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Numerical investigation of spatial inhomogeneities in gravity and quantum field theory Vincart-Emard, Alexandre 2017

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Numerical Investigation of SpatialInhomogeneities in Gravity and Quantum FieldTheorybyAlexandre Vincart-EmardB. Sc., Universite´ de Montre´al, 2011M. Sc., University of Waterloo, 2012a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Physics)The University of British Columbia(Vancouver)August 2017c© Alexandre Vincart-Emard, 2017AbstractMany interesting phenomena, such as high-temperature superconductivityand the quark-gluon plasma, still lack a satisfyingly predictive theoreticaldescription. However, recent advances have revealed a curious connectionbetween quantum field theories at strong coupling and classical gravity. Thiscorrespondence, known as the gauge/gravity duality or holographic corre-spondence, offers a promising perspective for investigating strongly corre-lated systems. In this thesis, we focus on using these new tools to examinethe consequences of breaking translational invariance in such systems.We first use this duality to study the holographic realization of a spatiallyinhomogeneous condensed matter device known as a Josephson junction. Wedo so by constructing the gravitational equivalent of two superconductorsseparated by a weak metallic link, from which we then extract various field-theoretic quantities of interest. These include the spontaneously generatedJosephson current, the superconducting order parameter, as well as a novelquantity we refer to as edge currents, which are indicative of gapless chiralmodes localized at the interfaces between phases.We then investigate the more abstract construct of entanglement en-tropy in holographic theories. We model the fast local injection of energyin a 2+1 dimensional field theory and study the resulting thermalization ofquantum entanglement. We achieve this objective by numerically evolvingthe geometry dual to a local quench from which we then compute the area ofvarious minimal surfaces, the holographic proxy for entanglement entropy.We observe the appearance of a lightcone featuring two distinct regimes ofentanglement propagation and provide a phenomenological explanation ofthe underlying mechanisms at play.Finally, we turn our attention to spatial inhomogeneities in gravitationalsystems themselves. We use an approximation of general relativity in whichthe number of spacetime dimensions is infinite to investigate the Gregory-Laflamme instability of higher-dimensional charged black branes. We arguethat charged branes are always unstable in this new language, and push theapproximation to next-to-leading order to compute the critical dimensionbelow which the instability results in horizon fragmentation. We also ex-amine the stability properties of two-dimensional black membranes and findthat the triangular lattice minimizes brane enthalpy.iiLay SummaryBlack holes are mysterious gravitational objects that have long captivatedthe collective imaginary. The rules that govern them, however, are highlyreminiscent of the mathematics describing the quantum dynamics of strongly-correlated electrons. This curious connection, dubbed the gauge/gravityduality, has allowed us to study unconventional materials such as high-temperature superconductors through a new lens with the hope of over-coming the stagnation currently impeding on theoretical progress.In this thesis we use the gauge/gravity duality to study spatially inhomo-geneous systems using the language of black holes. We examine the trans-port properties of arrays of superconductors and investigate the dynamics ofentanglement propagation after a system is locally injected with energy. Wealso explore the spatial instabilities experienced by higher-dimensional blackholes in a new framework where the number of spacetime dimensions is infi-nite. Our goal: understanding black holes better to harness their predictivepower more effectively.iiiPrefaceA version of chapter 2 has been published. Moshe Rozali & AlexandreVincart-Emard, Chiral edge currents in a holographic Josephson junction, A.J. High Energ. Phys. (2014) 2014: 3. My responsibilities were to establishthe notation used in the manuscript, to numerically investigate the model,and to communicate the technical results of sections 3 and 4 with the help ofaccompanying figures. Moshe Rozali fleshed out the physical interpretationof these results and contextualized them by writing the introduction andconclusion.A version of chapter 3 has been published. Mukund Rangamani, MosheRozali & Alexandre Vincart-Emard, Dynamics of holographic entanglemententropy following a local quench, A. J. High Energ. Phys. (2016) 2016: 69.My main contributions were to conduct and summarize the numerical imple-mentation of the local quench as described in sections 2 and 3, and to leadthe exploration of the model that resulted in section 4, which was draftedby all three authors. I also wrote the material found in the appendicesto outline the least obvious details of the implementation. Moshe Rozaliprovided efficient code for the computation of extremal surfaces as well asanalytical and technical guidance throughout. Our collaborator MukundRangamani was responsible for providing context to our results by writingthe introduction and conclusion.A version of chapter 4 is currently undergoing peer-reviewing. I was re-sponsible for the numerical exploration and optimization of the new modelunder consideration, and wrote sections 2 and 3 as well as the appendix,which has been merged with the one from chapter 2 in this thesis for con-ciseness. Moshe Rozali helped draft the results of section 3 and providedthe manuscript’s introduction.A version of chapter 5 has been published. Moshe Rozali & AlexandreVincart-Emard, On brane instabilities in the large D limit, A. J. High Energ.Phys. (2016) 2016: 166. I was in charge of overseeing all of the analytical andnumerical implementation details of this article, guided by Moshe Rozali. Ialso drafted about 95% of the manuscript, whereas Moshe Rozali added finaltouches to improve the quality of the publication further.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Introducing the Gauge/Gravity Duality . . . . . . . . . . . . 51.2.1 Statement of the Duality . . . . . . . . . . . . . . . . 61.2.2 Holographic Dictionary . . . . . . . . . . . . . . . . . 121.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 211.3.1 Holographic Superconductors . . . . . . . . . . . . . . 211.3.2 Entanglement Entropy . . . . . . . . . . . . . . . . . . 281.3.3 Large D Limit of General Relativity . . . . . . . . . . 351.3.4 Numerical Methods . . . . . . . . . . . . . . . . . . . 412 Chiral Edge Currents in a Holographic Josephson Junction 542.1 Introduction and Summary . . . . . . . . . . . . . . . . . . . 542.2 Setup and Solutions . . . . . . . . . . . . . . . . . . . . . . . 562.3 Chiral Edge Currents . . . . . . . . . . . . . . . . . . . . . . . 602.4 Josephson Currents . . . . . . . . . . . . . . . . . . . . . . . . 632.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 Dynamics of Holographic Entanglement Entropy Follow-ing a Local Quench . . . . . . . . . . . . . . . . . . . . . . . . 683.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2 Preliminaries: Holographic Local Quench . . . . . . . . . . . 733.2.1 Metric Ansatz . . . . . . . . . . . . . . . . . . . . . . 733.2.2 Asymptotic Geometry . . . . . . . . . . . . . . . . . . 743.2.3 Boundary Stress Tensor . . . . . . . . . . . . . . . . . 76v3.2.4 Holographic Entanglement Entropy . . . . . . . . . . . 773.3 The Quench Spacetime and Extremal Surfaces . . . . . . . . 783.3.1 Numerical Solutions . . . . . . . . . . . . . . . . . . . 783.3.2 Extremal Surfaces . . . . . . . . . . . . . . . . . . . . 813.4 Propagation of Entanglement Entropy . . . . . . . . . . . . . 833.4.1 An Emergent Light-cone . . . . . . . . . . . . . . . . . 843.4.2 Entanglement Decay . . . . . . . . . . . . . . . . . . . 903.5 Conclusions and Future Directions . . . . . . . . . . . . . . . 914 Comments on Entanglement Propagation . . . . . . . . . 934.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2 Holographic Local Quenches . . . . . . . . . . . . . . . . . . . 954.2.1 Setup for Charged Quenches . . . . . . . . . . . . . . 954.2.2 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . 974.2.3 Integration Strategy . . . . . . . . . . . . . . . . . . . 984.2.4 Holographic Entanglement Entropy . . . . . . . . . . . 994.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.3.1 Emergent Lightcone and Entanglement Velocity . . . 1014.3.2 Entanglement Maximum . . . . . . . . . . . . . . . . . 1054.3.3 Entanglement Decay . . . . . . . . . . . . . . . . . . . 1054.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 On Brane Instabilities in the Large D Limit . . . . . . . . . 1115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2 Charged p-brane Solutions . . . . . . . . . . . . . . . . . . . . 1135.2.1 Uniformly Charged p-Branes . . . . . . . . . . . . . . 1135.2.2 Characteristic Formulation for a Charged Black String 1145.2.3 Solutions and Effective Brane Equations at LeadingOrder . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.3 Phase Structure of the NUBS . . . . . . . . . . . . . . . . . . 1175.3.1 Solutions and Effective Brane Equations at Sublead-ing Order . . . . . . . . . . . . . . . . . . . . . . . . . 1185.3.2 Charged Black String Phase Diagram . . . . . . . . . 1215.3.3 Pinch-Off? . . . . . . . . . . . . . . . . . . . . . . . . 1235.4 Two Dimensional Non-Uniform Phases . . . . . . . . . . . . . 1255.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.1 Summary and Future Directions . . . . . . . . . . . . . . . . 1306.1.1 AdS/CMT . . . . . . . . . . . . . . . . . . . . . . . . 130vi6.1.2 Holographic Entanglement Entropy . . . . . . . . . . . 1326.1.3 Large D Limit of General Relativity . . . . . . . . . . 1346.2 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 136Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137A Apparent Horizons . . . . . . . . . . . . . . . . . . . . . . . . 151B Numerical Implementation of Characteristic Formulation . 153C Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . 156viiList of FiguresFigure 1.1 Double-line propagator for U(N) matrix gauge theories. . 7Figure 1.2 Illustration of power-counting for planar and non-planardiagrams.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.3 Stack of N coincident D3-branes. . . . . . . . . . . . . . . 9Figure 1.4 Diagram illustrating the commutativity of limits assump-tion in the initial derivation of the AdS/CFT correspon-dence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 1.5 The defining hyperboloid of AdSd+1. . . . . . . . . . . . . 14Figure 1.6 Comparison between global AdS and the Poincare´ patch. 14Figure 1.7 The UV/IR relation, illustrated via geodesics and mini-mal surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 1.8 Second-order phase transition of an holographic s-wavesuperconductor. . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 1.9 Conductivity of holographic superconductors as a func-tion of frequency. . . . . . . . . . . . . . . . . . . . . . . . 26Figure 1.10 Depiction of a light-sheet. . . . . . . . . . . . . . . . . . . 31Figure 1.11 Illustration of the entanglement-carrying wavefront ob-served in global quenches. . . . . . . . . . . . . . . . . . . 33Figure 1.12 Free-streaming model in two dimensions. . . . . . . . . . 35Figure 1.13 Geometric construction of the Chebyshev grid. . . . . . . 43Figure 1.14 Illustration of spectral accuracy for numerical differenti-ation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 1.15 Caustic formation and trapped surfaces in the bulk. . . . 49Figure 2.1 The gauge field M3y (r, x) and its associated boundary cur-rent Jy(x). . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 2.2 Edge current as a function of the junction’s depth. . . . . 62Figure 2.3 Edge current as a function of the superconducting orderparameter. . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 2.4 Temperature dependence of the edge current. . . . . . . . 63Figure 2.5 Josephson current. . . . . . . . . . . . . . . . . . . . . . . 64Figure 2.6 Exponential decay of the Josephson current and of theorder parameter as a function of the junction’s width. . . 65Figure 2.7 Temperature dependence of the coherence length. . . . . . 66Figure 3.1 Evolution profiles for the radial shift and the energy density. 80viiiFigure 3.2 Evolution of the total energy after a quench. . . . . . . . 80Figure 3.3 Growth of the apparent horizon following a quench, inboundary time. . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 3.4 Evolution of the geodesics’ radial depth following a quench. 83Figure 3.5 Dependence of the entanglement entropy’s maximum onthe entangling surface’s width for different scalar ampli-tudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 3.6 Dependence of the entanglement entropy’s maximum onthe entangling surface’s width for different initial temper-atures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 3.7 Lightcone at high temperature. . . . . . . . . . . . . . . . 88Figure 3.8 Lightcones and lightcone velocities for three different ini-tial states parametrized by initial temperature. . . . . . . 89Figure 3.9 Lightcones and lightcone velocities for larger scalar am-plitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Figure 3.10 Exponential decay of the entanglement entropy at latetimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 4.1 Lightcone for all charged configurations. . . . . . . . . . . 104Figure 4.2 Maximum of holographic entanglement entropy as a func-tion of strip width. . . . . . . . . . . . . . . . . . . . . . . 106Figure 4.3 Logarithmic decay of holographic entanglement entropyat late times. . . . . . . . . . . . . . . . . . . . . . . . . . 107Figure 4.4 Logarithmic decay of holographic entanglement entropyat late times for various strip widths. . . . . . . . . . . . . 108Figure 4.5 Transition in the scaling behaviour of the logarithmic decay.109Figure 5.1 Entropy differences per unit length as a function of chargedensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Figure 5.2 Phase diagram for the black string instability. . . . . . . . 124Figure 5.3 Change of coordinates for an oblique lattice. . . . . . . . 126Figure 5.4 Black string enthalpy for different oblique lattice angles. . 128ixAcknowledgementsFirst and foremost I would like to thank my supervisor Moshe Rozali, forwithout his continual guidance the projects presented in this dissertationwould never have seen the light of day. I will be eternally grateful for hispatience and optimism, and for always taking the time to discuss with mewhenever I would show up at his door unannounced with endless questions.My experience in grad school was exempt of the crippling pressure too of-ten felt by doctoral students thanks to his mentorship style, which had aprofound impact on my capacity to learn and grow over the past five years.I gratefully acknowledge the contributions of Mukund Rangamani, whohelped shape up our collaboration with his deep insights and his phenomenaltalent with words. My gratitude extends to the members of my supervisorycommittee — Gordon Semenoff, Joanna Karczmarek and Marcel Franz —for their helpful assistance throughout this process.A heartfelt thank you is also in order for Daniel, who has witnessed thejoys and frustrations of research firsthand despite not understanding a wordI said. You have my enduring gratitude for always inspiring me to reachhigher and do better. Thanks for sharing all of the important milestones ofthe past few years with me, and for all the wonderful times that made thisjourney memorable.Last but not least, I am forever indebted to my parents for always be-lieving in me and giving me the tools I needed to succeed. Venturing intothe unknown always felt safer knowing I had their unwavering trust andsupport, and I am proud to share this accomplishment with them.xChapter 1Introduction1.1 MotivationQuantum field theory (QFT) is the perfect union between special relativityand quantum mechanics. Originating in the 1920s in an attempt to quantizethe electromagnetic field, the resulting theory – quantum electrodynamics(QED) – quickly became the most accurate predictive framework that the-oretical physics had ever produced. Besides its high-precision predictions(today’s best theoretical and experimental values for the fine structure con-stant agree to eight significant figures, an unprecedented achievement in allof science), QED also initiated a paradigm shift in our understanding offundamental physics by introducing and formalizing important notions likethat of particles/antiparticles, renormalizability and symmetry breaking, toname only a few. Work in condensed matter, particle and statistical physicsall highly benefited from leveraging quantum field theory’s computationalpower, which helped establish it as the definite language of nature.Quantum field theories are generally formulated in terms of their under-lying symmetries. A particular type of symmetry, namely gauge symmetry,was found to have a profound impact in the way it dictates and limits thescope of possible physical theories. Unlike physical symmetries like thosearising from Noether’s theorem, gauge symmetries do not describe the prop-erties of physical systems. Rather they are indicative of a redundancy in thedescription of physics; two states related by a gauge transformation are to beunderstood as the same physical state. The consequence of requiring gaugeinvariance is a constraint on the form of the interactions between particles.In the case of a U(1) gauge theory like QED, the electron and the photonmay only interact in the way prescribed byLQED = −14FµνFµν − ψ(iγµ(∂µ − iAµ)−m)ψ (1.1)in order for the theory to be invariant under local U(1) rotations ψ →e−iα(x)ψ.The new machinery of gauge invariance ultimately led to quantum chro-1modynamics (QCD), a theory of strong interactions used to model compositeparticles known as hadrons, which include the proton and the neutron. Con-ceptually QCD is simply a field theory with fermionic fields named quarkstaken to be invariant under SU(3) gauge transformations. These funda-mental constituents interact via an octet of gauge bosons known as gluonssubjected to the quarks’ three gauge degrees of freedom (i.e. three typesof charges) known as colours. However, QED and QCD are vastly differ-ent theories. Whereas QED is a weakly-coupled theory (the fine structureconstant is small enough to allow a perturbative expansion of scatteringamplitudes), the theory of QCD is one of strong interactions1 where per-turbative methods fail miserably. This property greatly hinders our abilityto make predictions with traditional approaches, thus other techniques needto be used to expand our understanding of quantum dynamics in this newregime.Quantum chromodynamics is not the only theory that suffers from limi-tations due to strong interactions. In the realm of condensed matter physics,many new metallic materials discovered since the 1980s were found to ex-hibit thermodynamic and transport properties not described accurately byFermi liquid theory, which is widely used to model most interacting systemsof fermions, including metals, insulators, conventional superconductors andsuperfluids. All Fermi liquids have two things in common: their groundstates are characterized by a Fermi surface, and their low-energy excitationsabout that Fermi surface behave as weakly interacting quasiparticles andquasiholes of same electric charge and statistics. Non-Fermi liquids (NFL),in contrast, are strange metals made up of electrons so strongly correlatedthat they cannot propagate long enough to show their particle-like proper-ties. In other words, the decay rate of excitations in a NFL is too large fora quasiparticle interpretation of its dynamics. Novel phases of matter thatfall in that category, such as high-Tc cuprate superconductors and fermionsnear quantum critical points, have so far eluded a satisfactory theoreticaldescription and remain largely misunderstood.Alternative methods thus need to be developed. In practice Monte Carlomethods are often used to estimate the system’s partition function and itsderivatives. This is done by discretizing spacetime on a finite lattice, eval-uating the (real) Euclidean action via random sampling, and then extrap-olating the results to the infinite volume, vanishing lattice spacing limit.Unfortunately this approach has a few limitations. For one the lattice sizes1QCD is an asymptotically free theory, meaning that it becomes free at very highenergies but remains strongly interacting for energies below ΛQCD ≈ 220 MeV.2that can be simulated are very small [1]: a 483 × 64 lattice in QCD cor-responds approximately to a (4 fm)3 volume, which precludes the studyof long-wavelength physics. Such simulations also require prohibitively ex-pensive computational resources. Another conceptual issue known as thefermion sign problem arises when considering theories at non-zero fermiondensity. Such systems are described by a chemical potential that inducesa highly-oscillatory behaviour in the partition function (i.e. the Euclideanaction becomes complex), which effectively invalidates the use of MonteCarlo methods. In addition, lattice methods are limited in scope since theyare only applicable to compute quantities derived from the partition func-tion, therefore prohibiting the study of transport coefficients and far-from-equilibrium dynamics.In both cases, progress has been largely inhibited by the failure of thefield theory framework to be predictive when the fundamental constituentsinteract strongly with one another. As such different tools need to be de-veloped. In 1974, ’t Hooft had an idea: What if we considered an SU(N)gauge theory for some large value of N? Could it be that an expansion in1/N might yield satisfactory predictions for QCD when evaluated at N = 3?He showed that the gauge theory greatly simplified under the assumptionN → ∞, under which the relevant Feynman diagrams are sorted accord-ing to their degree of planarity [2]. Despite this simplification an analyticallarge N field-theoretic solution is still lacking, but progress was made on anentirely different front in 1997 when Maldacena used string theory to for-mulate the AdS/CFT correspondence [3]. Surprisingly, lingering questionsabout strongly coupled field theories would instead find their answers withinclassical general relativity.Maldacena’s conjecture, also known as the gauge/gravity duality, is astatement that relates the physics of conformal field theories (CFT) at largeN with the gravitational dynamics of anti-de Sitter (AdS) spacetimes in onemore dimension. The field theory is said to exist on the boundary of AdS,and the extra radial dimension which extends in the bulk is understood asits energy scale. Since both theories describe the same physics, they sharethe same fundamental degrees of freedom despite the field theory living inone fewer dimension. It is then said that the AdS/CFT correspondence is arealization of the holographic principle, which has warranted the use of thequalificative holographic in regards to the wealth of results obtained withinthis framework.The correspondence’s perhaps most important property is that it is astrong/weak duality that provides a computational bridge between the twotheories: when the field theory is at strong coupling and calculations are3notoriously difficult, they can instead be carried out in the much friendliersetup of classical gravity. For instance, the computation of expectation val-ues is instead mapped to the more approachable problem of solving theEinstein equations with appropriate boundary conditions, whereas quanti-ties like entanglement entropy are assigned a geometrical interpretation asthe area of a minimal surface. Put simply the two theories are two differentfaces of the same coin, and every physical phenomena on one side has a dualdescription on the other.The scope of application of the gauge/gravity duality is also quite vast.Systems at zero temperature correspond to pure AdS spacetimes with no IRcutoff (so as not to introduce a temperature scale), whereas the presence ofa black hole horizon in the gravitational bulk encodes the thermodynamicsof the boundary theory. Similarly, theories at finite charge density possessgravitational duals in the form of charged black holes. The addition ofmatter fields in such backgrounds has led to the study of strongly-correlatedsystems, including but not limited to NFL as a dual description of Diracfermions in AdS, quantum phase transitions via scalar condensation in thebulk, and holographic QCD.The gauge/gravity framework has also succeeded in relating fluid flowin the boundary theory to the dynamics of inhomogeneous dynamical blackholes in AdS, giving rise to what is now known as the fluid/gravity corre-spondence [4]. This holographic hydrodynamics point of view has provenparticularly useful in shaping our physical intuition about matter interac-tions given the absence of quasiparticle concept at strong coupling. More-over, it is possible to go beyond the long-wavelength approximation to studyfar-from-equilibrium dynamics directly by solving the time-dependent Ein-stein equations in all their glory. This regime is often inaccessible to currenttheoretical approaches (as in the case of lattice QCD), thus further addingto the duality’s usefulness and predictive potential.Finally the AdS/CFT correspondence has provided physicists with newtools to address fundamental questions about the nature of spacetime. Is-sues regarding the black hole information paradox and the properties ofentanglement entropy, to name only a few, are readily framed and studiedin this new language. The new geometric insights now possible thanks tothe duality have stimulated creativity and productivity in these areas morethan ever.All in all, the gauge/gravity duality has been a proficient tool for improv-ing our understanding of field theories in the strong coupling regime. Mostof the early results were found for static, spatially homogeneous settings inwhich only the radial dynamics was relevant. The reason, beyond developing4our intuition with simple toy models, was mainly that the dual gravitationaldescription amounted to solving ordinary differential equations that posedno significant computational challenge. However many interesting thingsstart to happen once we consider spatially inhomogeneous setups and time-dependence. For instance, in quantum mechanics the finite well potentialalready leads to novel phenomena like scattering and quantum tunnelling.On account of the complexity of interactions in large N gauge theories, itfollows that richer situations should arise when studying spatially inhomoge-neous systems. On the gravity side, the trade-off is the need to solve systemsof coupled non-linear partial differential equations numerically, a dauntingtask if one is not acquainted with effective techniques to do so.In this dissertation we undertook a numerical study of spatial inhomo-geneities in hopes of discovering the richer dynamics underlying stronglycoupled field theories via the tools of the gauge/gravity duality. Our firstforay led us to study the properties of chiral holographic Josephson junc-tions and the various currents that flow spontaneously as a consequence ofbroken translation invariance. We then took on the ambitious project ofgeneralizing the physics of global quenches – rapid injections of energy ina system – by considering variations of finite spatial extent, as well as theresulting dynamics of holographic entanglement entropy that characterizesthis non-equilibrium process. Finally, we veered away from the AdS/CFTframework to study gravity as an object of its own rather than as a proxy toolby investigating the stability properties of higher-dimensional black branesin asymptotically flat space.We start by motivating the validity of the correspondence in Section 1.2via a series of mostly qualitative arguments from string theory, the unifyingframework from which it initially emerged. The literature review of Section1.3 then serves to bring the reader up to speed in regards to the many top-ics present in this thesis. These topics include expositions to the physicsof holographic superconductors, to the holographic formulation of entangle-ment entropy, to an approximation of general relativity where the numberof spacetime dimensions is taken to be large, and to the many numericalmethods we have used throughout our analysis of these various systems.1.2 Introducing the Gauge/Gravity DualityThe gauge/gravity duality is a statement about the equivalence between cer-tain classes of quantum fields theories and theories of quantum gravity withan additional spacetime dimension. The first realization of this equivalence5was found in 1997 by Maldacena, who used string theory to argue thatN = 4 SU(N) SYM in 4D ←→ Type IIB SUGRA on AdS5 × S5.A few comments are now in order before proceeding further. First, despitebeing initially formulated in four spacetime dimensions, the conjecture hassince then been extended to unite d-dimensional QFTs and AdS gravityin d + 1 dimensions. Second, supersymmetry, which relates the N = 4multiplet of super-Yang-Mills to the compact manifold S5 in the particularexample above, is not an essential ingredient of the correspondence and maybe relayed to the background without affecting its validity. Although super-symmetry is useful in constraining the scope of possible interactions andby providing various quantitative checks on both sides, in this dissertationwe adopt a bottom-up approach to the gauge/gravity duality in which weonly consider the universal subsector of supergravity, namely the one thatdetermines the dynamics of Einstein gravity with a negative cosmologicalconstantSuniversal =116piGN∫dd+1x√−g (R− 2Λ) , (1.2)with possible inclusions of matter fields. Despite being exempt of supersym-metric fields, the action (1.2) is always a consistent supergravity truncationand its solutions may be uplifted to the full string theory if need be. Assuch we will restrict our attention to the study of this decoupled subsector,whose dynamics are understood to be universal across field theories withgravitational duals. Last, we seek not to prove the conjecture in what fol-lows but merely to motivate and justify its validity so that the reader mayfeel confident in the predictions made with this novel theoretical approach,while still being aware of its limitations.1.2.1 Statement of the DualityIn this section we lay out the necessary ingredients to understand the cor-respondence at a qualitative level via the unifying physics of D-branes. Inparticular, we clarify what it means for a gauge theory to be strongly coupledand review the conditions under which the stringy and quantum correctionsof supergravity are suppressed.Gauge theories at large NConsider a Yang-Mills theory with gauge coupling gYM and composed ofmatrix-valued fields Φ that transform in the adjoint representation of U(N).6ab cdhabcdi = adcb =Figure 1.1: The propagator of a U(N) matrix gauge theory in the double-linenotation. Each Kronecker δ represents a flow of colour.A schematic Lagrangian for this theory would be of the formL ∼ Tr [(dΦ)2 + Φ2 + gYMΦ3 + g2YMΦ4 + · · · ] . (1.3)The fields Φ, which may include scalars, fermions and/or gauge fields, areunderstood to be N × N matrices. Under these assumptions, the indexstructure of the propagator is found to be〈ΦabΦcd〉 = δadδcb , (1.4)which can be thought of as encoding colour flow between a fundamental-antifundamental pair2, as shown in Figure 1.1.Looking at vacuum-vacuum Feynman diagrams reveals something in-teresting about matrix-valued gauge theories: double-line graphs can havenon-planar topologies [2]. In fact, every graph can be thought of as thetriangulation of a two-dimensional surface with genus g, which intuitivelycorresponds to the number of “handles” a surface possesses. It can be shownthat the coefficient of a vacuum-vacuum diagram with genus g and 2k ver-tices is N2−2gλk, where we have introduced the ’t Hooft coupling λ ≡ g2YMN .In the case where λ is kept fixed but N is taken to be large, diagrams withhigher genus become suppressed, leaving only planar graphs at leading orderO(N2). In fact planarity affects the number of loops present in a diagram,with non-planar diagrams allowing for fewer colour loops, as illustrated inFigure 1.2. This observation suggests that the theory greatly simplifies whentaking the ’t Hooft limit (also called the planar limit)N →∞ and gYM → 0 with λ = g2YMN kept fixed. (1.5)We remark that the ’t Hooft limit is neither classical nor free despite thegauge coupling gYM going to 0; it can be seen that the diagrammatic ex-2Note that for SU(N) theories, the propagator instead reads 〈ΦabΦcd〉 = δadδcb − δab δcd/Nto account for the tracelessness of the matrix fields (Φaa = 0). However in the large Nlimit the dynamics of SU(N) is equivalent to that of U(N).7/ g2YMN3 = N2(a) A planar diagram./ g2YMN = N0(b) A non-planar diagram.Figure 1.2: An illustration of power-counting for planar and non-planar dia-grams. Figure (a) shows a planar diagram made with 2 three-point verticeseach contributing gYM, and 3 colour loops each contributing a factor of N .In contrast, figure (b) also has 2 three-point vertices but only 1 colour loop,which affects its power of N only.pansion for the partition function includes an infinite number of modes. Intheory one may use perturbation theory if λ is small enough, but the regimeof interest of AdS/CFT is when the coupling is strong, λ 1, and quantumloop corrections of all orders are included.The low-energy limit of D-branesLet us now move on to the topic of Dp-branes, which are topological defectsextended in p spatial dimensions in string theory (with string coupling gs)on which string endpoints can end [5]. D-branes also act as sources for closedstrings, which in this context are best thought of as the excitations of thevacuum. In what follows we adopt two different but equivalent points of viewregarding the low-energy limit of D-branes, meaning at energies lower thanthe one naturally set by the string length `s. For concreteness we considerthe case where p = 3 on which Maldacena’s conjecture was initially based,but note that the analysis can be extended to different brane configurationsas well.We first consider N