉. (1.16)In terms of the scalar field in the bulk, it is the value of ψ at the boundarythat determines the source ψ(0). By solving the equation of motion followingfrom (1.15) in the AdS background (1.9), for large r, and comparing to the181.2. Introduction to the gauge / gravity dualitydefinition for expectation values from partition functions, equation (1.12),we arrive at the expansionψ =ψ(0)rd−∆ + · · ·+〈Oψ〉r∆ + . . . , (1.17)where∆ = 12(d+√d2 + 4m2L2)(1.18)and . . . denotes terms higher order in 1/r (additional terms may appearat orders lower than the 〈Oψ〉 term, as indicated). As anticipated, thesource ψ(0) and expectation value of the operator Oψ are encoded in theasymptotics of the scalar field.16 ∆, which depends on the mass m, is thescaling dimension of the operator in the field theory, meaning that we canwrite the field theory correlation function of Oψ schematically as17〈Oψ(x)Oψ(y)〉 ∼1|x− y|2∆ . (1.19)In practice, the value of the source ψ(0) is part of the boundary conditionsimposed on the gravity theory. Posing the equations of motion for the clas-sical configuration of ψ in the AdS background as subject to the boundarycondition r∆ψ → ψ(0) as r → ∞ gives a well-defined problem, the solutionof which allows one to read off the expectation value 〈Oψ〉.Next, let us consider a U(1) vector field Aµ in the bulk, with MaxwellactionSA =∫dd+1x√−g 14F2, (1.20)where F is the field strength for the gauge field. The gauge field will coupleto an operator with one Lorentz index, which we will call jµ. The generatingfunctional for the dual field theory will beZCFT [A(0)µ] =〈ei∫A(0)µjµ〉, (1.21)where we are reminded that, as above for the scalar field, we must includeboundary conditions on the gauge field Aµ in the bulk. Once again we canwrite the solution for this field near the asymptotic boundary, findingAµ = A(0)µ +〈jµ〉rd−2 + . . . . (1.22)16Generically, the source is encoded in the non-normalizable mode of the field while theresponse is dual to the normalizable portion of the field.17To see this, one may again use the symmetry xµ → axµ, r → a−1r. Under this scaling,the scalar field ψ should be invariant. Using equation (1.17), we can then read off howOψ should scale under this transformation.191.2. Introduction to the gauge / gravity dualityThe allowed set of local U(1) gauge transformations on the gravity sidereduce to global U(1) transformations at the boundary r → ∞, implyingthat the current jµ is a global U(1) symmetry current. Notice also thatthe scaling dimension of jµ is ∆j = d − 1, which agrees with the scalingdimension of a conserved current in a d-dimensional conformal field theory[27].In particular, this correspondence provides a straightforward methodwith which to study a field theory at finite density. One may turn on achemical potential µ for a global U(1) charge by fixing the boundary con-dition A(0)t = µ; the corresponding response jt = ρ will be the conservedcharge density. Thus, to study strongly coupled field theory at finite density,one simply has to study the dynamics of an electric field (arising from thegauge potential At) in a gravitational background (see Figure 1.5).18~EFigure 1.5: Including an electric field in the bulk is dual to a field theory atfinite charge density. (This image was inspired by a similar figure in [14].)Finally, we consider the metric gµν , a tensor field in the bulk. Theminimal action for the metric is the Einstien-Hilbert action, given above inequation (1.10). To understand the field theory quantities that are encodedin the metric, recall that, in a field theory, the energy-momentum tensor Tµνarises as the conserved current associated with Lorentz transformations. Asfor the gauge field above, the local gauge freedom of the metric in the bulkwill reduce to a global transformation on the boundary: Diffeomorphism18The situation described here corresponds to studying the field theory in the grandcanonical ensemble, in which we fix the chemical potential. By using alternate boundaryconditions for the gauge field, we could study the canonical ensemble, in which we fix thecharge density.201.2. Introduction to the gauge / gravity dualityinvariance in the bulk becomes the freedom of global (Lorentz) coordinatetransformations in the field theory. Thus, it is natural to identify the (bound-ary value of the) metric with the source for the energy-momentum tensorTµν . Written in the same manner as the scalar and gauge fields above, thefield theory generating functional will beZCFT [g(0)µν ] =〈ei∫g(0)µνTµν〉, (1.23)where g(0)µν refers to the boundary value of the metric gµν . Once againwe may solve the equations perturbatively near the boundary, finding theschematic expansion [28]gµν ∼ r2g(0)µν + · · ·+〈Tµν〉rd−2 + . . . , (1.24)where . . . denotes terms higher order in 1/r (additional terms may appearat orders lower than the 〈Tµν〉 term, as indicated). In the field theory, Tµνis the symmetry current that results from a change of coordinates. As canbe seen in these expressions, the metric of the field theory (the tensor thatcouples to Tµν) is g(0)µν , conformal to the boundary value of the bulk metricgµν . It is this relationship that drives the notion that the field theory ‘liveson the boundary’ of the gravity bulk. As a final check, notice that thedimension of the Tµν is ∆T = d, as required by the conformal symmetry inthe field theory [27].1.2.4 Holography and numericsAs discussed above, a typical application of holography to the modelling ofsome strongly coupled physics involves solving the classical field equations ofa gravitating system. In low-dimensionality or in cases with high symmetry,analytic solutions are known. For example, the possible solutions for Ein-stein gravity (with negative cosmological constant) include AdS space andthe Schwarzschild-AdS black hole. However, restricting to analytic solutionsgreatly restricts the characteristics of the corresponding field theory, and ifwe wish to model more interesting types of field theories, we quickly arriveat systems for which no closed-form solutions exist. There are two ways wemight want to extend the known solutions, which we discuss here: we couldinclude more fields on the gravity side (scalar, gauge, fermion, or tensorfields, for example) or we could examine situations with reduced symmetry.In both cases, we must turn to numerical methods in order to make progress.211.2. Introduction to the gauge / gravity dualityAs discussed in the previous subsection, the correspondence betweenbulk fields and field theory operators means that if one wishes to add con-tent to the field theory, then one should consider a gravity side with morefields (the nature of which are determined by the desired operator content).Upon adding fields to the gravity side, one quickly arrives at systems forwhich analytic solutions are not known.19 In these cases, numerical tech-niques become useful tools with which to find solutions. If we maintainsymmetry in the field theory directions, the equations of motion reduce toordinary differential equations, which depend only on the holographic radialcoordinate r. For example, in the expression (1.17), this would imply thatδψ and 〈Oψ〉 are independent of the field theory directions. In this case, theequations of motion may be solved in a straightforward manner using, forexample, standard computer mathematics software. The majority of studiesin the literature have focussed in this way on homogeneous field theory.In order to provide contact with real experimental systems in which theassumption of homogeneity does not apply, one needs to introduce a de-pendence on the field theory directions (so that δψ = δψ(x) and 〈Oψ〉 =〈Oψ(x)〉). To use the correspondence to its full capacity, we should also applyit to studying strongly coupled theories in these less symmetric situations.Generically, if we introduce a dependence on the field theory directions, theequations of motion will take the form of coupled partial differential equa-tions, and any hope of an analytic solution is lost. Therefore, numericalmethods are necessary to address this entire new sector of problems con-cerning strongly coupled theories. In particular, for static problems, theequations take the form of well-defined boundary value problems, for whichstandard numerical techniques exist. Many of these may reasonably be im-plemented with only modest computational resources; thus, these problemsare accessible to many researchers with current technologies and equipment.However, only recently have some research groups began to study field the-ory in inhomogeneous situations in this way.A portion of this thesis (the projects described in chapters 3, 4, and 6) isdedicated to applying numerical techniques to certain holographic situationswith reduced symmetry. In appendix A, we briefly review the numericalprocedures used in these studies.19An exception is the Reissner-Nordstro¨m-AdS black hole, which solves the Einstein-Maxwell system (a gauge field in dynamical gravity), and is dual to a field theory at finitedensity.221.3. Thesis overview1.3 Thesis overviewThis thesis consists of four projects covering three distinct domains of appli-cability. In this section, for each project, we briefly describe the motivation,methods, and main results. First, summarized in section 1.3.1, we studyholographic field theories in generality, seeking results for generic finite den-sity theories. Next, we apply holography to the study of phases whichspontaneously break translation invariance (section 1.3.2). These have ap-plications in condensed matter physics. Finally, we turn to QCD, and studytwo separate problems. First, we model the existence of a colour supercon-ductivity phase at high densities (section 1.3.3). Second, we examine theconstruction of the baryon in a holographic model (section 1.3.4).1.3.1 Density versus chemical potential in holographicprobe theories20One difficult regime of strongly coupled field theory that holography is par-ticularly suited to study is that of finite charge density. Here, lattice tech-niques fail due to the ‘sign problem’, whereby at finite chemical potential,the Euclidean action becomes complex, resulting in a highly oscillatory pathintegral. We can avoid this difficulty by mapping the problem to a grav-ity dual. As reviewed above, in section 1.2.3, according to the holographicdictionary, in order to have a chemical potential (indicating a global U(1)symmetry) in the field theory, one must simply include a U(1) gauge field inthe gravity bulk. Given this simple access to finite density configurations,it is interesting to characterize the types of field theories which have a dualformulation and to extract qualitative (and, ideally, quantitive) results fromthe gravity approach.In this work, we study systems with the minimal structure of a con-served charge, finding in particular which relations between charge density(ρ) and chemical potential (µ) are possible in field theories with a gravitydual. We focus on Lorentz invariant 3 + 1 dimensional holographic fieldtheories with the goal of offering a survey of results in the context of holog-raphy. Comparing and contrasting the results of such a study can providean understanding of the behaviour of strongly interacting matter that iscommon across all models and those features that are particular to certainconstructions, and may provide qualitative results applicable to QCD andother strongly coupled systems.20This section is a summary of the work presented in chapter 2 and published in [1].231.3. Thesis overview~Edifferent gravitytheoriesFigure 1.6: In chapter 2, we study holographic field theories with the mini-mal structure of a conserved charge, which corresponds to having a gravitybulk with an electric field. We enumerate the results for the relationshipbetween chemical potential µ and charge density ρ across a large number of3 + 1 dimensional example field theories. These field theories differ in theirdual gravity description which is indicated by the shaded area at the left ofthe figure. (This image was inspired by a similar figure in [14].)We find that, at large µ, a large class of theories are well-modelled by apower law relationship of the formρ = cµα + . . . , (1.25)where the dots denote terms subdominant in powers of µ. By studyingvarious general and specific examples of holographic field theories, we mayenumerate the possible values of the parameters α and c. The particularsituations we examine in this work may be split into general results andspecific examples. We summarize the cases we consider:1. General considerations:(a) Using thermodynamic stability and causality as general field the-ory constraints, we derive the condition α ≥ 1, restricting thepower that can appear in the relationship.(b) For a holographic field theory in which the gauge field is governedby the probe Maxwell action (that is, in a fixed gravitationalbackground), we find that, analytically, α = 1.241.3. Thesis overview(c) For a holographic field theory using a Born-Infeld gauge fieldaction, we derive α > 1.2. Specific examples:(a) We consider Dp-Dq brane systems, given by a single probe Dqbrane embedded in the black brane background generated by astack of N Dp branes, for various (p, q). In these systems, theBorn-Infeld action describes the dynamics of the gauge field onthe probe brane and determines the possible behaviours (pow-ers α) that may arise. We analytically determine the differentpossible behaviours.(b) Next, we consider bottom-up models,21 both probe and backre-acted, in both black hole and soliton (horizon-less) geometries,and with and without a scalar field. Depending on the example,we use analytical or numerical approaches to solve the equationsof motion on the gravity side before using the correspondenceto interpret our results in terms of the dual field theory and toevaluate the dependence of α and c on both the model and theparameters within each model.These results, the main output of our study, are summarized in Ta-ble 2.1 (for Dp-Dq systems) and in Table 2.2 (for bottom-up models).1.3.2 Holographic stripes22High temperature superconductivity is one of the most interesting and tech-nologically relevant problems in condensed matter physics. Experimentshave recorded many novel results in materials that exhibit high temperaturesuperconductivity. One of these is the observation of translation-symmetrybreaking states, or stripes, which have been observed in the form of chargedensity waves (see Figure 1.7), in which the charge density varies with po-sition, and spin density waves, in which the spin density varies with posi-tion. Striped phases are believed to be due to strong coupling effects and atractable theoretical model is not yet available. By applying the holographic21‘Bottom-up’ models are those in which the theory does not arise from an explicitstring theory construction. Starting with the action (1.10) would be considered a bottom-up approach, while Dp-Dq brane systems are examples of ‘top-down’ models.22This section is a summary of the work presented in chapters 3 and 4 and publishedin [2] and [3]. Chapter 3 presents a concise version of the study while chapter 4 providesfull results and the complete details of the analysis.251.3. Thesis overviewxTemperaturex xSC SC SCCDW+SDW CDW+SDWCDW+SDWCDWCDWnematica) b) c)nematicFigure 1.7: A schematic phase diagram for the cuprates. The verticalaxis is temperature while the horizontal axis represents doping. Shown areschematic diagrams at: a) weak coupling; b) coupling that varies with x,and; c) strong coupling. Phases of the cuprates include the superconductingphase (SC) and inhomogeneous phases: the nematic phase, the charge den-sity wave (CDW), and the spin density wave (SDW). In chapters 3 and 4, westudy a holographic model of a strongly coupled system which exhibits aninhomogeneous phase. (Reprinted with permission from [29], c©2009 Taylor& Francis.)correspondence, one may study strongly coupled systems that share impor-tant features with these experimental materials. In this project, we seek tomodel the spontaneous transition to a translation-symmetry-breaking phase,in hopes that general lessons may be extracted from the results and appliedto the experimental systems. To this end, our goal is to find the gravitydual of a 2 + 1 dimensional model system with spontaneous striped order.To find the gravity dual of a system with stripes, we need a mechanismto break the translational invariance. One mechanism to introduce spatialinhomogeneities is the inclusion of a Chern-Simons-type term in the gravityLagrangian. The specific model we study, due to Donos and Gauntlett [30],isL = 12(R+12)−12∂µψ∂µψ−12m2ψ2− 14FµνFµν−1√−gc116√3ψ µνρσFµνFρσ.(1.26)The Chern-Simons coupling between the scalar field ψ and the gauge fieldAµ (the term proportional to c1) promotes the formation of stripes for largeenough chemical potential and for a range of wave-vectors. Through a per-turbative analysis, Donos and Gauntlett showed that instabilities of the ho-261.3. Thesis overviewmogeneous background towards the formation of stripes indeed exist in theabove model [30]. These stripes appear in the charge density, current den-sity, and energy-momentum tensor of the dual field theory. In this project,we perform the full analysis of the system, solving for the nonlinear inho-mogeneous solutions and characterizing the phase transition between thehomogeneous and the striped phases.The equations governing our gravitational model are the Einstein equa-tions, Gµν = Tµν , and the matter field equations. The fields will vary inboth the holographic radial direction and the inhomogeneous direction ofthe field theory, resulting in coupled nonlinear partial differential equations.To solve the equations, we discretize on a rectangular grid before applyinga Gauss-Seidel relaxation method on the resulting algebraic equations; seeappendix A.2 for more details about the numerical approach in this project.With the solutions for varying temperature and wave-vectors in hand,we can study the thermodynamics of the homogeneous and inhomogeneousphases in order to determine which phase dominates. For the field theoryon a domain of finite size, we make this comparison in each of the micro-canonical, canonical, and grand canonical ensembles, confirming that thestriped solution dominates (when it exists) in all cases, and finding a secondorder transition to the inhomogeneous phase as the temperature is lowered.For the experimentally interesting case of the field theory on an infinitedomain, we compare the free energy density of the homogeneous solution tothat of all stripes available at the given temperature in order to determinethe dominant stripe width. We find a second order transition to the stripedphase as the temperature is lowered and that, once inside the domain of thestriped phase, the width of the dominant stripe increases with decreasingtemperature. This main result is shown in Figure 3.5.1.3.3 Towards a holographic model of coloursuperconductivity23As discussed above, low energy QCD is the prototypical strongly coupledfield theory. Since the domains of applicability of typical field theory ap-proaches (perturbative quantum field theory and lattice simulations) arerestricted to limited regions of the phase diagram, the holographic cor-respondence, which provides access to the thermodynamics of the theoryacross the parameter space, offers a promising avenue for study. Unfor-tunately, the precise gravity dual of QCD is not know. However, certain23This section is a summary of the work presented in chapter 5 and published in [4].271.3. Thesis overviewfeatures of QCD may be modelled in the holographic approach, offering (atthe least) qualitative information about the phase diagram of QCD. Onesuch feature that is particularly amenable to the gravity approach is theexistence of a colour superconductivity quark matter phase at high density.(See Figure 1.8.)liqTµgasQGPCFLnuclearsuperfluidheavy ioncolliderneutron starnon−CFLhadronicFigure 1.8: A schematic of the conjectured QCD phase diagram. At smalldensities and temperatures is the hadronic phase while at larger tempera-tures and densities is the quark-gluon plasma (QGP). At very high densities,we find phases which exhibit colour superconductivity (shaded in yellow),labelled by whether or not they are expected to exhibit the phenomenonof colour-flavour-locking (CFL). In chapter 5, we study a phenomenologicalholographic model of the colour superconductivity phase. (Reprinted withpermission from [31], c©2008 the American Physical Society.)In this project, we employed a bottom-up, phenomenological approachto model a confining gauge theory on 3 + 1 dimensional Minkowski spacewhich displays a colour superconductivity phase at large densities. The keyfeatures in our model include a QCD (confinement) scale,24 a conservedbaryon current (dual to a U(1) gauge field Aµ), and an operator to char-acterize the colour superconductivity phase (dual to a scalar field ψ). The24This is provided by the addition of an extra periodic direction in the geometry. On thegravity side, this allows a phase transition between a geometry with a black hole horizonand one without (a soliton configuration). The soliton geometry cuts off the gravity bulkat a minimum radius, introducing a mass gap in the dual field theory and resulting in aconfining phase [32].281.3. Thesis overviewphenomenological gravity action describing this minimal set of ingredientsis given byS =∫d6x√−g{R+ 20L2 −14F2 − |∂µψ|2 −m2|ψ|2}. (1.27)The standard solution to this model is the charged AdS black hole (withψ = 0), which is dual to the deconfined phase of the field theory. Theexistence of a confinement scale admits a second, horizon-less gravity so-lution, which translates to the confined phase of the field theory. Finally,it is known that in this model, for large chemical potential, there is an in-stability towards a ‘hairy’ black hole, with non-zero scalar field ψ. In thisphase, the operator Oψ dual to the scalar field will acquire a non-zero ex-pectation value: 〈Oψ〉 6= 0. Within our model, Oψ is interpreted as somequark operator whose expectation value indicates the presence of a (colour)superconducting condensate. Thus, the field theory dual to the system de-scribed by the action (1.27) exhibits three distinct phases: a confined phase,a deconfined phase, and a colour superconducting phase.Solving the system of ordinary differential equations derived from theaction and computing the free energy of each phase at various temperaturesand chemical potentials allows us to construct the phase diagram given inFigure 5.1, which is the main result of this study. A particular interestingoutcome is that our model predicts a very small temperature for the onsetof the colour superconducting phase.1.3.4 Holographic baryons from oblate instantons25In this project, we again seek to apply the holographic correspondence in thestudy of particular aspects of QCD. This time, we turn our attention to theconstruction of the baryon in a holographic model of large-Nc two-flavourQCD.The model we consider is based on the Sakai-Sugimoto model of holo-graphic QCD [13], a top-down string theory construction. The Sakai-Sugimotoconstruction begins with a stack of Nc D4 branes,26 compactified on a circleand considered in the low energy limit. On the field theory side, this givespure massless SU(Nc) Yang-Mills theory, representing the gluonic degrees offreedom of QCD. In order to introduce flavour into the theory, one adds Nf25This section is a summary of the work presented in chapter 6 and published in [5].26The construction of the D4 brane background here closely resembles the D3 branesystem used in the original derivation of the correspondence, as reviewed in section 1.2.2above.291.3. Thesis overviewD8 branes to the construction; strings extending from the background D4branes to the flavour D8 branes give states transforming in the fundamentalof SU(Nc), corresponding to the quarks of the theory. The gravity dual ofthis theory is found by replacing the stack of D4 branes by the metric theysource. If the number of D8 branes is small, Nf  Nc, the effect of the D8branes on the geometry can be neglected, and we are left with D8 branesembedded in the background sourced by the D4 branes. In this model,baryons arise on the gravity side as configurations of the U(Nf ) gauge fieldon the flavour D8 branes that possess a non-trivial topological charge. SeeFigure 1.9 for a visualization of this situation.uu d← r →Figure 1.9: A visualization of the duality between the instanton in the bulk(at the left, viewed side-on) and the baryon in the boundary field theory.The instanton, a configuration of the bulk gauge fields with non-trivial topo-logical charge, is oblate: it is squashed in the gravity radial direction r andspherical in the field theory directions. In chapter 6, we numerically solvefor the minimal energy instanton configuration in the bulk.The action describing gauge field configurations on Nf = 2 D8 branes isgiven byS ∝∫d4xdr tr[12h(r)F2µν + k(r)F2µr]+ γ∫M5tr(AF2 − i2A3F − 110A5), (1.28)where A is a U(2) gauge field with field strength F , and h(r) and k(r) areknown functions. Then, our task of solving for a baryon in this theory re-duces to finding gauge field configurations with a non-zero topological charge301.3. Thesis overviewdescribed by the action (1.28). The solution to this is an instanton27 in thethree spatial field theory directions and the holographic radial direction r ofthe bulk.There are three effects which determine the size and shape of the instan-ton. Firstly, the coupling γ, the only parameter in the model (1.28), controlsthe strength of the Coulomb self-repulsion of the instanton. For large γ, theinstanton tends to spread out in all directions. Second, the gravitationalwell of the underlying geometry seeks to keep the instanton at small radialcoordinate r. For small γ, the instanton is small and the geometry does notplay a large role. However, for large γ, the geometry does become importantand the minimum energy configuration is an oblate instanton, squashed inthe r-direction and SO(3)-symmetric in the field theory directions. Finally,the form of the gauge field action plays a role in restricting the deformation.Since a spherical instanton minimizes the Yang-Mills action, this acts torestrict the squashing of the configuration.Previous studies have approximated the gravity dual of the baryon asan SO(4)-symmetric BPST instanton. However, at non-zero values of thecoupling γ, as detailed above, the solution will only possess SO(3) symmetry.It has been shown that the SO(4) approximation to the baryon fails tosatisfy certain model-independent form-factor relations [33]. In this project,we seek to find the precise, SO(3)-symmetric solution to the field equations,in order to improve upon the previous calculations of holographic baryonsin this model.We assume only an SO(3) symmetry and solve the resulting partial dif-ferential field equations using pseudospectral differentiation on a Chebyshevgrid and a Newton’s method for the resulting algebraic equations (see ap-pendix A.3 for a review of these methods). Our approach allows us to com-pute the properties of the baryon, including the charge profile and mass, at arange of couplings γ. Our main results may be found in Figures 6.4 and 6.7.In particular, we may evaluate our solution at the value of γ that has beenfound to best fit the mesonic spectrum of QCD, finding significantly morerealistic values than previous studies for the mass and size of the baryon.These results are found in equations (6.24) and (6.30).27This is a misnomer, due to the fact that the original studies used the Belavin-Polyakov-Schwarz-Tyupkin (BPST) instanton as the static solution for the four spatial directions;the solution is more precisely described as a soliton in the spatial directions, and static intime.31Chapter 2Density versus chemicalpotential in holographicprobe theories12.1 IntroductionThe AdS/CFT correspondence [7–9], which conjectures the equivalence ofa gravity theory in d + 1 dimensions and a gauge theory in d dimensions,has become a valuable tool for the study of strongly coupled field theories.Using the correspondence, many questions about quantum field theoriesmay be phrased in the context of a gravity theory; in the limit of strongcoupling, certain previously intractable field theory calculations are mappedto relatively simple classical gravity computations.Holography and finite densityOne difficult regime of strongly coupled field theory that gauge / gravityduality is particularly suited to study is that of finite charge density. Here,lattice techniques fail due to the ‘sign problem’: at finite chemical potential,the Euclidean action becomes complex which results in a highly oscillatorypath integral. We can avoid this difficulty by mapping the problem to agravity dual using the AdS/CFT dictionary. According to the dictionary,in order to have a global U(1) symmetry in the field theory, one needs toinclude a U(1) gauge field in the gravity bulk. The charge density andchemical potential are encoded in the asymptotic behaviour of the gaugefield. At strong coupling in the field theory, the bulk theory is well describedby classical gravity, and one may solve the classical equations of motion onthe gravity side to study the field theory at finite density.Given this relatively simple access to finite density configurations, wemight hope that some physically realistic strongly interacting systems may1A version of this chapter has been published [1].322.1. Introductionbe approximately described by a holographic dual. In this case, qualitativefeatures of the holographic theory would carry over to the exact theory. Itwould be useful to characterize the types of finite density field theories thathave a dual formulation and admit this type of study.In this paper, we seek to answer this question from the perspective of theholographic theory. Specializing to holographic probes, in which fields areconsidered as small fluctuations on fixed gravitational backgrounds, we studysystems with the minimal structure of a conserved charge and find the ρ−µrelations that are possible in the field theory duals. We attack this problemby first deriving constraints on the relationship based on general groundsbefore studying several specific examples of holographic field theories.Summary of resultsIn our study, we observe that, at large densities, the field theory dual to asubstantial class of gravity models can be described by a power law relationof the form2ρ = cµα. (2.1)Firstly, we look to understand the constraints on the the ρ − µ rela-tionship from the point of view of the field theory, using local stabilityand causality. Usually, results here depend on the particular form of thefree energy. In all cases with ρ − µ behaviour (2.1), local thermodynamicstability places the condition α > 0 on the exponent. In general, for a the-ory at low temperature, we may write the particular free energy expansionf ∝ −µα+1 − aµβT γ , with γ > 0 and a > 0, with corresponding chargedensity ρ ∝ (α+ 1)µα + aβµβ−1T γ . Combined, local stability and causalitydemand that α ≥ 1 and γ > 1.Next, we consider Born-Infeld and Maxwell actions for the gauge fieldin a generic background. Under mild assumptions, in both cases, the powerα is constrained. For the Born-Infeld action, the conditionα > 1 (Born-Infeld action) (2.2)arises,3 while, for the Maxwell action, the power law coefficient is fixed toα = 1. (Maxwell action) (2.3)2Here and throughout, α refers to the power in this form of ρ− µ relationship.3Naively, we could construct systems for which α ≤ 1, however, in these situations, thecontribution of the constant charge density to the total energy diverges, consequently wecan not say that there is a power law relation. This divergence signals a breakdown ofthe probe approximation rendering these systems outside the scope of these notes. Noticethat α > 1 is consistent with the bound derived from stability and causality.332.1. IntroductionInterestingly, these conditions are in agreement with those derived fromfield theory considerations, giving rise to the same range of possible valuesof α. In summary, all power law relationships consistent with stability andcausality can be realized in simple probe gauge field setups by varying thebackground metric.To see which values of α arise for backgrounds corresponding to specificmodels, we explore a variety of 3 + 1 Poincare´-invariant holographic fieldtheories dual to Dp-Dq brane systems and ‘bottom-up’ models with gaugeand scalar fields. The former have been used, for example, in studies ofholographic systems with fundamental matter [13, 34–37], producing manyfeatures of QCD, including confinement,4 chiral symmetry breaking, andthermal phase transitions [39–42]. Bottom-up, phenomenological modelshave been studied in various model-building applications including super-conductors5 [11, 18, 44–47] and superfluids [12, 48, 49].In the Dp-Dq systems, Table 2.1, a variety of powers α in the range1 < α ≤ 3 are realized, respecting the α > 1 constraint. Note that theseresults only involve the Born-Infeld action and neglect couplings of the braneto other background spacetime fields.Probe braned = 4 d = 5Background branes D9 D8 D7 D6 D5 D4 D8 D7 D6D3 3 3 3D4 5/2 2 3/2 3 5/2D5 2 2D6 3/2Table 2.1: The power α in the relationship ρ ∝ µα at large ρ for 3 +1 dimensional field theories dual to the given brane background with theindicated probe brane, with d− 1 shared spacelike directions. For d = 5 thetheory is considered to have a small periodic spacelike direction while forbackground Dp branes with p > 3, the background is compactified to 3 + 1dimensions.In the phenomenological probe models, Table 2.2, in all cases except one(the probe gauge field in the black hole background), the dominant power4It was recently pointed out that the usual identification of the black D4 brane as thestrong coupling continuation of the deconfined phase in the field theory is not valid [38].5A top-down realization of a gauge / gravity superconductor has been found in [43].342.1. Introductionα is determined by conformal invariance, since we consider asymptoticallyAdS backgrounds.6 Since µ and T are the only dimensionful parameters,the density must take the form ρ = µd−1h(T/µ), where the underlyingspace has d spacetime dimensions. At large µ and fixed T , we can expandh to see that µd−1 dominates the ρ − µ relationship. In systems with onesmall periodic spacelike direction, the dominant power α is larger than thecorresponding theory without a periodic direction since, at large densities,on the scale of the distance between charges, the theory is effectively higherdimensional.7 Our study of bottom-up models also includes an analysis ofthe gravity models in the full backreacted regime. As seen in Table 2.2,the power law α in these cases is also determined by the same conformalinvariance argument.In these bottom-up models we are more interested in the detailed be-haviour at intermediate values of µ. It is found that, in general, whenthe scalar field condenses in the bulk, the corresponding field theory is in adenser state than that without the scalar field. As well, the field theory dualto the gauge field and scalar field in the soliton background is in a denserstate than that dual to the same fields in the black hole background. In thesystems with a scalar field, at large µ, the ρ − µ relationship is well fit bythe form ρ = c(q,m2)µα,8 where q and m2 are the charge and mass-squaredof the scalar field. While the power α is fixed by the conformal invariance,we find that the scaling coefficient c(q,m2) increases with increasing q ordecreasing m2.6Different power laws can arise for holographic theories on different backgrounds, suchas Lifshitz spacetimes. However, these will not be considered here.7The phase transition that holographic theories with a periodic direction undergo asthe density increases was studied in [50].8In the probe cases we can scale q to 1, leaving c = c(m2).352.1. IntroductionRegime Background Fields d = 4 d = 5probe black holeφ 1 1φ, ψ 3 4soliton φ, ψ 4backreacted black holeφ 3 4φ, ψ 3 4soliton φ, ψ 4Table 2.2: The power α in the relationship ρ ∝ µα at large ρ for 3 + 1dimensional field theories dual to the given gravitational background withthe stated fields considered in either the probe or backreacted limits. φ isthe time component of the gauge field, ψ is a charged scalar field, and d isthe number of spacetime dimensions. For d = 5 the theory is considered tohave a small periodic spacelike direction.OrganizationIn section 2.2, we discuss some possible general examples of finite densityfield theories and attempt to establish bounds on the ρ− µ relationship byimposing thermodynamical constraints on these systems. In section 2.3 webriefly introduce holographic chemical potential and find, for Maxwell andBorn-Infeld types of action, under mild assumptions, to what extent theyreproduce the relationship found in section 2.2. In section 2.4 we investigatethe probe limit of both top-down and bottom-up theories; first we studyDp-Dq systems, then we move to gauge and scalar fields in both black holeand soliton (with one extra periodic dimension) backgrounds. Section 2.5extends the analysis of the bottom-up models to include the backreaction ofthe fields on the metric.Relation to previous workSome of the results presented in these notes have appeared previously in