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Holographic gauge/gravity duality and symmetry breaking in semimetals Kim, Namshik 2017

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Holographic Gauge/Gravity Dualityand Symmetry Breakingin SemimetalsbyNamshik KimM.Sc., The University of British Columbia, 2011A 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)February 2017c© Namshik Kim 2017AbstractWe use the AdS/CFT correspondence (the holographic duality of gauge/gravitytheory) to study exciton driven dynamical symmetry breaking in certain(2+1)-dimensional defect quantum field theories. These models can be ar-gued to be analogs of the electrons with Coulomb interactions which occurin Dirac semimetals and the results our study of these model systems areindicative of behaviours that might be expected in semimetal systems suchas monolayer and double monolayer graphene. The field theory models havesimple holographic duals, the D3-probe-D5 brane system and the D3-probe-D7 brane system. Analysis of those systems yields information about thestrong coupling planar limits of the defect quantum field theories. We studythe possible occurrence of exciton condensates in the strong coupling limitof single-defect theories as well as double monolayer theories where we finda rich and interesting phase diagram. The phenomena which we study in-clude the magnetic catalysis of chiral symmetry breaking in monolayers andinter-layer exciton condensation in double monolayers. In the latter case, wefind a solvable model where the current-current correlations functions in theplanar strongly coupled field theory can be computed explicitly and exhibitinteresting behavior. Although the models that we analyze differ in detailfrom real condensed matter systems, we identify some phenomena which canoccur at strong coupling in a generic system and which could well be rele-vant to the ongoing experiments on multi-monolayer Dirac semimetals. Anexample is the spontaneous nesting of Fermi surfaces in double monolayers.In particular, we suggest an easy to observe experimental signature of thisphenomenon.iiPrefaceThis thesis contains achievements published by the author and appeared inthe Journal of High Energy Physics, and Physics Letter B. All the papersare based on the one big project, finding a dual gravity model for doublemonolayer semimetals.A version of chapter 2 has been published : Gianluca Grignani, NamshikKim and Gordon W. Semenoff, D3- D5 Holography with Flux, Phys. Lett. B715, 225 (2012). The thesis author was responsible for solving the equationof the motion of the systems and double-checking and refining the numericalresults which my collaborators obtained.A version of chapter 3 has been published : Namshik Kim and JoshuaL. Davis, Flavor-symmetry Breaking with Charged Probes, JHEP 1206 064(2012) [54]. The thesis author contrived the critical idea to stabilize thejoined embedding, and was mainly responsible for numerical calculationsand its visualization.A version of chapter 4 has been published : Gianluca Grignani, NamshikKim and Gordon W. Semenoff, D7-anti-D7 bilayer: holographic dynamicalsymmetry breaking, Phys. Lett. B 722, 360 (2013) [91]. The thesis authorcontributed to all the main computations of correlation functions and writinga first draft for published paper. It was modified by Gianluca Grignani, andGordon Semenoff rewrote it compactly after the first draft.A version of chapter 5 and 6 are based on the same project. The versionof the chapter 5 has been published : Gianluca Grignani, Namshik Kim,Andrea Marini and Gordon W. Semenoff, D7-anti-D7 bilayer: holographicdynamical symmetry breaking, JHEP 1412, 091 (2014) [100]. The versionof the chapter 6 has been accepted for publication in Phys. Lett. B. Thethesis author studied the corresponding field theory model for both gravitydual systems, and contributed obtaining D7-branes dual gravity model andnumerically investigating it.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction and Summary of Results . . . . . . . . . . . . 11.1 Why AdS/CMT . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 D-brane and statement of AdS/CFT conjecture . . . . . . . 21.3 AdS/CFT dictionary . . . . . . . . . . . . . . . . . . . . . . 71.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 D3-probe-D5 Holography with Internal Flux . . . . . . . . 273 Dynamical Symmetry Breaking with Charged Probe Pair 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 D3-charged probes in AdS5 × S5 . . . . . . . . . . . . . . . . 423.2.1 D3-brane probe . . . . . . . . . . . . . . . . . . . . . 453.2.2 D5 probes . . . . . . . . . . . . . . . . . . . . . . . . 463.2.3 D7 probes . . . . . . . . . . . . . . . . . . . . . . . . 473.3 Solutions to effective Lagrangian . . . . . . . . . . . . . . . . 493.3.1 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . 52ivTable of Contents3.4 Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.5 Phase diagram and discussion . . . . . . . . . . . . . . . . . 564 D7-anti-D7 Double Monolayer : Holographic Dynamical Sym-metry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 Holographic D3-probe-D5 Model of a Double Layer DiracSemimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.1 Introduction and Summary . . . . . . . . . . . . . . . . . . . 675.2 Geometry of branes with magnetic field and density . . . . . 775.2.1 Length, Chemical Potential and Routhians . . . . . . 815.3 Double monolayers with a magnetic field . . . . . . . . . . . 845.3.1 Separation and free energy . . . . . . . . . . . . . . . 875.4 Double monolayer with a magnetic field and a charge-balancedchemical potential . . . . . . . . . . . . . . . . . . . . . . . . 895.4.1 Solutions for q 6= 0 . . . . . . . . . . . . . . . . . . . . 905.4.2 Separation and free energy . . . . . . . . . . . . . . . 935.4.3 Phase diagrams . . . . . . . . . . . . . . . . . . . . . 965.5 Double monolayers with un-matched charge densities . . . . 985.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036 Holographic D3-probe-D7 Model of a Double Layer DiracSemimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114AppendicesA Some Calculations for Chapter 4. . . . . . . . . . . . . . . . 126B The Phase Diagram of D3-probe-D7 System . . . . . . . . 129B.1 Geometry of branes . . . . . . . . . . . . . . . . . . . . . . . 129B.2 Double monolayers without charge density . . . . . . . . . . 134B.2.1 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . 137B.2.2 Free energy . . . . . . . . . . . . . . . . . . . . . . . . 138B.3 Double monolayers with charge density . . . . . . . . . . . . 139B.3.1 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . 142B.3.2 Black Hole embedding ρ dependent solution . . . . . 143vTable of ContentsB.4 Connected solutions . . . . . . . . . . . . . . . . . . . . . . . 147C BKT Quantum Phase Transition . . . . . . . . . . . . . . . . 152viList of Tables5.1 Types of possible solutions, where Mink stands for Minkowski em-beddings and BH for black hole embeddings. . . . . . . . . . . . . 855.2 Types of possible solutions for q 6= 0. . . . . . . . . . . . . . . . . 916.1 Types of possible solutions for the balanced charge (q,−q)case, where (Mink,BH) stand for (Minkowski,black-hole) em-beddings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110B.1 Types of possible solutions, where Mink stands for Minkowskiembeddings and BH for black-hole embeddings. . . . . . . . . 134B.2 Types of possible solutions, where Mink stands for Minkowski em-beddings and BH for black-hole embeddings. . . . . . . . . . . . . 141viiList of Figures1.1 D-branes and open strings on the D-branes. . . . . . . . . . . 31.2 Crystal structure of a graphene monolayer and double mono-layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Energy dispersion of graphene . . . . . . . . . . . . . . . . . . 101.4 Energy dispersion around K point. . . . . . . . . . . . . . . 111.5 Schematic illustration of a graphene double monolayers . . . 121.6 Energy band and Fermi level . . . . . . . . . . . . . . . . . . 141.7 Superfluid phase diagram . . . . . . . . . . . . . . . . . . . . 151.8 The model of defect quantum field theory in 2+1 dimension . 171.9 Probe D5 branes on D3 branes background geometry . . . . . 191.10 The solutions of probe D5 branes model with different internalfluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.11 The graph of c versus v . . . . . . . . . . . . . . . . . . . . . 201.12 Phase diagrams for the defect system . . . . . . . . . . . . . . 221.13 A D5 brane and an anti-D5 brane suspended in AdS5 . . . . 231.14 When the D5 brane and an anti-D5 brane are suspended as shown,their natural tendency is to join together. . . . . . . . . . . . . . 241.15 (color online)Phase diagram of the D3-probe-D5 brane system . . 251.16 (color online)Phase diagram of the D3-probe-D7 brane system 262.1 The interpolation of the funnel solution . . . . . . . . . . . . 332.2 The graph of c versus v . . . . . . . . . . . . . . . . . . . . . 342.3 The function r sin(ψ(r)) is plotted versus r for some embeddings 353.1 Straight embeddings and a joined embedding . . . . . . . . . 393.2 A D3-charged probe brane in (finite-temperature) AdS5 × S5. 403.3 The classes of the solutions of a brane/anti-brane pair . . . . 413.4 A cartoon representation of a probe brane . . . . . . . . . . . 433.5 The sign of the integration constant . . . . . . . . . . . . . . 513.6 Skinny and chubby joined embeddings . . . . . . . . . . . . . 513.7 The graph of asymptotic brane separation . . . . . . . . . . . 533.8 An unphysical solution with negative L. . . . . . . . . . . . . . . 54viiiList of Figures3.9 The renormalized free energy . . . . . . . . . . . . . . . . . . 553.10 Phase diagrams for the defect system . . . . . . . . . . . . . . 564.1 Double monolayer system and gauge groups . . . . . . . . . . . . 594.2 The z-position of the D7-branes depends on AdS-radius and withthe appropriate orientation the branes would always intersect. . . 604.3 Joined configuration. . . . . . . . . . . . . . . . . . . . . . . . 615.1 Phase diagram of the D3-probe-D5 brane system with bal-anced charge densities . . . . . . . . . . . . . . . . . . . . . . 705.2 D5 brane embedding without a magnetic field . . . . . . . . . 715.3 D5 brane embedding with a magnetic field . . . . . . . . . . . 725.4 The straight embedding of a D5 brane/an anti-D5 pair . . . . 745.5 The joined embedding of a D5 brane/an anti-D5 pair . . . . . 755.6 The Minkowski embedding of a D5 brane/an anti-D5 pair . . 765.7 The separation of the monolayers, L versus F with q = 0 . . . 885.8 Double monolayer in a magnetic field, where each monolayeris charge neutral . . . . . . . . . . . . . . . . . . . . . . . . . 895.9 The separation of the monolayers, L versus f with finite q . . 935.10 Plots of the free energies as a function of the chemical potential . . 955.11 Phase diagram of the D3-probe-D5 branes system with bal-anced charge densities . . . . . . . . . . . . . . . . . . . . . . 965.12 Phase diagram for large separation between the layers. . . . . . . 975.13 Free energy difference as a function of a brane separation withfinite charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.14 Phase diagram in terms of the brane separation L and the chargedensity q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.15 Energetically favored solution for unpaired charges when Q > Q¯. . 1015.16 Free energy of the solutions when one brane is disconnected andtwo branes are connected . . . . . . . . . . . . . . . . . . . . . . 1015.17 Free energy of the solutions when all the branes are disconnected1025.18 Free energy of the solutions when all the branes are connected 1026.1 Phase diagram of the charge balanced double monolayer (ex-actly nested Fermi surfaces) . . . . . . . . . . . . . . . . . . . 106A.1 ρm as a function of f2. . . . . . . . . . . . . . . . . . . . . . . 127A.2 Regularized energy . . . . . . . . . . . . . . . . . . . . . . . . 128B.1 Unconnected solution for ρ0 = 0 . . . . . . . . . . . . . . . . . 136B.2 Unconnected solution for ρ0 = 0 and q = 0. . . . . . . . . . . . . 136ixList of FiguresB.3 Brane separation L as a function of ρ0 with charge density . . 139B.4 Regularized energy density ∆E as a function of the braneseparation L . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140B.5 Black hole embedding solution with charge density . . . . . . 144B.6 Shooting technique for q = 1/50 . . . . . . . . . . . . . . . . . . 145B.7 Shooting technique for q = 1/30 . . . . . . . . . . . . . . . . . . 145B.8 Shooting technique for q = 1/28 . . . . . . . . . . . . . . . . . . 146B.9 Shooting technique for q = 2/53 ' 0.0377 . . . . . . . . . . . . . 146B.10 Energy of the ρ-dependent black-hole embedding solution . . 147B.11 Brane separation as a function of ρ0 . . . . . . . . . . . . . . 150B.12 Energy as a function of L . . . . . . . . . . . . . . . . . . . . 150B.13 Close around the transition point in Fig. B.12 . . . . . . . . . . . 151xGlossaryJHEP Journal of High Energy PhysicsPRL Physics Review LetterPLB Physics Letter BQFT Quantum field theoryQCD Quantum ChromodynamicsQED Quantum ElectrodynamicsAdS Anti de Sitter space-timeCFT Conformal field theorydCFT Defect conformal field theoryCMT Condensed matter theorySYM Supersymmetric Yang-Mills theoryEC Exciton condensateN The number of colors or D3-branesN The number of super chargesxiAcknowledgementsMy supervisor Gordon Semenoff is the first one should be appreciated forthis thesis. His passion, keen intuition and broad knowledge of physics havebeen always very impressive to me. I could be here as following the sight ofhis back. I like his jokes and laughs, too.I would like to thank Gianluca Grignani, professor at Universita` degliStudi di Perugia, Perugia. He is a true gentleman. I was so pleased at co-working with him. I also thank to Josh Davis, and Andrea Marini I workedtogether.At last, I express my gratitude to Ariel Zhitnisky, Joanna Karczmarek,Moshe Rozali, Fei Zhou, Janis McKenna, Seungjoon Hyun, and all the physi-cists and students that I was affected by.For my personal life, I appreciate the support and love of my girl friend,May Tang. I specially thank my parents, Tagil Kim and Jaeok Lee. Withouttheir unbelievable support and devotion, I could not reach this point. I loveyou.xiiTo my parents.xiiiChapter 1Introduction and Summaryof ResultsVLADIMIR :We’ll hang ourselves tomorrow. Unless Godot comes.ESTRAGON :And if he comes?VLADIMIR :We will be saved.- Waiting for Godot by Samuel Beckett1.1 Why AdS/CMTQuantum field theory (QFT) is an important tool for understanding thebehavior of condensed matter systems. Interacting quantum field theoriesare notoriously difficult to solve. In fact, there are very few if any exactlysolvable field theories which describe systems where the interactions betweenparticles are important. There are two main approaches to understandingan interacting quantum field theory. One is numerical where the idea issimply to solve the equations of motion by numerical means. This approachhas proven quite useful in many cases, however, it has severe limitations.Because the complexity of many condensed matter problems requires pro-hibitive amounts of computer power, there are also specific problems suchas descriptions of Fermi surface and finite density where the more powerfultechniques (e.g, Monte-Carlo simulations) cannot be used.The other main approach is perturbative. It assumes that interactionsare small, and systematically takes them into account using time-dependentperturbation theory. Of course, this can yield accurate results for manysystems where the interactions are indeed small. However, there are somecondensed matter systems where the interactions are much larger. Moreover,11.2. D-brane and statement of AdS/CFT conjectureperturbation theory is not a reliable technique. These systems are said tobe strongly coupled.A couple of intrinsically non-perturbative approaches to quantum fieldtheory using time dependent perturbation theory have been developed overtime. Perhaps the most exciting one is AdS/CMT. This approach usesthe duality between certain strongly interacting quantum field theories andcertain solutions of superstring theory to study the quantum field theoryin the strongly coupled regime. It holds the promise of providing us witha way of solving interesting quantum field theories in the limit where theirinteractions are very large. It also, in principle, gives a way of systematicallycorrecting this strong coupling limit.There are two common approaches to AdS/CMT. Currently, the mostcommon one is bottom-up. It searches for phenomena occurring in stronglycoupled field theories which are so universal that they should be exhibited bypractically any strongly coupled systems with similar environmental factors.The ultimate goal of this approach is to develop paradigms which enableunderstanding and classification of strongly coupled systems.The top-down approach, on the other hand, seeks quantum field theorieswhich have a resemblance to specific condensed matter systems, and which,simultaneously, have a string theory dual. It, then, uses the techniques ofstring theory to study the behavior of the field theory in the parametricregime where the string theory is tractable. This is usually the strong cou-pling regime of field theory. The top-down approach is the main one whichwill be used in this thesis.1.2 D-brane and statement of AdS/CFTconjectureThe key to understanding AdS/CFT are D-branes, which are extended ob-jects in string theory. Let us start from the dynamics of a string ,which aregoverned by Polyakov action. Polyakov action is classically equivalent toNambu-Goto action, the action of a relativistic string which has length andtension.S = − 14piα′∫d2σ ∂αX · ∂αX (1.1)21.2. D-brane and statement of AdS/CFT conjectureFigure 1.1: The picture is obtained from [8] which is also a good reference forAdS/CFT correspondence. (a) A single D-brane and a strings on it transforms inU(1) representation. The length of the open string should be very short becauseof the tension of it, so it is massless. (b)A stack of D-branes constitute U(1) ×U(1)×U(1) · · · gauge group. (c)The distance between D-branes is proportional tothe mass of the open strings between D-branes. For the coincident N D-branes,we obtain non-Abelian SU(N) gauge theory. The index of the gauge field in theadjoint representation is obtained from labeling N D-branes on which open stringsend.Xµ(σ, τ)1 We derive the equation of motion from the extrema of the action.Let us consider the variation of the action from τi to τf .δS = − 12piα′∫ τfτidτ∫ pi0dσ ∂αX · ∂αδX (1.2)The total derivative term becomes as follows from an integration by part.∫ pi0dσ X˙ · δX|τfτi −∫ τfτidτ X ′ · δX|pi0 (1.3)To satisfy the equation of the motion, this total derivative term should1σ, τ are worldsheet coordinates where a string sweeps out. For a periodic σ ∈ [0, 2pi),we have a closed string, and for σ ∈ [0, pi] we have an open string.31.2. D-brane and statement of AdS/CFT conjectureall vanish. The first term is destroyed by requiring δXµ = 0 at τ = τi, τf .To eliminate the second term of (1.3), we demand as follows :∂σXµδXµ = 0 at σ = 0, pi (1.4)There are various boundary conditions which satisfy the above equa-tion. For example, Dirichlet boundary conditions, Xµ, fix end points of theopen string to lie on some hypersurface called a Dp-brane. Moreover, thecondition breaks Lorentz and translation symmetry. The ‘p’ in Dp-branesindicates the number of spatial dimensions. Thus, Dp-branes are simplyp+1 dimensional hypersurfaces on which open strings are attached to beginand end. When quantum fluctuation of an open string is taken into account,the D-brane becomes a dynamical object.In Fig. 1.1, we have used the terminology, representation. we want tobriefly introduce about it and others which will be useful in this thesis. Arepresentation of G (group) is a mapping, D of the element of G onto aset of linear operator preserve the structure of group multiplications. If weassume the group element depends on the continuous parameter, we can ex-pand the linear operator of the representation, and can define the generator,Ta, D(dα) = 1 + i dαaTa + . . . , where dα is an infinitesimal change of thecontinuous parameter. The generators can have some commutation rela-tions and if the structure constants from commutators themselves generatea representation, then we call it an adjoint representation. The fundamentalrepresentation can be understood as the representation for a spin. We candefine raising and lowering operators (or creation and annihilation oprera-tors) from generators, and can define the highest spin eigenvalues. If wewant to interpret some quantum excitation as a spin or a gauge field. Theyshould transform in the proper representations. 2Some readers would have been wondering what AdS is. Instead of de-scribing AdS, here we will introduce its interesting properties. The curva-ture of the space-time is constant and negative, so it exerts a repulsive forceagainst the boundary. In Minkowski space-time, the Hawking thermal ra-diation from a black hole causes it to gradually evaporate. However, if youput a black hole at Poincare´ killing horizon of AdS space-time, the Hawk-ing radiation comes back to the black hole. For example, AdS3 space-timehas a Poincare´ killing horizon at r = 0 since it cannot cover all AdS withthe Poincare´ patch, ds2 = r2(−dt2 + dx2) + dr2r2[14]. There is no tempera-ture associated with it. However, if use AdS-Schwarzshild space-time, the2For the better and more deeper understanding, see the book written by HowardGeorgi. It is not too abstract and easier to read for the physics majored.[7].41.2. D-brane and statement of AdS/CFT conjecturetemperature is associated. By this mechanism, we can stabilize the embed-ding. As a result, if we allow for the thermodynamic property of the blackhole, we can utilize the AdS space-time as a perfect box for thermodynamicexperiments.A dual description of D-branes for AdS/CFT is one of the most suc-cessful achievements in string theory. It is also a black brane solution ofsupergravity or type II superstring theory. D3 branes are the most interest-ing D-branes because the solution of gauge theory is more precise. D3 branesprovide N = 4 SYM in four dimensions. D3 branes extend in time and threespatial dimensional directions. D3 branes have a mass per unit volume, andthey have a charge associated to a self-dual 5-form field strength. Its world-brane has 3+1 dimensional Poincare´ invariance, and has constant axion anddilation fields, Dilaton fields Φ and axion C are solutions of Type II stringtheory here. The dilaton-axion field τ ≡ C + i e−iΦ changes under a Moe-bius transformation. τ → aτ+bcτ+d , ad − bc = 1, a, b, c, d ∈ R. This propertywill be important in AdS/CFT as the reflection of duality in N = 4 SYMtheory. See references [3–5] for more details. This means that D3-branes areregarded as solutions of the ten-dimensional Einstein equations coupled tothe 5-form. When we consider a stack of N D3-branes, a factor of the stringcoupling g from the genus and a factor of N from the Chan-Paton trace,in string perturbation theory, are considered. Thus, perturbation theoryis good as gN  1 and bad as gN  1. g ≡ gstring = g2YM . The blackbranes and D-branes source the same Ramond-Ramond fluxes [13]. Thenear horizon geometrical solution of a black 3-brane is AdS5 × S5 :ds2D3 =r2L2(−dt2 + dx2 + dy2 + dz2) + L2dr2r2+ L2dΩ2S53. (1.5)L is the curvature radius of AdS5 and S5, and L4 = 4pigsNα′2 obtainedfrom the black brane solution. As gN  1, L is large in string units, so thelow energy supergravity is nicely described, but as gN  1, that is no moreeffective. Therefore, these two perturbative prescriptions are complementaryin each limit.D-branes have definite charges, and when a large number of D-branesare put, they become massive and the black hole is expected. This givestwo approaches to the dynamics of the theory at the low energy limit. Atthis limit, both gN  1 and gN  1 descriptions consist of massless strings(open and closed). As gN  1, only open strings remain interacting, andclosed strings are irrelevant and decoupled. The lowest energy excitations3Sn ≡ {x ∈ Rn+1 : ||x|| = 1}.51.2. D-brane and statement of AdS/CFT conjectureof the open strings are described by gauge fields. These provide a super-symmetric Yang-Mills gauge theory, which lives on the world-volume of thebranes. As gN  1, there are massive energy states in the curved geometry.The low energy dynamics is supergravity in the AdS5 × S5, which is thelow energy limit of closed string theory. The degrees of freedom are fluctua-tions of the supergravity fields about the black brane background and theylive in the bulk of ten dimensional space-time. There are some situationswhere these two descriptions have an overlapping domain of validity. Inthose cases, the same physical system is described by two different theorieswhich must therefore be dual to each other. Because the degrees of free-dom in these theories live on spaces of different dimensions (bulk and itsboundary), it is a holographic principle – a property of string theories anda supposed property of quantum gravity that describe the physics of a bulkand can be encoded on a boundary.The original conjectured statement of AdS/CFT [12] is that there is anexact duality between N = 4 supersymmetric SU(N) Yang-Mills (SYM)theory in flat 4D space-time (boundary of the AdS5) and the full quantumtype IIB string theory on AdS5 × S5. For the strongest conjecture, thereis an exact equivalence of the physical spectra at any value of the parame-ters between two theories, including operator observables, states, correlationfunctions and full dynamics, Wilson loop, thermal states, and so on, whichare also translated into the languages of strong theory in AdS. The weakerform of a conjecture is that N = 4 supersymmetric SU(N) Yang-Mills the-ory in flat 4D for t’Hooft limit, large fixed λ = g2YMN and classical type IIBsupergravity on AdS5 × S5. Moreover, the λ−1/2 expansion on the gaugetheory corresponds to α′ expansion on gravity side. There are many goodreviews and original papers to introduce AdS/CFT correspondences. Wefound [8]-[14] useful. More basic knowledge on string theory is availablein textbooks [3–5]. In addition, many videos and lectures on the personalwebsites of physicists are useful study tools for beginners to quickly graspthe whole picture.[6]There is an equivalence of the physical spectra at any value of the pa-rameters between two theories, including operator observables, states, cor-relation functions and full dynamics, Wilson loop, thermal states, and soon, which are also translated into the languages of string theory in AdS.Thus, we can utilize the dictionary to convert the physical quantities fromone theory to the other [8].61.3. AdS/CFT dictionary1.3 AdS/CFT dictionaryIn this section, we will briefly review the application of AdS/CFT corre-spondence : the mapping between contents of two corresponding theories.We will introduce some mapping which will be used in the thesis. Thegravitational partition function is as follows :ZAdS =∫φ0Dφe−S(φ, gµν)φ(t, xi, ; r =∞) = r∆−dφ0(t, xi) + 〈O〉2∆− 4r−∆ + · · · (1.6)φ0 is the boundary value of the field in the gravity side. ∆ is a conformaldimension. The concerned field is a scalar field in the equation, but we cangeneralize it to have indices. ZAdS is simply a classical action of supergravity.It is not difficult to obtain using perturbation.These partition functions are exactly the same asZCFT = 〈e∫∂M φ0O〉Thus, O is the corresponding operator of the source φ0 lives on theasymptotic boundary. Because ZAdS = ZCFT , the one point function isobtained as follows :〈O〉 = δ log ZAdS [φ]δφ0(1.7)Similarly, the two point function can be acquired. We present one ex-ample we will use in later chapters. Remember that the separation betweenD-branes is proportional to the mass of the string connected between thebranes. For a D3-probe-D5 system, the model has SO(3)×SO(3) symmetryand the non-zero separation breaks either of the SO(3) symmetries. Thatcorresponds to chiral symmetry breaking [76]. It occurs with an externalmagnetic field. We will go through this subject thoroughly in later chapters.In this section, let us accept this correspondence, and see how to utilize theholographic duality.For example, let us consider our probe D5 branes. The induced met-ric of probe D5 branes without an external field provides the near horizongeometry, AdS4 × S2.ds2D5 ∼ r2(−dt2 + dx2 + dy2) +dr2r2+ dΩ2S2 (1.8)71.4. MotivationWe expect Dirac fermionic fields and consider the action of the fields.The relevant coordinate along the separation is ψ of S2 of S5 in 1.5. If ψis non-zero, the rotational U(1) symmetry perpendicular to D5 branes isbroken, and it is dual to U(1) chiral symmetry in the gauge theory side.Here is the asymptotic expansion,ψ(r) =pi2+mr+cr2+ · · · , r →∞ (1.9)The length of the separations is r cos(ψ) which is asymptoticallym. m ∝ is the mass of the corresponding quark. By the equation 1.6, c is theexpectation value of the corresponding dual field operator and is linear tochiral condensation, c ∝ 〈q¯q〉. O should be q¯q from the mass term of DiracHamiltonian, mq¯q. ∆ = 1, 2 by ∆± = 12(d+√d2 + 4m2L2). d = 3 for thisdefect field theory. As expected, the source m and 〈q¯q〉 are encoded in theasymptotic expansion of the scalar field.1.4 MotivationGraphene is a single 2+1 dimensional layer of carbon, packed in a hexagonallattice. Carbon hybridization forms three sp2 orbitals, one Pz orbital. Thus,the sp2 orbitals arrange themselves in a plane at 120 angles, and the latticethus formed is the honeycomb lattice. It has been a subject of great interestas a 2+1 dimensional material and the origin of several fascinating predic-tions [98] since many decades ago. Recently, it has captured attentions incondensed matter physics and even in other theoretical physics since Geimand his collaborators acquired the material in laboratory [15]. A carbonatom has four valence electrons. Three of these electrons form strong co-valent σ-bonds with neighboring atoms. The fourth, pi-orbital is un-paired.This property of orbital bonding enables a honeycomb lattice structure in2+1 D. The Hamiltonian isH =∑~A,i(t b†~A+~sia ~A + t∗a†~Ab ~A+~si), t ∼ 2.7ev, |~si| ∼ 1.4A˚ (1.10)It is easy to find the energy dispersion of a single layer of graphene usingtight binding approximation. For the wave function of graphene, we take alinear combination of Bloch functions for sublattices A and B.a ~A = eiEt/~+i~k· ~Aa0, b ~B = eiEt/~+i~k· ~Bb0 (1.11)81.4. MotivationFigure 1.2: This figure and captions are taken from [17]. (a) Crystal structure ofmonolayer graphene with A (B) atoms shown as white (black) circles. The shadedarea is the conventional unit cell, a1 and a2 are primitive lattice vectors. (b) Recip-rocal lattice of monolayer and double-layer graphene with lattice points indicatedas crosses, b1 and b2 are primitive reciprocal lattice vectors. The shaded hexagonis the first Brillouin zone with Γ indicating the center, and K+ and K− showingtwo non-equivalent corners. (c) Above and (d) side view of the crystal structure ofdouble-layer graphene. Atoms A1 and B1 on the lower layer are shown as whiteand black circles, A2, B2 on the upper layer are black and grey, respectively. Theshaded area in (c) indicates the conventional unit cell.As a result, we obtain two energy bands.E(k) = ±t√(1 + 2 cos(3kys2) cos(√3kxs2))2 + sin2(3kys2) , (1.12)where s is the distance of between sites. It is plotted in Fig. 1.3. We obtaintwo bands which are degenerated at the K+ and K− points.Graphene has unique properties that derive from its honeycomb-like lat-tice structure such as the Dirac-like spectrum around the tip of the Diraccone. In other words, in the case of graphene, the low energy propertiesnear the Fermi energy can be described by Dirac equation by expanding theenergy dispersion around the K+ and