coincident D3-branes in Type IIB string theory atweak coupling gsN  1. At low energies, where only massless string statescan be excited, gravity in the bulk decouples entirely from the dynamics onthe branes. This leaves us with massless closed strings, which are identifiedas the sources for Type IIB supergravity in flat space, as well as with openstrings. As shown in Figure 1.3, the latter can end on any of the N coinci-dent D3-branes. It is argued in [6] that the resulting effective action on the8Figure 1.3: The low-energy limit of a theory of N coincident D3-branes is atheory of massless strings. The closed strings are responsible for flat spacegravity, whereas the open strings give rise to an SU(N) gauge theory.branes is that of a ten-dimensional SU(N) (supersymmetric) gauge theorydimensionally reduced to the p + 1 worldvolume of the Dp-branes. Equiv-alence between the couplings on the field theory side and in the D-branedescription identify g2YM = 4pigs. We therefore conclude that when gsN  1D3-branes = Free gravity +N = 4 SU(N) SYM in 3+1 dimensions.On the other hand, we can also think of Dp-branes as black p-brane solutionsof classical supergravity when the coupling is strong gsN  1 [7]. Thegeometry of the D3-branes is given by the metricds2 =ηµνdxµdxνf(r)1/2+ f(r)1/2(dr2 + r2dΩ25), f(r) = 1 +4pigsN`4sr4, (1.6)where xµ denotes the 4 coordinates along the worldvolume of the branes anddΩ25 is the metric of a unit five-dimensional sphere S5. For this solution tobe valid, a self-dual 5-form F5 = (1 + ∗)dtdx1dx2dx3df with flux on the S5∫S5∗F5 = N (1.7)9is required. Again there are two types of low-energy excitations that decou-ple from each other. The first corresponds to long-wavelength excitationsin the bulk, i.e. free gravity in flat space. The second corresponds to fi-nite energy excitations that become increasingly red-shifted (as seen froman observer at infinity) as they approach the horizon r = 0. The latter arecaptured by the near-horizon geometryds2 =r2L2ηµνdxµdxν +L2r2dr2 + L2dΩ25, L4 = 4pigsN`4s, (1.8)which corresponds to AdS5 × S5, with L acting both as the AdS radius ofcurvature and the radius of the unit 5-sphere. Thus we conclude that whengsN  1D3-branes = Free gravity + Type IIB supergravity on AdS5 × S5.We therefore have two descriptions of D-brane physics in two distinct regimes.The low-energy limit of the D3-brane system corresponds to an SU(N)gauge theory when gsN  1, whereas it can be described by supergravityon AdS5 × S5 when gsN  1. However we have seen that the gauge theorydescription is valid for all values of the ’t Hooft coupling. It is not too bigof a leap then to think of D-branes as the fundamental object that unifiessupergravity and gauge theories, regardless of their defining regime. In otherwords, the two theories are equivalent representations of the same physics,valid for all values of gs and N , as illustrated in Figure 1.4. This is embodiedin the famous dualityN = 4 SU(N) SYM in 4D ←→ Type IIB supergravity on AdS5 × S5which is most useful when gsN  1 on both sides.We have already reasoned that the gauge theory greatly simplifies whenN  λ  1, which also implies gYM → 0. On the gravity side, this isequivalent toL4`4s= 4pigsN = λ 1, (1.9)which is a statement about stringy corrections to the geometry being sup-pressed since the string scale `s is much smaller than the AdS radius L.Similarly, to suppress quantum corrections one needs to look at the ratioL/`P , where `P is the Planck length, which can be related to the stringlength via `4P = gs`4s in ten-dimensional string theory. We thus obtain the10conditionL4`4P=L4gs`4s=(L4`4s)NgsN∼ N  1, (1.10)which is saying that quantum corrections to the geometry are akin to 1/Ncorrections in gauge theory.SUGRA 3-braneN = 4 SYM (⌧ 1) N = 4 SYM ( 1)⇥ S5,SU(N)low-energylow-energystrong couplingstrong couplingFigure 1.4: This diagram illustrates how the AdS/CFT correspondence wasfirst derived. The starting point is a stack of N coincident D3-branes, inthe upper-left corner. The low-energy limit of this fundamental object re-sults in an SU(N) super-Yang-Mills theory at weak coupling, which is as-sumed valid at any value of the coupling λ. The strong coupling limit ofD3-branes instead yields a 3-brane solution of classical supergravity whosenear-horizon/low-energy limit yields Type IIB string theory on AdS5 × S5.Assuming that the low-energy and strong coupling limits commute, we iden-tify the two theories in the lower-right corner, resulting in the first knownexample of a field theory with a gravitational dual.11In this section we have argued that SU(N) super-Yang-Mills gauge the-ories at large N and strong coupling λ 1 describe the exact same physicsas supergravity on AdS5 × S5 provided it is classical, i.e. provided bothstringy and quantum corrections are suppressed. It is interesting to notethat the duality can be broadened to include spacetimes that are asymptot-ically AdSp+2×χ8−p, where χ8−p is an 8− p dimensional compact manifoldsubject to string theoretical constraints. The conjecture still stands evenif the bulk contains a black hole, with the only difference being that thedual field theory becomes thermal. Moreover, we mentioned that the choicep = 3 in the argument above was for the sake of simplicity; similar analysesfor other brane configurations reveal that, when applicable, strongly cou-pled gauge theories in d dimensions admit a dual gravitational descriptionin d+1 dimensions. In other words, the gauge/gravity duality is at its hearta particular realization of the holographic principle, a notion we will definein more detail in the next section.1.2.2 Holographic DictionaryWe now provide evidence supporting the validity of the AdS/CFT corre-spondence. By exploring the geometry of AdS, we show that its symmetriesare in perfect agreement with those of a CFT. We then explore the idea thatthe correspondence is in fact a realization of the holographic principle bymatching the degrees of freedom on both sides. We finish by formulatingthe field/operator correspondence to establish a direct relationship betweendual quantities. All of these non-trivial checks that form the holographicdictionary give strong foundations to the conjecture.Anti-de Sitter geometrySymmetries play an important and central role in our understanding ofquantum field theories. They constrain the form of interactions, impose theexistence of force carriers as gauge bosons, give us a powerful tool to classifyelementary particles, and so on. In fact, if two theories share the samesymmetries, then there are good reasons to believe that their underlyingmechanisms may well be the same. An early check of the gauge/gravityduality was to identify the symmetries of anti-de Sitter spacetimes with theconformal symmetries of the boundary theory.Let us start by examining d-dimensional conformal field theories, whichare invariant under coordinate transformations that preserve angles. Such12transformations effectively rescale the metric by an arbitrary positive factorgµν(x)→ Ω2(x)gµν(x). (1.11)In the particular case of Minkowski space, conformal transformations includetranslations, Lorentz transformations, dilatations xµ → axµ, and inversionsxµ → xµ/x2. It is a straightforward but tedious exercise to extract theconformal symmetry group from the generators of infinitesimal transforma-tions. It can nonetheless be shown that the conformal algebra obeyed bythese generators is isomorphic to the algebra of SO(2, d) [8].As it turns out, the conformal group SO(2, d) is precisely the symmetrygroup of anti-de Sitter spacetimes, which is easy to verify. The variation ofthe Einstein-Hilbert action with a negative cosmological constantS =116piGN∫dd+1x√−g (R− 2Λ) , Λ = −d(d− 1)2L2(1.12)yields the Einstein equations for which pure d+1 dimensional anti-de Sitterspace, AdSd+1, is a solution. The most straightforward way to study thesymmetries of this spacetime is by considering its defining hyperboloid withradius LX20 +X2d+1 −d∑i=1X2i = L2, (1.13)embedded in a flat d+ 2 dimensional geometry with two timelike directionsR2,dds2 = −dX20 − dX2d+1 +d∑i=1dX2i . (1.14)Its symmetries are therefore described by the group SO(2, d) by construc-tion, which is what we sought in the first place. We however note that thehyperboloid contains closed timelike curves that need to be unwrapped forthe space to be causal, as illustrated in Figure 1.5. Thus the equivalenceapplies only when we consider its universal cover, for which X0 ∈ R.There are many coordinate patches that satisfy the defining equation(1.13) (see Figure 1.6). One such solution is known as global AdS, whosemetric is given byds2 = −(1 +r2L2)dt2 +(1 +r2L2)−1dr2 + r2dΩ2d−1 (1.15)13X0Xd+1~XFigure 1.5: The defining hyperboloid of AdSd+1. The red circle illustratesan acasual closed timelike curve.trrR1,d1Sd1Figure 1.6: On the left is a representation of global AdS, which has a cylin-drical boundary topology. On the right is a representation of the Poincare´patch, which can be viewed as a collection of warped Minkowski spacetimesconnected along a radial direction.and whose coordinates cover the entire hyperboloid. Its boundary, locatedat r → ∞, is spatially compact and has topology R × Sd−1. Alternatively,14the Poincare´ patchds2 =L2r2dr2 +r2L2ηµνdxµdxν (1.16)is the set of coordinates {(r, t, ~x)| r > 0, x ∈ R1,d−1} that cover only halfof the defining hyperboloid (1.13). In contrast to the global chart, theboundary is simply a warped version of Minkowski spacetime R1,d−1. Bothcharts are equally valid, and the choice of which one to use depends solelyon the topology that we require of the boundary theory. Without loss ofgenerality, throughout this thesis we will focus on the non-compact Poincare´patch (1.16), whose metric makes manifest the decomposition of the isometrygroup SO(2, d) of AdS into its subgroups ISO(1, d− 1) (Poincare´ transfor-mations acting on (t, ~x)) and SO(1, 1) (rescaling symmetry). These symme-tries underlie many field theories of interest, whereas the compact nature ofglobal AdS is mathematically attractive but feels physically artificial.The rescaling symmetry of AdS actually provides us with an invaluableinsight in how to think of the extra radial dimension from a field theoryperspective. Indeed, notice how dilatations xµ → axµ on the boundary arebalanced with the bulk radial coordinate scaling as r → a−1r for a > 0.Since energy is conjugate to time, a process with energy E on the boundaryfield theory would necessarily scale as E → a−1E under this symmetry,i.e. exactly like the bulk radius r. This equivalence under rescaling tells usthat r can be interpreted as an energy scale from the field theory point ofview; high-energy/short-distance phenomena is mapped unto gravitationaldynamics at large radius r, whereas low-energy/long-wavelength physics isdescribed by the near-horizon (r = 0) geometry. We examine this generalproperty of AdS spacetimes further in the next section.Counting and matching the degrees of freedomIt is a straightforward exercise to show that the degrees of freedom on thefield theory side of the duality match with the ones allowed in a theoryof quantum gravity. Let us start investigating the latter situation with asimple thought experiment showing that a black hole is the most entropicconfiguration of space.Without loss of generality [9], consider the smallest spherical region ofspacetime with area A that completely encloses a matter system with massm right below the threshold M required for it to become a black hole. Alsoassume that the system has entropy Sm. Let there be a thin shell located15outside the region of interest with total mass δm = M−m such that a blackhole with area A would form upon collapse. The total initial entropy of thematter system is Sinitial = Sm+Sδm, whereas upon collapse the final entropyis that of a black hole, Sfinal = A/4GN , where GN is Newton’s constant. Inorder for the second law of thermodynamics to hold, the initial entropy of thematter system cannot exceed that of the associated circumscribed black hole:Sm + Sδm ≤ A/4GN . We thus learn that the maximal entropy associatedwith a region R of space is proportional to the area of its boundary ∂RSmax(R) = Area(∂R)4GN. (1.17)Equation (1.17) is known as the Bekenstein bound [10], and it is a funda-mental property of gravitational systems in that we have not considered themicroscopic properties of matter in deriving it. The Bekenstein bound is atthe core of the holographic principle since it states that, counter to intuition,any stable region of spacetime can be described by the degrees of freedomon its boundary rather than the ones in its volume.We can now make use of (1.17) to compute the degrees of freedom allowedin the AdS5×S5 geometry. Without loss of generality, let us work in thePoincare´ patchds2 = L2dz2 + ηµνdxµdxνz2+ L2dΩ25, (1.18)which has a boundary at z = 0. The area of this boundary formally diverges,but we can regularize it by introducing an IR cutoff3 at z = δ  1 whichacts as a boundary-like surface with a well-defined area. We findS =14GN∫z=δdΩ5d3x√−g∣∣∣t fixed=Vol(R3)Vol(S5)L84GNδ3∼ Vol(R3)N2δ3,(1.19)where we have used the fact that the ten-dimensional Newton constant isrelated to the Planck length via GN = `8P , by definition.In contrast, the degrees of freedom in a general quantum field theoryscale like the volume of the region of interest. Let’s consider a discretizedversion of an SU(N) matrix field theory by introducing an UV cutoff δdefining the length of the discretized volume cells. There are approximatelyVol(R3)/δ3 such lattice sites, each with N(N − 1) degrees of freedom thatfollow from the dimension of the adjoint representation of SU(N). We can3The new coordinate z is inversely proportional to the radial coordinate r of (1.8):z = L2/r. As such z = δ corresponds to a large distance regularizer in the bulk.16therefore estimate the total number of degrees of freedom in the volume tobeS ∼ Vol(R3)N2δ3, (1.20)in complete agreement with (1.19) up to numerical factors.This matching of degrees of freedom in the two theories suggests thatthe gauge theory living on the boundary of AdS5 is sufficient to describethe gravitational dynamics inside the bulk and vice-versa. The introductionof δ as a cutoff in the derivation above shows that the short-distance (UV)physics of the gauge theory is implemented near the boundary (IR) of AdS5,a phenomena known as the UV/IR relation [11]. Similarly, we may extendthis argument by considering geodesics and minimal surfaces anchored onthe boundary of AdS5 (see Figure 1.7). Geodesics that have endpoints closeto each other stay relatively close to the boundary and probe high-energyphenomena from the field theory point of view. Conversely, geodesics withendpoints distanced further apart and thus associated to long-wavelengthphysics on the boundary necessarily go deeper in the bulk, effectively map-ping the UV of the gravity side to the IR of the field theory. In additionto our previous discussion regarding the rescaling symmetry, this mappingserves as an additional argument for interpreting the extra radial dimensionas an energy scale.Figure 1.7: The gravitational dynamics deeper in the bulk influence thelengths and areas of geodesics and minimal surfaces with larger spatial sup-port on the boundary. Larger regions are associated with coarse-grainedlong-wavelength physical phenomena, which leads to the interpretation ofthe AdS radius r as an energy scale for the dual boundary theory.17Observables and correlation functionsPerhaps the most remarkable aspect of the gauge/gravity duality is the nat-ural prescription it provides for calculating gauge theory correlators. Cor-relation functions are notoriously difficult to compute without the help ofperturbation theory, which fails in the strong coupling regime we are inter-ested in. In the case of conformal field theories, conformal invariance is avaluable asset in determining the natural scaling of n-point functions, butthe conformal toolbox also fails when new scales that deform the CFT areintroduced. However, we have argued that the physical content of super-gravity on asymptotically anti-de Sitter spacetimes can be described entirelyby the gauge theory living on its boundary. As such, classical supergravityshould — and does — offer an equivalent framework in which to computegauge theory correlators at strong coupling. The prescription is implementedviaZCFT[φ0(x)] = ZSUGRA[limr→∞φ(r, x) ∼ φ0(x)], (1.21)which is a statement about the equivalence of the partition functions, andtherefore of the physical content, on both sides of the duality. It assertsthat every field in the bulk of the gravitational theory acts as a source fora local operator in the field theory. The sources themselves correspond tothe leading, usually non-normalizable falloff in the asymptotic behaviourof the bulk fields, in agreement with the fact that local operators describeshort-distance physics.Further simplifications occur when we consider the limit 1  λ  N .The supergravity action is inversely proportional to the 10-dimensional New-ton constant GN and thus scales proportionally to N2. Ignoring quantumand stringy corrections therefore allows us to use the saddle-point approxi-mation, yieldingZSUGRA ' extremum e−ISUGRA , (1.22)where ISUGRA is the on-shell supergravity action on AdS5 × S5. As suchonly the classical equations of motion derived from the on-shell supergravityaction are relevant in the description of field theory physics at leading orderin N .On the other hand, for a CFT governed by an action S[O], the partitionfunction acts as the generating functional of correlation functionsZCFT[φ0(x)] =∫DO e−S[O]+∫ddx φ0(x)O(x) =〈e∫ddx φ0(x)O(x)〉CFT.Thus a theory with Z[φ0(x)] corresponds to a deformation of the pure CFT18Z[0] by a source φ0(x). At strong coupling, where perturbation theory doesnot apply, one may instead compute expectation values, Green’s functionsand higher-order connected correlators in gauge theory by taking functionalderivatives of the generating functional of connected correlation functionsW [φ0(x)] = − log〈e∫ddx φ0(x)O(x)〉CFT' ISUGRA, (1.23)subject to appropriate boundary conditions.In particular, an important lesson one can learn from (1.23) is that ex-pectation values of sourced operators in the field theory can be recoveredeasily by looking at the asymptotic falloff of the bulk fields themselves. Forillustrative purposes, consider a scalar field φ = φ(r) in AdSd+1 subject tothe Klein-Gordon equation− 1√−g∂µ(√−ggµν∂νφ)+m2φ = 0. (1.24)Near the boundary, the ansatz φ ∼ r−∆ reveals that the above equation issatisfied only if the condition∆(∆− d) = m2L2 ⇐⇒ ∆± = d±√d2 + 4m2L22(1.25)is met. Defining ∆ = ∆+ as the largest root, we learn that the scalar fieldhas two independent falloffs near the boundaryφ ∼ φ0rd−∆+φ1r∆. (1.26)We thus identify the coefficient of the leading mode φ0 as the source (onwhich we impose Dirichlet boundary conditions in AdS), whereas holo-graphic renormalization [12, 13] informs us that φ1 measures the response〈O(x)〉 to that source on the boundary. In terms of the dual field theory,∆ (and thus the mass m) corresponds to the conformal dimension of theoperator O. Moreover, nothing prohibits m2 from taking negative values inAdS spacetimes since the normalizable mode proportional to φ1 decays inthe limit z → 0 instead of becoming unstable. In fact, taking m2L2 < 0corresponds to taking ∆ < d, thereby making O a relevant operator, andthe only requirement is that the mass squared is above the Breitenlohner-Freedman (BF) bound, m2 ≥ m2BF = −(d/2L)2, otherwise the dimension∆ becomes unphysically complex [14]. Note that the above discussion isgeneral in that non-normalizable modes correspond to sources, normaliz-19able ones are related to operator expectation values, and mass dictates theconformal dimension of the dual field theory operators4.Equation (1.23) therefore arms us with a useful prescription to easilycompute correlation functions of strongly interacting fields with geomet-ric bulk constructs. It also provides a non-trivial check on the correspon-dence since both quantities can be calculated on both sides. The agreementbetween the CFT calculations, where conformal symmetry constrains thestructure of n-point functions, and their counterpart in anti-de Sitter spacewas of great help in affirming the validity of the correspondence initially.The above prescription also yields a way to identify which bulk fields actas sources for which boundary operators. Let’s consider a charge densitydeformation for concreteness. The introduction of a vector field aµ coupledto a current jµ in the boundary theory amounts to inserting a termδS = −∫ddx aµ(x)jµ(x) (1.27)in the CFT actionZCFT =∫DOe−∫ddx(L−aµjµ). (1.28)The vector field aµ = (µ, ~νs), whose components are the chemical potentialµ and the superfluid velocity ~νs, is associated with the conserved Noethercurrent jµ due to the invariance of the theory under global U(1) rotationsO → e−iαO. Thus we identify j0 = ρ(x) as being the charge density of theboundary theory with conserved charge Q =∫dd−1x ρ(x), and ji as theelectric currents. It can be shown that for matter fields OS[O, ∂µO]− aµjµ ∼ S[O, DµO] (1.29)with Dµ = ∂µ − iaµ. In other words, the chemical potential and the super-fluid velocities enter the CFT as the components of a U(1) gauge field despiteactually being related to a global symmetry. This can be mimicked on thegravity side by introducing a bulk gauge field having aµ as its asymptoticvalue. The corresponding supergravity deformationδI = −14∫dd+1x√−gFµνFµν , Fµν = ∇µAν −∇νAµ (1.30)4Note that both falloffs for the scalar field are normalizable for −d2/4 < m2L2 <−d2/4 + 1, leading to two different CFTs with operators of dimensions ∆± depending onour choice of boundary conditions [15].20with the boundary condition Aµ(r →∞) = aµ satisfies this condition natu-rally. In this case the U(1) symmetry, previously associated to a conservedelectric current on the boundary, now has the interpretation of a gaugesymmetry in the bulk.The identification between bulk gauge invariance and global symmetrieson the boundary is in fact general and applies to all bulk gauge fields andtheir associated conserved currents, as long as both possess the same Lorentzstructure and quantum numbers. A particularly interesting instance of thisproperty is the natural coupling between the bulk metric and the bound-ary energy-momentum tensor, which effectively maps the diffeomorphisminvariance of the metric to global symmetries arising from Lorentz invari-ance. This is a deep connection that we will revisit in more detail when weintroduce the characteristic formulation of general relativity in AdS space-times.1.3 Literature ReviewNow that we have provided arguments that support the validity of thegauge/gravity, we turn our attention to specific applications. In particularwe examine the holographic realization of unconventional superconductorsand discuss entanglement propagation in strongly interacting many-bodysystems via the lens of gauge/gravity. We also introduce the perturbativeframework of general relativity when the number of spacetime dimensionsis large, and end with a short discussion on the numerical methods usedthroughout this thesis.1.3.1 Holographic SuperconductorsChapter 2 initiates the study of holographic p + ip Josephson junctions.Josephson junctions are quantum devices constructed by inserting a thinlayer of normal metal or insulator between two superconducting electrodesand exhibit various interesting properties that arise as a consequence of bro-ken gauge invariance in the superconducting phase. In this section we showhow to use the gauge/gravity duality to build a superconductor and thendiscuss interesting properties of Josephson junctions and their holographicrealization.21Conventional superconductorsConventional superconductors, which undergo a phase transition betweenmetallic and superconducting states at low temperatures, are best under-stood from the BCS theory point of view [16]. Consider a gas of electrons ona lattice made of positive ions. At high temperatures, the dominating forcebetween electrons is their Coulomb repulsion, resulting in a metalling state.However, as the temperature is lowered the interaction between electronsclose to the Fermi surface becomes attractive due to phonon interactionsfrom the lattice, resulting in the binding of electrons in Cooper pairs. Thekey discovery of [16] was to show that these pairs, being composite bosons,are prone to condense, which in turn changes the electronic behaviour andopens up a band gap Eg in the energy spectrum. It is this gap which isresponsible for superconductivity, effectively preventing electron scatteringand other small energy excitations to ensure non-dissipative electronic flow.At the level of symmetries, it is commonly said that the Cooper pair con-densate acquires a vacuum expectation value and spontaneously breaks thelocal U(1) electromagnetic gauge symmetry down to the Z2 of particle-holesymmetry, in accordance with the Anderson-Higgs mechanism [17, 18]. Putdifferently, the Higgs mechanism causes the electron phase to be absorbedinside a massive vector field aµ whose gauge-invariant components are thechemical potential and superfluid velocities [19]. This situation is reminis-cent of our earlier treatment of global U(1) symmetries in AdS/CFT, whichwe summarized via the equivalenceS[O, ∂µO]− aµjµ ∼ S[O, DµO]. (1.31)This similarity strongly suggests that spontaneously breaking gauge invari-ance in the bulk could generate superconductivity on the boundary. and itwas only a matter of time until all of the ingredients were assembled in thegauge/gravity duality to build a holographic superconductor.Holographic superconductorsThe model that we introduce in what follows provides a microscopic de-scription of superconductivity at strong coupling via its dual gravitationalinterpretation. Indeed, holographic superconductivity arises without theneed for quasiparticles; the onset of superconductivity below a critical tem-perature is diagnosed via the condensation of a charged scalar operator, andthe theory is formulated entirely in terms of conserved charges, currents, andexpectation values. The hope is that this first-principle model of supercon-22ductivity may help elucidate the more elusive properties of unconventionalmaterials, such as the high-Tc cuprates.A simple Abelian Higgs model in asymptotically anti-de Sitter space-time was first proposed in [20] to show that, under certain conditions, thenecessary symmetry-breaking ingredients could be achieved in the bulk toreproduce superconductivity on the boundary theory [21, 22]. Consider theLagrangianL = R− 6L2− 14FµνFµν − |∂µψ − iqAµψ|2 −m2|ψ|2. (1.32)The first two terms correspond to the Hilbert-Einstein action for asymptot-ically AdS4 spacetimes, while the remaining terms correspond to the Higgssector, composed of a charged scalar field ψ = ψ(r) and an Abelian gaugefield A = φ(r) dt. Let’s simplify things further by taking m2 = −2 andconsidering the probe limit. Taking q →∞ while keeping both ψ = qψ andφ = qφ fixed in the equations of motion resulting from (1.32) results in bothof them dropping out of Einstein’s equations, yielding the Schwarzschild-AdS4 black hole solutionds2 = −f(r)dt2 + dr2f(r)+ r2(dx2 + dy2), f(r) =r2L2(1− r30r3)(1.33)with Hawking temperatureT =3r04piL2(1.34)and effectively preventing matter fields from backreacting on the geometry.Note that we can use scaling symmetries to set L = r0 = 1, which we donow. For their part, the Maxwell and Klein-Gordon equationsφ′′ +2rφ′ − 2ψ2fφ = 0 (1.35a)ψ′′ +(f ′f+2r)ψ′ +(φ2f2+2f)ψ = 0 (1.35b)remain unchanged, and the asymptotic behaviour of ψ and φ reveals theproperties of the boundary field theory. Asymptotic analysis shows thatφ = µ− ρr+ · · · (1.36)ψ =ψ(1)r+ψ(2)r2+ · · · (1.37)23as r →∞. We choose not to source the scalar field by setting ψ(1) = 0; thisensures that solutions with 〈O〉 ≡ √2ψ(2) 6= 0 (and thus ψ 6= 0) sponta-neously break gauge invariance in the bulk and thus the U(1) symmetry onthe boundary, as is expected from Cooper pair condensation in conventionalsuperconductors. Given the isotropy of the scalar operator in the boundarytheory, such symmetry-breaking solutions correspond to s-wave holographicsuperconductors.As it turns out, the solutions to equations (1.35) do depend on the dimen-sionless ratio T/µ of the black hole solution. Note that there are only twoinequivalent temperatures in a conformal field theory: zero and non-zero. Inour case the chemical potential deforms the dual field theory by introducinga new scale, and maintaining scale invariance requires that all equilibriumquantities depend uniquely on T/µ. Thus changing µ amounts to changingthe temperature of the boundary theory, which we can do continuously. Toidentify the critical temperature Tc at which the phase transition occurs, itis possible to use either µ or ρ since Tc ∝ µ ∝ √ρ.At higher temperatures, which we control by keeping T = 3/4pi andtaking µ small, the only solution to (1.35) isφ = µ(1− 1r), ψ = 0, (1.38)which corresponds to a normal metal in the dual theory. However, low-ering the temperature yields a second-order phase transition in the orderparameter 〈O〉, as shown in Figure 0.2 0.4 0.6 0.8 1.0 1.202468TTcXO\TcFigure 1.8: The normalizable falloff ψ(2) ∼ 〈O〉 serves as an order parameterfor the superconducting phase transition of an s-wave superconductor.24The underlying mechanism explaining why ψ 6= 0 involves the effectivemass of the scalar due to its coupling to the gauge field via covariant deriva-tivesm2eff = m2 + gttq2φ2. (1.39)Very close to the horizon, there is a competition between gtt → −∞ andφ → 0. A stability analysis of black hole perturbations showed that theeffective mass (1.39) is driven below the BF bound at low enough tempera-tures due to the near-horizon profile of the gauge field, in such a way thatthe theory becomes unstable to scalar condensation. Thus not only doesthis gravitational solution provide a dual description of superconductivityat strong coupling, it also acts as one of the few counterexamples to thelong-standing conjectures that black holes cannot have stable hair.Indeed, it is worth mentioning that it is a remarkable discovery for solu-tions with ψ 6= 0 to exist in the bulk. In general relativity, such uncommonsolutions are called hairy black holes, where hair is defined as the free pa-rameters of a black hole not subject to a Gauss law [23]. In fact most blackhole solutions with hair were known to be unstable, including neutral AdSblack holes with neutral scalar hair [24], until [20] showed that Reissner-Nordstrom black holes in AdS are stable (i.e. are a minimum of the freeenergy) under condensation of highly charged scalars.We now turn attention to the study of the electric conductivity of thedual field theory to show that the above setup is indeed a holographic realiza-tion of a superconductor. This is done by linearizing the Maxwell equationsabout a perturbation Ax, for which we allow a time dependence e−iωt. Theconductivity can then be calculated from Ohm’s law (see Figure 1.9)σ(ω) =〈Jx〉Ex= − iA(1)xωA(0)x, where Ax = A(0)x +A(1)xr+ · · · (1.40)An interesting observation is the presence of an infinite DC conductivity(ω = 0) for T < Tc, as is expected from a superconducting phase. A gapfrequency ωg can be extracted from the electric conductivity by looking atwhere Re(σ) stops being exponentially suppressed. Note that the real partof the conductivity is related to dissipative processes, and the presence ofa gap signals non-dissipative electric currents for ω . ωg and implies thepresence of an energy gap in the band spectrum.A surprising feature of holographic superconductivity is the regularityωg ≈ 8Tc across all masses m2 > m2BF for both d = 3 and 4 [25]. Comparingto the BCS theory result ωg ≈ 3.5Tc, we conclude that holographic supercon-250 50 100 150 200 250 3000.