theliterature. Finite density studies for probe brane systems have appeared forthe Sakai-Sugimoto model [42, 51–53], the D3-D7 system [35, 36, 54–56], andthe D4-D6 system [57]. The bottom-up models we consider are naturallystudied at finite chemical potential (see, for example, [18] for the black holecase and [58] for the soliton dual to a 2 + 1 dimensional field theory) due tothe presence of the gauge field.Our work focusses on the ρ− µ relation at large chemical potential over362.2. CFT thermodynamicsa broad class of theories that are dual to 3+1 dimensional field theories. Wefind, on very general grounds, constraints on the ρ−µ relation in holographicmodels constructed from Maxwell and Born-Infeld actions. Additionally, weuse thermodynamical considerations to constrain the ρ−µ relation from thefield theory point of view and find that these constraints are in agreementwith those derived holographically. Further, we extend the analysis in theabove references to the large density regime and include additional examples,collecting the results of a large range of models.2.2 CFT thermodynamicsIn this section, by appealing to local thermodynamic stability and causalityin the field theory, we attempt to establish generic constraints satisfied bythe coefficient α from a purely field theory stand point. The results foundhere will lay ground for our intuition when approaching this problem fromthe holographic side.Generic system at large chemical potentialIn order to study the density and chemical potential from the field theoryperspective, we begin with a general ansatz for the free energy of a hypothet-ical system. In the large density limit, we expect that the chemical potentialwill dominate the expression, so we may write9f ∝ −µα+1 − aµβT γ + . . . , (2.4)where the dots denote corrections higher order in T/µ. For a positive,imposing a positive entropy density s = −(∂f/∂T )|µ > 0 implies γ > 0,consistent with the second term being subleading in the low temperatureexpansion.Considering the field theory as a thermodynamical system and imposinglocal stability demands that [59]10χ =(∂ρ∂µ)T> 0, (2.5)andCρ = T( ∂s∂T)ρ= −T ∂2f∂T 2 −( ∂2f∂T∂µ)2 1∂2f∂µ2 > 0. (2.6)9Recall ρ = −(∂f/∂µ)T so that, again, ρ ∝ µα.10χ is the charge susceptibility and Cρ is the specific heat at constant volume.372.2. CFT thermodynamicsApplying these to (2.4) in the T/µ → 0 limit gives the constraints α > 0and γ > 1.Examining the speed of sound vs of our system also allows us to establisha constraint. To ensure causality, we impose0 ≤ vs ≤ 1, (2.7)with the speed of sound given by [12]v2s = −[(∂2f∂T 2)ρ2 +(∂2f∂µ2)s2 − 2(∂2f∂T∂µ)ρs](sT + ρµ)[(∂2f∂T 2)(∂2f∂µ2)−(∂2f∂T∂µ)2] , (2.8)where ρ and s are the charge and entropy densities. For γ > 1, this impliesthe stronger bound of α ≥ 1. This is the same bound as derived in section 2.3from consideration of the bulk dual of field theories. It is interesting that itarises from very general circumstances in both cases.Zero temperatureIn the zero temperature limit of ansatz (2.4) only the first term survives, sothat f ∝ −µα+1. In this case, the only condition for local stability is givenby equation (2.5), which trivially leads to ρ ∝ µα with α > 0. Computingthe speed of sound and enforcing causality leads again to α ≥ 1.General conformal theoryFor a conformal field theory in d spacetime dimensions, the most generalfree energy density isf = −µdg(Tµ), (2.9)where g(x) is an arbitrary dimensionless function. Local stability dependson the details of the function g, and a general statement is not possible atthis point. To ensure causality, we compute equation (2.8), finding the speedof propagation to bev2s =1d− 1 , (2.10)from which it follows directly that a conformal theory obeys requirement(2.7) only in dimension d ≥ 2. This result is trivial, as sound waves are notpossible if there are no spacelike dimensions to propagate in.382.3. General holographic field theories at finite densityFree fermionsAs an example, we will compute the ρ− µ relationship for a system of freefermions. In the grand canonical ensemble, the partition function for spin1/2 particles of charge q in a 3 dimensional box and subjected to a largechemical potential isZ(µ, T ) =∏~n(1 + e−β(E~n−µq)), (2.11)where the product is over available momentum levels. The partition functionfor antiparticles follows with the replacement q → −q so we include antipar-ticles by considering the total partition function Z˜(µ, T ) = Z(µ, T )Z(−µ, T ).Passing to the continuum limit, approximating the fermions as massless, andsetting q = 1, the resultant charge density isρ = µ33pi2 +µT 23 . (2.12)The dominant power in this case is the same as is expected in a genericconformal field theory.2.3 General holographic field theories at finitedensityIt was shown in the previous section how local stability and causality leadto α ≥ 1. In this section, under mild assumptions, we investigate the Born-Infeld and Maxwell actions in the large µ regime and observe to what extentthey fall under the general results from section 2.2.2.3.1 Finite densityTo find constraints on the ρ − µ relation in holographic field theories, webegin by studying very general systems with the minimal structure of aconserved charge. The holographic dictionary gives that a conserved chargein the field theory is dual to a massless U(1) gauge field A in the bulk [60].If the gauge field is a function only of the radial coordinate r, the chemicalpotential and the charge density are encoded in the behaviour of A asµ = At(∞) (2.13)andρ = − ∂SE∂At(∞), (2.14)392.3. General holographic field theories at finite densitywhere SE is the Euclidean action evaluated on the saddle-point and thederivative is taken holding other sources fixed. As discussed in [36], anequivalent expression for the charge density is11ρ =( 1d− 2) ∂L∂(∂rAt), (2.15)where the normalization of ρ has been chosen for later convenience. Afterwriting down the gravitational Lagrangian, our prescription for computingthe charge density at a given chemical potential is to solve the equations ofmotion with a fixed boundary condition for the gauge field, equation (2.13),before reading off the density using equation (2.15).2.3.2 Gauge field actionsTo include a gauge field in our AdS/CFT construction, we simply includeit in the bulk action. Two gauge field Lagrangians that have appeared inholographic studies are the Maxwell and the Born-Infeld Lagrangians. Typi-cally, the Maxwell action is used in bottom-up holographic models while theBorn-Infeld action appears in the study of brane dynamics. Below, in sec-tion 2.4 we will consider holographic models using both types of Lagrangians.However, much insight can be gained by investigating these actions undergeneric conditions. Therefore, in this section, we study general versions ofthese two Lagrangians, at fixed temperature and large chemical potential, inthe probe approximation.12 Interpreting our results using (2.13) and (2.15),we will develop some constraints for the ρ − µ relationship in holographictheories described by these actions.The Maxwell actionConsider a gauge field described by the Maxwell action∫ √−gF 2 in a generalbackground of the formds2 = gFTµν (r)dxµdxν + grr(r)dr2. (2.16)If we assume homogeneity in the field theory directions and consider a purelyelectrical gauge field (keeping only its time-component), the Lagrangian is11Generically, At is a cyclic variable, so that the conjugate momentum is conserved, andwe may evaluate this expression at any r.12In the probe approximation, we assume there is no backreaction on the gravity metric.This is enforced in this case by studying the gauge field Lagrangian on a fixed backgroundgeometry.402.3. General holographic field theories at finite densitysimplyL = g(r) (∂rAt)2 , (2.17)for some function g(r). From this we findρ =( 2d− 2)g(r)∂rAt. (2.18)In the systems considered below, the spacetime either has a horizon orsmoothly cuts off at some radius rmin. The value of the gauge field at thispoint is a boundary condition for the problem. Below, At(rmin) is eitherzero or a constant, neither of which affect the ρ − µ behaviour; we takeAt(rmin) = 0 here. Integrating (2.18), we findµ = ρ(d− 22)∫ ∞rmindrg(r) . (2.19)Provided the integral is finite, we haveρ ∝ µ. (2.20)Thus, for any holographic field theory with the gauge field described onlyby the Maxwell Lagrangian in a fixed metric we have α = 1.The Born-Infeld actionThe Born-Infeld action is the non-linear generalization of Maxwell electro-dynamics and is the appropriate language in which to describe the dynamicsof gauge fields living on branes. Assuming homogeneity in the field theorydirections, so that the gauge potential varies only with the radial direction,these systems are governed by an action of the generic form13L =√g(r)− h(r)(∂rAt)2, (2.21)where again, we take At to be the only non-zero part of the gauge field. Thecharge density is given by the constant of motionρ =( 1d− 2) h(r)∂rAt(r)√g(r)− h(r)(∂rAt)2. (2.22)13g(r) and h(r) are arbitrary functions; g(r) is not related to the previous discussion.412.4. Holographic probesHere, we assume that the gauge field is sourced by a charged black holehorizon at r+.14 Euclidean regularity of the potential At fixes its value atthe horizon as At(r+) = 0 [36]. Then, we can integrate to findµ =∫ ∞r+dr√g(r)h(r)(d− 2)ρ√h(r) + (d− 2)2ρ2. (2.23)To extract the large ρ behaviour, we split the integral at Λ  1. Forρ Λ, the integral from r+ to Λ approaches a constant, while the functionsin the integral from Λ to ∞ can be approximated by their large r forms,which will be denoted with a ∞ subscript. The expression for the chemicalpotential now becomesµ ≈∫ Λr+dr√g(r)h(r) +∫ ∞Λdr√g∞(r)h∞(r)(d− 2)ρ√h∞(r) + (d− 2)2ρ2. (2.24)The ρ dependence of µ comes from the second term. If g∞(r)/h∞(r) ≈ r2mand h∞(r) ≈ rn, by putting x = r/ρ2/n we find thatµ ∼ ρ(2+2m)/n∫ ∞r+ρ2/ndx xm√xn + 1. (2.25)The convergence of the integral here requires that n/(2 + 2m) > 1, resultingin the relationshipρ ∝ µα with α > 1, (2.26)where the power α depends on the specific bulk geometry.2.4 Holographic probesWith the general constraints of the previous sections in hand, we move onto study particular holographic field theories in the probe approximation,to see which specific values of α are realized. Here, we study two commonprobe configurations that have arisen in previous holographic studies. Theseare extensions of the actions considered in section 2.3. First, we examineprobe branes in the black brane background using the Born-Infeld action.14To have a non-trivial field configuration, a source for the gauge field in the bulk isrequired. In the low temperature, horizon-free versions of these models, this source isgiven by lower dimensional branes wrapped in directions transverse to the probe branes[61].422.4. Holographic probesThen, we move on to the phenomenological perspective, in which we writedown an effective gravity action without appealing to the higher dimensionalstring theory. In this approximation, using the Maxwell action, we lookat the gauge field in both the planar Schwarzschild black hole and solitonbackgrounds, with and without a coupling to a scalar field.In both cases, in the systems we consider, the only sources in the fieldtheory are the temperature T and chemical potential µ. Below, we fix Tand work at large µ (such that µ/T  1). In this regime, we look for arelationship ρ ∝ µα + . . . , where the dots denote terms higher order in T/µ.2.4.1 Probe branes and the Born-Infeld actionIn the systems we will consider here, the background consists of Nc D-branes;in the largeNc limit, these branes are replaced with a classical gravity metric.In this regime, fundamental matter is added by placing Nf probe branes inthe geometry [62].The brane actionAssuming that the background spacetime metric Gµν is given, the actiongoverning the dynamics of a single Dq probe brane is the Born-Infeld actionS ∝∫dq+1σe−φ√−det(gab + 2piα′Fab). (2.27)Here, latin indices refer to brane coordinates and greek indices denote space-time coordinates, while Xµ(σa) describes the brane embedding. gab is theinduced metric on the probe brane given by gab = ∂aXµ∂bXνGµν , Fab isthe field strength for the U(1) gauge field on the brane, and φ is the dilatonfield. Following the previous discussion, the only component of the gaugefield we choose to turn on is At, additionally, we assume it depends onlyon the radial coordinate r, At = At(r). Considering that the probe braneis extended in the r direction and the spacetime metric is diagonal, theLagrangian simplifies toL ∝ e−φ√−det(gab)(1 + (∂rAt)2gttgrr), (2.28)432.4. Holographic probeswhere we rescaled At to absorb the 2piα′ term. In the notation of equation(2.21), we can writeg(r) = −det(gab)e−2φ, (2.29)h(r) = det(gab)e−2φgttgrr. (2.30)The backgroundFor Nc Dp branes, at large Nc, the high temperature background is the blackDp brane metric, given by15ds2 = H−1/2(−fdt2 + d~x2p) +H1/2(dr2f + r2dΩ28−p), (2.31)withH(r) =(Lr)7−p, f(r) = 1−(r+r)7−p, eφ = gsH(3−p)/4. (2.32)L is the characteristic length of the space, while gs is the string coupling.This metric has a horizon at r = r+.Our probe Dq brane is fixed to share d − 1 spacelike directions withthe Dp branes. If p > d − 1, the fundamental matter propagates on a ddimensional defect and we may consider the extra p−(d−1) directions alongthe background brane to be compactified, giving an effective d dimensionalgauge theory at low energies. Alternatively, we can build a d−1 dimensionalgauge theory by compactifying one or more of the directions shared by theprobe and background branes. Below, we will study field theories that areeffectively 3 + 1 dimensional using both methods.We stipulate that the Dq probe brane wraps an Sn inside the S8−pand extends along the radial direction r. These quantities are related byq = d+ n. The induced metric on the Dq brane isds2 = H−1/2(−fdt2 + d~x2d−1) +(η(r) + H1/2f)dr2 +H1/2r2dΩ2n, (2.33)whereη(r) = ∂rXµ∂rXνGµν −Grr. (2.34)15More details on this solution can be found in [34].442.4. Holographic probesCalculating equations (2.29) and (2.30) gives16g(r) = r2nfH 12 (p+n−d−3)(η(r) + H1/2f), (2.35)h(r) = r2nH 12 (p+n−d−2), (2.36)from which (2.23) gives the chemical potentialµ =∫ ∞r+dr (d− 2)ρ√r2n(Lr)( 7−p2 )(p+n−d−2) + (d− 2)2ρ2√fη(r)H1/2 + 1. (2.37)Now, η(r) will be some combination of (∂rχi)2, where the χi denote thedirections of transverse brane fluctuations. By writing down the equationsof motion we can observe that ∂rχi = 0 is a solution, in which case theprobe brane goes straight into the black hole along the radial direction r.This describes the high temperature, deconfined regime; we set η(r) = 0 inthe following.For large ρ we findρ ∝ µ 14 [(p−7)(p−d−2)+(p−3)(q−d)], (2.38)so that for the probe brane systems,α = 14[(p− 7)(p− d− 2) + (p− 3)(q − d)]. (2.39)As above, α is constrained as α > 1 for convergence of the integral. If α ≤ 1,the contribution of the constant charge density to the total energy diverges,signalling a breakdown of the probe approximation. At this point, we canuse equations (2.38) and (2.39) to investigate what type of ρ−µ behaviourscan arise from Dp-Dq brane constructions.Example: the Sakai-Sugimoto modelThe well-known Sakai-Sugimoto model [13] consists of Nf probe D8-D8branes in a background of Nc D4 branes compactified on a circle. We havep = 4, q = 8, and d = 4. Putting these numbers into (2.38) yieldsρ ∝ µ5/2, (2.40)consistent with previous results [42, 53].16We leave the constant factors of gs from eφ out of the Lagrangian, as our goal here isjust the power law dependence.452.4. Holographic probesρ− µ in 3 + 1 dimensional probe brane theoriesEquation (2.39) determines the dominant power law behaviour in all Dp-Dq configurations relevant to 3 + 1 dimensional field theory. As discussedabove, we can set the number of shared probe and background directions tobe d − 1 = 3 or put d − 1 = 4 and demand one of the the spacelike shareddirections to be periodic; see Table 2.1 for the results. The power α = 3 isan upper bound for the 3 + 1 dimensional probe brane gauge theories wehave considered.Our calculation above involves only the Born-Infeld action for the probebrane and in particular neglects any possible Chern-Simons terms that ap-pear due to the coupling between the brane and a spacetime tensor field.The Chern-Simons term is important in the D4-D4 system, for example [37].2.4.2 Bottom-up models and the Einstein-Maxwell actionWe now turn our attention to bottom-up AdS/CFT models in the proberegime. To construct a phenomenological gauge / gravity model, we beginwith a theory of gravity with a cosmological constant, such that the geometryis asymptotically AdS. To study the relationship between charge density andchemical potential in the dual field theory, we demand that there must bea gauge field in the bulk. At this point, our model has the ingredients forus to compute our desired result. But, one may ask what type of extensionsare possible. Motivated by superconductivity and superfluidity studies, wewill consider also a charged scalar field in our gravity theory. Adding ascalar field alters the dynamics of the system, notably resulting in differentphases [63, 64]. When the scalar field takes on a non-zero expectation value,this breaks the U(1) gauge symmetry in the bulk and corresponds to thepresence of a U(1) condensate in the boundary theory.The particular model we study is the Einstein-Maxwell system with acharged scalar field:S =∫dd+1x√−g{R+ d(d− 1)L2 −14FµνFµν − |∂µψ − iqAµψ|2 − V (|ψ|)}.(2.41)Different dual field theories may be obtained by considering this action indifferent regimes and with different parameters. Below, we make the follow-ing ansatz for the gauge and scalar fields:A = φ(r)dt, ψ = ψ(r). (2.42)462.4. Holographic probesThe r component of Maxwell’s equations will give that the phase of thecomplex field ψ is constant, so without loss of generality we take ψ real. Forthe remainder of the study, we choose units such that L = 1 and considerthe potential V (ψ) = m2ψ2.The probe limitTo get the probe approximation for the system described by (2.41), werescale ψ → ψ/q and A → A/q before taking q → ∞ while keeping theproduct qµ fixed (to maintain the same A − ψ coupling). The gauge andscalar fields decouple from the Einstein equations and we study the fields ina fixed gravitational background.The background is governed by the actionS =∫dd+1x√−g {R+ d(d− 1)} . (2.43)One solution here is the planar Schwarzschild-AdS black hole, given byds2bh = (−fbh(r)dt2 + r2dxidxi) +dr2fbh(r), (2.44)withfbh(r) = r2(1− rd+rd), (2.45)where r+ is the black hole horizon. Below, we consider two systems in theSchwarzschild-AdS background: the probe gauge field, and the probe gaugeand scalar fields.Computing µ and ρIf the kinetic term for the gauge theory on the gravity side is the MaxwellLagrangian,L = 14√−gFµνFµν , (2.46)then for an asymptotically AdS space the field equation for the time com-ponent of the gauge field isφ′′ + d− 1r φ′ + · · · = 0, (2.47)472.4. Holographic probeswhere ′ denotes an r derivative and . . . denotes terms that have higherpowers of 1/r. The solution isφ(r) = φ1 +φ2rd−2 + . . . . (2.48)Recalling that φ(∞) = µ determines that φ1 = µ, while we can plug (2.48)into (2.46) and compute, using (2.15), that φ2 = ρ. We have thatφ(r) = µ− ρrd−2 + . . . , (2.49)so that in practice, below, we just have to read off the coefficients of theleading and next to leading power of 1/r to find the chemical potential andthe charge density.The scalar fieldSolving the scalar field equation at large r in an asymptotically AdS spaceresults in the behaviourψ = ψ1rλ− +ψ2rλ+ + . . . , (2.50)whereλ± =12{d±√d2 + 4m2}. (2.51)For m2 near the Breitenlohner-Freedman (BF) bound [65, 66], in the range−(d−1)2/4 ≥ m2 ≥ −d2/4, the choice of either ψ1 = 0 or ψ2 = 0 results in anormalizable solution [63]. For m2 > −(d− 1)2/4, ψ1 is a non-normalizablemode and ψ2 is a normalizable mode. For the cases with the scalar field,we define our field theory by taking ψ1 = 0, so that we never introduce asource for the operator dual to the scalar field.The probe gauge fieldHere, we study the probe gauge field, without the scalar field, in the Schwarzschild-AdS background (2.44). The equation of motion for φ isφ′′ + d− 1r φ′ = 0. (2.52)Regularity at the horizon demands that φ(r+) = 0 and the AdS/CFTdictionary gives φ(∞) = µ, leading toφ(r) = µ(1− rd−2+rd−2). (2.53)482.4. Holographic probesThen, applying (2.49), we haveρ = µrd−2+ . (2.54)The horizon r+ depends only on the temperature, T = r+d/4pi,17 so this isa linear relationship between ρ and µ, in accordance with (2.20).Adding a scalar fieldWe now turn on the scalar field in (2.41), and consider the dynamics in theSchwarzschild-AdS background (2.44).The field equations becomeψ′′ +(f ′bhfbh+ d− 1r)ψ′ +(q2φ2f2bh− m2fbh)ψ = 0, (2.55)φ′′ + d− 1r φ′ − 2q2ψ2fbhφ = 0. (2.56)At this point, we can scale q to 1 by scaling φ and ψ, and so m is the onlyparameter here.The coupling allows the gauge field to act as a negative mass for thescalar field. At small chemical potentials, ψ = 0 is the solution. As weincrease µ, the effect of the gauge field on the scalar field becomes largeenough such that the effective mass of the scalar field drops below the BFbound of the near horizon limit of the geometry, so that a non-zero profilefor ψ is possible, and we have a phase transition to the field theory state withbroken U(1) symmetry. A smaller (more negative) squared mass results ina smaller critical chemical potential, at which the scalar field turns on.Using a simple shooting method, for d = 4 we numerically solve equations(2.55, 2.56) and arrive at the relationshipρ = cpbh(m2)µ3, (2.57)where cpbh(m2) is a scaling constant that depends on the mass of the scalarfield. The coupling to the scalar field has resulted in the larger power (α = 3)in the scaling of ρ. A smaller squared mass corresponds to a larger value ofcpbh and, for a given chemical potential, is dual to field theory with a highercharge density. In Figure 2.1, we can see the existence of a denser statewhen the scalar field turns on as well as the relative relation between themass of the scalar field and the charge density in the field theory.17For a Euclidean metric ds2 = α(r)dτ2 + dr2β(r) with periodic τ = it coordinate andα(r+) = β(r+) = 0, regularity at the horizon demands that the temperature (the inverseperiod of τ) be given by T =√α′(r+)β′(r+)/4pi.492.4. Holographic probes10 20 301510020050010002000Μ  TѐT3Figure 2.1: Charge density versus chemical potential for the probe gaugeand scalar fields in the d = 4 black hole background, on a log-log scale. Thethick dashed line is for the system with no scalar field for which, analytically,ρ ∝ µ. At a critical chemical potential, depending on the mass of thescalar field, configurations with non-zero scalar field become available. Thethin dotted line is a model power law ρ ∝ µ3, as described in equation(2.57). From left to right, the thick solid lines are for scalar field massesm2 = −15/4, −14/4, −13/4, and −3. A more negative scalar field massresults in a denser field theory state at a given chemical potential.The soliton probeMotivated by recent work [4, 58, 67], we now add more structure to thebulk theory in the form of an extra periodic dimension. To model a 3 + 1dimensional field theory, we set d = 5 and stipulate that this includes oneperiodic spacelike coordinate w of length 2piR. At energies much less thanthe scale set by this length, E  1/R, the dual field theory will be effectively3+1 dimensional. The extra dimension sets another scale for the field theoryand enables a richer phase structure in the system.18With the extra periodic direction, there is another solution to the back-18The phase diagram including both black hole and soliton solutions, was studied in[58] for a 2+1 dimensional field theory in the context of holographic superconductors andin [4] for a 3 + 1 dimensional field theory in the context of holographic QCD and coloursuperconductivity.502.4. Holographic probesground described by (2.43). This is the AdS-soliton, given as the double-analytic continuation of the Schwarzschild-AdS solution (2.44):ds2sol = (r2dxµdxµ + fsol(r)dw2) +dr2fsol(r), (2.58)withfsol = r2(1− r50r5). (2.59)Here, r0 is the location of the tip of the soliton. For regularity, it is fixed bythe length of the w dimension asr0 =25R. (2.60)By computing the free energy of the systems, it can be shown that the soli-ton background dominates over the black hole background for small enoughtemperatures and chemical potentials. As the temperature or chemical po-tential is increased, there is a first order phase transition to the black hole,which is the holographic version of a confinement / deconfinement transition.For zero scalar field, the soliton can be considered at any temperatureand chemical potential; the period of the Euclidean time direction defines thetemperature while φ = µ = constant is a solution to the field equations. Inthis case, ρ = 0 and we do not have an interesting ρ−µ relation. Consideringa non-zero scalar field provides a source for the gauge field and allows non-trivial configurations.In the soliton background (2.58), the equations of motion areψ′′ +(f ′solfsol+ 4r)ψ′ +( q2φ2r2fsol− m2fsol)ψ = 0, (2.61)φ′′ +(f ′solfsol+ 2r)φ′ − 2q2ψ2fsolφ = 0. (2.62)As in the black hole case, at this point we can set q = 1 by scaling the fields.After numerically integrating, we haveρ = cpsol(m2)µ4. (2.63)Compared to the black hole case, above, we find a larger power of µ. At largedensities, the average distance between charges becomes small compared tothe size R of the periodic direction. In this limit, the system becomes512.5. ρ− µ in backreacted systemseffectively higher dimensional and so we would expect a larger power α inthe ρ− µ relationship. The numerics were consistent with this result.As can be seen in Figure 2.2, a more negative mass squared results ina smaller critical chemical potential and a denser field theory state at agiven chemical potential. This is as expected by comparing the structure ofthe equations to those in the black hole case. Further, at a given chemicalpotential, the soliton solution corresponds to a denser field theory state thanthe black hole solution with the same scalar field mass.6 8 10 12 14 160100200300400500Μ  TѐT4Figure 2.2: Charge density versus chemical potential for the probe gauge andscalar fields in the soliton background and the d = 5 black hole background.The thin dashed line is the probe gauge field in the black hole backgroundfor which, analytically, ρ ∝ µ. The thick solid lines are the soliton results(from left to right, the squared mass of the scalar field is −22/4, −5, −18/4,and −4) while the thick dashed lines are the black hole results (again, fromleft to right, m2 = −22/4, −5, −18/4, and −4). Each of the thick linesapproaches the power law ρ ∝ µ4, equation (2.63). At a given chemicalpotential, the soliton background gives a field theory in a denser state.2.5 ρ− µ in backreacted systemsDespite our analysis in section 2.4 relying on the probe approximation, it isinteresting to ask how much of a difference allowing for backreaction on thebottom-up models could make to the ρ − µ relation and the bounds found522.5. ρ− µ in backreacted systemspreviously. Henceforth we generalize the bottom-up model introduced insection 2.4.2 and allow for the backreaction of the gauge and scalar field onthe metric. Recall that the action isS =∫dd+1x√−g{R+ d(d− 1)− 14FµνFµν − |∂µψ − iqAµψ|2 −m2ψ2}.(2.64)We start by studying the well-known Reissner-Nordstrom-AdS (RN-AdS)solution to the Einstein equation, in which ψ = 0. Later, we allow thescalar field to acquire a non-zero profile and investigate its consequences onthe ρ−µ profile. We finish with the investigation of the backreacted versionof the solitonic solution.2.5.1 Charged black holesThe backreacted solution with no scalar field is the planar RN-AdS blackhole, given byds2 = (−fRN(r)dt2 + r2dxidxi) +dr2fRN(r), (2.65)with19fRN(r) = r2(1−(1 + (d− 2)µ22(d− 1)r2+) rd+rd +(d− 2)µ22(d− 1)r2(d−2)+r2(d−1)). (2.66)The gauge potential isφ(r) = µ(1− rd−2+rd−2), (2.67)so that, using (2.49), we have ρ = µrd−2+ . Here, the horizon r+ can beexpressed as a function of the temperature and chemical potential throughthe Hawking temperatureT = 14pi(dr+ −(d− 2)2µ22(d− 1)r+). (2.68)19We parametrize this solution in terms of the location of the horizon r+ and theasymptotic value of the gauge field (the chemical potential µ) instead of the usual choicesof the charge and mass of the black hole.532.5. ρ− µ in backreacted systemsEliminating r+ in favour of ρ and µ in (2.68), we may solve for ρ to findρ =( (d− 2)22d(d− 1)) d−22µd−1[(2(d− 1)d) 12 2piT(d− 2)µ +√1 + 8pi2(d− 1)T 2d(d− 2)2µ2]d−2.(2.69)Notice that the dominant power in the ρ−µ relationship is µd−1, as expectedin a d dimensional conformal field theory. For d = 4, the particular large µexpansion isρ = 16µ3 + pi√6µ2T + 12pi2µT 2 + 14√32pi3T 3 + . . . . (2.70)2.5.2 Hairy black holesIf we turn on the scalar field, an analytic solution to the equations of motionis no longer possible and we turn to numerical calculation. We take as ourmetric ansatzds2 = −g(r)e−χ(r)dt2 + dr2g(r) + r2(dxidxi), (2.71)where g(r) will be fixed to have a zero at r+, giving a horizon. We arrive atthe following equations of motion:ψ′′ +(g′g −χ′2 +d− 1r)ψ′ + 1g(q2φ2eχg −m2)ψ = 0, (2.72)φ′′ +(χ′2 +d− 1r)φ′ − 2q2ψ2g φ = 0, (2.73)χ′ + 2rψ′2d− 1 +2rq2φ2ψ2eχ(d− 1)g2 = 0, (2.74)g′ +(d− 2r −χ′2)g + reχφ′22(d− 1) +rm2ψ2d− 1 − dr = 0. (2.75)The first two equations can be derived via the Euler-Lagrange equationsfor φ and ψ, while the final two equations are the tt and rr components ofEinstein’s equation.In this system, as in the probe case, section 2.4.2, at small chemicalpotentials the scalar field is identically zero. As we increase the chemi-cal potential above a critical value, the system undergoes a second order542.5. ρ− µ in backreacted systemsphase transition to a state with non-zero scalar field. When the scalar fieldcondenses, the corresponding field theory is in a denser state at the samechemical potential than for the system without scalar field.We solve the equations numerically for d = 4, to yield the result, in thephase with the scalar field,ρ = cbh(q,m2)µ3. (2.76)As we increase the charge or decrease the mass squared of the scalar field, thecritical chemical potential, at which the scalar condenses, decreases, whilethe scaling coefficient cbh increases. The scaling coefficient cbh(q,m2) is, inall cases we checked, larger than the coefficient of the µ3 term in the AdS-Reissner-Nordstrom black hole, equation (2.70), indicating that the densityscales faster with the chemical potential when the scalar field is present.When we include metric backreaction for the black hole, the dominantpower in the ρ − µ relationship is greater than the probe case when thereis no scalar field and is the same as the probe case when there is a scalarfield, indicating that, at least for the systems considered, the bounds foundfor the ρ− µ behaviour apply to the backreacted cases as well.2.5.3 Backreacted solitonMotivated by the form of the soliton background (2.58) we choose the metricansatzds2 = dr2r2B(r) + r2(eA(r)B(r)dw2 − eC(r)dt2 + dxidxi), (2.77)where we constrain B(r0) = 0 so that the tip of the soliton is at r0. Thefield and Einstein equations giveψ′′ +(6r +A′2 +B′B +C ′2)ψ′ + 1r2B(e−C(qφ)2r2 −m2)ψ = 0, (2.78)φ′′ +(4r +A′2 +B′B −C ′2)φ′ − 2ψ2q2φr2B = 0, (2.79)B′(4r −C ′2)+B(ψ′2 − 12A′C ′ + e−Cφ′22r2 +20r2)++ 1r2(e−C(qφ)2ψ2r2 +m2ψ2 − 20)= 0, (2.80)552.6. DiscussionC ′′ + 12C′2 +(6r +A′2 +B′B)C ′ −(φ′2 + 2(qφ)2ψ2r2B) e−Cr2 = 0, (2.81)A′ = 2r2C ′′ + r2C ′2 + 4rC ′ + 4r2ψ′2 − 2e−Cφ′2r(8 + rC ′) . (2.82)We solve equations (2.78-2.81) numerically with asymptotically AdSboundary conditions before integrating (2.82) to find A.20 The results areconsistent with a ρ− µ relationship of the formρ = csol(q,m2)µ4. (2.83)As in the probe case, the effective higher dimension of the space dictatesthe power in the relationship. The dependence of csol(q,m2) on q and m2is as in the backreacted black hole case, section 2.5.2. Like the black holewith scalar field, the backreacted soliton with scalar field gives the samedominant power α as the corresponding probe case.2.6 DiscussionIn these notes we studied the ρ−µ relation for a variety of holographic fieldtheories, and set conditions for physically consistent relationships based onlocal stability and causality. We observed that all of the examples consid-ered are well modelled by a power law ρ = cµα in the large µ regime andthat none of them fail to satisfy any of the general constraints establishedin sections 2.2 and 2.3. Except for the case of a probe gauge field in theSchwarzschild-AdS black hole background, the power α in all the bottom-upmodels obeyed the generic dimensional argument discussed in the introduc-tion, as can be seen in Table 2.2. This resulted in a larger power for themodels with an extra periodic dimension. The brane constructions, Ta-ble 2.1, displayed a larger variety of power laws, with the range 1 < α ≤ 3,where α depended on the particular dimensions of the probe and backgroundbranes.The study of bottom-up models led to the conclusion that, in general,the presence of a non-zero profile for the scalar field in the bulk inducesa larger charge density on the boundary. In most cases, this change wasrealized as an increase of the scaling coefficient c while the power law was20More details on the numerical process can be found in [4].562.6. Discussionkept unaltered. The only exception was the probe Einstein-Maxwell case,section 2.4.2. Here, in the absence of a scalar field, the probe Maxwellfield enjoys its standard linear equations of motion, and naturally we find alinear ρ−µ relationship. With a non-zero scalar field, the power law becomesρ ∝ µd−1, as expected for the underlying CFT. In systems with an extraperiodic direction, the numerical results displayed in Figure 2.2 support theconclusion that at a given (large enough) chemical potential, the solitonicphase is denser than the corresponding black hole phase.Despite our attempt to survey a large variety of holographic models, wedo not claim to have presented a complete report and we do not discard thepossibility of finding different ρ − µ relations in other types of bottom-upand top-down models. For example, one generalization would be to includeNf > 1 flavour branes in the Dp-Dq systems; this has been shown to changethe power α in the relation [53]. It would be interesting to extend this studyto other classes of systems and to see how the results compare to those givenhere.57Chapter 3Holographic stripes13.1 Introduction and summaryThe gauge / gravity duality describes phenomena in strongly coupled fieldtheories via their relation to classical or semi-classical gravitational systems.From the perspective of the boundary quantum field theory, this relation canbe used to construct and study models for ill-understood phenomena whicharise in such strongly coupled systems. On the other hand, the relationto local quantum field theory helps motivate and interpret new results inclassical and quantum gravity. In this chapter we apply holography to studythe spontaneous breaking of translation invariance and the formation ofstriped order.Stripes are known to form in a variety of strongly coupled systems, fromlarge N QCD [68, 69] to systems of strongly correlated electrons (for a reviewsee [29]). The formation of stripes and the associated reduced dimensionalityare speculated to be related to the mechanism of superconductivity in thecuprates [70]. It is therefore useful to study striped phases in the holographiccontext.Besides its interest in the boundary theory, this study has an intrinsicinterest in the bulk gravitational context.2 We describe striking bulk andboundary properties of our bulk solutions, including frame dragging effects,the magnetic field, the curvature and the geometry. Some of the featurescan be understood as the emergence of a near horizon region which acts asa bulk topological insulator. The magnetoelectric effect is then responsiblefor the patterns we observe for the bulk magnetic field and vorticity.Our study is facilitated by a numerical solution of the set of couplednonlinear Einstein and matter equations in the bulk, which exhibit a nor-1A version of this chapter has been published [2]. This chapter presents a conciseversion of the study of holographic stripes while chapter 4 provides full results and thecomplete details of the analysis.2Our model describes a black hole whose instability to the formation of inhomogeneousstructures resembles the black string instability [71] which is known to be of the secondorder for high enough dimensions [72].583.1. Introduction and summarymalizable inhomogeneous mode. Previous studies of inhomogeneous solu-tions in asymptotically AdS spacetimes concentrated on non-normalizablemodes [73] (i.e. explicit rather than spontaneous breaking of translation in-variance) or the study of co-homogeneity one solutions [74–77], where one ofthe translational Killing vectors is replaced by a helical Killing vector. Morerecently, such spontaneous breaking was exhibited in a probe model, whichwas shown to have a magnetic field induced lattice ground state [78]. Incontrast to the above, our solutions are co-homogeneity two, they backreacton the geometry, and exhibit spontaneous breaking of translation invariancebelow a critical temperature.These features are