K− points, and it is a direct conse-quence of graphene’s crystal symmetry. Let us consider a wave vector forvalley K as q = K− k, and expand with |k|  1. The Hamiltonian with91.4. MotivationFigure 1.3: Energy dispersion with two bands for the pz electrons; red planeindicates the Fermi level [18].Fermi velocity, vg ≡ 3s|t|2~ ∼ 106m/s4 is as following :HΨ = vg( −~σ∗  p 00 ~σ  p)Ψ, (1.13)where the ~σ = (σx, σy) is a vector of Pauli matrices, and Ψ is a bi-spinor ofK-points of the first Brillouin zone. The corners of the Brillouin zone formtwo inequivalent groups of K-points. Thus, Ψ = (ΨK+ ,ΨK−)T .HΨ = c( −~σ∗  p 00 ~σ  p)Ψ (1.14)From the comparison with Dirac equation, (1.14), we can see that Fermivelocity of the graphene, vg, behaves like the speed of light. This spinor iscalled a pseudospin. QED phenomena inversely proportional to the speedof light should be, in graphene, enhanced by a factor of c/vg ∼ 3005. Thisrelativistic behaviour of electrons in graphene in low energy could lead tonew possibilities for testing relativistic phenomena (e.g. Some QED effectscannot be tested in particle physics such as perfect tunnelling of relativis-tic electrons). The energy level is half-filled in the neutral graphene. The4vg is obtained from understanding of the Fig. 1.3. The yellow circle around the valley,K-point, provides Dirac-like spectrum around the tip of the Dirac cone, and we read thevelocity from comparison of the slope with Dirac cone.5Fine structure constant with Gauss’ unit is e2~c . Since vg behaves like speed of light,the effective fine structure constant of our model should be e2~ .101.4. MotivationFigure 1.4: As we look into the ~K point of Fig. 1.3, we find out the interactionsof electrons in graphenes are very strong. We will discuss about the energy bandand Fermi energy, EF , later in this section.conduction band and valence band meet at he Dirac chemical point, andgraphene is a zero gap semimetal, which is a material with a small overlapbetween the conduction band and the valence band. A semimetal or semi-conductor is an insulator at 0 K. Since the energy gap is almost none, thevalence band is slightly populated at room temperature. In other words,there can be small electrical conductivity in the semimetal at some finitetemperature.Because of the strength and specificity of its covalent bonds, grapheneis one of the strongest materials in nature, with literally no extrinsic substi-tutional impurities, yielding the highest electronic mobilities among metalsand semiconductors [15] [16]. Therefore, graphene is being considered formany applications that range from conducting paints and flexible displaysto high speed electronics.We model the 2+1 dimensional defect dual field theory in graphene dou-ble monolayers separated by a dielectric barrier which controls the strengthof interactions between an electron on one layer and a hole on the otherlayer. The double monolayer semimetal means two monolayers of semimet-als, each of them would be a Dirac semimetal in isolation, and they areseparated by an insulator, so we ignore direct transfer of electric chargebetween the layers. A double monolayer should be distinguished from bi-111.4. MotivationFigure 1.5: Schematic illustration of a graphene double monolayers exciton con-densate channel in which two single-layer graphene sheets are separated by a dielec-tric (SiO2 in this illustration) barrier. We predict that electron and hole carriersinduced by external gates will form a high temperature exciton condensate. Thisfigure and captions are modified from the figure and caption in [66]layer by whether allows hop or not between each monolayers, and hopping iscontrolled by dielectric materials between double layers. Then, the systemhas two conserved charges, the electric charge in each layer. The Coulombinteraction between an electron in one layer and a hole in the other layeris attractive. A bound state of an electron and a hole forms is called anexciton. Exciton is a boson and, at low temperatures they can condenseto form a charge-neutral superfluid. We will call this an inter-layer excitoncondensate. To distinguish the condensate can occur on the same layer, wewill call it an intra-layer condensate.Electric external gates on each layer adjust the chemical potentials ofeach layer. For the weakly coupled system (Fig. 1.5), when biased byexternal gates, the perfect nesting of the electron and hole Fermi surfacesin each layer tend to form inter-layer exciton condensates6(EC) [65, 66]. Incondensed matter physics, Fermi energy is an internal chemical potential atzero temperature. This is also the maximum kinetic energy of particles inFermi gas. Fermi-Dirac distribution is as follows :〈ni〉 = 1e(i−µ)/kBT + 1,where 〈ni〉 is the mean occupation number, and µ is the Fermi energy. At6The quantum coherent bound states of an electron on one layer and a hole on theother layer. The bound states from the same monolayer form intra-layer EC.121.4. MotivationT = 0 ,〈ni〉 ≈{1 (i < µ)0 (i > µ)By Pauli exclusion principle, the number of the states below µ is exactlythe same as that of the energy level below µ. In momentum space, theseparticles fill up a sphere of radius µ. The surface of the sphere is equivalentto Fermi surface. In condensed matter physics, we can consider a Fermisurface as a boundary in reciprocal space. It is very useful for predicting thethermal, electrical, magnetic, and optical properties of metals, semimetals,and doped semiconductors. The shape of the Fermi surface is derived fromthe periodicity and symmetry of the crystalline lattice and from the occupa-tion of electronic energy bands. The energy band shows the possible energyrange of electrons in the material. The presence of free electrons tells us ifthe material is a metal, a semiconductor or an insulator.because of the Pauli exclusive principle, electrons pack into the lowestavailable energy states and build up a Fermi sea of electron energy states.Fermi level is the highest energy of the electrons at zero temperatures. Theconduction band is the lowest range of vacant electronic states, and thevalence band is the lower band below the conduction band and the bandgap. Chemical potential is the required energy to add an electron in thesystem, and it is same as the Fermi energy at zero temperature. Due toPauli exclusion principle, the adding electron should have a new highestenergy.In conductors, electrons only partially fill the valence band and the va-lence and conduction bands are very close or overlap, thus electrons becomeconductive easily. On the other hand, insulators and semiconductors haveFermi levels lying in the forbidden band gap and have full valence bands,therefore insulators have electrons with nowhere to go or jump and semi-conductors become conductive only at certain temperatures since for T > 0,thermal excitation allow the particles to be found in the higher energy states.The nesting condition with ~k = 0 requires only that the Fermi surfaces beidentical in area and shape. It does not require the two layers to have alignedhoneycomb lattices and hence aligned Brillouin-zones. At weak coupling, abinding energy is maximized when the momenta are as ~p1 = −~p2, whichmeans the Fermi energies are nested, 1 = 2 =~p21,22m . Similarly, it is observedthat a Cooper pair is destroyed when an external magnetic field is exertedin the system since the Fermi energies are split by Zeeman effect.These studies of inter-layer EC are very interesting for understanding131.4. MotivationFigure 1.6: The energy band and Fermi level are very useful to see the physicalproperties of solids. We can classify solids from the size of the gap and where thegap and Fermi level are. These are related to the presence of the free electrons insolids.fundamental issues with quantum coherence over mesoscopic distance scalesand dynamical symmetry breaking. Moreover, the study accompanies nu-merous interesting applications in electronic devices because the result ofmean field theory calculations expects a room temperature superfluid [66].As a quantum phenomenon, the room temperature superfluid is intriguing.It is known that superfluidity occurs at very low temperature and weakcoupling [2]. Let us consider a finite density ρ of non-relativistic bosonsinteracting with a short ranged repulsive Mexican well potential. Then, aspontaneous symmetry breaking occurs; and a gapless mode in the fluid ofbosons with linear dispersion, (ω ∝ ~k) yields superfluidity. A fluid flow-ing down with a velocity v and mass m could lose its momentum by cre-ating a momentum excitation : mv = mv′ + ~k. The energy becomes12mv2 ≥ 12mv′2 + ~ω. By eliminating v′, we obtain the relation, v ≥ ω/k. Inother words, if the fluid is slower than that critical velocity vc = ω/k, it can-not lose its momentum, and so it gains superfluidity. It can be understoodeasily by comparing to a Fermi liquid having a continuum of gapless excita-tions. Thus, we should lower the temperature of the weakly coupled systemto gain the superfluidity. If the interaction is strong enough, we could expectthe energy gap between the lowest state and an excited state to become big-ger and might gain superfluidity with higher temperature. Therefore, it is141.4. MotivationFigure 1.7: Normal to superfluid phase diagram at weak coupling and captiontaken from [66]. The critical temperature Tc in Kelvin on distance is around roomtemperature with nanoscale distances. It is obtained using a coupling constantappropriate for a SiO2 dielectric.important to check if the system has an energy gap, which results in excitoncondensates and superfluidity (and superconductivity). We can investigatethe superfluidity by checking the gap and correlation functions in stronginteracting systems.In the model the temperature should be higher than that of usual Bose-Einstein condensates or typical superconductors because condensation isdriven by Coulomb interactions over the fill band width, rather than byphonon-mediated interactions among quasiparticles in a narrow shell aroundthe Fermi surface. In this sense, exciton condensation is more similar toferromagnetism, which is also driven by Coulomb interactions and appears tobe at high temperatures [66]. Besides, with an external gate, the model hasmore carriers, and graphene layers are so thin that the Coulomb interactionis not screened as much as in semiconductor quantum well bilayers. Thenumerical estimation is shown in Fig. 1.7.In fact, the potential applications of superfluids are not as exciting andwide as those of superconductors. However, it is still useful for various areas.The most interesting application from this model might be bilayer (or possi-bly double monolayers) pseudo spin field-effect transistor (BiSFET) [112] byusing transitions between superfluid mode and non-superfluid mode, whichis researched by the author of [66] and his collaborators.Moreover, there are some educational applications, showing how quan-151.4. Motivationtum effects can become macroscopic in scale under certain extreme con-ditions as illustrated by Bose-Einstein condensates. Many bosons can becondensed at the same time at the lowest level of quantum states by abosonization, which means it is quantum effect and can be seen in macro-scopic level as do LASERs and superconductors. Superfluids have someimpressive and unique properties that distinguish them from other forms ofmatter. As they have no internal viscosity, a quantum vortex formed withinone persists indefinitely and its angular velocity is quantized. A superfluidhas zero thermodynamic entropy and infinite thermal conductivity. Super-fluids can also climb up and out of a container in a one-atom-thick layer ifthe container is not sealed; this is called a capillary action and a creep phe-nomenon. 7. This superfluid quantum liquid prevails gravity of the Earth onthe surface and is able to climb out of any container in a thin film moving upto 35 cm per second. A conventional molecule embedded within a superfluidcan move with full rotational freedom, behaving like a gas. Other interest-ing properties may be discovered and designed to be applied for electronics,among others, in the future.Despite the usefulness of this model, the theoretical analysis of the weakinteracting model is limited in fully understanding the system because theCoulomb interaction at the distance (3nm) that causes room temperaturesuperfluid must be very strong. Therefore, the perturbation theory shouldbe re-summed in ad hoc way to take screening into account. We find non-perturbative model of very strongly coupled multi-monolayer systems8. Thismodel is a defect quantum field theory which is the holographic AdS/CFTdual of a D3-probe-D5/anti-D5-branes system and D3-probe-D7/anti-D7-branes system.We model defect quantum field theory in 2+1 dimensional space-timewith double monolayer defects. It is shown in Fig. 1.8. N = 4 SYM theorylives in a 3+1 dimensional bulk and there can be U(1) charged fermionicfields on defect. The interaction between them is mediated by N = 4 SYMgauge fields in bulk. In order to consider a holographic gravity dual of thesemimetals, e.g. graphene, we use evidence that D7 brane model resemblesgraphene in that it has relativistic fermions on defect. D7 branes sharethe coordinates t, x, y with D3 branes, and intersect probe branes along z.7Readers can watch the remarkable properties of liquid helium when cooled below thelambda point (the superfluid state) through the hyperlink provided in [24]. It is video-recorded in 1963 by Alfred Leitner8 Graphenes interact by exchanging photons. The double monolayers in the modelinteract by exchanging gluons of N = 4 supersymmetric Yang-Mills theory and truncateto planar limit. We do not construct the lattice structure.161.4. MotivationFigure 1.8: The infinite parallel planar (2+1)-dimensional defects bi-secting (3+1)-dimensional spacetime.This D7 brane model breaks supersymmetry so that it becomes a fermonictheory. On the other hand, D5 brane has supermultiplet of fermions andscalars on defect. Supersymmetry is not broken until an external magneticfield is introduced. Low energy physics is governed by fermions. This isknown from research on intra-layer condensates which are seen in graphene.As we mentioned earlier, graphenes interact by exchanging photons of U(1)theory, whereas our model interactions exchange N = 4 super Yang-Mills(SYM) gluons, and truncate to planar limit; N →∞, λ = g2YMN →∞ andgYM  1. As we have discussed briefly, large number of D3-branes at lowenergy limit cause the AdS background geometry, and we consider probeD7 and D5 branes embedded on the background geometry. We also takethe no backreaction limit, N ≡ N3  N5, N7. 9 The motivation of thedouble layer Dirac Semimetals is well discussed in the later section, 5.1. Werecommend reading this introductory section before proceeding.9If the mass of the sun is not heavy enough, it will be much more difficult to calculatethe motion of the earth because of the gravitational backreaction. We should consider themovement of the sun at every moment. Similarly, we can fix the geometrical backgroundwith the no-backreaction limit, N3  N5, N7.171.5. Overview1.5 OverviewThe rest of the thesis is summarized as follows. We study a monolayer systembefore studying a double monolayer system. We probe D5 branes on D3branes AdS geometry background, AdS5×S5. Fig. 1.9 is the figure to showthis brane model. D5 branes span in AdS4 on AdS5. D3 and D5 branes arein force equilibrium without an external magnetic field or charges, D5 branesjust stretch out toward a Poincare´ killing horizon of D3 branes (Left figure ofFig. 1.9). With an external magnetic field perpendicular to D5 branes, thesupersymmetry and conformal symmetry are both broken. Then, D5 branesreceive repulsive force from D3 branes, and the force becomes stronger ascloser as to the Poincare´ horizon10, so the branes pinch off and truncate ata finite AdS5-radius, before they reach the Poincare´ horizon. This is calledMinkowski embedding (Right figure of Fig. 1.9). It can be considered asdynamical symmetry breaking corresponding to intra-layer condensates insingle layer of graphene. This configuration has a charge gap. Chargeddegrees of freedom are open strings which stretch from the D5 brane tothe Poincare´ horizon. When, the D5 brane does not reach the Poincare´horizon,the open string has a minimum length and therefore a mass gap.It occurs at any value of an external magnetic field. This phenomenonis elaborated to understand a quantum Hall ferromagnetism in [85] and[77]. The quantum Hall system consists of a bunch of electrons moving in aplane in the presence of an external magnetic field B perpendicular to theplane. The magnetic field is assumed to be sufficiently strong so that theelectrons all have spin up, so they can be treated as spinless fermions. Thespinless electrons in a magnetic field have Landau quantized energy level.The Coulomb interaction has generated a small energy gap. This gap breakssome symmetry of the system. The phenomenon of interaction inducedgaps and broken symmetries at integer filling factors is known as quantumHall ferromagnetism. The four-fold degeneracy of graphene’s Landau levelsfollows from approximate spin-degeneracy and from Bloch state degeneracybetween two inequivalent points in the honeycomb lattice Brillouin zone [86].For D3/D5 branes model, the gap breaks SO(3) chiral symmetry and makesthe neutral state gapped.When U(1) charge density is introduced, it becomes more interesting.In this case, the tension of D3-D5 strings competes with D5 branes tension.The tension of the strings is so greater than that of D5 branes that the10In Fig. 1.9, r = 0 is the Poincare´ horizon, and r = ∞ is an asymptotic boundary ofAdS5.181.5. OverviewFigure 1.9: Probe D5 branes on D3 branes background geometry without and withan external magnetic field.strings have zero length and the mass gap vanishes. If the ratio of chargedensity to the magnetic field, filling fraction, qb11 is bigger than some criticalvalue, the system restores the symmetry. In other words, D5 branes stretchdown to the horizon directly, and the chiral condensate term vanishes with avanished mass gap. On the other hand, if the filling fraction is smaller thanthe critical value, the chiral symmetry is still broken without a mass gap.This is called a Berezinski-Kosterlitz-Thouless-like (BKT) phase transition.We have showed that an internal flux on the sphere of D5 branes also behaveslike the charge density. The newly defined parameter for the phase transitionis√f2 + ( qb )2, where f is a constant factor proportional to the internal flux.We present more detail in the chapter 2. Fig. 1.10 and Fig. 1.11 are the mainnumerical results obtained in [92] and introduced in chapter 2 of this thesis.We have found the solutions of systems in Fig. 1.10, which interpolate for theMinkowski embedding with an external magnetic field and internal fluxes.Fig. 1.11 is obtained from the data of multiple interpolate solutions as seenin Fig. 1.10. m is a parameter of a fermonic mass, and c is a parameterof an intra-layer chiral condensation in (1.9). We interpret it as a solutionwith dynamical symmetry breaking, i.e., a chiral condensate with no baremass term.In chapter 3, at last do we consider double layer probe branes. Forprobe D5 brane on D3 brane background geometry, we have a symmetric11q is rescaled charge density. b is rescaled magnetic field. f to be appeared below is arescaled internal flux on two sphere of five sphere.191.5. OverviewFigure 1.10: Fig. 2.1 in the chapter 2. The solutions of probe D5 branes modelwith different internal fluxes with the parameter f and the external magnetic field.Figure 1.11: Fig. 2.2 in the chapter 2. The graph of c versus v.201.5. Overviewtrivial phase before breaking supersymmetries. Turning on the B-field, bothbranes and anti-branes pinch off from the horizon and intra-layer excitontransitions occur for each layer.The each probe and anti probe brane have symmetries U(Nf ) and U(N¯f )respectively. When the brane pair meets, symmetries of the pair are brokento the symmetry of one set of branes geometrically down there. The Diracfermions lives on each probe brane and anti-probe brane. Thus, the sym-metry breaking corresponds condensate of the fermions and anti-fermion.Note that we do not consider the charge density yet. We have chronologi-cally studied this joined U-shaped model before studying how to introducea charge density in the model. The readers might wonder why we do notconsider the factor of temperature. It is easy to introduce temperature inthe model by putting a black hole in AdS. AdS itself is a covariant box inwhich we can put the black hole and do experiments on it. A black holepulls probe branes toward the horizon, so symmetry favors to be restored.It is known as symmetry restoration by temperature. Since we know whatwould happen when we introduce temperature in the model, without a lossof generality, we would consider zero temperature in the model in the laterchapters than chapter 3. We also investigate the thermodynamics of braneconfigurations and obtain a phase diagram of the configurations (symmetricphase/broken symmetric phase), and have found phase diagrams. We loadmultiple phase transition lines in Fig. 1.12 with respect to different fluxes,α ≡ ζf√f2+412. The dominant solution above any specific curve is given bythe two disconnected brane worldvolumes, i.e. the symmetric phase. Thejoined solutions, the broken symmetry phase, dominates below the curve.As a result, we newly see inverse magnetic catalysis in some range of internalfluxes of probe branes. For example, see the curve with α = −0.5, uppergreen solid line. The symmetric phase is favored as B increases.In chapter 4, we study a holographic model of dynamical symmetrybreaking in 2+1-dimenisons, where a parallel D7-anti-D7 brane pair fusesinto a single object, corresponding to the U(1) × U(1) → U(1) symmetrybreaking pattern. It is slightly different from the model we have studied inthe chapter 2. Simply saying, this model is more symmetrical. Therefore,we show that the current-current correlation functions can be computedanalytically and exhibit the low momentum structure that is expected whenglobal symmetries are spontaneously broken. Of interest is that we canchoose the U(1) as global or gauged by choice of boundary condition. We12f is a constant proportional to internal flux on S2. ζ is a coefficient of an orientation.See the detail in chapter 3.211.5. OverviewFigure 1.12: Fig. 3.10 in the chapter 3. Phase diagrams for the defect systemobtained from D3/probe D7 branes model with an external magnetic field andinternal fluxes. α ≡ ζf√f2+4. ζ is a parameter of an orientation, which are signfactors +1/− 1 of Wess-Zumino term. See chapter 3 for the detail.are able to see 3 different current-current two point functions from Dirchlet-Dirichlet, Neumann-Neumann, and Dirchlet-Neumann boundary conditionpair. The most interesting case is that one is global U(1) and the other isgauged (DN boundary condition pair). Then, the global U(1) symmetry isspontaneously broken and its current has a pole in its correlation function.The unbroken gauged U(1) has a massless pole corresponding to the photon.It is seen in bilayer graphenes. We also find that these correlation (2-point)functions have poles attributable to infinite towers of vector mesons withequally spaced masses.In chapter 5 and 6, we finalize the project we have discussed in thesection 1.4. Respectively, we study probe D5 branes in chapter 5, and studyprobe D7 branes in chapter 6. A D5 brane and an anti-D5 brane are aresuspended with a distance L apart at the AdS5 asymptotic boundary, asshown in Fig. 1.13. When the D5 brane and an anti-D5 brane are exposedto a magnetic field, and if the field is strong enough, they can pinch off andend before they join, Fig. 1.13. So far, it is not different from the modelof a single layer. However, the brane and anti-brane tend to join like aparticle-hole pair without an external magnetic field. The tendency to joincompetes with the tendency to pinch off, and the competitions provide thephase diagrams Fig. 1.15 and Fig. 1.16 in [100, 101] , which present the221.5. OverviewFigure 1.13: A D5 brane and an anti-D5 brane suspended in AdS5 withoutan external magnetic field, both branes stretch to Poincare´ horizon. Whenturning on an external magnetic field, they can pinch off before join.results of a thermodynamical preference between 3 possible phases in thephase diagrams [100, 101].Let us first see the phase diagram of probe D5 branes at Fig. 1.15.Electric field in the field theory model would effectively change the electriccharge density. It is known that a doped bilayer graphene also has a bandgap. That is why non-zero charge q is needed for inter-layer phases in theblue/green regions. For the degenerate gapless double monolayer graphene,the Dirac point chemical potential is a non-zero. It is also in agreement withD3/D5 phase diagram we obtained. For a fixed separation L, the red regionis gapless and no inter-layer condensation with non-zero charge density, andit corresponds to Minkowski embedding. There are two dotted lines in thephase diagram. One is for the first order phase transition between L = 1.357and 1.7, and the other line is asymptote to the spot BKT transition of singlelayer model between intra-layer phase and symmetric phase occurs when Lis infinite. For infinite separation, L, we can treat the system disconnected,and then recover the behavior of a single layer. For small L, the branes areconnected at any value of a chemical potential. It is also notable that thereis no symmetric phase corresponds to the left figure in Fig. 1.13. It makessense because we turned on an external magnetic field for the system.For the phase diagram of probe D7 branes, the vertical axis is layerseparation L in units of the inverse ultraviolet cutoff, R. The horizontalaxis is the charge density q in units of R2, where R4 = λα′2. The extremal231.5. OverviewFigure 1.14: When the D5 brane and an anti-D5 brane are suspended as shown,their natural tendency is to join D3 brane geometry is given as follows :ds2R2=r2(−dt2 + dx2 + dy2 + dz2)√1 +R4r4+√1 +R4r4(dr2r2+ dψ2 + sin2 ψ5∑i=1(dθi)2)(1.15)where Σ5i=1 (θi)2 = 1. Because R ∼ α1/2, 1/R can be regarded as a UVcutoff. The main merit of using this geometry compared to AdS5 × S5 weused for probe D5 branes is that we need an external magnetic field nomore in order to break the supersymmetry. The asymptotic large r limit isMinkowski spacetime. In the near horizon limit, rR  1, the geometry isAdS5 × S5.We can find a couple of differences below Fig. 1.15 with Fig. 1.16,made by the cutoff factor. The most outstanding one is by cutoff. WhenL is smaller than about R, inter-layer condensates never occur any more.The brane pair is disconnected shown the red and white area in Fig. 1.16.Moreover, there exists symmetric phase (white). The phase curve betweenblue and green regions is also a bit different.We have introduced chemical energy nesting in the weak interactingmodel. The reader might ask what would happen at strong coupling, andmaybe, could expect no need of the perfect nesting. However, inter-layercondensate occurs only if charge densities are balanced. It is sharper than in241.5. Overview0 0.5 1 1.5 2 2.5 3024681st order2ndorder2ndorderintraq = 0intra/interinterλ1/42pi√Bµ√2piBλ1/4L1.357Figure 1.15: (color online)Phase diagram of the D3-probe-D5 brane systemthe field theory. Moving away from nesting destroys the mass gap. We newlyfound that when there is more than one species of fermions, nesting can occurspontaneously. The tendency of interlayer condensate with perfect nesting isthe most, so even with unbalanced charges on each layer, they make a perfectnesting at first, and the remaining charged probe branes have an intra-layercondensate and the left non-charged probe branes remain symmetric. Thiskind of perfect nesting is hard to achieve in the lab. Graphenes or semimetalswe concern have crystal structure, whereas our model assume the density ofthe matter is uniform. Then it would be a bit difficult to match the atomicstructures of two layers in the lab. We need more work in this part.It is interesting to think why the green and blue region extend to aninfinite separation, L→∞. It also agrees with weakly coupled systems. Atweak coupling, the Coulomb force is a long-range interaction. For example,superfluidity can exist inside neutron stars. By analogy with electrons insidesuperconductors forming Cooper pairs due to electron-lattice interaction, itis expected that nucleons in a neutron star at sufficiently high density andlow temperature can also form Cooper pairs due to the long-range attractivenuclear force and lead to superfluidity and superconductivity. From thispoint of view we expect that the strong correlations to play an importantrole in delimiting the magnitude of the pairing gap [68]. As we mentioned251.5. Overview0 0.02 0.04 0.06 0.08 0.10246intraintra/interinterchiral symm.intraqLFigure 1.16: (color online)Phase diagram of the D3-probe-D7 brane system. Wehave used charge density, q, instead of chemical potential, µ as an x-axis. If we usechemical potential (x-axis), the diagram with L & 1 looks almost the same to Fig.1.15. The q and L are rescaled with√2piBλ1/4 = the previous section, the balanced Fermi surface seems more crucial toform condensates than the distance between layers. This also agrees withweakly interacting theory.26Chapter 2D3-probe-D5 Holographywith Internal FluxThree quarks for Muster Mark!Sure he hasn’t got much of a barkAnd sure any he has it’s all beside the mark.