ِTReHΣLFigure 1.9: This image shows the formation of a conductivity gap in thedissipative part of the conductivity. This gap increases as the temperatureis lowered; from left to right, these curves have µ = 5, 7, 9, 11.ductors are indeed strongly coupled as they require more energy to overcomethe conductivity gap. We also note that the relation ωg = 2Eg between theconductivity and band gaps is only applicable for weakly coupled systems inwhich there is a pairing mechanism at work. The band gap for holographicsuperconductors may be extracted via the relation Re(σ) ≈ e−Eg/T at smallfrequencies, and we generally observe a non-integer proportionality betweenthe two gaps, indicative of the absence of a quasiparticle interpretation [22].In addition to the symmetry breaking condensate and the properties ofthe conductivity, this gravitational model shares other similarities with realsuperconductors, namely in its behaviour when exposed to magnetic fields(generation of screening currents, existence of a magnetic penetration depth,formation of superconducting droplets, etc.) [22].Other symmetry-breaking solutions have also been found for which theorder parameter exhibits different symmetries (for a review, see [26]). Forinstance, rather than introducing a scalar charged under a U(1) gauge field,it was instead proposed to consider an SU(2) Yang-Mills theory in the bulkwhere the U(1) subgroup of SU(2) is identified with the electromagneticgauge symmetry [27, 28]. The generators of SU(2) satisfy the relation[τ b, τ c] = abcτa, where τa are related to the Pauli matrices and abc is theantisymmetry tensor. Choosing τ3 as the generator of the electromagnetic26U(1), the gauge field ansatz A = Aaµτadxµ withA = φ(r)τ3dt+ w(r)τ1dx (1.41)breaks rotational invariance by picking the x direction as special, therebygiving rise to a px-wave superconductor when the order parameter w con-denses. Similarly, the ansatzA = φ(r)τ3dt+ w(r)(τ1dx+ τ2dy)(1.42)preserves a combined gauge and spatial rotation such that the resultingholographic superconductor is of p+ip-type symmetry. Both solutions breaktime-reversal invariance since the condensate w appears as a magnetic fieldin F 3µν . These additional features allow us to make more precise statementsabout the connection between holographic models and real-life materialssuch as the cuprates, which exhibit similar properties [29, 30].In light of all the above, attempts to study exotic theories with the toolsof the gauge/gravity duality indeed look promising. Despite the deceivinglysimple nature of the models considered, one can hope that the underlyingfoundational principles may reveal generic features of strongly coupled fieldtheories.Josephson junctions and their holographic realizationsWe now turn our attention to Josephson junctions [31], a quantum devicemade of two superconductors (S) separated by a weak link, typically aninsulator (I) or a normal metal (N). SIS and SNS junctions exhibit a pecu-liar phenomenon called the Josephson effect in which a supercurrent mayflow from one superconducting electrode to the other even in the absenceof an externally applied voltage. This phenomena is due to the proximityeffect, explained microscopically at weak coupling by a charge-transfer pro-cess known as Andreev reflection whereby the electrons in the weak linktransfer the order of the superconducting condensate across the interface,giving rise to a supercurrent. The proximity effect can be thought of as amacroscopic quantum tunnelling of charge [32].Josephson junctions thus provide a clear example that lowering the de-gree of symmetry in a system can lead to novel phenomena. The consequenceof spontaneously breaking U(1) gauge invariance is that superconductors“pick up” a particular phase, and it is precisely the gauge-invariant phasedifference ∆ϕ between the two superconducting layers that is responsible27for the Josephson current, whose magnitude isJ = Jmax sin(∆ϕ). (1.43)At the level of symmetries, the Josephson effect occurs as a consequenceof spatial discontinuity of the superconducting material. Given that holo-graphic duals are sensitive to the underlying symmetry structure, the holo-graphic realization of such a junction should be possible.As a matter of fact, a holographic Josephson junction was first con-structed in [33] by letting the chemical potential of the Abelian Higgs modelvary spatially in such a way that two superconducting regions would be sep-arated by a normal metal. The resulting system of equations can be solvednumerically; an asymptotic analysis of Ax, whose presence is required sincetranslation invariance is broken, reveals that the behaviour (1.43) can bereproduced with remarkable precision. Moreover, it was found that themagnitude of the maximal current Jmax decayed exponentially with the sizeof the metallic link as the proximity effect weakens. Now then how doeshaving an order parameter with different symmetry properties affect thephysics of Josephson junctions? We answer this question in more detail inchapter Entanglement EntropyWe now steer away from the condensed matter applications of the holo-graphic duality to take a deeper look at a quite remarkable entry in theAdS/CFT dictionary, which relates the entanglement entropy of a subregionin the dual field theory to the area of a bulk extremal surface anchored onits boundary. In what follows we define the notion of entanglement entropy,outline how to compute its value geometrically, and discuss constraints onentanglement propagation in the context of holography.Holographic entanglement entropyWhen investigating the properties of a quantum field theory, a physicist’sfirst instinct is to look at the correlation functions of its local operators. Themotivation is simple: all dynamical quantities describing the system, suchas scattering cross sections and decay rates of particles, can be derived fromthese correlators. However, lessons from QFT at strong coupling teach usthat some non-local quantities are crucial in characterizing a theory’s phasestructure. For instance, the confinement-deconfinement transition of QCDemploys gauge-invariant Wilson loops as an order parameter.28Entanglement entropy is another example of a non-local physical quan-tity of great interest. As its name suggests, the entanglement entropy of aregion gives a measure of how the degrees of freedom localized within areentangled with the rest of the system. Beyond determining the number ofoperative degrees of freedom in a QFT, entanglement entropy is also usedas an order parameter for characterizing topological states of matter notreadily described by the symmetry breaking paradigm. Understanding thisquantity better may thus provide new perspectives in the methods we useto investigate field theories.Consider a system described by a pure quantum state |Ψ〉, with densitymatrix ρ = |Ψ〉〈Ψ|. The von Neumann entropy of the total systemS = −Tr ρ log ρ (1.44)is necessarily zero, by virtue of |Ψ〉 being a pure state. We now divide thesystem into two regions, A and its complement B. This partitioning amountsto separating the total Hilbert space of the system as H = HA ⊗ HB. Asa result, one may describe the state of the degrees of freedom within Awith total ignorance of what goes on in B via the reduced density matrixρA = TrB ρ, where the trace is taken over HB. The bipartite entanglemententropy is thus defined as the von Neumann entropy of the reduced densitymatrixSA = −TrA ρA log ρA. (1.45)For purely quantum system, SA = SB, whereas the above measure mixesboth entanglement and thermal entropies at finite temperature and SA 6=SB. Entanglement entropy in d dimensional QFTs is also known to obey anarea law5 [35, 36]SA = αArea(∂A)d−2+ finite terms, (1.46)where α is a context-dependent constant and the UV cutoff  is introduced tomodel the divergence of entanglement entropy in the continuum limit. Thisnon-extensive property has an intuitive interpretation: the quantum entan-glement between A and its complement B is strongest at their boundary ∂Aand thus scales as the area of the boundary rather than with the volumewithin. It also evokes a curious connection with the Bekenstein-Hawking5In the case of a 1+1 CFT at criticality, entanglement entropy scales logarithmicallywith the size of A [34].29entropy for black holesSBH =Area of horizon4GN. (1.47)Indeed, a black hole can be thought of as the inaccessible region B whosedegrees of freedom are traced out when computing the reduced density ma-trix ρA. However issues related to the microscopic origin of entanglement,namely the dependence of entanglement entropy on the number of matterfields and on an ultraviolet cutoff, prevent the full realization of this in-triguing connection [37]. Despite these subtletites, the intuitive similaritybetween entanglement entropy and black hole entropy still served as a strongmotivation for the holographic proposal of Ryu-Takayanagi (RT), which wenow discuss.For time-independent d+ 1 dimensional asymptotically AdS spacetimes,it was first proposed in [38, 39] that the entanglement entropy of a boundaryregion A is proportional to the area of the minimal surface γA anchored onits boundary ∂ASA =Area(γA)4G(d+1)N. (1.48)The minimal surface γA is a codimension-2 surface in the bulk that actsas the holographic screen6 with the most severe entropy bound on the lostinformation [40]. The Ryu-Takayanagi proposal has been very successfulin reproducing the analytical results found for the entanglement entropy of1+1 dimensional CFTs. As for higher dimensional CFTs, the computationof entropy is usually complicated and not amenable to analytic results, inwhich case the alternative method (1.48) is of great help in uncovering someof the more elusive properties typical of strongly coupled systems.Calculating entanglement entropy from the area of minimal surfaces is anotion that makes sense in static spacetimes, but the generalization to time-dependent settings is not an immediate one. Minimal surfaces are an ill-defined concept in Lorentzian spacetimes since perturbations in the timelikedirection can be made to decrease their area indefinitely. This difficultycan be avoided in the static case by either Wick-rotating time to obtain aEuclidean geometry, or by restricting our attention to slices of constant time.In other words, the notion of time on the boundary can be seen to extendnaturally in the bulk for static spacetimes such that there exists a canonicalfoliation by codimension-1 spacelike surfaces containing the minimal surfaces6By holographic screen we mean a bulk surface γ whose boundary ∂γ = ∂A isolatesregion A from its complement, “shielding” it from the latter’s degrees of freedom.30of interest. The addition of time-dependence is trivial from the field theorypoint of view since there still exists a natural Hamiltonian notion of timefor dynamical QFTs in fixed backgrounds. However, from a gravitationalperspective we need to ensure that the geometric construction (1.48) can begeneralized in a covariant way to account for diffeomorphism invariance inthe bulk.A covariant holographic entanglement entropy proposal was first outlinedin [41]. In Lorentzian spacetimes, the concept of minimal surface is insteadreplaced by that of extremal surface, i.e. a saddle point of the area functionalArea(E) =∫Edd−1y√det g˜, (1.49)where g˜αβ is the induced metric on E described by coordinates yα. The con-struction proposed in [41] involves light-sheets, which are trapped manifoldscorresponding to congruences of null geodesics with non-positive expansions(expansions measure the fractional rate of change of a geodesic congruence’scross-sectional area [42]) (see Figure 1.10). The extremal surface EA is thensimply the intersection of future- and past-directed light-sheets with van-Figure 1.10: Depiction of a light-sheet. A codimension-2 spacelike surfaceon the boundary (in blue) necessarily has four congruences of null geodesics:future/past directed outgoing/ingoing geodesics. The converging (i.e. non-positive expansion) light rays shown above form a light-sheet in the bulk.31ishing expansions, a condition that results in a stationary point of (1.49).With EA constructed this way, we obtain the time-dependent covariant en-tanglement entropySA(t) =Area(EA)4G(d+1)N. (1.50)Initially formulated as conjectures, the Ryu-Takayanagi proposal and itscovariant analogue have since both been derived from AdS/CFT first prin-ciples [43, 44], thereby firmly cementing their presence in the holographicdictionary. Furthermore, holographic entanglement entropy satisfies manynon-trivial properties, such as consistency with field theory causality [45] inthe covariant case, positivity, continuity, sub-additivity (SA1 +SA2 ≥ SA forA = A1 ∪ A2) and many other inequalities that follow from the propertiesof density matrices [46]. All of these results reveal intriguing connectionsbetween quantum information and geometry, and further investigation onboth sides of the duality are bound to reveal additional insights about thefundamental nature of entanglement.Far-from-equilibrium physics and entanglement propagationWith a time-dependent prescription to compute entanglement entropy inhand, we now find ourselves in a position to ask interesting questions aboutthe thermalization of entanglement for strongly coupled field theories in non-equilibrium settings. Methods for driving a system out of equilibrium thatcan be reproduced both in experimental setups and in holographic modelsinclude quantum quenches, which are deformations of the theory occurringover a relatively short timescale. In a field theory, one may quench a systemby modifying the Hamiltonian and letting the former eigenstate evolve toits new equilibrium. Alternatively, one may excite the ground state of asystem by turning on a collection of sources for a short period of time. Thework done by the sources will drive the system into an excited state thatwill eventually return to equilibrium according to the initial Hamiltonian’sdynamics.Understanding the dynamics of quantum matter out of equilibrium isno easy task, yet holography provides a simple toy model that may of-fer a universal characterization of non-equilibrium processes. One of themost common type of holographic quench in the AdS/CFT literature is aglobal quench modelled after the time-dependent Vaidya metric written in32Eddington-Finkelstein coordinatesds2 =L2z2(−f(v, z)dv2 − 2dvdz + d~x2) , f(v, z) = 1−Θ(v)g(z). (1.51)The Θ(v) function is such that spacetime is pure AdS for v < 0, whereas itis determined by the black hole geometry dictated by g(z) for v > 0. TheVaidya-AdS metric models an infinitely thin shell of null dust collapsing intoa black hole and is thus dual to a thermal quench where a field theory isuniformly and rapidly injected with energy from an external source.This setup has been studied extensively [47–61] in a wide variety of con-texts, including but not limited to the study of dynamical correlation func-tions and Wilson loops, observations of universal scaling laws of boundaryobservables, quenches with charged matter, thermalization of mutual andtripartite information, and generalizations to non-relativistic theories. Inparticular, [47, 61] thoroughly investigated the thermalization of entangle-ment entropy in the Vaidya-AdS geometry and found a universal characteri-zation of entanglement growth. For a spatial region bounded by a surface Σof size R, an analysis of entanglement entropy at macroscopic scales R `eqreveals that∆SΣ(t) = seq (VΣ − VΣ−vEt) + · · · , (1.52)which suggests that entanglement propagates locally, carried by a wave-frontdubbed entanglement tsunami travelling at velocity vE as in Figure 1.11.⌃⌃ vEtFigure 1.11: Illustration of entanglement growth via the entanglement-carrying wavefront ∆SΣ(t) = seq (VΣ − VΣ−vEt), also dubbed entanglementtsunami. The yellow region is entangled with the exterior of Σ while thewhite region has not been affected yet. Figure adapted from [47].33In this scenario, `eq ∼ zh ∼ s−1/(d−1)eq denotes the time scale after whichproduction of thermal entropy ceases to occur locally post-perturbation atthe microscopic level, and the condition R  `eq effectively precludes thestudy of narrow entangling regions, instead focusing on macroscopic regionswhose extremal surfaces probe the deepest part of the IR geometry.The tsunami velocity can be extracted from the metric (1.51); it is equaltovE =(zhzm)d−1√−g(zm), (1.53)where zm minimizes g(z)z−2(d−1). Note that vE is a property of the fieldtheory at equilibrium since it is derived from the black hole geometry. Assuch, it is independent of the details of the initial state. Moreover, it isnatural to expect that the addition of matter sources reduces the efficiencyof equilibration processes. In the bulk, this translates into the statementthat for general black hole geometries, vE ≤ v(S)E , where the latter is thetsunami velocity of a Schwarzschild black hole. Thus v(S)E effectively actsas an upper bound of entanglement growth in strongly coupled systems,which intuitively makes sense since entanglement generation in a gapless fieldtheory is most efficient in the limit of infinite coupling, which correspondsto the Schwarzschild geometry in the language of the correspondence.Nonetheless, the tsunami picture does not provide a description of theunderlying microscopic mechanisms responsible for the spread of entangle-ment in generic field theories. Consider for instance a 1+1 CFT, for whichthe entanglement entropy of an interval of length R grows linearly until itsaturates after a time t = R/2. This behaviour can be understood by con-sidering free-streaming quasiparticles carrying entanglement [62]; the satu-ration time is then understood to be the time required until all EPR pairsproduced locally within the interval meet, thus inducing correlations be-tween local observables (see Figure 1.12). Consequently, interactions maybe disregarded since the long-range entanglement of the final state finds itsorigins from the spread of short-distance correlations.The free-streaming model can be generalized to higher dimensions andcompared to the entanglement tsunami picture [63], but fails to capture im-portant aspects of entanglement growth for strongly interacting systems. In-deed, the tsunami wave-front is found to propagate faster than entanglement-carrying quasiparticles travelling at the speed of lightv(streaming)E =Γ(d−12 )√piΓ(d2)≤ v(S)E < 1. (1.54)34tR20RFigure 1.12: The free-streaming model asserts that entanglement is spreadvia quasiparticles (in blue) propagating at the speed of light. Saturation fora subregion of size R (in red) occurs at t = R/2, corresponding to the timeit takes for all quasiparticles to correlate with one another.This observation suggests that the simplified model of [63], despite repro-ducing many of the scaling behaviours of entanglement found in the Vaidya-AdS setup, is not an appropriate description of entanglement spread in thestrong coupling regime because it neglects the crucial role played by inter-actions. This result is not very surprising considering that the field theoriesunder consideration are not amenable to a quasiparticle description. Onthat account, the model of [47, 61] only provides predictions rather than ex-planations of the principles responsible for entanglement propagation, whichcalls for a closer inspection of entanglement thermalization in holographicsetups, which we investigate in chapters 3 and Large D Limit of General RelativityThe theory of general relativity is, at its heart, a geometric description ofgravitational dynamics. The Einstein equationsRµν − R2gµν + Λgµν =8piGNc4Tµν (1.55)relate the curvature of spacetime, captured by the Ricci tensor Rµν and itstrace R, to the presence of matter, embodied by the energy-momentum ten-sor Tµν . Despite their concise appearance, the Einstein equations are second-order coupled non-linear hyperbolic-elliptical partial differential equations –in other words, very hard to solve. Known analytical solutions usually boast35a high degree of symmetry, which in turn greatly simplifies the equations.However, a myriad of phenomena not prone to such simplifications are alsoencoded within (1.55), and analytical approaches are far and few between.Additionally, we have so far used gravity as a tool to study stronglycoupled field theories. Occasionally this programme led to novel discoveriesabout the physics of black holes, such as the possibility of supporting stablescalar hair in AdS spacetimes, but most of the insights were nonethelessgained the other way around. In this section we introduce a new formal-ism, the large D limit of general relativity, as an attempt at providing ananalytical framework to study gravitational systems themselves. Gravity isinteresting in its own right but its secrets are generally well-guarded, thusthe hope is that this new approximation may yield universal predictionsabout the classical physics of black holes.Large D formalismThe large D formalism, which gives a description of gravity when the numberof spacetime dimensions tends to infinity, is an approximation that greatlysimplifies gravitational dynamics by decoupling the near-horizon region fromthe rest of spacetime. As a result, large D black holes essentially behave likenon-interacting particles of vanishing collision cross-section [64]. Althoughunrealistic, taking D →∞ allows us to make a simplifying expansion in 1/Dabout non-perturbative solutions of Einstein’s equations (e.g. black branesolutions) with the expectation that sensible results may be obtained forintermediate values of D.Let us now examine some of the consequences of taking D to be large.One of the main considerations of this formalism is the introduction of ahierarchy of scales. Take for instance the Schwarzschild-Tangherlini solutionwith a horizon radius r0ds2 = −f(r)dt2 + dr2f(r)+ r2dΩ2D−2, f(r) = 1−(r0r)D−3. (1.56)Letting D →∞ results in a spacetime that is asymptotically flat everywhereoutside the horizon. Expanding f(r) in the neighbourhood of r = r0 for Dlarge, we obtainlimD→∞,r→r0f(r) =Dr0(r − r0), (1.57)informing us that the gravitational field is strongly localized in the near-36horizon regionr − r0 . r0D≡ `κ (1.58)and essentially non-existent outside. The new scale `κ is in fact related tothe black hole’s gravitational gradient∂f∂r∣∣∣∣r=r0→ Dr0= `−1κ (1.59)as well as to its surface gravity.This separation of scale is effectively responsible for the decoupling of thenear-horizon dynamics from the far-region’s; solving the Einstein equationsin the overlap region `κ  r − r0  r0 simply involves the matching ofasymptotic expansions. This is done by requiring regularity at the horizon,which in turn yields effective boundary conditions on large-distance fieldsfrom imposing continuity. In spirit, this matching procedure amounts tointegrating out the degrees of freedom at scales < `κ.In addition to this curvature length `κ, black holes are also characterizedby scales due to their geometry. The area of a unit SD−2 in the large Dlimit isΩD−2 ∼ D√2pi(2pieD)D/2, (1.60)which leads us to define an area/entropy length scale `A related to the eventhorizon’s area`A ∼ A1/(D−2)H ∼r0√D. (1.61)We thus find a hierarchy: `κ  `A  r0. A surprising aspect of the large Dformalism is that the notion of short distances arises from the parametricdependence of these length scales on D rather than from the usual compar-ison of distances with the horizon radius. This hierarchy is important sinceeach scale, once fixed, defines what physical regime we home in as we take Dto infinity. We can see this concretely by comparing the Gregory-Laflammeinstability with black hole quasinormal modes. On one hand, when D islarge black branes are unstable to perturbations with wavelengths largerthanλGL =2pikGL=2pir0√D(1 +O(D−1)) . (1.62)Consequently an appropriate rescaling of the spatial directions along thebrane d~x → d~x/√D is necessary to capture the physics of interest sincethe degrees of freedom < λGL are effectively irrelevant. On the other hand,37black hole quasinormal modes are characterized by very large frequenciesRe ΩQN ∼ Dr0, (1.63)which is an entirely different regime from the Gregory-Laflamme spectrumΩGLr0 ∼ O(D0). As such a study of quasinormal modes at large D willsimply ignore black brane instabilities.These geometric considerations need to be kept in mind when investigat-ing gravitational dynamics of large D systems, or else we might miss effectswe might be interested in. A similar argument applies in the presence ofmatter fields. For example, the addition of a U(1) gauge field necessarilyintroduces a new length scale – in this case a “charge radius” – which mea-sures the gravitational reach of electromagnetism away from the horizon.A corollary of this discussion is that non-geometric quantities like mass orangular momentum are not natural in the large D language; conversion con-stants such as Newton’s constant are needed to assign a conceptually clearmeaning to them.Finally, an important aspect of this formalism is that it makes no claimabout the range of validity and the accuracy of this expansion for interme-diate values of D. Such questions are best answered on a case-by-case basis.However, this approximation is surprisingly successful in capturing somerobust features of higher-dimensional black holes, such as their quasinor-mal modes and the generic Gregory-Laflamme instability that afflicts them,even at low values of D. In fact the largest deviations in the unstable blackstring’s spectrum when D = 7 is about 4%, decreasing to 1% when D = 8and much lower as D is increased [64, 65]. Such optimistic results are ofcourse encouraging the pursuit of new knowledge in this direction.Gregory-Laflamme instabilityWe now conclude this section with a short introduction to the Gregory-Laflamme instability [66, 67], a surprising phenomenon found in higher di-mensional (D > 4) gravity.As we have already discussed, four-dimensional static black holes in flatspace are stable to linearized perturbations and uniquely determined bytheir mass, charge and angular momentum. Their topology is also fixed;only spherical event horizon are allowed. These constraints are easily re-laxed when considering higher-dimensional black holes, as we can see fromcomparing two equally valid solutions when D = 5. The Schwarzschild38solution can be generalized tods2 = −V5(r)dt2 + dr2V5(r)+ r2dΩ23, V5(r) = 1−r25r2. (1.64)This black hole has an event horizon located at radiusr25 =8G(5)N M3pi. (1.65)We can alternatively consider the view where we add an extra linear di-mension z to the four-dimensional Schwarzschild solution. Assuming thatnothing depends on this new dimension, the D = 4 black hole solution isalso a solution to R5a = 0, by construction. This is the so-called black stringds2 = −V (r)dt2 + dr2V (r)+ r2dΩ22 + dz2, V (r) = 1− r0r, (1.66)and r0 = 2G(4)N M .Already we see that the topology of black holes can be much richer whenD > 4. Solution (1.64) features an hyperspherical horizon whose topology isS3, whereas the black string (1.66) is topologically R× S2, i.e. cylindrical.Moreover, the black string solution formally has infinite mass. To remedythis, we take the direction z to be compactified over a circle of length Lsuch that we have a string of finite length and mass, in accordance to theprinciples of Kaluza-Klein theory.Still, it is possible for these two solutions to have the same energy (in con-trast with the uniqueness theorem that prevent this in the four-dimensionalcase), which prevents us from using the principle of least energy to de-termine which configuration is the likeliest. Instead we look at states ofhighest entropy. The entropy of a black object is related to the area A of itsevent horizon in Planck units via S = A/4GN . At this point, an importantsubtlety of Kaluza-Klein theory needs to be taken into consideration. Com-pactifying extra dimensions always results in an effective change in Newton’sconstant, which we need to account for if we are to make sensible compar-isons between the black hole and black string. Indeed, dimensional analysisof the gravitational constant in D dimensions reveals that[G(D)N]= lengthD−2. (1.67)39In particular, we have G(5)N = LG(4)N . Thus the black hole (1.64) has entropySBH =2pi2r354G(5)N=pi2r352LG(4)N(1.68)whereas the black string’s (1.66) isSBS =4pir20L4G(5)N=pir20G(4)N. (1.69)Assuming both configurations have the same mass M and setting G(4)N = 1without loss of generality, we findSBH =√8L27piMSBS. (1.70)This thermodynamic argument shows that the black hole’s entropy surpassesthat of the string for large enough L, which suggests that the latter is subjectto a long-wavelength instability. This instability has been investigated in theseminal papers [66, 67] by Gregory and Laflamme in which they solved thelinearized Einstein equations numerically and confirmed the existence of aspectrum of exponentially growing modes near the horizon.The perturbative nature of this calculation reveals information about theonset of the instability, yet it remains unclear what its endpoint should be.Simply put, there are two possible scenarios: deformation into a non-uniformblack string (NUBS), or fragmentation of the event horizon. Arguments pro-posed in [68] claim that an event horizon can never classically pinch-off andshould settle the critical string into a NUBS. However, computer simulationsshowed evidence to the contrary: the authors of [69] discovered that thelate-time numerical evolution of the five-dimensional black string instabil-ity tended towards a fractal-like distribution of spherical black holes alongever-thinning string regions. The singularity therefore becomes “naked”,thus providing the first counter-example of a classical process that violatesthe cosmic censorship conjecture.Whether quantum mechanical processes take over and prevent the un-cloaking of the singularity as the black string pinches off remains unknownto this day. However it was later discovered that the fate of the black stringis more complex than previously thought as it depends on the number ofspacetime dimensions [70]. Perturbative NUBS solutions close to the criticalpoint of the instability were found to be thermodynamically favoured over40the critical string for D > 13, having both lower energy and higher entropy.In contrast, perturbative NUBS with D ≤ 13 are too massive and not en-tropic enough in comparison to be the final state of the evolution, and thefragmentation scenario is still the most likely candidate of the black string’s(classical) fate.In the large D formalism of general relativity, we thus expect the finalstate of the Gregory-Laflamme instability to be a NUBS at leading order [71].In fact, the procedure of matched asymptotic expansions described in theprevious section succeeded in reproducing the critical dimension D = 13.5of the instability for neutral strings [72] as well as its spectrum to surprisingaccuracy [64]. In light of these results, there is hope that we can learn muchmore about brane instabilities analytically by performing a perturbativeexpansion in 1/D at subleading order, which is the topic of chapter Numerical MethodsThroughout this thesis, we will encounter various non-linear boundary valueproblems and systems of initial value ODEs that do not admit an analyticsolution and therefore require a numerical approach. In this section, wediscuss spectral methods for solving boundary value problems when thedomain is bounded by regular singular points; we outline the celebratedNewton-Raphson root-finding algorithm to deal with non-linear equations;we introduce the characteristic formulation of general relativity, a frameworkparticularly well-suited for the time-evolution of Einstein’s equations in thepresence of a negative cosmological constant; finally, we present the Runge-Kutta-Fehlberg time-stepping algorithm of order 5, ideal for solving non-stiffinitial value problems.Spectral MethodsDue to its nature as a “confining box”, the equations of motion for matterfields in asymptotically AdS static spacetimes are classified as elliptic partialdifferential equations, which require the imposition of boundary conditionson the edges of the domain of dependence. The two radial boundaries,namely the AdS boundary and the horizon, are typically regular singularpoints of the differential equations around which finite differences methodusually perform badly. Spectral methods, on the other hand, have specialconvergence properties not affected as much by such singularities.Spectral methods assume that the solution to a differential equation canbe approximated as a sum of Chebyshev polynomials Tn(x) (well-suited for41non-periodic domains)u(x) ≈ uN (x) =N∑n=0αnTn(x), (1.71)where the coefficients αn are chosen such that the error made by using thisapproximation is minimized. However, in practice it is much simpler to workwith a collocation grid {xi}Ni=0 and to find the unknowns {u(xi)}Ni=0 directlyinstead of solving the system of equations for the αn. The two concepts areequivalent and can be related with the help of Lagrange interpolation, whichwe denote symbolically as{αn}Nn=0 ⇐⇒ {u(xi)}Ni=0. (1.72)It can be proven that Lagrange interpolation is optimal, i.e. that its Cauchyerroru(x)− uN (x) = 1(N + 1)!u(N+1)(ξ)N∏i=0(x− xi) (1.73)is minimized, when the collocation points are chosen to be the maxima ofTN+1(x)xi = cos(ipiN), i = 0, · · · , N. (1.74)Notice how these points are denser near the endpoints x = ±1 than atthe center, as shown in Figure 1.13. This non-uniform choice of grid turnsout to cure the Runge phenomenon afflicting Lagrange interpolation withequidistant interpolation points. In fact, many of the important propertiesof spectral methods are a consequence of this non-uniform density of points,although the related proofs are beyond the scope of this section. The lessonto bear in mind however is that uniform discretization is in fact the worstchoice one can make when it comes to numerical accuracy, and spectralmethods offer an optimal alternative.Now that we are in possession of a collocation grid, it is a straightfor-ward task to define differentiation matrices acting on discretized functions~u with components {u(xi)}Ni=0. Indeed, the same way a centered differenceapproximation for first derivativesu′(x) ≈ u(x+ h)− u(x− h)2h(1.75)may be written in matrix form as D · ~u, where the components of D are42Figure 1.13: The maxima of TN+1(x) (shown at the bottom for N = 15) areuniformly distributed along the unit circle, which results in the boundary-dense grid xi = cos(ipi/N) when projected on the x-axis.Di,i±1 = ±1/2h (with all other entries 0 if we assume periodicity), thespectral grid (1.74) leads to an (N+1)× (N+1)-dimensional differentiationmatrix DN whose components are(DN )00 =2N2 + 16, (DN )NN = −2N2 + 16(1.76)(DN )ii =−xi2(1− x2i, i = 1, · · · , N − 1, (1.77)(DN )ij =cicj(−1)i−jxi − xj , i 6= j, i, j = 1, · · · , N − 1, (1.78)where ci = 2 if i = 0 or N , and ci = 1 otherwise.DN may be thought of as a global differentiation matrix in the sensethat it uses the information at every grid point to compute each deriva-tive u′(xi). This may result in poor accuracy when applied to non-analyticfunctions since local divergences may propagate and “infect” the derivativeeverywhere. However in practice spectral methods are most useful whensmooth functions are involved. As a matter of fact, perhaps the most re-430 10 20 30 4010-1110-810-50.01Nlog Ε(a) Numerical differentiation of f(x) = e−x sin(pix).0 20 40 60 8010-1410-1110-810-50.01Nlog Ε(b) Numerical convergence of order parameter ψ(2) for µ = 9.Figure 1.14: The upper plot illustrates the spectral accuracy of numericaldifferentiation for a smooth function, with error measure N = |f ′(x)−DNf |.The lower plot shows the exponential convergence of the numerical solutionof the holographic superconductor differential equation (1.35). The errormeasure in this case is N = |ψ(2)N − ψ(2)N−1|, the difference in the order pa-rameter for successive values of N . In both cases machine precision preventsfurther improvements starting around N = 20 and N = 40 respectively.44markable property of these differentiation matrices is their spectral accuracy,the very rapid decrease of the error made from numerical differentiation asthe grid size increases (illustrated in Figure 1.14). The convergence rate ofspectral methods typically goes like O(N−m) ∀m for smooth functions, andas fast as O(cN ) for 0 < c < 1 for analytic functions. This is to be contrastedwith the convergence rate of finite difference and finite element schemes, forwhich the error decreases like O(N−m) for a specific m depending on theorder of approximation and smoothness of the function under consideration.As a result, spectral methods can reach an accuracy on the level of machineprecision for moderate values of N , which make them very powerful whenemployed on modest desktop computers.Newton-Raphson MethodThe equations of general relativity are generally coupled and non-linear andoften require a numerical approach. After discretizing differential equationson a collocation grid, they effectively become systems of algebraic equa-tions for which many root-finding techniques already exist. The easiest andmost widely used technique is the Newton-Raphson method, which we nowproceed to describe.Consider the algebraic system of equationsF(u) = 0, (1.79)which for our