analyzed as a function of temperature. In particular,we find that the horizon of the black hole develops a ‘neck’ and a ‘bulge’ inthe transverse direction which shrink with temperature, such that the ratioof their sizes contracts as fast as ∼ T σ, with an order σ ∼ 0.1 exponent.Simultaneously, the proper length of the horizon in the transverse directiongrows at a rate ∼ 1/T 0.1. However, the curvature remains finite, and itsmaximal value, occurring at the bulge, tends to a constant in the limitT → 0.The bulk black hole solutions give rise to the holographic stripes onthe boundary, characterized by non vanishing momentum and electric cur-rent and modulations in charge and mass density. Starting small near Tc,the amplitudes of the modulations grow steadily at lower temperatures, ap-proaching finite values at T → 0.Finally, we study the thermodynamics of the system by constructingphase diagrams in various ensembles. For small values of the axion coupling,where the thermodynamic potentials in both phases are nearly degenerate,our numerical method is not accurate enough to sharply distinguish betweenweak first order and second order transitions. However, for sufficiently largevalues of the axion coupling we discover a clear second order phase transitionin the canonical (fixed charge), the grand canonical (fixed chemical poten-tial) and the micro-canonical ensembles. We describe both the finite system(of fixed length) and the infinite system, where we find that the dominantstripe width changes as function of temperature.593.2. The holographic setup3.2 The holographic setupThe Lagrangian describing our coupled system is [30]L = 12R−12∂µψ∂µψ −14FµνFµν − V (ψ)− Lint,V (ψ) = −6 + 12m2ψ2,Lint =1√−gc116√3ψ µνρσFµνFρσ, (3.1)where R is the Ricci scalar, Fµν is the Faraday tensor, Lint describes theaxion coupling and g is the determinant of the metric. We use units in whichthe AdS radius l2 = 1/2, Newton’s constant 8piGN = 1, and c = ~ = 1, andchoose m2 = −4 and several values of c1.Perturbative instabilities towards the formation of charge and currentdensity waves were identified in [30] for a range of wave numbers and tem-peratures.3 We note the appearance of axion electrodynamics in the bulktheory. It is curious that here, as in several examples of inhomogeneousinstabilities (see also [80]), the topology of the bulk fields seems to play animportant role, though the analysis performed to discover the instability islocal in nature.4In this chapter we investigate the end-point of the instability. Part ofthe boundary data is the spatial periodicity, and we focus mostly on thewave number with the largest critical temperature Tc [30]. This state isa co-homogeneity two solution, thus we construct the family of stationarysolutions that emerge from the critical point, assuming all the fields to befunctions of the radial coordinate r and one spatial coordinate x.Our ansatz includes the scalar field ψ(r, x), the gauge field componentsAt(r, x) and Ay(r, x) and the metricds2 = −2r2f(r)e2A(r,x)dt2 + 2r2e2C(r,x)(dy −W (r, x)dt)2+ e2B(r,x)( dr22r2f(r) + 2r2dx2), (3.2)where for the sake of convenience we included in the definition of the metricfunctions the factor f(r) characterizing the metric of the AdS Reissner-3An interesting application of the instability in this model has appeared very recently[79].4It is not generic, however, that the topology of the bulk fields is essential for inho-mogeneous instabilities. See [81] for an example system whose bulk does not involve anaxion.603.2. The holographic setupNordstro¨m (RN for short) solution, with horizon at r = r0:f(r) = 1−(1 + µ24r20)(r0r)3+ µ24r20(r0r)4.The inhomogeneous solutions reduce to the RN solution above the criticaltemperature.The conformal in r, x plane ansatz (3.2) is convenient in constructingco-homogeneity two solutions. With this ansatz, the Einstein and mat-ter equations reduce to seven coupled elliptic equations and two constraintequations. Moreover, the constraint system can be solved elegantly usingits similarity to a Cauchy-Riemann problem [82].The boundary conditions we impose correspond to regularity conditionsat the horizon and asymptotically AdS conditions at the conformal bound-ary. With these boundary conditions, the set of solutions we find dependon three parameters: the temperature T , the chemical potential µ and theperiodicity in the x direction L. Using the conformal symmetry inherent inasymptotically AdS spaces, the moduli space of solution depends only onthe two dimensionless combinations of these parameters. To focus on thedominant critical mode that becomes unstable at the largest temperatureTc we choose L = 2pi/kc.On the spatial boundaries it is useful to impose ‘staggered’ periodicityconditions. Using two reflection symmetries which are preserved by theform of the unstable perturbation, one can reduce the numerical domain toa quarter period and impose5 ∂xψ(x = 0) = 0, ψ(x = L/4) = 0, h(x = 0) =0, ∂xh(x = L/4) = 0 and ∂xg(x = 0) = 0, ∂xg(x = L/4) = 0, where hrepresents the fields Ay and W , and g refers collectively to A,B,C and At.The elliptic equations derived from (3.1) are discretized using finite dif-ference methods and are solved numerically by a straightforward relaxationwith the specified boundary conditions. In this method the equations areiterated starting with an initial guess for all fields, until successive changesin the functions drop below the desired tolerance. We verify that the re-maining two constraints are satisfied by those solutions. More details of thisnumerical procedure are given in chapter 4 and appendix A.2.5Our boundary conditions do not exclude the homogeneous solution, but since thatsolution is unstable we find that in practice our numerical procedure converges to theinhomogeneous solution unless we are very close to the critical point.613.3. The solutionsFigure 3.1: Metric functions for θ ' 0.11 and c1 = 4.5. Note that the metricfunctions A,B and C have half the period of W . The variation is maximalnear the horizon, located at ρ = 0, and it decays as the conformal boundaryis approached, when ρ → ∞. The matter fields (not shown) behave in aqualitatively similar manner.3.3 The solutionsA convenient way to parametrize our inhomogeneous solutions is by thedimensionless temperature θ = T/Tc, relative to the critical temperatureTc. Our method allows us to find solutions in the range 0.003 . θ . 0.9 forc1 = 4.5 and the range 0.00016 . θ . 0.96 for c1 = 8, for fixed µ.Bulk Geometry. For subcritical temperatures, as we descend into theinhomogeneous regime, the metric and the matter fields start developingincreasing variation in x. Figure 3.1 displays the metric functions for θ '0.11, over a full period in the x direction, in the case c1 = 4.5. The matterfields have qualitatively similar behaviour. The variation of all fields ismaximal near the horizon of the black hole at ρ ≡√r2 − r20 = 0, and itgradually decreases toward the conformal boundary, ρ→∞.Many of the special features of the solutions we find are related to thepresence of axion electrodynamics, the effective description of the electro-magnetic response of a topological insulator, in the gravity action. In thebroken phase we have an axion gradient in the near horizon geometry, which623.3. The solutionstherefore realizes a topological insulator interface.6 The presence and thepattern of a near horizon magnetic field, summarized in the field Ay, can berelated to the magnetoelectric effect in such interfaces.In curved space the magnetic field is accompanied by vorticity, whichis manifested by the function W . This causes frame dragging effects in they direction. Test particles will be pushed along y with speeds W (r, x), inparticular the direction of the flow reverses every half the period along x.The drag vanishes at the horizon and at the location of the nodes of W wherex = Ln/2, for integer n (see Figure 3.1). In general, the dragging effectremains bounded, the vector ∂t is everywhere timelike, and no ergoregionforms.The Ricci scalar of the RN solution is RRN = −24, constant in r andindependent of the parameters of the black hole. This is no longer true forthe inhomogeneous phases, where the Ricci scalar becomes position depen-dent. The right panel of Figure 3.2 illustrates the spatial variation of theRicci scalar, relative to the RRN for θ ' 0.003. The plot corresponds toc1 = 4.5, however we observe qualitatively similar results for other values ofthe coupling.The maximal curvature is always along the horizon at x = nL/2 forinteger n. It grows when the temperature decreases and approaches thefinite value of R ' −94 in the small temperature limit.The left panel in Figure 3.2 shows the variation of transverse extentof the horizon in the y direction, ry(x) ≡√2 r0 exp[C(r0, x)], along x forθ ' 0.003. Typically there is a ‘bulge’ occurring at x = nL/2 and a ‘neck’ atx = (2n+1)L/4, for integer n. Note that Ricci scalar curvature is maximalat the bulge and not at the neck as would happen, for instance, in the spher-ically symmetric black string case. The size of both the neck and the bulgemonotonically decrease with temperature, however, the neck is shrinkingfaster. We find that the ratio scales as a power law rnecky /rbulgey ∼ θσ nearthe lower end of the range of θ’s that we investigated. The exponent σ de-pends on the coupling, ranging from about 0.5 for c1 = 4.5 to approximately0.1 for c1 = 8.Another aspect of the geometry is the proper size of the stripe in the x di-rection at fixed r, lx(r) ≡∫ L0 exp[B(r, x)] dx. The proper length tends to thecoordinate length as 1/r3 asymptotically as r → ∞, but it exceeds that asthe horizon is approached. Namely, the inhomogeneous phase ‘pushes space’around it along x, resembling the ‘Archimedes effect’. The proper length of6It would be interesting to discuss localized matter excitations on the interface, espe-cially fermions, along the lines of [83].633.3. The solutionsFigure 3.2: Left panel: The variation along x of the size of the horizon inthe y direction includes alternating ‘necks’ and ‘bulges’. Right panel: Ricciscalar relative to that of RN black hole, R/RRN − 1 for θ ' 0.003 over halfthe period. The scalar curvature is maximal along the horizon at the bulgex = nL/2 for integer n. The axion coupling here is c1 = 4.5 and similarresults appear for other c1’s.the horizon is maximal and it grows as the temperature decreases. We findthat at small θ the proper length of the horizon diverges approximately as∼ θ−0.1.Boundary Observables. Near the conformal boundary the fields decayto their AdS values, and the subleading terms in their variation are usedto define the asymptotic charge densities of our solutions. The subleadingfall-offs of the metric functions in our ansatz determine the boundary stress-energy tensor, whereas the fall-offs of the gauge field determine the chargeand current densities of the boundary theory. Finally, the subleading termof the scalar field near infinity determines the scalar condensate.For our inhomogeneous solutions we find that all charge and current den-sities are spatially modulated, except for 〈Txx〉, which is constant, consistentwith the conservation of boundary energy-momentum. We define the totalcharges of a single stripe by integrating the charge densities over the fullperiod L. These integrated quantities are charge densities per unit lengthin the translationally invariant direction y.643.4. Thermodynamics-0.3-0.2-0.10.0HF-FRNLN20.0 0.2 0.4 0.6 0.8 1.0TTc-0.04-0.020.00HW-WRNLΜ2Figure 3.3: Difference in the thermodynamic potentials between the inho-mogeneous phase and the RN solution for c1 = 8, plotted against the tem-perature. In both ensembles there is a second order phase transition, withthe inhomogeneous solution dominating below the critical temperature.3.4 ThermodynamicsWe demonstrated that below the critical temperature Tc there exists a newbranch of solutions which are spatially inhomogeneous. The question ofwhich solution dominates the thermodynamics depends on the ensembleused. We start our discussion by fixing the boundary periodicity, corre-sponding to working in a finite system of length L.7 We discuss the systemwith infinite length in the inhomogeneous x-direction below.The canonical ensemble corresponds to fixing the temperature and thetotal charge. This describes the physical situation in which the system isimmersed in a heat bath consisting of uncharged particles. In the upperpanel of Figure 3.3 we plot the difference of the normalized total free en-ergy, F = M − TS, between the two classical solutions as function of thetemperature T , for c1 = 8. In our ensemble the total charge N is fixed, andwe use the scaling symmetry of the boundary theory to set N = 1, or inother words measure all quantities in terms of N . As a result the free energyis a function of one parameter, the temperature T . The figure displays asecond order phase transition, where the inhomogeneous solution dominates7Here, we mostly discuss the case L = 2pi/kc, where kc is the wavelength of thedominant instability, that with the highest critical temperature. Results for other valuesof L appear in chapter 4, and are qualitatively similar.653.4. ThermodynamicsExtremal RN0.72 0.74 0.76 0.78 0.80 0.82 0.84MN20.00.51.01.52.02.5SNFigure 3.4: The entropy of the inhomogeneous solution for c1 = 8 (pointswith dotted line) and of the RN solution (solid line). Below the criticaltemperature, the striped solution has higher entropy than the RN. The RNbranch terminates at the extremal RN black hole, while the striped solutionpersists to smaller energies.the thermodynamics below the critical temperature Tc, the temperature atwhich inhomogeneities first develop.If we fix the chemical potential instead of the charge, we discuss a sit-uation where the system is immersed in a plasma made of charged parti-cles. To study the thermodynamics we use the grand canonical free energyΩ = M−TS−µN , displayed in the lower panel of Figure 3.3. In this ensem-ble it is convenient to measure all quantities in units of the fixed chemicalpotential µ. Then, again, the free energy is a function of only the temper-ature T . In the fixed chemical potential ensemble we find a similar secondorder transition, where the inhomogeneous charge distribution starts dom-inating the thermodynamics at the temperature where the inhomogeneousinstability develops.The physical situation relevant to the study of the real time dynam-ics of the instability corresponds to fixing the mass and the charge. Thisis the microcanonical ensemble, describing an isolated system in which allconserved quantities are fixed. In this ensemble it is convenient to measureall quantities in terms of the (fixed) charge, and the remaining control pa-rameter is then the mass M . We find that in this ensemble as well, thestriped solutions dominate the thermodynamics (have higher entropy) forall temperature below the critical temperature Tc, at least when the axion663.4. Thermodynamics1.0 1.2 1.4 1.6 1.8 2.0 2.20.000.020.040.060.080.10L̐4TΜHΩ-ΩRNLΜ3-0.04-0.03-0.02-0.010.00Figure 3.5: A contour plot of the free energy density, relative to the ho-mogenous solution. The red line shows the variation of the dominant stripewidth as function of the temperature for c1 = 8.coupling c1 is sufficiently large. This is shown in Figure 3.4.Finally, we can also study the infinite system in the inhomogeneous x-direction, which we choose to look at in the canonical ensemble. In this casewe are in a position to compare the free energy density of different stripes, ofdifferent lengths in the x-direction. This comparison is shown in Figure 3.5,where we see that the qualitative picture is the same as in the finite sys-tem – a second order transition with striped solutions dominating at everytemperature below the critical temperature. Just below the critical temper-ature, the dominant stripe is that corresponding to the critical wavelengthkc. However, for lower temperature different stripes will dominate, in factwe see in Figure 3.5 that the dominant stripe width tends to increase withdecreasing temperature.67Chapter 4Striped order in theAdS/CFT correspondence14.1 Introduction and summaryThe gauge / gravity duality is a relationship between a strongly coupled fieldtheory and a gravity system in one higher dimension. This correspondencehas been fruitful in studying various field theory phenomena by translatingthe problem to the gravitational context. In particular, the duality hasshone new light on many condensed matter systems - see [18, 21, 84, 85] forreviews.Early models in this area, such as the holographic superconductor [11],focused on homogeneous phases of field theories. In this case, the fields onthe gravity side depend only on the radial coordinate in the bulk and theproblem reduces to the solution of ordinary differential equations. However,many interesting phenomena occur in less symmetric situations. Generi-cally, the problem of finding the gravity dual to an inhomogeneous bound-ary system will necessitate solving relatively more difficult partial differentialequations, almost always resulting in the need for numerical methods. Whilethese become technically hard problems, there exist established numericalapproaches. Due to the success of the holographic method in studying ho-mogeneous situations, it is worthwhile to push the correspondence to theseless symmetric situations in order to describe more general phenomena inthis context.One particular area of condensed matter that appears to be amenableto a holographic description is the appearance of striped phases in certainmaterials.2 These phases are characterized by the spontaneous breakingof translational invariance in the system. Examples include charge densitywaves and spin density waves in strongly correlated electron systems, where1A version of this chapter has been published [3]. A concise presentation of this studyis given in chapter 3.2Stripes are also known to form in large-N QCD [68, 69].684.1. Introduction and summaryeither the charge and/or the spin densities become spatially modulated (fora review see [29]). The formation of stripes is conjectured to be related tothe mechanism of superconductivity in the cuprates [70]. To approach thisstriking phenomenon from the holographic perspective, one would look foran asymptotically AdS gravity system which allows a spontaneous transitionto a modulated phase.Recently, several interesting spatially modulated holographic systemshave been studied. One way to study stripes on the boundary is to sourcethem by imposing spatial modulation in the non-normalizable modes of somefields, explicitly breaking the translation invariance, as in [73, 86].3 How-ever, if one wishes to make contact with the context described above, itis important that the inhomogeneity emerges spontaneously rather than beintroduced explicitly.In some cases, the spatially modulated phase has an extra symmetry,allowing the situation to be posed as a co-homogeneity one problem on thegravity side. Examples include systems in which one of the translationalKilling vectors is replaced by a helical Killing vector [74–76, 80, 90, 91].More general inhomogeneous instabilities, in which one of the translationsymmetries is fully broken, have been described in phenomenological model[30, 92] and in certain #ND = 6 brane systems [93–95].4In this chapter, we study the full non-linear co-homogeneity two stripedsolutions to the Einstein-Maxwell-axion model that stem from the normal-izable, inhomogeneous modes of the Reissner-Nordstro¨m-AdS solution de-tailed in [30]. In this model, below a critical temperature, stripes sponta-neously form in the bulk and on the boundary. We study the propertiesof the stripes in both the fixed length system, in which the wavenumber isset by the size of the domain and charges are integrated over the stripe,and the infinite system, in which the corresponding thermodynamic den-sities are studied. For the black hole at fixed length, we examine the be-haviour in different thermodynamic ensembles as we vary the temperatureand wavenumber.The study is facilitated by a numerical solution to the set of coupledEinstein and matter equations in the bulk. Inspired by the black stringcase [82, 98], we fix the metric in the conformal gauge, resulting in a setof field equations and a set of constraint equations. Then, as described in[82], the resulting constraint equations can be solved by imposing particular3In a similar vein, more recently, lattice-deformed black branes have been of interestin studies of conductivity in holographic models [79, 87–89].4Other studies of inhomogeneity in the context of holography include [78, 83, 96, 97].694.1. Introduction and summaryboundary conditions on the fields.As well as being of interest from the holographic perspective these nu-merical solutions are important as they represent new inhomogeneous blackhole solutions in AdS. We find strong evidence that the unstable homo-geneous branes transition smoothly to the striped state below the criticaltemperature.5 As we approach zero temperature the relative inhomogeneityis seen to grow without bound and the black hole horizon tends to pinch off,signalling the formation of a spacetime singularity in this limit.A subset of our results has already been reported in chapter 3; in thischapter we provide full details. The summary of the results follow:Boundary field theory• We calculate the fully back-reacted normalizable inhomogeneous modes.• The stripes have momentum, electric current and modulations in chargeand mass density (see [100] for a recent study of angular momentumgeneration).• As a function of temperature, the modulations start small, then growand saturate as T → 0.• We study the stripe of fixed length in various ensembles, finding asecond order phase transition, for sufficiently large axion coupling,in each of the grand canonical (temperature T , chemical potential µfixed), canonical (T , charge N fixed) and microcanonical (mass M , Nfixed) ensembles. We compute corresponding critical exponents.• For the infinite length system, there is a second order transition to astriped phase. The width of the dominant stripe grows as the temper-ature is decreased.• In the zero temperature limit, within the accuracy of our numerics,the entropy appears to approach a non-zero value.Bulk geometryThe new inhomogeneous black brane solutions that we find have peculiarfeatures, including5The instability to the formation of the striped black branes resembles the black stringinstability [71] which is known to be of the second order for high enough dimensions[72, 99].704.2. Numerical set-up: Einstein-Maxwell-axion model• The inhomogeneities are localized near the horizon, and die off asymp-totically following a power law decay.• The phenomena of vorticity, frame dragging and the magneto-electriceffect similar to one produced by a near horizon topological insulatorare observed.• The inhomogeneous black brane has a neck and a bulge. In the cur-vature at the horizon, the maximum is at the bulge. In the limit ofsmall temperatures, the neck shrinks to zero size.• The proper length of the horizon grows when temperature is decreas-ing, and diverges as 1/T 0.1 in the limit T → 0. The proper length inthe stripe direction increases from the boundary to the horizon, whichcan be thought of as a manifestation of an ‘Archimedes effect’.In section 4.2, we define our model and set up our numerical approach,describing our ansatz, boundary conditions and solving procedure. Then,in section 4.3, we report on interesting geometrical features of the bulk so-lutions. Section 4.4 studies the solutions at fixed length from the point ofview of the boundary theory. There, we make the comparison to the homo-geneous solution and find a second order transition, in addition to describingthe observables in the theory. In section 4.5, we relax the fixed length con-dition and find the striped solution that dominates the thermodynamics forthe infinite system. Appendix B.1 provides details about computing theobservables of the inhomogeneous solutions while appendix B.2 gives moredetails on the numerics, including checks of the solutions and validations ofour numerical method.Note added: As the manuscript that forms the basis for this chapterwas being completed, [101] and [102, 103] appeared, which use a differentmethod and have some overlap with this work.4.2 Numerical set-up: Einstein-Maxwell-axionmodelIn [30], perturbative instabilities of the Reissner-Nordstro¨m-AdS (RN forshort) black brane were found within the Einstein-Maxwell-axion model. In[2] and here, we construct the full non-linear branch of stationary solutionsfollowing this zero mode.714.2. Numerical set-up: Einstein-Maxwell-axion model4.2.1 The model and ansatzThe Lagrangian describing our coupled system can be written as [30]L = 12(R+12)−12∂µψ∂µψ−12m2ψ2− 14FµνFµν−1√−gc116√3ψ µνρσFµνFρσ,(4.1)where R is the Ricci scalar, Fµν is the Faraday tensor, ψ is a pseudo-scalarfield and g is the determinant of the metric. We use units in which the AdSradius l2 = 1/2, Newton’s constant 8piGN = 1 and c = ~ = 1, and choosem2 = −4. The constant c1 controls the strength of the axion coupling.For this choice of scalar field mass, instabilities exist for all choices ofc1. For c1 = 0, the instability is towards a black hole with neutral scalarhair. For c1 > 0, inhomogeneous instabilities along one field theory directionexist for a range of wavenumbers k. The critical temperature at which eachmode becomes unstable depends on the wavenumber: Tc(k). For a given c1,there is a maximum critical temperature, above which there are no unstablemodes. As one increases c1, the critical temperature of a given mode kincreases, such that for a fixed temperature a larger range of wavenumberswill be unstable. See appendix B.2.1 for more details on the perturbativeanalysis.One may consider generalizations of this action, including higher ordercouplings between the scalar field and the gauge field. In particular, asdiscussed in [30], generalizing the Maxwell term as − τ(ψ)4 FµνFµν , whereτ(ψ) is a function of the scalar field, results in a model that can be upliftedto a D = 11 supergravity solution (for particular choices of c1, m, andthe parameters in τ(ψ)). In this study, we wish to study the formation ofholographic stripes phenomenologically. The existence of the axion-couplingterm (c1 6= 0) is a sufficient condition for the inhomogeneous solutions andso we set τ(ψ) = 1 here.We are looking for stationary black hole solutions that can be describedby an ansatz of the formds2 = −2r2f(r)e2A(r,x)dt2 + 2r2e2C(r,x)(dy −W (r, x)dt)2+ e2B(r,x)( dr22r2f(r) + 2r2dx2),ψ = ψ(r, x), A = At(r, x)dt+Ay(r, x)dy, (4.2)where r is the radial direction in AdS and x is the field theory directionalong which inhomogeneities form. We term the scalar field and gauge fields724.2. Numerical set-up: Einstein-Maxwell-axion modelcollectively as the matter fields. f(r) is a given function whose zero definesthe black brane horizon. We take f(r) to be that of the RN solution,f(r) = 1−(1 + µ24r20)(r0r)3+ µ24r20(r0r)4, (4.3)so that the horizon is located at r = r0. The homogeneous solution is theRN black brane, given byA = B = C = W = ψ = Ay = 0, At(r) = µ(1− r0/r), (4.4)where µ is the chemical potential. Above the maximum critical temperature,this is the only solution to the system.To find the non-linear inhomogeneous solutions, we numerically solve theequations of motion derived from the ansatz (4.2). The Einstein equationresults in four second order elliptic equations, formed from combinations ofGtt−T tt = 0, Gty−T ty = 0, Gyy−T yy = 0, and Grr+Gxx−(T rr +T xx ) = 0, and twohyperbolic constraint equations, Grx−T rx = 0 and Grr−Gxx−(T rr−T xx ) = 0, forthe metric functions. The gauge field equations and scalar field equation givesecond order elliptic equations for the matter fields. For completeness, thefull equations are given in appendix B.2.2. Our strategy will be to solve theseseven elliptic equations subject to boundary conditions that ensure that theconstraint equations will be satisfied on a solution. Below, we describe theconstraint system and our boundary conditions. For more details about thenumerical approach, we refer to appendix B.2.4.2.2 The constraintsThe two equations Grx−T rx = 0 and Grr−Gxx−(T rr −T xx ) = 0, which we do notexplicitly solve, are the constraint equations. Using the Bianchi identities[82], we see that the constraints satisfy∂x(√−g(Grx − T rx ))+ 2r2√f∂r(r2√f√−g(Grr −Gxx − (T rr − T xx )))= 0,(4.5)2r2√f∂r(√−g(Grx − T rx ))− ∂x(r2√f√−g(Grr −Gxx − (T rr − T xx )))= 0.(4.6)Defining rˆ by ∂rˆ = 2r2√f∂r gives Cauchy-Riemann relations∂x(√−g(Grx − T rx ))+ ∂rˆ(r2√f√−g(Grr −Gxx − (T rr − T xx )))= 0, (4.7)734.2. Numerical set-up: Einstein-Maxwell-axion model∂rˆ(√−g(Grx − T rx ))− ∂x(r2√f√−g(Grr −Gxx − (T rr − T xx )))= 0, (4.8)showing that the weighted constraints satisfy Laplace equations. Then, sat-isfying one constraint on the entire boundary and the other at one pointon the boundary implies that they will both vanish on the entire domain.In practice we will take either zero data or Neumann boundary conditionsat the boundaries in the x-direction. The unique solution to Laplace’sequation with zero data on the horizon and the boundary at infinity andthese conditions in the x-direction is zero. Therefore, as long as we fulfillone constraint at the horizon and the asymptotic boundary and the otherat one point (on the horizon or boundary), the constraints will be satis-fied if the elliptic equations are. Our boundary conditions will be suchthat √−g(Grx − T rx ) = 0 at the horizon and conformal infinity and thatr2√f√−g(Grr −Gxx − (T rr − T xx )) = 0 at one point on the horizon.4.2.3 Boundary conditionsThe elliptic equations to be solved are subject to physical boundary condi-tions. There are four boundaries of our domain (see Figure 4.1): the hori-zon, the conformal boundary, and the periodic boundaries in the x-direction,which are described next.Staggered periodicityTo specify the boundary conditions in the x direction we look at the formof the linearized perturbation which becomes unstable (see appendix B.2.1).To leading order in the perturbation parameter λ, they are of the form:ψ(x) ∼ λ cos(kx),Ay(x) ∼ λ sin(kx),gty(x) ∼ λ sin(kx), (4.9)where k is the wavenumber of the unstable mode. To second order in theperturbation parameter, the functions gtt, gxx, gyy and At (which we denotecollectively as h) are turned on, with the schematic behaviourh(x) ∼ λ2(cos(2kx) + C), (4.10)where C are independent of x.744.2. Numerical set-up: Einstein-Maxwell-axion model6-regularity,pgGrx = 0A,B,C,W / 1r3 ,At  µ,Ay / 1r , / 1r2 ,pgGrx = 0 = @xAy = @xgty = @xh = 0@x = Ay = gty = @xh = 0rr = rcutr = r0xx = L4x = 0Figure 1: A summary of the boundary conditions on our domain. At the horizon,r = r0, we impose regularity conditions. At the conformal boundary, r ! 1, wehave fall o↵ conditions on the fields (imposed at large but finite r = rcut) such that wedo not source the inhomogeneity. In the x-direction, we use symmetries to reduce thedomain to a quarter period L/4. Then, we impose either periodic or zero conditionson the fields, according to their behavior under the discrete symmetries discussed inthe text. (h collectively denotes the fields {gtt, gxx, gyy, At}.) In addition to these,we explicitly satisfy the constraint equation pgGrx = 0 on the horizon and theconformal boundary.where k is the wavenumber of the unstable mode. To second order in the perturbationparameter, the functions gtt, gxx, gyy and At (which we denote collectively as h) areturned on, with the schematic behaviorh(x) ⇠ 2(cos(2kx) + C), (2.10)where C are independent of x.All these functions are periodic with period L = 2⇡k . However, they are notthe most general periodic functions with period L. For numerical stability it isworthwhile to specify their properties further and encode those properties in theboundary conditions we impose on the full solution. We concentrate on the behaviorof the perturbation with respect to two independent Z2 reflection symmetries.8Figure 4.1: A summary of the boundary conditions on our domain. Atthe orizon, r = r0, we impo e regularity conditions. At the conformalboundary, r →∞, we have fall off con itions on the fields (imposed at largebut finite r = rcut) such that we do not source the inhomogeneity. In the x-direction, we use symmetries to reduce the domain to a quarter period L/4.Then, we impo e either periodic or zero conditions on fields, ccordingto their behaviour under the discrete symmetries discussed in the text. (hcollectively denotes the fields {gtt, gxx, gyy, At}.) In addition to these, weexplicitly satisfy the constraint equation √−gGrx = 0 on the horizon andthe conformal boundary.All these functions are periodic with period L = 2pi/k. However, theyare not the most general periodic functions with period L. For numeri-cal stability it is worthwhile to specify their properties further and encodethose properties in the boundary conditions we impose on the full solution.We c ncentrate on the behaviour of the perturbation wi h respect to twoindependent Z2 reflection symmetries.The first Z2 symmetry is that of x → −x, y → −y, which is a rotationin the x, y plane. This is a symmetry of the action and of the linearizedperturbation (keeping in mind that Ay and gty change sign under reflectionof the y coordinate). We conclude therefore that this is a symmetry of thefull solution.Similarly, the Z2 operation x → L2 − x, y → −y is a symmetry of theaction, which is also a symmetry of the linearized system when accompanied754.2. Numerical set-up: Einstein-Maxwell-axion modelby λ→ −λ. In other words the functions ψ,Ay, gty are restricted to be oddwith respect to this Z2 operation, while the rest of the functions, which wecollectively denoted as h, are even.The two symmetries defined here restrict the form of the functions thatcan appear in the perturbative expansions for each of the functions above.For example, it is easy to see that the function ψ(x) gets corrected only inodd powers of λ and the most general form of the harmonic that can appearin the perturbative expansion is cos(nkx), for n odd. Similar commentsapply to the other functions above.We restrict ourselves to those harmonics which may appear in the fullsolution. The most efficient way to do so is to work with a quarter ofthe full period L (reconstructing the full periodic solution using the knownbehaviour of each function with respect to the two Z2 operations definedabove). The specific properties of each function appearing in our solutionsare imposed by demanding the following boundary conditions:∂xψ(x = 0) = 0, ψ(x = L4)= 0,Ay(x = 0) = 0, ∂xAy(x = L4)= 0,gty(x = 0) = 0, ∂xgty(x = L4)= 0,∂xh(x = 0) = 0, ∂xh(x = L4)= 0. (4.11)At the horizonIn our coordinates (4.2) the horizon is at fixed r = r0. For numerical con-venience we introduce another radial coordinate ρ =√r2 − r20, such thatthe horizon is at ρ = 0.6 Expanding the equations of motion around ρ = 0yields a set of Neumann regularity conditions,∂ρA = ∂ρC = ∂ρW = ∂ρψ = ∂ρAt = ∂ρAy = 0, (4.12)and two conditions in the inhomogeneous direction along the horizon,∂xW = ∂x(At +WAy) = 0. (4.13)Thus, bothW and the combination At+WAy are constant along the horizon.The boundary conditions in the x direction (4.11) imply that W = 0. Then,6In the rest of the paper, we use r and ρ interchangeably as our radial coordinate. Weuse the coordinate ρ in the numerics.764.2. Numerical set-up: Einstein-Maxwell-axion modelthe second condition together with regularity of the vector field A on theEuclidean section give that At = 0 on the horizon.The regularity conditions give eight conditions for the six functionsA,C,W,ψ,At and Ay. In principle, we would choose any six of these toimpose at the horizon. If we find a non-singular solution to the equations,then the other two conditions should also be satisfied. In practice, some ofthese conditions work better than others for finding the numerical solution.We find that using Neumann conditions for A,C, ψ, and Ay and Dirichletconditions for W and At results in a more stable relaxation.7The conditions for B are determined using the constraint equations.Expanding the weighted constraints at the horizon, we find√−g(Grx − T rx ) ∝ ∂x(A−B) +O(ρ), (4.14)r2√f√−g(Grr −Gxx − (T rr − T xx )) ∝ ∂ρB +O(ρ). (4.15)The first condition gives constant surface gravity (or temperature) along thehorizon. As discussed above, we will impose one constraint at the horizonand the boundary, and the other at one point. In practice, we will satisfyr2√f√−g(Grr−Gxx−(T rr −T xx )) at (ρ, x) = (0, 0), updating the value of B atthis point using the Neumann condition ∂ρB = 0. This will set the difference(B−A)|(ρ,x)=(0,0) ≡ d0, which we will then use to update B using a Dirichletcondition along the rest of the horizon, satisfying √−g(Grx − T rx ) = 0.At the conformal boundaryIn our coordinates, the boundary is at r = ∞. Since we are looking forspontaneous breaking of homogeneities, our boundary conditions will besuch that the field theory sources are homogeneous. This implies that thenon-normalizable modes of the bulk fields are homogeneous. The inhomo-geneity of the striped solutions will be imprinted on the normalizable modesof the fields, or the coefficient of the next-to-leading fall-off term in theasymptotic expansions.The form of our metric ansatz is such that the metric functions A,B,Cand W represent the normalizable modes of the metric. Imposing