- Finngans wake by James JoyceThe AdS/CFT duality of an appropriately oriented probe D5-brane em-bedded in AdS5 × S5 space-time and a supersymmetric defect conformalfield theory is a well-studied example of holography [87]-[99]. In the limit oflarge N and large radius of curvature, the D5-brane geometry is found as anextremum of the Dirac-Born-Infeld action with appropriate Wess-Zuminoterms added. Its world-volume is the product space AdS4(⊂ AdS5)× S2(⊂S5) which preserves an OSp(4|4) subgroup of the SU(2, 2|4) superconformalsymmetry of the AdS5 × S5 background. The superconformal field theorywhich is dual to this D3-D5 system, and which is described by it in the strongcoupling limit, has a flat co-dimension one membrane that is embedded in3+1-dimensional flat space. The bulk of the 3+1-dimensional space is occu-pied by N = 4 supersymmetric Yang-Mills theory with SU(N) gauge group.A bi-fundamental chiral hypermultiplet lives on the membrane defect andits field theory is dual to the low energy modes of open strings connectingthe D5-branes and the D3-branes. These fields transform in the fundamen-tal representation of the SU(N) bulk gauge group and in the fundamentalrepresentation of the global U(N5), where N5 is the number of D5-branes (inthe probe limit, N5 << N and we will take N5 = 1). The defect field theorypreserves half of the supersymmetries of the bulk N = 4 theory, resultingin the residual OSp(4|4) super-conformal symmetry. It is massless with ahypermultiplet mass operator which breaks an SU(2) R-symmetry [89].An external magnetic field has a profound effect on this system. In thequantum field theory, the magnetic field is constant and is perpendicular to27Chapter 2. D3-probe-D5 Holography with Internal Fluxthe membrane defect. In the string theory, the magnetic field destabilizesthe conformal symmetric state to one which spontaneously breaks the SU(2)R-symmetry and generates a mass gap for the D3-D5 strings [76]. The onlysolution for the D5-brane embedding has it pinching off before it reachesthe Poincare´ horizon of AdS5. As a result, the D3-D5 strings which, whenexcited, must reach from the D5-brane to the Poincare´ horizon, have a min-imum length and an energy gap. This occurs for any value of the magneticfield, in fact, since the theory has conformal invariance, the magnetic fieldis the only dimensional parameter and there is no distinction between largefield and small field. A mass and a mass operator condensate for the D3-D5strings can readily be identified (their conformal dimensions are protectedby supersymmetry) and there is simply no solution of the probe D5-braneembedding problem with a magnetic field when both the mass and the con-densate are zero. There can be a solution when one of those parametersvanishes and the other does not vanish. Such a solution can be interpretedas presence of a condensate in the absence of a mass operator, that is, as dy-namical symmetry breaking. This phenomenon is regarded as a holographicrealization of the “magnetic catalysis” of chiral symmetry breaking that hasbeen studied in 2+1-dimensional quantum field theories [71]-[78] . The fieldtheory studies rely on weak coupling expansions and re-summation of Feyn-man diagrams. Whether the phenomenon can persist at strong coupling isan interesting question which appears to have an affirmative answer in thecontext of this construction. It and many other aspects of the phase dia-gram of the D5-brane have been well studied in what is by now an extensiveliterature [36, 76, 99, 102].This interesting behavior becomes more complex when a U(1) chargedensity is introduced. The state then has a non-zero density of D3-D5strings as well as a magnetic field. There is also a tuneable dimensionlessparameter, the ratio of charge density to the field, the “filling fraction”ν = 2piρB . In this case, there is no charge gap. The D5-brane must necessarilyreach the Poincare´ horizon. This is due to the fact that, to have a nonzerocharge density, there must be a density of fundamental strings suspendedbetween the D5-brane and the Poincare´ horizon. However, the fundamentalstring tension is always greater than the D5-brane tension [36] and suchstrings would therefore pull the D5-brane to the horizon. The result is agapless state: the D3-D5 strings could have zero length, and therefore haveno energy gap. At weak coupling, the analogous process is the formationof a fermi surface and a gapless metallic state when the charge density isnonzero.What is more, if the filling fraction is large enough, the state with no28Chapter 2. D3-probe-D5 Holography with Internal Fluxmass term and mass operator condensate equal to zero exists and is stable.In this state, the SU(2) R-symmetry is not broken. As the filling frac-tion is lowered from large values where the system takes up this symmetricphase, as pointed out in the beautiful paper [102], the system undergoesa Berezinski-Kosterlitz-Thouless-like (BKT) phase transition. This phasetransition has BKT scaling and is one of the rare examples on non-meanfield phase transitions in holographic systems. When the filling fraction isless than the critical value, again, even though the D5-brane world-volumenow reaches the Poincare´ horizon, there is no solution of the theory unlesseither the mass operator or mass operator condensate or both are turnedon.In this chapter, we shall observe that, as well as density, there is asecond parameter which can drive the BKT transition. The parameter isthe value of a magnetic flux which forms a U(1) monopole bundle on theD5-brane world-volume 2-sphere. The possibility of adding this flux wassuggested by Myers and Wapler [36]. They found that the idea could beused to construct stable D3-D7 systems, in particular, and a modificationof their idea was subsequently used to study holography in D3-D7 systems[37, 39, 54] In the limit where the string theory is classical, the problem ofembedding a D5-brane in the AdS5×S5 geometry reduces to that of findingan extremum of the Dirac-Born-Infeld and Wess-Zumino actions,S =T5gs∫d6σ[−√−det(g + 2piα′F ) + C(4) ∧ 2piα′F](2.1)where gs is the closed string coupling constant, which is related to the N = 4Yang-Mills coupling by 4pigs = g2YM , gab(σ) is the induced metric of the D5brane, C(4) is the 4-form of the AdS5×S5 background, F is the world-volumegauge field and T5 =1(2pi)5α′3 . We shall use the metric of AdS5 × S5ds2 = L2[r2(−dt2 + dx2 + dy2 + dx2) + dr2r2++dψ2 + cos2 ψ(dθ2 + sin2 θdφ2) + sin2 ψ(dθ˜2 + sin2 θ˜dφ˜2)](2.2)Here the 5-sphere is represented by two 2-spheres fibered over the intervalψ ∈ [0, pi2 ]. The 4-form isC(4) = L4r4dt ∧ dx ∧ dy ∧ dz + L4 c(ψ)2d cos θ ∧ dφ ∧ d cos θ˜ ∧ dφ˜ (2.3)with ∂ψc(ψ) = 8 sin2 ψ cos2 ψ The radius of curvature of AdS is L andL2 =√λα′ with λ = g2YMN . The embedding of the D5-brane is mostly29Chapter 2. D3-probe-D5 Holography with Internal Fluxdetermined by symmetry. The dynamical variables are{x(σ), y(σ), z(σ), t(σ), r(σ), ψ(σ), θ(σ), φ(σ), θ˜(σ), φ˜(σ)}We look for a solution of the formσ1 = x, σ2 = y, σ3 = t, σ4 = r, σ5 = θ, σ6 = φ, θ˜ = 0, φ˜ = 0 (2.4)and the remaining coordinates depending only on σ4 = r, (z(r), ψ(r)).13With this Ansa¨tz, the D5-brane world-volume metric isds2 = L2[r2(−dt2 + dx2 + dy2) + dr2r2(1 + r2ψ′2 + r4z′2) + cos2 ψ(dθ2 + sin2 θdφ2)](2.7)where prime denotes derivative by r and the world-volume gauge fields areF =L22piα′a′(r) dr ∧ dt+ L22piα′b dx ∧ dy + L22piα′f2d cos θ ∧ dφ (2.8)Here, f is the strength of the monopole bundle.14 b is a constant magneticfield which is proportional to a constant magnetic field in the field theorydual. a(r) is the temporal world-volume gauge field which must be non-zero in order to have a uniform charge density in the field theory dual. Thebosonic part of the R-symmetry is SU(2)×SU(2). One SU(2) is the isometry13 This ansatz is symmetric under spacetime parity which can be defined for the Wess-Zumino terms∫d6σµ1µ2...µ6∂µ1x(σ)∂µ2y(σ)∂µ3z(σ)∂µ4t(σ)r4(σ) ∂µ5Aµ6(σ) (2.5)∫d6σµ1µ2...µ6∂µ1 cos θ(σ)∂µ2φ(σ) ∂µ3 cos θ˜(σ)∂µ4 φ˜(σ)c(ψ)∂µ5 Aµ6(σ) (2.6)in the following way. The world-volume coordinates transform as {σ′1, σ′2, . . . , σ′6} ={−σ1, σ2, . . . , σ6} and the embedding functions as x′(σ′) = −x(σ), θ˜′(σ′) = pi − θ˜(σ),A′1(σ′) = −A1(σ) with all other variables obeying χ(σ′) = χ(σ). This is a symmetry of theWess-Zumino terms and the solution (2.4) is invariant. Charge conjugation flips the signof all gauge fields, A → −A and we augment it by {σ′1, . . . , σ′5, σ′6} → {σ1, . . . ,−σ5, σ6}.The Wess-Zumino terms are invariant. The background field fd cos θ∧dφ is also invariantonce we choose σ5 =pi2− θ. The fields a(r) breaks C and preserves P. b breaks C and Pand preserves CP.14A monopole bundle has quantized flux. Here the number of quanta is very large inthe strong coupling limit nD ∼√λ, so that it is to a good approximation a continuouslyvariable parameter. b and q are related to the physical magnetic field and charge densityas b = 2pi√λB, q = 4pi3√λNρ so that qb= piN2piρB≡ piNν where the dimensionless parameter ν isthe filling fraction. A Landau level would have degeneracy N . so filling fraction of a set ofN degenerate levels naturally scales like N to give order one b and q in the large N limit.30Chapter 2. D3-probe-D5 Holography with Internal Fluxof the S2 which is wrapped by the D5 brane (2.7) and is also a symmetryof the background fields (2.8). The other is the rotation in the transverseS2 ⊂ S5 with S5 coordinates θ˜, φ˜. This is a symmetry of the embeddingonly when the former S2 is maximal, that is, when ψ(r) = 0 for all r. Ifψ(r) deviates from zero, it must choose a direction in the transverse space,and the choice breaks the second SU(2). The hypermultiplet mass shows upin the D5 brane embedding asM ∼ m ≡ limr→∞ r sinψ(r) (2.9)and deviation of ψ(r) from the constant ψ = 0 so that the parameter m isnonzero is a signal of having switched on a hypermultiplet mass operator inthe dual field theory.With (2.7) and (2.8), the action (4.5) isS = N∫d3xdr[−√(f2 + 4 cos4 ψ)(b2 + r4)(1 + r2ψ′2 + r4z′2)− a′2 + fr4z′](2.10)where N = 2piT5L6gs =√λN4pi3. The factor of 2pi in the numerator comes fromhalf of the volume of the unit 2-sphere (the other factor of 2 is still in theaction). The Wess-Zumino term gives a source for z(r).Now, we must solve the equations of motion for the functions ψ(r), a(r)and z(r) which result from (2.10) and the variational principle. The variablesa(r) and z(r) are cyclic and they can be eliminated using their equations ofmotion,ddrδSδz′(r)= 0 →√(f2 + 4 cos4 ψ)(b2 + r4)r4z′√1 + r2ψ′2 + r4z′2 − a′2− fr4 = pz (2.11)ddrδSδa′(r)= 0 →√(f2 + 4 cos4 ψ)(b2 + r4)a′√1 + r2ψ′2 + r4z′2 − a′2= −q (2.12)where pz and q are constants of integration. If these equations are to holdnear r → 0, we must set pz = 0. q is proportional to the charge density inthe field theory dual. Then, we can solve for z′ and a′,z′ =f√1 + r2ψ′2√4 cos4 ψ(b2 + r4) + f2b2 + q2(2.13)a′ =−q√1 + r2ψ′2√4 cos4 ψ(b2 + r4) + f2b2 + q2(2.14)31Chapter 2. D3-probe-D5 Holography with Internal FluxWe must then use the Legendre transformationR = S −∫a′(r)∂L∂a′(r)−∫z′(r)∂L∂z′(r)to eliminate z′ and a′. We obtain the RouthianR = N∫d3xdr√4 cos4 ψ(b2 + r4) + b2f2 + q2√1 + r2ψ′2 (2.15)which must now be used to find an equation of motion for ψ(r),ψ¨1 + ψ˙2+ ψ˙[1 +8r4 cos4 ψ4(b2 + r4) cos4 ψ + f2b2 + q2]+8(b2 + r4) cos3 ψ sinψ4(b2 + r4) cos4 ψ + f2b2 + q2= 0(2.16)where the overdot is the logarithmic derivative ψ˙ = r ddrψ.First, we note that, if ψ(r) is to be finite at r → ∞, its logarithmicderivatives should vanish. Then, the only boundary condition which is com-patible with the equation of motion is ψ(r →∞) = 0.If we set b = 0, f does not appear in the Routhian (2.15) or in theequation of motion (2.16). ψ(r) which is then f -independent. In fact, theconstant solution, ψ = 0 is a stable solution of (2.16). z(r) is f and r-dependent. Equation (2.13) has the solution z(r) =∫dr f√4r4+f2. Theworldvolume metric is still that of AdS4 × S2,ds2 = L2[r2(−dt2 + dx2 + dy2) + dr2r2(1 +f24)]+ L2[dθ2 + sin2 θdφ2](2.17)where, now, the radii of the two spaces differ, the S2 still has radius Lwhereas AdS4 has radius L√1 + f24 . The field theory dual of this system wasdiscussed in reference [36]. It has a planar defect dividing three dimensionalspace into two half-spaces with N = 4 Yang-Mills theory with gauge groupSU(N + nD) on one side of the defect and N = 4 Yang-Mills theory withgauge group SU(N) on the other side. Here nD is the number of Diracmonopole quanta in f . The r-dependence of the embedding function z(r)can be viewed as an energy-scale dependent position of the defect in thefield theory.When b is not zero, scaling r → √br, removes b from most of equation(2.16), the dependence which remains is only inthe parameter f2 +( qb)2.If this parameter is large enough, the solution ψ(r) = 0 is still a stable32Chapter 2. D3-probe-D5 Holography with Internal FluxFigure 2.1: We integrate equation (2.16) with q = 0, f2 = 0.01 and f2 = 100.The solution interpolates between the correct asymptotic values, ψ(r = ∞) = 0and ψ(r = 0) = pi2 . With larger f2, it clearly has a slower approach to ψ = pi2 . TheAdS radius r is measured in units of 1/√b.solution of (2.16). The BKT phase transition found in reference [102] wasdriven by the change in behavior of the equation for ψ(r) in the small rregion and, in that paper, it was found by adjusting qb (they had f = 0)with the critical value being( qb)2∣∣∣crit.= 28. At that point, the symmetricsolution ψ = 0 becomes unstable. This is easily seen by looking at solutionsof the linearized equation which, at small r, must be ψ ∼ c1rν+ +c2rν− withν± = −12 ± 12√1− 32/ (4 + f2 + qb)2 and the instability sets in when theexponents become complex, that is, at[f2 +( qb)2]crit.= 28. The complexexponents are due to the fact that, in the r ∼ 0 regime, the fluctuationsobey a wave equation for AdS2 with a mass that violates the Breitenholder-Freedman bound. Since, in the stable regime, f2 +( qb)2> 28 both of theexponents in the fluctuations are negative, deviation from ψ(r) = 0 is notallowed, it is an isolated solution. We can find this solution and the phasetransition even when qb vanishes by varying f , stability where f2 > 28 andthe phase transition at f2crit = 28. In particular, this allows us to study thetheory in the charge neutral state where q = 0. From the point of view of thespace-time symmetry, the flux f is charge conjugation symmetric, whereasthe finite charge density state is not. In fact f itself does not violate any2+1-dimensional spacetime symmetries associated with Lorentz, C, P or Tinvariance.33Chapter 2. D3-probe-D5 Holography with Internal FluxFigure 2.2: The constants c versus m are plotted for a sequence of embeddingsof the D5-brane in the region where the constant ψ solutions are unstable. Heref2 = .01. Note that there is a special value of the condensate c where m = 0.We interpret this as a solution with dynamical symmetry breaking, i.e. as a chiralcondensate with no bare mass term.When the symmetric solution ψ = 0 is unstable, we must find anothersolution of equation (2.16) for ψ(r), where we now assume that it dependson r. ψ = 0 was an isolated solution, there are no other solutions closeby.As soon as it depends on r, if ψ(r) is to remain finite in the small r region,it must go to the other solution of (2.16) at small r, ψ(r → 0) = pi2 .When either or both of q and f are nonzero, the D5-brane must reach thePoincare´ horizon. Otherwise, the charge density q and magnetic monopoleflux f would have to have sources on the D5-brane worldvolume. q wouldbe sourced by a uniform density of fundamental strings and the magneticmonopole flux f by nD D3-branes which are suspended between the world-volume and the Poincare´ horizon. However, as in the case of fundamentalstrings, it is possible to show that the D3-brane tension is always greater thanthe D5-brane tension. Like the fundamental string, a suspended D3-branewould drag the D5-brane to the horizon. The D5-brane world-volume couldstill reflect this behavior with a spike or funnel-like configuration. Whenthere are both suspended fundamental strings and D3-branes, it is interest-ing that the embedding problem depends on the combination√f2 +( qb)2,reminiscent of bound states of F-strings and D-branes. We then expect tofind solutions of (2.16) which interpolate between ψ = 0 at r →∞ to ψ = pi2at r → 0. Indeed, for generic asymptotic behavior, such solutions are easyto find by a shooting technique. Examples are given in figure 2.1.34Chapter 2. D3-probe-D5 Holography with Internal FluxFigure 2.3: The function r sin(ψ(r)) is plotted versus r for some embeddingspa-rameterized by the asymptotic m and c, including the one which is close the solutionwith m = 0 which is associated with dynamical symmetry breaking.It is also possible to find solutions that can be interpreted as chiralsymmetry breaking, although the D5-brane still reaches the Poincare´ horizonand we expect that the D3-D5 strings are still gapless. In the region of larger, the linearized equation for ψ(r) is solved byψ(r) =mr+cr2+ . . .The two asymptotic behaviors have power laws associated with the ultravi-olet conformal dimension of the mass and the chiral condensate in the dualfield theory. These are the same as their classical dimensions since they areprotected by supersymmetry. A symmetry breaking solution would haveone of these equal to zero (and the other one interpreted as a condensate).Indeed, it is easy to find a family of solutions of (2.16) which, as we tunem, still exists and has nonzero c in the limit where m goes to zero. Thec versus m behavior of this family of solutions is shown in figure 2.2. Thebehavior if r sin(ψ(r)) which can be interpreted as the separation of the D5and D3-branes is plotted in figure 2.3 for some values of m and c.As an extension of our results here, it would be interesting to analyzethe electromagnetic properties of the solution with finite f and q = 0. Thisis a charge neutral state and it has a mass operator condensate. It is pos-sible to study Maxwell equations for fluctuations of the worldvolume gaugefield and though it is difficult to obtain an analytic solution, it is relativelystraightforward to show that they have no solution when the field strength35Chapter 2. D3-probe-D5 Holography with Internal Fluxis a constant. This implies that the charged matter is still gapless and pro-vides the singularities in response functions which make the theory singularat low energy and momentum.36Chapter 3Dynamical SymmetryBreaking with ChargedProbe PairBut you’ve ceased to believe in your theory already,what will you run away with?- Crime and Punishment by Fydor Dostoevsky3.1 IntroductionThe AdS/CFT correspondence [12], and holographic duality in general, isa powerful, conjectured technique for the analysis of strongly coupled fieldtheories. While originally pursued to address questions about low-energyQCD, it has expanded to include studies of a variety of strongly coupledfield theories in diverse dimensions.15Of much interest in recent years has been the study of defect theories andthe interaction of defects. Such defects can be constructed holographicallyby the intersection of different stacks of D-branes, one of the earliest knownexamples being the supersymmetric (2 + 1)-dimensional intersection of theD3/D5 system [89], a defect in the ambient (3+1)-dimensional N = 4 superYang-Mills native to the D3 worldvolume. A common technique for studyingthese systems is to consider the quenched approximation of the field theory,where one stack, say of Dp-branes, has parametrically more branes than theother, say of Dq-branes. The gravity description of this scenario can thenbe reliably computed at strong coupling by using a probe Dq-brane actionin the near-horizon region of a classical p-brane supergravity solution [26].15For older review articles see [8, 14], while [19, 21] are more recent with an emphasison applications for condensed matter.373.1. IntroductionThe full dual field theory lives at the asymptotic boundary of this spacetimeand the defect theory lives where the probe brane intersects the boundary.Multiple defects may be studied by allowing several stacks of Dq-branesto intersect the boundary. As discussed first in [27, 28], a coherent state ofspatially separated defects can be achieved by a continuous probe brane con-figuration with a multiply connected intersection with the boundary. Sincethe boundary components must have opposite orientation in this scenario,it can be understood as brane/anti-brane recombination. In the scenarioof [27, 28], the defect degrees of freedom were d = 3 + 1 chiral fermions,with those on the brane component of opposite chirality from those on theanti-brane. The coherent state where the worldvolumes join in the bulk thusdescribes chiral symmetry breaking. In [30, 31], this scenario was general-ized to allow for intersections of other dimension and brane species as well asfor the joining process to occur dynamically.16 Further generalizations haveincluded adding external magnetic and electric fields as well as chemicalpotential [29, 32–34].In this chapter, we consider scenarios of bulk brane/anti-brane recombi-nation in AdS5 × S5,ds2 ∼ r2 (−dt2 + dx2 + dy2 + dz2)+ dr2r2+ dΩ25 . (3.1)As an additional ingredient to previous studies, we consider probes whichare electrically charged under the background F5 Ramond-Ramond field.The probe branes form two stacks, each spanning some cycle in S5, the non-compact directions (t, x, y) and some curve z(r). The stacks have oppositeorientation and are separated in the z direction along the boundary.An uncharged probe brane – such as in the studies cited above – expe-riences no force in the non-compact directions from the F5. For such a casethere are then two qualitative classes of solutions, depicted in Fig. 3.1. Thefirst solution is the so-called “black hole embedding” which reaches all theway down to the spacetime horizon. These embeddings are “straight” in thesense that dzdr = 0. The second solution is a joined embedding which has twodisconnected boundaries of opposite orientation although the entire world-volume is a simply connected and oriented manifold. Only these solutionshave dzdr 6= 0. Note that since there is no Ramond-Ramond force, the braneorientation does not play a role.On the other hand, if the probe branes are charged under the spacetimeRamond-Ramond field, the situation is somewhat different. This can occur16In [27, 28], topological considerations force the branes to join while in [30, 31] and laterworks there are multiple consistent solutions and only the minimum energy one dominates.383.1. IntroductionFigure 3.1: Straight embeddings and a joined embedding where there is no forcefrom the background Ramond-Ramond flux. The arrows represent worldvolumeorientation. There would be no change in the embedding if the arrows were reversed.either because the probe itself is a D3-brane, or the charge could be inducedby worldvolume fluxes on the probe. The D3/D5 system where the D5brane carries q unit of D3-brane charge was first introduced in [87]. In[36], black-hole embeddings of D5 and D7 probe branes with induced D3-brane charge were studied in AdS5 × S5. Additional D7 brane embeddingscarrying D3 charge were introduced in [37] and studied further in [38, 39].These probes are affected by the background F5 and even the black holeembeddings have dzdr 6= 0. In Fig. 3.2, we see such a black-hole embedding.The brane orientation plays a major role in this situation; an oppositelyoriented probe would bend in the opposite z-direction.These electrically charged probe branes have a richer space of joinedsolutions than their uncharged cousins. Due to the force in the z-direction,the qualitative features of the solution depend strongly on the orientation,specifically the left-right ordering of the boundary components. The choiceof orientation gives rise to the classes of solutions seen in Fig. 3.3. The topleft figure pictures a brane/anti-brane pair which tend toward each otherdespite not actually connecting, while the top right figure pictures a joinedpair. These two solutions have the same boundary conditions and so it isa dynamical question which has the lower energy and is therefore stable.The figures in the bottom row also depict solutions with the same boundaryconditions, but with the worldvolume orientations all opposite of the figuresabove. Note the surprising feature in the bottom right figure, where thejoined embedding becomes wider in the bulk than at the boundary. We393.1. IntroductionFigure 3.2: A D3-charged probe brane in (finite-temperature) AdS5 × S5.The probe bends in the z-direction as it descends from the boundary (thesolid line at the top) to enter the horizon represented by the dotted line atthe bottom. The arrow represents the orientation of the D3 charge. Anoppositely oriented brane would bend in the opposite z-direction.will call these joined solutions “chubby” and conversely the more typicalsolutions in the top right (which are widest at the boundary) we will call“skinny.”There are multiple perspectives on what these brane systems are holo-graphically dual to. Firstly, the (2 + 1)-dimensional intersection of a probebrane with the boundary is conventionally associated with a defect in N = 4super-Yang-Mills gauge theory. The field content of the defect is given bythe lowest level open string modes which are localized at the D-brane inter-section. For a D5-brane probe, the defect theory is supersymmetric sincethe intersection is #ND = 4; this is the spectrum studied in [89]. For theD7-brane probe, the intersection is #ND = 6 and the spectrum is simplymassless fermions [40], in fact T-dual to the D4/D8 intersections of theSakai-Sugimoto model [27, 28]. As a caveat, it should be mentioned that itis not clear if this picture of the spectrum still holds when internal fluxesexist on the probe, but is often nonetheless used to guide intuition.A defect dual to a stack of N branes or anti-branes is associated witha U(N) global flavor symmetry inherited from the gauge field living on thebrane worldvolume. Thus the recombination of an equal number of branesand anti-branes describes a breaking of symmetry U(N) × U(N) → U(N).Since the defects are separated in space, the duals of these scenarios canbe considered interacting (2 + 1)-dimensional defect bi-layer systems or asdiscussed in [31], (2 + 1)-dimensional effective field theories with non-localinteractions.403.1. IntroductionFigure 3.3: The top row pictures possible solutions of a brane/anti-brane pair inthe presence of a Ramond-Ramond force. Note that the branes bend toward eachother as they extend into the bulk even if they don’t join. If the orientations arereversed, we have instead the bottom set of solutions. These always bend awayfrom each other when initially leaving the boundary even if they eventually join inthe bulk.413.2. D3-charged probes in AdS5 × S5The dual interpretation above holds for probes with or without D3-charge. However, for D3-charged probes there are some other interestingproperties of these solutions. A D3-charged probe brane – even a higherdimensional brane with an induced D3 charge – contributes to the overallRamond-Ramond flux of the system. This flux is in turn related by theAdS/CFT dictionary to the rank of the dual gauge group. Therefore adefect of D3-charge k forms a domain wall in the dual gauge theory withSU(N) gauge symmetry on one side and SU(N + k) on the other [36]. Acartoon representation of this situation is depicted in Fig. 3.4. Once theprobe enters the horizon, it is effectively parallel to the original stack ofD3-branes sourcing the geometry, adding to the overall D3-brane charge asmeasured by a Gaussian surface outside the horizon. This is interpretedas a larger gauge symmetry existing in the region to the left. It followsthat if there are multiple D3-charged defects, that we have a spatially non-trivial pattern of symmetry breaking in the dual theory, with a gauge groupbetween the defects which is different from that outside. Thus the joinedsolutions should be considered dual to finite-width domain walls.In this chapter, we will study the thermodynamics of these domain walls,mostly from the bulk perspective. In Section 2, we introduce a class of D3-charged probe branes and derive a one-dimensional effective particle me-chanics action that describes the entire class. The solutions of the equationof motion of this effective action are studied in Section 3 and a renormal-ized free energy computed in Section 4. Finally, in Section 5, we examinethe phase diagram of this system in the space of external magnetic field andasymptotic separation with some comments on the phenomenon of magneticcatalysis.3.2 D3-charged probes in AdS5 × S5Consider the background IIB supergravity solution thermal AdS5×S5, thenear-horizon geometry of N3 D3-branes at finite temperature. The line-element is given byL−2ds2 = r2(−h(r)dt2 + dx2 + dy2 + dz2)+ dr2h(r)r2+ dΩ25 . (3.2)The S5 line element is represented as a bundle over S2 × S2,dΩ25 = dψ2 + sin2 ψdΩ22 + cos2 ψdΩ˜22 , (3.3)423.2. D3-charged probes in AdS5 × S5Figure 3.4: A cartoon representation of a probe brane (dashed line) carrying D3-charge k bending to become parallel with the stack of N3 D3-branes sourcing theAdS geometry, represented by the solid lines at the bottom. The arrows representbrane worldvolume orientation. The dual gauge group is SU(N3) towards the rightwhile it is enhanced to SU(N3 + k) to the left.where ψ ∈ (0, pi2 ) anddΩ22 = dθ2 + sin2 θdφ2 , (3.4)is the line-element for a unit S2. The blackening function ish(r) = 1− r4hr4. (3.5)At zero-temperature, rh = 0. However, any non-zero value of rh can berescaled by a coordinate transformation. Therefore, for finite temperature,we can choose without loss of generality rh = 1. The scale of the geometryis related to the microscopic string theory parameters viaL4 = 4pigsN3(α′)2 . (3.6)There is also a self-dual five-form Ramond-Ramond fluxF5 =4L4gs(r3dt ∧ dx ∧ dy ∧ dz ∧ dr + ω5). (3.7)Here ω5 is the volume form on the unit five-sphereω5 = sin2 ψ cos2 ψdψ ∧ ω2 ∧ ω˜2 , (3.8)433.2. D3-charged probes in AdS5 × S5where ω2 = sin θdθ ∧ dφ is the S2 volume form. We encode this flux withthe four-form potentialgsL4C4 = r4h(r)dt ∧ dx ∧ dy ∧ dz + 12c (ψ)ω2 ∧ ω˜2 . (3.9)The function c (ψ) isc(ψ) = ψ − 14sin (4ψ) + c0 (3.10)where c0 is an arbitrary constant, a residual ambiguity due to the gaugesymmetry of the Ramond-Ramond field. A similar constant could be addedto the coefficient of dt ∧ dx ∧ dy ∧ dz, but we have chosen to partially fixthe gauge by requiring that the first term in C4 vanish at the horizon. Thisensures that the term is well-defined on the Euclidean section of (3.2) whichsimplifies the treatment of the Wess-Zumino terms.We will now consider the following set of branest x y z r ψ Ω2 Ω˜2D3′ − − − − · · · ·D3 − − − ∼ ∼ · · ·D5 − − − ∼ ∼ · − ·D7 − − − ∼ ∼ · − −(3.11)The D3′ row refers to the large stack of D3 branes which source the AdSgeometry while the other rows record the configurations of the probes. Adash indicates the brane is extended in that direction, with support overthe entire range of the coordinate. A dot indicates the respective braneis completely localized in that coordinate. Finally the ∼ symbols indicatethat the brane traces a curve in those directions. For example, the D5-braneextends along the non-compact (t, x, y) directions, wraps one of the two S2factors in the S5, is localized in ψ and on the other S2, and finally, lies alonga curve in the (z, r) space.These probes all intersect the boundary on some 2 + 1 dimensional sub-space at a fixed value of z (although for the D3 probes, this will turn outto be z = ±∞). In order to induce D3-brane charge,17 the probe D5 andD7-branes will carry internal flux topologically supported on one or bothS2 factors, respectively. We will also allow for magnetic field in the threedimensional defect on the boundary, i.e. a non-zero Fxy component. In [37],17Such flux is actually required to stabilize the D7 probe at a non-trivial value of ψ atthe AdS boundary [37].443.2. D3-charged probes in AdS5 × S5D7 branes with a more general ansatz were studied. However, our focus willbe a class of solutions with different boundary conditions.The D5 and D7 probes outlined above have 3+1 non-compact directionsand wrap some compact cycles. If one imagines integrating over these cycles,one would obtain an effective 3 + 1 dimensional object which carries D3charge in AdS5. In other words, the higher-dimensional D3-charged branesact as effective D3-branes. These effective branes are much like excitedstates of a proper D3, they carry D3 charge but the effective tension isgreater than the charge. This will become clearer in the next few sections.First, we will calculate an effective action for a D3 probe with the ansatz(3.11). We will then see that D5 and D7 probes will yield an effective actionof the same form.3.2.1 D3-brane probeFirst, let us introduce a D3-brane probe as a model system. The actioncomprises the familiar DBI and Wess-Zumino termsS3 = −T3∫d3+1ξe−φ√−det (g + 2piα′F )− T3∫C4 . (3.12)The three-brane tension isT3 =1(2pi)31α′2. (3.13)We choose a static gauge where ξa = {t, x, y, r} are brane coordinates andthe embedding is given by the function z(r). The induced metric is thusds23L2= r2(−hdt2 + dx2 + dy2)+ (1 + r4hz˙2) dr2r2h, (3.14)where a dot indicates differentiation by r. We also allow a magnetic fieldnormalized as2piα′L2F = Bdx ∧ dy . (3.15)This information is sufficient to compute the Born-Infeld termSDBI = N3∫dr√(r4 +B2) (1 + r4hz˙2) , (3.16)where the overall constant isN3 = T3L4V2+1gs, (3.17)453.2. D3-charged probes in AdS5 × S5with V2+1 the infinite volume factor of the (t, x, y) directions.To compute the Wess-Zumino term we also need to specify an orienta-tion, which we encode via an orientation parameter ζ = ±1. Evaluating,∫C4 = V2+1ζ∫r4hz˙dr . (3.18)Note that while orientation is an invariant geometric feature intrinsic tothe entire D-brane worldvolume, the parameter ζ is partly an artifact ofthe coordinates we use. Therefore ζ may take different values on separatebranches of the same continuous brane. For example, in a brane/anti-branerecombination, the left branch has ζ = 1 and the right branch ζ = −1, yetthe worldvolume is continuous.Putting together the terms above – and dropping an overall