purposes includes both the discretized differential equation(s)and boundary conditions of interest. The Newton-Raphson method is aniterative process in which an initial guess is improved repeatedly until aconvergence condition is met. For the sake of the argument, let’s assumethat u + ∆u is one of the solutions to (1.79), in which caseF(u + ∆u) = 0 = F(u) + J(u) ·∆u +O(∆u)2, (1.80)provided that the displacement ∆u away from u is small enough. Note thatwe have introduced the Jacobian matrix, whose components are given byJab =∂Fa∂ub. (1.81)Equation (1.80) informs us about the optimal direction ∆u towards a root.The Newton-Raphson algorithm simply turns the information contained inthis displacement into an iterative process that works even if we are (mod-45erately) far away from a root, as each iteration takes us closer to a solution.Given an initial guess u0, improved iterates are found according toui+1 = ui − J−1(ui) · F(ui), i ≥ 0, (1.82)and progress is stopped when a root is found.There are two ways to determine convergence. We may look at |F(ui)|directly, which should approach zero as the algorithm converge. It is alsopossible to monitor the norm of each individual displacement; if |ui+1−ui| <δ for a specified tolerance parameter δ, then successive iterates no longerchange significantly and we conclude that convergence has been reached.Convergence depends strongly on the guess provided initially; differentinitial guesses may converge towards different solutions. If so, we say thatu0 is in the basin of convergence of a root. In practice, the choice of u0 isoften informed by our physical intuition or our understanding of the prob-lem at hand, with better choices resulting in a significant improvement inperformance.Despite its simplicity, this root-finding algorithm is very powerful. Inthe case of spectral methods, Newton-Raphson’s quadratic convergence ratetogether with spectral accuracy add up to compensate for the cost of invert-ing a dense Jacobian, thus yielding an efficient way to solve coupled ellipticPDEs at high precision with modest computational resources.Characteristic FormulationGauge theories with gravitational duals have been used in a wide array ofapplications so far, mostly in systems found in or near equilibrium. For in-stance, they have been most useful in extracting the transport coefficients ofstrongly interacting theories via linear response theory, and have also givenrise to an hydrodynamic approach when studying their IR physics. Thegauge/gravity duality also provides a framework making it possible to go be-yond these limited regimes, enabling us to describe the far-from-equilibriumdynamics of quantum field theories by investigating gravitational infall inAdS. The most interesting problems usually involve the evolution of non-trivial inhomogeneous boundary conditions resulting in a non-static geom-etry, which in turn require the Einstein equations to be solved numericallyin all their glory. As one would expect, this is no easy task; neverthelessframing the AdS initial value problem in a characteristic formulation allowsfor a systematic and stable approach to this problem, even in the absenceof a high degree of symmetry.46The characteristic formulation of general relativity requires one to framethe gravitational infall problem in a coordinate system based on a null slicingof spacetime along infalling null geodesics. The resulting metric ansatz is ageneralization of ingoing Eddington-Finkelstein coordinatesds2 =r2L2gµν(x, r)dxµdxν − 2 ωµ(x)dxµdr. (1.83)In the above r is a radial coordinate such that the AdS boundary is located atr =∞, {xµ} denote the d boundary coordinates where t = x0 is a null timecoordinate in the bulk which coincides with field theory time at r =∞, andthe vector ωµ is assumed to be timelike and physically represents fluid flowon the boundary. This is indeed a null slicing of spacetime since keeping allcoordinates but r fixed results in ds2 = 0, and we say that ∂r is a directionalderivative along infalling null geodesics.The metric (1.83) restricts diffeomorphism invariance by allowing onlytwo types of residual diffeomorphisms, which are easily fixed. The metric isinvariant under d-dimensional changes of coordinates as well as under radialshiftsxµ → x¯µ = f(xµ), (1.84)r → r¯ = r + λ(x). (1.85)The former can be used to set ωµ(x) = −δ0µ, whereas the latter can be usedto gauge-fix the apparent horizon’s location7. This is important becausegravitational infall problems involve excitations falling into a black hole, thuscreating ripples on the horizon such that its location varies in time and space:rh = rh(t,x). The radial shift field λ(x) encodes the dependence on x ofthese deformations, and gauge-fixing radial diffeomorphisms lets us lock theIR boundary of the radial domain in place throughout the time-evolution.Doing so not only eases numerical implementation since the computationaldomain of the Einstein equations remains fixed in size, but also lets us avoidstability issues by excising the unphysical region behind the horizon.The ansatz (1.83) provides suitable coordinates throughout the domainas long as coordinate singularities do not develop in the bulk, and providedthe apparent horizon remains planar. On one hand gravity’s attractive na-ture may result in the focusing and intersection of infalling radial geodesics,7The event horizon of a black hole depends on the full history – including both pastand future – of the spacetime geometry and as such cannot be determined locally, but theapparent horizon can. See Appendix A for details.47a phenomenon called caustic formation. Caustics are coordinate singulari-ties for which an event (x, r) is not uniquely determined anymore, and thenull foliation of spacetime breaks down. As a result, the ansatz (1.83) inthe presence of caustics requires different coordinate patches for different re-gions in order to preserve regularity. On the other hand apparent horizonsare defined as the boundary of trapped outgoing null geodesic congruences.If such a compact trapped surface formed inside the bulk, it would invalidatethe global regularity of our ansatz and potentially lead to caustic formationas well. As shown in Figure 1.15, it is possible to cure these singularities inboth cases simply by increasing the IR cutoff in a way such that the causticsform behind it, at the cost of losing information on low-energy physics onthe boundary. Thankfully such pathologies do not arise often, if at all, inpractice.Let us now turn our attention to the special structure of Einstein’s equa-tions under the metric ansatz (1.83), which we may rewrite asds2 = Gij(X)dxidxj + 2dt [dr − Eµ(X)dxµ] , Eµ = (A,Fi), (1.86)where X = (r, t, xi) denotes all bulk coordinates. Under radial shifts (1.85),the components of Eµ transform as a gauge fieldEµ(r, x)→ Eµ(x, r¯) = Eµ(x, r¯ − λ) + ∂µλ(x). (1.87)It is natural to implement this gauge invariance in the Einstein equationsby defining the derivativesd+ = ∂t +A(X)∂r, (1.88a)di = ∂i + Fi(X)∂r, (1.88b)which transform covariantly under radial shifts and d-dimensional diffeomor-phisms. The modified temporal derivative (1.88a) is a directional derivativealong outgoing null geodesics, whereas (1.88b) has an analogous geometri-cal interpretation as a derivative along spacelike directions orthogonal toingoing and outgoing radial null geodesics.An extensive analysis reveals that rewriting the Einstein field equationsin terms of the new time derivative d+ effectively separates them into twoclasses. The first contains radial equations for auxiliary fields, which can besolved sequentially on each t = constant slice, whereas the second containsthe dynamical equations that let us evolve the geometry from one null timeslice to the next. This distinction however is not unique and different choices48xrxr(a) On the left: the region of spacetime coloured in grey focuses infalling geodesics(in blue), leading to caustic formation in the bulk. On the right: the caustics formbehind the apparent horizon (in red).rxxr(b) On the left: a compact trapped surface discontinuously develops in the bulk.On the right: the apparent horizon maintains its planar topology. Note: lightershades correspond to later times.Figure 1.15: These figures, adapted from [73], illustrate the two types ofpathologies leading to the breakdown of the characteristic formulation.49result in different schemes of variable stability. Nonetheless, the sequentialnature of the different equations involved is what makes the characteristicformulation powerful. Expressing the field equations in terms of ingoing andoutgoing null vectors, namely ∂r and d+, allows the repackaging of their non-linearities in a sorted sequence of radial ODEs easily handled by spectralmethods.All we need to complete our discussion is a set of initial conditions thatencode the phenomenon of interest. Initial conditions for this type of grav-itational infall problem in AdSd+1 are provided partly from the particulardetails of the modelization and partly from boundary data, namely fromthe boundary stress tensor. In particular, the asymptotic behaviour of thespatial part Gij of the metric (1.86) ultimately determine the initial stateof the system (e.g. anisotropy, shape of colliding shockwaves, etc.) whereasA and Fi fix the energy and momentum densities of the field theory andneed to be specified on each time slice. To extract this information, we firstrewrite the metric in Fefferman-Graham formds2 =L2ρ2(gµν(x, ρ)dxµdxν + dρ2)(1.89)withgµν(x, ρ→ 0) ∼ ηµν +∞∑n=dg(n)µν (x)ρn. (1.90)In these coordinates, holographic renormalization tells us that part of themetric at order d remains undetermined by the equations of motion andrequires additional information from the CFT via the identification (up toa normalizing constant)〈Tµν(x)〉 ∼ g(d)µν (x), (1.91)plus possible additions depending on the matter content of the theory [12,13]. With this information put in by hand, the evolution scheme can thenbe implemented straightforwardly due to its sequential nature. We havefound that the Runge-Kutta-Fehlberg time-stepping algorithm of order 5was particularly useful in evolving the geometry along null time. As such itis the topic of the following section.50Runge-Kutta-Fehlberg with Adaptive Step SizeThe Runge-Kutta (RK) methods are powerful explicit time-stepping algo-rithms useful in solving initial value differential equations of the formdydt= f(t, y); y(t0) = y0. (1.92)The key idea behind RK-type algorithms is to express the solution y(t) interms of derivative terms f(t, y) sampled at many intermediate values in away that minimizes truncation error. The standard, most used method is byfar the fourth-order method known as RK4, which uses the time-evolutionschemey(t0 + h) = y(t0) +h6(f1 + 2f2 + 2f3 + f4) , (1.93)wheref1 = f(t0, y0);f2 = f(t0 +h2, y0 +h2f1);f3 = f(t0 +h2, y0 +h2f2);f4 = f(t0 + h, y0 + hf3). (1.94)This algorithm performs wonderfully in many cases, but keeping the stepsize h fixed may be inefficient. Consider for instance a solution with bothslow- and fast-changing regions. If one were to employ RK4 to solve thecorresponding differential equation, one would have to set h equal to thesmallest numerical resolution required in regions of most rapid variationsto maintain accuracy throughout the domain. Doing so may prove wastefulin better-behaved regions, where a larger step size would yield an equallysatisfying error.The easiest way to resolve this issue would be to compute the solutiontwice with step sizes h and h/2, and then evaluate the difference as a measureof the error. However this comparison involves twice the amount of compu-tational work and is thus an inefficient solution to our efficiency problem.The insight of Fehlberg was to realize that the fourth- and fifth-order RKmethods both use the same intermediate function evaluations f(t, y), but51weighted differently. They are given respectively byy(4)(t0 + h) = y0 + h ~b(4)i · ~f, ~b(4) =(25216, 0,14082565,21974104,−15, 0), (1.95)y(5)(t0 + h) = y0 + h ~b(5)i · ~f, ~b(5) =(16135, 0,665612825,2856156430,− 950,255),(1.96)in which the function samples are sequentially given viafi = ft0 + cih, y0 + h 6∑j=1aijfj , 1 ≤ i ≤ 6 (1.97)with~c =(0,14,38,1213, 1,12)(1.98)anda =0 0 0 0 0 014 0 0 0 0 0332932 0 0 0 019322197 −72002197 72962197 0 0 0439216 −8 3680513 − 8454104 0 0− 827 2 −35442565 18594104 −1140 0 . (1.99)The error can then be evaluated by taking the difference between the twoapproximationsE = |y(5) − y(4)| = h |~r · ~f |, ~r =(1360, 0,− 1284275,− 219775240,150,255).(1.100)Our last task is to determine a condition for the step size to continuouslyadapt to our error tolerance. The truncation error of the difference betweenfourth- and fifth-order RK methods is|y(5) − y(4)| ≈Mh5 (1.101)for some constant M . If we denote  as our tolerance in the derivative’serror, then the function itself must have an error E ∼ h. We can thenfind an optimal step size h¯ that sets the truncation error to be equal to the52prescribed tolerance by imposing the conditionh¯ = Mh¯5 =h¯5h5E (1.102)and therefore obtain the improved step sizeh¯ = h 4√|~r · ~f |. (1.103)If the calculated error E is higher than the tolerance, then we deduce thath > h¯ and a smaller step size is required. The prior step then needs to berejected and calculated anew with hnew < h. Conversely, if the calculatederror is lower than the tolerance, then it is possible to take larger stepswithout any penalty. In that case h offers a good estimate of the nextiteration’s step size. As a result, the time-stepping algorithm is alwaysadjusting h in a way that optimizes both resources and errors.53Chapter 2Chiral Edge Currents in a Holo-graphic Josephson JunctionWe discuss the Josephson effect and the appearance of dissipationless edgecurrents in a holographic Josephson junction configuration involving a chiral,time-reversal breaking superconductor in 2+1 dimensions. Such a supercon-ductor is expected to be topological, thereby supporting topologically pro-tected gapless Majorana-Weyl edge modes. Such modes can manifest them-selves in chiral dissipationless edge currents, which we exhibit and investi-gate in the context of our construction. The physics of the Josephson currentitself, though expected to be unconventional in some non-equilibrium set-tings, is shown to be conventional in our setup which takes place in thermalequilibrium. We comment on various ways in which the expected Majorananature of the edge excitations, and relatedly the unconventional nature oftopological Josephson junctions, can be verified in the holographic context.2.1 Introduction and SummaryThe physics of topological insulators and superconductors has become acentral topic in modern condensed matter physics (for reviews see [74, 75]).Many of the interesting phenomena exhibited in such materials follow fromthe existence of topologically protected gapless edge modes. For topologicalsuperconductors, these are expected to be chiral Majorana modes. Thesearch for such Majorana excitations in various condensed matter systemsis currently an intense experimental effort (for a review see [76]).Topological superconductivity is expected to arise in time-reversal break-ing superconductors, with a “p+ip” order parameter symmetry, which werefer to here as chiral superconductors. Experimentally, such topologicalsuperconductivity might arise intrinsically, for example in the StrontiumRuthenate Sr2RuO4 (see e.g. [29, 30] for reviews), or by proximity effect(following the suggestion of Fu and Kane [77]). That system was analyzedby Green and Read [78], who demonstrated the existence of Majorana-Weylfermions propagating on the edges of a two-dimensional chiral superconduc-54tor1. A particularly clear construction of the edge modes as Andreev boundstates can be found in [80].In this note we use the tools of gauge-gravity duality to investigate thetopological nature of the holographic superconductor. As we will see, themanifestation of topology comes in the form of chiral edge excitations whichmanifest themselves as chiral currents localized at edges of the superconduc-tor. To this end, we construct a gravity solution that exhibits the basic phe-nomena associated with topological superconductivity, namely topologicallyprotected edge modes and spontaneously generated edge currents. Indeed,after the observation of [20] that black holes can be unstable to scalar con-densation, an s-wave superconductor was constructed in [22]. Holographicduals to p-wave superconductors were constructed in [27], and the modelwe are using here, an holographic dual to a chiral superconductor, was con-structed in [81]. We review that construction in section 2 below2.Since much of the new interesting physics associated with topological su-perconductivity has to do with edges and interfaces, we construct a Joseph-son junction involving the holographic chiral superconductor. HolographicJosephson junctions were constructed first in [33] (see also [82–84] for otherconfigurations). Our work will focus on building an S-N-S holographicJosephson junction for the holographic chiral superconductor (S) for whichthe weak link is a normal metal (N). The construction of the gravity solu-tion involves the numerical solution of a set of partial differential equations,details of which are presented in section 2.One of the dramatic manifestations of the topologically protected gap-less modes are spontaneously generated dissipationless currents, localizedat the edges of a topological superconductor. The relation between theedge currents, edge states and gauge invariance is explained in [80]. SinceJosephson junctions involve two such interfaces between topological andnon-topological materials, we expect to find counter-propagating currents,one on each interface. Such currents are clearly visible in our setting andwe discuss their features in section 3 below. We find that up to small cor-rections, the strength of the edge currents is determined by the jump of theorder parameter amplitude across the interface between the superconducting1See however [79] for a recent null experimental result in Sr2RuO4.2The model we discuss here supports competing orders, indeed the p-wave order pa-rameter [27] is thermodynamically preferred in this model. Our Josephson junction istherefore an idealized configuration, but is nevertheless an interesting probe of the time-reversal breaking holographic superconductor. We expect that the features we uncoverhere, to do with topological structures, are insensitive to the phase structure of the fullmodel.55material and the weak link.The counter-propagating edge currents we observe are independent ofeach other (for wide enough junctions), and would exist for a single isolatedinterface as well. They are indicative of chiral gapless edge modes localizedon such interfaces3. The full Josephson junction has a pair of these modes,a feature which is speculated to be responsible for some unusual propertiesof the topological Josephson junction. Therefore, in section 4 we turn toexamine the Josephson current in our junction.Anomalies in the current-phase relation in such “unconventional” Joseph-son junctions were reported in [85], but a more recent direct measurementreveals a conventional relation [86]. While the physics of such junction isexpected to be unconventional, in that it is 4pi periodic in the phase acrossthe weak link [87], equilibrium configurations might still exhibit the conven-tional 2pi periodicity. Other attempts to discover unconventional periodicityas a signature of the aforementioned pair of Majorana bound states includethe AC Josephson effect [88], Josephson junctions in magnetic fields [89],current noise measurements [90] or unconventional Shapiro steps [91].In section 4 we exhibit the details of the Josephson effect in our holo-graphic construction. We find conventional results, fairly similar to thes-wave case reported in [33]. The current-phase relation is 2pi periodic andthe maximal current decays exponentially as the width of the junction in-creases (as opposed to the power law decay observed in [85]). Furthermore,the temperature dependence of the critical current and the coherence lengthare also found to be fairly conventional. We conclude that our setup, whichtakes place in thermal equilibrium, is thus insensitive to the unconventionalfeatures expected to arise from the presence of gapless Majorana modes.We conclude in section 5 with outlook and directions for future work.In particular, we outline some calculations that would verify the existenceof gapless Majorana modes and exhibit the expected doubled periodicity ofthe physics in the Josephson junction. We hope to report on such results inthe near future.2.2 Setup and SolutionsOur discussion of the time-reversal breaking holographic superconductor [81]follows the conventions of [28]. Let us consider the following action:3For the existence of a charge current, at least two Majorana-Weyl fermions are re-quired.56S =∫d4x√−g[R+6L2− 14g2(F aµν)2](2.1)where F aµν = ∂µAaν − ∂νAaµ + abcAbµAcν is the field strength tensor for anSU(2) gauge field, and abc is the totally antisymmetric tensor, with 123 = 1.The gauge field can be conveniently expressed as a matrix-valued one form:A = Aaµτadxµ, where τa = σa/2i, σa being the usual Pauli matrices. Itfollows that [τa, τ b] = abcτ c.We will be working in the probe approximation, thereby neglecting thebackreaction of the gauge field on the metric. In the current model, theprobe approximation, controlled by the ratio of the Newton’s constant tothe gauge coupling, breaks down at sufficiently low temperatures. However,though adding backreaction should be straightforward, this is unnecessaryfor the effects we are interested in, and we will restrict ourselves to workingin a fixed gravitational background.Specifically, we choose the metric to be the asymptotically AdS4 planarSchwarzschild black hole: and L and r0 are the AdS and horizon radii,respectively. Such a black hole has Hawking temperatureT =14pidhdr∣∣∣∣r=r0=3r04piL2(2.2)Scaling symmetries further enable us to work in units in which L = 1 and setr0 = 1. This corresponds to measuring all dimensional quantities in units oftemperature.To understand the symmetry structure of our ansatz, it is useful to definecomplex coordinatesζ =x+ iy√2and τ± =τ1 ± iτ2√2(2.3)The ansatz for the spatially homogeneous p+ ip superconductor is given by[28, 81]A = Φ τ3dt+ wτ−dζ + w∗τ+dζ¯ (2.4)Here Φ breaks the SU(2) symmetry explicitly to an Abelian subgroup at theshort distance scale of the chemical potential µ, and w is the order parameterwhich breaks the U(1) symmetry spontaneously at a much longer distancescale. Note that in these conventions a U(1) gauge transformation is a phaserotation of the complex order parameter w. In the homogeneous case w canbe chosen to be everywhere real, but with inhomogeneities this is no longer57the case. In particular the phase difference across the Josephson junction isan interesting gauge-invariant quantity directly responsible for driving theJosephson current.The order parameter w is invariant under a combination of spatial rota-tions and gauge transformations, thus the superconductor is isotropic. Sincew is complex, time-reversal is broken spontaneously. As we see below, thishas interesting consequences for the physics probed by the Josephson junc-tion in this system.In order to build a holographic Josephson junction, the fields must havespatial dependence. We model a Josephson junction by choosing an appro-priate profile for the chemical potential µ(x), as described below. The fieldsthen all depend on the spatial coordinate x and the radial coordinate r. Ouransatz for a p+ ip Josephson junction is thenA = Φ τ3dt+ wτ−dζ + w∗τ+dζ¯+Axτ3dx+Myτ3dy +Arτ3dr (2.5)We are using a somewhat mixed notation for the spatial directions, wherethe chiral nature of the order parameter is most clearly exhibited using the ζcoordinate defined in (2.3). Note the presence of the field My which, unlikeother instances of holographic Josephson junctions, cannot be eliminatedusing symmetries. This field will encode the presence of dissipationless edgecurrents, which we discuss below.Following [33] we choose to work in terms of gauge invariant combina-tions. If w = |w|eiθ, those are w ≡ |w|, My and Mµ = Aµ−∂µθ for µ = x, r.Our ansatz yields the following system of 5 coupled non-linear elliptic PDEs,∂2rΦ +2r∂rΦ +∂2xΦr2h− 2w2Φr2h= 0∂2rw +h′h∂rw +∂2xw2r2h− 3w∂xMy2r2h+wΦ2h2− w3r2h− (M2x +M2y )w2r2h−M2rw = 0∂2rMx +h′h(∂rMx − ∂xMr)− ∂r∂xMr − Mxw2r2h= 0∂2rMy +h′h∂rMx +∂2xMyr2h+3∂x(w2)2r2h− Myw2r2h= 0∂2xMr − ∂r∂xMx − 2Mrw2 = 0(2.6)58and an additional constraint:∂r(hMrw2)+12r2∂x(w2Mx)= 0 (2.7)We thus need to choose boundary conditions such that the constraint issatisfied at the boundaries of the integration domain.Next we discuss the boundary conditions satisfied by our fields. Theboundary conditions at the horizon are determined by requiring regularityand satisfying the constraint. That is, when expanding the equations of mo-tion and constraint near the horizon, divergent terms arise which we requireto cancel. At the spatial boundaries we impose Neumann boundary condi-tions on all fields. Near conformal infinity the fields behave asymptoticallyasΦ(r, x) = µ(x)− ρ(x)r+ · · ·w(r, x) = w(1)(x) +w(2)(x)r+ · · ·Mx(r, x) = vx(x) +Jxr+ · · ·My(r, x) = vy(x) +Jy(x)r+ · · ·Mr(r, x) = O(1r3)We will input µ(x) to model a Josephson junction, and choose the conden-sate w to be normalizable, w(1)(x) = 0. The current in the x-direction Jxis constant by the continuity equation and we choose it to be one of theparameters of our solutions. The conjugate quantity vx(x) is then read fromthe solution and encodes the phase difference across the junction. Finally,we set vy(x) = 0 as there is no applied voltage in the transverse direction y,and read off the spontaneously generated transverse current Jy(x) from thesolution.To model a Josephson junction we need to choose the profile µ(x) appro-priately. In the case of homogeneous superconductors, the scale invariantquantity to consider is T/µ, i.e. changing the temperature is equivalentto changing µ. This is no longer the case in the spatially inhomogeneouscase: while our chemical potential is spatially varying, the temperature isstill constant. Instead, to measure the temperature in a scale invariant waywe use the scale invariant quantity T/Tc, where the critical temperature59Tc is proportional to µ∞ ≡ µ(∞). Our simulations set the proportionalityrelation to beTc ≈ 0.065 µ∞ (2.8)Since we now change the temperature by varying µ, there is a correspondingcritical chemical potential µc below which the condensate vanishes. We thusneed a profile with the following features:{µ(x) < µc in the normal metal phase, − `2 < x < `2 ;µ(x) > µc otherwise for the superconducting phases.As in [33], a profile that satisfies these conditions isµ(x) = µ∞[1− 1− 2 tanh( `2σ ){tanh(x+ `2σ)− tanh(x− `2σ)}](2.9)where µ∞ > µc is the maximal height of the chemical potential. The pa-rameter σ controls the steepness of the profile, whereas  controls its depth– the chemical potential inside of the normal phase is typically µ0 ≡ µ∞.Moreover, it is convenient to work with compactified variables z = 1/r andx¯ = tanh( x4σ )/ tanh(p4σ ), where p is the length of the x-direction.Pseuso-spectral collocation methods on a Chebyshev grid were used todiscretize the above equations. The resulting equations were solved usingthe Newton iterative method. One key characteristic of the pseudo-spectralmethods is their exponential convergence in the size of the grid used, whichwe have confirmed explicitly for our solutions. The solutions used in thispaper were produced using a grid of 41 points in both the radial and spatialdirection, yielding an estimated maximal error of about 10−4 in the localvalue of all functions.2.3 Chiral Edge CurrentsWe start by discussing the phenomenon unique to the time-reversal breakingchiral superconductor: the existence of edge currents. The existence ofdissipationless chiral edge currents is indicative of gapless chiral modes livingon an interface between the superconductor and the normal state. In aJosephson junction configuration there are two such interfaces and thereforetwo independent counter-propagating modes. In this section we focus onaspects of these modes that are localized at each interface separately, whichwould exist in a simpler domain wall geometry. In the next section we60turn to discuss aspects of the physics more specific to a Josephson junctionconfiguration with two such interfaces.To be more concrete, the introduction of a gauge field M3y makes itpossible to measure a current Jy(x) propagating in the y-direction. We havespecified M3y (r =∞, x) = 0 so that the system has no applied voltage thatwould drive a current in the y-direction, thus this is a dissipationless currentflowing without resistance. Under such conditions, the fieldM3y would vanisheverywhere for a p-wave order parameter, but has a non-trivial profile inthe p + ip case. As shown in figure 2.1, this current is localized at theinterfaces of the superconducting and normal phases, travelling in oppositedirections with equal magnitude. We have checked that the strength of thecurrent is independent of the Josephson phase (or equivalently, the strengthof the Josephson current) and the width of the junction (for sufficientlywide junctions). Therefore the currents on both interfaces are local effectsindependent of each other.The quantity of interest is the total current per unit area, defined asfollows:Jedge ≡∫ ∞0Jy(x)dx (2.10)We focus on this quantity as it is independent of details of the interfaceprofile such as the steepness, parametrized by σ above. Furthermore, we findthat the edge current is essentially constant when the weak link is a normalmetal, independent of the relative depth parameter . However, when theweak link becomes superconducting, the current decreases as we increase ,-10 -5 0 5 10-1.0- yHxLFigure 2.1: The left plot displays the solution for the gauge field M3y . Theright plot shows the resulting boundary current Jy(x) (in blue), as wellas the rescaled chemical potential profile µ(x)/µ∞ (in red). The solutioncorresponds to µ∞ = 7, ` = 4, σ = 0.6 and  = 0.3.61eventually vanishing when the solution is perfectly homogeneous.This dependence on the relative depth is shown in Figure 2.2. It indicatesthat the current is controlled by the jump in the amplitude of the orderparameter across the interface between the superconducting material and theweak link. Indeed, in Figure 2.3 we plot the dependence on the edge currenton the order parameter in the superconducting phase 〈OS〉 (choosing  suchthat the weak link is always at the normal state, i.e. has approximatelyzero condensate). The edge current depends on the magnitude of the orderparameter through a power law relationship Jedge ∼ 〈OS〉α with α rangingbetween 2.04 and 2.12 for different choices of parameters.It is also natural to examine the temperature dependence of the edgecurrent, which we plot in Figure 2.4. We see that the dominant change inthe edge current as we change the temperature comes through the change inthe amplitude of the order parameter. As expected the edge current vanishesat the critical temperature Tc.0.0 0.2 0.4 0.6 0.8 1.0012345eJ edgeêT cS-N-S S-S’-SFigure 2.2: The data above corresponds to µ∞ = 10. As we increase thedepth , we observe a linear decrease in the edge current value happeningaround the dashed line at c = µc/µ, which is the critical depth at whichthe weak link becomes superconducting. At  = 1, Jedge goes to 0 since µ(x)becomes homogeneous.620 10 20 30 400123456XOS\Tc2J edgeTcFigure 2.3: This plot illustrates the dependence of the edge current on theamplitude of the order parameter in the superconducting phase for ` = 4,σ = 0.6 and  = 0.3. The curve fits Jedge ∼ 〈OS〉α with α ' 2.0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00123456TTcJ edgeTcFigure 2.4: The temperature dependence of the edge current is pictured for` = 4, σ = 0.6 and  = Josephson CurrentsThe Josephson effect is a macroscopic quantum phenomenon in which adissipationless current flows across a weak link between two superconducting63electrodes, in the absence of an external applied voltage. Rather, it is thegauge invariant phase difference across the junction that is responsible forthe current. Following [33], we will consider S-N-S Josephson junctions, forwhich the weak link is a non-superconducting (“normal”) metal, as describedabove. We also make some comments on the S-S’-S case, in which the weaklink is superconducting. For a discussion of an S-I-S weak link, see [84].The Josephson current flowing across the junction has the expected formJx = Jmax sin γ (2.11)where the gauge invariant phase difference across the junction is obtainedfrom the solution asγ = −∫ ∞−∞dx [vx(x)− vx(±∞)] (2.12)Figure 2.5 has been obtained by computing γ for multiple solutions corre-sponding to different inputs Jx; it clearly demonstrates the expected depen-dence of the Josephson current on the phase difference.Another interesting feature of the critical current Jmax is that it decays-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-0.0004-0.00020.00000.00020.0004ΓJ xTc2Figure 2.5: This graph, produced with µ∞ = 10, ` = 4, σ = 0.4 and = 0.05, shows the agreement of our data with the expectation (2.11). Thesolid line, describing the best fit curve to our data (in red), is the curveJx = 5.2× 10−4 sin(0.9986γ).642.0 2.5 3.0 3.5 4.0 4.5{J maxêT c22.0 2.5 3.0 3.5 4.0 4.5{XO N\êT c2Figure 2.6: The relationship of the critical current and of the order parame-ter as the weak link length grows larger is illustrated in these two plots. Bothsets of data are fitted to a decaying exponential and independently yield thesame coherence length, up to a difference of 1.8%. We used µ∞ = 10, σ = 0.4and  = 0.05 for both plots.exponentially when the width ` of the weak link increases, i.e. it obeys arelation of the form4JmaxT 2c= AJe− `ξ (2.13)for ξ  `. Additionally, the order parameter at the center of the junctionhas a similar behaviour: 〈ON 〉T 2c= AOe− `2ξ (2.14)where 〈ON 〉 is the magnitude of the order parameter in the normal phase4Note that we have switched from the numerically convenient conventions of measuringdimensional quantities in units of the (varying) temperature T , to the more physicalconventions of measuring those quantities in terms of the fixed temperature Tc.650.5 0.6 0.7 0.8 0.9 1.00.0000.0010.0020.0030.0040.005TêTcJ maxêT c20.5 0.6 0.7 0.8 0.9êTcXO N\êT c2Figure 2.7: The dependence of the coherence length ξ, expressed throughthe critical current or order parameter 〈ON 〉 (in the inset), on temperature.Near the critical point, the critical current follows a power law characteristicof the S-N-S junction, but with fairly large corrections. The parameters forthe chemical potential used to produce these figures are ` = 4, σ = 0.6 and = 0.3.