that thegeometry is asymptotically AdS with Minkowski space on the boundaryimplies that these four metric perturbations must vanish as r → ∞. Byexpanding the equations of motion near the boundary, one can show thatA,B,C and W fall off as 1/r3. In practice, we place the outer boundary ofour domain at large but finite rcut and impose the fall-off conditions there.7Using Neumann conditions at the horizon for W and At results in values at the horizonthat converge to zero with step-size, consistent with the above analysis.774.2. Numerical set-up: Einstein-Maxwell-axion modelAs in the RN solution, we source the field theory charge density witha homogeneous chemical potential, corresponding to a Dirichlet conditionfor the gauge field At at the boundary. In the inhomogeneous solutions,we expect the spontaneous generation of a modulated field theory currentjy(x), dual to the normalizable mode of Ay. Solving the equations near theboundary with these conditions reveals the expansions At = µ+O(1/r) andAy = O(1/r), which we impose numerically at rcut.The scalar field equation of motion gives the asymptotic solutionψ = ψ(1)rλ− +ψ(2)rλ+ + . . . , (4.16)whereλ± =12(3±√9 + 4(lm)2). (4.17)For the range of scalar field masses −9/2 ≤ m2 ≤ −5/2, both modes arenormalizable, and fixing one mode gives a source for the other. In our studywe will choose m2 = −4, giving λ− = 1, λ+ = 2. Since we are looking forspontaneous symmetry breaking, in this case we must choose either ψ(1) = 0or ψ(2) = 0. We choose the former, so that ψ falls off as 1/r2.Now, consider the weighted constraint √−gGrx. As discussed above, inorder to solve the constraint system, we require this to disappear at theconformal boundary. Near the boundary, √−g ∝ r2 + . . . , so for √−gGrx todisappear we must have Grx = O(1/r3). Expanding the equations near theboundary we haveGrx − T rx ∝3∂xA(3)(x) + 2∂xB(3)(x) + 3∂xC(3)(x)r2 +O( 1r3), (4.18)where X = X(3)(x)/r3 + . . . for X = {A,B,C}. Therefore, in addition tothe boundary conditions mentioned above, for √−gGrx = 0 to be satisfied atr =∞, it appears that we should have that 3A(3)(x)+2B(3)(x)+3C(3)(x) =const. The means to impose this addition condition comes from the factthat our metric (4.2) has an unfixed residual gauge freedom [104], allow-ing one to transform to new r˜ = r˜(r, x), x˜ = x˜(r, x) coordinates which areharmonic functions of r and x. Performing such a transformation generatesan additional function in (4.18), which can then be chosen to ensure thatthe constraint is satisfied (in appendix B.2 we describe how). This condi-tion implies the conservation of the boundary energy momentum tensor, seeappendix B.1.784.2. Numerical set-up: Einstein-Maxwell-axion model4.2.4 Parameters and algorithmThe physical data specifying each solution is the chemical potential µ, thetemperature T , and the periodicity L.8 Since the boundary theory is confor-mal, it will only depend on dimensionless ratios of these parameters. Thismanifests itself in the following scaling symmetry of the equations:r → λr, (t, x, y)→ 1λ(t, x, y), Aµ → λAµ. (4.19)We use this to select µ = 1. Then, our results are functions of the dimen-sionless temperature T/µ and the dimensionless periodicity Lµ.The temperature is controlled by the coordinate location of the horizon.For a given r0, the temperature of the RN phase is T0 = (1/8pir0)(12r20 −1) while the temperature of the inhomogeneous solution is T = e−d0T0.Recall that (B − A)|r0 = d0 is dynamically generated by satisfying theconstraints at the horizon. From our numerical solutions, we find that d0monotonically increases as we lower the temperature, so that T0 gives areliable parametrization of the physical temperature T . In practice, wegenerate solutions by choosing values of T0 below the critical temperatureTc(k).We solve the equations by finite-difference approximation techniques. Weuse second order finite-differencing on the equations (B.27) – (B.33) beforeusing a point-wise Gauss-Seidel relaxation method on the resulting algebraicequations.9 For the results in this paper, for c1 = 4.5, a cutoff of ρcut ={6, 8} was used while for c1 = 5.5 and c1 = 8, for which the modulationswere larger, a cutoffs of ρcut = 10 and ρcut = 12 correspondingly wereused. Grid spacings used for the finite-difference scheme were in the rangedρ, dx = 0.04 − 0.005. Neumann boundary conditions are differenced tosecond order using one-sided finite-difference stencils in order to update theboundary values at each step. At the asymptotic boundary ρcut we imposethe boundary conditions by second order differencing a differential equationbased on the fall-off (for example, ∂rA = −3A/r) to obtain an update rulefor the boundary value. As a result we find quadratic convergence as afunction of grid-spacing for our method, see appendix B.2.5.8Fixing µ, T and L gives the system in the grand canonical ensemble. Once the phasespace has been mapped in one ensemble other ensembles can be considered via appropriatereinterpretation of the numerical data. See section 4.4 for a description of this process.9See appendix A.2 for a description of this procedure.794.3. The solutions4.3 The solutionsThe system of equations (B.27) – (B.33) is solved subject to the boundaryconditions described in the previous sections. The details of our numeri-cal algorithm are found in appendices A.2 and B.2. Here we focus on theproperties of the solutions and their geometry.Unless otherwise specified the following plots were obtained using theaxion coupling of c1 = 4.5. In this section, we consider solutions for whichthe periodicity is determined by the dominant critical wavenumber kc; forc1 = 4.5, this gives Lµ/4 ' 2.08, see Table B.1. We found that the geometryand most of the other features are qualitatively similar for the couplingsc1 = 5.5 and c1 = 8. A convenient way to parametrize our inhomogeneoussolutions is by the dimensionless temperature T/Tc, relative to the criticaltemperature Tc, below which the translation invariance along x is broken.For c1 = 4.5, our method allows us find solutions in the range 0.003 .T/Tc . 0.9.4.3.1 Metric and fieldsFor subcritical temperatures, as we descend into inhomogeneous regime,the metric and the matter fields start developing increasing variation inx. Figure 4.2 displays the metric functions, and Figure 4.3 shows the nonvanishing components of the vector potential field and of the scalar field forT/Tc ' 0.11 over a full period in the x direction. The variation of all fieldsis maximal near the horizon of the black hole at ρ =√r2 − r20 = 0, and itgradually decreases toward the conformal boundary, ρ→∞.Many of the special features of the solutions we find may be explainedvia axion electrodynamics as seen in the effective description of the elec-tromagnetic response of a topological insulator. This effect is mediated bythe interaction term in our Lagrangian (4.1). In the broken phase we havean axion gradient in the near horizon geometry, which realizes a topologicalinsulator interface, see Figure 4.3. The characteristic patterning of the nearhorizon magnetic field, B = ∇×A, shown in Figure 4.4, is reminiscent of themagnetoelectric effect at such interfaces. The magnetic vortices are local-ized near the black hole horizon and have alternating direction of magneticfield lines.In curved space the magnetic field is accompanied by vorticity, which ismanifested by the function W . This causes frame dragging effects in they direction. Test particles will be pushed along y with speeds W (r, x), inparticular the direction of the flow reverses every half the period along x.804.3. The solutionsFigure 4.2: Metric functions for T/Tc ' 0.11. Note the metric functionsA,B and C have half the period of W . The variation is maximal nearthe horizon, located at ρ = 0, and it decays as the conformal boundary isapproached, when ρ→∞.The drag vanishes at the horizon and at the location of the nodes of W wherex = nL/2, for integer n, see Figure 4.2. In general, the dragging effectremains bounded, and no ergoregion forms, where the vector ∂t becomesspacelike.4.3.2 The geometryThere are several ways to envisage the geometry of our solutions, we discussthem in turn.The Ricci scalar of the RN solution is RRN = −24, constant in r andindependent of the parameters of the black hole. This is no longer true forthe inhomogeneous phases, where the Ricci scalar becomes position depen-dent. Figure 4.5 illustrates the spatial variation of the Ricci scalar, relativeto RRN for T/Tc ' 0.054. The maximal curvature is always along the hori-zon at x = nL/2 for integer n. It grows when the temperature decreases814.3. The solutionsFigure 4.3: At relative to the corresponding RN solutions, Ay and ψ forT/Tc ' 0.11. The period of At is twice that of ψ and Ay. The x-dependencedies off gradually as the conformal boundary is approached, at ρ→∞.−4 −3 −2 −1 0 1 2 3 400.20.40.60.81ρxFigure 4.4: Magnetic field lines for solution with T/Tc ' 0.07. The patternof vortices of alternating field directions form at the horizon (located atρ = 0).and approaches the finite value of R ' −94 in the small temperature limit.Embedding in a given background space is a convenient way to illustratecurved geometry. We consider the embedding of 2-dimensional spatial slices824.3. The solutionsFigure 4.5: Ricci scalar relative to that of RN black hole, R/RRN − 1,RRN = −24, for T/Tc ' 0.054 over half the period. The scalar curvature ismaximal along the horizon at x = nL/2 for integer n.of constant x of the full geometry (4.2)ds22 =e2B(r,x)2 r2 f(r)dr2 + 2 r2 e2C(r,x)dy2 (4.20)as a surface in 3-dimensional AdS spaceds23 = 2 r˜2 dz2 +dr˜22 r˜2 + 2 r˜2dy2. (4.21)We are looking for a hypersurface parametrized by z = z(r˜). Then themetric on such a hypersurface readsds22 =[1 + 2 r˜2(dzdr˜)2] dr˜22 r˜2 + 2 r˜2dy2. (4.22)Comparing (4.22) and (4.20) we obtain set of the relationsr˜ = r eC ,[12 r˜2 + 2 r˜2(dzdr˜)2](dr˜dr)2= e2B(r,x)2 r2 f(r) , (4.23)834.3. The solutionsFigure 4.6: The embedding diagram of constant x spatial slices, as a functionof x at given y for T/Tc ' 0.035. The geometry of ρ = const slices ismaximally curved at x = nL/2 for integer n.resulting in the embedding equationdzdr =12 r2√f(r)−1 e2B(r,x)−2C(r,x) − (1 + r ∂rC(r, x))2. (4.24)We integrate this equation for a given x, and in Figure 4.6 show the embed-ding at constant y. The maximal curvature along ρ = const slices occurs atx = nL/2 for integer n, which is consistent with Figure 4.5.The proper length of the stripe along x relative to the background AdSspacetime at given r islx(r)/lx(r =∞) =∫ L/40eB(r, x) dx. (4.25)Figure 4.7 shows the dependence of the normalized proper length on theradial distance from the horizon. The proper length tends to the coordinatelength as 1/r3 asymptotically as r →∞, but it exceeds that as the horizon isapproached. Namely, the inhomogeneous black brane ‘pushes space’ aroundit along x, in a manner resembling the ‘Archimedes effect’.The proper length of the horizon in x direction is obtained calculating(4.25) at r0. Figure 4.8 demonstrates the dependence of this quantity on844.3. The solutions10−2 10−1 10000.10.20.30.40.50.6ρl x(ρ)/l x(∞)−1Figure 4.7: Radial dependence of the normalized proper length along x forT/Tc ' 0.054. While asymptotically the proper length coincides with thecoordinate size of the strip, it grows as the horizon is approached. This is amanifestation of the ‘Archimedes effect’.the temperature. For high temperatures the length of the horizon resemblesthat of the homogeneous RN solution, however, it grows when temperaturedecreases. We find that at small T/Tc the proper length of the horizondiverges approximately as (T/Tc)−0.1.The transverse extent of the horizon, per unit coordinate length y, isgiven byry(x) =√2 r0 eC(r0,x). (4.26)Figure 4.9 shows the variation of ry(x) along the horizon for T/Tc ' 0.054.Typically there is a ‘bulge’ occurring at x = nL/2 and a ‘neck’ at x =(2n + 1)L/4, for integer n. Comparing this with Figure 4.5 we note thatRicci scalar curvature is maximal at the bulge and not at the neck aswould happen, for instance, in the cylindrical geometry in black stringcase [98]. Figure 4.10 displays the dependence of the sizes of the neckand bulge on T/Tc. Both sizes monotonically decrease with temperature,however the rate at which the neck is shrinking exceeds that of the bulge.This is demonstrated in Figure 4.11. In fact, we find that for c1 = 4.5,rnecky /rbulgey ∼ (T/Tc)1/2 near the lower end of the range of temperaturesthat we investigated. For other values of the axion coupling the scaling ofthe ratio is again power-law, with an exponent of the same order of magni-854.4. Thermodynamics at finite length0 0.1 0.2 0.3 0.411.11.21.31.41.51.61.71.81.9T /TcL h/LRN h  datafit to Lh/LhRN ∼ (T/Tc)−0.1Figure 4.8: Temperature dependence of the proper length of the horizonalong the stripe. Starting from as low as L at high temperatures, the properlength grows monotonically and for small T/Tc the growth is well approxi-mated by the power-law dependence ∼ (T/Tc)−0.1.tude, e.g. for c1 = 8, the exponent is about 0.12. This signals a pinch-off ofthe horizon in the limit T → 0.4.4 Thermodynamics at finite lengthIn this section we consider the thermodynamics and phase transitions in thesystem, assuming that the stripe length is kept fixed. For the finite systemthe length of the interval is part of the specification of the ensemble and iskept fixed. In the next section we discuss the infinite system, for which thestripe width can adjust dynamically.4.4.1 The first lawWe demonstrated that below the critical temperature there exists a newbranch of solutions which are spatially inhomogeneous. In the microcanon-ical ensemble the control variables of the field theory are the entropy S, thecharge density N , and the length of the x-direction L, with correspondingconjugate variables temperature T , chemical potential µ, and tension in the864.4. Thermodynamics at finite length−2 −1 0 1 200.050.1r yxFigure 4.9: The extent of the horizon in the transverse direction, ry, asa function of x for T/Tc ' 0.054 in x ∈ [−L/2, L/2]. The characteristicpattern of alternating ‘necks’ and ‘bulges’ forms along x.−2.5 −2 −1.5 −1 −0.50.050.10.150.20.250.30.350.4r ylog(T/Tc)  neckbulge homogeneous RNFigure 4.10: The dependence of the size of the neck and the bulge on tem-perature.874.4. Thermodynamics at finite length10−3 10−2 10−1 10010−1100T /Tcrnecky/rbulge y  datafit to ryneck/rybulge  = 2 (T/Tc)0.5Figure 4.11: The ratio of the transverse extents of the neck and the bulgeshrinks as rnecky /rbulgey ∼ (T/Tc)1/2 at small temperatures, indicating apinch-off of the horizon in the limit T → 0.x-direction τx.10 The usual first law is augmented by a term correspondingto expansions and contractions in the x-direction and is given bydM = TdS + µdN + τxdL. (4.27)where M , S, and N are quantities per unit length in the trivial y direction,but are integrated over the stripe.Our system has a scaling symmetry given by (4.19). In the field theory,this corresponds to a change of energy scale. Under this transformation, thethermodynamic quantities scale asM → λ2M, T → λT, S → λS, µ→ λµ,N → λN, τx → λ3τx, L→1λL. (4.28)Using these in (4.27) with λ = 1 + , for  small, yields2M = TS + µN − τxL, (4.29)the Smarr’s-like expression that our solutions must satisfy and that can beused as a check of our numerics. For all of our solutions, we have verifiedthat this identity is satisfied to one percent.10Explicit expressions for these quantities in terms of our ansatz are given in appendixB.1.884.4. Thermodynamics at finite length4.4.2 Phase transitionsThe question of which solution dominates the thermodynamics depends onthe ensemble considered. In the holographic context the choice of thermo-dynamic ensemble is expressed through the choice of boundary conditions.The corresponding thermodynamic potential is computed as the on-shellbulk action, appropriately renormalized and with boundary terms renderingthe variational problem well-defined. We examine each ensemble in turn.The grand canonical ensembleIn our numerical approach, the natural ensemble to consider is the grandcanonical ensemble, fixing the temperature T , the chemical potential µ,and the periodicity of the asymptotic x direction as L. The correspondingthermodynamic potential is the grand free energy densityΩ(T, µ, L) = M − TS − µN. (4.30)Different solutions of the bulk equations with the same values of T, µ, Lcorrespond to different saddle point contributions to the partition function.The solution with smallest grand free energy Ω is the dominant configura-tion, determining the thermodynamics in the fixed T, µ, L ensemble. In ourcase we have two solutions for each choice of T, µ, L, one homogeneous andone striped. Exactly how one one saddle point comes to dominate over theother at temperatures below the critical temperature determines the orderof the phase transition.In this ensemble it is convenient to measure all quantities in units of thefixed chemical potential µ. Then, after fixing L from the critical mode ap-pearing at the highest Tc (see Figure B.1 and Table B.1 in appendix B.2.1),we have that Ω/µ2 is a function only of the dimensionless temperature T/µ.In the fixed chemical potential ensemble for large enough axion coupling wefind a second order transition, where the inhomogeneous charge distributionstarts dominating the thermodynamics immediately below the temperatureat which the inhomogeneous instability develops. Near the critical tem-perature, the behaviour of the grand free energy difference is consistentwith (Ω − ΩRN )/µ2 ∝ (1 − T/Tc)2, while the entropy difference goes as(S − SRN )/µ ∝ T/Tc − 1. This is as expected from a second order tran-sition. As can be seen in Figure 4.12 and Figure 4.13, we find this secondorder transition for a range of lengths, L, and for a variety of values of theaxion coupling c1. With the current accuracy of our numerical procedure,we find it increasingly difficult to resolve the order of the phase transition894.4. Thermodynamics at finite lengthL̐4=1.400.0 0.2 0.4 0.6 0.8 1.0-0.05-0.04-0.03-0.02-0.010.00TTcHW-WRNLΜ2L̐4=1.210.0 0.2 0.4 0.6 0.8 1.0-0.04-0.03-0.02-0.010.00TTcHW-WRNLΜ2L̐4=1.040.0 0.2 0.4 0.6 0.8 1.0-0.020-0.015-0.010-0.0050.000TTcHW-WRNLΜ2L̐4=0.920.0 0.2 0.4 0.6 0.8 1.0-0.015-0.010-0.0050.000TTcHW-WRNLΜ2Figure 4.12: The grand free energy relative to the RN solution for severalsolutions of different fixed lengths at c1 = 8. In all cases shown we observea second order phase transition. The critical exponents determined nearthe critical points in each case are consistent with the quadratic behaviour(Ω− ΩRN )/µ2 ∝ (1− T/Tc)2.for smaller values of c1. In fact, for c1 = 4.5 the grand free energies ofthe homogeneous and inhomogeneous phases are nearly degenerate but stillallow us to determine the phase transition as second order. It would beinteresting to see if the phase transition remains of second order or changesto the first order for smaller values of the axion coupling.To examine the observables in the striped phase further, we focus onc1 = 8 and the corresponding dominant critical mode, Lµ/4 ' 1.21, andconsider solutions for the temperatures 0.00016 . T/Tc . 0.96. Variousquantities are plotted with the corresponding homogeneous results in Fig-ure 4.14. Along this branch of solutions, the mass of the stripes is more thanthe RN solution and the entropy is always less. We plot the maximum of theboundary current density 〈jy〉, momentum density 〈Ty0〉 and pseudo-scalaroperator vev 〈Oψ〉. Fitting the data near the critical point to the func-tion (1 − T/Tc)α, we find the approximate critical exponents αjy = 0.40,αTy0 = 0.41 and αOψ = 0.38 with relative fitting error of about 10%.We find evidence that the entropy of the striped black branes does nottend to zero in the small temperature limit, see Figure 4.14. This is further904.4. Thermodynamics at finite length0.0 0.2 0.4 0.6 0.8 1.0-0.008-0.006-0.004-0.0020.000TTcHW-WRNLΜ2Figure 4.13: The grand free energy relative to the RN solution for c1 =4.5 and fixed Lµ/4 = 2.08. The grand free energies of the homogeneousand inhomogeneous phases are nearly degenerate, such that their maximalfractional difference is about 1%.supported by the behaviour of the transverse size of the horizon (4.26).Here the bulge seems to contract at a much slower rate than the neck,which evidently shrinks to zero size in the limit T → 0. However, strictlyspeaking, this conclusion is based on extrapolation of the finite temperaturedata to T = 0. Checking whether the entropy asymptotes to a finite valueor goes to zero in this limit, as suggested in [102, 103], will require furtherinvestigation with a method of higher numerical accuracy.The canonical ensembleTo study the system in the canonical ensemble we fix the temperature,total charge and length of the system. This describes the physical situationin which the system is immersed in a heat bath consisting of unchargedparticles. The relevant thermodynamic potential in this ensemble is the freeenergy densityF (T,N,L) = M − TS. (4.31)If we measure all quantities in units of the fixed charge N , then, again, thefree energy F/N2 is only a function of the dimensionless temperature T/N .To solve our system with a fixed charge, we would need to fix the integralin x of the coefficient of the 1/r term in the asymptotic expansion of thegauge field At. Numerically, it is much easier to fix the chemical potential,as this gives a Dirichlet condition on At at the boundary. In the grand914.4. Thermodynamics at finite length0.0 0.2 0.4 0.6 0.8 1.00.40.50.60.70.80.9TTcMΜ20.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.02.5TTcSΜ0.0 0.2 0.4 0.6 0.8 1.00.70.80.91.0TTcNΜ0.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.080.100.12TTc<jy>maxΜ0.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.08TTc<OΨ>maxΜ0.0 0.2 0.4 0.6 0.8 1.00.000.050.100.150.20TTc<Ty0>maxΜ2Figure 4.14: The observables in the grand canonical ensemble for c1 = 8and Lµ/4 = 1.21 (points with dotted line) plotted with the correspondingquantities for the RN black hole (solid line). Fitting the data near thecritical point to the function (1 − T/Tc)α, we find the approximate criticalexponents αjy = 0.40, αTy0 = 0.41 and αOψ = 0.38 with relative fitting errorof about 10%.924.4. Thermodynamics at finite length0.0 0.2 0.4 0.6 0.8 1.0-0.30-0.25-0.20-0.15-0.10-0.050.00TTcHF-FRNLN2Figure 4.15: The difference in canonical free energy, at c1 = 8 and fixedlength LN/4 = 1.25, between the striped solution and the RN black hole.The striped solution dominates immediately below the critical temperature,signalling a second order phase transition.canonical ensemble, we solved for one-parameter families of solutions atfixed Lµ, labelled by the dimensionless temperature T/µ. Equivalently, inthe (Lµ, T/µ) plane, we solve along the line of fixed Lµ. Translated to thesituation in which we measure quantities in terms of the charge density N ,these solutions become one-parameter families of solutions with varying LN ,or a curve in the (LN, T/N) plane with LN a function of T/N . By varyingthe length Lµ (or solving with µ = 1 and varying L), we can find a collectionof solutions that intersect the desired fixed LN line. By interpolating thesesolutions and evaluating the interpolants at fixed LN , we can study thestripes in the canonical ensemble.In this ensemble we find a similar second order transition, in which theinhomogeneous solution dominates the thermodynamics below the criticaltemperature (Figure 4.15). The scaling of the free energy below the criticaltemperature is nearly quadratic in |T − Tc|, a mean field theory exponentas is common in large N models.The microcanonical ensembleThe microcanonical ensemble describes an isolated system in which all con-served charges (in this case the mass and the charge) are fixed. This ensem-ble describes the physical situation relevant to the study of the real timedynamics of an isolated black brane at fixed length. In this case, the state934.4. Thermodynamics at finite lengthExtremal RN0.72 0.74 0.76 0.78 0.80 0.82 0.84MN20.00.51.01.52.02.5SNFigure 4.16: The entropy of the inhomogeneous solution for c1 = 8 (pointswith dotted line) and of the RN solution (solid line). Below the criticaltemperature, the striped solution has higher entropy than the RN. The RNbranch terminates at the extremal RN black hole, while the striped solutionpersists to smaller energies.that maximizes entropy is the dominant solution. As shown in Figure 4.16,we find that the entropy of our inhomogeneous solutions is always greaterthan that of the RN black hole of the same mass. Furthermore, the massof the inhomogeneous solutions is always smaller than that of the criticalRN black hole. Therefore, at fixed LN , the unstable RN black holes belowcritical temperature are expected to decay smoothly to our inhomogeneoussolution.Fixing the tensionAlternatively, one could attempt to compare solutions with different valuesof L. The meaningful comparison is in an ensemble fixing the tension τx. Forexample, one could compare the Legendre transformed grand free energyG(T, µ, τx) = M − TS − µN − τxL (4.32)where the additional terms comes from boundary terms in the action ren-dering the new variational problem (fixing τx) well-defined. The candidatesaddle points are the solutions we find with various periodicities L, and theirrelative importance in the thermodynamic limit is determined byG(T, µ, τx).In particular the solution which is thermodynamically dominant depends on944.5. Thermodynamics for the infinite systemthe value of τx we hold fixed. In this study we concentrate on the thermo-dynamics in the fixed L ensemble and we leave the study of the fixed τxensemble to future work.4.5 Thermodynamics for the infinite systemIn this section we lift the assumption of the finite extent of the system inthe x-direction and consider the thermodynamics of the formation of thestripes below the critical temperature. For the infinite system we can definedensities of thermodynamic quantities along x:m = ML , s =SL, n =NL . (4.33)In terms of these, the first law for the system becomesdm = Tds+ µdn (4.34)and the conformal identity is3m = 2(Ts+ µn). (4.35)In the infinite system, we compare stripes of different lengths, at fixedT/µ, to each other and to the homogeneous solution. The solution thatdominates the thermodynamics is the one with the smallest free energydensity ω, whereω = m− Ts− µn. (4.36)This comparison is shown in Figure 4.17 for c1 = 8, where we see that thefree energy density of the stripes is negative relative to the RN black hole,indicating that the striped phase is preferred at every temperature below thecritical temperature.11 Very close to the critical temperature, the dominantstripe is that with the critical wavelength kc. As we lower the temperature,the minimum of the free energy density traces out a curve in the (Lµ, T )plane, and the dominant stripe width increases to Lµ/4 ≈ 2.One can also study the observables of the system along this line of min-imum free energy density. The results are qualitatively similar to those forthe fixed L system (Figure 4.14). In particular, the free energy density scalesas (ω − ωRN )/µ3 ∝ (1− T/Tc)2 near the critical point, indicating a secondorder transition in the infinite system as well.11In appendix B.2.4, we describe the generation of Figure 4.17.954.5. Thermodynamics for the infinite system1.0 1.2 1.4 1.6 1.8 2.0 2.20.000.020.040.060.080.10L̐4TΜHΩ-ΩRNLΜ3-0.04-0.03-0.02-0.010.00Figure 4.17: Action density for c1 = 8 system relative to the RN solution.The red line denotes the approximate line of minimum free energy.96Chapter 5Towards a holographic modelof colour superconductivity15.1 IntroductionBackgroundQuantum chromodynamics is believed to display a rich phase structure atfinite temperature and chemical potential, with phase transitions associatedwith deconfinement, nuclear matter condensation, the breaking of (approx-imate) flavour symmetries (which are exact in generalizations with equalquark masses and/or massless quarks), and the onset at high density ofquark matter phases displaying colour superconductivity (for reviews seefor example [31, 105–107]). However, apart from the regimes of asymptoti-cally large temperature or chemical potential, a direct analytic study of thethermodynamic properties of the theory is not possible.Even using numerical simulations, only the physics at zero chemical po-tential is currently accessible, since at finite µ the Euclidean action becomescomplex, and the resulting oscillatory path integral cannot reliably be sim-ulated using standard Monte-Carlo techniques. Current proposals for thephase diagram of QCD and related theories are largely based on qualitativearguments and phenomenological models. While these provide a plausiblepicture, it is possible that they miss important features of the physics. Itwould certainly be satisfying to have examples of theories similar to QCDin which the full phase diagram could be explored directly.The holographic approachA modern route to understanding properties of strongly coupled gaugetheories, that would be otherwise inaccessible, is via the AdS/CFT cor-respondence, or gauge theory / gravity duality. This suggests that cer-tain quantum field theories (usually called ‘holographic theories’), generally1A version of this chapter has been published [4].975.1. Introductionwith large-rank gauge groups, are equivalent to gravitational systems. Bythis correspondence, calculations of physical observables in the field theoryare mapped to gravitational calculations; in many cases difficult strongly-coupled quantum mechanical calculations in the field theory (such as thoserequired to understand the thermodynamic properties of QCD) are mappedto relatively simple classical gravity calculations. Optimistically, it maythen be possible to find a theory qualitatively similar to QCD for which thephysics at arbitrary temperature and chemical potential can be understoodexactly via simple calculations in a dual gravitational system.By now, there are well-known examples in gauge-theory / gravity dualityfor which the field theory shares many of the qualitative features of QCD(see, for example [13]). Further, many of these theories have been studiedat finite temperature and chemical potential, revealing phase transitionsassociated with deconfinement, chiral symmetry breaking, meson melting,and the condensation of nuclear matter. However, to date, most of thetheories that can be studied reliably using dual gravity calculations havethe restriction that the number of flavours is kept fixed in the large Nclimit. In such theories, the physics at large chemical potentials is known tobe qualitatively different than in real QCD. For example, at asymptoticallylarge chemical potential, theories with large Nc and fixed Nf are believedto exhibit an inhomogeneous ‘chiral density wave’ behaviour [68, 69], ratherthan the homogenous quark matter phases predicted for finite Nc and Nf . Inorder to find examples of holographic theories which most closely resemblereal QCD at finite chemical potential, one should therefore attempt to findexamples of calculable gravitational systems corresponding to theories withfinite Nf/Nc. This situation presents some technical challenges, as we nowreview.In the well-known examples of holographic gauge theories, the additionof flavour fields in the field theory corresponds to adding D-branes on thegravity side [62]. Quarks correspond to strings which have one endpoint onthese D-branes, while mesons correspond to the quantized modes of openstrings which begin and end on the branes. The configurations of these D-branes in theories with finite Nf and large Nc are determined by findingaction-minimizing configurations of the branes on a fixed background geom-etry. On the other hand, in order to have Nf of order Nc in a large Nc theory,we need a large number of these flavour branes, and these will back-reacton the spacetime geometry itself. For Nf ∼ Nc, there are as many degreesof freedom in the flavour fields as there are in the colour fields (gauge fieldsand adjoints), so it is natural to expect that the back-reaction will be sosignificant that in the final description the flavour branes themselves will be985.1. Introductioncompletely replaced by a modified geometry with fluxes (in the same waythat the branes whose low-energy excitations give rise to the adjoint degreesof freedom do not appear explicitly in the gravity dual description of thefield theory).There has been significant progress in understanding the back-reactionof flavour branes, with some fully-back reacted analytic solutions available(for a review see [108]), but so far, there has not been enough progress tofully explore the phase structure of a QCD-like theory with finite Nf/Nc.In particular, as far as we are aware, colour superconductivity phases havenot been identified previously in holographic field theories.2Quark matter from the bottom upIn this chapter, we aim to come up with a holographic system describing aconfining gauge theory that does exhibit a quark-matter phase with coloursuperconductivity at large chemical potential. However, motivated by recentcondensed matter applications of gauge/gravity duality (see, for example [18,21, 44, 45, 64, 84]), we will avoid many of the technical challenges describedabove by taking what is known as a ‘bottom up’ approach. Rather thanworking in a specific string theoretical model which takes into account theback-reaction of flavour branes, we will make an ansatz for the ingredientsnecessary for such a model to describe the relevant physics. We study thesimplest possible gravitational theory with this minimal set of features, withthe hope that it captures the qualitative physics of interest. We will indeedfind that even this simple theory exhibits many of the expected features.IngredientsWe wish to construct a gravitational theory to provide a holographic de-scription of a four-dimensional confining gauge theory on Minkowski spacewith Nf ∼ Nc flavours. On the gravity side, the Minkowski space will ap-pear as the fixed boundary geometry of our spacetime, but we must haveat least one extra dimension corresponding to the energy scale in the fieldtheory. Since the field theory has a scale (the QCD or confinement scale),the asymptotic behaviour of the solution must exhibit an additional scalerelative to the asymptotically AdS geometries that appear in gravity dualsof conformal field theories. In the simplest examples of gravity duals forconfining gauge theories, this scale is provided by the size of an additional2However, see [109] for a possible manifestation of the related colour-flavour lockingphase in a holographic system.995.1. Introductioncircular direction in the geometry.3 Thus, we will work with a gravitationalsystem in six dimensions whose boundary geometry is R3,1 × S1. We willassume that the asymptotic geometry is locally Anti-de-Sitter space, so theconfining gauge theory we consider arises from a five-dimensional conformalfield theory compactified on a circle. When we study the theory at finitetemperature, there will be an additional circle in the asymptotic (Euclidean)geometry, the Euclidean time direction whose period is 1/T .The gauge theories we are interested in have at least one other con-served current, corresponding to baryon (or quark) number. By the usualAdS/CFT dictionary, this operator corresponds on the gravity side to a U(1)gauge field in the bulk. The asymptotic value of the time component forthis gauge field corresponds to the chemical potential in our theory, whilethe asymptotic value of the radial electric flux corresponds to the baryoncharge density in the field theory. For a given chemical potential, the min-imum action solution will have some specific value for the flux, allowing usto relate density and chemical potential.The colour superconductivity phases believed to exist at large density inQCD and related theories are usually characterized by condensates of theform 〈ψψ〉, bilinear in the quark fields ψ, which spontaneously break theU(N) gauge symmetry, and the U(1)B global symmetry. Naively, we wouldwant to model such operators by a bulk charged scalar field corresponding tothe condensate. However, bulk fields always correspond to gauge-invariantoperators, while by definition the ψψ bilinears which break the gauge sym-metry are not gauge-invariant (in fact, there is no way to make a singletfrom two fundamental fields, except in the case of SU(2)). Additionally, thesimplest gauge-invariant operators charged under U(1)B involve N ψ fieldsand have dimension of order N , thus our holographic dual theory shouldhave no light scalar fields charged under the U(1)B gauge field.The correct way to understand the condensation of the ψψ bilinears isas an example of spontaneously broken gauge symmetry (as in the Higgsmechanism), rather than as a phase transition characterized by some gauge-invariant order parameter. Nevertheless, the transition to colour supercon-ductivity can be characterized by the discontinuous behaviour of gauge-invariant operators, which are of the form ψψ(ψψ)†. Such operators aregauge invariant and neutral under the U(1)B, and therefore should corre-spond to an uncharged scalar field in the bulk with dimension of order 1.43There are other possibilities here, as we mention briefly in the discussion section.4As emphasized by Andreas Karch, a gauge invariant operator of the form O4 =ψψ(ψψ)† can be written as a sum of terms OαOα where