constantfactor – yields an effective particle mechanics LagrangianL3 =√(r4 +B2) (1 + r4hz˙2) + ζr4hz˙ . (3.19)We will find similar effective Lagrangians for the D5 and D7 probes, differingonly in the coefficient of the second term. Here that coefficient is of unitmagnitude since physically it is the D3-brane charge per tension.3.2.2 D5 probesThe probe action for D5-branes isS5 = −T5∫d5+1ξe−φ√−det (g + 2piα′F )− 2piα′T5∫C4 ∧ F , (3.20)where the tension isT5 =1(2pi)51α′3. (3.21)We choose a static gauge with coordinates ξa = {t, x, y, r, θ, φ} and embed-ding function z(r). The induced metric isds25L2= r2(−hdt2 + dx2 + dy2)+ (1 + r4hz˙2) dr2r2h+ sin2 ψdΩ22 . (3.22)The ansatz for worldvolume flux is2piα′L2F = Bdx ∧ dy + f2ω2 . (3.23)463.2. D3-charged probes in AdS5 × S5The magnetic field is a continuous quantity but the flux on the compactsphere is, of course, quantizedf =2piα′L2n , n ∈ Z . (3.24)Substituting all this into the action yieldsS5 = −N5∫dr[√(r4 +B2)(f2 + 4 sin4 ψ)(1 + r4hz˙2) + ζfr4hz˙],(3.25)with the normalizationN5 = 2piT5L6V2+1gs, (3.26)and once again we have introduced an orientation parameter ζ = ±1. Ouransatz is for constant ψ but we see that ψ has a potential. The ψ equationof motion isddψ√f2 + 4 sin4 ψ = 0 , (3.27)yielding18ψ =pi2. (3.28)We insert this back into the D5 action. Up to an overall constant we againobtain an effective particle Lagrangian for z(r),L5 =√(r4 +B2) (1 + r4hz˙2) +ζf√f2 + 4r4hz˙ . (3.29)The only difference from the D3 is in the coefficient of the second term, theeffective D3-brane charge per unit tension. The magnitude of this ratio isless than unity here, in keeping with the picture that this D5 probe is aD3-brane in an excited state.3.2.3 D7 probesThe D7-brane action isS7 = −T7∫d7+1ξe−φ√−det (g + 2piα′F )− (2piα′)22T7∫C4∧F∧F , (3.30)18Another solution is ψ = 0 but it is physically trivial since the brane volume is thenexactly zero.473.2. D3-charged probes in AdS5 × S5with tensionT7 =1(2pi)71α′4. (3.31)In a static gauge with coordinates ξa ={t, x, y, r, θ, φ, θ˜, φ˜}, we describe theembedding with the function z(r). The induced metric isds27L2= r2(−hdt2 + dx2 + dy2)+(1 + r4hz˙2) dr2r2h+sin2 ψdΩ22 +cos2 ψdΩ˜22 . (3.32)For the worldvolume flux we use the ansatz2piα′L2F = Bdx ∧ dy + f12ω2 +f22ω˜2 . (3.33)The fluxes on the S2 factors are quantizedfi =2piα′L2ni , ni ∈ Z . (3.34)The DBI portion of the action isSDBI = −N7∫dr√(r4 +B2)(f21 + 4 sin4 ψ)(f22 + 4 cos4 ψ) (1 + r4hz˙2) (3.35)withN7 = 4pi2T7L8V2,1gs. (3.36)The Wess-Zumino term is given byS = −N7ζf1f2∫drr4hz˙ , (3.37)with ζ the orientation parameter. We minimize the ψ potentialddψ√(f21 + 4 sin4 ψ) (f22 + 4 cos4 ψ)= 0 , (3.38)yielding the implicit equation19f22 sin2 ψ − f21 cos2 ψ + 4 cos2 ψ sin2 ψ(cos2 ψ − sin2 ψ) = 0 . (3.39)19While this can be solved for general fi, it can be seen that fluctuations δψ around thesolution can violate the BF bound [42, 43]. In particular, for absolutely no internal fluxesfi = 0, the D7 will be unstable [44]. See [37] for more discussion of stabilizing this D7brane embedding.483.3. Solutions to effective LagrangianSubstituting this back into the action yields, up to an overall constant, aneffective particle Lagrangian for the D7-braneL7 =√(r4 +B2) (1 + r4hz˙2) +ζf1f2√(f21 + 4 sin4 ψ0)(f22 + 4 cos4 ψ0)r4hz˙ , (3.40)where ψ0 is a constant that solves (3.39). This again takes the form of theeffective D3 Lagrangian with a charge per tension smaller than unity.3.3 Solutions to effective LagrangianWe found that all three of the D3-charged probes under consideration aredescribed by an effective particle Lagrangian of the formSeff =∫dr√r4 +B2√1 + r4hz˙2 + α∫drr4hz˙ . (3.41)The parameter α is the effective D3-brane charge per tension and is givenbyα =ζ D3−braneζf√f2+4D5−braneζf1f2√(f21 +4 sin4 ψ0)(f22 +4 cos4 ψ0)D7−brane(3.42)with ψ0 solving (3.39) in the case of the D7. Note that |α| < 1 for both theD5 and D7 probes.The equation of motion derived from (3.41) can be immediately inte-grated since the variable z(r) is cyclicP ≡√r4 +B21 + r4hz˙2r4hz˙ + αr4h = constant . (3.43)Define the intermediate functiong(r) =Pr4h− α , (3.44)then solve for z˙ to obtainz˙ =g(r)√r4 +B2 − r4hg(r)2 . (3.45)The full profile z(r) is obtained by integration. This cannot be done analyt-ically in general, but for any choice of B, P and α the integration of (3.45)is easily evaluated numerically.493.3. Solutions to effective LagrangianThese solutions are completely specified by the integration constant P .For any brane profile that enters the black hole horizon, substituting r = 1into (3.43) shows that P must vanish since h (r = rh = 1) = 0,P = 0 for solutions with support at r = 1 (3.46)In keeping with the literature we call these solutions black hole embeddings.Since P = 0, we have g(r) = −α. Thus, we see from (3.45) that for theblack hole embeddings z(r) is single-valued and monotonic.The other possibility is that the profile has a minimum value of r. With-out loss of generality, we can choose this minimum to be located at z = 0.The signal of a minimum would be z˙ diverging at some r = r0. This yieldsthe expression for the integration constantP = r40√h0(√1 +B2r40sign (z˙0) + α√h0), (3.47)where h0 = h(r0) and z˙0 = z˙(r → r0). The presence of an absolute minimumrequires that the brane bends back up to the boundary. This other leg of thebrane will have opposite orientation parameter ζ so this solution is a joinedbrane/anti-brane pair. We thus call the P 6= 0 solutions joined embeddings.The magnitude of the first term in the parentheses of (3.47) is greaterthan unity while that of the second term is less than unity. Thereforesign (z˙ (r → r0)) = sign (P ) . (3.48)However, z˙ → −∞ when approaching from the left of the minimum whilez˙ → +∞ when approaching from the right. Furthermore, the orientation pa-rameter ζ changes sign from one branch to the other. Therefore, P changessign as well,20 with P < 0 for z < 0 and P > 0 for z > 0 (see Fig. 3.5).The joined configuration is clearly symmetric under parity z → −z, so wecan without loss of generality focus our attention to a single branch. Wewill therefore restrict our attention to P ≥ 0, which includes the black holeembedding and the “right branch” with z˙0 > 0 of the joined solutions.At the boundarysign (z˙ (r →∞)) = −sign (α) , (3.49)20The reader may find this disconcerting, since P is playing the role of a conservedquantity. The resolution lies in the multi-valuedness of the function z(r). P need onlybe constant on a given single-valued branch. The minimum is precisely where the single-valued parameterization z(r) breaks down and so consequently does the definition of P .That the magnitude of P is constant follows from the continuity of the embedding.503.3. Solutions to effective LagrangianFigure 3.5: The sign of the integration constant P is the same as that of z˙ as r0is approached and flips accordingly as the minimum at z = 0 is crossed.indicating that the direction in which the brane bends initially on its descentfrom infinity is given entirely by the sign of the D3-brane charge. Comparing(3.48) and (3.49) we see there are thus two qualitative classes of joinedsolutions, given by the relative sign of P and α. For sign(P ) = −sign(α),the sign of z˙ remains the same throughout the branch, i.e. each branchof the brane is separately monotonic. On the other hand, for sign(P ) =sign(α) even a given branch is not monotonic. We call these two possibilities“skinny” and “chubby,” respectively. See Fig. 3.6.Figure 3.6: “Skinny” and “chubby” joined embeddings.Physically, we know that the brane and anti-brane have an attractiondue to exchange of gravitons and Ramond-Ramond quanta. Further, thebackground F5 also deflects branes and anti-branes in opposite directions.In the skinny solutions, the background F5 pushes the two stacks togetherwhile in the chubby solutions the Ramond-Ramond field forces them apart.513.3. Solutions to effective Lagrangian3.3.1 AsymptoticsThe asymptotic separation in z of a joined brane/anti-brane pair is notindependent of r0. Define L asL(r0) = 2∫ ∞r0z˙(r) , (3.50)where the factor of two arises since the integral is only over one branch ofthe brane system. For a joined solution, i.e. any solution with r0 > 1, L isthe asymptotic separation in the z direction of the two ends of the solution.For r0 = 1 however, the brane and anti-brane are disconnected black holeembeddings. In this case the asymptotic separation is truly a free parameterand L(r0 = 1) simply records (twice) the range in z that each branch of theembedding spans.The probe branes for generic α have the large r behaviorz˙ (r  1) = − α√1− α21r2+O(1r6), (|α| < 1) . (3.51)The case |α| = 1 is non-generic. Indeed, expanding (3.45) in yieldsz˙ (r  1) = − α√1 +B2 + 2αP+O(1r4), (|α| = 1) . (3.52)It follows that L converges for |α| < 1 and diverges for |α| = 1, whichmeans that |α| = 1 branes (i.e. D3-brane probes) do not intersect the AdSboundary at finite z while those with generic α do. The impossibility of theD3 probe to intersect the AdS boundary at finite z may be a symptom ofthe open string tachyon present at weak coupling at the (2 + 1)-dimensionalintersection of D3-branes.21 Whatever the explanation, we will now restrictour attention to D5-branes and D7-branes so that we can study probeswhich intersect the boundary at a finite location.The right-hand side of (3.50) is a complicated function of r0 since z˙depends on it through the integration constant P . We do not have ananalytic expression but can plot it numerically. As an example, see Fig. 3.7,which plots L(r0) for a D7-brane probe with B = 0 and f1 = f2 =1√2.Note that L(r0) is not monotonic and has a maximum. Therefore, when thebrane/anti-brane pair are sufficiently separated at the boundary (with anL & 1.3) there are no joined solutions, only black hole-embeddings. Further,21Since such a system has #ND = 2. See [40].523.4. Free energydue to the maximum there is a range of L where there are two r0, that istwo solutions with the same boundary condition.Another feature worth noting is the abrupt end of the curve at r0 = 1.The r0 = 1 solution is a black-hole embedding and L(1) is (twice) the ∆zspanned by a single branch of that embedding. The curve L(r0) does notcontinue past this point.Figure 3.7: Asymptotic brane separation L of a joined solution (α = − 13 and nomagnetic field) as a function of minimum radius r0.There is a class of unphysical solutions lurking within the family thatwe have been discussing. Some of the non-monotonic branches, i.e. thosewith sign (P ) = sign (α), will turn out to have negative L. Qualitativelythese solutions appear as in Fig. 3.8. Note that they have the same bound-ary conditions as a “skinny” solution. These solutions are clearly unstableto brane reconnection at the intersection point and will not be consideredfurther.3.4 Free energyNow that we have classified the solutions, we investigate the phases of a pairof brane/anti-brane probes. The dynamical problem is to find the solution ina given ensemble, with given boundary conditions, which has the lowest freeenergy. This solution will dominate and be thermodynamically stable. Inthe present case, the boundary conditions are given by the asymptotic branepositions and orientations and the values of the fluxes, including magnetic533.4. Free energyFigure 3.8: An unphysical solution with negative L.field. Without loss of generality, we can assume22 that the center of thepair is at z = 0, i.e. if they join, they join at z = 0. Then the boundaryconditions are given by L, B, and α.The free energy is conventionally given as the negative of the on-shellaction. This is, up to a positive constant, simply the effective action (3.41)F (r0) =∫ ∞r0dr{√(r4 +B2) (1 + r4hz˙2) + αr4hz˙}, (3.53)This is the free energy of a single leg of the brane/anti-brane system. Inthe case of r0 = 1, (3.53) is the energy of one entire worldvolume, fromhorizon to boundary. For r0 > 1, it computes the free energy of one half ofthe joined brane/anti-brane system. In all cases since the other branch isobtained by symmetry, the true free energy is just twice (3.53). Substitutingin the general solution (3.45) we getF (r0) =∫ ∞r0dr√r4 +B21− r4r4+B2hg2(1 +r4r4 +B2αhg),=∫ 1r00duu41 +B2u4 + αh(1u)g(1u)√1 +B2u4 − h ( 1u) g ( 1u)2 , (3.54)where we changed integration variables to u = r−1 in the second line.22Due to translation invariance in the z direction.543.4. Free energyFigure 3.9: The renormalized free energy as a function of r0 for B = 0 andα = − 13 .Note that (3.54) is generically infinite. Indeed, placing a cut-off at thelower end of the u integral yieldsF (r0) =∫duu41 +B2u4 + αh(1u)g(1u)√1 +B2u4 − h ( 1u) g ( 1u)2 ∼√1− α233+ finite . (3.55)Since this divergence is independent of r0, the difference in free energy be-tween any two embeddings will be finite and numerically computable. Wewill thus compute a renormalized free energy∆F (r0) ≡ F (r0)− F0 , (3.56)with F0 the divergent free energy of the black hole embedding with r0 = 1,F0 =∫ 10duu4√1 +B2u4 − α2h(1u). (3.57)When ∆F < 0, the joined solution has less energy than the black holeembeddings and so it dominates, indicating flavor symmetry breaking in thebi-layer description.In Fig. 3.9 we plot the renormalized free energy as a function of r0 for thecase B = 0 and α = −13 . This is the same set of solutions whose asymptoticseparation versus r0 is plotted in Fig. 3.7. The only joined solutions withnegative free energy are those with r0 & 1.19 which corresponds to L . 1.2.For any larger L, the black hole embedding is less energetic or the joinedembedding does not exist.553.5. Phase diagram and discussionFigure 3.10: Phase diagrams for the defect system. Above any fixed α curve,the dominant solution is given by the two disconnected brane worldvolumes, i.e.the symmetric phase. The joined solutions, the broken symmetry phase, dominatesbelow the curve.3.5 Phase diagram and discussionIn Fig. 3.10, we plot the phase diagram of the brane/anti-brane system inthe L-B plane. Each curve is at fixed α, above the curve being the flavorsymmetric phase where the stacks do not join while below the curve thesymmetry is broken to the diagonal subgroup by brane recombination. Wecan see that for α negative, the stacks always join at small enough L. Thisis quite intuitive since the background F5 assists the native attraction ofthe brane and anti-brane so there is no effect to prevent their joining. Onthe other hand, we see that for large enough positive α, the stacks do notjoin at small L unless there is also a strong enough external magnetic field.Intuitively, the force from the background F5 is strong enough to overcomethe brane/anti-brane attraction even at arbitrarily small separation.In these types of studies, there is a general expectation of magnetic catal-ysis, that an external magnetic field favors the breaking of flavor symmetry.This effect has been seen both in perturbative and large-N calculations inquantum field theory [48]. It is also known to be a common feature inholographic scenarios of various dimensional brane intersections [29, 53, 76].However, in [51] the Sakai-Sugimoto model was studied at finite chemicalpotential and magnetic field and an inverse magnetic catalysis was found in acertain region of the phase diagram, i.e. at zero temperature and fixed finite563.5. Phase diagram and discussionchemical potential, an increase in magnetic field can prompt a transition toa symmetric state.We see in Fig. 3.10 both catalysis and inverse catalysis, depending onthe value of α and the region of the curve in question. One can see thatall positive α embeddings exhibit catalysis, i.e. all of the chubby solutions.In these cases, it appears that external magnetic field always enhances theattraction of the brane/anti-brane pair. However, for 0 & α & −.2 thecurves are similar to positive α so the sign of the induced D3 charge isnot sufficient to determine the behavior with respect to magnetic field. Forα ≈ −.2, we see a maximum, indicating a region of inverse catalysis for smallB. This region expands as α is decreased until there is inverse catalysis forall B.It is not clear from the point of view of the field theory what dictateswhether the system exhibits catalysis or inverse catalysis. We will refrainfrom speculating on the exact mechanism here and leave this question tofuture work.57Chapter 4D7-anti-D7 DoubleMonolayer : HolographicDynamical SymmetryBreakingNow they have beaten me, he thought.I am too old to club sharks to death.But I will try it as long as I have the oarsand the short club and the tiller- The Old Man and the Sea by Ernest HemingwayNote that the title of the original paper was Bilayer instead of Dou-ble Monolayers. Double monolyer should be distinguished from bilayer bywhether allows hop or not between each monolayers.In weakly coupled quantum field theory, spontaneous symmetry break-ing is a familiar paradigm. It is based on formation of a condensate, usuallyan order parameter obtaining a nonzero expectation value and the resultingfeatures of the spectrum such as goldstone bosons and a Higgs field. Stringtheory holography has given an alternative picture of dynamical symmetrybreaking in terms of geometry. Particularly with probe branes, the sym-metry breaking corresponds to the branes favoring a less symmetric world-volume geometry over a more symmetric one. This is seen in the Sakai-Sugimoto model of holographic quantum chromodynamics [27]. There, chi-ral symmetry breaking corresponds to the fact that a D8-D8 brane pairprefer to fuse into a cigar-like geometry, rather than remaining in a moresymmetric independent configuration. In this chapter, we shall study amodel which is close in spirit to the Sakai-Sugimoto model, the D7-D7 sys-tem which has a 2+1-dimensional overlap with a stack of D3-branes. It canbe considered a toy model of chiral symmetry breaking in strongly coupled58Chapter 4. D7-anti-D7 Double Monolayer : Holographic Dynamical Symmetry BreakingFigure 4.1: Two parallel 2-dimensional spaces, depicted by the vertical darklines, are inhabited by fundamental representation fermions which interactvia fields of N = 4 supersymmetric Yang-Mills theory in the bulk. TheYang-Mills theory in the region between the layers has a different rank gaugegroup than that in the regions external to the double monolayer.2+1-dimensional quantum field theories containing fermions and it is explic-itly solvable. The symmetry breaking pattern is U(N)×U(N)→ U(N) and,at least in principle, it is possible to gauge various subgroups of the globalsymmetry group and to study the Higgs mechanism at strong coupling. Inthe following we shall concentrate on the case U(1) × U(1) → U(1) whichdisplays the essential features of the mechanism.Before analyzing the D7-D7 system, let us discuss its quantum fieldtheory dual, the double monolayer system depicted in figure 4.1. Masslessrelativistic 2+1-dimensional fermions are confined to each of two parallel butspatially separated layers. They are two-component spinor representationsof the SO(2,1) Lorentz group with a U(1) global symmetry for the fermionsinhabiting each layer. The overall global symmetry is thus U(1) × U(1).The 3+1-dimensional bulk contains N = 4 supersymmetric Yang-Mills the-ory. The fermions transform in the fundamental representation of the gaugegroups of the Yang-Mills theories. As shown in figure 1, the rank of theYang-Mills gauge groups differ in the interior and exterior of the doublemonolayer by an integer k which arises from the worldvolume flux in theD7-D7 system. The D-brane system which we shall discuss studies this the-ory in the strong coupling planar limit where, first, the Yang-Mills couplinggYM is taken to zero and N to infinity while holding λ ≡ g2YMN fixed and,59Chapter 4. D7-anti-D7 Double Monolayer : Holographic Dynamical Symmetry BreakingD7 D7boundary (r=infinity)horizon (r=0)zrb)Figure 4.2: The z-position of the D7-branes depends on AdS-radius and with theappropriate orientation the branes would always intersect.subsequently, a strong coupling limit of large λ is taken. The field theorymechanism for the symmetry breaking which we shall analyze is an excitoncondensate which binds a fermion on one layer to an anti-fermion on theother layer and breaks the U(1)× U(1) symmetry to a diagonal U(1).There has been significant recent interest in graphene double monolayersystems where formation of an exciton driven dynamical symmetry breakingof the kind that we are discussing has been conjectured [12]. The geome-try is similar, with the layers in figure 4.1 replaced by graphene sheets andthe space in between with a dielectric insulator. In spite of some differences:graphene is a relativistic electron gas with a strong non-relativistic Coulombinteraction, whereas what we describe is an entirely relativistic non-Abeliangauge theory, there are also similarities and perhaps lessons to be learned.For example, we find that the exciton condensate forms in the strong cou-pling limit even in the absence of fermion density whereas the weak couplingcomputations that analyze graphene need nonzero electron and hole densi-ties in the sheets to create an instability. We also find “coulomb drag”,where the existence of an electric current in one layer induces a current inthe other[47]. In the holographic model, the drag would vanish in the ab-sence of a condensate, whereas it is large when a condensate is present. Thecorrelator between the electric current in the two sheets (from (4.10) below)60Chapter 4. D7-anti-D7 Double Monolayer : Holographic Dynamical Symmetry BreakingFigure 4.3: Joined< ja(k)j˜b(`) >=4λ(1 + f2)|k|(2pi)2 sinh 2|k|ρm(δab − kakbk2)δ(k + `) (4.1)where |k| =√~k2 − ω2/v2F , there is a factor of 4 from the degeneracy ofgraphene, vF is the electron fermi velocity and λ and f2 are parameters andρm, given in (A.5), is proportional to the interlayer spacing. Aside fromthe superfluid pole at k2 = 0, this correlator has an infinite series of polesat k2 = (npi/ρm)2, n = 1, 2, ... due to vector mesons. Parameters partiallycancel in the ratio of the current-current correlator in (4.1) to the singlelayer correlator, < jj˜ > / < jj >= csch2|k|ρm.Symmetry breaking in the D7-D7 system has already been studied inreference [54]. The mechanism is a joining of the D7 and D7 worldvol-umes as depicted in figure 4.3. The D7 and D7 are probe branes [87] inthe AdS5 × S5 geometry which is the holographic dual of 3+1-dimensionalN = 4 supersymmetric Yang-Mills theory. A single probe D7-brane is stablewhen it has magnetic flux added to its worldvolume [37]. Its most symmet-ric configuration is dual to a defect conformal field theory [37][39] wherethe flux (f in the following) is an important parameter which determines,for example, the conformal dimension of the fermion mass operator. TheD7-D7 pair would tend to annihilate and are prevented from doing so by61Chapter 4. D7-anti-D7 Double Monolayer : Holographic Dynamical Symmetry Breakingboundary conditions that contain a pressure (the parameter P in the fol-lowing) which holds them apart. The problem to be solved is that of findingthe configuration of the D7 and D¯7 in the AdS5 × S5 background, subjectto the appropriate boundary conditions. We shall impose the parity andtime-reversal invariant boundary conditions that were discussed in reference[39]. We differ from reference [54] in that we use the zero temperature limit,a simplification that allows us to obtain our main result, explicit current-current correlation functions for the theory described by the joined solution(4.11)-(4.13). The AdS5 × S5 metric isds2 = R2[r2(−dt2 + dx2 + dy2 + dz2) + dr2r2+dψ2 + sin2 ψdΩ22 + cos2 ψdΩ˜22] (4.2)where dΩ22 and dΩ˜22 are metrics of unit 2-spheres and ψ ∈ [0, pi2 ]. The radiusof curvature is R2 =√4pigsNα′, where gs is the closed string couplingconstant and N the number of units of Ramond-Ramond 4-form flux of theIIB string background. The holographic dictionary sets g2YM = 4pigs, and Nbecomes the rank of the Yang-Mills gauge group. The embedding of the D7in this space is mostly determined by symmetry. We take the D7 and D7embeddings to wrap (t, x, y), S2 and S˜2 and to sit at the parity symmetricpoint ψ = pi4 . To solve embedding equations, the transverse coordinate zmust depend on the radius r. At the boundary of AdS5 (r → ∞), weimpose the boundary condition that the D7 is located at z = −L/2 and D7at z = L/2. The worldvolume metric of one of the branes is thendσ2 = R2[r2(−dt2 + dx2 + dy2) + dr2r2(1 + r4z˙(r)2)+12dΩ22 +12d˜Ω22] (4.3)where z˙ = dz/dr. The field strength of the world-volume gauge fields areF =R22piα′f2Ω2 +R22piα′f2Ω˜2 (4.4)where Ω2 and Ω˜2 are the volume forms of the unit 2-spheres. The fluxforms two Dirac monopole bundles with monopole number nD =√λf2.Stability and other properties of the theory [37][39] require that 23/50 <f2 < 1, otherwise it is a tunable parameter. The embedding is determined62Chapter 4. D7-anti-D7 Double Monolayer : Holographic Dynamical Symmetry Breakingby extremizing the Dirac-Born-Infeld plus Wess-Zumino actions,S = −T7Ngs∫d8σ[√−det(g(σ) + 2piα′F )∓(2piα′)22F ∧ F ∧ C4](4.5)T7 = 1/(2pi)7α′4 is the brane tension, C4 is the Ramond-Ramond 4-form ofthe IIB string background and the ∓ refer to the D7 and D7, respectively.With our Ansatz, this reduces to a variational problem with LagrangianL = (1 + f2)r2√1 + r4z˙(r)2 ∓ f2r4z˙(r) (4.6)z(r) is a cyclic variable whose equation of motion is solved by z±(r) =±L2 ∓∫∞r drz˙+(r) is the position of the brane to the right (upper sign) orleft (lower sign) of z = 0 and z˙±(r) = ± f2r4+Pr2√(r4−P )((1+2f2)r4+P ) . P is anintegration constant proportional to the pressure needed to hold the braneswith their asymptotic separation L. When they are not joined, they do notinteract, at least in this classical limit, and P must be zero. Then z±(r) =±L2 ∓ f2√1+2f2 ras depicted in figure 4.2. When they are joined, as depictedin figure 4.3, P must be nonzero and they are joined at a minimum radiusr0 = P14 and L and P are related by LP14 = 2∫∞1 drf2r4+1r2√(r4−1)((1+2f2)r4+1) .The joined solution will always be the lower energy solution when thebranes are oriented as in figures 4.2 and 4.3. They are also stable for anyvalue of L when the brane and antibrane are interchanged, the “chubbysolutions” discussed in reference [54], only when 23/50 ≤ f2 . .56. Whenf2 > .56 the chubby solutions are unstable for any L. (As noted in ref-erence [54], there can be a much richer phase structure when temperature,density or external magnetic fields are introduced.) For the chubby solution,the gauge group ranks N and N + k in figure 4.1 trade positions.A simple diagnostic of the properties of the fermion system in the stronglycoupled quantum field theory which is dual to the joined branes is thecurrent-current correlation function. It is obtained by solving the classi-cal dynamics of the gauge field on the world-volume of the branes with theDirichlet boundary condition. The quadratic form in boundary data in theon-shell action yields the current-current correlator. Here, the brane geome-try is simple enough that, to quadratic order, AdS components of the vectorfield decouple from the fluctuations of the worldvolume geometry, as well asfrom those components on S2, S˜2. To find them, we simply need to solve63Chapter 4. D7-anti-D7 Double Monolayer : Holographic Dynamical Symmetry BreakingMaxwell’s equations on the worldvolume,∂A[√ggBCgDE(∂CAE − ∂EAC)]= 0where the worldvolume metric is given in equation (4.3) above and the gaugefields have indices A,B... = (t, x, y, r). In the Ar = 0 gauge,∂r(∂aAa) = 0 , ∂2ρAa + ∂b(∂bAa − ∂aAb) = 0 (4.7)where indices a, b, ... = (t, x, y) and we have redefined the radial coordinateas ρ =∫∞rdrr2√1 + r4z˙2. In the simpler case of a single D7-brane, say thebrane which originates on the right in figure 4.2, whose geometry is AdS4,these equations are solved by [39]Aa(ρ, k) = Aa(k) cosh |k|ρ+ 1|k|A′a(k) sinh |k|ρwhere Aa(k, ρ) =∫d3xeikxAa(x, ρ), kaAa(k) = 0 = kaA′(k) and |k| =√~k2 − k20. Regularity at the Poincare horizon (ρ → ∞) requires A′a(k) =−|k|Aa(k). Moreover, with the on-shell action,S = −N(f2 + 1)4pi2∫d3k|k|Aa(−k)(δab − kakb/k2)Ab(k) + . . .e−S is a generating function for current-current correlators in the dual con-formal field theory where the U(1) symmetry is global, (ja(k) = gYMδ/δAa(−k))< ja(k)jb(`) >=λ(f2 + 1)2pi2|k| (δab − kakb/k2) δ(k + `) (4.8)Alternatively, if instead of the Dirichlet boundary conditions used above, weimpose the Neuman boundary condition that ∂ρAa(k, ρ) approaches A′a(k)as ρ → 0, we can write the on-shell action as a functional of A′(k) and itgenerates correlators of the gauge field in a different conformal field theorywhere the U(1) symmetry is gauged and the gauge field is dynamical. Ityields the Landau gauge 2-point function of the photon field in that theory[56] (aa(k) = δ/δA′(−k)),< aa(k)ab(`) >=N(f2 + 1)2pi21|k|(δab − kakb/k2)δ(k + `)The momentum dependence of these correlation functions is consistent withconformal symmetry.64Chapter 4. D7-anti-D7 Double Monolayer : Holographic Dynamical Symmetry BreakingTo analyze the joined configuration, we note that in that case ρ reachesa maximumρm =L2∫ 10dx(1+f2)√(1−x4)((1+2f2)−x4)∫ 10dx(f2+x4)√(1−x4)((1+2f2)−x4)(4.9)We use a variable s = ρ for the left branch and s = 2ρm − ρ for the rightbranch of figure 4.3. With the Dirichlet boundary conditions Aa(s = 0, k) =Aa(k) and Aa(s = 2ρm, k) = A˜a(k) the on-shell action isS˜ = −N(f2 + 1)4pi2∫d3k[(|Aa(k)|2 + |A˜a(k)|2) coth 2|k|ρm−2Aa(−k)Aa(k)csch2|k|ρm] + . . . (4.10)The current-current correlation functions can are diagonalized by j+ ≡ j+ j˜,j− ≡ j − j˜, so that< ja+jb− > = 0 (4.11)< ja+jb+ > =λ(f2 + 1)2pi2k tanh kρm(δab − kakbk2)(4.12)< ja−jb− > =λ(f2 + 1)2pi2k coth kρm(δab − kakbk2)(4.13)At large Euclidean momenta, (4.12) and (4.13) revert to the conformal fieldtheory correlators in (4.8). At time-like momenta the correlator < ja−jb− >has a pole at k2 = 0 which is the signature of dynamical breaking of adiagonal U(1) subgroup of the U(1) × U(1) symmetry and gives rise tosuperfluid linear response. On the other hand, the correlator < ja+jb+ >∼k2 for small k, which indicates that the system is an insulator in the channelwhich couples to the other diagonal U(1) subgroup with current ja+. Inaddition, both correlators have an interesting analytic structure. They haveno cut singularities. < ja+jb+ > has poles at the (Minkowski signature)energiesk20 = k21 + k22 +(pi(2n+ 1)2ρm)2, n = 0, 1, . . . (4.14)and < ja−jb− > has poles atk20 = k21 + k22 +(pinρm)2, n = 0, 1, . . . (4.15)65Chapter 4. D7-anti-D7 Double Monolayer : Holographic Dynamical Symmetry Breakingindicating two infinite towers of massive spin-one particles. These would benarrow bound state resonances with decay widths that vanish as N →∞, asone expects in the large-N limit that we are studying here [57]. The currentoperators create these single-particle states from the vacuum. Their creationof multi-particle states, which would normally result in cut singularities, issuppressed in the large N planar limit. The resonances are simply the towerof vector mesons whose masses (4.14) and (4.15) occur at eigenvalues of−∂2s with Dirichlet boundary conditions on the interval s ∈ [0, 2ρm]. Thefact that currents create either even or odd harmonics is due to L → −Lreflection symmetry.In the above, we used Dirichlet boundary conditions for the worldvolumegauge field. It is possible, alternatively, to select Neumann boundary con-ditions by choosing ∂sAa rather than Aa on the asymptotic boundary. Theresult is dual to a field theory where the U(1) symmetries are gauged andthe on-shell action generates photon correlation functions [56]. Most rele-vant are mixed Neuman and Dirichlet boundary conditions. For example,in graphene, a diagonal electromagnetic U(1) is gauged whereas the orthog-onal U(1) is a global symmetry. This is obtained by applying the Dirichletcondition to A(s = 0, k) − A(s = 2ρm, k) and the Neuman condition to∂sA(s = 0, k)− ∂sA(s = 2ρm, k). In this case, the correlation functions are< jaab > = 0 (4.16)< jajb > =λ(f2 + 1)4pi2k coth kρm(δab − kakbk2)(4.17)< aaab > =N(f2 + 1)4pi21kcoth kρm(δab − kakbk2)(4.18)The global U(1) symmetry is spontaneously broken and its current ja has apole in its correlation function. The unbroken gauged U(1) has a masslesspole corresponding to the photon. In addition, the two towers of interme-diate states have the same masses with values (4.15). There is a family ofmore general mixed boundary conditions which are interesting and whichwill be examined in detail elsewhere.66Chapter 5Holographic D3-probe-D5Model of a Double LayerDirac SemimetalsI have ugly lips.It was all my fault.I sobbed standing next to the coat left behind.None of the laughs could raise my heavy heart.I want to forget the tavern.There is none like you in this world.I lost my love in that narrow place.