(x = 0), and the junction has no current. Both of these relations are dueto the proximity effect, the leakage of the superconducting order into thenormal state. Note then that the coherence length ξ should be the same inboth cases. The results of our numerics show remarkable precision, yield-ing ξ ≈ 0.4547 for the Josephson current, and ξ ≈ 0.4468 for the orderparameter: a difference of only 1.8%. See for instance Figure 2.6.The coherence length has an interesting temperature dependence, plot-ted in Figure 2.7. While the plot does not have a simple fit, it has theexpected behaviour at T → Tc, where it vanishes due to the disappearanceof superconductivity. Near the critical temperature the critical current isexpected to vanish as [92]:Jmax(T ) ∝ (Tc − T )β near Tc (2.15)For a conventional s-wave superconductor, and junctions wide compared66to the coherence length, a quadratic dependence (β = 2) is characteristic ofthe S-N-S junction, whereas different critical exponent are expected for othertypes of weak links (for example for the S-I-S junction β = 1). Interestingly,our results presented in Figure 2.7 indicate that β ∼ 2.52; furthermore theexponent β also depends on the steepness and depth of the chemical poten-tial profile. While β is closest to the critical exponent of the S-N-S junction,the corrections are fairly large. Those corrections are probably related toour setup having a varying chemical potential. It would be interesting to re-produce the expected scaling with a more conventional setup of a Josephsonjunction for which the chemical potential is spatially homogeneous.2.5 ConclusionsIn this paper we have started the investigation of Majorana bound states inthe holographic context. We have discussed the dissipationless edge currentswhich are an indirect evidence for such modes. Additionally, we have con-structed a Josephson junction involving a topological chiral superconductor,and probed the physics of the Josephson effect. The results we obtained areconsistent with a conventional effect, with 2pi periodic current-phase rela-tion, and exponential decay of the current with the junction width.These results support the expectation that though the physics is 4piperiodic, an unconventional periodicity will not be visible in thermal equi-librium. The presence of Majorana modes corresponds to having two stateswhich are exchanged upon a 2pi phase rotation. However, in equilibrium theJosephson current receive contribution from both states, weighted accordingto their Boltzmann weight. This, thermal equilibrium results are expectedto exhibits conventional periodicity, consistent with what we find,It would be interesting to continue this investigation with the goal ofdisplaying more direct signatures of the Majorana bound states. One suchdirect signature would be in the Andreev scattering off a superconductinginterface – bound states can be then seen in analyzing the phase shift. Fur-thermore, one can construct holographically a non-equilibrium configurationwhich is expected to exhibit the unconventional periodicity associated withMajorana bound states. We hope to report on the result of such investiga-tions in the near future.67Chapter 3Dynamics of Holographic En-tanglement Entropy Followinga Local QuenchWe discuss the behaviour of holographic entanglement entropy following alocal quench in 2+1 dimensional strongly coupled CFTs. The entanglementgenerated by the quench propagates along an emergent light-cone, remi-niscent of the Lieb-Robinson light-cone propagation of correlations in non-relativistic systems. We find the speed of propagation is bounded from belowby the entanglement tsunami velocity obtained earlier for global quenchesin holographic systems, and from above by the speed of light. The formeris realized for sufficiently broad quenches, while the latter pertains for welllocalized quenches. The non-universal behaviour in the intermediate regimeappears to stem from finite-size effects. We also note that the entanglemententropy of subsystems reverts to the equilibrium value exponentially fast, incontrast to a much slower equilibration seen in certain spin models.3.1 IntroductionIn recent years we have seen enormous progress in qualitative and quanti-tative understanding of out-of-equilibrium quantum dynamics. Theoreticaland numerical methods have been very effective to unearth the generic be-haviour of a variety of observables in such systems. Coupled with the rapidgrowth of experimental techniques in cold atom and many-body systemsto probe such dynamics, one can furthermore ratify our theoretical under-standing. Motivated by these considerations we continue our explorationsof dynamics of strongly coupled non-equilibrium quantum systems usingholographic methods.One simple scenario of interest in many circumstances is a situationwhere we start with a QFT in global equilibrium and deform it by turningon external sources for relevant operators. The sources provide external dialswhich can serve to do work on the system and drive it out of equilibrium.We could consider sources that act homogeneously in space (but localized68in time), which is often referred to as global quench, or have it act locally inspacetime, which corresponds to a local quench. Both types of protocols arewell studied in literature in the past decade or thereabouts. In either casewe are considering deformations of the formSQFT 7→ SQFT +∫ddxJ (x)O(x) , (3.1)where O(x) is a (composite) operator of the QFT and J the classical sourcewe dial. The distinction at this level between local and global quenches issimply in the spacetime support of the source J (x).Much of the analytic progress in this front has been in 1+1 dimensionalCFTs, where the quench protocols of the form (3.1) can be incorporated intoa Euclidean path integral, and studied efficiently by computing correlationfunctions of the deforming operator O(x) in the unperturbed state of theCFT, cf., [62, 93] for the original discussion and [94] for a review.Our primary interest is in exploring the dynamics of strongly coupledQFTs subject to such protocols in higher dimensions. A natural frameworkto explore this question is provided by the holographic AdS/CFT dualitywhich maps the QFT problem onto the dynamics of a gravitational systemin asymptotically AdS spacetime. For concreteness we will focus on 2+1CFTs which are originally in global thermal equilibrium and subject themto a quench by a local scalar operator O of dimension ∆. The gravitationalproblem then comprises of Einstein-Hilbert gravity coupled to a massivescalar, whose mass m is related to the conformal dimension by the standardformula, viz., ∆ = 32 +√94 +m2 `2AdS.1 The initial global equilibrium statemaps onto a planar Schwarzschild-AdS4 black hole and the problem at handinvolves analyzing the deformation of this said black hole consequent toturning on a boundary source for the scalar field. This then amounts to agravitational infall problem. The pulse of scalar on the boundary propagatesinto the bulk and dissipates through the black hole horizon. Of interest tous are the observables in the interim process.While there are many quantities that could be, and indeed have been[41, 47, 49–53, 55–59, 61, 95–121], studied in this context, we will for def-initeness focus our attention on entanglement entropy. While strictly notan observable, the entanglement entropy for a particularly chosen spatialregion of the QFT captures important aspects of the field theory dynam-ics. Not only does it provide a measure of how correlations in the system1 We will only consider deformations by operators which are well separated from theunitarity bound – our focus will be on conformally coupled scalars with ∆ = 2.69evolve following the quench, but it furthermore is also a simple quantity tocompute in the holographic context. The holographic entanglement entropyproposals of [38, 39] and their covariant generalization [41] provide an ex-tremely simple route to its computation. All we are required to do is solve aclassical problem of finding areas of extremal surfaces anchored on the saidregion of interest.In what follows we will explore how holographic entanglement entropyevolves following a local quench. We will restrict our attention to a veryspecific scenario, wherein we quench a CFT3 with a ∆ = 2 operator. Thedisturbance will be taken to be localized in space and time – we pick expo-nential damping in space and an inverse Po¨schl-Teller switch on/off in time,cf., (4.26). We retain translational invariance in one spatial direction, break-ing homogeneity in the other. We study entanglement entropy for strip-likespatial regions that are aligned with the symmetry we retain, so that theproblem of finding extremal surfaces can be mapped to effectively findinggeodesics in an auxiliary three dimensional spacetime. Of interest to us arehow the entanglement entropy growth is correlated with the position andsize of the strip relative to the quench location.To appreciate the question, let us recall some well known facts. Theclassic analysis of [62] of entanglement entropy growth following a globalquench in CFT2 has spurred lots of activity on the subject. While the twodimensional case can effectively be described by a quasiparticle picture, sincethe entanglement growth is linear due to left and right movers decoupling(following an initial quadratic ramp up [47, 61, 103]), the holographic modelspresent a much different picture in higher dimensions.2The results of various analyses of global quenches have been beautifullyencapsulated in the ‘entanglement tsunami’ picture developed by Liu-Suh in[47, 61] and further explored recently in [63]. Following an initial quadraticgrowth in time, the entanglement entropy for any region grows linearly ata rate dictated by the tsunami velocity vE . To define this quantity unam-biguously the authors chose to normalize the local value of entanglemententropy relative to the final thermal entropy expected for the same regiononce equilibration is complete. This does leave a single parameter which2 We note here that oftentimes global quenches are holographically modeled by con-sidering a Vaidya-AdS geometry (see [41, 122] for early discussions) that corresponds toinfalling null matter in the bulk, which does not accord a clean CFT interpretation. Acleaner perspective is offered by either solving the non-linear dynamics of gravity cou-pled to realistic matter like a scalar field, or more simply by implementing an end of theworld brane boundary state [106] explicitly in holography. The results for the growth ofentanglement entropy are however independent of the particularities of the modeling.70is the aforementioned velocity. It was found not only vE ≤ 1 as requiredby causality with equality in d = 2 consistent with the CFT2 analysis, butone could further bound it by a universal dimensional dependent constantv∗E(d).3 This upper bound on velocity was attained holographically for mat-ter that collapsed into a Schwarzschild-AdSd black hole at late times.Given this rather clear situation for global quenches, we are interestedin ascertaining the behaviour when we localize the quench protocol to a fi-nite spatial domain. We in principle could focus on deformations by sourcesdelta-function supported at point. This is natural when studying this prob-lem in QFT as one can map the computation to that of computing corre-lation functions on some background, however for our purposes of carryingout numerical investigations we choose to smear out the source. We expectfirstly that the underlying locality of the QFT forces entanglement entropyto behave causally; as explained in [45, 123] this means that the source makesits presence felt only when it acts in the causal past of the entangling sur-face (the boundary of the region of interest). This is indeed what one seesin explicit computations in CFT2. The entanglement entropy only startschanging after a time lag set by the time it takes for the quench disturbanceto propagate between the region of interest and its complement. As long asthe quench front is localized either in the region or in the complement, weonly have the initial state entanglement.Previous analyses of holographic local quenches by [104] involved mod-eling the system by the infall of a massive particle – this is effectively aneikonal approximation wherein one is assuming that the wavepackets of thequench are tightly collimated. Moreover, the authors chose to work withvery heavy operators ∆ 1 which could then be approximated in terms ofworldlines of a small black holes. The relevant geometry can be obtained byapplying a suitable symmetry transformation to the global Schwarzschild-AdS black hole and with it in hand properties of holographic entanglemententropy were explored. This picture was further supported by field theoryanalysis of such deformations at large central charge [124, 125]. Our aim totackle this problem from a different perspective by studying the entangle-ment evolution in a quenched gravitational background as explained above.We will recover most of the results mentioned above in our analysis.We can moreover explore quantitative features of the entanglement evo-3 This statement as far as we are aware is robust for QFTs whose holographic dualsare given in terms of two derivative Einstein-Hilbert gravity coupled to sensible matter.There is a-priori no reason for them to hold when the gravitational dynamics includeshigher derivative corrections and we in particular are not aware of any statement of thiskind.71lution. We see that the propagation of entanglement is confined to an effec-tive light-cone. We extract an entanglement velocity vE from this emergentcausal structure. Unlike the case of the global quench, the velocity dependson the details of the quench. It appears to grow monotonically with increasein the amplitude of the quench source as well as with the increase of theinitial temperature. For a certain range of parameters is appears to trackthe tsunami velocity bound v∗E(3) of [47], while for others it reaches close tothe speed of light.There is a somewhat annoying fact that the tsunami velocity v∗E(3) =0.687 in three spacetime dimensions is marginally lower than the speed ofsound vs = 0.707, making it somewhat hard to convincingly point to preciseorigin of the effect. We also see contamination from edge effects both fromfinite size of A and the finite width of the quench source. We have notexamined the detailed non-linear effects that cause the velocity to grow fromthe tsunami bound towards the speed of light, but display some exampleswhich illustrate the pattern.While our numerical results are constrained to probing small spatial re-gions relative to thermal scale,4 we nevertheless are able to extract boththis entanglement velocity as well as examine the return to the equilibrium.In contrast to studies in lattice models in low dimensions which display alogarithmic return of entanglement entropy to its equilibrium value after thequench, we find that the holographic systems prefer to equilibrate exponen-tially.The outline of the paper is as follows. In §4.2 we describe the basic set-upfor holographic local quenches, describing the general methodology and thedetermination of entanglement entropy from the gravitational background.In §3.3 we give the basic numerical results for the quench spacetime andextremal surfaces therein. The key statements regarding the behaviour ofentanglement entropy in a locally quenched CFT are then extracted in §3.4,where we describe the growth velocity vE and the return to equilibrium.We end with some open questions in §3.5. Some details of the numericalmethods are collected in Appendices A and B.4This constraint arises because our numerical solutions only determine the geometryto the exterior of the apparent horizon. For small regions A the extremal surfaces stay inthis domain, but for larger regions, they do penetrate the apparent horizon – see [50, 126].723.2 Preliminaries: Holographic Local QuenchWe are interested in the behaviour of entanglement entropy in a 2 + 1 di-mensional field theory that has been driven out of equilibrium locally by aninhomogeneous relevant scalar operator. Holographically, this amounts tosolving the gravitational dynamics of a 3+1 dimensional asymptotically AdSspacetime and its consequences for the area of extremal surfaces anchoredon the boundary.3.2.1 Metric AnsatzIn order to dynamically evolve a spacetime geometry following a local quench,it is convenient to choose our metric ansatz to be a generalization of the in-falling Eddington-Finkelstein coordinates for black holes. We choose to workin an asymptotically AdS4 spacetime, dual to a 2 + 1 dimensional CFT,ds2 = −2Ae2χ dt2 + 2 e2χ dt dr − 2Fx dtdx+ Σ2(eB dx2 + e−B dy2),(3.2)where r denotes the radial bulk coordinate, with the boundary lying atr =∞, and t is a null coordinate that coincides with time on the boundary.We have chosen our quench to be localized in the x-direction and transla-tionally invariant in the y direction. Hence all the fields appearing above{A,χ, Fx,Σ, B} depend only on the coordinates {r, t, x} with ∂y being anisometry.This choice for the metric has many advantages: it provides us withcoordinates that remain regular throughout the entire domain as the space-time equilibrates, it leads to a characteristic formulation of our gravitationalinfall problem, and it comes with a residual radial diffeomorphism that isof great computational help [73]. Indeed, the metric (3.2) remains invariantunder radial shifts,5r → r = r + λ(xµ) . (3.3)On physical grounds, we anticipate that the black hole’s horizon will growlocally as the effects of matter from the boundary are felt in the interior ofthe bulk. Hence a sensible gauge choice is to dynamically determine λ sothat the coordinate location of the black hole’s apparent horizon6 remains5 For notational clarity, we use upper case Latin indices {M,N, ...} to represent bulkcoordinates, and lower case Greek indices {µ, ν, ...} to refer to boundary coordinates.6 See Appendix A for further details about our numerical scheme73fixed. This keeps the calculational domain simple.Einstein’s equations in the presence of a scalar field are given byRMN − R2GMN − d(d− 1)2 `2AdSGMN = TMNTMN = ∇MΦ∇NΦ +GMNLΦ, LΦ = −12(GMN∇MΦ∇NΦ +m2Φ2).(3.4)For simplicity, we restrict our attention to m2 `2AdS = −2 so that the asymp-totic expansion of the scalar field near the boundary is analytic in powersof 1/r:Φ(r, t, x) =φ0(t, x)r+φ1(t, x)r2+ · · · (3.5)We note that since t is a null coordinate, φ1(t, x) will have contributionscoming from both the source and the response of the scalar field, as will beexplained below.3.2.2 Asymptotic GeometryIn a theory of gravity on asymptotically AdS spacetimes, asymptotic analysisalone is not sufficient to determine the bulk metric [13]. Indeed, the missingpiece in the asymptotic analysis is the boundary stress tensor, determinedby solving the full bulk equations:Tµν ∼ g(3)µν , (3.6)where g(3)µν is the part of the metric undetermined by the equations of motionfor d = 3.While our infalling coordinate chart (3.2) differs from the standardFefferman-Graham chart typically used for asymptotic expansions, it is astraightforward exercise to carry out an asymptotic analysis. Demandingthat the field equations are obeyed in the near-boundary r →∞ domain we74findA(r, t, x) =(r + λ(t, x))22− ∂tλ(t, x)− 14φ0(t, x)2 +a(3)(t, x)r+ · · · (3.7)χ(r, t, x) =c(3)(t, x)r3+ · · · (3.8)Fx(r, t, x) = − ∂xλ(t, x) + f(3)(t, x)r+ · · · (3.9)Σ(r, t, x) = r + λ(t, x)− 14φ0(t, x)2 + · · · (3.10)B(r, t, x) =b(3)(t, x)r3+ · · · . (3.11)One may also show that the explicit map to the Fefferman-Graham coordi-nates {τ, ρ, ξ} takes the asymptotic formτ(r, t, x) = t+1r− λ(t, x)r2+ · · · , (3.12)ρ(r, t, x) = r + λ(t, x)− 14φ0(t, x)2r+ · · · , (3.13)ξ(r, t, x) = x+O(r−3). (3.14)Additional care needs to be taken when dealing with scalar fields in a theoryof gravity formulated in terms of null coordinates. Indeed, the falloff ofscalar fields with m2`2AdS = −2 is known to behave in Fefferman-Grahamcoordinates as:Φ(ρ, τ, ξ) =φsource(τ, ξ)ρ+φresponse(τ, ξ)ρ2+ · · · (3.15)as we approach ρ→∞. By using the coordinate expansion above, we obtainΦ(r, t, x) =φsource(t, x)r+φresponse(t, x) + ∂tφsource(t, x)− λ(t, x)φsource(t, x)r2+ · · · ,(3.16)thus confirming our earlier claim that φ1 = φresponse + ∂tφ0−λφ0 mixes thesource and the expectation value of the scalar.753.2.3 Boundary Stress TensorIn order to solve Einstein’s equations as efficiently as possible, we foundit useful to use the boundary stress tensor and its conservation equationsto find and propagate the undetermined fields a(3) and f (3) accurately intime (in our scheme, b(3) and c(3) need to be read off from the solutionsdirectly). For asymptotically AdS4 spacetimes, the boundary stress tensor inthe presence of a scalar field of mass squared m2`2AdS = −2 can be expressedin the Brown-York form asTµν = Kµν −Kγµν + 2 γµν −(γRµν − 12γRγµν)+12γµν φ2, (3.17)where we have introduced some boundary data: γµν is the induced metricon the boundary, Kµν ,K ≡ γµνKµν its extrinsic curvatures, and γRµν , γRits intrinsic curvatures. Explicitly in terms of the asymptotic expansioncoefficients we find that the energy-momentum tensor takes the formT00 = 2a(3) + 4c(3) + φ0φresponse, (3.18)Ttx =32f (3) − 12φ0∂xφ0 , (3.19)while the conservation equations in the presence of the scalar source φ0(x, t)read∂tT00 = ∂xTtx + ∂tφ0 φresponse, (3.20)∂tTtx =12(∂xT00 − 3 ∂xb(3) + ∂xφ0 φresponse − φ0 ∂xφresponse). (3.21)We take our initial state to be in thermal equilibrium, which translatesto an initial condition on the bulk metric, which is then the planar staticSchwarzschild-AdS4 black hole spacetime with temperatureT =3M134pi. (3.22)The initial boundary stress tensor is then simply Tµν = diag{1, 12 , 12}. Tomodel our local quench, we simply need to specify a source function φ0(t, x)and let the system evolve according to the Einstein equations, all while mak-ing sure that λ is gauge-chosen to fix the location of the apparent horizon.763.2.4 Holographic Entanglement EntropyOnce we have obtained solutions for the local quench, we can study thesubsequent dynamics of the entanglement entropy of a region A on theboundary using the covariant holographic entanglement entropy prescription[41]. The latter requires us to determine extremal surfaces anchored on theentangling surface on the boundary.For simplicity, we exploit the translational invariance, and restrict ourattention to a strip-regionA = {(x, y) | x ∈ (−L,L), y ∈ R} , ∂A = {(x, y) | x = ±L, y ∈ R} .(3.23)The extremal surfaces EA anchored on ∂A are straightforwardly determinedby solving a set of ODEs. Using coordinates adapted to the ∂y isometry, weparameterize the surface by coordinates y, τ . Consequentially, EA is thenobtained by solving the geodesic equations in an auxiliary three dimen-sional spacetime with metric g˜MNdXN dXM = gyy gMN dXN dXM , withthe restriction to y = constant understood, i.e., XM (τ) = {t(τ), r(τ), x(τ)}.Equivalently we solve the Euler-Lagrange equations obtained from the La-grangian L = gyy gMNX˙MX˙N .While we have phrased the determination of EA as a boundary valueproblem, it is practical to switch to an initial value formulation. We param-eterize the solutions by specifying the turning point, or tip, of the geodesicin the bulk, XM∗ (τ) = {t∗, r∗, x = 0}, and evolve towards the boundaryusing an ODE solver (for instance the Matlab solver ode45 ) until both ∂Aand a specified UV cutoff are reached.To this end, we have chosen to transform our system of 3 second orderODEs into a system of 6 first order ODEs in the variables{t, Pt ≡ Σ2 t˙, r, P+ ≡ e2χ(r˙ −A t˙) , x, Px ≡ Σ2 x˙− e−BFx t˙} .(3.24)With these new variables,7 L = 2P+Pt + P 2x . The boundary conditions atthe turning point are{t = t∗, Pt = 0, r = r∗, P+ = 0, x = 0, Px = ±1} . (3.25)The conditions on Pt and P+ are a consequence that, because of symmetry,7 These definitions for the momenta ensure that all quantities are of order O(1) fornumerical stability.77we expect t˙ = r˙ = 0 at X∗, whereas the condition for Px has been chosento normalize the action by setting L = 1. The sign determines whether thegeodesic will go towards the positive or negative x-axis.To translate from the length of the geodesic to the actual entanglemententropy SA we pick an IR regulator Ly along the translationally invariantdirection and a UV cutoff . We choose to present the results for the regu-lated entanglement entropy by subtracting off the corresponding answer inthe unperturbed theory. There are two natural regularizations we can use:Regulator 1: We subtract the entanglement in the ‘instantaneous thermalstate’ obtained by taking the Schwarzschild-AdS4 metric with a horizonlocated at r+(x, t) = M13 +λ(t, x). This choice allows clean matching of theasymptotic coordinate chart.Regulator 2: We alternately can choose to subtract of the vacuum entangle-ment entropy for the same region, with a dynamical UV cut-off vac(x, t).This gives∆SA = Ly∫ dτ − 2− 2λ(t, x) + 4piL(Γ(34)Γ(14))2 . (3.26)The two regulators differ by a finite amount that is invariant temporally,allowing us to cross-check our numerical results. In what follows we willsimply quote ∆SA normalized by Ly.3.3 The Quench Spacetime and ExtremalSurfacesWe now turn to describing the results of solving Einstein’s equations sourcesby the scalar field boundary condition. We then describe properties of theextremal surfaces of interest in these geometries.3.3.1 Numerical SolutionsWe use the characteristic formulation of Einstein’s equations resulting fromthe null slicing of spacetime outlined in [73] to numerically find the geome-try. Even though we start with a complicated set of PDEs, the characteristicformulation simplifies the equations of motion into two categories: the equa-tions for the auxiliary fields that are local in time and reduce to a nestedset of radial ODEs, and the equations for dynamical quantities that encodethe evolution of the geometry.78To numerically integrate the Einstein and Klein-Gordon equations, wediscretize the radial direction using a Chebyshev collocation grid. Thischoice of discretization for the extra dimension is particularly well suitedto find smooth solutions to boundary value problems while ensuring theirexponential convergence as the grid size is increased. We opted to choosea rational Chebyshev basis to deal with the non-compact spatial direction.The main advantage of working with a rational Chebyshev grid is that theboundary conditions at x = ±∞ are already implemented behaviourally ; aslong as the solution decays at least algebraically fast or asymptotes to a con-stant, we can avoid specifying the boundary conditions explicitly [127]. Weuse a grid of 41 points in both directions. To propagate in time, we use anexplicit fifth-order Runge-Kutta-Fehlberg method with adaptive step size.We also avoid aliasing in both the radial and spatial directions by applying alow-pass filter at each time step that gets rid of the top third of the Fouriermodes.We chose the source function to be φ0(t, x) = f(x)g(t) withf(x) =α2[tanh(x+ σ4s)− tanh(x− σ4s)], g(t) = sech2(t− tq∆tq).(3.27)With it, we can ramp up the scalar field to reach its maximum value αat time t = tq∆ before it vanishes again. The parameters {s, tq,∆} arechosen to facilitate the numerics, whereas σ determines the width of theperturbation. In practice, we found s = 0.15, tq = 0.25 and ∆ = 8 to giveus satisfying accuracy for the late-time behaviour of the scalar field whilepreserving a nicely localized shape for the pulse. So we therefore study thequench protocols parametrized by two parameters: an amplitude α and awidth σ. Along with the initial temperature of the system which we take tobe parametrized by M , we have three parameters at our disposal.φ0(x, t) =α2[tanh(53(x+ σ))− tanh(53(x− σ))]sech2 (4 t− 8) ,Protocol parameters: {α, σ,M}(3.28)The evolution of the spacetime following our quench is fairly simple.The injection of local excitation results in hydrodynamical evolution almostfrom the very beginning (cf., [95, 96] for analogous statements with spatialhomogeneity). Since our perturbation excites the sound mode of the system,79(a) λ(t, x) (b) T00(t, x)Figure 3.1: Evolution profile of the (a) radial shift λ(x, t), and (b) T00(x, t)component of the stress tensor, for α = 0.5, M = 0.1, σ = 2. The field λdetermines the evolution of the entropy in our solution.we have the initial energy-momentum perturbation dispersing at the speedof sound. The presence of shear viscosity results in entropy production,manifested in the solution by the local growth of the horizon area element.0 2 4 6 8 3.2: Evolution of the total energy on the boundary E =∫T00 dxafter a quench described by parameters α = 0.5, M = 0.1, σ = 2.In Figure 3.1a we display the spatial and temporal profile of the functionλ(x, t), related to the area element of the horizon. We see that the initial80perturbation indeed results in entropy production, as expected. Curiously,the initial perturbation splits to two localized perturbations after some time;those follow the expected hydrodynamic evolution. Figure 3.1b shows theequivalent evolution of the energy density for the same set of parameters.Finally, Figure 3.2 shows that following the conclusion of the quench thetotal energy is conserved. These features verify the intuitive picture of hy-drodynamical evolution following a local excitation of the system.To quantify the entropy production, we can monitor the growth of thearea of the apparent horizon as a function of time. In order to express theresult in physical units, we need to convert from the natural time scale onthe horizon to the time measured in the boundary. Recall that our solutionsfor the metric components are obtained on a slice of constant ingoing timecoordinate t. We could, following [128], map the horizon data along ingoingnull geodesics to the boundary. We will refrain from doing so explicitlyand instead work directly in the chosen coordinates leaving implicit thistranslation.8Using the induced metric hab on a constant t slice we obtain the areaelement on the horizon which can be integrated directly. Since the naiveanswer is infinite, we regulate it by removing the contribution from the initialequilibrium state (i.e. subtract off the static Schwarzchild-AdS answer) toobtain:∆Areah = Ly∫r=rh(Σ2√1 + 2λ′e−2χ−B − r2h)dx (3.29)The numerical results are expressed in Figure 3.3, where we also show thetotal energy for comparison. Notice the striking resemblance of the horizon’sarea evolution with that of the total energy injected into the system by thequenching scalar field. This seems to indicate that the growth of the horizonis dictated by processes governed by the speed of sound, such as energy andmomentum transport. This is indeed the intuition we would have from thehydrodynamic regime of slow variations and it is a reassuring check of theset-up that this indeed is upheld.3.3.2 Extremal SurfacesHaving the solution at hand we can compute the extremal surfaces as de-scribed in §3.2.4. In Figure 3.4 we display the radial depth of the turningpoint for the extremal surfaces, as function of (boundary) time. Different8 We also note that λ(t, x) is defined on a constant ingoing time slice, and as such theradial shifts affect the horizon “instantaneously” rather than causally.810 2 4 6 3.3: I The growth of the apparent horizon (in blue) as a function ofboundary time, for α = 0.3 and M = 0.1. We also overlay the plot for thetotal energy∫T00dx produced by quenching the system in red for directcomparison.points correspond to different extremal surfaces, which contribute to entan-glement entropy of surfaces of varying lengths. We have plotted the radialdepth both in the computational coordinate (in which the horizon is at fixedradial distance) and in coordinates in which the horizon grows.Since our computational domain ends at the apparent horizon, we cannotprobe extremal surfaces that extend past into the trapped region. These areknown to exist in various explicit simulations (cf., [126] for a comprehensivesurvey in Vaidya-AdS spacetimes). Pragmatically, this restricts our atten-tion to small regions A. We will nevertheless see that despite this restrictionwe can still extract interesting physical features of SA using surfaces thatlie outside the apparent horizon.One of the interesting features to notice from Figure 3.4 is that thegeodesics never go beyond their initial depth in the bulk when we considertheir position in the ungauged radial coordinate, i.e., where the radial depth820 2 4 6 3.4: Evolution of the geodesics’ radial depth for a quench; α = 0.5,M = 0.1, L = 0.8, σ = 2. The blue data points represent the radial depth u∗in the fixed, gauged coordinate system, whereas the red data points representthe ungauged radial depth U∗.isU∗ ≡ u∗1 + λ(t∗, x = 0)u∗, (3.30)with u∗ = 1/r∗ being the radial position of the tip in the coordinate systemwhere the apparent horizon is at a fixed coordinate locus.3.4 Propagation of Entanglement EntropyArmed with the numerical results for the spacetime geometry and the ex-tremal surfaces therein, we are now in a position to extract some physicallessons for the evolution of entanglement entropy following a local quench.We restrict our attention to regions A centered around the source of theinitial excitation which is taken to be w.l.o.g. at x = 0. We will examinethe behaviour of ∆SA as a function of the width L of the strip and time tafter the quench.We note that the region of parameter space that we can explore numer-ically is limited. The amplitude α of the scalar field cannot be too large,83otherwise the time-evolution of the quench solution does not converge. Sim-ilarly, the evolution code becomes unstable if the spatial discretization fallsbelow a critical grid size, which has for consequence that we cannot resolvequenches with width σ below a certain threshold. The width L of the en-tangling surface is in turn constrained by the initial values we can pick forM , which determines the position of the event horizon of the initial configu-ration: if M is taken to be large, then we cannot find extremal surfaces thatgo deep enough in the bulk to probe larger regions A, whereas if M is takentoo small, then it becomes increasingly harder to quench the spacetime witha scalar source. We found