each Oα ∼ (ψ†ψ)α is gaugeinvariant (and α represents flavour/Lorentz indices). Thus, O4 is something like a double-1005.1. IntroductionCombining everything so far, we want to study gravity in six dimensionswith negative cosmological constant and boundary geometry R3,1×S1 witha U(1) gauge field and a neutral scalar field. The simplest action for thissystem is5∫d6x√−g{R+ 20L2 −14F2 − |∂µψ|2 −m2|ψ|2}, (5.1)where we include one tunable parameter, the mass m of the scalar field,which determines the dimension of the corresponding operator in the dualfield theory. More generally, we could consider other potentials for the scalarfield, or a more complicated action (e.g. with a Chern-Simons term or ofBorn-Infeld type) for the gauge field, but we restrict here to this simplestpossible model.6ResultsStarting with the model (5.1), we have explored the phase structure by min-imizing the gravitational action for specific values of temperature (corre-sponding to the asymptotic size of the Euclidean S1 direction) and chemicalpotential (corresponding to the asymptotic value of A0). Our results for thephase diagrams are shown in Figures 5.1, 5.2, and 5.3. For small µ, we find aconfined phase at low-temperature and a deconfined phase at high temper-ature, with the scalar field uncondensed in each case. However, increasingµ at zero temperature, we find (setting LAdS = 1) for −254 ≤ m2 ≤ −5 atransition to a phase with nonzero scalar condensate (on a geometry withhorizon) and finite homogeneous quark density, as expected for a coloursuperconductivity phase. Increasing the temperature from zero, we find atransition back to the deconfined phase at a remarkably low temperature; forexample, at m2 = −6, the critical temperature at which superconductivitydisappears isT/µ ∼ .00006333 . (5.2)trace operator. In a large N theory, factorization of correlators implies that the expecta-tion value of O4 can be calculated classically from the Oα expectation values (up to 1/Ncorrections). Thus, discontinuous behaviour of O4 should be directly related to discontin-uous behaviour in the simpler gauge-invariant operators Oα (which also have no baryoncharge), so it may be more appropriate to think of the scalar field in our model as beingdual to one of these simpler operators.5Since we will also consider the case of a charged scalar field, we have written the actionusing standard normalizations for a complex scalar, but we will take the scalar to be realin the uncharged case.6For another approach to modeling the QCD phase diagram by an effective holographicapproach, see for example [59, 110].1015.1. Introduction0.20.30.40.5 Tdeconfined00.10.2 00.511.522.5μconfinedsuperconductingFigure 5.1: Phase diagram of our model gauge theory with m2 = −6,R = 2/5. Region in dashed box is expanded in next figure.The tendency for the scalar field to condense at low temperatures for therange of masses above can be understood in a simple way, as explained forexample in [11, 111]. In d+ 1 dimensional anti-de Sitter space with anti-deSitter radius L, the minimum mass for a scalar field to avoid instability ism2BF = −d2/(4L2). The minimum action solution for large chemical poten-tial in the absence of any scalar field is a planar Reissner-Nordstrom blackhole solution with one of the isometry directions periodically identified. Inthe limit of zero temperature, the near horizon region of this black hole hasgeometry AdS2 × R4, with the radius of the AdS2 equal to L2 = L/√20.Thus, in the near-horizon region, there will be an instability toward conden-sation of the scalar field if m2 < −1/(4L22) = −5/L2. We thus have a range(setting L = 1) of −25/4 ≤ m2 ≤ −5 for which the scalar field tends tocondense in the near-horizon region but is stable in the asymptotic region.Numerical simulations verify that we indeed have scalar field condensationfor precisely this range of masses.While there is no guarantee that the gravitational system we study has alegitimate field theory dual, ‘top-down’ gravitational systems correspondingto fully consistent field theories must have the same basic elements (usuallywith additional fields and a more complicated Lagrangian). The fact thatthe expected physics emerges even in our stripped-down version suggests1025.1. Introduction0.00010.0002Tdeconfinedconfined00.0001 012µsuperconductingFigure 5.2: Phase diagram of our model gauge theory with m2 = −6,R = 2/5. Region in dashed box is expanded in next figure.0.00010.0002Tdeconfined00.0001 1.73231.73251.7327µconfinedsuperconductingFigure 5.3: Phase diagram of our model gauge theory with m2 = −6,R = 2/5. The dashed curve represents the phase boundary in theory withouta scalar field.1035.2. Basic setupthat quark-matter phases will be found also in the complete models, onceback-reaction effects are under control. Optimistically, qualitative featuresthat we find in the bottom-up model (such as the extremely low transi-tion temperature between superconducting and deconfined phases) may bepresent also in more complete holographic theories. In this case, our simplemodel may provide novel qualitative insights into fully consistent QCD-liketheories.Charged scalarWhile less relevant to colour superconductivity, it is also interesting to ex-plore the physics of our model when we make the scalar field charged underthe gauge field. In this case, the scalar field corresponds to a gauge-invariantoperator in the field theory that is charged under the U(1) associated withA, and the kinetic term for the scalar field is modified in the usual way as∂µψ → ∂µψ − iqAµψ. As we have argued above, this symmetry cannot beU(1)B, but could be another flavour symmetry, such as isospin in a modelwith two or more flavours. The flavour superconductivity associated withmeson condensation was studied previously in the holographic context (withfinite Nf ), for example in [43, 112, 113]. Our results are qualitatively similarto the ones obtained in those studies, and we leave more detailed comparisonfor future work.In section 5.5 below, we determine the phase diagram for various valuesof q and m. The same system was studied for the 2 + 1 dimensional casein [58] and originally in [67] for the case of large q. The application therewas to holographic insulator/superconductor systems, but the intriguingresemblance of the phase diagrams in those papers to QCD phase diagramspartially motivated the present study.5.2 Basic setupIn this chapter, we consider holographic field theories with a conserved cur-rent Jµ, assumed to be a baryon current (or isospin current when we considercharged scalar fields) and some gauge-invariant operator O whose condensa-tion indicates the onset of (colour or flavour) superconductivity. We wouldlike to explore the phase structure of the theory for finite temperature T andchemical potential µ; that is, we would like to find the phase that minimizesthe Gibbs free energy density g = e − Ts − µρ, where e, s, and ρ are theenergy density, entropy density, and charge density in the field theory. We1045.2. Basic setupcan also ask about the values of e, s, ρ, and 〈O〉 as a function of temperatureand chemical potential.As discussed in the introduction, our holographic theories are defined bya dual gravitational background which involves a metric, U(1) gauge field,and scalar field, with a simple action∫d6x√−g{R+ 20L2 −14F2 − |∂µψ|2 −m2|ψ|2}. (5.3)We choose coordinates (t, x, y, z) for the non-compact field theory directions,w for the compact field theory direction, and r for the radial direction. Wetake boundary conditions for which the asymptotic (large r) behaviour ofthe metric isds2 →( rL)2 (−dt2 + dx2 + dy2 + dz2 + dw2)+(Lr)2dr2 , (5.4)where w is taken to be periodic with period R. To study the theory at finitetemperature, we take the period of τ = it in the Euclidean solution to be1/T .The equations of motion constrain the gauge field to behave asymptoti-cally asAν = aν −jν3r3 + . . . . (5.5)Since Aν is assumed to be the field corresponding to the conserved baryoncurrent operator Jν , in the field theory, the usual AdS/CFT dictionarytells us that aν is interpreted as the coefficient of the Jν in the Lagrangian(i.e. an external source for the baryon current) while jµ is interpreted asthe expectation value of baryon current for the state corresponding to theparticular solution we are looking at. To study the theory at finite chemicalpotential µ without any external source for the spatial components of thebaryon current, we want to takeaν = (µ, 0, 0, 0) . (5.6)The scalar field equations of motion imply that asymptoticallyψ = ψ1rλ− +ψ2rλ+ + · · · , (5.7)whereλ∓ =12(d∓√d2 + 4m2) . (5.8)1055.2. Basic setupThe holographic field theories we consider are defined by assuming ψ1 = 0.In this case, λ+ gives the dimension of the operator dual to ψ.7 In this case,ψ2 (which will be different for solutions corresponding to different states ofthe field theory) gives us the expectation value of the operator O in the fieldtheory.By the AdS/CFT correspondence, the field theory free energy corre-sponds to the Euclidean action of the solution. Thus, to investigate thefield theory state which minimizes free-energy for given T and µ, we need tofind the gravitational solution with boundary conditions given above whichminimizes the Euclidean action. Note that we only consider solutions withtranslation invariance in t, x, y, z, and w. It would be interesting to investi-gate the possibility of inhomogeneous phases (or at least the stability of oursolutions to inhomogeneous perturbations) but we leave this as a questionfor future work.Calculating the actionIn order to obtain finite results when calculating the gravitational action fora solution, it is important to include boundary contributions to the action.In terms of the Lorentzian metric, gauge field and scalar, the fully regulatedexpression that we require is [64]S = limrM→∞[−∫r<rMdd+1x√−g{R+ d(d− 1)L2 −14F2 − |Dµψ|2 −m2|ψ|2}+∫r=rMddx√−γ{−2K + 2(d− 1)L −1Lλ−|ψ|2}], (5.9)whereλ− =d2 −12√d2 + 4m2 . (5.10)Here, γ is the metric induced on the boundary surface r = rM , and K isdefined asK = γµν∇µnν , (5.11)where nµ is the outward unit normal vector at r = rM . The scalar countert-erm here is the appropriate one assuming that our boundary condition is to7For a certain range scalar field masses in the range −d2/4 ≤ m2 ≤ −d2/4 + 1, it isalso consistent to define a theory by fixing ψ2 = 0. In this case, the dimension of the dualoperator is λ−. We consider this case briefly in section 4.2.1065.2. Basic setupfix the coefficient of the leading term in the large r expansion of ψ. Sincewe are setting this term to zero, it turns out that the counterterm vanishesin the rM →∞ limit.For all cases we consider, the metric takes the formds2 = r2L2dx2i + g00(r)dt2 + grr(r)dr2 + gww(r)dw2 . (5.12)Assuming the Einstein equations are satisfied, we can show (by subtractinga term proportional to the xx component of the equation of motion) thatthe integrand in the first term may be written as a total derivative withrespect to r−√−g{R+ d(d− 1)L2 −14F2 − |Dµψ|2 −m2|ψ|2}= ∂r( 2rgrr√−g).(5.13)Usingnµ = (0, . . . , 0,√grr) , (5.14)we haveK = γµν∇µnν= γµν{−Γrµνnr}= γµν{12grr ∂gµν∂r√grr}= 12√grrγµν ∂γµν∂r= 1√grr∂ ln(√−γ)∂r (5.15)so that√−γ(−2K) = − 2√grr∂√−γ∂r . (5.16)Our final expression for the action density isS/Vd =2rgrr√−g∣∣∣∣rMr0+{− 2√grr∂√−γ∂r +2(d− 1)L√−γ}r=rM. (5.17)1075.2. Basic setupAction in terms of asymptotic fieldsIt is convenient to rewrite the expression (5.17), in terms of the asymptoticexpansion of the fields. For the ansatz (5.12), and the boundary conditionsappropriate to our case, we findgtt = −r2 +g(3)ttr3 + . . . ,grr =1r2 +g(7)rrr7 + . . . ,gww = r2 +g(3)wwr3 + . . . ,ψ = ψ(3)r3 + . . . ,φ = µ− ρ3r3 + . . . . (5.18)Inserting these expansions into our expression above for the action we findthat (assuming the term at r = r0 vanishes)S = 5g(3)ww + 4g(7)rr − 5g(3)tt . (5.19)However, using the equations of motion, we find that g(3)ww + g(7)rr − g(3)tt = 0,so we can simplify to:S = −g(7)rr . (5.20)Numerically, it can be a bit tricky to read off g(7)rr because there is also a1/r8 term in the expansion of grr. But using the equations of motion, wecan findg(8)rr =34(7 +m2)(ψ(3))2 . (5.21)From this, it follows that the combination−r7grr(r) + r5 −34(7 +m2)r5ψ2(r) (5.22)behaves like−g(7)rr +O(1/r3) . (5.23)So, we can numerically evaluate the action by takingS ≈ −r7∗grr(r∗) + r5∗ −34(7 +m2)r5∗ψ2(r∗) , (5.24)where r∗ is taken to be large but not too close to the cutoff value.1085.3. Review: ψ = 0 solutions5.3 Review: ψ = 0 solutionsWe begin by considering the solutions for which the scalar field is set tozero.5.3.1 AdS soliton solutionAt zero temperature and chemical potential, the simplest solution with ourboundary conditions is pure AdS with periodically identified w. However,assuming antiperiodic boundary conditions for any fermions around the wcircle, there is another solution with lower action. This is the AdS soliton[114], described by the metric (setting L = 1)ds2 = r2(−dt2 + dx2 + dy2 + dz2 + f(r) dw2)+ dr2r2f(r) , (5.25)wheref(r) = 1− r50r5 . (5.26)As long as we choose the period 2piR for w such thatr0 =25R (5.27)the solution smoothly caps off at r = r0. This IR end of the spacetimecorresponds in the field theory to the fact that we have a confined phasewith a mass gap. The fluctuation spectrum about this solution correspondsto a discrete spectrum of glueball states in the field theory.Starting from this solution, we can obtain a solution valid for any tem-perature and chemical potential, by periodically identifying the Euclideantime direction and setting A0 = µ everywhere. Using (5.20) we find thatthe action for this solution isSsol = −r50 = −( 25R)5. (5.28)The negative value indicates that this solution is preferred over the pureAdS solution with action zero.5.3.2 Reissner-Nordstrom black hole solutionFor sufficiently large temperature and/or chemical potential, the AdS solitonis no longer the ψ = 0 solution with minimum action. The preferred solution1095.3. Review: ψ = 0 solutionsis the planar Reissner-Nordstrom black hole, with metricds2 = r2(−dt2f(r) + dx2 + dy2 + dz2 + dw2)+ dr2r2f(r) , (5.29)wheref(r) = 1−(1 + 3µ28r2+) r5+r5 +3µ2r6+8r8 , (5.30)the scalar potential isφ(r) = µ(1− r3+r3), (5.31)and w is periodically identified as before.This solution has a horizon at r = r+. The temperature of the solu-tion (determined as the inverse period of the Euclidean time for which theEuclidean solution is smooth) is given in terms of r+ byT = 14pi(5r+ −9µ28r+). (5.32)From (5.20), we find that the action for this solution isSRN = −r5+(1 + 38µ2r2+). (5.33)Thus, we find that the black hole solution has lower action than the solitonforr+(1 + 38µ2r2+) 15> 25R , (5.34)where r+ is determined in terms of T and µ by (5.32). This defines a curvein the T −µ plane that begins on the µ = 0 axis at T = 1/(2piR) and curvesdown to the T = 0 axis at µ = 219/10/(51/234/5R) ≈ 4.3547/(2piR), as shownin Figure 5.4.As usual, the existence of a horizon in this solution indicates that thecorresponding field theory state is in a deconfined phase [32].In the next sections, we consider solutions with nonzero scalar field.We will find that for large µ there exist solutions with nonzero scalar fieldthat have lower action than the solutions we have considered, so the phasediagram of Figure 5.4 will be modified.1105.4. Neutral scalar field: Colour superconductivity0.20.30.40.5 TReissner-Nordstrom(deconfined)00.10.2 00.511.522.5μAdS soliton(confined)Figure 5.4: Phase diagram without scalar field, in units where R = 2/5.5.4 Neutral scalar field: ColoursuperconductivityIn the case of a neutral scalar field, our simple model has no explicit sourcefor the gauge field in the bulk, so homogeneous solutions with a non-trivialstatic electric field (corresponding to a non-zero baryon number density inthe field theory) necessarily have a horizon from which the flux can emerge8.To look for solutions of this form, we consider the ansatz9ds2 = −g(r)e−χ(r)dt2 + dr2g(r) + r2(dw2 + dx2 + dy2 + dz2) ,At = φ(r) ,ψ = ψ(r) . (5.35)8In a more complete model, the source might be provided by some non-perturbativedegrees of freedom in the theory, such as the wrapped D-branes that give rise to baryonsin the Sakai-Sugimoto model.9We could have considered a more complicated ansatz, with an extra undeterminedfunction in front of dw2. However, it is plausible that as for the ψ = 0 solution, theminimum action solution for the case where the w circle does not contract in the bulk isa periodic identification of the solution with non compact w and rotational invariance inthe x, y, z, w directions.1115.4. Neutral scalar field: Colour superconductivityThe scalar and Maxwell’s equations that follow from the action (5.3) areψ′′ +(4r −χ′2 +g′g)ψ′ − m2g ψ = 0 , (5.36)φ′′ +(4r +χ′2)φ′ = 0 , (5.37)while the Einstein equations are satisfied ifχ′ + rψ′22 = 0 , (5.38)g′ +(3r −χ′2)g + reχφ′28 +m2rψ24 − 5r = 0 . (5.39)These have two symmetries:ψ˜(r) = ψ(ar) , φ˜(r) = 1aφ(ar) , χ˜(r) = χ(ar) , g˜(r) =1a2 g(ar) ,(5.40)arising from the underlying conformal invariance, andχ˜ = χ+ ∆ , φ˜ = e−∆2 φ . (5.41)We would like to find solutions with a horizon at some r = r+. Theelectric potential must also vanish at the horizon, and we are looking forsolutions for which the leading falloff ψ1 in (5.7) vanishes for the scalar.Also, multiplying the first equation (5.36) by g and evaluating at r = r+ fixesψ′(r+) in terms of ψ(r+) and g′(r+). Altogether, our boundary conditionsareg(r+) = 0 , φ(r+) = 0 , χ(∞) = 0 , ψ1 = 0 , (5.42)andψ′(r+) =8m2ψ(r+)40r+ − 2m2r2+ψ2(r+)− r+eχ(r+)(φ′(r+))2. (5.43)The remaining freedom to choose r+ and φ′(r+) leads to a family of solutionswith different T and µ. Explicitly, we haveµ = φ(∞) , T = 14pig′(r+)e−χ(r+)/2 . (5.44)Note that solutions with the same T/µ are simply related by the scalingsymmetry (5.40).1125.4. Neutral scalar field: Colour superconductivity5.4.1 Numerical evaluation of solutionsTo find solutions in practice, we can make use of the symmetries (5.40) toinitially set r+ = 1 and χ(0) = 0 and solve the equations with boundaryconditionsg(1) = 0 , χ(0) = 0 , φ(1) = 0 , φ′(1) = E0 , ψ(1) = ψ0 , (5.45)andψ′(1) = 8m2ψ040− 2m2ψ20 − E20. (5.46)We can integrate the φ and χ equations explicitly to obtainχ(r) = −∫ r0dr˜12 r˜(∂ψ∂r)2,φ(r) = E0∫ r1dr˜r˜4 e− 12χ(r˜) , (5.47)leaving the remaining equationsψ′′ + (4r +rψ′24 +g′g )ψ′ − m2g ψ = 0 ,g′ + 3gr +gr4 ψ′2 + E208r7 +m2rψ24 − 5r = 0 . (5.48)We use E0 as a shooting parameter to enforce ψ1 = 0, and find onesolution for each ψ0. From these solutions, we apply the symmetry (5.41)with ∆ = −χ(∞) to restore χ(∞) = 0 and finally use the symmetry (5.40)to scale to the desired temperature or chemical potential.Using this method, we find that solutions exist for scalar mass in therange −25/4 ≤ m2 ≤ 5, which is exactly the range of masses for which thescalar is stable in the asymptotic region but unstable in the near-horizonregion.10 For a given m2 in this range, solutions exist in the region T/µ <γ(m2), where γ(m2) is a dimensionless number depending on m2 (which weevaluate in the next section). The value of γ(m2) is remarkably small forall m2 in the allowed range. For example, with m2 = −6 (not particularlyclose to the limiting value m2 = −5), we have γ ≈ .00006333. It wouldbe interesting to understand better how this small dimensionless numberemerges since the setup has no small parameters. From the bulk point of10Solutions of this form were first found in lower dimensions in [11]. The zero-temperature limit of such solutions were considered in [47].1135.4. Neutral scalar field: Colour superconductivityview it is presumably related to the warping between IR and UV regions ofthe geometry.11 From the boundary viewpoint, the low critical temperaturemay be explained by the BKL scaling [111, 115, 116] near a quantum criticalpoint.For a given T and µ, we can use (5.20) to evaluate the action for thesolution and compare this with the action for the soliton and/or Reissner-Nordstrom solution with the same T and µ. We find that the action forthe new solutions is always less than the action for the Reissner-Nordstromsolutions, and is also less than the action for the soliton solutions for chem-ical potential in a region µ > µc(T ). Thus, the solutions with scalar fieldrepresent the equilibrium phase in the region T/µ < γ, µ > µc(T ), as shownin Figures 5.1 – 5.3 above.The transition between the deconfined and superconducting phases issecond order, while the transition between confined and superconductingphases is first order. The place where these phase boundaries meet repre-sents a triple point for the phase diagram where the three phases (confined,deconfined, superconducting) can coexist.5.4.2 Critical temperatureFor fixed m2, the value of ψ(0) in the solutions increases from zero at T/µ =γ, diverging as T/µ → 0. Since ψ is small everywhere near T/µ = γ, thecritical value of T/µ will be the value where the ψ equation, linearizedaround the Reissner-Nordstrom background, has a solution with the correctboundary conditions. Thus, we consider the equationψ′′ + (4r +g′g )ψ′ − m2g ψ = 0 , (5.49)where (setting r+ = 1)g(r) = r2 −(1 + 3µ28) 1r3 +3µ28r6 , (5.50)and find the value µ = µc for which the equation admits a solution withboundary conditions ψ(1) = 1 (we are free to choose this), ψ′(1) = m2/g′(1)and the right falloff (ψ1 = 0) at infinity.1211By considering the alternate quantization mentioned in section 2 and fine-tuning themass so that the dual operator has the smallest possible dimension consistent with uni-tarity in the dual field theory, we can obtain γ as large as 0.0151, so even under the mostfavorable circumstances, the critical T/µ is quite small.12To obtain a very accurate result, we first find a series solution ψlow near r = 1 withψ(1) = 1 (we are free to choose this) and ψ′(1) = m2/g′(1) and find a series solution ψhigh1145.4. Neutral scalar field: Colour superconductivity0.00010.011-6.5-6-5.5-5m21E-121E-101E-080.0000010.0001(T/μ) CFigure 5.5: Critical T/µ vs m2 of neutral scalar (filled circles). Massis above BF bound asymptotically but below BF bound in near-horizonregion of zero-temperature background solution in the range −6.25 ≤ m2 <−5. Unfilled circles represent critical values in the theory with alternatequantization of the scalar field, possible in the range −6.25 ≤ m2 < −5.25.The choice r+ = 1 implies that T = (5 − 9µ2/8)/(4pi), so we haveγ = (5 − 9µ2c/8)/(4piµc). The results for γ(m2) are plotted in Figure 5.5.For comparison, we also considered the theory defined with the alternatequantization (ψ∞2 = 0) of the bulk scalar field (mentioned in section 2). Aswe see in Figure 5.5, the critical temperatures are somewhat larger in thiscase, but still much smaller than 1 relative to µ.5.4.3 Properties of the superconducting phaseIn the superconducting phase, it is interesting to ask how the charge den-sity and free energy behave as a function of chemical potential. Since thesolutions (as for the planar RN-black hole solutions) are trivially related tosolutions where the w direction is non-compact, and since the underlyingtheory has a conformal symmetry, physical quantities in this phase (or infor large r with the correct fall-off (ψ1 = 0) at infinity. Starting with ψlow and ψ′low atsome r = r1 where the low r series solution is still very accurate, we then numericallyintegrate up to r = r2 where the large r series is very accurate and then find µ for whichψ′num(r2)/ψnum(r2) = ψ′high/ψhigh.1155.5. Charged scalar field: Flavour superconductivitythe RN phase) behave as µnF (T/µ) for some non-trivial function F anda power n.13 At the critical value of T/µ, we have a second order transi-tion from the RN phase to the phase with scalar, so the free energy and itsderivatives, and other physical quantities such as the density, are continuousacross the transition. Thus, the relevant function F in these cases will bethe same for the two phases across the transition. We find that the functionF for either the charge density or the free energy changes very little betweenthe very small value of T/µ where the the transition occurs and the T → 0limit. Thus, to a good approximation, we find that the density and freeenergy behave in the superconducting phase in the same way as for the zerotemperature limit of the RN phase. For R=2/5, we haveρ ≈ 0.320µ4 , (5.51)whileG ≈ −.064µ5 . (5.52)In both cases, the behaviour at large µ is governed by the underlying 4+1dimensional conformal field theory.5.5 Charged scalar field: FlavoursuperconductivityIn this section, we generalize our holographic model to the case where thescalar field is charged under the gauge field in the bulk. As we discussed inthe introduction, this implies that the dual field theory includes some low-dimension gauge-invariant operator with charge, so the charge in this caseis more naturally thought of as some isospin-type charge (since the smallestgauge-invariant operators carrying baryon charge have dimensions of orderN).A significant qualitative difference in this case is that a scalar field con-densate acts as a source for the electric field in the bulk, so it is possible tohave solutions with no horizon carrying a finite charge density in the fieldtheory. This gives the possibility of a fourth phase in which the scalar fieldcondenses in the soliton background.To obtain the action for the charged scalar case, we begin with the action(5.3) and make the replacement ∂µψ → ∂µψ − iqAµψ. The results of theprevious section correspond to q = 0.13If the solutions instead depended on the circle direction in a non-trivial way, we mighthave a general function of RT and Rµ.1165.5. Charged scalar field: Flavour superconductivity5.5.1 Low-temperature horizon free solutions with scalarAbove some critical value of µ, there exist horizon-free geometries with ascalar field condensate. The solutions may be parameterized by the magni-tude of the scalar at the IR tip of the geometry, and we will find a singlesolution for each such value. To determine these geometries, we need to takeinto account back-reaction on the metric. The most general solution withthe desired properties can be described by the ansatzds2 = r2(eA(r)B(r)dw2 + dx2 + dy2 + dz2 − eC(r)dt2) + dr2r2B(r) ,At = φ(r) ,ψ = ψ(r) , (5.53)where we demand A(∞) = C(∞) = 0 and B(∞) = 1. As for the solitongeometry, we expect that the w circle is contractible in the bulk so thatB(r0) = 0 for some r0. For the geometry to be smooth at this point, theperiodicity of the w direction must be chosen so that2piR = 4pie−A(r0)/2r20B′(r0). (5.54)Starting from the action (5.3) with scalar derivatives replaced by covari-ant derivatives, the scalar and Maxwell equations are:ψ′′ +(6r +A′2 +B′B +C ′2)ψ′ + 1r2B(e−C(qφ)2r2 −m2)ψ = 0 , (5.55)φ′′ +(4r +A′2 +B′B −C ′2)φ′ − 2ψ2q2φr2B = 0 . (5.56)Following [58], we find that the Einstein equations give:A′ = 2r2C ′′ + r2C ′2 + 4rC ′ + 4r2ψ′2 − 2e−Cφ′2r(8 + rC ′) , (5.57)C ′′ + 12C′2 +(6r +A′2 +B′B)C ′ −(φ′2 + 2(qφ)2ψ2r2B) e−Cr2 = 0 , (5.58)B′(4r −C ′2)+B(ψ′2 − 12A′C ′ + e−Cφ′22r2 +20r2)+ 1r2(e−C(qφ)2ψ2r2 +m2ψ2 − 20)= 0 . (5.59)1175.5. Charged scalar field: Flavour superconductivityThese equations have two scaling symmetries,ψ˜(r) = ψ(ar) , φ˜(r) = 1aφ(ar) , A˜(r) = A(ar),B˜(r) = B(ar) , C˜(r) = C(ar) , (5.60)andC˜ = C + ∆ , φ˜ = e∆2 φ . (5.61)Numerical evaluation of solutionsTo find solutions, we first use the scaling symmetries to fix r0 = 1 andC(r0) = 0. For each value of ψ(1), we use φ(1) as a shooting parameter,choosing the value so that ψ has the desired behaviour for large r. Fromthe solution obtained in this way, we can use (5.61) with ∆ = −C(∞) toobtain the desired boundary condition C(∞) = 0 in the rescaled solution.From (5.54), we see that the choice r0 = 1 corresponds to a periodicity forthe w direction equal to2piR = 4pie−A(1)/2B′(1) . (5.62)which will generally be different for solutions corresponding to different val-ues of ψ(1). In order to obtain solutions corresponding to our chosen valueR = 2/5 (such that the action for the soliton solution is -1) we use the scal-ing (5.60), taking a = B′(1)/5e−A(∞)/2. After all the scalings, we calculatethe chemical potential and action (making use of (5.20)) asµ = φ(∞) , S = [B] 1r5. (5.63)The action is plotted against chemical potential for various values of qin Figures 5.6, 5.7, and 5.8 taking the example of a mass just above the BFbound, m2 = −6.We find that for large enough values of q, the chemical potential increasesmonotonically and the action decreases monotonically as we increase ψ(r0).This implies that we have a second order transition to the superconductingphase at a critical value, which can be determined by a linearized analysis(see appendix C.1) to be µ ≈ 1.0125/q.Below q ≈ 1.35, the chemical potential is no longer monotonic in ψ(r0).We see that for q = 1.3, this results in a second order phase transition atµ ≈ 1.558, followed by a first order phase transition at µ ≈ 1.616 (takingR = 2/5). For smaller q (e.g. q = 1.2 in Figure 5.8), we simply have a1185.5. Charged scalar field: Flavour superconductivity-10.911.11.21.31.4Sμ-2Sq=2Figure 5.6: Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 2.011.11.21.31.41.51.61.71.81.92Sμ-2-1Sq = 1.3Figure 5.7: Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 1.3.1195.5. Charged scalar field: Flavour superconductivity00.50.70.91.11.31.51.71.92.1μ-2-1 Sq=1.2Figure 5.8: Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 1.2.first order transition to the superconducting phase at a value of chemicalpotential that is less than the value for the solution with infinitesimal scalarfield. All of these results are completely analogous to the lower-dimensionalresults of [58].5.5.2 Hairy black hole solutionsAt high temperatures, the w circle is no longer contractible, and we assumethat (as for the solutions without scalar field) the solution can be obtained byperiodic identification of a solution with boundary R4,1 instead of R3,1×S1.Thus, we take the ansatzds2 = −g(r)e−χ(r)dt2 + dr2g(r) + r2(dw2 + dx2 + dy2 + dz2) ,At = φ(r) ,ψ = ψ(r) .The scalar and Maxwell’s equations areψ′′ +(4r −χ′2 +g′g)ψ′ + 1g(eχq2φ2g −m2)ψ = 0 , (5.64)1205.5. Charged scalar field: Flavour superconductivityφ′′ +(4r +χ′2)φ′ − 2q2ψ2g φ = 0 , (5.65)while the Einstein equations are satisfied ifχ′ + rψ′22 +reχq2φ2ψ22g2 = 0 , (5.66)g′ +(3r −χ′2)g + reχφ′28 +m2rψ24 − 5r = 0 . (5.67)These have two symmetries:ψ˜(r) = ψ(ar) , φ˜(r) = 1aφ(ar) , χ˜(r) = χ(ar) , g˜(r) =1a2 g(ar) ,(5.68)andχ˜ = χ+ ∆ , φ˜ = e−∆2 φ . (5.69)As we did for q = 0, we would like to find solutions with a horizon at somer = r+. The electric potential must also vanish at the horizon, and we arelooking for solutions for which the leading falloff ψ1 in (5.7) vanishes for thescalar. Also, multiplying the first equation (5.64) by g and evaluating atr = r+ fixes ψ′(r+) in terms of ψ(r+) and g′(r+). Altogether, our boundaryconditions areg(r+) = 0 , φ(r+) = 0 , χ(∞) = 0 , ψ1 = 0 , (5.70)andψ′(r+) =8m2ψ(r+)40r+ − 2m2r2+ψ2(r+)− r+eχ(r+)(φ′(r+))2. (5.71)The remaining freedom to choose r+ and φ′(r+) leads to a family of solutionswith different T and µ. Explicitly, we haveµ = φ(∞) , T = 14pig′(r+)e−χ(r+)/2 . (5.72)Solutions with the same T/µ are simply related by the scaling symmetry(5.68).1215.6. DiscussionNumerical evaluation of solutionsTo find solutions in practice, we can make use of the symmetries (5.68),(5.69) to initially set r+ = 1 and χ(0) = 0 and solve the equations withboundary conditionsg(1) = 0 , χ(0) = 0 , φ(1) = 0 , φ′(1) = E0 , ψ(1) = ψ0 , (5.73)andψ′(1) = 8m2ψ040− 2m2ψ20 − E20. (5.74)We use E0 as a shooting parameter to enforce ψ1 = 0, and find onesolution for each ψ0. From these solutions, we apply the symmetry (5.69)with ∆ = −χ(∞) to restore χ(∞) = 0 and finally use the symmetry (5.68)to scale to the desired temperature or chemical potential.5.5.3 Phase diagramsAt a generic point in the phase diagram, we can have up to four solutions(AdS soliton, planar RN black hole, soliton with scalar, black hole withscalar), or more in cases where there is more than one solution of a giventype.To map out the phase diagram, we evaluate the action for the varioussolutions using the methods of section 2. The equilibrium phase correspondsto the solution with lowest action. The phase diagrams for q = 1.3 and q = 2(in the case m2 = −6) are shown in Figures 5.9 and 5.10/5.11.For large q, the condensation of the scalar field occurs in a region of thephase diagram where the back-reaction is negligible, so the phase diagrammay be understood here (for µ ∼ 1/q) by treating the gauge field and scalaron a fixed background (the Schwarzschild black hole). The resulting phasediagram is shown in Figure 5.12.5.6 DiscussionIn this chapter, we have investigated the phase structure for a simple classof holographic systems which we have argued have the minimal set of in-gredients to holographically describe the phenomenon of colour supercon-ductivity. Even in these simple models, we find a rich phase structure withfeatures similar to the conjectured behaviour of QCD at finite temperature1225.6. Discussion0.0 0.5 1.0 1.5 2.0Μ0.00.10.20.30.40.5TFigure 5.9: Phase diagram for m2 = −6 and q = 2. Clockwise from theorigin, the phases correspond to the AdS soliton (confined), RN black hole,black hole with scalar, and soliton with scalar.0.0 0.5 1.0 1.5 2.0Μ0.00.10.20.30.40.5TFigure 5.10: Phase diagram for m2 = −6 and q = 1.3. Clockwise from theorigin, the phases correspond to the AdS soliton (confined), RN black hole,black hole with scalar, and soliton with scalar.1235.6. Discussion1.5 1.6 1.7 1.8 1.9 2.0Μ0.000000.000050.000100.000150.00020TFigure 5.11: Small temperature region of phase diagram for m2 = −6 andq = 1.3. Dashed line represents a first order transition within the solitonwith scalar phase.1 2 3 4Μq0.20.40.60.8TFigure 5.12: Phase diagram for large q, m2 = −6.1245.6. Discussionand baryon chemical potential. It would be useful to verify the thermody-namic stability (and also the stability towards gravitational perturbations)of the phases that we have identified. This could indicate regions of thephase diagram where we have not yet identified the true equilibrium phasefor the model, for example since our ansatz might be too symmetric.We have calculated some of the basic thermodynamic observables, butit would be interesting to investigate more fully the physical properties ofthe various phases and establish more definitively a connection between thephase we find at large µ and small temperature and the physics of coloursuperconductivity.Apart from the ψψψ†ψ† condensate that we can see directly using theingredients of our model, there are various other features that character-ize a colour superconductivity phase [31]. Typically, the breaking of gaugesymmetry is accompanied by some breaking of exact or approximate flavoursymmetries. Thus, the superconducting phase has a low-energy spectrumcharacterized by Goldstone bosons or pseudo-Goldstone bosons associatedwith the broken flavour symmetries, together with massive vector bosonsassociated with the spontaneously broken gauge symmetry. It would there-fore be interesting to analyze the spectrum of fluctuations in our model tocompare with these expectations.A caveat related to looking for features associated with the global flavoursymmetries (and their breaking) in our model is that we may not haveincluded enough ingredients in our bottom-up approach for all these featuresto be present. In simple models where the flavour degrees of freedom areassociated with probe branes, there are explicit gauge fields in the bulk dualto the global symmetry current operators. However, in fully back-reactedsolutions (appropriate for studying Nf ∼ Nc), these branes are replaced bya modified geometry with additional fluxes (for an explicit example of suchsolutions, see [117]). In these solutions (which we are trying to model in ourapproach), it is less clear how to identify the global symmetry group from thegravity solution, but presumably it has to do with some detailed propertiesof the geometry. Thus, it is possible that the Goldstone modes associatedwith broken flavour symmetries correspond to fluctuations in some fields(e.g. form-fields) that we have not included.The colour superconducting condensate also breaks the global baryonnumber symmetry, so there should be an associated Goldstone boson relatedto the phase of the condensate, and associated superfluidity phenomena.In other holographic models with superfluidity, the condensate is dual toa charged scalar field in the bulk and the Goldstone mode is related tofluctuations in the phase of this field. However, as we mentioned in the1255.6. Discussionintroduction, the baryon operator has dimension of order N , so we do notexpect a light charged scalar field in the bulk. In a more complete top-downmodel, the baryon operator may be related to some non-perturbative degreesof freedom (such as D-branes) in the bulk, and it may be necessary to havea model with these degrees of freedom included in order to directly see theGoldstone mode from the bulk physics. Related to these observations, itmay be interesting to probe our model with D-branes (put in by hand), inorder to make the relation to microscopic physics more manifest, and tohelp gain a better understanding of the phenomenological parameters of ourmodel.There are a number of variants on the model that would be interestingto study. First, the breaking of scale-invariance, implemented in our modelby the varying circle direction in the