- In Front of the House by Ki, Hyung-do5.1 Introduction and SummaryThe possibility of Coulomb drag-mediated exciton condensation in dou-ble monolayer graphene or other multi-layer heterostructures has recentlyreceived considerable attention [63]-[106]. The term “double monolayergraphene” refers to two monolayers of graphene23, each of which would be aDirac semi-metal in isolation, and which are brought into close proximity butare still separated by an insulator so that direct transfer of electric chargecarriers between the layers is negligible. The system then has two conservedcharges, the electric charge in each layer. The Coulomb interaction betweenan electron in one layer and a hole in the other layer is attractive. A boundstate of an electron and a hole that forms due to this attraction is called anexciton. Excitons are bosons and, at low temperatures they can condense toa form a charge-neutral superfluid. We will call this an inter-layer exciton23 It should be distinguished from bilayer graphene where electrons are allowed to hopbetween the layers.675.1. Introduction and Summarycondensate. Electrons and holes in the same monolayer can also form anexciton bound state, which we will call this an intra-layer exciton and itsBose condensate an intra-layer condensate.Inter-layer excitons have been observed in some cold atom analogs ofdouble monolayers [107]-[109] and as a transient phenomenon in GalliumArsenide/ Aluminium-Gallium-Arsenide double quantum wells, albeit onlyat low temperatures and in the presence of magnetic fields[109]-[111]. Theirstudy is clearly of interest for understanding fundamental issues with quan-tum coherence over mesoscopic distance scales and dynamical symmetrybreaking. Recent interest in this possibility in graphene double layers hasbeen inspired by some theoretical modelling which seemed to indicate thatthe exciton condensate could occur at room temperature [66]. A room tem-perature superfluid would have interesting applications in electronic deviceswhere proposals include ultra-fast switches and dispersionless field-effecttransistors [112]-[117]. This has motivated some recent experimental studiesof double monolayers of graphene separated by ultra-thin insulators, downto the nanometer scale [118][119]. These experiments have revealed inter-esting features of the phenomenon of Coulomb drag. However, to this date,coherence between monolayers has yet to be observed in a stationary stateof a double monolayer.One impediment to a truly quantitative analysis of inter-layer coher-ence is the fact that the Coulomb interaction at sub-nanoscale distances isstrong and perturbation theory must be re-summed in an ad hoc way totake screening into account [105][120]. In fact, inter-layer coherence willlikely always require strong interactions. The purpose of this chapter isto point out the existence of an inherently nonperturbative model of verystrongly coupled multi-monolayer systems. This model is a defect quantumfield theory which is the holographic AdS/CFT dual of a D3-probe-D5 branesystem. It is simple to analyze and exactly solvable in the limit where thequantum field theory interactions are strong. External magnetic field andcharge density can be incorporated into the solution and it exhibits a richphase diagram where it has phases with inter-layer exciton condensates.It might be expected that, with a sufficiently strong attractive electron-hole interaction, an inter-layer condensate would always form. One of thelessons of our work will be that this is not necessarily so. In fact, it wasalready suggested in reference [94] that, when both monolayers are chargeneutral, and in a constant external magnetic field, there can be an inter-layeror an intra-layer condensate but there were no phases where the two kindsof condensate both occur at the same time. What is more, the inter-layercondensate only appears for small separations of the monolayers, up to a685.1. Introduction and Summarycritical separation. As the spacing between the monolayers is increased tothe critical distance, there is a phase transition where an intra-layer con-densate takes over. Intra-layer condensates in a strong magnetic field arealready well known to occur in monolayer graphene in the integer quantumHall regime [124]-[128]. They are thought to be a manifestation of “quantumHall ferromagnetism” [129]-[134] or the “magnetic catalysis of chiral sym-metry breaking” [71]-[83] which involve symmetry breaking with an intra-layer exciton condensate. It has been argued that the latter phenomenon,intra-layer exciton condensation, in a single monolayer is also reflected insymmetry breaking behaviour of the D3-probe-D5 brane system [84]-[85].Another striking conclusion that we will come to is that, even in thestrong coupling limit, there is no inter-layer exciton condensate unless thecharge densities of the monolayers are fine-tuned in such a way that, at weakcoupling, the electron Fermi surface on one monolayer and the hole Fermisurface in the other monolayer are perfectly nested, that is, they have iden-tical Fermi energies. In this particle-hole symmetric theory, this means thatthe charge densities on the monolayers are of equal magnitude and oppositesign. It is surprising that this need for charge balance is even sharper in thestrong coupling limit than what is seen at weak coupling, where the infraredsingularity from nesting does provide the instability needed for exciton con-densation, but where, also, there is a narrow window near perfect nestingwhere condensation is still possible [65]. In our model, at strong coupling,there is inter-layer condensate only in the perfectly nested (or charge bal-anced) case. This need for such fine tuning of charge densities could help toexplain why such a condensate is hard to find in experiments where chargedimpurities would disturb the charge balance.When the charge densities of the monolayers are non-zero, and when theyare balanced, there can be an inter-layer condensate at any separation of themonolayers. The phase diagram which we shall find for the D3-probe-D5brane system in a magnetic field and with nonzero, balanced charge densitiesis depicted in figure 5.1. The blue region has an inter-layer condensate andno intra-layer condensate. The green region has both inter-layer and intra-layer condensates. The red region has only an intra-layer condensate. Fromthe vertical axis in figure 5.1 we see that, in the charge neutral case. theinter-layer condensate exists only for separation less than a critical one.It has recently been suggested [101] that there is another possible be-haviour which can lead to inter-layer condensates when the charges of themonolayers are not balanced. This can occur when the material of the mono-layers contain more than one species of fermions. For example, graphene hasfour species and emergent SU(4) symmetry [98]. In that case, the most sym-695.1. Introduction and Summary0 0.5 1 1.5 2 2.5 3024681st order2ndorder2ndorderintraq = 0intra/interinterλ1/42pi√Bµ√2piBλ1/4L1.357Figure 5.1: Phase diagram of the D3-probe-D5 brane system with balanced chargedensities. Layer separation is plotted on the vertical axis and the chemical potentialµ for electrons in one monolayer and holes in the other monolayer is plotted onthe horizontal axis. The units employed set the length scale√ √λ2piB equal to one.The blue region is a phase with an inter-layer condensate and with no intra-layercondensate. The green region is a phase with both an inter-layer and an intra-layercondensate. The red region has only an intra-layer condensate. In that region, thechemical potential is too small to induce a density of the massive electrons ( µ isin the charge gap) and the charge densities on both of the monolayers vanishes.The electrons and holes are massive in that phase due to the intra-layer excitoncondensate. The dotted line, separating a pure inter-layer from a pure intra-layercondensate, is a line of first order phase transitions. The solid lines, on the otherhand, indicate second order transitions.metric state of a monolayer has the charge of that monolayer shared equallyby each of the four species of electrons. Other less symmetric states arepossible.Consider, for example, the double monolayer with one monolayer havingelectron charge density Q and the other monolayer having hole density Q¯(or electron charge density −Q¯), with Q > Q¯ > 0. On the hole-chargedmonolayer, some subset, which must be one, two or three of the fermionspecies could take up all of the hole charge density, Q¯. Then, in the electronmonolayer, the same number, one, two or three species of electrons would705.1. Introduction and Summarytake up electron charge density Q¯ and the remainder of the species will takeelectron charge density Q−Q¯. The (one, two or three) species with matchedcharge densities will then spawn an inter-layer exciton condensate. Theremaining species on the hole monolayer is charge neutral. A charge-neutralmonolayer will have an intra-layer condensate. The remaining species in theelectron monolayer, with charge density Q − Q¯, would also have an intra-layer condensate and it would not have a charge gap (all of the fermionsare massive, but this species has a finite density and it does not have acharge gap). A simple signature of this state would be that one of themonolayers is charge gapped, whereas the other one is not. The implicationis that perfect fine-tuning of Fermi surfaces is not absolutely necessary forinter-layer condensation. We will show that, in a few examples, this type ofspontaneous nesting can occur. However, some important questions, such ashow unbalanced the charge densities can be so that there is still a condensateare left for future work.D5r =∞AdS5r = 0rxytzFigure 5.2: A D5 brane is embedded in AdS5 × S5 where the metric of AdS5 isds2 =√λα′[dr2r2 + r2(dx2 + dy2 + dz2 − dt2)] and the D5 brane world-volume is anAdS4 subspace which fills r, x, y, t and sits at a point in z. The AdS5 boundary islocated at r = ∞ and the Poincare` horizon at r = 0. The D5 brane also wrapsa maximal, contractible S2 subspace of S5 of the AdS5 × S5 background. Theinternal bosonic symmetries of the configuration are SO(3) of the wrapped S2 anda further SO(3) symmetry of the position of the maximal S2 in S5.715.1. Introduction and SummaryD5r =∞AdS5r = 0rxytz~BFigure 5.3: When the D5 brane is exposed to a magnetic field, it pinches off beforeit reaches the Poincare` horizon. It does so for any value of the magnetic field, withthe radius at which it pinches off proportional to√2piB√λ. In this configuration, theembedding of the S2 ⊂ S5 depends on the AdS5 radius. It is still the maximal onewhich can be embedded in S5 at the boundary, but it shrinks and collapses to apoint at the radius where the D5 brane pinches off.We will model a double monolayer system with a relativistic defect quan-tum field theory consisting of two parallel, infinite, planar 2+1-dimensionaldefects embedded in 3+1-dimensional Minkowski space. The defects areseparated by a length, L. Some U(1) charged degrees of freedom inhabit thedefects and play the role of the two dimensional relativistic electron gases.We can consider states with charge densities on the monolayers. As well, wecan expose them to a constant external magnetic field. We could also turnon a temperature and study them in a thermal state, however, we will notdo so in this chapter.The theory that we use has an AdS/CFT dual, the D3-probe-D5 branesystem where the D5 and anti-D5 branes are probes embedded in the AdS5×S5 background of the type IIB superstring theory. The AdS5×S5 is sourcedby N D3 branes and it is tractable in the large N limit where we si-multaneously scale the string theory coupling constant gs to zero so thatλ ≡ gsN/4pi = g2YMN is held constant. Here, gYM is the coupling constantof the gauge fields in the defect quantum field theory. The D5 and anti-725.1. Introduction and SummaryD5 branes are semi-classical when the quantum field theory on the doublemonolayer is strongly coupled, that is, where λ is large. It is solved by em-bedding a D5 brane and an anti-D5 brane in the AdS5 × S5 background.The boundary conditions of the embedding are such that, as they approachthe boundary of AdS5, the world volumes approach the two parallel 2+1-dimensional monolayers. The dynamical equations which we shall use areidentical for the brane and the anti-brane. The reason why we use a brane-anti-brane pair is that they can partially annihilate. This annihilation willbe the string theory dual of the formation of an inter-layer exciton conden-sate.The phase diagram of a single D5 brane or a stack of coincident D5branes is well known [99], with an important modification in the integerquantum Hall regime [84],[85]. In the absence of a magnetic field or chargedensity, a single charge neutral D5 brane takes up a supersymmetric andconformally invariant configuration. The D5 brane world-volume is itselfAdS4 and it stretches from the boundary of AdS5 to the Poincare` horizon,as depicted in figure 5.2. It also wraps an S2 ⊂ S5. This is a maximallysymmetric solution of the theory. It has a well-established quantum fieldtheory dual whose Lagrangian is known explicitly [87]-[90]. The latter isa conformally symmetric phase of a defect super-conformal quantum fieldtheory 24.Now, let us introduce a magnetic field on the D5 brane world volume.This is dual to the 2+1-dimensional field theory in a background constantmagnetic field. As soon as an external magnetic field is introduced, thesingle D5 brane changes its geometry drastically [76]. The brane pinches offand truncates at a finite AdS5-radius, before it reaches the Poincare` horizon.This is called a “Minkowski embedding” and is depicted in figure 5.3. This24Of course a supersymmetric conformal field theory is not a realistic model of asemimetal. Here, we will use this model with a strong magnetic field. It was observed inreferences [84], [85] that the supersymmetry and conformal symmetry are both broken byan external magnetic field, and that the low energy states of the weakly coupled systemwere states with partial fillings of the fermion zero modes which occur in the magneticfield (the charge neutral point Landau level). The dynamical problem to be solved is thatof deciding which partial fillings of zero modes have the lowest energy. It is a direct analogof the same problem in graphene or other Dirac semimetals. It is in this regime that D3-D5 system exhibits quantum Hall ferromagnetism and other interesting phenomena whichcan argued to be a strong coupling extrapolation of universal features of a semimetal ina similar environment. It is for this reason that we will concentrate on the system with amagnetic field, with the assumption that the very low energy states of the theory are themost important for the physics of exciton formation, and that this situation persists tostrong coupling. There have been a number of works which have used D branes to modeldouble monolayers [54]-[96].735.1. Introduction and Summaryconfiguration has a charge gap. Charged degrees of freedom are open stringswhich stretch from the D5 brane to the Poincare` horizon. When, the D5brane does not reach the Poincare` horizon, these strings have a minimumlength and therefore a mass gap. This is the gravity dual of the massgeneration that accompanies exciton condensation in a single monolayer.D5 D5r =∞AdS5r = 0LFigure 5.4: A D5 brane and an anti-D5 brane are are suspended in AdS5 as shown.They are held a distance L apart at the AdS5 boundary.Let us now consider the double monolayer system. We will begin withthe case where both of the monolayers are charge neutral and there is nomagnetic field. We will model the strong coupled system by a pair whichconsists of a probe D5 brane and a probe anti-D5 brane suspended in theAdS5 background as depicted in figure 5.4. Like a particle-hole pair, theD5 brane and the anti-D5 brane have a tendency to annihilate. However,we can impose boundary conditions which prevent their annihilation. Werequire that, as the D5 brane approaches the boundary of the AdS5 space,it is parallel to the anti-D5 brane and it is separated from the anti-D5 braneby a distance L. Then, as each brane hangs down into the bulk of AdS5,they can still lower their energy by partially annihilating as depicted inthe joined configuration in figure 5.4. This joining of the brane and anti-brane is the AdS/CFT dual of inter-layer exciton condensation. The inthis case, when they are both charge neutral, the branes will join for anyvalue of the separation L. In this strongly coupled defect quantum field745.1. Introduction and Summarytheory, with vanishing magnetic field and vanishing charge density on bothmonolayers, the inter-layer exciton condensate exists for any value of theinter-layer distance.D5 D5r =∞AdS5r = 0LFigure 5.5: When the D5 brane and an anti-D5 brane are suspended as shown,their natural tendency is to join together. This is the configuration with the lowestenergy when L is fixed. It is also the configuration which describes the quantumfield theory with an inter-layer exciton condensate.If we now turn on a magnetic field B so that the dimensionless param-eter BL2 is small, the branes join as they did in the absence of the field.However, in a stronger field, as BL2 is increased, there is a competitionbetween the branes joining and, alternatively, each of the branes pinchingoff and truncating, as they would do if there were isolated. The pinched offbranes are depicted in figure 5.6. This configuration has intra-layer excitoncondensates on each monolayer but no inter-layer condensate. We thus seethat, in a magnetic field, the charge neutral double monolayer always hasa charge gap due to exciton condensation. However, it has an inter-layercondensate only when the branes are close enough.Now, we can also introduce a charge density on both the D5 brane andthe anti-D5 brane. We shall find a profound difference between the caseswhere the overall density, the sum of the density on the two branes is zeroand where it is nonzero. In the first case, when it is zero, joined configura-tions of branes exist for all separations. Within those configurations, there755.1. Introduction and SummaryD5 D5r =∞AdS5r = 0L~BFigure 5.6: When the D5 brane and an anti-D5 brane are exposed to a magneticfield, and if the field is strong enough, they can pinch off and end before they join.This tendency to pinch off competes with their tendency to join and in a strongenough field they will take up the phase that is shown where they pinch off beforethey can join.are regions where the exciton condensate is inter-layer only and a regionwhere it is a mixture of intra-layer and inter-layer. These are seen in thephase diagram in figure 5.1. The blue region has only an inter-layer excitoncondensate. The green region has a mixed inter-layer and intra-layer con-densate. In the red region, the chemical potential is of too small a magnitudeto induce a charge density (it is in the charge gap) and the phase is iden-tical to the neutral one, with an intra-layer condensate and no inter-layercondensate.In the case where the D5 and anti-D5 brane are not overall neutral,they cannot join. There is never an inter-layer condensate. They can haveintra-layer condensates if their separation is small enough. However, there isanother possibility, which occurs if we have stacks of multiple D5 branes. Inthat case, there is the possibility that the D5 branes in a stack do not sharethe electric charge equally. Instead some of them take on electric chargesthat matches the charge of the anti-D5 branes, so that some of them can join,and the others absorb the remainder of the unbalanced charge and do notjoin. At weak coupling this would correspond to a spontaneous nesting of the765.2. Geometry of branes with magnetic field and densityFermi surfaces of some species of fermions in the monolayers, with the otherspecies taking up the difference of the charges. At weak coupling, as wellas in our strong coupling limit, the question is whether the spontaneouslynested system is energetically favored over one with a uniform distributionof charge. We shall find that, for the few values of the charge where we havebeen able to compare the energies, this is indeed the case.In the remainder of the chapter, we will describe the quantitative analysiswhich leads to the above description of the behaviour of the D3-probe-D5brane system. In section 2 we will discuss the mathematical problem offinding the geometry of probe D5 branes embedded in AdS5 × S5 in theconfigurations which give us the gravity dual of the double monolayer. Insection 3 we will discuss the behaviour of the double monolayer where eachlayer is charge neutral and they are in a magnetic field. In section 4 we willdiscuss the double monolayer in a magnetic field and with balanced chargedensities. In section 5 we will explore the behaviour of double monolayerswith un-matched charge densities. Section 6 contains some discussion of theconclusions.5.2 Geometry of branes with magnetic field anddensityWe will consider a pair of probe branes, a D5 brane and an anti-D5 branesuspended in AdS5 × S5. They are both constrained to reach the boundaryof AdS5 with their world volume geometries approaching AdS4 × S2 wherethe AdS4 is a subspace of AdS5 with one coordinate direction suppressedand S2 is a maximal two-sphere embedded in S5. What is more, whenthey reach the boundary, we impose the boundary condition that they areseparated from each other by a distance L.We shall use coordinates where the metric of the AdS5×S5 backgroundisds2 =dr2r2+ r2(−dt2 + dx2 + dy2 + dz2)+ dψ2 + sin2 ψd2Ω2 + cos2 ψd2Ω˜2(5.1)where d2Ω2 = dθ2 + sin2 θdφ2 and d2Ω˜2 = dθ˜2 + sin2 θ˜dφ˜2 are the metrics oftwo 2-spheres, S2 and S˜2. The world volume geometry of the D5 brane is forthe most part determined by symmetry. We require Lorentz and translationinvariance in 2+1-dimensions. This is achieved by both the D5 and the anti-D5 brane wrapping the subspace of AdS5 with coordinates t, x, y. We willalso assume that all solutions have an SO(3) symmetry. This is achieved775.2. Geometry of branes with magnetic field and densitywhen both the D5 and anti-D5 brane world volumes wrap the 2-sphereS2 with coordinates θ, φ. Symmetry requires that none of the remainingvariables depend on t, x, y, θ, φ. For the remaining internal coordinate ofthe D5 or anti-D5 brane, it is convenient to use the projection of the AdS5radius, r onto the brane world-volume. The D5 and anit- D5 branes willsit at points in the remainder of the AdS5 × S5 directions, z, ψ, θ˜, φ˜. Thepoints z(r) and ψ(r) generally depend on r and these functions become thedynamical variables of the embedding (along with world volume gauge fieldswhich we will introduce shortly). The wrapped S2 has an SO(3) symmetry.What is more, the point ψ = pi2 where the wrapped sphere is maximal hasan additional SO(3) symmetry25. The geometry of the D5 brane and theanti-D5 brane are both given by the ansatzds2 =dr2r2(1 + (r2z′)2 + (rψ′)2)+r2(−dt2 + dx2 + dy2)+sin2 ψd2Ω2 (5.2)The introduction of a charge density and external magnetic field will requireD5 world-volume gauge fields. In the ar = 0 gauge, the field strength 2-formF is given by2pi√λF = a′0(r)dr ∧ dt+ bdx ∧ dy (5.3)In this expression, b is a constant which will give a constant magnetic fieldin the holographic dual and a0(r) will result in the world volume electricfield which is needed in order to have a nonzero U(1) charge density in thequantum field theory. The magnetic field B and temporal gauge field A0are defined in terms of them asb =2pi√λB , a0 =2pi√λA0 (5.4)In this Section, we will use the field strength (5.3) for both the D5 braneand the anti-D5 brane.The asymptotic behavior at r →∞ for the embedding functions in (5.2)and the gauge field (5.3) are such that the sphere S2 becomes maximal,ψ(r)→ pi2+c1r+c2r2+ . . . (5.5)25At that point where S2 is maximal, sinψ = sin pi2= 1 and cosψ = 0, that is, thevolume of S˜2 vanishes. The easiest way to see that this embedding has an SO(3) symmetryis to parameterize the S5 by (x1, ..., x6) with x21 + . . . x26 = 1. S2 is the space x21 +x22 +x23 =sin2 ψ and S˜2 is x24 + x25 + x26 = cos2 ψ. The point cosψ = 0 with x4 = x5 = x6 = 0requires no choice of position on S˜2 and it thus has SO(3) symmetry. On the other hand,if cosψ 6= 0 and therefore some of the coordinates (x4, x5, x6) are nonzero, the symmetryis reduced to an SO(2) rotation about the direction chosen by the vector (x4, x5, x6).785.2. Geometry of branes with magnetic field and densityand the D5 brane and anti-D5 brane are separated by a distance L,z(r)→ L2− fr5+ . . . (5.6)for the D5 brane andz(r)→ −L2+fr5+ . . . (5.7)for the anti-D5 brane. The asymptotic behaviour of the gauge field isa0(r) = µ− qr+ . . . (5.8)with µ and q related to the chemical potential and the charge density, re-spectively. There are two constants which specify the asymptotic behaviorin each of the above equations. In all cases, we are free to choose one of thetwo constants as a boundary condition, for example we could choose c1, q, f .Then, the other constants, c2, µ, L, are fixed by requiring that the solutionis non-singular.In this chapter, we will only consider solutions where the boundary con-dition is c1 = 0. This is the boundary condition that is needed for the Diracfermions in the double monolayer quantum field theory to be massless atthe fundamental level. Of course they will not remain massless when thereis an exciton condensate. In the case where they are massless, we say thatthere is “chiral symmetry”, or that c1 = 0 is a chiral symmetric boundarycondition. Then, when we solve the equation of motion for ψ(r), there aretwo possibilities. The first possibility is that c2 = 0 and ψ =pi2 , a constantfor all values of r. This is the phase with good chiral symmetry. Secondly,c2 6= 0 and ψ is a non-constant function of r. This describes the phase withspontaneously broken chiral symmetry. The constant c2 is proportional tothe strength of the intra-layer chiral exciton condensate the D5 brane or theanti-D5 brane. The constant f instead is proportional to the strength of theinter-layer condensate.To be more general, we could replace the single D5 brane by a stackof N5 coincident D5 branes and the single anti-D5 brane by another stackof N¯5 coincident anti-D5 branes. Then, the main complication is that theworld volume theories of the D5 and anti-D5 branes become non-Abelian inthe sense that the embedding coordinates become matrices and the world-volume gauge fields also have non-Abelian gauge symmetry. The Born-Infeld action must also be generalized to be, as well as an integral overcoordinates, a trace over the matrix indices. For now, we will assume that795.2. Geometry of branes with magnetic field and densitythe non-Abelian structure plays no significant role. Then, all of the matrixdegrees of freedom are proportional to unit matrices and the trace in thenon-Abelian Born-Infeld action simply produces a factor of the number ofbranes, N5 or N¯5 (see equation (5.9) below). We will also take N5 = N¯5and leave the interesting possibility that N5 6= N¯5 for future work. (Thisgeneralization could, for example, describe the interesting situation where adouble monolayer consists of a layer of graphene and a layer of topologicalinsulator.) We also have not searched for interesting non-Abelian solutionsof the world volume theories which would provide other competing phasesof the double monolayer system. Some such phases are already known toexist. For example, it was shown in references [84] and [85] that, whenthe Landau level filling fraction, which is proportional to Q/B, is greaterthan approximately 0.5, there is a competing non-Abelian solution whichresembles a D7 brane and which plays in important role in matching thereinteger quantum Hall states which are expected to appear at integer fillingfractions. In the present work, we will avoid this region by assuming thatthe filling fraction is sub-critical. Some other aspects of the non-Abelianstructure will be important to us in section 5.The Born-Infeld action for either the stack of D5 branes or the stack ofanti-D5 branes is given byS = N5∫dr sin2 ψ√r4 + b2√1 + (rψ′)2 + (r2z′)2 − (a′0)2 (5.9)whereN5 =√λNN52pi3V2+1with V2+1 the volume of the 2+1-dimensional space-time, N the numberof D3 branes, N5 the number of D5 branes. The Wess-Zumino terms thatoccur in the D brane action will not play a role in the D5 brane problem.The variational problem of extremizing the Born-Infeld action (5.9) in-volves two cyclic variables, a0(r) and z(r). Being cyclic, their canonicalmomenta must be constants,Q = − δSδA′0≡ 2piN5√λq , q =sin2 ψ√r4 + b2a′0√1 + (rψ′)2 + (r2z′)2 − (a′0)2(5.10)Πz =δSδz′≡ N5f , f = sin2 ψ√r4 + b2r4z′√1 + (rψ′)2 + (r2z′)2 − (a′0)2, (5.11)805.2. Geometry of branes with magnetic field and densitySolving (5.10) and (5.11) for a′0(r) and z′(r) in terms of q and f we geta′0 =qr2√1 + r2ψ′2√r4 (b2 + r4) sin4 ψ + q2r4 − f2(5.12)z′ =f√1 + r2ψ′2r2√r4 (b2 + r4) sin4 ψ + q2r4 − f2(5.13)The Euler-Lagrange equation can be derived by varying the action (5.9).We eliminate a′0(r) and z′(r) from that equation using equations (5.12) and(5.13). Then the equation of motion for ψ readsrψ′′ + ψ′1 + r2ψ′2−ψ′ (f2 + q2r4 + r4 (b2 + 3r4) sin4 ψ)− 2r3 (b2 + r4) sin3 ψ cosψf2 − q2r4 − r4 (b2 + r4) sin4 ψ = 0(5.14)This equation must be solved with the boundary conditions in equation(5.5)-(5.8) (remembering that we can choose only one of the integrationconstants, the other being fixed by regularity of the solution) in order to findthe function ψ(r). Once we know that function, we can integrate equations(5.12) and (5.13) to find a0(r) and z(r).Clearly, ψ = pi2 , a constant, for all values of r, is always a solutionof equation (5.14), even when the magnetic field and charge density arenonzero. However, for some range of the parameters, it will not be the moststable solution.5.2.1 Length, Chemical Potential and RouthiansThe solutions of the equations of motion are implicitly functions of the inte-gration constants. We can consider a variation of the integration constantsin such a way that the functions ψ(r), a0(r), z(r) remain solutions as theconstants vary. Then, the on-shell action varies in a specific way. Considerthe action (5.9) evaluated on solutions of the equations of motion. We callthe on-shell action the free energy F1 = S[ψ, z, a0]/N5. If we take a variationof the parameters in the solution, here, specifically µ and L, while keepingc1 = 0, and assuming that the equations of motion are obeyed, we obtainδF1 =∫ ∞0dr(δψ∂L∂ψ′+ δa0∂L∂a′0+ δz∂L∂z′)′= −qδµ+ fδL (5.15)The first term, with δψ vanishes because δψ ∼ δc2/r2. We see that F1 is afunction of the chemical potential µ and the distance L and the conjugate815.2. Geometry of branes with magnetic field and densityvariables, the charge density and the force needed to hold the D5 brane andanti-D5 brane apart are gotten by taking partial derivatives,q = −∂F1∂µ∣∣∣∣L, f =∂F1∂L∣∣∣∣µ(5.16)When the dynamical system relaxes to its ground state, with the parametersµ and L held constant, it relaxes to a minimum of F1.There are other possibilities for free energies. For example, the quan-tity which is minimum when the charge density, rather than the chemicalpotential, is fixed, is obtained from F1[L, µ] by a Legendre transform,F2[L, q] = F1[L, µ] + qµ (5.17)If we formally consider F2 off-shell as an action from which, for fixed q andf , we can derive equations of motion for ψ(r) and z(r),F2 = SN5 +∫qa′0dr =∫dr√sin4 ψ(r4 + b2) + q2√1 + (rψ′)2 + (r2z′)2(5.18)where we have useda′0 =q√1 + r4z′2 + r2ψ′2√(b2 + r4) sin4 ψ + q2obtained by solving equation (5.10) for a′0. The equation of motion for ψ(r),equation (5.14), can be derived from (5.18) by varying ψ(r). Moreover, westill havef =sin2 ψ√r4 + b2r4z′√1 + (rψ′)2 + (r2z′)2 − (a′0)2=√(b2 + r4) sin4 ψ + q2 r4z′√1 + r4z′2 + r2ψ′2(5.19)which was originally derived from (5.9) by varying z and then finding a firstintegral of the resulting equation of motion. It can also be derived from(5.18).Once the function ψ(r) is known, we can solve equation (5.19) for z′(r)and then integrate to compute the separation of the D5 and anti-D5 branes,L = 2∫ ∞r0dr z′(r) = 2f∫ ∞r0dr√1 + r2ψ′2r2√r4 (b2 + r4) sin4 ψ + q2r4 − f2(5.20)825.2. Geometry of branes with magnetic field and densitywhere ψ(r) is a solution of (5.14) and r0 is the turning point, that is theplace where the denominator in the integrand vanishes. This turning pointdepends on the value of ψ(r0). When ψ is the constant solution ψ = pi/2,r0 =4√√(b2 + q2)2 + 4f2 − b2 − q24√2(5.21)and the integral in (5.20) can be done analytically. It readsL = 2f∫ ∞r0dr1r2√r4 (b2 + r4) + q2r4 − f2 =f√piΓ(54)2F1(12 ,54 ;74 ;− f2r08)2r05Γ(74)√b2 + q2(5.22)For b = q = 0, f = r40, we getL =2√piΓ(58)r0Γ(18)in agreement with the result quoted in reference [94] .Analogously, the chemical potential is related to the integral of the gaugefield strength on the brane in the (r, 0) directions, (5.12),µ =∫ ∞r0a′0(r) dr = q∫ ∞r0drr2√1 + r2ψ′2√r4 (b2 + r4) sin4 ψ + q2r4 − f2(5.23)When ψ is the constant solution ψ = pi/2 the integral in (5.23) can againbe done analytically and readsµ = q∫ ∞r0drr2√r4 (b2 + r4) + q2r4 − f2 =q√piΓ(54)2F1(14 ,12 ;34 ;− f2r08)r0Γ(34)(5.24)Through equations (5.20) and (5.23), L and µ are viewed as functionsof f and q, this equations can in principle be inverted to have f and q asfunctions of L and µ.We can now use (5.12) and (5.13) to eliminate a′0 and z′ from the action(5.9) to get the expression of the free energy F1F1[L, µ] =∫ ∞r0dr(b2 + r4)sin4 ψr2√1 + r2ψ′2√r4 (b2 + r4) sin4 ψ + q2r4 − f2(5.25)this has to be thought of as a function of L and µ, where f and q are consid-ered as functions of L and µ, given implicitly by (5.20) and (5.23). Note that835.3. Double monolayers with a magnetic fieldwe do not do a Legendre transform here since we need the variational func-tional which is a function of L and µ the D5 brane separation and chemicalpotential that are the physically relevant parameters.Using (5.13) to eliminate z′ in the Routhian (5.18), we now get a functionof L and qF2[L, q] =∫ ∞r0dr((b2 + r4)sin4 ψ + q2) r2√1 + r2ψ′2√r4 (b2 + r4) sin4 ψ + q2r4 − f2(5.26)The Routhian (5.26) is a function of L through the fact that it is a functionof f and f is a function of L given implicitly by (5.20). Of course had weperformed the Legendre transform of the Routhian also with respect to L,the result would beF3[f, q] = F2[L, q]−∫fz′dr =∫ ∞r0dr√1 + r2ψ′2r2√r4 (b2 + r4) sin4 ψ + q2r4 − f2(5.27)which is the variational functional appropriate for variations which holdboth q and f fixed.Note that, for convenience, from now on we shall scale the magnetic fieldb to 1 in all the equations and formulas we wrote: This can be easily doneimplementing the following rescalingsr →√br , f → b2f , q → b q , L→√bL , µ→ µ√b, Fi → b3/2Fi .