that using quenches with width σ = 2, togetherwith M ranging from 0.005 and 0.2 and α between 0.1 and 0.5, yielded in-teresting results that remained mostly the same, albeit delayed in time, asthose with σ chosen larger.Before proceeding we remind the reader that for regions A which aremuch wider than the width of the quench source profile, there is a timedelay before the entanglement entropy starts to change. This is consistentwith the causal properties one would required of entanglement. Only whenthe quench can affect both the region and its complement (by being in thepast of the entangling surface) would we expect a change in the entanglementfor A. This is clearly borne out in our simulations and is used to benchmarkthat we are on the right track.3.4.1 An Emergent Light-coneWe first note that the entanglement generated by the local quench is linearlydispersing, i.e., it traces an effective light-cone. This is quite reminiscent ofthe Lieb-Robinson bound [129] in non-relativistic theories, where correla-tions follow an effective information light-cone. The speed of entanglementpropagation is then denoted by vE below.The velocity vE we find is bounded from below. A-priori one mightguess whether the lower bound is given by the speed of sound, which is thespeed in which the initial pulse spreads, thereby further exciting the systemand generating additional entanglement on larger scales. The true speedis however a bit lower, as we shall see, suggesting that the mechanism ofentanglement propagation differs from that which drives physical transportof energy and other conserved charges in the system.99 A-priori this statement statement appears reasonable, since the propagation of energyin the system is governed by the ability of the system to homogenize, which per se is notthe same as becoming quantum entangled. There is thus far no clear mechanism forintuiting entanglement transport in quantum field theories, though the attempts of [63]84We therefore interpret the velocity vE as the speed in which the initialentanglement, generated locally by the quench, propagates in time. Theentanglement velocity can be extracted from the emergent light-cone definedalong the curve where ∆SA(t) reaches a maximum for every L in the L− tplane. We remark that unlike the results of [104], the height of this peakdoes not remain constant in our setup. Instead, we find that the maximumvalue of SA(t) increases as we increase L.This behaviour of the entanglement entropy can be quantified ratherexplicitly. We find that dependence is strongest when the amplitude ofthe scalar field is varied. For small sizes L, the maximum of SA increaseslinearly with L. If we denote the slope of these curves by s, then we findthe interesting relation∂∂LSA(L, tmax, α) = s(α) ∼ α2 for small/intermediate regions. (3.31)The actual scaling for the slopes obtained from our numerical data are:• s(α) ∼ α1.92 for α = {0.1, 0.2, 0.3, 0.4, 0.5} and M = 0.1• s(α) ∼ α2.0043 for α = {0.05, 0.1, 0.15, 0.2} and M = 0.01In the first case, the linear behaviour is shown in Figure 3.5. In the secondinstance (not pictured), while the linear nature breaks down when L is large,the slopes for small to intermediate regions still depend quadratically on theamplitudes. The dependence on temperature is less interesting. When thetemperature M changes, the maximum of the entanglement entropy shiftsslightly, as can be seen in Figure 3.6.For general values of parameters, the entanglement velocity vE changeswith parameters, always bounded from below by the tsunami velocity (4.31),and above by the speed of light. We do however find two universal resultswhich we now turn to.Universal Behaviour at High TemperatureIn the limit of an approximate global quench where the region A is containedwithin the local quench, i.e., L . σ, and at high temperatures, we find auniversal light-cone velocity vE = 1 (to very high accuracy), regardless of theamplitude of driving scalar field (including values well within the non-linearregime)10. This is depicted in Figure 3.7. We note that for some values ofsuggest potentially interesting mechanisms for the same.10 It is worth noting that previous results for global quenches could not have seen thisfeature since the entanglement entropy saturates for strip geometries.850.4 0.5 0.6 0.7 0.8 0.9 AHtL<Figure 3.5: Maximum of the entanglement entropy SA(t) as a function of L,for α = {0.1, 0.2, 0.3, 0.4, 0.5} starting from the bottom, and M = 0.1. Theslopes of these curves depend quadratically on the amplitude α of the scalarfield.parameters, this universal behaviour can be affected by edge effects of thelocal quench, and is seen for small enough surfaces only.As we decrease the black hole temperature, the velocity at the small sur-faces becomes lower than 1. This confirms that vE = 1 is a high temperatureeffect only.Wide Quench ProfilesAn interesting feature of the emergent light-cone is the abrupt change ofvelocity as the width of the region A, L, is increased. When the size of theregion A becomes of the same order as the width of the local quench, thecurve traced by the peak of the entanglement entropy goes from one linearregime to another, as shown in Figs. 3.8a, 3.8b, and 3.8c.Interestingly, for the first two data sets (for which α = 0.1, M ={0.005, 0.01, 0.02}, σ = 2), the light-cone velocities vE = {0.678, 0.688, 0.706}are very close to the tsunami velocity of a Schwarzschild-AdS4 black hole860.5 1.0 1.5 2.0 AHtL<Figure 3.6: Maximum of the entanglement entropy SA(t) as a function of L,for a range of masses of the initial black hole M = {0.005, 0.01, 0.02, 0.1},starting from the top, and α = 0.1. Lowering the temperature (decreas-ing M) slightly increases the maximum of SA(t). The same phenomena isobserved for α = 0.2.found in [47], given byv∗E(3) =(η − 1) 12 (η−1)η12η∣∣∣∣∣d=3=√3243= 0.687, with η =2(d− 1)d. (3.32)We note that temperature does not seem to have an effect on vE , whichis consistent with the above formula. For these parameters, the evolution isdescribed by linear response to good approximation, and in that regime thetsunami velocity seems to capture the spatial propagation of entanglementto very good accuracy.This behaviour should be anticipated on physical grounds. When theregion A is completely immersed in the quench source, we are back to thecase where we may approximately think of the situation as a global quenchproblem. The fact that the source is not homogeneous in Ac is irrelevantbecause all that matters is that the excitations produced by the quench873.0 3.2 3.4 3.7: Position of the maximum of SA(t) in the L−t plane for a quenchdescribed by α = 0.5, M = 0.1. The light-cone velocity extracted from theslope of this line is vLC = 1, and is independent of the value of α.are in the causal past of the entangling surface ∂A. With this in mind weimmediately anticipate that the results for the Vaidya quench explore in[47, 61] should apply and one see a linear growth with the tsunami velocity.The story of the local quench however should be a lot richer than thehomogeneous global quench. For one, we can encounter an interplay betweenthe size of A and the width of the pulse. We also expect that the non-linearities of gravity will play a role as we try to increase the amplitude.Indeed we see that velocity vE increases as we increase the strength of thenon-linearities in the bulk evolution – this is illustrated in Figs. 3.9a and3.9b (where the scalar field amplitude was doubled from 0.1 to 0.2). Thisgoes against the idea of the tsunami velocity as an upper bound on the speedpropagation of the entanglement propagation, at least when that evolutionis spatially resolved. Coupled with the earlier observation regarding theupper bound on vE ≤ 1, we find it natural to conjecture thatv∗E(3) = 0.687 ≤ vE ≤ 1 (3.33)The details of deviation from the two extreme limits appear to depend on883.0 3.5 4.0 4.5 5.0 5.5 M = 0.005, vE = 0.678(0.818)3.0 3.5 4.0 4.5 5.0 5.5 M = 0.01, vE = 0.688(0.834)3.0 3.5 4.0 4.5 M = 0.02, vE = 0.706(0.859)Figure 3.8: Position of the maximum of SA(t) in the L − t plane for aquenches described by α = 0.1, starting from different initial states param-eterized by M shown above. The light-cone velocities for large L for thethree scenarios are also indicated, as are the corresponding values for smallregion sizes (in parenthesis). While we give the values of the velocity vEfor small regions, this data should be interpreted with care, for we typicallyfind that edge effects contaminate the data, and these slopes should not betaken at face value in the small L regime.various effects which we have not yet disentangled. While the upper boundfollows form causality, it is unclear at present whether the tsunami velocityencountered (herein and before) is a fundamental bound on informationprocessing in strongly coupled systems. It would be interesting to come upwith a model which allows us to explore the different propagation velocities893.0 3.5 4.0 4.5 M = 0.01, vE = 0.764(0.887)3.0 3.5 4.0 4.5 M = 0.02, vE = 0.776(0.934)Figure 3.9: Position of the maximum of SA(t) in the L − t plane for aquenches described by α = 0.2, starting from different initial states parame-terized by M shown above. The light-cone velocities for large L for the threescenarios are also indicated. Conventions are the same as in Figure 3.8.perhaps along the lines of [63].3.4.2 Entanglement DecayThe process of return to equilibrium is characterized by universal behaviourand critical exponents. Therefore, an interesting quantity in our model is thedecay of the entanglement entropy after it has reached a local maximum. Toour knowledge this is the first time this decay has been calculated in eitherholographic theories or in higher dimensional conformal field theories.From our numerical data we find that the profile for the decay is bestfitted by an exponential damping∆SA(t) ∼ a1e−a2(t−a3) + a4 , (3.34)where the parameters ai depend on the specifics of the sources chosen toimplement the quench protocol. In Figure 3.10 we depict the behaviourfor a particular simulation (parameters in the caption). Note also the timedelay in the initial growth, which illustrates the causality feature discussedearlier.It is interesting to contrast our result for the exponential return to equi-librium against a more slow return seen in some spin chain models. Forinstance, in [130] the authors study free electrons in a half-filled chain and900 2 4 6 3.10: Exponential decay of the entanglement entropy evolution atlate times; α = 0.5, M = 0.1, L = 0.8. The fit parameters for the particularchoice of quench parameters turns out to be a1 = 2.5335, a2 = 0.5277,a3 = 0.6049, and a4 = 0.0454. Note that we evolve the solution for late butfinite time, which explains why a4 6= 0. In the infinite time limit we expecta4 = 0.determined the growth and decay of the entanglement entropy after a localquench. In that set-up they find a very slow return to the unperturbedvalue. In two dimensions the decay is characterized by SA(t) ∼ a1 log(t)+a2tas t → ∞. The parameters a1, a2.are again obtained by fitting and dependon the specific details of the quench.It is somewhat intriguing that the holographic computations relax muchfaster. This is reminiscent of features of scrambling in black hole physics,which we comment on in our discussion § Conclusions and Future DirectionsThe main focus of the present paper was to describe the dynamics of theholographic entanglement entropy following a local quench. While this prob-lem has been studied in the past using various known exact solutions tomodel the quench, we have carried out a full numerical simulation of Ein-stein’s equations in the presence of a perturbing external source on theboundary of AdS. Given the explicit numerical solution to the quench ge-ometry, we can study the dynamics of entanglement entropy by exploring91the behaviour of extremal surfaces that are anchored on the boundary.The upshot of our analysis was a clear signal that entanglement entropydisperses linearly, in a manner reminiscent of the Lieb-Robinson light-cone.The dispersion velocity appears to depend on the details of the quench,though we were able to bound the result between two interesting bounds thathave been discussed in the literature earlier. On the one hand we found thatfor wide quench profile, the propagation speed saturated a putative lowerbound, given by the entanglement tsunami velocity obtained by [47] in thecontext of global quenches (modeled using the Vaidya-AdS spacetime). Onthe other hand well localized quenches appear to propagate entanglementat the speed of light. It is rather curious that we have results very similarto the Vaidya-AdS quench, for the geometry we construct is not the same.This lends support to thesis of [47, 61] that the holographic tsunami velocityought to be a generic phenomenon.The second aspect of holographic entanglement entropy which is inter-esting in our study is the rather rapid reversion of result to the equilibriumvalue. In various simulations we have tested, the reversion is exponentiallyfast, in contrast to the much slower logarithmic decay seen in spin models.This suggests again, as has been suspected in the past, that black holes arevery efficient at information processing, cf., [131, 132].There are many other interesting areas for further investigation. It wouldbe interesting to study other quench protocols and other theories, includ-ing massive models, primarily to extract a more detailed dependence ofthe entanglement velocity and the rate of equilibration. A particularly in-teresting direction is the study of (global and local) quenches past criticalpoints, generalizing the results of [133] to higher dimensions. It would alsobe interesting to study other non-local measures besides the entanglemententropy, which are more sensitive to the spatial structure of entanglementin quantum field theory, and to the differences between strongly coupledholographic CFTs and CFTs of small central charge. In particular, the mu-tual information of disjoint intervals would be interesting to calculate in oursetup for local quenches. Finally, one can make a direct connection to thestudy of entanglement entropy following a local quench in two-dimensionalCFTs, for which we have analytic results to explain the behaviour at largecentral charge [124]. We hope to report on these results in the near future[134].92Chapter 4Comments on EntanglementPropagationWe extend our work on entanglement propagation following a local quenchin 2+1 dimensional holographic conformal field theories. We find that entan-glement propagates along an emergent lightcone, whose speed of propagationvE seems distinct from other measures of quantum information spreading.We compare the relations we find to information and hydrodynamic veloci-ties in strongly coupled 2+1 dimensional theories. While early-time entan-glement velocities corresponding to small entangling regions are numericallyclose to the butterfly velocity, late-time entanglement velocities for largeregions show less regularity. We also generalize and extend our previousresults regarding the late-time decay of the entanglement entropy back toits equilibrium value.4.1 IntroductionThe generation and propagation of quantum information is a fascinatingsubject, bringing together insights from quantum information theory, many-body physics and perhaps most surprisingly, studies of the quantum mechan-ics of black holes. Here we focus on entanglement as a measure of quantuminformation.One way to generate entanglement is by quenching the system, i.e. start-ing the evolution from an atypical excited state of the Hamiltonian, usuallygenerated as the ground state of another, closely related Hamiltonian. Thequenching process generates short range entanglement which then evolvesand propagates as the system reaches the typical, thermal state1.1We note in passing that much of the work on holographic quenches has been doneat finite temperature, for example quenching past thermal critical points. Such studiesmix quantum entanglement and classical correlations. To directly probe the quantumentanglement of the ground state one needs to work at zero temperature, for examplequenching past quantum critical points. While some work in that direction has beendone, much more remains to be explored. On the holographic side, the bulk geometry atzero temperature does not involve a regular horizon, which makes both the mechanics andphysics quite different from the thermal case.93In the holographic context, quenching the system can be achieved bystarting at equilibrium and turning on external sources (non-normalizablemodes) for marginal or relevant operators, which drive the system out ofequilibrium for a finite duration of time. Much attention has been givento global, i.e. spatially homogeneous, quenches. In this case the time-dependence of the entanglement entropy is the observable of interest, andmany insights have been gained both in the holographic context, as well as inmore traditional approaches to many body physics. Models of entanglementevolution, based on those results, are put forward in [47, 61, 63, 135, 136].It would be interesting to incorporate the spatially-resolved holographic re-sults, discussed here and in [137], into such models.Indeed, the setup of local quenches, whereby the system is excited locallyin the spatial domain, provides a spatially-resolved probe of the generationand propagation of entanglement. In [137] we initiated the study of suchquenches, and here we continue that study in a more general set of holo-graphic theories involving a charged black hole horizon, corresponding tostrongly coupled conformal field theories in 2+1 dimensions at finite chargedensity. We focus on testing our previous results concerning entanglementpropagation in this more general, yet quite similar, context. We are thusable to generalize and improve our original discussion, to test which of ourprevious results are robust, and to investigate which of our conjectures holdin a more general context2.Similarly to our previous work, we find that entanglement propagationdefines an emergent lightcone structure for the theory. The maximal valueof entanglement defines a lightcone, except for narrow transition regimes.We typically find two associated lightcone velocities, one to do with shorttimes, and one with longer times3. The associated lightcone velocity vE inthose regimes depends on various parameters, and we have previously foundsome regularities in the neutral quenches.Here we extend that analysis: we find that the early-time velocity seemsto be related to the butterfly velocity, while late-time velocities have morecomplicated phenomenology. We discuss the phenomenology of vE in thismore general setup, and compare our results to other measures of entangle-ment propagation in that regime. We also discuss the return of the entan-glement entropy to its equilibrium value, where we are able to give moreprecise results than previously due to improved numerics.2This is similar in spirit to [138, 139], where it was found that breaking conformalinvariance has only a limited effect on holographic results.3Due to numerical limitations, these not asymptotically long times.94The outline of this paper goes as follows: In Section 4.2 we discuss oursetup for local quenches in charged spacetimes, our numerical integrationstrategy using the characteristic formulation of general relativity, and ourholographic calculation of the extremal surfaces encoding the entanglemententropy of regions on the boundary. Section 4.3 contains analysis of the dy-namics of holographic entanglement entropy. We continue our investigationof the emergent lightcone structure that encodes the spatial propagationof entanglement entropy, by including the effects of charge and discussingvarious mechanisms that may underlie the phenomenology of entanglementdynamics. We also extend our description of entanglement thermalization,for which an improved numerical implementation of the quenches’ evolutionat late times reveals a logarithmic return to equilibrium rather than an ex-ponential damping. We provide a brief summary of our results in Section 4.4as well as further details on the numerical aspects of this work in AppendixB.4.2 Holographic Local QuenchesIn this section we introduce our setup for local quenches in charged space-times. The local quench is generated by an inhomogeneous scalar sourcewhich is turned on for a finite duration. The resulting bulk solution is foundnumerically, and the extremal surfaces in that geometry encode the dynam-ics of the entanglement entropy. Here we describe that setup, before turningto the results in the next section. We focus mostly on differences from [137],and the reader may wish to consult that reference for additional details.4.2.1 Setup for Charged QuenchesWe choose our metric to be a generalization of the infalling Eddington-Finkelstein coordinates for black holes in an asymptotically AdS4 spacetime[73, 140]ds2 = −2Ae2χ dt2 + 2 e2χ dt dr − 2Fx dtdx+ Σ2(eB dx2 + e−B dy2),(4.1)and we introduce a gauge field V in the radial gaugeV = V0 dt+ Vx dx. (4.2)The coordinate r denotes the radial bulk coordinate, with the boundarylocated at r = ∞, and t is a null coordinate that coincides with time on95the boundary. Our quench, controlled by a relevant scalar on the bound-ary, will have local support in x while being translationally invariant in they direction. Hence all the fields under consideration depend only on thecoordinates {r, t, x} with ∂y being an isometry.This null slicing of spacetime, known as the characteristic formulation,is well adapted to treat gravitational infall problems since the coordinatesremain regular everywhere as the quench propagates through the bulk. Ouransatz also provides us with a residual radial diffeomorphismr → r = r + λ(xµ) , (4.3)which we use to fix the coordinate location of the apparent horizon and thuskeep the computational domain rectangular.The Einstein-Maxwell equations in the presence of a scalar field are givenbyRMN − R2GMN − 3`2AdSGMN = TΦMN + TVMN , (4.4)∇MFMN = 0 (4.5)where the matter stress tensors are given byTΦMN = ∇MΦ∇NΦ +GMNLΦ, LΦ = −12(GMN∇MΦ∇NΦ +m2Φ2),(4.6)T VMN = GABFMAFNB − 14F 2GMN , F = dV. (4.7)Before the quench, the spacetime geometry obeys the vacuum Maxwell-Einstein equations and is described by the RNAdS4 black hole of mass Mand charge Qds2 = −r2f(r) dt2 +2 dt dr+r2 (dx2 + dy2) , f(r) = 1−Mr3+Q22r4, (4.8)and the time-component of the gauge field isV0 = µ− Qr, µ ≡ Qr+. (4.9)The chemical potential µ is chosen so that V0 vanishes at the event horizon.In fact, RN black holes typically possess two horizons r±, which correspond96to the two real solutions of f(r) = 0. The black hole’s Hawking temperatureis given byT =r2+f′(r+)4pi, (4.10)and extremality occurs when T = 0, i.e. when Q =√3Mr+/2 and the twohorizons coincide.4.2.2 Asymptotic AnalysisWe now turn our attention to the asymptotic behaviour of our system. Wefirst make a simplifying choice and take m2 `2AdS = −2 in order to ensurethat the near-boundary expansion of the bulk scalar field is in integer powersof 1/rΦ(r, t, x) =φ0(t, x)r+φ1(t, x)r2+ · · · . (4.11)Requiring that the Einstein-Maxwell equations in the presence of Φ aresatisfied as r →∞ informs us that the gauge field behaves likeV0(r, t, x) = µ(t, x)− ρ(t, x)r+ · · · (4.12)Vx(r, t, x) = µx(t, x) +jx(t, x)r+V(2)x (t, x)r2· · · (4.13)whereas the metric components have the asymptotic expansionA(r, t, x) =(r + λ(t, x))22− ∂tλ(t, x)− 14φ0(t, x)2 +a(3)(t, x)r+ · · ·(4.14)χ(r, t, x) =c(3)(t, x)r3+ · · · (4.15)Fx(r, t, x) = − ∂xλ(t, x) + f(3)(t, x)r+ · · · (4.16)Σ(r, t, x) = r + λ(t, x)− 14φ0(t, x)2 + · · · (4.17)B(r, t, x) =b(3)(t, x)r3+ · · · . (4.18)The functions G(3)µν are undetermined by the equations of motion and requirethe input of boundary data via the stress tensor Tµν [13], defined in its97Brown-York form as [141]Tµν = Kµν −Kγµν + 2 γµν −(γRµν − 12γRγµν)+12γµν φ20, (4.19)where γµν is the induced metric on the boundary, Kµν ,K ≡ γµνKµν itsextrinsic curvatures, and γRµν ,γR its intrinsic curvatures. It is straightfor-ward to show thatT00 = 2a(3) + 4c(3) + φ0φresponse, (4.20)Ttx =32f (3) − 12φ0∂xφ0 , (4.21)and that these components obey the conservation equations∂tT00 = ∂xTtx + ∂tφ0 φresponse − (∂tµx − ∂xµ)2 − jx(∂xµ− ∂tµx), (4.22)∂tTtx =12(∂xT00 − 3 ∂xb(3) + ∂xφ0 φresponse − φ0 ∂xφresponse)+ ρ(∂xµ− ∂tµx). (4.23)In addition to energy and momentum, the electric charge and current arealso conserved∂tρ = − jx − ∂2xµ+ ∂t∂xµx, (4.24)∂tjx = V(2)x + jxλ−12∂xρ. (4.25)4.2.3 Integration StrategyThe characteristic formulation of the Maxwell-Einstein and Klein-Gordonequations conveniently reorganizes the coupled PDEs in two simpler cate-gories: equations for auxiliary fields that are local in time and that obeynested radial ODEs, and equations for dynamical fields that propagate thegeometry from one null slice to the next [73, 140]. Here we outline ournumerical integration strategy, and refer the reader to Appendix B for adiscussion on the more technical aspects of our implementation.We modelled the quench source function as φ0(t, x) = f(x)g(t), withf(x) =α2[tanh(x+ σ4s)− tanh(x− σ4s)], g(t) = sech2(t− tq∆tq).(4.26)98We let the scalar field profile reach a maximum value α at time t = tq∆. Weset tq = 0.25 and ∆ = 8, and chose the steepness s according to the widthσ of the perturbation in order to have a smooth profile. By t = 3, φ0 ≈ 0,and the quench has concluded.We performed domain decomposition in the radial direction, using 4 do-mains each discretized by a Chebyshev collocation grid containing 11 points.In doing so, errors located near the boundary or near the apparent horizonremain localized within their respective subdomain [127], thus improving thesolutions for auxiliary fields over the entire radial domain. We discretizedthe spatial direction using a uniformly-spaced Fourier grid over the interval[−30, 30] and used 121 points for σ = 2, and 173 points for σ = 0.5 tomaintain an acceptable spatial resolution as the quench profile propagatesfurther away at later times.As for the time evolution, we used an explicit fifth-order Runge-Kutta-Fehlberg (RKF) method with adaptive step size to propagate dynamicalquantities. Note that we evolved each quench until t = 20, the approximatetime at which the fields perturbations reach the spatial boundaries. We alsogot rid of high-frequency modes that contaminated our solutions by applyinga smooth low-pass filter that discarded the top third of the Fourier modes.However, we remark that it is important not to filter the bulk scalar fieldΦ if we want its RKF-propagated boundary profile to agree with the sourceφ0 at all times.4.2.4 Holographic Entanglement EntropyThe next step after obtaining numerical solutions for our local quench isto study the evolution of the holographic entanglement entropy (HEE) of aregion A on the boundary. For simplicity, we consider a strip that extendsinfinitely in the y directionA = {(x, y) | x ∈ (−L,L), y ∈ R} , ∂A = {(x, y) | x = ±L, y ∈ R} .(4.27)The covariant HEE prescription [41] tells us that the entanglement entropyis determined by the area of extremal surfaces anchored on ∂A. It is naturalto use the quench’s translational invariance to parametrize the extremalsurfaces by the coordinates τ and y. The extremal surfaces we are lookingfor will also be translationally invariant in y, and the problem of calculatingtheir area reduces to that of calculating the proper length of the geodesics99XM (τ) = {t(τ), r(τ), x(τ)} arising from the LagrangianL = Gyy GMNX˙MX˙N . (4.28)The resulting system of 3 second order ODEs can be transformed into asystem of 6 first order ODEs in the variables{t, Pt ≡ Σ2 t˙, r, P+ ≡ e2χ(r˙ −A t˙) , x, Px ≡ Σ2 x˙− e−BFx t˙} ,(4.29)for which L = 2P+Pt + P 2x .Keeping in mind that the length of a geodesic in an asymptotically AdSspacetime is formally infinite, we introduce a UV cutoff r = −1 and usea regularization scheme in which we subtract the entanglement entropy ofa RNAdS4 geometry expressed with the radial coordinate r¯ = r + λ(t, x),thus effectively matching asymptotic coordinate charts in both setups andsetting ∆SA = 0 prior to the quench4.To solve the Euler-Lagrange equations derived from (4.28), we adopt aninitial value problem point of view in which the initial conditions at theturning point are{t = t∗, Pt = 0, r = r∗, P+ = 0, x = 0, Px = ±1} , (4.30)and we use a shooting method in r∗ so that x = L when r = −1. Note thatthe tolerance parameters of the ODE solver must be chosen so that L = 1along the geodesic, which in turn provides us with a safety check for oursolutions.4.3 ResultsHaving described our setup and methods of calculation, we now turn tosummarizing the patterns observed in our extended framework. In eachcase, we provide context by starting our discussion with a brief reminder ofour observations for neutral local quenches before broadening the scope ofour analysis to account for the effects of charge.4This regularization procedure is equivalent to subtracting the vacuum entanglemententropy for the region A with a dynamical cutoff vac(t, x) related to the radial shift λ(t, x).1004.3.1 Emergent Lightcone and Entanglement VelocityEntanglement lightconeThe local nature of the quenches (having finite energy at infinite volume, i.e.zero energy density) implies that the entanglement entropy of any region Ainitially grows with time, reaching a maximum, before inevitably decayingto its pre-quench value as the perturbation dissipates away. Much of ouranalysis has to do with the spatial structure of that maximum, as a functionof the spatial extent L of A and the time t. We find that, except for narrowtransition ranges, the curve traced by the maximum in the L − t planeis linear: the spatial propagation of entanglement defines a new lightconestructure, distinct from the causal structure of both bulk and boundarytheories.We note that a similar observation was made in [104], in which localquenches are implemented as a perturbative approximation to the backreac-tion caused by a massive infalling particle in pure AdS. In that context, thetrajectory traced by ∆SA(tmax, L) in the L− t plane always follows a slopeof vE = 1 (additionally, the amplitude of that maximum remains constantthroughout).It turns out that the structure of our results is much richer since ournumerical scheme accounts for the full backreaction of the quench on thegeometry. While our data reveals the appearance of an emergent lightcone,this result emerges from the analysis rather than being one of the assump-tions put in by hand. Indeed, as we will detail below, we typically find twolinear regimes separated by a narrow transition, with distinct velocities atearly and late times.The slope of the curve traced by the maximum, vE , is a natural measureof how fast entanglement propagates spatially. Much of our analysis has todo with analyzing this velocity vE . We find a rich structure in the depen-dence of the emergent lightcone velocity on parameters. In particular, whileit is conceptually similar to other measures of quantum information spread-ing such as the butterfly or tsunami velocities, we find that it is numericallydistinct from them under most circumstances.Let us now turn to describing the regularities found in the entanglementvelocity vE .101Entanglement velocityAs is expected from a relativistic theory, we found that vE was bounded fromabove by the speed of light, with the bound being saturated universally inthe high temperature regime.Perhaps more interesting was the discovery of a lower bound on vE dif-ferent from the speed of sound of a three-dimensional CFT, vsound = 1/√2 =0.707. Indeed, the speed of sound, which underpins the thermalization ofenergy and momentum on the boundary theory, seemed a likely candidate totrack the generation and propagation of entanglement. However, our initialanalysis showed that this lower bound lied slightly below vsound, and in factwas consistently very close to v∗E(3), the tsunami velocity of a Schwarzschild-AdS4 black hole [47]v∗E(d = 3) =(η − 1) 12 (η−1)η12η∣∣∣∣∣d=3=√3243= 0.687, with η =2(d− 1)d.