bulk, could be achieved in other ways,replacing gww with a more general scalar field, as in the model of [59].In the setup of that paper, the transition between confined and deconfinedphases was found to exhibit crossover behaviour at small chemical potential,a feature expected in the real QCD phase diagram and expected generallyfor massive quarks with sufficiently large Nf/Nc. It would be interesting tolook for an even more realistic holographic model by incorporating featuresof the model we have studied here and the model of [59].It would also be interesting to look at the effects of a Chern-Simons termfor the bulk gauge field. In [80] and [90], it was shown that such a term(with sufficiently large coefficient) gives rise to an instability toward inho-mogeneous phases, perhaps associated with the chiral density wave phasebelieved to exist at large density in QCD with Nf  Nc [68, 69]. It isinteresting to investigate the interplay between these inhomogeneous insta-bilities and the superconducting instabilities discussed in the present paper.It would also be interesting to consider more general actions (such as Born-Infeld) for the gauge field, interaction terms for the scalar field in the bulk,or other couplings between the scalar field and gauge field.Finally, once the technical challenges of writing down fully back-reactedsolutions for top-down models of holographic QCD with Nf ∼ Nc havebeen overcome, it will be interesting to see whether the basic features wefind here are manifested in the more complete string-theoretic models. Ifcertain features are found to be universal, these might taken as qualitativepredictions for the QCD phase diagram, or at least motivate an effort tounderstand whether these features are also present in the phase diagram ofreal-world QCD.126Chapter 6Holographic baryons fromoblate instantons16.1 IntroductionPerhaps the most successful holographic model of QCD has been the Sakai-Sugimoto model [13, 118], defined by the physics of Nf probe D8-branes inthe background dual to the decoupling limit ofNc D4-branes compactified ona circle with antiperiodic boundary conditions for the fermions. This modelreproduces many features of real QCD, including chiral symmetry breaking,a deconfinement transition [32, 41], and a realistic meson spectrum.The description of baryons in the Sakai-Sugimoto model involves soli-tonic configurations of the Yang-Mills field on the D8-brane.2 In a sim-plified ansatz where the Yang-Mills field is taken to depend only on thefour non-compact spatial directions in the bulk, configurations with baryoncharge are precisely those configurations with non-zero instanton numberfor this reduced 4D Yang-Mills field [13, 119–121]. This connection betweenbaryon charge and bulk instanton number stems from a Chern-Simons terms tr (F ∧ F ) in the reduced D8-brane action. Here, tr (F ∧ F ) is the instan-ton density for the SU(2) part of the Yang-Mills field, and s is the U(1)part of the Yang-Mills field, dual to the baryon current operator in the fieldtheory.To date, the study of baryons in the Sakai-Sugimoto model has beensomewhat unsatisfactory, for several reasons: I) While the action for thegauge field is of Born-Infeld type, only the leading Yang-Mills terms aretypically used when studying the instantons. II) For large ’t Hooft couplingwhere the model can be studied most reliably, the size of the instanton inthe bulk has been argued to be much smaller than the size of the compactdirections in the bulk. In this case, the assumption that the gauge field doesnot depend on the compact directions is questionable. III) Rather than1A version of this chapter has been published [5].2Mesons correspond to pertubative excitations of the D8-branes.1276.1. Introductionsolving the bulk equations to determine the precise solitonic configurationof the Yang-Mills field, the form has been taken to be that of a flat-spaceSO(4) symmetric instanton, with the size of the instanton as the only freeparameter.The assumptions in I) and II) here amount to replacing the originaltop-down Sakai-Sugimoto model with a phenomenological (bottom-up) holo-graphic model that retains many of the same successes as the Sakai-Sugimotomodel. For the present chapter, we continue to make these assumptions,though we hope to relax them in future work in order to better understandbaryons in the fully-consistent top-down model. Our goal in the presentchapter is to overcome the third deficiency, by setting up and solving nu-merically a set of partial differential equations that determine the properform of the soliton.3 Using these solutions, we are able to calculate themass and baryon charge distribution of the baryons as a function of themodel parameter γ (proportional to the inverse ’t Hooft coupling λ) thatcontrols the strength of the Chern-Simons term relative to the Yang-Millsterm.One motivation for our study is the work of [125], which points out thatthe flat-space instanton approximation used previously does not give thecorrect large radius asymptotic behaviour (known from model-independentconstraints) for the baryon form factors (computed for example in [126–128]). Via a perturbative expansion of the equations at large radius, it waslater shown [33] that by relaxing the assumption of SO(4) symmetry, theproper asymptotic behaviour can be recovered.4 Thus, we expect that byconstructing and studying the complete solutions, we can obtain a signifi-cantly improved picture of the properties of baryons in holographic QCD.The solutions that we find take the form of ‘oblate instantons’: comparedwith the SO(4) symmetric configurations, the correct solutions are deformedto configurations with SO(3) symmetry that are spread out more in the fieldtheory directions than in the radial direction. This shape is expected. TheCoulomb repulsion between instanton charge density at different locations(induced by the Chern-Simons coupling to the Abelian gauge field) actssymmetrically in all directions, impelling the instanton to spread out bothin the radial and field theory directions. Gravitational forces in the bulk limitthe spreading in the radial direction, but there are no equivalent forces actingto radially compress the instanton in the field theory directions. Thus, the3[122–124] have used a similar numerical approach in other phenomenological holo-graphic QCD models.4In the earlier work [129], a similar expansion was used in a phenomenological holo-graphic QCD model. See also [130] for a recent related study.1286.1. Introductioninstanton is oblate, compressed in one direction relative to the other three.The anisotropy is limited by the Yang-Mills action for the SU(2) gauge field,which in flat space is minimized (in the one-instanton sector) for sphericallysymmetric configurations.The size and anisotropy of the instantons is controlled by the parameterγ (related to the inverse ’t Hooft coupling in the original model). For small γ,the spreading effects of the Chern-Simons term are small, and the instantonsbecome small and approximately symmetrical near their core. For larger γ,the instantons become significantly larger and more anisotropic. Using ournumerics, we are able to construct solutions up to γ of order 100 and evaluatethe mass and baryon charge profiles of the corresponding baryons.While our model is not expected to quantitatively match real-world QCDmeasurements, previous studies have found that the meson spectrum agreesreasonably well with the spectrum in QCD for a suitable choice of the pa-rameter γ. Thus, it is interesting to compare the mass and size of thebaryons in our model to the QCD values for the light nucleons. Using thevalue γ = 2.55 that gives the best fit to the meson spectrum [127], we findthat the mass and baryon charge radius of the baryon are 1.19 GeV and0.90 fm. This mass is significantly closer to realistic values (∼ 0.94 GeVfor the proton and neutron) than the previous value of 1.60 GeV based onthe SO(4) symmetric ansatz. The baryon charge radius is quite similar tomeasured values for the size of the proton and neutron. For example, theelectric charge radius of a proton has been measured to be in the range 0.84fm – 0.88 fm [131], while the magnetic radii of the proton and neutron arelisted in [131] as 0.78 fm and 0.86 fm respectively.An outline for the remainder of the chapter is as follows: In section 6.2,we briefly review the description of baryons in the Sakai-Sugimoto model andset up the problem. In section 6.3, we describe our numerical approach tothe equations. In section 6.4, we describe physical properties of the solution,focusing on the baryon mass and the distribution of baryon charge (chargedensity as a function of radius), as a function of γ. Our main results maybe found in Figures 6.4 and 6.7. We conclude in section 6.5 with a briefdiscussion of directions for future work.Note: While this work was being completed, [132] appeared, which alsopresents a numerical solution of the Sakai-Sugimoto NB = 1 soliton, usingdifferent methods, and which has some overlap with this paper.1296.2. Baryons as solitons in the Sakai-Sugimoto model6.2 Baryons as solitons in the Sakai-SugimotomodelIn this section, we give a brief review of the Sakai-Sugimoto model and setup the construction of a baryon in this model.The Sakai-Sugimoto model consists of Nf probe D8 branes in the nearhorizon geometry of Nc D4 branes wrapped on a circle with anti-periodicboundary conditions for the fermions. The metric of the D4 background is[32]ds2 = λ3 l2s(49u32(ηµνdxµdxν + f(u)dx24)+ 1u 32( du2f(u) + u2dΩ24)),eΦ =(λ3) 32 u 34piNc, f(u) = 1− 1u3 , F4 = dC3 =2piNcV44, (6.1)where 4 is the volume form on S4 and V4 is the volume of the unit 4-sphere.The direction x4, with radius 2pi, corresponds to the direction on whichthe D4-branes are compactified. The u and x4 directions form a cigar-typegeometry and the space pinches off at u = 1. The four dimensional SU(Nc)gauge theory dual to this metric has a dimensionless coupling λ.The flavor degrees of freedom are provided by Nf probe D8 branes inthe background (6.1). The action for a single D8 brane isSD8 = −µ8∫d9σe−Φ√−det(gab + 2piα′Fab) + SCS , (6.2)with µ8 = 1/(2pi)8l9s and where SCS is the Chern-Simons term. Below, weexpand this action around a particular embedding and take the non-Abeliangeneralization of the result to define the action we consider. We take theprobe branes to wrap the sphere directions and fill the 3 + 1 field theorydirections. Then, the embedding is described by a curve x4(u) in the cigargeometry, with boundary conditions fixing the position of the probe branesas u→∞.In this chapter, we consider only the antipodal case, in which the ends ofthe probe branes are held at opposite sides of the x4 circle. The minimumenergy configuration with these boundary conditions is that in which theprobe branes extend down the cigar at constant angle x4, meeting at u = 1.Going to the radial coordinate z defined by u3 = 1 + z2, and expanding theaction (6.2) for small gauge fields around the antipodal embedding gives the1306.2. Baryons as solitons in the Sakai-Sugimoto modelmodel we consider [119]:S = −κ∫d4xdz tr[12h(z)F2µν + k(z)F2µz]+ Nc24pi2∫M5tr(AF2 − i2A3F − 110A5), (6.3)where κ = λNc/(216pi3), h(z) = (1 + z2)−1/3 and k(z) = 1 + z2. A is aU(Nf ) gauge field with field strength F = dA + iA ∧ A. In this paper, wefocus on the case Nf = 2. We split the gauge field into SU(2) and U(1)parts as A = A+ 1212Aˆ.5The competing forces that determine the size of the soliton are evidentin the effective action (6.3). First, the gravitational potential of the curvedbackground will work to localize the soliton near the tip of the cigar, atz = 0. This will be counterbalanced by the repulsive potential due to thecoupling between the U(1) part of the gauge field and the instanton chargein the Chern-Simons term. At large λ, the effect of the Chern-Simons term issuppressed, and the result is a small instanton, which was previously approx-imated by the flat-space SO(4) symmetric BPST instanton. As discussed in[33], this approach fails to properly describe several aspects of the baryon.Due to the curved background, the actual solution will only be invariantunder SO(3) rotations in the field theory directions. This distinction is es-pecially important if we wish to use this model away from the strict large λlimit, as in that case, the soliton can become large such that the effects ofthe curved background are important for more than just the asymptotics ofthe solution.The most general field configuration invariant under combined SO(3)rotations and SU(2) gauge transformations may be written as [133, 134]6Aaj =φ2 + 1r2 jakxk +φ1r3 [δjar2 − xjxa] +Arxjxar2 ,Aaz = Azxar , Aˆ0 = sˆ. (6.4)where each of the fields are functions of the boundary radial coordinater2 = xaxa and the holographic radial coordinate z. The ranges of thesecoordinates are 0 < r <∞ and −∞ < z <∞. With these definitions, thereis a residual gauge symmetry under which Aµ transforms as a U(1) gauge5We define the SU(2) generators to satisfy [τa, τ b] = iεabcτ c.6This ansatz has also been used in the study of holographic QCD in a phenomenologicalmodel [122–124] and was applied to the Sakai-Sugimoto model in [33].1316.2. Baryons as solitons in the Sakai-Sugimoto modelfield in the r−z plane and φ = φ1 + iφ2 transforms as a complex scalar fieldwith charge (−1), so that Dµφ = ∂µφ− iAµφ.The free energy of the system is given by the Euclidean action evaluatedon the solution. Since we work at zero temperature and consider only staticsolutions, the mass-energy equals the free energy, and we only pick up aminus sign from the analytic continuation. Then, in terms of the aboveansatz, the mass of the system is written asM = MYM +MCS , (6.5)where∫dtM = −S,MYM = 4piκ∫drdz[h(z)|Drφ|2 + k(z)|Dzφ|2 +14r2k(z)F 2µν+ 12r2h(z)(1− |φ|2)2 − 12r2 (h(z)(∂rsˆ)2 + k(z)(∂z sˆ)2)](6.6)andMCS = −2piκγ∫drdz sˆ µν [∂µ(−iφ∗Dνφ+ h.c.) + Fµν ] , (6.7)with γ = Nc/(16pi2κ) = 27pi/(2λ) and Fµν = ∂µAν−∂νAµ. For the classicalsolution, γ is the only parameter in the system. It controls the relativestrength of the Chern-Simons term; a larger γ will increase the size of thesoliton.The equations of motion that follow from extremizing the mass-energyare given by0 = Dr (h(z)Drφ) +Dz (k(z)Dzφ) +h(z)r2 φ(1− |φ|2) + iγµν∂µsˆDνφ,0 = ∂r(r2k(z)Frz)− k(z) (iφ∗Dzφ+ h.c.)− γrz∂rsˆ(1− |φ|2),0 = ∂z(r2k(z)Fzr)− h(z) (iφ∗Drφ+ h.c.)− γzr∂z sˆ(1− |φ|2),0 = ∂r(h(z)r2∂rsˆ)+ ∂z(k(z)r2∂z sˆ)− γ2 µν [∂µ(−iφ∗Dνφ+ h.c) + Fµν ] .(6.8)The baryon number is given by the instanton number of the non-Abelian1326.3. Numerical setup and boundary conditionspart of the gauge field,NB =18pi2∫d4x trF ∧ F= 14pi∫drdz µν [∂µ(−iφ∗Dνφ+ h.c.) + Fµν ]= 14pi∫drdz (∂rqr + ∂zqz), (6.9)where F is the field strength of the SU(2) gauge field A andqr = (−iφ∗Dzφ+ h.c.) + 2Az, qz = (iφ∗Drφ+ h.c.)− 2Ar. (6.10)Since the expression is a total derivative, the boundary conditions on ourSU(2) gauge field will set the baryon charge. We study configurations withNB = 1.6.3 Numerical setup and boundary conditionsIn this section we describe our setup, including our boundary conditions,gauge fixing, and details about the numerical procedure we use.6.3.1 Gauge fixingThere is a residual U(1) gauge freedom in the above ansatz, and we chooseto use the Lorentz gauge χ ≡ ∂µAµ = 0. Our gauge fixing is achieved byadding a gauge fixing term to the equations of motion, analogous to theEinstein-DeTurck method developed in [135]. Alternatively, one can viewthis procedure as adding a gauge fixing term to the action, and working inthe Feynman gauge.As a result one obtains modified equations of motion in which the prin-cipal part of the equations is simply the standard elliptic operator ∂2r + ∂2z .Once a solution is obtained, one has to make sure it is also a solution to theoriginal, unmodified equations, i.e that χ = 0. This has to be checked nu-merically, but can be expected to be satisfied since χ is a harmonic function,so with suitably chosen boundary conditions (for example such that χ = 0on the boundaries of the integration domain) uniqueness of the solution tothe Laplace equation guarantees that χ = 0. For the solutions presentedhere, the gauge condition is well satisfied as the L2 norm of χ, normalizedby the number of grid points N , satisfies |χ|/N < 10−5.1336.3. Numerical setup and boundary conditions6.3.2 Ansatz and boundary conditionsFor small γ, the soliton solution is well localized near the origin (r, z) =(0, 0). For small z, k(z) ∼ h(z) ∼ 1 and the SU(2) part of the actionreduces to that of the Witten model [133] for instantons. Then, in thisregime, we expect the solution to possess an approximate SO(4) symmetry,and thus we find it convenient to use the spherical coordinatesR =√r2 + z2, θ = arctan(r/z) (6.11)for our numerical calculation. The inverse transformation is r = R sin θ, z =R cos θ. One can show that by restricting the ansatz (6.4) to SO(4) sym-metry,7 the solution can be written in terms of two spherically symmetricfunctions f(R) and g(R) asφ1 = −rzf(R), φ2 = r2f(R)−1, Ar = −zf(R), Az = rf(R), sˆ = g(R).(6.12)In this parametrization, the BPST instanton is given byf(R) = 2ρ2 +R2 , g(R) = 0, (6.13)where ρ determines the size of the energy distribution. The non-trivialwinding of the instanton is built into the expressions in (6.12) through theappropriate factors of r and z and the factor of 2 in the numerator of f(R)fixes the winding number to be NB = 1. The BPST solution has a scalingsymmetry in that it admits solutions of arbitrary scale ρ.The factors of k(z) and h(z) in the Sakai-Sugimoto model break theSO(4) symmetry. This has two effects on the SO(4) ansatz. First, thefunctions φ1, φ2, Ar, and Az will not be related to each other through thecommon function f(R). Second, the functions appearing in the ansatz mustbe promoted to functions of both the radial coordinate R and the angle θ.These considerations motivate our reduced ansatz asφ1 = −(R2 sin θ cos θ1 +R2)ψ1(R, θ), φ2 =(R2 sin2 θ1 +R2)ψ2(R, θ)− 1,Ar = −(R cos θ1 +R2)ar(R, θ), Az =(R sin θ1 +R2)az(R, θ), sˆ =s(R, θ)R sin θ .(6.14)7This assumption would be valid if k and h were spherically symmetric. The Chern-Simons term does not break the SO(4) symmetry.1346.3. Numerical setup and boundary conditionsIn each of the non-Abelian gauge field functions we include a factor of (1 +R2)−1 such that we may use Dirichlet boundary conditions at R = ∞ tofix the baryon number. We rescale s by a factor of r−1 = (R sin θ)−1 inorder to have better control over the behaviour of the gauge field near ther = 0 boundary. We numerically solve for the five functions {ψ1, ψ2, ar, az, s}on the domain (0 ≤ R < ∞, 0 ≤ θ ≤ pi/2) corresponding to (0 ≤ r <∞, 0 ≤ z < ∞). In practice, we use a finite cutoff at R = R∞, chosensuch that the physical data extracted from the solution does not depend onit. The symmetries of the solution around z = 0 are used to extend it to(−∞ < z ≤ 0).In terms of the coordinates (R, θ), the baryon charge becomesNB =14pi∫dRdθ (∂RqR + ∂θqθ), (6.15)where we have definedqR = R(sin θqr + cos θqz), qθ = cos θqr − sin θqz. (6.16)The baryon number is given by the boundary integralsNB =14pi(∫ ∞0dR qθ∣∣∣θ=0+∫ pi0dθ qR∣∣∣R=∞+∫ 0∞dR qθ∣∣∣θ=pi+∫ 0pidθ qR∣∣∣R=0).(6.17)Plugging our ansatz into qR and qθ and evaluating on the boundaries showsthat the only contribution to the winding is from the boundary at R =∞.Thus, the baryon number reduces toNB =12pi∫ pi/20dθ qR∣∣∣R=∞, (6.18)and we use boundary conditions at the cutoff R∞ to impose that NB = 1.The boundary conditions we use are as follows. At θ = pi/2 (whichmaps back to z = 0), we have Neumann conditions on all the fields, asthe odd/even characteristics of the functions about z = 0 are built into theansatz (6.14). At this boundary χ = 0 implies ∂θaz = 0 so that this bound-ary condition satisfies the gauge choice. To obtain boundary conditions atθ = 0 (r = 0), we expand the equations of motion for small θ. Satisfyingthese order by order in θ gives a set of conditions on the fields. A subset of1356.3. Numerical setup and boundary conditionsthese conditions that results in a convergent solution is given by8θ = 0 : ∂θψ1 = 0, ∂θψ2 = 0, ar = ψ1, ∂θaz = 0, s = 0. (6.19)The gauge condition at θ = 0 can be shown to be satisfied on a solutiongiven these boundary conditions. At the origin R = 0, a similar procedureyieldsR = 0 : ∂Rψ1 = 0, ∂Rψ2 = 0, ∂Rar = 0, ∂Raz = 0, s = 0. (6.20)We do not explicitly satisfy the gauge condition at R = 0.9 At the cutoffR∞, the boundary conditions are determined by behaviour of the gaugefield Aˆ0 and the winding number NB = 1. As discussed below, in section6.4.2, the field theory density of baryon charge ρB(r) (defined below) isproportional to the coefficient of the z−1 falloff of the Abelian gauge fieldAˆ0, at large z. In order to reliably calculate ρB(r), we therefore imposethat s falls off as z−1 by using the boundary condition s = −z∂zs, suitablytranslated into (R, θ) coordinates, at the cutoff R∞. Since we rescaled theSU(2) gauge fields by (1 + R2)−1, we are left with Dirichlet conditions onthe other functions, givingR = R∞ : ψ1 = ψ2 = ar = az = 2, s = −R cos2 θ ∂Rs+ sin θ cos θ ∂θs.(6.21)Given the asymptotic boundary behaviour of the fields, the gauge choice issatisfied for large R∞. With these large R conditions, we have qR = 4 andso NB = 1, as desired.6.3.3 Numerical procedureWe solve the equations of motion by using spectral methods on a Cheby-shev grid, using Newton’s method to solve the resulting non-linear algebraicequations.10 For the results presented here, we take the number of gridpoints to be (NR, Nθ) = (50, 25). We introduce a cutoff at large R = R∞.For a large enough cutoff we can reliably read off the z−1 falloff in order toobtain information about the baryon charge density. However, if the cutoff8In practice, we use the boundary condition ∂θaz = 12R∂θ∂Rψ1 during the solvingprocedure, as we found empirically that this results in a more stable Newton iteration.Once the numerical procedure converges, the solution satisfies the boundary conditionsgiven here.9We check that the gauge condition χ = 0 is numerically satisfied on our solutionsacross the domain. See section 6.3.1.10See appendix A.3 for a description of this procedure.1366.4. Solutionsis too large, the total mass-energy of the solution becomes dependent onR∞. In practice, we take R∞ to vary with γ, such that we can computeboth the mass-energy and the baryon charge density with confidence acrossmost of our domain. We find that while the charge density can be computedto good accuracy for large γ, the mass-energy becomes unreliable for γ & 70.To generate a solution, we continue the Newton method until the residualsreach a very small value (∼ 10−9). For generic values of γ, we can solve forthe configuration from a trivial initial guess (zero for all the fields), whilefor very large or very small γ, we solve by using a nearby solution as theinitial guess. Finally, the convergence of our solutions is demonstrated inFigure 6.1.25 30 35 40 45 50 555 ´ 10-61 ´ 10-55 ´ 10-51 ´ 10-45 ´ 10-40.001NRDuFigure 6.1: The convergence of the value ∆u = |u(NR)−u(NR− 2)|/NRNθ,where u(NR) denotes the solution for the five fields {ψ1, ψ2, ar, az, s} on thegrid with NR points in the R direction and Nθ = NR/2 points in the θdirection. These runs are for γ = 10 and R∞ = 60. The dashed line is thebest linear fit, showing the exponential convergence ∆u ∝ e−0.18N .6.4 SolutionsWe focus on two observables of the baryon in the Sakai-Sugimoto model:the mass-energy and the baryon charge density. We examine each of thesein turn.1376.4. Solutions0 1 2 3 4 5012345  renergy density z50100150200(a) γ = 0.2.0 5 10 15 2005101520  renergy density z0.050.10.150.20.250.3(b) γ = 10.Figure 6.2: The energy density ρE(r, z) in the (r, z) plane. For small γ, thesolution appears approximately spherically symmetric. As the coupling γincreases, the soliton expands and deforms, becoming elongated along z = 0.6.4.1 The mass-energyThe energy distribution of the soliton tells us how the structure is deformedas we increase the repulsion of the instanton charges by tuning the couplingγ. Writing the mass-energy as11M = 14pi∫d4x ρE(r, z), (6.22)we plot the energy density ρE(r, z) of the soliton in Figures 6.2 and 6.3. Forsmall γ, the core of the soliton appears spherically symmetric in the (r, z)plane. A closer inspection reveals a skewed tail with a slower falloff of energydensity in the z direction; compare Figures 6.2a and 6.3a. As we increaseγ, the core of the soliton expands and deforms, smearing along the z-axis.In [13], the mass of the baryon was approximated as the energy of a D4brane wrapping the S4, giving M0 = 8pi2κ. The mass of the wrapped D4brane coincides with the mass of a point-like SO(4) instanton at γ = 0.By allowing a finite size spherical instanton, [119] computed a correction tothis, findingMSO(4) = M0 +√215Nc. (6.23)In Figure 6.4, we plot the total mass-energy, normalized by M0, of the soli-ton found here using the more general SO(3) ansatz. As γ decreases and the11We define ρE(r, z)/4pi as the integrand of equation (6.5) multiplied by a suitableJacobian factor.1386.4. Solutions0 1 2 3 4 5012345  rlog energy density z−10−505(a) γ = 0.2.0 5 10 15 2005101520  rlog energy density z−14−12−10−8−6−4−2(b) γ = 10.Figure 6.3: The logarithm of the energy density ρE(r, z) in the (r, z) plane,on the same domain as the corresponding plots in Figure 6.2. A large portionof the energy away from the soliton core is contained in the tail at largeholographic radial coordinate z and small field theory coordinate r.soliton shrinks, the effect of the curved background becomes less importantand the energy approaches that of the point-like spherical instanton. As γincreases and the soliton becomes more deformed, the energy of the config-uration also increases. For γ > 10, we notice that the mass-energy appearsto be controlled by a power law. The best fit in this region gives M ∝ γ0.53.By fitting the Sakai-Sugimoto model to the experimental values for theρ meson mass and the pion decay constant, one can fix both the parameterκ and the energy scale in the field theory. In [127], this procedure yieldsκ = 0.00745 and an energy scale such that 1 in the dimensionless units wehave been using corresponds to 949 MeV. With Nc = 3, this gives γ = 2.55.We can compare our numerical results for the baryon mass to those of theSO(4) approximation for these values of the parameters. We findMSO(4) ' 1.60 GeV,MSO(3) ' 1.19 GeV. (6.24)There is a large difference in the results of the two approaches. Interestingly,the SO(3) result is a much better approximation of the true mass of thenucleons.6.4.2 The baryon chargeThe baryon charge in the field theory is related to the instanton numberdensity 18pi2 trF ∧F in the bulk. In Figures 6.5 and 6.6 we plot the instanton1396.4. Solutions0.1 0.5 1.0 5.0 10.0 50.01.010.05.02.03.01.57.0ΓMM0Figure 6.4: The total mass of the soliton as a function of γ, normalized by themass M0 = 8pi2κ of a D4 brane wrapping the sphere directions (equivalentlythe mass of a point-like SO(4) instanton at γ = 0 in the effective theory).As γ decreases, the mass of the numerical solution approaches that of thepoint-like instanton. For γ > 10, our results can be approximated by therelation M ∝ γ0.53.charge density for two representative solutions. The result closely matchesthe energy density of the soliton.The baryon charge density can be found from the baryon number current,as defined for example in [127]:JµB = −2Ncκ(k(z)Fˆµz) ∣∣∣z=∞z=−∞. (6.25)Writing the Abelian gauge field near the boundary asAˆ0 =Aˆ(1)0 (r)z + . . . , (6.26)where . . . denotes terms at higher order in 1/z, we find that the baryondensity isρB(r) = J0B(r) =Aˆ(1)0 (r)8pi2γ . (6.27)In terms of the density, the total baryon charge isNB =∫ ∞0dr 4pir2 ρB(r). (6.28)1406.4. Solutions0 1 2 3 4 5012345  rinstanton charge density z51015202530(a) γ = 0.2.0 5 10 15 2005101520  rinstanton charge density z1234567x 10−3(b) γ = 10.Figure 6.5: The instanton number density 18pi2 trF ∧ F in the (r, z) plane.The distribution of the instanton charge closely mimics the distribution ofenergy density, as shown in Figures 6.2 and 6.3.0 1 2 3 4 5012345  rlog instanton charge density z−12−10−8−6−4−202(a) γ = 0.2.0 5 10 15 2005101520  rlog instanton charge density z−20−15−10−5(b) γ = 10.Figure 6.6: The logarithm of the instanton number density 18pi2 trF∧F in the(r, z) plane, on the same domain as the corresponding plots in Figure 6.5.1416.4. SolutionsWe fit our numerical solutions to the functional form in equation (6.26)and read off the coefficient Aˆ(1)0 (r) in order to find ρB(r). This fit is onlyrobust up to a value of r that depends on the coupling γ: r = r¯(γ). Asdemonstrated in [33], the charge density ρB(r) decays as 1/r9. Thus thefield Aˆ0 is decaying much faster in the field theory r direction than theholographic radial z direction. Since we solve in the coordinate R = (r2 +z2)1/2, and choose a large cutoff R∞ such that the z falloff is reliable, wemight expect the fit to break down at some point, after ρB(r) has decayedto a very small value. Numerically, we determine r¯(γ) as the point at whichthe error in the fit reaches ten times the error in the fit at r = 0.In Figure 6.7, we plot the baryon charge ρB(r) up to the cutoff r¯(γ)for various values of γ. As γ increases, the baryon density at the originρB(0) decreases and the charge moves toward the tail of the distribution. Inthe log-log plot, the 1/r9 falloff of the charge density can clearly be seen.Figure 6.8 shows the behaviour of the baryon charge density across our entirerange of γ.0 2 4 6 8 10 12 140.0000.0010.0020.0030.004rΡBHrL0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.010-610-510-40.001rΡBHrLFigure 6.7: Left: The charge density ρB(r) for γ = 4, 12, 20, 28, from topto bottom. Right: The same data on a log-log axis. As γ increases, thecharge density becomes less peaked near the origin. The 1/r9 falloff ofρB(r) behaviour can be seen in the tail of the charge distributions.As a check of our solution, we can compute NB by both formulas (6.9)and (6.28). We find that, across the range of γ and using both formulas,NB = 1 to good precision.Lastly, with the charge density ρB(r), we can compute the baryon chargeradius〈r2〉 =∫ ∞0r2(4pir2ρB(r))dr. (6.29)To integrate past the cutoff r¯(γ), we approximate the tail of the distributionas ρB(r; γ) ∼ c(γ)/r9, where c(γ) is approximated from the value of the1426.4. SolutionsFigure 6.8: The charge density ρB(r) for varying γ.density at the integration cutoff. The baryon charge radius is plotted inFigure 6.9. For γ > 35, the relation appears to obey a power law, with bestfit given by 〈r2〉 ∝ γ0.93.As above, it is interesting to compare the result to that obtained fromthe SO(4) approximation, evaluated at the parameters defined by the fit tomeson physics. The result is12〈r2〉1/2SO(4) ' 0.785 fm,〈r2〉1/2SO(3) ' 0.90 fm. (6.30)In this model, the baryon charge radius equals the electric charge radius ofthe proton [127]. The result from our numerics is very close to the exper-imental value for the electric charge radius of the proton, which has beenmeasured to be in the range 0.84 fm – 0.88 fm.12We compare to the result from the classical analysis of the SO(4) baryon, given inequation (3.11) of [127].1436.5. Conclusion1 2 5 10 20 50 100 200151050100500Γ<r2>Figure 6.9: The baryon charge radius 〈r2〉 =∫r2(4pir2ρB(r))dr as a func-tion of γ. For γ > 35, the relation can be approximated by 〈r2〉 ∝ γ0.93.6.5 ConclusionWe have studied properties of baryons in a holographic model of QCD re-lated to the Sakai-Sugimoto model by simplifying the Born-Infeld part ofthe D8-brane action to a 5D Yang-Mills plus Chen-Simons action for thegauge fields in the non-compact directions. By dropping the assumptionof SO(4) symmetry and finding direct solutions to the bulk field equationsfor the gauge field, we have found that various properties of the baryons inthe holographic QCD model change significantly. In particular, the baryonmass gives substantially better agreement with measured values. There areseveral interesting directions for future work.Within the present model, it would be interesting to calculate otherobservables such as the form-factors associated with the isospin currents(associated with the SU(2) flavour symmetry) and compare these to re-sults calculated using the SO(4) symmetric ansatz [127]. It would also beinteresting to consider interactions between two baryons. This requires aless-symmetric ansatz, but the numerics should still be feasible. Again, itwould be interesting to compare with previous results calculated assumingflat-space instanton configurations [136]. For higher baryon charge, it shouldbe feasible to consider the question of nuclear masses as a function of baryonnumber, at least within the space of SO(3)-symmetric configurations. Theactual ground states for higher baryon number may not be so symmetric1446.5. Conclusionhowever. In addition, it would be interesting to investigate solutions witha finite baryon charge density (e.g. at finite baryon chemical potential).Such configurations were considered with various simplifying assumptionsin [42, 50, 52, 53, 137, 138]. As shown in [53], these are necessarily inhomo-geneous in the field theory directions, so a numerical approach similar to theone used in this paper is likely necessary to investigate detailed propertiesof the ground state at various densities.Finally, it is interesting to investigate effects of replacing the Yang-Millsaction used here with the full D8-brane Born-Infeld action. This is incom-pletely known, but one could work for example with the Abelian Born-Infeldaction promoted to a non-Abelian action via the symmetrized trace prescrip-tion that has been shown to be correct for the F 4 terms. While the equa-tions in this case will be significantly more complicated, they should poseno serious obstacle for the numerical approach that we are using. An inter-esting difference between the Born-Infeld and Maxwell actions for Abeliangauge fields is that the Maxwell action associates an infinite energy to pointcharges, while this energy is finite in the Born-Infeld case. Thus, we mightexpect that the tendency for the instantons to spread out is somewhat lesswith the Born-Infeld action. In this case, we may expect a somewhat smaller,less massive baryon. Thus, the baryon mass in the model using the Born-Infeld action may be even closer to the experimental value than we havefound here.145Chapter 7Conclusion7.1 SummaryIn this thesis, we applied the holographic correspondence to the study ofvarious strongly coupled phenomena. The projects comprising the thesisfell into three domains of applicability: i. General holographic field theories;ii. Holographic condensed matter, and; iii. Holographic QCD. Technically,this work involved posing and solving classical field equations in curvedbackgrounds (with and without backreaction). To facilitate these studies,we made extensive use of numerical methods, overcoming some technicalobstacles which may have discouraged previous researchers.In chapter 2, we studied strongly coupled field theories with the minimalstructure of a conserved charge, focussing in particular on the relationshipbetween charge density and chemical potential at large density. At finitecharge density, the typical lattice field theory approach to strong couplingdynamics fails, rendering holography as the most convenient and reliablecalculational method. After some general thermodynamic considerations,we applied the holographic approach to study a wide range of model fieldtheories, including those built via explicit string theory constructions andthose developed through a phenomenological approach. We enumerated ourresults across these theories, providing a useful guide to a subset of thebehaviours available in holographic theories.In chapters 3 and 4, we applied the holographic correspondence to the ex-perimentally observed phenomenon of the spontaneous formation of stripedphases. We studied a phenomenological model dual to a strongly coupledfield theory that undergoes a spontaneous transition to a phase with stripedorder as we lower the temperature. Building on previous work that showeda perturbative instability towards a striped phase, we applied numerical rel-ativity techniques in order to find the full nonlinear striped solutions acrossthe parameter range. The geometries we found exhibited novel character-istics including a charge density wave, a momentum density wave, and amodulated black brane horizon that tends to pinch off as we lower the tem-perature. Given the solutions, we constructed the phase diagram of the1467.2. Future directionssystem, showing that the field theory undergoes a second order phase tran-sition to the striped phase.In chapter 5, we used holography to study the colour superconductivityphase of QCD, which is expected to exist at large density. To facilitate this,we constructed a phenomenological model of QCD, designed to mimic cer-tain aspects of the phase diagram. Using results from previous studies, weincluded in our holographic model the ingredients necessary for three QCDphases: a confining phase, a deconfining phase, and a colour superconduc-tivity phase. By analyzing the thermodynamics of the different phases, weconstructed the phase diagram at all values of temperature and chemicalpotential, showing that, indeed, our model qualitatively resembled the ex-pected phase diagram of QCD.Finally, in chapter 6, we applied numerical