(5.28)5.3 Double monolayers with a magnetic fieldIn reference [94] the case of a double monolayer where both of the monolayersare charge neutral was considered with an external magnetic field. In thissection, we will re-examine their results within our framework and using ournotation. The equation of motion for ψ(r) in this case isrψ′′ + ψ′1 + r2ψ′2− ψ′ (f2 + r4 (1 + 3r4) sin4 ψ)− 2r3 (1 + r4) sin3 ψ cosψf2 − r4 (1 + r4) sin4 ψ = 0(5.29)There are in principle four type of solutions for which c1 = 0 in (5.5) [94]:1. An unconnected, constant solution that reaches the Poincare´ horizon.An embedding of the D5 brane which reaches the Poincare´ horizonis called a “black hole (BH) embedding”. Being a constant solution,845.3. Double monolayers with a magnetic fieldthis corresponds to a state of the double monolayer where both theintra-layer and inter-layer condensates vanish.2. A connected constant ψ = pi2 solution. Since this is a connected so-lution, z(r) has a non trivial profile in r and its boundary behaviouris given by equation (5.7) with f non-zero. This solution correspondsto a double monolayer with a non-zero inter-layer condensate and avanishing intra-layer condensate.3. An unconnected solution with zero force between the branes, withf = 0 and z(r) constant functions for both the D5 brane and the anti-D5 brane, but where the branes pinch off before reaching the Poincare´horizon. An embedding of a single D brane which does not reach thePoincaee´ horizon is called a “Minkowski embedding”. Since ψ(r) mustbe r-dependent, its asymptotic behaviour is given in (5.5) with a non-vanishing c2. This embedding corresponds to a double monolayer witha non-zero intra-layer condensate and a vanishing inter-layer conden-sate.4. A connected r-dependent solution, where both z(r) and ψ(r) are non-trivial functions of r. This solution corresponds to the double mono-layer with both an intra-layer and an inter-layer condensate.This classification of the solutions is summarized in table 5.1.f = 0 f 6= 0c2 = 0Type 1 Type 2unconnected, ψ = pi/2 connected, ψ = pi/2BH, chiral symm. interc2 6= 0Type 3 Type 4unconnected, r-dependent ψ connected, r-dependent ψMink, intra intra/interTable 5.1: Types of possible solutions, where Mink stands for Minkowski embed-dings and BH for black hole embeddings.For type 2 and 4 solutions the D5 and the anti-D5 world-volumes haveto join smoothly at a finite r = r0. For these solution the charge density onthe brane and on the anti-brane, as well as the value of the constant f that855.3. Double monolayers with a magnetic fieldgives the interaction between the brane and the anti-brane, are equal andopposite.Consider now the solutions of the type 3, types 1 and 2 are just ψ = pi/2.The equation for ψ (5.29) with f = 0 simplifies further torψ′′ + ψ′1 + r2ψ′2− ψ′r(1 + 3r4)sin4 ψ − 2 (1 + r4) sin3 ψ cosψr (1 + r4) sin4 ψ= 0 (5.30)In this case it is obvious from (5.13) that z(r) is a constant. Solutions oftype 3 are those for which ψ(r) goes to zero at a finite value of r, rmin, sothat the two-sphere in the world-volume of the D5 brane shrinks to zero atrmin.A solution to (5.30) of this type can be obtained by a shooting technique.The differential equation can be solved from either direction: from rmin orfrom the boundary at r = ∞. In either case, there is a one-parameterfamily of solutions, from rmin the parameter is rmin, from infinity it is thevalue of the modulus c2 in (5.5), which can be used to impose the boundaryconditions at r →∞ with c1 = 0. The parameters at the origin rmin and atinfinity can be varied to find the unique solution that interpolates betweenthe Poincare´ horizon and the boundary at r =∞.Consider now the solution of equation (5.29) of type 4. In this case welook for a D5 that joins at some given r0 the corresponding anti-D5. Atr0, z′(r0)→∞ and r0 can be determined by imposing this condition, that,from (5.13) with q = 0 and b scaled out, readsf2 − r40(1 + r40)sin4 ψ(r0) = 0 (5.31)which yieldsψ(r0) = sin−1(4√f2r40(1 + r40)) (5.32)The lowest possible value of r0 is obtained when ψ =pi2and is given byr0,min(f) =4√√1 + 4f2 − 14√2Note that r0,min grows when f grows.Using (5.32) we can derive from the equation of motion (5.29) the con-dition on ψ′(r0), it readsψ′(r0) =(r40 + 1)√√r40+1f − 1r202r40 + 1(5.33)865.3. Double monolayers with a magnetic fieldTo find the solution let us fix some r¯ between r0 and r = ∞. Start withshooting from the origin with boundary conditions (5.32) and (5.33). (5.32)leads to z′(r0) → ∞, but for a generic choice of r0 the solution for ψ(r)does not encounter the solution coming from infinity that has c1 = 0, wethen need to vary the two parameters r0 and c2 in such a way that the twosolutions, coming from r0 and from r =∞ meet at some intermediate point.For ψ and ψ′ given by (5.32) and (5.33) at the origin, integrate thesolution outwards to r¯ and compute ψ and its derivative at r¯. For eachsolution, put a point on a plot of ψ′(r¯) vs ψ(r¯), then do the same thingstarting from the boundary, r = ∞, and varying the coefficient c2 of theexpansion around infinity. Where the two curves intersect the r-dependentsolutions from the two sides match and give the values of the moduli forwhich there is a solution.5.3.1 Separation and free energyThere are then four types of solutions of the equation of motion (5.29)representing double monolayers with a magnetic field, of type 1, 2, 3 and 4.Solutions 1 and 3 are identical to two independent copies of a single mono-layer with B field solution, sitting at a separation L. The brane separationfor the solutions of type 2 and 4 is given in (5.20) (for q = 0 in this case) andit is plotted in fig. 5.7, the blue line gives the analytic curve (5.22), keepinginto account that also r0 is a function of f through equation (5.21). For ther-dependent solution, green line, instead, r0 is defined as a function of f byequation (5.31), once the solution ψ(r) is known numerically.We shall now compare the free energies of these solutions as a functionof the separation to see at which separation one becomes preferred withrespect to the other.Since we want to compare solutions at fixed values of L the correctquantity that provides the free energy for each configuration is given by theaction evaluated on the corresponding solutionF1[L] =∫ ∞r0dr(1 + r4)sin4 ψr2√1 + r2ψ′2√r4 (1 + r4) sin4 ψ − f2. (5.34)Note that this formula is obtained from (5.25), by setting q = 0 and per-forming the rescaling (5.28). The dependence of F1 on L is implicit (recallthat we can in principle trade f for L). This free energy is divergent sincein the large r limit the argument of the integral goes as ∼ r2. However, inorder to find the energetically favored configuration, we are only interestedin the difference between the free energies of two solutions, which is always875.3. Double monolayers with a magnetic field0.5 1.0 1.5 2.0 f0. 5.7: The separation of the monolayers, L, is plotted on the vertical axisand the force parameter f is plotted on the horizontal axis. The branch indicatedby the blue line is for the constant connected (type 2) solution. (It is a graph ofequation (5.22).) The green line is for the r-dependent connected (type 4) solution.finite. We then choose the free energy of the unconnected (f = 0) constantψ = pi/2 solution, type 1, as the reference free energy (zero level), so thatany other (finite) free energy can be defined as∆F1(ψ; f) = F1(ψ; f)−F1(ψ = pi/2; f = 0) =∫ ∞r0dr((1 + r4)sin4 ψr2√1 + r2ψ′2√r4 (1 + r4) sin4 ψ − f2−√1 + r4)− r0 2F1(−12,14;54;−r04). (5.35)where the last term is a constant that keeps into account that the ψ =pi/2 disconnected solution reaches the Poincare´ horizon, whereas the othersolutions do not. It turns out that, in this particular case where the D5brane and the anti-D5 brane are both charge neutral, the solutions of type 1and 4 always have a higher free energy than solutions 2 and 3. By means ofnumerical computations we obtain for the free energy ∆F1 of the solutions2, 3 and 4, the behaviours depicted in Figure 5.8. This shows that thedominant configuration is the connected one with an inter-layer condensatefor small brane separation L and the disconnected one, with only an intra-layer condensate, for large L. The first order transition between the two885.4. Double monolayer with a magnetic field and a charge-balanced chemical potentialphases takes place at L ' 1.357 in agreement with the value quoted inreference [94].0.5 1.0 1.5 2.0 2.5-0.8-0.6-0.4-0.2L∆F1Figure 5.8: Double monolayer in a magnetic field, where each monolayer is chargeneutral. The regularized free energy ∆F1 is plotted on the vertical axis, and theinter-layer separation L (in units of 1/√b), which is plotted on the horizontal axis.The blue line corresponds to the connected solution (type 2), the red line to theunconnected solution (type 3) and the green line to the connected r-dependentsolutions (type 4). All solutions are regulated by subtracting the free energy of theconstant unconnected solution of type 1. The latter is the black line at the top ofthe diagram. The type 1 and type 4 solutions exist but they never have the lowestenergy. For large L, the type 3 solution is preferred and small L the type 2 solutionis more stable. This reproduces results quoted in reference [94].5.4 Double monolayer with a magnetic field anda charge-balanced chemical potentialWe shall now study the possible configurations for the D5-anti D5 probebranes in the AdS5 × S5 background, with a magnetic field and a chemicalpotential. The chemical potentials are balanced in such a way that the895.4. Double monolayer with a magnetic field and a charge-balanced chemical potentialchemical potential on one monolayer induces a density of electrons and thechemical potential on the other monolayer induces a density of holes whichhas identical magnitude to the density of electrons. Moreover, the chemicalpotentials are exactly balanced so that the density of electrons and thedensity of holes in the respective monolayers are exactly equal. Due to theparticle-hole symmetry of the quantum field theory, it is sufficient that thechemical potentials have identical magnitudes. The parameters that we keepfixed in our analysis are the magnetic field b, the monolayer separation Land the chemical potential µ.In order to derive the allowed configurations we have to solve equa-tion (5.14) for ψ as well as equation (5.12) for the gauge potential a0 andequation (5.13) for z. In practice the difficult part is to find all the solutionsof the equation of motion for ψ, which is a non-linear ordinary differentialequation. Once one has a solution for ψ it is straightforward to build thecorresponding solutions for z and a0, simply by plugging the solution for ψinto the equations (5.12)-(5.13) and integrating them.It should be noted that any solution of the equation (5.12) for the gaugepotential a0(r) always has an ambiguity in that a0(r)+constant is also a so-lution. The constant is fixed by remembering that a0(r) is the time compo-nent of a vector field and it should therefore vanish at the Poincare´ horizon.When the charge goes to zero, a0 =constant is the only solution of equation(5.12) and this condition puts the constant to zero. Of course, this is inline with particle-hole symmetry which tells us that the state with chemicalpotential set equal to zero has equal numbers of particles and holes. Theresults of the previous section, where µ and q were equal to zero, care aspecial case of what we will derive below.5.4.1 Solutions for q 6= 0Now we consider the configurations with a charge density different fromzero. The differential equation for ψ in this case isrψ′′ + ψ′1 + r2ψ′2−ψ′ (f2 + q2r4 + r4 (1 + 3r4) sin4 ψ)− 2r3 (1 + r4) sin3 ψ cosψf2 − q2r4 − r4 (1 + r4) sin4 ψ = 0 .(5.36)As usual, we shall look for solutions with c1 = 0 in equation (5.5). Wecan again distinguish four types of solutions according to the classification oftable 5.2. The main difference between the solutions summarized in table 5.2and those in table 1 are that the type 3 solution now has a black hole, ratherthan a Minkowski embedding. This is a result of the fact that, as explained905.4. Double monolayer with a magnetic field and a charge-balanced chemical potentialin section 1, the world-volume of a D5 brane that carries electric chargedensity must necessarily reach the Poincare´ horizon if it does not join withthe anti-D5 brane. The latter, where it reaches the Poincare´ horizon, is anun-gapped state and it must be so even when there is an intra-layer excitoncondensate. It is, however, incompatible with an inter-layer condensate.f = 0 f 6= 0c2 = 0Type 1 Type 2unconnected, ψ = pi/2 connected, ψ = pi/2BH, chiral symm. interc2 6= 0Type 3 Type 4unconnected, r-dependent ψ connected, r-dependent ψBH, intra intra/interTable 5.2: Types of possible solutions for q 6= 0.Type 1 solutions are trivial both in ψ and z (they are both constants).They correspond to two parallel black hole (BH) embeddings for the D5and the anti-D5. This configuration is the chiral symmetric one. In type 2solutions the chiral symmetry is broken by the inter-layer condensate (f 6=0): In this case the branes have non flat profiles in the z direction. Solutionsof type 3 and 4 are r-dependent and consequently are the really non-trivialones to find. Type 3 solutions have non-zero expectation value of the intra-layer condensate and they can be only black hole embeddings, this is themost significant difference with the zero charge case. Type 4 solutions breakchiral symmetry in both the inter- and intra-layer channel. For type 2 and4 solutions the D5 and the anti-D5 world-volumes have to join smoothly ata finite r = r0.Now we look for the non-trivial solutions of equation (5.36). We startconsidering the solutions of type 4. We can build such a solution requiringthat the D5 profile smoothly joins at some given r0 the corresponding anti-D5 profile. The condition that has to be satisfied in order to have a smoothsolution for the connected D5/anti-D5 world-volumes is z′(r0)→∞ which,from (5.13) (with b scaled to 1), corresponds to the conditionf2 − r40[q2 +(1 + r40)sin4 ψ(r0)]= 0 . (5.37)915.4. Double monolayer with a magnetic field and a charge-balanced chemical potentialFrom this we can determine the boundary value ψ(r0)ψ(r0) = arcsin(4√f2 − q2r40r40(1 + r40)) . (5.38)Note that the request that 0 ≤ sinψ(r0) ≤ 1 fixes both a lower and an upperbound on r0r0,min(f, q) =4√√(1 + q2)2 + 4f2 − 1− q24√2, r0,max(f, q) =√fq.Using (5.32) we can derive from the equation of motion (5.36) the conditionon ψ′(r0), which readsψ′(r0) =(r40 + 1) (f2 − q2r40)√√ r40+1f2−q2r20− 1r20f2(2r40 + 1)− q2r80 (5.39)We can then build such solutions imposing the conditions (5.38) and (5.39)at r0, where r0 is the modulus. With the usual shooting technique we thenlook for solutions that also have the desired behavior at infinity, i.e. thosethat match the boundary conditions (5.5) with c1 = 0. It turns out that inthe presence of a charge density there are solutions of type 4 for any valueof q, in this case, however, these solutions will play an important role in thephase diagram.Next we consider the solutions of type 3. These solutions can be inprinciple either BH or Minkowski embeddings. However when there is acharge density different from zero only BH embeddings are allowed. A chargedensity on the D5 world-volume is indeed provided by fundamental stringsstretched between the D5 and the Poincare´ horizon. These strings have atension that is always greater than the D5 brane tension and thus they pullthe D5 down to the Poincare´ horizon [104]. For this reason when q 6= 0 theonly disconnected solutions we will look for are BH embedding. Solutionsof this kind with c2 6= 0 can be built numerically along the lines of ref. [92].Note that because of the equation of motion they must necessarily haveψ(0) = 026.26Actually also the condition ψ(0) = pi/2 is allowed by the equation of motion, but thiswould correspond to the constant solution with c2 = 0, namely the type 1 solution.925.4. Double monolayer with a magnetic field and a charge-balanced chemical potential5.4.2 Separation and free energyThe brane separation is given in (5.20) and for the solutions of type 2 and 4is plotted in fig. 5.9 for q = 0.01, the blue line gives the curve (5.22), keepinginto account that also r0 is a function of f through equation (5.21). For ther-dependent solution instead r0 is defined as a function of f by equation(5.37), once the solution ψ(r) is known numerically. The r-dependent con-nected solution, green line, has two branches one in which L decreases withincreasing f and the other one in which L increases as f increases. It is clearfrom the picture that when q → 0 one of the branches of the green solutiondisappears and fig. 5.9 will become identical to fig. 5.7.1 2 3 4 f0. 5.9: The separation of the monolayers, L, is plotted on the vertical axisand the force parameter f is plotted on the horizontal axis, in the case wherethe monolayers have charge densities and q = 0.01. The branch indicated by theblue line is for the constant connected (type 2) solution. The green line is for ther-dependent connected (type 4) solution.Once we have determined all the possible solutions, it is necessary tostudy which configuration is energetically favored. We shall compare thefree energy of the solutions at fixed values of L and µ, since this is the mostnatural experimental condition for the double monolayer system. Thus theright quantity to define the free energy is the action (5.25), which after the935.4. Double monolayer with a magnetic field and a charge-balanced chemical potentialrescaling (5.28) is given byF1[L, µ] =∫ ∞r0dr(1 + r4)sin4 ψr2√1 + r2ψ′2√r4 (1 + r4) sin4 ψ + q2r4 − f2(5.40)As usual we regularize the divergence in the free energy by considering thedifference of free energies of pairs of solutions, which is really what we areinterested in. So we define a regularized free energy ∆F1 by subtracting toeach free energy that of the unconnected (f = 0) constant ψ = pi/2 solution,∆F1(ψ; f, q) ≡ F1(ψ; f, q)−F1(ψ = pi/2; f = 0, qˆ). (5.41)As we already noticed, the free energy (5.40) and consequently ∆F1 areimplicit functions of L and µ, via f and q, which are the parameters thatwe really have under control in the calculations. Thus when computing theregularized free energy ∆F1 we have to make sure that the two solutionsinvolved have the same chemical potential.27 This is the reason why in thedefinition of ∆F1 (5.41) we subtract the free energy of two solutions withdifferent values of q: the qˆ in (5.41) is in fact the value of the charge such thatthe chemical potential of the regulating solution (ψ = pi/2; f = 0) equalsthat of the solution we are considering (ψ; f, q). To be more specific, fora solution with chemical potential µ, which can be computed numericallythrough (5.23) (or through (5.24) for the ψ = pi/2 case), qˆ must satisfyµ(ψ = pi/2; f = 0, qˆ) ≡ 4Γ(54)2qˆ√pi(1 + qˆ2)1/4= µand therefore it is given byqˆ =√J +√J (J + 4)2, J ≡ pi2µ4(2Γ(54))8 .For the type 2 solution the regularized free energy density can be com-27In principle we also have to make sure that the two solutions have the same L. How-ever this is not necessary in practice, since in ∆F1 we use as reference free energy thatof an unconnected solution, which therefore is completely degenerate in L. Indeed theunconnected configuration is given just by two copies of the single D5 brane solution withzero force between them, which can then be placed at any distance L.945.4. Double monolayer with a magnetic field and a charge-balanced chemical potentialputed analytically. Reintroducing back the magnetic filed, it reads∆F1[L, µ] =∫ ∞r0dr(b2 + r4) r2r20√(r4 − r40) (f2 + r4r40) − 1√b2 + qˆ2 + r4−∫ r00drb2 + r4√b2 + qˆ2 + r4=√piΓ(−34)16r04√b2 + qˆ2Γ(34) [ 4√b2 + qˆ2(2r40 2F1(−34 , 12; 34;−f2r80)−3 (b2 + r40) 2F1(14 , 12; 34;−f2r80))+√2r0(2b2 − qˆ2)](5.42)where r0 is given in (5.21).A comparison of the free energies of the various solutions for L = 1.5and L = 5 is given in figure 5.10. The chirally symmetric solution wouldbe along the µ-axis since it is the solution we used to regularize all thefree energies. It always has a higher free energy, consequently, the chirallysymmetric phase is always metastable.0.5 1.0 1.5 2.0-0.7-0.6-0.5-0.4-0.3-0.2-0.10.0∆F1µL = 1.50.5 1.0 1.5 2.0-0.5-0.4-0.3-0.2-0.10.0∆F1µL = 5Figure 5.10: Plots of the free energies as a function of the chemical potential:type 2 (blue line), type 3 (red-line) and type 4 (green line) solutions forL = 1.5 and L = 5.955.4. Double monolayer with a magnetic field and a charge-balanced chemical potential5.4.3 Phase diagramsWorking on a series of constant L slices we are then able to draw the phasediagram (µ,L) for the system. For the reader convenience we reproducethe phase diagram that we showed in the introduction in fig. 5.11 (here thelabels are rescaled however). We see that the dominant phases are three:• The connected configuration with c2 = 0 (type 2 solution) where theflavor symmetry is broken by the inter-layer condensate (blue area);• The connected configuration with c2 6= 0 and f 6= 0 (type 4 solutionwith q 6= 0) where the chiral symmetry is broken by the intra-layercondensate and the flavor symmetry is broken by the inter-layer con-densates (green area);• The unconnected Minkowski embedding configuration with c2 6= 0(type 3 solution with q = 0) where the chiral symmetry is broken bythe intra-layer condensate (red area);Note that in all these three phases chiral symmetry is broken.0 0.5 1 1.5 2 2.5 3024681st order2ndorder2ndorderf = 0c 6= 0q = 0Minkf 6= 0c 6= 0q 6= 0f 6= 0 c = 0 q 6= 01.357µLFigure 5.11: Phase diagram of the D3-probe-D5 branes system with balancedcharge densities. Layer separation is plotted on the vertical axis and chemicalpotential µ for electrons in one monolayer and holes in the other monolayer isplotted on the horizontal axis. The units are the same as in figure 5.1.965.4. Double monolayer with a magnetic field and a charge-balanced chemical potential0 0.5 1 1.5 2 2.5 3 3.510−1100101102103intraq = 0intra/interinterµLFigure 5.12: Phase diagram for large separation between the layers.As expected, for small enough L the connected configuration is the dom-inant one. We note that for L . 1.357, which, as we already pointed out,is the critical value for L in the zero-chemical potential case, the connectedconfiguration is always preferred for any value of µ. When 1.357 . L . 1.7the system faces a second order phase transition from the unconnectedMinkowski embedding phase – favored for small values of µ – to the con-nected phase – favored at higher values of µ. When L & 1.7 as the chemicalpotential varies the system undergoes two phase transitions: The first hap-pens at µ ' 0.76 and it is a second order transition from the unconnectedMinkowski embedding phase to the connected phase with both condensates.Increasing further the chemical potential the system switches to the con-nected phase with only an intra-layer condensate again via a second ordertransition.Therefore it is important to stress that, with a charge density, a phasewith coexisting inter-layer and intra-layer condensates can be the energeti-cally preferred state. Indeed, it is the energetically favored solution in thegreen area and corresponds to states of the double monolayer with both theinter-layer and intra-layer condensates.The behavior of the system at large separation between the layers is givenin fig. 5.12. For L→∞ the phase transition line between the green and theblue area approaches a vertical asymptote at µ ' 2.9. The connected solu-975.5. Double monolayers with un-matched charge densitiestions for L → ∞ become the corresponding, r-dependent or r-independentdisconnected solutions. Thus for an infinite distance between the layers werecover exactly the behavior of a single layer [99] where at µ ' 2.9 thesystem undergoes a BKT transition between the intra-layer BH embeddingphase to the chiral symmetric one [102].It is interesting to consider also the phase diagram in terms of the braneseparation and charge density fig. 5.14. In this case the relevant free en-ergy function that has to be considered is the Legendre transformation ofthe action with respect to q, namely the Routhian F2[L, q] defined in equa-tion (5.26). For the regularization of the free energy we choose proceed inanalogy as before: For each solution of given q and L we subtract the freeenergy of the constant disconnected (type 1) solution with the same chargeq, obtaining the following regularized free energy∆F2[L, q] ≡ F2[L, q]−F2(ψ = pi/2; f = 0)[q]. (5.43)In fig. 5.14 the phase represented by the red region in fig. 5.11, is just givenby a line along the q = 0 axis. By computing the explicit form of the freeenergies as function of the brane separation L, it is possible to see in factthat the r-dependent connected solution has two branches. These branchesreflect the fact that also the separation L has two branches as a function off , as illustrated in fig. 5.9. In the limit q → 0 one of these branches tends tooverlap to the r-dependent disconnected solution and for q = 0 disappears.This is illustrated in fig. 5.13 where one can see that in the q → 0 limit thefree energy difference as a function of L goes back to that represented infig. Double monolayers with un-matched chargedensitiesWe now consider a more general system of two coincident D5 branes andtwo coincident anti-D5 branes, with total charges Q = q1 + q2 > 0 and −Q¯where Q¯ = q3 + q4 > 0. Then, unlike before, this corresponds to a doublemonolayer with unpaired charge on the two layers. For such a system weare interested in determining the most favored configuration, i.e. to find outhow the charges Q and Q¯ distribute among the branes and which types ofsolutions give rise to the least free energy for the whole system. Since theparameter that we take under control is the charge, and not the chemicalpotential, we shall use the free energy F2, defined in equation (5.17), inorder to compare the different solutions.985.5. Double monolayers with un-matched charge densities0.5 1.0 1.5 2.0 2.5-0.8-0.6-0.4-0.2L∆F2Figure 5.13: Free energy difference ∆F2 as a function of L for q = 0.01, the twobranches of the r-dependent connected solution, green line, tend to become just theone of fig. 5.8.What we keep fixed in this setup are the overall charges Q and Q¯ inthe two layers, while we let the charge on each brane vary: Namely the qivary with the constraints that Q = q1 + q2 and Q¯ = q3 + q4 are fixed. Thenwe want to compare configurations with different values for the charges qion the single branes. For this reason we must choose a regularization ofthe free energy that does not depend on the charge on the single brane,and clearly the one that we used in the previous section is not suitable.The most simple choice of such a regularization consists in subtracting tothe integrand of the free energy only its divergent part in the large r limit,which is r2. We denote this regularized free energy as ∆F2,r.Without loss of generality we suppose that Q > Q¯. Then for simplicitywe fix the values of the charges to Q = 0.15 and Q¯ = 0.1 and the separationbetween the layers to L = 1. There are two possible cases.(i) A configuration in which the D5 brane with charge q1 is describedby a black hole embedding whereas the D5 brane with charge q2 isconnected with the anti-D5 brane with charge q3, so that q2 = q3.995.5. Double monolayers with un-matched charge densities0 0.5 1 1.5 2 2.5 301234intra/interinterintraqLFigure 5.14: Phase diagram in terms of the brane separation L and the chargedensity q.Then we haveq2 = q3 = Q¯− q4 , q1 = Q− q2 = Q− Q¯+ q4The free energy of this solution as a function of the parameter q4 isgiven by the plot in fig. 5.16 for Q = 0.15 and Q¯ = 0.1.From fig. 5.16 it is clear that the lowest free energy is achieved whenq4 = 0 which corresponds to the fact that one anti-D5 brane is repre-sented by a Minkowski embedding.