(4.31)The tsunami velocity is a holographic measure of how fast entanglementpropagates spatially when spacetime is globally quenched and depends uniquelyon a black hole’s conserved charges. Given the naturalness of this velocity inmatters related to entanglement entropy propagation, we conjectured thatvE should be found within the boundsv∗E(3) = 0.687 ≤ vE ≤ 1. (4.32)This situation is in a way reminiscent of quantum spin systems, which admitan upper Lieb-Robinson bound on the speed at which information can traveldespite the absence of relativistic constraints [129]. However our holographiccalculation also provides us with an unexpected lower bound on informationprocessing based on the properties of spacetime itself.We now extend our analysis of the structure of the entanglement light-cone and the velocity vE by including the effects of charge. Our main resultpersists: in all the cases we examined, the entanglement traces a lightconestructure. We can therefore look more closely at the relation between theentanglement propagation velocity vE defined by our emergent lightconestructure, and other closely related velocities. We note that while thosevelocities are conceptually similar, and numerically close to each other forneutral black holes, their dependence on charge is distinct. We can thereforehope to make better distinction between them by examining our results fordifferent parameter ranges, in particular focussing on the charge dependence.102Relation to other velocitiesIn our simulations for wide quenches we find two stages for entanglementpropagation, both exhibiting a lightcone structure, and a narrow transitionregime between them. For the early-time results, governing the evolutionof small entangling surfaces, it is natural to suspect some relation to thebutterfly velocity, quantifying the spatial spread of chaos [142–144]. Wenote that the presence of charge does not affect its value: vbutterfly =√3/2 =0.866.Indeed, this velocity seems to play a role in our results for the spatialpropagation of entanglement entropy: early-time velocities are in the rangevE ∈ [0.8, 0.9], numerically close to the butterfly velocity. In fact, it wasshown that the butterfly velocity naturally characterizes the saturation timefor large strip regions in the case of global quenches [145]. Since the L < σregime under consideration approximates a global quench for which tmaxcan be thought of as the saturation time’s counterpart, vbutterfly seems alikely candidate to quantify the initial spread of quantum information thatwe observe.For the late-time velocities, governing the evolution of larger entanglingsurfaces, we had previously found a relation to the tsunami velocity, whichappears as a lower bound of entanglement propagation in the neutral case.It turns out that the tsunami velocity of RNAdS4 black hole decreases as itscharge increases, ultimately vanishing at extremality. If the tsunami veloc-ity serves as a lower bound for all values of the charge, then the addition ofcharge should change the measured slopes vE in a predictable way. In partic-ular, we should find that the spatial propagation of the entropy significantlyslows down near extremality.Note however a subtle order of limits issue. Our numerics, performedoutside the apparent horizon, are restricted to sufficiently narrow entanglingsurfaces. This is sufficient for discovering the emergent lightcone structure,which we investigate here. However, the asymptotic IR limit L → ∞ is apriori distinct and may exhibit different regularities. In particular, even inthe extremal limit, the entangling surfaces relevant for the emerging light-cone are not deformed much in the near-horizon region. It may be the casethat infinitely wide surfaces are more sensitive to the near-horizon geometry,and thus exhibit a more dramatic behaviour in the near-extremal limit.As it turns out, the inclusion of charge does not affect our results in adramatic way, in this regime. Figure 4.1 shows the small effect charge hason the lower bound for entanglement propagation speeds; the slopes vE allfall within the same range for all charged configurations. In the case where1033.0 3.5 4.0 4.5 M = 0.1, σ = 23 4 5 6 M = 0.01, σ = 2Figure 4.1: The curves traced by the maximum of ∆SA(t) in the L−t plane.Note that all charged configurations have been included in the same figureto illustrate the weak dependence of the lightcone behaviour with respect tocharge. In both cases, the early-time velocities are found in close proximityto the butterfly velocity (vE ∈ [0.8, 0.9]), whereas the late-time velocitiesare found within vE ∈ [0.65, 0.71], an interval containing various velocitiesof interest.the minimal surfaces can penetrate deeper in the bulk, we still observe twolinear regimes (as in Fig. 4.1b) corresponding approximately to L < σ andL > σ. The tsunami velocity originally appeared in the large L limit, andwe observe that charge only marginally decreases the slope vE .The range of the lightcone velocities found at large L in our simula-tions, vE ∈ [0.65, 0.71], is also fairly close to other hydrodynamic velocities:vsound = 0.707 and vshear = 0.665. The latter is obtained from second-orderhydrodynamics results interpreted in terms of the phenomenological Muller-Israel-Stewart theory. This shear velocity, which encodes the velocity of thewavefront of momentum relaxation, is defined as [146]vshear =√DητΠ≈ 0.665, (4.33)where Dη is the effective shear “diffusion” constant obtained from an anal-ysis of the sound pole, and the hydrodynamic parameter τΠ is the shearrelaxation time, which can be calculated from AdS/CFT [147]Dη =14piT, and τΠ =34piT[1− 12(log 3− pi3√3)]. (4.34)104As this velocity has to do with entropy production, it can naturally affectthe evolution of holographic entanglement entropy in our setup.In summary, it remains unclear exactly what phenomena come into playto influence entanglement propagation in the late-time regime, where we findan emergent lightcone. On one hand, we have seen that the slope tracedby ∆SA(tmax, L) does not decrease as we approach extremality, which sug-gests that the charged tsunami velocity does not provide an appropriatedescription of the lower bound for vE . Additionally, our analysis remainsinconclusive as to the relevance of the neutral tsunami velocity v∗E(3). Wealso see that the entanglement velocity is fairly close to hydrodynamical ve-locities, related to entropy production. As such we are unable to disentanglethe various effects which may influence entanglement propagation, and it isentirely possible that different mechanisms may compete to influence theshape of the entanglement lightcone in the late-time regime, resulting in thevariations observed in vE .4.3.2 Entanglement MaximumIn the neutral case, we found that the value of the entanglement entropymaximum ∆SA(tmax) increased linearly with the size L of the entanglingregion for small L. This increase was also quantified as a function of thescalar source’s maximal amplitude α∂∂L∆SA(tmax, L;α) ∼ α2. (4.35)For fixed amplitudes, we observe that the maximum ∆SA(tmax) was notaffected by the addition of charge for small L, and increased marginallywhen changing Q, even as we approach extremality (see Figure 4.2). Thus,our previously discovered regularities seem robust to the addition of charge.4.3.3 Entanglement DecayWe now turn our attention to the late-time behaviour of holographic entan-glement entropy. Our earlier work on neutral quenches showed evidence thatthe process of return to equilibrium was best described by an exponentialdamping∆SA(t) = a1e−a2(t−a3) + a4. (4.36)The parameters ai depended on the particular features of the quench butdid not seem to follow any discernible pattern. However, our analysis was1050.5 1.0 1.5 M = 0.1, σ = 20.5 1.0 1.5 2.0 2.5 M = 0.01, σ = 2Figure 4.2: The maximum of ∆SA(t) as a function of strip width L forα = 0.1. Note that all charged configurations have been included in thesame figure to illustrate the weak dependence of the entanglement entropywith respect to charge.limited by the quality of our numerical quench solutions. In particular, thebulk fields could not be propagated past t = 9 without loss of accuracyat large x and large memory requirements. We managed to evolve thequenches up until t = 20 in a reasonable time by making a few modifications,106including increasing the spatial resolution by discretizing the x directionwith a uniform Fourier grid and by solving the radial ODEs for the auxiliaryfields independently for each discretized xj .These improvements allowed us to investigate the late-time behaviour ofthe entanglement entropy over much larger time intervals. This additionalinformation revealed that the exponential decay we observed previously wasdue to fitting the late-time data over too short of a time interval. In fact,the new data instead suggests that∆SA(t) ∼ a1 log t+ a2tδ, (4.37)is a much better fit, as illustrated in Figure 4.3. This result is more in linewith those derived from spin chain models [130].0 5 10 15 4.3: The late-time behaviour of the entanglement entropy closelyfollows the logarithmic decay (4.37) for L = {1, 2, 3}, from top to bottom,for α = 0.1, M = 0.01, σ = 2 and Q = 0.04. In this particular case, the bestfit exponents are, respectively, δ = {1.36, 1.48, 1.49}.Interestingly, the best-fit exponents δ, obtained by a least-square fit, aregenerally clustered around either δ = 1 or δ = 1.5, which marks a departure1070 5 10 15 M = 0.1, σ = 2, L = 3.10 5 10 15 M = 0.01, σ = 2, L = 3.4Figure 4.4: The decay of ∆SA(t) and its best logarithmic fit for Q =0.99 Qext and α = 0.1. The sizes L have been chosen such that the ex-tremal surfaces probe the near-horizon geometry at one point during thequench’s evolution, i.e. L is taken as large as the quench allows it to be. Wefind δ = 1.5 for the figure on the right.from the prediction ∆SA(t) ∼ t−6 made in the perturbative analysis of [104].Our findings show that there is a complex interplay between the size L, theinitial energy density M , the initial charge density Q, and the amount ofinjected energy in the characterization of entanglement entropy’s return toequilibrium. When M = 0.1, the logarithmic decay fits the data with δ = 1.5at low Q and small L for both σ = 0.5 and σ = 2. However, (4.37) becomes abad fit as either the charge and/or the size of A are increased, as showcasedin Figure 4.4a. We find that the breakdown occurs around Q ∼ Qext/2.In contrast, the logarithmic return to equilibrium fits the data for allvalues of Q and L when M = 0.01. When σ = 0.5, thermalization isdominated by δ = 1 except for near-extremal black holes Q = 0.99 Qext, forwhich δ = 1.5 no matter the size of the entangling surface. Taking σ = 2reveals an even richer picture in which we observe a sharp transition betweendecays characterized by δ = 1 and δ = 1.5. As illustrated in Figure 4.5, thelate-time evolution of holographic entanglement entropy in the neutral, largesize limit is fitted best with δ = 1. The exponent δ = 1.5 appears either asextremality is approached, as in the σ = 0.5 case, or in the small L limit, asin the M = 0.1 case.These observations lead us to believe that the late-time behaviour of∆SA(t) is influenced not only by the parameters characterizing the geodesicsand the geometry of the unquenched spacetime, but also by the amountof energy injected by the scalar quench. As such it is hard to disentangle108and generalize our findings when the underlying competing processes harborinherently different length scales.Figure 4.5: This figure illustrates the sharp transition between δ = 1 andδ = 1.5 in the logarithmic decay of the HEE as a function of Q and L forM = 0.01 and σ = 2.4.4 SummaryWe have studied the spatial propagation of entanglement entropy followinga local excitation of the system. We find that the entanglement genericallypropagates along an emergent lightcone, whose velocity may change overa narrow transition regime. In our simulation we find early and late-timevelocities, and look at their dependence on parameters and relation to otherinteresting information and hydrodynamical velocities.The early-time entanglement velocity for small strips seems similar tothe butterfly velocity. As both have to do with the initial propagation ofquantum information, we find that relation plausible, especially as it mirrorsan analytical result derived in an analogous global quench scenario. We arehowever unable to disentangle the various effects that could influence thelate-time entanglement velocity: the propagation in that regime seems likelyto be controlled by a combination of many mechanisms.109We are also able to exhibit some universality in the logarithmic returnof the entanglement to its equilibrium value. In particular, the relation toknown results for spin chains in 1+1 dimensional CFTs is intriguing.There are very few avenues to investigate the propagation of quantuminformation in higher-dimensional strongly coupled conformal field theories.We hope that the phenomenology we present can illuminate that difficultsubject: in particular it would be instructive to have a simple model in-corporating the regularities we find in the holographic results. We hope toreturn to these issues in the future.110Chapter 5On Brane Instabilities in theLarge D LimitUsing an expansion in large number of dimensions, taken to subleadingorders, we discuss several issues concerning the Gregory-Laflamme instabil-ities. We map out the phase diagram of neutral and charged black strings,and comment on the possible transition in the nature of the final state of theinstability at higher order in the 1/D expansion. We also discuss unstableblack membranes, and show that in certain limits the preferred shape of thenon-uniform phase is a triangular lattice.5.1 IntroductionSince its discovery, the Gregory-Laflamme instability [66, 67] has been asource of many insights into General Relativity and its extended black branesolutions in higher dimensions. The fate of the instability for a string ismuch studied (for a comprehensive review, see [148]): there is now strongevidence that the end-point of the black string instability depends on thenumber of spacetime dimensions. It was shown in [70] that there exists acritical dimension D∗ = 13.5 above which non-uniform black strings (NUBS)become stable and have larger horizon areas than their uniform counterparts,thus making them natural candidates as the end-point of the GL instability.BelowD∗, numerical simulations [69, 149] have presented evidence that blackstring horizons bifurcate in a self-similar cascade of black holes pinching offto arbitrary small scales along the string direction, thus violating the cosmiccensorship hypothesis (despite the arguments proposed by [68]). A numericalevolution beyond the critical dimension would be a welcome addition, butthe high-performance computing resources required would be an obstacle tothis endeavour.A different approach — general relativity in the limit where the numberof dimensions is large [64] — offers a promising framework in which onecan address such questions analytically, or numerically with only modestresources. Despite the theory being formally valid only when D → ∞, itspotential even at finite D was highlighted in a range of applications, ranging111from the striking agreement of large D black holes quasinormal modes forboth large and small values of D, to the alternative derivation of the criticaldimension D∗ found in [72].In this paper we discuss different aspects of the phase structure of thenon-uniform black objects. Following in the footsteps of [71, 150], we per-form our analysis by promoting the mass, charge and momentum densitieson the string to be collective variables, and solve the resulting equationsnumerically, observing their conserved charges at asymptotic infinity. Us-ing this approach, we discuss several issues concerning the end-state of theGregory-Laflamme instability of extended black objects.Discussing the general charged black string, we find that the entropy dif-ference between the non-uniform configuration and its uniform counterpartremains finite and positive for all such charged black strings, even in theextremal limit. Indeed, in the extremal limit we are able to show that factanalytically. Thus we conclude that there is a second order transition to anon-uniform phase for all charged NUBS, which are entropically favoureddespite the weakening of the GL instability due to the addition of electriccharge.We also investigate the physics of the neutral string to next-to-leadingorder (NLO). Our goal is to find the signal, in the large D expansion, of thetransition of the instability end-point from a non-uniform black string to apinch-off scenario. Indeed, below the critical dimension D∗ where the NUBShave lower entropy than the uniform string, there is a different end-pointto the instability, which is expected to be a pinch-off. While we find signsthat this is indeed the case, we are unable to find a universal value for theassociated critical dimension (which may be different from D∗) from ouranalysis.Lastly, we turn our attention to the phase structure of two-dimensionalunstable membranes on oblique lattices. By comparing brane solutions ofdifferent shapes, we find that the triangular lattice configuration is the onethat minimizes the corresponding thermodynamic potential for localized 2-branes.The outline of the paper is as follows: In Section 5.2, we summarize howto obtain charged (and neutral) black string solutions in the characteristicformulation of general relativity, at leading order in D. This serves to setup our notations and explains our numerical method. In Section 5.3, wediscuss the phase structure of charged black string. To that end, we find thesubleading corrections to the metric and gauge field, necessary to discussthe entropy difference (in the micro-canonical ensemble) between the uni-form and non-uniform solutions. We find that a charged non-uniform string112always have a larger horizon area than uniform configurations, even in theextremal limit. We also obtain the equations governing the dynamics of the1/D corrections to the mass and momentum densities, and discuss stabil-ity conditions of the neutral string to next-to-leading order, and the signalsthat there may be a transition to pinch-off as the final state as we lower D.Lastly, we explore the thermodynamics of unstable two-dimensional braneson general oblique lattices in Section 5.4. We find that the preferred shapeof the lattice is triangular, up to small deviations, likely due to finite sizeeffects.5.2 Charged p-brane SolutionsWe start our analysis by finding static charged black string solutions. Keep-ing in mind that we are interested in finding non-uniform black string solu-tions, we must allow redistribution of mass and charge to occur along thespatial direction. To this end, we introduce a local Galilean boost velocityand promote the mass and charge densities to vary along the string. Wethen solve the Einstein-Maxwell equations at leading and subleading orders,from which we extract the effective equations that describe the nonlinearfluctuations of the string horizon.5.2.1 Uniformly Charged p-BranesThe equations of motion that charged, spherically symmetric p-branes satisfyin the limit where D → ∞ can be obtained by considering the Einstein-Maxwell action in D = n+ p+ 3 dimensions,IEM =∫dDx√G(RG − F24), (5.1)where F = dV is a Maxwell potential. Performing dimensional reduction on(5.1) so that the metric becomes of the formds2G = gµν(x)dxµdxν + eφ(x)dΩ2n+1, (5.2)where the coordinates xµ = (t, r, zA) span a p + 2 dimensional space, weobtain the action (see Appendix C for details)IEM =∫dp+2x√g e(n+1)φ2(Rg + n(n+ 1)e−φ +n(n+ 1)4(∇φ)2 − F24).(5.3)113The equations of motion that follow from (5.3) are [151]Rµν − n+ 12∇µ∇νφ− n+ 14∇µφ∇νφ−12(FµαFαν −12(n+ p+ 1)F 2gµν)= 0∇αFαµ + n+ 12∇αφ Fαµ = 0n e−φ − n+ 14(∇φ)2 − 12∇2φ+ 14(n+ p+ 1)F 2 = 0.(5.4)The solution to these equations is known [152]: non-dilatonic black p-branesin the presence of an electric potential have the metricds2 = − fh2dt2 + hB(dr2f+ r2dΩ2n+1 + d~z2), (5.5)wheref(r) = 1−(r0r)n, h(r) = 1 +(r0r)nsinh2 α, B =2n+ p, (5.6)whereas the gauge field V has solutionV = −√Nh(r0r)nsinhα coshα dt, N ≡ B + 2. (5.7)In these coordinates, the outer horizon is located at r = r0, whereas theinner horizon coincides with the singularity at r = Characteristic Formulation for a Charged BlackStringTo describe a non-uniform charged black string, we start by a more gen-eral ansatz where the black string is locally boosted along its worldvolumeza = (t, zA). Doing so is easier in the characteristic formulation of generalrelativity, where the metric is expressed in terms of the ingoing Eddington-Finkelstein (EF) coordinates. For a general Lorentz boost ua, the EF coor-dinates σa = (v, xA) take the form [153, 154]σa = za + uar∗, r∗(r) = r +∫ ∞rf − hN/2fdr. (5.8)114The boosted metric for the charged string becomesds2 = hB(− fhNuaubdσadσb − 2h−N/2uadσadr + ∆abdσadσb + r2dΩ2n+1),(5.9)where ∆ab = ηab + uaub is the orthogonal projector defined by the boostvector. Similarly, the gauge potential V becomesV = −√Nh(r0r)nsinhα coshα uadσa, (5.10)and we take the radial gauge in order to set Vr = 0.Our aim is to find solutions to the Einstein-Maxwell equations in whichthe black string’s energy and charge densities, as well as the boost velocityalong the string, are promoted to collective coordinates that vary in timealong the x-direction. Given the hierarchy of scales present in the large Dlimit of black holes, we must specify the length scale relevant to the physicswe wish to explore. It is known that black branes are unstable when sub-jected to perturbations of wavelength ∼ r0/√D. As such, we need to rescalethe direction x along the string, dx→ dx/√n, thus making the boost non-relativistic. Additionally, the quasinormal modes under consideration scalelike ω ∼ O(D0), implying that the dynamics of the near-horizon geometry isdecoupled from the asymptotic region [155]. Consequently, we will requirethe metric components to be asymptotically flat and the potentials to vanishat infinity at all orders in the perturbative expansion.We write the metric and the gauge potential in terms of unknown fieldsds2 = −Adv2 + 2uvdvdr + 2uadxadr − 2Cadxadv +Gabdxadxb, (5.11)V = V0 dv + Va dxa, (5.12)for which we allow the following 1/n expansion:A =∑k≥0A(k)(v, x,R)nk, uv =∑k≥0u(k)v (v, x,R)nk, Ca =∑k≥0C(k)a (v, x,R)nk+1,(5.13)Gab =1n1 +∑k≥0G(k)ab (v, x,R)nk+1 , ua = u(0)an+∑k≥1u(k)a (v, x,R)nk+1, (5.14)115V0 =∑k≥0V(k)0 (v, x,R)nk, Va =∑k≥0V(k)a (v, x,R)nk+1, (5.15)where the new radial coordinate R = (r/r0)n is well-suited for near-horizonanalysis. For the scalar field, we make the choice to keep φ = log(r2hB)at all orders in the expansion in order to maintain the spherical symmetryof the solution. Also note that demanding ua to be a constant at leadingorder is simply a gauge choice; we will set u(0)a = 0 to fix the rest frame ofthe black string.5.2.3 Solutions and Effective Brane Equations at LeadingOrderAt leading order, the solutions to the Einstein-Maxwell equations are givenbyA(0) = 1− mR+q22R2, C(0)a =(1− q22mR)paR, u(0)v = 1, (5.16)G(0)ab =(1− q22mR)papbmR− log(1− R−R)(2δab + ∂apbm+ ∂bpam), (5.17)V(0)0 = −qR, V (0)a =qpamR. (5.18)Note that the radial coordinate appearing above has been shifted so thatthe outer and inner horizons of the charged black hole are now located atR± =12(m±√m2 − 2q2). (5.19)The collective variables m and q are directly related to the energy and chargedensities M and Q of uniform p-branesM = rn0 (n+ 1 + nN sinh2 α), Q = n√Nrn0 sinhα coshα. (5.20)The large n expansion of these conserved quantities shows a correspondencebetween the old and new effective fields on the branes at leading order:m ≡ rn0 cosh 2α and q ≡rn0√2sinh 2α. (5.21)As for p, it is related to the momentum density on the black brane; the gaugechoice u(0)a = 0 ensures that the total momentum on the brane vanishes.116The equations that govern the dynamics of the collective variables forgeneral p-branes are1 [150]:∂tm− ∂i∂im = −∂ipi, (5.22a)∂tq − ∂i∂iq = −∂i(piqm), (5.22b)∂tpi − ∂j∂jpi = ∂i (R+ − R−)− ∂j[pipjm+ R−(∂ipjm+ ∂jpim)]. (5.22c)These equations, which describe the fluctuations of black branes on thecompactified string directions xi in the large D limit, can also be written asconservation equations ∂µτ(0)µν = 0 for a quasilocal stress tensor at R→∞,whose components areτ(0)00 = m, τ(0)0i = ∂im− pi, (5.23)τ(0)ij = ∂i∂jm− (R+ − R−) δij +pipjm− (∂ipj + ∂jpi)(1− R−m). (5.24)These equations are very easy to solve numerically. In doing so we dis-cover the stable end-point of the charged string instability, as well as thetime-dependent process leading to that end-point. To discuss the thermo-dynamics, to which we turn next, we need to discuss the next order in thelarge D expansion.5.3 Phase Structure of the NUBSHaving set up our equations and non-uniform solutions, we now turn ourattention to the phase structure of the charged black string, compactifiedalong x ∈ [−L/2, L/2]. A proper analysis first requires us to examine theproperties of NUBS to subleading order in the large D expansion.1In the large n limit, ∂t = ∂v and the dynamics of the collective variables take placein Schwarzschild time.1175.3.1 Solutions and Effective Brane Equations atSubleading OrderAt subleading order for the charged black string (p = 1), we find the solutionsA(1) = − δmR+qδqR2− q22R2+ log(1− R−R)[−mR+q22R2]( pm)′− log R[p′R− qR2(pqm)′](5.25)V (1) = − δqR− qRlog(1− R−R)( pm)′ − log RR(pqm)′(5.26)u(1)v =p22m2[−mR+q22R2]−(1− R−R)−1 R−R( pm)′(5.27)where a prime denotes differentiation with respect to x. It is straightfor-ward to verify that δm(σ) and δq(σ), which appear as integration constants,indeed correspond to the 1/n corrections to the mass and charge densitiesby computing the ADM mass M and the electric flux at infinity viaM = −∮Sn+1∞∇µξν(v)dSµν =∫ (m(σ) +δm(σ)n+ · · ·)dx, (5.28)Q = 12∮Sn+1∞FµνdSµν =∫ (q(σ) +δq(σ)n+ · · ·)dx, (5.29)where ξµ(v) = δµv is a timelike Killing vector, and the integration is performedover Sn+1 at spatial infinity. Some ambiguity remains when using (5.28)and (5.29) to define the mass and charge density corrections since a shift ineither quantity by the derivative of a periodic function results in identicalADM mass and electrix flux. We thus examine the multipole expansion ofg00 and V0 about asymptotic infinity to identify δm(σ) and δq(σ) as theappropriate corrections. Let us remark that M and Q remain conservedat all orders in the 1/n expansion, and as such the corrections δm(σ) andδq(σ) have vanishing integrals over the string direction.The above solutions will be useful in Section 5.3.2. For the remainderof this section, we will focus our efforts on the neutral case. The string’smomentum correction is found in the dvdx component of the metricC(1)x =δpR+log RR(p2m)′; (5.30)118it is a quantity associated with the asymptotic Killing vector ξµ(x) = δµxP =∮Sn+1∞∇µξν(x)dSµν =∫ (p(σ) +δp(σ)n+ · · ·)dx. (5.31)The equations that govern the dynamics of δm and δp are2∂tδm − δm′′ + δp′ = F ′δm, (5.32a)∂tδp − δp′′ −[(1 +p2m2)δm− 2pmδp]′= F ′δp, (5.32b)with the source functions Fδm and Fδp given byFδm = p+(m+ 2p′ − 32p2m)′, (5.33)Fδp = F0 logm+F02− p2m− 32(p3m2)′+4pm′m− 2p2m′′m2+4pp′′m, (5.34)andF0 = 2m[1 +( pm)′]( pm)′. (5.35)As is the case at leading order, these equations can also be rewritten ascorrection terms for the asymptotic stress tensor τij :τ(1)00 = δm, τ(1)0x = δp− δm′ − Fδm, (5.36)τ (1)xx = − F0(logm− 3) +m− δm(1 +p2m2)+ 2δppm+(δm+ 4m+ 4p′ − 7p2m)′′− 2(δp+ 3p− 32p3m2)′. (5.37)The left-hand sides of equations (5.32) correspond to the differential opera-tors one would find at order 1/n by letting m→ m+δm/n and p→ p+δp/nin the collective equations (5.22). However, the presence of the source termsbreaks Galilean invariance. Moreover, whereas the leading order equationsare invariant under a rescaling of the mass and momentum, the collective2The large D equations at NLO were first obtained in [156] for asymptotically AdSspacetimes, albeit in a gauge different from ours. We have confirmed that the equations weobtain agree with their AdS counterpart up to a redefinition of the momentum correctionand a few sign flips.119equations for the correction terms are not. This can be understood as aconsequence of the dependence of the black string temperature on its massdensity m0 at NLO. Indeed, when the NUBS is stationary, one can calculatethe surface gravity κ via the relationκ2 =√−12∇µξν(v)∇µξ(v)ν , (5.38)and evaluation at the Killing horizon is understood. Rescaling the surfacegravity such that the temperature is of O(1) at leading order, we obtainT =κ2pin=14pi− 14pin(logm− (m′)22m2+m′′m)= T (0) +T (1)n, (5.39)with T (1) = − 14pi(logm− m′22m2+m′′m). (5.40)As a consequence of this, we find that the shapes of δm and δp dependon the additional parameter m0. However, m0 should not be regarded asan independent parameter of our solutions. The initial state of our systemis uniquely characterized by D and the ratio L/r0, and as such mass andmomentum profiles at different initial temperatures contain the same in-formation packaged differently. It is easier to work in units where r0 = 1(for which m0 = 1 also), but we can equivalently rescale all dimensionfulquantities by an appropriate power of r0 to obtain the same information.Let us now turn our attention to the next-to-leading order correctionto the dispersion relation for the black string. Linearized perturbationsaround the uniform black string solution m(x) = m0 + ∆me−iωt+ikx, withmomentum k = 2pi/L aligned along the string direction x, allow for a non-trivial solution only if the condition∣∣∣∣∣ −k2 + iω − k2n −ik + ik(1−2k2)nik −k2 + iω − k2(1+2 logm0)n∣∣∣∣∣ = 0 (5.41)is satisfied. Letting Ω = −iω, the dispersion relation for the black stringreadsΩ(k) = k − k2 − k2n(1 + 2k + 2k logm0 − 2k2)+O(n−2). (5.42)120This yields the corrected threshold mode Ω(kGL) = 0 to bekGL = 1− 1 + 2 logm02n+O(n−2). (5.43)Rewriting the above equations as Ωˆ(kˆ) ≡ Ω(kr0)r0 so that they becomedimensionless eliminates the dependence on the mass density m0Ωˆ(kˆ) = kˆ − kˆ2 − kˆ2n(1 + 2kˆ − 2kˆ2)+O(n−2), kˆGL = 1− 12n+O(n−2),(5.44)and thus we recover the expected result [64], regarding the shift of the criticalwavelength of the Gregory-Laflamme instability at subleading order.5.3.2 Charged Black String Phase DiagramWhen k < kGL, neutral NUBS always have a lower event horizon surfacearea than uniform string solutions. Since the addition of charge weakens theGL instability of black strings, it is natural to wonder if the NUBS remainsentropically favoured, especially as we approach extremality.The entropy S of a black string is related to the area of its horizonS = Ω(n+1)4G∫horizon√gxxe(n+1)φ2 dx =Ω(n+1)4G√n(S0 + S1n+ · · ·). (5.45)At leading order, the area of a boosted string is given by the integral of theouter horizon radiusS0 =∫R+ dx. (5.46)Due to conservation of energy and charge, this integral is the same for theUBS and NUBS, and we need to examine S1 to witness an entropy differencebetween the two phases. However, it does not suffice to know the expansionof√gxx at subleading order; one also has to take into consideration the 1/ncorrection to the Killing horizon. We obtain the latter by requiringg00 = A(0)(v, x,Rh) +A(1)(v, x,Rh)n= 0, with Rh = R+ +R(1)hn. (5.47)Thus we findS1 = R(1)h + R+ log (R+ − R−) +R+2G(0)xx (R+) (5.48)121At equilibrium, charge density diffuses until it becomes proportional to themass profile of the non-uniform black string, which enables us to write q =ρm, with ρ = q0/m0 being the (conserved) charge-to-mass ratio of the blackstring. This yieldsSNUBS1 =∫1 +√1− 2ρ22{(m′′ +m)log((1 +√1− 2ρ2)m2)+12(2ρ21− 2ρ2 +√1− 2ρ2m+m′2m)}dx (5.49)SUBS1 =2pik(1 +√1− 2ρ2){m02log((1 +√1− 2ρ2)m02)(5.50)+ρ21− 2ρ2 +√1− 2ρ2m02}(5.51)where we have taken advantage of the fact that the coefficients multiplyingδm and δq were constant to integrate them away. It is easy to check thatthe difference in the two phases’ horizon area ∆S1 ≡ SNUBS1 − SUBS1 isalways positive no matter the ratio ρ. In particular, the entropy differenceat extremality is half that of neutral strings∆S1|ρ= 1√2=12∆S1|ρ=0 = −2pim0k(T(1)NUBS − T (1)UBS)> 0. (5.52)This result indicates that the NUBS is always the preferred phase, and thusthe instability persists for all charged brane configurations (as illustrated inFigure 5.1). However, despite the effective theory (5.22) admitting a smoothlimit when ρ → 1√2, we need to keep in mind that the large D expansionformally breaks down at extremality. Nevertheless, this result corroboratesthe ones obtained via hydrodynamics [154]. This and the exact cancellationin (5.52) of the pathologic divergences typically encountered at extremalityboth offer a positive outlook on the validity of results beyond the limits ofour approximation.We note that the numerical results of this section and the next have beenobtained by evolving small periodic perturbations around a uniform blackstring solution using a Runge-Kutta-Fehlberg method on a periodic Fouriergrid made of 41 points. The conserved quantities M, Q and P, as wellas the charge-to-mass ratio ρ, all remained constant during the evolution.Likewise, the integrals of δm and δp along the string direction were both122øøø0.0 0.1 0.2 0.3 0.4 0.5 0.6ΡDSFigure 5.1: Entropy difference per unit length as a function of the chargedensity for k = {0.75, 0.85, 0.95}, starting from the top. The numericalevolution breaks down at extremality, hence we use a star plot marker todistinguish the analytical result at ρ = 1/√2 from the others.zero to very good numerical accuracy until the final state was reached.5.3.3 Pinch-Off?Let us now turn our attention back to the mass and momentum correctionsδm and δp. In principle, the contents of equations (5.32) should provideus with a method for identifying the critical dimension D∗ below which theblack string would pinch-off, rather than settle on a smooth non-uniformfinal state. Such a transition in the nature of the final state is expected atlow enough D. It is interesting to see how this manifests itself in the largeD expansion.Below we present criteria we impose on the solutions, and the time evolu-tion towards those solutions, to investigate that question. While the variouscriteria we impose clearly indicate a tendency towards a pinch-off, we werenot able to find a universal value for the critical dimension D∗.To determine the critical dimension n∗, we first define the total correctedmass densityMn(x) = m(x) +δm(x)n, (5.53)1230.70 0.75 0.80 0.85 0.90 0.95 1.0001020304050kkGLn*Figure 5.2: The critical curves n∗ for m0 = {1, 3, 5}, from top to bottom,obtained at NLO. Black strings with parameters above the critical curvescorrespond to stable NUBS, whereas we conjecture a pinch-off scenario forthe those below, which exhibit negative tension.and the corrected tensionTn = −∫ (τ (0)xx +τ(1)xxn)dx, (5.54)which are both gauge invariant quantities.One attempt at diagnosing the stability of the black string at subleadingorder using our knowledge of m(x) and δm(x) is to find n∗ such that Mn∗(x)becomes locally negative. One can do so either by looking at the dynamicalevolution, or by examining the properties of the final state only. It turns outthat the dynamical evolution of the collective variables is highly sensitive tothe size of the initial perturbations around the uniform solution m(x) = m0.As such, this method does not provide a reliable stability diagnostic. Asfor the shape of the end-point of the dynamical evolution, the final shape ofδm(x) does depends solely on the static profile m(x). This makes it possibleto find n∗ such that Mn∗(x) < 0 locally, but this method has not yieldedaccurate results.124An alternate, more successful, method to identify the critical dimensionn∗ uses the corrected tension (5.54). Indeed, it is possible to find a criticalcurve n∗ as a function of the dimensionless wavenumber k/kGL by assumingthat the fate of a NUBS with negative tension is to pinch-off. Our resultsare summarized Figure 5.2 for three different initial configurations m0. Notethat as we vary m0 we change kGL, so this is an alternate way to scan thethe “thickness” k/kGL . However, while the qualitative features are similar,we still see slight differences between the three curves, which are either anartifact of early truncation in the 1/n expansion or a sign that negativebrane tension is a sufficient but not necessary condition for pinching-off.Indeed, nothing stops a pinch-off from happening at positive tension, andas such our critical curves may be thought of as an approximate probe untilfurther investigation.While the results we obtain preclude us from assigning an unambiguousvalue for the critical dimension, it can serve as a bound on the dimensionin which the final state pinches off, and it illustrates the dependence of thecritical dimension on the brane thickness. It is interesting to note that thedimensions near the critical point k = kGL are quite close to the expectedvalue n∗ = 9.5.4 Two Dimensional Non-Uniform PhasesWe now move to discuss the case of unstable membranes, for which thereare two independent modes of Gregory-Laflamme instabilities. Similar tothe one-dimensional case we find that in the leading order in the large Dlimit, the final state is a smooth and non-uniform configuration which wecall a lattice. Using the tools developed above, we study the phase diagramand determine the preferred size and shape of this lattice configuration.In two dimensions, it is possible to construct periodic black brane config-urations over oblique lattices. The lattices can be described by two vectors,describing the periodicities of the system:kx =2piLx(cosα,− sinα), ky = 2piLy(0, 1), with 0 ≤ α ≤ pi/2. (5.55)Thus we parametrize possible non-uniform solutions solutions by the threeparameters (Lx, Ly, α). Furthermore, since we are mostly interested in thepreferred shape of the non-uniform configuration, we take Lx = Ly = L.The angle α characterizes then the shape: special cases include α = 0 forcheckerboard lattices, α = pi/6 for triangular lattices, and α = pi/2 for125xyLx sec↵Ly↵(a) Lattice cell in (x, y) coordinatesvuLyLx(b) Lattice cell in (u, v) coordinatesFigure 5.3: The change of coordinates from (x, y) to (u, v) maps obliquelattice cells (a) to rectangular ones (b). Their area is Acell = LxLy secα.stripes.For the purpose of constructing the solutions, it is easier to work withthe coordinates (u, v) defined byu = x cosα− y sinα, v = y. (5.56)In these coordinates (illustrated in Figure 5.3), periodic boundary conditionsare simply(u, v) ≡ (u+ Lxnx, v + Lyny). (5.57)for any integers nx, ny.In order to make meaningful comparison between lattices of differentsize and shape, we need to work with the right thermodynamic potential.Instead of fixing the size and the shape of the unit cell, we instead fix theconjugate variables: the tensions in different directions. See [157] for ageneral discussion, and [158] for a recent application closely related to thecurrent discussion.The first law of black brane dynamics, in the micro-canonical ensemble,can be written asdM = κdA+ T abdVab, (5.58)where T ab are related to the tensions along brane directions, and Vab is amatrix of periodicities. To fix the conjugate variables instead of the size and126shape of the brane configuration, we define the “enthalpy” H of the brane3as the Legendre transformH ≡M − T abVab. (5.59)Our goal is to minimize this new potential. But first, we need to find an ex-pression for the tensions T ab. These are usually obtained from the quasilocalstress tensor at R→∞, which we have already found in (5.24):τab = ∂a∂bm−mδab + papbm− (∂apb + ∂bpa) . (5.60)We identify this boundary stress tensor as the source for the tensions, suchthatT ab = 〈τab〉 =∫∫cellτabdxdy/∫∫celldxdy =1L2∫∫cellτabdudv. (5.61)In the orthogonal coordinates (u, v), only the pressures T uu and T vv shouldcontribute, and as such we take Vmn = L2δmn for m,n = u, v. Since we areworking at constant mass, the quantity we must minimize is the tension βgiven byβ(α) = − T mnδmn (5.62)= − 1L2∫∫cell(cos2 α τxx − 2 cosα sinα τxy + (1 + sin2 α) τyy) dudv.