techniques to the constructionof the baryon in the well-known Sakai-Sugimoto model of large-Nc QCD. Asreviewed above, the gravity dual of the field theory baryon in this model is agauge field configuration on the probe D8 branes with non-trivial topologicalcharge. Previous studies of the baryon in this model assumed a sphericalsymmetry for this gauge field configuration, an assumption which was shownto produce results that failed certain model-independent tests of baryons.We relaxed this spherical symmetry, formulating and solving the full nonlin-ear partial differential equations using numerical techniques in order to findan oblate instanton, the true minimum energy configuration correspondingto the baryon. We studied the dependence of the mass and baryon charge ofthe baryon on a parameter γ, which determines the baryon self-repulsion. Atthe value of γ dictated by the best fit of parameters to the meson spectrumof QCD, our solution was found to give significantly more realistic valuesfor the mass and charge radius of the proton than previous studies.7.2 Future directionsIn this section, we briefly describe some of the most promising and interest-ing studies that would comprise extensions to the work presented here andthat would depend on the numerical techniques used in this thesis. Some ofthese and more have already been discussed in the bulk of the thesis.7.2.1 Inhomogeneous holography and condensed matterThere are many possible directions to follow up on the holographic stripesproject, discussed in chapters 3 and 4. Firstly, we offer advice for undertak-ing future projects in this area. As opposed to using the numerical method1477.2. Future directionsoutlined in those chapters (using a conformal ansatz, with finite differencetechniques), it is the thesis author’s recommendation that a more efficientapproach would be to use the deTurck method [135] combined with pseu-dospectral differentiation. This method has been successfully applied tosimilar problems in, for example, [88, 101, 102]. The deTurck approachavoids the gauge fixing issues that we encountered, described in section B.2.3and, anecdotally, works very well for problems of this type. Furthermore,pseudospectral differentiation combined with the Newton’s method, as de-scribed in appendix A.3, offers a compact algorithm with running times ona desktop computer that may be measured in minutes. This is in contrastto the Gauss-Seidel relaxation used in those chapters, which required hoursof processing time on a parallel computer system for each solution.A direct extension of the work described here involves the zero temper-ature state. The question of what happens to the stripes as we go to zerotemperature was unresolved in our study, as our numerics broke down atvery small temperatures. However, solutions of the black brane at finite tem-perature could direct the search for the zero temperature geometry. This isinteresting both theoretically and in terms of experimental results. Theoret-ically, it has been demonstrated that the homogeneous Reissner-Nordstromblack brane has a non-zero entropy at zero temperature, implying that thedual field theory is in violation of the third law of thermodynamics. This isone of the original motivations that prompted the search for instabilities ofthe theory, leading to the discovery of these striped phases. An interestingquestion then is to find the true ground state of the charged black branein particular holographic theories. Given these, one could make qualitativeor, possibly, quantitative comparisons to the low temperature behaviour ofexperimental systems. In addition, it has been speculated that the groundstate of QCD may also display lattice behaviour. If true, it would be veryinteresting if these holographic models displayed a similar behaviour.Secondly, in order to again make connection to the condensed matterliterature, it would be interesting to compute the optical conductivity in thismodel of holographic stripes. Such a project would be along the lines of therecent work by Horowitz, Santos, and Tong [87, 88], in which they introducedan inhomogeneity ‘by hand’, thereby breaking the translation invariance.This is important because in examples of homogeneous holographic fieldtheories, the DC conductivity is always infinite. It has been pointed outthat breaking the translation invariance will remove the infinite peak atzero frequency and widen it into a Drude peak, typical of condensed mattersystems, and allowing the holographic theory to better model experimental1487.2. Future directionsmaterials.1 The analysis required in such a study would present a new levelof technical difficulty, as the conductivity computation would require thesolving of a system of partial differential equations linearized around thenumerical striped background solution. However, these results would beextremely interesting in the push to closer align holographic models withexperimental phenomena.An alternative direction would be to apply these techniques to find thegeometries dual to translation-symmetry breaking phases in theories exhibit-ing hyperscaling violation [139, 140].2 Systems with hyperscaling violationhave been suggested to describe theories with a Fermi surface [141], mak-ing them interesting as models of condensed matter systems. Several recentstudies have shown the existence of instabilities towards striped phases inmodels dual to field theories with this behaviour [142, 143]. As above, forstriped phases, it would be useful to have examples of holographic theoriesthat at least qualitatively mimic these experimental results.A final interesting extension of the stripes program would be to allowtranslation symmetry breaking in both spatial field theory directions, re-sulting in a checkerboard-type field theory configuration. Checkerboards,or lattices, have shown up in condensed matter situations (see, for example,[144]) and it would be interesting to have a concrete theoretical model whichrealizes their formation.3 Technically, this would require solving partial dif-ferential equations with dependence on three directions, increasing the scaleof the problem. However, the techniques described above should render thisproblem tractable, even with relatively modest computational resources.7.2.2 Baryons in holographic QCDThe application of numerical methods to the area of holographic QCD of-fers many interesting directions for future research. In particular, in theSakai-Sugimoto model discussed in chapter 6 (or even other models of holo-graphic QCD) there are many unresolved questions regarding the behaviourof baryons and how they interact.1Indeed, [87, 88] did make comparisons between conductivity in their model and inthe cuprates, finding a striking agreement for the power-law describing behaviour at mid-infrared frequencies.2These theories posses a dynamical critical exponent z and a hyperscaling violatingexponent θ, which alter the thermodynamics of the system. In particular, the scaling ofentropy with temperature in such a theory is given by S ∼ T d−θ/z. (In scale invarianttheories without hyperscaling violation, z = 1 and θ = 0.)3A holographic example of such a situation was recently studied in [78].1497.2. Future directionsAs discussed in chapter 6, in the Sakai-Sugimoto model the gravity dualof a field theory baryon is an instanton configuration of the gauge fieldson the probe flavour branes. In this model, the usual spherical instantonis deformed to an oblate spheroid, the shape of which is determined by acompetition between the coupling γ (controlling the self-repulsion of theinstanton), the background geometry, and the restriction towards sphericalsymmetry due to the Yang-Mills form of the action. Motivated by this, afirst extension of this work, which would allow a better understanding ofthese holographic baryons and would be interesting more generally, wouldbe to study the deformation of instantons governed by the standard Yang-Mills action. It is known that a spherical instanton minimizes the energy ofthis system; of particular interest in terms of holographic baryons would bethe energy dependence of the state on the deformation of the configuration.Operationally, this could be studied by including Lagrange multiplier termsfor the quadrupole moment of the instanton in the action. A technicaldifficulty in solving this numerically is the scaling symmetry of the sphericalinstanton: all sizes of spherical instantons possess the same total energy.Some care must then be taken in setting up the numerical problem suchthat there is a unique solution.4In our study of baryons, the action, equation (6.3), was derived by a seriesexpansion of the DBI action for the gauge fields on the probe flavour branes.This expansion is based on the assumption that the variation of the gaugefields is small on the order of the string scale. However, this assumption isnot strictly satisfied at strong coupling in the model. In fact, in [13], it wasshown that the characteristic size of the instanton in this model is on theorder of the string scale, violating the original assumption. Then, in orderto more precisely describe baryons in this model, it would be interestingto strip away this assumption and study the full DBI action governing thegauge fields on the brane. Technically, this is difficult, due to the non-Abelian nature of the gauge fields: a correct procedure for evaluating thetrace over the gauge structure has not been established [145]. However, onemay study simplifications of the DBI action that may more closely resemblethe actual system of interest. One very simple way to do this would beto modify the structure of the U(1) part of the action, from the standardYang-Mills to the Born-Infeld type. Since the Born-Infeld action softensthe infinities associated with point charges, resulting in a smaller repulsiveforce for two charges brought very close together, one would expect that this4Some preliminary attempts by the thesis author to study the Yang-Mills instanton inthis way were unsuccessful due to difficulties in setting up the numerical problem.1507.2. Future directionsmodification would allow the instanton in our model to relax somewhat toa smaller, less energetic configuration.5 A second, more realistic approachwould be to include more gauge field terms from the expansion of the DBIaction. The form of this expansion is known up to fourth order [146, 147].By including further terms in the expansion, one may be able to glean someindications of how the baryon configuration would be expected to changeupon using the full DBI action. This type of project would only necessitatea minor modification of the numerical procedure used in our previous studyand thus offers a tantalizing possibility.The above two suggestions were related to better understanding the con-struction of the baryon in the Sakai-Sugimoto model. One could also extendthe above work by studying further questions related to the baryon. A firstinteresting direction here would be to study configurations of higher baryonnumber; these would be holographic nuclei. A conceptually straightforwardquestion along these lines would be to examine how the masses of the nucleidepend on the baryon number in this model. Given the fit of the modelparameters to meson physics, it would be very interesting to evaluate themasses of the holographic nuclei and to make a comparison to observed re-sults. As shown in chapter 6, this model provides a quantitatively realisticrelationship between meson masses and the mass of the baryons; it would beinteresting if the higher baryon number configurations also matched exper-imental results. In the field theory, a deuterium state (two baryons boundtogether) would have only a cylindrical symmetry. To find the holographicdual of this state would then require a three-dimensional code, increasingthe numerical complexity of the problem.6 However, as discussed above forholographic checkerboards, this should still be tractable with the numericaltechniques used here.Finally, given a three-dimensional solver and the construction of a cylin-drically symmetric deuterium state, one could investigate the force betweenbaryons. This problem has been studied in various approximations in holo-graphic models, including under the assumption of SO(4)-symmetric instan-tons representing baryons [136]. By studying the deuterium state, one could5Indeed, a preliminary investigation by the thesis author, using a Born-Infeld actionfor the U(1) part of the gauge field, showed that the minimum energy solution was slightlysmaller and less massive than the solution for the model with Yang-Mills U(1) action.6One simplification could be to look for holographic nuclei with the field theory SO(3)symmetry of the single baryon. A preliminary study by the author within the ansatzof chapter 6 found a large negative binding energy for the two baryon configuration,indicating that this configuration, with the two baryons ‘on top’ of each other in the fieldtheory, will certainly not be preferred.1517.3. Final remarksprecisely compute the dependence of the force between the baryons on thedistance between them (by, for example, using again Lagrange multipliersfor the quadrupole moment of the state). It would be interesting to comparethe less-symmetric case to previous results in order to further understandinteractions between holographic baryons.7.3 Final remarksThe work presented in this thesis consists of several important examplesof applied holography. While the research presented in chapters 2 and 5provided interesting new results for holographic field theories in general andwithin a particular phenomenological model of holographic QCD, the mainsignificance and contribution of this thesis comes from the application of nu-merical techniques to holographic situations with reduced symmetry. Thestriped phases work of chapters 3 and 4 represented the first black hole so-lutions of that kind7 and was an important step in the progress of applyingthe holographic correspondence to find the full gravitational bulk (includingthe geometry) to systems with a broken translation symmetry. Meanwhile,the holographic baryons of chapter 6 were results of one of the first studiesapplying numerical techniques to this problem,8 thus pushing the field intonew territory. Both projects required dedicated efforts to overcome signifi-cant technical obstacles that may have discouraged other researchers fromundertaking such work. Overall, these studies are part of a recent thrust,led by a subset of researchers in the field, towards combining numericaltechniques (including those used in numerical relativity) with holography;examples of these works include [83, 87–89, 101, 102, 104, 148–151].9This marriage of numerical and holographic techniques offers tremen-dous promise in terms of connecting the existing literature on holographictheories to experimentally observed strongly coupled systems. This includesboth the theory that underlies our world (QCD) and those theories thatdescribe the novel materials that will be the cornerstone of tomorrow’s tech-nological advances (condensed matter). It is the thesis author’s expectationthat the approaches used in this work and in related studies will becomestandard tools of practitioners in the field and will allow the study of new7These were the first full solutions for planar black holes that spontaneously breaktranslation symmetry.8The study [132] appeared while the work on holographic baryons was being completed,while the earlier studies [122–124] used similar numerical techniques.9Even more recently, some time-dependent holographic problems have been solvedusing numerics. See [152–154].1527.3. Final remarksand more realistic classes of models. 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Thermodynamics of asymptoticallylocally AdS spacetimes. JHEP 0508, 004 (2005). hep-th/0505190.166Appendix AReview of numericaltechniques usedIn this thesis, in the work described in chapters 3, 4, and 6, we employedstandard numerical methods for the solution of the partial differential equa-tions that arose in problems in holography applied to situations with in-homogeneity in the field theory. These were the construction of a stripedphase in a strongly coupled field theory and the gravity dual of a baryon ina large-Nc field theory.In this appendix, we briefly review the implementation of the two majortechniques used. Firstly, in section A.1, we sketch our general approach forsolving partial differential equations numerically. Following this discussion,we describe the two specific approaches employed: finite differencing with aGauss-Seidel relaxation method (section A.2) and pseudospectral differenti-ation with a Newton iteration (section A.3). This appendix is not intendedto be a complete guide to these approaches. Rather, it is to provide a (hope-fully enlightening) caricature of the techniques used. The interested readeris directed to the useful references [155–157] for more information.Although our studies involved partial differential equations, with depen-dence on two variables, for clarity of presentation below we will restrict ourdescriptions of the numerical techniques to the case of an ordinary differ-ential equation for the field ϕ(x), with dependence on one coordinate x.Specifically, we will consider as our model equation Poisson’s equation inone dimension,d2ϕ(x)dx2 = ρ(x), (A.1)on the domain x ∈ [−1, 1]. We will use this model equation for illustrativepurposes in our discussion of the numerical techniques.167A.1. Generic numerics for elliptic partial differential equationsA.1 Generic numerics for elliptic partialdifferential equationsIn this thesis, we applied the holographic correspondence to the study ofcertain time-independent problems. The partial differential equations thatresulted from these static situations were boundary value problems. In aproblem of this type, data specified on the boundaries of the domain1 deter-mines the solution on the interior. The main task for solving these problemsis, given a set of boundary conditions, to find the configuration that satisfiesboth the boundary conditions and the equations of motion in the interiorof the domain. The prototypical boundary value problem is the Poissonequation (A.1), which appears in various contexts in physics. The par-tial differential equations encountered in this thesis are comparable to morecomplicated, nonlinear versions of the Poisson equation.In order to solve these more complicated boundary value problems, forwhich closed-form analytic solutions are not known (and may not even bepossible), one must turn to numerical methods. Typically, numerical meth-ods create some approximation of the continuous field ϕ(x) in order to trans-late the differential equation into a number of coupled algebraic equations,which can be solved numerically. Perhaps the most conceptually straight-forward approach, which is undertaken in this thesis, is to discretize thedomain on a grid, so that the continuous function ϕ(x) becomes a set ofvalues ϕi at each grid point. By choosing a suitable discretization of thederivatives, one arrives at a set of coupled algebraic equations for the fieldvalues ϕi. If we write the field as the vector~ϕ =...ϕi... , (A.2)then, after the discretization, we can view this problem as the matrix equa-tionA · ~ϕ = ~ρ, (A.3)where ~ρ is the vector created by evaluating the source ρ(x) at the gridpoints and, in our model problem (A.1), A is the matrix that represents thediscretized operator d2dx2 .1This could be, for example, the value of the fields (Dirichlet conditions), a zero deriva-tive condition (Neumann conditions), or some more complicated condition.168A.2. The numerical approach for holographic stripesWithin this general framework, there is some choice as to how one maysolve their problem. First, in terms of the domain grid and the discretiza-tion of the derivatives, different choices can affect the performance of thealgorithms and the accuracy of the numerical approximations. Two commonmethods, used in this thesis and reviewed below, are finite differencing (inthis case, on a rectangular grid) and pseudospectral methods on a Cheby-shev grid. In addition, upon approximating the continuous equation as amatrix equation, different options are available to solve the coupled alge-braic equations. In our two cases here, we use a Gauss-Seidel relaxationmethod and a Newton iteration.A.2 The numerical approach for holographicstripesFor the study on holographic stripes reported in chapters 3 and 4, the tech-nique used was a second-order finite differencing on a rectangular grid fol-lowed by Gauss-Seidel relaxation to solve the resulting algebraic equations.We provide a basic description of each of these components in turn below.A.2.1 Finite differencing on a rectangular gridFinite differencing is one of the most familiar methods with which to ap-proximate a derivative. Indeed, the definition of the derivative relies on thelimit of a finite difference equation. To use this method on a constant grid,one first splits the domain into N sections of equal length h = 2/N , withgrid points given by xi = −1 + ih. The continuous field is transformedinto a discrete vector by taking its values at these grid points, ϕi = ϕ(xi),and organizing them according to equation (A.2). Derivatives of ϕ are com-puted using the standard finite difference equations. In our work, we usedsecond-order finite differences, which, for our model problem, consists of thereplacementd2ϕ(xi)dx2 →ϕi+1 − 2ϕi + ϕi−1h2 . (A.4)By using a series expansion for the field ϕ, one can show that the error inthis expression is of order h2. Therefore, using this method, one expectsquadratic convergence of our approximate solution as we increase the griddensity (decrease h).Upon making the replacement (A.4), at each grid point xi we now havean equation involving the value of the field at that point, ϕi, and the values169A.2. The numerical approach for holographic stripesof the field at neighbouring points, ϕi−1 and ϕi+1, resulting in N+1 coupledalgebraic equations. We can rewrite this as the matrix equation A · ~ϕ = ~ρ,where now A is the tri-diagonal matrixA = 1h2. . .1 −2 11 −2 11 −2 1. . .(A.5)with zeros above and below the listed entries.2 At this stage, solving theoriginal differential equation becomes the task of solving this algebraic ma-trix equation.A.2.2 Relaxation iterationTo solve the coupled algebraic equations resulting from our finite differencediscretization, we use a standard pointwise Gauss-Seidel-style relaxation it-eration, which we review here. This type of iteration can be understood intwo ways: Firstly, it is an algorithm which exploits the sparse structure ofthe matrix A, so that each iteration can be computed quickly. Secondly, itcan be rephrased as solving a related diffusion problem to find an equilib-rium configuration. (This motivates the label ‘relaxation’.) After definingthe iteration, we will discuss the second, more relaxing interpretation.After differencing, our model equation readsA · ~ϕ = ~ρ. (A.6)Following [155], we split the matrix A as A = L+D+U , where L is the lowertriangular part of A, D is the diagonal of A, and U is the upper triangularpart of A. We set up the iteration asD · ~ϕ(n+1) = −(L+ U) · ~ϕ(n) + ~ρ, (A.7)where the superscript (n) labels the nth iteration. The value for our fieldat the next iteration, ~ϕ(n+1), is calculated with low computational expense,as D, being diagonal, is easily inverted. One can see that if the algorithmconverges, such that the difference between ~ϕ(n) and ~ϕ(n+1) becomes very2Boundary conditions can be incorporated into this structure by altering the rowswhich update the boundary values of the field.170A.2. The numerical approach for holographic stripesclose and below some threshold, we will have solved our original equationup to the defined numerical accuracy. This is a standard iterative methodfor solving a matrix equation of this type.Now, consider a related diffusion problem, defined as∂~ϕ∂λ = −A · ~ϕ+ ~ρ, (A.8)where λ labels an auxiliary ‘time’ dimension. Finding an equilibrium so-lution to (A.8), such that ∂~ϕ/∂λ = 0, will give a solution to our modelproblem. If we discretize the time variable λ using a first-order forwardfinite difference equation,∂~ϕ∂λ →~ϕ(n+1) − ~ϕ(n)∆λ, (A.9)numerically solving this diffusion equation amounts to the update schemegiven by~ϕ(n+1) = ~ϕ(n) −∆λ(A · ~ϕ(n) − ~ρ). (A.10)Compare this to the iteration above: adding and subtracting ~ϕ(n) on theright-hand side of (A.7) gives~ϕ(n+1) = ~ϕ(n) −D−1 ·(A · ~ϕ(n) − ~ρ). (A.11)Thus, in our model problem, with the identification ∆λ = (D−1)ii, (A.7)is identical to the scheme that solves the diffusion equation (A.8). Findinga fixed point of the iteration is equivalent to finding an equilibrium solu-tion to the diffusion problem, and ultimately gives a solution to the desireddifferential equation.In terms of performance of the scheme described above, a simple way toincrease the speed is to perform the updates ‘in place’. Typically, the fieldvalues ϕ(n)i are updated to ϕ(n+1)i one at a time (from i = 0 to i = N) asthe algorithm loops over the grid domain. According to our second-orderfinite differencing and the iteration (A.7), in our model problem ϕ(n+1)i willdepend on ϕ(n)i+1 and ϕ(n)i−1. In this setup, by the time the loop arrives at thepoint i, ϕ(n+1)i−1 will have been computed. If, instead of computing the entirevector ~ϕ(n+1) based on the previous vector ~ϕ(n), one updates each ϕ(n)i usingupdated values at neighbouring grid points as soon as they become available(uses ϕ(n+1)i−1 instead of ϕ(n)i−1), the algorithm experiences a constant factor171A.3. The numerical approach for holographic baryonsspeedup. This is called a Gauss-Seidel method. In our solver, we implementsuch an update procedure.For linear problems, such as our model Poisson equation, this type ofiteration can be shown to converge, based on the structure of the matricesthat result from finite differencing. For nonlinear problems, such as thoseencountered in this thesis, no general results are available. Practically, thisdoes not impede the implementation of the relaxation. For each problem,one can simply apply the method and observe if convergence occurs. If theresidual ~ξ ≡ A · ~ϕ − ~ρ converges to zero, one can be sure that they havefound a solution.A.3 The numerical approach for holographicbaryonsFor the study on holographic baryons in the Sakai-Sugimoto model reportedin chapter 6, we used pseudospectral differentiation on a Chebyshev gridbefore applying Newton’s method to the resulting algebraic equations. In thefollowing subsections, we motivate and describe these technical components.A.3.1 Pseudospectral differentiation and Chebyshev gridsIn this section, we will summarize the development of pseudospectral meth-ods as outlined in [156]. The main goal of this section is to motivate theuse of the Chebyshev differentiation matrix DN , whose action on the vec-tor of grid points ~ϕ will give an approximation to the derivative, and theChebyshev grid. The main idea we will encounter is that pseudospectraldifferentiation can be considered as an order N finite difference scheme,where N is the number of grid points. The method is ‘spectral’ becauseapproximations to the derivatives use information from every grid point inthe domain. When using pseudospectral methods, increasing the number ofgrid points increases the accuracy of the approximations in two ways: byreducing the grid spacing, h ∝ N−1, and by increasing the power of h thatcontrols the error. The result is exponential convergence in N , of the orderO(1/NN ).To begin our discussion, we recast the second-order finite differencingreviewed in section A.2.1 as a polynomial interpolation. Let ψi denote theapproximation to ϕ′(xi). To find ψi (with second-order accuracy) via aninterpolation, let p(x) be the unique polynomial (of degree 2 or less) thatgoes through the function ϕ at xi and at its two neighbouring points (such172A.3. The numerical approach for holographic baryonsthat p(xi−1) = ϕi−1, p(xi) = ϕi, and p(xi+1) = ϕi+1). On a constant grid,with xi+1 − xi = h, this polynomial isp(x) =((x− xi)(x− xi+1)2h2)ϕi−1−((x− xi−1)(x− xi+1)h2)ϕi+((x− xi−1)(x− xi)2h2)ϕi+1. (A.12)Given this interpolant, we find ψi by differentiating p(x) and evaluating theresult at xi. Carrying out this procedure givesψi =ϕi+1 − ϕi−12h . (A.13)This is precisely the expression for a centred second-order finite difference.By using a series of (second-order) polynomial approximations through eachgrid point, one can derive the linear transformation D2 corresponding to thesecond-order first derivative on a constant grid, such that ~ψ = D2 · ~ϕ. Theresult isD2 =12h. . .−1 0 1−1 0 1−1 0 1. . ., (A.14)with other entries being zero.The expressions for higher order finite difference schemes follow in thesame manner. To derive the (centred) finite difference approximation toϕ′(xi) of degree n, one first finds the unique polynomial p(x) (of degree n orless) that interpolates ϕ(x) at the set of points (xi−n/2, . . . , xi, . . . , xi+n/2).Then, taking the derivative of p(x) and evaluating at xi will give the nthorder finite difference expression. In this manner, we can build up the dif-ferentiation matrix Dn for any n. The error in the approximate derivativeψi (the difference ϕ′(xi)− ψi) will be of order O(hn).The matrix for our pseudospectral differentiation is found by taking thisscheme to limit. On the grid with N + 1 grid points, we find the uniquepolynomial of degree less than or equal to N , before differentiating it and173A.3. The numerical approach for holographic baryonsevaluating it at each grid point. This gives a linear transformation3~ψ = D˜N · ~ϕ (A.15)which produces an approximation of the derivative ϕ′(x). The error in thisapproximation will be roughly of order O(hN ) = O(N−N ). By using a poly-nomial of degree N , the matrix D˜N will be dense, as compared to the sparsestructure of the matrix used above for the finite difference scheme. Thus,solving a differential equation using pseudospectral differentiation involvesmore costly matrix inversions than using a low-order finite difference scheme.However, the exponential convergence with N means that in practice, whenusing pseudospectral methods, one has to use many fewer grid points thanfor a finite difference scheme, resulting in savings in vector and matrix stor-age space. Computational savings in terms of fewer iterations also resultsand is described more in section A.3.2.The polynomial one finds through the interpolation process depends onthe grid points xi. Above, we assumed the constant grid xi = −1 + ih. Ithas been proven that, for polynomial interpolation on the interval [−1, 1],the optimal interpolation points (the grid points at which the interpolant istaken to match the original function) are given by the Chebyshev pointsxi = cos( ipiN), i = 0, 1, . . . , N. (A.16)These points are less dense in the interior of the domain, around x = 0,and cluster near the boundaries at x = ±1. This allows one to avoid theRunge phenomenon, in which, for polynomial interpolation on a constantgrid, errors accumulate near the boundary of the domain. The Chebyshevdifferentiation matrices DN are defined as the matrices that enact the lineartransformation of differentiating, according to the above polynomial inter-polation procedure, when using the Chebyshev points (A.16) as the inter-polation points. The explicit expressions for the entries of the matrix DN ,as found in, for example, [156], are(DN )00 =2N2 + 16 , (DN )NN = −2N2 + 16 ,(DN )ii =−xi2(1− x2i ), i = 1, . . . , N − 1,(DN )ij =cicj(−1)i+j(xi − xj), i 6= j, i, j = 1, . . . , N − 1, (A.17)3We write this differentiation matrix as D˜N , to distinguish it from the Chebyshevdifferentiation matrix DN defined below.174A.3. The numerical approach for holographic baryonswhereci ={2, i = 0, N,1, otherwise.