(ii) Then we can consider the configuration in which all the branes aredisconnected. In this caseq1 = Q− q2 , q3 = Q¯− q4In fig. 5.17 we give the free energy of the D5 branes, and of the theanti-D5 brane. It is clear from fig. 5.17 that the lowest free energyconfiguration is when both branes on the same layer have the samecharge. The free energy of the complete configuration will be then thesum of the free energy of the D5 and of the anti-D5 layers each withcharge evenly distributed over the branes. For the case considered we1005.5. Double monolayers with un-matched charge densitiesr =∞r = 0q2 = q3 = Q¯q1 = Q− Q¯q4 = 0q1 q2 −q3 −q4Q −Q¯Figure 5.15: Energetically favored solution for unpaired charges when Q > Q¯.0.02 0.04 0.06 0.08 0.102.882.902.922.942.962.983.00∆F2,rq4Figure 5.16: Free energy of the solutions when one brane is disconnected and twobranes are connectedobtain a free energy ∆F2,r ' 3.26, which however is higher then thefree energy of the configuration (i).In the special case in which Q and Q¯ are equal, e.g. Q = Q¯ = 0.15, thereare four possible configurations: Either the branes are all disconnected, or1015.5. Double monolayers with un-matched charge densities0.02 0.04 0.06 0.08 0.10 0.12 0.141.65151.65201.65251.65301.65351.6540∆F2,rq2D50.02 0.04 0.06 0.08 0.101.61161.61181.61201.61221.61241.6126∆F2,rq4D5Figure 5.17: Free energy of the solutions when all the branes are disconnected:The energetically favored solution is when they have the same charge.the two pairs branes are both connected, or a brane and an anti-brane areconnected and the other are black hole embeddings, or, finally, a brane andan anti-brane are connected and have all the charges Q and Q¯, so the restare Minkowski embeddings.When they are all disconnected the physical situation is described initem ii and the energetically favored solution is that with the same charge.When they are all connected the configuration has the following charges.q1 = Q− q2 , q3 = q1 , q2 = q4 , q3 = Q¯− q4 = Q− q2Fig. 5.18 shows the free energy of the system of branes and anti-branes when0.02 0.04 0.06 0.08 0.10 0.12 0.142.5642.5662.5682.5702.5722.5742.576∆F2,rq2Figure 5.18: Free energy of the solutions when all the branes are connected, theenergetically favored solution is when the charge is distributed evenly between thebranes and anti-branes.they are all connected, clearly the energetically most favored solution is that1025.6. Discussionwith the charge distributed evenly. This solution has a lower free energy withrespect to the one of all disconnected branes and charges distributed evenlyand also with respect to a solution with one brane anti-brane connectedsystem and two black hole embeddings. Since the connected solutions forL . 1.357 and q = 0 are always favored with respect to the unconnectedones, also the solution with two Minkowski embeddings and one connectedsolution has higher free energy with respect to the one with two connectedpairs.Summarizing for Q = Q¯ the energetically favored solution is the onewith two connected pairs and all the charges are evenly distributed q1 =q2 = q3 = q4 = Q/2.5.6 DiscussionWe have summarized the results of our investigations in section 1. Here,we note that there are many problems that are left for further work. Forexample, in analogy with the computations in reference [65] which useda non-relativistic Coulomb potential, it would be interesting to study thedouble monolayer quantum field theory model that we have examined here,but at weak coupling, in perturbation theory. At weak coupling, and inthe absence of magnetic field or charge density, an individual monolayer isa defect conformal field theory. The double monolayer which has nestedfermi surfaces should have an instability to pairing. It would be interestingto understand this instability better. What we expect to find is an inter-layer condensate which forms in the perfect system at weak coupling andgives the spectrum a charge gap. The condensate would break conformalsymmetry and it would be interesting to understand how it behaves underrenormalization.The spontaneous nesting deserves further study. It would be interestingto find a phase diagram for it to, for example, understand how large a chargemiss-match can be.Everything that we have done is at zero temperature. Of course, thetemperature dependence of various quantities could be of interest and itwould be interesting (and straightforward) to study this aspect of the model.It would be interesting to check whether the qualitative features whichwe have described could be used to find a bottom-up holographic model ofdouble monolayers, perhaps on the lines of the one constructed by Sonner[93].103Chapter 6Holographic D3-probe-D7Model of a Double LayerDirac SemimetalsOn the rumbling bus, somewhere,I saw a white notice board on the street.It is written with clear letters thatYou are leaving away the county of Mujin.Good bye.I was heavily ashamed.- Record of a Journey to Mujin by Seung-Ok KimThe possibility that an inter-layer exciton condensate can form in a dou-ble monolayer of two-dimensional electron gases has been of interest for along time [64]. A double monolayer contains two layers, each containing anelectron gas, separated by an insulator so that electrons cannot be trans-ferred between the layers. Electrons and holes in the two layers can stillinteract via the Coulomb interaction. The exciton which would condenseis a bound state of an electron in one layer and a hole in the other layer.This idea has recently seen a revival with some theoretical computationsfor emergent relativistic systems such as graphene or some topological in-sulators which suggested that a condensate could form at relatively hightemperatures, even at room temperature [15]. A room temperature super-fluid would have applications in electronic devices where proposals includeultra-fast switches and dispersionless field-effect transistors [112].An exciton condensate might be more readily achievable in a doublemonolayer with relativistic electrons due to particle-hole symmetry and thepossibility of engineering nested Fermi surfaces of electrons in one layer andthe holes in the other layer. This nesting would enhance the effects of theattractive Coulomb interaction between an electron and a hole. Even at veryweak coupling, it can be shown to produce an instability to exciton conden-104Chapter 6. Holographic D3-probe-D7 Model of a Double Layer Dirac Semimetalssation [65]. However, in spite of this optimism, an inter-layer condensatehas yet to be observed in a relativistic material, even in experiments usingclean graphene sheets with separations down to the nanometer scale [118].The difficulty with theoretical computations, where the Coulomb interac-tion is strong, is the necessity of ad-hoc inclusion of screening, to which theproperties of the strongly coupled system have been argued to be sensitive[120].In this chapter we will study a model of a double monolayer of relativistictwo-dimensional electron gases. This model has a known AdS/CFT dualwhich is easy to study and it can be solved exactly in the strong couplinglimit. We shall learn that, in this model. the only condensates which formare excitons, bound states of electrons with holes in the same layer (intra-layer) or bound states of electrons in one layer with holes in the other layer(inter-layer). Moreover, even though at very strong coupling, the idea of aFermi surface loses its meaning, we find that the tendency to form an inter-layer condensate is indeed greatly enhanced by the charge balance which, atweak coupling, would give nested particle and hole Fermi surfaces. We shallsee that, in the strong coupling limit, and when the charges are balanced, aninter-layer condensate can form for any separation of the layers. As well asthe inter-layer condensate, such a strong interaction will also form an intra-layer condensate. We find that a mixture of the two condensates is favouredfor small charge densities and larger layer separations. For sufficiently largecharge densities, on the other hand, the only condensate is the inter-layercondensate. These results for charge balanced layers are summarized infigure 6.1. When the charges are not balanced, so that at weak couplingthe Fermi surfaces would not be nested, no inter-layer condensate forms,regardless of the layer separation. This dramatic difference is similar to andeven sharper than what is seen at weak coupling [65] where condensationoccurs in only a narrow window of densities near nesting.However, even in the non-nested case, we can find a novel symmetrybreaking mechanism where an inter-layer condensate can form. If each elec-tron gas contains more than one species of relativistic electrons (for example,graphene has four species of massless Dirac electrons and some topologi-cal insulators have two species), the electric charge can redistribute itselfamongst the species to spontaneously nest one or more pairs of Fermi sur-faces, with the unbalanced charge taken up by the other electron species.Then the energy is lowered by formation of a condensate of the nested elec-trons, the others remaining un-condensed. To our knowledge, this possibilityhas not been studied before. The result is a new kind of symmetry break-ing where Fermi surfaces nest spontaneously and break some of the internal105Chapter 6. Holographic D3-probe-D7 Model of a Double Layer Dirac Semimetals0 0.02 0.04 0.06 0.08 0.10246intraintra/interinterchiral symm.intraqLFigure 6.1: (color online) Phase diagram of the charge balanced double monolayer(exactly nested Fermi surfaces). The vertical axis is layer separation L in units ofthe inverse ultraviolet cutoff, R. The horizontal axis is the charge density q in unitsof R−2. The green region has both inter- and intra-layer condensates. The blueregion has only an inter-layer condensate. The red region has only an intra-layercondensate. The white region has no condensates.symmetry of the electron gas in each layer. We demonstrate that, for someexamples of the charge density, this type of condensate indeed exists as thelowest energy solution.The model which we shall consider is a defect quantum field theory con-sisting of a pair of parallel, infinite, planar 2+1-dimensional defects in 3+1-dimensional Minkowski space and separated by a distance L. The defects areeach inhabited by NF species of relativistic massless Dirac Fermions. TheFermions interact by exchanging massless quanta of maximally supersym-metric Yang-Mills theory which inhabits the surrounding 3+1-dimensionalbulk. In the absence of the defects, the latter would be a conformal fieldtheory. The interactions which it mediates have a 1/r fall-off, similar to theCoulomb interaction and, in the large N planar limit which we will con-sider, like the Coulomb force, the electron-hole interaction is attractive inall channels.106Chapter 6. Holographic D3-probe-D7 Model of a Double Layer Dirac SemimetalsThe field theory action isS =∫d4x1g2YMTr[−12FµνFµν −6∑b=1DµΦbDµΦb + . . .]+∫d3x2∑a=1NF∑i=1ψ¯ai[iγµ∂µ + γµAµ + Φ6]ψai (6.1)The first term is the action of N = 4 supersymmetric Yang-Mills theorywhere Aµ is the Yang-Mills gauge field and Φ6 is one of the scalar fields andthe second term is the action of the defect Fermions. In the second term, thesubscript a labels the defects and i the Fermion species. The action includesall of the marginal operators which are compatible with the symmetries. Ithas a global U(1) symmetry which we associate with electric charge.The defect field theory (6.1) is already interesting with one layer. Itis thought to have a conformally symmetric weak coupling phase for 0 ≤λ ≤ λc. When λ > λc, chiral symmetry is broken by an intra-layer excitoncondensate [69]. Near the critical point, the order parameter is thoughtto scale as〈ψ¯1iψ1i〉 ∼ Λ2 exp (−b/√λ− λc) where Λ is an ultraviolet (UV)cutoff. In the strong coupling phase, the condensate and therefore the chargegap are finite only when the coupling is tuned to be close to its critical value.The holographic construction examines this theory in the strong couplinglimit, where λ λc. In that limit, it is cutoff dependent and it can only bedefined by introducing a systematic UV cutoff. We will find a string-inspiredway to do this, tantamount to defining the model (6.1) as a limit of the IIBstring theory which is finite and resolves the singularities. It will allow us tostudy the strong coupling limit using the string theory dual of this system.When there are two monolayers, the field theory (6.1) can also have aninter-layer exciton condensate with order parameter〈ψ¯1iψ2i〉. The results ofreference [65] suggest that, with balanced charge densities and nested Fermisurfaces, the inter-layer condensate occurs even for very weak coupling. Notmuch is known as to how it would behave at strong coupling. It is the strongcoupling limit of this model which we will now solve using its string theorydual.The string theory dual of the defect field theory is the D3-probe-D7 branesystem of IIB string theory[70]. A monolayer is a single stack of NF D7 co-incident branes. A double monolayer has two parallel stacks, one of NF D7branes and another of NF anti-D7 branes separabilityted by a distance L.In both cases, the D7 brane stacks overlap N >> NF coincident D3 branes.With the appropriate orientation, the lowest energy states of open strings107Chapter 6. Holographic D3-probe-D7 Model of a Double Layer Dirac Semimetalswhich connect the D3 to the D7 branes are massless two-component relativis-tic Fermions that propagate on 2+1-dimensions and are the defect fields in(6.1). In the large N and strong coupling limits, the D3 branes are replacedby the AdS5 × S5 background and solving the theory reduces to extrem-izing the classical Born-Infeld action S ∼ NFTD7∫d8σ√−det(γ + 2piα′F )for the D7 brane embedded with world-volume gauge field strength F andmetric γab in AdS5 × S5. However, there is an immediate problem withthis setup. Any D7 brane geometry which approaches the appropriate D7brane boundary conditions at the boundary of AdS5 is unstable. This isa reflection of the fact that the strong coupling limit of the quantum fieldtheory on a single D7 brane is not conformally symmetric. We shall use asuggestion by Davis [44] who regulated the D7 brane by embedding itin the extremal black D3 brane geometry, with metricds2R2=r2(−dt2 + dx2 + dy2 + dz2)√1 +R4r4+√1 +R4r4(dr2r2+ dψ2 + sin2 ψ5∑i=1(dθi)2)(6.2)where∑5i=1(θi)2 = 1. and R4 = λα′2. The asymptotic, large r limit of thismetric is 10-dimensional Minkowski space. It has a horizon at r = 0. In thenear horizon limit, which produces the IIB string on AdS5 × S5, rR  1,it approaches the Poincare´ patch of AdS5 × S5. Since R contains the stringscale α′, 1/R can be regarded as a (UV) cutoff.The D7 and anti-D7 world-volumes are almost entirely determined bysymmetry. They have 2+1-dimensional Poincare´ invariance and wrap (t, x, y).The model (6.1) has an SO(5) R-symmetry. The D7’s must therefore wrap(θ1, . . . , θ5) to form an S4. For the remaining world-volume coordinate, weuse the radius r in (6.2). The dynamical variables are then ψ(r) and thepositions z1(r) and z2(r) of D7 and anti-D7, which by symmetry can onlybe functions of r. ψ = pi2 is a point of higher symmetry, corresponding toparity in the defect field theory with massless Fermions. ψ(r) = pi2 +cr2+ . . .is required to approach pi2 at r → ∞ and, if it becomes r-dependent at all(c 6= 0), parity is broken by an intra-layer condensate. Parity can be re-stored if pairs of branes have condensates of opposite signs. This wouldbreak flavour symmetry when NF is even, U(NF )→ U(NF /2)× U(NF /2).Whether this sort of flavour symmetry breaking or parity and time reversalbreaking takes place is an interesting dynamical question which will be stud-ied elsewhere. Finally, it will turn out that, either z1,2(r) are constants, or108Chapter 6. Holographic D3-probe-D7 Model of a Double Layer Dirac Semimetalsthe D7 and anti-D7 meet and smoothly join together at a minimum radius,r0. Asymptotically, z1,2(r) = ±L/2∓Rr40/r4 + . . ..We have performed numerical computations to determine the lowest en-ergy embeddings of the D7 (and anti-D7) branes as a function of the chargedensity (q) and the brane-anti-brane separation L. In the following weoutline the results of these computations. The formalism for studying theembeddings of the probe D branes is already well-known in the literatureand we refer the reader there for details. Examples for double monolayerscan be found in references [54, 91, 93–96, 100]When we suspend a single D7 brane in the black D3 brane metric (6.2),we find that the lowest energy solution truncates before it reaches the hori-zon. This is called a “Minkowski embedding”. The function ψ(r) movesfrom ψ = pi2 at r ∼ ∞ to ψ = 0 or ψ = pi at the r where the brane pinchesoff. The S4 which the world-volume wraps shrinks to a point there and thiscollapsing cycle is what makes the truncation smooth. This brane geometryis interpreted as a charge-gapped state. The lowest energy charged exci-tation is a fundamental string which would be suspended between the D7brane and the horizon. In this case, that string has a minimum length andtherefore a mass gap.We can introduce a charge density q on the single monolayer. Whenthe D7 brane carries a charge density, its world volume must necessarilyreach the horizon. This is called a “black hole embedding”. Charge in thequantum field theory corresponds to D7 brane world-volume electric fieldEr ∼ q. This hedgehog-like electric field points outward from the centreof the brane. The radial lines of flux of the electric field can only end ifthere are sources. Such sources would be fundamental strings, suspendedbetween the D7 brane and the horizon. However, the strings have a largertension than the D7 brane and they pull the the D7 brane to the horizonresulting in a gapless state. This is confirmed by numerical solutions of theembedding equation of a single brane and, indeed, we find that the S4 whichis wrapped by the world volume shrinks to a point as it enters the horizon.This state no longer has a charge gap. Even in the absence of a charge gap,we find that, for small charge densities, there is still an intra-layer excitoncondensate. Our numerical studies show that it persists up to a quantumphase transition at a critical density qcrit. ≈ 0.0377/R2. At densities greaterthan the critical one, ψ = pi2 , is a constant.Now, consider the double monolayer with D7 and anti-D7 branes. A D7-anti-D7 pair of branes would tend to annihilate. We prevent this annihilationby requiring that they be separated by a distance L as they approach theboundary at r → ∞. When their world volume enters the bulk, they can109Chapter 6. Holographic D3-probe-D7 Model of a Double Layer Dirac Semimetalsstill come together and annihilate – their world volumes fusing together at aminimal radius r0. This competes with the tendency of a monolayer braneto pinch off at some radius. Indeed, when the charge density is zero, wesee both behaviours. When the stacks of branes are near enough, that is,L < Lc ' 2.31R is small enough, they join. This state has an inter-layercondensate. When they are farther apart, they remain un-joined. Instead,they pinch off to form Minkowski embedding, corresponding to a state withintra-layer condensates.When we introduce balanced charges q and −q on the D7 and anti-D7,respectively, there are four modes of behaviour which are summarized intable 6.1. Each of these behaviours occurs in the phase diagram in figure6.1. Type 1 solutions are maximally symmetric with ψ = pi2 and z1,2 =z2 − z1 = L, const. z2(r)− z1(r)→ 0 at r0c = 0Type 1 Type 2un-joined, ψ = pi2 joined, ψ =pi2BH, no condensate interc 6= 0Type 3 Type 4un-joined, ψ(r) r-dependent joined, ψ(r) r-dependentMink (q = 0) intra intra+interBH (q 6= 0) intra only when q 6= 0Table 6.1: Types of possible solutions for the balanced charge (q,−q) case,where (Mink,BH) stand for (Minkowski,black-hole) embeddings.±L/2. They occur in the white region of figure 6.1. They have no excitoncondensates at all. Type 3 solutions occur in the red region. They have ψ(r)a nontrivial function, but z1,2 = ±L/2. The branes do not join. They areMinkowski embeddings when q = 0 and black hole embeddings when q 6= 0.Type 3 has an intra-layer exciton condensate only. There is a quantum phasetransition between type 1 and type 3 solutions at qc = 0.0377. Both type1 and type 3 solutions occur only for very small layer separations, or orderthe UV cutoff. Type 2 solutions occupy the blue region. They have ψ = pi2 ,constant, z1,2(r) are nontrivial functions. The D7 and anti-D7 branes joinat a radius, r0 6= 0. The intra-layer condensate vanishes and there is a non-zero inter-layer condensate. In type 4 solutions, both ψ(r) and z1,2(r) havenontrivial profiles. The D7 and anti-D7 branes join and ψ(r) also varies110Chapter 6. Holographic D3-probe-D7 Model of a Double Layer Dirac Semimetalswith radius. This phase has both and inter- and intra-layer condensate.This solution exists only when q is nonzero and, then, only for small valuesof q. For r0 ' 0 we have q < 0.0377, when r0 grows, the allowed values of qdecrease.Consider a double monolayer with un-balanced charges, Q > 0 on theD7 and −Q¯ < 0 on the anti-D7 brane. The same argument as to why asingle charged D7 brane must have a Minkowski embedding applies and,on the face of it, it is impossible for the branes to join before they reachthe horizon. There is, however, another possibility which arises when thereare more than one species of Fermions on each brane, that is, NF > 1.In that case, one or more of the Fermion species can nest spontaneously,with the deficit of charge residing in the other species. This would breakinternal symmetry. For example, if Q > Q¯ > 0, k branes take up chargeQ¯ and the remaining NF − k take up the remainder Q − Q¯, this wouldbreak U(NF )×U(NF )→ U(NF −k)×U(k)×U(NF −k)×U(k). Then thebranes with matched charges (Q¯) would join, further breaking the symmetryU(NF − k)×U(k)×U(NF − k)×U(k)→ U(NF − k)×U(k)×U(NF − k).Then, NF−k charged D7 branes and NF−k uncharged anti-D7 branes eitherbreak parity or some of the remaining U(NF − k) × U(NF − k) symmetry.The uncharged branes must take up a Minkowski embedding. We havecomputed the energies of some of these symmetry breaking states for thecase where NF = 2. We find a range of charge densities where spontaneousnesting is energetically preferred. The implications of this idea for doublemonolayer physics is clear. The Fermion and hole densities of individualmonolayers would not necessarily have to be fine tuned in order to nest theFermi surfaces. It could happen spontaneously.The intra-layer and inter-layer condensates discussed here have not beenseen in graphene to date (with a possible exception [97]), presumably be-cause the coupling is not strong enough. Our results show that the inter-layer condensation is extremely sensitive to Fermi surface nesting, even inthe strong coupling limit. It would be interesting to better explore spon-taneous nesting, since creating favourable conditions for it could be a wayforward with graphene.111Chapter 7ConclusionP.S. please if you get a chanse put some flowerson Algernons grave in the bak yard- Flowers for Algernon by Daniel KeyesIn this thesis, we introduced why we need to study AdS/CFT correspon-dence and the key idea of the conjecture. We also showed how to apply theholographic duality to the concerned dual field theory, and presented themotivation of our project during the PhD program. I will summarize theprevious chapters by exhibiting the research procedures and how and whatwe achieved as follows.The first candidate we considered for dual model of double monolayergraphene is D3-probe-D7 model based on [37]. Because supersymmetry andconformal symmetry are completely broken for this model, the embedding isnot stable. We turn on the internal fluxes to stabilize the model, and turn onan external magnetic field as well. I led the research to seeking a condition ofstability of the double layer model and simulating and analyzing behaviorsof the model. We finally analyzed magnetic catalysis and inverse magneticcatalysis for double monolayers [54]. This is presented in the chapter 3This set-up in [54, 91] itself is very interesting, and recently it inversemagnetic catalysis research in QCD is active. However, there is a criticalweak point that the model cannot contain intra-layer EC at all for oursemimetal perspective. Thus, we leave the research for the future work.Instead, we employ an extremal black D3-brane geometry instead of AdS toimbed D7-branes [101]. It is presented in the chapter 6 and the appendix B.We also introduce an external magnetic field28 on the D3-probe-D5-branes to break supersymmetry and conformal symmetry and make fermiondefects probe branes. D3 and probe D5-branes are no more in the BPS28The dual field theory does not contain an external magnetic field. We could regardit as two form B gauge field in the supergravity background. They are equivalent in DBIaction of the probe branes.112Chapter 7. Conclusionstates, they act repulsive forces each other. The strength grows towardthe Poincare´ Killing horizon, D5-branes pinch off somewhere (Minkowskiembedding). This has a mass gap. This is the gravity dual of the mass gen-eration that accompanies intra-layer EC in a single monolayer. In the Fig.1.15, the electric external gates would change the effective charge density.It is also known that a doped bilayer graphene also has a band gap. In asingle monolayer to have a nonzero charge density, there must be a densityof fundamental strings suspended between the D5-brane and the Poincare´horizon. However, the fundamental string tension is always greater thanthe D5-brane tension and such strings would therefore pull the D5-brane tothe horizon(BH embedding). The result is a gapless state. For an infinitedistance L between the layers in the phase diagram, we recover exactly thebehavior of a single layer [102] where at µ ∼ 2.9(Asymptotic dotted line inthe Fig. 1.15) the system undergoes a BKT transition between the intra-layer BH embedding phase to the chiral symmetric one. Note that the otherdotted line is where the 1st order phase transition occurs. 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Macdonald, “Quantum Hall effects in graphene-based two-dimensional electron systems”, Nanotechnology 12, 052001(2012).125Appendix ASome Calculations forChapter 4.The Lagrangian we are intrested in isL = (1 + f2)r2√1 + r4z˙(r)2 ∓ f2.r4z˙(r) (A.1)z(r) is a cyclic variable, and the equation of the motion isddr[(1 + f2)r6z˙√1 + r4z˙(r)2∓ f2r4]. = 0 (A.2)The solution for the equationA.2 isz±(r) = ±L2∓∫ ∞rdrz˙+(r).This z± is the position of the brane to the right (+) or left (-) of z = 0, andz˙± isz˙±(r) = ± f2r4 + Pr2√(r4 − P )((1 + 2f2)r4 + P ) .P is an integration constant proportional to the pressure needed to hold thebranes with their asymptotic separation L.(1 + f2)r6z˙√1 + r4z˙(r)2∓ f2r4 = ±P (A.3)When they are not joined, they do not interact, at least in this classicallimit, and P must be zero. When they are joined, P must be nonzero andthey are joined at a minimum radius r0 = P14 and L and P are related byLP14 = 2∫ ∞1drf2r4 + 1r2√(r4 − 1)((1 + 2f2)r4 + 1)=pi3/2(3f2 2F1(14 ,12 ;34 ;1−2f2−1)+ 2F1(12 ,54 ;74 ;1−2f2−1))4√2 + 4f2Γ(34)Γ(74) (A.4)126Appendix A. Some Calculations for Chapter 4.0.5 0.6 0.7 0.8 0.9 1.0 fΡmP14Figure A.1: ρm as a function of f2.We have redefined the radial coordinate asρ =∫ ∞rdrr2√1 + r4z˙2To analyze the joined configuration, we note that in that case ρ reachesa maximumρm =(1 + f2)r0∫ 10dx√(1− x4)((1 + 2f2)− x4) (A.5)=√pi(f2 + 1)Γ(54)2F1(14 ,12 ;34 ;1−2f2−1)r0√2f2 + 1Γ(34) (A.6)=L2∫ 10dx(1+f2)√(1−x4)((1+2f2)−x4)∫ 10dx(f2+x4)√(1−x4)((1+2f2)−x4)(A.7)=3L(f2 + 1)2(2F1(12, 54; 74; 1−2f2−1)2F1(14, 12; 34; 1−2f2−1) + 3f2) (A.8)A plot of eq.(A.6) is given in Fig.A.5.The regularized energy can be computed as the difference between theon-shell actions of the joined solution and that of the disjoined one. It is127Appendix A. Some Calculations for Chapter 4.0.5 0.6 0.7 0.8 0.9 1.0 f 2-0.30-0.25-0.20-0.15-0.10-0.05Sreg P-34Figure A.2: Regularized energy, it is always negative, the joined solution hasalways lower energy.given bySreg =∫ ∞r0dr( (2f2 + 1)r6 − f2r2r40√(r4 − r40) ((2f2 + 1) r4 + r40)−√2f2 + 1r2)−∫ r00dr√2f2 + 1r2This can be computed exactlySreg = −pi3/2r30(3f2 2F1(14 ,12 ;34 ;1−2f2−1)+ 2F1(12 ,54 ;74 ;1−2f2−1))18√4f2 + 2Γ(34)2and from its plot in Fig.A.2 it seems smooth and always negative for 23/50 <f2 < 1.From these results it is simple to compute the energy as a function of L,which is simply given by the confining potentialSreg = −P6L (A.9)128Appendix BThe Phase Diagram ofD3-probe-D7 SystemWe present the detail numerical procedure to obtain the phase diagram 6.1in the chapter 6.B.1 Geometry of branesWe consider a pair of probe branes, a D7 brane and an anti-D7 suspendedin the asymptotically flat D3-brane solution. We use coordinates where themetric background isds2 =(1 +R4r4)−1/2 (−dt2 + dx2 + dy2 + dz2)+(1 + R4r4)1/2 (dr2 + r2d2Ω5)(B.1)andd2Ω5 = dψ2 + sin2 ψd2Ω4 (B.2)where we have written the geometry to display the directions the D3 lie in(t, x, y, z). We then introduce the coordinatesρ = r sinψ , l = r cosψso that the metric (6.2) becomesds2 =(1 +R4(ρ2 + l2)2)−1/2 (−dt2 + dx2 + dy2 + dz2)+(1 +R4(ρ2 + l2)2)1/2 (dρ2 + dl2 + ρ2d2Ω4)(B.3)We make an ansatz for the D7 brane and anti-D7 brane geometries wherethey both fill the directions x, y, t, ρ and wrap the S4 ⊂ S5 according tot x y z ρ l θ1 θ2 θ3 θ4D3 × × × ×D7/D7 × × × z(ρ) × l(ρ) × × × ×(B.4)129B.1. Geometry of branesThey sit at points in the remainder of the directions z, l and the pointsz(ρ), l(ρ) generally depend on ρ. The wrapped S4 has an SO(5) symmetry.The geometry of the D7 brane and anti-D7 brane are both given by theansatzds2 =(1 +R4(ρ2 + l2)2)−1/2 (−dt2 + dx2 + dy2)+(1 +R4(ρ2 + l2)2)1/2dρ21 + l′(ρ)2 + z′(ρ)21 +R4(ρ2 + l2)2+ ρ2d2Ω4(B.5)The introduction of a charge density requires D7 world-volume gaugefields. In the aρ = 0 gauge, the field strength 2-form F is given by2pil2sF = a′0(ρ)dρ ∧ dt (B.6)In this expression, a0(ρ) will result in the world volume electric field whichis needed in order to have a nonzero U(1) charge density in the quantumfield theory. The temporal gauge field A0 are defined in terms of it asa0 = 2pil2sA0 (B.7)In this Section, we will use the field strength (B.6) for both the D7 braneand the anti-D7 brane.The asymptotic behavior at ρ→∞ for the embedding functions in (B.5)isl(ρ)→ m+ cρ3+ . . . (B.8)and the D7 brane and anti-D7 brane are separated by a distance L,z(ρ)→ L2− ρ40ρ4+ . . . (B.9)for the D7 brane andz(ρ)→ −L2+ρ40ρ4+ . . . (B.10)for the anti-D7 brane. The asymptotic behavior of the gauge field (B.6) isa0(ρ) = µ− qρ4+ . . . (B.11)130B.1. Geometry of braneswith µ and q related to the chemical potential and the charge density, re-spectively.There are two constants which specify the asymptotic behavior in eachof the above equations. In all cases, we are free to choose one of the twoconstants as a boundary condition, for example we could choose m, q, ρ0.Then, the other constants, c, µ, L, are fixed by requiring that the solution isnon-singular.We will only consider solutions where the boundary condition is m = 0.This is the chiral symmetric boundary condition. Its equivalent in the dualgauge theory is obtained by specifying that the hypermultiplet is masslessand the theory has chiral symmetry. Then, when we solve the equation ofmotion for l(ρ), there are two possibilities. First c = 0 and l(ρ) = 0, aconstant for all values of ρ. This is the chirally symmetric phase. Secondly,c 6= 0 and l is a non-constant function of ρ. This describes the phase withspontaneously broken chiral symmetry. The constant c is proportional tothe intra-layer chiral condensate〈ψ¯1ψ1〉for the D7 brane and〈ψ¯2ψ2〉forthe anti-D7 brane.To be more general, we will consider consider a stack of Nf D5-branesand Nf anti-D7 branes. The Born Infeld action for either the stack of D7or anti-D7 branes is given byS = N7∫dρρ4√(1 +R4(l2 + ρ2)2)(1− a0′2 + l′2) + z′2 (B.12)whereN7 = Nf48pi5l8sgsV2+1with V2+1 the volume of the 2+1-dimensional space-time, ls the stringlength, gs the string coupling and NF the number of D7 branes. The Wess-Zumino terms that occur in the D brane action will not play a role in theD7 brane problem.The variational problem of extremizing the Born-Infeld action (B.12)involves two cyclic variables, a0(ρ) and z(ρ). Being cyclic, their canonical131B.1. Geometry of branesmomenta must be constants,Q = − δSδa′0≡ 2pil2sN7q , q =ρ4a0′ (2ρ2l2 + l4 + ρ4 +R4)(l2 + ρ2)2√(2ρ2l2+l4+ρ4+R4)(1−a0′2+l′2)(l2+ρ2)2+ z′2(B.13)Πz =δSδz′≡ N7ρ40 , ρ40 =ρ4z′√(2ρ2l2+l4+ρ4+R4)(1−a0′2+l′2)(l2+ρ2)2+ z′2, (B.14)Solving (B.13) and (B.14) for a′0(ρ) and z′(ρ) in terms of q and ρ0 wegeta′0 = −q(l2 + ρ2)√1 + l′2√l2 (l2 + 2ρ2)(ρ8 + q2 − ρ80)+ q2ρ4 +(ρ8 − ρ80)(ρ4 +R4)(B.15)z′ =ρ40(l4 + 2l2ρ2 + ρ4 +R4)√1 + l′2(l2 + ρ2)√l2 (l2 + 2ρ2)(ρ8 + q2 − ρ80)+ q2ρ4 +(ρ8 − ρ80)(ρ4 +R4)(B.16)We can now use the Legendre transformed action to find l(ρ) for fixed ρ0and fixed charge q, eliminating a′0(ρ) and z′(ρ) using equations (B.15) and(B.16).132B.1.GeometryofbranesAfter scaling out the parameter R, (ρ, l, z, a0)→ R(ρ, l, z, a0), such a Routhian readsR = SN7 − ρ40∫dρz′(ρ)− q∫dρa′0(ρ)=∫dρ√(1 + l′2)(l4(ρ8 + q2 − ρ80)+ 2l2ρ2(ρ8 + q2 − ρ80)+ q2ρ4 + (ρ4 + 1)(ρ8 − ρ80))l2 + ρ2(B.17)The Euler-Lagrange equation for l(ρ) can be derived by varying the Routhian (B.17) and reads(l2 + ρ2)l′′(l2(l2 + 2ρ2) (ρ8 + q2 − ρ80)+ q2ρ4 +(ρ4 + 1) (ρ8 − ρ80))+ 2(1 + l′2) (ρl′(2l2ρ6(l4 + 3l2ρ2 + 3ρ4 + 1)+ 2ρ12 + ρ8 + ρ80)+ l(ρ8 − ρ80))= 0 (B.18)This equation must be solved with the boundary conditions in equation (B.8)-(B.11) (recalling that we can chooseonly one of the integration constants, the other being fixed by regularity of the solution) in order to find thefunction l(ρ). Once we know that function, we can integrate equations (B.15) and (B.16) to find a0(ρ) and z(ρ).Clearly, l = 0, a null constant, for all values of ρ, is always a solution of equation (B.18), even when ρ0 and thecharge density are nonzero. However, for some range of the parameters, it will not be the most stable solution.Using the asymptotic behavior (B.8) with m = 0 and c = c, the solution, up to the order 23 in the 1ρ expansion,has the forml(ρ) =cρ3− 2c7ρ7+c(189c2 − 21q2 + 21ρ80 + 22)154ρ11− c(807c2 − 86q2 + 31ρ80 + 44)495ρ15+c(2525985c4 + c2(−561330q2 + 561330ρ80 + 930444)+ 31185q4 − 2q2 (31185ρ80 + 48241)+ 31185ρ160 + 20032ρ80 + 32648)526680ρ19−c(2476927215c4 − 174c2(3057600q2 − 2010225ρ80 − 2193536)+ 7(4086075q4 − 2q2(2508030ρ80 + 2776603)+ 929985ρ160 + 798676ρ80 + 1403864))211988700ρ23+ O((1ρ)27)in terms of the modulus c.133B.2. Double monolayers without charge densityB.2 Double monolayers without charge densityThe equation of motion for l in this case is(l2 + ρ2)l′′(l2(l2 + 2ρ2) (ρ8 − ρ80)+(ρ4 + 1) (ρ8 − ρ80))+ 2(1 + l′2) (ρl′(2l2ρ6(l4 + 3l2ρ2 + 3ρ4 + 1)+ 2ρ12 + ρ8 + ρ80)+ l(ρ8 − ρ80))= 0(B.19)There are in principle four type of solutions for which m = 0 in (B.8):1. Unconnected, constant l = 0 solution that reaches the Poincare´ hori-zon, l = 0 and ρ = 0, i.e. a black-hole (BH) embedding.2. A connected constant l = 0 solution.3. A ρ-dependent unconnected solution with ρ0 = 0, where the branepinches off (ρ = 0) before reaching the Poincare´ horizon, i.e. a Minkowskiembedding.4. A connected ρ-dependent solution.These solutions are summarized in the following table. We look for solutionwith m = 0 in (B.8). We can distinguish four types of solutions accordingto the classification of table B.1.