(5.63)As expected, the enthalpy of oblique lattices includes contributions from theshear components of the stress tensor τxy.It is straightforward to apply the change of variables (5.56) to (5.22) inorder to find the inhomogeneous solutions on the oblique lattice numerically.For that purpose we discretize the lattices on a 31×31 periodic Fourier spec-tral grid, and we used the fifth order Runge-Kutta-Fehlberg time-steppingalgorithm to perform the time evolution towards the stable inhomogeneoussolution.Once we obtain the solutions for different shapes, we can find whichshape is preferred — our results are illustrated in Figure 5.4 for two differentchoices of lattice size. We find that the minimum of H is reached for latticeswith opening angles close to α = pi/6, which corresponds to the triangular3The conventional entalphy is obtained by Legendre transform with respect to the totalvolume, to work with fixed pressure instead.127lattice. Unsurprisingly, the position of the minimum depends on the sizeof the cell. Based on these results, we expect that for asymptotically largelattices the triangular lattice is the preferred configuration, and the slightdeviations we see are due to finite size effects.One can repeat the exercise with respect to the size of the preferredconfiguration. Indeed, in the one-dimensional case, where there is a size butnot shape parameter, the tension β of a black string with mass density m0decays exponentially with the string length Lβ(L) = −〈τxx〉 ∼ m0 e−a(L−2pi), with a ≈ 1.827, (5.64)meaning that the size of the preferred configuration is asymptotically large.0.48 0.50 0.52 0.54 0.56 0.58 0.601.Α105ΒHΑL(a) The tension as a function of theopening angle for k = 0.6. The mini-mum is located at α ≈ 0.548.0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.641.Α102ΒHΑL(b) The tension as a function of theopening angle for k = 0.8. The mini-mum is located at α ≈ 0.579.Figure 5.4: The enthalpy of the Bravais lattices reaches its minimum closerto α = pi/6 ≈ 0.524 as the size of the cell increases. Note that the numericalevolution becomes unstable for large α and large L, thus preventing us fromprobing larger oblique cells.5.5 ConclusionThe principal focus of our work was to determine the fate of extended blackobjects, in the approximation where the number of dimensions is large. Thetools of general relativity at large D have proven useful at unveiling robustproperties of higher dimensional black holes to surprising accuracy, and ourhope is that likewise the results presented in this paper hold up beyond the128asymptotic limit of that approximation.The main loose end left in this work is determining whether negativebrane tension is an appropriate test to accurately determine the fate of theblack string instability. While we see indications that the pinch-off scenariois likely as the final state at sufficiently low D, as well as a non-trivialdependence of the associated critical dimension on the string thickness, wehave not obtained a precise unambiguous result nor succeeded in reconcilingthese features of our solution with the current results in the literature. Wehope to return to this in the future.It would also be an interesting endeavour to explore the dynamics ofcharged dilatonic Kaluza-Klein black holes given the existence of exact uni-form solutions. Such a direct comparison would expand our understandingof the effects of charge on the stability of black strings in more generalscenarios.129Chapter 6Conclusion6.1 Summary and Future DirectionsIn this dissertation we investigated the consequences of breaking spatialtranslation invariance in gravitational systems. We used the tools of thegauge/gravity duality to construct a Josephson junction holographically andto examine entanglement propagation in strongly coupled systems. We alsomade use of the so-called large D approximation of general relativity toaddress black brane instabilities at next-to-leading order for asymptoticallyflat spacetimes.6.1.1 AdS/CMTIn chapter 2 we constructed the holographic equivalent of a chiral Josephsonjunction as part of the AdS/CMT (condensed matter theory) programme.Chiral superconductivity occurs at low temperatures when an operator withp + ip-wave symmetry acquires a vacuum expectation value that explicitlybreaks the U(1) subgroup of SU(2), which we identified with electromag-netism. To model a Josephson junction, we considered a chemical potentialvarying spatially in such a way that the order parameter was non-zero every-where except for a narrow region in the centre, effectively corresponding toa metallic weak link surrounded by two infinite p+ip-wave superconductors.Our setup showed the expected current-phase sinusoidal relation knownas the Josephson current as well as various relationships obeyed by this crit-ical current and the order parameter. However, the p+ ip symmetry of oursetup revealed a curious feature absent from p- and s-wave superconduc-tors: gapless chiral counter-propagating currents localized at the boundarybetween phases. We found that the intensity of these edge currents re-mained approximately constant for S-N-S junctions regardless of the chargedensity of the normal phase, whereas it decreased linearly as a functionof the chemical potential for S-S′-S junctions. In addition we observed aquadratic relationship between the edge currents and the magnitude of theorder parameter.Gapless chiral signatures such as these edge currents are usually indica-130tive of Majorana topological modes, albeit indirectly. Our holographic setupis unfortunately insensitive to other unconventional features of topologicalsuperconductivity, such as the 4pi-periodicity of the Josephson current. Inthe condensed matter literature this 4pi-periodicity anomaly is typically ob-served in the AC Josephson effect, which can be constructed holographicallyby having a time-dependent phase for the bulk order parameter. Such asetup should be relatively easy to study now that we have developed thetools to treat with non-static geometries effectively.Alternatively, the presence of Majorana zero-modes may be revealed byinvestigating fermion scattering across a chiral interface. The conventionalJosephson effect is attributed to Cooper pair tunnelling, but is believed tobe transported by single electrons in topological superconductors. As suchstudying a fermionic action with a Majorana mass term [159]Sη =∫dd+1x√−gφ∗ψc(η∗ + η∗5Γ5)ψ + h.c. (6.1)in the presence of a chiral superconductor is akin to studying Andreev re-flection at strong coupling, which has the potential to reveal unconven-tional signatures indicative of topological zero-modes. Complications aris-ing from studying the Dirac equation in curved spacetimes with spatialinhomogeneities prevented a preliminary analysis to be carried out, but wenonetheless believe that (6.1) may hold answers to address topological su-perconductivity with the gauge/gravity duality.The AdS/CMT programme is also quite vast and offers various other op-portunities to investigate spatial inhomogeneities. For instance, holographicsystems intrinsically lack the lattice structure responsible for momentumrelaxation in conventional systems. Indeed, translation invariance in fieldtheories at finite density results in the appearance of an anomalous deltafunction in the optical conductivity at zero frequency. This infinite metal-lic Drude peak is due to the inability for charge carriers to dissipate theirmomentum in homogeneous holographic theories. The introduction of asymmetry-breaking lattice in the form of a periodic neutral scalar field thatcan backreact of the geometry broadens this Drude peak [160], a featuremore in line with experimental observations. An alternate method for in-corporating momentum dissipation without compromising the homogeneityof bulk solutions is to introduce spatially-dependent massless scalar sourcesthat effectively act as a channel for relaxation [161]. Such a model alsoresults in a widened Drude peak at zero frequency.Regardless of the method, the addition of momentum dissipation can131help broaden our understanding of current models: the transport propertiesof holographic superconductors may change in the presence of a lattice; thephonon spectrum of the field theory may be calculated via the fluctuationsof the bulk field responsible for the lattice; Fermi surfaces calculated fromprobe fermions may have more realistic features, etc. Breaking translationinvariance invariably results in a more accurate description of the modelsat the cost of computational complexity, but holography makes this costbearable compared to traditional methods. It is our hope that the numericaltechniques used throughout this thesis help bridge the complexity gap inaddressing this type of problem in the future.6.1.2 Holographic Entanglement EntropyChapters 3 and 4 were devoted to the study of entanglement propagation infield theories admitting a gravitational dual. Entanglement was producedby the intermediary of a scalar source responsible for a localized injection ofenergy into a system in thermal equilibrium. The resulting thermalizationof spacetime was then probed via the dynamics of bulk extremal surfaces an-chored on the boundary whose area acts as a dual measure of entanglemententropy. The two most striking results were the appearance of an entan-glement lightcone structure in the L− t plane and the logarithmic decay ofentanglement entropy.The emergent lightcone featured two distinct regimes of entanglementpropagation: early-time propagation for wide quenches (L < σ), and late-time propagation for large entangling surfaces (L > σ). Our analysis sug-gests that the mechanisms responsible for the spatial spread of chaos in holo-graphic theories, characterized by the butterfly velocity vbutterfly =√3/2 =0.866, are very likely to influence the thermalization of entanglement degreesof freedom at early-times. This result also echoes those found in the liter-ature for global quenches, where entanglement saturation occurs at timest ∼ L/vbutterfly for large strip regions. Local quenches for large entanglingsurfaces were instead characterized by a lightcone velocity vE ∈ [0.65, 0.7],an interval that comprises many velocities of interest. These include thespeed of sound, the shear velocity, and the tsunami velocity. Our initialanalysis suggested that the latter could play a role in quantifying the spreadof entanglement at late-times given that it is an intrinsic property of theequilibrium thermal state, but the addition of charge quickly dispelled thisnotion. Extremal black holes are characterized by a vanishing tsunami ve-locity, but vE was found to be robust against the addition of charge, whichled us to speculate that the mechanisms underlying momentum diffusion in132the sound and shear channels would be responsible for entanglement prop-agation instead.As for the decaying behaviour of entanglement entropy, improved nu-merics helped us revise our initial conclusions indicative of an exponentialdamping at late-times. Modifications including domain decomposition inthe radial direction, higher spatial resolution, and other “under the hood”changes, allowed us to extend the evolution of the quench for twice as longin a reasonable computational time. This extra data instead revealed that alogarithmic decay ∼ log t/tδ provided a much better fit. This result mirrorssimilar conclusions obtained in the context of spin chains in two-dimensionalCFTs (which have δ = 1), but we also found evidence for a transition be-tween the δ = 1 and δ = 1.5 regimes as a function of charge and strip width,indicating a richer story that we have yet to discover.These two projects were very ambitious but technical limitations com-plicated the analysis. For instance the space of parameters that could beexplored was huge; the scalar quench amplitude and width, the strip width,the initial mass density and the initial charge density could all be variedindependently, and each typically introduced an inherent length scale of itsown. Given the competition between these different scales and our relianceon a numerical approach, it became difficult to thoroughly characterize thedefining regimes in order to compare and contrast our results with thosein the literature. We were also confronted to an order of limits issue whenstudying theories at finite charge densities. The near-horizon geometry ofextremal black holes is AdS2×R2, which indicates an emergent scale invari-ance in the IR. Our original motivation for studying RN backgrounds wasthen to investigate the dynamics of extremal surfaces as we progressivelychanged the near-horizon topology. Unfortunately our numerics restrictedus to the study of narrow entangling surfaces, which did not allow for deepenough probes of the geometry. As such the L → ∞ and extremal limits,as well as their potentially distinct regularities, remained out of reach withthis setup.A proper numerical analysis of large entangling regions and their IRdynamics in near-extremal backgrounds would be a welcome addition tothe current analysis. Moreover, the technology we have developed to solvethe dynamical Einstein and geodesic equations could be used to investigateshapes other than the infinite strip, such as spherical regions or off-centergeometries. Other measures of quantum information would also be interest-ing to compute in our setup. In particular, the mutual information between133two regions A and B, defined asI(A,B) = SA + SB − SA∪B, (6.2)is UV-finite and provides a measure of the total quantum correlations be-tween A and B without contributions from thermal entropy [162].The connection between the gauge/gravity duality and quantum infor-mation is only just starting to be unraveled. Recent developments haveshown that AdS/CFT provides a natural setting to study tensor networksworking as encoders for quantum error-correction code [163, 164]. More-over, a tensor network known as Multi-scale Entanglement RenormalizationAnsatz (MERA), used to estimate the ground state of quantum systemswith long-range entanglement, was found to resemble the hyperbolic geom-etry of anti-de Sitter spacetimes, suggesting there might be an underlyingAdS/MERA correspondence at work [165]. Work in that direction has thepotential to provide answers to fundamental questions about the nature ofentanglement.Lastly, other future directions include studies of thermalization in stronglycoupled field theories far-from-equilibrium. The gauge/gravity duality al-lows us to investigate QCD-like theories, which may reveal insights aboutthe dynamics of the quark-gluon plasma and the confinement/deconfine-ment phase transition characteristic of asymptotic freedom. The power ofnumerical relativity has already been harnessed to model heavy ion collisionsand jet quenching, and the gauge/gravity duality has provided a languageto discuss hydrodynamic quantities such as shear viscosity at strong cou-pling [166]. The current models boast a high degree of symmetry, and morerealistic features are bound to be extracted by considering inhomogeneoussettings as we have done throughout this dissertation.6.1.3 Large D Limit of General RelativityIn chapter 5 we set our sights on brane instabilities in asymptotically flatspacetimes. Solving the full dynamics of the inhomogeneous Einstein equa-tions is typically very difficult in flat space, but the formalism of generalrelativity when the number of dimensions D is large reduces the computa-tion to an asymptotic matching problem. This simplification occurs becausethe gravitational field of a black hole becomes localized within a distance∼ 1/D of the horizon, thus decoupling it from the outside dynamics.The effective conservation equations describing the fluctuations in thebrane’s energy, momentum and charge densities at leading and next-to-134leading order in a 1/D expansion were both easily solvable numerically andamenable to a linearized perturbation analysis. Examining these equationsin turn allowed us to recover the instability spectrum at next-to-leadingorder and improved our understanding of the black string’s phase struc-ture. In particular, we confirmed that unstable branes remain unstable atextremality, established a novel condition to determine the critical dimen-sion below which the string is prone to fragmentation, and discovered thattwo-dimensional membranes minimize their enthalpy on triangular lattices.Our investigation thus adds to the wealth of results already supporting thevalidity of the large D approximation.A notable aspect of this formalism is its hydro-elastic complementarity,which describes the equivalence between the black branes’ elastic theorydescription and their hydrodynamical features at large D. To paraphrase theauthors of [150], ripples on a black brane can be interpreted both as pressurewaves on a fluid and as wrinkles on a membrane. The equivalence is manifestin that the effective equations can be described in terms of curvatures andsurface gravity (the elastic point of view), as well as via the dynamics ofa stress tensor with a truncated gradient expansion (as in hydrodynamics).Note however that the large D formalism has more predictive power thana naive hydrodynamical approach since it captures phenomena the lattercannot (e.g. static NUBS) and typically remains valid even when gradientsbecome steep. We made use of this alternative point of view to computebrane tension in our work, which provided us with a novel criteria to predictbrane instability. However a proper analysis of the effective equations andtheir hydro-elastic formulation at next-to-leading order is still lacking andwould definitely be worth pursuing.The large D approximation can also be used in anti-de Sitter geome-tries, which opens up endless possibilities when combining it with the toolsof the gauge/gravity correspondence. In fact holographic superconductorshave already been subjected to this formalism [167], and an analysis of thedynamics between rarefaction waves and shockwaves using the tools of thefluid/gravity correspondence at large D was conducted in [156]. Problemsdue to backreaction of matter on the geometry are trivialized in this con-text, and extensions to include different symmetries and conserved chargescould be pursued straightforwardly. We could also investigate condensedmatter systems admitting a gravitational dual and their properties robustto taking D →∞, extract the generic features of thermalization of stronglycoupled field theories in a 1/D expansion, and so on. The large D formalismis highly flexible and its many applications have the promising potential tounravel the mysteries of black hole dynamics further than ever before.1356.2 Concluding RemarksThe principal contribution of the work presented in this dissertation is thestudy of gravitational systems with a reduced degree of symmetry due tothe presence of spatial inhomogeneities. The loss of translation invarianceadds a layer of complexity compared to the homogeneous case since thephysical content of general relativity comes packaged in coupled non-linearpartial differential equations requiring a numerical approach. We have how-ever repeatedly seen that this compromise in complexity typically resultsin richer dynamics. Indeed, through the lens of the gauge/gravity dualitywe were able to reproduce the physics of Josephson junctions and observedunconventional gapless chiral currents at the normal metal/superconductorinterface. We also observed an entanglement lightcone by studying localquenches in holographic theories, which allowed us to speculate about themechanisms responsible for entanglement propagation. This structure hadnot been fully appreciated in the study of global quenches due to entan-glement entropy saturation, an issue that we circumvented by injecting afinite amount of energy via a localized excitation. Finally, we succeeded inreproducing many salient features of black string dynamics with the formal-ism of general relativity with infinite dimensions. Technical issues, such asnumerical instabilities typically found in asymptotically flat spacetimes andgauge redundancies in gravity perturbation theory, were eschewed in favourof simpler effective brane equations that captured many universal aspects ofnon-uniform black strings. In view of all this, the central lesson from chap-ters 2 through 5 is that there is still much to learn by adding more realisticfeatures to the current models.Despite its technical challenges, numerical holography is currently one ofthe most promising tools at our disposal to learn about phenomena defyingour current understanding. It offers surprising connections between vastlydifferent systems via the unifying language of black holes and provides awealth of opportunities to study systems of reduced symmetry without thesevere limitations of more standard approaches. Similarly to the large Nlimit of strongly coupled field theories, the large D limit of general relativitybrings about important simplifications of gravitational dynamics, which inturn provides a method to investigate a wide variety of higher-dimensionalblack objects without relying on prohibitively expensive simulations. 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The apparent horizon isthen defined as the boundary of this trapped region, on which the geodesiccongruences have vanishing expansions.In our case, a planar spacelike surface like the apparent horizon canbe parametrized by the two orthogonal vector fields spanning the x and ydirections:eMx = (0, 0, 1, 0), eMy = (0, 0, 0, 1). (A.1)We now construct future-directed null geodesic congruences orthogonal toboth ex and ey. Ingoing geodesic congruences can be parametrized by thetangent null vector field kM = (0,−1, 0, 0), whereas outgoing geodesic con-gruences haveNM = e−2χ(1, A+ e−2χe−B F 2x2 Σ2,e−B FxΣ2, 0). (A.2)The normalization is chosen such that gMN kM NN = −1. Since we areinterested in the rate of change of the cross-sectional area of null geodesiccongruences along their transverse directions, we need to define the trans-verse metrichMN = gMN + kMNN +NMkN . (A.3)With this in hand, we can calculate the expansion1 θ ≡ hMN∇MNN . Setting1The expansion for the ingoing geodesic congruences is always negative, so we needonly worry about the congruences along N .151θ = 0 yields a condition on the dynamics of the field Σ:[d+Σ− e−B2 Σ(Fx ∂xB − ∂xFx − e−2χ F 2x∂rΣΣ)]r=rh= 0 , (A.4)where d+ ≡ ∂t +A ∂r. In addition, taking a time derivative of this relationyields a stationarity condition that ensures that the horizon condition holdsfor all times. One can show that the resulting constraint can be expressedas a second order spatial ODE that determines the value of A(r, t, x) at the(fixed) apparent horizon.152Appendix BNumerical Implementation ofCharacteristic FormulationThe characteristic formulation of Einstein’s equations in the presence ofmatter reorganizes all the fields into two categories: auxiliary fields obeyingradial ODEs that can be solved sequentially, and dynamical fields whichare used to evolve the geometry from one null slice to the next. This sep-aration of fields can be achieved by expressing time derivatives in terms ofthe directional derivative along outgoing null geodesics, d+ = ∂t + A ∂r,thereby completely eliminating the presence of A from the auxiliary equa-tions. Changing to a compact variable u = 1/r, we rewrite the fields ap-pearing in our equations asΦ(u, t, x) ≡ φ(u, t, x)u,Er(u, t, x) ≡ er(u, t, x)u2,Σ(u, t, x) ≡ 1 + λ(t, x)uu− 14φ(u, t, x)2u,B(u, t, x) ≡ b(u, t, x)u2,χ(u, t, x) ≡ c(u, t, x)u2,Fx(u, t, x) ≡ − ∂xλ(t, x) + fx(u, t, x),d+Σ(u, t, x) ≡ (1 + λ(t, x)u)22u2+ Σ˜(u, t, x),d+Φ(u, t, x) ≡ − 12φ(u, t, x) +(Φ˜(u, t, x) +12∂uφ(u, t, x)),d+B(u, t, x) ≡ B˜(u, t, x)u2,A(u, t, x) ≡ (1 + λ(t, x)u)22u2+ a(u, t, x),(B.1)in order to subtract the divergent parts. The field Er(r, t, x) above is definedasEr = ∂rV0 +e−BΣ2Fx ∂rVx ∼ F tr (B.2)153in order to decouple the radial equations satisfied by V0 and Fx. Howeverwe note that the equations for d+B and d+Vx form a linear system of radialODEs that cannot be decoupled.Given initial conditions specified by φ, λ, b and Vx all being 0, as well asthe CFT data T00 and Ttx, we can solve the radial ODEs for the auxiliaryfields c, er, fx, V0, Σ˜, Φ˜, and for the coupled system B˜ and d+Vx, in thatorder. These fields obey the boundary conditions∂uc(u = 0) = − 112λφ20 +16φ0φ1, (B.3)er(u = 0) = ρ, (B.4)fx(u = 0) = 0 and ∂ufx(u = 0) = f(3) =23Ttx +13φ0∂xφ0, (B.5)V0(u = 0) = µ, (B.6)Φ˜(u = 0) = − φ1 − λφ0 + ∂tφ0, (B.7)B˜(u = 0) =16((∂xφ0)2 − 12φ0∂2xφ0 − ∂xTtx)− 12jx(∂tµx − ∂xµ− 12jx)− 12∂2ub∣∣∣u=0, (B.8)d+Vx(u = 0) = ∂tµx − 12jx. (B.9)There are two options when treating with the field Σ˜, one of which is toimpose the condition[d+Σ− e−B2 Σ(Fx ∂xB − ∂xFx − e−2χ F 2x∂rΣΣ)]r=rh= 0, (B.10)which determines the location of the apparent horizon as the boundary oftrapped surfaces, as derived in Appendix A. Our second option is to set∂uΣ˜(u = 0) =12T00 − 13φ0φ1 − 112λφ20 (B.11)on the boundary, as required by self-consistency of the equations of mo-tion. Either conditions imply the other; imposing the latter should yieldthe former and vice-versa, and we can use this as a safety check for ournumerics.Now that we have solved for the necessary auxiliary fields, we have topropagate the solutions along null slices. In order to propagate λ, we re-154quire a horizon stationarity condition, obtained by differentiating (B.10)with respect to time, thus ensuring that the location of the apparent hori-zon remains fixed on all null slices. This procedure yields a boundary valueproblem in x for the field A(uh, t, x). We can then extract ∂tλ from therelationd+Σ = ∂tλ+A− d+(14φ2)(B.12)evaluated at the horizon. The same equation in turn enables us to solvefor A everywhere in the bulk since λ does not depend on the radial coordi-nate. With A in hand, it now becomes straightforward to extract the timederivatives for b, φ and Vx from the solutions for d+B, d+Φ and d+Vx, andfrom the definition of d+ = ∂t +A ∂r. At this point, all that is left to do ispropagate these fields with a time-stepping algorithm, along with T00, Ttxand ρ using the conservation equations (4.22), (4.23), and (4.24), and torepeat the process on new null time slices until satisfied with the evolutionof the quench.155Appendix CDimensional ReductionThe framework of general relativity when the number of spacetime dimen-sions D is large is a novel way to investigate non-perturbative objects likeblack branes in a 1/D expansion that trivializes the gravitational dynamicsaway from their event horizons. For this expansion to take place, the de-pendence on D needs to be extracted explicitly from Einstein’s equations.For a spherically symmetric spacetime described byds2 = gµν(x)dxµdxν + eφ(x)dΩ2d, (C.1)where gµν(x) is the metric of the object of interest and dΩ2d the one on theunit d-sphere, the dependence on d can be made explicit via dimensionalreduction.Cartan’s formalism is particularly well-suited for the task at hand [168].The first step is to reexpress the coordinate basis metric (C.1) in terms ofa vierbein basis eα(x) by relating the basis vectors via dxµ = eµαeα, whereeµα denotes the (coordinate-dependent) transformation matrix between thetwo basis. This leads us to rewrite1gµν(x)dxµdxν = ηab ea(x)eb(x), (C.2)andeφ(x)dΩ2d = δij ei(x, θ)ej(x, θ), (C.3)where we have used the inner-product constraint gµν(x)eµa(x)eνb(x) = ηab(or equivalently gµν(x) = eaµ (x)ebν (x)ηab) that defines the vierbein fieldseµa(x) as the square root of the metric. Also note that ei(x, θ) = eφ(x)/2eiγ(θ),with eiγ(θ) our basis on Sd.Another important element of this formalism is the spin connection ω,1We use lowercase latin letters (a, b, ...) at the start of the alphabet to denote the basisassociated with the sector described by gµν , and the letters in the middle (i, j, ...) to identifythat of the d-sphere. These latin indices are raised and lowered by the corresponding flatmetric.156which plays the role of the affine connection in the vierbein basis∇µ,cXa = ∂µ,cXa + (ωµ,c)abXb, (C.4)∇µ,cXa = ∂µ,cXa − (ωµ,c)baXb. (C.5)In non-coordinate form, ωab = (ωµ)abdxµ = (ωc)abec. Additionally, the spinconnection is antisymmetric: ωab = −ωba.We now have all the necessary elements to compute the Riemann tensorin the vierbein basis, which is more tractable analytically. Our first task is tofind the spin connection for the metric (C.1) by making use of Cartan’s firststructure equation, which relates the exterior derivatives2 of the vierbeinbasis de with the spin connection on the manifold via the antisymmetricwedge productdeα = −ωαβ ∧ eβ. (C.6)On the d-sphere, we expect the spin connection to act separately on bothsectors because of the scale factor eφdei = −ωi a ∧ ea − ωi j ∧ ej . (C.7)Extracting the dependence on φ explicitly leads us todei = d(eφ(x)/2eiγ)= d(eφ/2) ∧ eiγ + eφ/2deiγ (C.8)=12φ,a ea ∧ ei − eφ/2 (ω iγ j ∧ ejγ) (C.9)= − 12φ,a ei ∧ ea − ω iγ j ∧ ej , (C.10)from which we learn that ωi a =12φ,a ei.In contrast, the gµν sector is self-containeddea = −ωab ∧ eb − ωai ∧ ei = −ωab ∧ eb, (C.11)where we have used the fact that the wedge product is antisymmetric, ei ∧ei = 0.With the spin connection in hand, we can use Cartan’s second structureequation to compute the Riemann tensor RRαβ = dωαβ + ωαρ ∧ ωρβ. (C.12)2The exterior derivative of a scalar function is simply dφ = φ,aea.157Note that its components may be read from the vierbein basisRαβ =12Rαβρσ eρ ∧ eσ. (C.13)It is useful to separate the calculation in three:• For the Riemann tensor pertaining to the metric gµν only,Rab = Rag b (C.14)since this sector is self-contained. The corresponding Ricci scalar isthen simply R1 = Rg, the one computed from the metric gµν by usualmeans.• For the Riemann tensor pertaining to the d-sphere only,Ri j =(dωi j + ωik ∧ ωkj)+ ωi a ∧ ωaj (C.15)= R iγ j + ωia ∧(−ηabωjb)(C.16)= R iγ j −14(∇φ)2 ei ∧ ej (C.17)=12(R iγ jmn −14(∇φ)2 (δimδjn − δinδjm))em ∧ en. (C.18)Note that we have explicitly antisymmetrized the term proportionalto (∇φ)2, in accordance to the Riemann tensor’s definition.To compute the Ricci scalar, we sum the Riemann tensor over its firstand third indices and then take the traceR2 = Rγ + δijRmimj (C.19)= Rγ − 14(∇φ)2 (δmmδij − δmjδim)δij (C.20)= Rγ − d(d− 1)4(∇φ)2 . (C.21)For a sphere of radius eφ, the Ricci scalar is simply Rγ = d(d+ 1)e−φ.158• For the mixed sector,Ri a = dωia + ωib ∧ ωba + ωi j ∧ ωja (C.22)= d(12φ,a ei)+(12φ,b ei)∧ ω bg a + ω iγ j ∧(12φ,a ej)(C.23)=12φ,ab eb ∧ ei + 14φ,aφ,b eb ∧ ei − 12φ,a ωiγ j ∧ ej (C.24)+12φ,b ei ∧ (ωc)baec +12φ,a ωiγ j ∧ ej (C.25)=12∇a∇bφ eb ∧ ei + 14φ,aφ,beb ∧ ei (C.26)= − 12(∇a∇bφ+ 12φ,aφ,b)δimem ∧ eb. (C.27)The components of the Riemann tensor are already antisymmetric inm and b since ∇mφ = 0.The associated Ricci scalar is thereforeR3 = ηabRi aib = −d(∇2φ+ 12(∇φ)2). (C.28)Not forgetting that√−G = edφ/2√−g for the total metric Gµν , theEinstein-Hilbert action S can be rewritten asS =∫dDx√−G (R1 +R2 +R3) (C.29)=∫dDx√−g edφ/2[Rg + d(d+ 1)e−φ − d(d− 1)4(∇φ)2− d(∇2φ+ 12(∇φ)2)](C.30)=∫dDx√−g edφ/2[Rg + d(d+ 1)e−φ +d(d− 1)4(∇φ)2], (C.31)where we have used integration by parts in the last step.With the dependence on d now explicit, the Einstein equations can beobtained straightforwardly by varying (C.31) with respect to both the metricand φ. In the presence of an Abelian gauge field A, the dimensionally-159reduced action instead reads [152]S =∫dDx√−g edφ/2[Rg + d(d+ 1)e−φ +d(d− 1)4(∇φ)2 − 14F 2](C.32)with F = dA, and the Maxwell equations can be obtained in the usual wayby varying the above with respect to A.160


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