(A.18)To use the Chebyshev method on our model problem, we discretize thedomain on a Chebyshev grid and replace the derivatives with the Chebyshevmatrices DN , to getD2N · ~ϕ = ~ρ. (A.19)We have now reduced our original differential equation to an algebraic matrixequation, the solution of which we describe in the following section.As a final note, we provide an alternate description of the Chebyshevspectral method. Although we motivated this method through finite dif-ferencing and polynomial interpolation, this approach has a different in-terpretation in terms of an expansion in the Chebyshev polynomials. TheChebyshev polynomials are a set of polynomials defined byTn(x) = cosnθ(x), (A.20)where θ(x) is defined throughx = cos θ. (A.21)In this latter method, one expands the field in terms of the Chebyshevpolynomials asϕ(x) =N∑n=0anTn(x). (A.22)Explicitly following this approach, one would insert the expansion (A.22)into the original equation and use orthogonality of the Chebyshev poly-nomials to derive algebraic equations for the vector of coefficients ~a. It isequivalent to use the values of the function ~ϕ as the unknowns and explicitlysolve for these, as we do above. These considerations show that, in additionto finding the value of the function at the grid points (finding the vector~ϕ), one can compute the vector of coefficients ~a in order to find an entirefunction, (A.22), which approximates our solution.A.3.2 Newton’s method for matrix equationsIn this section, we review the use of Newton’s method in solving the matrixequations that result after discretizing the problem on a Chebyshev grid175A.3. The numerical approach for holographic baryonsand replacing the derivatives with the Chebyshev differentiation matrices.In this section, we will change notation to write our problem asL(ϕ) = 0, (A.23)where L is the operator which represents our differential equation. In thecase of our model Poisson equation, we would have L(ϕ) ≡ d2ϕdx2 − ρ. Ourtask is to find the field ϕ which satisfies equation (A.23).Upon discretizing according to the prescription in the previous section,we get one equation of motion at each grid point, and thus arrive at a vectorof equations, which we write as~L(~ϕ) = 0. (A.24)The Newton’s method for this system of equations can be derived by a seriesexpansion in the usual way. Letting subscripts denote components of vectorsas above, we can expand our operator near a vector ~ϕ∗ asLi(~ϕ∗) = Li(~ϕ) +(∂Li∂ϕj)(ϕ∗j − ϕj) + . . . , (A.25)where . . . denote terms higher order in (ϕ∗i − ϕi). Now, if we assume that~ϕ∗ satisfies the equations of motion, we have Li(~ϕ∗) = 0. Then, truncatingthe series and rearranging terms, we arrive at a prescription for finding ~ϕ∗based on the nearby vector ~ϕ:ϕ∗i = ϕi −(∂Lj∂ϕi)−1Lj(~ϕ). (A.26)Starting from an initial guess for our fields, we iterate according to equa-tion (A.26) until the original equation (A.24) is satisfied to some numericaltolerance.From the iteration (A.26) and the definition of the Chebyshev matri-ces DN , one can motivate that the numerical approach described here willconverge in fewer iterations than that described in section A.2. The Gauss-Seidel relaxation is a local relaxation procedure, as the update procedurefor the field at each grid point ϕi depends only on neighbouring points. Themethod described in this section is spectral, in that the update procedurefor the field at each point ϕi depends on the value of the field at every otherpoint. Thus, the iterations are able to quickly propagate changes in the fieldto all corners of the domain, resulting in faster convergence. In practice, thenumber of iterations needed for numerical convergence is several orders of176A.3. The numerical approach for holographic baryonsmagnitude smaller for the Chebyshev spectral method with Newton itera-tion. Therefore, even though each iteration is slower due to the need fora more complex matrix inversion, the amount of total computational timeneeded for solving using pseudospectral methods is much smaller than forthe finite difference method. Indeed, the solution of the partial differentialequations encountered in chapter 6 was accessible with only modest desktopresources.177Appendix BStriped order supplementarymaterialIn this appendix to chapter 4, we provide details of the asymptotic chargesin our model (section B.1) and about our numerical procedure (section B.2).B.1 Asymptotic chargesB.1.1 Deriving the chargesSince our ansatz is inhomogeneous and includes off-diagonal terms in themetric, and our action is not standard (in that it includes the axion coupling)we have re-derived the expressions for the charges and other observables inour geometry. In deriving the asymptotic charges of our spacetime, for thefour dimensional Einstein-Maxwell-Higgs theory we discuss in the main text,we follow the covariant treatment of [158, 159]. We refer the reader to thosepapers for details of the method used.The bulk action has to be supplemented by boundary terms of two types.First, there are boundary terms needed to ensure that the variational prob-lem is well-defined. Then there are counter-terms, terms depending onlyon the boundary values (leading non-normalizable modes) of fields on thecutoff surface, which are added to render the on-shell action and the con-served charges finite. Both kinds of boundary terms are the standard onesfor Einstein-Maxwell-Higgs theory; the additional axion coupling does notnecessitate an additional boundary terms of either kind as long as the scalarmass satisfies m2 < 0.We find it convenient to study the first variation of the on-shell action,which always reduces to boundary terms. The expression for the regulatedfirst variation of the on-shell action can be differentiated with respect tothe boundary values of the bulk fields, to give finite expressions for the con-served charges. We write those expressions below in terms of the asymptoticexpansion of the fields occurring in our ansatz, carefully taking into accountthe differences between our coordinate system and the standard Fefferman-178B.1. Asymptotic chargesGraham form of the asymptotic metric, which is used to derive the standardexpressions in the literature.Having explained our procedure, we now display the expressions for theobservables used in the main text. We first assume the radial coordinate isin the standard Fefferman-Graham form, and then discuss additional termsarising from change of coordinate necessary to bring our asymptotic metricinto the standard form.For the scalar fields ψ, one can write asymptoticallyψ(x, r) = ψ(0)(x)r−λ− + ψ(1)(x)r−λ+ (B.1)withλ± =32 ±√94 +m2. (B.2)We set ψ(0)(x) = 0 as part of our boundary conditions, then the coefficientψ(1)(x) is the spatially modulated VEV of the scalar operator dual to ψ.Similarly, the gauge field can be expanded near the boundary asAµ(x, r) = A(0)µ (x)−A(1)µ (x)r . (B.3)The functions A(1)µ (x) correspond to the charge and current density of theboundary theory.As for the boundary energy-momentum tensor, the expression is fairlysimple in odd number of boundary dimensions, and we have checked thatit is not modified by the matter action. With our normalization conventionone can writeTij = 6g(3)ij , (B.4)where the superscripts of the metric functions denote the order in the asymp-totic expansion.Since our metric ansatz is not of the Fefferman-Graham form, we needto perform a change of coordinate (in the x, r plane, for which we usedthe conformal ansatz) to put the metric is such a form. The details of thetransformation are straightforward and the process results in the followingshifts in the asymptotic metric quantities:∆g(3)ij =23g(x), (B.5)for every i, j, where g(x) is the leading asymptotic correction to the metriccomponent grr. That is, at large r that metric component becomesgrr(r, x)→12r2 +g(x)r5 . (B.6)179B.1. Asymptotic chargesFinally, since the metric becomes diagonal asymptotically, the non-vanishingtime components of the energy-momentum tensor Ttt and Tyt have a simpleinterpretation as energy and momentum density, respectively. The con-served charges are given by integrating those densities over a spatial slice.B.1.2 Explicit expressions for the chargesHomogeneous solutionFor reference, in this subsection we give the explicit expressions for thehomogeneous RN solution in our conventions. The radius of the horizon isgiven in terms of the temperature byr0 =16(2piT +√3µ2 + 4pi2T 2). (B.7)The mass, entropy and charge of the RN solution of fixed length L areMRN =(4r30 + µ2r0)L, (B.8)SRN = 4pir20L, (B.9)NRN = 2r0µL. (B.10)The corresponding densities in the infinite system are given by dividingthrough by L.Inhomogeneous solutionHere we list explicit expressions for the thermodynamic quantities in oursystem in terms of our solution ansatz. Conserved charges are given byintegrating over the inhomogeneous direction. We define f (3) = −(4r30 +µ2r0)/4, the 1/r3 term from the function f(r) (equation (4.3)), and X(3)(x),for X = {R,S, T}, as the coefficient of the 1/r3 term of the correspondingmetric function. The energy-momentum tensor yields the mass4M =∫ L0〈T tt(x˜)〉dx˜ = 4∫ L0ξ(x)2(−f (3) + 5S(3)(x) + 3T (3)(x))dx, (B.11)the tension in the x directionτx = −∫ L0〈T xx(x˜)〉dx˜ = 2∫ L0ξ(x)2(f (3)+6R(3)(x)+4S(3)(x)+6T (3)(x))dx,(B.12)4See appendix B.2.3 for details about the numerical process, including the definitionsof the x˜ coordinate and ξ(x). The functions {R,S, T} are defined on the UV grid; theyare analogous to {A,B,C} in the original ansatz.180B.1. Asymptotic chargesand the pressure in the y directionPy =∫ L0〈T yy(x˜)〉dx˜ = −2∫ L0ξ(x)2(f (3) +6R(3)(x)+10S(3)(x))dx. (B.13)Now, expanding the equations of motion at the asymptotic boundary, weget the relation R(3)(x) + 2S(3)(x) + T (3)(x) = 0. Using this, we see that〈Tµν(z)〉 is traceless, as necessary. Conservation of the energy momentumtensor requires ∂xτx = 0. This is related to the constraint equation (4.18)and we explain our strategy to ensure it is satisfied in appendix B.2.3.The coefficient of the 1/r falloff of the gauge field gives the chargeN = −2∫ L0A(1)t (x). (B.14)At the horizon, we read the (constant) temperature asT = 18pir0(12r20 − µ2)e−(B−A)|r=r0 (B.15)and the entropy is proportional to the area of the event horizon, given byS = 4pir20∫ L/40e(B(r0,x)+C(r0,x))dx. (B.16)B.1.3 Consistency of the first lawsHere, we discuss the first laws for both the finite length stripe and the stripeon the infinite domain.Finite systemIn our system, as described above, we have unequal bulk stresses τx5 andPy. Then, if we have a rectangle of side lengths (L,Ly), the work done bythe expansion or compression of this region will differ depending on whichdirection the stress is in. The usual −PdV term in the first law is replacedand we havedMˆ = TdSˆ + µdNˆ + τxLydL− PyLdLy, (B.17)5We define τx = −Px, where Px is the pressure in the x direction. For our solutions,τx > 0.181B.1. Asymptotic chargeswhere the hatted variables represent thermodynamic quantities integratedover the entire system. Defining densities (in the trivial y-direction) byM = MˆLy, S = SˆLy, N = NˆLy, (B.18)we can write the first law asdM = TdS + µdN + τxdL+dLyLy(−M + TS + µN − PyL). (B.19)Tracelessness of the energy-momentum tensor implies M = L(Px + Py), sothat the term proportional to dLy can be rewritten as the conformal identity(4.29), which disappears for a conformal system described by the first law(4.27). Therefore, the first law (4.27) and the conformal identity (4.29) areconsistent.Infinite systemFor the infinite system, we define densities in both the x and y directions asequation (4.33). Under the scaling symmetry (4.19), these scale asm→ λ3m, s→ λ2s, n→ λ2n. (B.20)Using the first law (4.34), we derive the conformal identity (4.35). Again,we can see this from the first law for the system with integrated charges.Plugging the densities m, s, n into the first law of the finite length system(4.27), we arrive atdm = Tds+ µdn+ dLL (−m+ Ts+ µs+ τx). (B.21)Using the conformal identity of the finite length system (4.29), we see thatthe term proportional to dL is just the conformal identity for the infinitesystem, which is satisfied for a system described by (4.34).182B.2. Further details about the numericsB.2 Further details about the numericsB.2.1 The linearized analysisFollowing [30], we look for static normalizable modes around the Reissner-Nordstrom background. We consider the fluctuation6δgty = λ((r − r0)r w(r) sin(kx)),δAy = λ(a(r) sin(kx)),δψ = λ(φ(r) cos(kx)), (B.22)where λ is a small parameter in which we can expand the equations. Puttingthis ansatz into (B.27) - (B.33) and expanding to linear order in λ, we arriveat the linearized systemw′′(r)− r0a′(r)r3(r − r0)+ (4r − 2r0)w′(r)r(r − r0)+ w(r)(2r0(4r3 + 4r2r0 + 4rr02 − r0)− k2r2)r2 (4r4 − r (4r03 + r0) + r02)= 0,a′′(r) +(8r4 + r(4r03 + r0)− 2r02)a′(r)r (4r4 − r (4r03 + r0) + r02)− k2a(r)4r4 − r (4r03 + r0) + r02+ c1kr0φ(r)√3 (4r4 − r (4r03 + r0) + r02)− 4rr0w′(r)4r3 + 4r2r0 + 4rr02 − r0− 4r02w(r)4r4 − r (4r03 + r0) + r02= 0,φ′′(r) + c1kr0a(r)2√3r2 (4r4 − r (4r03 + r0) + r02)− φ(r)(k2 + 2m2r2)4r4 − r (4r03 + r0) + r02−(−16r3 + 4r03 + r0)φ′(r)4r4 − r (4r03 + r0) + r02= 0.(B.23)Fixing the scalar field mass as m2 = −4, there are three parameters in theseequations: the temperature of the black brane T0 (equivalently the locationof the horizon r0), the wavenumber k, and the strength of the axion coupling6Regularity at the black hole horizon enforces that δgty(r0) = 0.183B.2. Further details about the numericsc1. In this analysis, we will choose c1 and k and then use a shooting methodto find the T0 at which normalizable modes appear.Due to the linearity of the equations, the scale of our solutions is arbi-trary. We use this to fix a Dirichlet condition on w at the horizon. Changingcoordinates to ρ =√r2 − r20, and expanding the equations near ρ = 0 givesregularity conditions on the fluctuations at the horizon in terms of Neumannboundary conditions. Our horizon boundary conditions are thenw(ρ)|ρ=0 = 1, w′(ρ)|ρ=0 = a′(ρ)|ρ=0 = φ′(ρ)|ρ=0 = 0, (B.24)Namely, that the fields are quadratic in ρ near the horizon. In order tosearch for normalizable modes, we set the sources in the field theory to zeroby imposing leading order fall-off conditions near the AdS boundary:w(ρ) = w3ρ3 + . . . , a(ρ) =a1ρ + . . . , φ(ρ) =φ2ρ2 + . . . . (B.25)In practice, after fixing c1 and k, we use T0 as a shooting parameter tofind the solution with the correct w fall-off and the corresponding criticaltemperature Tc.For each c1, we find a range of unstable momenta. By adjusting thestrength of the axion coupling, one can find a large variation in the sizeof this unstable region in the (k/µ, T0/µ) plane (see Figure B.1). The re-lationship between c1 and the maximum critical temperature is well fit byTmaxc (c1)/µ = 0.025c1 − 0.091. The wavenumbers for the dominant criticalmodes, corresponding to Tmaxc (c1), for select c1 are found in Table B.1.c1 Tmaxc /µ kc/µ Lµ/4 = pi/2kc4.5 0.012 0.75 2.085.5 0.037 0.92 1.718 0.11 1.3 1.2118 0.37 2.85 0.5536 0.80 5.65 0.28Table B.1: The maximum critical temperatures and corresponding criticalwavenumbers for varying c1.B.2.2 The equations of motionFor completeness, here we present the equations of motion derived from theLagrangian (4.1). The Einstein equations in our case are four second order184B.2. Further details about the numericsc1=8c1=36c1=4.5c1=180 2 4 6 8 10 120.00.20.40.60.8k ΜTcΜFigure B.1: The critical temperatures at which the Reissner Nordstromblack brane becomes unstable, for varying axion coupling c1. As the strengthof the axion coupling increases, the size of the unstable region (the area underthe critical temperature curve) also increases.elliptic equations for the metric components and two constraint equations.For the compactness of the expressions, we defineOˆU · OˆV = ∂rU∂rV +14r4f ∂xU∂xV, Oˆ2U = ∂2rU +14r4f ∂2xU. (B.26)The four elliptic equations, formed from combinations of Gtt− T tt = 0, Gty −T ty = 0, Gyy − T yy = 0, and Grr +Gxx − (T rr + T xx ) = 0, then take the formOˆ2A+ (OˆA)2 + OˆA · OˆC − e−2A+2C2f (OˆW )2 − e−2A4r2f (OˆAt)2− 14r2(e−2AW 2f + e−2C)(OˆAy)2 −e−2AW2r2f OˆAt · OˆAy+(5r +3f ′2f)∂rA+(1r +f ′2f)∂rC +3r2 −3e2Br2f +e2Bm2ψ24r2f+ 3f′rf +f ′′2f = 0, (B.27)185B.2. Further details about the numericsOˆ2B + 12(Oˆψ)2 − e−2A+2C4f (OˆW )2 − OˆA · OˆC − 1r ∂rA+(2r +f ′2f)∂rB −(1r +f ′2f)∂rC = 0, (B.28)Oˆ2C + (OˆC)2 + OˆA · OˆC + e−2A+2C2f (OˆW )2 + e−2A4r2f (OˆAt)2+ 14r2(e−2AW 2f + e−2C)(OˆAy)2 +e−2AW2r2f OˆAt · OˆAy+ 1r ∂rA+(5r +f ′f)∂rC +3r2 −3e2Br2f +e2Bm2ψ24r2f +f ′rf = 0,(B.29)andOˆ2W − OˆA · OˆW + 3OˆC · OˆW − e−2CWr2 (OˆAy)2− e−2Cr2 OˆAt · OˆAy +4r ∂rW = 0. (B.30)The matter field equations areOˆ2ψ + OˆA · Oˆψ + OˆC · Oˆψ + c1e−A−C8√3r4f(∂rAt∂xAy − ∂xAt∂rAy)+(4r +f ′f)∂rψ −e2Bm2ψ2r2f = 0, (B.31)Oˆ2At − OˆA · OˆAt + OˆC · OˆAt +e−2A+2CWf OˆW · OˆAt + OˆW · OˆAy+ 2WOˆC · OˆAy − 2WOˆA · OˆAy +e−2A+2CW 2f OˆW · OˆAy+ c14√3r2(eA−C − e−A+CW 2f)(∂rψ∂xAy − ∂xψ∂rAy)− c1e−A+CW4√3r2f(∂rψ∂xAt − ∂xψ∂rAt) +2r ∂rAt −Wf ′f ∂rAy = 0,(B.32)186B.2. Further details about the numericsandOˆ2Ay + OˆA · OˆAy − OˆC · OˆAy −e−2A+2CWf OˆW · OˆAy −e−2A+2Cf OˆW · OˆAt+ c1e−A+C4√3r2f(∂rψ∂xAt − ∂xψ∂rAt) +c1e−A+CW4√3r2f(∂rψ∂xAy − ∂xψ∂rAy)+(2r +f ′f)∂rAy = 0. (B.33)Finally, the constraint equations are∂x∂rA+ ∂x∂rC − ∂rA (∂xB − ∂xA)− (∂xA+ ∂xC) ∂rB− (∂xB − ∂xC) ∂rC +f ′2f ∂xA−( f ′2f +2r)∂xB− e−2A2fr2 (∂xAt +W∂xAy) (∂rAt +W∂rAy)−e−2(A−C)2f ∂xW∂rW+ e−2C2r2 ∂xAy∂rAy + ∂xψ∂rψ = 0(B.34)and∂2rA+ ∂2rC −14fr4 (∂2xA+ ∂2xC) +(1− 14fr4)(∂rA)2 +(1− 14fr4)(∂rC)2+ 12fr4 (∂xA+ ∂xC) ∂xB − 2 (∂rA+ ∂rC) ∂rB +(3f ′2f +2r)∂rA−(f ′f +4r)∂rB +( f ′2f +2r)∂rC +e−2A8f2r6 (∂xAt +W∂xAy)2− e−2A2fr2 (∂rAt +W∂rAy)2 − e−2(A−C)2f((∂rW )2 −14fr4 (∂xW )2)+ e−2C2r2((∂rAy)2 −14fr4 (∂xAy)2)+ (∂rψ)2 −14fr4 (∂xψ)2 + f′′2f+ 2f′fr = 0. (B.35)B.2.3 ConstraintsThe constraint equations, Grx−T rx = 0 and Grr−Gxx− (T rr −T xx ) = 0, are thenon-trivial Einstein equations that are not part of the system of second-order187B.2. Further details about the numericselliptic equations that we numerically solve. As discussed in section 4.2, theweighted constraints can be shown to solve Laplace equations on the domain.If we satisfy one of the constraints on all boundaries and the other at onepoint, they will be satisfied everywhere. At the black hole horizon, we chooseto impose r2√f√−g(Grr −Gxx − (T rr − T xx )) = 0 at the point (ρ, x) = (0, 0)and √−g(Grx − T rx ) = 0 across the horizon. Since we use periodic boundaryconditions in the inhomogeneous direction, the boundaries at x = 0 andx = xmax are trivial if√−g(Grx− T rx ) = 0 at the horizon and the conformalboundary. Then, we are left with the task of satisfying √−g(Grx − T rx ) = 0at the boundary.In section 4.2, we found the asymptotic expansion of this constraint asGrx − T rx ∝3∂xA(3)(x) + 2∂xB(3)(x) + 3∂xC(3)(x)r2 +O(r−3), (B.36)where A(3)(x), B(3)(x) and C(3)(x) come from solving the elliptic equations.It appears that, within our problem, we do not have the ability to make theweighted constraint disappear. The key lies in an unfixed gauge symmetryin our original metric that is related to conformal transformations of the(r, x) plane.7 Essentially, within our metric ansatz, we have the freedom totransform to any plane (r′, x′) that is conformally related to (r, x). Demand-ing that the weighted constraint √−g(Grx − T rx ) vanishes at the conformalboundary uniquely identifies the correct coordinates (r˜, x˜).Our procedure is to split the domain at some intermediate radial valueρint. On the IR portion of the grid, 0 < ρ < ρint, the equations are asabove. On the UV portion of the grid, ρint < ρ < ρcut, we use the coordinatefreedom to select the correct asymptotic radial coordinate. We can writethe metric in the UV asds2 = −2r˜2f˜(r˜, x˜)e2Rdt2 + e2S( dr˜22r˜2f˜(r˜, x˜)+ 2r˜2dx˜2)+ 2r˜2e2T (dy − Udt)2,(B.37)where f˜(r˜, x˜) ≡ f(r(r˜, x˜)). Under a transformation in the (r˜, x˜) plane suchthat r˜ and x˜ satisfy Cauchy-Riemann-like relations∂r˜(r, x)∂r =r˜(r, x)2r2∂x˜(r, x)∂x ,∂x˜(r, x)∂r = −14r2r˜(r, x)2f(r)∂r˜(r, x)∂x ,(B.38)7See [104] for a discussion of the same issue in a different context.188B.2. Further details about the numericsthe metric becomesds2 = −2r˜(r, x)2f(r)e2Rdt2 + e2S |∇r˜(r, x)|2( dr22r2f(r) + 2r2dx2)+ 2r˜(r, x)2e2T (dy − Udt)2 (B.39)with|∇r˜(r, x)|2 = r2r˜(r, x)2(∂r˜(r, x)∂r)2+ 14r2r˜(r, x)2f(r)(∂r˜(r, x)∂x)2. (B.40)We now have an extra function r˜(r, x) in our system which we may use tosatisfy the constraint and fix the residual gauge freedom, as we will now see.The Cauchy-Riemann-like conditions give the Laplace-like equation∂∂r( r2r˜(r, x)2∂r˜(r, x)∂r)+ ∂∂x( 14r2r˜(r, x)2f(r)∂r˜(r, x)∂x)= 0. (B.41)We can solve this asymptotically, findingr˜(r, x) = ξ(x)r + 2ξ′(x)2 − ξ(x)ξ′′(x)24ξ(x)r + . . . , (B.42)where ξ(x) is an arbitrary function that encodes the coordinate freedom wehave.Expanding the constraint asymptotically, we haveGrx − T rx ∝1r2(2(3∂xR(3)(x) + 2∂xS(3)(x) + 3∂xT (3)(x))ξ(x)+ 3(f (3) + 2R(3)(x)− 4S(3)(x) + 2T (3)(x))ξ′(x))+O(r−3),(B.43)where X = X(3)(x)/r3 + . . . asymptotically, for X = {R,S, T}. Demandingthat the constraint (B.43) vanishes at the leading order yields a differen-tial equation we can solve for ξ(x), giving us a boundary condition for thefunction r˜(r, x), such that the weighted constraint will disappear at the con-formal boundary. However, we have found that the code is unstable if wedirectly use this solution for ξ(x). Instead of directly integrating the con-straint, we use the freedom in ξ(x) to fix the tension τx to be constant.This enforces the same effect on the tension as if we had used the explicitsolution for ξ(x) but is much more stable numerically. Below, we check that189B.2. Further details about the numericsthe constraints are suitably satisfied even though our boundary conditionsdo not exactly fix them. To this end, we setξ(x) = K(f (3) + 6R(3)(x) + 4S(3)(x) + 6T (3)(x))1/3 . (B.44)Expanding the equations asymptotically gives the expressionR(3)(x)+2S(3)(x)+T (3)(x) = 0; if this is satisfied on our solutions our definition of ξ(x) coincideswith that found by integrating the constraint (B.43).The constant K appearing in ξ(x) sets the scale of the boundary theory.We use it to fix the length of the inhomogeneous direction in the field theoryto be Lµ/4. The correct coordinate in the inhomogeneous direction of thefield theory is x˜. From the Cauchy-Riemann conditions, we can find thelarge r expansion of x˜(r, x) asx˜(r, x) =∫ x0dx′ξ(x′) +ξ′(x)8ξ(x)2r2 + . . . . (B.45)Integrating to find the proper length of one cycle in the boundary, we solvefor K at leading order in r to findK = 4L∫ L/40(f (3) + 6R(3)(x) + 4S(3)(x) + 6T (3)(x))1/3dx′. (B.46)When integrating the charges over the inhomogeneous direction in the fieldtheory, one must remember to integrate over the correct coordinate, dx˜ =dx/ξ(x).Our corrected numerical procedure is as follows. On the IR grid, wesolve the elliptic equations (B.27) - (B.33) for the metric functions A,B,Cand W . On the UV grid, we solve the equivalent elliptic equations fromthe metric (B.39) in the variables R,S, T and U plus equation (B.41) forthe new field r˜(r, x). At the horizon, we enforce the boundary conditionsdiscussed in §4.2. At the interface ρ = ρint, we impose matching conditionson the four metric functions and that r˜(ρint, x) = r(ρint). Asymptotically,R,S, T and U all fall off as 1/r˜3. To set boundary conditions on r˜, we noticethat∂rr˜(r, x) +r˜(r, x)r = 2ξ(x) +O( 1r3). (B.47)We truncate this expression at O(r−2) and finite difference to find an updateprocedure for r˜(ρcut, x). This boundary condition is updated iteratively asthe functions R,S, T are updated in our solving procedure such that oncewe find a solution with small residuals we can be sure that the tension isconstant and the constraint is satisfied.190B.2. Further details about the numerics1.0 1.2 1.4 1.6 1.8 2.0 2.20.000.020.040.060.080.10L̐4TΜFigure B.2: The data underlying Figure 4.17. The points represent solutionswe computed. These were interpolated to find the free energy density overthe domain. The solid blue line is the edge of the unstable region and thethick red line is the approximate line of minimum free energy density.B.2.4 Generating the action density plotTo generate the relative action density plot, Figure 4.17, we find the solutionson a grid of lengths L and temperatures T0, as shown in Figure B.2. Byinterpolating these solutions on the domain, we can map the thermodynamicquantities across the unstable region and determine the approximate line ofminimum free energy, or the dominant solution in the infinite size system.B.2.5 Convergence and independence of numericalparametersPerformance of the method and convergence of physical dataAs discussed above, to solve the equations numerically, we use a second or-der finite differencing approximation before using a point-wise Gauss-Seidelrelaxation method on the resulting algebraic equations. The method, in-cluding the UV procedure described above, performs well for this system.The UV procedure is unstable for a generic initial guess, resulting in adivergent norm. To find a solution from a generic initial guess, we can runthe relaxation without the UV procedure until the norm is small enoughthat the result approximates the true solution, before activating the UV191B.2. Further details about the numerics0 500 0001.0 ´ 1061.5 ´ 10610-710-50.0010.110iterationsÈresidualL2Figure B.3: The behaviour of the L2 norm of the residual during the relax-ation iterations for c1 = 8, T0 = 0.04 and Lµ/4 = 0.75. From top to bottom(at the left of the plot) the grid spacing is dρ, dx = 0.04, 0.02, 0.01. The UVprocedure is unstable unless the solution is close enough to correct solution.For grid spacing dρ, dx = 0.04, the UV procedure was activated after 3×105iterations while for the others, the initial guess was taken to be a solutionwith slightly different parameters such that the UV procedure could be usedimmediately.procedure to find the true solution. Once we have these first solutions,by using these as an initial guess for solutions nearby in parameter spaceand by interpolating to a finer grid, we can generate further solutions byrelaxing with the UV procedure. In Figure B.3, we plot the L2 norm ofthe total residual during the relaxation of the c1 = 8 solution at T0 = 0.04and Lµ/4 = 0.75 for the grid spacings dρ, dx = 0.04, 0.02, 0.01, showingthe expected exponential behaviour of the Gauss-Seidel relaxation. Thephysical data extracted from our solutions is consistent with the expectedsecond order convergence of our finite-difference scheme, see Figure B.4.Asymptotic versus first law massA useful check of the numerics is to compare the mass of the system read offfrom the asymptotics of the metric, equation (B.11), to that computed byintegrating the first law, equation (4.27). Since the temperature and entropyare read off from the horizon, comparing these two methods of finding themass provides a non-trivial global consistency check on our results. We192B.2. Further details about the numerics0.0 0.2 0.4 0.6 0.80.01250.01300.01350.01400.01450.01500.01550.0160TTc<OΨ>Μ2Figure B.4: The value of the scalar field condensate for varying grid sizesfor c1 = 8 and Lµ/4 = 0.75. From top to bottom, the grid spacing isdρ, dx = 0.01, 0.02, 0.04. The results are consistent with second order scalingas expected from our numerical approach.verify that the difference between the asymptotic mass and the first law massremains smaller than 0.5% across our set of trials, indicating consistency ofour results.A related check of the numerics is the conformal identity or the Smarr-like relation, 2M = TS + µN − τxL, derived above from the first law forthe finite length system. To evaluate how well our solutions satisfy thisequation, we examine the ratio2Mfall−off − TS − µN + τxLmax(Mfall−off , TS, µN, τxL), (B.48)since the largest term in the expression sets a scale for the cancellation weexpect. This ratio is very small for our solutions near the critical tempera-ture. As we lower the temperature, this ratio increases, but stays small. Theprecise value depends on the parameters of the solution, but is not largerthan order 1%. Moreover, this ratio decreases as we move the position ofthe finite cutoff of the conformal boundary to a larger radius.Finite ρcut boundary checkFor the c1 = 8 trials reported in the paper, we use ρcut = 12 as our conformalboundary. In Table B.2 we present results for varying ρcut, showing that193B.2. Further details about the numericsour choice is large enough such that the physical results are insensitive tothe cutoff. Although the physical results presented in the table appear verystable, at small ρcut, the results for the mass and charge depend significantlyon the fitting procedure for the asymptotic metric functions and gauge field.By running our simulations at ρcut = 12, we are both well within the theregion where the solutions do not change with the conformal boundary andwithin a region where our fitting procedure to the asymptotics behaves well.ρcut S M N1 0.758504 0.305774 0.5274062 0.767913 0.342327 0.4905243 0.768211 0.341928 0.4905934 0.768285 0.342043 0.4905835 0.768311 0.342136 0.4905776 0.768322 0.34221 0.4905747 0.768328 0.342277 0.4905728 0.768332 0.342324 0.490579 0.768334 0.342367 0.49056910 0.768335 0.342402 0.49056811 0.768336 0.342434 0.49056812 0.768336 0.342459 0.490567Table B.2: Behaviour of physical quantities with the cutoff for c1 = 8 andLµ/4 = 0.75 and for fixed grid resolution dρ, dx ∼ 0.02. The entropy S isread off at the horizon, while the mass M and the charge N are read off atthe conformal boundary. Both the entropy and the charge are very robustagainst the location of the conformal boundary. The mass takes slightlylonger to settle down, but is well within the convergent range for ρcut = 12.Behaviour of the constraintsOne of the most important checks for our numerical solution is the behaviourof the constraints. For numerical homogeneous solutions found with ourmethod, the L2 norm of the constraints is very small, on the order of 10−4.For the inhomogeneous solutions, the constraints are small near the criticaltemperature, but grow and saturate as we lower to the temperature, tohave a maximum L2 norm on the order of 10−2: see Figure B.5. Since ourboundary conditions explicitly fix the weighted constraints on the horizon,they disappear there. The weighted constraints then increase towards theconformal boundary, approaching a modulated profile of constant amplitude.194B.2. Further details about the numericsr2pfpg(Grr Gxx  (T rr  T xx ))pg(Grx  T rx )Figure 22: The weighted constraints for c1 = 8 and Lµ/4 = 1.21. The top plots arenear the critical point, T/Tc = 0.97, while the bottom plots are at small temperature,T/Tc = 0.00016. By our boundary conditions, the constraints disappear at thehorizon. They approach a finite value as they approach the asymptotic boundary.plitude. The amplitude near the conformal boundary controls the overall L2 normof the constraints.The constraint violation improves marginally with step size and with movingthe interface closer to the horizon, but does not improve as we take the conformalboundary to a larger radius. To check that the constraints are well satisfied on oursolution, we compare them to the sum of the absolute value of the terms that makeup the constraints. That is, if the constraints are given by Pi hi, we compare thisto Pi |hi|. This procedure gives us an idea of the scale of the cancellation amongthe individual terms hi. We find that the sum Pi |hi| diverges approximately as r4towards the asymptotic boundary, such that the approach of the constraint violationto a constant is a good indicator that the constraints are satisfied on the solution. InTable 3, we compare the L2 norm of these two sums on the entire domain, showing46Figure B.5: The weighted constraints for c1 = 8 and Lµ/4 = 1.21. The topplots are near the critical point, T/Tc = 0.97, while the bottom plots areat small temperature, T/Tc = 0.00016. By our boundary conditions, theconstraints disappear at the horizon. They approach a finite value as theyapproach the asymptotic boundary.The amplitude near the conformal boundary controls the overall L2 normof the constraints.The constraint violati n improves m rginally with step size nd withmoving the interface closer to the ho izon, but does not improve as we takethe conformal boundary to a larger radius. To check that the constraints arewell satisfied on our solution, we compare them to the sum of the absolutevalue of the terms that make up the constraints. That is, if the constraintsare given by∑i hi, we compare this to∑i |hi|. This proce ure gives us anidea of the scale of the cancellation among the individual terms hi. We findthat the sum∑i |hi| diverges approximately as r4 towards the asymptoticboundary, such that the approach of the constraint violation to a constantis a good indicator that the constraints are satisfied on the solution. InTable B.3, we compare the L2 norm of these two sums on the entire domain,showing that the constraint violation for the inhomogeneous solutions isgenerally about four orders of magnitude less than the scale set by∑i |hi|.195B.2. Further details about the numericsInterestingly, the relative constraint improves marginally as we go to lowertemperatures.Parameters T0 L2(∑i hi)/L2(∑i |hi|)c1 = 8, Lµ/4 = 2.00 (RN solution) 0.105 9.12 · 10−7c1 = 8, Lµ/4 = 1.21 (striped solution) 0.075 2.02 · 10−40.05 1.84 · 10−40.025 1.58 · 10−40.005 1.37 · 10−40.001 1.32 · 10−4Table B.3: Comparison of the constraint violation, measured by theschematic constraint equation∑i hi, to the scale set by the individual terms,∑i |hi|, for grid size dρ, dx ∼ 0.01. We take the L2 norm of the measureson the entire domain. The c1 = 8, Lµ/4 = 2.00 solution is a homogeneousRN solution found numerically with our code, for which the constraints arevery well satisfied. The constraints for the striped solutions are satisfiedcompared to the scale set by∑i |hi| by four orders of magnitude and therelative constraint improves marginally as we lower the temperature.The asymptotic equation of motionExpanding the equations of motion asymptotically gives the relationR(3)(x) + 2S(3)(x) + T (3)(x) = 0, (B.49)which can be used to give another check of the numerics. As explained inB.1.2, this condition implies the tracelessness of the energy-momentum ten-sor. For the inhomogeneous solutions near the critical temperature we findthat this expression is on the order of the individual metric functions X(3),where X = {R,S, T}, but generally decreases as we lower the temperature.As well, we find that homogeneous solutions found using our numerical tech-niques satisfy (B.49) well. There seems to be an unidentified systematic er-ror here that may deserve further attention in the future. Possible problemsmay occur in the implementation of the UV procedure or in our procedureto read off the coefficients of the falloffs of the metric functions. However,our physical results are robust under changes to the boundary conditions,so that we are confident in our results despite this possible systematic. Inparticular, the physical quantities extracted from the horizon are indepen-dent of the different boundary constraint fixing schemes we implemented.196B.2. Further details about the numericsTherefore, we advocate using the mass derived from the integrated first law,which uses no asymptotic metric functions.197Appendix CColour superconductivitysupplementary materialIn this appendix to chapter 5, we provide more details about our model witha charged scalar field, analyzing both the large charge limit (section C.1)and finding the critical chemical potential for scalar field condensation (sec-tion C.2).C.1 Large charge limitIn this section, we analyze the case of large q. This is particularly simple,since in this limit, the back-reaction of the scalar and the gauge field on themetric go to zero in the region of the phase diagram where transitions tothe superconducting phases occur. Explicitly, we can show that in the limitq → ∞ with qµ fixed, the gauge field and scalar field decouple from theequations for the metric, but still give rise to a nontrivial phase structure.To investigate this, we need only consider the scalar field and gauge fieldequations on the fixed background spacetimes corresponding to low tem-peratures (the soliton geometry) and high temperatures (the Schwarzschildblack hole).Low TemperatureStarting from the action (5.3) for the scalar field and gauge field on thesoliton background (5.25), we find that the equations of motion are (settingL = 1)φ′′ +(f ′f +4r)φ′ − 2q2r2f ψ2φ = 0 ,ψ′′ +(f ′f +6r)ψ′ + q2r4f φ2ψ − m2r2f ψ = 0 , (C.1)where f is defined in (5.26).198C.1. Large charge limitThese equations have two scaling symmetries related to the conformalsymmetry of the boundary field theory and to the absence of back-reactionin our large charge limit. Given a solution (φ(r), ψ(r), r0, q,m), we can checkthat the scaling(φ(r), ψ(r), r0, q,m)→ (βφ(αr), βαψ(αr),r0α ,qαβ ,m) (C.2)sends solutions to solutions. For our calculations, we will use this to setr0 = q = 1.Multiplying these equations by f and taking the limit r → r0 = 1, wefind that regular solutions must obeyφ′(1) = 2ψ2(1)φ(1)5 ,ψ′(1) = ψ(1)5(m2 − φ2(1)). (C.3)We have two remaining parameters, ψ(0) and φ(0). One of these can be fixedby demanding that the ‘non-normalizible’ mode of ψ vanishes at infinity,while different values of the remaining parameter correspond to differentvalues of µ.Employing numerics, we find that for a fixed value of m2, there is somecritical value of µ above which solutions with a condensed scalar field exist.In order to determine the critical value µc(m2), we use the fact that thefield values go to zero as we approach the critical µ from above. Thus, atthe critical µ, the equations above linearized around the background solutionφ = µ should admit a solution with the correct boundary conditions. Thelinearized equations decouple from each other, so we need only study the ψequation. This becomesψ′′ +( 6r5 − 1r(r5 − 1))ψ′ + r(µ2 −m2r2)r5 − 1 ψ = 0 . (C.4)We can take ψ(1) = 1 without loss of generality, so the boundary conditionfor ψ′ becomesψ′(1) = 15(m2 − µ2) . (C.5)Given m2, we now find µ2 by demanding that the leading asymptotic mode(ψ1) of ψ vanishes. Our results for the critical µ as a function of m2 areshown in Figure C.1.199C.1. Large charge limitHigh temperatureThe high temperature geometry relevant to the limit of large q with µq fixedis the µ → 0 limit of the Reissner-Nordstrom geometry (5.29), which givesthe planar AdS-Schwarzschild black hole (with one of the spatial directionscompactified). This is the relevant background for T > 1/(2piR).Explicitly, we haveds2 = r2(−dt2f(r) + dx2 + dy2 + dz2 + dw2)+ dr2r2f(r) , (C.6)wheref(r) = 1− r5+r5 . (C.7)Here, r+ is related to the temperature byr+ =4piT5 . (C.8)The equations of motion in this background areψ′′ +(f ′f +6r)ψ′ + q2r4f2φ2ψ − m2r2f ψ = 0 . (C.9)The equations have the same scaling symmetry as before, so we can set r+ =q = 1 for numerics. Here, the choice r+ = 1 corresponds to T = 1/(2piR),where R is the radius chosen in the previous section by setting r0 = 1. Inthis case, the boundary conditions areφ(1) = 0 , ψ′(1) = m2L2ψ(1)5 . (C.10)To determine the physics at other temperatures, we can fix q and R and usethe scaling to adjust the temperature.For any values of parameters, we have a solutionψ = 0 , φ(r) = µ(1− 1r3 ) . (C.11)corresponding to the pure Reissner-Nordstrom background in the probelimit.As in the low temperature phase, we find a critical value µc = F (m2)(or, restoring temperature dependence, µc = TTcF (m2)) for each choice of200C.1. Large charge limitFigure C.1: Critical values of µq vs m2 for scalar condensation in largeq limit. The top curve is the critical value for µ in black hole phase (justabove the transition temperature), while the bottom curve is the critical µin low temperature phase.m2, above which there is another solution with nonzero ψ. This critical µmay again be determined by a linearized analysis, from which we obtain theequationψ′′ +( 6r5 − 1r(r5 − 1))ψ′ +(µ2(r3 − 1)2r4(r5 − 1)2 −m2r3r5 − 1)ψ = 0 . (C.12)We can set ψ(1) = 1 without loss of generality, and this requiresψ′(1) = m25 . (C.13)These can be solved numerically to find F (m2), and our results (with thelow temperature results) are plotted in Figure C.1.A sample phase diagram, for the case m2 = −6 is shown in Figure 5.12.Order of phase transitions in the probe limitTo complete this section, we verify analytically that the action for solutionswith scalar field in the probe limit is always less than the correspondingunperturbed solution. In this limit we neglect the gravity back reaction ofthe gauge fields and scalar. The on-shell action in this approximation isgiven byST d =∫dd+1x √−ggttgrrA′2t2 . (C.14)201C.2. Critical µ for solutions with infinitesimal charged scalarWe have used the fact that the scalar action is quadratic and vanishes on-shell once the boundary value of scalar is kept to zero [49]. Writing the actionin this simple form gives us information about the relative free energy of thedifferent phases.The solution for At in the superconducting phase may be written asASt = A0t + δAt , (C.15)where δAt → 0 in the IR region of the bulk and near the boundary. A0t isthe value of At in the normal phase. Then, from equation (C.14) we getSnewT dV =SoldT dV + 2∫dr√−ggrrgtt∂rA0t∂r(δAt) +∫ √−ggrrgtt (δAt)′22 dr .(C.16)The cross term between A0t and δAt vanishes after integrating by parts andthen using the eom of A0t . HenceδS = Snew − Sold = (T dV )∫ √−ggrrgtt (δAt)′22 dr < 0, (C.17)as gtt < 0. Therefore if a phase with non-trivial scalar condensate exists itwill always have a lower free energy than the normal phase and the associatedtransition will be of second order.The introduction of gravity may give rise to a positive term in the on-shell action and the nature of phase transition may change.C.2 Critical µ for solutions with infinitesimalcharged scalarTo find the critical µ at which solutions with infinitesimal scalar field exist,we find the value of µ for which the linearized scalar equation about the ap-propriate background admits a solution with the right boundary conditionsat infinity.At low temperatures, this gives (setting r0 = 1)ψ′′ +(g′g + 4r)ψ′ + 1g(q2φ2r2 −m2)ψ ,g(r) = r2 − 1r3 , φ = µ , (C.18)while for the RN black hole background (setting r+ = 1) we haveψ′′ +(g′g + 4r)ψ′ + 1g(q2φ2g −m2)ψ ,202C.2. Critical µ for solutions with infinitesimal charged scalar0.010.1100.20.40.60.811.21.41.61.82(T/μ)c0.000010.00010.001(T/μ)cqFigure C.2: Critical T/µ vs charge q for condensation of m2 = −6 scalarfield in Reissner-Nordstrom background.g(r) = r2 −(1 + 3µ28)1r3 +3µ28r6 ,φ = µ(1− 1r3). (C.19)More general values of r0 or r+ can be restored by the scaling symmetry.For m2 = −6, we find a critical value of µ in the low-temperature casegiven by µlowq = 5.089/(2piR). At high temperatures, the critical solutionsexist T/µ when has a critical value as plotted in Figure C.2.203

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