ρ0 = 0 ρ0 6= 0c = 0Type 1 Type 2unconnected, l = 0 connected, l = 0BH, chiral symm. Mink, interc 6= 0Type 3 Type 4unconnected, ρ-dependent l connected, ρ-dependent lMink, intra Mink, intra/interTable B.1: Types of possible solutions, where Mink stands for Minkowskiembeddings and BH for black-hole embeddings.Type 1 solutions are trivial both in l and z (they are both constants).They correspond to two parallel black hole (BH) embeddings for the D7and the anti-D7. This configuration is the chiral symmetric one. However,134B.2. Double monolayers without charge densityby studying the fluctuations around this solution we will show in the nextsection that it is unstable,In type 2 solutions the chiral symmetry is broken by the inter-layer con-densate (ρ0 6= 0). In this case the branes have non flat profiles in the zdirection. Solutions of type 3 and 4 are the really non-trivial ones to find.Type 3 solutions have non-zero expectation value of the intra-layer conden-sate and there are only Minkowski (Mink) embeddings. Type 4 solutionsbreak chiral symmetry in both the inter- and intra-layer channel. For type 2and 4 solutions the D7 and the anti-D7 world-volumes have to join smoothlyat a finite ρ = ρ0. These are indeed connected solutions and of course theyare Minkowski embeddings.Let’s consider the solutions of the type 3, types 1 and 2 are just l = 0.The equation for l (B.19) with ρ0 = 0 simplifies further tol′′ + 2(l′2 + 1) (2l2 (l4 + 3l2ρ2 + 3ρ4 + 1)+ 2ρ6 + ρ2) l′ + lρρ (l2 + ρ2) (l4 + 2l2ρ2 + ρ4 + 1)= 0 (B.20)In this case it is obvious from (B.16) that z(ρ) is a constant. Solutions oftype 3 are those for which l(ρ) goes to a constant, l0, at ρ = 0 so that thefour-sphere in the world-volume of the D7-brane shrinks to zero.A solution to (B.20) of this type can be obtained by a shooting technique.The differential equation can be solved from either direction: from ρ = 0or from the boundary at ρ = ∞. In either case, there is a one-parameterfamily of solutions, from ρ = 0 the parameter is l0, from infinity it is thevalue of the modulus c in (B.19), which can be used to impose the boundaryconditions at ρ→∞ with c = 0.To find the solution fix some ρ¯ in the middle (say ρ¯ = 2). Start withshooting from the origin. For constant values of l at the origin ρ = 0,integrate the solution outwards to ρ¯ and compute l and its derivative at ρ¯.For each solution, put a point on a plot of l′(ρ¯) vs. l(ρ¯) (the red curve inFigure B.1). Then, do the same thing starting from the boundary at ρ =∞,and varying the coefficient c of the expansion around infinity. This is givenby the blue curve in Figure B.1.To find a condition on the derivative at ρ = 0 we found an asymptoticexpansion of the solution around the origin, the first few terms readl(ρ) = l0 − ρ25l0(l40 + 1) + (375l80 + 325l40 + 46) ρ41750l30(l40 + 1)3 +O (ρ6) (B.21)When the two solutions, coming from ρ = 0 and from ρ =∞ meet at theintermediate point, then there is a solution. The only value of l0 at which135B.2. Double monolayers without charge density-0.0001 -0.00005 0.00005 0.0001 lHΡ=2L-0.00003-0.000025-0.00002-0.000015-0.00001-5.´10-65.´10-6l'HΡ=2LFigure B.1: Unconnected solution for ρ0 = 0, there is a unique solution when theblue and red curves cross.this happens is l0 ' 0.254 and for the parameter c we get c ' 0.051. Withthese values l(ρ¯ = 2) = 0.00628566, l′(ρ¯ = 2) = −0.00921382. The precisionof these numbers could easily be improved by scanning the region aroundthe point where the two curves meet. The solution is plotted in Figure B.2.0.5 1.0 1.5 2.0 2.5 3.0 Ρ0.ΡLFigure B.2: Unconnected solution for ρ0 = 0 and q = 0.Consider now the solution of equation (B.19) of type 4. In this case welook for a D7 that joins at some given ρ0 the corresponding anti-D7. Atρ0, z′(ρ0) → ∞. At ρ = ρ0 we choose l′(ρ0) = 0 and l(ρ0) = l0 and weuse l0 for the shooting technique at the turning point ρ0. At infinity we use136B.2. Double monolayers without charge densitythe expansion (B.19) with q = 0 to define the boundary conditions. Theshooting technique provides a unique solution for each value of ρ0.B.2.1 FluctuationsConsider first the constant disconnected solution, the fluctuation around itcan be easily found by writing l(ρ) = δl(ρ) and expanding eq.(B.20) at thefirst order in . We getρ(ρ(ρ4 + 1)δl′′ +(4ρ4 + 2)δl′)+ 2δl = 0whose general solution readsδl = a1ρ12i(√7+i)2F1(−18+i√78,58+i√78; 1 +i√74;−ρ4)+a2ρ− 12i(√7−i)2F1(−18− i√78,58− i√78; 1− i√74;−ρ4)(B.22)for small ρ this solution goes likeδl ∼ a1ρ− 12 + i√72 + a2ρ− 12− i√72the complex exponents is a sign of the instability of the solution l = 0 forq = ρ0 = 0.When ρ0 is non zero instead, we can show, with a similar argument,that the constant connected solution is actually stable. The equation forthe fluctuation now is2δl(ρ80 − ρ8)− ρ ((ρ5 + ρ) δl′′ (ρ8 − ρ80)+ 2δl′ (2ρ12 + ρ8 + ρ80)) = 0which expanded around ρ0 becomes4 (ρ− ρ0) δl + ρ20(1 + ρ40) (δl′ + 2 (ρ− ρ0) δl′′)= 0The general solution of this equation isδl = a14√4ρ− 4ρ0J 14 √2ρ√ρ20(ρ40 + 1) − √2ρ0√ρ20(ρ40 + 1)+a24√4ρ− 4ρ0Y 14 √2ρ√ρ20(ρ40 + 1) − √2ρ0√ρ20(ρ40 + 1) (B.23)137B.2. Double monolayers without charge densitywhose expansion around ρ0 givesδl ∼ a1[const√ρ− ρ0 +O((ρ− ρ0)5/2)]+a2[const + const√ρ− ρ0 +O((ρ− ρ0)2)]thus it is always possible to take a linear combination of these solutions thathas a derivative with a correct behavior, i.e the solution with integer powersof ρ− ρ0. Thus when ρ0 6= 0 the solution l = 0 is stable.There are then three types of solutions of the equation of motion (B.19)representing bi-layers. Solutions of type 2, 3 and 4.B.2.2 Free energyWe want now to compare the energies of these solutions for fixed separationsof the layers. By setting the asymptotic behavior of z(ρ) (B.9) and (B.10)we implicitly defined the layer separation for a connected solution asL = 2∫ ∞ρ0dρz′ = 2∫ ∞ρ0dρρ40√l4 + 2l2ρ2 + ρ4 + 1√1 + l′2(l2 + ρ2)√ρ8 − ρ80(B.24)for zero charge density. This separation is plotted in Fig.B.3 for the con-stant connected and the ρ-dependent connected solutions. The first has twobranches and a minimum at ρ0 ' 0.84 where L ' 1.166, the latter does notexist for ρ0 > 0.265, but joins with the constant connected at that point.The energy of the solutions at fixed separation L is given by the on shellaction (B.12), S/N7 where we plug in the three possible solutions we found.E(l; ρ0) =∫ ∞ρ0dρρ8√1 +1(l2 + ρ2)2√1 + l′2ρ8 − ρ80(B.25)This energy is divergent since in the large ρ limit the argument of the in-tegral goes as ∼ ρ4. However, in order to find the energetically favoredconfiguration, we are only interested in the difference between the energyof two solutions, which is always finite. We then choose the energy of theunconnected (ρ0 = 0) constant l = 0 solution, type 1, as the reference energy(zero energy level), so that any other (finite) energy density can be definedas ∆E(l; ρ0) = E(l; ρ0)−E(l = 0; ρ0 = 0). The solution of type 1 always hasa higher energy with respect to solutions 2, 3 and 4 and indeed it is alwaysunstable.By means of numerical computations we obtain for the energy density∆E of the solutions 2, 3 and 4, the behaviors depicted in Figure B.4. At138B.3. Double monolayers with charge density0.5 1.0 1.5 2.0 2.5 3.0Ρ01234LFigure B.3: Brane separation L as a function of ρ0, which in this case represents thedistance from the Poincare` horizon reached by the brane. The blue line correspondsto the connected solution (type 2) and the green line to the connected non constantsolution (type 4).small separation, at variance with the D5-D¯5 case, the configuration whichis energetically favored is the Minkowski embedding disconnected solution.At L = 1.1674 the constant connected becomes energetically favored witha jump in energy. Then, up to L = 2.31 the constant connected solutionwith an inter-layer condensate is preferred. For larger separation L > 2.31the disconnected configuration is preferred and the condensation is in theintra-layer channel.The green lines in the figures show the value of the energy and separationfor the connected solution with non zero z(ρ) and l(ρ). These configurationshave both intra and inter layer condensates. As can be seen from Fig.5.8they are never energetically favored.B.3 Double monolayers with charge densityNow we consider the configurations with a charge density different from zero.The equation of motion in this case is given by the most general equation(B.18).139B.3. Double monolayers with charge density1 2 3 4 5L-0.030-0.025-0.020-0.015-0.010-0.005DEFigure B.4: Regularized energy density ∆E as a function of the brane separationL. The blue line corresponds to the connected solution (type 2), the red line to theunconnected one (type 3) and the green line to the connected non constant solution(type 4). All the solutions are regulated with respect to the constant unconnectedsolution of type 1.We look for solution with m = 0 in (B.8). We can distinguish four typesof solutions according to the classification of table B.2.Type 1 solutions are trivial both in l and z (they are both constants).They correspond to two parallel black hole (BH) embeddings for the D7and the anti-D7. This configuration is the chiral symmetric one. In type 2solutions the chiral symmetry is broken by the inter-layer condensate (ρ0 6=0): In this case the branes have non flat profiles in the z direction. Solutionsof type 3 and 4 are the really non-trivial ones to find. Type 3 solutions havenon-zero expectation value of the intra-layer condensate and they can be onlyblack hole embeddings, this is the true difference with the zero charge case,where this type of solutions were Minkowski embeddings, when the charge140B.3. Double monolayers with charge densityρ0 = 0 ρ0 6= 0c = 0Type 1 Type 2unconnected, l = 0 connected, l = 0BH, chiral symm. Mink, interc 6= 0Type 3 Type 4unconnected, ρ-dependent l connected, ρ-dependent lBH, intra Mink, intra/interTable B.2: Types of possible solutions, where Mink stands for Minkowski embed-dings and BH for black-hole embeddings.density is non-zero only BH embeddings are allowed. A charge density onthe D7 world-volume is indeed provided by fundamental strings stretchedbetween the D7 and the Poincare´ horizon. These strings have a tension thatis always greater than the D7-brane tension and thus they pull the D7 downto the Poincare´ horizon. For this reason when q 6= 0 the only disconnectedsolutions we will look for are BH embedding.Type 4 solutions break chiral symmetry in both the inter- and intra-layerchannel. For type 2 and 4 solutions the D7 and the anti-D7 world-volumeshave to join smoothly at a finite turning point ρ = ρt. These are indeedconnected solutions and of course they are Minkowski embeddings.We now state what we have found, we will then provide support to ourstatements.1. The constant l = 0 disconnected solution ρ0 = 0, when a charge ispresent, is stable, this can be shown by an analysis of the fluctuationsabout it.2. The constant l = 0 connected solution with ρ0 6= 0 joins at the pointρt where z′(ρ) in (B.16) diverges. This is given by the real solution ofq2ρ4 +(ρ8 − ρ80) (ρ4 + 1)= 0 (B.26)141B.3. Double monolayers with charge densitywhich isρt =16 2 3√2(−3q2 + 3ρ80 + 1)3√√4(3q2 − 3ρ80 − 1)3+(9q2 + 18ρ80 − 2)2+ 9q2 + 18ρ80 − 2+22/33√√4(3q2 − 3ρ80 − 1)3+(9q2 + 18ρ80 − 2)2+ 9q2 + 18ρ80 − 2− 2)]1/4(B.27)as can be checked numerically for any value of q and ρ0.For this solution the brane separation L, as a function of ρ0, has aminimum and two branches, as for the blue line in the q = 0 case ofFig.B.3, a branch where it decreases for increasing ρ0 and a branchwhere it increases for increasing ρ0. Consequently the energy, mea-sured with respect to the constant disconnected solution, has a formsimilar to the one of the energy for q = 0, i.e. the blue line in Fig.B.4.3. The BH-embedding solution, which is non-constant and has ρ0 = 0,exists only for q < 0.038, as can be show by means of the shootingtechnique described in the previous section.4. The connected non-constant solution joins at the point ρt where z′(ρ)in (B.16) diverges. This is given by the real solution ofl2(l2 + 2ρ2) (ρ8 + q2 − ρ80)+ q2ρ4 +(ρ8 − ρ80) (ρ4 + 1)= 0 (B.28)which in turn depends on the solution l itself.This solution exists only for small values of q. For ρ0 ' 0 we haveq < 0.038, when ρ0 grows, the allowed values of q decrease.B.3.1 FluctuationsThe equation of motion for disconnected solutions with charge density isgiven by (B.18) with ρ0 = 0, namely(l2 + ρ2) (l2(l2 + 2ρ2) (ρ8 + q2)+ ρ12 + ρ8 + q2ρ4)l′′+((2l2(3ρ2l2 + l4 + 3ρ4 + 1)+ 2ρ6 + ρ2)l′ + ρl) (l′2 + 1)= 0 (B.29)142B.3. Double monolayers with charge densityClearly l = 0 is a solution and fluctuations around it can be easily found bywriting l(ρ) = δl(ρ) and expanding at the first order in . The equation forδl(ρ) reads2ρ3δl′ + 2ρ2δl + q2δl′′ = 0whose general solution isδl = a1 1F1(14;34;− ρ42q2)+ a2ρ 1F1(12;54;− ρ42q2)For small ρ this goes likeδl ' a1 − a1 ρ46q2+ a2ρ+O(ρ5)Being all the power of ρ non negative integers, we only have normalizablemodes and this shows that the l = 0 solution becomes stable when there isa charge density.When ρ0 is non vanishing, i.e. when considering connected solutionswith a non trivial profile in the z direction, the equation for the fluctuationsaround the l = 0 solution reads−2δl (ρ8 − ρ80)−ρ (2 (2ρ12 + ρ8 + ρ80) δl′ + ρδl′′ (q2ρ4 + (ρ4 + 1) (ρ8 − ρ80))) = 0Expanded around ρ0 it becomes4(ρ40 + 1)ρ40(2(ρ− ρ0)δl′′ + δl′)+ 16ρ20δl(ρ− ρ0)− q2ρ0δl′′ = 0and this has a solution for ρ ' ρ0 of the formδl = a1+a2(ρ−ρ0)+2a2ρ30(ρ40 + 1)(ρ− ρ0)2q2+8ρ0(ρ− ρ0)3(a1q2 + 3a2(ρ40 + 1)2ρ50)3q4+O((ρ− ρ0)4)in terms of integer powers of ρ−ρ0 and two moduli. The l = 0 connectedsolution therefore is stable.B.3.2 Black Hole embedding ρ dependent solutionThe black hole embedding ρ-dependent profile is a solution of Eq.(B.29) andit can be found by means of the usual shooting technique. The boundaryconditions that can be imposed for ρ→∞ are defined by the expansion in(B.19) where the modulus c can be varied to match the solution coming fromρ = 0. The latter can be obtained imposing boundary conditions derived143B.3. Double monolayers with charge densityby expanding the solution around ρ ∼ 0 and demanding that l(ρ) → 0 forρ→ 0. The first terms of this expansion up to ρ17 arel(ρ) =αρ− αρ55 (α2 + 1) q2+ρ9(−5α9q2 − 20α7q2 − 30α5q2 − 20α3q2 + 11α3 − 5αq2 + 8α)90 (α2 + 1)3 q4+ρ13(385α9q2 + 1450α7q2 + 2040α5q2 + 1270α3q2 − 343α3 + 295αq2 − 184α)3510 (α2 + 1)4 q6+ρ17α5967000 (α2 + 1)7 q8(131625α18q4 + 1184625α16q4 + 4738500α14q4+50α12q2(221130q2 − 20173)+ 50α10q2 (331695q2 − 111499)+50α8q2(331695q2 − 255263)+ 25α6 (442260q4 − 619044q2 + 21209)+25α4(189540q4 − 418766q2 + 50213)+ α2 (1184625q4 − 3739750q2 + 931552)+25(5265q4 − 21994q2 + 8464))+O (ρ21) (B.30)The expansion is written in terms of the modulus α which is the parameterthat can be varied to adjust the shooting technique.A typical solution is presented in Fig.B.5, where it is clear that it reachesthe Poincare` horizon at l = ρ = 0 Some examples of the results of the1 2 3 4 5Ρ0.050.100.15lHΡLFigure B.5: Black hole embedding solution for q = 1/100, it clearly reaches thePoincare` horizon.shooting technique are given in Figs.B.6, B.7, B.8 B.9.It is quite clear from these graphs that the solution tends to disappearincreasing the charge q, we checked that for q > 0.0377 the curves coming144B.3. Double monolayers with charge density0.05 0.10 0.15 0.20lH2L-0.025-0.020-0.015-0.010-0.005l'H2LFigure B.6: Shooting technique for q = 1/500.01 0.02 0.03 0.04 0.05lH2L-0.005-0.004-0.003-0.002-0.0010.000l'H2LFigure B.7: Shooting technique for q = 1/30from infinity and from zero meet only at l(2) = l′(2) = 0, which correspondsto the trivial chirally symmetric solution.Energy of the ρ-dependent black hole embedding solutionWe can now compute the energy of the ρ-dependent black hole embeddingsolutions that we have found by varying the value of the charge. This isgiven by the Routhian in (B.17) with ρ0 = 0, computed on the solutions145B.3. Double monolayers with charge density0.01 0.02 0.03 0.04 0.05lH2L-0.005-0.004-0.003-0.002-0.0010.000l'H2LFigure B.8: Shooting technique for q = 1/280.0005 0.0010 0.0015 0.0020lH2L-0.0030-0.0025-0.0020-0.0015-0.0010-0.00050.0000l'H2LFigure B.9: Shooting technique for q = 2/53 ' 0.0377found numerically.R = SN7 − q∫dρa′0(ρ)=∫dρ√(1 + l′2) (l4 (ρ8 + q2) + 2l2ρ2 (ρ8 + q2) + q2ρ4 + (ρ4 + 1) ρ8)l2 + ρ2(B.31)146B.4. Connected solutionsAgain this energy is divergent and has to be regulated by subtracting to itthe energy of the constant l = 0 disconnected solution with the same charge,namely∆E =∫ ∞0dρ√(1 + l′2) (l4(ρ8 + q2)+ 2l2ρ2(ρ8 + q2)+ q2ρ4 +(ρ4 + 1)ρ8)l2 + ρ2−√q2 +(ρ4 + 1)ρ4(B.32)This quantity is plotted in Fig.B.10 as a function of the charge q. It is clearfrom this plot that the energy of the ρ-dependent black-hole embeddingsolution is always lower then its ρ-independent counterpart. For q ' 0.0377the two solutions actually merge, so that for higher values of the charge theonly existing solution is the chirally symmetric one.0.005 0.010 0.015 0.020 0.025 0.030 0.035q-0.0025-0.0020-0.0015-0.0010-0.0005DEFigure B.10: Energy of the ρ-dependent black-hole embedding solution, red line,measured with respect to the energy of the constant chirally symmetric solution,blue line.B.4 Connected solutionsWhen ρ0 in (5.13) is non zero there is a non trivial profile in the z directionand it is possible to construct connected solutions even when there is acharge density. In this case the inter-layer condensate is stable only whenthe layers have charge densities of equal magnitude and opposite sign, sothat the total system is charge neutral.According to the classification shown in table B.2 we can distinguish twotypes of connected solutions, i.e. type 2 and type 4. Let us consider first147B.4. Connected solutionsthe type 2 solutions. These have a flat profile in the l direction, l = 0, andthus for such solutions we can easily write the brane separation asL = 2∫ ∞ρtdρρ40(ρ4 + 1)ρ2√q2ρ4 +(ρ8 − ρ80)(ρ4 + 1)(B.33)where ρt is given in (B.27).Since we want the energy of the solution as a function of the layer sepa-ration L and the charge q we need to perform a Legendre transform of theaction S only with respect to the charge, namelyRq = SN7 − q∫dρa′0(ρ) =∫dρ(l2(l2 + 2ρ2) (ρ8 + q2)+ ρ4(ρ8 + ρ4 + q2))l2 + ρ2√l′2 + 1l2 (l2 + 2ρ2)(ρ8 + q2 − ρ80)+ q2ρ4 + (ρ4 + 1)(ρ8 − ρ80) (B.34)This needs to be regularized and we do it with respect to the constantdisconnected solution with the same charge. For the l = 0 solution then weget∆E =∫ ∞ρtdρ ρ2(ρ8 + ρ4 + q2)√q2ρ4 +(ρ4 + 1) (ρ8 − ρ80) −√ρ8 + ρ4 + q2− ∫ ρt0dρ√ρ8 + ρ4 + q2 (B.35)There exist also a ρ-dependent connected solution that corresponds tothe existence of both intra-layers and inter-layers condensates. These arenon trivial solutions of the most general equation of motion (B.18) with theadditional condition that there must be a point ρt that solves the equation(B.28) where z′(ρ) in (B.16) diverges. Thus to find this solution we imposethe boundary condition that l is at ρt a solution of (B.28), namely it is givenbyl(ρt) =√√√√√ ρ80 − ρ8tq2 − ρ80 + ρ8t− ρ2t (B.36)ρt has a maximum value that is given by (B.27), namely it is reached whenl = 0, i.e. for the constant connected solution. Then we can get the boundarycondition on the first derivative of l, l′, computing the equation of motionat ρt and using (B.36) to get the value of l at ρt. This readsl′(ρt) =(ρ8t − ρ80)√√ q2q2−ρ80+ρ8t− 1− ρ2t2q2ρ7t√ρ80−ρ8t(q2−ρ80+ρ8t )3/2 + ρ80ρt − ρ9t(B.37)148B.4. Connected solutionsThen these boundary conditions can be used to apply the shooting techniqueby varying the parameter ρt in order to match the solution coming frominfinity, where the value of the chiral condensate (c in (B.19)) is the modulusthat must be varied to obtain the matching.Solutions of this type exist only for small charges q < 0.0377 exactlythe same value we found as a threshold for the existence of the black-holeembedding solution. When the charge decrease the range of values of ρ0 forwhich the solution exists, increases.The brane separation L for this solution is defined by twice the integralof z′ in (B.16)L = 2∫ ∞ρtdρρ40(l4 + 2l2ρ2 + ρ4 + 1)√1 + l′2(l2 + ρ2)√l2 (l2 + 2ρ2)(ρ8 + q2 − ρ80)+ q2ρ4 +(ρ8 − ρ80)(ρ4 + 1)(B.38)The energy is given by (B.34) computed on the solution and again we regu-larize it by subtracting the contribution of the constant disconnected solu-tion with the same charge.Figure B.11 shows the behavior of the brane separation as a function ofthe parameter ρ0 for both type 2 and type 4 solutions when the charge isfixed to q = 0.001. The solution of type 2 has two branches as in the zerocharge case. The solution of type 4 instead as a different behavior comparedto the neutral case. Analogous figures can be obtained for different values ofthe charge, provided q < 0.0377, increasing the charge the growing branchin the green solution becomes smaller and the range of allowed ρ0 for thissolution decreases.The energy of the connected solutions can be compared to that of thecorresponding black-hole solution with the same charge. The q = 0.001 caseis represented in Fig.B.12 and B.13 where the region around the transitionpoint is enhanced.From Fig.B.13 it is clear that for L < 1.16 the lowest energy solutionis the black-hole embedding one with only an inter-layer condensate, atL = 1.16 there is a jump in energy to the constant connected which hasonly an inter-layer condensate and for L > 2.2 the energetically favoredsolution becomes the connected ρ-dependent with both inter and intra layercondensates.We finally have performed this analysis also for other charges and thishas allowed us to finally draw the the phase diagram in chapter 6.149B.4. Connected solutions0.5 1.0 1.5 2.0 2.5 3.0Ρ0123LFigure B.11: Brane separation as a function of ρ0. The blue line is obtained fromthe constant solution, the green line from the ρ-dependent one.0.5 1.0 1.5 2.0 2.5 3.0 3.5L-0.03-0.02-0.01EFigure B.12: Energy as a function of L. The blue line is obtained from the constantsolution, the green line from the ρ-dependent one and the red line the black holeembedding solution with the same charge q = 0.001.150B.4. Connected solutions1 2 3L-0.005-0.004-0.003-0.002-0.001EFigure B.13: Close around the transition point in Fig. B.12151Appendix CBKT Quantum PhaseTransitionThis chapter is the review of the part of the paper [102] about BKT transi-tion. We will start from the Lagrangian, the equation (2.10).L˜ ∼ L− ∂L∂a′0a′0 −∂L∂z′z′, (C.1)where ′ = ddr .u ≡√r2 + l(r)2The equation of motion with respect to z(r) is :∂L˜∂z′=√(r4 + f2u4)(1 + H2u4) + d2√1 + l′2 + u4z′2u4z′ + fu4 (C.2)=(√(r4 + f2u4)(1 + H2u4) + d2√1 + l′2 + u4z′2z′ + f)u4 = const. ≡ C (C.3)z′2 =(1 + l′2)(C − fu4)2u4(u4((r4 + f2u4)(1 + H2u4) + d2)− (C − fu4)2) (C.4)z′ =(C − fu4)√1 + l′2u2√u4((r4 + f2u4)(1 + H2u4) + d2)− (C − fu4)2(C.5)L˜ =√1 + l′2√(r4 + f2u4)(1 +H2u4) + d2 − (C − fu4)2u4(C.6)C should be zero, because of regularity at r = 0 and r =∞. Then,L˜ =√1 + l′2√r4(1 +H2u4) + f2H2 + d2 (C.7)152Appendix C. BKT Quantum Phase TransitionWithout Wess-Zumino (WZ) term, the Lagrangian was:L˜′ =√1 + l′2√(r4 + f2u4)(1 +H2u4) + d2 (C.8)As the result of [103] , it givesm2 = − 2H2d2 +H2 +H2f2(C.9)at at IR, r  1, and BF bound is violated asd2H2+ f2 < 7 (C.10)By this result, even for d = 0, violation condition is provided. We comparethis with our Rauthian with the WZ term. The Lagrangian up to quadraticorder of l(r),L ∼ −√r4 + d2 +H2(1 + f2)l′22+H2√r4 + d2 +H2(1 + f2)l2r2. (C.11)The equation of motion is:0 = ∂r(√r4 + d2 +H2(1 + f2)l′) +H2√r4 + d2 +H2(1 + f2)2lr2=√r4 + d2 +H2(1 + f2)l′′+2r3l′√r4 + d2 +H2(1 + f2)+H2√r4 + d2 +H2(1 + f2)2lr2We are interested in IR, r  1. Then the equation of motion is:√d2 +H2(1 + f2)l′′ +H2√d2 +H2(1 + f2)2lr2= 0 (C.12)l′′ +2H2d2 +H2(1 + f2)lr2= 0 (C.13)This equation of motion gives the result of [103] without WZ term. For thedeeper understanding, let us confirm the result without WZ term.L ∼ −√1 + l′2√(r4 + f2u4)(1 +H2u4) + d2 (C.14)153Appendix C. BKT Quantum Phase TransitionExpand up to quadratic order of l(r),L ∼ −(1 + l′22)((1 + f2)(r4 +H2) + d2 + 2l2(f2r2 − H2r2))1/2(C.15)L ∼ −√(1 + f2)(r4 +H2) + d2l′22− (f2r2 −H2/r2)l2√(1 + f2)(r4 +H2) + d2. (C.16)Let us check it carefully with the equation of motion. The equation ofmotion is:0 = ∂r(√(1 + f2)(r4 +H2) + d2l′)− 2(f2r2 −H2/r2)l√(1 + f2)(r4 +H2) + d2=√(1 + f2)(r4 +H2) + d2l′′+2(1 + f2)r3l′√(1 + f2)(r4 +H2) + d2− 2(f2r2 −H2/r2)l√(1 + f2)(r4 +H2) + d2The same result as above is obtained at IR.√d2 +H2(1 + f2)l′′ +H2√d2 +H2(1 + f2)2lr2= 0 (C.17)On the other hand, UV limited case presents different results betweenwith or without WZ term. Without WZ, if r  1, the equation of motionis: √1 + f2r2l′′ + 2√1 + f2rl′ − 2f2l√1 + f2= 0 (C.18)√1 + f2rl′′ + 2√1 + f2l′ − 2f2√1 + f2lr= 0 (C.19)(AdSp+2 −m2)lr=1√−g∂µgµν√−g∂ν lr−m2 lr=1rp∂rrp+2∂r(lr)−m2 lr= rl′′ + pl′ − prl −m2 lr= 0 (C.20)2929We will explain in more detail in the later section154Appendix C. BKT Quantum Phase TransitionThus, for p = 2, The mass of this scalar particle, l/r on AdS4 is:m2 = −2 + 2f21 + f2=−21 + f2≥ −2 (C.21)The mass varies by changing S2 flux, but is larger than -2, which neverviolates the Breitenlohner-Freedman (BF) bound. With WZ, it is simplerand m2 = −2 as f = 0.To check C = 0 case, consider the blackhole embedding at finite temper-ature.ds2D5 =w2L2(−gtdt2 + gxdx2 + gxdy2) + L2w2(dρ2 + ρ2dΩ22 + dl2 + l2ρ2dΩ˜22)wheregt :=(w4 − w40)22w4(w4 + w40), gx :=w4 + w402w4, (C.22)We required this modified metric to keep SO(3) × SO(3) isometry. Withprofiling ansatz, l := l(r), z := z(r) and dΩ˜ = 0,ds2D5 =w2L2(−gtdt2+gxdx2+gxdy2)+(L2w2+L2l′(ρ)2w2+w2L2z′(ρ)2)dρ2+L2ρ2w2dΩ22w2 = ρ2 + l2,L = LWZ + LDBI∼ −f(w4 + w40w2)2z′ − (1− w40w4)√1 + l′2 + (w4 + w40)z′2√√√√(ρ4 + f2w4)( 1w4 + w40+w4 + w40w4) +w4w4 + w40d2The equation of motion with respect to cyclic z, ∂z′L = const.−f(w4 + w40w2)2 − (1− w40w4)√√√√(ρ4 + f2w4)( 1w4 + w40+w4 + w40w4) +w4w4 + w40d2(w4 + w40)z′√1 + l′2 + (w4 + w40)z′2= const. = −4w40fThe consistent constant at the horizon is −4w40f , thusz′ = −f(w4 − w40w2)2√√√√ 1 + l′2(1− w40w4)b(w4 + w40)2 − f2(w4 + w40)(w4−w40w2)4b := (ρ4 + f2w4)(1w4 + w40+w4 + w40w4) +w4w4 + w40d2155Appendix C. BKT Quantum Phase TransitionNow take w0 = 0, then above constant C = 0, and z′ is the same as abovezero temperature case.Consider f 6= 0(f > 0), H 6= 0. The Lagrangian is:L ∼ −√1 +H2u4√r4 + f2u4√1 + l′2 + u4z′2 − fu4z′ (C.23)The equation of motion with respect to l(r), ∂r(∂L∂l′ ) =∂L∂l is following. Weplugged z′ by using the equation for C = 0 into the equation of motion ofl(r).∂r(√r4 + f2u4√1 +H2u4l′√1 + l′2 + u4z′2) = 2u2l((r4+f2u4)(−H2u8)√1 + l′2r4 +H2(f2 + r4u4 )+√1 + l′2r4 +H2(f2 + r4u4 )f2(1 +H2u4) +√r4 +H2(f2 + r4u4 )1 + l′2f2(1 + l′2r4 +H2(f2 + r4u4 ))−2f2√1 + l′2r4 +H2(f2 + r4u4 )= −2r4lu6H2√1 + l′2r4 +H2(f2 + r4u4 )< 0WZ term is generic if we consider S2 flux, and it cancels the attractiveeffect from DBI term. If H = 0, the Chern-Simons (CS) term cancels allnonzero forces by flux term. For H 6= 0, it also vanishes all attraction byflux.Let us start with d 6= 0, H 6= 0 and l = l(r). Then the rescaled andLegendre transformed Lagrangian is:L ∼ −√1 + l′2√r4(1 +1(r2 + l2)2) + d2, (C.24)and it can be expanded up to the quadratic order asL ∼ 12√1 + r4 + d2l′2 − l2r2√1 + r4 + d2(C.25)The equation of motion is:∂r(√1 + r4 + d2l′) +2lr2√1 + r4 + d2= 0 (C.26)156Appendix C. BKT Quantum Phase Transition√1 + r4 + d2l′′ +2r3l′√1 + r4 + d2+2lr2√1 + r4 + d2= 0 (C.27)For AdS metric, the mass opeartor with scalar field is defined as:AdSp+2φ =1√−g∂µgµν√−g∂νφ = r2∂2r + (p+ 2)r∂rφ+ . . . (C.28)At r  1, the equation of motion (45) is:r2l′′ + 2rl′ +2lr4= r2l′′ + 2rl′ + O(1r4) = 0, (C.29)and it is the same as the equation of motion of the AdS4 scalar fieldlr .(AdSp+2 −m2)lr=1√−g∂µgµν√−g∂ν lr−m2 lr=1rp∂rrp+2∂r(lr)−m2 lr= rl′′ + pl′ − prl −m2 lr= 0For p = 2, and m2 = −2 both equations correspond. At r  1, the equationof motion is:l′′ +2r3l′1 + d2+21 + d2lr2= 0, (C.30)and it corresponds to the equation of motion for p = 0, and m2 = − 21+d2.To see the critical phenomena, let us see the solution of the equation ofmotion with CS term.l(r) ∼ c+g+ + c−g− (C.31)ν =dH√1 + f2; d2 +H2(1 + f2) = H2(1 + f2)(1 + ν2)g± = (H2(1+f2)(1+ν2))− 18 (1∓√−7+ν2+f2(1+ν2)(1+f2)(1+ν2))r12 (1∓√−7+ν2+f2(1+ν2)(1+f2)(1+ν2))2F1[x±, y±, z±, w]x± =18(1∓√−7 + ν2 + f2(1 + ν2)(1 + f2)(1 + ν2))y± =18(3∓√−7 + ν2 + f2(1 + ν2)(1 + f2)(1 + ν2))157Appendix C. BKT Quantum Phase Transitionz± = 1∓ 14√−7 + ν2 + f2(1 + ν2)(1 + f2)(1 + ν2)w = − r4H2(1 + f2)(1 + ν2)It seems more appropriate than the Rauthian without CS in [103].α =√d2c − d2H2(1 + f2)(1 + ν2); dc = H√7− f2; δ = ( 1H2(1 + f2)(1 + ν2))1/4; u = δrUsing the convention of [103], our g± is:g± = u1±α2 2F1[1± iα8,3± iα8, 1± iα4,−u4] (C.32)158


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